Mathematical Modeling of Mathematical Modeling of Cellular Signaling in MacrophagesCellular Signaling in Macrophages

Lipid Signaling KineticsLipid Signaling Kinetics

Mathematical Modeling of Mathematical Modeling of Cellular Signaling in MacrophagesCellular Signaling in Macrophages

Lipid Signaling KineticsLipid Signaling Kinetics

Mary Ann HornMary Ann Horn

National Science Foundation amp Vanderbilt UniversityNational Science Foundation amp Vanderbilt University

Joint work with Joint work with

Hannah L Callender Department of MathematicsHannah L Callender Department of Mathematics

amp H Alex Brown amp the Brown Lab Department of Pharmacologyamp H Alex Brown amp the Brown Lab Department of Pharmacology

Mary Ann HornMary Ann Horn

National Science Foundation amp Vanderbilt UniversityNational Science Foundation amp Vanderbilt University

Joint work with Joint work with

Hannah L Callender Department of MathematicsHannah L Callender Department of Mathematics

amp H Alex Brown amp the Brown Lab Department of Pharmacologyamp H Alex Brown amp the Brown Lab Department of Pharmacology

Collaboration with Alliance for Cellular Collaboration with Alliance for Cellular SignalingSignaling

Collaboration with Alliance for Cellular Collaboration with Alliance for Cellular SignalingSignaling

bull Alliance for Cellular Signaling (AfCS) ldquoDetermine how cells Alliance for Cellular Signaling (AfCS) ldquoDetermine how cells process signaling information in a context dependent mannerrdquoprocess signaling information in a context dependent mannerrdquo

bull AfCS Screen a large number of ligands (signaling molecules) AfCS Screen a large number of ligands (signaling molecules) and analyze their effects on the and analyze their effects on the RAW 2647 cellRAW 2647 cell (a (a macrophage-like cell line derived from mice a macrophage-like cell line derived from mice a macrophagemacrophage is is a type of white blood cell that surrounds and kills a type of white blood cell that surrounds and kills microorganisms removes dead cells and stimulates the action microorganisms removes dead cells and stimulates the action of other immune system cells) of other immune system cells)

bull FocusFocus Construction of a comprehensive mathematical model Construction of a comprehensive mathematical model for the signaling pathway of one ligand uridine 5rsquo-diphosphate for the signaling pathway of one ligand uridine 5rsquo-diphosphate ((UDPUDP) in the RAW 2647 cell) in the RAW 2647 cellndash UDP is a common signaling nucleotide often released when cells UDP is a common signaling nucleotide often released when cells

undergo some type of trauma or mechanical stimulusundergo some type of trauma or mechanical stimulus

bull Modeling approachModeling approach Utilize a system of nonlinear ordinary Utilize a system of nonlinear ordinary differential equations designed to model the time-dependent differential equations designed to model the time-dependent behavior of key players in the pathway behavior of key players in the pathway

Basics of Cellular SignalingBasics of Cellular SignalingBasics of Cellular SignalingBasics of Cellular Signaling

Signaling molecules bind to cell Signaling molecules bind to cell surface receptors surface receptors

Initiate a series of intracellular Initiate a series of intracellular reactions reactions

These reactions regulate virtually These reactions regulate virtually all aspects of cell behavior (including all aspects of cell behavior (including metabolism movement metabolism movement proliferation survival and proliferation survival and differentiation)differentiation)

bull Breakdown in the signaling pathways that control normal cell Breakdown in the signaling pathways that control normal cell proliferation and survival result in many types of cancerproliferation and survival result in many types of cancer Need for better understandingNeed for better understanding Mathematical modeling can provide insight into these signaling Mathematical modeling can provide insight into these signaling

pathwayspathways

Canonical UDP signaling pathwayCanonical UDP signaling pathwayCanonical UDP signaling pathwayCanonical UDP signaling pathway

11 Ligand or stimulus Ligand or stimulus ((UDPUDP) binds to a ) binds to a specific G-protein specific G-protein coupled receptor coupled receptor ((P2YP2Y66))

33 GGqq GTPGTP activates activates phospholipase C-phospholipase C-3 (3 (PLC-PLC-33))

44 Active Active PLC-PLC-33 cleaves cleaves phosphatidylinositol 45-phosphatidylinositol 45-bisphosphate (bisphosphate (PIPPIP22) into ) into inositol 145-trisphosphate inositol 145-trisphosphate ((IPIP33) and diacylglycerol ) and diacylglycerol ((DAGDAG))

22 The ligand bound receptor The ligand bound receptor causes the exchange of causes the exchange of GDPGDP for for GTPGTP on the on the GGqq subunit of the G-proteinsubunit of the G-protein

55 IPIP33 binds to a specific receptor binds to a specific receptor on the endoplasmic reticulum on the endoplasmic reticulum releasing sequestered releasing sequestered CaCa2+2+

UDP

P2Y

6

GDPG PLC

Ca2+

PIP

2

DAG

IP3

GGTP

IP3

IP3-R

IP3 -R

Plasma Membrane

Endoplasmic Reticulum

Ca2+

Ca2+

Ca2+

UDP

GGTP

Ca2+ Ca2+

Ca2+

Ca2+

Ca2+

Ca2+

Mathematical ModelMathematical ModelMathematical ModelMathematical Model

bull The model consists of a base set of 10 coupled nonlinear The model consists of a base set of 10 coupled nonlinear ordinary differential equations (ODEs)ordinary differential equations (ODEs)

bull The ODEs are constructed using mass action kinetics to The ODEs are constructed using mass action kinetics to approximate diffusion binding and unbinding of molecular approximate diffusion binding and unbinding of molecular species in various compartments of the RAW cellspecies in various compartments of the RAW cell effective rates of production and degradation are proportional to effective rates of production and degradation are proportional to

the concentrations (or number of molecules) of participating the concentrations (or number of molecules) of participating reactantsreactants

bull Four modulesFour modulesndash Receptor regulationReceptor regulationndash G-protein casadeG-protein casadendash Species-specific DAG dynamicsSpecies-specific DAG dynamicsndash Cytosolic CaCytosolic Ca2+2+ dynamics dynamics

bull Receptor-ligandReceptor-ligand binding (based on work by Lemon 2003) binding (based on work by Lemon 2003)

ndash Number of ldquoactivatedrdquo surface receptors at time tNumber of ldquoactivatedrdquo surface receptors at time tndash Number of ldquoinactivatedrdquo surface receptors at time tNumber of ldquoinactivatedrdquo surface receptors at time tndash Concentration of ligand (constant)Concentration of ligand (constant)

Model Equations Receptor Model Equations Receptor regulationregulation

Model Equations Receptor Model Equations Receptor regulationregulation

Nondimensionalized equationsNondimensionalized equations

Model Equations G-protein Model Equations G-protein activationactivation

Model Equations G-protein Model Equations G-protein activationactivation

bull G-proteinG-protein activation activation

ndash Number of activated G-proteins (GNumber of activated G-proteins (GGTP) at time tGTP) at time t

Nondimensionalized equationsNondimensionalized equations

Model Equations PIPModel Equations PIP22 hydrolysis and hydrolysis and replenishmentreplenishment

Model Equations PIPModel Equations PIP22 hydrolysis and hydrolysis and replenishmentreplenishment

bull PIPPIP2 2 hydrolysis and replenishmenthydrolysis and replenishment

Nondimensionalized equationsNondimensionalized equations

ndash Number of PIPNumber of PIP22 molecules at time t molecules at time tndash Initial number of ldquofreerdquo PIPInitial number of ldquofreerdquo PIP22

PLC

PIP

2

DAG

IP3

GGTPCa2+

GGTP

bull IPIP33

Model Equations IPModel Equations IP33 production and production and degradationdegradation

Model Equations IPModel Equations IP33 production and production and degradationdegradation

Nondimensionalized equationsNondimensionalized equations

PLC

PIP

2

DA

G

IP3

G

GTPCa2+

G

GTP

bull Concentration of IPConcentration of IP33 at time t (in at time t (in M)M)

Model Equations DAGModel Equations DAGModel Equations DAGModel Equations DAG

bull DAG DAG (separate ODEs for each species considered)(separate ODEs for each species considered)

Nondimensionalized equationsNondimensionalized equations

bull Concentration of DAG at time tConcentration of DAG at time t

PLC

PIP

2

DA

G

IP3

G

GTPCa2+G

GTP

Model Equations Li and Rinzel (1994) Model Equations Li and Rinzel (1994) CaCa2+2+

Model Equations Li and Rinzel (1994) Model Equations Li and Rinzel (1994) CaCa2+2+

bull Free cytosolicFree cytosolic Ca Ca2+2+

bull hh = fraction of IP = fraction of IP33 channels not yet inactivated by Ca channels not yet inactivated by Ca2+2+

bull cc00 = concentration of total free Ca = concentration of total free Ca2+2+ per cytosolic volume per cytosolic volume

Timeseries data for the modelTimeseries data for the modelTimeseries data for the modelTimeseries data for the model

bull Data from collaborators at UTSW (Sternweis Lab)Data from collaborators at UTSW (Sternweis Lab)ndash Inositol 145-trisphosphate (Inositol 145-trisphosphate (IPIP33) production post 25) production post 25M M

UDP stimulation in the RAW cellUDP stimulation in the RAW cellndash Cytosolic calcium (Cytosolic calcium (CaCa2+2+) release post stimulation with ) release post stimulation with

2525M UDP in the RAW cellM UDP in the RAW cell

bull Diacylglycerol (Diacylglycerol (DAGDAG) data H Callender collected from a ) data H Callender collected from a novel method of quantitative analysis of multiple novel method of quantitative analysis of multiple species of DAG developed in the Brown Lab species of DAG developed in the Brown Lab

Reference Reference

Callender H L et al Quantification of Diacylglycerol Species from Callender H L et al Quantification of Diacylglycerol Species from Cellular Extracts by Electrospray Ionization Mass Spectrometry Using a Cellular Extracts by Electrospray Ionization Mass Spectrometry Using a Linear Regression Algorithm Linear Regression Algorithm Anal Chem Anal Chem 7979 (2007) 263-272 (2007) 263-272

IPIP33 and Ca and Ca2+2+ data from AfCS data from AfCSIPIP33 and Ca and Ca2+2+ data from AfCS data from AfCS

AA

bull IPIP33 response (in pmoles per response (in pmoles per 100 100 L lysate)in RAW 2647 L lysate)in RAW 2647 cells to 25 cells to 25 M UDPM UDP

bull Points represent the average Points represent the average of four experimentsof four experiments

bull Cytosolic calcium response Cytosolic calcium response (in (in M) in the RAW 2647 M) in the RAW 2647 cell to 25 cell to 25 M UDP M UDP

bull The graph displays 43 The graph displays 43 experimental repeatsexperimental repeats

B

What is DAGWhat is DAGWhat is DAGWhat is DAG

bull DAG is a cellular second messenger molecule which plays an DAG is a cellular second messenger molecule which plays an important role in initiating various changes in cell behavior important role in initiating various changes in cell behavior including cell activation differentiation proliferation and including cell activation differentiation proliferation and tumor promotion tumor promotion

bull There are many different species of DAG depending on the There are many different species of DAG depending on the number of carbons and number of double bonds in the fatty number of carbons and number of double bonds in the fatty acyl chains and different species can have different cellular acyl chains and different species can have different cellular functionsfunctions

bull ExampleExample 320 DAG320 DAG

Two fatty acyl Two fatty acyl (hydrocarbon) (hydrocarbon) chains in every DAG chains in every DAG speciesspecies

Each corner Each corner represents carbonrepresents carbon

320 DAG 18 carbons in first chain 14 in the second no double bonds 320 DAG 18 carbons in first chain 14 in the second no double bonds in either chainin either chain

Kinetics of monodi unsaturated DAGsKinetics of monodi unsaturated DAGsKinetics of monodi unsaturated DAGsKinetics of monodi unsaturated DAGs

bull Time based behavior of four monodi unsaturated DAG species after addition of 25 Time based behavior of four monodi unsaturated DAG species after addition of 25 M M (solid red squares) and 025 (solid red squares) and 025 M (solid green triangles) UDPM (solid green triangles) UDP

bull Time points contain a minimum of nine replicates performed on three different Time points contain a minimum of nine replicates performed on three different experimental daysexperimental days

- 25 M UDP

- 025M UDP

0 5 10 15 20 25 30

0 5 10 15 20 25 30 0 5 10 15 20 25 30

0 5 10 15 20 25 30

321 DAG 341 DAG

362 DAG341 DAGep

150

100

50-5

00

300

200

100

-100

0

150

100

50-5

00

-100

200

6040

20-2

00

-40

80

Time (min) Time (min)

Time (min) Time (min)

D

AG

(n

g)

D

AG

(n

g)

D

AG

(n

g)

D

AG

(n

g)

- 25 M UDP

- 025M UDP

- 25 M UDP

- 025M UDP

- 25 M UDP

- 025M UDP

Kinetics of PUFA containing DAGsKinetics of PUFA containing DAGsKinetics of PUFA containing DAGsKinetics of PUFA containing DAGsbull Time based behavior of four polyunsaturated fatty acid (PUFA) containing DAG species Time based behavior of four polyunsaturated fatty acid (PUFA) containing DAG species

after addition of 25 after addition of 25 M (solid red squares) and 025 M (solid red squares) and 025 M (solid green triangles) UDPM (solid green triangles) UDPbull Time points contain a minimum of nine replicates performed on three different Time points contain a minimum of nine replicates performed on three different

experimental daysexperimental daysbull Monodi unsaturated DAG species give a larger increase than polyunsaturated fatty acid Monodi unsaturated DAG species give a larger increase than polyunsaturated fatty acid

(PUFA) containing DAGs(PUFA) containing DAGs

0 5 10 15 20 25 30

0 5 10 15 20 25 30 0 5 10 15 20 25 30

0 5 10 15 20 25 30

385 DAG 383 DAG

364 DAG384 DAG6

42

-40

30

20

10

0

15

10

50

-56

40

2-2

8

Time (min) Time (min)

Time (min) Time (min)

D

AG

(n

g)

D

AG

(n

g)

D

AG

(n

g)

D

AG

(n

g)

-28

40

- 25 M UDP

- 025M UDP

- 25 M UDP

- 025M UDP

- 25 M UDP

- 025M UDP

- 25 M UDP

- 025M UDP

Differential DAG KineticsDifferential DAG KineticsDifferential DAG KineticsDifferential DAG Kinetics

bull Time based behavior of three DAG species with varying degrees of unsaturation after Time based behavior of three DAG species with varying degrees of unsaturation after addition of addition of 25 25 MM (solid red triangles) and (solid red triangles) and 025 025 MM (solid green squares) UDP (solid green squares) UDP

bull Time points contain nine replicates performed on three different experimental daysTime points contain nine replicates performed on three different experimental days

341 DAG

Best fit with current model structure

Overall Objectives of Modeling EffortOverall Objectives of Modeling EffortOverall Objectives of Modeling EffortOverall Objectives of Modeling Effort

bull Predict quantitative changes in lipid species after stimulation by Predict quantitative changes in lipid species after stimulation by various ligands and ligand concentrations in the RAW 2647 various ligands and ligand concentrations in the RAW 2647 macrophagemacrophage

bull Comparison and refinement of model output with AfCS IPComparison and refinement of model output with AfCS IP33 measurements and Cameasurements and Ca2+2+ traces as well as DAG data generated in traces as well as DAG data generated in the Brown labthe Brown lab

bull Predict in silico effects such as the effect of knock-downs etc on Predict in silico effects such as the effect of knock-downs etc on given pathwaygiven pathway

bull Suggest modifications to current pathway structuresSuggest modifications to current pathway structures

Modifications to the modelModifications to the modelModifications to the modelModifications to the model

bull Include an additional branch in the pathway Include an additional branch in the pathway for a second pool of DAGfor a second pool of DAG

bull Simplify CaSimplify Ca2+2+ equations for mathematical equations for mathematical analysis purposesanalysis purposes

Proposed PathwayProposed PathwayProposed PathwayProposed Pathway

bull Note We measure total cellular DAG levelsNote We measure total cellular DAG levelsbull Initial production of DAG from the hydrolysis of PIPInitial production of DAG from the hydrolysis of PIP22 in pool 1 (plasma in pool 1 (plasma

membrane) is offset by phosphorylation of DAG by DAG kinase in pool 2 membrane) is offset by phosphorylation of DAG by DAG kinase in pool 2 (Endoplasmic Reticulum Nucleus) to aid in the PI replacement pathway(Endoplasmic Reticulum Nucleus) to aid in the PI replacement pathway

bull Second wave of DAG is a result of resynthesis of PIPSecond wave of DAG is a result of resynthesis of PIP22 which is then which is then hydrolyzed to form DAG and IPhydrolyzed to form DAG and IP33

PI PIP PIP2 DAG

IP3

IP2IP

Ins

PI

CDP-DAGPADAG

PA

PCPE

PCPE

PITP

PI4K PI5K PLC DGK

DGK CDSPLC PIS

POOL 2

POOL 1

IP4

PI PIP PIP2 DAG

IP3

IP2IP

Ins

PI

CDP-DAGPADAG

PA

PCPE

PC

PITP

PI4K PI5K PLC

DGK CDSPLC PIS

IP4

LPP

LPP

Model Equations DAG (pool 1)Model Equations DAG (pool 1)Model Equations DAG (pool 1)Model Equations DAG (pool 1)

bull DAG (pool 1)DAG (pool 1) (separate ODEs for each species (separate ODEs for each species considered)considered)

bull Concentration of DAG from pool 1 at time tConcentration of DAG from pool 1 at time t

Nondimensionalized equationsNondimensionalized equations

Model Equations DAG (pool 2)Model Equations DAG (pool 2)Model Equations DAG (pool 2)Model Equations DAG (pool 2)

bull DAG (pool 2)DAG (pool 2) (separate ODEs for each species (separate ODEs for each species considered)considered)

bull Concentration of pool 2 DAG molecules at time tConcentration of pool 2 DAG molecules at time t

bull Baseline concentration of pool 2 DAGBaseline concentration of pool 2 DAG

Nondimensionalized equationsNondimensionalized equations

Simplified CaSimplified Ca2+2+ Equations EquationsSimplified CaSimplified Ca2+2+ Equations Equations

bull CaCa2+2+ modulemodule (to match experimental AfCS (to match experimental AfCS trace)trace)

0 100 200 300 400 500 6000

002

004

006

008

01

012

014

Concentration of Ca

2+ (uM)

Ca2+ response to 25uM UDP

time (s)

wherewhere

Theoretical AnalysisTheoretical AnalysisTheoretical AnalysisTheoretical Analysis

bull Existence and UniquenessExistence and Uniqueness

bull Positivity and Boundedness Positivity and Boundedness

(for biological relevance)(for biological relevance)

bull Analysis of steady state behaviorAnalysis of steady state behavior

Existence of Solutions (full model)Existence of Solutions (full model)Existence of Solutions (full model)Existence of Solutions (full model)

First we write our system of ODEs in the formFirst we write our system of ODEs in the form (1)

(2)

(1)

Uniqueness of SolutionsUniqueness of SolutionsUniqueness of SolutionsUniqueness of Solutions

Next we denote a solution of Next we denote a solution of (1)(1) by by with initial condition with initial condition

Since our system satisfies the hypotheses of Theorems 1 and 2 Since our system satisfies the hypotheses of Theorems 1 and 2 on our set of interest (for all positive time and on a positive on our set of interest (for all positive time and on a positive bounded set in space) we know bounded set in space) we know there exists a unique there exists a unique (local) solution(local) solution (ie on some finite time interval possibly (ie on some finite time interval possibly small)small)

QuestionQuestion Do the solutions remain positive and bounded (for Do the solutions remain positive and bounded (for biological significance)biological significance)

(3)

(1) (3)

Positivity and boundedness of Positivity and boundedness of solutionssolutions

Positivity and boundedness of Positivity and boundedness of solutionssolutions

bull We first use Theorem 3 to show positivity and boundedness of xWe first use Theorem 3 to show positivity and boundedness of x11 and and xx22

bull Next we use these results and the Fundamental Theorem of Calculus to Next we use these results and the Fundamental Theorem of Calculus to show positivity and boundedness for the remaining equations show positivity and boundedness for the remaining equations

bull This then ensures a global solutionThis then ensures a global solution

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical Analysis

bull Parameter Estimation in SIMULINKParameter Estimation in SIMULINK

bull Sensitivity AnalysisSensitivity Analysis

Parameter Estimation in SIMULINK (for Parameter Estimation in SIMULINK (for FULL model)FULL model)

Parameter Estimation in SIMULINK (for Parameter Estimation in SIMULINK (for FULL model)FULL model)

bull Total number of model parameters (for full model with Li and Rinzel CaTotal number of model parameters (for full model with Li and Rinzel Ca2+2+ module) = module) = 3434ndash From the literature = From the literature = 2020ndash Estimated = Estimated = 1414

bull Receptor moduleReceptor module Total = Total = 77ndash From literature = From literature = 66ndash Estimated = Estimated = 11 (k (kpp rate of receptor phosphorylation) rate of receptor phosphorylation)

bull G-protein cascadeG-protein cascade Total = Total = 99ndash From literature = From literature = 66ndash Estimated = Estimated = 3 3 (k(khydhyd k kreprep k kd3d3))

bull DAG kineticsDAG kinetics Total = Total = 5 5 (for each DAG species considered)(for each DAG species considered)ndash From literature = From literature = 00ndash Estimated = Estimated = 5 5 (all DAG parameters)(all DAG parameters)

bull CaCa2+2+ module module Total = Total = 1313ndash From literature = From literature = 88ndash Estimated = Estimated = 55

SIMULINK DetailsSIMULINK DetailsSIMULINK DetailsSIMULINK Details

bull Unknown rate parameters were estimated using SIMULINK

ndash Minimizes a user-specified cost function via a user-specified optimization method

ndash Nonlinear least squares optimization method of Levenberg-Marquardt was used to minimize a sum of squared errors cost function of the empirical observations and model predictions for IP3 Ca2+ and multiple species of DAG

bull Note Although the Gauss-Newton method is often more efficient the method of Levenberg-Marquardt has proved to be more robust

Measured vs Simulated 384 DAG and Measured vs Simulated 384 DAG and 341 DAG341 DAG

Measured vs Simulated 384 DAG and Measured vs Simulated 384 DAG and 341 DAG341 DAG

bull Timecourse of stimulation with 25 Timecourse of stimulation with 25 M UDP in RAW 2647 cellsM UDP in RAW 2647 cells

bull Solid black lines represent model simulations Solid black lines represent model simulations

bull ((aa) 384 DAG response (representative of the response of most poly ) 384 DAG response (representative of the response of most poly unsaturated fatty acid-containing DAG species) unsaturated fatty acid-containing DAG species)

bull ((bb) 341 DAG response (representative of the response of most mono- ) 341 DAG response (representative of the response of most mono- and di-unsaturated fatty acid-containing DAG species) and di-unsaturated fatty acid-containing DAG species)

bull Data points contain nine replicates performed on three different Data points contain nine replicates performed on three different experimental days with error bars = 1 SEM Units are total change in ng experimental days with error bars = 1 SEM Units are total change in ng over baseline levels in ~8x10over baseline levels in ~8x1066 cells cells

Measured vs Simulated IPMeasured vs Simulated IP33 and Ca and Ca2+2+Measured vs Simulated IPMeasured vs Simulated IP33 and Ca and Ca2+2+

bull Timecourse of stimulation with 25 Timecourse of stimulation with 25 M UDP in RAW 2647 cellsM UDP in RAW 2647 cells

bull Solid black lines represent model simulationsSolid black lines represent model simulations

bull ((cc) IP) IP33 response in pmols per ~35x10 response in pmols per ~35x1055 cells cells

bull Points in (c) represent the average of four experiments and error Points in (c) represent the average of four experiments and error bars are 1 SEM bars are 1 SEM

bull ((dd) Ca) Ca2+2+ response in response in M Red curve is a representative CaM Red curve is a representative Ca2+2+ trace trace taken from the UDP experiments within the AfCS single ligand taken from the UDP experiments within the AfCS single ligand screenscreen

Model Simulations P2YModel Simulations P2Y66 G GGTP GTP PIPPIP22

Model Simulations P2YModel Simulations P2Y66 G GGTP GTP PIPPIP22

bull ((aa) = total P2Y) = total P2Y66 activated (solid activated (solid line) and inactivated (dashed line) and inactivated (dashed line) surface receptorsline) surface receptors

bull ((bb) = total G) = total GGTP GTP

bull ((cc) = total PIP) = total PIP22 available for available for hydrolysishydrolysis

P2Y6 from 25M UDP GGTP from 25M UDP

PIP2 from 25M UDP

Sensitivity AnalysisSensitivity AnalysisSensitivity AnalysisSensitivity Analysis

bull Sensitivity analysis techniques are valuable tools Sensitivity analysis techniques are valuable tools designed to answer questions regarding which of the designed to answer questions regarding which of the uncertain input variables is more important in uncertain input variables is more important in determining the uncertainty in our output determining the uncertainty in our output

bull Likewise sensitivity analysis can provide insight into Likewise sensitivity analysis can provide insight into which parameter should be studied in more detail in which parameter should be studied in more detail in order to reduce the most variance in the model output order to reduce the most variance in the model output

bull The ability to answer these types of questions could The ability to answer these types of questions could lead to important insight into the design of new lead to important insight into the design of new experiments and in determining which experiments experiments and in determining which experiments would give us the most valuable informationwould give us the most valuable information

Sampling MethodSampling MethodSampling MethodSampling Method

bull Generate a random sample of our space of input Generate a random sample of our space of input variables over a ten percent variation from each variables over a ten percent variation from each parameters nominal value using the parameters nominal value using the Latin Hypercube Latin Hypercube Sampling (LHS) methodSampling (LHS) method

bull Uses Uses Standardized Regression Coefficients (SRCs) Standardized Regression Coefficients (SRCs) obtained by performing multiple linear regression obtained by performing multiple linear regression analysisanalysisndash offers a measure of sensitivity that is multi-dimensionally offers a measure of sensitivity that is multi-dimensionally

averaged over the entire space of parameter values averaged over the entire space of parameter values ndash SRCs give insight into degree of nonlinearity in the model SRCs give insight into degree of nonlinearity in the model

Computing RComputing Ryy22 values valuesComputing RComputing Ryy22 values values

bull SRCs are only reliable measures of sensitivity when SRCs are only reliable measures of sensitivity when degree of nonlinearity is ldquosmallrdquodegree of nonlinearity is ldquosmallrdquo

bull Use model coefficients of determination RUse model coefficients of determination Ryy22 given by given by

bull where ywhere yii is the estimate of y is the estimate of yii obtained from the regression obtained from the regression modelmodel

bull RRyy22 ge 07 ensures SRCs are good sensitivity measures ge 07 ensures SRCs are good sensitivity measures

SRCs (as sensitivity measures) for SRCs (as sensitivity measures) for DAGDAG

SRCs (as sensitivity measures) for SRCs (as sensitivity measures) for DAGDAG

SRCs (as sensitivity measures) for IPSRCs (as sensitivity measures) for IP33SRCs (as sensitivity measures) for IPSRCs (as sensitivity measures) for IP33

Effects of 3 most sensitive DAG Effects of 3 most sensitive DAG parametersparameters

Effects of 3 most sensitive DAG Effects of 3 most sensitive DAG parametersparameters

A B

C

AA k kdp2dp2 degradation of pool 2 DAG degradation of pool 2 DAG

BB k kdp1dp1 degradation of pool 1 DAG degradation of pool 1 DAG

CC k kap2ap2 production of pool 2 DAG production of pool 2 DAG

ConclusionsConclusionsConclusionsConclusions

bull We have developed a model of the UDP signaling We have developed a model of the UDP signaling pathway in RAW 2647 macrophages which can predict pathway in RAW 2647 macrophages which can predict the responses of multiple species of DAG as well as the the responses of multiple species of DAG as well as the responses of IPresponses of IP33 Ca Ca2+2+ receptor dynamics G-protein receptor dynamics G-protein activation and PIPactivation and PIP22 hydrolysis hydrolysis

bull Simplified model resultsSimplified model resultsndash We have obtained global existence uniqueness positivity We have obtained global existence uniqueness positivity

and boundedness of solutionsand boundedness of solutionsndash We have proven global stability of a unique steady state We have proven global stability of a unique steady state

within our region of interestwithin our region of interestbull Full model analysisFull model analysis

ndash Using SIMULINK we have estimated unknown rate Using SIMULINK we have estimated unknown rate parameters to obtain best fits to multiple DAG traces IPparameters to obtain best fits to multiple DAG traces IP33 and Caand Ca2+2+ all in response to 25 all in response to 25M UDPM UDP

ndash We have performed sensitivity analysis using the Latin We have performed sensitivity analysis using the Latin Hypercube sampling technique in combination with Hypercube sampling technique in combination with standardized regression coefficients to determine which standardized regression coefficients to determine which model parameters are responsible for most of the model model parameters are responsible for most of the model output uncertaintyoutput uncertainty

Future DirectionsFuture DirectionsFuture DirectionsFuture Directions

bull We have conducted multiple experiments to We have conducted multiple experiments to pharmacologically inhibit several different enzymes we pharmacologically inhibit several different enzymes we believe to play a role in this signaling pathway as believe to play a role in this signaling pathway as suggested by current known metabolic pathways and by suggested by current known metabolic pathways and by modeling resultsmodeling results

bull The next step is to perform gene knockdowns on The next step is to perform gene knockdowns on specific enzymes to verify results of inhibitor dataspecific enzymes to verify results of inhibitor data

bull The model output also suggests a time delay from The model output also suggests a time delay from receptor activation to PIPreceptor activation to PIP22 hydrolysis An upcoming task hydrolysis An upcoming task is to investigate the outcomes of adding such a delay is to investigate the outcomes of adding such a delay termterm

bull The model could be greatly enhanced by incorporating The model could be greatly enhanced by incorporating spatial dynamics so this will also be a major focus for spatial dynamics so this will also be a major focus for future researchfuture research

AcknowledgementsAcknowledgementsAcknowledgementsAcknowledgements

Mathematics DepartmentMathematics Departmentbull Hannah L Callender Hannah L Callender

PhDPhD

CollaboratorsCollaboratorsbull Alliance for Cellular SignalingAlliance for Cellular Signalingbull UT SouthwesternUT Southwestern

ndash Paul Sternweis PhDPaul Sternweis PhDndash Dianne DeCamp PhDDianne DeCamp PhD

Brown LabBrown Labbull H Alex Brown PhDH Alex Brown PhDbull Jeffrey S Forrester PhDJeffrey S Forrester PhDbull Mark Byrne PhDMark Byrne PhDbull Anita Preininger PhDAnita Preininger PhDbull Michelle ArmstrongMichelle Armstrongbull Andrew GoodmanAndrew Goodmanbull Pavlina Ivanova PhDPavlina Ivanova PhDbull Steve Milne PhDSteve Milne PhD

Collaboration with Alliance for Cellular Collaboration with Alliance for Cellular SignalingSignaling

Collaboration with Alliance for Cellular Collaboration with Alliance for Cellular SignalingSignaling

bull Alliance for Cellular Signaling (AfCS) ldquoDetermine how cells Alliance for Cellular Signaling (AfCS) ldquoDetermine how cells process signaling information in a context dependent mannerrdquoprocess signaling information in a context dependent mannerrdquo

bull AfCS Screen a large number of ligands (signaling molecules) AfCS Screen a large number of ligands (signaling molecules) and analyze their effects on the and analyze their effects on the RAW 2647 cellRAW 2647 cell (a (a macrophage-like cell line derived from mice a macrophage-like cell line derived from mice a macrophagemacrophage is is a type of white blood cell that surrounds and kills a type of white blood cell that surrounds and kills microorganisms removes dead cells and stimulates the action microorganisms removes dead cells and stimulates the action of other immune system cells) of other immune system cells)

bull FocusFocus Construction of a comprehensive mathematical model Construction of a comprehensive mathematical model for the signaling pathway of one ligand uridine 5rsquo-diphosphate for the signaling pathway of one ligand uridine 5rsquo-diphosphate ((UDPUDP) in the RAW 2647 cell) in the RAW 2647 cellndash UDP is a common signaling nucleotide often released when cells UDP is a common signaling nucleotide often released when cells

undergo some type of trauma or mechanical stimulusundergo some type of trauma or mechanical stimulus

bull Modeling approachModeling approach Utilize a system of nonlinear ordinary Utilize a system of nonlinear ordinary differential equations designed to model the time-dependent differential equations designed to model the time-dependent behavior of key players in the pathway behavior of key players in the pathway

Basics of Cellular SignalingBasics of Cellular SignalingBasics of Cellular SignalingBasics of Cellular Signaling

Signaling molecules bind to cell Signaling molecules bind to cell surface receptors surface receptors

Initiate a series of intracellular Initiate a series of intracellular reactions reactions

These reactions regulate virtually These reactions regulate virtually all aspects of cell behavior (including all aspects of cell behavior (including metabolism movement metabolism movement proliferation survival and proliferation survival and differentiation)differentiation)

bull Breakdown in the signaling pathways that control normal cell Breakdown in the signaling pathways that control normal cell proliferation and survival result in many types of cancerproliferation and survival result in many types of cancer Need for better understandingNeed for better understanding Mathematical modeling can provide insight into these signaling Mathematical modeling can provide insight into these signaling

pathwayspathways

Canonical UDP signaling pathwayCanonical UDP signaling pathwayCanonical UDP signaling pathwayCanonical UDP signaling pathway

11 Ligand or stimulus Ligand or stimulus ((UDPUDP) binds to a ) binds to a specific G-protein specific G-protein coupled receptor coupled receptor ((P2YP2Y66))

33 GGqq GTPGTP activates activates phospholipase C-phospholipase C-3 (3 (PLC-PLC-33))

44 Active Active PLC-PLC-33 cleaves cleaves phosphatidylinositol 45-phosphatidylinositol 45-bisphosphate (bisphosphate (PIPPIP22) into ) into inositol 145-trisphosphate inositol 145-trisphosphate ((IPIP33) and diacylglycerol ) and diacylglycerol ((DAGDAG))

22 The ligand bound receptor The ligand bound receptor causes the exchange of causes the exchange of GDPGDP for for GTPGTP on the on the GGqq subunit of the G-proteinsubunit of the G-protein

55 IPIP33 binds to a specific receptor binds to a specific receptor on the endoplasmic reticulum on the endoplasmic reticulum releasing sequestered releasing sequestered CaCa2+2+

UDP

P2Y

6

GDPG PLC

Ca2+

PIP

2

DAG

IP3

GGTP

IP3

IP3-R

IP3 -R

Plasma Membrane

Endoplasmic Reticulum

Ca2+

Ca2+

Ca2+

UDP

GGTP

Ca2+ Ca2+

Ca2+

Ca2+

Ca2+

Ca2+

Mathematical ModelMathematical ModelMathematical ModelMathematical Model

bull The model consists of a base set of 10 coupled nonlinear The model consists of a base set of 10 coupled nonlinear ordinary differential equations (ODEs)ordinary differential equations (ODEs)

bull The ODEs are constructed using mass action kinetics to The ODEs are constructed using mass action kinetics to approximate diffusion binding and unbinding of molecular approximate diffusion binding and unbinding of molecular species in various compartments of the RAW cellspecies in various compartments of the RAW cell effective rates of production and degradation are proportional to effective rates of production and degradation are proportional to

the concentrations (or number of molecules) of participating the concentrations (or number of molecules) of participating reactantsreactants

bull Four modulesFour modulesndash Receptor regulationReceptor regulationndash G-protein casadeG-protein casadendash Species-specific DAG dynamicsSpecies-specific DAG dynamicsndash Cytosolic CaCytosolic Ca2+2+ dynamics dynamics

bull Receptor-ligandReceptor-ligand binding (based on work by Lemon 2003) binding (based on work by Lemon 2003)

ndash Number of ldquoactivatedrdquo surface receptors at time tNumber of ldquoactivatedrdquo surface receptors at time tndash Number of ldquoinactivatedrdquo surface receptors at time tNumber of ldquoinactivatedrdquo surface receptors at time tndash Concentration of ligand (constant)Concentration of ligand (constant)

Model Equations Receptor Model Equations Receptor regulationregulation

Model Equations Receptor Model Equations Receptor regulationregulation

Nondimensionalized equationsNondimensionalized equations

Model Equations G-protein Model Equations G-protein activationactivation

Model Equations G-protein Model Equations G-protein activationactivation

bull G-proteinG-protein activation activation

ndash Number of activated G-proteins (GNumber of activated G-proteins (GGTP) at time tGTP) at time t

Nondimensionalized equationsNondimensionalized equations

Model Equations PIPModel Equations PIP22 hydrolysis and hydrolysis and replenishmentreplenishment

Model Equations PIPModel Equations PIP22 hydrolysis and hydrolysis and replenishmentreplenishment

bull PIPPIP2 2 hydrolysis and replenishmenthydrolysis and replenishment

Nondimensionalized equationsNondimensionalized equations

ndash Number of PIPNumber of PIP22 molecules at time t molecules at time tndash Initial number of ldquofreerdquo PIPInitial number of ldquofreerdquo PIP22

PLC

PIP

2

DAG

IP3

GGTPCa2+

GGTP

bull IPIP33

Model Equations IPModel Equations IP33 production and production and degradationdegradation

Model Equations IPModel Equations IP33 production and production and degradationdegradation

Nondimensionalized equationsNondimensionalized equations

PLC

PIP

2

DA

G

IP3

G

GTPCa2+

G

GTP

bull Concentration of IPConcentration of IP33 at time t (in at time t (in M)M)

Model Equations DAGModel Equations DAGModel Equations DAGModel Equations DAG

bull DAG DAG (separate ODEs for each species considered)(separate ODEs for each species considered)

Nondimensionalized equationsNondimensionalized equations

bull Concentration of DAG at time tConcentration of DAG at time t

PLC

PIP

2

DA

G

IP3

G

GTPCa2+G

GTP

Model Equations Li and Rinzel (1994) Model Equations Li and Rinzel (1994) CaCa2+2+

Model Equations Li and Rinzel (1994) Model Equations Li and Rinzel (1994) CaCa2+2+

bull Free cytosolicFree cytosolic Ca Ca2+2+

bull hh = fraction of IP = fraction of IP33 channels not yet inactivated by Ca channels not yet inactivated by Ca2+2+

bull cc00 = concentration of total free Ca = concentration of total free Ca2+2+ per cytosolic volume per cytosolic volume

Timeseries data for the modelTimeseries data for the modelTimeseries data for the modelTimeseries data for the model

bull Data from collaborators at UTSW (Sternweis Lab)Data from collaborators at UTSW (Sternweis Lab)ndash Inositol 145-trisphosphate (Inositol 145-trisphosphate (IPIP33) production post 25) production post 25M M

UDP stimulation in the RAW cellUDP stimulation in the RAW cellndash Cytosolic calcium (Cytosolic calcium (CaCa2+2+) release post stimulation with ) release post stimulation with

2525M UDP in the RAW cellM UDP in the RAW cell

bull Diacylglycerol (Diacylglycerol (DAGDAG) data H Callender collected from a ) data H Callender collected from a novel method of quantitative analysis of multiple novel method of quantitative analysis of multiple species of DAG developed in the Brown Lab species of DAG developed in the Brown Lab

Reference Reference

Callender H L et al Quantification of Diacylglycerol Species from Callender H L et al Quantification of Diacylglycerol Species from Cellular Extracts by Electrospray Ionization Mass Spectrometry Using a Cellular Extracts by Electrospray Ionization Mass Spectrometry Using a Linear Regression Algorithm Linear Regression Algorithm Anal Chem Anal Chem 7979 (2007) 263-272 (2007) 263-272

IPIP33 and Ca and Ca2+2+ data from AfCS data from AfCSIPIP33 and Ca and Ca2+2+ data from AfCS data from AfCS

AA

bull IPIP33 response (in pmoles per response (in pmoles per 100 100 L lysate)in RAW 2647 L lysate)in RAW 2647 cells to 25 cells to 25 M UDPM UDP

bull Points represent the average Points represent the average of four experimentsof four experiments

bull Cytosolic calcium response Cytosolic calcium response (in (in M) in the RAW 2647 M) in the RAW 2647 cell to 25 cell to 25 M UDP M UDP

bull The graph displays 43 The graph displays 43 experimental repeatsexperimental repeats

B

What is DAGWhat is DAGWhat is DAGWhat is DAG

bull DAG is a cellular second messenger molecule which plays an DAG is a cellular second messenger molecule which plays an important role in initiating various changes in cell behavior important role in initiating various changes in cell behavior including cell activation differentiation proliferation and including cell activation differentiation proliferation and tumor promotion tumor promotion

bull There are many different species of DAG depending on the There are many different species of DAG depending on the number of carbons and number of double bonds in the fatty number of carbons and number of double bonds in the fatty acyl chains and different species can have different cellular acyl chains and different species can have different cellular functionsfunctions

bull ExampleExample 320 DAG320 DAG

Two fatty acyl Two fatty acyl (hydrocarbon) (hydrocarbon) chains in every DAG chains in every DAG speciesspecies

Each corner Each corner represents carbonrepresents carbon

320 DAG 18 carbons in first chain 14 in the second no double bonds 320 DAG 18 carbons in first chain 14 in the second no double bonds in either chainin either chain

Kinetics of monodi unsaturated DAGsKinetics of monodi unsaturated DAGsKinetics of monodi unsaturated DAGsKinetics of monodi unsaturated DAGs

bull Time based behavior of four monodi unsaturated DAG species after addition of 25 Time based behavior of four monodi unsaturated DAG species after addition of 25 M M (solid red squares) and 025 (solid red squares) and 025 M (solid green triangles) UDPM (solid green triangles) UDP

bull Time points contain a minimum of nine replicates performed on three different Time points contain a minimum of nine replicates performed on three different experimental daysexperimental days

- 25 M UDP

- 025M UDP

0 5 10 15 20 25 30

0 5 10 15 20 25 30 0 5 10 15 20 25 30

0 5 10 15 20 25 30

321 DAG 341 DAG

362 DAG341 DAGep

150

100

50-5

00

300

200

100

-100

0

150

100

50-5

00

-100

200

6040

20-2

00

-40

80

Time (min) Time (min)

Time (min) Time (min)

D

AG

(n

g)

D

AG

(n

g)

D

AG

(n

g)

D

AG

(n

g)

- 25 M UDP

- 025M UDP

- 25 M UDP

- 025M UDP

- 25 M UDP

- 025M UDP

Kinetics of PUFA containing DAGsKinetics of PUFA containing DAGsKinetics of PUFA containing DAGsKinetics of PUFA containing DAGsbull Time based behavior of four polyunsaturated fatty acid (PUFA) containing DAG species Time based behavior of four polyunsaturated fatty acid (PUFA) containing DAG species

after addition of 25 after addition of 25 M (solid red squares) and 025 M (solid red squares) and 025 M (solid green triangles) UDPM (solid green triangles) UDPbull Time points contain a minimum of nine replicates performed on three different Time points contain a minimum of nine replicates performed on three different

experimental daysexperimental daysbull Monodi unsaturated DAG species give a larger increase than polyunsaturated fatty acid Monodi unsaturated DAG species give a larger increase than polyunsaturated fatty acid

(PUFA) containing DAGs(PUFA) containing DAGs

0 5 10 15 20 25 30

0 5 10 15 20 25 30 0 5 10 15 20 25 30

0 5 10 15 20 25 30

385 DAG 383 DAG

364 DAG384 DAG6

42

-40

30

20

10

0

15

10

50

-56

40

2-2

8

Time (min) Time (min)

Time (min) Time (min)

D

AG

(n

g)

D

AG

(n

g)

D

AG

(n

g)

D

AG

(n

g)

-28

40

- 25 M UDP

- 025M UDP

- 25 M UDP

- 025M UDP

- 25 M UDP

- 025M UDP

- 25 M UDP

- 025M UDP

Differential DAG KineticsDifferential DAG KineticsDifferential DAG KineticsDifferential DAG Kinetics

bull Time based behavior of three DAG species with varying degrees of unsaturation after Time based behavior of three DAG species with varying degrees of unsaturation after addition of addition of 25 25 MM (solid red triangles) and (solid red triangles) and 025 025 MM (solid green squares) UDP (solid green squares) UDP

bull Time points contain nine replicates performed on three different experimental daysTime points contain nine replicates performed on three different experimental days

341 DAG

Best fit with current model structure

Overall Objectives of Modeling EffortOverall Objectives of Modeling EffortOverall Objectives of Modeling EffortOverall Objectives of Modeling Effort

bull Predict quantitative changes in lipid species after stimulation by Predict quantitative changes in lipid species after stimulation by various ligands and ligand concentrations in the RAW 2647 various ligands and ligand concentrations in the RAW 2647 macrophagemacrophage

bull Comparison and refinement of model output with AfCS IPComparison and refinement of model output with AfCS IP33 measurements and Cameasurements and Ca2+2+ traces as well as DAG data generated in traces as well as DAG data generated in the Brown labthe Brown lab

bull Predict in silico effects such as the effect of knock-downs etc on Predict in silico effects such as the effect of knock-downs etc on given pathwaygiven pathway

bull Suggest modifications to current pathway structuresSuggest modifications to current pathway structures

Modifications to the modelModifications to the modelModifications to the modelModifications to the model

bull Include an additional branch in the pathway Include an additional branch in the pathway for a second pool of DAGfor a second pool of DAG

bull Simplify CaSimplify Ca2+2+ equations for mathematical equations for mathematical analysis purposesanalysis purposes

Proposed PathwayProposed PathwayProposed PathwayProposed Pathway

bull Note We measure total cellular DAG levelsNote We measure total cellular DAG levelsbull Initial production of DAG from the hydrolysis of PIPInitial production of DAG from the hydrolysis of PIP22 in pool 1 (plasma in pool 1 (plasma

membrane) is offset by phosphorylation of DAG by DAG kinase in pool 2 membrane) is offset by phosphorylation of DAG by DAG kinase in pool 2 (Endoplasmic Reticulum Nucleus) to aid in the PI replacement pathway(Endoplasmic Reticulum Nucleus) to aid in the PI replacement pathway

bull Second wave of DAG is a result of resynthesis of PIPSecond wave of DAG is a result of resynthesis of PIP22 which is then which is then hydrolyzed to form DAG and IPhydrolyzed to form DAG and IP33

PI PIP PIP2 DAG

IP3

IP2IP

Ins

PI

CDP-DAGPADAG

PA

PCPE

PCPE

PITP

PI4K PI5K PLC DGK

DGK CDSPLC PIS

POOL 2

POOL 1

IP4

PI PIP PIP2 DAG

IP3

IP2IP

Ins

PI

CDP-DAGPADAG

PA

PCPE

PC

PITP

PI4K PI5K PLC

DGK CDSPLC PIS

IP4

LPP

LPP

Model Equations DAG (pool 1)Model Equations DAG (pool 1)Model Equations DAG (pool 1)Model Equations DAG (pool 1)

bull DAG (pool 1)DAG (pool 1) (separate ODEs for each species (separate ODEs for each species considered)considered)

bull Concentration of DAG from pool 1 at time tConcentration of DAG from pool 1 at time t

Nondimensionalized equationsNondimensionalized equations

Model Equations DAG (pool 2)Model Equations DAG (pool 2)Model Equations DAG (pool 2)Model Equations DAG (pool 2)

bull DAG (pool 2)DAG (pool 2) (separate ODEs for each species (separate ODEs for each species considered)considered)

bull Concentration of pool 2 DAG molecules at time tConcentration of pool 2 DAG molecules at time t

bull Baseline concentration of pool 2 DAGBaseline concentration of pool 2 DAG

Nondimensionalized equationsNondimensionalized equations

Simplified CaSimplified Ca2+2+ Equations EquationsSimplified CaSimplified Ca2+2+ Equations Equations

bull CaCa2+2+ modulemodule (to match experimental AfCS (to match experimental AfCS trace)trace)

0 100 200 300 400 500 6000

002

004

006

008

01

012

014

Concentration of Ca

2+ (uM)

Ca2+ response to 25uM UDP

time (s)

wherewhere

Theoretical AnalysisTheoretical AnalysisTheoretical AnalysisTheoretical Analysis

bull Existence and UniquenessExistence and Uniqueness

bull Positivity and Boundedness Positivity and Boundedness

(for biological relevance)(for biological relevance)

bull Analysis of steady state behaviorAnalysis of steady state behavior

Existence of Solutions (full model)Existence of Solutions (full model)Existence of Solutions (full model)Existence of Solutions (full model)

First we write our system of ODEs in the formFirst we write our system of ODEs in the form (1)

(2)

(1)

Uniqueness of SolutionsUniqueness of SolutionsUniqueness of SolutionsUniqueness of Solutions

Next we denote a solution of Next we denote a solution of (1)(1) by by with initial condition with initial condition

Since our system satisfies the hypotheses of Theorems 1 and 2 Since our system satisfies the hypotheses of Theorems 1 and 2 on our set of interest (for all positive time and on a positive on our set of interest (for all positive time and on a positive bounded set in space) we know bounded set in space) we know there exists a unique there exists a unique (local) solution(local) solution (ie on some finite time interval possibly (ie on some finite time interval possibly small)small)

QuestionQuestion Do the solutions remain positive and bounded (for Do the solutions remain positive and bounded (for biological significance)biological significance)

(3)

(1) (3)

Positivity and boundedness of Positivity and boundedness of solutionssolutions

Positivity and boundedness of Positivity and boundedness of solutionssolutions

bull We first use Theorem 3 to show positivity and boundedness of xWe first use Theorem 3 to show positivity and boundedness of x11 and and xx22

bull Next we use these results and the Fundamental Theorem of Calculus to Next we use these results and the Fundamental Theorem of Calculus to show positivity and boundedness for the remaining equations show positivity and boundedness for the remaining equations

bull This then ensures a global solutionThis then ensures a global solution

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical Analysis

bull Parameter Estimation in SIMULINKParameter Estimation in SIMULINK

bull Sensitivity AnalysisSensitivity Analysis

Parameter Estimation in SIMULINK (for Parameter Estimation in SIMULINK (for FULL model)FULL model)

Parameter Estimation in SIMULINK (for Parameter Estimation in SIMULINK (for FULL model)FULL model)

bull Total number of model parameters (for full model with Li and Rinzel CaTotal number of model parameters (for full model with Li and Rinzel Ca2+2+ module) = module) = 3434ndash From the literature = From the literature = 2020ndash Estimated = Estimated = 1414

bull Receptor moduleReceptor module Total = Total = 77ndash From literature = From literature = 66ndash Estimated = Estimated = 11 (k (kpp rate of receptor phosphorylation) rate of receptor phosphorylation)

bull G-protein cascadeG-protein cascade Total = Total = 99ndash From literature = From literature = 66ndash Estimated = Estimated = 3 3 (k(khydhyd k kreprep k kd3d3))

bull DAG kineticsDAG kinetics Total = Total = 5 5 (for each DAG species considered)(for each DAG species considered)ndash From literature = From literature = 00ndash Estimated = Estimated = 5 5 (all DAG parameters)(all DAG parameters)

bull CaCa2+2+ module module Total = Total = 1313ndash From literature = From literature = 88ndash Estimated = Estimated = 55

SIMULINK DetailsSIMULINK DetailsSIMULINK DetailsSIMULINK Details

bull Unknown rate parameters were estimated using SIMULINK

ndash Minimizes a user-specified cost function via a user-specified optimization method

ndash Nonlinear least squares optimization method of Levenberg-Marquardt was used to minimize a sum of squared errors cost function of the empirical observations and model predictions for IP3 Ca2+ and multiple species of DAG

bull Note Although the Gauss-Newton method is often more efficient the method of Levenberg-Marquardt has proved to be more robust

Measured vs Simulated 384 DAG and Measured vs Simulated 384 DAG and 341 DAG341 DAG

Measured vs Simulated 384 DAG and Measured vs Simulated 384 DAG and 341 DAG341 DAG

bull Timecourse of stimulation with 25 Timecourse of stimulation with 25 M UDP in RAW 2647 cellsM UDP in RAW 2647 cells

bull Solid black lines represent model simulations Solid black lines represent model simulations

bull ((aa) 384 DAG response (representative of the response of most poly ) 384 DAG response (representative of the response of most poly unsaturated fatty acid-containing DAG species) unsaturated fatty acid-containing DAG species)

bull ((bb) 341 DAG response (representative of the response of most mono- ) 341 DAG response (representative of the response of most mono- and di-unsaturated fatty acid-containing DAG species) and di-unsaturated fatty acid-containing DAG species)

bull Data points contain nine replicates performed on three different Data points contain nine replicates performed on three different experimental days with error bars = 1 SEM Units are total change in ng experimental days with error bars = 1 SEM Units are total change in ng over baseline levels in ~8x10over baseline levels in ~8x1066 cells cells

Measured vs Simulated IPMeasured vs Simulated IP33 and Ca and Ca2+2+Measured vs Simulated IPMeasured vs Simulated IP33 and Ca and Ca2+2+

bull Timecourse of stimulation with 25 Timecourse of stimulation with 25 M UDP in RAW 2647 cellsM UDP in RAW 2647 cells

bull Solid black lines represent model simulationsSolid black lines represent model simulations

bull ((cc) IP) IP33 response in pmols per ~35x10 response in pmols per ~35x1055 cells cells

bull Points in (c) represent the average of four experiments and error Points in (c) represent the average of four experiments and error bars are 1 SEM bars are 1 SEM

bull ((dd) Ca) Ca2+2+ response in response in M Red curve is a representative CaM Red curve is a representative Ca2+2+ trace trace taken from the UDP experiments within the AfCS single ligand taken from the UDP experiments within the AfCS single ligand screenscreen

Model Simulations P2YModel Simulations P2Y66 G GGTP GTP PIPPIP22

Model Simulations P2YModel Simulations P2Y66 G GGTP GTP PIPPIP22

bull ((aa) = total P2Y) = total P2Y66 activated (solid activated (solid line) and inactivated (dashed line) and inactivated (dashed line) surface receptorsline) surface receptors

bull ((bb) = total G) = total GGTP GTP

bull ((cc) = total PIP) = total PIP22 available for available for hydrolysishydrolysis

P2Y6 from 25M UDP GGTP from 25M UDP

PIP2 from 25M UDP

Sensitivity AnalysisSensitivity AnalysisSensitivity AnalysisSensitivity Analysis

bull Sensitivity analysis techniques are valuable tools Sensitivity analysis techniques are valuable tools designed to answer questions regarding which of the designed to answer questions regarding which of the uncertain input variables is more important in uncertain input variables is more important in determining the uncertainty in our output determining the uncertainty in our output

bull Likewise sensitivity analysis can provide insight into Likewise sensitivity analysis can provide insight into which parameter should be studied in more detail in which parameter should be studied in more detail in order to reduce the most variance in the model output order to reduce the most variance in the model output

bull The ability to answer these types of questions could The ability to answer these types of questions could lead to important insight into the design of new lead to important insight into the design of new experiments and in determining which experiments experiments and in determining which experiments would give us the most valuable informationwould give us the most valuable information

Sampling MethodSampling MethodSampling MethodSampling Method

bull Generate a random sample of our space of input Generate a random sample of our space of input variables over a ten percent variation from each variables over a ten percent variation from each parameters nominal value using the parameters nominal value using the Latin Hypercube Latin Hypercube Sampling (LHS) methodSampling (LHS) method

bull Uses Uses Standardized Regression Coefficients (SRCs) Standardized Regression Coefficients (SRCs) obtained by performing multiple linear regression obtained by performing multiple linear regression analysisanalysisndash offers a measure of sensitivity that is multi-dimensionally offers a measure of sensitivity that is multi-dimensionally

averaged over the entire space of parameter values averaged over the entire space of parameter values ndash SRCs give insight into degree of nonlinearity in the model SRCs give insight into degree of nonlinearity in the model

Computing RComputing Ryy22 values valuesComputing RComputing Ryy22 values values

bull SRCs are only reliable measures of sensitivity when SRCs are only reliable measures of sensitivity when degree of nonlinearity is ldquosmallrdquodegree of nonlinearity is ldquosmallrdquo

bull Use model coefficients of determination RUse model coefficients of determination Ryy22 given by given by

bull where ywhere yii is the estimate of y is the estimate of yii obtained from the regression obtained from the regression modelmodel

bull RRyy22 ge 07 ensures SRCs are good sensitivity measures ge 07 ensures SRCs are good sensitivity measures

SRCs (as sensitivity measures) for SRCs (as sensitivity measures) for DAGDAG

SRCs (as sensitivity measures) for SRCs (as sensitivity measures) for DAGDAG

SRCs (as sensitivity measures) for IPSRCs (as sensitivity measures) for IP33SRCs (as sensitivity measures) for IPSRCs (as sensitivity measures) for IP33

Effects of 3 most sensitive DAG Effects of 3 most sensitive DAG parametersparameters

Effects of 3 most sensitive DAG Effects of 3 most sensitive DAG parametersparameters

A B

C

AA k kdp2dp2 degradation of pool 2 DAG degradation of pool 2 DAG

BB k kdp1dp1 degradation of pool 1 DAG degradation of pool 1 DAG

CC k kap2ap2 production of pool 2 DAG production of pool 2 DAG

ConclusionsConclusionsConclusionsConclusions

bull We have developed a model of the UDP signaling We have developed a model of the UDP signaling pathway in RAW 2647 macrophages which can predict pathway in RAW 2647 macrophages which can predict the responses of multiple species of DAG as well as the the responses of multiple species of DAG as well as the responses of IPresponses of IP33 Ca Ca2+2+ receptor dynamics G-protein receptor dynamics G-protein activation and PIPactivation and PIP22 hydrolysis hydrolysis

bull Simplified model resultsSimplified model resultsndash We have obtained global existence uniqueness positivity We have obtained global existence uniqueness positivity

and boundedness of solutionsand boundedness of solutionsndash We have proven global stability of a unique steady state We have proven global stability of a unique steady state

within our region of interestwithin our region of interestbull Full model analysisFull model analysis

ndash Using SIMULINK we have estimated unknown rate Using SIMULINK we have estimated unknown rate parameters to obtain best fits to multiple DAG traces IPparameters to obtain best fits to multiple DAG traces IP33 and Caand Ca2+2+ all in response to 25 all in response to 25M UDPM UDP

ndash We have performed sensitivity analysis using the Latin We have performed sensitivity analysis using the Latin Hypercube sampling technique in combination with Hypercube sampling technique in combination with standardized regression coefficients to determine which standardized regression coefficients to determine which model parameters are responsible for most of the model model parameters are responsible for most of the model output uncertaintyoutput uncertainty

Future DirectionsFuture DirectionsFuture DirectionsFuture Directions

bull We have conducted multiple experiments to We have conducted multiple experiments to pharmacologically inhibit several different enzymes we pharmacologically inhibit several different enzymes we believe to play a role in this signaling pathway as believe to play a role in this signaling pathway as suggested by current known metabolic pathways and by suggested by current known metabolic pathways and by modeling resultsmodeling results

bull The next step is to perform gene knockdowns on The next step is to perform gene knockdowns on specific enzymes to verify results of inhibitor dataspecific enzymes to verify results of inhibitor data

bull The model output also suggests a time delay from The model output also suggests a time delay from receptor activation to PIPreceptor activation to PIP22 hydrolysis An upcoming task hydrolysis An upcoming task is to investigate the outcomes of adding such a delay is to investigate the outcomes of adding such a delay termterm

bull The model could be greatly enhanced by incorporating The model could be greatly enhanced by incorporating spatial dynamics so this will also be a major focus for spatial dynamics so this will also be a major focus for future researchfuture research

AcknowledgementsAcknowledgementsAcknowledgementsAcknowledgements

Mathematics DepartmentMathematics Departmentbull Hannah L Callender Hannah L Callender

PhDPhD

CollaboratorsCollaboratorsbull Alliance for Cellular SignalingAlliance for Cellular Signalingbull UT SouthwesternUT Southwestern

ndash Paul Sternweis PhDPaul Sternweis PhDndash Dianne DeCamp PhDDianne DeCamp PhD

Brown LabBrown Labbull H Alex Brown PhDH Alex Brown PhDbull Jeffrey S Forrester PhDJeffrey S Forrester PhDbull Mark Byrne PhDMark Byrne PhDbull Anita Preininger PhDAnita Preininger PhDbull Michelle ArmstrongMichelle Armstrongbull Andrew GoodmanAndrew Goodmanbull Pavlina Ivanova PhDPavlina Ivanova PhDbull Steve Milne PhDSteve Milne PhD

Basics of Cellular SignalingBasics of Cellular SignalingBasics of Cellular SignalingBasics of Cellular Signaling

Signaling molecules bind to cell Signaling molecules bind to cell surface receptors surface receptors

Initiate a series of intracellular Initiate a series of intracellular reactions reactions

These reactions regulate virtually These reactions regulate virtually all aspects of cell behavior (including all aspects of cell behavior (including metabolism movement metabolism movement proliferation survival and proliferation survival and differentiation)differentiation)

bull Breakdown in the signaling pathways that control normal cell Breakdown in the signaling pathways that control normal cell proliferation and survival result in many types of cancerproliferation and survival result in many types of cancer Need for better understandingNeed for better understanding Mathematical modeling can provide insight into these signaling Mathematical modeling can provide insight into these signaling

pathwayspathways

Canonical UDP signaling pathwayCanonical UDP signaling pathwayCanonical UDP signaling pathwayCanonical UDP signaling pathway

11 Ligand or stimulus Ligand or stimulus ((UDPUDP) binds to a ) binds to a specific G-protein specific G-protein coupled receptor coupled receptor ((P2YP2Y66))

33 GGqq GTPGTP activates activates phospholipase C-phospholipase C-3 (3 (PLC-PLC-33))

44 Active Active PLC-PLC-33 cleaves cleaves phosphatidylinositol 45-phosphatidylinositol 45-bisphosphate (bisphosphate (PIPPIP22) into ) into inositol 145-trisphosphate inositol 145-trisphosphate ((IPIP33) and diacylglycerol ) and diacylglycerol ((DAGDAG))

22 The ligand bound receptor The ligand bound receptor causes the exchange of causes the exchange of GDPGDP for for GTPGTP on the on the GGqq subunit of the G-proteinsubunit of the G-protein

55 IPIP33 binds to a specific receptor binds to a specific receptor on the endoplasmic reticulum on the endoplasmic reticulum releasing sequestered releasing sequestered CaCa2+2+

UDP

P2Y

6

GDPG PLC

Ca2+

PIP

2

DAG

IP3

GGTP

IP3

IP3-R

IP3 -R

Plasma Membrane

Endoplasmic Reticulum

Ca2+

Ca2+

Ca2+

UDP

GGTP

Ca2+ Ca2+

Ca2+

Ca2+

Ca2+

Ca2+

Mathematical ModelMathematical ModelMathematical ModelMathematical Model

bull The model consists of a base set of 10 coupled nonlinear The model consists of a base set of 10 coupled nonlinear ordinary differential equations (ODEs)ordinary differential equations (ODEs)

bull The ODEs are constructed using mass action kinetics to The ODEs are constructed using mass action kinetics to approximate diffusion binding and unbinding of molecular approximate diffusion binding and unbinding of molecular species in various compartments of the RAW cellspecies in various compartments of the RAW cell effective rates of production and degradation are proportional to effective rates of production and degradation are proportional to

the concentrations (or number of molecules) of participating the concentrations (or number of molecules) of participating reactantsreactants

bull Four modulesFour modulesndash Receptor regulationReceptor regulationndash G-protein casadeG-protein casadendash Species-specific DAG dynamicsSpecies-specific DAG dynamicsndash Cytosolic CaCytosolic Ca2+2+ dynamics dynamics

bull Receptor-ligandReceptor-ligand binding (based on work by Lemon 2003) binding (based on work by Lemon 2003)

ndash Number of ldquoactivatedrdquo surface receptors at time tNumber of ldquoactivatedrdquo surface receptors at time tndash Number of ldquoinactivatedrdquo surface receptors at time tNumber of ldquoinactivatedrdquo surface receptors at time tndash Concentration of ligand (constant)Concentration of ligand (constant)

Model Equations Receptor Model Equations Receptor regulationregulation

Model Equations Receptor Model Equations Receptor regulationregulation

Nondimensionalized equationsNondimensionalized equations

Model Equations G-protein Model Equations G-protein activationactivation

Model Equations G-protein Model Equations G-protein activationactivation

bull G-proteinG-protein activation activation

ndash Number of activated G-proteins (GNumber of activated G-proteins (GGTP) at time tGTP) at time t

Nondimensionalized equationsNondimensionalized equations

Model Equations PIPModel Equations PIP22 hydrolysis and hydrolysis and replenishmentreplenishment

Model Equations PIPModel Equations PIP22 hydrolysis and hydrolysis and replenishmentreplenishment

bull PIPPIP2 2 hydrolysis and replenishmenthydrolysis and replenishment

Nondimensionalized equationsNondimensionalized equations

ndash Number of PIPNumber of PIP22 molecules at time t molecules at time tndash Initial number of ldquofreerdquo PIPInitial number of ldquofreerdquo PIP22

PLC

PIP

2

DAG

IP3

GGTPCa2+

GGTP

bull IPIP33

Model Equations IPModel Equations IP33 production and production and degradationdegradation

Model Equations IPModel Equations IP33 production and production and degradationdegradation

Nondimensionalized equationsNondimensionalized equations

PLC

PIP

2

DA

G

IP3

G

GTPCa2+

G

GTP

bull Concentration of IPConcentration of IP33 at time t (in at time t (in M)M)

Model Equations DAGModel Equations DAGModel Equations DAGModel Equations DAG

bull DAG DAG (separate ODEs for each species considered)(separate ODEs for each species considered)

Nondimensionalized equationsNondimensionalized equations

bull Concentration of DAG at time tConcentration of DAG at time t

PLC

PIP

2

DA

G

IP3

G

GTPCa2+G

GTP

Model Equations Li and Rinzel (1994) Model Equations Li and Rinzel (1994) CaCa2+2+

Model Equations Li and Rinzel (1994) Model Equations Li and Rinzel (1994) CaCa2+2+

bull Free cytosolicFree cytosolic Ca Ca2+2+

bull hh = fraction of IP = fraction of IP33 channels not yet inactivated by Ca channels not yet inactivated by Ca2+2+

bull cc00 = concentration of total free Ca = concentration of total free Ca2+2+ per cytosolic volume per cytosolic volume

Timeseries data for the modelTimeseries data for the modelTimeseries data for the modelTimeseries data for the model

bull Data from collaborators at UTSW (Sternweis Lab)Data from collaborators at UTSW (Sternweis Lab)ndash Inositol 145-trisphosphate (Inositol 145-trisphosphate (IPIP33) production post 25) production post 25M M

UDP stimulation in the RAW cellUDP stimulation in the RAW cellndash Cytosolic calcium (Cytosolic calcium (CaCa2+2+) release post stimulation with ) release post stimulation with

2525M UDP in the RAW cellM UDP in the RAW cell

bull Diacylglycerol (Diacylglycerol (DAGDAG) data H Callender collected from a ) data H Callender collected from a novel method of quantitative analysis of multiple novel method of quantitative analysis of multiple species of DAG developed in the Brown Lab species of DAG developed in the Brown Lab

Reference Reference

Callender H L et al Quantification of Diacylglycerol Species from Callender H L et al Quantification of Diacylglycerol Species from Cellular Extracts by Electrospray Ionization Mass Spectrometry Using a Cellular Extracts by Electrospray Ionization Mass Spectrometry Using a Linear Regression Algorithm Linear Regression Algorithm Anal Chem Anal Chem 7979 (2007) 263-272 (2007) 263-272

IPIP33 and Ca and Ca2+2+ data from AfCS data from AfCSIPIP33 and Ca and Ca2+2+ data from AfCS data from AfCS

AA

bull IPIP33 response (in pmoles per response (in pmoles per 100 100 L lysate)in RAW 2647 L lysate)in RAW 2647 cells to 25 cells to 25 M UDPM UDP

bull Points represent the average Points represent the average of four experimentsof four experiments

bull Cytosolic calcium response Cytosolic calcium response (in (in M) in the RAW 2647 M) in the RAW 2647 cell to 25 cell to 25 M UDP M UDP

bull The graph displays 43 The graph displays 43 experimental repeatsexperimental repeats

B

What is DAGWhat is DAGWhat is DAGWhat is DAG

bull DAG is a cellular second messenger molecule which plays an DAG is a cellular second messenger molecule which plays an important role in initiating various changes in cell behavior important role in initiating various changes in cell behavior including cell activation differentiation proliferation and including cell activation differentiation proliferation and tumor promotion tumor promotion

bull There are many different species of DAG depending on the There are many different species of DAG depending on the number of carbons and number of double bonds in the fatty number of carbons and number of double bonds in the fatty acyl chains and different species can have different cellular acyl chains and different species can have different cellular functionsfunctions

bull ExampleExample 320 DAG320 DAG

Two fatty acyl Two fatty acyl (hydrocarbon) (hydrocarbon) chains in every DAG chains in every DAG speciesspecies

Each corner Each corner represents carbonrepresents carbon

320 DAG 18 carbons in first chain 14 in the second no double bonds 320 DAG 18 carbons in first chain 14 in the second no double bonds in either chainin either chain

Kinetics of monodi unsaturated DAGsKinetics of monodi unsaturated DAGsKinetics of monodi unsaturated DAGsKinetics of monodi unsaturated DAGs

bull Time based behavior of four monodi unsaturated DAG species after addition of 25 Time based behavior of four monodi unsaturated DAG species after addition of 25 M M (solid red squares) and 025 (solid red squares) and 025 M (solid green triangles) UDPM (solid green triangles) UDP

bull Time points contain a minimum of nine replicates performed on three different Time points contain a minimum of nine replicates performed on three different experimental daysexperimental days

- 25 M UDP

- 025M UDP

0 5 10 15 20 25 30

0 5 10 15 20 25 30 0 5 10 15 20 25 30

0 5 10 15 20 25 30

321 DAG 341 DAG

362 DAG341 DAGep

150

100

50-5

00

300

200

100

-100

0

150

100

50-5

00

-100

200

6040

20-2

00

-40

80

Time (min) Time (min)

Time (min) Time (min)

D

AG

(n

g)

D

AG

(n

g)

D

AG

(n

g)

D

AG

(n

g)

- 25 M UDP

- 025M UDP

- 25 M UDP

- 025M UDP

- 25 M UDP

- 025M UDP

Kinetics of PUFA containing DAGsKinetics of PUFA containing DAGsKinetics of PUFA containing DAGsKinetics of PUFA containing DAGsbull Time based behavior of four polyunsaturated fatty acid (PUFA) containing DAG species Time based behavior of four polyunsaturated fatty acid (PUFA) containing DAG species

after addition of 25 after addition of 25 M (solid red squares) and 025 M (solid red squares) and 025 M (solid green triangles) UDPM (solid green triangles) UDPbull Time points contain a minimum of nine replicates performed on three different Time points contain a minimum of nine replicates performed on three different

experimental daysexperimental daysbull Monodi unsaturated DAG species give a larger increase than polyunsaturated fatty acid Monodi unsaturated DAG species give a larger increase than polyunsaturated fatty acid

(PUFA) containing DAGs(PUFA) containing DAGs

0 5 10 15 20 25 30

0 5 10 15 20 25 30 0 5 10 15 20 25 30

0 5 10 15 20 25 30

385 DAG 383 DAG

364 DAG384 DAG6

42

-40

30

20

10

0

15

10

50

-56

40

2-2

8

Time (min) Time (min)

Time (min) Time (min)

D

AG

(n

g)

D

AG

(n

g)

D

AG

(n

g)

D

AG

(n

g)

-28

40

- 25 M UDP

- 025M UDP

- 25 M UDP

- 025M UDP

- 25 M UDP

- 025M UDP

- 25 M UDP

- 025M UDP

Differential DAG KineticsDifferential DAG KineticsDifferential DAG KineticsDifferential DAG Kinetics

bull Time based behavior of three DAG species with varying degrees of unsaturation after Time based behavior of three DAG species with varying degrees of unsaturation after addition of addition of 25 25 MM (solid red triangles) and (solid red triangles) and 025 025 MM (solid green squares) UDP (solid green squares) UDP

bull Time points contain nine replicates performed on three different experimental daysTime points contain nine replicates performed on three different experimental days

341 DAG

Best fit with current model structure

Overall Objectives of Modeling EffortOverall Objectives of Modeling EffortOverall Objectives of Modeling EffortOverall Objectives of Modeling Effort

bull Predict quantitative changes in lipid species after stimulation by Predict quantitative changes in lipid species after stimulation by various ligands and ligand concentrations in the RAW 2647 various ligands and ligand concentrations in the RAW 2647 macrophagemacrophage

bull Comparison and refinement of model output with AfCS IPComparison and refinement of model output with AfCS IP33 measurements and Cameasurements and Ca2+2+ traces as well as DAG data generated in traces as well as DAG data generated in the Brown labthe Brown lab

bull Predict in silico effects such as the effect of knock-downs etc on Predict in silico effects such as the effect of knock-downs etc on given pathwaygiven pathway

bull Suggest modifications to current pathway structuresSuggest modifications to current pathway structures

Modifications to the modelModifications to the modelModifications to the modelModifications to the model

bull Include an additional branch in the pathway Include an additional branch in the pathway for a second pool of DAGfor a second pool of DAG

bull Simplify CaSimplify Ca2+2+ equations for mathematical equations for mathematical analysis purposesanalysis purposes

Proposed PathwayProposed PathwayProposed PathwayProposed Pathway

bull Note We measure total cellular DAG levelsNote We measure total cellular DAG levelsbull Initial production of DAG from the hydrolysis of PIPInitial production of DAG from the hydrolysis of PIP22 in pool 1 (plasma in pool 1 (plasma

membrane) is offset by phosphorylation of DAG by DAG kinase in pool 2 membrane) is offset by phosphorylation of DAG by DAG kinase in pool 2 (Endoplasmic Reticulum Nucleus) to aid in the PI replacement pathway(Endoplasmic Reticulum Nucleus) to aid in the PI replacement pathway

bull Second wave of DAG is a result of resynthesis of PIPSecond wave of DAG is a result of resynthesis of PIP22 which is then which is then hydrolyzed to form DAG and IPhydrolyzed to form DAG and IP33

PI PIP PIP2 DAG

IP3

IP2IP

Ins

PI

CDP-DAGPADAG

PA

PCPE

PCPE

PITP

PI4K PI5K PLC DGK

DGK CDSPLC PIS

POOL 2

POOL 1

IP4

PI PIP PIP2 DAG

IP3

IP2IP

Ins

PI

CDP-DAGPADAG

PA

PCPE

PC

PITP

PI4K PI5K PLC

DGK CDSPLC PIS

IP4

LPP

LPP

Model Equations DAG (pool 1)Model Equations DAG (pool 1)Model Equations DAG (pool 1)Model Equations DAG (pool 1)

bull DAG (pool 1)DAG (pool 1) (separate ODEs for each species (separate ODEs for each species considered)considered)

bull Concentration of DAG from pool 1 at time tConcentration of DAG from pool 1 at time t

Nondimensionalized equationsNondimensionalized equations

Model Equations DAG (pool 2)Model Equations DAG (pool 2)Model Equations DAG (pool 2)Model Equations DAG (pool 2)

bull DAG (pool 2)DAG (pool 2) (separate ODEs for each species (separate ODEs for each species considered)considered)

bull Concentration of pool 2 DAG molecules at time tConcentration of pool 2 DAG molecules at time t

bull Baseline concentration of pool 2 DAGBaseline concentration of pool 2 DAG

Nondimensionalized equationsNondimensionalized equations

Simplified CaSimplified Ca2+2+ Equations EquationsSimplified CaSimplified Ca2+2+ Equations Equations

bull CaCa2+2+ modulemodule (to match experimental AfCS (to match experimental AfCS trace)trace)

0 100 200 300 400 500 6000

002

004

006

008

01

012

014

Concentration of Ca

2+ (uM)

Ca2+ response to 25uM UDP

time (s)

wherewhere

Theoretical AnalysisTheoretical AnalysisTheoretical AnalysisTheoretical Analysis

bull Existence and UniquenessExistence and Uniqueness

bull Positivity and Boundedness Positivity and Boundedness

(for biological relevance)(for biological relevance)

bull Analysis of steady state behaviorAnalysis of steady state behavior

Existence of Solutions (full model)Existence of Solutions (full model)Existence of Solutions (full model)Existence of Solutions (full model)

First we write our system of ODEs in the formFirst we write our system of ODEs in the form (1)

(2)

(1)

Uniqueness of SolutionsUniqueness of SolutionsUniqueness of SolutionsUniqueness of Solutions

Next we denote a solution of Next we denote a solution of (1)(1) by by with initial condition with initial condition

Since our system satisfies the hypotheses of Theorems 1 and 2 Since our system satisfies the hypotheses of Theorems 1 and 2 on our set of interest (for all positive time and on a positive on our set of interest (for all positive time and on a positive bounded set in space) we know bounded set in space) we know there exists a unique there exists a unique (local) solution(local) solution (ie on some finite time interval possibly (ie on some finite time interval possibly small)small)

QuestionQuestion Do the solutions remain positive and bounded (for Do the solutions remain positive and bounded (for biological significance)biological significance)

(3)

(1) (3)

Positivity and boundedness of Positivity and boundedness of solutionssolutions

Positivity and boundedness of Positivity and boundedness of solutionssolutions

bull We first use Theorem 3 to show positivity and boundedness of xWe first use Theorem 3 to show positivity and boundedness of x11 and and xx22

bull Next we use these results and the Fundamental Theorem of Calculus to Next we use these results and the Fundamental Theorem of Calculus to show positivity and boundedness for the remaining equations show positivity and boundedness for the remaining equations

bull This then ensures a global solutionThis then ensures a global solution

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical Analysis

bull Parameter Estimation in SIMULINKParameter Estimation in SIMULINK

bull Sensitivity AnalysisSensitivity Analysis

Parameter Estimation in SIMULINK (for Parameter Estimation in SIMULINK (for FULL model)FULL model)

Parameter Estimation in SIMULINK (for Parameter Estimation in SIMULINK (for FULL model)FULL model)

bull Total number of model parameters (for full model with Li and Rinzel CaTotal number of model parameters (for full model with Li and Rinzel Ca2+2+ module) = module) = 3434ndash From the literature = From the literature = 2020ndash Estimated = Estimated = 1414

bull Receptor moduleReceptor module Total = Total = 77ndash From literature = From literature = 66ndash Estimated = Estimated = 11 (k (kpp rate of receptor phosphorylation) rate of receptor phosphorylation)

bull G-protein cascadeG-protein cascade Total = Total = 99ndash From literature = From literature = 66ndash Estimated = Estimated = 3 3 (k(khydhyd k kreprep k kd3d3))

bull DAG kineticsDAG kinetics Total = Total = 5 5 (for each DAG species considered)(for each DAG species considered)ndash From literature = From literature = 00ndash Estimated = Estimated = 5 5 (all DAG parameters)(all DAG parameters)

bull CaCa2+2+ module module Total = Total = 1313ndash From literature = From literature = 88ndash Estimated = Estimated = 55

SIMULINK DetailsSIMULINK DetailsSIMULINK DetailsSIMULINK Details

bull Unknown rate parameters were estimated using SIMULINK

ndash Minimizes a user-specified cost function via a user-specified optimization method

ndash Nonlinear least squares optimization method of Levenberg-Marquardt was used to minimize a sum of squared errors cost function of the empirical observations and model predictions for IP3 Ca2+ and multiple species of DAG

bull Note Although the Gauss-Newton method is often more efficient the method of Levenberg-Marquardt has proved to be more robust

Measured vs Simulated 384 DAG and Measured vs Simulated 384 DAG and 341 DAG341 DAG

Measured vs Simulated 384 DAG and Measured vs Simulated 384 DAG and 341 DAG341 DAG

bull Timecourse of stimulation with 25 Timecourse of stimulation with 25 M UDP in RAW 2647 cellsM UDP in RAW 2647 cells

bull Solid black lines represent model simulations Solid black lines represent model simulations

bull ((aa) 384 DAG response (representative of the response of most poly ) 384 DAG response (representative of the response of most poly unsaturated fatty acid-containing DAG species) unsaturated fatty acid-containing DAG species)

bull ((bb) 341 DAG response (representative of the response of most mono- ) 341 DAG response (representative of the response of most mono- and di-unsaturated fatty acid-containing DAG species) and di-unsaturated fatty acid-containing DAG species)

bull Data points contain nine replicates performed on three different Data points contain nine replicates performed on three different experimental days with error bars = 1 SEM Units are total change in ng experimental days with error bars = 1 SEM Units are total change in ng over baseline levels in ~8x10over baseline levels in ~8x1066 cells cells

Measured vs Simulated IPMeasured vs Simulated IP33 and Ca and Ca2+2+Measured vs Simulated IPMeasured vs Simulated IP33 and Ca and Ca2+2+

bull Timecourse of stimulation with 25 Timecourse of stimulation with 25 M UDP in RAW 2647 cellsM UDP in RAW 2647 cells

bull Solid black lines represent model simulationsSolid black lines represent model simulations

bull ((cc) IP) IP33 response in pmols per ~35x10 response in pmols per ~35x1055 cells cells

bull Points in (c) represent the average of four experiments and error Points in (c) represent the average of four experiments and error bars are 1 SEM bars are 1 SEM

bull ((dd) Ca) Ca2+2+ response in response in M Red curve is a representative CaM Red curve is a representative Ca2+2+ trace trace taken from the UDP experiments within the AfCS single ligand taken from the UDP experiments within the AfCS single ligand screenscreen

Model Simulations P2YModel Simulations P2Y66 G GGTP GTP PIPPIP22

Model Simulations P2YModel Simulations P2Y66 G GGTP GTP PIPPIP22

bull ((aa) = total P2Y) = total P2Y66 activated (solid activated (solid line) and inactivated (dashed line) and inactivated (dashed line) surface receptorsline) surface receptors

bull ((bb) = total G) = total GGTP GTP

bull ((cc) = total PIP) = total PIP22 available for available for hydrolysishydrolysis

P2Y6 from 25M UDP GGTP from 25M UDP

PIP2 from 25M UDP

Sensitivity AnalysisSensitivity AnalysisSensitivity AnalysisSensitivity Analysis

bull Sensitivity analysis techniques are valuable tools Sensitivity analysis techniques are valuable tools designed to answer questions regarding which of the designed to answer questions regarding which of the uncertain input variables is more important in uncertain input variables is more important in determining the uncertainty in our output determining the uncertainty in our output

bull Likewise sensitivity analysis can provide insight into Likewise sensitivity analysis can provide insight into which parameter should be studied in more detail in which parameter should be studied in more detail in order to reduce the most variance in the model output order to reduce the most variance in the model output

bull The ability to answer these types of questions could The ability to answer these types of questions could lead to important insight into the design of new lead to important insight into the design of new experiments and in determining which experiments experiments and in determining which experiments would give us the most valuable informationwould give us the most valuable information

Sampling MethodSampling MethodSampling MethodSampling Method

bull Generate a random sample of our space of input Generate a random sample of our space of input variables over a ten percent variation from each variables over a ten percent variation from each parameters nominal value using the parameters nominal value using the Latin Hypercube Latin Hypercube Sampling (LHS) methodSampling (LHS) method

bull Uses Uses Standardized Regression Coefficients (SRCs) Standardized Regression Coefficients (SRCs) obtained by performing multiple linear regression obtained by performing multiple linear regression analysisanalysisndash offers a measure of sensitivity that is multi-dimensionally offers a measure of sensitivity that is multi-dimensionally

averaged over the entire space of parameter values averaged over the entire space of parameter values ndash SRCs give insight into degree of nonlinearity in the model SRCs give insight into degree of nonlinearity in the model

Computing RComputing Ryy22 values valuesComputing RComputing Ryy22 values values

bull SRCs are only reliable measures of sensitivity when SRCs are only reliable measures of sensitivity when degree of nonlinearity is ldquosmallrdquodegree of nonlinearity is ldquosmallrdquo

bull Use model coefficients of determination RUse model coefficients of determination Ryy22 given by given by

bull where ywhere yii is the estimate of y is the estimate of yii obtained from the regression obtained from the regression modelmodel

bull RRyy22 ge 07 ensures SRCs are good sensitivity measures ge 07 ensures SRCs are good sensitivity measures

SRCs (as sensitivity measures) for SRCs (as sensitivity measures) for DAGDAG

SRCs (as sensitivity measures) for SRCs (as sensitivity measures) for DAGDAG

SRCs (as sensitivity measures) for IPSRCs (as sensitivity measures) for IP33SRCs (as sensitivity measures) for IPSRCs (as sensitivity measures) for IP33

Effects of 3 most sensitive DAG Effects of 3 most sensitive DAG parametersparameters

Effects of 3 most sensitive DAG Effects of 3 most sensitive DAG parametersparameters

A B

C

AA k kdp2dp2 degradation of pool 2 DAG degradation of pool 2 DAG

BB k kdp1dp1 degradation of pool 1 DAG degradation of pool 1 DAG

CC k kap2ap2 production of pool 2 DAG production of pool 2 DAG

ConclusionsConclusionsConclusionsConclusions

bull We have developed a model of the UDP signaling We have developed a model of the UDP signaling pathway in RAW 2647 macrophages which can predict pathway in RAW 2647 macrophages which can predict the responses of multiple species of DAG as well as the the responses of multiple species of DAG as well as the responses of IPresponses of IP33 Ca Ca2+2+ receptor dynamics G-protein receptor dynamics G-protein activation and PIPactivation and PIP22 hydrolysis hydrolysis

bull Simplified model resultsSimplified model resultsndash We have obtained global existence uniqueness positivity We have obtained global existence uniqueness positivity

and boundedness of solutionsand boundedness of solutionsndash We have proven global stability of a unique steady state We have proven global stability of a unique steady state

within our region of interestwithin our region of interestbull Full model analysisFull model analysis

ndash Using SIMULINK we have estimated unknown rate Using SIMULINK we have estimated unknown rate parameters to obtain best fits to multiple DAG traces IPparameters to obtain best fits to multiple DAG traces IP33 and Caand Ca2+2+ all in response to 25 all in response to 25M UDPM UDP

ndash We have performed sensitivity analysis using the Latin We have performed sensitivity analysis using the Latin Hypercube sampling technique in combination with Hypercube sampling technique in combination with standardized regression coefficients to determine which standardized regression coefficients to determine which model parameters are responsible for most of the model model parameters are responsible for most of the model output uncertaintyoutput uncertainty

Future DirectionsFuture DirectionsFuture DirectionsFuture Directions

bull We have conducted multiple experiments to We have conducted multiple experiments to pharmacologically inhibit several different enzymes we pharmacologically inhibit several different enzymes we believe to play a role in this signaling pathway as believe to play a role in this signaling pathway as suggested by current known metabolic pathways and by suggested by current known metabolic pathways and by modeling resultsmodeling results

bull The next step is to perform gene knockdowns on The next step is to perform gene knockdowns on specific enzymes to verify results of inhibitor dataspecific enzymes to verify results of inhibitor data

bull The model output also suggests a time delay from The model output also suggests a time delay from receptor activation to PIPreceptor activation to PIP22 hydrolysis An upcoming task hydrolysis An upcoming task is to investigate the outcomes of adding such a delay is to investigate the outcomes of adding such a delay termterm

bull The model could be greatly enhanced by incorporating The model could be greatly enhanced by incorporating spatial dynamics so this will also be a major focus for spatial dynamics so this will also be a major focus for future researchfuture research

AcknowledgementsAcknowledgementsAcknowledgementsAcknowledgements

Mathematics DepartmentMathematics Departmentbull Hannah L Callender Hannah L Callender

PhDPhD

CollaboratorsCollaboratorsbull Alliance for Cellular SignalingAlliance for Cellular Signalingbull UT SouthwesternUT Southwestern

ndash Paul Sternweis PhDPaul Sternweis PhDndash Dianne DeCamp PhDDianne DeCamp PhD

Brown LabBrown Labbull H Alex Brown PhDH Alex Brown PhDbull Jeffrey S Forrester PhDJeffrey S Forrester PhDbull Mark Byrne PhDMark Byrne PhDbull Anita Preininger PhDAnita Preininger PhDbull Michelle ArmstrongMichelle Armstrongbull Andrew GoodmanAndrew Goodmanbull Pavlina Ivanova PhDPavlina Ivanova PhDbull Steve Milne PhDSteve Milne PhD

Canonical UDP signaling pathwayCanonical UDP signaling pathwayCanonical UDP signaling pathwayCanonical UDP signaling pathway

11 Ligand or stimulus Ligand or stimulus ((UDPUDP) binds to a ) binds to a specific G-protein specific G-protein coupled receptor coupled receptor ((P2YP2Y66))

33 GGqq GTPGTP activates activates phospholipase C-phospholipase C-3 (3 (PLC-PLC-33))

44 Active Active PLC-PLC-33 cleaves cleaves phosphatidylinositol 45-phosphatidylinositol 45-bisphosphate (bisphosphate (PIPPIP22) into ) into inositol 145-trisphosphate inositol 145-trisphosphate ((IPIP33) and diacylglycerol ) and diacylglycerol ((DAGDAG))

22 The ligand bound receptor The ligand bound receptor causes the exchange of causes the exchange of GDPGDP for for GTPGTP on the on the GGqq subunit of the G-proteinsubunit of the G-protein

55 IPIP33 binds to a specific receptor binds to a specific receptor on the endoplasmic reticulum on the endoplasmic reticulum releasing sequestered releasing sequestered CaCa2+2+

UDP

P2Y

6

GDPG PLC

Ca2+

PIP

2

DAG

IP3

GGTP

IP3

IP3-R

IP3 -R

Plasma Membrane

Endoplasmic Reticulum

Ca2+

Ca2+

Ca2+

UDP

GGTP

Ca2+ Ca2+

Ca2+

Ca2+

Ca2+

Ca2+

Mathematical ModelMathematical ModelMathematical ModelMathematical Model

bull The model consists of a base set of 10 coupled nonlinear The model consists of a base set of 10 coupled nonlinear ordinary differential equations (ODEs)ordinary differential equations (ODEs)

bull The ODEs are constructed using mass action kinetics to The ODEs are constructed using mass action kinetics to approximate diffusion binding and unbinding of molecular approximate diffusion binding and unbinding of molecular species in various compartments of the RAW cellspecies in various compartments of the RAW cell effective rates of production and degradation are proportional to effective rates of production and degradation are proportional to

the concentrations (or number of molecules) of participating the concentrations (or number of molecules) of participating reactantsreactants

bull Four modulesFour modulesndash Receptor regulationReceptor regulationndash G-protein casadeG-protein casadendash Species-specific DAG dynamicsSpecies-specific DAG dynamicsndash Cytosolic CaCytosolic Ca2+2+ dynamics dynamics

bull Receptor-ligandReceptor-ligand binding (based on work by Lemon 2003) binding (based on work by Lemon 2003)

ndash Number of ldquoactivatedrdquo surface receptors at time tNumber of ldquoactivatedrdquo surface receptors at time tndash Number of ldquoinactivatedrdquo surface receptors at time tNumber of ldquoinactivatedrdquo surface receptors at time tndash Concentration of ligand (constant)Concentration of ligand (constant)

Model Equations Receptor Model Equations Receptor regulationregulation

Model Equations Receptor Model Equations Receptor regulationregulation

Nondimensionalized equationsNondimensionalized equations

Model Equations G-protein Model Equations G-protein activationactivation

Model Equations G-protein Model Equations G-protein activationactivation

bull G-proteinG-protein activation activation

ndash Number of activated G-proteins (GNumber of activated G-proteins (GGTP) at time tGTP) at time t

Nondimensionalized equationsNondimensionalized equations

Model Equations PIPModel Equations PIP22 hydrolysis and hydrolysis and replenishmentreplenishment

Model Equations PIPModel Equations PIP22 hydrolysis and hydrolysis and replenishmentreplenishment

bull PIPPIP2 2 hydrolysis and replenishmenthydrolysis and replenishment

Nondimensionalized equationsNondimensionalized equations

ndash Number of PIPNumber of PIP22 molecules at time t molecules at time tndash Initial number of ldquofreerdquo PIPInitial number of ldquofreerdquo PIP22

PLC

PIP

2

DAG

IP3

GGTPCa2+

GGTP

bull IPIP33

Model Equations IPModel Equations IP33 production and production and degradationdegradation

Model Equations IPModel Equations IP33 production and production and degradationdegradation

Nondimensionalized equationsNondimensionalized equations

PLC

PIP

2

DA

G

IP3

G

GTPCa2+

G

GTP

bull Concentration of IPConcentration of IP33 at time t (in at time t (in M)M)

Model Equations DAGModel Equations DAGModel Equations DAGModel Equations DAG

bull DAG DAG (separate ODEs for each species considered)(separate ODEs for each species considered)

Nondimensionalized equationsNondimensionalized equations

bull Concentration of DAG at time tConcentration of DAG at time t

PLC

PIP

2

DA

G

IP3

G

GTPCa2+G

GTP

Model Equations Li and Rinzel (1994) Model Equations Li and Rinzel (1994) CaCa2+2+

Model Equations Li and Rinzel (1994) Model Equations Li and Rinzel (1994) CaCa2+2+

bull Free cytosolicFree cytosolic Ca Ca2+2+

bull hh = fraction of IP = fraction of IP33 channels not yet inactivated by Ca channels not yet inactivated by Ca2+2+

bull cc00 = concentration of total free Ca = concentration of total free Ca2+2+ per cytosolic volume per cytosolic volume

Timeseries data for the modelTimeseries data for the modelTimeseries data for the modelTimeseries data for the model

bull Data from collaborators at UTSW (Sternweis Lab)Data from collaborators at UTSW (Sternweis Lab)ndash Inositol 145-trisphosphate (Inositol 145-trisphosphate (IPIP33) production post 25) production post 25M M

UDP stimulation in the RAW cellUDP stimulation in the RAW cellndash Cytosolic calcium (Cytosolic calcium (CaCa2+2+) release post stimulation with ) release post stimulation with

2525M UDP in the RAW cellM UDP in the RAW cell

bull Diacylglycerol (Diacylglycerol (DAGDAG) data H Callender collected from a ) data H Callender collected from a novel method of quantitative analysis of multiple novel method of quantitative analysis of multiple species of DAG developed in the Brown Lab species of DAG developed in the Brown Lab

Reference Reference

Callender H L et al Quantification of Diacylglycerol Species from Callender H L et al Quantification of Diacylglycerol Species from Cellular Extracts by Electrospray Ionization Mass Spectrometry Using a Cellular Extracts by Electrospray Ionization Mass Spectrometry Using a Linear Regression Algorithm Linear Regression Algorithm Anal Chem Anal Chem 7979 (2007) 263-272 (2007) 263-272

IPIP33 and Ca and Ca2+2+ data from AfCS data from AfCSIPIP33 and Ca and Ca2+2+ data from AfCS data from AfCS

AA

bull IPIP33 response (in pmoles per response (in pmoles per 100 100 L lysate)in RAW 2647 L lysate)in RAW 2647 cells to 25 cells to 25 M UDPM UDP

bull Points represent the average Points represent the average of four experimentsof four experiments

bull Cytosolic calcium response Cytosolic calcium response (in (in M) in the RAW 2647 M) in the RAW 2647 cell to 25 cell to 25 M UDP M UDP

bull The graph displays 43 The graph displays 43 experimental repeatsexperimental repeats

B

What is DAGWhat is DAGWhat is DAGWhat is DAG

bull DAG is a cellular second messenger molecule which plays an DAG is a cellular second messenger molecule which plays an important role in initiating various changes in cell behavior important role in initiating various changes in cell behavior including cell activation differentiation proliferation and including cell activation differentiation proliferation and tumor promotion tumor promotion

bull There are many different species of DAG depending on the There are many different species of DAG depending on the number of carbons and number of double bonds in the fatty number of carbons and number of double bonds in the fatty acyl chains and different species can have different cellular acyl chains and different species can have different cellular functionsfunctions

bull ExampleExample 320 DAG320 DAG

Two fatty acyl Two fatty acyl (hydrocarbon) (hydrocarbon) chains in every DAG chains in every DAG speciesspecies

Each corner Each corner represents carbonrepresents carbon

320 DAG 18 carbons in first chain 14 in the second no double bonds 320 DAG 18 carbons in first chain 14 in the second no double bonds in either chainin either chain

Kinetics of monodi unsaturated DAGsKinetics of monodi unsaturated DAGsKinetics of monodi unsaturated DAGsKinetics of monodi unsaturated DAGs

bull Time based behavior of four monodi unsaturated DAG species after addition of 25 Time based behavior of four monodi unsaturated DAG species after addition of 25 M M (solid red squares) and 025 (solid red squares) and 025 M (solid green triangles) UDPM (solid green triangles) UDP

bull Time points contain a minimum of nine replicates performed on three different Time points contain a minimum of nine replicates performed on three different experimental daysexperimental days

- 25 M UDP

- 025M UDP

0 5 10 15 20 25 30

0 5 10 15 20 25 30 0 5 10 15 20 25 30

0 5 10 15 20 25 30

321 DAG 341 DAG

362 DAG341 DAGep

150

100

50-5

00

300

200

100

-100

0

150

100

50-5

00

-100

200

6040

20-2

00

-40

80

Time (min) Time (min)

Time (min) Time (min)

D

AG

(n

g)

D

AG

(n

g)

D

AG

(n

g)

D

AG

(n

g)

- 25 M UDP

- 025M UDP

- 25 M UDP

- 025M UDP

- 25 M UDP

- 025M UDP

Kinetics of PUFA containing DAGsKinetics of PUFA containing DAGsKinetics of PUFA containing DAGsKinetics of PUFA containing DAGsbull Time based behavior of four polyunsaturated fatty acid (PUFA) containing DAG species Time based behavior of four polyunsaturated fatty acid (PUFA) containing DAG species

after addition of 25 after addition of 25 M (solid red squares) and 025 M (solid red squares) and 025 M (solid green triangles) UDPM (solid green triangles) UDPbull Time points contain a minimum of nine replicates performed on three different Time points contain a minimum of nine replicates performed on three different

experimental daysexperimental daysbull Monodi unsaturated DAG species give a larger increase than polyunsaturated fatty acid Monodi unsaturated DAG species give a larger increase than polyunsaturated fatty acid

(PUFA) containing DAGs(PUFA) containing DAGs

0 5 10 15 20 25 30

0 5 10 15 20 25 30 0 5 10 15 20 25 30

0 5 10 15 20 25 30

385 DAG 383 DAG

364 DAG384 DAG6

42

-40

30

20

10

0

15

10

50

-56

40

2-2

8

Time (min) Time (min)

Time (min) Time (min)

D

AG

(n

g)

D

AG

(n

g)

D

AG

(n

g)

D

AG

(n

g)

-28

40

- 25 M UDP

- 025M UDP

- 25 M UDP

- 025M UDP

- 25 M UDP

- 025M UDP

- 25 M UDP

- 025M UDP

Differential DAG KineticsDifferential DAG KineticsDifferential DAG KineticsDifferential DAG Kinetics

bull Time based behavior of three DAG species with varying degrees of unsaturation after Time based behavior of three DAG species with varying degrees of unsaturation after addition of addition of 25 25 MM (solid red triangles) and (solid red triangles) and 025 025 MM (solid green squares) UDP (solid green squares) UDP

bull Time points contain nine replicates performed on three different experimental daysTime points contain nine replicates performed on three different experimental days

341 DAG

Best fit with current model structure

Overall Objectives of Modeling EffortOverall Objectives of Modeling EffortOverall Objectives of Modeling EffortOverall Objectives of Modeling Effort

bull Predict quantitative changes in lipid species after stimulation by Predict quantitative changes in lipid species after stimulation by various ligands and ligand concentrations in the RAW 2647 various ligands and ligand concentrations in the RAW 2647 macrophagemacrophage

bull Comparison and refinement of model output with AfCS IPComparison and refinement of model output with AfCS IP33 measurements and Cameasurements and Ca2+2+ traces as well as DAG data generated in traces as well as DAG data generated in the Brown labthe Brown lab

bull Predict in silico effects such as the effect of knock-downs etc on Predict in silico effects such as the effect of knock-downs etc on given pathwaygiven pathway

bull Suggest modifications to current pathway structuresSuggest modifications to current pathway structures

Modifications to the modelModifications to the modelModifications to the modelModifications to the model

bull Include an additional branch in the pathway Include an additional branch in the pathway for a second pool of DAGfor a second pool of DAG

bull Simplify CaSimplify Ca2+2+ equations for mathematical equations for mathematical analysis purposesanalysis purposes

Proposed PathwayProposed PathwayProposed PathwayProposed Pathway

bull Note We measure total cellular DAG levelsNote We measure total cellular DAG levelsbull Initial production of DAG from the hydrolysis of PIPInitial production of DAG from the hydrolysis of PIP22 in pool 1 (plasma in pool 1 (plasma

membrane) is offset by phosphorylation of DAG by DAG kinase in pool 2 membrane) is offset by phosphorylation of DAG by DAG kinase in pool 2 (Endoplasmic Reticulum Nucleus) to aid in the PI replacement pathway(Endoplasmic Reticulum Nucleus) to aid in the PI replacement pathway

bull Second wave of DAG is a result of resynthesis of PIPSecond wave of DAG is a result of resynthesis of PIP22 which is then which is then hydrolyzed to form DAG and IPhydrolyzed to form DAG and IP33

PI PIP PIP2 DAG

IP3

IP2IP

Ins

PI

CDP-DAGPADAG

PA

PCPE

PCPE

PITP

PI4K PI5K PLC DGK

DGK CDSPLC PIS

POOL 2

POOL 1

IP4

PI PIP PIP2 DAG

IP3

IP2IP

Ins

PI

CDP-DAGPADAG

PA

PCPE

PC

PITP

PI4K PI5K PLC

DGK CDSPLC PIS

IP4

LPP

LPP

Model Equations DAG (pool 1)Model Equations DAG (pool 1)Model Equations DAG (pool 1)Model Equations DAG (pool 1)

bull DAG (pool 1)DAG (pool 1) (separate ODEs for each species (separate ODEs for each species considered)considered)

bull Concentration of DAG from pool 1 at time tConcentration of DAG from pool 1 at time t

Nondimensionalized equationsNondimensionalized equations

Model Equations DAG (pool 2)Model Equations DAG (pool 2)Model Equations DAG (pool 2)Model Equations DAG (pool 2)

bull DAG (pool 2)DAG (pool 2) (separate ODEs for each species (separate ODEs for each species considered)considered)

bull Concentration of pool 2 DAG molecules at time tConcentration of pool 2 DAG molecules at time t

bull Baseline concentration of pool 2 DAGBaseline concentration of pool 2 DAG

Nondimensionalized equationsNondimensionalized equations

Simplified CaSimplified Ca2+2+ Equations EquationsSimplified CaSimplified Ca2+2+ Equations Equations

bull CaCa2+2+ modulemodule (to match experimental AfCS (to match experimental AfCS trace)trace)

0 100 200 300 400 500 6000

002

004

006

008

01

012

014

Concentration of Ca

2+ (uM)

Ca2+ response to 25uM UDP

time (s)

wherewhere

Theoretical AnalysisTheoretical AnalysisTheoretical AnalysisTheoretical Analysis

bull Existence and UniquenessExistence and Uniqueness

bull Positivity and Boundedness Positivity and Boundedness

(for biological relevance)(for biological relevance)

bull Analysis of steady state behaviorAnalysis of steady state behavior

Existence of Solutions (full model)Existence of Solutions (full model)Existence of Solutions (full model)Existence of Solutions (full model)

First we write our system of ODEs in the formFirst we write our system of ODEs in the form (1)

(2)

(1)

Uniqueness of SolutionsUniqueness of SolutionsUniqueness of SolutionsUniqueness of Solutions

Next we denote a solution of Next we denote a solution of (1)(1) by by with initial condition with initial condition

Since our system satisfies the hypotheses of Theorems 1 and 2 Since our system satisfies the hypotheses of Theorems 1 and 2 on our set of interest (for all positive time and on a positive on our set of interest (for all positive time and on a positive bounded set in space) we know bounded set in space) we know there exists a unique there exists a unique (local) solution(local) solution (ie on some finite time interval possibly (ie on some finite time interval possibly small)small)

QuestionQuestion Do the solutions remain positive and bounded (for Do the solutions remain positive and bounded (for biological significance)biological significance)

(3)

(1) (3)

Positivity and boundedness of Positivity and boundedness of solutionssolutions

Positivity and boundedness of Positivity and boundedness of solutionssolutions

bull We first use Theorem 3 to show positivity and boundedness of xWe first use Theorem 3 to show positivity and boundedness of x11 and and xx22

bull Next we use these results and the Fundamental Theorem of Calculus to Next we use these results and the Fundamental Theorem of Calculus to show positivity and boundedness for the remaining equations show positivity and boundedness for the remaining equations

bull This then ensures a global solutionThis then ensures a global solution

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical Analysis

bull Parameter Estimation in SIMULINKParameter Estimation in SIMULINK

bull Sensitivity AnalysisSensitivity Analysis

Parameter Estimation in SIMULINK (for Parameter Estimation in SIMULINK (for FULL model)FULL model)

Parameter Estimation in SIMULINK (for Parameter Estimation in SIMULINK (for FULL model)FULL model)

bull Total number of model parameters (for full model with Li and Rinzel CaTotal number of model parameters (for full model with Li and Rinzel Ca2+2+ module) = module) = 3434ndash From the literature = From the literature = 2020ndash Estimated = Estimated = 1414

bull Receptor moduleReceptor module Total = Total = 77ndash From literature = From literature = 66ndash Estimated = Estimated = 11 (k (kpp rate of receptor phosphorylation) rate of receptor phosphorylation)

bull G-protein cascadeG-protein cascade Total = Total = 99ndash From literature = From literature = 66ndash Estimated = Estimated = 3 3 (k(khydhyd k kreprep k kd3d3))

bull DAG kineticsDAG kinetics Total = Total = 5 5 (for each DAG species considered)(for each DAG species considered)ndash From literature = From literature = 00ndash Estimated = Estimated = 5 5 (all DAG parameters)(all DAG parameters)

bull CaCa2+2+ module module Total = Total = 1313ndash From literature = From literature = 88ndash Estimated = Estimated = 55

SIMULINK DetailsSIMULINK DetailsSIMULINK DetailsSIMULINK Details

bull Unknown rate parameters were estimated using SIMULINK

ndash Minimizes a user-specified cost function via a user-specified optimization method

ndash Nonlinear least squares optimization method of Levenberg-Marquardt was used to minimize a sum of squared errors cost function of the empirical observations and model predictions for IP3 Ca2+ and multiple species of DAG

bull Note Although the Gauss-Newton method is often more efficient the method of Levenberg-Marquardt has proved to be more robust

Measured vs Simulated 384 DAG and Measured vs Simulated 384 DAG and 341 DAG341 DAG

Measured vs Simulated 384 DAG and Measured vs Simulated 384 DAG and 341 DAG341 DAG

bull Timecourse of stimulation with 25 Timecourse of stimulation with 25 M UDP in RAW 2647 cellsM UDP in RAW 2647 cells

bull Solid black lines represent model simulations Solid black lines represent model simulations

bull ((aa) 384 DAG response (representative of the response of most poly ) 384 DAG response (representative of the response of most poly unsaturated fatty acid-containing DAG species) unsaturated fatty acid-containing DAG species)

bull ((bb) 341 DAG response (representative of the response of most mono- ) 341 DAG response (representative of the response of most mono- and di-unsaturated fatty acid-containing DAG species) and di-unsaturated fatty acid-containing DAG species)

bull Data points contain nine replicates performed on three different Data points contain nine replicates performed on three different experimental days with error bars = 1 SEM Units are total change in ng experimental days with error bars = 1 SEM Units are total change in ng over baseline levels in ~8x10over baseline levels in ~8x1066 cells cells

Measured vs Simulated IPMeasured vs Simulated IP33 and Ca and Ca2+2+Measured vs Simulated IPMeasured vs Simulated IP33 and Ca and Ca2+2+

bull Timecourse of stimulation with 25 Timecourse of stimulation with 25 M UDP in RAW 2647 cellsM UDP in RAW 2647 cells

bull Solid black lines represent model simulationsSolid black lines represent model simulations

bull ((cc) IP) IP33 response in pmols per ~35x10 response in pmols per ~35x1055 cells cells

bull Points in (c) represent the average of four experiments and error Points in (c) represent the average of four experiments and error bars are 1 SEM bars are 1 SEM

bull ((dd) Ca) Ca2+2+ response in response in M Red curve is a representative CaM Red curve is a representative Ca2+2+ trace trace taken from the UDP experiments within the AfCS single ligand taken from the UDP experiments within the AfCS single ligand screenscreen

Model Simulations P2YModel Simulations P2Y66 G GGTP GTP PIPPIP22

Model Simulations P2YModel Simulations P2Y66 G GGTP GTP PIPPIP22

bull ((aa) = total P2Y) = total P2Y66 activated (solid activated (solid line) and inactivated (dashed line) and inactivated (dashed line) surface receptorsline) surface receptors

bull ((bb) = total G) = total GGTP GTP

bull ((cc) = total PIP) = total PIP22 available for available for hydrolysishydrolysis

P2Y6 from 25M UDP GGTP from 25M UDP

PIP2 from 25M UDP

Sensitivity AnalysisSensitivity AnalysisSensitivity AnalysisSensitivity Analysis

bull Sensitivity analysis techniques are valuable tools Sensitivity analysis techniques are valuable tools designed to answer questions regarding which of the designed to answer questions regarding which of the uncertain input variables is more important in uncertain input variables is more important in determining the uncertainty in our output determining the uncertainty in our output

bull Likewise sensitivity analysis can provide insight into Likewise sensitivity analysis can provide insight into which parameter should be studied in more detail in which parameter should be studied in more detail in order to reduce the most variance in the model output order to reduce the most variance in the model output

bull The ability to answer these types of questions could The ability to answer these types of questions could lead to important insight into the design of new lead to important insight into the design of new experiments and in determining which experiments experiments and in determining which experiments would give us the most valuable informationwould give us the most valuable information

Sampling MethodSampling MethodSampling MethodSampling Method

bull Generate a random sample of our space of input Generate a random sample of our space of input variables over a ten percent variation from each variables over a ten percent variation from each parameters nominal value using the parameters nominal value using the Latin Hypercube Latin Hypercube Sampling (LHS) methodSampling (LHS) method

bull Uses Uses Standardized Regression Coefficients (SRCs) Standardized Regression Coefficients (SRCs) obtained by performing multiple linear regression obtained by performing multiple linear regression analysisanalysisndash offers a measure of sensitivity that is multi-dimensionally offers a measure of sensitivity that is multi-dimensionally

averaged over the entire space of parameter values averaged over the entire space of parameter values ndash SRCs give insight into degree of nonlinearity in the model SRCs give insight into degree of nonlinearity in the model

Computing RComputing Ryy22 values valuesComputing RComputing Ryy22 values values

bull SRCs are only reliable measures of sensitivity when SRCs are only reliable measures of sensitivity when degree of nonlinearity is ldquosmallrdquodegree of nonlinearity is ldquosmallrdquo

bull Use model coefficients of determination RUse model coefficients of determination Ryy22 given by given by

bull where ywhere yii is the estimate of y is the estimate of yii obtained from the regression obtained from the regression modelmodel

bull RRyy22 ge 07 ensures SRCs are good sensitivity measures ge 07 ensures SRCs are good sensitivity measures

SRCs (as sensitivity measures) for SRCs (as sensitivity measures) for DAGDAG

SRCs (as sensitivity measures) for SRCs (as sensitivity measures) for DAGDAG

SRCs (as sensitivity measures) for IPSRCs (as sensitivity measures) for IP33SRCs (as sensitivity measures) for IPSRCs (as sensitivity measures) for IP33

Effects of 3 most sensitive DAG Effects of 3 most sensitive DAG parametersparameters

Effects of 3 most sensitive DAG Effects of 3 most sensitive DAG parametersparameters

A B

C

AA k kdp2dp2 degradation of pool 2 DAG degradation of pool 2 DAG

BB k kdp1dp1 degradation of pool 1 DAG degradation of pool 1 DAG

CC k kap2ap2 production of pool 2 DAG production of pool 2 DAG

ConclusionsConclusionsConclusionsConclusions

bull We have developed a model of the UDP signaling We have developed a model of the UDP signaling pathway in RAW 2647 macrophages which can predict pathway in RAW 2647 macrophages which can predict the responses of multiple species of DAG as well as the the responses of multiple species of DAG as well as the responses of IPresponses of IP33 Ca Ca2+2+ receptor dynamics G-protein receptor dynamics G-protein activation and PIPactivation and PIP22 hydrolysis hydrolysis

bull Simplified model resultsSimplified model resultsndash We have obtained global existence uniqueness positivity We have obtained global existence uniqueness positivity

and boundedness of solutionsand boundedness of solutionsndash We have proven global stability of a unique steady state We have proven global stability of a unique steady state

within our region of interestwithin our region of interestbull Full model analysisFull model analysis

ndash Using SIMULINK we have estimated unknown rate Using SIMULINK we have estimated unknown rate parameters to obtain best fits to multiple DAG traces IPparameters to obtain best fits to multiple DAG traces IP33 and Caand Ca2+2+ all in response to 25 all in response to 25M UDPM UDP

ndash We have performed sensitivity analysis using the Latin We have performed sensitivity analysis using the Latin Hypercube sampling technique in combination with Hypercube sampling technique in combination with standardized regression coefficients to determine which standardized regression coefficients to determine which model parameters are responsible for most of the model model parameters are responsible for most of the model output uncertaintyoutput uncertainty

Future DirectionsFuture DirectionsFuture DirectionsFuture Directions

bull We have conducted multiple experiments to We have conducted multiple experiments to pharmacologically inhibit several different enzymes we pharmacologically inhibit several different enzymes we believe to play a role in this signaling pathway as believe to play a role in this signaling pathway as suggested by current known metabolic pathways and by suggested by current known metabolic pathways and by modeling resultsmodeling results

bull The next step is to perform gene knockdowns on The next step is to perform gene knockdowns on specific enzymes to verify results of inhibitor dataspecific enzymes to verify results of inhibitor data

bull The model output also suggests a time delay from The model output also suggests a time delay from receptor activation to PIPreceptor activation to PIP22 hydrolysis An upcoming task hydrolysis An upcoming task is to investigate the outcomes of adding such a delay is to investigate the outcomes of adding such a delay termterm

bull The model could be greatly enhanced by incorporating The model could be greatly enhanced by incorporating spatial dynamics so this will also be a major focus for spatial dynamics so this will also be a major focus for future researchfuture research

AcknowledgementsAcknowledgementsAcknowledgementsAcknowledgements

Mathematics DepartmentMathematics Departmentbull Hannah L Callender Hannah L Callender

PhDPhD

CollaboratorsCollaboratorsbull Alliance for Cellular SignalingAlliance for Cellular Signalingbull UT SouthwesternUT Southwestern

ndash Paul Sternweis PhDPaul Sternweis PhDndash Dianne DeCamp PhDDianne DeCamp PhD

Brown LabBrown Labbull H Alex Brown PhDH Alex Brown PhDbull Jeffrey S Forrester PhDJeffrey S Forrester PhDbull Mark Byrne PhDMark Byrne PhDbull Anita Preininger PhDAnita Preininger PhDbull Michelle ArmstrongMichelle Armstrongbull Andrew GoodmanAndrew Goodmanbull Pavlina Ivanova PhDPavlina Ivanova PhDbull Steve Milne PhDSteve Milne PhD

Mathematical ModelMathematical ModelMathematical ModelMathematical Model

bull The model consists of a base set of 10 coupled nonlinear The model consists of a base set of 10 coupled nonlinear ordinary differential equations (ODEs)ordinary differential equations (ODEs)

bull The ODEs are constructed using mass action kinetics to The ODEs are constructed using mass action kinetics to approximate diffusion binding and unbinding of molecular approximate diffusion binding and unbinding of molecular species in various compartments of the RAW cellspecies in various compartments of the RAW cell effective rates of production and degradation are proportional to effective rates of production and degradation are proportional to

the concentrations (or number of molecules) of participating the concentrations (or number of molecules) of participating reactantsreactants

bull Four modulesFour modulesndash Receptor regulationReceptor regulationndash G-protein casadeG-protein casadendash Species-specific DAG dynamicsSpecies-specific DAG dynamicsndash Cytosolic CaCytosolic Ca2+2+ dynamics dynamics

bull Receptor-ligandReceptor-ligand binding (based on work by Lemon 2003) binding (based on work by Lemon 2003)

ndash Number of ldquoactivatedrdquo surface receptors at time tNumber of ldquoactivatedrdquo surface receptors at time tndash Number of ldquoinactivatedrdquo surface receptors at time tNumber of ldquoinactivatedrdquo surface receptors at time tndash Concentration of ligand (constant)Concentration of ligand (constant)

Model Equations Receptor Model Equations Receptor regulationregulation

Model Equations Receptor Model Equations Receptor regulationregulation

Nondimensionalized equationsNondimensionalized equations

Model Equations G-protein Model Equations G-protein activationactivation

Model Equations G-protein Model Equations G-protein activationactivation

bull G-proteinG-protein activation activation

ndash Number of activated G-proteins (GNumber of activated G-proteins (GGTP) at time tGTP) at time t

Nondimensionalized equationsNondimensionalized equations

Model Equations PIPModel Equations PIP22 hydrolysis and hydrolysis and replenishmentreplenishment

Model Equations PIPModel Equations PIP22 hydrolysis and hydrolysis and replenishmentreplenishment

bull PIPPIP2 2 hydrolysis and replenishmenthydrolysis and replenishment

Nondimensionalized equationsNondimensionalized equations

ndash Number of PIPNumber of PIP22 molecules at time t molecules at time tndash Initial number of ldquofreerdquo PIPInitial number of ldquofreerdquo PIP22

PLC

PIP

2

DAG

IP3

GGTPCa2+

GGTP

bull IPIP33

Model Equations IPModel Equations IP33 production and production and degradationdegradation

Model Equations IPModel Equations IP33 production and production and degradationdegradation

Nondimensionalized equationsNondimensionalized equations

PLC

PIP

2

DA

G

IP3

G

GTPCa2+

G

GTP

bull Concentration of IPConcentration of IP33 at time t (in at time t (in M)M)

Model Equations DAGModel Equations DAGModel Equations DAGModel Equations DAG

bull DAG DAG (separate ODEs for each species considered)(separate ODEs for each species considered)

Nondimensionalized equationsNondimensionalized equations

bull Concentration of DAG at time tConcentration of DAG at time t

PLC

PIP

2

DA

G

IP3

G

GTPCa2+G

GTP

Model Equations Li and Rinzel (1994) Model Equations Li and Rinzel (1994) CaCa2+2+

Model Equations Li and Rinzel (1994) Model Equations Li and Rinzel (1994) CaCa2+2+

bull Free cytosolicFree cytosolic Ca Ca2+2+

bull hh = fraction of IP = fraction of IP33 channels not yet inactivated by Ca channels not yet inactivated by Ca2+2+

bull cc00 = concentration of total free Ca = concentration of total free Ca2+2+ per cytosolic volume per cytosolic volume

Timeseries data for the modelTimeseries data for the modelTimeseries data for the modelTimeseries data for the model

bull Data from collaborators at UTSW (Sternweis Lab)Data from collaborators at UTSW (Sternweis Lab)ndash Inositol 145-trisphosphate (Inositol 145-trisphosphate (IPIP33) production post 25) production post 25M M

UDP stimulation in the RAW cellUDP stimulation in the RAW cellndash Cytosolic calcium (Cytosolic calcium (CaCa2+2+) release post stimulation with ) release post stimulation with

2525M UDP in the RAW cellM UDP in the RAW cell

bull Diacylglycerol (Diacylglycerol (DAGDAG) data H Callender collected from a ) data H Callender collected from a novel method of quantitative analysis of multiple novel method of quantitative analysis of multiple species of DAG developed in the Brown Lab species of DAG developed in the Brown Lab

Reference Reference

Callender H L et al Quantification of Diacylglycerol Species from Callender H L et al Quantification of Diacylglycerol Species from Cellular Extracts by Electrospray Ionization Mass Spectrometry Using a Cellular Extracts by Electrospray Ionization Mass Spectrometry Using a Linear Regression Algorithm Linear Regression Algorithm Anal Chem Anal Chem 7979 (2007) 263-272 (2007) 263-272

IPIP33 and Ca and Ca2+2+ data from AfCS data from AfCSIPIP33 and Ca and Ca2+2+ data from AfCS data from AfCS

AA

bull IPIP33 response (in pmoles per response (in pmoles per 100 100 L lysate)in RAW 2647 L lysate)in RAW 2647 cells to 25 cells to 25 M UDPM UDP

bull Points represent the average Points represent the average of four experimentsof four experiments

bull Cytosolic calcium response Cytosolic calcium response (in (in M) in the RAW 2647 M) in the RAW 2647 cell to 25 cell to 25 M UDP M UDP

bull The graph displays 43 The graph displays 43 experimental repeatsexperimental repeats

B

What is DAGWhat is DAGWhat is DAGWhat is DAG

bull DAG is a cellular second messenger molecule which plays an DAG is a cellular second messenger molecule which plays an important role in initiating various changes in cell behavior important role in initiating various changes in cell behavior including cell activation differentiation proliferation and including cell activation differentiation proliferation and tumor promotion tumor promotion

bull There are many different species of DAG depending on the There are many different species of DAG depending on the number of carbons and number of double bonds in the fatty number of carbons and number of double bonds in the fatty acyl chains and different species can have different cellular acyl chains and different species can have different cellular functionsfunctions

bull ExampleExample 320 DAG320 DAG

Two fatty acyl Two fatty acyl (hydrocarbon) (hydrocarbon) chains in every DAG chains in every DAG speciesspecies

Each corner Each corner represents carbonrepresents carbon

320 DAG 18 carbons in first chain 14 in the second no double bonds 320 DAG 18 carbons in first chain 14 in the second no double bonds in either chainin either chain

Kinetics of monodi unsaturated DAGsKinetics of monodi unsaturated DAGsKinetics of monodi unsaturated DAGsKinetics of monodi unsaturated DAGs

bull Time based behavior of four monodi unsaturated DAG species after addition of 25 Time based behavior of four monodi unsaturated DAG species after addition of 25 M M (solid red squares) and 025 (solid red squares) and 025 M (solid green triangles) UDPM (solid green triangles) UDP

bull Time points contain a minimum of nine replicates performed on three different Time points contain a minimum of nine replicates performed on three different experimental daysexperimental days

- 25 M UDP

- 025M UDP

0 5 10 15 20 25 30

0 5 10 15 20 25 30 0 5 10 15 20 25 30

0 5 10 15 20 25 30

321 DAG 341 DAG

362 DAG341 DAGep

150

100

50-5

00

300

200

100

-100

0

150

100

50-5

00

-100

200

6040

20-2

00

-40

80

Time (min) Time (min)

Time (min) Time (min)

D

AG

(n

g)

D

AG

(n

g)

D

AG

(n

g)

D

AG

(n

g)

- 25 M UDP

- 025M UDP

- 25 M UDP

- 025M UDP

- 25 M UDP

- 025M UDP

Kinetics of PUFA containing DAGsKinetics of PUFA containing DAGsKinetics of PUFA containing DAGsKinetics of PUFA containing DAGsbull Time based behavior of four polyunsaturated fatty acid (PUFA) containing DAG species Time based behavior of four polyunsaturated fatty acid (PUFA) containing DAG species

after addition of 25 after addition of 25 M (solid red squares) and 025 M (solid red squares) and 025 M (solid green triangles) UDPM (solid green triangles) UDPbull Time points contain a minimum of nine replicates performed on three different Time points contain a minimum of nine replicates performed on three different

experimental daysexperimental daysbull Monodi unsaturated DAG species give a larger increase than polyunsaturated fatty acid Monodi unsaturated DAG species give a larger increase than polyunsaturated fatty acid

(PUFA) containing DAGs(PUFA) containing DAGs

0 5 10 15 20 25 30

0 5 10 15 20 25 30 0 5 10 15 20 25 30

0 5 10 15 20 25 30

385 DAG 383 DAG

364 DAG384 DAG6

42

-40

30

20

10

0

15

10

50

-56

40

2-2

8

Time (min) Time (min)

Time (min) Time (min)

D

AG

(n

g)

D

AG

(n

g)

D

AG

(n

g)

D

AG

(n

g)

-28

40

- 25 M UDP

- 025M UDP

- 25 M UDP

- 025M UDP

- 25 M UDP

- 025M UDP

- 25 M UDP

- 025M UDP

Differential DAG KineticsDifferential DAG KineticsDifferential DAG KineticsDifferential DAG Kinetics

bull Time based behavior of three DAG species with varying degrees of unsaturation after Time based behavior of three DAG species with varying degrees of unsaturation after addition of addition of 25 25 MM (solid red triangles) and (solid red triangles) and 025 025 MM (solid green squares) UDP (solid green squares) UDP

bull Time points contain nine replicates performed on three different experimental daysTime points contain nine replicates performed on three different experimental days

341 DAG

Best fit with current model structure

Overall Objectives of Modeling EffortOverall Objectives of Modeling EffortOverall Objectives of Modeling EffortOverall Objectives of Modeling Effort

bull Predict quantitative changes in lipid species after stimulation by Predict quantitative changes in lipid species after stimulation by various ligands and ligand concentrations in the RAW 2647 various ligands and ligand concentrations in the RAW 2647 macrophagemacrophage

bull Comparison and refinement of model output with AfCS IPComparison and refinement of model output with AfCS IP33 measurements and Cameasurements and Ca2+2+ traces as well as DAG data generated in traces as well as DAG data generated in the Brown labthe Brown lab

bull Predict in silico effects such as the effect of knock-downs etc on Predict in silico effects such as the effect of knock-downs etc on given pathwaygiven pathway

bull Suggest modifications to current pathway structuresSuggest modifications to current pathway structures

Modifications to the modelModifications to the modelModifications to the modelModifications to the model

bull Include an additional branch in the pathway Include an additional branch in the pathway for a second pool of DAGfor a second pool of DAG

bull Simplify CaSimplify Ca2+2+ equations for mathematical equations for mathematical analysis purposesanalysis purposes

Proposed PathwayProposed PathwayProposed PathwayProposed Pathway

bull Note We measure total cellular DAG levelsNote We measure total cellular DAG levelsbull Initial production of DAG from the hydrolysis of PIPInitial production of DAG from the hydrolysis of PIP22 in pool 1 (plasma in pool 1 (plasma

membrane) is offset by phosphorylation of DAG by DAG kinase in pool 2 membrane) is offset by phosphorylation of DAG by DAG kinase in pool 2 (Endoplasmic Reticulum Nucleus) to aid in the PI replacement pathway(Endoplasmic Reticulum Nucleus) to aid in the PI replacement pathway

bull Second wave of DAG is a result of resynthesis of PIPSecond wave of DAG is a result of resynthesis of PIP22 which is then which is then hydrolyzed to form DAG and IPhydrolyzed to form DAG and IP33

PI PIP PIP2 DAG

IP3

IP2IP

Ins

PI

CDP-DAGPADAG

PA

PCPE

PCPE

PITP

PI4K PI5K PLC DGK

DGK CDSPLC PIS

POOL 2

POOL 1

IP4

PI PIP PIP2 DAG

IP3

IP2IP

Ins

PI

CDP-DAGPADAG

PA

PCPE

PC

PITP

PI4K PI5K PLC

DGK CDSPLC PIS

IP4

LPP

LPP

Model Equations DAG (pool 1)Model Equations DAG (pool 1)Model Equations DAG (pool 1)Model Equations DAG (pool 1)

bull DAG (pool 1)DAG (pool 1) (separate ODEs for each species (separate ODEs for each species considered)considered)

bull Concentration of DAG from pool 1 at time tConcentration of DAG from pool 1 at time t

Nondimensionalized equationsNondimensionalized equations

Model Equations DAG (pool 2)Model Equations DAG (pool 2)Model Equations DAG (pool 2)Model Equations DAG (pool 2)

bull DAG (pool 2)DAG (pool 2) (separate ODEs for each species (separate ODEs for each species considered)considered)

bull Concentration of pool 2 DAG molecules at time tConcentration of pool 2 DAG molecules at time t

bull Baseline concentration of pool 2 DAGBaseline concentration of pool 2 DAG

Nondimensionalized equationsNondimensionalized equations

Simplified CaSimplified Ca2+2+ Equations EquationsSimplified CaSimplified Ca2+2+ Equations Equations

bull CaCa2+2+ modulemodule (to match experimental AfCS (to match experimental AfCS trace)trace)

0 100 200 300 400 500 6000

002

004

006

008

01

012

014

Concentration of Ca

2+ (uM)

Ca2+ response to 25uM UDP

time (s)

wherewhere

Theoretical AnalysisTheoretical AnalysisTheoretical AnalysisTheoretical Analysis

bull Existence and UniquenessExistence and Uniqueness

bull Positivity and Boundedness Positivity and Boundedness

(for biological relevance)(for biological relevance)

bull Analysis of steady state behaviorAnalysis of steady state behavior

Existence of Solutions (full model)Existence of Solutions (full model)Existence of Solutions (full model)Existence of Solutions (full model)

First we write our system of ODEs in the formFirst we write our system of ODEs in the form (1)

(2)

(1)

Uniqueness of SolutionsUniqueness of SolutionsUniqueness of SolutionsUniqueness of Solutions

Next we denote a solution of Next we denote a solution of (1)(1) by by with initial condition with initial condition

Since our system satisfies the hypotheses of Theorems 1 and 2 Since our system satisfies the hypotheses of Theorems 1 and 2 on our set of interest (for all positive time and on a positive on our set of interest (for all positive time and on a positive bounded set in space) we know bounded set in space) we know there exists a unique there exists a unique (local) solution(local) solution (ie on some finite time interval possibly (ie on some finite time interval possibly small)small)

QuestionQuestion Do the solutions remain positive and bounded (for Do the solutions remain positive and bounded (for biological significance)biological significance)

(3)

(1) (3)

Positivity and boundedness of Positivity and boundedness of solutionssolutions

Positivity and boundedness of Positivity and boundedness of solutionssolutions

bull We first use Theorem 3 to show positivity and boundedness of xWe first use Theorem 3 to show positivity and boundedness of x11 and and xx22

bull Next we use these results and the Fundamental Theorem of Calculus to Next we use these results and the Fundamental Theorem of Calculus to show positivity and boundedness for the remaining equations show positivity and boundedness for the remaining equations

bull This then ensures a global solutionThis then ensures a global solution

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical Analysis

bull Parameter Estimation in SIMULINKParameter Estimation in SIMULINK

bull Sensitivity AnalysisSensitivity Analysis

Parameter Estimation in SIMULINK (for Parameter Estimation in SIMULINK (for FULL model)FULL model)

Parameter Estimation in SIMULINK (for Parameter Estimation in SIMULINK (for FULL model)FULL model)

bull Total number of model parameters (for full model with Li and Rinzel CaTotal number of model parameters (for full model with Li and Rinzel Ca2+2+ module) = module) = 3434ndash From the literature = From the literature = 2020ndash Estimated = Estimated = 1414

bull Receptor moduleReceptor module Total = Total = 77ndash From literature = From literature = 66ndash Estimated = Estimated = 11 (k (kpp rate of receptor phosphorylation) rate of receptor phosphorylation)

bull G-protein cascadeG-protein cascade Total = Total = 99ndash From literature = From literature = 66ndash Estimated = Estimated = 3 3 (k(khydhyd k kreprep k kd3d3))

bull DAG kineticsDAG kinetics Total = Total = 5 5 (for each DAG species considered)(for each DAG species considered)ndash From literature = From literature = 00ndash Estimated = Estimated = 5 5 (all DAG parameters)(all DAG parameters)

bull CaCa2+2+ module module Total = Total = 1313ndash From literature = From literature = 88ndash Estimated = Estimated = 55

SIMULINK DetailsSIMULINK DetailsSIMULINK DetailsSIMULINK Details

bull Unknown rate parameters were estimated using SIMULINK

ndash Minimizes a user-specified cost function via a user-specified optimization method

ndash Nonlinear least squares optimization method of Levenberg-Marquardt was used to minimize a sum of squared errors cost function of the empirical observations and model predictions for IP3 Ca2+ and multiple species of DAG

bull Note Although the Gauss-Newton method is often more efficient the method of Levenberg-Marquardt has proved to be more robust

Measured vs Simulated 384 DAG and Measured vs Simulated 384 DAG and 341 DAG341 DAG

Measured vs Simulated 384 DAG and Measured vs Simulated 384 DAG and 341 DAG341 DAG

bull Timecourse of stimulation with 25 Timecourse of stimulation with 25 M UDP in RAW 2647 cellsM UDP in RAW 2647 cells

bull Solid black lines represent model simulations Solid black lines represent model simulations

bull ((aa) 384 DAG response (representative of the response of most poly ) 384 DAG response (representative of the response of most poly unsaturated fatty acid-containing DAG species) unsaturated fatty acid-containing DAG species)

bull ((bb) 341 DAG response (representative of the response of most mono- ) 341 DAG response (representative of the response of most mono- and di-unsaturated fatty acid-containing DAG species) and di-unsaturated fatty acid-containing DAG species)

bull Data points contain nine replicates performed on three different Data points contain nine replicates performed on three different experimental days with error bars = 1 SEM Units are total change in ng experimental days with error bars = 1 SEM Units are total change in ng over baseline levels in ~8x10over baseline levels in ~8x1066 cells cells

Measured vs Simulated IPMeasured vs Simulated IP33 and Ca and Ca2+2+Measured vs Simulated IPMeasured vs Simulated IP33 and Ca and Ca2+2+

bull Timecourse of stimulation with 25 Timecourse of stimulation with 25 M UDP in RAW 2647 cellsM UDP in RAW 2647 cells

bull Solid black lines represent model simulationsSolid black lines represent model simulations

bull ((cc) IP) IP33 response in pmols per ~35x10 response in pmols per ~35x1055 cells cells

bull Points in (c) represent the average of four experiments and error Points in (c) represent the average of four experiments and error bars are 1 SEM bars are 1 SEM

bull ((dd) Ca) Ca2+2+ response in response in M Red curve is a representative CaM Red curve is a representative Ca2+2+ trace trace taken from the UDP experiments within the AfCS single ligand taken from the UDP experiments within the AfCS single ligand screenscreen

Model Simulations P2YModel Simulations P2Y66 G GGTP GTP PIPPIP22

Model Simulations P2YModel Simulations P2Y66 G GGTP GTP PIPPIP22

bull ((aa) = total P2Y) = total P2Y66 activated (solid activated (solid line) and inactivated (dashed line) and inactivated (dashed line) surface receptorsline) surface receptors

bull ((bb) = total G) = total GGTP GTP

bull ((cc) = total PIP) = total PIP22 available for available for hydrolysishydrolysis

P2Y6 from 25M UDP GGTP from 25M UDP

PIP2 from 25M UDP

Sensitivity AnalysisSensitivity AnalysisSensitivity AnalysisSensitivity Analysis

bull Sensitivity analysis techniques are valuable tools Sensitivity analysis techniques are valuable tools designed to answer questions regarding which of the designed to answer questions regarding which of the uncertain input variables is more important in uncertain input variables is more important in determining the uncertainty in our output determining the uncertainty in our output

bull Likewise sensitivity analysis can provide insight into Likewise sensitivity analysis can provide insight into which parameter should be studied in more detail in which parameter should be studied in more detail in order to reduce the most variance in the model output order to reduce the most variance in the model output

bull The ability to answer these types of questions could The ability to answer these types of questions could lead to important insight into the design of new lead to important insight into the design of new experiments and in determining which experiments experiments and in determining which experiments would give us the most valuable informationwould give us the most valuable information

Sampling MethodSampling MethodSampling MethodSampling Method

bull Generate a random sample of our space of input Generate a random sample of our space of input variables over a ten percent variation from each variables over a ten percent variation from each parameters nominal value using the parameters nominal value using the Latin Hypercube Latin Hypercube Sampling (LHS) methodSampling (LHS) method

bull Uses Uses Standardized Regression Coefficients (SRCs) Standardized Regression Coefficients (SRCs) obtained by performing multiple linear regression obtained by performing multiple linear regression analysisanalysisndash offers a measure of sensitivity that is multi-dimensionally offers a measure of sensitivity that is multi-dimensionally

averaged over the entire space of parameter values averaged over the entire space of parameter values ndash SRCs give insight into degree of nonlinearity in the model SRCs give insight into degree of nonlinearity in the model

Computing RComputing Ryy22 values valuesComputing RComputing Ryy22 values values

bull SRCs are only reliable measures of sensitivity when SRCs are only reliable measures of sensitivity when degree of nonlinearity is ldquosmallrdquodegree of nonlinearity is ldquosmallrdquo

bull Use model coefficients of determination RUse model coefficients of determination Ryy22 given by given by

bull where ywhere yii is the estimate of y is the estimate of yii obtained from the regression obtained from the regression modelmodel

bull RRyy22 ge 07 ensures SRCs are good sensitivity measures ge 07 ensures SRCs are good sensitivity measures

SRCs (as sensitivity measures) for SRCs (as sensitivity measures) for DAGDAG

SRCs (as sensitivity measures) for SRCs (as sensitivity measures) for DAGDAG

SRCs (as sensitivity measures) for IPSRCs (as sensitivity measures) for IP33SRCs (as sensitivity measures) for IPSRCs (as sensitivity measures) for IP33

Effects of 3 most sensitive DAG Effects of 3 most sensitive DAG parametersparameters

Effects of 3 most sensitive DAG Effects of 3 most sensitive DAG parametersparameters

A B

C

AA k kdp2dp2 degradation of pool 2 DAG degradation of pool 2 DAG

BB k kdp1dp1 degradation of pool 1 DAG degradation of pool 1 DAG

CC k kap2ap2 production of pool 2 DAG production of pool 2 DAG

ConclusionsConclusionsConclusionsConclusions

bull We have developed a model of the UDP signaling We have developed a model of the UDP signaling pathway in RAW 2647 macrophages which can predict pathway in RAW 2647 macrophages which can predict the responses of multiple species of DAG as well as the the responses of multiple species of DAG as well as the responses of IPresponses of IP33 Ca Ca2+2+ receptor dynamics G-protein receptor dynamics G-protein activation and PIPactivation and PIP22 hydrolysis hydrolysis

bull Simplified model resultsSimplified model resultsndash We have obtained global existence uniqueness positivity We have obtained global existence uniqueness positivity

and boundedness of solutionsand boundedness of solutionsndash We have proven global stability of a unique steady state We have proven global stability of a unique steady state

within our region of interestwithin our region of interestbull Full model analysisFull model analysis

ndash Using SIMULINK we have estimated unknown rate Using SIMULINK we have estimated unknown rate parameters to obtain best fits to multiple DAG traces IPparameters to obtain best fits to multiple DAG traces IP33 and Caand Ca2+2+ all in response to 25 all in response to 25M UDPM UDP

ndash We have performed sensitivity analysis using the Latin We have performed sensitivity analysis using the Latin Hypercube sampling technique in combination with Hypercube sampling technique in combination with standardized regression coefficients to determine which standardized regression coefficients to determine which model parameters are responsible for most of the model model parameters are responsible for most of the model output uncertaintyoutput uncertainty

Future DirectionsFuture DirectionsFuture DirectionsFuture Directions

bull We have conducted multiple experiments to We have conducted multiple experiments to pharmacologically inhibit several different enzymes we pharmacologically inhibit several different enzymes we believe to play a role in this signaling pathway as believe to play a role in this signaling pathway as suggested by current known metabolic pathways and by suggested by current known metabolic pathways and by modeling resultsmodeling results

bull The next step is to perform gene knockdowns on The next step is to perform gene knockdowns on specific enzymes to verify results of inhibitor dataspecific enzymes to verify results of inhibitor data

bull The model output also suggests a time delay from The model output also suggests a time delay from receptor activation to PIPreceptor activation to PIP22 hydrolysis An upcoming task hydrolysis An upcoming task is to investigate the outcomes of adding such a delay is to investigate the outcomes of adding such a delay termterm

bull The model could be greatly enhanced by incorporating The model could be greatly enhanced by incorporating spatial dynamics so this will also be a major focus for spatial dynamics so this will also be a major focus for future researchfuture research

AcknowledgementsAcknowledgementsAcknowledgementsAcknowledgements

Mathematics DepartmentMathematics Departmentbull Hannah L Callender Hannah L Callender

PhDPhD

CollaboratorsCollaboratorsbull Alliance for Cellular SignalingAlliance for Cellular Signalingbull UT SouthwesternUT Southwestern

ndash Paul Sternweis PhDPaul Sternweis PhDndash Dianne DeCamp PhDDianne DeCamp PhD

Brown LabBrown Labbull H Alex Brown PhDH Alex Brown PhDbull Jeffrey S Forrester PhDJeffrey S Forrester PhDbull Mark Byrne PhDMark Byrne PhDbull Anita Preininger PhDAnita Preininger PhDbull Michelle ArmstrongMichelle Armstrongbull Andrew GoodmanAndrew Goodmanbull Pavlina Ivanova PhDPavlina Ivanova PhDbull Steve Milne PhDSteve Milne PhD

bull Receptor-ligandReceptor-ligand binding (based on work by Lemon 2003) binding (based on work by Lemon 2003)

ndash Number of ldquoactivatedrdquo surface receptors at time tNumber of ldquoactivatedrdquo surface receptors at time tndash Number of ldquoinactivatedrdquo surface receptors at time tNumber of ldquoinactivatedrdquo surface receptors at time tndash Concentration of ligand (constant)Concentration of ligand (constant)

Model Equations Receptor Model Equations Receptor regulationregulation

Model Equations Receptor Model Equations Receptor regulationregulation

Nondimensionalized equationsNondimensionalized equations

Model Equations G-protein Model Equations G-protein activationactivation

Model Equations G-protein Model Equations G-protein activationactivation

bull G-proteinG-protein activation activation

ndash Number of activated G-proteins (GNumber of activated G-proteins (GGTP) at time tGTP) at time t

Nondimensionalized equationsNondimensionalized equations

Model Equations PIPModel Equations PIP22 hydrolysis and hydrolysis and replenishmentreplenishment

Model Equations PIPModel Equations PIP22 hydrolysis and hydrolysis and replenishmentreplenishment

bull PIPPIP2 2 hydrolysis and replenishmenthydrolysis and replenishment

Nondimensionalized equationsNondimensionalized equations

ndash Number of PIPNumber of PIP22 molecules at time t molecules at time tndash Initial number of ldquofreerdquo PIPInitial number of ldquofreerdquo PIP22

PLC

PIP

2

DAG

IP3

GGTPCa2+

GGTP

bull IPIP33

Model Equations IPModel Equations IP33 production and production and degradationdegradation

Model Equations IPModel Equations IP33 production and production and degradationdegradation

Nondimensionalized equationsNondimensionalized equations

PLC

PIP

2

DA

G

IP3

G

GTPCa2+

G

GTP

bull Concentration of IPConcentration of IP33 at time t (in at time t (in M)M)

Model Equations DAGModel Equations DAGModel Equations DAGModel Equations DAG

bull DAG DAG (separate ODEs for each species considered)(separate ODEs for each species considered)

Nondimensionalized equationsNondimensionalized equations

bull Concentration of DAG at time tConcentration of DAG at time t

PLC

PIP

2

DA

G

IP3

G

GTPCa2+G

GTP

Model Equations Li and Rinzel (1994) Model Equations Li and Rinzel (1994) CaCa2+2+

Model Equations Li and Rinzel (1994) Model Equations Li and Rinzel (1994) CaCa2+2+

bull Free cytosolicFree cytosolic Ca Ca2+2+

bull hh = fraction of IP = fraction of IP33 channels not yet inactivated by Ca channels not yet inactivated by Ca2+2+

bull cc00 = concentration of total free Ca = concentration of total free Ca2+2+ per cytosolic volume per cytosolic volume

Timeseries data for the modelTimeseries data for the modelTimeseries data for the modelTimeseries data for the model

bull Data from collaborators at UTSW (Sternweis Lab)Data from collaborators at UTSW (Sternweis Lab)ndash Inositol 145-trisphosphate (Inositol 145-trisphosphate (IPIP33) production post 25) production post 25M M

UDP stimulation in the RAW cellUDP stimulation in the RAW cellndash Cytosolic calcium (Cytosolic calcium (CaCa2+2+) release post stimulation with ) release post stimulation with

2525M UDP in the RAW cellM UDP in the RAW cell

bull Diacylglycerol (Diacylglycerol (DAGDAG) data H Callender collected from a ) data H Callender collected from a novel method of quantitative analysis of multiple novel method of quantitative analysis of multiple species of DAG developed in the Brown Lab species of DAG developed in the Brown Lab

Reference Reference

Callender H L et al Quantification of Diacylglycerol Species from Callender H L et al Quantification of Diacylglycerol Species from Cellular Extracts by Electrospray Ionization Mass Spectrometry Using a Cellular Extracts by Electrospray Ionization Mass Spectrometry Using a Linear Regression Algorithm Linear Regression Algorithm Anal Chem Anal Chem 7979 (2007) 263-272 (2007) 263-272

IPIP33 and Ca and Ca2+2+ data from AfCS data from AfCSIPIP33 and Ca and Ca2+2+ data from AfCS data from AfCS

AA

bull IPIP33 response (in pmoles per response (in pmoles per 100 100 L lysate)in RAW 2647 L lysate)in RAW 2647 cells to 25 cells to 25 M UDPM UDP

bull Points represent the average Points represent the average of four experimentsof four experiments

bull Cytosolic calcium response Cytosolic calcium response (in (in M) in the RAW 2647 M) in the RAW 2647 cell to 25 cell to 25 M UDP M UDP

bull The graph displays 43 The graph displays 43 experimental repeatsexperimental repeats

B

What is DAGWhat is DAGWhat is DAGWhat is DAG

bull DAG is a cellular second messenger molecule which plays an DAG is a cellular second messenger molecule which plays an important role in initiating various changes in cell behavior important role in initiating various changes in cell behavior including cell activation differentiation proliferation and including cell activation differentiation proliferation and tumor promotion tumor promotion

bull There are many different species of DAG depending on the There are many different species of DAG depending on the number of carbons and number of double bonds in the fatty number of carbons and number of double bonds in the fatty acyl chains and different species can have different cellular acyl chains and different species can have different cellular functionsfunctions

bull ExampleExample 320 DAG320 DAG

Two fatty acyl Two fatty acyl (hydrocarbon) (hydrocarbon) chains in every DAG chains in every DAG speciesspecies

Each corner Each corner represents carbonrepresents carbon

320 DAG 18 carbons in first chain 14 in the second no double bonds 320 DAG 18 carbons in first chain 14 in the second no double bonds in either chainin either chain

Kinetics of monodi unsaturated DAGsKinetics of monodi unsaturated DAGsKinetics of monodi unsaturated DAGsKinetics of monodi unsaturated DAGs

bull Time based behavior of four monodi unsaturated DAG species after addition of 25 Time based behavior of four monodi unsaturated DAG species after addition of 25 M M (solid red squares) and 025 (solid red squares) and 025 M (solid green triangles) UDPM (solid green triangles) UDP

bull Time points contain a minimum of nine replicates performed on three different Time points contain a minimum of nine replicates performed on three different experimental daysexperimental days

- 25 M UDP

- 025M UDP

0 5 10 15 20 25 30

0 5 10 15 20 25 30 0 5 10 15 20 25 30

0 5 10 15 20 25 30

321 DAG 341 DAG

362 DAG341 DAGep

150

100

50-5

00

300

200

100

-100

0

150

100

50-5

00

-100

200

6040

20-2

00

-40

80

Time (min) Time (min)

Time (min) Time (min)

D

AG

(n

g)

D

AG

(n

g)

D

AG

(n

g)

D

AG

(n

g)

- 25 M UDP

- 025M UDP

- 25 M UDP

- 025M UDP

- 25 M UDP

- 025M UDP

Kinetics of PUFA containing DAGsKinetics of PUFA containing DAGsKinetics of PUFA containing DAGsKinetics of PUFA containing DAGsbull Time based behavior of four polyunsaturated fatty acid (PUFA) containing DAG species Time based behavior of four polyunsaturated fatty acid (PUFA) containing DAG species

after addition of 25 after addition of 25 M (solid red squares) and 025 M (solid red squares) and 025 M (solid green triangles) UDPM (solid green triangles) UDPbull Time points contain a minimum of nine replicates performed on three different Time points contain a minimum of nine replicates performed on three different

experimental daysexperimental daysbull Monodi unsaturated DAG species give a larger increase than polyunsaturated fatty acid Monodi unsaturated DAG species give a larger increase than polyunsaturated fatty acid

(PUFA) containing DAGs(PUFA) containing DAGs

0 5 10 15 20 25 30

0 5 10 15 20 25 30 0 5 10 15 20 25 30

0 5 10 15 20 25 30

385 DAG 383 DAG

364 DAG384 DAG6

42

-40

30

20

10

0

15

10

50

-56

40

2-2

8

Time (min) Time (min)

Time (min) Time (min)

D

AG

(n

g)

D

AG

(n

g)

D

AG

(n

g)

D

AG

(n

g)

-28

40

- 25 M UDP

- 025M UDP

- 25 M UDP

- 025M UDP

- 25 M UDP

- 025M UDP

- 25 M UDP

- 025M UDP

Differential DAG KineticsDifferential DAG KineticsDifferential DAG KineticsDifferential DAG Kinetics

bull Time based behavior of three DAG species with varying degrees of unsaturation after Time based behavior of three DAG species with varying degrees of unsaturation after addition of addition of 25 25 MM (solid red triangles) and (solid red triangles) and 025 025 MM (solid green squares) UDP (solid green squares) UDP

bull Time points contain nine replicates performed on three different experimental daysTime points contain nine replicates performed on three different experimental days

341 DAG

Best fit with current model structure

Overall Objectives of Modeling EffortOverall Objectives of Modeling EffortOverall Objectives of Modeling EffortOverall Objectives of Modeling Effort

bull Predict quantitative changes in lipid species after stimulation by Predict quantitative changes in lipid species after stimulation by various ligands and ligand concentrations in the RAW 2647 various ligands and ligand concentrations in the RAW 2647 macrophagemacrophage

bull Comparison and refinement of model output with AfCS IPComparison and refinement of model output with AfCS IP33 measurements and Cameasurements and Ca2+2+ traces as well as DAG data generated in traces as well as DAG data generated in the Brown labthe Brown lab

bull Predict in silico effects such as the effect of knock-downs etc on Predict in silico effects such as the effect of knock-downs etc on given pathwaygiven pathway

bull Suggest modifications to current pathway structuresSuggest modifications to current pathway structures

Modifications to the modelModifications to the modelModifications to the modelModifications to the model

bull Include an additional branch in the pathway Include an additional branch in the pathway for a second pool of DAGfor a second pool of DAG

bull Simplify CaSimplify Ca2+2+ equations for mathematical equations for mathematical analysis purposesanalysis purposes

Proposed PathwayProposed PathwayProposed PathwayProposed Pathway

bull Note We measure total cellular DAG levelsNote We measure total cellular DAG levelsbull Initial production of DAG from the hydrolysis of PIPInitial production of DAG from the hydrolysis of PIP22 in pool 1 (plasma in pool 1 (plasma

membrane) is offset by phosphorylation of DAG by DAG kinase in pool 2 membrane) is offset by phosphorylation of DAG by DAG kinase in pool 2 (Endoplasmic Reticulum Nucleus) to aid in the PI replacement pathway(Endoplasmic Reticulum Nucleus) to aid in the PI replacement pathway

bull Second wave of DAG is a result of resynthesis of PIPSecond wave of DAG is a result of resynthesis of PIP22 which is then which is then hydrolyzed to form DAG and IPhydrolyzed to form DAG and IP33

PI PIP PIP2 DAG

IP3

IP2IP

Ins

PI

CDP-DAGPADAG

PA

PCPE

PCPE

PITP

PI4K PI5K PLC DGK

DGK CDSPLC PIS

POOL 2

POOL 1

IP4

PI PIP PIP2 DAG

IP3

IP2IP

Ins

PI

CDP-DAGPADAG

PA

PCPE

PC

PITP

PI4K PI5K PLC

DGK CDSPLC PIS

IP4

LPP

LPP

Model Equations DAG (pool 1)Model Equations DAG (pool 1)Model Equations DAG (pool 1)Model Equations DAG (pool 1)

bull DAG (pool 1)DAG (pool 1) (separate ODEs for each species (separate ODEs for each species considered)considered)

bull Concentration of DAG from pool 1 at time tConcentration of DAG from pool 1 at time t

Nondimensionalized equationsNondimensionalized equations

Model Equations DAG (pool 2)Model Equations DAG (pool 2)Model Equations DAG (pool 2)Model Equations DAG (pool 2)

bull DAG (pool 2)DAG (pool 2) (separate ODEs for each species (separate ODEs for each species considered)considered)

bull Concentration of pool 2 DAG molecules at time tConcentration of pool 2 DAG molecules at time t

bull Baseline concentration of pool 2 DAGBaseline concentration of pool 2 DAG

Nondimensionalized equationsNondimensionalized equations

Simplified CaSimplified Ca2+2+ Equations EquationsSimplified CaSimplified Ca2+2+ Equations Equations

bull CaCa2+2+ modulemodule (to match experimental AfCS (to match experimental AfCS trace)trace)

0 100 200 300 400 500 6000

002

004

006

008

01

012

014

Concentration of Ca

2+ (uM)

Ca2+ response to 25uM UDP

time (s)

wherewhere

Theoretical AnalysisTheoretical AnalysisTheoretical AnalysisTheoretical Analysis

bull Existence and UniquenessExistence and Uniqueness

bull Positivity and Boundedness Positivity and Boundedness

(for biological relevance)(for biological relevance)

bull Analysis of steady state behaviorAnalysis of steady state behavior

Existence of Solutions (full model)Existence of Solutions (full model)Existence of Solutions (full model)Existence of Solutions (full model)

First we write our system of ODEs in the formFirst we write our system of ODEs in the form (1)

(2)

(1)

Uniqueness of SolutionsUniqueness of SolutionsUniqueness of SolutionsUniqueness of Solutions

Next we denote a solution of Next we denote a solution of (1)(1) by by with initial condition with initial condition

Since our system satisfies the hypotheses of Theorems 1 and 2 Since our system satisfies the hypotheses of Theorems 1 and 2 on our set of interest (for all positive time and on a positive on our set of interest (for all positive time and on a positive bounded set in space) we know bounded set in space) we know there exists a unique there exists a unique (local) solution(local) solution (ie on some finite time interval possibly (ie on some finite time interval possibly small)small)

QuestionQuestion Do the solutions remain positive and bounded (for Do the solutions remain positive and bounded (for biological significance)biological significance)

(3)

(1) (3)

Positivity and boundedness of Positivity and boundedness of solutionssolutions

Positivity and boundedness of Positivity and boundedness of solutionssolutions

bull We first use Theorem 3 to show positivity and boundedness of xWe first use Theorem 3 to show positivity and boundedness of x11 and and xx22

bull Next we use these results and the Fundamental Theorem of Calculus to Next we use these results and the Fundamental Theorem of Calculus to show positivity and boundedness for the remaining equations show positivity and boundedness for the remaining equations

bull This then ensures a global solutionThis then ensures a global solution

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical Analysis

bull Parameter Estimation in SIMULINKParameter Estimation in SIMULINK

bull Sensitivity AnalysisSensitivity Analysis

Parameter Estimation in SIMULINK (for Parameter Estimation in SIMULINK (for FULL model)FULL model)

Parameter Estimation in SIMULINK (for Parameter Estimation in SIMULINK (for FULL model)FULL model)

bull Total number of model parameters (for full model with Li and Rinzel CaTotal number of model parameters (for full model with Li and Rinzel Ca2+2+ module) = module) = 3434ndash From the literature = From the literature = 2020ndash Estimated = Estimated = 1414

bull Receptor moduleReceptor module Total = Total = 77ndash From literature = From literature = 66ndash Estimated = Estimated = 11 (k (kpp rate of receptor phosphorylation) rate of receptor phosphorylation)

bull G-protein cascadeG-protein cascade Total = Total = 99ndash From literature = From literature = 66ndash Estimated = Estimated = 3 3 (k(khydhyd k kreprep k kd3d3))

bull DAG kineticsDAG kinetics Total = Total = 5 5 (for each DAG species considered)(for each DAG species considered)ndash From literature = From literature = 00ndash Estimated = Estimated = 5 5 (all DAG parameters)(all DAG parameters)

bull CaCa2+2+ module module Total = Total = 1313ndash From literature = From literature = 88ndash Estimated = Estimated = 55

SIMULINK DetailsSIMULINK DetailsSIMULINK DetailsSIMULINK Details

bull Unknown rate parameters were estimated using SIMULINK

ndash Minimizes a user-specified cost function via a user-specified optimization method

ndash Nonlinear least squares optimization method of Levenberg-Marquardt was used to minimize a sum of squared errors cost function of the empirical observations and model predictions for IP3 Ca2+ and multiple species of DAG

bull Note Although the Gauss-Newton method is often more efficient the method of Levenberg-Marquardt has proved to be more robust

Measured vs Simulated 384 DAG and Measured vs Simulated 384 DAG and 341 DAG341 DAG

Measured vs Simulated 384 DAG and Measured vs Simulated 384 DAG and 341 DAG341 DAG

bull Timecourse of stimulation with 25 Timecourse of stimulation with 25 M UDP in RAW 2647 cellsM UDP in RAW 2647 cells

bull Solid black lines represent model simulations Solid black lines represent model simulations

bull ((aa) 384 DAG response (representative of the response of most poly ) 384 DAG response (representative of the response of most poly unsaturated fatty acid-containing DAG species) unsaturated fatty acid-containing DAG species)

bull ((bb) 341 DAG response (representative of the response of most mono- ) 341 DAG response (representative of the response of most mono- and di-unsaturated fatty acid-containing DAG species) and di-unsaturated fatty acid-containing DAG species)

bull Data points contain nine replicates performed on three different Data points contain nine replicates performed on three different experimental days with error bars = 1 SEM Units are total change in ng experimental days with error bars = 1 SEM Units are total change in ng over baseline levels in ~8x10over baseline levels in ~8x1066 cells cells

Measured vs Simulated IPMeasured vs Simulated IP33 and Ca and Ca2+2+Measured vs Simulated IPMeasured vs Simulated IP33 and Ca and Ca2+2+

bull Timecourse of stimulation with 25 Timecourse of stimulation with 25 M UDP in RAW 2647 cellsM UDP in RAW 2647 cells

bull Solid black lines represent model simulationsSolid black lines represent model simulations

bull ((cc) IP) IP33 response in pmols per ~35x10 response in pmols per ~35x1055 cells cells

bull Points in (c) represent the average of four experiments and error Points in (c) represent the average of four experiments and error bars are 1 SEM bars are 1 SEM

bull ((dd) Ca) Ca2+2+ response in response in M Red curve is a representative CaM Red curve is a representative Ca2+2+ trace trace taken from the UDP experiments within the AfCS single ligand taken from the UDP experiments within the AfCS single ligand screenscreen

Model Simulations P2YModel Simulations P2Y66 G GGTP GTP PIPPIP22

Model Simulations P2YModel Simulations P2Y66 G GGTP GTP PIPPIP22

bull ((aa) = total P2Y) = total P2Y66 activated (solid activated (solid line) and inactivated (dashed line) and inactivated (dashed line) surface receptorsline) surface receptors

bull ((bb) = total G) = total GGTP GTP

bull ((cc) = total PIP) = total PIP22 available for available for hydrolysishydrolysis

P2Y6 from 25M UDP GGTP from 25M UDP

PIP2 from 25M UDP

Sensitivity AnalysisSensitivity AnalysisSensitivity AnalysisSensitivity Analysis

bull Sensitivity analysis techniques are valuable tools Sensitivity analysis techniques are valuable tools designed to answer questions regarding which of the designed to answer questions regarding which of the uncertain input variables is more important in uncertain input variables is more important in determining the uncertainty in our output determining the uncertainty in our output

bull Likewise sensitivity analysis can provide insight into Likewise sensitivity analysis can provide insight into which parameter should be studied in more detail in which parameter should be studied in more detail in order to reduce the most variance in the model output order to reduce the most variance in the model output

bull The ability to answer these types of questions could The ability to answer these types of questions could lead to important insight into the design of new lead to important insight into the design of new experiments and in determining which experiments experiments and in determining which experiments would give us the most valuable informationwould give us the most valuable information

Sampling MethodSampling MethodSampling MethodSampling Method

bull Generate a random sample of our space of input Generate a random sample of our space of input variables over a ten percent variation from each variables over a ten percent variation from each parameters nominal value using the parameters nominal value using the Latin Hypercube Latin Hypercube Sampling (LHS) methodSampling (LHS) method

bull Uses Uses Standardized Regression Coefficients (SRCs) Standardized Regression Coefficients (SRCs) obtained by performing multiple linear regression obtained by performing multiple linear regression analysisanalysisndash offers a measure of sensitivity that is multi-dimensionally offers a measure of sensitivity that is multi-dimensionally

averaged over the entire space of parameter values averaged over the entire space of parameter values ndash SRCs give insight into degree of nonlinearity in the model SRCs give insight into degree of nonlinearity in the model

Computing RComputing Ryy22 values valuesComputing RComputing Ryy22 values values

bull SRCs are only reliable measures of sensitivity when SRCs are only reliable measures of sensitivity when degree of nonlinearity is ldquosmallrdquodegree of nonlinearity is ldquosmallrdquo

bull Use model coefficients of determination RUse model coefficients of determination Ryy22 given by given by

bull where ywhere yii is the estimate of y is the estimate of yii obtained from the regression obtained from the regression modelmodel

bull RRyy22 ge 07 ensures SRCs are good sensitivity measures ge 07 ensures SRCs are good sensitivity measures

SRCs (as sensitivity measures) for SRCs (as sensitivity measures) for DAGDAG

SRCs (as sensitivity measures) for SRCs (as sensitivity measures) for DAGDAG

SRCs (as sensitivity measures) for IPSRCs (as sensitivity measures) for IP33SRCs (as sensitivity measures) for IPSRCs (as sensitivity measures) for IP33

Effects of 3 most sensitive DAG Effects of 3 most sensitive DAG parametersparameters

Effects of 3 most sensitive DAG Effects of 3 most sensitive DAG parametersparameters

A B

C

AA k kdp2dp2 degradation of pool 2 DAG degradation of pool 2 DAG

BB k kdp1dp1 degradation of pool 1 DAG degradation of pool 1 DAG

CC k kap2ap2 production of pool 2 DAG production of pool 2 DAG

ConclusionsConclusionsConclusionsConclusions

bull We have developed a model of the UDP signaling We have developed a model of the UDP signaling pathway in RAW 2647 macrophages which can predict pathway in RAW 2647 macrophages which can predict the responses of multiple species of DAG as well as the the responses of multiple species of DAG as well as the responses of IPresponses of IP33 Ca Ca2+2+ receptor dynamics G-protein receptor dynamics G-protein activation and PIPactivation and PIP22 hydrolysis hydrolysis

bull Simplified model resultsSimplified model resultsndash We have obtained global existence uniqueness positivity We have obtained global existence uniqueness positivity

and boundedness of solutionsand boundedness of solutionsndash We have proven global stability of a unique steady state We have proven global stability of a unique steady state

within our region of interestwithin our region of interestbull Full model analysisFull model analysis

ndash Using SIMULINK we have estimated unknown rate Using SIMULINK we have estimated unknown rate parameters to obtain best fits to multiple DAG traces IPparameters to obtain best fits to multiple DAG traces IP33 and Caand Ca2+2+ all in response to 25 all in response to 25M UDPM UDP

ndash We have performed sensitivity analysis using the Latin We have performed sensitivity analysis using the Latin Hypercube sampling technique in combination with Hypercube sampling technique in combination with standardized regression coefficients to determine which standardized regression coefficients to determine which model parameters are responsible for most of the model model parameters are responsible for most of the model output uncertaintyoutput uncertainty

Future DirectionsFuture DirectionsFuture DirectionsFuture Directions

bull We have conducted multiple experiments to We have conducted multiple experiments to pharmacologically inhibit several different enzymes we pharmacologically inhibit several different enzymes we believe to play a role in this signaling pathway as believe to play a role in this signaling pathway as suggested by current known metabolic pathways and by suggested by current known metabolic pathways and by modeling resultsmodeling results

bull The next step is to perform gene knockdowns on The next step is to perform gene knockdowns on specific enzymes to verify results of inhibitor dataspecific enzymes to verify results of inhibitor data

bull The model output also suggests a time delay from The model output also suggests a time delay from receptor activation to PIPreceptor activation to PIP22 hydrolysis An upcoming task hydrolysis An upcoming task is to investigate the outcomes of adding such a delay is to investigate the outcomes of adding such a delay termterm

bull The model could be greatly enhanced by incorporating The model could be greatly enhanced by incorporating spatial dynamics so this will also be a major focus for spatial dynamics so this will also be a major focus for future researchfuture research

AcknowledgementsAcknowledgementsAcknowledgementsAcknowledgements

Mathematics DepartmentMathematics Departmentbull Hannah L Callender Hannah L Callender

PhDPhD

CollaboratorsCollaboratorsbull Alliance for Cellular SignalingAlliance for Cellular Signalingbull UT SouthwesternUT Southwestern

ndash Paul Sternweis PhDPaul Sternweis PhDndash Dianne DeCamp PhDDianne DeCamp PhD

Brown LabBrown Labbull H Alex Brown PhDH Alex Brown PhDbull Jeffrey S Forrester PhDJeffrey S Forrester PhDbull Mark Byrne PhDMark Byrne PhDbull Anita Preininger PhDAnita Preininger PhDbull Michelle ArmstrongMichelle Armstrongbull Andrew GoodmanAndrew Goodmanbull Pavlina Ivanova PhDPavlina Ivanova PhDbull Steve Milne PhDSteve Milne PhD

Model Equations G-protein Model Equations G-protein activationactivation

Model Equations G-protein Model Equations G-protein activationactivation

bull G-proteinG-protein activation activation

ndash Number of activated G-proteins (GNumber of activated G-proteins (GGTP) at time tGTP) at time t

Nondimensionalized equationsNondimensionalized equations

Model Equations PIPModel Equations PIP22 hydrolysis and hydrolysis and replenishmentreplenishment

Model Equations PIPModel Equations PIP22 hydrolysis and hydrolysis and replenishmentreplenishment

bull PIPPIP2 2 hydrolysis and replenishmenthydrolysis and replenishment

Nondimensionalized equationsNondimensionalized equations

ndash Number of PIPNumber of PIP22 molecules at time t molecules at time tndash Initial number of ldquofreerdquo PIPInitial number of ldquofreerdquo PIP22

PLC

PIP

2

DAG

IP3

GGTPCa2+

GGTP

bull IPIP33

Model Equations IPModel Equations IP33 production and production and degradationdegradation

Model Equations IPModel Equations IP33 production and production and degradationdegradation

Nondimensionalized equationsNondimensionalized equations

PLC

PIP

2

DA

G

IP3

G

GTPCa2+

G

GTP

bull Concentration of IPConcentration of IP33 at time t (in at time t (in M)M)

Model Equations DAGModel Equations DAGModel Equations DAGModel Equations DAG

bull DAG DAG (separate ODEs for each species considered)(separate ODEs for each species considered)

Nondimensionalized equationsNondimensionalized equations

bull Concentration of DAG at time tConcentration of DAG at time t

PLC

PIP

2

DA

G

IP3

G

GTPCa2+G

GTP

Model Equations Li and Rinzel (1994) Model Equations Li and Rinzel (1994) CaCa2+2+

Model Equations Li and Rinzel (1994) Model Equations Li and Rinzel (1994) CaCa2+2+

bull Free cytosolicFree cytosolic Ca Ca2+2+

bull hh = fraction of IP = fraction of IP33 channels not yet inactivated by Ca channels not yet inactivated by Ca2+2+

bull cc00 = concentration of total free Ca = concentration of total free Ca2+2+ per cytosolic volume per cytosolic volume

Timeseries data for the modelTimeseries data for the modelTimeseries data for the modelTimeseries data for the model

bull Data from collaborators at UTSW (Sternweis Lab)Data from collaborators at UTSW (Sternweis Lab)ndash Inositol 145-trisphosphate (Inositol 145-trisphosphate (IPIP33) production post 25) production post 25M M

UDP stimulation in the RAW cellUDP stimulation in the RAW cellndash Cytosolic calcium (Cytosolic calcium (CaCa2+2+) release post stimulation with ) release post stimulation with

2525M UDP in the RAW cellM UDP in the RAW cell

bull Diacylglycerol (Diacylglycerol (DAGDAG) data H Callender collected from a ) data H Callender collected from a novel method of quantitative analysis of multiple novel method of quantitative analysis of multiple species of DAG developed in the Brown Lab species of DAG developed in the Brown Lab

Reference Reference

Callender H L et al Quantification of Diacylglycerol Species from Callender H L et al Quantification of Diacylglycerol Species from Cellular Extracts by Electrospray Ionization Mass Spectrometry Using a Cellular Extracts by Electrospray Ionization Mass Spectrometry Using a Linear Regression Algorithm Linear Regression Algorithm Anal Chem Anal Chem 7979 (2007) 263-272 (2007) 263-272

IPIP33 and Ca and Ca2+2+ data from AfCS data from AfCSIPIP33 and Ca and Ca2+2+ data from AfCS data from AfCS

AA

bull IPIP33 response (in pmoles per response (in pmoles per 100 100 L lysate)in RAW 2647 L lysate)in RAW 2647 cells to 25 cells to 25 M UDPM UDP

bull Points represent the average Points represent the average of four experimentsof four experiments

bull Cytosolic calcium response Cytosolic calcium response (in (in M) in the RAW 2647 M) in the RAW 2647 cell to 25 cell to 25 M UDP M UDP

bull The graph displays 43 The graph displays 43 experimental repeatsexperimental repeats

B

What is DAGWhat is DAGWhat is DAGWhat is DAG

bull DAG is a cellular second messenger molecule which plays an DAG is a cellular second messenger molecule which plays an important role in initiating various changes in cell behavior important role in initiating various changes in cell behavior including cell activation differentiation proliferation and including cell activation differentiation proliferation and tumor promotion tumor promotion

bull There are many different species of DAG depending on the There are many different species of DAG depending on the number of carbons and number of double bonds in the fatty number of carbons and number of double bonds in the fatty acyl chains and different species can have different cellular acyl chains and different species can have different cellular functionsfunctions

bull ExampleExample 320 DAG320 DAG

Two fatty acyl Two fatty acyl (hydrocarbon) (hydrocarbon) chains in every DAG chains in every DAG speciesspecies

Each corner Each corner represents carbonrepresents carbon

320 DAG 18 carbons in first chain 14 in the second no double bonds 320 DAG 18 carbons in first chain 14 in the second no double bonds in either chainin either chain

Kinetics of monodi unsaturated DAGsKinetics of monodi unsaturated DAGsKinetics of monodi unsaturated DAGsKinetics of monodi unsaturated DAGs

bull Time based behavior of four monodi unsaturated DAG species after addition of 25 Time based behavior of four monodi unsaturated DAG species after addition of 25 M M (solid red squares) and 025 (solid red squares) and 025 M (solid green triangles) UDPM (solid green triangles) UDP

bull Time points contain a minimum of nine replicates performed on three different Time points contain a minimum of nine replicates performed on three different experimental daysexperimental days

- 25 M UDP

- 025M UDP

0 5 10 15 20 25 30

0 5 10 15 20 25 30 0 5 10 15 20 25 30

0 5 10 15 20 25 30

321 DAG 341 DAG

362 DAG341 DAGep

150

100

50-5

00

300

200

100

-100

0

150

100

50-5

00

-100

200

6040

20-2

00

-40

80

Time (min) Time (min)

Time (min) Time (min)

D

AG

(n

g)

D

AG

(n

g)

D

AG

(n

g)

D

AG

(n

g)

- 25 M UDP

- 025M UDP

- 25 M UDP

- 025M UDP

- 25 M UDP

- 025M UDP

Kinetics of PUFA containing DAGsKinetics of PUFA containing DAGsKinetics of PUFA containing DAGsKinetics of PUFA containing DAGsbull Time based behavior of four polyunsaturated fatty acid (PUFA) containing DAG species Time based behavior of four polyunsaturated fatty acid (PUFA) containing DAG species

after addition of 25 after addition of 25 M (solid red squares) and 025 M (solid red squares) and 025 M (solid green triangles) UDPM (solid green triangles) UDPbull Time points contain a minimum of nine replicates performed on three different Time points contain a minimum of nine replicates performed on three different

experimental daysexperimental daysbull Monodi unsaturated DAG species give a larger increase than polyunsaturated fatty acid Monodi unsaturated DAG species give a larger increase than polyunsaturated fatty acid

(PUFA) containing DAGs(PUFA) containing DAGs

0 5 10 15 20 25 30

0 5 10 15 20 25 30 0 5 10 15 20 25 30

0 5 10 15 20 25 30

385 DAG 383 DAG

364 DAG384 DAG6

42

-40

30

20

10

0

15

10

50

-56

40

2-2

8

Time (min) Time (min)

Time (min) Time (min)

D

AG

(n

g)

D

AG

(n

g)

D

AG

(n

g)

D

AG

(n

g)

-28

40

- 25 M UDP

- 025M UDP

- 25 M UDP

- 025M UDP

- 25 M UDP

- 025M UDP

- 25 M UDP

- 025M UDP

Differential DAG KineticsDifferential DAG KineticsDifferential DAG KineticsDifferential DAG Kinetics

bull Time based behavior of three DAG species with varying degrees of unsaturation after Time based behavior of three DAG species with varying degrees of unsaturation after addition of addition of 25 25 MM (solid red triangles) and (solid red triangles) and 025 025 MM (solid green squares) UDP (solid green squares) UDP

bull Time points contain nine replicates performed on three different experimental daysTime points contain nine replicates performed on three different experimental days

341 DAG

Best fit with current model structure

Overall Objectives of Modeling EffortOverall Objectives of Modeling EffortOverall Objectives of Modeling EffortOverall Objectives of Modeling Effort

bull Predict quantitative changes in lipid species after stimulation by Predict quantitative changes in lipid species after stimulation by various ligands and ligand concentrations in the RAW 2647 various ligands and ligand concentrations in the RAW 2647 macrophagemacrophage

bull Comparison and refinement of model output with AfCS IPComparison and refinement of model output with AfCS IP33 measurements and Cameasurements and Ca2+2+ traces as well as DAG data generated in traces as well as DAG data generated in the Brown labthe Brown lab

bull Predict in silico effects such as the effect of knock-downs etc on Predict in silico effects such as the effect of knock-downs etc on given pathwaygiven pathway

bull Suggest modifications to current pathway structuresSuggest modifications to current pathway structures

Modifications to the modelModifications to the modelModifications to the modelModifications to the model

bull Include an additional branch in the pathway Include an additional branch in the pathway for a second pool of DAGfor a second pool of DAG

bull Simplify CaSimplify Ca2+2+ equations for mathematical equations for mathematical analysis purposesanalysis purposes

Proposed PathwayProposed PathwayProposed PathwayProposed Pathway

bull Note We measure total cellular DAG levelsNote We measure total cellular DAG levelsbull Initial production of DAG from the hydrolysis of PIPInitial production of DAG from the hydrolysis of PIP22 in pool 1 (plasma in pool 1 (plasma

membrane) is offset by phosphorylation of DAG by DAG kinase in pool 2 membrane) is offset by phosphorylation of DAG by DAG kinase in pool 2 (Endoplasmic Reticulum Nucleus) to aid in the PI replacement pathway(Endoplasmic Reticulum Nucleus) to aid in the PI replacement pathway

bull Second wave of DAG is a result of resynthesis of PIPSecond wave of DAG is a result of resynthesis of PIP22 which is then which is then hydrolyzed to form DAG and IPhydrolyzed to form DAG and IP33

PI PIP PIP2 DAG

IP3

IP2IP

Ins

PI

CDP-DAGPADAG

PA

PCPE

PCPE

PITP

PI4K PI5K PLC DGK

DGK CDSPLC PIS

POOL 2

POOL 1

IP4

PI PIP PIP2 DAG

IP3

IP2IP

Ins

PI

CDP-DAGPADAG

PA

PCPE

PC

PITP

PI4K PI5K PLC

DGK CDSPLC PIS

IP4

LPP

LPP

Model Equations DAG (pool 1)Model Equations DAG (pool 1)Model Equations DAG (pool 1)Model Equations DAG (pool 1)

bull DAG (pool 1)DAG (pool 1) (separate ODEs for each species (separate ODEs for each species considered)considered)

bull Concentration of DAG from pool 1 at time tConcentration of DAG from pool 1 at time t

Nondimensionalized equationsNondimensionalized equations

Model Equations DAG (pool 2)Model Equations DAG (pool 2)Model Equations DAG (pool 2)Model Equations DAG (pool 2)

bull DAG (pool 2)DAG (pool 2) (separate ODEs for each species (separate ODEs for each species considered)considered)

bull Concentration of pool 2 DAG molecules at time tConcentration of pool 2 DAG molecules at time t

bull Baseline concentration of pool 2 DAGBaseline concentration of pool 2 DAG

Nondimensionalized equationsNondimensionalized equations

Simplified CaSimplified Ca2+2+ Equations EquationsSimplified CaSimplified Ca2+2+ Equations Equations

bull CaCa2+2+ modulemodule (to match experimental AfCS (to match experimental AfCS trace)trace)

0 100 200 300 400 500 6000

002

004

006

008

01

012

014

Concentration of Ca

2+ (uM)

Ca2+ response to 25uM UDP

time (s)

wherewhere

Theoretical AnalysisTheoretical AnalysisTheoretical AnalysisTheoretical Analysis

bull Existence and UniquenessExistence and Uniqueness

bull Positivity and Boundedness Positivity and Boundedness

(for biological relevance)(for biological relevance)

bull Analysis of steady state behaviorAnalysis of steady state behavior

Existence of Solutions (full model)Existence of Solutions (full model)Existence of Solutions (full model)Existence of Solutions (full model)

First we write our system of ODEs in the formFirst we write our system of ODEs in the form (1)

(2)

(1)

Uniqueness of SolutionsUniqueness of SolutionsUniqueness of SolutionsUniqueness of Solutions

Next we denote a solution of Next we denote a solution of (1)(1) by by with initial condition with initial condition

Since our system satisfies the hypotheses of Theorems 1 and 2 Since our system satisfies the hypotheses of Theorems 1 and 2 on our set of interest (for all positive time and on a positive on our set of interest (for all positive time and on a positive bounded set in space) we know bounded set in space) we know there exists a unique there exists a unique (local) solution(local) solution (ie on some finite time interval possibly (ie on some finite time interval possibly small)small)

QuestionQuestion Do the solutions remain positive and bounded (for Do the solutions remain positive and bounded (for biological significance)biological significance)

(3)

(1) (3)

Positivity and boundedness of Positivity and boundedness of solutionssolutions

Positivity and boundedness of Positivity and boundedness of solutionssolutions

bull We first use Theorem 3 to show positivity and boundedness of xWe first use Theorem 3 to show positivity and boundedness of x11 and and xx22

bull Next we use these results and the Fundamental Theorem of Calculus to Next we use these results and the Fundamental Theorem of Calculus to show positivity and boundedness for the remaining equations show positivity and boundedness for the remaining equations

bull This then ensures a global solutionThis then ensures a global solution

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical Analysis

bull Parameter Estimation in SIMULINKParameter Estimation in SIMULINK

bull Sensitivity AnalysisSensitivity Analysis

Parameter Estimation in SIMULINK (for Parameter Estimation in SIMULINK (for FULL model)FULL model)

Parameter Estimation in SIMULINK (for Parameter Estimation in SIMULINK (for FULL model)FULL model)

bull Total number of model parameters (for full model with Li and Rinzel CaTotal number of model parameters (for full model with Li and Rinzel Ca2+2+ module) = module) = 3434ndash From the literature = From the literature = 2020ndash Estimated = Estimated = 1414

bull Receptor moduleReceptor module Total = Total = 77ndash From literature = From literature = 66ndash Estimated = Estimated = 11 (k (kpp rate of receptor phosphorylation) rate of receptor phosphorylation)

bull G-protein cascadeG-protein cascade Total = Total = 99ndash From literature = From literature = 66ndash Estimated = Estimated = 3 3 (k(khydhyd k kreprep k kd3d3))

bull DAG kineticsDAG kinetics Total = Total = 5 5 (for each DAG species considered)(for each DAG species considered)ndash From literature = From literature = 00ndash Estimated = Estimated = 5 5 (all DAG parameters)(all DAG parameters)

bull CaCa2+2+ module module Total = Total = 1313ndash From literature = From literature = 88ndash Estimated = Estimated = 55

SIMULINK DetailsSIMULINK DetailsSIMULINK DetailsSIMULINK Details

bull Unknown rate parameters were estimated using SIMULINK

ndash Minimizes a user-specified cost function via a user-specified optimization method

ndash Nonlinear least squares optimization method of Levenberg-Marquardt was used to minimize a sum of squared errors cost function of the empirical observations and model predictions for IP3 Ca2+ and multiple species of DAG

bull Note Although the Gauss-Newton method is often more efficient the method of Levenberg-Marquardt has proved to be more robust

Measured vs Simulated 384 DAG and Measured vs Simulated 384 DAG and 341 DAG341 DAG

Measured vs Simulated 384 DAG and Measured vs Simulated 384 DAG and 341 DAG341 DAG

bull Timecourse of stimulation with 25 Timecourse of stimulation with 25 M UDP in RAW 2647 cellsM UDP in RAW 2647 cells

bull Solid black lines represent model simulations Solid black lines represent model simulations

bull ((aa) 384 DAG response (representative of the response of most poly ) 384 DAG response (representative of the response of most poly unsaturated fatty acid-containing DAG species) unsaturated fatty acid-containing DAG species)

bull ((bb) 341 DAG response (representative of the response of most mono- ) 341 DAG response (representative of the response of most mono- and di-unsaturated fatty acid-containing DAG species) and di-unsaturated fatty acid-containing DAG species)

bull Data points contain nine replicates performed on three different Data points contain nine replicates performed on three different experimental days with error bars = 1 SEM Units are total change in ng experimental days with error bars = 1 SEM Units are total change in ng over baseline levels in ~8x10over baseline levels in ~8x1066 cells cells

Measured vs Simulated IPMeasured vs Simulated IP33 and Ca and Ca2+2+Measured vs Simulated IPMeasured vs Simulated IP33 and Ca and Ca2+2+

bull Timecourse of stimulation with 25 Timecourse of stimulation with 25 M UDP in RAW 2647 cellsM UDP in RAW 2647 cells

bull Solid black lines represent model simulationsSolid black lines represent model simulations

bull ((cc) IP) IP33 response in pmols per ~35x10 response in pmols per ~35x1055 cells cells

bull Points in (c) represent the average of four experiments and error Points in (c) represent the average of four experiments and error bars are 1 SEM bars are 1 SEM

bull ((dd) Ca) Ca2+2+ response in response in M Red curve is a representative CaM Red curve is a representative Ca2+2+ trace trace taken from the UDP experiments within the AfCS single ligand taken from the UDP experiments within the AfCS single ligand screenscreen

Model Simulations P2YModel Simulations P2Y66 G GGTP GTP PIPPIP22

Model Simulations P2YModel Simulations P2Y66 G GGTP GTP PIPPIP22

bull ((aa) = total P2Y) = total P2Y66 activated (solid activated (solid line) and inactivated (dashed line) and inactivated (dashed line) surface receptorsline) surface receptors

bull ((bb) = total G) = total GGTP GTP

bull ((cc) = total PIP) = total PIP22 available for available for hydrolysishydrolysis

P2Y6 from 25M UDP GGTP from 25M UDP

PIP2 from 25M UDP

Sensitivity AnalysisSensitivity AnalysisSensitivity AnalysisSensitivity Analysis

bull Sensitivity analysis techniques are valuable tools Sensitivity analysis techniques are valuable tools designed to answer questions regarding which of the designed to answer questions regarding which of the uncertain input variables is more important in uncertain input variables is more important in determining the uncertainty in our output determining the uncertainty in our output

bull Likewise sensitivity analysis can provide insight into Likewise sensitivity analysis can provide insight into which parameter should be studied in more detail in which parameter should be studied in more detail in order to reduce the most variance in the model output order to reduce the most variance in the model output

bull The ability to answer these types of questions could The ability to answer these types of questions could lead to important insight into the design of new lead to important insight into the design of new experiments and in determining which experiments experiments and in determining which experiments would give us the most valuable informationwould give us the most valuable information

Sampling MethodSampling MethodSampling MethodSampling Method

bull Generate a random sample of our space of input Generate a random sample of our space of input variables over a ten percent variation from each variables over a ten percent variation from each parameters nominal value using the parameters nominal value using the Latin Hypercube Latin Hypercube Sampling (LHS) methodSampling (LHS) method

bull Uses Uses Standardized Regression Coefficients (SRCs) Standardized Regression Coefficients (SRCs) obtained by performing multiple linear regression obtained by performing multiple linear regression analysisanalysisndash offers a measure of sensitivity that is multi-dimensionally offers a measure of sensitivity that is multi-dimensionally

averaged over the entire space of parameter values averaged over the entire space of parameter values ndash SRCs give insight into degree of nonlinearity in the model SRCs give insight into degree of nonlinearity in the model

Computing RComputing Ryy22 values valuesComputing RComputing Ryy22 values values

bull SRCs are only reliable measures of sensitivity when SRCs are only reliable measures of sensitivity when degree of nonlinearity is ldquosmallrdquodegree of nonlinearity is ldquosmallrdquo

bull Use model coefficients of determination RUse model coefficients of determination Ryy22 given by given by

bull where ywhere yii is the estimate of y is the estimate of yii obtained from the regression obtained from the regression modelmodel

bull RRyy22 ge 07 ensures SRCs are good sensitivity measures ge 07 ensures SRCs are good sensitivity measures

SRCs (as sensitivity measures) for SRCs (as sensitivity measures) for DAGDAG

SRCs (as sensitivity measures) for SRCs (as sensitivity measures) for DAGDAG

SRCs (as sensitivity measures) for IPSRCs (as sensitivity measures) for IP33SRCs (as sensitivity measures) for IPSRCs (as sensitivity measures) for IP33

Effects of 3 most sensitive DAG Effects of 3 most sensitive DAG parametersparameters

Effects of 3 most sensitive DAG Effects of 3 most sensitive DAG parametersparameters

A B

C

AA k kdp2dp2 degradation of pool 2 DAG degradation of pool 2 DAG

BB k kdp1dp1 degradation of pool 1 DAG degradation of pool 1 DAG

CC k kap2ap2 production of pool 2 DAG production of pool 2 DAG

ConclusionsConclusionsConclusionsConclusions

bull We have developed a model of the UDP signaling We have developed a model of the UDP signaling pathway in RAW 2647 macrophages which can predict pathway in RAW 2647 macrophages which can predict the responses of multiple species of DAG as well as the the responses of multiple species of DAG as well as the responses of IPresponses of IP33 Ca Ca2+2+ receptor dynamics G-protein receptor dynamics G-protein activation and PIPactivation and PIP22 hydrolysis hydrolysis

bull Simplified model resultsSimplified model resultsndash We have obtained global existence uniqueness positivity We have obtained global existence uniqueness positivity

and boundedness of solutionsand boundedness of solutionsndash We have proven global stability of a unique steady state We have proven global stability of a unique steady state

within our region of interestwithin our region of interestbull Full model analysisFull model analysis

ndash Using SIMULINK we have estimated unknown rate Using SIMULINK we have estimated unknown rate parameters to obtain best fits to multiple DAG traces IPparameters to obtain best fits to multiple DAG traces IP33 and Caand Ca2+2+ all in response to 25 all in response to 25M UDPM UDP

ndash We have performed sensitivity analysis using the Latin We have performed sensitivity analysis using the Latin Hypercube sampling technique in combination with Hypercube sampling technique in combination with standardized regression coefficients to determine which standardized regression coefficients to determine which model parameters are responsible for most of the model model parameters are responsible for most of the model output uncertaintyoutput uncertainty

Future DirectionsFuture DirectionsFuture DirectionsFuture Directions

bull We have conducted multiple experiments to We have conducted multiple experiments to pharmacologically inhibit several different enzymes we pharmacologically inhibit several different enzymes we believe to play a role in this signaling pathway as believe to play a role in this signaling pathway as suggested by current known metabolic pathways and by suggested by current known metabolic pathways and by modeling resultsmodeling results

bull The next step is to perform gene knockdowns on The next step is to perform gene knockdowns on specific enzymes to verify results of inhibitor dataspecific enzymes to verify results of inhibitor data

bull The model output also suggests a time delay from The model output also suggests a time delay from receptor activation to PIPreceptor activation to PIP22 hydrolysis An upcoming task hydrolysis An upcoming task is to investigate the outcomes of adding such a delay is to investigate the outcomes of adding such a delay termterm

bull The model could be greatly enhanced by incorporating The model could be greatly enhanced by incorporating spatial dynamics so this will also be a major focus for spatial dynamics so this will also be a major focus for future researchfuture research

AcknowledgementsAcknowledgementsAcknowledgementsAcknowledgements

Mathematics DepartmentMathematics Departmentbull Hannah L Callender Hannah L Callender

PhDPhD

CollaboratorsCollaboratorsbull Alliance for Cellular SignalingAlliance for Cellular Signalingbull UT SouthwesternUT Southwestern

ndash Paul Sternweis PhDPaul Sternweis PhDndash Dianne DeCamp PhDDianne DeCamp PhD

Brown LabBrown Labbull H Alex Brown PhDH Alex Brown PhDbull Jeffrey S Forrester PhDJeffrey S Forrester PhDbull Mark Byrne PhDMark Byrne PhDbull Anita Preininger PhDAnita Preininger PhDbull Michelle ArmstrongMichelle Armstrongbull Andrew GoodmanAndrew Goodmanbull Pavlina Ivanova PhDPavlina Ivanova PhDbull Steve Milne PhDSteve Milne PhD

Model Equations PIPModel Equations PIP22 hydrolysis and hydrolysis and replenishmentreplenishment

Model Equations PIPModel Equations PIP22 hydrolysis and hydrolysis and replenishmentreplenishment

bull PIPPIP2 2 hydrolysis and replenishmenthydrolysis and replenishment

Nondimensionalized equationsNondimensionalized equations

ndash Number of PIPNumber of PIP22 molecules at time t molecules at time tndash Initial number of ldquofreerdquo PIPInitial number of ldquofreerdquo PIP22

PLC

PIP

2

DAG

IP3

GGTPCa2+

GGTP

bull IPIP33

Model Equations IPModel Equations IP33 production and production and degradationdegradation

Model Equations IPModel Equations IP33 production and production and degradationdegradation

Nondimensionalized equationsNondimensionalized equations

PLC

PIP

2

DA

G

IP3

G

GTPCa2+

G

GTP

bull Concentration of IPConcentration of IP33 at time t (in at time t (in M)M)

Model Equations DAGModel Equations DAGModel Equations DAGModel Equations DAG

bull DAG DAG (separate ODEs for each species considered)(separate ODEs for each species considered)

Nondimensionalized equationsNondimensionalized equations

bull Concentration of DAG at time tConcentration of DAG at time t

PLC

PIP

2

DA

G

IP3

G

GTPCa2+G

GTP

Model Equations Li and Rinzel (1994) Model Equations Li and Rinzel (1994) CaCa2+2+

Model Equations Li and Rinzel (1994) Model Equations Li and Rinzel (1994) CaCa2+2+

bull Free cytosolicFree cytosolic Ca Ca2+2+

bull hh = fraction of IP = fraction of IP33 channels not yet inactivated by Ca channels not yet inactivated by Ca2+2+

bull cc00 = concentration of total free Ca = concentration of total free Ca2+2+ per cytosolic volume per cytosolic volume

Timeseries data for the modelTimeseries data for the modelTimeseries data for the modelTimeseries data for the model

bull Data from collaborators at UTSW (Sternweis Lab)Data from collaborators at UTSW (Sternweis Lab)ndash Inositol 145-trisphosphate (Inositol 145-trisphosphate (IPIP33) production post 25) production post 25M M

UDP stimulation in the RAW cellUDP stimulation in the RAW cellndash Cytosolic calcium (Cytosolic calcium (CaCa2+2+) release post stimulation with ) release post stimulation with

2525M UDP in the RAW cellM UDP in the RAW cell

bull Diacylglycerol (Diacylglycerol (DAGDAG) data H Callender collected from a ) data H Callender collected from a novel method of quantitative analysis of multiple novel method of quantitative analysis of multiple species of DAG developed in the Brown Lab species of DAG developed in the Brown Lab

Reference Reference

Callender H L et al Quantification of Diacylglycerol Species from Callender H L et al Quantification of Diacylglycerol Species from Cellular Extracts by Electrospray Ionization Mass Spectrometry Using a Cellular Extracts by Electrospray Ionization Mass Spectrometry Using a Linear Regression Algorithm Linear Regression Algorithm Anal Chem Anal Chem 7979 (2007) 263-272 (2007) 263-272

IPIP33 and Ca and Ca2+2+ data from AfCS data from AfCSIPIP33 and Ca and Ca2+2+ data from AfCS data from AfCS

AA

bull IPIP33 response (in pmoles per response (in pmoles per 100 100 L lysate)in RAW 2647 L lysate)in RAW 2647 cells to 25 cells to 25 M UDPM UDP

bull Points represent the average Points represent the average of four experimentsof four experiments

bull Cytosolic calcium response Cytosolic calcium response (in (in M) in the RAW 2647 M) in the RAW 2647 cell to 25 cell to 25 M UDP M UDP

bull The graph displays 43 The graph displays 43 experimental repeatsexperimental repeats

B

What is DAGWhat is DAGWhat is DAGWhat is DAG

bull DAG is a cellular second messenger molecule which plays an DAG is a cellular second messenger molecule which plays an important role in initiating various changes in cell behavior important role in initiating various changes in cell behavior including cell activation differentiation proliferation and including cell activation differentiation proliferation and tumor promotion tumor promotion

bull There are many different species of DAG depending on the There are many different species of DAG depending on the number of carbons and number of double bonds in the fatty number of carbons and number of double bonds in the fatty acyl chains and different species can have different cellular acyl chains and different species can have different cellular functionsfunctions

bull ExampleExample 320 DAG320 DAG

Two fatty acyl Two fatty acyl (hydrocarbon) (hydrocarbon) chains in every DAG chains in every DAG speciesspecies

Each corner Each corner represents carbonrepresents carbon

320 DAG 18 carbons in first chain 14 in the second no double bonds 320 DAG 18 carbons in first chain 14 in the second no double bonds in either chainin either chain

Kinetics of monodi unsaturated DAGsKinetics of monodi unsaturated DAGsKinetics of monodi unsaturated DAGsKinetics of monodi unsaturated DAGs

bull Time based behavior of four monodi unsaturated DAG species after addition of 25 Time based behavior of four monodi unsaturated DAG species after addition of 25 M M (solid red squares) and 025 (solid red squares) and 025 M (solid green triangles) UDPM (solid green triangles) UDP

bull Time points contain a minimum of nine replicates performed on three different Time points contain a minimum of nine replicates performed on three different experimental daysexperimental days

- 25 M UDP

- 025M UDP

0 5 10 15 20 25 30

0 5 10 15 20 25 30 0 5 10 15 20 25 30

0 5 10 15 20 25 30

321 DAG 341 DAG

362 DAG341 DAGep

150

100

50-5

00

300

200

100

-100

0

150

100

50-5

00

-100

200

6040

20-2

00

-40

80

Time (min) Time (min)

Time (min) Time (min)

D

AG

(n

g)

D

AG

(n

g)

D

AG

(n

g)

D

AG

(n

g)

- 25 M UDP

- 025M UDP

- 25 M UDP

- 025M UDP

- 25 M UDP

- 025M UDP

Kinetics of PUFA containing DAGsKinetics of PUFA containing DAGsKinetics of PUFA containing DAGsKinetics of PUFA containing DAGsbull Time based behavior of four polyunsaturated fatty acid (PUFA) containing DAG species Time based behavior of four polyunsaturated fatty acid (PUFA) containing DAG species

after addition of 25 after addition of 25 M (solid red squares) and 025 M (solid red squares) and 025 M (solid green triangles) UDPM (solid green triangles) UDPbull Time points contain a minimum of nine replicates performed on three different Time points contain a minimum of nine replicates performed on three different

experimental daysexperimental daysbull Monodi unsaturated DAG species give a larger increase than polyunsaturated fatty acid Monodi unsaturated DAG species give a larger increase than polyunsaturated fatty acid

(PUFA) containing DAGs(PUFA) containing DAGs

0 5 10 15 20 25 30

0 5 10 15 20 25 30 0 5 10 15 20 25 30

0 5 10 15 20 25 30

385 DAG 383 DAG

364 DAG384 DAG6

42

-40

30

20

10

0

15

10

50

-56

40

2-2

8

Time (min) Time (min)

Time (min) Time (min)

D

AG

(n

g)

D

AG

(n

g)

D

AG

(n

g)

D

AG

(n

g)

-28

40

- 25 M UDP

- 025M UDP

- 25 M UDP

- 025M UDP

- 25 M UDP

- 025M UDP

- 25 M UDP

- 025M UDP

Differential DAG KineticsDifferential DAG KineticsDifferential DAG KineticsDifferential DAG Kinetics

bull Time based behavior of three DAG species with varying degrees of unsaturation after Time based behavior of three DAG species with varying degrees of unsaturation after addition of addition of 25 25 MM (solid red triangles) and (solid red triangles) and 025 025 MM (solid green squares) UDP (solid green squares) UDP

bull Time points contain nine replicates performed on three different experimental daysTime points contain nine replicates performed on three different experimental days

341 DAG

Best fit with current model structure

Overall Objectives of Modeling EffortOverall Objectives of Modeling EffortOverall Objectives of Modeling EffortOverall Objectives of Modeling Effort

bull Predict quantitative changes in lipid species after stimulation by Predict quantitative changes in lipid species after stimulation by various ligands and ligand concentrations in the RAW 2647 various ligands and ligand concentrations in the RAW 2647 macrophagemacrophage

bull Comparison and refinement of model output with AfCS IPComparison and refinement of model output with AfCS IP33 measurements and Cameasurements and Ca2+2+ traces as well as DAG data generated in traces as well as DAG data generated in the Brown labthe Brown lab

bull Predict in silico effects such as the effect of knock-downs etc on Predict in silico effects such as the effect of knock-downs etc on given pathwaygiven pathway

bull Suggest modifications to current pathway structuresSuggest modifications to current pathway structures

Modifications to the modelModifications to the modelModifications to the modelModifications to the model

bull Include an additional branch in the pathway Include an additional branch in the pathway for a second pool of DAGfor a second pool of DAG

bull Simplify CaSimplify Ca2+2+ equations for mathematical equations for mathematical analysis purposesanalysis purposes

Proposed PathwayProposed PathwayProposed PathwayProposed Pathway

bull Note We measure total cellular DAG levelsNote We measure total cellular DAG levelsbull Initial production of DAG from the hydrolysis of PIPInitial production of DAG from the hydrolysis of PIP22 in pool 1 (plasma in pool 1 (plasma

membrane) is offset by phosphorylation of DAG by DAG kinase in pool 2 membrane) is offset by phosphorylation of DAG by DAG kinase in pool 2 (Endoplasmic Reticulum Nucleus) to aid in the PI replacement pathway(Endoplasmic Reticulum Nucleus) to aid in the PI replacement pathway

bull Second wave of DAG is a result of resynthesis of PIPSecond wave of DAG is a result of resynthesis of PIP22 which is then which is then hydrolyzed to form DAG and IPhydrolyzed to form DAG and IP33

PI PIP PIP2 DAG

IP3

IP2IP

Ins

PI

CDP-DAGPADAG

PA

PCPE

PCPE

PITP

PI4K PI5K PLC DGK

DGK CDSPLC PIS

POOL 2

POOL 1

IP4

PI PIP PIP2 DAG

IP3

IP2IP

Ins

PI

CDP-DAGPADAG

PA

PCPE

PC

PITP

PI4K PI5K PLC

DGK CDSPLC PIS

IP4

LPP

LPP

Model Equations DAG (pool 1)Model Equations DAG (pool 1)Model Equations DAG (pool 1)Model Equations DAG (pool 1)

bull DAG (pool 1)DAG (pool 1) (separate ODEs for each species (separate ODEs for each species considered)considered)

bull Concentration of DAG from pool 1 at time tConcentration of DAG from pool 1 at time t

Nondimensionalized equationsNondimensionalized equations

Model Equations DAG (pool 2)Model Equations DAG (pool 2)Model Equations DAG (pool 2)Model Equations DAG (pool 2)

bull DAG (pool 2)DAG (pool 2) (separate ODEs for each species (separate ODEs for each species considered)considered)

bull Concentration of pool 2 DAG molecules at time tConcentration of pool 2 DAG molecules at time t

bull Baseline concentration of pool 2 DAGBaseline concentration of pool 2 DAG

Nondimensionalized equationsNondimensionalized equations

Simplified CaSimplified Ca2+2+ Equations EquationsSimplified CaSimplified Ca2+2+ Equations Equations

bull CaCa2+2+ modulemodule (to match experimental AfCS (to match experimental AfCS trace)trace)

0 100 200 300 400 500 6000

002

004

006

008

01

012

014

Concentration of Ca

2+ (uM)

Ca2+ response to 25uM UDP

time (s)

wherewhere

Theoretical AnalysisTheoretical AnalysisTheoretical AnalysisTheoretical Analysis

bull Existence and UniquenessExistence and Uniqueness

bull Positivity and Boundedness Positivity and Boundedness

(for biological relevance)(for biological relevance)

bull Analysis of steady state behaviorAnalysis of steady state behavior

Existence of Solutions (full model)Existence of Solutions (full model)Existence of Solutions (full model)Existence of Solutions (full model)

First we write our system of ODEs in the formFirst we write our system of ODEs in the form (1)

(2)

(1)

Uniqueness of SolutionsUniqueness of SolutionsUniqueness of SolutionsUniqueness of Solutions

Next we denote a solution of Next we denote a solution of (1)(1) by by with initial condition with initial condition

Since our system satisfies the hypotheses of Theorems 1 and 2 Since our system satisfies the hypotheses of Theorems 1 and 2 on our set of interest (for all positive time and on a positive on our set of interest (for all positive time and on a positive bounded set in space) we know bounded set in space) we know there exists a unique there exists a unique (local) solution(local) solution (ie on some finite time interval possibly (ie on some finite time interval possibly small)small)

QuestionQuestion Do the solutions remain positive and bounded (for Do the solutions remain positive and bounded (for biological significance)biological significance)

(3)

(1) (3)

Positivity and boundedness of Positivity and boundedness of solutionssolutions

Positivity and boundedness of Positivity and boundedness of solutionssolutions

bull We first use Theorem 3 to show positivity and boundedness of xWe first use Theorem 3 to show positivity and boundedness of x11 and and xx22

bull Next we use these results and the Fundamental Theorem of Calculus to Next we use these results and the Fundamental Theorem of Calculus to show positivity and boundedness for the remaining equations show positivity and boundedness for the remaining equations

bull This then ensures a global solutionThis then ensures a global solution

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical Analysis

bull Parameter Estimation in SIMULINKParameter Estimation in SIMULINK

bull Sensitivity AnalysisSensitivity Analysis

Parameter Estimation in SIMULINK (for Parameter Estimation in SIMULINK (for FULL model)FULL model)

Parameter Estimation in SIMULINK (for Parameter Estimation in SIMULINK (for FULL model)FULL model)

bull Total number of model parameters (for full model with Li and Rinzel CaTotal number of model parameters (for full model with Li and Rinzel Ca2+2+ module) = module) = 3434ndash From the literature = From the literature = 2020ndash Estimated = Estimated = 1414

bull Receptor moduleReceptor module Total = Total = 77ndash From literature = From literature = 66ndash Estimated = Estimated = 11 (k (kpp rate of receptor phosphorylation) rate of receptor phosphorylation)

bull G-protein cascadeG-protein cascade Total = Total = 99ndash From literature = From literature = 66ndash Estimated = Estimated = 3 3 (k(khydhyd k kreprep k kd3d3))

bull DAG kineticsDAG kinetics Total = Total = 5 5 (for each DAG species considered)(for each DAG species considered)ndash From literature = From literature = 00ndash Estimated = Estimated = 5 5 (all DAG parameters)(all DAG parameters)

bull CaCa2+2+ module module Total = Total = 1313ndash From literature = From literature = 88ndash Estimated = Estimated = 55

SIMULINK DetailsSIMULINK DetailsSIMULINK DetailsSIMULINK Details

bull Unknown rate parameters were estimated using SIMULINK

ndash Minimizes a user-specified cost function via a user-specified optimization method

ndash Nonlinear least squares optimization method of Levenberg-Marquardt was used to minimize a sum of squared errors cost function of the empirical observations and model predictions for IP3 Ca2+ and multiple species of DAG

bull Note Although the Gauss-Newton method is often more efficient the method of Levenberg-Marquardt has proved to be more robust

Measured vs Simulated 384 DAG and Measured vs Simulated 384 DAG and 341 DAG341 DAG

Measured vs Simulated 384 DAG and Measured vs Simulated 384 DAG and 341 DAG341 DAG

bull Timecourse of stimulation with 25 Timecourse of stimulation with 25 M UDP in RAW 2647 cellsM UDP in RAW 2647 cells

bull Solid black lines represent model simulations Solid black lines represent model simulations

bull ((aa) 384 DAG response (representative of the response of most poly ) 384 DAG response (representative of the response of most poly unsaturated fatty acid-containing DAG species) unsaturated fatty acid-containing DAG species)

bull ((bb) 341 DAG response (representative of the response of most mono- ) 341 DAG response (representative of the response of most mono- and di-unsaturated fatty acid-containing DAG species) and di-unsaturated fatty acid-containing DAG species)

bull Data points contain nine replicates performed on three different Data points contain nine replicates performed on three different experimental days with error bars = 1 SEM Units are total change in ng experimental days with error bars = 1 SEM Units are total change in ng over baseline levels in ~8x10over baseline levels in ~8x1066 cells cells

Measured vs Simulated IPMeasured vs Simulated IP33 and Ca and Ca2+2+Measured vs Simulated IPMeasured vs Simulated IP33 and Ca and Ca2+2+

bull Timecourse of stimulation with 25 Timecourse of stimulation with 25 M UDP in RAW 2647 cellsM UDP in RAW 2647 cells

bull Solid black lines represent model simulationsSolid black lines represent model simulations

bull ((cc) IP) IP33 response in pmols per ~35x10 response in pmols per ~35x1055 cells cells

bull Points in (c) represent the average of four experiments and error Points in (c) represent the average of four experiments and error bars are 1 SEM bars are 1 SEM

bull ((dd) Ca) Ca2+2+ response in response in M Red curve is a representative CaM Red curve is a representative Ca2+2+ trace trace taken from the UDP experiments within the AfCS single ligand taken from the UDP experiments within the AfCS single ligand screenscreen

Model Simulations P2YModel Simulations P2Y66 G GGTP GTP PIPPIP22

Model Simulations P2YModel Simulations P2Y66 G GGTP GTP PIPPIP22

bull ((aa) = total P2Y) = total P2Y66 activated (solid activated (solid line) and inactivated (dashed line) and inactivated (dashed line) surface receptorsline) surface receptors

bull ((bb) = total G) = total GGTP GTP

bull ((cc) = total PIP) = total PIP22 available for available for hydrolysishydrolysis

P2Y6 from 25M UDP GGTP from 25M UDP

PIP2 from 25M UDP

Sensitivity AnalysisSensitivity AnalysisSensitivity AnalysisSensitivity Analysis

bull Sensitivity analysis techniques are valuable tools Sensitivity analysis techniques are valuable tools designed to answer questions regarding which of the designed to answer questions regarding which of the uncertain input variables is more important in uncertain input variables is more important in determining the uncertainty in our output determining the uncertainty in our output

bull Likewise sensitivity analysis can provide insight into Likewise sensitivity analysis can provide insight into which parameter should be studied in more detail in which parameter should be studied in more detail in order to reduce the most variance in the model output order to reduce the most variance in the model output

bull The ability to answer these types of questions could The ability to answer these types of questions could lead to important insight into the design of new lead to important insight into the design of new experiments and in determining which experiments experiments and in determining which experiments would give us the most valuable informationwould give us the most valuable information

Sampling MethodSampling MethodSampling MethodSampling Method

bull Generate a random sample of our space of input Generate a random sample of our space of input variables over a ten percent variation from each variables over a ten percent variation from each parameters nominal value using the parameters nominal value using the Latin Hypercube Latin Hypercube Sampling (LHS) methodSampling (LHS) method

bull Uses Uses Standardized Regression Coefficients (SRCs) Standardized Regression Coefficients (SRCs) obtained by performing multiple linear regression obtained by performing multiple linear regression analysisanalysisndash offers a measure of sensitivity that is multi-dimensionally offers a measure of sensitivity that is multi-dimensionally

averaged over the entire space of parameter values averaged over the entire space of parameter values ndash SRCs give insight into degree of nonlinearity in the model SRCs give insight into degree of nonlinearity in the model

Computing RComputing Ryy22 values valuesComputing RComputing Ryy22 values values

bull SRCs are only reliable measures of sensitivity when SRCs are only reliable measures of sensitivity when degree of nonlinearity is ldquosmallrdquodegree of nonlinearity is ldquosmallrdquo

bull Use model coefficients of determination RUse model coefficients of determination Ryy22 given by given by

bull where ywhere yii is the estimate of y is the estimate of yii obtained from the regression obtained from the regression modelmodel

bull RRyy22 ge 07 ensures SRCs are good sensitivity measures ge 07 ensures SRCs are good sensitivity measures

SRCs (as sensitivity measures) for SRCs (as sensitivity measures) for DAGDAG

SRCs (as sensitivity measures) for SRCs (as sensitivity measures) for DAGDAG

SRCs (as sensitivity measures) for IPSRCs (as sensitivity measures) for IP33SRCs (as sensitivity measures) for IPSRCs (as sensitivity measures) for IP33

Effects of 3 most sensitive DAG Effects of 3 most sensitive DAG parametersparameters

Effects of 3 most sensitive DAG Effects of 3 most sensitive DAG parametersparameters

A B

C

AA k kdp2dp2 degradation of pool 2 DAG degradation of pool 2 DAG

BB k kdp1dp1 degradation of pool 1 DAG degradation of pool 1 DAG

CC k kap2ap2 production of pool 2 DAG production of pool 2 DAG

ConclusionsConclusionsConclusionsConclusions

bull We have developed a model of the UDP signaling We have developed a model of the UDP signaling pathway in RAW 2647 macrophages which can predict pathway in RAW 2647 macrophages which can predict the responses of multiple species of DAG as well as the the responses of multiple species of DAG as well as the responses of IPresponses of IP33 Ca Ca2+2+ receptor dynamics G-protein receptor dynamics G-protein activation and PIPactivation and PIP22 hydrolysis hydrolysis

bull Simplified model resultsSimplified model resultsndash We have obtained global existence uniqueness positivity We have obtained global existence uniqueness positivity

and boundedness of solutionsand boundedness of solutionsndash We have proven global stability of a unique steady state We have proven global stability of a unique steady state

within our region of interestwithin our region of interestbull Full model analysisFull model analysis

ndash Using SIMULINK we have estimated unknown rate Using SIMULINK we have estimated unknown rate parameters to obtain best fits to multiple DAG traces IPparameters to obtain best fits to multiple DAG traces IP33 and Caand Ca2+2+ all in response to 25 all in response to 25M UDPM UDP

ndash We have performed sensitivity analysis using the Latin We have performed sensitivity analysis using the Latin Hypercube sampling technique in combination with Hypercube sampling technique in combination with standardized regression coefficients to determine which standardized regression coefficients to determine which model parameters are responsible for most of the model model parameters are responsible for most of the model output uncertaintyoutput uncertainty

Future DirectionsFuture DirectionsFuture DirectionsFuture Directions

bull We have conducted multiple experiments to We have conducted multiple experiments to pharmacologically inhibit several different enzymes we pharmacologically inhibit several different enzymes we believe to play a role in this signaling pathway as believe to play a role in this signaling pathway as suggested by current known metabolic pathways and by suggested by current known metabolic pathways and by modeling resultsmodeling results

bull The next step is to perform gene knockdowns on The next step is to perform gene knockdowns on specific enzymes to verify results of inhibitor dataspecific enzymes to verify results of inhibitor data

bull The model output also suggests a time delay from The model output also suggests a time delay from receptor activation to PIPreceptor activation to PIP22 hydrolysis An upcoming task hydrolysis An upcoming task is to investigate the outcomes of adding such a delay is to investigate the outcomes of adding such a delay termterm

bull The model could be greatly enhanced by incorporating The model could be greatly enhanced by incorporating spatial dynamics so this will also be a major focus for spatial dynamics so this will also be a major focus for future researchfuture research

AcknowledgementsAcknowledgementsAcknowledgementsAcknowledgements

Mathematics DepartmentMathematics Departmentbull Hannah L Callender Hannah L Callender

PhDPhD

CollaboratorsCollaboratorsbull Alliance for Cellular SignalingAlliance for Cellular Signalingbull UT SouthwesternUT Southwestern

ndash Paul Sternweis PhDPaul Sternweis PhDndash Dianne DeCamp PhDDianne DeCamp PhD

Brown LabBrown Labbull H Alex Brown PhDH Alex Brown PhDbull Jeffrey S Forrester PhDJeffrey S Forrester PhDbull Mark Byrne PhDMark Byrne PhDbull Anita Preininger PhDAnita Preininger PhDbull Michelle ArmstrongMichelle Armstrongbull Andrew GoodmanAndrew Goodmanbull Pavlina Ivanova PhDPavlina Ivanova PhDbull Steve Milne PhDSteve Milne PhD

bull IPIP33

Model Equations IPModel Equations IP33 production and production and degradationdegradation

Model Equations IPModel Equations IP33 production and production and degradationdegradation

Nondimensionalized equationsNondimensionalized equations

PLC

PIP

2

DA

G

IP3

G

GTPCa2+

G

GTP

bull Concentration of IPConcentration of IP33 at time t (in at time t (in M)M)

Model Equations DAGModel Equations DAGModel Equations DAGModel Equations DAG

bull DAG DAG (separate ODEs for each species considered)(separate ODEs for each species considered)

Nondimensionalized equationsNondimensionalized equations

bull Concentration of DAG at time tConcentration of DAG at time t

PLC

PIP

2

DA

G

IP3

G

GTPCa2+G

GTP

Model Equations Li and Rinzel (1994) Model Equations Li and Rinzel (1994) CaCa2+2+

Model Equations Li and Rinzel (1994) Model Equations Li and Rinzel (1994) CaCa2+2+

bull Free cytosolicFree cytosolic Ca Ca2+2+

bull hh = fraction of IP = fraction of IP33 channels not yet inactivated by Ca channels not yet inactivated by Ca2+2+

bull cc00 = concentration of total free Ca = concentration of total free Ca2+2+ per cytosolic volume per cytosolic volume

Timeseries data for the modelTimeseries data for the modelTimeseries data for the modelTimeseries data for the model

bull Data from collaborators at UTSW (Sternweis Lab)Data from collaborators at UTSW (Sternweis Lab)ndash Inositol 145-trisphosphate (Inositol 145-trisphosphate (IPIP33) production post 25) production post 25M M

UDP stimulation in the RAW cellUDP stimulation in the RAW cellndash Cytosolic calcium (Cytosolic calcium (CaCa2+2+) release post stimulation with ) release post stimulation with

2525M UDP in the RAW cellM UDP in the RAW cell

bull Diacylglycerol (Diacylglycerol (DAGDAG) data H Callender collected from a ) data H Callender collected from a novel method of quantitative analysis of multiple novel method of quantitative analysis of multiple species of DAG developed in the Brown Lab species of DAG developed in the Brown Lab

Reference Reference

Callender H L et al Quantification of Diacylglycerol Species from Callender H L et al Quantification of Diacylglycerol Species from Cellular Extracts by Electrospray Ionization Mass Spectrometry Using a Cellular Extracts by Electrospray Ionization Mass Spectrometry Using a Linear Regression Algorithm Linear Regression Algorithm Anal Chem Anal Chem 7979 (2007) 263-272 (2007) 263-272

IPIP33 and Ca and Ca2+2+ data from AfCS data from AfCSIPIP33 and Ca and Ca2+2+ data from AfCS data from AfCS

AA

bull IPIP33 response (in pmoles per response (in pmoles per 100 100 L lysate)in RAW 2647 L lysate)in RAW 2647 cells to 25 cells to 25 M UDPM UDP

bull Points represent the average Points represent the average of four experimentsof four experiments

bull Cytosolic calcium response Cytosolic calcium response (in (in M) in the RAW 2647 M) in the RAW 2647 cell to 25 cell to 25 M UDP M UDP

bull The graph displays 43 The graph displays 43 experimental repeatsexperimental repeats

B

What is DAGWhat is DAGWhat is DAGWhat is DAG

bull DAG is a cellular second messenger molecule which plays an DAG is a cellular second messenger molecule which plays an important role in initiating various changes in cell behavior important role in initiating various changes in cell behavior including cell activation differentiation proliferation and including cell activation differentiation proliferation and tumor promotion tumor promotion

bull There are many different species of DAG depending on the There are many different species of DAG depending on the number of carbons and number of double bonds in the fatty number of carbons and number of double bonds in the fatty acyl chains and different species can have different cellular acyl chains and different species can have different cellular functionsfunctions

bull ExampleExample 320 DAG320 DAG

Two fatty acyl Two fatty acyl (hydrocarbon) (hydrocarbon) chains in every DAG chains in every DAG speciesspecies

Each corner Each corner represents carbonrepresents carbon

320 DAG 18 carbons in first chain 14 in the second no double bonds 320 DAG 18 carbons in first chain 14 in the second no double bonds in either chainin either chain

Kinetics of monodi unsaturated DAGsKinetics of monodi unsaturated DAGsKinetics of monodi unsaturated DAGsKinetics of monodi unsaturated DAGs

bull Time based behavior of four monodi unsaturated DAG species after addition of 25 Time based behavior of four monodi unsaturated DAG species after addition of 25 M M (solid red squares) and 025 (solid red squares) and 025 M (solid green triangles) UDPM (solid green triangles) UDP

bull Time points contain a minimum of nine replicates performed on three different Time points contain a minimum of nine replicates performed on three different experimental daysexperimental days

- 25 M UDP

- 025M UDP

0 5 10 15 20 25 30

0 5 10 15 20 25 30 0 5 10 15 20 25 30

0 5 10 15 20 25 30

321 DAG 341 DAG

362 DAG341 DAGep

150

100

50-5

00

300

200

100

-100

0

150

100

50-5

00

-100

200

6040

20-2

00

-40

80

Time (min) Time (min)

Time (min) Time (min)

D

AG

(n

g)

D

AG

(n

g)

D

AG

(n

g)

D

AG

(n

g)

- 25 M UDP

- 025M UDP

- 25 M UDP

- 025M UDP

- 25 M UDP

- 025M UDP

Kinetics of PUFA containing DAGsKinetics of PUFA containing DAGsKinetics of PUFA containing DAGsKinetics of PUFA containing DAGsbull Time based behavior of four polyunsaturated fatty acid (PUFA) containing DAG species Time based behavior of four polyunsaturated fatty acid (PUFA) containing DAG species

after addition of 25 after addition of 25 M (solid red squares) and 025 M (solid red squares) and 025 M (solid green triangles) UDPM (solid green triangles) UDPbull Time points contain a minimum of nine replicates performed on three different Time points contain a minimum of nine replicates performed on three different

experimental daysexperimental daysbull Monodi unsaturated DAG species give a larger increase than polyunsaturated fatty acid Monodi unsaturated DAG species give a larger increase than polyunsaturated fatty acid

(PUFA) containing DAGs(PUFA) containing DAGs

0 5 10 15 20 25 30

0 5 10 15 20 25 30 0 5 10 15 20 25 30

0 5 10 15 20 25 30

385 DAG 383 DAG

364 DAG384 DAG6

42

-40

30

20

10

0

15

10

50

-56

40

2-2

8

Time (min) Time (min)

Time (min) Time (min)

D

AG

(n

g)

D

AG

(n

g)

D

AG

(n

g)

D

AG

(n

g)

-28

40

- 25 M UDP

- 025M UDP

- 25 M UDP

- 025M UDP

- 25 M UDP

- 025M UDP

- 25 M UDP

- 025M UDP

Differential DAG KineticsDifferential DAG KineticsDifferential DAG KineticsDifferential DAG Kinetics

bull Time based behavior of three DAG species with varying degrees of unsaturation after Time based behavior of three DAG species with varying degrees of unsaturation after addition of addition of 25 25 MM (solid red triangles) and (solid red triangles) and 025 025 MM (solid green squares) UDP (solid green squares) UDP

bull Time points contain nine replicates performed on three different experimental daysTime points contain nine replicates performed on three different experimental days

341 DAG

Best fit with current model structure

Overall Objectives of Modeling EffortOverall Objectives of Modeling EffortOverall Objectives of Modeling EffortOverall Objectives of Modeling Effort

bull Predict quantitative changes in lipid species after stimulation by Predict quantitative changes in lipid species after stimulation by various ligands and ligand concentrations in the RAW 2647 various ligands and ligand concentrations in the RAW 2647 macrophagemacrophage

bull Comparison and refinement of model output with AfCS IPComparison and refinement of model output with AfCS IP33 measurements and Cameasurements and Ca2+2+ traces as well as DAG data generated in traces as well as DAG data generated in the Brown labthe Brown lab

bull Predict in silico effects such as the effect of knock-downs etc on Predict in silico effects such as the effect of knock-downs etc on given pathwaygiven pathway

bull Suggest modifications to current pathway structuresSuggest modifications to current pathway structures

Modifications to the modelModifications to the modelModifications to the modelModifications to the model

bull Include an additional branch in the pathway Include an additional branch in the pathway for a second pool of DAGfor a second pool of DAG

bull Simplify CaSimplify Ca2+2+ equations for mathematical equations for mathematical analysis purposesanalysis purposes

Proposed PathwayProposed PathwayProposed PathwayProposed Pathway

bull Note We measure total cellular DAG levelsNote We measure total cellular DAG levelsbull Initial production of DAG from the hydrolysis of PIPInitial production of DAG from the hydrolysis of PIP22 in pool 1 (plasma in pool 1 (plasma

membrane) is offset by phosphorylation of DAG by DAG kinase in pool 2 membrane) is offset by phosphorylation of DAG by DAG kinase in pool 2 (Endoplasmic Reticulum Nucleus) to aid in the PI replacement pathway(Endoplasmic Reticulum Nucleus) to aid in the PI replacement pathway

bull Second wave of DAG is a result of resynthesis of PIPSecond wave of DAG is a result of resynthesis of PIP22 which is then which is then hydrolyzed to form DAG and IPhydrolyzed to form DAG and IP33

PI PIP PIP2 DAG

IP3

IP2IP

Ins

PI

CDP-DAGPADAG

PA

PCPE

PCPE

PITP

PI4K PI5K PLC DGK

DGK CDSPLC PIS

POOL 2

POOL 1

IP4

PI PIP PIP2 DAG

IP3

IP2IP

Ins

PI

CDP-DAGPADAG

PA

PCPE

PC

PITP

PI4K PI5K PLC

DGK CDSPLC PIS

IP4

LPP

LPP

Model Equations DAG (pool 1)Model Equations DAG (pool 1)Model Equations DAG (pool 1)Model Equations DAG (pool 1)

bull DAG (pool 1)DAG (pool 1) (separate ODEs for each species (separate ODEs for each species considered)considered)

bull Concentration of DAG from pool 1 at time tConcentration of DAG from pool 1 at time t

Nondimensionalized equationsNondimensionalized equations

Model Equations DAG (pool 2)Model Equations DAG (pool 2)Model Equations DAG (pool 2)Model Equations DAG (pool 2)

bull DAG (pool 2)DAG (pool 2) (separate ODEs for each species (separate ODEs for each species considered)considered)

bull Concentration of pool 2 DAG molecules at time tConcentration of pool 2 DAG molecules at time t

bull Baseline concentration of pool 2 DAGBaseline concentration of pool 2 DAG

Nondimensionalized equationsNondimensionalized equations

Simplified CaSimplified Ca2+2+ Equations EquationsSimplified CaSimplified Ca2+2+ Equations Equations

bull CaCa2+2+ modulemodule (to match experimental AfCS (to match experimental AfCS trace)trace)

0 100 200 300 400 500 6000

002

004

006

008

01

012

014

Concentration of Ca

2+ (uM)

Ca2+ response to 25uM UDP

time (s)

wherewhere

Theoretical AnalysisTheoretical AnalysisTheoretical AnalysisTheoretical Analysis

bull Existence and UniquenessExistence and Uniqueness

bull Positivity and Boundedness Positivity and Boundedness

(for biological relevance)(for biological relevance)

bull Analysis of steady state behaviorAnalysis of steady state behavior

Existence of Solutions (full model)Existence of Solutions (full model)Existence of Solutions (full model)Existence of Solutions (full model)

First we write our system of ODEs in the formFirst we write our system of ODEs in the form (1)

(2)

(1)

Uniqueness of SolutionsUniqueness of SolutionsUniqueness of SolutionsUniqueness of Solutions

Next we denote a solution of Next we denote a solution of (1)(1) by by with initial condition with initial condition

Since our system satisfies the hypotheses of Theorems 1 and 2 Since our system satisfies the hypotheses of Theorems 1 and 2 on our set of interest (for all positive time and on a positive on our set of interest (for all positive time and on a positive bounded set in space) we know bounded set in space) we know there exists a unique there exists a unique (local) solution(local) solution (ie on some finite time interval possibly (ie on some finite time interval possibly small)small)

QuestionQuestion Do the solutions remain positive and bounded (for Do the solutions remain positive and bounded (for biological significance)biological significance)

(3)

(1) (3)

Positivity and boundedness of Positivity and boundedness of solutionssolutions

Positivity and boundedness of Positivity and boundedness of solutionssolutions

bull We first use Theorem 3 to show positivity and boundedness of xWe first use Theorem 3 to show positivity and boundedness of x11 and and xx22

bull Next we use these results and the Fundamental Theorem of Calculus to Next we use these results and the Fundamental Theorem of Calculus to show positivity and boundedness for the remaining equations show positivity and boundedness for the remaining equations

bull This then ensures a global solutionThis then ensures a global solution

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical Analysis

bull Parameter Estimation in SIMULINKParameter Estimation in SIMULINK

bull Sensitivity AnalysisSensitivity Analysis

Parameter Estimation in SIMULINK (for Parameter Estimation in SIMULINK (for FULL model)FULL model)

Parameter Estimation in SIMULINK (for Parameter Estimation in SIMULINK (for FULL model)FULL model)

bull Total number of model parameters (for full model with Li and Rinzel CaTotal number of model parameters (for full model with Li and Rinzel Ca2+2+ module) = module) = 3434ndash From the literature = From the literature = 2020ndash Estimated = Estimated = 1414

bull Receptor moduleReceptor module Total = Total = 77ndash From literature = From literature = 66ndash Estimated = Estimated = 11 (k (kpp rate of receptor phosphorylation) rate of receptor phosphorylation)

bull G-protein cascadeG-protein cascade Total = Total = 99ndash From literature = From literature = 66ndash Estimated = Estimated = 3 3 (k(khydhyd k kreprep k kd3d3))

bull DAG kineticsDAG kinetics Total = Total = 5 5 (for each DAG species considered)(for each DAG species considered)ndash From literature = From literature = 00ndash Estimated = Estimated = 5 5 (all DAG parameters)(all DAG parameters)

bull CaCa2+2+ module module Total = Total = 1313ndash From literature = From literature = 88ndash Estimated = Estimated = 55

SIMULINK DetailsSIMULINK DetailsSIMULINK DetailsSIMULINK Details

bull Unknown rate parameters were estimated using SIMULINK

ndash Minimizes a user-specified cost function via a user-specified optimization method

ndash Nonlinear least squares optimization method of Levenberg-Marquardt was used to minimize a sum of squared errors cost function of the empirical observations and model predictions for IP3 Ca2+ and multiple species of DAG

bull Note Although the Gauss-Newton method is often more efficient the method of Levenberg-Marquardt has proved to be more robust

Measured vs Simulated 384 DAG and Measured vs Simulated 384 DAG and 341 DAG341 DAG

Measured vs Simulated 384 DAG and Measured vs Simulated 384 DAG and 341 DAG341 DAG

bull Timecourse of stimulation with 25 Timecourse of stimulation with 25 M UDP in RAW 2647 cellsM UDP in RAW 2647 cells

bull Solid black lines represent model simulations Solid black lines represent model simulations

bull ((aa) 384 DAG response (representative of the response of most poly ) 384 DAG response (representative of the response of most poly unsaturated fatty acid-containing DAG species) unsaturated fatty acid-containing DAG species)

bull ((bb) 341 DAG response (representative of the response of most mono- ) 341 DAG response (representative of the response of most mono- and di-unsaturated fatty acid-containing DAG species) and di-unsaturated fatty acid-containing DAG species)

bull Data points contain nine replicates performed on three different Data points contain nine replicates performed on three different experimental days with error bars = 1 SEM Units are total change in ng experimental days with error bars = 1 SEM Units are total change in ng over baseline levels in ~8x10over baseline levels in ~8x1066 cells cells

Measured vs Simulated IPMeasured vs Simulated IP33 and Ca and Ca2+2+Measured vs Simulated IPMeasured vs Simulated IP33 and Ca and Ca2+2+

bull Timecourse of stimulation with 25 Timecourse of stimulation with 25 M UDP in RAW 2647 cellsM UDP in RAW 2647 cells

bull Solid black lines represent model simulationsSolid black lines represent model simulations

bull ((cc) IP) IP33 response in pmols per ~35x10 response in pmols per ~35x1055 cells cells

bull Points in (c) represent the average of four experiments and error Points in (c) represent the average of four experiments and error bars are 1 SEM bars are 1 SEM

bull ((dd) Ca) Ca2+2+ response in response in M Red curve is a representative CaM Red curve is a representative Ca2+2+ trace trace taken from the UDP experiments within the AfCS single ligand taken from the UDP experiments within the AfCS single ligand screenscreen

Model Simulations P2YModel Simulations P2Y66 G GGTP GTP PIPPIP22

Model Simulations P2YModel Simulations P2Y66 G GGTP GTP PIPPIP22

bull ((aa) = total P2Y) = total P2Y66 activated (solid activated (solid line) and inactivated (dashed line) and inactivated (dashed line) surface receptorsline) surface receptors

bull ((bb) = total G) = total GGTP GTP

bull ((cc) = total PIP) = total PIP22 available for available for hydrolysishydrolysis

P2Y6 from 25M UDP GGTP from 25M UDP

PIP2 from 25M UDP

Sensitivity AnalysisSensitivity AnalysisSensitivity AnalysisSensitivity Analysis

bull Sensitivity analysis techniques are valuable tools Sensitivity analysis techniques are valuable tools designed to answer questions regarding which of the designed to answer questions regarding which of the uncertain input variables is more important in uncertain input variables is more important in determining the uncertainty in our output determining the uncertainty in our output

bull Likewise sensitivity analysis can provide insight into Likewise sensitivity analysis can provide insight into which parameter should be studied in more detail in which parameter should be studied in more detail in order to reduce the most variance in the model output order to reduce the most variance in the model output

bull The ability to answer these types of questions could The ability to answer these types of questions could lead to important insight into the design of new lead to important insight into the design of new experiments and in determining which experiments experiments and in determining which experiments would give us the most valuable informationwould give us the most valuable information

Sampling MethodSampling MethodSampling MethodSampling Method

bull Generate a random sample of our space of input Generate a random sample of our space of input variables over a ten percent variation from each variables over a ten percent variation from each parameters nominal value using the parameters nominal value using the Latin Hypercube Latin Hypercube Sampling (LHS) methodSampling (LHS) method

bull Uses Uses Standardized Regression Coefficients (SRCs) Standardized Regression Coefficients (SRCs) obtained by performing multiple linear regression obtained by performing multiple linear regression analysisanalysisndash offers a measure of sensitivity that is multi-dimensionally offers a measure of sensitivity that is multi-dimensionally

averaged over the entire space of parameter values averaged over the entire space of parameter values ndash SRCs give insight into degree of nonlinearity in the model SRCs give insight into degree of nonlinearity in the model

Computing RComputing Ryy22 values valuesComputing RComputing Ryy22 values values

bull SRCs are only reliable measures of sensitivity when SRCs are only reliable measures of sensitivity when degree of nonlinearity is ldquosmallrdquodegree of nonlinearity is ldquosmallrdquo

bull Use model coefficients of determination RUse model coefficients of determination Ryy22 given by given by

bull where ywhere yii is the estimate of y is the estimate of yii obtained from the regression obtained from the regression modelmodel

bull RRyy22 ge 07 ensures SRCs are good sensitivity measures ge 07 ensures SRCs are good sensitivity measures

SRCs (as sensitivity measures) for SRCs (as sensitivity measures) for DAGDAG

SRCs (as sensitivity measures) for SRCs (as sensitivity measures) for DAGDAG

SRCs (as sensitivity measures) for IPSRCs (as sensitivity measures) for IP33SRCs (as sensitivity measures) for IPSRCs (as sensitivity measures) for IP33

Effects of 3 most sensitive DAG Effects of 3 most sensitive DAG parametersparameters

Effects of 3 most sensitive DAG Effects of 3 most sensitive DAG parametersparameters

A B

C

AA k kdp2dp2 degradation of pool 2 DAG degradation of pool 2 DAG

BB k kdp1dp1 degradation of pool 1 DAG degradation of pool 1 DAG

CC k kap2ap2 production of pool 2 DAG production of pool 2 DAG

ConclusionsConclusionsConclusionsConclusions

bull We have developed a model of the UDP signaling We have developed a model of the UDP signaling pathway in RAW 2647 macrophages which can predict pathway in RAW 2647 macrophages which can predict the responses of multiple species of DAG as well as the the responses of multiple species of DAG as well as the responses of IPresponses of IP33 Ca Ca2+2+ receptor dynamics G-protein receptor dynamics G-protein activation and PIPactivation and PIP22 hydrolysis hydrolysis

bull Simplified model resultsSimplified model resultsndash We have obtained global existence uniqueness positivity We have obtained global existence uniqueness positivity

and boundedness of solutionsand boundedness of solutionsndash We have proven global stability of a unique steady state We have proven global stability of a unique steady state

within our region of interestwithin our region of interestbull Full model analysisFull model analysis

ndash Using SIMULINK we have estimated unknown rate Using SIMULINK we have estimated unknown rate parameters to obtain best fits to multiple DAG traces IPparameters to obtain best fits to multiple DAG traces IP33 and Caand Ca2+2+ all in response to 25 all in response to 25M UDPM UDP

ndash We have performed sensitivity analysis using the Latin We have performed sensitivity analysis using the Latin Hypercube sampling technique in combination with Hypercube sampling technique in combination with standardized regression coefficients to determine which standardized regression coefficients to determine which model parameters are responsible for most of the model model parameters are responsible for most of the model output uncertaintyoutput uncertainty

Future DirectionsFuture DirectionsFuture DirectionsFuture Directions

bull We have conducted multiple experiments to We have conducted multiple experiments to pharmacologically inhibit several different enzymes we pharmacologically inhibit several different enzymes we believe to play a role in this signaling pathway as believe to play a role in this signaling pathway as suggested by current known metabolic pathways and by suggested by current known metabolic pathways and by modeling resultsmodeling results

bull The next step is to perform gene knockdowns on The next step is to perform gene knockdowns on specific enzymes to verify results of inhibitor dataspecific enzymes to verify results of inhibitor data

bull The model output also suggests a time delay from The model output also suggests a time delay from receptor activation to PIPreceptor activation to PIP22 hydrolysis An upcoming task hydrolysis An upcoming task is to investigate the outcomes of adding such a delay is to investigate the outcomes of adding such a delay termterm

bull The model could be greatly enhanced by incorporating The model could be greatly enhanced by incorporating spatial dynamics so this will also be a major focus for spatial dynamics so this will also be a major focus for future researchfuture research

AcknowledgementsAcknowledgementsAcknowledgementsAcknowledgements

Mathematics DepartmentMathematics Departmentbull Hannah L Callender Hannah L Callender

PhDPhD

CollaboratorsCollaboratorsbull Alliance for Cellular SignalingAlliance for Cellular Signalingbull UT SouthwesternUT Southwestern

ndash Paul Sternweis PhDPaul Sternweis PhDndash Dianne DeCamp PhDDianne DeCamp PhD

Brown LabBrown Labbull H Alex Brown PhDH Alex Brown PhDbull Jeffrey S Forrester PhDJeffrey S Forrester PhDbull Mark Byrne PhDMark Byrne PhDbull Anita Preininger PhDAnita Preininger PhDbull Michelle ArmstrongMichelle Armstrongbull Andrew GoodmanAndrew Goodmanbull Pavlina Ivanova PhDPavlina Ivanova PhDbull Steve Milne PhDSteve Milne PhD

Model Equations DAGModel Equations DAGModel Equations DAGModel Equations DAG

bull DAG DAG (separate ODEs for each species considered)(separate ODEs for each species considered)

Nondimensionalized equationsNondimensionalized equations

bull Concentration of DAG at time tConcentration of DAG at time t

PLC

PIP

2

DA

G

IP3

G

GTPCa2+G

GTP

Model Equations Li and Rinzel (1994) Model Equations Li and Rinzel (1994) CaCa2+2+

Model Equations Li and Rinzel (1994) Model Equations Li and Rinzel (1994) CaCa2+2+

bull Free cytosolicFree cytosolic Ca Ca2+2+

bull hh = fraction of IP = fraction of IP33 channels not yet inactivated by Ca channels not yet inactivated by Ca2+2+

bull cc00 = concentration of total free Ca = concentration of total free Ca2+2+ per cytosolic volume per cytosolic volume

Timeseries data for the modelTimeseries data for the modelTimeseries data for the modelTimeseries data for the model

bull Data from collaborators at UTSW (Sternweis Lab)Data from collaborators at UTSW (Sternweis Lab)ndash Inositol 145-trisphosphate (Inositol 145-trisphosphate (IPIP33) production post 25) production post 25M M

UDP stimulation in the RAW cellUDP stimulation in the RAW cellndash Cytosolic calcium (Cytosolic calcium (CaCa2+2+) release post stimulation with ) release post stimulation with

2525M UDP in the RAW cellM UDP in the RAW cell

bull Diacylglycerol (Diacylglycerol (DAGDAG) data H Callender collected from a ) data H Callender collected from a novel method of quantitative analysis of multiple novel method of quantitative analysis of multiple species of DAG developed in the Brown Lab species of DAG developed in the Brown Lab

Reference Reference

Callender H L et al Quantification of Diacylglycerol Species from Callender H L et al Quantification of Diacylglycerol Species from Cellular Extracts by Electrospray Ionization Mass Spectrometry Using a Cellular Extracts by Electrospray Ionization Mass Spectrometry Using a Linear Regression Algorithm Linear Regression Algorithm Anal Chem Anal Chem 7979 (2007) 263-272 (2007) 263-272

IPIP33 and Ca and Ca2+2+ data from AfCS data from AfCSIPIP33 and Ca and Ca2+2+ data from AfCS data from AfCS

AA

bull IPIP33 response (in pmoles per response (in pmoles per 100 100 L lysate)in RAW 2647 L lysate)in RAW 2647 cells to 25 cells to 25 M UDPM UDP

bull Points represent the average Points represent the average of four experimentsof four experiments

bull Cytosolic calcium response Cytosolic calcium response (in (in M) in the RAW 2647 M) in the RAW 2647 cell to 25 cell to 25 M UDP M UDP

bull The graph displays 43 The graph displays 43 experimental repeatsexperimental repeats

B

What is DAGWhat is DAGWhat is DAGWhat is DAG

bull DAG is a cellular second messenger molecule which plays an DAG is a cellular second messenger molecule which plays an important role in initiating various changes in cell behavior important role in initiating various changes in cell behavior including cell activation differentiation proliferation and including cell activation differentiation proliferation and tumor promotion tumor promotion

bull There are many different species of DAG depending on the There are many different species of DAG depending on the number of carbons and number of double bonds in the fatty number of carbons and number of double bonds in the fatty acyl chains and different species can have different cellular acyl chains and different species can have different cellular functionsfunctions

bull ExampleExample 320 DAG320 DAG

Two fatty acyl Two fatty acyl (hydrocarbon) (hydrocarbon) chains in every DAG chains in every DAG speciesspecies

Each corner Each corner represents carbonrepresents carbon

320 DAG 18 carbons in first chain 14 in the second no double bonds 320 DAG 18 carbons in first chain 14 in the second no double bonds in either chainin either chain

Kinetics of monodi unsaturated DAGsKinetics of monodi unsaturated DAGsKinetics of monodi unsaturated DAGsKinetics of monodi unsaturated DAGs

bull Time based behavior of four monodi unsaturated DAG species after addition of 25 Time based behavior of four monodi unsaturated DAG species after addition of 25 M M (solid red squares) and 025 (solid red squares) and 025 M (solid green triangles) UDPM (solid green triangles) UDP

bull Time points contain a minimum of nine replicates performed on three different Time points contain a minimum of nine replicates performed on three different experimental daysexperimental days

- 25 M UDP

- 025M UDP

0 5 10 15 20 25 30

0 5 10 15 20 25 30 0 5 10 15 20 25 30

0 5 10 15 20 25 30

321 DAG 341 DAG

362 DAG341 DAGep

150

100

50-5

00

300

200

100

-100

0

150

100

50-5

00

-100

200

6040

20-2

00

-40

80

Time (min) Time (min)

Time (min) Time (min)

D

AG

(n

g)

D

AG

(n

g)

D

AG

(n

g)

D

AG

(n

g)

- 25 M UDP

- 025M UDP

- 25 M UDP

- 025M UDP

- 25 M UDP

- 025M UDP

Kinetics of PUFA containing DAGsKinetics of PUFA containing DAGsKinetics of PUFA containing DAGsKinetics of PUFA containing DAGsbull Time based behavior of four polyunsaturated fatty acid (PUFA) containing DAG species Time based behavior of four polyunsaturated fatty acid (PUFA) containing DAG species

after addition of 25 after addition of 25 M (solid red squares) and 025 M (solid red squares) and 025 M (solid green triangles) UDPM (solid green triangles) UDPbull Time points contain a minimum of nine replicates performed on three different Time points contain a minimum of nine replicates performed on three different

experimental daysexperimental daysbull Monodi unsaturated DAG species give a larger increase than polyunsaturated fatty acid Monodi unsaturated DAG species give a larger increase than polyunsaturated fatty acid

(PUFA) containing DAGs(PUFA) containing DAGs

0 5 10 15 20 25 30

0 5 10 15 20 25 30 0 5 10 15 20 25 30

0 5 10 15 20 25 30

385 DAG 383 DAG

364 DAG384 DAG6

42

-40

30

20

10

0

15

10

50

-56

40

2-2

8

Time (min) Time (min)

Time (min) Time (min)

D

AG

(n

g)

D

AG

(n

g)

D

AG

(n

g)

D

AG

(n

g)

-28

40

- 25 M UDP

- 025M UDP

- 25 M UDP

- 025M UDP

- 25 M UDP

- 025M UDP

- 25 M UDP

- 025M UDP

Differential DAG KineticsDifferential DAG KineticsDifferential DAG KineticsDifferential DAG Kinetics

bull Time based behavior of three DAG species with varying degrees of unsaturation after Time based behavior of three DAG species with varying degrees of unsaturation after addition of addition of 25 25 MM (solid red triangles) and (solid red triangles) and 025 025 MM (solid green squares) UDP (solid green squares) UDP

bull Time points contain nine replicates performed on three different experimental daysTime points contain nine replicates performed on three different experimental days

341 DAG

Best fit with current model structure

Overall Objectives of Modeling EffortOverall Objectives of Modeling EffortOverall Objectives of Modeling EffortOverall Objectives of Modeling Effort

bull Predict quantitative changes in lipid species after stimulation by Predict quantitative changes in lipid species after stimulation by various ligands and ligand concentrations in the RAW 2647 various ligands and ligand concentrations in the RAW 2647 macrophagemacrophage

bull Comparison and refinement of model output with AfCS IPComparison and refinement of model output with AfCS IP33 measurements and Cameasurements and Ca2+2+ traces as well as DAG data generated in traces as well as DAG data generated in the Brown labthe Brown lab

bull Predict in silico effects such as the effect of knock-downs etc on Predict in silico effects such as the effect of knock-downs etc on given pathwaygiven pathway

bull Suggest modifications to current pathway structuresSuggest modifications to current pathway structures

Modifications to the modelModifications to the modelModifications to the modelModifications to the model

bull Include an additional branch in the pathway Include an additional branch in the pathway for a second pool of DAGfor a second pool of DAG

bull Simplify CaSimplify Ca2+2+ equations for mathematical equations for mathematical analysis purposesanalysis purposes

Proposed PathwayProposed PathwayProposed PathwayProposed Pathway

bull Note We measure total cellular DAG levelsNote We measure total cellular DAG levelsbull Initial production of DAG from the hydrolysis of PIPInitial production of DAG from the hydrolysis of PIP22 in pool 1 (plasma in pool 1 (plasma

membrane) is offset by phosphorylation of DAG by DAG kinase in pool 2 membrane) is offset by phosphorylation of DAG by DAG kinase in pool 2 (Endoplasmic Reticulum Nucleus) to aid in the PI replacement pathway(Endoplasmic Reticulum Nucleus) to aid in the PI replacement pathway

bull Second wave of DAG is a result of resynthesis of PIPSecond wave of DAG is a result of resynthesis of PIP22 which is then which is then hydrolyzed to form DAG and IPhydrolyzed to form DAG and IP33

PI PIP PIP2 DAG

IP3

IP2IP

Ins

PI

CDP-DAGPADAG

PA

PCPE

PCPE

PITP

PI4K PI5K PLC DGK

DGK CDSPLC PIS

POOL 2

POOL 1

IP4

PI PIP PIP2 DAG

IP3

IP2IP

Ins

PI

CDP-DAGPADAG

PA

PCPE

PC

PITP

PI4K PI5K PLC

DGK CDSPLC PIS

IP4

LPP

LPP

Model Equations DAG (pool 1)Model Equations DAG (pool 1)Model Equations DAG (pool 1)Model Equations DAG (pool 1)

bull DAG (pool 1)DAG (pool 1) (separate ODEs for each species (separate ODEs for each species considered)considered)

bull Concentration of DAG from pool 1 at time tConcentration of DAG from pool 1 at time t

Nondimensionalized equationsNondimensionalized equations

Model Equations DAG (pool 2)Model Equations DAG (pool 2)Model Equations DAG (pool 2)Model Equations DAG (pool 2)

bull DAG (pool 2)DAG (pool 2) (separate ODEs for each species (separate ODEs for each species considered)considered)

bull Concentration of pool 2 DAG molecules at time tConcentration of pool 2 DAG molecules at time t

bull Baseline concentration of pool 2 DAGBaseline concentration of pool 2 DAG

Nondimensionalized equationsNondimensionalized equations

Simplified CaSimplified Ca2+2+ Equations EquationsSimplified CaSimplified Ca2+2+ Equations Equations

bull CaCa2+2+ modulemodule (to match experimental AfCS (to match experimental AfCS trace)trace)

0 100 200 300 400 500 6000

002

004

006

008

01

012

014

Concentration of Ca

2+ (uM)

Ca2+ response to 25uM UDP

time (s)

wherewhere

Theoretical AnalysisTheoretical AnalysisTheoretical AnalysisTheoretical Analysis

bull Existence and UniquenessExistence and Uniqueness

bull Positivity and Boundedness Positivity and Boundedness

(for biological relevance)(for biological relevance)

bull Analysis of steady state behaviorAnalysis of steady state behavior

Existence of Solutions (full model)Existence of Solutions (full model)Existence of Solutions (full model)Existence of Solutions (full model)

First we write our system of ODEs in the formFirst we write our system of ODEs in the form (1)

(2)

(1)

Uniqueness of SolutionsUniqueness of SolutionsUniqueness of SolutionsUniqueness of Solutions

Next we denote a solution of Next we denote a solution of (1)(1) by by with initial condition with initial condition

Since our system satisfies the hypotheses of Theorems 1 and 2 Since our system satisfies the hypotheses of Theorems 1 and 2 on our set of interest (for all positive time and on a positive on our set of interest (for all positive time and on a positive bounded set in space) we know bounded set in space) we know there exists a unique there exists a unique (local) solution(local) solution (ie on some finite time interval possibly (ie on some finite time interval possibly small)small)

QuestionQuestion Do the solutions remain positive and bounded (for Do the solutions remain positive and bounded (for biological significance)biological significance)

(3)

(1) (3)

Positivity and boundedness of Positivity and boundedness of solutionssolutions

Positivity and boundedness of Positivity and boundedness of solutionssolutions

bull We first use Theorem 3 to show positivity and boundedness of xWe first use Theorem 3 to show positivity and boundedness of x11 and and xx22

bull Next we use these results and the Fundamental Theorem of Calculus to Next we use these results and the Fundamental Theorem of Calculus to show positivity and boundedness for the remaining equations show positivity and boundedness for the remaining equations

bull This then ensures a global solutionThis then ensures a global solution

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical Analysis

bull Parameter Estimation in SIMULINKParameter Estimation in SIMULINK

bull Sensitivity AnalysisSensitivity Analysis

Parameter Estimation in SIMULINK (for Parameter Estimation in SIMULINK (for FULL model)FULL model)

Parameter Estimation in SIMULINK (for Parameter Estimation in SIMULINK (for FULL model)FULL model)

bull Total number of model parameters (for full model with Li and Rinzel CaTotal number of model parameters (for full model with Li and Rinzel Ca2+2+ module) = module) = 3434ndash From the literature = From the literature = 2020ndash Estimated = Estimated = 1414

bull Receptor moduleReceptor module Total = Total = 77ndash From literature = From literature = 66ndash Estimated = Estimated = 11 (k (kpp rate of receptor phosphorylation) rate of receptor phosphorylation)

bull G-protein cascadeG-protein cascade Total = Total = 99ndash From literature = From literature = 66ndash Estimated = Estimated = 3 3 (k(khydhyd k kreprep k kd3d3))

bull DAG kineticsDAG kinetics Total = Total = 5 5 (for each DAG species considered)(for each DAG species considered)ndash From literature = From literature = 00ndash Estimated = Estimated = 5 5 (all DAG parameters)(all DAG parameters)

bull CaCa2+2+ module module Total = Total = 1313ndash From literature = From literature = 88ndash Estimated = Estimated = 55

SIMULINK DetailsSIMULINK DetailsSIMULINK DetailsSIMULINK Details

bull Unknown rate parameters were estimated using SIMULINK

ndash Minimizes a user-specified cost function via a user-specified optimization method

ndash Nonlinear least squares optimization method of Levenberg-Marquardt was used to minimize a sum of squared errors cost function of the empirical observations and model predictions for IP3 Ca2+ and multiple species of DAG

bull Note Although the Gauss-Newton method is often more efficient the method of Levenberg-Marquardt has proved to be more robust

Measured vs Simulated 384 DAG and Measured vs Simulated 384 DAG and 341 DAG341 DAG

Measured vs Simulated 384 DAG and Measured vs Simulated 384 DAG and 341 DAG341 DAG

bull Timecourse of stimulation with 25 Timecourse of stimulation with 25 M UDP in RAW 2647 cellsM UDP in RAW 2647 cells

bull Solid black lines represent model simulations Solid black lines represent model simulations

bull ((aa) 384 DAG response (representative of the response of most poly ) 384 DAG response (representative of the response of most poly unsaturated fatty acid-containing DAG species) unsaturated fatty acid-containing DAG species)

bull ((bb) 341 DAG response (representative of the response of most mono- ) 341 DAG response (representative of the response of most mono- and di-unsaturated fatty acid-containing DAG species) and di-unsaturated fatty acid-containing DAG species)

bull Data points contain nine replicates performed on three different Data points contain nine replicates performed on three different experimental days with error bars = 1 SEM Units are total change in ng experimental days with error bars = 1 SEM Units are total change in ng over baseline levels in ~8x10over baseline levels in ~8x1066 cells cells

Measured vs Simulated IPMeasured vs Simulated IP33 and Ca and Ca2+2+Measured vs Simulated IPMeasured vs Simulated IP33 and Ca and Ca2+2+

bull Timecourse of stimulation with 25 Timecourse of stimulation with 25 M UDP in RAW 2647 cellsM UDP in RAW 2647 cells

bull Solid black lines represent model simulationsSolid black lines represent model simulations

bull ((cc) IP) IP33 response in pmols per ~35x10 response in pmols per ~35x1055 cells cells

bull Points in (c) represent the average of four experiments and error Points in (c) represent the average of four experiments and error bars are 1 SEM bars are 1 SEM

bull ((dd) Ca) Ca2+2+ response in response in M Red curve is a representative CaM Red curve is a representative Ca2+2+ trace trace taken from the UDP experiments within the AfCS single ligand taken from the UDP experiments within the AfCS single ligand screenscreen

Model Simulations P2YModel Simulations P2Y66 G GGTP GTP PIPPIP22

Model Simulations P2YModel Simulations P2Y66 G GGTP GTP PIPPIP22

bull ((aa) = total P2Y) = total P2Y66 activated (solid activated (solid line) and inactivated (dashed line) and inactivated (dashed line) surface receptorsline) surface receptors

bull ((bb) = total G) = total GGTP GTP

bull ((cc) = total PIP) = total PIP22 available for available for hydrolysishydrolysis

P2Y6 from 25M UDP GGTP from 25M UDP

PIP2 from 25M UDP

Sensitivity AnalysisSensitivity AnalysisSensitivity AnalysisSensitivity Analysis

bull Sensitivity analysis techniques are valuable tools Sensitivity analysis techniques are valuable tools designed to answer questions regarding which of the designed to answer questions regarding which of the uncertain input variables is more important in uncertain input variables is more important in determining the uncertainty in our output determining the uncertainty in our output

bull Likewise sensitivity analysis can provide insight into Likewise sensitivity analysis can provide insight into which parameter should be studied in more detail in which parameter should be studied in more detail in order to reduce the most variance in the model output order to reduce the most variance in the model output

bull The ability to answer these types of questions could The ability to answer these types of questions could lead to important insight into the design of new lead to important insight into the design of new experiments and in determining which experiments experiments and in determining which experiments would give us the most valuable informationwould give us the most valuable information

Sampling MethodSampling MethodSampling MethodSampling Method

bull Generate a random sample of our space of input Generate a random sample of our space of input variables over a ten percent variation from each variables over a ten percent variation from each parameters nominal value using the parameters nominal value using the Latin Hypercube Latin Hypercube Sampling (LHS) methodSampling (LHS) method

bull Uses Uses Standardized Regression Coefficients (SRCs) Standardized Regression Coefficients (SRCs) obtained by performing multiple linear regression obtained by performing multiple linear regression analysisanalysisndash offers a measure of sensitivity that is multi-dimensionally offers a measure of sensitivity that is multi-dimensionally

averaged over the entire space of parameter values averaged over the entire space of parameter values ndash SRCs give insight into degree of nonlinearity in the model SRCs give insight into degree of nonlinearity in the model

Computing RComputing Ryy22 values valuesComputing RComputing Ryy22 values values

bull SRCs are only reliable measures of sensitivity when SRCs are only reliable measures of sensitivity when degree of nonlinearity is ldquosmallrdquodegree of nonlinearity is ldquosmallrdquo

bull Use model coefficients of determination RUse model coefficients of determination Ryy22 given by given by

bull where ywhere yii is the estimate of y is the estimate of yii obtained from the regression obtained from the regression modelmodel

bull RRyy22 ge 07 ensures SRCs are good sensitivity measures ge 07 ensures SRCs are good sensitivity measures

SRCs (as sensitivity measures) for SRCs (as sensitivity measures) for DAGDAG

SRCs (as sensitivity measures) for SRCs (as sensitivity measures) for DAGDAG

SRCs (as sensitivity measures) for IPSRCs (as sensitivity measures) for IP33SRCs (as sensitivity measures) for IPSRCs (as sensitivity measures) for IP33

Effects of 3 most sensitive DAG Effects of 3 most sensitive DAG parametersparameters

Effects of 3 most sensitive DAG Effects of 3 most sensitive DAG parametersparameters

A B

C

AA k kdp2dp2 degradation of pool 2 DAG degradation of pool 2 DAG

BB k kdp1dp1 degradation of pool 1 DAG degradation of pool 1 DAG

CC k kap2ap2 production of pool 2 DAG production of pool 2 DAG

ConclusionsConclusionsConclusionsConclusions

bull We have developed a model of the UDP signaling We have developed a model of the UDP signaling pathway in RAW 2647 macrophages which can predict pathway in RAW 2647 macrophages which can predict the responses of multiple species of DAG as well as the the responses of multiple species of DAG as well as the responses of IPresponses of IP33 Ca Ca2+2+ receptor dynamics G-protein receptor dynamics G-protein activation and PIPactivation and PIP22 hydrolysis hydrolysis

bull Simplified model resultsSimplified model resultsndash We have obtained global existence uniqueness positivity We have obtained global existence uniqueness positivity

and boundedness of solutionsand boundedness of solutionsndash We have proven global stability of a unique steady state We have proven global stability of a unique steady state

within our region of interestwithin our region of interestbull Full model analysisFull model analysis

ndash Using SIMULINK we have estimated unknown rate Using SIMULINK we have estimated unknown rate parameters to obtain best fits to multiple DAG traces IPparameters to obtain best fits to multiple DAG traces IP33 and Caand Ca2+2+ all in response to 25 all in response to 25M UDPM UDP

ndash We have performed sensitivity analysis using the Latin We have performed sensitivity analysis using the Latin Hypercube sampling technique in combination with Hypercube sampling technique in combination with standardized regression coefficients to determine which standardized regression coefficients to determine which model parameters are responsible for most of the model model parameters are responsible for most of the model output uncertaintyoutput uncertainty

Future DirectionsFuture DirectionsFuture DirectionsFuture Directions

bull We have conducted multiple experiments to We have conducted multiple experiments to pharmacologically inhibit several different enzymes we pharmacologically inhibit several different enzymes we believe to play a role in this signaling pathway as believe to play a role in this signaling pathway as suggested by current known metabolic pathways and by suggested by current known metabolic pathways and by modeling resultsmodeling results

bull The next step is to perform gene knockdowns on The next step is to perform gene knockdowns on specific enzymes to verify results of inhibitor dataspecific enzymes to verify results of inhibitor data

bull The model output also suggests a time delay from The model output also suggests a time delay from receptor activation to PIPreceptor activation to PIP22 hydrolysis An upcoming task hydrolysis An upcoming task is to investigate the outcomes of adding such a delay is to investigate the outcomes of adding such a delay termterm

bull The model could be greatly enhanced by incorporating The model could be greatly enhanced by incorporating spatial dynamics so this will also be a major focus for spatial dynamics so this will also be a major focus for future researchfuture research

AcknowledgementsAcknowledgementsAcknowledgementsAcknowledgements

Mathematics DepartmentMathematics Departmentbull Hannah L Callender Hannah L Callender

PhDPhD

CollaboratorsCollaboratorsbull Alliance for Cellular SignalingAlliance for Cellular Signalingbull UT SouthwesternUT Southwestern

ndash Paul Sternweis PhDPaul Sternweis PhDndash Dianne DeCamp PhDDianne DeCamp PhD

Brown LabBrown Labbull H Alex Brown PhDH Alex Brown PhDbull Jeffrey S Forrester PhDJeffrey S Forrester PhDbull Mark Byrne PhDMark Byrne PhDbull Anita Preininger PhDAnita Preininger PhDbull Michelle ArmstrongMichelle Armstrongbull Andrew GoodmanAndrew Goodmanbull Pavlina Ivanova PhDPavlina Ivanova PhDbull Steve Milne PhDSteve Milne PhD

Model Equations Li and Rinzel (1994) Model Equations Li and Rinzel (1994) CaCa2+2+

Model Equations Li and Rinzel (1994) Model Equations Li and Rinzel (1994) CaCa2+2+

bull Free cytosolicFree cytosolic Ca Ca2+2+

bull hh = fraction of IP = fraction of IP33 channels not yet inactivated by Ca channels not yet inactivated by Ca2+2+

bull cc00 = concentration of total free Ca = concentration of total free Ca2+2+ per cytosolic volume per cytosolic volume

Timeseries data for the modelTimeseries data for the modelTimeseries data for the modelTimeseries data for the model

bull Data from collaborators at UTSW (Sternweis Lab)Data from collaborators at UTSW (Sternweis Lab)ndash Inositol 145-trisphosphate (Inositol 145-trisphosphate (IPIP33) production post 25) production post 25M M

UDP stimulation in the RAW cellUDP stimulation in the RAW cellndash Cytosolic calcium (Cytosolic calcium (CaCa2+2+) release post stimulation with ) release post stimulation with

2525M UDP in the RAW cellM UDP in the RAW cell

bull Diacylglycerol (Diacylglycerol (DAGDAG) data H Callender collected from a ) data H Callender collected from a novel method of quantitative analysis of multiple novel method of quantitative analysis of multiple species of DAG developed in the Brown Lab species of DAG developed in the Brown Lab

Reference Reference

Callender H L et al Quantification of Diacylglycerol Species from Callender H L et al Quantification of Diacylglycerol Species from Cellular Extracts by Electrospray Ionization Mass Spectrometry Using a Cellular Extracts by Electrospray Ionization Mass Spectrometry Using a Linear Regression Algorithm Linear Regression Algorithm Anal Chem Anal Chem 7979 (2007) 263-272 (2007) 263-272

IPIP33 and Ca and Ca2+2+ data from AfCS data from AfCSIPIP33 and Ca and Ca2+2+ data from AfCS data from AfCS

AA

bull IPIP33 response (in pmoles per response (in pmoles per 100 100 L lysate)in RAW 2647 L lysate)in RAW 2647 cells to 25 cells to 25 M UDPM UDP

bull Points represent the average Points represent the average of four experimentsof four experiments

bull Cytosolic calcium response Cytosolic calcium response (in (in M) in the RAW 2647 M) in the RAW 2647 cell to 25 cell to 25 M UDP M UDP

bull The graph displays 43 The graph displays 43 experimental repeatsexperimental repeats

B

What is DAGWhat is DAGWhat is DAGWhat is DAG

bull DAG is a cellular second messenger molecule which plays an DAG is a cellular second messenger molecule which plays an important role in initiating various changes in cell behavior important role in initiating various changes in cell behavior including cell activation differentiation proliferation and including cell activation differentiation proliferation and tumor promotion tumor promotion

bull There are many different species of DAG depending on the There are many different species of DAG depending on the number of carbons and number of double bonds in the fatty number of carbons and number of double bonds in the fatty acyl chains and different species can have different cellular acyl chains and different species can have different cellular functionsfunctions

bull ExampleExample 320 DAG320 DAG

Two fatty acyl Two fatty acyl (hydrocarbon) (hydrocarbon) chains in every DAG chains in every DAG speciesspecies

Each corner Each corner represents carbonrepresents carbon

320 DAG 18 carbons in first chain 14 in the second no double bonds 320 DAG 18 carbons in first chain 14 in the second no double bonds in either chainin either chain

Kinetics of monodi unsaturated DAGsKinetics of monodi unsaturated DAGsKinetics of monodi unsaturated DAGsKinetics of monodi unsaturated DAGs

bull Time based behavior of four monodi unsaturated DAG species after addition of 25 Time based behavior of four monodi unsaturated DAG species after addition of 25 M M (solid red squares) and 025 (solid red squares) and 025 M (solid green triangles) UDPM (solid green triangles) UDP

bull Time points contain a minimum of nine replicates performed on three different Time points contain a minimum of nine replicates performed on three different experimental daysexperimental days

- 25 M UDP

- 025M UDP

0 5 10 15 20 25 30

0 5 10 15 20 25 30 0 5 10 15 20 25 30

0 5 10 15 20 25 30

321 DAG 341 DAG

362 DAG341 DAGep

150

100

50-5

00

300

200

100

-100

0

150

100

50-5

00

-100

200

6040

20-2

00

-40

80

Time (min) Time (min)

Time (min) Time (min)

D

AG

(n

g)

D

AG

(n

g)

D

AG

(n

g)

D

AG

(n

g)

- 25 M UDP

- 025M UDP

- 25 M UDP

- 025M UDP

- 25 M UDP

- 025M UDP

Kinetics of PUFA containing DAGsKinetics of PUFA containing DAGsKinetics of PUFA containing DAGsKinetics of PUFA containing DAGsbull Time based behavior of four polyunsaturated fatty acid (PUFA) containing DAG species Time based behavior of four polyunsaturated fatty acid (PUFA) containing DAG species

after addition of 25 after addition of 25 M (solid red squares) and 025 M (solid red squares) and 025 M (solid green triangles) UDPM (solid green triangles) UDPbull Time points contain a minimum of nine replicates performed on three different Time points contain a minimum of nine replicates performed on three different

experimental daysexperimental daysbull Monodi unsaturated DAG species give a larger increase than polyunsaturated fatty acid Monodi unsaturated DAG species give a larger increase than polyunsaturated fatty acid

(PUFA) containing DAGs(PUFA) containing DAGs

0 5 10 15 20 25 30

0 5 10 15 20 25 30 0 5 10 15 20 25 30

0 5 10 15 20 25 30

385 DAG 383 DAG

364 DAG384 DAG6

42

-40

30

20

10

0

15

10

50

-56

40

2-2

8

Time (min) Time (min)

Time (min) Time (min)

D

AG

(n

g)

D

AG

(n

g)

D

AG

(n

g)

D

AG

(n

g)

-28

40

- 25 M UDP

- 025M UDP

- 25 M UDP

- 025M UDP

- 25 M UDP

- 025M UDP

- 25 M UDP

- 025M UDP

Differential DAG KineticsDifferential DAG KineticsDifferential DAG KineticsDifferential DAG Kinetics

bull Time based behavior of three DAG species with varying degrees of unsaturation after Time based behavior of three DAG species with varying degrees of unsaturation after addition of addition of 25 25 MM (solid red triangles) and (solid red triangles) and 025 025 MM (solid green squares) UDP (solid green squares) UDP

bull Time points contain nine replicates performed on three different experimental daysTime points contain nine replicates performed on three different experimental days

341 DAG

Best fit with current model structure

Overall Objectives of Modeling EffortOverall Objectives of Modeling EffortOverall Objectives of Modeling EffortOverall Objectives of Modeling Effort

bull Predict quantitative changes in lipid species after stimulation by Predict quantitative changes in lipid species after stimulation by various ligands and ligand concentrations in the RAW 2647 various ligands and ligand concentrations in the RAW 2647 macrophagemacrophage

bull Comparison and refinement of model output with AfCS IPComparison and refinement of model output with AfCS IP33 measurements and Cameasurements and Ca2+2+ traces as well as DAG data generated in traces as well as DAG data generated in the Brown labthe Brown lab

bull Predict in silico effects such as the effect of knock-downs etc on Predict in silico effects such as the effect of knock-downs etc on given pathwaygiven pathway

bull Suggest modifications to current pathway structuresSuggest modifications to current pathway structures

Modifications to the modelModifications to the modelModifications to the modelModifications to the model

bull Include an additional branch in the pathway Include an additional branch in the pathway for a second pool of DAGfor a second pool of DAG

bull Simplify CaSimplify Ca2+2+ equations for mathematical equations for mathematical analysis purposesanalysis purposes

Proposed PathwayProposed PathwayProposed PathwayProposed Pathway

bull Note We measure total cellular DAG levelsNote We measure total cellular DAG levelsbull Initial production of DAG from the hydrolysis of PIPInitial production of DAG from the hydrolysis of PIP22 in pool 1 (plasma in pool 1 (plasma

membrane) is offset by phosphorylation of DAG by DAG kinase in pool 2 membrane) is offset by phosphorylation of DAG by DAG kinase in pool 2 (Endoplasmic Reticulum Nucleus) to aid in the PI replacement pathway(Endoplasmic Reticulum Nucleus) to aid in the PI replacement pathway

bull Second wave of DAG is a result of resynthesis of PIPSecond wave of DAG is a result of resynthesis of PIP22 which is then which is then hydrolyzed to form DAG and IPhydrolyzed to form DAG and IP33

PI PIP PIP2 DAG

IP3

IP2IP

Ins

PI

CDP-DAGPADAG

PA

PCPE

PCPE

PITP

PI4K PI5K PLC DGK

DGK CDSPLC PIS

POOL 2

POOL 1

IP4

PI PIP PIP2 DAG

IP3

IP2IP

Ins

PI

CDP-DAGPADAG

PA

PCPE

PC

PITP

PI4K PI5K PLC

DGK CDSPLC PIS

IP4

LPP

LPP

Model Equations DAG (pool 1)Model Equations DAG (pool 1)Model Equations DAG (pool 1)Model Equations DAG (pool 1)

bull DAG (pool 1)DAG (pool 1) (separate ODEs for each species (separate ODEs for each species considered)considered)

bull Concentration of DAG from pool 1 at time tConcentration of DAG from pool 1 at time t

Nondimensionalized equationsNondimensionalized equations

Model Equations DAG (pool 2)Model Equations DAG (pool 2)Model Equations DAG (pool 2)Model Equations DAG (pool 2)

bull DAG (pool 2)DAG (pool 2) (separate ODEs for each species (separate ODEs for each species considered)considered)

bull Concentration of pool 2 DAG molecules at time tConcentration of pool 2 DAG molecules at time t

bull Baseline concentration of pool 2 DAGBaseline concentration of pool 2 DAG

Nondimensionalized equationsNondimensionalized equations

Simplified CaSimplified Ca2+2+ Equations EquationsSimplified CaSimplified Ca2+2+ Equations Equations

bull CaCa2+2+ modulemodule (to match experimental AfCS (to match experimental AfCS trace)trace)

0 100 200 300 400 500 6000

002

004

006

008

01

012

014

Concentration of Ca

2+ (uM)

Ca2+ response to 25uM UDP

time (s)

wherewhere

Theoretical AnalysisTheoretical AnalysisTheoretical AnalysisTheoretical Analysis

bull Existence and UniquenessExistence and Uniqueness

bull Positivity and Boundedness Positivity and Boundedness

(for biological relevance)(for biological relevance)

bull Analysis of steady state behaviorAnalysis of steady state behavior

Existence of Solutions (full model)Existence of Solutions (full model)Existence of Solutions (full model)Existence of Solutions (full model)

First we write our system of ODEs in the formFirst we write our system of ODEs in the form (1)

(2)

(1)

Uniqueness of SolutionsUniqueness of SolutionsUniqueness of SolutionsUniqueness of Solutions

Next we denote a solution of Next we denote a solution of (1)(1) by by with initial condition with initial condition

Since our system satisfies the hypotheses of Theorems 1 and 2 Since our system satisfies the hypotheses of Theorems 1 and 2 on our set of interest (for all positive time and on a positive on our set of interest (for all positive time and on a positive bounded set in space) we know bounded set in space) we know there exists a unique there exists a unique (local) solution(local) solution (ie on some finite time interval possibly (ie on some finite time interval possibly small)small)

QuestionQuestion Do the solutions remain positive and bounded (for Do the solutions remain positive and bounded (for biological significance)biological significance)

(3)

(1) (3)

Positivity and boundedness of Positivity and boundedness of solutionssolutions

Positivity and boundedness of Positivity and boundedness of solutionssolutions

bull We first use Theorem 3 to show positivity and boundedness of xWe first use Theorem 3 to show positivity and boundedness of x11 and and xx22

bull Next we use these results and the Fundamental Theorem of Calculus to Next we use these results and the Fundamental Theorem of Calculus to show positivity and boundedness for the remaining equations show positivity and boundedness for the remaining equations

bull This then ensures a global solutionThis then ensures a global solution

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical Analysis

bull Parameter Estimation in SIMULINKParameter Estimation in SIMULINK

bull Sensitivity AnalysisSensitivity Analysis

Parameter Estimation in SIMULINK (for Parameter Estimation in SIMULINK (for FULL model)FULL model)

Parameter Estimation in SIMULINK (for Parameter Estimation in SIMULINK (for FULL model)FULL model)

bull Total number of model parameters (for full model with Li and Rinzel CaTotal number of model parameters (for full model with Li and Rinzel Ca2+2+ module) = module) = 3434ndash From the literature = From the literature = 2020ndash Estimated = Estimated = 1414

bull Receptor moduleReceptor module Total = Total = 77ndash From literature = From literature = 66ndash Estimated = Estimated = 11 (k (kpp rate of receptor phosphorylation) rate of receptor phosphorylation)

bull G-protein cascadeG-protein cascade Total = Total = 99ndash From literature = From literature = 66ndash Estimated = Estimated = 3 3 (k(khydhyd k kreprep k kd3d3))

bull DAG kineticsDAG kinetics Total = Total = 5 5 (for each DAG species considered)(for each DAG species considered)ndash From literature = From literature = 00ndash Estimated = Estimated = 5 5 (all DAG parameters)(all DAG parameters)

bull CaCa2+2+ module module Total = Total = 1313ndash From literature = From literature = 88ndash Estimated = Estimated = 55

SIMULINK DetailsSIMULINK DetailsSIMULINK DetailsSIMULINK Details

bull Unknown rate parameters were estimated using SIMULINK

ndash Minimizes a user-specified cost function via a user-specified optimization method

ndash Nonlinear least squares optimization method of Levenberg-Marquardt was used to minimize a sum of squared errors cost function of the empirical observations and model predictions for IP3 Ca2+ and multiple species of DAG

bull Note Although the Gauss-Newton method is often more efficient the method of Levenberg-Marquardt has proved to be more robust

Measured vs Simulated 384 DAG and Measured vs Simulated 384 DAG and 341 DAG341 DAG

Measured vs Simulated 384 DAG and Measured vs Simulated 384 DAG and 341 DAG341 DAG

bull Timecourse of stimulation with 25 Timecourse of stimulation with 25 M UDP in RAW 2647 cellsM UDP in RAW 2647 cells

bull Solid black lines represent model simulations Solid black lines represent model simulations

bull ((aa) 384 DAG response (representative of the response of most poly ) 384 DAG response (representative of the response of most poly unsaturated fatty acid-containing DAG species) unsaturated fatty acid-containing DAG species)

bull ((bb) 341 DAG response (representative of the response of most mono- ) 341 DAG response (representative of the response of most mono- and di-unsaturated fatty acid-containing DAG species) and di-unsaturated fatty acid-containing DAG species)

bull Data points contain nine replicates performed on three different Data points contain nine replicates performed on three different experimental days with error bars = 1 SEM Units are total change in ng experimental days with error bars = 1 SEM Units are total change in ng over baseline levels in ~8x10over baseline levels in ~8x1066 cells cells

Measured vs Simulated IPMeasured vs Simulated IP33 and Ca and Ca2+2+Measured vs Simulated IPMeasured vs Simulated IP33 and Ca and Ca2+2+

bull Timecourse of stimulation with 25 Timecourse of stimulation with 25 M UDP in RAW 2647 cellsM UDP in RAW 2647 cells

bull Solid black lines represent model simulationsSolid black lines represent model simulations

bull ((cc) IP) IP33 response in pmols per ~35x10 response in pmols per ~35x1055 cells cells

bull Points in (c) represent the average of four experiments and error Points in (c) represent the average of four experiments and error bars are 1 SEM bars are 1 SEM

bull ((dd) Ca) Ca2+2+ response in response in M Red curve is a representative CaM Red curve is a representative Ca2+2+ trace trace taken from the UDP experiments within the AfCS single ligand taken from the UDP experiments within the AfCS single ligand screenscreen

Model Simulations P2YModel Simulations P2Y66 G GGTP GTP PIPPIP22

Model Simulations P2YModel Simulations P2Y66 G GGTP GTP PIPPIP22

bull ((aa) = total P2Y) = total P2Y66 activated (solid activated (solid line) and inactivated (dashed line) and inactivated (dashed line) surface receptorsline) surface receptors

bull ((bb) = total G) = total GGTP GTP

bull ((cc) = total PIP) = total PIP22 available for available for hydrolysishydrolysis

P2Y6 from 25M UDP GGTP from 25M UDP

PIP2 from 25M UDP

Sensitivity AnalysisSensitivity AnalysisSensitivity AnalysisSensitivity Analysis

bull Sensitivity analysis techniques are valuable tools Sensitivity analysis techniques are valuable tools designed to answer questions regarding which of the designed to answer questions regarding which of the uncertain input variables is more important in uncertain input variables is more important in determining the uncertainty in our output determining the uncertainty in our output

bull Likewise sensitivity analysis can provide insight into Likewise sensitivity analysis can provide insight into which parameter should be studied in more detail in which parameter should be studied in more detail in order to reduce the most variance in the model output order to reduce the most variance in the model output

bull The ability to answer these types of questions could The ability to answer these types of questions could lead to important insight into the design of new lead to important insight into the design of new experiments and in determining which experiments experiments and in determining which experiments would give us the most valuable informationwould give us the most valuable information

Sampling MethodSampling MethodSampling MethodSampling Method

bull Generate a random sample of our space of input Generate a random sample of our space of input variables over a ten percent variation from each variables over a ten percent variation from each parameters nominal value using the parameters nominal value using the Latin Hypercube Latin Hypercube Sampling (LHS) methodSampling (LHS) method

bull Uses Uses Standardized Regression Coefficients (SRCs) Standardized Regression Coefficients (SRCs) obtained by performing multiple linear regression obtained by performing multiple linear regression analysisanalysisndash offers a measure of sensitivity that is multi-dimensionally offers a measure of sensitivity that is multi-dimensionally

averaged over the entire space of parameter values averaged over the entire space of parameter values ndash SRCs give insight into degree of nonlinearity in the model SRCs give insight into degree of nonlinearity in the model

Computing RComputing Ryy22 values valuesComputing RComputing Ryy22 values values

bull SRCs are only reliable measures of sensitivity when SRCs are only reliable measures of sensitivity when degree of nonlinearity is ldquosmallrdquodegree of nonlinearity is ldquosmallrdquo

bull Use model coefficients of determination RUse model coefficients of determination Ryy22 given by given by

bull where ywhere yii is the estimate of y is the estimate of yii obtained from the regression obtained from the regression modelmodel

bull RRyy22 ge 07 ensures SRCs are good sensitivity measures ge 07 ensures SRCs are good sensitivity measures

SRCs (as sensitivity measures) for SRCs (as sensitivity measures) for DAGDAG

SRCs (as sensitivity measures) for SRCs (as sensitivity measures) for DAGDAG

SRCs (as sensitivity measures) for IPSRCs (as sensitivity measures) for IP33SRCs (as sensitivity measures) for IPSRCs (as sensitivity measures) for IP33

Effects of 3 most sensitive DAG Effects of 3 most sensitive DAG parametersparameters

Effects of 3 most sensitive DAG Effects of 3 most sensitive DAG parametersparameters

A B

C

AA k kdp2dp2 degradation of pool 2 DAG degradation of pool 2 DAG

BB k kdp1dp1 degradation of pool 1 DAG degradation of pool 1 DAG

CC k kap2ap2 production of pool 2 DAG production of pool 2 DAG

ConclusionsConclusionsConclusionsConclusions

bull We have developed a model of the UDP signaling We have developed a model of the UDP signaling pathway in RAW 2647 macrophages which can predict pathway in RAW 2647 macrophages which can predict the responses of multiple species of DAG as well as the the responses of multiple species of DAG as well as the responses of IPresponses of IP33 Ca Ca2+2+ receptor dynamics G-protein receptor dynamics G-protein activation and PIPactivation and PIP22 hydrolysis hydrolysis

bull Simplified model resultsSimplified model resultsndash We have obtained global existence uniqueness positivity We have obtained global existence uniqueness positivity

and boundedness of solutionsand boundedness of solutionsndash We have proven global stability of a unique steady state We have proven global stability of a unique steady state

within our region of interestwithin our region of interestbull Full model analysisFull model analysis

ndash Using SIMULINK we have estimated unknown rate Using SIMULINK we have estimated unknown rate parameters to obtain best fits to multiple DAG traces IPparameters to obtain best fits to multiple DAG traces IP33 and Caand Ca2+2+ all in response to 25 all in response to 25M UDPM UDP

ndash We have performed sensitivity analysis using the Latin We have performed sensitivity analysis using the Latin Hypercube sampling technique in combination with Hypercube sampling technique in combination with standardized regression coefficients to determine which standardized regression coefficients to determine which model parameters are responsible for most of the model model parameters are responsible for most of the model output uncertaintyoutput uncertainty

Future DirectionsFuture DirectionsFuture DirectionsFuture Directions

bull We have conducted multiple experiments to We have conducted multiple experiments to pharmacologically inhibit several different enzymes we pharmacologically inhibit several different enzymes we believe to play a role in this signaling pathway as believe to play a role in this signaling pathway as suggested by current known metabolic pathways and by suggested by current known metabolic pathways and by modeling resultsmodeling results

bull The next step is to perform gene knockdowns on The next step is to perform gene knockdowns on specific enzymes to verify results of inhibitor dataspecific enzymes to verify results of inhibitor data

bull The model output also suggests a time delay from The model output also suggests a time delay from receptor activation to PIPreceptor activation to PIP22 hydrolysis An upcoming task hydrolysis An upcoming task is to investigate the outcomes of adding such a delay is to investigate the outcomes of adding such a delay termterm

bull The model could be greatly enhanced by incorporating The model could be greatly enhanced by incorporating spatial dynamics so this will also be a major focus for spatial dynamics so this will also be a major focus for future researchfuture research

AcknowledgementsAcknowledgementsAcknowledgementsAcknowledgements

Mathematics DepartmentMathematics Departmentbull Hannah L Callender Hannah L Callender

PhDPhD

CollaboratorsCollaboratorsbull Alliance for Cellular SignalingAlliance for Cellular Signalingbull UT SouthwesternUT Southwestern

ndash Paul Sternweis PhDPaul Sternweis PhDndash Dianne DeCamp PhDDianne DeCamp PhD

Brown LabBrown Labbull H Alex Brown PhDH Alex Brown PhDbull Jeffrey S Forrester PhDJeffrey S Forrester PhDbull Mark Byrne PhDMark Byrne PhDbull Anita Preininger PhDAnita Preininger PhDbull Michelle ArmstrongMichelle Armstrongbull Andrew GoodmanAndrew Goodmanbull Pavlina Ivanova PhDPavlina Ivanova PhDbull Steve Milne PhDSteve Milne PhD

Timeseries data for the modelTimeseries data for the modelTimeseries data for the modelTimeseries data for the model

bull Data from collaborators at UTSW (Sternweis Lab)Data from collaborators at UTSW (Sternweis Lab)ndash Inositol 145-trisphosphate (Inositol 145-trisphosphate (IPIP33) production post 25) production post 25M M

UDP stimulation in the RAW cellUDP stimulation in the RAW cellndash Cytosolic calcium (Cytosolic calcium (CaCa2+2+) release post stimulation with ) release post stimulation with

2525M UDP in the RAW cellM UDP in the RAW cell

bull Diacylglycerol (Diacylglycerol (DAGDAG) data H Callender collected from a ) data H Callender collected from a novel method of quantitative analysis of multiple novel method of quantitative analysis of multiple species of DAG developed in the Brown Lab species of DAG developed in the Brown Lab

Reference Reference

Callender H L et al Quantification of Diacylglycerol Species from Callender H L et al Quantification of Diacylglycerol Species from Cellular Extracts by Electrospray Ionization Mass Spectrometry Using a Cellular Extracts by Electrospray Ionization Mass Spectrometry Using a Linear Regression Algorithm Linear Regression Algorithm Anal Chem Anal Chem 7979 (2007) 263-272 (2007) 263-272

IPIP33 and Ca and Ca2+2+ data from AfCS data from AfCSIPIP33 and Ca and Ca2+2+ data from AfCS data from AfCS

AA

bull IPIP33 response (in pmoles per response (in pmoles per 100 100 L lysate)in RAW 2647 L lysate)in RAW 2647 cells to 25 cells to 25 M UDPM UDP

bull Points represent the average Points represent the average of four experimentsof four experiments

bull Cytosolic calcium response Cytosolic calcium response (in (in M) in the RAW 2647 M) in the RAW 2647 cell to 25 cell to 25 M UDP M UDP

bull The graph displays 43 The graph displays 43 experimental repeatsexperimental repeats

B

What is DAGWhat is DAGWhat is DAGWhat is DAG

bull DAG is a cellular second messenger molecule which plays an DAG is a cellular second messenger molecule which plays an important role in initiating various changes in cell behavior important role in initiating various changes in cell behavior including cell activation differentiation proliferation and including cell activation differentiation proliferation and tumor promotion tumor promotion

bull There are many different species of DAG depending on the There are many different species of DAG depending on the number of carbons and number of double bonds in the fatty number of carbons and number of double bonds in the fatty acyl chains and different species can have different cellular acyl chains and different species can have different cellular functionsfunctions

bull ExampleExample 320 DAG320 DAG

Two fatty acyl Two fatty acyl (hydrocarbon) (hydrocarbon) chains in every DAG chains in every DAG speciesspecies

Each corner Each corner represents carbonrepresents carbon

320 DAG 18 carbons in first chain 14 in the second no double bonds 320 DAG 18 carbons in first chain 14 in the second no double bonds in either chainin either chain

Kinetics of monodi unsaturated DAGsKinetics of monodi unsaturated DAGsKinetics of monodi unsaturated DAGsKinetics of monodi unsaturated DAGs

bull Time based behavior of four monodi unsaturated DAG species after addition of 25 Time based behavior of four monodi unsaturated DAG species after addition of 25 M M (solid red squares) and 025 (solid red squares) and 025 M (solid green triangles) UDPM (solid green triangles) UDP

bull Time points contain a minimum of nine replicates performed on three different Time points contain a minimum of nine replicates performed on three different experimental daysexperimental days

- 25 M UDP

- 025M UDP

0 5 10 15 20 25 30

0 5 10 15 20 25 30 0 5 10 15 20 25 30

0 5 10 15 20 25 30

321 DAG 341 DAG

362 DAG341 DAGep

150

100

50-5

00

300

200

100

-100

0

150

100

50-5

00

-100

200

6040

20-2

00

-40

80

Time (min) Time (min)

Time (min) Time (min)

D

AG

(n

g)

D

AG

(n

g)

D

AG

(n

g)

D

AG

(n

g)

- 25 M UDP

- 025M UDP

- 25 M UDP

- 025M UDP

- 25 M UDP

- 025M UDP

Kinetics of PUFA containing DAGsKinetics of PUFA containing DAGsKinetics of PUFA containing DAGsKinetics of PUFA containing DAGsbull Time based behavior of four polyunsaturated fatty acid (PUFA) containing DAG species Time based behavior of four polyunsaturated fatty acid (PUFA) containing DAG species

after addition of 25 after addition of 25 M (solid red squares) and 025 M (solid red squares) and 025 M (solid green triangles) UDPM (solid green triangles) UDPbull Time points contain a minimum of nine replicates performed on three different Time points contain a minimum of nine replicates performed on three different

experimental daysexperimental daysbull Monodi unsaturated DAG species give a larger increase than polyunsaturated fatty acid Monodi unsaturated DAG species give a larger increase than polyunsaturated fatty acid

(PUFA) containing DAGs(PUFA) containing DAGs

0 5 10 15 20 25 30

0 5 10 15 20 25 30 0 5 10 15 20 25 30

0 5 10 15 20 25 30

385 DAG 383 DAG

364 DAG384 DAG6

42

-40

30

20

10

0

15

10

50

-56

40

2-2

8

Time (min) Time (min)

Time (min) Time (min)

D

AG

(n

g)

D

AG

(n

g)

D

AG

(n

g)

D

AG

(n

g)

-28

40

- 25 M UDP

- 025M UDP

- 25 M UDP

- 025M UDP

- 25 M UDP

- 025M UDP

- 25 M UDP

- 025M UDP

Differential DAG KineticsDifferential DAG KineticsDifferential DAG KineticsDifferential DAG Kinetics

bull Time based behavior of three DAG species with varying degrees of unsaturation after Time based behavior of three DAG species with varying degrees of unsaturation after addition of addition of 25 25 MM (solid red triangles) and (solid red triangles) and 025 025 MM (solid green squares) UDP (solid green squares) UDP

bull Time points contain nine replicates performed on three different experimental daysTime points contain nine replicates performed on three different experimental days

341 DAG

Best fit with current model structure

Overall Objectives of Modeling EffortOverall Objectives of Modeling EffortOverall Objectives of Modeling EffortOverall Objectives of Modeling Effort

bull Predict quantitative changes in lipid species after stimulation by Predict quantitative changes in lipid species after stimulation by various ligands and ligand concentrations in the RAW 2647 various ligands and ligand concentrations in the RAW 2647 macrophagemacrophage

bull Comparison and refinement of model output with AfCS IPComparison and refinement of model output with AfCS IP33 measurements and Cameasurements and Ca2+2+ traces as well as DAG data generated in traces as well as DAG data generated in the Brown labthe Brown lab

bull Predict in silico effects such as the effect of knock-downs etc on Predict in silico effects such as the effect of knock-downs etc on given pathwaygiven pathway

bull Suggest modifications to current pathway structuresSuggest modifications to current pathway structures

Modifications to the modelModifications to the modelModifications to the modelModifications to the model

bull Include an additional branch in the pathway Include an additional branch in the pathway for a second pool of DAGfor a second pool of DAG

bull Simplify CaSimplify Ca2+2+ equations for mathematical equations for mathematical analysis purposesanalysis purposes

Proposed PathwayProposed PathwayProposed PathwayProposed Pathway

bull Note We measure total cellular DAG levelsNote We measure total cellular DAG levelsbull Initial production of DAG from the hydrolysis of PIPInitial production of DAG from the hydrolysis of PIP22 in pool 1 (plasma in pool 1 (plasma

membrane) is offset by phosphorylation of DAG by DAG kinase in pool 2 membrane) is offset by phosphorylation of DAG by DAG kinase in pool 2 (Endoplasmic Reticulum Nucleus) to aid in the PI replacement pathway(Endoplasmic Reticulum Nucleus) to aid in the PI replacement pathway

bull Second wave of DAG is a result of resynthesis of PIPSecond wave of DAG is a result of resynthesis of PIP22 which is then which is then hydrolyzed to form DAG and IPhydrolyzed to form DAG and IP33

PI PIP PIP2 DAG

IP3

IP2IP

Ins

PI

CDP-DAGPADAG

PA

PCPE

PCPE

PITP

PI4K PI5K PLC DGK

DGK CDSPLC PIS

POOL 2

POOL 1

IP4

PI PIP PIP2 DAG

IP3

IP2IP

Ins

PI

CDP-DAGPADAG

PA

PCPE

PC

PITP

PI4K PI5K PLC

DGK CDSPLC PIS

IP4

LPP

LPP

Model Equations DAG (pool 1)Model Equations DAG (pool 1)Model Equations DAG (pool 1)Model Equations DAG (pool 1)

bull DAG (pool 1)DAG (pool 1) (separate ODEs for each species (separate ODEs for each species considered)considered)

bull Concentration of DAG from pool 1 at time tConcentration of DAG from pool 1 at time t

Nondimensionalized equationsNondimensionalized equations

Model Equations DAG (pool 2)Model Equations DAG (pool 2)Model Equations DAG (pool 2)Model Equations DAG (pool 2)

bull DAG (pool 2)DAG (pool 2) (separate ODEs for each species (separate ODEs for each species considered)considered)

bull Concentration of pool 2 DAG molecules at time tConcentration of pool 2 DAG molecules at time t

bull Baseline concentration of pool 2 DAGBaseline concentration of pool 2 DAG

Nondimensionalized equationsNondimensionalized equations

Simplified CaSimplified Ca2+2+ Equations EquationsSimplified CaSimplified Ca2+2+ Equations Equations

bull CaCa2+2+ modulemodule (to match experimental AfCS (to match experimental AfCS trace)trace)

0 100 200 300 400 500 6000

002

004

006

008

01

012

014

Concentration of Ca

2+ (uM)

Ca2+ response to 25uM UDP

time (s)

wherewhere

Theoretical AnalysisTheoretical AnalysisTheoretical AnalysisTheoretical Analysis

bull Existence and UniquenessExistence and Uniqueness

bull Positivity and Boundedness Positivity and Boundedness

(for biological relevance)(for biological relevance)

bull Analysis of steady state behaviorAnalysis of steady state behavior

Existence of Solutions (full model)Existence of Solutions (full model)Existence of Solutions (full model)Existence of Solutions (full model)

First we write our system of ODEs in the formFirst we write our system of ODEs in the form (1)

(2)

(1)

Uniqueness of SolutionsUniqueness of SolutionsUniqueness of SolutionsUniqueness of Solutions

Next we denote a solution of Next we denote a solution of (1)(1) by by with initial condition with initial condition

Since our system satisfies the hypotheses of Theorems 1 and 2 Since our system satisfies the hypotheses of Theorems 1 and 2 on our set of interest (for all positive time and on a positive on our set of interest (for all positive time and on a positive bounded set in space) we know bounded set in space) we know there exists a unique there exists a unique (local) solution(local) solution (ie on some finite time interval possibly (ie on some finite time interval possibly small)small)

QuestionQuestion Do the solutions remain positive and bounded (for Do the solutions remain positive and bounded (for biological significance)biological significance)

(3)

(1) (3)

Positivity and boundedness of Positivity and boundedness of solutionssolutions

Positivity and boundedness of Positivity and boundedness of solutionssolutions

bull We first use Theorem 3 to show positivity and boundedness of xWe first use Theorem 3 to show positivity and boundedness of x11 and and xx22

bull Next we use these results and the Fundamental Theorem of Calculus to Next we use these results and the Fundamental Theorem of Calculus to show positivity and boundedness for the remaining equations show positivity and boundedness for the remaining equations

bull This then ensures a global solutionThis then ensures a global solution

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical Analysis

bull Parameter Estimation in SIMULINKParameter Estimation in SIMULINK

bull Sensitivity AnalysisSensitivity Analysis

Parameter Estimation in SIMULINK (for Parameter Estimation in SIMULINK (for FULL model)FULL model)

Parameter Estimation in SIMULINK (for Parameter Estimation in SIMULINK (for FULL model)FULL model)

bull Total number of model parameters (for full model with Li and Rinzel CaTotal number of model parameters (for full model with Li and Rinzel Ca2+2+ module) = module) = 3434ndash From the literature = From the literature = 2020ndash Estimated = Estimated = 1414

bull Receptor moduleReceptor module Total = Total = 77ndash From literature = From literature = 66ndash Estimated = Estimated = 11 (k (kpp rate of receptor phosphorylation) rate of receptor phosphorylation)

bull G-protein cascadeG-protein cascade Total = Total = 99ndash From literature = From literature = 66ndash Estimated = Estimated = 3 3 (k(khydhyd k kreprep k kd3d3))

bull DAG kineticsDAG kinetics Total = Total = 5 5 (for each DAG species considered)(for each DAG species considered)ndash From literature = From literature = 00ndash Estimated = Estimated = 5 5 (all DAG parameters)(all DAG parameters)

bull CaCa2+2+ module module Total = Total = 1313ndash From literature = From literature = 88ndash Estimated = Estimated = 55

SIMULINK DetailsSIMULINK DetailsSIMULINK DetailsSIMULINK Details

bull Unknown rate parameters were estimated using SIMULINK

ndash Minimizes a user-specified cost function via a user-specified optimization method

ndash Nonlinear least squares optimization method of Levenberg-Marquardt was used to minimize a sum of squared errors cost function of the empirical observations and model predictions for IP3 Ca2+ and multiple species of DAG

bull Note Although the Gauss-Newton method is often more efficient the method of Levenberg-Marquardt has proved to be more robust

Measured vs Simulated 384 DAG and Measured vs Simulated 384 DAG and 341 DAG341 DAG

Measured vs Simulated 384 DAG and Measured vs Simulated 384 DAG and 341 DAG341 DAG

bull Timecourse of stimulation with 25 Timecourse of stimulation with 25 M UDP in RAW 2647 cellsM UDP in RAW 2647 cells

bull Solid black lines represent model simulations Solid black lines represent model simulations

bull ((aa) 384 DAG response (representative of the response of most poly ) 384 DAG response (representative of the response of most poly unsaturated fatty acid-containing DAG species) unsaturated fatty acid-containing DAG species)

bull ((bb) 341 DAG response (representative of the response of most mono- ) 341 DAG response (representative of the response of most mono- and di-unsaturated fatty acid-containing DAG species) and di-unsaturated fatty acid-containing DAG species)

bull Data points contain nine replicates performed on three different Data points contain nine replicates performed on three different experimental days with error bars = 1 SEM Units are total change in ng experimental days with error bars = 1 SEM Units are total change in ng over baseline levels in ~8x10over baseline levels in ~8x1066 cells cells

Measured vs Simulated IPMeasured vs Simulated IP33 and Ca and Ca2+2+Measured vs Simulated IPMeasured vs Simulated IP33 and Ca and Ca2+2+

bull Timecourse of stimulation with 25 Timecourse of stimulation with 25 M UDP in RAW 2647 cellsM UDP in RAW 2647 cells

bull Solid black lines represent model simulationsSolid black lines represent model simulations

bull ((cc) IP) IP33 response in pmols per ~35x10 response in pmols per ~35x1055 cells cells

bull Points in (c) represent the average of four experiments and error Points in (c) represent the average of four experiments and error bars are 1 SEM bars are 1 SEM

bull ((dd) Ca) Ca2+2+ response in response in M Red curve is a representative CaM Red curve is a representative Ca2+2+ trace trace taken from the UDP experiments within the AfCS single ligand taken from the UDP experiments within the AfCS single ligand screenscreen

Model Simulations P2YModel Simulations P2Y66 G GGTP GTP PIPPIP22

Model Simulations P2YModel Simulations P2Y66 G GGTP GTP PIPPIP22

bull ((aa) = total P2Y) = total P2Y66 activated (solid activated (solid line) and inactivated (dashed line) and inactivated (dashed line) surface receptorsline) surface receptors

bull ((bb) = total G) = total GGTP GTP

bull ((cc) = total PIP) = total PIP22 available for available for hydrolysishydrolysis

P2Y6 from 25M UDP GGTP from 25M UDP

PIP2 from 25M UDP

Sensitivity AnalysisSensitivity AnalysisSensitivity AnalysisSensitivity Analysis

bull Sensitivity analysis techniques are valuable tools Sensitivity analysis techniques are valuable tools designed to answer questions regarding which of the designed to answer questions regarding which of the uncertain input variables is more important in uncertain input variables is more important in determining the uncertainty in our output determining the uncertainty in our output

bull Likewise sensitivity analysis can provide insight into Likewise sensitivity analysis can provide insight into which parameter should be studied in more detail in which parameter should be studied in more detail in order to reduce the most variance in the model output order to reduce the most variance in the model output

bull The ability to answer these types of questions could The ability to answer these types of questions could lead to important insight into the design of new lead to important insight into the design of new experiments and in determining which experiments experiments and in determining which experiments would give us the most valuable informationwould give us the most valuable information

Sampling MethodSampling MethodSampling MethodSampling Method

bull Generate a random sample of our space of input Generate a random sample of our space of input variables over a ten percent variation from each variables over a ten percent variation from each parameters nominal value using the parameters nominal value using the Latin Hypercube Latin Hypercube Sampling (LHS) methodSampling (LHS) method

bull Uses Uses Standardized Regression Coefficients (SRCs) Standardized Regression Coefficients (SRCs) obtained by performing multiple linear regression obtained by performing multiple linear regression analysisanalysisndash offers a measure of sensitivity that is multi-dimensionally offers a measure of sensitivity that is multi-dimensionally

averaged over the entire space of parameter values averaged over the entire space of parameter values ndash SRCs give insight into degree of nonlinearity in the model SRCs give insight into degree of nonlinearity in the model

Computing RComputing Ryy22 values valuesComputing RComputing Ryy22 values values

bull SRCs are only reliable measures of sensitivity when SRCs are only reliable measures of sensitivity when degree of nonlinearity is ldquosmallrdquodegree of nonlinearity is ldquosmallrdquo

bull Use model coefficients of determination RUse model coefficients of determination Ryy22 given by given by

bull where ywhere yii is the estimate of y is the estimate of yii obtained from the regression obtained from the regression modelmodel

bull RRyy22 ge 07 ensures SRCs are good sensitivity measures ge 07 ensures SRCs are good sensitivity measures

SRCs (as sensitivity measures) for SRCs (as sensitivity measures) for DAGDAG

SRCs (as sensitivity measures) for SRCs (as sensitivity measures) for DAGDAG

SRCs (as sensitivity measures) for IPSRCs (as sensitivity measures) for IP33SRCs (as sensitivity measures) for IPSRCs (as sensitivity measures) for IP33

Effects of 3 most sensitive DAG Effects of 3 most sensitive DAG parametersparameters

Effects of 3 most sensitive DAG Effects of 3 most sensitive DAG parametersparameters

A B

C

AA k kdp2dp2 degradation of pool 2 DAG degradation of pool 2 DAG

BB k kdp1dp1 degradation of pool 1 DAG degradation of pool 1 DAG

CC k kap2ap2 production of pool 2 DAG production of pool 2 DAG

ConclusionsConclusionsConclusionsConclusions

bull We have developed a model of the UDP signaling We have developed a model of the UDP signaling pathway in RAW 2647 macrophages which can predict pathway in RAW 2647 macrophages which can predict the responses of multiple species of DAG as well as the the responses of multiple species of DAG as well as the responses of IPresponses of IP33 Ca Ca2+2+ receptor dynamics G-protein receptor dynamics G-protein activation and PIPactivation and PIP22 hydrolysis hydrolysis

bull Simplified model resultsSimplified model resultsndash We have obtained global existence uniqueness positivity We have obtained global existence uniqueness positivity

and boundedness of solutionsand boundedness of solutionsndash We have proven global stability of a unique steady state We have proven global stability of a unique steady state

within our region of interestwithin our region of interestbull Full model analysisFull model analysis

ndash Using SIMULINK we have estimated unknown rate Using SIMULINK we have estimated unknown rate parameters to obtain best fits to multiple DAG traces IPparameters to obtain best fits to multiple DAG traces IP33 and Caand Ca2+2+ all in response to 25 all in response to 25M UDPM UDP

ndash We have performed sensitivity analysis using the Latin We have performed sensitivity analysis using the Latin Hypercube sampling technique in combination with Hypercube sampling technique in combination with standardized regression coefficients to determine which standardized regression coefficients to determine which model parameters are responsible for most of the model model parameters are responsible for most of the model output uncertaintyoutput uncertainty

Future DirectionsFuture DirectionsFuture DirectionsFuture Directions

bull We have conducted multiple experiments to We have conducted multiple experiments to pharmacologically inhibit several different enzymes we pharmacologically inhibit several different enzymes we believe to play a role in this signaling pathway as believe to play a role in this signaling pathway as suggested by current known metabolic pathways and by suggested by current known metabolic pathways and by modeling resultsmodeling results

bull The next step is to perform gene knockdowns on The next step is to perform gene knockdowns on specific enzymes to verify results of inhibitor dataspecific enzymes to verify results of inhibitor data

bull The model output also suggests a time delay from The model output also suggests a time delay from receptor activation to PIPreceptor activation to PIP22 hydrolysis An upcoming task hydrolysis An upcoming task is to investigate the outcomes of adding such a delay is to investigate the outcomes of adding such a delay termterm

bull The model could be greatly enhanced by incorporating The model could be greatly enhanced by incorporating spatial dynamics so this will also be a major focus for spatial dynamics so this will also be a major focus for future researchfuture research

AcknowledgementsAcknowledgementsAcknowledgementsAcknowledgements

Mathematics DepartmentMathematics Departmentbull Hannah L Callender Hannah L Callender

PhDPhD

CollaboratorsCollaboratorsbull Alliance for Cellular SignalingAlliance for Cellular Signalingbull UT SouthwesternUT Southwestern

ndash Paul Sternweis PhDPaul Sternweis PhDndash Dianne DeCamp PhDDianne DeCamp PhD

Brown LabBrown Labbull H Alex Brown PhDH Alex Brown PhDbull Jeffrey S Forrester PhDJeffrey S Forrester PhDbull Mark Byrne PhDMark Byrne PhDbull Anita Preininger PhDAnita Preininger PhDbull Michelle ArmstrongMichelle Armstrongbull Andrew GoodmanAndrew Goodmanbull Pavlina Ivanova PhDPavlina Ivanova PhDbull Steve Milne PhDSteve Milne PhD

IPIP33 and Ca and Ca2+2+ data from AfCS data from AfCSIPIP33 and Ca and Ca2+2+ data from AfCS data from AfCS

AA

bull IPIP33 response (in pmoles per response (in pmoles per 100 100 L lysate)in RAW 2647 L lysate)in RAW 2647 cells to 25 cells to 25 M UDPM UDP

bull Points represent the average Points represent the average of four experimentsof four experiments

bull Cytosolic calcium response Cytosolic calcium response (in (in M) in the RAW 2647 M) in the RAW 2647 cell to 25 cell to 25 M UDP M UDP

bull The graph displays 43 The graph displays 43 experimental repeatsexperimental repeats

B

What is DAGWhat is DAGWhat is DAGWhat is DAG

bull DAG is a cellular second messenger molecule which plays an DAG is a cellular second messenger molecule which plays an important role in initiating various changes in cell behavior important role in initiating various changes in cell behavior including cell activation differentiation proliferation and including cell activation differentiation proliferation and tumor promotion tumor promotion

bull There are many different species of DAG depending on the There are many different species of DAG depending on the number of carbons and number of double bonds in the fatty number of carbons and number of double bonds in the fatty acyl chains and different species can have different cellular acyl chains and different species can have different cellular functionsfunctions

bull ExampleExample 320 DAG320 DAG

Two fatty acyl Two fatty acyl (hydrocarbon) (hydrocarbon) chains in every DAG chains in every DAG speciesspecies

Each corner Each corner represents carbonrepresents carbon

320 DAG 18 carbons in first chain 14 in the second no double bonds 320 DAG 18 carbons in first chain 14 in the second no double bonds in either chainin either chain

Kinetics of monodi unsaturated DAGsKinetics of monodi unsaturated DAGsKinetics of monodi unsaturated DAGsKinetics of monodi unsaturated DAGs

bull Time based behavior of four monodi unsaturated DAG species after addition of 25 Time based behavior of four monodi unsaturated DAG species after addition of 25 M M (solid red squares) and 025 (solid red squares) and 025 M (solid green triangles) UDPM (solid green triangles) UDP

bull Time points contain a minimum of nine replicates performed on three different Time points contain a minimum of nine replicates performed on three different experimental daysexperimental days

- 25 M UDP

- 025M UDP

0 5 10 15 20 25 30

0 5 10 15 20 25 30 0 5 10 15 20 25 30

0 5 10 15 20 25 30

321 DAG 341 DAG

362 DAG341 DAGep

150

100

50-5

00

300

200

100

-100

0

150

100

50-5

00

-100

200

6040

20-2

00

-40

80

Time (min) Time (min)

Time (min) Time (min)

D

AG

(n

g)

D

AG

(n

g)

D

AG

(n

g)

D

AG

(n

g)

- 25 M UDP

- 025M UDP

- 25 M UDP

- 025M UDP

- 25 M UDP

- 025M UDP

Kinetics of PUFA containing DAGsKinetics of PUFA containing DAGsKinetics of PUFA containing DAGsKinetics of PUFA containing DAGsbull Time based behavior of four polyunsaturated fatty acid (PUFA) containing DAG species Time based behavior of four polyunsaturated fatty acid (PUFA) containing DAG species

after addition of 25 after addition of 25 M (solid red squares) and 025 M (solid red squares) and 025 M (solid green triangles) UDPM (solid green triangles) UDPbull Time points contain a minimum of nine replicates performed on three different Time points contain a minimum of nine replicates performed on three different

experimental daysexperimental daysbull Monodi unsaturated DAG species give a larger increase than polyunsaturated fatty acid Monodi unsaturated DAG species give a larger increase than polyunsaturated fatty acid

(PUFA) containing DAGs(PUFA) containing DAGs

0 5 10 15 20 25 30

0 5 10 15 20 25 30 0 5 10 15 20 25 30

0 5 10 15 20 25 30

385 DAG 383 DAG

364 DAG384 DAG6

42

-40

30

20

10

0

15

10

50

-56

40

2-2

8

Time (min) Time (min)

Time (min) Time (min)

D

AG

(n

g)

D

AG

(n

g)

D

AG

(n

g)

D

AG

(n

g)

-28

40

- 25 M UDP

- 025M UDP

- 25 M UDP

- 025M UDP

- 25 M UDP

- 025M UDP

- 25 M UDP

- 025M UDP

Differential DAG KineticsDifferential DAG KineticsDifferential DAG KineticsDifferential DAG Kinetics

bull Time based behavior of three DAG species with varying degrees of unsaturation after Time based behavior of three DAG species with varying degrees of unsaturation after addition of addition of 25 25 MM (solid red triangles) and (solid red triangles) and 025 025 MM (solid green squares) UDP (solid green squares) UDP

bull Time points contain nine replicates performed on three different experimental daysTime points contain nine replicates performed on three different experimental days

341 DAG

Best fit with current model structure

Overall Objectives of Modeling EffortOverall Objectives of Modeling EffortOverall Objectives of Modeling EffortOverall Objectives of Modeling Effort

bull Predict quantitative changes in lipid species after stimulation by Predict quantitative changes in lipid species after stimulation by various ligands and ligand concentrations in the RAW 2647 various ligands and ligand concentrations in the RAW 2647 macrophagemacrophage

bull Comparison and refinement of model output with AfCS IPComparison and refinement of model output with AfCS IP33 measurements and Cameasurements and Ca2+2+ traces as well as DAG data generated in traces as well as DAG data generated in the Brown labthe Brown lab

bull Predict in silico effects such as the effect of knock-downs etc on Predict in silico effects such as the effect of knock-downs etc on given pathwaygiven pathway

bull Suggest modifications to current pathway structuresSuggest modifications to current pathway structures

Modifications to the modelModifications to the modelModifications to the modelModifications to the model

bull Include an additional branch in the pathway Include an additional branch in the pathway for a second pool of DAGfor a second pool of DAG

bull Simplify CaSimplify Ca2+2+ equations for mathematical equations for mathematical analysis purposesanalysis purposes

Proposed PathwayProposed PathwayProposed PathwayProposed Pathway

bull Note We measure total cellular DAG levelsNote We measure total cellular DAG levelsbull Initial production of DAG from the hydrolysis of PIPInitial production of DAG from the hydrolysis of PIP22 in pool 1 (plasma in pool 1 (plasma

membrane) is offset by phosphorylation of DAG by DAG kinase in pool 2 membrane) is offset by phosphorylation of DAG by DAG kinase in pool 2 (Endoplasmic Reticulum Nucleus) to aid in the PI replacement pathway(Endoplasmic Reticulum Nucleus) to aid in the PI replacement pathway

bull Second wave of DAG is a result of resynthesis of PIPSecond wave of DAG is a result of resynthesis of PIP22 which is then which is then hydrolyzed to form DAG and IPhydrolyzed to form DAG and IP33

PI PIP PIP2 DAG

IP3

IP2IP

Ins

PI

CDP-DAGPADAG

PA

PCPE

PCPE

PITP

PI4K PI5K PLC DGK

DGK CDSPLC PIS

POOL 2

POOL 1

IP4

PI PIP PIP2 DAG

IP3

IP2IP

Ins

PI

CDP-DAGPADAG

PA

PCPE

PC

PITP

PI4K PI5K PLC

DGK CDSPLC PIS

IP4

LPP

LPP

Model Equations DAG (pool 1)Model Equations DAG (pool 1)Model Equations DAG (pool 1)Model Equations DAG (pool 1)

bull DAG (pool 1)DAG (pool 1) (separate ODEs for each species (separate ODEs for each species considered)considered)

bull Concentration of DAG from pool 1 at time tConcentration of DAG from pool 1 at time t

Nondimensionalized equationsNondimensionalized equations

Model Equations DAG (pool 2)Model Equations DAG (pool 2)Model Equations DAG (pool 2)Model Equations DAG (pool 2)

bull DAG (pool 2)DAG (pool 2) (separate ODEs for each species (separate ODEs for each species considered)considered)

bull Concentration of pool 2 DAG molecules at time tConcentration of pool 2 DAG molecules at time t

bull Baseline concentration of pool 2 DAGBaseline concentration of pool 2 DAG

Nondimensionalized equationsNondimensionalized equations

Simplified CaSimplified Ca2+2+ Equations EquationsSimplified CaSimplified Ca2+2+ Equations Equations

bull CaCa2+2+ modulemodule (to match experimental AfCS (to match experimental AfCS trace)trace)

0 100 200 300 400 500 6000

002

004

006

008

01

012

014

Concentration of Ca

2+ (uM)

Ca2+ response to 25uM UDP

time (s)

wherewhere

Theoretical AnalysisTheoretical AnalysisTheoretical AnalysisTheoretical Analysis

bull Existence and UniquenessExistence and Uniqueness

bull Positivity and Boundedness Positivity and Boundedness

(for biological relevance)(for biological relevance)

bull Analysis of steady state behaviorAnalysis of steady state behavior

Existence of Solutions (full model)Existence of Solutions (full model)Existence of Solutions (full model)Existence of Solutions (full model)

First we write our system of ODEs in the formFirst we write our system of ODEs in the form (1)

(2)

(1)

Uniqueness of SolutionsUniqueness of SolutionsUniqueness of SolutionsUniqueness of Solutions

Next we denote a solution of Next we denote a solution of (1)(1) by by with initial condition with initial condition

Since our system satisfies the hypotheses of Theorems 1 and 2 Since our system satisfies the hypotheses of Theorems 1 and 2 on our set of interest (for all positive time and on a positive on our set of interest (for all positive time and on a positive bounded set in space) we know bounded set in space) we know there exists a unique there exists a unique (local) solution(local) solution (ie on some finite time interval possibly (ie on some finite time interval possibly small)small)

QuestionQuestion Do the solutions remain positive and bounded (for Do the solutions remain positive and bounded (for biological significance)biological significance)

(3)

(1) (3)

Positivity and boundedness of Positivity and boundedness of solutionssolutions

Positivity and boundedness of Positivity and boundedness of solutionssolutions

bull We first use Theorem 3 to show positivity and boundedness of xWe first use Theorem 3 to show positivity and boundedness of x11 and and xx22

bull Next we use these results and the Fundamental Theorem of Calculus to Next we use these results and the Fundamental Theorem of Calculus to show positivity and boundedness for the remaining equations show positivity and boundedness for the remaining equations

bull This then ensures a global solutionThis then ensures a global solution

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical Analysis

bull Parameter Estimation in SIMULINKParameter Estimation in SIMULINK

bull Sensitivity AnalysisSensitivity Analysis

Parameter Estimation in SIMULINK (for Parameter Estimation in SIMULINK (for FULL model)FULL model)

Parameter Estimation in SIMULINK (for Parameter Estimation in SIMULINK (for FULL model)FULL model)

bull Total number of model parameters (for full model with Li and Rinzel CaTotal number of model parameters (for full model with Li and Rinzel Ca2+2+ module) = module) = 3434ndash From the literature = From the literature = 2020ndash Estimated = Estimated = 1414

bull Receptor moduleReceptor module Total = Total = 77ndash From literature = From literature = 66ndash Estimated = Estimated = 11 (k (kpp rate of receptor phosphorylation) rate of receptor phosphorylation)

bull G-protein cascadeG-protein cascade Total = Total = 99ndash From literature = From literature = 66ndash Estimated = Estimated = 3 3 (k(khydhyd k kreprep k kd3d3))

bull DAG kineticsDAG kinetics Total = Total = 5 5 (for each DAG species considered)(for each DAG species considered)ndash From literature = From literature = 00ndash Estimated = Estimated = 5 5 (all DAG parameters)(all DAG parameters)

bull CaCa2+2+ module module Total = Total = 1313ndash From literature = From literature = 88ndash Estimated = Estimated = 55

SIMULINK DetailsSIMULINK DetailsSIMULINK DetailsSIMULINK Details

bull Unknown rate parameters were estimated using SIMULINK

ndash Minimizes a user-specified cost function via a user-specified optimization method

ndash Nonlinear least squares optimization method of Levenberg-Marquardt was used to minimize a sum of squared errors cost function of the empirical observations and model predictions for IP3 Ca2+ and multiple species of DAG

bull Note Although the Gauss-Newton method is often more efficient the method of Levenberg-Marquardt has proved to be more robust

Measured vs Simulated 384 DAG and Measured vs Simulated 384 DAG and 341 DAG341 DAG

Measured vs Simulated 384 DAG and Measured vs Simulated 384 DAG and 341 DAG341 DAG

bull Timecourse of stimulation with 25 Timecourse of stimulation with 25 M UDP in RAW 2647 cellsM UDP in RAW 2647 cells

bull Solid black lines represent model simulations Solid black lines represent model simulations

bull ((aa) 384 DAG response (representative of the response of most poly ) 384 DAG response (representative of the response of most poly unsaturated fatty acid-containing DAG species) unsaturated fatty acid-containing DAG species)

bull ((bb) 341 DAG response (representative of the response of most mono- ) 341 DAG response (representative of the response of most mono- and di-unsaturated fatty acid-containing DAG species) and di-unsaturated fatty acid-containing DAG species)

bull Data points contain nine replicates performed on three different Data points contain nine replicates performed on three different experimental days with error bars = 1 SEM Units are total change in ng experimental days with error bars = 1 SEM Units are total change in ng over baseline levels in ~8x10over baseline levels in ~8x1066 cells cells

Measured vs Simulated IPMeasured vs Simulated IP33 and Ca and Ca2+2+Measured vs Simulated IPMeasured vs Simulated IP33 and Ca and Ca2+2+

bull Timecourse of stimulation with 25 Timecourse of stimulation with 25 M UDP in RAW 2647 cellsM UDP in RAW 2647 cells

bull Solid black lines represent model simulationsSolid black lines represent model simulations

bull ((cc) IP) IP33 response in pmols per ~35x10 response in pmols per ~35x1055 cells cells

bull Points in (c) represent the average of four experiments and error Points in (c) represent the average of four experiments and error bars are 1 SEM bars are 1 SEM

bull ((dd) Ca) Ca2+2+ response in response in M Red curve is a representative CaM Red curve is a representative Ca2+2+ trace trace taken from the UDP experiments within the AfCS single ligand taken from the UDP experiments within the AfCS single ligand screenscreen

Model Simulations P2YModel Simulations P2Y66 G GGTP GTP PIPPIP22

Model Simulations P2YModel Simulations P2Y66 G GGTP GTP PIPPIP22

bull ((aa) = total P2Y) = total P2Y66 activated (solid activated (solid line) and inactivated (dashed line) and inactivated (dashed line) surface receptorsline) surface receptors

bull ((bb) = total G) = total GGTP GTP

bull ((cc) = total PIP) = total PIP22 available for available for hydrolysishydrolysis

P2Y6 from 25M UDP GGTP from 25M UDP

PIP2 from 25M UDP

Sensitivity AnalysisSensitivity AnalysisSensitivity AnalysisSensitivity Analysis

bull Sensitivity analysis techniques are valuable tools Sensitivity analysis techniques are valuable tools designed to answer questions regarding which of the designed to answer questions regarding which of the uncertain input variables is more important in uncertain input variables is more important in determining the uncertainty in our output determining the uncertainty in our output

bull Likewise sensitivity analysis can provide insight into Likewise sensitivity analysis can provide insight into which parameter should be studied in more detail in which parameter should be studied in more detail in order to reduce the most variance in the model output order to reduce the most variance in the model output

bull The ability to answer these types of questions could The ability to answer these types of questions could lead to important insight into the design of new lead to important insight into the design of new experiments and in determining which experiments experiments and in determining which experiments would give us the most valuable informationwould give us the most valuable information

Sampling MethodSampling MethodSampling MethodSampling Method

bull Generate a random sample of our space of input Generate a random sample of our space of input variables over a ten percent variation from each variables over a ten percent variation from each parameters nominal value using the parameters nominal value using the Latin Hypercube Latin Hypercube Sampling (LHS) methodSampling (LHS) method

bull Uses Uses Standardized Regression Coefficients (SRCs) Standardized Regression Coefficients (SRCs) obtained by performing multiple linear regression obtained by performing multiple linear regression analysisanalysisndash offers a measure of sensitivity that is multi-dimensionally offers a measure of sensitivity that is multi-dimensionally

averaged over the entire space of parameter values averaged over the entire space of parameter values ndash SRCs give insight into degree of nonlinearity in the model SRCs give insight into degree of nonlinearity in the model

Computing RComputing Ryy22 values valuesComputing RComputing Ryy22 values values

bull SRCs are only reliable measures of sensitivity when SRCs are only reliable measures of sensitivity when degree of nonlinearity is ldquosmallrdquodegree of nonlinearity is ldquosmallrdquo

bull Use model coefficients of determination RUse model coefficients of determination Ryy22 given by given by

bull where ywhere yii is the estimate of y is the estimate of yii obtained from the regression obtained from the regression modelmodel

bull RRyy22 ge 07 ensures SRCs are good sensitivity measures ge 07 ensures SRCs are good sensitivity measures

SRCs (as sensitivity measures) for SRCs (as sensitivity measures) for DAGDAG

SRCs (as sensitivity measures) for SRCs (as sensitivity measures) for DAGDAG

SRCs (as sensitivity measures) for IPSRCs (as sensitivity measures) for IP33SRCs (as sensitivity measures) for IPSRCs (as sensitivity measures) for IP33

Effects of 3 most sensitive DAG Effects of 3 most sensitive DAG parametersparameters

Effects of 3 most sensitive DAG Effects of 3 most sensitive DAG parametersparameters

A B

C

AA k kdp2dp2 degradation of pool 2 DAG degradation of pool 2 DAG

BB k kdp1dp1 degradation of pool 1 DAG degradation of pool 1 DAG

CC k kap2ap2 production of pool 2 DAG production of pool 2 DAG

ConclusionsConclusionsConclusionsConclusions

bull We have developed a model of the UDP signaling We have developed a model of the UDP signaling pathway in RAW 2647 macrophages which can predict pathway in RAW 2647 macrophages which can predict the responses of multiple species of DAG as well as the the responses of multiple species of DAG as well as the responses of IPresponses of IP33 Ca Ca2+2+ receptor dynamics G-protein receptor dynamics G-protein activation and PIPactivation and PIP22 hydrolysis hydrolysis

bull Simplified model resultsSimplified model resultsndash We have obtained global existence uniqueness positivity We have obtained global existence uniqueness positivity

and boundedness of solutionsand boundedness of solutionsndash We have proven global stability of a unique steady state We have proven global stability of a unique steady state

within our region of interestwithin our region of interestbull Full model analysisFull model analysis

ndash Using SIMULINK we have estimated unknown rate Using SIMULINK we have estimated unknown rate parameters to obtain best fits to multiple DAG traces IPparameters to obtain best fits to multiple DAG traces IP33 and Caand Ca2+2+ all in response to 25 all in response to 25M UDPM UDP

ndash We have performed sensitivity analysis using the Latin We have performed sensitivity analysis using the Latin Hypercube sampling technique in combination with Hypercube sampling technique in combination with standardized regression coefficients to determine which standardized regression coefficients to determine which model parameters are responsible for most of the model model parameters are responsible for most of the model output uncertaintyoutput uncertainty

Future DirectionsFuture DirectionsFuture DirectionsFuture Directions

bull We have conducted multiple experiments to We have conducted multiple experiments to pharmacologically inhibit several different enzymes we pharmacologically inhibit several different enzymes we believe to play a role in this signaling pathway as believe to play a role in this signaling pathway as suggested by current known metabolic pathways and by suggested by current known metabolic pathways and by modeling resultsmodeling results

bull The next step is to perform gene knockdowns on The next step is to perform gene knockdowns on specific enzymes to verify results of inhibitor dataspecific enzymes to verify results of inhibitor data

bull The model output also suggests a time delay from The model output also suggests a time delay from receptor activation to PIPreceptor activation to PIP22 hydrolysis An upcoming task hydrolysis An upcoming task is to investigate the outcomes of adding such a delay is to investigate the outcomes of adding such a delay termterm

bull The model could be greatly enhanced by incorporating The model could be greatly enhanced by incorporating spatial dynamics so this will also be a major focus for spatial dynamics so this will also be a major focus for future researchfuture research

AcknowledgementsAcknowledgementsAcknowledgementsAcknowledgements

Mathematics DepartmentMathematics Departmentbull Hannah L Callender Hannah L Callender

PhDPhD

CollaboratorsCollaboratorsbull Alliance for Cellular SignalingAlliance for Cellular Signalingbull UT SouthwesternUT Southwestern

ndash Paul Sternweis PhDPaul Sternweis PhDndash Dianne DeCamp PhDDianne DeCamp PhD

Brown LabBrown Labbull H Alex Brown PhDH Alex Brown PhDbull Jeffrey S Forrester PhDJeffrey S Forrester PhDbull Mark Byrne PhDMark Byrne PhDbull Anita Preininger PhDAnita Preininger PhDbull Michelle ArmstrongMichelle Armstrongbull Andrew GoodmanAndrew Goodmanbull Pavlina Ivanova PhDPavlina Ivanova PhDbull Steve Milne PhDSteve Milne PhD

What is DAGWhat is DAGWhat is DAGWhat is DAG

bull DAG is a cellular second messenger molecule which plays an DAG is a cellular second messenger molecule which plays an important role in initiating various changes in cell behavior important role in initiating various changes in cell behavior including cell activation differentiation proliferation and including cell activation differentiation proliferation and tumor promotion tumor promotion

bull There are many different species of DAG depending on the There are many different species of DAG depending on the number of carbons and number of double bonds in the fatty number of carbons and number of double bonds in the fatty acyl chains and different species can have different cellular acyl chains and different species can have different cellular functionsfunctions

bull ExampleExample 320 DAG320 DAG

Two fatty acyl Two fatty acyl (hydrocarbon) (hydrocarbon) chains in every DAG chains in every DAG speciesspecies

Each corner Each corner represents carbonrepresents carbon

320 DAG 18 carbons in first chain 14 in the second no double bonds 320 DAG 18 carbons in first chain 14 in the second no double bonds in either chainin either chain

Kinetics of monodi unsaturated DAGsKinetics of monodi unsaturated DAGsKinetics of monodi unsaturated DAGsKinetics of monodi unsaturated DAGs

bull Time based behavior of four monodi unsaturated DAG species after addition of 25 Time based behavior of four monodi unsaturated DAG species after addition of 25 M M (solid red squares) and 025 (solid red squares) and 025 M (solid green triangles) UDPM (solid green triangles) UDP

bull Time points contain a minimum of nine replicates performed on three different Time points contain a minimum of nine replicates performed on three different experimental daysexperimental days

- 25 M UDP

- 025M UDP

0 5 10 15 20 25 30

0 5 10 15 20 25 30 0 5 10 15 20 25 30

0 5 10 15 20 25 30

321 DAG 341 DAG

362 DAG341 DAGep

150

100

50-5

00

300

200

100

-100

0

150

100

50-5

00

-100

200

6040

20-2

00

-40

80

Time (min) Time (min)

Time (min) Time (min)

D

AG

(n

g)

D

AG

(n

g)

D

AG

(n

g)

D

AG

(n

g)

- 25 M UDP

- 025M UDP

- 25 M UDP

- 025M UDP

- 25 M UDP

- 025M UDP

Kinetics of PUFA containing DAGsKinetics of PUFA containing DAGsKinetics of PUFA containing DAGsKinetics of PUFA containing DAGsbull Time based behavior of four polyunsaturated fatty acid (PUFA) containing DAG species Time based behavior of four polyunsaturated fatty acid (PUFA) containing DAG species

after addition of 25 after addition of 25 M (solid red squares) and 025 M (solid red squares) and 025 M (solid green triangles) UDPM (solid green triangles) UDPbull Time points contain a minimum of nine replicates performed on three different Time points contain a minimum of nine replicates performed on three different

experimental daysexperimental daysbull Monodi unsaturated DAG species give a larger increase than polyunsaturated fatty acid Monodi unsaturated DAG species give a larger increase than polyunsaturated fatty acid

(PUFA) containing DAGs(PUFA) containing DAGs

0 5 10 15 20 25 30

0 5 10 15 20 25 30 0 5 10 15 20 25 30

0 5 10 15 20 25 30

385 DAG 383 DAG

364 DAG384 DAG6

42

-40

30

20

10

0

15

10

50

-56

40

2-2

8

Time (min) Time (min)

Time (min) Time (min)

D

AG

(n

g)

D

AG

(n

g)

D

AG

(n

g)

D

AG

(n

g)

-28

40

- 25 M UDP

- 025M UDP

- 25 M UDP

- 025M UDP

- 25 M UDP

- 025M UDP

- 25 M UDP

- 025M UDP

Differential DAG KineticsDifferential DAG KineticsDifferential DAG KineticsDifferential DAG Kinetics

bull Time based behavior of three DAG species with varying degrees of unsaturation after Time based behavior of three DAG species with varying degrees of unsaturation after addition of addition of 25 25 MM (solid red triangles) and (solid red triangles) and 025 025 MM (solid green squares) UDP (solid green squares) UDP

bull Time points contain nine replicates performed on three different experimental daysTime points contain nine replicates performed on three different experimental days

341 DAG

Best fit with current model structure

Overall Objectives of Modeling EffortOverall Objectives of Modeling EffortOverall Objectives of Modeling EffortOverall Objectives of Modeling Effort

bull Predict quantitative changes in lipid species after stimulation by Predict quantitative changes in lipid species after stimulation by various ligands and ligand concentrations in the RAW 2647 various ligands and ligand concentrations in the RAW 2647 macrophagemacrophage

bull Comparison and refinement of model output with AfCS IPComparison and refinement of model output with AfCS IP33 measurements and Cameasurements and Ca2+2+ traces as well as DAG data generated in traces as well as DAG data generated in the Brown labthe Brown lab

bull Predict in silico effects such as the effect of knock-downs etc on Predict in silico effects such as the effect of knock-downs etc on given pathwaygiven pathway

bull Suggest modifications to current pathway structuresSuggest modifications to current pathway structures

Modifications to the modelModifications to the modelModifications to the modelModifications to the model

bull Include an additional branch in the pathway Include an additional branch in the pathway for a second pool of DAGfor a second pool of DAG

bull Simplify CaSimplify Ca2+2+ equations for mathematical equations for mathematical analysis purposesanalysis purposes

Proposed PathwayProposed PathwayProposed PathwayProposed Pathway

bull Note We measure total cellular DAG levelsNote We measure total cellular DAG levelsbull Initial production of DAG from the hydrolysis of PIPInitial production of DAG from the hydrolysis of PIP22 in pool 1 (plasma in pool 1 (plasma

membrane) is offset by phosphorylation of DAG by DAG kinase in pool 2 membrane) is offset by phosphorylation of DAG by DAG kinase in pool 2 (Endoplasmic Reticulum Nucleus) to aid in the PI replacement pathway(Endoplasmic Reticulum Nucleus) to aid in the PI replacement pathway

bull Second wave of DAG is a result of resynthesis of PIPSecond wave of DAG is a result of resynthesis of PIP22 which is then which is then hydrolyzed to form DAG and IPhydrolyzed to form DAG and IP33

PI PIP PIP2 DAG

IP3

IP2IP

Ins

PI

CDP-DAGPADAG

PA

PCPE

PCPE

PITP

PI4K PI5K PLC DGK

DGK CDSPLC PIS

POOL 2

POOL 1

IP4

PI PIP PIP2 DAG

IP3

IP2IP

Ins

PI

CDP-DAGPADAG

PA

PCPE

PC

PITP

PI4K PI5K PLC

DGK CDSPLC PIS

IP4

LPP

LPP

Model Equations DAG (pool 1)Model Equations DAG (pool 1)Model Equations DAG (pool 1)Model Equations DAG (pool 1)

bull DAG (pool 1)DAG (pool 1) (separate ODEs for each species (separate ODEs for each species considered)considered)

bull Concentration of DAG from pool 1 at time tConcentration of DAG from pool 1 at time t

Nondimensionalized equationsNondimensionalized equations

Model Equations DAG (pool 2)Model Equations DAG (pool 2)Model Equations DAG (pool 2)Model Equations DAG (pool 2)

bull DAG (pool 2)DAG (pool 2) (separate ODEs for each species (separate ODEs for each species considered)considered)

bull Concentration of pool 2 DAG molecules at time tConcentration of pool 2 DAG molecules at time t

bull Baseline concentration of pool 2 DAGBaseline concentration of pool 2 DAG

Nondimensionalized equationsNondimensionalized equations

Simplified CaSimplified Ca2+2+ Equations EquationsSimplified CaSimplified Ca2+2+ Equations Equations

bull CaCa2+2+ modulemodule (to match experimental AfCS (to match experimental AfCS trace)trace)

0 100 200 300 400 500 6000

002

004

006

008

01

012

014

Concentration of Ca

2+ (uM)

Ca2+ response to 25uM UDP

time (s)

wherewhere

Theoretical AnalysisTheoretical AnalysisTheoretical AnalysisTheoretical Analysis

bull Existence and UniquenessExistence and Uniqueness

bull Positivity and Boundedness Positivity and Boundedness

(for biological relevance)(for biological relevance)

bull Analysis of steady state behaviorAnalysis of steady state behavior

Existence of Solutions (full model)Existence of Solutions (full model)Existence of Solutions (full model)Existence of Solutions (full model)

First we write our system of ODEs in the formFirst we write our system of ODEs in the form (1)

(2)

(1)

Uniqueness of SolutionsUniqueness of SolutionsUniqueness of SolutionsUniqueness of Solutions

Next we denote a solution of Next we denote a solution of (1)(1) by by with initial condition with initial condition

Since our system satisfies the hypotheses of Theorems 1 and 2 Since our system satisfies the hypotheses of Theorems 1 and 2 on our set of interest (for all positive time and on a positive on our set of interest (for all positive time and on a positive bounded set in space) we know bounded set in space) we know there exists a unique there exists a unique (local) solution(local) solution (ie on some finite time interval possibly (ie on some finite time interval possibly small)small)

QuestionQuestion Do the solutions remain positive and bounded (for Do the solutions remain positive and bounded (for biological significance)biological significance)

(3)

(1) (3)

Positivity and boundedness of Positivity and boundedness of solutionssolutions

Positivity and boundedness of Positivity and boundedness of solutionssolutions

bull We first use Theorem 3 to show positivity and boundedness of xWe first use Theorem 3 to show positivity and boundedness of x11 and and xx22

bull Next we use these results and the Fundamental Theorem of Calculus to Next we use these results and the Fundamental Theorem of Calculus to show positivity and boundedness for the remaining equations show positivity and boundedness for the remaining equations

bull This then ensures a global solutionThis then ensures a global solution

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical Analysis

bull Parameter Estimation in SIMULINKParameter Estimation in SIMULINK

bull Sensitivity AnalysisSensitivity Analysis

Parameter Estimation in SIMULINK (for Parameter Estimation in SIMULINK (for FULL model)FULL model)

Parameter Estimation in SIMULINK (for Parameter Estimation in SIMULINK (for FULL model)FULL model)

bull Total number of model parameters (for full model with Li and Rinzel CaTotal number of model parameters (for full model with Li and Rinzel Ca2+2+ module) = module) = 3434ndash From the literature = From the literature = 2020ndash Estimated = Estimated = 1414

bull Receptor moduleReceptor module Total = Total = 77ndash From literature = From literature = 66ndash Estimated = Estimated = 11 (k (kpp rate of receptor phosphorylation) rate of receptor phosphorylation)

bull G-protein cascadeG-protein cascade Total = Total = 99ndash From literature = From literature = 66ndash Estimated = Estimated = 3 3 (k(khydhyd k kreprep k kd3d3))

bull DAG kineticsDAG kinetics Total = Total = 5 5 (for each DAG species considered)(for each DAG species considered)ndash From literature = From literature = 00ndash Estimated = Estimated = 5 5 (all DAG parameters)(all DAG parameters)

bull CaCa2+2+ module module Total = Total = 1313ndash From literature = From literature = 88ndash Estimated = Estimated = 55

SIMULINK DetailsSIMULINK DetailsSIMULINK DetailsSIMULINK Details

bull Unknown rate parameters were estimated using SIMULINK

ndash Minimizes a user-specified cost function via a user-specified optimization method

ndash Nonlinear least squares optimization method of Levenberg-Marquardt was used to minimize a sum of squared errors cost function of the empirical observations and model predictions for IP3 Ca2+ and multiple species of DAG

bull Note Although the Gauss-Newton method is often more efficient the method of Levenberg-Marquardt has proved to be more robust

Measured vs Simulated 384 DAG and Measured vs Simulated 384 DAG and 341 DAG341 DAG

Measured vs Simulated 384 DAG and Measured vs Simulated 384 DAG and 341 DAG341 DAG

bull Timecourse of stimulation with 25 Timecourse of stimulation with 25 M UDP in RAW 2647 cellsM UDP in RAW 2647 cells

bull Solid black lines represent model simulations Solid black lines represent model simulations

bull ((aa) 384 DAG response (representative of the response of most poly ) 384 DAG response (representative of the response of most poly unsaturated fatty acid-containing DAG species) unsaturated fatty acid-containing DAG species)

bull ((bb) 341 DAG response (representative of the response of most mono- ) 341 DAG response (representative of the response of most mono- and di-unsaturated fatty acid-containing DAG species) and di-unsaturated fatty acid-containing DAG species)

bull Data points contain nine replicates performed on three different Data points contain nine replicates performed on three different experimental days with error bars = 1 SEM Units are total change in ng experimental days with error bars = 1 SEM Units are total change in ng over baseline levels in ~8x10over baseline levels in ~8x1066 cells cells

Measured vs Simulated IPMeasured vs Simulated IP33 and Ca and Ca2+2+Measured vs Simulated IPMeasured vs Simulated IP33 and Ca and Ca2+2+

bull Timecourse of stimulation with 25 Timecourse of stimulation with 25 M UDP in RAW 2647 cellsM UDP in RAW 2647 cells

bull Solid black lines represent model simulationsSolid black lines represent model simulations

bull ((cc) IP) IP33 response in pmols per ~35x10 response in pmols per ~35x1055 cells cells

bull Points in (c) represent the average of four experiments and error Points in (c) represent the average of four experiments and error bars are 1 SEM bars are 1 SEM

bull ((dd) Ca) Ca2+2+ response in response in M Red curve is a representative CaM Red curve is a representative Ca2+2+ trace trace taken from the UDP experiments within the AfCS single ligand taken from the UDP experiments within the AfCS single ligand screenscreen

Model Simulations P2YModel Simulations P2Y66 G GGTP GTP PIPPIP22

Model Simulations P2YModel Simulations P2Y66 G GGTP GTP PIPPIP22

bull ((aa) = total P2Y) = total P2Y66 activated (solid activated (solid line) and inactivated (dashed line) and inactivated (dashed line) surface receptorsline) surface receptors

bull ((bb) = total G) = total GGTP GTP

bull ((cc) = total PIP) = total PIP22 available for available for hydrolysishydrolysis

P2Y6 from 25M UDP GGTP from 25M UDP

PIP2 from 25M UDP

Sensitivity AnalysisSensitivity AnalysisSensitivity AnalysisSensitivity Analysis

bull Sensitivity analysis techniques are valuable tools Sensitivity analysis techniques are valuable tools designed to answer questions regarding which of the designed to answer questions regarding which of the uncertain input variables is more important in uncertain input variables is more important in determining the uncertainty in our output determining the uncertainty in our output

bull Likewise sensitivity analysis can provide insight into Likewise sensitivity analysis can provide insight into which parameter should be studied in more detail in which parameter should be studied in more detail in order to reduce the most variance in the model output order to reduce the most variance in the model output

bull The ability to answer these types of questions could The ability to answer these types of questions could lead to important insight into the design of new lead to important insight into the design of new experiments and in determining which experiments experiments and in determining which experiments would give us the most valuable informationwould give us the most valuable information

Sampling MethodSampling MethodSampling MethodSampling Method

bull Generate a random sample of our space of input Generate a random sample of our space of input variables over a ten percent variation from each variables over a ten percent variation from each parameters nominal value using the parameters nominal value using the Latin Hypercube Latin Hypercube Sampling (LHS) methodSampling (LHS) method

bull Uses Uses Standardized Regression Coefficients (SRCs) Standardized Regression Coefficients (SRCs) obtained by performing multiple linear regression obtained by performing multiple linear regression analysisanalysisndash offers a measure of sensitivity that is multi-dimensionally offers a measure of sensitivity that is multi-dimensionally

averaged over the entire space of parameter values averaged over the entire space of parameter values ndash SRCs give insight into degree of nonlinearity in the model SRCs give insight into degree of nonlinearity in the model

Computing RComputing Ryy22 values valuesComputing RComputing Ryy22 values values

bull SRCs are only reliable measures of sensitivity when SRCs are only reliable measures of sensitivity when degree of nonlinearity is ldquosmallrdquodegree of nonlinearity is ldquosmallrdquo

bull Use model coefficients of determination RUse model coefficients of determination Ryy22 given by given by

bull where ywhere yii is the estimate of y is the estimate of yii obtained from the regression obtained from the regression modelmodel

bull RRyy22 ge 07 ensures SRCs are good sensitivity measures ge 07 ensures SRCs are good sensitivity measures

SRCs (as sensitivity measures) for SRCs (as sensitivity measures) for DAGDAG

SRCs (as sensitivity measures) for SRCs (as sensitivity measures) for DAGDAG

SRCs (as sensitivity measures) for IPSRCs (as sensitivity measures) for IP33SRCs (as sensitivity measures) for IPSRCs (as sensitivity measures) for IP33

Effects of 3 most sensitive DAG Effects of 3 most sensitive DAG parametersparameters

Effects of 3 most sensitive DAG Effects of 3 most sensitive DAG parametersparameters

A B

C

AA k kdp2dp2 degradation of pool 2 DAG degradation of pool 2 DAG

BB k kdp1dp1 degradation of pool 1 DAG degradation of pool 1 DAG

CC k kap2ap2 production of pool 2 DAG production of pool 2 DAG

ConclusionsConclusionsConclusionsConclusions

bull We have developed a model of the UDP signaling We have developed a model of the UDP signaling pathway in RAW 2647 macrophages which can predict pathway in RAW 2647 macrophages which can predict the responses of multiple species of DAG as well as the the responses of multiple species of DAG as well as the responses of IPresponses of IP33 Ca Ca2+2+ receptor dynamics G-protein receptor dynamics G-protein activation and PIPactivation and PIP22 hydrolysis hydrolysis

bull Simplified model resultsSimplified model resultsndash We have obtained global existence uniqueness positivity We have obtained global existence uniqueness positivity

and boundedness of solutionsand boundedness of solutionsndash We have proven global stability of a unique steady state We have proven global stability of a unique steady state

within our region of interestwithin our region of interestbull Full model analysisFull model analysis

ndash Using SIMULINK we have estimated unknown rate Using SIMULINK we have estimated unknown rate parameters to obtain best fits to multiple DAG traces IPparameters to obtain best fits to multiple DAG traces IP33 and Caand Ca2+2+ all in response to 25 all in response to 25M UDPM UDP

ndash We have performed sensitivity analysis using the Latin We have performed sensitivity analysis using the Latin Hypercube sampling technique in combination with Hypercube sampling technique in combination with standardized regression coefficients to determine which standardized regression coefficients to determine which model parameters are responsible for most of the model model parameters are responsible for most of the model output uncertaintyoutput uncertainty

Future DirectionsFuture DirectionsFuture DirectionsFuture Directions

bull We have conducted multiple experiments to We have conducted multiple experiments to pharmacologically inhibit several different enzymes we pharmacologically inhibit several different enzymes we believe to play a role in this signaling pathway as believe to play a role in this signaling pathway as suggested by current known metabolic pathways and by suggested by current known metabolic pathways and by modeling resultsmodeling results

bull The next step is to perform gene knockdowns on The next step is to perform gene knockdowns on specific enzymes to verify results of inhibitor dataspecific enzymes to verify results of inhibitor data

bull The model output also suggests a time delay from The model output also suggests a time delay from receptor activation to PIPreceptor activation to PIP22 hydrolysis An upcoming task hydrolysis An upcoming task is to investigate the outcomes of adding such a delay is to investigate the outcomes of adding such a delay termterm

bull The model could be greatly enhanced by incorporating The model could be greatly enhanced by incorporating spatial dynamics so this will also be a major focus for spatial dynamics so this will also be a major focus for future researchfuture research

AcknowledgementsAcknowledgementsAcknowledgementsAcknowledgements

Mathematics DepartmentMathematics Departmentbull Hannah L Callender Hannah L Callender

PhDPhD

CollaboratorsCollaboratorsbull Alliance for Cellular SignalingAlliance for Cellular Signalingbull UT SouthwesternUT Southwestern

ndash Paul Sternweis PhDPaul Sternweis PhDndash Dianne DeCamp PhDDianne DeCamp PhD

Brown LabBrown Labbull H Alex Brown PhDH Alex Brown PhDbull Jeffrey S Forrester PhDJeffrey S Forrester PhDbull Mark Byrne PhDMark Byrne PhDbull Anita Preininger PhDAnita Preininger PhDbull Michelle ArmstrongMichelle Armstrongbull Andrew GoodmanAndrew Goodmanbull Pavlina Ivanova PhDPavlina Ivanova PhDbull Steve Milne PhDSteve Milne PhD

Kinetics of monodi unsaturated DAGsKinetics of monodi unsaturated DAGsKinetics of monodi unsaturated DAGsKinetics of monodi unsaturated DAGs

bull Time based behavior of four monodi unsaturated DAG species after addition of 25 Time based behavior of four monodi unsaturated DAG species after addition of 25 M M (solid red squares) and 025 (solid red squares) and 025 M (solid green triangles) UDPM (solid green triangles) UDP

bull Time points contain a minimum of nine replicates performed on three different Time points contain a minimum of nine replicates performed on three different experimental daysexperimental days

- 25 M UDP

- 025M UDP

0 5 10 15 20 25 30

0 5 10 15 20 25 30 0 5 10 15 20 25 30

0 5 10 15 20 25 30

321 DAG 341 DAG

362 DAG341 DAGep

150

100

50-5

00

300

200

100

-100

0

150

100

50-5

00

-100

200

6040

20-2

00

-40

80

Time (min) Time (min)

Time (min) Time (min)

D

AG

(n

g)

D

AG

(n

g)

D

AG

(n

g)

D

AG

(n

g)

- 25 M UDP

- 025M UDP

- 25 M UDP

- 025M UDP

- 25 M UDP

- 025M UDP

Kinetics of PUFA containing DAGsKinetics of PUFA containing DAGsKinetics of PUFA containing DAGsKinetics of PUFA containing DAGsbull Time based behavior of four polyunsaturated fatty acid (PUFA) containing DAG species Time based behavior of four polyunsaturated fatty acid (PUFA) containing DAG species

after addition of 25 after addition of 25 M (solid red squares) and 025 M (solid red squares) and 025 M (solid green triangles) UDPM (solid green triangles) UDPbull Time points contain a minimum of nine replicates performed on three different Time points contain a minimum of nine replicates performed on three different

experimental daysexperimental daysbull Monodi unsaturated DAG species give a larger increase than polyunsaturated fatty acid Monodi unsaturated DAG species give a larger increase than polyunsaturated fatty acid

(PUFA) containing DAGs(PUFA) containing DAGs

0 5 10 15 20 25 30

0 5 10 15 20 25 30 0 5 10 15 20 25 30

0 5 10 15 20 25 30

385 DAG 383 DAG

364 DAG384 DAG6

42

-40

30

20

10

0

15

10

50

-56

40

2-2

8

Time (min) Time (min)

Time (min) Time (min)

D

AG

(n

g)

D

AG

(n

g)

D

AG

(n

g)

D

AG

(n

g)

-28

40

- 25 M UDP

- 025M UDP

- 25 M UDP

- 025M UDP

- 25 M UDP

- 025M UDP

- 25 M UDP

- 025M UDP

Differential DAG KineticsDifferential DAG KineticsDifferential DAG KineticsDifferential DAG Kinetics

bull Time based behavior of three DAG species with varying degrees of unsaturation after Time based behavior of three DAG species with varying degrees of unsaturation after addition of addition of 25 25 MM (solid red triangles) and (solid red triangles) and 025 025 MM (solid green squares) UDP (solid green squares) UDP

bull Time points contain nine replicates performed on three different experimental daysTime points contain nine replicates performed on three different experimental days

341 DAG

Best fit with current model structure

Overall Objectives of Modeling EffortOverall Objectives of Modeling EffortOverall Objectives of Modeling EffortOverall Objectives of Modeling Effort

bull Predict quantitative changes in lipid species after stimulation by Predict quantitative changes in lipid species after stimulation by various ligands and ligand concentrations in the RAW 2647 various ligands and ligand concentrations in the RAW 2647 macrophagemacrophage

bull Comparison and refinement of model output with AfCS IPComparison and refinement of model output with AfCS IP33 measurements and Cameasurements and Ca2+2+ traces as well as DAG data generated in traces as well as DAG data generated in the Brown labthe Brown lab

bull Predict in silico effects such as the effect of knock-downs etc on Predict in silico effects such as the effect of knock-downs etc on given pathwaygiven pathway

bull Suggest modifications to current pathway structuresSuggest modifications to current pathway structures

Modifications to the modelModifications to the modelModifications to the modelModifications to the model

bull Include an additional branch in the pathway Include an additional branch in the pathway for a second pool of DAGfor a second pool of DAG

bull Simplify CaSimplify Ca2+2+ equations for mathematical equations for mathematical analysis purposesanalysis purposes

Proposed PathwayProposed PathwayProposed PathwayProposed Pathway

bull Note We measure total cellular DAG levelsNote We measure total cellular DAG levelsbull Initial production of DAG from the hydrolysis of PIPInitial production of DAG from the hydrolysis of PIP22 in pool 1 (plasma in pool 1 (plasma

membrane) is offset by phosphorylation of DAG by DAG kinase in pool 2 membrane) is offset by phosphorylation of DAG by DAG kinase in pool 2 (Endoplasmic Reticulum Nucleus) to aid in the PI replacement pathway(Endoplasmic Reticulum Nucleus) to aid in the PI replacement pathway

bull Second wave of DAG is a result of resynthesis of PIPSecond wave of DAG is a result of resynthesis of PIP22 which is then which is then hydrolyzed to form DAG and IPhydrolyzed to form DAG and IP33

PI PIP PIP2 DAG

IP3

IP2IP

Ins

PI

CDP-DAGPADAG

PA

PCPE

PCPE

PITP

PI4K PI5K PLC DGK

DGK CDSPLC PIS

POOL 2

POOL 1

IP4

PI PIP PIP2 DAG

IP3

IP2IP

Ins

PI

CDP-DAGPADAG

PA

PCPE

PC

PITP

PI4K PI5K PLC

DGK CDSPLC PIS

IP4

LPP

LPP

Model Equations DAG (pool 1)Model Equations DAG (pool 1)Model Equations DAG (pool 1)Model Equations DAG (pool 1)

bull DAG (pool 1)DAG (pool 1) (separate ODEs for each species (separate ODEs for each species considered)considered)

bull Concentration of DAG from pool 1 at time tConcentration of DAG from pool 1 at time t

Nondimensionalized equationsNondimensionalized equations

Model Equations DAG (pool 2)Model Equations DAG (pool 2)Model Equations DAG (pool 2)Model Equations DAG (pool 2)

bull DAG (pool 2)DAG (pool 2) (separate ODEs for each species (separate ODEs for each species considered)considered)

bull Concentration of pool 2 DAG molecules at time tConcentration of pool 2 DAG molecules at time t

bull Baseline concentration of pool 2 DAGBaseline concentration of pool 2 DAG

Nondimensionalized equationsNondimensionalized equations

Simplified CaSimplified Ca2+2+ Equations EquationsSimplified CaSimplified Ca2+2+ Equations Equations

bull CaCa2+2+ modulemodule (to match experimental AfCS (to match experimental AfCS trace)trace)

0 100 200 300 400 500 6000

002

004

006

008

01

012

014

Concentration of Ca

2+ (uM)

Ca2+ response to 25uM UDP

time (s)

wherewhere

Theoretical AnalysisTheoretical AnalysisTheoretical AnalysisTheoretical Analysis

bull Existence and UniquenessExistence and Uniqueness

bull Positivity and Boundedness Positivity and Boundedness

(for biological relevance)(for biological relevance)

bull Analysis of steady state behaviorAnalysis of steady state behavior

Existence of Solutions (full model)Existence of Solutions (full model)Existence of Solutions (full model)Existence of Solutions (full model)

First we write our system of ODEs in the formFirst we write our system of ODEs in the form (1)

(2)

(1)

Uniqueness of SolutionsUniqueness of SolutionsUniqueness of SolutionsUniqueness of Solutions

Next we denote a solution of Next we denote a solution of (1)(1) by by with initial condition with initial condition

Since our system satisfies the hypotheses of Theorems 1 and 2 Since our system satisfies the hypotheses of Theorems 1 and 2 on our set of interest (for all positive time and on a positive on our set of interest (for all positive time and on a positive bounded set in space) we know bounded set in space) we know there exists a unique there exists a unique (local) solution(local) solution (ie on some finite time interval possibly (ie on some finite time interval possibly small)small)

QuestionQuestion Do the solutions remain positive and bounded (for Do the solutions remain positive and bounded (for biological significance)biological significance)

(3)

(1) (3)

Positivity and boundedness of Positivity and boundedness of solutionssolutions

Positivity and boundedness of Positivity and boundedness of solutionssolutions

bull We first use Theorem 3 to show positivity and boundedness of xWe first use Theorem 3 to show positivity and boundedness of x11 and and xx22

bull Next we use these results and the Fundamental Theorem of Calculus to Next we use these results and the Fundamental Theorem of Calculus to show positivity and boundedness for the remaining equations show positivity and boundedness for the remaining equations

bull This then ensures a global solutionThis then ensures a global solution

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical Analysis

bull Parameter Estimation in SIMULINKParameter Estimation in SIMULINK

bull Sensitivity AnalysisSensitivity Analysis

Parameter Estimation in SIMULINK (for Parameter Estimation in SIMULINK (for FULL model)FULL model)

Parameter Estimation in SIMULINK (for Parameter Estimation in SIMULINK (for FULL model)FULL model)

bull Total number of model parameters (for full model with Li and Rinzel CaTotal number of model parameters (for full model with Li and Rinzel Ca2+2+ module) = module) = 3434ndash From the literature = From the literature = 2020ndash Estimated = Estimated = 1414

bull Receptor moduleReceptor module Total = Total = 77ndash From literature = From literature = 66ndash Estimated = Estimated = 11 (k (kpp rate of receptor phosphorylation) rate of receptor phosphorylation)

bull G-protein cascadeG-protein cascade Total = Total = 99ndash From literature = From literature = 66ndash Estimated = Estimated = 3 3 (k(khydhyd k kreprep k kd3d3))

bull DAG kineticsDAG kinetics Total = Total = 5 5 (for each DAG species considered)(for each DAG species considered)ndash From literature = From literature = 00ndash Estimated = Estimated = 5 5 (all DAG parameters)(all DAG parameters)

bull CaCa2+2+ module module Total = Total = 1313ndash From literature = From literature = 88ndash Estimated = Estimated = 55

SIMULINK DetailsSIMULINK DetailsSIMULINK DetailsSIMULINK Details

bull Unknown rate parameters were estimated using SIMULINK

ndash Minimizes a user-specified cost function via a user-specified optimization method

ndash Nonlinear least squares optimization method of Levenberg-Marquardt was used to minimize a sum of squared errors cost function of the empirical observations and model predictions for IP3 Ca2+ and multiple species of DAG

bull Note Although the Gauss-Newton method is often more efficient the method of Levenberg-Marquardt has proved to be more robust

Measured vs Simulated 384 DAG and Measured vs Simulated 384 DAG and 341 DAG341 DAG

Measured vs Simulated 384 DAG and Measured vs Simulated 384 DAG and 341 DAG341 DAG

bull Timecourse of stimulation with 25 Timecourse of stimulation with 25 M UDP in RAW 2647 cellsM UDP in RAW 2647 cells

bull Solid black lines represent model simulations Solid black lines represent model simulations

bull ((aa) 384 DAG response (representative of the response of most poly ) 384 DAG response (representative of the response of most poly unsaturated fatty acid-containing DAG species) unsaturated fatty acid-containing DAG species)

bull ((bb) 341 DAG response (representative of the response of most mono- ) 341 DAG response (representative of the response of most mono- and di-unsaturated fatty acid-containing DAG species) and di-unsaturated fatty acid-containing DAG species)

bull Data points contain nine replicates performed on three different Data points contain nine replicates performed on three different experimental days with error bars = 1 SEM Units are total change in ng experimental days with error bars = 1 SEM Units are total change in ng over baseline levels in ~8x10over baseline levels in ~8x1066 cells cells

Measured vs Simulated IPMeasured vs Simulated IP33 and Ca and Ca2+2+Measured vs Simulated IPMeasured vs Simulated IP33 and Ca and Ca2+2+

bull Timecourse of stimulation with 25 Timecourse of stimulation with 25 M UDP in RAW 2647 cellsM UDP in RAW 2647 cells

bull Solid black lines represent model simulationsSolid black lines represent model simulations

bull ((cc) IP) IP33 response in pmols per ~35x10 response in pmols per ~35x1055 cells cells

bull Points in (c) represent the average of four experiments and error Points in (c) represent the average of four experiments and error bars are 1 SEM bars are 1 SEM

bull ((dd) Ca) Ca2+2+ response in response in M Red curve is a representative CaM Red curve is a representative Ca2+2+ trace trace taken from the UDP experiments within the AfCS single ligand taken from the UDP experiments within the AfCS single ligand screenscreen

Model Simulations P2YModel Simulations P2Y66 G GGTP GTP PIPPIP22

Model Simulations P2YModel Simulations P2Y66 G GGTP GTP PIPPIP22

bull ((aa) = total P2Y) = total P2Y66 activated (solid activated (solid line) and inactivated (dashed line) and inactivated (dashed line) surface receptorsline) surface receptors

bull ((bb) = total G) = total GGTP GTP

bull ((cc) = total PIP) = total PIP22 available for available for hydrolysishydrolysis

P2Y6 from 25M UDP GGTP from 25M UDP

PIP2 from 25M UDP

Sensitivity AnalysisSensitivity AnalysisSensitivity AnalysisSensitivity Analysis

bull Sensitivity analysis techniques are valuable tools Sensitivity analysis techniques are valuable tools designed to answer questions regarding which of the designed to answer questions regarding which of the uncertain input variables is more important in uncertain input variables is more important in determining the uncertainty in our output determining the uncertainty in our output

bull Likewise sensitivity analysis can provide insight into Likewise sensitivity analysis can provide insight into which parameter should be studied in more detail in which parameter should be studied in more detail in order to reduce the most variance in the model output order to reduce the most variance in the model output

bull The ability to answer these types of questions could The ability to answer these types of questions could lead to important insight into the design of new lead to important insight into the design of new experiments and in determining which experiments experiments and in determining which experiments would give us the most valuable informationwould give us the most valuable information

Sampling MethodSampling MethodSampling MethodSampling Method

bull Generate a random sample of our space of input Generate a random sample of our space of input variables over a ten percent variation from each variables over a ten percent variation from each parameters nominal value using the parameters nominal value using the Latin Hypercube Latin Hypercube Sampling (LHS) methodSampling (LHS) method

bull Uses Uses Standardized Regression Coefficients (SRCs) Standardized Regression Coefficients (SRCs) obtained by performing multiple linear regression obtained by performing multiple linear regression analysisanalysisndash offers a measure of sensitivity that is multi-dimensionally offers a measure of sensitivity that is multi-dimensionally

averaged over the entire space of parameter values averaged over the entire space of parameter values ndash SRCs give insight into degree of nonlinearity in the model SRCs give insight into degree of nonlinearity in the model

Computing RComputing Ryy22 values valuesComputing RComputing Ryy22 values values

bull SRCs are only reliable measures of sensitivity when SRCs are only reliable measures of sensitivity when degree of nonlinearity is ldquosmallrdquodegree of nonlinearity is ldquosmallrdquo

bull Use model coefficients of determination RUse model coefficients of determination Ryy22 given by given by

bull where ywhere yii is the estimate of y is the estimate of yii obtained from the regression obtained from the regression modelmodel

bull RRyy22 ge 07 ensures SRCs are good sensitivity measures ge 07 ensures SRCs are good sensitivity measures

SRCs (as sensitivity measures) for SRCs (as sensitivity measures) for DAGDAG

SRCs (as sensitivity measures) for SRCs (as sensitivity measures) for DAGDAG

SRCs (as sensitivity measures) for IPSRCs (as sensitivity measures) for IP33SRCs (as sensitivity measures) for IPSRCs (as sensitivity measures) for IP33

Effects of 3 most sensitive DAG Effects of 3 most sensitive DAG parametersparameters

Effects of 3 most sensitive DAG Effects of 3 most sensitive DAG parametersparameters

A B

C

AA k kdp2dp2 degradation of pool 2 DAG degradation of pool 2 DAG

BB k kdp1dp1 degradation of pool 1 DAG degradation of pool 1 DAG

CC k kap2ap2 production of pool 2 DAG production of pool 2 DAG

ConclusionsConclusionsConclusionsConclusions

bull We have developed a model of the UDP signaling We have developed a model of the UDP signaling pathway in RAW 2647 macrophages which can predict pathway in RAW 2647 macrophages which can predict the responses of multiple species of DAG as well as the the responses of multiple species of DAG as well as the responses of IPresponses of IP33 Ca Ca2+2+ receptor dynamics G-protein receptor dynamics G-protein activation and PIPactivation and PIP22 hydrolysis hydrolysis

bull Simplified model resultsSimplified model resultsndash We have obtained global existence uniqueness positivity We have obtained global existence uniqueness positivity

and boundedness of solutionsand boundedness of solutionsndash We have proven global stability of a unique steady state We have proven global stability of a unique steady state

within our region of interestwithin our region of interestbull Full model analysisFull model analysis

ndash Using SIMULINK we have estimated unknown rate Using SIMULINK we have estimated unknown rate parameters to obtain best fits to multiple DAG traces IPparameters to obtain best fits to multiple DAG traces IP33 and Caand Ca2+2+ all in response to 25 all in response to 25M UDPM UDP

ndash We have performed sensitivity analysis using the Latin We have performed sensitivity analysis using the Latin Hypercube sampling technique in combination with Hypercube sampling technique in combination with standardized regression coefficients to determine which standardized regression coefficients to determine which model parameters are responsible for most of the model model parameters are responsible for most of the model output uncertaintyoutput uncertainty

Future DirectionsFuture DirectionsFuture DirectionsFuture Directions

bull We have conducted multiple experiments to We have conducted multiple experiments to pharmacologically inhibit several different enzymes we pharmacologically inhibit several different enzymes we believe to play a role in this signaling pathway as believe to play a role in this signaling pathway as suggested by current known metabolic pathways and by suggested by current known metabolic pathways and by modeling resultsmodeling results

bull The next step is to perform gene knockdowns on The next step is to perform gene knockdowns on specific enzymes to verify results of inhibitor dataspecific enzymes to verify results of inhibitor data

bull The model output also suggests a time delay from The model output also suggests a time delay from receptor activation to PIPreceptor activation to PIP22 hydrolysis An upcoming task hydrolysis An upcoming task is to investigate the outcomes of adding such a delay is to investigate the outcomes of adding such a delay termterm

bull The model could be greatly enhanced by incorporating The model could be greatly enhanced by incorporating spatial dynamics so this will also be a major focus for spatial dynamics so this will also be a major focus for future researchfuture research

AcknowledgementsAcknowledgementsAcknowledgementsAcknowledgements

Mathematics DepartmentMathematics Departmentbull Hannah L Callender Hannah L Callender

PhDPhD

CollaboratorsCollaboratorsbull Alliance for Cellular SignalingAlliance for Cellular Signalingbull UT SouthwesternUT Southwestern

ndash Paul Sternweis PhDPaul Sternweis PhDndash Dianne DeCamp PhDDianne DeCamp PhD

Brown LabBrown Labbull H Alex Brown PhDH Alex Brown PhDbull Jeffrey S Forrester PhDJeffrey S Forrester PhDbull Mark Byrne PhDMark Byrne PhDbull Anita Preininger PhDAnita Preininger PhDbull Michelle ArmstrongMichelle Armstrongbull Andrew GoodmanAndrew Goodmanbull Pavlina Ivanova PhDPavlina Ivanova PhDbull Steve Milne PhDSteve Milne PhD

Kinetics of PUFA containing DAGsKinetics of PUFA containing DAGsKinetics of PUFA containing DAGsKinetics of PUFA containing DAGsbull Time based behavior of four polyunsaturated fatty acid (PUFA) containing DAG species Time based behavior of four polyunsaturated fatty acid (PUFA) containing DAG species

after addition of 25 after addition of 25 M (solid red squares) and 025 M (solid red squares) and 025 M (solid green triangles) UDPM (solid green triangles) UDPbull Time points contain a minimum of nine replicates performed on three different Time points contain a minimum of nine replicates performed on three different

experimental daysexperimental daysbull Monodi unsaturated DAG species give a larger increase than polyunsaturated fatty acid Monodi unsaturated DAG species give a larger increase than polyunsaturated fatty acid

(PUFA) containing DAGs(PUFA) containing DAGs

0 5 10 15 20 25 30

0 5 10 15 20 25 30 0 5 10 15 20 25 30

0 5 10 15 20 25 30

385 DAG 383 DAG

364 DAG384 DAG6

42

-40

30

20

10

0

15

10

50

-56

40

2-2

8

Time (min) Time (min)

Time (min) Time (min)

D

AG

(n

g)

D

AG

(n

g)

D

AG

(n

g)

D

AG

(n

g)

-28

40

- 25 M UDP

- 025M UDP

- 25 M UDP

- 025M UDP

- 25 M UDP

- 025M UDP

- 25 M UDP

- 025M UDP

Differential DAG KineticsDifferential DAG KineticsDifferential DAG KineticsDifferential DAG Kinetics

bull Time based behavior of three DAG species with varying degrees of unsaturation after Time based behavior of three DAG species with varying degrees of unsaturation after addition of addition of 25 25 MM (solid red triangles) and (solid red triangles) and 025 025 MM (solid green squares) UDP (solid green squares) UDP

bull Time points contain nine replicates performed on three different experimental daysTime points contain nine replicates performed on three different experimental days

341 DAG

Best fit with current model structure

Overall Objectives of Modeling EffortOverall Objectives of Modeling EffortOverall Objectives of Modeling EffortOverall Objectives of Modeling Effort

bull Predict quantitative changes in lipid species after stimulation by Predict quantitative changes in lipid species after stimulation by various ligands and ligand concentrations in the RAW 2647 various ligands and ligand concentrations in the RAW 2647 macrophagemacrophage

bull Comparison and refinement of model output with AfCS IPComparison and refinement of model output with AfCS IP33 measurements and Cameasurements and Ca2+2+ traces as well as DAG data generated in traces as well as DAG data generated in the Brown labthe Brown lab

bull Predict in silico effects such as the effect of knock-downs etc on Predict in silico effects such as the effect of knock-downs etc on given pathwaygiven pathway

bull Suggest modifications to current pathway structuresSuggest modifications to current pathway structures

Modifications to the modelModifications to the modelModifications to the modelModifications to the model

bull Include an additional branch in the pathway Include an additional branch in the pathway for a second pool of DAGfor a second pool of DAG

bull Simplify CaSimplify Ca2+2+ equations for mathematical equations for mathematical analysis purposesanalysis purposes

Proposed PathwayProposed PathwayProposed PathwayProposed Pathway

bull Note We measure total cellular DAG levelsNote We measure total cellular DAG levelsbull Initial production of DAG from the hydrolysis of PIPInitial production of DAG from the hydrolysis of PIP22 in pool 1 (plasma in pool 1 (plasma

membrane) is offset by phosphorylation of DAG by DAG kinase in pool 2 membrane) is offset by phosphorylation of DAG by DAG kinase in pool 2 (Endoplasmic Reticulum Nucleus) to aid in the PI replacement pathway(Endoplasmic Reticulum Nucleus) to aid in the PI replacement pathway

bull Second wave of DAG is a result of resynthesis of PIPSecond wave of DAG is a result of resynthesis of PIP22 which is then which is then hydrolyzed to form DAG and IPhydrolyzed to form DAG and IP33

PI PIP PIP2 DAG

IP3

IP2IP

Ins

PI

CDP-DAGPADAG

PA

PCPE

PCPE

PITP

PI4K PI5K PLC DGK

DGK CDSPLC PIS

POOL 2

POOL 1

IP4

PI PIP PIP2 DAG

IP3

IP2IP

Ins

PI

CDP-DAGPADAG

PA

PCPE

PC

PITP

PI4K PI5K PLC

DGK CDSPLC PIS

IP4

LPP

LPP

Model Equations DAG (pool 1)Model Equations DAG (pool 1)Model Equations DAG (pool 1)Model Equations DAG (pool 1)

bull DAG (pool 1)DAG (pool 1) (separate ODEs for each species (separate ODEs for each species considered)considered)

bull Concentration of DAG from pool 1 at time tConcentration of DAG from pool 1 at time t

Nondimensionalized equationsNondimensionalized equations

Model Equations DAG (pool 2)Model Equations DAG (pool 2)Model Equations DAG (pool 2)Model Equations DAG (pool 2)

bull DAG (pool 2)DAG (pool 2) (separate ODEs for each species (separate ODEs for each species considered)considered)

bull Concentration of pool 2 DAG molecules at time tConcentration of pool 2 DAG molecules at time t

bull Baseline concentration of pool 2 DAGBaseline concentration of pool 2 DAG

Nondimensionalized equationsNondimensionalized equations

Simplified CaSimplified Ca2+2+ Equations EquationsSimplified CaSimplified Ca2+2+ Equations Equations

bull CaCa2+2+ modulemodule (to match experimental AfCS (to match experimental AfCS trace)trace)

0 100 200 300 400 500 6000

002

004

006

008

01

012

014

Concentration of Ca

2+ (uM)

Ca2+ response to 25uM UDP

time (s)

wherewhere

Theoretical AnalysisTheoretical AnalysisTheoretical AnalysisTheoretical Analysis

bull Existence and UniquenessExistence and Uniqueness

bull Positivity and Boundedness Positivity and Boundedness

(for biological relevance)(for biological relevance)

bull Analysis of steady state behaviorAnalysis of steady state behavior

Existence of Solutions (full model)Existence of Solutions (full model)Existence of Solutions (full model)Existence of Solutions (full model)

First we write our system of ODEs in the formFirst we write our system of ODEs in the form (1)

(2)

(1)

Uniqueness of SolutionsUniqueness of SolutionsUniqueness of SolutionsUniqueness of Solutions

Next we denote a solution of Next we denote a solution of (1)(1) by by with initial condition with initial condition

Since our system satisfies the hypotheses of Theorems 1 and 2 Since our system satisfies the hypotheses of Theorems 1 and 2 on our set of interest (for all positive time and on a positive on our set of interest (for all positive time and on a positive bounded set in space) we know bounded set in space) we know there exists a unique there exists a unique (local) solution(local) solution (ie on some finite time interval possibly (ie on some finite time interval possibly small)small)

QuestionQuestion Do the solutions remain positive and bounded (for Do the solutions remain positive and bounded (for biological significance)biological significance)

(3)

(1) (3)

Positivity and boundedness of Positivity and boundedness of solutionssolutions

Positivity and boundedness of Positivity and boundedness of solutionssolutions

bull We first use Theorem 3 to show positivity and boundedness of xWe first use Theorem 3 to show positivity and boundedness of x11 and and xx22

bull Next we use these results and the Fundamental Theorem of Calculus to Next we use these results and the Fundamental Theorem of Calculus to show positivity and boundedness for the remaining equations show positivity and boundedness for the remaining equations

bull This then ensures a global solutionThis then ensures a global solution

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical Analysis

bull Parameter Estimation in SIMULINKParameter Estimation in SIMULINK

bull Sensitivity AnalysisSensitivity Analysis

Parameter Estimation in SIMULINK (for Parameter Estimation in SIMULINK (for FULL model)FULL model)

Parameter Estimation in SIMULINK (for Parameter Estimation in SIMULINK (for FULL model)FULL model)

bull Total number of model parameters (for full model with Li and Rinzel CaTotal number of model parameters (for full model with Li and Rinzel Ca2+2+ module) = module) = 3434ndash From the literature = From the literature = 2020ndash Estimated = Estimated = 1414

bull Receptor moduleReceptor module Total = Total = 77ndash From literature = From literature = 66ndash Estimated = Estimated = 11 (k (kpp rate of receptor phosphorylation) rate of receptor phosphorylation)

bull G-protein cascadeG-protein cascade Total = Total = 99ndash From literature = From literature = 66ndash Estimated = Estimated = 3 3 (k(khydhyd k kreprep k kd3d3))

bull DAG kineticsDAG kinetics Total = Total = 5 5 (for each DAG species considered)(for each DAG species considered)ndash From literature = From literature = 00ndash Estimated = Estimated = 5 5 (all DAG parameters)(all DAG parameters)

bull CaCa2+2+ module module Total = Total = 1313ndash From literature = From literature = 88ndash Estimated = Estimated = 55

SIMULINK DetailsSIMULINK DetailsSIMULINK DetailsSIMULINK Details

bull Unknown rate parameters were estimated using SIMULINK

ndash Minimizes a user-specified cost function via a user-specified optimization method

ndash Nonlinear least squares optimization method of Levenberg-Marquardt was used to minimize a sum of squared errors cost function of the empirical observations and model predictions for IP3 Ca2+ and multiple species of DAG

bull Note Although the Gauss-Newton method is often more efficient the method of Levenberg-Marquardt has proved to be more robust

Measured vs Simulated 384 DAG and Measured vs Simulated 384 DAG and 341 DAG341 DAG

Measured vs Simulated 384 DAG and Measured vs Simulated 384 DAG and 341 DAG341 DAG

bull Timecourse of stimulation with 25 Timecourse of stimulation with 25 M UDP in RAW 2647 cellsM UDP in RAW 2647 cells

bull Solid black lines represent model simulations Solid black lines represent model simulations

bull ((aa) 384 DAG response (representative of the response of most poly ) 384 DAG response (representative of the response of most poly unsaturated fatty acid-containing DAG species) unsaturated fatty acid-containing DAG species)

bull ((bb) 341 DAG response (representative of the response of most mono- ) 341 DAG response (representative of the response of most mono- and di-unsaturated fatty acid-containing DAG species) and di-unsaturated fatty acid-containing DAG species)

bull Data points contain nine replicates performed on three different Data points contain nine replicates performed on three different experimental days with error bars = 1 SEM Units are total change in ng experimental days with error bars = 1 SEM Units are total change in ng over baseline levels in ~8x10over baseline levels in ~8x1066 cells cells

Measured vs Simulated IPMeasured vs Simulated IP33 and Ca and Ca2+2+Measured vs Simulated IPMeasured vs Simulated IP33 and Ca and Ca2+2+

bull Timecourse of stimulation with 25 Timecourse of stimulation with 25 M UDP in RAW 2647 cellsM UDP in RAW 2647 cells

bull Solid black lines represent model simulationsSolid black lines represent model simulations

bull ((cc) IP) IP33 response in pmols per ~35x10 response in pmols per ~35x1055 cells cells

bull Points in (c) represent the average of four experiments and error Points in (c) represent the average of four experiments and error bars are 1 SEM bars are 1 SEM

bull ((dd) Ca) Ca2+2+ response in response in M Red curve is a representative CaM Red curve is a representative Ca2+2+ trace trace taken from the UDP experiments within the AfCS single ligand taken from the UDP experiments within the AfCS single ligand screenscreen

Model Simulations P2YModel Simulations P2Y66 G GGTP GTP PIPPIP22

Model Simulations P2YModel Simulations P2Y66 G GGTP GTP PIPPIP22

bull ((aa) = total P2Y) = total P2Y66 activated (solid activated (solid line) and inactivated (dashed line) and inactivated (dashed line) surface receptorsline) surface receptors

bull ((bb) = total G) = total GGTP GTP

bull ((cc) = total PIP) = total PIP22 available for available for hydrolysishydrolysis

P2Y6 from 25M UDP GGTP from 25M UDP

PIP2 from 25M UDP

Sensitivity AnalysisSensitivity AnalysisSensitivity AnalysisSensitivity Analysis

bull Sensitivity analysis techniques are valuable tools Sensitivity analysis techniques are valuable tools designed to answer questions regarding which of the designed to answer questions regarding which of the uncertain input variables is more important in uncertain input variables is more important in determining the uncertainty in our output determining the uncertainty in our output

bull Likewise sensitivity analysis can provide insight into Likewise sensitivity analysis can provide insight into which parameter should be studied in more detail in which parameter should be studied in more detail in order to reduce the most variance in the model output order to reduce the most variance in the model output

bull The ability to answer these types of questions could The ability to answer these types of questions could lead to important insight into the design of new lead to important insight into the design of new experiments and in determining which experiments experiments and in determining which experiments would give us the most valuable informationwould give us the most valuable information

Sampling MethodSampling MethodSampling MethodSampling Method

bull Generate a random sample of our space of input Generate a random sample of our space of input variables over a ten percent variation from each variables over a ten percent variation from each parameters nominal value using the parameters nominal value using the Latin Hypercube Latin Hypercube Sampling (LHS) methodSampling (LHS) method

bull Uses Uses Standardized Regression Coefficients (SRCs) Standardized Regression Coefficients (SRCs) obtained by performing multiple linear regression obtained by performing multiple linear regression analysisanalysisndash offers a measure of sensitivity that is multi-dimensionally offers a measure of sensitivity that is multi-dimensionally

averaged over the entire space of parameter values averaged over the entire space of parameter values ndash SRCs give insight into degree of nonlinearity in the model SRCs give insight into degree of nonlinearity in the model

Computing RComputing Ryy22 values valuesComputing RComputing Ryy22 values values

bull SRCs are only reliable measures of sensitivity when SRCs are only reliable measures of sensitivity when degree of nonlinearity is ldquosmallrdquodegree of nonlinearity is ldquosmallrdquo

bull Use model coefficients of determination RUse model coefficients of determination Ryy22 given by given by

bull where ywhere yii is the estimate of y is the estimate of yii obtained from the regression obtained from the regression modelmodel

bull RRyy22 ge 07 ensures SRCs are good sensitivity measures ge 07 ensures SRCs are good sensitivity measures

SRCs (as sensitivity measures) for SRCs (as sensitivity measures) for DAGDAG

SRCs (as sensitivity measures) for SRCs (as sensitivity measures) for DAGDAG

SRCs (as sensitivity measures) for IPSRCs (as sensitivity measures) for IP33SRCs (as sensitivity measures) for IPSRCs (as sensitivity measures) for IP33

Effects of 3 most sensitive DAG Effects of 3 most sensitive DAG parametersparameters

Effects of 3 most sensitive DAG Effects of 3 most sensitive DAG parametersparameters

A B

C

AA k kdp2dp2 degradation of pool 2 DAG degradation of pool 2 DAG

BB k kdp1dp1 degradation of pool 1 DAG degradation of pool 1 DAG

CC k kap2ap2 production of pool 2 DAG production of pool 2 DAG

ConclusionsConclusionsConclusionsConclusions

bull We have developed a model of the UDP signaling We have developed a model of the UDP signaling pathway in RAW 2647 macrophages which can predict pathway in RAW 2647 macrophages which can predict the responses of multiple species of DAG as well as the the responses of multiple species of DAG as well as the responses of IPresponses of IP33 Ca Ca2+2+ receptor dynamics G-protein receptor dynamics G-protein activation and PIPactivation and PIP22 hydrolysis hydrolysis

bull Simplified model resultsSimplified model resultsndash We have obtained global existence uniqueness positivity We have obtained global existence uniqueness positivity

and boundedness of solutionsand boundedness of solutionsndash We have proven global stability of a unique steady state We have proven global stability of a unique steady state

within our region of interestwithin our region of interestbull Full model analysisFull model analysis

ndash Using SIMULINK we have estimated unknown rate Using SIMULINK we have estimated unknown rate parameters to obtain best fits to multiple DAG traces IPparameters to obtain best fits to multiple DAG traces IP33 and Caand Ca2+2+ all in response to 25 all in response to 25M UDPM UDP

ndash We have performed sensitivity analysis using the Latin We have performed sensitivity analysis using the Latin Hypercube sampling technique in combination with Hypercube sampling technique in combination with standardized regression coefficients to determine which standardized regression coefficients to determine which model parameters are responsible for most of the model model parameters are responsible for most of the model output uncertaintyoutput uncertainty

Future DirectionsFuture DirectionsFuture DirectionsFuture Directions

bull We have conducted multiple experiments to We have conducted multiple experiments to pharmacologically inhibit several different enzymes we pharmacologically inhibit several different enzymes we believe to play a role in this signaling pathway as believe to play a role in this signaling pathway as suggested by current known metabolic pathways and by suggested by current known metabolic pathways and by modeling resultsmodeling results

bull The next step is to perform gene knockdowns on The next step is to perform gene knockdowns on specific enzymes to verify results of inhibitor dataspecific enzymes to verify results of inhibitor data

bull The model output also suggests a time delay from The model output also suggests a time delay from receptor activation to PIPreceptor activation to PIP22 hydrolysis An upcoming task hydrolysis An upcoming task is to investigate the outcomes of adding such a delay is to investigate the outcomes of adding such a delay termterm

bull The model could be greatly enhanced by incorporating The model could be greatly enhanced by incorporating spatial dynamics so this will also be a major focus for spatial dynamics so this will also be a major focus for future researchfuture research

AcknowledgementsAcknowledgementsAcknowledgementsAcknowledgements

Mathematics DepartmentMathematics Departmentbull Hannah L Callender Hannah L Callender

PhDPhD

CollaboratorsCollaboratorsbull Alliance for Cellular SignalingAlliance for Cellular Signalingbull UT SouthwesternUT Southwestern

ndash Paul Sternweis PhDPaul Sternweis PhDndash Dianne DeCamp PhDDianne DeCamp PhD

Brown LabBrown Labbull H Alex Brown PhDH Alex Brown PhDbull Jeffrey S Forrester PhDJeffrey S Forrester PhDbull Mark Byrne PhDMark Byrne PhDbull Anita Preininger PhDAnita Preininger PhDbull Michelle ArmstrongMichelle Armstrongbull Andrew GoodmanAndrew Goodmanbull Pavlina Ivanova PhDPavlina Ivanova PhDbull Steve Milne PhDSteve Milne PhD

Differential DAG KineticsDifferential DAG KineticsDifferential DAG KineticsDifferential DAG Kinetics

bull Time based behavior of three DAG species with varying degrees of unsaturation after Time based behavior of three DAG species with varying degrees of unsaturation after addition of addition of 25 25 MM (solid red triangles) and (solid red triangles) and 025 025 MM (solid green squares) UDP (solid green squares) UDP

bull Time points contain nine replicates performed on three different experimental daysTime points contain nine replicates performed on three different experimental days

341 DAG

Best fit with current model structure

Overall Objectives of Modeling EffortOverall Objectives of Modeling EffortOverall Objectives of Modeling EffortOverall Objectives of Modeling Effort

bull Predict quantitative changes in lipid species after stimulation by Predict quantitative changes in lipid species after stimulation by various ligands and ligand concentrations in the RAW 2647 various ligands and ligand concentrations in the RAW 2647 macrophagemacrophage

bull Comparison and refinement of model output with AfCS IPComparison and refinement of model output with AfCS IP33 measurements and Cameasurements and Ca2+2+ traces as well as DAG data generated in traces as well as DAG data generated in the Brown labthe Brown lab

bull Predict in silico effects such as the effect of knock-downs etc on Predict in silico effects such as the effect of knock-downs etc on given pathwaygiven pathway

bull Suggest modifications to current pathway structuresSuggest modifications to current pathway structures

Modifications to the modelModifications to the modelModifications to the modelModifications to the model

bull Include an additional branch in the pathway Include an additional branch in the pathway for a second pool of DAGfor a second pool of DAG

bull Simplify CaSimplify Ca2+2+ equations for mathematical equations for mathematical analysis purposesanalysis purposes

Proposed PathwayProposed PathwayProposed PathwayProposed Pathway

bull Note We measure total cellular DAG levelsNote We measure total cellular DAG levelsbull Initial production of DAG from the hydrolysis of PIPInitial production of DAG from the hydrolysis of PIP22 in pool 1 (plasma in pool 1 (plasma

membrane) is offset by phosphorylation of DAG by DAG kinase in pool 2 membrane) is offset by phosphorylation of DAG by DAG kinase in pool 2 (Endoplasmic Reticulum Nucleus) to aid in the PI replacement pathway(Endoplasmic Reticulum Nucleus) to aid in the PI replacement pathway

bull Second wave of DAG is a result of resynthesis of PIPSecond wave of DAG is a result of resynthesis of PIP22 which is then which is then hydrolyzed to form DAG and IPhydrolyzed to form DAG and IP33

PI PIP PIP2 DAG

IP3

IP2IP

Ins

PI

CDP-DAGPADAG

PA

PCPE

PCPE

PITP

PI4K PI5K PLC DGK

DGK CDSPLC PIS

POOL 2

POOL 1

IP4

PI PIP PIP2 DAG

IP3

IP2IP

Ins

PI

CDP-DAGPADAG

PA

PCPE

PC

PITP

PI4K PI5K PLC

DGK CDSPLC PIS

IP4

LPP

LPP

Model Equations DAG (pool 1)Model Equations DAG (pool 1)Model Equations DAG (pool 1)Model Equations DAG (pool 1)

bull DAG (pool 1)DAG (pool 1) (separate ODEs for each species (separate ODEs for each species considered)considered)

bull Concentration of DAG from pool 1 at time tConcentration of DAG from pool 1 at time t

Nondimensionalized equationsNondimensionalized equations

Model Equations DAG (pool 2)Model Equations DAG (pool 2)Model Equations DAG (pool 2)Model Equations DAG (pool 2)

bull DAG (pool 2)DAG (pool 2) (separate ODEs for each species (separate ODEs for each species considered)considered)

bull Concentration of pool 2 DAG molecules at time tConcentration of pool 2 DAG molecules at time t

bull Baseline concentration of pool 2 DAGBaseline concentration of pool 2 DAG

Nondimensionalized equationsNondimensionalized equations

Simplified CaSimplified Ca2+2+ Equations EquationsSimplified CaSimplified Ca2+2+ Equations Equations

bull CaCa2+2+ modulemodule (to match experimental AfCS (to match experimental AfCS trace)trace)

0 100 200 300 400 500 6000

002

004

006

008

01

012

014

Concentration of Ca

2+ (uM)

Ca2+ response to 25uM UDP

time (s)

wherewhere

Theoretical AnalysisTheoretical AnalysisTheoretical AnalysisTheoretical Analysis

bull Existence and UniquenessExistence and Uniqueness

bull Positivity and Boundedness Positivity and Boundedness

(for biological relevance)(for biological relevance)

bull Analysis of steady state behaviorAnalysis of steady state behavior

Existence of Solutions (full model)Existence of Solutions (full model)Existence of Solutions (full model)Existence of Solutions (full model)

First we write our system of ODEs in the formFirst we write our system of ODEs in the form (1)

(2)

(1)

Uniqueness of SolutionsUniqueness of SolutionsUniqueness of SolutionsUniqueness of Solutions

Next we denote a solution of Next we denote a solution of (1)(1) by by with initial condition with initial condition

Since our system satisfies the hypotheses of Theorems 1 and 2 Since our system satisfies the hypotheses of Theorems 1 and 2 on our set of interest (for all positive time and on a positive on our set of interest (for all positive time and on a positive bounded set in space) we know bounded set in space) we know there exists a unique there exists a unique (local) solution(local) solution (ie on some finite time interval possibly (ie on some finite time interval possibly small)small)

QuestionQuestion Do the solutions remain positive and bounded (for Do the solutions remain positive and bounded (for biological significance)biological significance)

(3)

(1) (3)

Positivity and boundedness of Positivity and boundedness of solutionssolutions

Positivity and boundedness of Positivity and boundedness of solutionssolutions

bull We first use Theorem 3 to show positivity and boundedness of xWe first use Theorem 3 to show positivity and boundedness of x11 and and xx22

bull Next we use these results and the Fundamental Theorem of Calculus to Next we use these results and the Fundamental Theorem of Calculus to show positivity and boundedness for the remaining equations show positivity and boundedness for the remaining equations

bull This then ensures a global solutionThis then ensures a global solution

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical Analysis

bull Parameter Estimation in SIMULINKParameter Estimation in SIMULINK

bull Sensitivity AnalysisSensitivity Analysis

Parameter Estimation in SIMULINK (for Parameter Estimation in SIMULINK (for FULL model)FULL model)

Parameter Estimation in SIMULINK (for Parameter Estimation in SIMULINK (for FULL model)FULL model)

bull Total number of model parameters (for full model with Li and Rinzel CaTotal number of model parameters (for full model with Li and Rinzel Ca2+2+ module) = module) = 3434ndash From the literature = From the literature = 2020ndash Estimated = Estimated = 1414

bull Receptor moduleReceptor module Total = Total = 77ndash From literature = From literature = 66ndash Estimated = Estimated = 11 (k (kpp rate of receptor phosphorylation) rate of receptor phosphorylation)

bull G-protein cascadeG-protein cascade Total = Total = 99ndash From literature = From literature = 66ndash Estimated = Estimated = 3 3 (k(khydhyd k kreprep k kd3d3))

bull DAG kineticsDAG kinetics Total = Total = 5 5 (for each DAG species considered)(for each DAG species considered)ndash From literature = From literature = 00ndash Estimated = Estimated = 5 5 (all DAG parameters)(all DAG parameters)

bull CaCa2+2+ module module Total = Total = 1313ndash From literature = From literature = 88ndash Estimated = Estimated = 55

SIMULINK DetailsSIMULINK DetailsSIMULINK DetailsSIMULINK Details

bull Unknown rate parameters were estimated using SIMULINK

ndash Minimizes a user-specified cost function via a user-specified optimization method

ndash Nonlinear least squares optimization method of Levenberg-Marquardt was used to minimize a sum of squared errors cost function of the empirical observations and model predictions for IP3 Ca2+ and multiple species of DAG

bull Note Although the Gauss-Newton method is often more efficient the method of Levenberg-Marquardt has proved to be more robust

Measured vs Simulated 384 DAG and Measured vs Simulated 384 DAG and 341 DAG341 DAG

Measured vs Simulated 384 DAG and Measured vs Simulated 384 DAG and 341 DAG341 DAG

bull Timecourse of stimulation with 25 Timecourse of stimulation with 25 M UDP in RAW 2647 cellsM UDP in RAW 2647 cells

bull Solid black lines represent model simulations Solid black lines represent model simulations

bull ((aa) 384 DAG response (representative of the response of most poly ) 384 DAG response (representative of the response of most poly unsaturated fatty acid-containing DAG species) unsaturated fatty acid-containing DAG species)

bull ((bb) 341 DAG response (representative of the response of most mono- ) 341 DAG response (representative of the response of most mono- and di-unsaturated fatty acid-containing DAG species) and di-unsaturated fatty acid-containing DAG species)

bull Data points contain nine replicates performed on three different Data points contain nine replicates performed on three different experimental days with error bars = 1 SEM Units are total change in ng experimental days with error bars = 1 SEM Units are total change in ng over baseline levels in ~8x10over baseline levels in ~8x1066 cells cells

Measured vs Simulated IPMeasured vs Simulated IP33 and Ca and Ca2+2+Measured vs Simulated IPMeasured vs Simulated IP33 and Ca and Ca2+2+

bull Timecourse of stimulation with 25 Timecourse of stimulation with 25 M UDP in RAW 2647 cellsM UDP in RAW 2647 cells

bull Solid black lines represent model simulationsSolid black lines represent model simulations

bull ((cc) IP) IP33 response in pmols per ~35x10 response in pmols per ~35x1055 cells cells

bull Points in (c) represent the average of four experiments and error Points in (c) represent the average of four experiments and error bars are 1 SEM bars are 1 SEM

bull ((dd) Ca) Ca2+2+ response in response in M Red curve is a representative CaM Red curve is a representative Ca2+2+ trace trace taken from the UDP experiments within the AfCS single ligand taken from the UDP experiments within the AfCS single ligand screenscreen

Model Simulations P2YModel Simulations P2Y66 G GGTP GTP PIPPIP22

Model Simulations P2YModel Simulations P2Y66 G GGTP GTP PIPPIP22

bull ((aa) = total P2Y) = total P2Y66 activated (solid activated (solid line) and inactivated (dashed line) and inactivated (dashed line) surface receptorsline) surface receptors

bull ((bb) = total G) = total GGTP GTP

bull ((cc) = total PIP) = total PIP22 available for available for hydrolysishydrolysis

P2Y6 from 25M UDP GGTP from 25M UDP

PIP2 from 25M UDP

Sensitivity AnalysisSensitivity AnalysisSensitivity AnalysisSensitivity Analysis

bull Sensitivity analysis techniques are valuable tools Sensitivity analysis techniques are valuable tools designed to answer questions regarding which of the designed to answer questions regarding which of the uncertain input variables is more important in uncertain input variables is more important in determining the uncertainty in our output determining the uncertainty in our output

bull Likewise sensitivity analysis can provide insight into Likewise sensitivity analysis can provide insight into which parameter should be studied in more detail in which parameter should be studied in more detail in order to reduce the most variance in the model output order to reduce the most variance in the model output

bull The ability to answer these types of questions could The ability to answer these types of questions could lead to important insight into the design of new lead to important insight into the design of new experiments and in determining which experiments experiments and in determining which experiments would give us the most valuable informationwould give us the most valuable information

Sampling MethodSampling MethodSampling MethodSampling Method

bull Generate a random sample of our space of input Generate a random sample of our space of input variables over a ten percent variation from each variables over a ten percent variation from each parameters nominal value using the parameters nominal value using the Latin Hypercube Latin Hypercube Sampling (LHS) methodSampling (LHS) method

bull Uses Uses Standardized Regression Coefficients (SRCs) Standardized Regression Coefficients (SRCs) obtained by performing multiple linear regression obtained by performing multiple linear regression analysisanalysisndash offers a measure of sensitivity that is multi-dimensionally offers a measure of sensitivity that is multi-dimensionally

averaged over the entire space of parameter values averaged over the entire space of parameter values ndash SRCs give insight into degree of nonlinearity in the model SRCs give insight into degree of nonlinearity in the model

Computing RComputing Ryy22 values valuesComputing RComputing Ryy22 values values

bull SRCs are only reliable measures of sensitivity when SRCs are only reliable measures of sensitivity when degree of nonlinearity is ldquosmallrdquodegree of nonlinearity is ldquosmallrdquo

bull Use model coefficients of determination RUse model coefficients of determination Ryy22 given by given by

bull where ywhere yii is the estimate of y is the estimate of yii obtained from the regression obtained from the regression modelmodel

bull RRyy22 ge 07 ensures SRCs are good sensitivity measures ge 07 ensures SRCs are good sensitivity measures

SRCs (as sensitivity measures) for SRCs (as sensitivity measures) for DAGDAG

SRCs (as sensitivity measures) for SRCs (as sensitivity measures) for DAGDAG

SRCs (as sensitivity measures) for IPSRCs (as sensitivity measures) for IP33SRCs (as sensitivity measures) for IPSRCs (as sensitivity measures) for IP33

Effects of 3 most sensitive DAG Effects of 3 most sensitive DAG parametersparameters

Effects of 3 most sensitive DAG Effects of 3 most sensitive DAG parametersparameters

A B

C

AA k kdp2dp2 degradation of pool 2 DAG degradation of pool 2 DAG

BB k kdp1dp1 degradation of pool 1 DAG degradation of pool 1 DAG

CC k kap2ap2 production of pool 2 DAG production of pool 2 DAG

ConclusionsConclusionsConclusionsConclusions

bull We have developed a model of the UDP signaling We have developed a model of the UDP signaling pathway in RAW 2647 macrophages which can predict pathway in RAW 2647 macrophages which can predict the responses of multiple species of DAG as well as the the responses of multiple species of DAG as well as the responses of IPresponses of IP33 Ca Ca2+2+ receptor dynamics G-protein receptor dynamics G-protein activation and PIPactivation and PIP22 hydrolysis hydrolysis

bull Simplified model resultsSimplified model resultsndash We have obtained global existence uniqueness positivity We have obtained global existence uniqueness positivity

and boundedness of solutionsand boundedness of solutionsndash We have proven global stability of a unique steady state We have proven global stability of a unique steady state

within our region of interestwithin our region of interestbull Full model analysisFull model analysis

ndash Using SIMULINK we have estimated unknown rate Using SIMULINK we have estimated unknown rate parameters to obtain best fits to multiple DAG traces IPparameters to obtain best fits to multiple DAG traces IP33 and Caand Ca2+2+ all in response to 25 all in response to 25M UDPM UDP

ndash We have performed sensitivity analysis using the Latin We have performed sensitivity analysis using the Latin Hypercube sampling technique in combination with Hypercube sampling technique in combination with standardized regression coefficients to determine which standardized regression coefficients to determine which model parameters are responsible for most of the model model parameters are responsible for most of the model output uncertaintyoutput uncertainty

Future DirectionsFuture DirectionsFuture DirectionsFuture Directions

bull We have conducted multiple experiments to We have conducted multiple experiments to pharmacologically inhibit several different enzymes we pharmacologically inhibit several different enzymes we believe to play a role in this signaling pathway as believe to play a role in this signaling pathway as suggested by current known metabolic pathways and by suggested by current known metabolic pathways and by modeling resultsmodeling results

bull The next step is to perform gene knockdowns on The next step is to perform gene knockdowns on specific enzymes to verify results of inhibitor dataspecific enzymes to verify results of inhibitor data

bull The model output also suggests a time delay from The model output also suggests a time delay from receptor activation to PIPreceptor activation to PIP22 hydrolysis An upcoming task hydrolysis An upcoming task is to investigate the outcomes of adding such a delay is to investigate the outcomes of adding such a delay termterm

bull The model could be greatly enhanced by incorporating The model could be greatly enhanced by incorporating spatial dynamics so this will also be a major focus for spatial dynamics so this will also be a major focus for future researchfuture research

AcknowledgementsAcknowledgementsAcknowledgementsAcknowledgements

Mathematics DepartmentMathematics Departmentbull Hannah L Callender Hannah L Callender

PhDPhD

CollaboratorsCollaboratorsbull Alliance for Cellular SignalingAlliance for Cellular Signalingbull UT SouthwesternUT Southwestern

ndash Paul Sternweis PhDPaul Sternweis PhDndash Dianne DeCamp PhDDianne DeCamp PhD

Brown LabBrown Labbull H Alex Brown PhDH Alex Brown PhDbull Jeffrey S Forrester PhDJeffrey S Forrester PhDbull Mark Byrne PhDMark Byrne PhDbull Anita Preininger PhDAnita Preininger PhDbull Michelle ArmstrongMichelle Armstrongbull Andrew GoodmanAndrew Goodmanbull Pavlina Ivanova PhDPavlina Ivanova PhDbull Steve Milne PhDSteve Milne PhD

Overall Objectives of Modeling EffortOverall Objectives of Modeling EffortOverall Objectives of Modeling EffortOverall Objectives of Modeling Effort

bull Predict quantitative changes in lipid species after stimulation by Predict quantitative changes in lipid species after stimulation by various ligands and ligand concentrations in the RAW 2647 various ligands and ligand concentrations in the RAW 2647 macrophagemacrophage

bull Comparison and refinement of model output with AfCS IPComparison and refinement of model output with AfCS IP33 measurements and Cameasurements and Ca2+2+ traces as well as DAG data generated in traces as well as DAG data generated in the Brown labthe Brown lab

bull Predict in silico effects such as the effect of knock-downs etc on Predict in silico effects such as the effect of knock-downs etc on given pathwaygiven pathway

bull Suggest modifications to current pathway structuresSuggest modifications to current pathway structures

Modifications to the modelModifications to the modelModifications to the modelModifications to the model

bull Include an additional branch in the pathway Include an additional branch in the pathway for a second pool of DAGfor a second pool of DAG

bull Simplify CaSimplify Ca2+2+ equations for mathematical equations for mathematical analysis purposesanalysis purposes

Proposed PathwayProposed PathwayProposed PathwayProposed Pathway

bull Note We measure total cellular DAG levelsNote We measure total cellular DAG levelsbull Initial production of DAG from the hydrolysis of PIPInitial production of DAG from the hydrolysis of PIP22 in pool 1 (plasma in pool 1 (plasma

membrane) is offset by phosphorylation of DAG by DAG kinase in pool 2 membrane) is offset by phosphorylation of DAG by DAG kinase in pool 2 (Endoplasmic Reticulum Nucleus) to aid in the PI replacement pathway(Endoplasmic Reticulum Nucleus) to aid in the PI replacement pathway

bull Second wave of DAG is a result of resynthesis of PIPSecond wave of DAG is a result of resynthesis of PIP22 which is then which is then hydrolyzed to form DAG and IPhydrolyzed to form DAG and IP33

PI PIP PIP2 DAG

IP3

IP2IP

Ins

PI

CDP-DAGPADAG

PA

PCPE

PCPE

PITP

PI4K PI5K PLC DGK

DGK CDSPLC PIS

POOL 2

POOL 1

IP4

PI PIP PIP2 DAG

IP3

IP2IP

Ins

PI

CDP-DAGPADAG

PA

PCPE

PC

PITP

PI4K PI5K PLC

DGK CDSPLC PIS

IP4

LPP

LPP

Model Equations DAG (pool 1)Model Equations DAG (pool 1)Model Equations DAG (pool 1)Model Equations DAG (pool 1)

bull DAG (pool 1)DAG (pool 1) (separate ODEs for each species (separate ODEs for each species considered)considered)

bull Concentration of DAG from pool 1 at time tConcentration of DAG from pool 1 at time t

Nondimensionalized equationsNondimensionalized equations

Model Equations DAG (pool 2)Model Equations DAG (pool 2)Model Equations DAG (pool 2)Model Equations DAG (pool 2)

bull DAG (pool 2)DAG (pool 2) (separate ODEs for each species (separate ODEs for each species considered)considered)

bull Concentration of pool 2 DAG molecules at time tConcentration of pool 2 DAG molecules at time t

bull Baseline concentration of pool 2 DAGBaseline concentration of pool 2 DAG

Nondimensionalized equationsNondimensionalized equations

Simplified CaSimplified Ca2+2+ Equations EquationsSimplified CaSimplified Ca2+2+ Equations Equations

bull CaCa2+2+ modulemodule (to match experimental AfCS (to match experimental AfCS trace)trace)

0 100 200 300 400 500 6000

002

004

006

008

01

012

014

Concentration of Ca

2+ (uM)

Ca2+ response to 25uM UDP

time (s)

wherewhere

Theoretical AnalysisTheoretical AnalysisTheoretical AnalysisTheoretical Analysis

bull Existence and UniquenessExistence and Uniqueness

bull Positivity and Boundedness Positivity and Boundedness

(for biological relevance)(for biological relevance)

bull Analysis of steady state behaviorAnalysis of steady state behavior

Existence of Solutions (full model)Existence of Solutions (full model)Existence of Solutions (full model)Existence of Solutions (full model)

First we write our system of ODEs in the formFirst we write our system of ODEs in the form (1)

(2)

(1)

Uniqueness of SolutionsUniqueness of SolutionsUniqueness of SolutionsUniqueness of Solutions

Next we denote a solution of Next we denote a solution of (1)(1) by by with initial condition with initial condition

Since our system satisfies the hypotheses of Theorems 1 and 2 Since our system satisfies the hypotheses of Theorems 1 and 2 on our set of interest (for all positive time and on a positive on our set of interest (for all positive time and on a positive bounded set in space) we know bounded set in space) we know there exists a unique there exists a unique (local) solution(local) solution (ie on some finite time interval possibly (ie on some finite time interval possibly small)small)

QuestionQuestion Do the solutions remain positive and bounded (for Do the solutions remain positive and bounded (for biological significance)biological significance)

(3)

(1) (3)

Positivity and boundedness of Positivity and boundedness of solutionssolutions

Positivity and boundedness of Positivity and boundedness of solutionssolutions

bull We first use Theorem 3 to show positivity and boundedness of xWe first use Theorem 3 to show positivity and boundedness of x11 and and xx22

bull Next we use these results and the Fundamental Theorem of Calculus to Next we use these results and the Fundamental Theorem of Calculus to show positivity and boundedness for the remaining equations show positivity and boundedness for the remaining equations

bull This then ensures a global solutionThis then ensures a global solution

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical Analysis

bull Parameter Estimation in SIMULINKParameter Estimation in SIMULINK

bull Sensitivity AnalysisSensitivity Analysis

Parameter Estimation in SIMULINK (for Parameter Estimation in SIMULINK (for FULL model)FULL model)

Parameter Estimation in SIMULINK (for Parameter Estimation in SIMULINK (for FULL model)FULL model)

bull Total number of model parameters (for full model with Li and Rinzel CaTotal number of model parameters (for full model with Li and Rinzel Ca2+2+ module) = module) = 3434ndash From the literature = From the literature = 2020ndash Estimated = Estimated = 1414

bull Receptor moduleReceptor module Total = Total = 77ndash From literature = From literature = 66ndash Estimated = Estimated = 11 (k (kpp rate of receptor phosphorylation) rate of receptor phosphorylation)

bull G-protein cascadeG-protein cascade Total = Total = 99ndash From literature = From literature = 66ndash Estimated = Estimated = 3 3 (k(khydhyd k kreprep k kd3d3))

bull DAG kineticsDAG kinetics Total = Total = 5 5 (for each DAG species considered)(for each DAG species considered)ndash From literature = From literature = 00ndash Estimated = Estimated = 5 5 (all DAG parameters)(all DAG parameters)

bull CaCa2+2+ module module Total = Total = 1313ndash From literature = From literature = 88ndash Estimated = Estimated = 55

SIMULINK DetailsSIMULINK DetailsSIMULINK DetailsSIMULINK Details

bull Unknown rate parameters were estimated using SIMULINK

ndash Minimizes a user-specified cost function via a user-specified optimization method

ndash Nonlinear least squares optimization method of Levenberg-Marquardt was used to minimize a sum of squared errors cost function of the empirical observations and model predictions for IP3 Ca2+ and multiple species of DAG

bull Note Although the Gauss-Newton method is often more efficient the method of Levenberg-Marquardt has proved to be more robust

Measured vs Simulated 384 DAG and Measured vs Simulated 384 DAG and 341 DAG341 DAG

Measured vs Simulated 384 DAG and Measured vs Simulated 384 DAG and 341 DAG341 DAG

bull Timecourse of stimulation with 25 Timecourse of stimulation with 25 M UDP in RAW 2647 cellsM UDP in RAW 2647 cells

bull Solid black lines represent model simulations Solid black lines represent model simulations

bull ((aa) 384 DAG response (representative of the response of most poly ) 384 DAG response (representative of the response of most poly unsaturated fatty acid-containing DAG species) unsaturated fatty acid-containing DAG species)

bull ((bb) 341 DAG response (representative of the response of most mono- ) 341 DAG response (representative of the response of most mono- and di-unsaturated fatty acid-containing DAG species) and di-unsaturated fatty acid-containing DAG species)

bull Data points contain nine replicates performed on three different Data points contain nine replicates performed on three different experimental days with error bars = 1 SEM Units are total change in ng experimental days with error bars = 1 SEM Units are total change in ng over baseline levels in ~8x10over baseline levels in ~8x1066 cells cells

Measured vs Simulated IPMeasured vs Simulated IP33 and Ca and Ca2+2+Measured vs Simulated IPMeasured vs Simulated IP33 and Ca and Ca2+2+

bull Timecourse of stimulation with 25 Timecourse of stimulation with 25 M UDP in RAW 2647 cellsM UDP in RAW 2647 cells

bull Solid black lines represent model simulationsSolid black lines represent model simulations

bull ((cc) IP) IP33 response in pmols per ~35x10 response in pmols per ~35x1055 cells cells

bull Points in (c) represent the average of four experiments and error Points in (c) represent the average of four experiments and error bars are 1 SEM bars are 1 SEM

bull ((dd) Ca) Ca2+2+ response in response in M Red curve is a representative CaM Red curve is a representative Ca2+2+ trace trace taken from the UDP experiments within the AfCS single ligand taken from the UDP experiments within the AfCS single ligand screenscreen

Model Simulations P2YModel Simulations P2Y66 G GGTP GTP PIPPIP22

Model Simulations P2YModel Simulations P2Y66 G GGTP GTP PIPPIP22

bull ((aa) = total P2Y) = total P2Y66 activated (solid activated (solid line) and inactivated (dashed line) and inactivated (dashed line) surface receptorsline) surface receptors

bull ((bb) = total G) = total GGTP GTP

bull ((cc) = total PIP) = total PIP22 available for available for hydrolysishydrolysis

P2Y6 from 25M UDP GGTP from 25M UDP

PIP2 from 25M UDP

Sensitivity AnalysisSensitivity AnalysisSensitivity AnalysisSensitivity Analysis

bull Sensitivity analysis techniques are valuable tools Sensitivity analysis techniques are valuable tools designed to answer questions regarding which of the designed to answer questions regarding which of the uncertain input variables is more important in uncertain input variables is more important in determining the uncertainty in our output determining the uncertainty in our output

bull Likewise sensitivity analysis can provide insight into Likewise sensitivity analysis can provide insight into which parameter should be studied in more detail in which parameter should be studied in more detail in order to reduce the most variance in the model output order to reduce the most variance in the model output

bull The ability to answer these types of questions could The ability to answer these types of questions could lead to important insight into the design of new lead to important insight into the design of new experiments and in determining which experiments experiments and in determining which experiments would give us the most valuable informationwould give us the most valuable information

Sampling MethodSampling MethodSampling MethodSampling Method

bull Generate a random sample of our space of input Generate a random sample of our space of input variables over a ten percent variation from each variables over a ten percent variation from each parameters nominal value using the parameters nominal value using the Latin Hypercube Latin Hypercube Sampling (LHS) methodSampling (LHS) method

bull Uses Uses Standardized Regression Coefficients (SRCs) Standardized Regression Coefficients (SRCs) obtained by performing multiple linear regression obtained by performing multiple linear regression analysisanalysisndash offers a measure of sensitivity that is multi-dimensionally offers a measure of sensitivity that is multi-dimensionally

averaged over the entire space of parameter values averaged over the entire space of parameter values ndash SRCs give insight into degree of nonlinearity in the model SRCs give insight into degree of nonlinearity in the model

Computing RComputing Ryy22 values valuesComputing RComputing Ryy22 values values

bull SRCs are only reliable measures of sensitivity when SRCs are only reliable measures of sensitivity when degree of nonlinearity is ldquosmallrdquodegree of nonlinearity is ldquosmallrdquo

bull Use model coefficients of determination RUse model coefficients of determination Ryy22 given by given by

bull where ywhere yii is the estimate of y is the estimate of yii obtained from the regression obtained from the regression modelmodel

bull RRyy22 ge 07 ensures SRCs are good sensitivity measures ge 07 ensures SRCs are good sensitivity measures

SRCs (as sensitivity measures) for SRCs (as sensitivity measures) for DAGDAG

SRCs (as sensitivity measures) for SRCs (as sensitivity measures) for DAGDAG

SRCs (as sensitivity measures) for IPSRCs (as sensitivity measures) for IP33SRCs (as sensitivity measures) for IPSRCs (as sensitivity measures) for IP33

Effects of 3 most sensitive DAG Effects of 3 most sensitive DAG parametersparameters

Effects of 3 most sensitive DAG Effects of 3 most sensitive DAG parametersparameters

A B

C

AA k kdp2dp2 degradation of pool 2 DAG degradation of pool 2 DAG

BB k kdp1dp1 degradation of pool 1 DAG degradation of pool 1 DAG

CC k kap2ap2 production of pool 2 DAG production of pool 2 DAG

ConclusionsConclusionsConclusionsConclusions

bull We have developed a model of the UDP signaling We have developed a model of the UDP signaling pathway in RAW 2647 macrophages which can predict pathway in RAW 2647 macrophages which can predict the responses of multiple species of DAG as well as the the responses of multiple species of DAG as well as the responses of IPresponses of IP33 Ca Ca2+2+ receptor dynamics G-protein receptor dynamics G-protein activation and PIPactivation and PIP22 hydrolysis hydrolysis

bull Simplified model resultsSimplified model resultsndash We have obtained global existence uniqueness positivity We have obtained global existence uniqueness positivity

and boundedness of solutionsand boundedness of solutionsndash We have proven global stability of a unique steady state We have proven global stability of a unique steady state

within our region of interestwithin our region of interestbull Full model analysisFull model analysis

ndash Using SIMULINK we have estimated unknown rate Using SIMULINK we have estimated unknown rate parameters to obtain best fits to multiple DAG traces IPparameters to obtain best fits to multiple DAG traces IP33 and Caand Ca2+2+ all in response to 25 all in response to 25M UDPM UDP

ndash We have performed sensitivity analysis using the Latin We have performed sensitivity analysis using the Latin Hypercube sampling technique in combination with Hypercube sampling technique in combination with standardized regression coefficients to determine which standardized regression coefficients to determine which model parameters are responsible for most of the model model parameters are responsible for most of the model output uncertaintyoutput uncertainty

Future DirectionsFuture DirectionsFuture DirectionsFuture Directions

bull We have conducted multiple experiments to We have conducted multiple experiments to pharmacologically inhibit several different enzymes we pharmacologically inhibit several different enzymes we believe to play a role in this signaling pathway as believe to play a role in this signaling pathway as suggested by current known metabolic pathways and by suggested by current known metabolic pathways and by modeling resultsmodeling results

bull The next step is to perform gene knockdowns on The next step is to perform gene knockdowns on specific enzymes to verify results of inhibitor dataspecific enzymes to verify results of inhibitor data

bull The model output also suggests a time delay from The model output also suggests a time delay from receptor activation to PIPreceptor activation to PIP22 hydrolysis An upcoming task hydrolysis An upcoming task is to investigate the outcomes of adding such a delay is to investigate the outcomes of adding such a delay termterm

bull The model could be greatly enhanced by incorporating The model could be greatly enhanced by incorporating spatial dynamics so this will also be a major focus for spatial dynamics so this will also be a major focus for future researchfuture research

AcknowledgementsAcknowledgementsAcknowledgementsAcknowledgements

Mathematics DepartmentMathematics Departmentbull Hannah L Callender Hannah L Callender

PhDPhD

CollaboratorsCollaboratorsbull Alliance for Cellular SignalingAlliance for Cellular Signalingbull UT SouthwesternUT Southwestern

ndash Paul Sternweis PhDPaul Sternweis PhDndash Dianne DeCamp PhDDianne DeCamp PhD

Brown LabBrown Labbull H Alex Brown PhDH Alex Brown PhDbull Jeffrey S Forrester PhDJeffrey S Forrester PhDbull Mark Byrne PhDMark Byrne PhDbull Anita Preininger PhDAnita Preininger PhDbull Michelle ArmstrongMichelle Armstrongbull Andrew GoodmanAndrew Goodmanbull Pavlina Ivanova PhDPavlina Ivanova PhDbull Steve Milne PhDSteve Milne PhD

Modifications to the modelModifications to the modelModifications to the modelModifications to the model

bull Include an additional branch in the pathway Include an additional branch in the pathway for a second pool of DAGfor a second pool of DAG

bull Simplify CaSimplify Ca2+2+ equations for mathematical equations for mathematical analysis purposesanalysis purposes

Proposed PathwayProposed PathwayProposed PathwayProposed Pathway

bull Note We measure total cellular DAG levelsNote We measure total cellular DAG levelsbull Initial production of DAG from the hydrolysis of PIPInitial production of DAG from the hydrolysis of PIP22 in pool 1 (plasma in pool 1 (plasma

membrane) is offset by phosphorylation of DAG by DAG kinase in pool 2 membrane) is offset by phosphorylation of DAG by DAG kinase in pool 2 (Endoplasmic Reticulum Nucleus) to aid in the PI replacement pathway(Endoplasmic Reticulum Nucleus) to aid in the PI replacement pathway

bull Second wave of DAG is a result of resynthesis of PIPSecond wave of DAG is a result of resynthesis of PIP22 which is then which is then hydrolyzed to form DAG and IPhydrolyzed to form DAG and IP33

PI PIP PIP2 DAG

IP3

IP2IP

Ins

PI

CDP-DAGPADAG

PA

PCPE

PCPE

PITP

PI4K PI5K PLC DGK

DGK CDSPLC PIS

POOL 2

POOL 1

IP4

PI PIP PIP2 DAG

IP3

IP2IP

Ins

PI

CDP-DAGPADAG

PA

PCPE

PC

PITP

PI4K PI5K PLC

DGK CDSPLC PIS

IP4

LPP

LPP

Model Equations DAG (pool 1)Model Equations DAG (pool 1)Model Equations DAG (pool 1)Model Equations DAG (pool 1)

bull DAG (pool 1)DAG (pool 1) (separate ODEs for each species (separate ODEs for each species considered)considered)

bull Concentration of DAG from pool 1 at time tConcentration of DAG from pool 1 at time t

Nondimensionalized equationsNondimensionalized equations

Model Equations DAG (pool 2)Model Equations DAG (pool 2)Model Equations DAG (pool 2)Model Equations DAG (pool 2)

bull DAG (pool 2)DAG (pool 2) (separate ODEs for each species (separate ODEs for each species considered)considered)

bull Concentration of pool 2 DAG molecules at time tConcentration of pool 2 DAG molecules at time t

bull Baseline concentration of pool 2 DAGBaseline concentration of pool 2 DAG

Nondimensionalized equationsNondimensionalized equations

Simplified CaSimplified Ca2+2+ Equations EquationsSimplified CaSimplified Ca2+2+ Equations Equations

bull CaCa2+2+ modulemodule (to match experimental AfCS (to match experimental AfCS trace)trace)

0 100 200 300 400 500 6000

002

004

006

008

01

012

014

Concentration of Ca

2+ (uM)

Ca2+ response to 25uM UDP

time (s)

wherewhere

Theoretical AnalysisTheoretical AnalysisTheoretical AnalysisTheoretical Analysis

bull Existence and UniquenessExistence and Uniqueness

bull Positivity and Boundedness Positivity and Boundedness

(for biological relevance)(for biological relevance)

bull Analysis of steady state behaviorAnalysis of steady state behavior

Existence of Solutions (full model)Existence of Solutions (full model)Existence of Solutions (full model)Existence of Solutions (full model)

First we write our system of ODEs in the formFirst we write our system of ODEs in the form (1)

(2)

(1)

Uniqueness of SolutionsUniqueness of SolutionsUniqueness of SolutionsUniqueness of Solutions

Next we denote a solution of Next we denote a solution of (1)(1) by by with initial condition with initial condition

Since our system satisfies the hypotheses of Theorems 1 and 2 Since our system satisfies the hypotheses of Theorems 1 and 2 on our set of interest (for all positive time and on a positive on our set of interest (for all positive time and on a positive bounded set in space) we know bounded set in space) we know there exists a unique there exists a unique (local) solution(local) solution (ie on some finite time interval possibly (ie on some finite time interval possibly small)small)

QuestionQuestion Do the solutions remain positive and bounded (for Do the solutions remain positive and bounded (for biological significance)biological significance)

(3)

(1) (3)

Positivity and boundedness of Positivity and boundedness of solutionssolutions

Positivity and boundedness of Positivity and boundedness of solutionssolutions

bull We first use Theorem 3 to show positivity and boundedness of xWe first use Theorem 3 to show positivity and boundedness of x11 and and xx22

bull Next we use these results and the Fundamental Theorem of Calculus to Next we use these results and the Fundamental Theorem of Calculus to show positivity and boundedness for the remaining equations show positivity and boundedness for the remaining equations

bull This then ensures a global solutionThis then ensures a global solution

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical Analysis

bull Parameter Estimation in SIMULINKParameter Estimation in SIMULINK

bull Sensitivity AnalysisSensitivity Analysis

Parameter Estimation in SIMULINK (for Parameter Estimation in SIMULINK (for FULL model)FULL model)

Parameter Estimation in SIMULINK (for Parameter Estimation in SIMULINK (for FULL model)FULL model)

bull Total number of model parameters (for full model with Li and Rinzel CaTotal number of model parameters (for full model with Li and Rinzel Ca2+2+ module) = module) = 3434ndash From the literature = From the literature = 2020ndash Estimated = Estimated = 1414

bull Receptor moduleReceptor module Total = Total = 77ndash From literature = From literature = 66ndash Estimated = Estimated = 11 (k (kpp rate of receptor phosphorylation) rate of receptor phosphorylation)

bull G-protein cascadeG-protein cascade Total = Total = 99ndash From literature = From literature = 66ndash Estimated = Estimated = 3 3 (k(khydhyd k kreprep k kd3d3))

bull DAG kineticsDAG kinetics Total = Total = 5 5 (for each DAG species considered)(for each DAG species considered)ndash From literature = From literature = 00ndash Estimated = Estimated = 5 5 (all DAG parameters)(all DAG parameters)

bull CaCa2+2+ module module Total = Total = 1313ndash From literature = From literature = 88ndash Estimated = Estimated = 55

SIMULINK DetailsSIMULINK DetailsSIMULINK DetailsSIMULINK Details

bull Unknown rate parameters were estimated using SIMULINK

ndash Minimizes a user-specified cost function via a user-specified optimization method

ndash Nonlinear least squares optimization method of Levenberg-Marquardt was used to minimize a sum of squared errors cost function of the empirical observations and model predictions for IP3 Ca2+ and multiple species of DAG

bull Note Although the Gauss-Newton method is often more efficient the method of Levenberg-Marquardt has proved to be more robust

Measured vs Simulated 384 DAG and Measured vs Simulated 384 DAG and 341 DAG341 DAG

Measured vs Simulated 384 DAG and Measured vs Simulated 384 DAG and 341 DAG341 DAG

bull Timecourse of stimulation with 25 Timecourse of stimulation with 25 M UDP in RAW 2647 cellsM UDP in RAW 2647 cells

bull Solid black lines represent model simulations Solid black lines represent model simulations

bull ((aa) 384 DAG response (representative of the response of most poly ) 384 DAG response (representative of the response of most poly unsaturated fatty acid-containing DAG species) unsaturated fatty acid-containing DAG species)

bull ((bb) 341 DAG response (representative of the response of most mono- ) 341 DAG response (representative of the response of most mono- and di-unsaturated fatty acid-containing DAG species) and di-unsaturated fatty acid-containing DAG species)

bull Data points contain nine replicates performed on three different Data points contain nine replicates performed on three different experimental days with error bars = 1 SEM Units are total change in ng experimental days with error bars = 1 SEM Units are total change in ng over baseline levels in ~8x10over baseline levels in ~8x1066 cells cells

Measured vs Simulated IPMeasured vs Simulated IP33 and Ca and Ca2+2+Measured vs Simulated IPMeasured vs Simulated IP33 and Ca and Ca2+2+

bull Timecourse of stimulation with 25 Timecourse of stimulation with 25 M UDP in RAW 2647 cellsM UDP in RAW 2647 cells

bull Solid black lines represent model simulationsSolid black lines represent model simulations

bull ((cc) IP) IP33 response in pmols per ~35x10 response in pmols per ~35x1055 cells cells

bull Points in (c) represent the average of four experiments and error Points in (c) represent the average of four experiments and error bars are 1 SEM bars are 1 SEM

bull ((dd) Ca) Ca2+2+ response in response in M Red curve is a representative CaM Red curve is a representative Ca2+2+ trace trace taken from the UDP experiments within the AfCS single ligand taken from the UDP experiments within the AfCS single ligand screenscreen

Model Simulations P2YModel Simulations P2Y66 G GGTP GTP PIPPIP22

Model Simulations P2YModel Simulations P2Y66 G GGTP GTP PIPPIP22

bull ((aa) = total P2Y) = total P2Y66 activated (solid activated (solid line) and inactivated (dashed line) and inactivated (dashed line) surface receptorsline) surface receptors

bull ((bb) = total G) = total GGTP GTP

bull ((cc) = total PIP) = total PIP22 available for available for hydrolysishydrolysis

P2Y6 from 25M UDP GGTP from 25M UDP

PIP2 from 25M UDP

Sensitivity AnalysisSensitivity AnalysisSensitivity AnalysisSensitivity Analysis

bull Sensitivity analysis techniques are valuable tools Sensitivity analysis techniques are valuable tools designed to answer questions regarding which of the designed to answer questions regarding which of the uncertain input variables is more important in uncertain input variables is more important in determining the uncertainty in our output determining the uncertainty in our output

bull Likewise sensitivity analysis can provide insight into Likewise sensitivity analysis can provide insight into which parameter should be studied in more detail in which parameter should be studied in more detail in order to reduce the most variance in the model output order to reduce the most variance in the model output

bull The ability to answer these types of questions could The ability to answer these types of questions could lead to important insight into the design of new lead to important insight into the design of new experiments and in determining which experiments experiments and in determining which experiments would give us the most valuable informationwould give us the most valuable information

Sampling MethodSampling MethodSampling MethodSampling Method

bull Generate a random sample of our space of input Generate a random sample of our space of input variables over a ten percent variation from each variables over a ten percent variation from each parameters nominal value using the parameters nominal value using the Latin Hypercube Latin Hypercube Sampling (LHS) methodSampling (LHS) method

bull Uses Uses Standardized Regression Coefficients (SRCs) Standardized Regression Coefficients (SRCs) obtained by performing multiple linear regression obtained by performing multiple linear regression analysisanalysisndash offers a measure of sensitivity that is multi-dimensionally offers a measure of sensitivity that is multi-dimensionally

averaged over the entire space of parameter values averaged over the entire space of parameter values ndash SRCs give insight into degree of nonlinearity in the model SRCs give insight into degree of nonlinearity in the model

Computing RComputing Ryy22 values valuesComputing RComputing Ryy22 values values

bull SRCs are only reliable measures of sensitivity when SRCs are only reliable measures of sensitivity when degree of nonlinearity is ldquosmallrdquodegree of nonlinearity is ldquosmallrdquo

bull Use model coefficients of determination RUse model coefficients of determination Ryy22 given by given by

bull where ywhere yii is the estimate of y is the estimate of yii obtained from the regression obtained from the regression modelmodel

bull RRyy22 ge 07 ensures SRCs are good sensitivity measures ge 07 ensures SRCs are good sensitivity measures

SRCs (as sensitivity measures) for SRCs (as sensitivity measures) for DAGDAG

SRCs (as sensitivity measures) for SRCs (as sensitivity measures) for DAGDAG

SRCs (as sensitivity measures) for IPSRCs (as sensitivity measures) for IP33SRCs (as sensitivity measures) for IPSRCs (as sensitivity measures) for IP33

Effects of 3 most sensitive DAG Effects of 3 most sensitive DAG parametersparameters

Effects of 3 most sensitive DAG Effects of 3 most sensitive DAG parametersparameters

A B

C

AA k kdp2dp2 degradation of pool 2 DAG degradation of pool 2 DAG

BB k kdp1dp1 degradation of pool 1 DAG degradation of pool 1 DAG

CC k kap2ap2 production of pool 2 DAG production of pool 2 DAG

ConclusionsConclusionsConclusionsConclusions

bull We have developed a model of the UDP signaling We have developed a model of the UDP signaling pathway in RAW 2647 macrophages which can predict pathway in RAW 2647 macrophages which can predict the responses of multiple species of DAG as well as the the responses of multiple species of DAG as well as the responses of IPresponses of IP33 Ca Ca2+2+ receptor dynamics G-protein receptor dynamics G-protein activation and PIPactivation and PIP22 hydrolysis hydrolysis

bull Simplified model resultsSimplified model resultsndash We have obtained global existence uniqueness positivity We have obtained global existence uniqueness positivity

and boundedness of solutionsand boundedness of solutionsndash We have proven global stability of a unique steady state We have proven global stability of a unique steady state

within our region of interestwithin our region of interestbull Full model analysisFull model analysis

ndash Using SIMULINK we have estimated unknown rate Using SIMULINK we have estimated unknown rate parameters to obtain best fits to multiple DAG traces IPparameters to obtain best fits to multiple DAG traces IP33 and Caand Ca2+2+ all in response to 25 all in response to 25M UDPM UDP

ndash We have performed sensitivity analysis using the Latin We have performed sensitivity analysis using the Latin Hypercube sampling technique in combination with Hypercube sampling technique in combination with standardized regression coefficients to determine which standardized regression coefficients to determine which model parameters are responsible for most of the model model parameters are responsible for most of the model output uncertaintyoutput uncertainty

Future DirectionsFuture DirectionsFuture DirectionsFuture Directions

bull We have conducted multiple experiments to We have conducted multiple experiments to pharmacologically inhibit several different enzymes we pharmacologically inhibit several different enzymes we believe to play a role in this signaling pathway as believe to play a role in this signaling pathway as suggested by current known metabolic pathways and by suggested by current known metabolic pathways and by modeling resultsmodeling results

bull The next step is to perform gene knockdowns on The next step is to perform gene knockdowns on specific enzymes to verify results of inhibitor dataspecific enzymes to verify results of inhibitor data

bull The model output also suggests a time delay from The model output also suggests a time delay from receptor activation to PIPreceptor activation to PIP22 hydrolysis An upcoming task hydrolysis An upcoming task is to investigate the outcomes of adding such a delay is to investigate the outcomes of adding such a delay termterm

bull The model could be greatly enhanced by incorporating The model could be greatly enhanced by incorporating spatial dynamics so this will also be a major focus for spatial dynamics so this will also be a major focus for future researchfuture research

AcknowledgementsAcknowledgementsAcknowledgementsAcknowledgements

Mathematics DepartmentMathematics Departmentbull Hannah L Callender Hannah L Callender

PhDPhD

CollaboratorsCollaboratorsbull Alliance for Cellular SignalingAlliance for Cellular Signalingbull UT SouthwesternUT Southwestern

ndash Paul Sternweis PhDPaul Sternweis PhDndash Dianne DeCamp PhDDianne DeCamp PhD

Brown LabBrown Labbull H Alex Brown PhDH Alex Brown PhDbull Jeffrey S Forrester PhDJeffrey S Forrester PhDbull Mark Byrne PhDMark Byrne PhDbull Anita Preininger PhDAnita Preininger PhDbull Michelle ArmstrongMichelle Armstrongbull Andrew GoodmanAndrew Goodmanbull Pavlina Ivanova PhDPavlina Ivanova PhDbull Steve Milne PhDSteve Milne PhD

Proposed PathwayProposed PathwayProposed PathwayProposed Pathway

bull Note We measure total cellular DAG levelsNote We measure total cellular DAG levelsbull Initial production of DAG from the hydrolysis of PIPInitial production of DAG from the hydrolysis of PIP22 in pool 1 (plasma in pool 1 (plasma

membrane) is offset by phosphorylation of DAG by DAG kinase in pool 2 membrane) is offset by phosphorylation of DAG by DAG kinase in pool 2 (Endoplasmic Reticulum Nucleus) to aid in the PI replacement pathway(Endoplasmic Reticulum Nucleus) to aid in the PI replacement pathway

bull Second wave of DAG is a result of resynthesis of PIPSecond wave of DAG is a result of resynthesis of PIP22 which is then which is then hydrolyzed to form DAG and IPhydrolyzed to form DAG and IP33

PI PIP PIP2 DAG

IP3

IP2IP

Ins

PI

CDP-DAGPADAG

PA

PCPE

PCPE

PITP

PI4K PI5K PLC DGK

DGK CDSPLC PIS

POOL 2

POOL 1

IP4

PI PIP PIP2 DAG

IP3

IP2IP

Ins

PI

CDP-DAGPADAG

PA

PCPE

PC

PITP

PI4K PI5K PLC

DGK CDSPLC PIS

IP4

LPP

LPP

Model Equations DAG (pool 1)Model Equations DAG (pool 1)Model Equations DAG (pool 1)Model Equations DAG (pool 1)

bull DAG (pool 1)DAG (pool 1) (separate ODEs for each species (separate ODEs for each species considered)considered)

bull Concentration of DAG from pool 1 at time tConcentration of DAG from pool 1 at time t

Nondimensionalized equationsNondimensionalized equations

Model Equations DAG (pool 2)Model Equations DAG (pool 2)Model Equations DAG (pool 2)Model Equations DAG (pool 2)

bull DAG (pool 2)DAG (pool 2) (separate ODEs for each species (separate ODEs for each species considered)considered)

bull Concentration of pool 2 DAG molecules at time tConcentration of pool 2 DAG molecules at time t

bull Baseline concentration of pool 2 DAGBaseline concentration of pool 2 DAG

Nondimensionalized equationsNondimensionalized equations

Simplified CaSimplified Ca2+2+ Equations EquationsSimplified CaSimplified Ca2+2+ Equations Equations

bull CaCa2+2+ modulemodule (to match experimental AfCS (to match experimental AfCS trace)trace)

0 100 200 300 400 500 6000

002

004

006

008

01

012

014

Concentration of Ca

2+ (uM)

Ca2+ response to 25uM UDP

time (s)

wherewhere

Theoretical AnalysisTheoretical AnalysisTheoretical AnalysisTheoretical Analysis

bull Existence and UniquenessExistence and Uniqueness

bull Positivity and Boundedness Positivity and Boundedness

(for biological relevance)(for biological relevance)

bull Analysis of steady state behaviorAnalysis of steady state behavior

Existence of Solutions (full model)Existence of Solutions (full model)Existence of Solutions (full model)Existence of Solutions (full model)

First we write our system of ODEs in the formFirst we write our system of ODEs in the form (1)

(2)

(1)

Uniqueness of SolutionsUniqueness of SolutionsUniqueness of SolutionsUniqueness of Solutions

Next we denote a solution of Next we denote a solution of (1)(1) by by with initial condition with initial condition

Since our system satisfies the hypotheses of Theorems 1 and 2 Since our system satisfies the hypotheses of Theorems 1 and 2 on our set of interest (for all positive time and on a positive on our set of interest (for all positive time and on a positive bounded set in space) we know bounded set in space) we know there exists a unique there exists a unique (local) solution(local) solution (ie on some finite time interval possibly (ie on some finite time interval possibly small)small)

QuestionQuestion Do the solutions remain positive and bounded (for Do the solutions remain positive and bounded (for biological significance)biological significance)

(3)

(1) (3)

Positivity and boundedness of Positivity and boundedness of solutionssolutions

Positivity and boundedness of Positivity and boundedness of solutionssolutions

bull We first use Theorem 3 to show positivity and boundedness of xWe first use Theorem 3 to show positivity and boundedness of x11 and and xx22

bull Next we use these results and the Fundamental Theorem of Calculus to Next we use these results and the Fundamental Theorem of Calculus to show positivity and boundedness for the remaining equations show positivity and boundedness for the remaining equations

bull This then ensures a global solutionThis then ensures a global solution

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical Analysis

bull Parameter Estimation in SIMULINKParameter Estimation in SIMULINK

bull Sensitivity AnalysisSensitivity Analysis

Parameter Estimation in SIMULINK (for Parameter Estimation in SIMULINK (for FULL model)FULL model)

Parameter Estimation in SIMULINK (for Parameter Estimation in SIMULINK (for FULL model)FULL model)

bull Total number of model parameters (for full model with Li and Rinzel CaTotal number of model parameters (for full model with Li and Rinzel Ca2+2+ module) = module) = 3434ndash From the literature = From the literature = 2020ndash Estimated = Estimated = 1414

bull Receptor moduleReceptor module Total = Total = 77ndash From literature = From literature = 66ndash Estimated = Estimated = 11 (k (kpp rate of receptor phosphorylation) rate of receptor phosphorylation)

bull G-protein cascadeG-protein cascade Total = Total = 99ndash From literature = From literature = 66ndash Estimated = Estimated = 3 3 (k(khydhyd k kreprep k kd3d3))

bull DAG kineticsDAG kinetics Total = Total = 5 5 (for each DAG species considered)(for each DAG species considered)ndash From literature = From literature = 00ndash Estimated = Estimated = 5 5 (all DAG parameters)(all DAG parameters)

bull CaCa2+2+ module module Total = Total = 1313ndash From literature = From literature = 88ndash Estimated = Estimated = 55

SIMULINK DetailsSIMULINK DetailsSIMULINK DetailsSIMULINK Details

bull Unknown rate parameters were estimated using SIMULINK

ndash Minimizes a user-specified cost function via a user-specified optimization method

ndash Nonlinear least squares optimization method of Levenberg-Marquardt was used to minimize a sum of squared errors cost function of the empirical observations and model predictions for IP3 Ca2+ and multiple species of DAG

bull Note Although the Gauss-Newton method is often more efficient the method of Levenberg-Marquardt has proved to be more robust

Measured vs Simulated 384 DAG and Measured vs Simulated 384 DAG and 341 DAG341 DAG

Measured vs Simulated 384 DAG and Measured vs Simulated 384 DAG and 341 DAG341 DAG

bull Timecourse of stimulation with 25 Timecourse of stimulation with 25 M UDP in RAW 2647 cellsM UDP in RAW 2647 cells

bull Solid black lines represent model simulations Solid black lines represent model simulations

bull ((aa) 384 DAG response (representative of the response of most poly ) 384 DAG response (representative of the response of most poly unsaturated fatty acid-containing DAG species) unsaturated fatty acid-containing DAG species)

bull ((bb) 341 DAG response (representative of the response of most mono- ) 341 DAG response (representative of the response of most mono- and di-unsaturated fatty acid-containing DAG species) and di-unsaturated fatty acid-containing DAG species)

bull Data points contain nine replicates performed on three different Data points contain nine replicates performed on three different experimental days with error bars = 1 SEM Units are total change in ng experimental days with error bars = 1 SEM Units are total change in ng over baseline levels in ~8x10over baseline levels in ~8x1066 cells cells

Measured vs Simulated IPMeasured vs Simulated IP33 and Ca and Ca2+2+Measured vs Simulated IPMeasured vs Simulated IP33 and Ca and Ca2+2+

bull Timecourse of stimulation with 25 Timecourse of stimulation with 25 M UDP in RAW 2647 cellsM UDP in RAW 2647 cells

bull Solid black lines represent model simulationsSolid black lines represent model simulations

bull ((cc) IP) IP33 response in pmols per ~35x10 response in pmols per ~35x1055 cells cells

bull Points in (c) represent the average of four experiments and error Points in (c) represent the average of four experiments and error bars are 1 SEM bars are 1 SEM

bull ((dd) Ca) Ca2+2+ response in response in M Red curve is a representative CaM Red curve is a representative Ca2+2+ trace trace taken from the UDP experiments within the AfCS single ligand taken from the UDP experiments within the AfCS single ligand screenscreen

Model Simulations P2YModel Simulations P2Y66 G GGTP GTP PIPPIP22

Model Simulations P2YModel Simulations P2Y66 G GGTP GTP PIPPIP22

bull ((aa) = total P2Y) = total P2Y66 activated (solid activated (solid line) and inactivated (dashed line) and inactivated (dashed line) surface receptorsline) surface receptors

bull ((bb) = total G) = total GGTP GTP

bull ((cc) = total PIP) = total PIP22 available for available for hydrolysishydrolysis

P2Y6 from 25M UDP GGTP from 25M UDP

PIP2 from 25M UDP

Sensitivity AnalysisSensitivity AnalysisSensitivity AnalysisSensitivity Analysis

bull Sensitivity analysis techniques are valuable tools Sensitivity analysis techniques are valuable tools designed to answer questions regarding which of the designed to answer questions regarding which of the uncertain input variables is more important in uncertain input variables is more important in determining the uncertainty in our output determining the uncertainty in our output

bull Likewise sensitivity analysis can provide insight into Likewise sensitivity analysis can provide insight into which parameter should be studied in more detail in which parameter should be studied in more detail in order to reduce the most variance in the model output order to reduce the most variance in the model output

bull The ability to answer these types of questions could The ability to answer these types of questions could lead to important insight into the design of new lead to important insight into the design of new experiments and in determining which experiments experiments and in determining which experiments would give us the most valuable informationwould give us the most valuable information

Sampling MethodSampling MethodSampling MethodSampling Method

bull Generate a random sample of our space of input Generate a random sample of our space of input variables over a ten percent variation from each variables over a ten percent variation from each parameters nominal value using the parameters nominal value using the Latin Hypercube Latin Hypercube Sampling (LHS) methodSampling (LHS) method

bull Uses Uses Standardized Regression Coefficients (SRCs) Standardized Regression Coefficients (SRCs) obtained by performing multiple linear regression obtained by performing multiple linear regression analysisanalysisndash offers a measure of sensitivity that is multi-dimensionally offers a measure of sensitivity that is multi-dimensionally

averaged over the entire space of parameter values averaged over the entire space of parameter values ndash SRCs give insight into degree of nonlinearity in the model SRCs give insight into degree of nonlinearity in the model

Computing RComputing Ryy22 values valuesComputing RComputing Ryy22 values values

bull SRCs are only reliable measures of sensitivity when SRCs are only reliable measures of sensitivity when degree of nonlinearity is ldquosmallrdquodegree of nonlinearity is ldquosmallrdquo

bull Use model coefficients of determination RUse model coefficients of determination Ryy22 given by given by

bull where ywhere yii is the estimate of y is the estimate of yii obtained from the regression obtained from the regression modelmodel

bull RRyy22 ge 07 ensures SRCs are good sensitivity measures ge 07 ensures SRCs are good sensitivity measures

SRCs (as sensitivity measures) for SRCs (as sensitivity measures) for DAGDAG

SRCs (as sensitivity measures) for SRCs (as sensitivity measures) for DAGDAG

SRCs (as sensitivity measures) for IPSRCs (as sensitivity measures) for IP33SRCs (as sensitivity measures) for IPSRCs (as sensitivity measures) for IP33

Effects of 3 most sensitive DAG Effects of 3 most sensitive DAG parametersparameters

Effects of 3 most sensitive DAG Effects of 3 most sensitive DAG parametersparameters

A B

C

AA k kdp2dp2 degradation of pool 2 DAG degradation of pool 2 DAG

BB k kdp1dp1 degradation of pool 1 DAG degradation of pool 1 DAG

CC k kap2ap2 production of pool 2 DAG production of pool 2 DAG

ConclusionsConclusionsConclusionsConclusions

bull We have developed a model of the UDP signaling We have developed a model of the UDP signaling pathway in RAW 2647 macrophages which can predict pathway in RAW 2647 macrophages which can predict the responses of multiple species of DAG as well as the the responses of multiple species of DAG as well as the responses of IPresponses of IP33 Ca Ca2+2+ receptor dynamics G-protein receptor dynamics G-protein activation and PIPactivation and PIP22 hydrolysis hydrolysis

bull Simplified model resultsSimplified model resultsndash We have obtained global existence uniqueness positivity We have obtained global existence uniqueness positivity

and boundedness of solutionsand boundedness of solutionsndash We have proven global stability of a unique steady state We have proven global stability of a unique steady state

within our region of interestwithin our region of interestbull Full model analysisFull model analysis

ndash Using SIMULINK we have estimated unknown rate Using SIMULINK we have estimated unknown rate parameters to obtain best fits to multiple DAG traces IPparameters to obtain best fits to multiple DAG traces IP33 and Caand Ca2+2+ all in response to 25 all in response to 25M UDPM UDP

ndash We have performed sensitivity analysis using the Latin We have performed sensitivity analysis using the Latin Hypercube sampling technique in combination with Hypercube sampling technique in combination with standardized regression coefficients to determine which standardized regression coefficients to determine which model parameters are responsible for most of the model model parameters are responsible for most of the model output uncertaintyoutput uncertainty

Future DirectionsFuture DirectionsFuture DirectionsFuture Directions

bull We have conducted multiple experiments to We have conducted multiple experiments to pharmacologically inhibit several different enzymes we pharmacologically inhibit several different enzymes we believe to play a role in this signaling pathway as believe to play a role in this signaling pathway as suggested by current known metabolic pathways and by suggested by current known metabolic pathways and by modeling resultsmodeling results

bull The next step is to perform gene knockdowns on The next step is to perform gene knockdowns on specific enzymes to verify results of inhibitor dataspecific enzymes to verify results of inhibitor data

bull The model output also suggests a time delay from The model output also suggests a time delay from receptor activation to PIPreceptor activation to PIP22 hydrolysis An upcoming task hydrolysis An upcoming task is to investigate the outcomes of adding such a delay is to investigate the outcomes of adding such a delay termterm

bull The model could be greatly enhanced by incorporating The model could be greatly enhanced by incorporating spatial dynamics so this will also be a major focus for spatial dynamics so this will also be a major focus for future researchfuture research

AcknowledgementsAcknowledgementsAcknowledgementsAcknowledgements

Mathematics DepartmentMathematics Departmentbull Hannah L Callender Hannah L Callender

PhDPhD

CollaboratorsCollaboratorsbull Alliance for Cellular SignalingAlliance for Cellular Signalingbull UT SouthwesternUT Southwestern

ndash Paul Sternweis PhDPaul Sternweis PhDndash Dianne DeCamp PhDDianne DeCamp PhD

Brown LabBrown Labbull H Alex Brown PhDH Alex Brown PhDbull Jeffrey S Forrester PhDJeffrey S Forrester PhDbull Mark Byrne PhDMark Byrne PhDbull Anita Preininger PhDAnita Preininger PhDbull Michelle ArmstrongMichelle Armstrongbull Andrew GoodmanAndrew Goodmanbull Pavlina Ivanova PhDPavlina Ivanova PhDbull Steve Milne PhDSteve Milne PhD

Model Equations DAG (pool 1)Model Equations DAG (pool 1)Model Equations DAG (pool 1)Model Equations DAG (pool 1)

bull DAG (pool 1)DAG (pool 1) (separate ODEs for each species (separate ODEs for each species considered)considered)

bull Concentration of DAG from pool 1 at time tConcentration of DAG from pool 1 at time t

Nondimensionalized equationsNondimensionalized equations

Model Equations DAG (pool 2)Model Equations DAG (pool 2)Model Equations DAG (pool 2)Model Equations DAG (pool 2)

bull DAG (pool 2)DAG (pool 2) (separate ODEs for each species (separate ODEs for each species considered)considered)

bull Concentration of pool 2 DAG molecules at time tConcentration of pool 2 DAG molecules at time t

bull Baseline concentration of pool 2 DAGBaseline concentration of pool 2 DAG

Nondimensionalized equationsNondimensionalized equations

Simplified CaSimplified Ca2+2+ Equations EquationsSimplified CaSimplified Ca2+2+ Equations Equations

bull CaCa2+2+ modulemodule (to match experimental AfCS (to match experimental AfCS trace)trace)

0 100 200 300 400 500 6000

002

004

006

008

01

012

014

Concentration of Ca

2+ (uM)

Ca2+ response to 25uM UDP

time (s)

wherewhere

Theoretical AnalysisTheoretical AnalysisTheoretical AnalysisTheoretical Analysis

bull Existence and UniquenessExistence and Uniqueness

bull Positivity and Boundedness Positivity and Boundedness

(for biological relevance)(for biological relevance)

bull Analysis of steady state behaviorAnalysis of steady state behavior

Existence of Solutions (full model)Existence of Solutions (full model)Existence of Solutions (full model)Existence of Solutions (full model)

First we write our system of ODEs in the formFirst we write our system of ODEs in the form (1)

(2)

(1)

Uniqueness of SolutionsUniqueness of SolutionsUniqueness of SolutionsUniqueness of Solutions

Next we denote a solution of Next we denote a solution of (1)(1) by by with initial condition with initial condition

Since our system satisfies the hypotheses of Theorems 1 and 2 Since our system satisfies the hypotheses of Theorems 1 and 2 on our set of interest (for all positive time and on a positive on our set of interest (for all positive time and on a positive bounded set in space) we know bounded set in space) we know there exists a unique there exists a unique (local) solution(local) solution (ie on some finite time interval possibly (ie on some finite time interval possibly small)small)

QuestionQuestion Do the solutions remain positive and bounded (for Do the solutions remain positive and bounded (for biological significance)biological significance)

(3)

(1) (3)

Positivity and boundedness of Positivity and boundedness of solutionssolutions

Positivity and boundedness of Positivity and boundedness of solutionssolutions

bull We first use Theorem 3 to show positivity and boundedness of xWe first use Theorem 3 to show positivity and boundedness of x11 and and xx22

bull Next we use these results and the Fundamental Theorem of Calculus to Next we use these results and the Fundamental Theorem of Calculus to show positivity and boundedness for the remaining equations show positivity and boundedness for the remaining equations

bull This then ensures a global solutionThis then ensures a global solution

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical Analysis

bull Parameter Estimation in SIMULINKParameter Estimation in SIMULINK

bull Sensitivity AnalysisSensitivity Analysis

Parameter Estimation in SIMULINK (for Parameter Estimation in SIMULINK (for FULL model)FULL model)

Parameter Estimation in SIMULINK (for Parameter Estimation in SIMULINK (for FULL model)FULL model)

bull Total number of model parameters (for full model with Li and Rinzel CaTotal number of model parameters (for full model with Li and Rinzel Ca2+2+ module) = module) = 3434ndash From the literature = From the literature = 2020ndash Estimated = Estimated = 1414

bull Receptor moduleReceptor module Total = Total = 77ndash From literature = From literature = 66ndash Estimated = Estimated = 11 (k (kpp rate of receptor phosphorylation) rate of receptor phosphorylation)

bull G-protein cascadeG-protein cascade Total = Total = 99ndash From literature = From literature = 66ndash Estimated = Estimated = 3 3 (k(khydhyd k kreprep k kd3d3))

bull DAG kineticsDAG kinetics Total = Total = 5 5 (for each DAG species considered)(for each DAG species considered)ndash From literature = From literature = 00ndash Estimated = Estimated = 5 5 (all DAG parameters)(all DAG parameters)

bull CaCa2+2+ module module Total = Total = 1313ndash From literature = From literature = 88ndash Estimated = Estimated = 55

SIMULINK DetailsSIMULINK DetailsSIMULINK DetailsSIMULINK Details

bull Unknown rate parameters were estimated using SIMULINK

ndash Minimizes a user-specified cost function via a user-specified optimization method

ndash Nonlinear least squares optimization method of Levenberg-Marquardt was used to minimize a sum of squared errors cost function of the empirical observations and model predictions for IP3 Ca2+ and multiple species of DAG

bull Note Although the Gauss-Newton method is often more efficient the method of Levenberg-Marquardt has proved to be more robust

Measured vs Simulated 384 DAG and Measured vs Simulated 384 DAG and 341 DAG341 DAG

Measured vs Simulated 384 DAG and Measured vs Simulated 384 DAG and 341 DAG341 DAG

bull Timecourse of stimulation with 25 Timecourse of stimulation with 25 M UDP in RAW 2647 cellsM UDP in RAW 2647 cells

bull Solid black lines represent model simulations Solid black lines represent model simulations

bull ((aa) 384 DAG response (representative of the response of most poly ) 384 DAG response (representative of the response of most poly unsaturated fatty acid-containing DAG species) unsaturated fatty acid-containing DAG species)

bull ((bb) 341 DAG response (representative of the response of most mono- ) 341 DAG response (representative of the response of most mono- and di-unsaturated fatty acid-containing DAG species) and di-unsaturated fatty acid-containing DAG species)

bull Data points contain nine replicates performed on three different Data points contain nine replicates performed on three different experimental days with error bars = 1 SEM Units are total change in ng experimental days with error bars = 1 SEM Units are total change in ng over baseline levels in ~8x10over baseline levels in ~8x1066 cells cells

Measured vs Simulated IPMeasured vs Simulated IP33 and Ca and Ca2+2+Measured vs Simulated IPMeasured vs Simulated IP33 and Ca and Ca2+2+

bull Timecourse of stimulation with 25 Timecourse of stimulation with 25 M UDP in RAW 2647 cellsM UDP in RAW 2647 cells

bull Solid black lines represent model simulationsSolid black lines represent model simulations

bull ((cc) IP) IP33 response in pmols per ~35x10 response in pmols per ~35x1055 cells cells

bull Points in (c) represent the average of four experiments and error Points in (c) represent the average of four experiments and error bars are 1 SEM bars are 1 SEM

bull ((dd) Ca) Ca2+2+ response in response in M Red curve is a representative CaM Red curve is a representative Ca2+2+ trace trace taken from the UDP experiments within the AfCS single ligand taken from the UDP experiments within the AfCS single ligand screenscreen

Model Simulations P2YModel Simulations P2Y66 G GGTP GTP PIPPIP22

Model Simulations P2YModel Simulations P2Y66 G GGTP GTP PIPPIP22

bull ((aa) = total P2Y) = total P2Y66 activated (solid activated (solid line) and inactivated (dashed line) and inactivated (dashed line) surface receptorsline) surface receptors

bull ((bb) = total G) = total GGTP GTP

bull ((cc) = total PIP) = total PIP22 available for available for hydrolysishydrolysis

P2Y6 from 25M UDP GGTP from 25M UDP

PIP2 from 25M UDP

Sensitivity AnalysisSensitivity AnalysisSensitivity AnalysisSensitivity Analysis

bull Sensitivity analysis techniques are valuable tools Sensitivity analysis techniques are valuable tools designed to answer questions regarding which of the designed to answer questions regarding which of the uncertain input variables is more important in uncertain input variables is more important in determining the uncertainty in our output determining the uncertainty in our output

bull Likewise sensitivity analysis can provide insight into Likewise sensitivity analysis can provide insight into which parameter should be studied in more detail in which parameter should be studied in more detail in order to reduce the most variance in the model output order to reduce the most variance in the model output

bull The ability to answer these types of questions could The ability to answer these types of questions could lead to important insight into the design of new lead to important insight into the design of new experiments and in determining which experiments experiments and in determining which experiments would give us the most valuable informationwould give us the most valuable information

Sampling MethodSampling MethodSampling MethodSampling Method

bull Generate a random sample of our space of input Generate a random sample of our space of input variables over a ten percent variation from each variables over a ten percent variation from each parameters nominal value using the parameters nominal value using the Latin Hypercube Latin Hypercube Sampling (LHS) methodSampling (LHS) method

bull Uses Uses Standardized Regression Coefficients (SRCs) Standardized Regression Coefficients (SRCs) obtained by performing multiple linear regression obtained by performing multiple linear regression analysisanalysisndash offers a measure of sensitivity that is multi-dimensionally offers a measure of sensitivity that is multi-dimensionally

averaged over the entire space of parameter values averaged over the entire space of parameter values ndash SRCs give insight into degree of nonlinearity in the model SRCs give insight into degree of nonlinearity in the model

Computing RComputing Ryy22 values valuesComputing RComputing Ryy22 values values

bull SRCs are only reliable measures of sensitivity when SRCs are only reliable measures of sensitivity when degree of nonlinearity is ldquosmallrdquodegree of nonlinearity is ldquosmallrdquo

bull Use model coefficients of determination RUse model coefficients of determination Ryy22 given by given by

bull where ywhere yii is the estimate of y is the estimate of yii obtained from the regression obtained from the regression modelmodel

bull RRyy22 ge 07 ensures SRCs are good sensitivity measures ge 07 ensures SRCs are good sensitivity measures

SRCs (as sensitivity measures) for SRCs (as sensitivity measures) for DAGDAG

SRCs (as sensitivity measures) for SRCs (as sensitivity measures) for DAGDAG

SRCs (as sensitivity measures) for IPSRCs (as sensitivity measures) for IP33SRCs (as sensitivity measures) for IPSRCs (as sensitivity measures) for IP33

Effects of 3 most sensitive DAG Effects of 3 most sensitive DAG parametersparameters

Effects of 3 most sensitive DAG Effects of 3 most sensitive DAG parametersparameters

A B

C

AA k kdp2dp2 degradation of pool 2 DAG degradation of pool 2 DAG

BB k kdp1dp1 degradation of pool 1 DAG degradation of pool 1 DAG

CC k kap2ap2 production of pool 2 DAG production of pool 2 DAG

ConclusionsConclusionsConclusionsConclusions

bull We have developed a model of the UDP signaling We have developed a model of the UDP signaling pathway in RAW 2647 macrophages which can predict pathway in RAW 2647 macrophages which can predict the responses of multiple species of DAG as well as the the responses of multiple species of DAG as well as the responses of IPresponses of IP33 Ca Ca2+2+ receptor dynamics G-protein receptor dynamics G-protein activation and PIPactivation and PIP22 hydrolysis hydrolysis

bull Simplified model resultsSimplified model resultsndash We have obtained global existence uniqueness positivity We have obtained global existence uniqueness positivity

and boundedness of solutionsand boundedness of solutionsndash We have proven global stability of a unique steady state We have proven global stability of a unique steady state

within our region of interestwithin our region of interestbull Full model analysisFull model analysis

ndash Using SIMULINK we have estimated unknown rate Using SIMULINK we have estimated unknown rate parameters to obtain best fits to multiple DAG traces IPparameters to obtain best fits to multiple DAG traces IP33 and Caand Ca2+2+ all in response to 25 all in response to 25M UDPM UDP

ndash We have performed sensitivity analysis using the Latin We have performed sensitivity analysis using the Latin Hypercube sampling technique in combination with Hypercube sampling technique in combination with standardized regression coefficients to determine which standardized regression coefficients to determine which model parameters are responsible for most of the model model parameters are responsible for most of the model output uncertaintyoutput uncertainty

Future DirectionsFuture DirectionsFuture DirectionsFuture Directions

bull We have conducted multiple experiments to We have conducted multiple experiments to pharmacologically inhibit several different enzymes we pharmacologically inhibit several different enzymes we believe to play a role in this signaling pathway as believe to play a role in this signaling pathway as suggested by current known metabolic pathways and by suggested by current known metabolic pathways and by modeling resultsmodeling results

bull The next step is to perform gene knockdowns on The next step is to perform gene knockdowns on specific enzymes to verify results of inhibitor dataspecific enzymes to verify results of inhibitor data

bull The model output also suggests a time delay from The model output also suggests a time delay from receptor activation to PIPreceptor activation to PIP22 hydrolysis An upcoming task hydrolysis An upcoming task is to investigate the outcomes of adding such a delay is to investigate the outcomes of adding such a delay termterm

bull The model could be greatly enhanced by incorporating The model could be greatly enhanced by incorporating spatial dynamics so this will also be a major focus for spatial dynamics so this will also be a major focus for future researchfuture research

AcknowledgementsAcknowledgementsAcknowledgementsAcknowledgements

Mathematics DepartmentMathematics Departmentbull Hannah L Callender Hannah L Callender

PhDPhD

CollaboratorsCollaboratorsbull Alliance for Cellular SignalingAlliance for Cellular Signalingbull UT SouthwesternUT Southwestern

ndash Paul Sternweis PhDPaul Sternweis PhDndash Dianne DeCamp PhDDianne DeCamp PhD

Brown LabBrown Labbull H Alex Brown PhDH Alex Brown PhDbull Jeffrey S Forrester PhDJeffrey S Forrester PhDbull Mark Byrne PhDMark Byrne PhDbull Anita Preininger PhDAnita Preininger PhDbull Michelle ArmstrongMichelle Armstrongbull Andrew GoodmanAndrew Goodmanbull Pavlina Ivanova PhDPavlina Ivanova PhDbull Steve Milne PhDSteve Milne PhD

Model Equations DAG (pool 2)Model Equations DAG (pool 2)Model Equations DAG (pool 2)Model Equations DAG (pool 2)

bull DAG (pool 2)DAG (pool 2) (separate ODEs for each species (separate ODEs for each species considered)considered)

bull Concentration of pool 2 DAG molecules at time tConcentration of pool 2 DAG molecules at time t

bull Baseline concentration of pool 2 DAGBaseline concentration of pool 2 DAG

Nondimensionalized equationsNondimensionalized equations

Simplified CaSimplified Ca2+2+ Equations EquationsSimplified CaSimplified Ca2+2+ Equations Equations

bull CaCa2+2+ modulemodule (to match experimental AfCS (to match experimental AfCS trace)trace)

0 100 200 300 400 500 6000

002

004

006

008

01

012

014

Concentration of Ca

2+ (uM)

Ca2+ response to 25uM UDP

time (s)

wherewhere

Theoretical AnalysisTheoretical AnalysisTheoretical AnalysisTheoretical Analysis

bull Existence and UniquenessExistence and Uniqueness

bull Positivity and Boundedness Positivity and Boundedness

(for biological relevance)(for biological relevance)

bull Analysis of steady state behaviorAnalysis of steady state behavior

Existence of Solutions (full model)Existence of Solutions (full model)Existence of Solutions (full model)Existence of Solutions (full model)

First we write our system of ODEs in the formFirst we write our system of ODEs in the form (1)

(2)

(1)

Uniqueness of SolutionsUniqueness of SolutionsUniqueness of SolutionsUniqueness of Solutions

Next we denote a solution of Next we denote a solution of (1)(1) by by with initial condition with initial condition

Since our system satisfies the hypotheses of Theorems 1 and 2 Since our system satisfies the hypotheses of Theorems 1 and 2 on our set of interest (for all positive time and on a positive on our set of interest (for all positive time and on a positive bounded set in space) we know bounded set in space) we know there exists a unique there exists a unique (local) solution(local) solution (ie on some finite time interval possibly (ie on some finite time interval possibly small)small)

QuestionQuestion Do the solutions remain positive and bounded (for Do the solutions remain positive and bounded (for biological significance)biological significance)

(3)

(1) (3)

Positivity and boundedness of Positivity and boundedness of solutionssolutions

Positivity and boundedness of Positivity and boundedness of solutionssolutions

bull We first use Theorem 3 to show positivity and boundedness of xWe first use Theorem 3 to show positivity and boundedness of x11 and and xx22

bull Next we use these results and the Fundamental Theorem of Calculus to Next we use these results and the Fundamental Theorem of Calculus to show positivity and boundedness for the remaining equations show positivity and boundedness for the remaining equations

bull This then ensures a global solutionThis then ensures a global solution

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical Analysis

bull Parameter Estimation in SIMULINKParameter Estimation in SIMULINK

bull Sensitivity AnalysisSensitivity Analysis

Parameter Estimation in SIMULINK (for Parameter Estimation in SIMULINK (for FULL model)FULL model)

Parameter Estimation in SIMULINK (for Parameter Estimation in SIMULINK (for FULL model)FULL model)

bull Total number of model parameters (for full model with Li and Rinzel CaTotal number of model parameters (for full model with Li and Rinzel Ca2+2+ module) = module) = 3434ndash From the literature = From the literature = 2020ndash Estimated = Estimated = 1414

bull Receptor moduleReceptor module Total = Total = 77ndash From literature = From literature = 66ndash Estimated = Estimated = 11 (k (kpp rate of receptor phosphorylation) rate of receptor phosphorylation)

bull G-protein cascadeG-protein cascade Total = Total = 99ndash From literature = From literature = 66ndash Estimated = Estimated = 3 3 (k(khydhyd k kreprep k kd3d3))

bull DAG kineticsDAG kinetics Total = Total = 5 5 (for each DAG species considered)(for each DAG species considered)ndash From literature = From literature = 00ndash Estimated = Estimated = 5 5 (all DAG parameters)(all DAG parameters)

bull CaCa2+2+ module module Total = Total = 1313ndash From literature = From literature = 88ndash Estimated = Estimated = 55

SIMULINK DetailsSIMULINK DetailsSIMULINK DetailsSIMULINK Details

bull Unknown rate parameters were estimated using SIMULINK

ndash Minimizes a user-specified cost function via a user-specified optimization method

ndash Nonlinear least squares optimization method of Levenberg-Marquardt was used to minimize a sum of squared errors cost function of the empirical observations and model predictions for IP3 Ca2+ and multiple species of DAG

bull Note Although the Gauss-Newton method is often more efficient the method of Levenberg-Marquardt has proved to be more robust

Measured vs Simulated 384 DAG and Measured vs Simulated 384 DAG and 341 DAG341 DAG

Measured vs Simulated 384 DAG and Measured vs Simulated 384 DAG and 341 DAG341 DAG

bull Timecourse of stimulation with 25 Timecourse of stimulation with 25 M UDP in RAW 2647 cellsM UDP in RAW 2647 cells

bull Solid black lines represent model simulations Solid black lines represent model simulations

bull ((aa) 384 DAG response (representative of the response of most poly ) 384 DAG response (representative of the response of most poly unsaturated fatty acid-containing DAG species) unsaturated fatty acid-containing DAG species)

bull ((bb) 341 DAG response (representative of the response of most mono- ) 341 DAG response (representative of the response of most mono- and di-unsaturated fatty acid-containing DAG species) and di-unsaturated fatty acid-containing DAG species)

bull Data points contain nine replicates performed on three different Data points contain nine replicates performed on three different experimental days with error bars = 1 SEM Units are total change in ng experimental days with error bars = 1 SEM Units are total change in ng over baseline levels in ~8x10over baseline levels in ~8x1066 cells cells

Measured vs Simulated IPMeasured vs Simulated IP33 and Ca and Ca2+2+Measured vs Simulated IPMeasured vs Simulated IP33 and Ca and Ca2+2+

bull Timecourse of stimulation with 25 Timecourse of stimulation with 25 M UDP in RAW 2647 cellsM UDP in RAW 2647 cells

bull Solid black lines represent model simulationsSolid black lines represent model simulations

bull ((cc) IP) IP33 response in pmols per ~35x10 response in pmols per ~35x1055 cells cells

bull Points in (c) represent the average of four experiments and error Points in (c) represent the average of four experiments and error bars are 1 SEM bars are 1 SEM

bull ((dd) Ca) Ca2+2+ response in response in M Red curve is a representative CaM Red curve is a representative Ca2+2+ trace trace taken from the UDP experiments within the AfCS single ligand taken from the UDP experiments within the AfCS single ligand screenscreen

Model Simulations P2YModel Simulations P2Y66 G GGTP GTP PIPPIP22

Model Simulations P2YModel Simulations P2Y66 G GGTP GTP PIPPIP22

bull ((aa) = total P2Y) = total P2Y66 activated (solid activated (solid line) and inactivated (dashed line) and inactivated (dashed line) surface receptorsline) surface receptors

bull ((bb) = total G) = total GGTP GTP

bull ((cc) = total PIP) = total PIP22 available for available for hydrolysishydrolysis

P2Y6 from 25M UDP GGTP from 25M UDP

PIP2 from 25M UDP

Sensitivity AnalysisSensitivity AnalysisSensitivity AnalysisSensitivity Analysis

bull Sensitivity analysis techniques are valuable tools Sensitivity analysis techniques are valuable tools designed to answer questions regarding which of the designed to answer questions regarding which of the uncertain input variables is more important in uncertain input variables is more important in determining the uncertainty in our output determining the uncertainty in our output

bull Likewise sensitivity analysis can provide insight into Likewise sensitivity analysis can provide insight into which parameter should be studied in more detail in which parameter should be studied in more detail in order to reduce the most variance in the model output order to reduce the most variance in the model output

bull The ability to answer these types of questions could The ability to answer these types of questions could lead to important insight into the design of new lead to important insight into the design of new experiments and in determining which experiments experiments and in determining which experiments would give us the most valuable informationwould give us the most valuable information

Sampling MethodSampling MethodSampling MethodSampling Method

bull Generate a random sample of our space of input Generate a random sample of our space of input variables over a ten percent variation from each variables over a ten percent variation from each parameters nominal value using the parameters nominal value using the Latin Hypercube Latin Hypercube Sampling (LHS) methodSampling (LHS) method

bull Uses Uses Standardized Regression Coefficients (SRCs) Standardized Regression Coefficients (SRCs) obtained by performing multiple linear regression obtained by performing multiple linear regression analysisanalysisndash offers a measure of sensitivity that is multi-dimensionally offers a measure of sensitivity that is multi-dimensionally

averaged over the entire space of parameter values averaged over the entire space of parameter values ndash SRCs give insight into degree of nonlinearity in the model SRCs give insight into degree of nonlinearity in the model

Computing RComputing Ryy22 values valuesComputing RComputing Ryy22 values values

bull SRCs are only reliable measures of sensitivity when SRCs are only reliable measures of sensitivity when degree of nonlinearity is ldquosmallrdquodegree of nonlinearity is ldquosmallrdquo

bull Use model coefficients of determination RUse model coefficients of determination Ryy22 given by given by

bull where ywhere yii is the estimate of y is the estimate of yii obtained from the regression obtained from the regression modelmodel

bull RRyy22 ge 07 ensures SRCs are good sensitivity measures ge 07 ensures SRCs are good sensitivity measures

SRCs (as sensitivity measures) for SRCs (as sensitivity measures) for DAGDAG

SRCs (as sensitivity measures) for SRCs (as sensitivity measures) for DAGDAG

SRCs (as sensitivity measures) for IPSRCs (as sensitivity measures) for IP33SRCs (as sensitivity measures) for IPSRCs (as sensitivity measures) for IP33

Effects of 3 most sensitive DAG Effects of 3 most sensitive DAG parametersparameters

Effects of 3 most sensitive DAG Effects of 3 most sensitive DAG parametersparameters

A B

C

AA k kdp2dp2 degradation of pool 2 DAG degradation of pool 2 DAG

BB k kdp1dp1 degradation of pool 1 DAG degradation of pool 1 DAG

CC k kap2ap2 production of pool 2 DAG production of pool 2 DAG

ConclusionsConclusionsConclusionsConclusions

bull We have developed a model of the UDP signaling We have developed a model of the UDP signaling pathway in RAW 2647 macrophages which can predict pathway in RAW 2647 macrophages which can predict the responses of multiple species of DAG as well as the the responses of multiple species of DAG as well as the responses of IPresponses of IP33 Ca Ca2+2+ receptor dynamics G-protein receptor dynamics G-protein activation and PIPactivation and PIP22 hydrolysis hydrolysis

bull Simplified model resultsSimplified model resultsndash We have obtained global existence uniqueness positivity We have obtained global existence uniqueness positivity

and boundedness of solutionsand boundedness of solutionsndash We have proven global stability of a unique steady state We have proven global stability of a unique steady state

within our region of interestwithin our region of interestbull Full model analysisFull model analysis

ndash Using SIMULINK we have estimated unknown rate Using SIMULINK we have estimated unknown rate parameters to obtain best fits to multiple DAG traces IPparameters to obtain best fits to multiple DAG traces IP33 and Caand Ca2+2+ all in response to 25 all in response to 25M UDPM UDP

ndash We have performed sensitivity analysis using the Latin We have performed sensitivity analysis using the Latin Hypercube sampling technique in combination with Hypercube sampling technique in combination with standardized regression coefficients to determine which standardized regression coefficients to determine which model parameters are responsible for most of the model model parameters are responsible for most of the model output uncertaintyoutput uncertainty

Future DirectionsFuture DirectionsFuture DirectionsFuture Directions

bull We have conducted multiple experiments to We have conducted multiple experiments to pharmacologically inhibit several different enzymes we pharmacologically inhibit several different enzymes we believe to play a role in this signaling pathway as believe to play a role in this signaling pathway as suggested by current known metabolic pathways and by suggested by current known metabolic pathways and by modeling resultsmodeling results

bull The next step is to perform gene knockdowns on The next step is to perform gene knockdowns on specific enzymes to verify results of inhibitor dataspecific enzymes to verify results of inhibitor data

bull The model output also suggests a time delay from The model output also suggests a time delay from receptor activation to PIPreceptor activation to PIP22 hydrolysis An upcoming task hydrolysis An upcoming task is to investigate the outcomes of adding such a delay is to investigate the outcomes of adding such a delay termterm

bull The model could be greatly enhanced by incorporating The model could be greatly enhanced by incorporating spatial dynamics so this will also be a major focus for spatial dynamics so this will also be a major focus for future researchfuture research

AcknowledgementsAcknowledgementsAcknowledgementsAcknowledgements

Mathematics DepartmentMathematics Departmentbull Hannah L Callender Hannah L Callender

PhDPhD

CollaboratorsCollaboratorsbull Alliance for Cellular SignalingAlliance for Cellular Signalingbull UT SouthwesternUT Southwestern

ndash Paul Sternweis PhDPaul Sternweis PhDndash Dianne DeCamp PhDDianne DeCamp PhD

Brown LabBrown Labbull H Alex Brown PhDH Alex Brown PhDbull Jeffrey S Forrester PhDJeffrey S Forrester PhDbull Mark Byrne PhDMark Byrne PhDbull Anita Preininger PhDAnita Preininger PhDbull Michelle ArmstrongMichelle Armstrongbull Andrew GoodmanAndrew Goodmanbull Pavlina Ivanova PhDPavlina Ivanova PhDbull Steve Milne PhDSteve Milne PhD

Simplified CaSimplified Ca2+2+ Equations EquationsSimplified CaSimplified Ca2+2+ Equations Equations

bull CaCa2+2+ modulemodule (to match experimental AfCS (to match experimental AfCS trace)trace)

0 100 200 300 400 500 6000

002

004

006

008

01

012

014

Concentration of Ca

2+ (uM)

Ca2+ response to 25uM UDP

time (s)

wherewhere

Theoretical AnalysisTheoretical AnalysisTheoretical AnalysisTheoretical Analysis

bull Existence and UniquenessExistence and Uniqueness

bull Positivity and Boundedness Positivity and Boundedness

(for biological relevance)(for biological relevance)

bull Analysis of steady state behaviorAnalysis of steady state behavior

Existence of Solutions (full model)Existence of Solutions (full model)Existence of Solutions (full model)Existence of Solutions (full model)

First we write our system of ODEs in the formFirst we write our system of ODEs in the form (1)

(2)

(1)

Uniqueness of SolutionsUniqueness of SolutionsUniqueness of SolutionsUniqueness of Solutions

Next we denote a solution of Next we denote a solution of (1)(1) by by with initial condition with initial condition

Since our system satisfies the hypotheses of Theorems 1 and 2 Since our system satisfies the hypotheses of Theorems 1 and 2 on our set of interest (for all positive time and on a positive on our set of interest (for all positive time and on a positive bounded set in space) we know bounded set in space) we know there exists a unique there exists a unique (local) solution(local) solution (ie on some finite time interval possibly (ie on some finite time interval possibly small)small)

QuestionQuestion Do the solutions remain positive and bounded (for Do the solutions remain positive and bounded (for biological significance)biological significance)

(3)

(1) (3)

Positivity and boundedness of Positivity and boundedness of solutionssolutions

Positivity and boundedness of Positivity and boundedness of solutionssolutions

bull We first use Theorem 3 to show positivity and boundedness of xWe first use Theorem 3 to show positivity and boundedness of x11 and and xx22

bull Next we use these results and the Fundamental Theorem of Calculus to Next we use these results and the Fundamental Theorem of Calculus to show positivity and boundedness for the remaining equations show positivity and boundedness for the remaining equations

bull This then ensures a global solutionThis then ensures a global solution

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical Analysis

bull Parameter Estimation in SIMULINKParameter Estimation in SIMULINK

bull Sensitivity AnalysisSensitivity Analysis

Parameter Estimation in SIMULINK (for Parameter Estimation in SIMULINK (for FULL model)FULL model)

Parameter Estimation in SIMULINK (for Parameter Estimation in SIMULINK (for FULL model)FULL model)

bull Total number of model parameters (for full model with Li and Rinzel CaTotal number of model parameters (for full model with Li and Rinzel Ca2+2+ module) = module) = 3434ndash From the literature = From the literature = 2020ndash Estimated = Estimated = 1414

bull Receptor moduleReceptor module Total = Total = 77ndash From literature = From literature = 66ndash Estimated = Estimated = 11 (k (kpp rate of receptor phosphorylation) rate of receptor phosphorylation)

bull G-protein cascadeG-protein cascade Total = Total = 99ndash From literature = From literature = 66ndash Estimated = Estimated = 3 3 (k(khydhyd k kreprep k kd3d3))

bull DAG kineticsDAG kinetics Total = Total = 5 5 (for each DAG species considered)(for each DAG species considered)ndash From literature = From literature = 00ndash Estimated = Estimated = 5 5 (all DAG parameters)(all DAG parameters)

bull CaCa2+2+ module module Total = Total = 1313ndash From literature = From literature = 88ndash Estimated = Estimated = 55

SIMULINK DetailsSIMULINK DetailsSIMULINK DetailsSIMULINK Details

bull Unknown rate parameters were estimated using SIMULINK

ndash Minimizes a user-specified cost function via a user-specified optimization method

ndash Nonlinear least squares optimization method of Levenberg-Marquardt was used to minimize a sum of squared errors cost function of the empirical observations and model predictions for IP3 Ca2+ and multiple species of DAG

bull Note Although the Gauss-Newton method is often more efficient the method of Levenberg-Marquardt has proved to be more robust

Measured vs Simulated 384 DAG and Measured vs Simulated 384 DAG and 341 DAG341 DAG

Measured vs Simulated 384 DAG and Measured vs Simulated 384 DAG and 341 DAG341 DAG

bull Timecourse of stimulation with 25 Timecourse of stimulation with 25 M UDP in RAW 2647 cellsM UDP in RAW 2647 cells

bull Solid black lines represent model simulations Solid black lines represent model simulations

bull ((aa) 384 DAG response (representative of the response of most poly ) 384 DAG response (representative of the response of most poly unsaturated fatty acid-containing DAG species) unsaturated fatty acid-containing DAG species)

bull ((bb) 341 DAG response (representative of the response of most mono- ) 341 DAG response (representative of the response of most mono- and di-unsaturated fatty acid-containing DAG species) and di-unsaturated fatty acid-containing DAG species)

bull Data points contain nine replicates performed on three different Data points contain nine replicates performed on three different experimental days with error bars = 1 SEM Units are total change in ng experimental days with error bars = 1 SEM Units are total change in ng over baseline levels in ~8x10over baseline levels in ~8x1066 cells cells

Measured vs Simulated IPMeasured vs Simulated IP33 and Ca and Ca2+2+Measured vs Simulated IPMeasured vs Simulated IP33 and Ca and Ca2+2+

bull Timecourse of stimulation with 25 Timecourse of stimulation with 25 M UDP in RAW 2647 cellsM UDP in RAW 2647 cells

bull Solid black lines represent model simulationsSolid black lines represent model simulations

bull ((cc) IP) IP33 response in pmols per ~35x10 response in pmols per ~35x1055 cells cells

bull Points in (c) represent the average of four experiments and error Points in (c) represent the average of four experiments and error bars are 1 SEM bars are 1 SEM

bull ((dd) Ca) Ca2+2+ response in response in M Red curve is a representative CaM Red curve is a representative Ca2+2+ trace trace taken from the UDP experiments within the AfCS single ligand taken from the UDP experiments within the AfCS single ligand screenscreen

Model Simulations P2YModel Simulations P2Y66 G GGTP GTP PIPPIP22

Model Simulations P2YModel Simulations P2Y66 G GGTP GTP PIPPIP22

bull ((aa) = total P2Y) = total P2Y66 activated (solid activated (solid line) and inactivated (dashed line) and inactivated (dashed line) surface receptorsline) surface receptors

bull ((bb) = total G) = total GGTP GTP

bull ((cc) = total PIP) = total PIP22 available for available for hydrolysishydrolysis

P2Y6 from 25M UDP GGTP from 25M UDP

PIP2 from 25M UDP

Sensitivity AnalysisSensitivity AnalysisSensitivity AnalysisSensitivity Analysis

bull Sensitivity analysis techniques are valuable tools Sensitivity analysis techniques are valuable tools designed to answer questions regarding which of the designed to answer questions regarding which of the uncertain input variables is more important in uncertain input variables is more important in determining the uncertainty in our output determining the uncertainty in our output

bull Likewise sensitivity analysis can provide insight into Likewise sensitivity analysis can provide insight into which parameter should be studied in more detail in which parameter should be studied in more detail in order to reduce the most variance in the model output order to reduce the most variance in the model output

bull The ability to answer these types of questions could The ability to answer these types of questions could lead to important insight into the design of new lead to important insight into the design of new experiments and in determining which experiments experiments and in determining which experiments would give us the most valuable informationwould give us the most valuable information

Sampling MethodSampling MethodSampling MethodSampling Method

bull Generate a random sample of our space of input Generate a random sample of our space of input variables over a ten percent variation from each variables over a ten percent variation from each parameters nominal value using the parameters nominal value using the Latin Hypercube Latin Hypercube Sampling (LHS) methodSampling (LHS) method

bull Uses Uses Standardized Regression Coefficients (SRCs) Standardized Regression Coefficients (SRCs) obtained by performing multiple linear regression obtained by performing multiple linear regression analysisanalysisndash offers a measure of sensitivity that is multi-dimensionally offers a measure of sensitivity that is multi-dimensionally

averaged over the entire space of parameter values averaged over the entire space of parameter values ndash SRCs give insight into degree of nonlinearity in the model SRCs give insight into degree of nonlinearity in the model

Computing RComputing Ryy22 values valuesComputing RComputing Ryy22 values values

bull SRCs are only reliable measures of sensitivity when SRCs are only reliable measures of sensitivity when degree of nonlinearity is ldquosmallrdquodegree of nonlinearity is ldquosmallrdquo

bull Use model coefficients of determination RUse model coefficients of determination Ryy22 given by given by

bull where ywhere yii is the estimate of y is the estimate of yii obtained from the regression obtained from the regression modelmodel

bull RRyy22 ge 07 ensures SRCs are good sensitivity measures ge 07 ensures SRCs are good sensitivity measures

SRCs (as sensitivity measures) for SRCs (as sensitivity measures) for DAGDAG

SRCs (as sensitivity measures) for SRCs (as sensitivity measures) for DAGDAG

SRCs (as sensitivity measures) for IPSRCs (as sensitivity measures) for IP33SRCs (as sensitivity measures) for IPSRCs (as sensitivity measures) for IP33

Effects of 3 most sensitive DAG Effects of 3 most sensitive DAG parametersparameters

Effects of 3 most sensitive DAG Effects of 3 most sensitive DAG parametersparameters

A B

C

AA k kdp2dp2 degradation of pool 2 DAG degradation of pool 2 DAG

BB k kdp1dp1 degradation of pool 1 DAG degradation of pool 1 DAG

CC k kap2ap2 production of pool 2 DAG production of pool 2 DAG

ConclusionsConclusionsConclusionsConclusions

bull We have developed a model of the UDP signaling We have developed a model of the UDP signaling pathway in RAW 2647 macrophages which can predict pathway in RAW 2647 macrophages which can predict the responses of multiple species of DAG as well as the the responses of multiple species of DAG as well as the responses of IPresponses of IP33 Ca Ca2+2+ receptor dynamics G-protein receptor dynamics G-protein activation and PIPactivation and PIP22 hydrolysis hydrolysis

bull Simplified model resultsSimplified model resultsndash We have obtained global existence uniqueness positivity We have obtained global existence uniqueness positivity

and boundedness of solutionsand boundedness of solutionsndash We have proven global stability of a unique steady state We have proven global stability of a unique steady state

within our region of interestwithin our region of interestbull Full model analysisFull model analysis

ndash Using SIMULINK we have estimated unknown rate Using SIMULINK we have estimated unknown rate parameters to obtain best fits to multiple DAG traces IPparameters to obtain best fits to multiple DAG traces IP33 and Caand Ca2+2+ all in response to 25 all in response to 25M UDPM UDP

ndash We have performed sensitivity analysis using the Latin We have performed sensitivity analysis using the Latin Hypercube sampling technique in combination with Hypercube sampling technique in combination with standardized regression coefficients to determine which standardized regression coefficients to determine which model parameters are responsible for most of the model model parameters are responsible for most of the model output uncertaintyoutput uncertainty

Future DirectionsFuture DirectionsFuture DirectionsFuture Directions

bull We have conducted multiple experiments to We have conducted multiple experiments to pharmacologically inhibit several different enzymes we pharmacologically inhibit several different enzymes we believe to play a role in this signaling pathway as believe to play a role in this signaling pathway as suggested by current known metabolic pathways and by suggested by current known metabolic pathways and by modeling resultsmodeling results

bull The next step is to perform gene knockdowns on The next step is to perform gene knockdowns on specific enzymes to verify results of inhibitor dataspecific enzymes to verify results of inhibitor data

bull The model output also suggests a time delay from The model output also suggests a time delay from receptor activation to PIPreceptor activation to PIP22 hydrolysis An upcoming task hydrolysis An upcoming task is to investigate the outcomes of adding such a delay is to investigate the outcomes of adding such a delay termterm

bull The model could be greatly enhanced by incorporating The model could be greatly enhanced by incorporating spatial dynamics so this will also be a major focus for spatial dynamics so this will also be a major focus for future researchfuture research

AcknowledgementsAcknowledgementsAcknowledgementsAcknowledgements

Mathematics DepartmentMathematics Departmentbull Hannah L Callender Hannah L Callender

PhDPhD

CollaboratorsCollaboratorsbull Alliance for Cellular SignalingAlliance for Cellular Signalingbull UT SouthwesternUT Southwestern

ndash Paul Sternweis PhDPaul Sternweis PhDndash Dianne DeCamp PhDDianne DeCamp PhD

Brown LabBrown Labbull H Alex Brown PhDH Alex Brown PhDbull Jeffrey S Forrester PhDJeffrey S Forrester PhDbull Mark Byrne PhDMark Byrne PhDbull Anita Preininger PhDAnita Preininger PhDbull Michelle ArmstrongMichelle Armstrongbull Andrew GoodmanAndrew Goodmanbull Pavlina Ivanova PhDPavlina Ivanova PhDbull Steve Milne PhDSteve Milne PhD

Theoretical AnalysisTheoretical AnalysisTheoretical AnalysisTheoretical Analysis

bull Existence and UniquenessExistence and Uniqueness

bull Positivity and Boundedness Positivity and Boundedness

(for biological relevance)(for biological relevance)

bull Analysis of steady state behaviorAnalysis of steady state behavior

Existence of Solutions (full model)Existence of Solutions (full model)Existence of Solutions (full model)Existence of Solutions (full model)

First we write our system of ODEs in the formFirst we write our system of ODEs in the form (1)

(2)

(1)

Uniqueness of SolutionsUniqueness of SolutionsUniqueness of SolutionsUniqueness of Solutions

Next we denote a solution of Next we denote a solution of (1)(1) by by with initial condition with initial condition

Since our system satisfies the hypotheses of Theorems 1 and 2 Since our system satisfies the hypotheses of Theorems 1 and 2 on our set of interest (for all positive time and on a positive on our set of interest (for all positive time and on a positive bounded set in space) we know bounded set in space) we know there exists a unique there exists a unique (local) solution(local) solution (ie on some finite time interval possibly (ie on some finite time interval possibly small)small)

QuestionQuestion Do the solutions remain positive and bounded (for Do the solutions remain positive and bounded (for biological significance)biological significance)

(3)

(1) (3)

Positivity and boundedness of Positivity and boundedness of solutionssolutions

Positivity and boundedness of Positivity and boundedness of solutionssolutions

bull We first use Theorem 3 to show positivity and boundedness of xWe first use Theorem 3 to show positivity and boundedness of x11 and and xx22

bull Next we use these results and the Fundamental Theorem of Calculus to Next we use these results and the Fundamental Theorem of Calculus to show positivity and boundedness for the remaining equations show positivity and boundedness for the remaining equations

bull This then ensures a global solutionThis then ensures a global solution

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical Analysis

bull Parameter Estimation in SIMULINKParameter Estimation in SIMULINK

bull Sensitivity AnalysisSensitivity Analysis

Parameter Estimation in SIMULINK (for Parameter Estimation in SIMULINK (for FULL model)FULL model)

Parameter Estimation in SIMULINK (for Parameter Estimation in SIMULINK (for FULL model)FULL model)

bull Total number of model parameters (for full model with Li and Rinzel CaTotal number of model parameters (for full model with Li and Rinzel Ca2+2+ module) = module) = 3434ndash From the literature = From the literature = 2020ndash Estimated = Estimated = 1414

bull Receptor moduleReceptor module Total = Total = 77ndash From literature = From literature = 66ndash Estimated = Estimated = 11 (k (kpp rate of receptor phosphorylation) rate of receptor phosphorylation)

bull G-protein cascadeG-protein cascade Total = Total = 99ndash From literature = From literature = 66ndash Estimated = Estimated = 3 3 (k(khydhyd k kreprep k kd3d3))

bull DAG kineticsDAG kinetics Total = Total = 5 5 (for each DAG species considered)(for each DAG species considered)ndash From literature = From literature = 00ndash Estimated = Estimated = 5 5 (all DAG parameters)(all DAG parameters)

bull CaCa2+2+ module module Total = Total = 1313ndash From literature = From literature = 88ndash Estimated = Estimated = 55

SIMULINK DetailsSIMULINK DetailsSIMULINK DetailsSIMULINK Details

bull Unknown rate parameters were estimated using SIMULINK

ndash Minimizes a user-specified cost function via a user-specified optimization method

ndash Nonlinear least squares optimization method of Levenberg-Marquardt was used to minimize a sum of squared errors cost function of the empirical observations and model predictions for IP3 Ca2+ and multiple species of DAG

bull Note Although the Gauss-Newton method is often more efficient the method of Levenberg-Marquardt has proved to be more robust

Measured vs Simulated 384 DAG and Measured vs Simulated 384 DAG and 341 DAG341 DAG

Measured vs Simulated 384 DAG and Measured vs Simulated 384 DAG and 341 DAG341 DAG

bull Timecourse of stimulation with 25 Timecourse of stimulation with 25 M UDP in RAW 2647 cellsM UDP in RAW 2647 cells

bull Solid black lines represent model simulations Solid black lines represent model simulations

bull ((aa) 384 DAG response (representative of the response of most poly ) 384 DAG response (representative of the response of most poly unsaturated fatty acid-containing DAG species) unsaturated fatty acid-containing DAG species)

bull ((bb) 341 DAG response (representative of the response of most mono- ) 341 DAG response (representative of the response of most mono- and di-unsaturated fatty acid-containing DAG species) and di-unsaturated fatty acid-containing DAG species)

bull Data points contain nine replicates performed on three different Data points contain nine replicates performed on three different experimental days with error bars = 1 SEM Units are total change in ng experimental days with error bars = 1 SEM Units are total change in ng over baseline levels in ~8x10over baseline levels in ~8x1066 cells cells

Measured vs Simulated IPMeasured vs Simulated IP33 and Ca and Ca2+2+Measured vs Simulated IPMeasured vs Simulated IP33 and Ca and Ca2+2+

bull Timecourse of stimulation with 25 Timecourse of stimulation with 25 M UDP in RAW 2647 cellsM UDP in RAW 2647 cells

bull Solid black lines represent model simulationsSolid black lines represent model simulations

bull ((cc) IP) IP33 response in pmols per ~35x10 response in pmols per ~35x1055 cells cells

bull Points in (c) represent the average of four experiments and error Points in (c) represent the average of four experiments and error bars are 1 SEM bars are 1 SEM

bull ((dd) Ca) Ca2+2+ response in response in M Red curve is a representative CaM Red curve is a representative Ca2+2+ trace trace taken from the UDP experiments within the AfCS single ligand taken from the UDP experiments within the AfCS single ligand screenscreen

Model Simulations P2YModel Simulations P2Y66 G GGTP GTP PIPPIP22

Model Simulations P2YModel Simulations P2Y66 G GGTP GTP PIPPIP22

bull ((aa) = total P2Y) = total P2Y66 activated (solid activated (solid line) and inactivated (dashed line) and inactivated (dashed line) surface receptorsline) surface receptors

bull ((bb) = total G) = total GGTP GTP

bull ((cc) = total PIP) = total PIP22 available for available for hydrolysishydrolysis

P2Y6 from 25M UDP GGTP from 25M UDP

PIP2 from 25M UDP

Sensitivity AnalysisSensitivity AnalysisSensitivity AnalysisSensitivity Analysis

bull Sensitivity analysis techniques are valuable tools Sensitivity analysis techniques are valuable tools designed to answer questions regarding which of the designed to answer questions regarding which of the uncertain input variables is more important in uncertain input variables is more important in determining the uncertainty in our output determining the uncertainty in our output

bull Likewise sensitivity analysis can provide insight into Likewise sensitivity analysis can provide insight into which parameter should be studied in more detail in which parameter should be studied in more detail in order to reduce the most variance in the model output order to reduce the most variance in the model output

bull The ability to answer these types of questions could The ability to answer these types of questions could lead to important insight into the design of new lead to important insight into the design of new experiments and in determining which experiments experiments and in determining which experiments would give us the most valuable informationwould give us the most valuable information

Sampling MethodSampling MethodSampling MethodSampling Method

bull Generate a random sample of our space of input Generate a random sample of our space of input variables over a ten percent variation from each variables over a ten percent variation from each parameters nominal value using the parameters nominal value using the Latin Hypercube Latin Hypercube Sampling (LHS) methodSampling (LHS) method

bull Uses Uses Standardized Regression Coefficients (SRCs) Standardized Regression Coefficients (SRCs) obtained by performing multiple linear regression obtained by performing multiple linear regression analysisanalysisndash offers a measure of sensitivity that is multi-dimensionally offers a measure of sensitivity that is multi-dimensionally

averaged over the entire space of parameter values averaged over the entire space of parameter values ndash SRCs give insight into degree of nonlinearity in the model SRCs give insight into degree of nonlinearity in the model

Computing RComputing Ryy22 values valuesComputing RComputing Ryy22 values values

bull SRCs are only reliable measures of sensitivity when SRCs are only reliable measures of sensitivity when degree of nonlinearity is ldquosmallrdquodegree of nonlinearity is ldquosmallrdquo

bull Use model coefficients of determination RUse model coefficients of determination Ryy22 given by given by

bull where ywhere yii is the estimate of y is the estimate of yii obtained from the regression obtained from the regression modelmodel

bull RRyy22 ge 07 ensures SRCs are good sensitivity measures ge 07 ensures SRCs are good sensitivity measures

SRCs (as sensitivity measures) for SRCs (as sensitivity measures) for DAGDAG

SRCs (as sensitivity measures) for SRCs (as sensitivity measures) for DAGDAG

SRCs (as sensitivity measures) for IPSRCs (as sensitivity measures) for IP33SRCs (as sensitivity measures) for IPSRCs (as sensitivity measures) for IP33

Effects of 3 most sensitive DAG Effects of 3 most sensitive DAG parametersparameters

Effects of 3 most sensitive DAG Effects of 3 most sensitive DAG parametersparameters

A B

C

AA k kdp2dp2 degradation of pool 2 DAG degradation of pool 2 DAG

BB k kdp1dp1 degradation of pool 1 DAG degradation of pool 1 DAG

CC k kap2ap2 production of pool 2 DAG production of pool 2 DAG

ConclusionsConclusionsConclusionsConclusions

bull We have developed a model of the UDP signaling We have developed a model of the UDP signaling pathway in RAW 2647 macrophages which can predict pathway in RAW 2647 macrophages which can predict the responses of multiple species of DAG as well as the the responses of multiple species of DAG as well as the responses of IPresponses of IP33 Ca Ca2+2+ receptor dynamics G-protein receptor dynamics G-protein activation and PIPactivation and PIP22 hydrolysis hydrolysis

bull Simplified model resultsSimplified model resultsndash We have obtained global existence uniqueness positivity We have obtained global existence uniqueness positivity

and boundedness of solutionsand boundedness of solutionsndash We have proven global stability of a unique steady state We have proven global stability of a unique steady state

within our region of interestwithin our region of interestbull Full model analysisFull model analysis

ndash Using SIMULINK we have estimated unknown rate Using SIMULINK we have estimated unknown rate parameters to obtain best fits to multiple DAG traces IPparameters to obtain best fits to multiple DAG traces IP33 and Caand Ca2+2+ all in response to 25 all in response to 25M UDPM UDP

ndash We have performed sensitivity analysis using the Latin We have performed sensitivity analysis using the Latin Hypercube sampling technique in combination with Hypercube sampling technique in combination with standardized regression coefficients to determine which standardized regression coefficients to determine which model parameters are responsible for most of the model model parameters are responsible for most of the model output uncertaintyoutput uncertainty

Future DirectionsFuture DirectionsFuture DirectionsFuture Directions

bull We have conducted multiple experiments to We have conducted multiple experiments to pharmacologically inhibit several different enzymes we pharmacologically inhibit several different enzymes we believe to play a role in this signaling pathway as believe to play a role in this signaling pathway as suggested by current known metabolic pathways and by suggested by current known metabolic pathways and by modeling resultsmodeling results

bull The next step is to perform gene knockdowns on The next step is to perform gene knockdowns on specific enzymes to verify results of inhibitor dataspecific enzymes to verify results of inhibitor data

bull The model output also suggests a time delay from The model output also suggests a time delay from receptor activation to PIPreceptor activation to PIP22 hydrolysis An upcoming task hydrolysis An upcoming task is to investigate the outcomes of adding such a delay is to investigate the outcomes of adding such a delay termterm

bull The model could be greatly enhanced by incorporating The model could be greatly enhanced by incorporating spatial dynamics so this will also be a major focus for spatial dynamics so this will also be a major focus for future researchfuture research

AcknowledgementsAcknowledgementsAcknowledgementsAcknowledgements

Mathematics DepartmentMathematics Departmentbull Hannah L Callender Hannah L Callender

PhDPhD

CollaboratorsCollaboratorsbull Alliance for Cellular SignalingAlliance for Cellular Signalingbull UT SouthwesternUT Southwestern

ndash Paul Sternweis PhDPaul Sternweis PhDndash Dianne DeCamp PhDDianne DeCamp PhD

Brown LabBrown Labbull H Alex Brown PhDH Alex Brown PhDbull Jeffrey S Forrester PhDJeffrey S Forrester PhDbull Mark Byrne PhDMark Byrne PhDbull Anita Preininger PhDAnita Preininger PhDbull Michelle ArmstrongMichelle Armstrongbull Andrew GoodmanAndrew Goodmanbull Pavlina Ivanova PhDPavlina Ivanova PhDbull Steve Milne PhDSteve Milne PhD

Existence of Solutions (full model)Existence of Solutions (full model)Existence of Solutions (full model)Existence of Solutions (full model)

First we write our system of ODEs in the formFirst we write our system of ODEs in the form (1)

(2)

(1)

Uniqueness of SolutionsUniqueness of SolutionsUniqueness of SolutionsUniqueness of Solutions

Next we denote a solution of Next we denote a solution of (1)(1) by by with initial condition with initial condition

Since our system satisfies the hypotheses of Theorems 1 and 2 Since our system satisfies the hypotheses of Theorems 1 and 2 on our set of interest (for all positive time and on a positive on our set of interest (for all positive time and on a positive bounded set in space) we know bounded set in space) we know there exists a unique there exists a unique (local) solution(local) solution (ie on some finite time interval possibly (ie on some finite time interval possibly small)small)

QuestionQuestion Do the solutions remain positive and bounded (for Do the solutions remain positive and bounded (for biological significance)biological significance)

(3)

(1) (3)

Positivity and boundedness of Positivity and boundedness of solutionssolutions

Positivity and boundedness of Positivity and boundedness of solutionssolutions

bull We first use Theorem 3 to show positivity and boundedness of xWe first use Theorem 3 to show positivity and boundedness of x11 and and xx22

bull Next we use these results and the Fundamental Theorem of Calculus to Next we use these results and the Fundamental Theorem of Calculus to show positivity and boundedness for the remaining equations show positivity and boundedness for the remaining equations

bull This then ensures a global solutionThis then ensures a global solution

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical Analysis

bull Parameter Estimation in SIMULINKParameter Estimation in SIMULINK

bull Sensitivity AnalysisSensitivity Analysis

Parameter Estimation in SIMULINK (for Parameter Estimation in SIMULINK (for FULL model)FULL model)

Parameter Estimation in SIMULINK (for Parameter Estimation in SIMULINK (for FULL model)FULL model)

bull Total number of model parameters (for full model with Li and Rinzel CaTotal number of model parameters (for full model with Li and Rinzel Ca2+2+ module) = module) = 3434ndash From the literature = From the literature = 2020ndash Estimated = Estimated = 1414

bull Receptor moduleReceptor module Total = Total = 77ndash From literature = From literature = 66ndash Estimated = Estimated = 11 (k (kpp rate of receptor phosphorylation) rate of receptor phosphorylation)

bull G-protein cascadeG-protein cascade Total = Total = 99ndash From literature = From literature = 66ndash Estimated = Estimated = 3 3 (k(khydhyd k kreprep k kd3d3))

bull DAG kineticsDAG kinetics Total = Total = 5 5 (for each DAG species considered)(for each DAG species considered)ndash From literature = From literature = 00ndash Estimated = Estimated = 5 5 (all DAG parameters)(all DAG parameters)

bull CaCa2+2+ module module Total = Total = 1313ndash From literature = From literature = 88ndash Estimated = Estimated = 55

SIMULINK DetailsSIMULINK DetailsSIMULINK DetailsSIMULINK Details

bull Unknown rate parameters were estimated using SIMULINK

ndash Minimizes a user-specified cost function via a user-specified optimization method

ndash Nonlinear least squares optimization method of Levenberg-Marquardt was used to minimize a sum of squared errors cost function of the empirical observations and model predictions for IP3 Ca2+ and multiple species of DAG

bull Note Although the Gauss-Newton method is often more efficient the method of Levenberg-Marquardt has proved to be more robust

Measured vs Simulated 384 DAG and Measured vs Simulated 384 DAG and 341 DAG341 DAG

Measured vs Simulated 384 DAG and Measured vs Simulated 384 DAG and 341 DAG341 DAG

bull Timecourse of stimulation with 25 Timecourse of stimulation with 25 M UDP in RAW 2647 cellsM UDP in RAW 2647 cells

bull Solid black lines represent model simulations Solid black lines represent model simulations

bull ((aa) 384 DAG response (representative of the response of most poly ) 384 DAG response (representative of the response of most poly unsaturated fatty acid-containing DAG species) unsaturated fatty acid-containing DAG species)

bull ((bb) 341 DAG response (representative of the response of most mono- ) 341 DAG response (representative of the response of most mono- and di-unsaturated fatty acid-containing DAG species) and di-unsaturated fatty acid-containing DAG species)

bull Data points contain nine replicates performed on three different Data points contain nine replicates performed on three different experimental days with error bars = 1 SEM Units are total change in ng experimental days with error bars = 1 SEM Units are total change in ng over baseline levels in ~8x10over baseline levels in ~8x1066 cells cells

Measured vs Simulated IPMeasured vs Simulated IP33 and Ca and Ca2+2+Measured vs Simulated IPMeasured vs Simulated IP33 and Ca and Ca2+2+

bull Timecourse of stimulation with 25 Timecourse of stimulation with 25 M UDP in RAW 2647 cellsM UDP in RAW 2647 cells

bull Solid black lines represent model simulationsSolid black lines represent model simulations

bull ((cc) IP) IP33 response in pmols per ~35x10 response in pmols per ~35x1055 cells cells

bull Points in (c) represent the average of four experiments and error Points in (c) represent the average of four experiments and error bars are 1 SEM bars are 1 SEM

bull ((dd) Ca) Ca2+2+ response in response in M Red curve is a representative CaM Red curve is a representative Ca2+2+ trace trace taken from the UDP experiments within the AfCS single ligand taken from the UDP experiments within the AfCS single ligand screenscreen

Model Simulations P2YModel Simulations P2Y66 G GGTP GTP PIPPIP22

Model Simulations P2YModel Simulations P2Y66 G GGTP GTP PIPPIP22

bull ((aa) = total P2Y) = total P2Y66 activated (solid activated (solid line) and inactivated (dashed line) and inactivated (dashed line) surface receptorsline) surface receptors

bull ((bb) = total G) = total GGTP GTP

bull ((cc) = total PIP) = total PIP22 available for available for hydrolysishydrolysis

P2Y6 from 25M UDP GGTP from 25M UDP

PIP2 from 25M UDP

Sensitivity AnalysisSensitivity AnalysisSensitivity AnalysisSensitivity Analysis

bull Sensitivity analysis techniques are valuable tools Sensitivity analysis techniques are valuable tools designed to answer questions regarding which of the designed to answer questions regarding which of the uncertain input variables is more important in uncertain input variables is more important in determining the uncertainty in our output determining the uncertainty in our output

bull Likewise sensitivity analysis can provide insight into Likewise sensitivity analysis can provide insight into which parameter should be studied in more detail in which parameter should be studied in more detail in order to reduce the most variance in the model output order to reduce the most variance in the model output

bull The ability to answer these types of questions could The ability to answer these types of questions could lead to important insight into the design of new lead to important insight into the design of new experiments and in determining which experiments experiments and in determining which experiments would give us the most valuable informationwould give us the most valuable information

Sampling MethodSampling MethodSampling MethodSampling Method

bull Generate a random sample of our space of input Generate a random sample of our space of input variables over a ten percent variation from each variables over a ten percent variation from each parameters nominal value using the parameters nominal value using the Latin Hypercube Latin Hypercube Sampling (LHS) methodSampling (LHS) method

bull Uses Uses Standardized Regression Coefficients (SRCs) Standardized Regression Coefficients (SRCs) obtained by performing multiple linear regression obtained by performing multiple linear regression analysisanalysisndash offers a measure of sensitivity that is multi-dimensionally offers a measure of sensitivity that is multi-dimensionally

averaged over the entire space of parameter values averaged over the entire space of parameter values ndash SRCs give insight into degree of nonlinearity in the model SRCs give insight into degree of nonlinearity in the model

Computing RComputing Ryy22 values valuesComputing RComputing Ryy22 values values

bull SRCs are only reliable measures of sensitivity when SRCs are only reliable measures of sensitivity when degree of nonlinearity is ldquosmallrdquodegree of nonlinearity is ldquosmallrdquo

bull Use model coefficients of determination RUse model coefficients of determination Ryy22 given by given by

bull where ywhere yii is the estimate of y is the estimate of yii obtained from the regression obtained from the regression modelmodel

bull RRyy22 ge 07 ensures SRCs are good sensitivity measures ge 07 ensures SRCs are good sensitivity measures

SRCs (as sensitivity measures) for SRCs (as sensitivity measures) for DAGDAG

SRCs (as sensitivity measures) for SRCs (as sensitivity measures) for DAGDAG

SRCs (as sensitivity measures) for IPSRCs (as sensitivity measures) for IP33SRCs (as sensitivity measures) for IPSRCs (as sensitivity measures) for IP33

Effects of 3 most sensitive DAG Effects of 3 most sensitive DAG parametersparameters

Effects of 3 most sensitive DAG Effects of 3 most sensitive DAG parametersparameters

A B

C

AA k kdp2dp2 degradation of pool 2 DAG degradation of pool 2 DAG

BB k kdp1dp1 degradation of pool 1 DAG degradation of pool 1 DAG

CC k kap2ap2 production of pool 2 DAG production of pool 2 DAG

ConclusionsConclusionsConclusionsConclusions

bull We have developed a model of the UDP signaling We have developed a model of the UDP signaling pathway in RAW 2647 macrophages which can predict pathway in RAW 2647 macrophages which can predict the responses of multiple species of DAG as well as the the responses of multiple species of DAG as well as the responses of IPresponses of IP33 Ca Ca2+2+ receptor dynamics G-protein receptor dynamics G-protein activation and PIPactivation and PIP22 hydrolysis hydrolysis

bull Simplified model resultsSimplified model resultsndash We have obtained global existence uniqueness positivity We have obtained global existence uniqueness positivity

and boundedness of solutionsand boundedness of solutionsndash We have proven global stability of a unique steady state We have proven global stability of a unique steady state

within our region of interestwithin our region of interestbull Full model analysisFull model analysis

ndash Using SIMULINK we have estimated unknown rate Using SIMULINK we have estimated unknown rate parameters to obtain best fits to multiple DAG traces IPparameters to obtain best fits to multiple DAG traces IP33 and Caand Ca2+2+ all in response to 25 all in response to 25M UDPM UDP

ndash We have performed sensitivity analysis using the Latin We have performed sensitivity analysis using the Latin Hypercube sampling technique in combination with Hypercube sampling technique in combination with standardized regression coefficients to determine which standardized regression coefficients to determine which model parameters are responsible for most of the model model parameters are responsible for most of the model output uncertaintyoutput uncertainty

Future DirectionsFuture DirectionsFuture DirectionsFuture Directions

bull We have conducted multiple experiments to We have conducted multiple experiments to pharmacologically inhibit several different enzymes we pharmacologically inhibit several different enzymes we believe to play a role in this signaling pathway as believe to play a role in this signaling pathway as suggested by current known metabolic pathways and by suggested by current known metabolic pathways and by modeling resultsmodeling results

bull The next step is to perform gene knockdowns on The next step is to perform gene knockdowns on specific enzymes to verify results of inhibitor dataspecific enzymes to verify results of inhibitor data

bull The model output also suggests a time delay from The model output also suggests a time delay from receptor activation to PIPreceptor activation to PIP22 hydrolysis An upcoming task hydrolysis An upcoming task is to investigate the outcomes of adding such a delay is to investigate the outcomes of adding such a delay termterm

bull The model could be greatly enhanced by incorporating The model could be greatly enhanced by incorporating spatial dynamics so this will also be a major focus for spatial dynamics so this will also be a major focus for future researchfuture research

AcknowledgementsAcknowledgementsAcknowledgementsAcknowledgements

Mathematics DepartmentMathematics Departmentbull Hannah L Callender Hannah L Callender

PhDPhD

CollaboratorsCollaboratorsbull Alliance for Cellular SignalingAlliance for Cellular Signalingbull UT SouthwesternUT Southwestern

ndash Paul Sternweis PhDPaul Sternweis PhDndash Dianne DeCamp PhDDianne DeCamp PhD

Brown LabBrown Labbull H Alex Brown PhDH Alex Brown PhDbull Jeffrey S Forrester PhDJeffrey S Forrester PhDbull Mark Byrne PhDMark Byrne PhDbull Anita Preininger PhDAnita Preininger PhDbull Michelle ArmstrongMichelle Armstrongbull Andrew GoodmanAndrew Goodmanbull Pavlina Ivanova PhDPavlina Ivanova PhDbull Steve Milne PhDSteve Milne PhD

Uniqueness of SolutionsUniqueness of SolutionsUniqueness of SolutionsUniqueness of Solutions

Next we denote a solution of Next we denote a solution of (1)(1) by by with initial condition with initial condition

Since our system satisfies the hypotheses of Theorems 1 and 2 Since our system satisfies the hypotheses of Theorems 1 and 2 on our set of interest (for all positive time and on a positive on our set of interest (for all positive time and on a positive bounded set in space) we know bounded set in space) we know there exists a unique there exists a unique (local) solution(local) solution (ie on some finite time interval possibly (ie on some finite time interval possibly small)small)

QuestionQuestion Do the solutions remain positive and bounded (for Do the solutions remain positive and bounded (for biological significance)biological significance)

(3)

(1) (3)

Positivity and boundedness of Positivity and boundedness of solutionssolutions

Positivity and boundedness of Positivity and boundedness of solutionssolutions

bull We first use Theorem 3 to show positivity and boundedness of xWe first use Theorem 3 to show positivity and boundedness of x11 and and xx22

bull Next we use these results and the Fundamental Theorem of Calculus to Next we use these results and the Fundamental Theorem of Calculus to show positivity and boundedness for the remaining equations show positivity and boundedness for the remaining equations

bull This then ensures a global solutionThis then ensures a global solution

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical Analysis

bull Parameter Estimation in SIMULINKParameter Estimation in SIMULINK

bull Sensitivity AnalysisSensitivity Analysis

Parameter Estimation in SIMULINK (for Parameter Estimation in SIMULINK (for FULL model)FULL model)

Parameter Estimation in SIMULINK (for Parameter Estimation in SIMULINK (for FULL model)FULL model)

bull Total number of model parameters (for full model with Li and Rinzel CaTotal number of model parameters (for full model with Li and Rinzel Ca2+2+ module) = module) = 3434ndash From the literature = From the literature = 2020ndash Estimated = Estimated = 1414

bull Receptor moduleReceptor module Total = Total = 77ndash From literature = From literature = 66ndash Estimated = Estimated = 11 (k (kpp rate of receptor phosphorylation) rate of receptor phosphorylation)

bull G-protein cascadeG-protein cascade Total = Total = 99ndash From literature = From literature = 66ndash Estimated = Estimated = 3 3 (k(khydhyd k kreprep k kd3d3))

bull DAG kineticsDAG kinetics Total = Total = 5 5 (for each DAG species considered)(for each DAG species considered)ndash From literature = From literature = 00ndash Estimated = Estimated = 5 5 (all DAG parameters)(all DAG parameters)

bull CaCa2+2+ module module Total = Total = 1313ndash From literature = From literature = 88ndash Estimated = Estimated = 55

SIMULINK DetailsSIMULINK DetailsSIMULINK DetailsSIMULINK Details

bull Unknown rate parameters were estimated using SIMULINK

ndash Minimizes a user-specified cost function via a user-specified optimization method

ndash Nonlinear least squares optimization method of Levenberg-Marquardt was used to minimize a sum of squared errors cost function of the empirical observations and model predictions for IP3 Ca2+ and multiple species of DAG

bull Note Although the Gauss-Newton method is often more efficient the method of Levenberg-Marquardt has proved to be more robust

Measured vs Simulated 384 DAG and Measured vs Simulated 384 DAG and 341 DAG341 DAG

Measured vs Simulated 384 DAG and Measured vs Simulated 384 DAG and 341 DAG341 DAG

bull Timecourse of stimulation with 25 Timecourse of stimulation with 25 M UDP in RAW 2647 cellsM UDP in RAW 2647 cells

bull Solid black lines represent model simulations Solid black lines represent model simulations

bull ((aa) 384 DAG response (representative of the response of most poly ) 384 DAG response (representative of the response of most poly unsaturated fatty acid-containing DAG species) unsaturated fatty acid-containing DAG species)

bull ((bb) 341 DAG response (representative of the response of most mono- ) 341 DAG response (representative of the response of most mono- and di-unsaturated fatty acid-containing DAG species) and di-unsaturated fatty acid-containing DAG species)

bull Data points contain nine replicates performed on three different Data points contain nine replicates performed on three different experimental days with error bars = 1 SEM Units are total change in ng experimental days with error bars = 1 SEM Units are total change in ng over baseline levels in ~8x10over baseline levels in ~8x1066 cells cells

Measured vs Simulated IPMeasured vs Simulated IP33 and Ca and Ca2+2+Measured vs Simulated IPMeasured vs Simulated IP33 and Ca and Ca2+2+

bull Timecourse of stimulation with 25 Timecourse of stimulation with 25 M UDP in RAW 2647 cellsM UDP in RAW 2647 cells

bull Solid black lines represent model simulationsSolid black lines represent model simulations

bull ((cc) IP) IP33 response in pmols per ~35x10 response in pmols per ~35x1055 cells cells

bull Points in (c) represent the average of four experiments and error Points in (c) represent the average of four experiments and error bars are 1 SEM bars are 1 SEM

bull ((dd) Ca) Ca2+2+ response in response in M Red curve is a representative CaM Red curve is a representative Ca2+2+ trace trace taken from the UDP experiments within the AfCS single ligand taken from the UDP experiments within the AfCS single ligand screenscreen

Model Simulations P2YModel Simulations P2Y66 G GGTP GTP PIPPIP22

Model Simulations P2YModel Simulations P2Y66 G GGTP GTP PIPPIP22

bull ((aa) = total P2Y) = total P2Y66 activated (solid activated (solid line) and inactivated (dashed line) and inactivated (dashed line) surface receptorsline) surface receptors

bull ((bb) = total G) = total GGTP GTP

bull ((cc) = total PIP) = total PIP22 available for available for hydrolysishydrolysis

P2Y6 from 25M UDP GGTP from 25M UDP

PIP2 from 25M UDP

Sensitivity AnalysisSensitivity AnalysisSensitivity AnalysisSensitivity Analysis

bull Sensitivity analysis techniques are valuable tools Sensitivity analysis techniques are valuable tools designed to answer questions regarding which of the designed to answer questions regarding which of the uncertain input variables is more important in uncertain input variables is more important in determining the uncertainty in our output determining the uncertainty in our output

bull Likewise sensitivity analysis can provide insight into Likewise sensitivity analysis can provide insight into which parameter should be studied in more detail in which parameter should be studied in more detail in order to reduce the most variance in the model output order to reduce the most variance in the model output

bull The ability to answer these types of questions could The ability to answer these types of questions could lead to important insight into the design of new lead to important insight into the design of new experiments and in determining which experiments experiments and in determining which experiments would give us the most valuable informationwould give us the most valuable information

Sampling MethodSampling MethodSampling MethodSampling Method

bull Generate a random sample of our space of input Generate a random sample of our space of input variables over a ten percent variation from each variables over a ten percent variation from each parameters nominal value using the parameters nominal value using the Latin Hypercube Latin Hypercube Sampling (LHS) methodSampling (LHS) method

bull Uses Uses Standardized Regression Coefficients (SRCs) Standardized Regression Coefficients (SRCs) obtained by performing multiple linear regression obtained by performing multiple linear regression analysisanalysisndash offers a measure of sensitivity that is multi-dimensionally offers a measure of sensitivity that is multi-dimensionally

averaged over the entire space of parameter values averaged over the entire space of parameter values ndash SRCs give insight into degree of nonlinearity in the model SRCs give insight into degree of nonlinearity in the model

Computing RComputing Ryy22 values valuesComputing RComputing Ryy22 values values

bull SRCs are only reliable measures of sensitivity when SRCs are only reliable measures of sensitivity when degree of nonlinearity is ldquosmallrdquodegree of nonlinearity is ldquosmallrdquo

bull Use model coefficients of determination RUse model coefficients of determination Ryy22 given by given by

bull where ywhere yii is the estimate of y is the estimate of yii obtained from the regression obtained from the regression modelmodel

bull RRyy22 ge 07 ensures SRCs are good sensitivity measures ge 07 ensures SRCs are good sensitivity measures

SRCs (as sensitivity measures) for SRCs (as sensitivity measures) for DAGDAG

SRCs (as sensitivity measures) for SRCs (as sensitivity measures) for DAGDAG

SRCs (as sensitivity measures) for IPSRCs (as sensitivity measures) for IP33SRCs (as sensitivity measures) for IPSRCs (as sensitivity measures) for IP33

Effects of 3 most sensitive DAG Effects of 3 most sensitive DAG parametersparameters

Effects of 3 most sensitive DAG Effects of 3 most sensitive DAG parametersparameters

A B

C

AA k kdp2dp2 degradation of pool 2 DAG degradation of pool 2 DAG

BB k kdp1dp1 degradation of pool 1 DAG degradation of pool 1 DAG

CC k kap2ap2 production of pool 2 DAG production of pool 2 DAG

ConclusionsConclusionsConclusionsConclusions

bull We have developed a model of the UDP signaling We have developed a model of the UDP signaling pathway in RAW 2647 macrophages which can predict pathway in RAW 2647 macrophages which can predict the responses of multiple species of DAG as well as the the responses of multiple species of DAG as well as the responses of IPresponses of IP33 Ca Ca2+2+ receptor dynamics G-protein receptor dynamics G-protein activation and PIPactivation and PIP22 hydrolysis hydrolysis

bull Simplified model resultsSimplified model resultsndash We have obtained global existence uniqueness positivity We have obtained global existence uniqueness positivity

and boundedness of solutionsand boundedness of solutionsndash We have proven global stability of a unique steady state We have proven global stability of a unique steady state

within our region of interestwithin our region of interestbull Full model analysisFull model analysis

ndash Using SIMULINK we have estimated unknown rate Using SIMULINK we have estimated unknown rate parameters to obtain best fits to multiple DAG traces IPparameters to obtain best fits to multiple DAG traces IP33 and Caand Ca2+2+ all in response to 25 all in response to 25M UDPM UDP

ndash We have performed sensitivity analysis using the Latin We have performed sensitivity analysis using the Latin Hypercube sampling technique in combination with Hypercube sampling technique in combination with standardized regression coefficients to determine which standardized regression coefficients to determine which model parameters are responsible for most of the model model parameters are responsible for most of the model output uncertaintyoutput uncertainty

Future DirectionsFuture DirectionsFuture DirectionsFuture Directions

bull We have conducted multiple experiments to We have conducted multiple experiments to pharmacologically inhibit several different enzymes we pharmacologically inhibit several different enzymes we believe to play a role in this signaling pathway as believe to play a role in this signaling pathway as suggested by current known metabolic pathways and by suggested by current known metabolic pathways and by modeling resultsmodeling results

bull The next step is to perform gene knockdowns on The next step is to perform gene knockdowns on specific enzymes to verify results of inhibitor dataspecific enzymes to verify results of inhibitor data

bull The model output also suggests a time delay from The model output also suggests a time delay from receptor activation to PIPreceptor activation to PIP22 hydrolysis An upcoming task hydrolysis An upcoming task is to investigate the outcomes of adding such a delay is to investigate the outcomes of adding such a delay termterm

bull The model could be greatly enhanced by incorporating The model could be greatly enhanced by incorporating spatial dynamics so this will also be a major focus for spatial dynamics so this will also be a major focus for future researchfuture research

AcknowledgementsAcknowledgementsAcknowledgementsAcknowledgements

Mathematics DepartmentMathematics Departmentbull Hannah L Callender Hannah L Callender

PhDPhD

CollaboratorsCollaboratorsbull Alliance for Cellular SignalingAlliance for Cellular Signalingbull UT SouthwesternUT Southwestern

ndash Paul Sternweis PhDPaul Sternweis PhDndash Dianne DeCamp PhDDianne DeCamp PhD

Brown LabBrown Labbull H Alex Brown PhDH Alex Brown PhDbull Jeffrey S Forrester PhDJeffrey S Forrester PhDbull Mark Byrne PhDMark Byrne PhDbull Anita Preininger PhDAnita Preininger PhDbull Michelle ArmstrongMichelle Armstrongbull Andrew GoodmanAndrew Goodmanbull Pavlina Ivanova PhDPavlina Ivanova PhDbull Steve Milne PhDSteve Milne PhD

Positivity and boundedness of Positivity and boundedness of solutionssolutions

Positivity and boundedness of Positivity and boundedness of solutionssolutions

bull We first use Theorem 3 to show positivity and boundedness of xWe first use Theorem 3 to show positivity and boundedness of x11 and and xx22

bull Next we use these results and the Fundamental Theorem of Calculus to Next we use these results and the Fundamental Theorem of Calculus to show positivity and boundedness for the remaining equations show positivity and boundedness for the remaining equations

bull This then ensures a global solutionThis then ensures a global solution

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical Analysis

bull Parameter Estimation in SIMULINKParameter Estimation in SIMULINK

bull Sensitivity AnalysisSensitivity Analysis

Parameter Estimation in SIMULINK (for Parameter Estimation in SIMULINK (for FULL model)FULL model)

Parameter Estimation in SIMULINK (for Parameter Estimation in SIMULINK (for FULL model)FULL model)

bull Total number of model parameters (for full model with Li and Rinzel CaTotal number of model parameters (for full model with Li and Rinzel Ca2+2+ module) = module) = 3434ndash From the literature = From the literature = 2020ndash Estimated = Estimated = 1414

bull Receptor moduleReceptor module Total = Total = 77ndash From literature = From literature = 66ndash Estimated = Estimated = 11 (k (kpp rate of receptor phosphorylation) rate of receptor phosphorylation)

bull G-protein cascadeG-protein cascade Total = Total = 99ndash From literature = From literature = 66ndash Estimated = Estimated = 3 3 (k(khydhyd k kreprep k kd3d3))

bull DAG kineticsDAG kinetics Total = Total = 5 5 (for each DAG species considered)(for each DAG species considered)ndash From literature = From literature = 00ndash Estimated = Estimated = 5 5 (all DAG parameters)(all DAG parameters)

bull CaCa2+2+ module module Total = Total = 1313ndash From literature = From literature = 88ndash Estimated = Estimated = 55

SIMULINK DetailsSIMULINK DetailsSIMULINK DetailsSIMULINK Details

bull Unknown rate parameters were estimated using SIMULINK

ndash Minimizes a user-specified cost function via a user-specified optimization method

ndash Nonlinear least squares optimization method of Levenberg-Marquardt was used to minimize a sum of squared errors cost function of the empirical observations and model predictions for IP3 Ca2+ and multiple species of DAG

bull Note Although the Gauss-Newton method is often more efficient the method of Levenberg-Marquardt has proved to be more robust

Measured vs Simulated 384 DAG and Measured vs Simulated 384 DAG and 341 DAG341 DAG

Measured vs Simulated 384 DAG and Measured vs Simulated 384 DAG and 341 DAG341 DAG

bull Timecourse of stimulation with 25 Timecourse of stimulation with 25 M UDP in RAW 2647 cellsM UDP in RAW 2647 cells

bull Solid black lines represent model simulations Solid black lines represent model simulations

bull ((aa) 384 DAG response (representative of the response of most poly ) 384 DAG response (representative of the response of most poly unsaturated fatty acid-containing DAG species) unsaturated fatty acid-containing DAG species)

bull ((bb) 341 DAG response (representative of the response of most mono- ) 341 DAG response (representative of the response of most mono- and di-unsaturated fatty acid-containing DAG species) and di-unsaturated fatty acid-containing DAG species)

bull Data points contain nine replicates performed on three different Data points contain nine replicates performed on three different experimental days with error bars = 1 SEM Units are total change in ng experimental days with error bars = 1 SEM Units are total change in ng over baseline levels in ~8x10over baseline levels in ~8x1066 cells cells

Measured vs Simulated IPMeasured vs Simulated IP33 and Ca and Ca2+2+Measured vs Simulated IPMeasured vs Simulated IP33 and Ca and Ca2+2+

bull Timecourse of stimulation with 25 Timecourse of stimulation with 25 M UDP in RAW 2647 cellsM UDP in RAW 2647 cells

bull Solid black lines represent model simulationsSolid black lines represent model simulations

bull ((cc) IP) IP33 response in pmols per ~35x10 response in pmols per ~35x1055 cells cells

bull Points in (c) represent the average of four experiments and error Points in (c) represent the average of four experiments and error bars are 1 SEM bars are 1 SEM

bull ((dd) Ca) Ca2+2+ response in response in M Red curve is a representative CaM Red curve is a representative Ca2+2+ trace trace taken from the UDP experiments within the AfCS single ligand taken from the UDP experiments within the AfCS single ligand screenscreen

Model Simulations P2YModel Simulations P2Y66 G GGTP GTP PIPPIP22

Model Simulations P2YModel Simulations P2Y66 G GGTP GTP PIPPIP22

bull ((aa) = total P2Y) = total P2Y66 activated (solid activated (solid line) and inactivated (dashed line) and inactivated (dashed line) surface receptorsline) surface receptors

bull ((bb) = total G) = total GGTP GTP

bull ((cc) = total PIP) = total PIP22 available for available for hydrolysishydrolysis

P2Y6 from 25M UDP GGTP from 25M UDP

PIP2 from 25M UDP

Sensitivity AnalysisSensitivity AnalysisSensitivity AnalysisSensitivity Analysis

bull Sensitivity analysis techniques are valuable tools Sensitivity analysis techniques are valuable tools designed to answer questions regarding which of the designed to answer questions regarding which of the uncertain input variables is more important in uncertain input variables is more important in determining the uncertainty in our output determining the uncertainty in our output

bull Likewise sensitivity analysis can provide insight into Likewise sensitivity analysis can provide insight into which parameter should be studied in more detail in which parameter should be studied in more detail in order to reduce the most variance in the model output order to reduce the most variance in the model output

bull The ability to answer these types of questions could The ability to answer these types of questions could lead to important insight into the design of new lead to important insight into the design of new experiments and in determining which experiments experiments and in determining which experiments would give us the most valuable informationwould give us the most valuable information

Sampling MethodSampling MethodSampling MethodSampling Method

bull Generate a random sample of our space of input Generate a random sample of our space of input variables over a ten percent variation from each variables over a ten percent variation from each parameters nominal value using the parameters nominal value using the Latin Hypercube Latin Hypercube Sampling (LHS) methodSampling (LHS) method

bull Uses Uses Standardized Regression Coefficients (SRCs) Standardized Regression Coefficients (SRCs) obtained by performing multiple linear regression obtained by performing multiple linear regression analysisanalysisndash offers a measure of sensitivity that is multi-dimensionally offers a measure of sensitivity that is multi-dimensionally

averaged over the entire space of parameter values averaged over the entire space of parameter values ndash SRCs give insight into degree of nonlinearity in the model SRCs give insight into degree of nonlinearity in the model

Computing RComputing Ryy22 values valuesComputing RComputing Ryy22 values values

bull SRCs are only reliable measures of sensitivity when SRCs are only reliable measures of sensitivity when degree of nonlinearity is ldquosmallrdquodegree of nonlinearity is ldquosmallrdquo

bull Use model coefficients of determination RUse model coefficients of determination Ryy22 given by given by

bull where ywhere yii is the estimate of y is the estimate of yii obtained from the regression obtained from the regression modelmodel

bull RRyy22 ge 07 ensures SRCs are good sensitivity measures ge 07 ensures SRCs are good sensitivity measures

SRCs (as sensitivity measures) for SRCs (as sensitivity measures) for DAGDAG

SRCs (as sensitivity measures) for SRCs (as sensitivity measures) for DAGDAG

SRCs (as sensitivity measures) for IPSRCs (as sensitivity measures) for IP33SRCs (as sensitivity measures) for IPSRCs (as sensitivity measures) for IP33

Effects of 3 most sensitive DAG Effects of 3 most sensitive DAG parametersparameters

Effects of 3 most sensitive DAG Effects of 3 most sensitive DAG parametersparameters

A B

C

AA k kdp2dp2 degradation of pool 2 DAG degradation of pool 2 DAG

BB k kdp1dp1 degradation of pool 1 DAG degradation of pool 1 DAG

CC k kap2ap2 production of pool 2 DAG production of pool 2 DAG

ConclusionsConclusionsConclusionsConclusions

bull We have developed a model of the UDP signaling We have developed a model of the UDP signaling pathway in RAW 2647 macrophages which can predict pathway in RAW 2647 macrophages which can predict the responses of multiple species of DAG as well as the the responses of multiple species of DAG as well as the responses of IPresponses of IP33 Ca Ca2+2+ receptor dynamics G-protein receptor dynamics G-protein activation and PIPactivation and PIP22 hydrolysis hydrolysis

bull Simplified model resultsSimplified model resultsndash We have obtained global existence uniqueness positivity We have obtained global existence uniqueness positivity

and boundedness of solutionsand boundedness of solutionsndash We have proven global stability of a unique steady state We have proven global stability of a unique steady state

within our region of interestwithin our region of interestbull Full model analysisFull model analysis

ndash Using SIMULINK we have estimated unknown rate Using SIMULINK we have estimated unknown rate parameters to obtain best fits to multiple DAG traces IPparameters to obtain best fits to multiple DAG traces IP33 and Caand Ca2+2+ all in response to 25 all in response to 25M UDPM UDP

ndash We have performed sensitivity analysis using the Latin We have performed sensitivity analysis using the Latin Hypercube sampling technique in combination with Hypercube sampling technique in combination with standardized regression coefficients to determine which standardized regression coefficients to determine which model parameters are responsible for most of the model model parameters are responsible for most of the model output uncertaintyoutput uncertainty

Future DirectionsFuture DirectionsFuture DirectionsFuture Directions

bull We have conducted multiple experiments to We have conducted multiple experiments to pharmacologically inhibit several different enzymes we pharmacologically inhibit several different enzymes we believe to play a role in this signaling pathway as believe to play a role in this signaling pathway as suggested by current known metabolic pathways and by suggested by current known metabolic pathways and by modeling resultsmodeling results

bull The next step is to perform gene knockdowns on The next step is to perform gene knockdowns on specific enzymes to verify results of inhibitor dataspecific enzymes to verify results of inhibitor data

bull The model output also suggests a time delay from The model output also suggests a time delay from receptor activation to PIPreceptor activation to PIP22 hydrolysis An upcoming task hydrolysis An upcoming task is to investigate the outcomes of adding such a delay is to investigate the outcomes of adding such a delay termterm

bull The model could be greatly enhanced by incorporating The model could be greatly enhanced by incorporating spatial dynamics so this will also be a major focus for spatial dynamics so this will also be a major focus for future researchfuture research

AcknowledgementsAcknowledgementsAcknowledgementsAcknowledgements

Mathematics DepartmentMathematics Departmentbull Hannah L Callender Hannah L Callender

PhDPhD

CollaboratorsCollaboratorsbull Alliance for Cellular SignalingAlliance for Cellular Signalingbull UT SouthwesternUT Southwestern

ndash Paul Sternweis PhDPaul Sternweis PhDndash Dianne DeCamp PhDDianne DeCamp PhD

Brown LabBrown Labbull H Alex Brown PhDH Alex Brown PhDbull Jeffrey S Forrester PhDJeffrey S Forrester PhDbull Mark Byrne PhDMark Byrne PhDbull Anita Preininger PhDAnita Preininger PhDbull Michelle ArmstrongMichelle Armstrongbull Andrew GoodmanAndrew Goodmanbull Pavlina Ivanova PhDPavlina Ivanova PhDbull Steve Milne PhDSteve Milne PhD

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical Analysis

bull Parameter Estimation in SIMULINKParameter Estimation in SIMULINK

bull Sensitivity AnalysisSensitivity Analysis

Parameter Estimation in SIMULINK (for Parameter Estimation in SIMULINK (for FULL model)FULL model)

Parameter Estimation in SIMULINK (for Parameter Estimation in SIMULINK (for FULL model)FULL model)

bull Total number of model parameters (for full model with Li and Rinzel CaTotal number of model parameters (for full model with Li and Rinzel Ca2+2+ module) = module) = 3434ndash From the literature = From the literature = 2020ndash Estimated = Estimated = 1414

bull Receptor moduleReceptor module Total = Total = 77ndash From literature = From literature = 66ndash Estimated = Estimated = 11 (k (kpp rate of receptor phosphorylation) rate of receptor phosphorylation)

bull G-protein cascadeG-protein cascade Total = Total = 99ndash From literature = From literature = 66ndash Estimated = Estimated = 3 3 (k(khydhyd k kreprep k kd3d3))

bull DAG kineticsDAG kinetics Total = Total = 5 5 (for each DAG species considered)(for each DAG species considered)ndash From literature = From literature = 00ndash Estimated = Estimated = 5 5 (all DAG parameters)(all DAG parameters)

bull CaCa2+2+ module module Total = Total = 1313ndash From literature = From literature = 88ndash Estimated = Estimated = 55

SIMULINK DetailsSIMULINK DetailsSIMULINK DetailsSIMULINK Details

bull Unknown rate parameters were estimated using SIMULINK

ndash Minimizes a user-specified cost function via a user-specified optimization method

ndash Nonlinear least squares optimization method of Levenberg-Marquardt was used to minimize a sum of squared errors cost function of the empirical observations and model predictions for IP3 Ca2+ and multiple species of DAG

bull Note Although the Gauss-Newton method is often more efficient the method of Levenberg-Marquardt has proved to be more robust

Measured vs Simulated 384 DAG and Measured vs Simulated 384 DAG and 341 DAG341 DAG

Measured vs Simulated 384 DAG and Measured vs Simulated 384 DAG and 341 DAG341 DAG

bull Timecourse of stimulation with 25 Timecourse of stimulation with 25 M UDP in RAW 2647 cellsM UDP in RAW 2647 cells

bull Solid black lines represent model simulations Solid black lines represent model simulations

bull ((aa) 384 DAG response (representative of the response of most poly ) 384 DAG response (representative of the response of most poly unsaturated fatty acid-containing DAG species) unsaturated fatty acid-containing DAG species)

bull ((bb) 341 DAG response (representative of the response of most mono- ) 341 DAG response (representative of the response of most mono- and di-unsaturated fatty acid-containing DAG species) and di-unsaturated fatty acid-containing DAG species)

bull Data points contain nine replicates performed on three different Data points contain nine replicates performed on three different experimental days with error bars = 1 SEM Units are total change in ng experimental days with error bars = 1 SEM Units are total change in ng over baseline levels in ~8x10over baseline levels in ~8x1066 cells cells

Measured vs Simulated IPMeasured vs Simulated IP33 and Ca and Ca2+2+Measured vs Simulated IPMeasured vs Simulated IP33 and Ca and Ca2+2+

bull Timecourse of stimulation with 25 Timecourse of stimulation with 25 M UDP in RAW 2647 cellsM UDP in RAW 2647 cells

bull Solid black lines represent model simulationsSolid black lines represent model simulations

bull ((cc) IP) IP33 response in pmols per ~35x10 response in pmols per ~35x1055 cells cells

bull Points in (c) represent the average of four experiments and error Points in (c) represent the average of four experiments and error bars are 1 SEM bars are 1 SEM

bull ((dd) Ca) Ca2+2+ response in response in M Red curve is a representative CaM Red curve is a representative Ca2+2+ trace trace taken from the UDP experiments within the AfCS single ligand taken from the UDP experiments within the AfCS single ligand screenscreen

Model Simulations P2YModel Simulations P2Y66 G GGTP GTP PIPPIP22

Model Simulations P2YModel Simulations P2Y66 G GGTP GTP PIPPIP22

bull ((aa) = total P2Y) = total P2Y66 activated (solid activated (solid line) and inactivated (dashed line) and inactivated (dashed line) surface receptorsline) surface receptors

bull ((bb) = total G) = total GGTP GTP

bull ((cc) = total PIP) = total PIP22 available for available for hydrolysishydrolysis

P2Y6 from 25M UDP GGTP from 25M UDP

PIP2 from 25M UDP

Sensitivity AnalysisSensitivity AnalysisSensitivity AnalysisSensitivity Analysis

bull Sensitivity analysis techniques are valuable tools Sensitivity analysis techniques are valuable tools designed to answer questions regarding which of the designed to answer questions regarding which of the uncertain input variables is more important in uncertain input variables is more important in determining the uncertainty in our output determining the uncertainty in our output

bull Likewise sensitivity analysis can provide insight into Likewise sensitivity analysis can provide insight into which parameter should be studied in more detail in which parameter should be studied in more detail in order to reduce the most variance in the model output order to reduce the most variance in the model output

bull The ability to answer these types of questions could The ability to answer these types of questions could lead to important insight into the design of new lead to important insight into the design of new experiments and in determining which experiments experiments and in determining which experiments would give us the most valuable informationwould give us the most valuable information

Sampling MethodSampling MethodSampling MethodSampling Method

bull Generate a random sample of our space of input Generate a random sample of our space of input variables over a ten percent variation from each variables over a ten percent variation from each parameters nominal value using the parameters nominal value using the Latin Hypercube Latin Hypercube Sampling (LHS) methodSampling (LHS) method

bull Uses Uses Standardized Regression Coefficients (SRCs) Standardized Regression Coefficients (SRCs) obtained by performing multiple linear regression obtained by performing multiple linear regression analysisanalysisndash offers a measure of sensitivity that is multi-dimensionally offers a measure of sensitivity that is multi-dimensionally

averaged over the entire space of parameter values averaged over the entire space of parameter values ndash SRCs give insight into degree of nonlinearity in the model SRCs give insight into degree of nonlinearity in the model

Computing RComputing Ryy22 values valuesComputing RComputing Ryy22 values values

bull SRCs are only reliable measures of sensitivity when SRCs are only reliable measures of sensitivity when degree of nonlinearity is ldquosmallrdquodegree of nonlinearity is ldquosmallrdquo

bull Use model coefficients of determination RUse model coefficients of determination Ryy22 given by given by

bull where ywhere yii is the estimate of y is the estimate of yii obtained from the regression obtained from the regression modelmodel

bull RRyy22 ge 07 ensures SRCs are good sensitivity measures ge 07 ensures SRCs are good sensitivity measures

SRCs (as sensitivity measures) for SRCs (as sensitivity measures) for DAGDAG

SRCs (as sensitivity measures) for SRCs (as sensitivity measures) for DAGDAG

SRCs (as sensitivity measures) for IPSRCs (as sensitivity measures) for IP33SRCs (as sensitivity measures) for IPSRCs (as sensitivity measures) for IP33

Effects of 3 most sensitive DAG Effects of 3 most sensitive DAG parametersparameters

Effects of 3 most sensitive DAG Effects of 3 most sensitive DAG parametersparameters

A B

C

AA k kdp2dp2 degradation of pool 2 DAG degradation of pool 2 DAG

BB k kdp1dp1 degradation of pool 1 DAG degradation of pool 1 DAG

CC k kap2ap2 production of pool 2 DAG production of pool 2 DAG

ConclusionsConclusionsConclusionsConclusions

bull We have developed a model of the UDP signaling We have developed a model of the UDP signaling pathway in RAW 2647 macrophages which can predict pathway in RAW 2647 macrophages which can predict the responses of multiple species of DAG as well as the the responses of multiple species of DAG as well as the responses of IPresponses of IP33 Ca Ca2+2+ receptor dynamics G-protein receptor dynamics G-protein activation and PIPactivation and PIP22 hydrolysis hydrolysis

bull Simplified model resultsSimplified model resultsndash We have obtained global existence uniqueness positivity We have obtained global existence uniqueness positivity

and boundedness of solutionsand boundedness of solutionsndash We have proven global stability of a unique steady state We have proven global stability of a unique steady state

within our region of interestwithin our region of interestbull Full model analysisFull model analysis

ndash Using SIMULINK we have estimated unknown rate Using SIMULINK we have estimated unknown rate parameters to obtain best fits to multiple DAG traces IPparameters to obtain best fits to multiple DAG traces IP33 and Caand Ca2+2+ all in response to 25 all in response to 25M UDPM UDP

ndash We have performed sensitivity analysis using the Latin We have performed sensitivity analysis using the Latin Hypercube sampling technique in combination with Hypercube sampling technique in combination with standardized regression coefficients to determine which standardized regression coefficients to determine which model parameters are responsible for most of the model model parameters are responsible for most of the model output uncertaintyoutput uncertainty

Future DirectionsFuture DirectionsFuture DirectionsFuture Directions

bull We have conducted multiple experiments to We have conducted multiple experiments to pharmacologically inhibit several different enzymes we pharmacologically inhibit several different enzymes we believe to play a role in this signaling pathway as believe to play a role in this signaling pathway as suggested by current known metabolic pathways and by suggested by current known metabolic pathways and by modeling resultsmodeling results

bull The next step is to perform gene knockdowns on The next step is to perform gene knockdowns on specific enzymes to verify results of inhibitor dataspecific enzymes to verify results of inhibitor data

bull The model output also suggests a time delay from The model output also suggests a time delay from receptor activation to PIPreceptor activation to PIP22 hydrolysis An upcoming task hydrolysis An upcoming task is to investigate the outcomes of adding such a delay is to investigate the outcomes of adding such a delay termterm

bull The model could be greatly enhanced by incorporating The model could be greatly enhanced by incorporating spatial dynamics so this will also be a major focus for spatial dynamics so this will also be a major focus for future researchfuture research

AcknowledgementsAcknowledgementsAcknowledgementsAcknowledgements

Mathematics DepartmentMathematics Departmentbull Hannah L Callender Hannah L Callender

PhDPhD

CollaboratorsCollaboratorsbull Alliance for Cellular SignalingAlliance for Cellular Signalingbull UT SouthwesternUT Southwestern

ndash Paul Sternweis PhDPaul Sternweis PhDndash Dianne DeCamp PhDDianne DeCamp PhD

Brown LabBrown Labbull H Alex Brown PhDH Alex Brown PhDbull Jeffrey S Forrester PhDJeffrey S Forrester PhDbull Mark Byrne PhDMark Byrne PhDbull Anita Preininger PhDAnita Preininger PhDbull Michelle ArmstrongMichelle Armstrongbull Andrew GoodmanAndrew Goodmanbull Pavlina Ivanova PhDPavlina Ivanova PhDbull Steve Milne PhDSteve Milne PhD

Parameter Estimation in SIMULINK (for Parameter Estimation in SIMULINK (for FULL model)FULL model)

Parameter Estimation in SIMULINK (for Parameter Estimation in SIMULINK (for FULL model)FULL model)

bull Total number of model parameters (for full model with Li and Rinzel CaTotal number of model parameters (for full model with Li and Rinzel Ca2+2+ module) = module) = 3434ndash From the literature = From the literature = 2020ndash Estimated = Estimated = 1414

bull Receptor moduleReceptor module Total = Total = 77ndash From literature = From literature = 66ndash Estimated = Estimated = 11 (k (kpp rate of receptor phosphorylation) rate of receptor phosphorylation)

bull G-protein cascadeG-protein cascade Total = Total = 99ndash From literature = From literature = 66ndash Estimated = Estimated = 3 3 (k(khydhyd k kreprep k kd3d3))

bull DAG kineticsDAG kinetics Total = Total = 5 5 (for each DAG species considered)(for each DAG species considered)ndash From literature = From literature = 00ndash Estimated = Estimated = 5 5 (all DAG parameters)(all DAG parameters)

bull CaCa2+2+ module module Total = Total = 1313ndash From literature = From literature = 88ndash Estimated = Estimated = 55

SIMULINK DetailsSIMULINK DetailsSIMULINK DetailsSIMULINK Details

bull Unknown rate parameters were estimated using SIMULINK

ndash Minimizes a user-specified cost function via a user-specified optimization method

ndash Nonlinear least squares optimization method of Levenberg-Marquardt was used to minimize a sum of squared errors cost function of the empirical observations and model predictions for IP3 Ca2+ and multiple species of DAG

bull Note Although the Gauss-Newton method is often more efficient the method of Levenberg-Marquardt has proved to be more robust

Measured vs Simulated 384 DAG and Measured vs Simulated 384 DAG and 341 DAG341 DAG

Measured vs Simulated 384 DAG and Measured vs Simulated 384 DAG and 341 DAG341 DAG

bull Timecourse of stimulation with 25 Timecourse of stimulation with 25 M UDP in RAW 2647 cellsM UDP in RAW 2647 cells

bull Solid black lines represent model simulations Solid black lines represent model simulations

bull ((aa) 384 DAG response (representative of the response of most poly ) 384 DAG response (representative of the response of most poly unsaturated fatty acid-containing DAG species) unsaturated fatty acid-containing DAG species)

bull ((bb) 341 DAG response (representative of the response of most mono- ) 341 DAG response (representative of the response of most mono- and di-unsaturated fatty acid-containing DAG species) and di-unsaturated fatty acid-containing DAG species)

bull Data points contain nine replicates performed on three different Data points contain nine replicates performed on three different experimental days with error bars = 1 SEM Units are total change in ng experimental days with error bars = 1 SEM Units are total change in ng over baseline levels in ~8x10over baseline levels in ~8x1066 cells cells

Measured vs Simulated IPMeasured vs Simulated IP33 and Ca and Ca2+2+Measured vs Simulated IPMeasured vs Simulated IP33 and Ca and Ca2+2+

bull Timecourse of stimulation with 25 Timecourse of stimulation with 25 M UDP in RAW 2647 cellsM UDP in RAW 2647 cells

bull Solid black lines represent model simulationsSolid black lines represent model simulations

bull ((cc) IP) IP33 response in pmols per ~35x10 response in pmols per ~35x1055 cells cells

bull Points in (c) represent the average of four experiments and error Points in (c) represent the average of four experiments and error bars are 1 SEM bars are 1 SEM

bull ((dd) Ca) Ca2+2+ response in response in M Red curve is a representative CaM Red curve is a representative Ca2+2+ trace trace taken from the UDP experiments within the AfCS single ligand taken from the UDP experiments within the AfCS single ligand screenscreen

Model Simulations P2YModel Simulations P2Y66 G GGTP GTP PIPPIP22

Model Simulations P2YModel Simulations P2Y66 G GGTP GTP PIPPIP22

bull ((aa) = total P2Y) = total P2Y66 activated (solid activated (solid line) and inactivated (dashed line) and inactivated (dashed line) surface receptorsline) surface receptors

bull ((bb) = total G) = total GGTP GTP

bull ((cc) = total PIP) = total PIP22 available for available for hydrolysishydrolysis

P2Y6 from 25M UDP GGTP from 25M UDP

PIP2 from 25M UDP

Sensitivity AnalysisSensitivity AnalysisSensitivity AnalysisSensitivity Analysis

bull Sensitivity analysis techniques are valuable tools Sensitivity analysis techniques are valuable tools designed to answer questions regarding which of the designed to answer questions regarding which of the uncertain input variables is more important in uncertain input variables is more important in determining the uncertainty in our output determining the uncertainty in our output

bull Likewise sensitivity analysis can provide insight into Likewise sensitivity analysis can provide insight into which parameter should be studied in more detail in which parameter should be studied in more detail in order to reduce the most variance in the model output order to reduce the most variance in the model output

bull The ability to answer these types of questions could The ability to answer these types of questions could lead to important insight into the design of new lead to important insight into the design of new experiments and in determining which experiments experiments and in determining which experiments would give us the most valuable informationwould give us the most valuable information

Sampling MethodSampling MethodSampling MethodSampling Method

bull Generate a random sample of our space of input Generate a random sample of our space of input variables over a ten percent variation from each variables over a ten percent variation from each parameters nominal value using the parameters nominal value using the Latin Hypercube Latin Hypercube Sampling (LHS) methodSampling (LHS) method

bull Uses Uses Standardized Regression Coefficients (SRCs) Standardized Regression Coefficients (SRCs) obtained by performing multiple linear regression obtained by performing multiple linear regression analysisanalysisndash offers a measure of sensitivity that is multi-dimensionally offers a measure of sensitivity that is multi-dimensionally

averaged over the entire space of parameter values averaged over the entire space of parameter values ndash SRCs give insight into degree of nonlinearity in the model SRCs give insight into degree of nonlinearity in the model

Computing RComputing Ryy22 values valuesComputing RComputing Ryy22 values values

bull SRCs are only reliable measures of sensitivity when SRCs are only reliable measures of sensitivity when degree of nonlinearity is ldquosmallrdquodegree of nonlinearity is ldquosmallrdquo

bull Use model coefficients of determination RUse model coefficients of determination Ryy22 given by given by

bull where ywhere yii is the estimate of y is the estimate of yii obtained from the regression obtained from the regression modelmodel

bull RRyy22 ge 07 ensures SRCs are good sensitivity measures ge 07 ensures SRCs are good sensitivity measures

SRCs (as sensitivity measures) for SRCs (as sensitivity measures) for DAGDAG

SRCs (as sensitivity measures) for SRCs (as sensitivity measures) for DAGDAG

SRCs (as sensitivity measures) for IPSRCs (as sensitivity measures) for IP33SRCs (as sensitivity measures) for IPSRCs (as sensitivity measures) for IP33

Effects of 3 most sensitive DAG Effects of 3 most sensitive DAG parametersparameters

Effects of 3 most sensitive DAG Effects of 3 most sensitive DAG parametersparameters

A B

C

AA k kdp2dp2 degradation of pool 2 DAG degradation of pool 2 DAG

BB k kdp1dp1 degradation of pool 1 DAG degradation of pool 1 DAG

CC k kap2ap2 production of pool 2 DAG production of pool 2 DAG

ConclusionsConclusionsConclusionsConclusions

bull We have developed a model of the UDP signaling We have developed a model of the UDP signaling pathway in RAW 2647 macrophages which can predict pathway in RAW 2647 macrophages which can predict the responses of multiple species of DAG as well as the the responses of multiple species of DAG as well as the responses of IPresponses of IP33 Ca Ca2+2+ receptor dynamics G-protein receptor dynamics G-protein activation and PIPactivation and PIP22 hydrolysis hydrolysis

bull Simplified model resultsSimplified model resultsndash We have obtained global existence uniqueness positivity We have obtained global existence uniqueness positivity

and boundedness of solutionsand boundedness of solutionsndash We have proven global stability of a unique steady state We have proven global stability of a unique steady state

within our region of interestwithin our region of interestbull Full model analysisFull model analysis

ndash Using SIMULINK we have estimated unknown rate Using SIMULINK we have estimated unknown rate parameters to obtain best fits to multiple DAG traces IPparameters to obtain best fits to multiple DAG traces IP33 and Caand Ca2+2+ all in response to 25 all in response to 25M UDPM UDP

ndash We have performed sensitivity analysis using the Latin We have performed sensitivity analysis using the Latin Hypercube sampling technique in combination with Hypercube sampling technique in combination with standardized regression coefficients to determine which standardized regression coefficients to determine which model parameters are responsible for most of the model model parameters are responsible for most of the model output uncertaintyoutput uncertainty

Future DirectionsFuture DirectionsFuture DirectionsFuture Directions

bull We have conducted multiple experiments to We have conducted multiple experiments to pharmacologically inhibit several different enzymes we pharmacologically inhibit several different enzymes we believe to play a role in this signaling pathway as believe to play a role in this signaling pathway as suggested by current known metabolic pathways and by suggested by current known metabolic pathways and by modeling resultsmodeling results

bull The next step is to perform gene knockdowns on The next step is to perform gene knockdowns on specific enzymes to verify results of inhibitor dataspecific enzymes to verify results of inhibitor data

bull The model output also suggests a time delay from The model output also suggests a time delay from receptor activation to PIPreceptor activation to PIP22 hydrolysis An upcoming task hydrolysis An upcoming task is to investigate the outcomes of adding such a delay is to investigate the outcomes of adding such a delay termterm

bull The model could be greatly enhanced by incorporating The model could be greatly enhanced by incorporating spatial dynamics so this will also be a major focus for spatial dynamics so this will also be a major focus for future researchfuture research

AcknowledgementsAcknowledgementsAcknowledgementsAcknowledgements

Mathematics DepartmentMathematics Departmentbull Hannah L Callender Hannah L Callender

PhDPhD

CollaboratorsCollaboratorsbull Alliance for Cellular SignalingAlliance for Cellular Signalingbull UT SouthwesternUT Southwestern

ndash Paul Sternweis PhDPaul Sternweis PhDndash Dianne DeCamp PhDDianne DeCamp PhD

Brown LabBrown Labbull H Alex Brown PhDH Alex Brown PhDbull Jeffrey S Forrester PhDJeffrey S Forrester PhDbull Mark Byrne PhDMark Byrne PhDbull Anita Preininger PhDAnita Preininger PhDbull Michelle ArmstrongMichelle Armstrongbull Andrew GoodmanAndrew Goodmanbull Pavlina Ivanova PhDPavlina Ivanova PhDbull Steve Milne PhDSteve Milne PhD

SIMULINK DetailsSIMULINK DetailsSIMULINK DetailsSIMULINK Details

bull Unknown rate parameters were estimated using SIMULINK

ndash Minimizes a user-specified cost function via a user-specified optimization method

ndash Nonlinear least squares optimization method of Levenberg-Marquardt was used to minimize a sum of squared errors cost function of the empirical observations and model predictions for IP3 Ca2+ and multiple species of DAG

bull Note Although the Gauss-Newton method is often more efficient the method of Levenberg-Marquardt has proved to be more robust

Measured vs Simulated 384 DAG and Measured vs Simulated 384 DAG and 341 DAG341 DAG

Measured vs Simulated 384 DAG and Measured vs Simulated 384 DAG and 341 DAG341 DAG

bull Timecourse of stimulation with 25 Timecourse of stimulation with 25 M UDP in RAW 2647 cellsM UDP in RAW 2647 cells

bull Solid black lines represent model simulations Solid black lines represent model simulations

bull ((aa) 384 DAG response (representative of the response of most poly ) 384 DAG response (representative of the response of most poly unsaturated fatty acid-containing DAG species) unsaturated fatty acid-containing DAG species)

bull ((bb) 341 DAG response (representative of the response of most mono- ) 341 DAG response (representative of the response of most mono- and di-unsaturated fatty acid-containing DAG species) and di-unsaturated fatty acid-containing DAG species)

bull Data points contain nine replicates performed on three different Data points contain nine replicates performed on three different experimental days with error bars = 1 SEM Units are total change in ng experimental days with error bars = 1 SEM Units are total change in ng over baseline levels in ~8x10over baseline levels in ~8x1066 cells cells

Measured vs Simulated IPMeasured vs Simulated IP33 and Ca and Ca2+2+Measured vs Simulated IPMeasured vs Simulated IP33 and Ca and Ca2+2+

bull Timecourse of stimulation with 25 Timecourse of stimulation with 25 M UDP in RAW 2647 cellsM UDP in RAW 2647 cells

bull Solid black lines represent model simulationsSolid black lines represent model simulations

bull ((cc) IP) IP33 response in pmols per ~35x10 response in pmols per ~35x1055 cells cells

bull Points in (c) represent the average of four experiments and error Points in (c) represent the average of four experiments and error bars are 1 SEM bars are 1 SEM

bull ((dd) Ca) Ca2+2+ response in response in M Red curve is a representative CaM Red curve is a representative Ca2+2+ trace trace taken from the UDP experiments within the AfCS single ligand taken from the UDP experiments within the AfCS single ligand screenscreen

Model Simulations P2YModel Simulations P2Y66 G GGTP GTP PIPPIP22

Model Simulations P2YModel Simulations P2Y66 G GGTP GTP PIPPIP22

bull ((aa) = total P2Y) = total P2Y66 activated (solid activated (solid line) and inactivated (dashed line) and inactivated (dashed line) surface receptorsline) surface receptors

bull ((bb) = total G) = total GGTP GTP

bull ((cc) = total PIP) = total PIP22 available for available for hydrolysishydrolysis

P2Y6 from 25M UDP GGTP from 25M UDP

PIP2 from 25M UDP

Sensitivity AnalysisSensitivity AnalysisSensitivity AnalysisSensitivity Analysis

bull Sensitivity analysis techniques are valuable tools Sensitivity analysis techniques are valuable tools designed to answer questions regarding which of the designed to answer questions regarding which of the uncertain input variables is more important in uncertain input variables is more important in determining the uncertainty in our output determining the uncertainty in our output

bull Likewise sensitivity analysis can provide insight into Likewise sensitivity analysis can provide insight into which parameter should be studied in more detail in which parameter should be studied in more detail in order to reduce the most variance in the model output order to reduce the most variance in the model output

bull The ability to answer these types of questions could The ability to answer these types of questions could lead to important insight into the design of new lead to important insight into the design of new experiments and in determining which experiments experiments and in determining which experiments would give us the most valuable informationwould give us the most valuable information

Sampling MethodSampling MethodSampling MethodSampling Method

bull Generate a random sample of our space of input Generate a random sample of our space of input variables over a ten percent variation from each variables over a ten percent variation from each parameters nominal value using the parameters nominal value using the Latin Hypercube Latin Hypercube Sampling (LHS) methodSampling (LHS) method

bull Uses Uses Standardized Regression Coefficients (SRCs) Standardized Regression Coefficients (SRCs) obtained by performing multiple linear regression obtained by performing multiple linear regression analysisanalysisndash offers a measure of sensitivity that is multi-dimensionally offers a measure of sensitivity that is multi-dimensionally

averaged over the entire space of parameter values averaged over the entire space of parameter values ndash SRCs give insight into degree of nonlinearity in the model SRCs give insight into degree of nonlinearity in the model

Computing RComputing Ryy22 values valuesComputing RComputing Ryy22 values values

bull SRCs are only reliable measures of sensitivity when SRCs are only reliable measures of sensitivity when degree of nonlinearity is ldquosmallrdquodegree of nonlinearity is ldquosmallrdquo

bull Use model coefficients of determination RUse model coefficients of determination Ryy22 given by given by

bull where ywhere yii is the estimate of y is the estimate of yii obtained from the regression obtained from the regression modelmodel

bull RRyy22 ge 07 ensures SRCs are good sensitivity measures ge 07 ensures SRCs are good sensitivity measures

SRCs (as sensitivity measures) for SRCs (as sensitivity measures) for DAGDAG

SRCs (as sensitivity measures) for SRCs (as sensitivity measures) for DAGDAG

SRCs (as sensitivity measures) for IPSRCs (as sensitivity measures) for IP33SRCs (as sensitivity measures) for IPSRCs (as sensitivity measures) for IP33

Effects of 3 most sensitive DAG Effects of 3 most sensitive DAG parametersparameters

Effects of 3 most sensitive DAG Effects of 3 most sensitive DAG parametersparameters

A B

C

AA k kdp2dp2 degradation of pool 2 DAG degradation of pool 2 DAG

BB k kdp1dp1 degradation of pool 1 DAG degradation of pool 1 DAG

CC k kap2ap2 production of pool 2 DAG production of pool 2 DAG

ConclusionsConclusionsConclusionsConclusions

bull We have developed a model of the UDP signaling We have developed a model of the UDP signaling pathway in RAW 2647 macrophages which can predict pathway in RAW 2647 macrophages which can predict the responses of multiple species of DAG as well as the the responses of multiple species of DAG as well as the responses of IPresponses of IP33 Ca Ca2+2+ receptor dynamics G-protein receptor dynamics G-protein activation and PIPactivation and PIP22 hydrolysis hydrolysis

bull Simplified model resultsSimplified model resultsndash We have obtained global existence uniqueness positivity We have obtained global existence uniqueness positivity

and boundedness of solutionsand boundedness of solutionsndash We have proven global stability of a unique steady state We have proven global stability of a unique steady state

within our region of interestwithin our region of interestbull Full model analysisFull model analysis

ndash Using SIMULINK we have estimated unknown rate Using SIMULINK we have estimated unknown rate parameters to obtain best fits to multiple DAG traces IPparameters to obtain best fits to multiple DAG traces IP33 and Caand Ca2+2+ all in response to 25 all in response to 25M UDPM UDP

ndash We have performed sensitivity analysis using the Latin We have performed sensitivity analysis using the Latin Hypercube sampling technique in combination with Hypercube sampling technique in combination with standardized regression coefficients to determine which standardized regression coefficients to determine which model parameters are responsible for most of the model model parameters are responsible for most of the model output uncertaintyoutput uncertainty

Future DirectionsFuture DirectionsFuture DirectionsFuture Directions

bull We have conducted multiple experiments to We have conducted multiple experiments to pharmacologically inhibit several different enzymes we pharmacologically inhibit several different enzymes we believe to play a role in this signaling pathway as believe to play a role in this signaling pathway as suggested by current known metabolic pathways and by suggested by current known metabolic pathways and by modeling resultsmodeling results

bull The next step is to perform gene knockdowns on The next step is to perform gene knockdowns on specific enzymes to verify results of inhibitor dataspecific enzymes to verify results of inhibitor data

bull The model output also suggests a time delay from The model output also suggests a time delay from receptor activation to PIPreceptor activation to PIP22 hydrolysis An upcoming task hydrolysis An upcoming task is to investigate the outcomes of adding such a delay is to investigate the outcomes of adding such a delay termterm

bull The model could be greatly enhanced by incorporating The model could be greatly enhanced by incorporating spatial dynamics so this will also be a major focus for spatial dynamics so this will also be a major focus for future researchfuture research

AcknowledgementsAcknowledgementsAcknowledgementsAcknowledgements

Mathematics DepartmentMathematics Departmentbull Hannah L Callender Hannah L Callender

PhDPhD

CollaboratorsCollaboratorsbull Alliance for Cellular SignalingAlliance for Cellular Signalingbull UT SouthwesternUT Southwestern

ndash Paul Sternweis PhDPaul Sternweis PhDndash Dianne DeCamp PhDDianne DeCamp PhD

Brown LabBrown Labbull H Alex Brown PhDH Alex Brown PhDbull Jeffrey S Forrester PhDJeffrey S Forrester PhDbull Mark Byrne PhDMark Byrne PhDbull Anita Preininger PhDAnita Preininger PhDbull Michelle ArmstrongMichelle Armstrongbull Andrew GoodmanAndrew Goodmanbull Pavlina Ivanova PhDPavlina Ivanova PhDbull Steve Milne PhDSteve Milne PhD

Measured vs Simulated 384 DAG and Measured vs Simulated 384 DAG and 341 DAG341 DAG

Measured vs Simulated 384 DAG and Measured vs Simulated 384 DAG and 341 DAG341 DAG

bull Timecourse of stimulation with 25 Timecourse of stimulation with 25 M UDP in RAW 2647 cellsM UDP in RAW 2647 cells

bull Solid black lines represent model simulations Solid black lines represent model simulations

bull ((aa) 384 DAG response (representative of the response of most poly ) 384 DAG response (representative of the response of most poly unsaturated fatty acid-containing DAG species) unsaturated fatty acid-containing DAG species)

bull ((bb) 341 DAG response (representative of the response of most mono- ) 341 DAG response (representative of the response of most mono- and di-unsaturated fatty acid-containing DAG species) and di-unsaturated fatty acid-containing DAG species)

bull Data points contain nine replicates performed on three different Data points contain nine replicates performed on three different experimental days with error bars = 1 SEM Units are total change in ng experimental days with error bars = 1 SEM Units are total change in ng over baseline levels in ~8x10over baseline levels in ~8x1066 cells cells

Measured vs Simulated IPMeasured vs Simulated IP33 and Ca and Ca2+2+Measured vs Simulated IPMeasured vs Simulated IP33 and Ca and Ca2+2+

bull Timecourse of stimulation with 25 Timecourse of stimulation with 25 M UDP in RAW 2647 cellsM UDP in RAW 2647 cells

bull Solid black lines represent model simulationsSolid black lines represent model simulations

bull ((cc) IP) IP33 response in pmols per ~35x10 response in pmols per ~35x1055 cells cells

bull Points in (c) represent the average of four experiments and error Points in (c) represent the average of four experiments and error bars are 1 SEM bars are 1 SEM

bull ((dd) Ca) Ca2+2+ response in response in M Red curve is a representative CaM Red curve is a representative Ca2+2+ trace trace taken from the UDP experiments within the AfCS single ligand taken from the UDP experiments within the AfCS single ligand screenscreen

Model Simulations P2YModel Simulations P2Y66 G GGTP GTP PIPPIP22

Model Simulations P2YModel Simulations P2Y66 G GGTP GTP PIPPIP22

bull ((aa) = total P2Y) = total P2Y66 activated (solid activated (solid line) and inactivated (dashed line) and inactivated (dashed line) surface receptorsline) surface receptors

bull ((bb) = total G) = total GGTP GTP

bull ((cc) = total PIP) = total PIP22 available for available for hydrolysishydrolysis

P2Y6 from 25M UDP GGTP from 25M UDP

PIP2 from 25M UDP

Sensitivity AnalysisSensitivity AnalysisSensitivity AnalysisSensitivity Analysis

bull Sensitivity analysis techniques are valuable tools Sensitivity analysis techniques are valuable tools designed to answer questions regarding which of the designed to answer questions regarding which of the uncertain input variables is more important in uncertain input variables is more important in determining the uncertainty in our output determining the uncertainty in our output

bull Likewise sensitivity analysis can provide insight into Likewise sensitivity analysis can provide insight into which parameter should be studied in more detail in which parameter should be studied in more detail in order to reduce the most variance in the model output order to reduce the most variance in the model output

bull The ability to answer these types of questions could The ability to answer these types of questions could lead to important insight into the design of new lead to important insight into the design of new experiments and in determining which experiments experiments and in determining which experiments would give us the most valuable informationwould give us the most valuable information

Sampling MethodSampling MethodSampling MethodSampling Method

bull Generate a random sample of our space of input Generate a random sample of our space of input variables over a ten percent variation from each variables over a ten percent variation from each parameters nominal value using the parameters nominal value using the Latin Hypercube Latin Hypercube Sampling (LHS) methodSampling (LHS) method

bull Uses Uses Standardized Regression Coefficients (SRCs) Standardized Regression Coefficients (SRCs) obtained by performing multiple linear regression obtained by performing multiple linear regression analysisanalysisndash offers a measure of sensitivity that is multi-dimensionally offers a measure of sensitivity that is multi-dimensionally

averaged over the entire space of parameter values averaged over the entire space of parameter values ndash SRCs give insight into degree of nonlinearity in the model SRCs give insight into degree of nonlinearity in the model

Computing RComputing Ryy22 values valuesComputing RComputing Ryy22 values values

bull SRCs are only reliable measures of sensitivity when SRCs are only reliable measures of sensitivity when degree of nonlinearity is ldquosmallrdquodegree of nonlinearity is ldquosmallrdquo

bull Use model coefficients of determination RUse model coefficients of determination Ryy22 given by given by

bull where ywhere yii is the estimate of y is the estimate of yii obtained from the regression obtained from the regression modelmodel

bull RRyy22 ge 07 ensures SRCs are good sensitivity measures ge 07 ensures SRCs are good sensitivity measures

SRCs (as sensitivity measures) for SRCs (as sensitivity measures) for DAGDAG

SRCs (as sensitivity measures) for SRCs (as sensitivity measures) for DAGDAG

SRCs (as sensitivity measures) for IPSRCs (as sensitivity measures) for IP33SRCs (as sensitivity measures) for IPSRCs (as sensitivity measures) for IP33

Effects of 3 most sensitive DAG Effects of 3 most sensitive DAG parametersparameters

Effects of 3 most sensitive DAG Effects of 3 most sensitive DAG parametersparameters

A B

C

AA k kdp2dp2 degradation of pool 2 DAG degradation of pool 2 DAG

BB k kdp1dp1 degradation of pool 1 DAG degradation of pool 1 DAG

CC k kap2ap2 production of pool 2 DAG production of pool 2 DAG

ConclusionsConclusionsConclusionsConclusions

bull We have developed a model of the UDP signaling We have developed a model of the UDP signaling pathway in RAW 2647 macrophages which can predict pathway in RAW 2647 macrophages which can predict the responses of multiple species of DAG as well as the the responses of multiple species of DAG as well as the responses of IPresponses of IP33 Ca Ca2+2+ receptor dynamics G-protein receptor dynamics G-protein activation and PIPactivation and PIP22 hydrolysis hydrolysis

bull Simplified model resultsSimplified model resultsndash We have obtained global existence uniqueness positivity We have obtained global existence uniqueness positivity

and boundedness of solutionsand boundedness of solutionsndash We have proven global stability of a unique steady state We have proven global stability of a unique steady state

within our region of interestwithin our region of interestbull Full model analysisFull model analysis

ndash Using SIMULINK we have estimated unknown rate Using SIMULINK we have estimated unknown rate parameters to obtain best fits to multiple DAG traces IPparameters to obtain best fits to multiple DAG traces IP33 and Caand Ca2+2+ all in response to 25 all in response to 25M UDPM UDP

ndash We have performed sensitivity analysis using the Latin We have performed sensitivity analysis using the Latin Hypercube sampling technique in combination with Hypercube sampling technique in combination with standardized regression coefficients to determine which standardized regression coefficients to determine which model parameters are responsible for most of the model model parameters are responsible for most of the model output uncertaintyoutput uncertainty

Future DirectionsFuture DirectionsFuture DirectionsFuture Directions

bull We have conducted multiple experiments to We have conducted multiple experiments to pharmacologically inhibit several different enzymes we pharmacologically inhibit several different enzymes we believe to play a role in this signaling pathway as believe to play a role in this signaling pathway as suggested by current known metabolic pathways and by suggested by current known metabolic pathways and by modeling resultsmodeling results

bull The next step is to perform gene knockdowns on The next step is to perform gene knockdowns on specific enzymes to verify results of inhibitor dataspecific enzymes to verify results of inhibitor data

bull The model output also suggests a time delay from The model output also suggests a time delay from receptor activation to PIPreceptor activation to PIP22 hydrolysis An upcoming task hydrolysis An upcoming task is to investigate the outcomes of adding such a delay is to investigate the outcomes of adding such a delay termterm

bull The model could be greatly enhanced by incorporating The model could be greatly enhanced by incorporating spatial dynamics so this will also be a major focus for spatial dynamics so this will also be a major focus for future researchfuture research

AcknowledgementsAcknowledgementsAcknowledgementsAcknowledgements

Mathematics DepartmentMathematics Departmentbull Hannah L Callender Hannah L Callender

PhDPhD

CollaboratorsCollaboratorsbull Alliance for Cellular SignalingAlliance for Cellular Signalingbull UT SouthwesternUT Southwestern

ndash Paul Sternweis PhDPaul Sternweis PhDndash Dianne DeCamp PhDDianne DeCamp PhD

Brown LabBrown Labbull H Alex Brown PhDH Alex Brown PhDbull Jeffrey S Forrester PhDJeffrey S Forrester PhDbull Mark Byrne PhDMark Byrne PhDbull Anita Preininger PhDAnita Preininger PhDbull Michelle ArmstrongMichelle Armstrongbull Andrew GoodmanAndrew Goodmanbull Pavlina Ivanova PhDPavlina Ivanova PhDbull Steve Milne PhDSteve Milne PhD

Measured vs Simulated IPMeasured vs Simulated IP33 and Ca and Ca2+2+Measured vs Simulated IPMeasured vs Simulated IP33 and Ca and Ca2+2+

bull Timecourse of stimulation with 25 Timecourse of stimulation with 25 M UDP in RAW 2647 cellsM UDP in RAW 2647 cells

bull Solid black lines represent model simulationsSolid black lines represent model simulations

bull ((cc) IP) IP33 response in pmols per ~35x10 response in pmols per ~35x1055 cells cells

bull Points in (c) represent the average of four experiments and error Points in (c) represent the average of four experiments and error bars are 1 SEM bars are 1 SEM

bull ((dd) Ca) Ca2+2+ response in response in M Red curve is a representative CaM Red curve is a representative Ca2+2+ trace trace taken from the UDP experiments within the AfCS single ligand taken from the UDP experiments within the AfCS single ligand screenscreen

Model Simulations P2YModel Simulations P2Y66 G GGTP GTP PIPPIP22

Model Simulations P2YModel Simulations P2Y66 G GGTP GTP PIPPIP22

bull ((aa) = total P2Y) = total P2Y66 activated (solid activated (solid line) and inactivated (dashed line) and inactivated (dashed line) surface receptorsline) surface receptors

bull ((bb) = total G) = total GGTP GTP

bull ((cc) = total PIP) = total PIP22 available for available for hydrolysishydrolysis

P2Y6 from 25M UDP GGTP from 25M UDP

PIP2 from 25M UDP

Sensitivity AnalysisSensitivity AnalysisSensitivity AnalysisSensitivity Analysis

bull Sensitivity analysis techniques are valuable tools Sensitivity analysis techniques are valuable tools designed to answer questions regarding which of the designed to answer questions regarding which of the uncertain input variables is more important in uncertain input variables is more important in determining the uncertainty in our output determining the uncertainty in our output

bull Likewise sensitivity analysis can provide insight into Likewise sensitivity analysis can provide insight into which parameter should be studied in more detail in which parameter should be studied in more detail in order to reduce the most variance in the model output order to reduce the most variance in the model output

bull The ability to answer these types of questions could The ability to answer these types of questions could lead to important insight into the design of new lead to important insight into the design of new experiments and in determining which experiments experiments and in determining which experiments would give us the most valuable informationwould give us the most valuable information

Sampling MethodSampling MethodSampling MethodSampling Method

bull Generate a random sample of our space of input Generate a random sample of our space of input variables over a ten percent variation from each variables over a ten percent variation from each parameters nominal value using the parameters nominal value using the Latin Hypercube Latin Hypercube Sampling (LHS) methodSampling (LHS) method

bull Uses Uses Standardized Regression Coefficients (SRCs) Standardized Regression Coefficients (SRCs) obtained by performing multiple linear regression obtained by performing multiple linear regression analysisanalysisndash offers a measure of sensitivity that is multi-dimensionally offers a measure of sensitivity that is multi-dimensionally

averaged over the entire space of parameter values averaged over the entire space of parameter values ndash SRCs give insight into degree of nonlinearity in the model SRCs give insight into degree of nonlinearity in the model

Computing RComputing Ryy22 values valuesComputing RComputing Ryy22 values values

bull SRCs are only reliable measures of sensitivity when SRCs are only reliable measures of sensitivity when degree of nonlinearity is ldquosmallrdquodegree of nonlinearity is ldquosmallrdquo

bull Use model coefficients of determination RUse model coefficients of determination Ryy22 given by given by

bull where ywhere yii is the estimate of y is the estimate of yii obtained from the regression obtained from the regression modelmodel

bull RRyy22 ge 07 ensures SRCs are good sensitivity measures ge 07 ensures SRCs are good sensitivity measures

SRCs (as sensitivity measures) for SRCs (as sensitivity measures) for DAGDAG

SRCs (as sensitivity measures) for SRCs (as sensitivity measures) for DAGDAG

SRCs (as sensitivity measures) for IPSRCs (as sensitivity measures) for IP33SRCs (as sensitivity measures) for IPSRCs (as sensitivity measures) for IP33

Effects of 3 most sensitive DAG Effects of 3 most sensitive DAG parametersparameters

Effects of 3 most sensitive DAG Effects of 3 most sensitive DAG parametersparameters

A B

C

AA k kdp2dp2 degradation of pool 2 DAG degradation of pool 2 DAG

BB k kdp1dp1 degradation of pool 1 DAG degradation of pool 1 DAG

CC k kap2ap2 production of pool 2 DAG production of pool 2 DAG

ConclusionsConclusionsConclusionsConclusions

bull We have developed a model of the UDP signaling We have developed a model of the UDP signaling pathway in RAW 2647 macrophages which can predict pathway in RAW 2647 macrophages which can predict the responses of multiple species of DAG as well as the the responses of multiple species of DAG as well as the responses of IPresponses of IP33 Ca Ca2+2+ receptor dynamics G-protein receptor dynamics G-protein activation and PIPactivation and PIP22 hydrolysis hydrolysis

bull Simplified model resultsSimplified model resultsndash We have obtained global existence uniqueness positivity We have obtained global existence uniqueness positivity

and boundedness of solutionsand boundedness of solutionsndash We have proven global stability of a unique steady state We have proven global stability of a unique steady state

within our region of interestwithin our region of interestbull Full model analysisFull model analysis

ndash Using SIMULINK we have estimated unknown rate Using SIMULINK we have estimated unknown rate parameters to obtain best fits to multiple DAG traces IPparameters to obtain best fits to multiple DAG traces IP33 and Caand Ca2+2+ all in response to 25 all in response to 25M UDPM UDP

ndash We have performed sensitivity analysis using the Latin We have performed sensitivity analysis using the Latin Hypercube sampling technique in combination with Hypercube sampling technique in combination with standardized regression coefficients to determine which standardized regression coefficients to determine which model parameters are responsible for most of the model model parameters are responsible for most of the model output uncertaintyoutput uncertainty

Future DirectionsFuture DirectionsFuture DirectionsFuture Directions

bull We have conducted multiple experiments to We have conducted multiple experiments to pharmacologically inhibit several different enzymes we pharmacologically inhibit several different enzymes we believe to play a role in this signaling pathway as believe to play a role in this signaling pathway as suggested by current known metabolic pathways and by suggested by current known metabolic pathways and by modeling resultsmodeling results

bull The next step is to perform gene knockdowns on The next step is to perform gene knockdowns on specific enzymes to verify results of inhibitor dataspecific enzymes to verify results of inhibitor data

bull The model output also suggests a time delay from The model output also suggests a time delay from receptor activation to PIPreceptor activation to PIP22 hydrolysis An upcoming task hydrolysis An upcoming task is to investigate the outcomes of adding such a delay is to investigate the outcomes of adding such a delay termterm

bull The model could be greatly enhanced by incorporating The model could be greatly enhanced by incorporating spatial dynamics so this will also be a major focus for spatial dynamics so this will also be a major focus for future researchfuture research

AcknowledgementsAcknowledgementsAcknowledgementsAcknowledgements

Mathematics DepartmentMathematics Departmentbull Hannah L Callender Hannah L Callender

PhDPhD

CollaboratorsCollaboratorsbull Alliance for Cellular SignalingAlliance for Cellular Signalingbull UT SouthwesternUT Southwestern

ndash Paul Sternweis PhDPaul Sternweis PhDndash Dianne DeCamp PhDDianne DeCamp PhD

Brown LabBrown Labbull H Alex Brown PhDH Alex Brown PhDbull Jeffrey S Forrester PhDJeffrey S Forrester PhDbull Mark Byrne PhDMark Byrne PhDbull Anita Preininger PhDAnita Preininger PhDbull Michelle ArmstrongMichelle Armstrongbull Andrew GoodmanAndrew Goodmanbull Pavlina Ivanova PhDPavlina Ivanova PhDbull Steve Milne PhDSteve Milne PhD

Model Simulations P2YModel Simulations P2Y66 G GGTP GTP PIPPIP22

Model Simulations P2YModel Simulations P2Y66 G GGTP GTP PIPPIP22

bull ((aa) = total P2Y) = total P2Y66 activated (solid activated (solid line) and inactivated (dashed line) and inactivated (dashed line) surface receptorsline) surface receptors

bull ((bb) = total G) = total GGTP GTP

bull ((cc) = total PIP) = total PIP22 available for available for hydrolysishydrolysis

P2Y6 from 25M UDP GGTP from 25M UDP

PIP2 from 25M UDP

Sensitivity AnalysisSensitivity AnalysisSensitivity AnalysisSensitivity Analysis

bull Sensitivity analysis techniques are valuable tools Sensitivity analysis techniques are valuable tools designed to answer questions regarding which of the designed to answer questions regarding which of the uncertain input variables is more important in uncertain input variables is more important in determining the uncertainty in our output determining the uncertainty in our output

bull Likewise sensitivity analysis can provide insight into Likewise sensitivity analysis can provide insight into which parameter should be studied in more detail in which parameter should be studied in more detail in order to reduce the most variance in the model output order to reduce the most variance in the model output

bull The ability to answer these types of questions could The ability to answer these types of questions could lead to important insight into the design of new lead to important insight into the design of new experiments and in determining which experiments experiments and in determining which experiments would give us the most valuable informationwould give us the most valuable information

Sampling MethodSampling MethodSampling MethodSampling Method

bull Generate a random sample of our space of input Generate a random sample of our space of input variables over a ten percent variation from each variables over a ten percent variation from each parameters nominal value using the parameters nominal value using the Latin Hypercube Latin Hypercube Sampling (LHS) methodSampling (LHS) method

bull Uses Uses Standardized Regression Coefficients (SRCs) Standardized Regression Coefficients (SRCs) obtained by performing multiple linear regression obtained by performing multiple linear regression analysisanalysisndash offers a measure of sensitivity that is multi-dimensionally offers a measure of sensitivity that is multi-dimensionally

averaged over the entire space of parameter values averaged over the entire space of parameter values ndash SRCs give insight into degree of nonlinearity in the model SRCs give insight into degree of nonlinearity in the model

Computing RComputing Ryy22 values valuesComputing RComputing Ryy22 values values

bull SRCs are only reliable measures of sensitivity when SRCs are only reliable measures of sensitivity when degree of nonlinearity is ldquosmallrdquodegree of nonlinearity is ldquosmallrdquo

bull Use model coefficients of determination RUse model coefficients of determination Ryy22 given by given by

bull where ywhere yii is the estimate of y is the estimate of yii obtained from the regression obtained from the regression modelmodel

bull RRyy22 ge 07 ensures SRCs are good sensitivity measures ge 07 ensures SRCs are good sensitivity measures

SRCs (as sensitivity measures) for SRCs (as sensitivity measures) for DAGDAG

SRCs (as sensitivity measures) for SRCs (as sensitivity measures) for DAGDAG

SRCs (as sensitivity measures) for IPSRCs (as sensitivity measures) for IP33SRCs (as sensitivity measures) for IPSRCs (as sensitivity measures) for IP33

Effects of 3 most sensitive DAG Effects of 3 most sensitive DAG parametersparameters

Effects of 3 most sensitive DAG Effects of 3 most sensitive DAG parametersparameters

A B

C

AA k kdp2dp2 degradation of pool 2 DAG degradation of pool 2 DAG

BB k kdp1dp1 degradation of pool 1 DAG degradation of pool 1 DAG

CC k kap2ap2 production of pool 2 DAG production of pool 2 DAG

ConclusionsConclusionsConclusionsConclusions

bull We have developed a model of the UDP signaling We have developed a model of the UDP signaling pathway in RAW 2647 macrophages which can predict pathway in RAW 2647 macrophages which can predict the responses of multiple species of DAG as well as the the responses of multiple species of DAG as well as the responses of IPresponses of IP33 Ca Ca2+2+ receptor dynamics G-protein receptor dynamics G-protein activation and PIPactivation and PIP22 hydrolysis hydrolysis

bull Simplified model resultsSimplified model resultsndash We have obtained global existence uniqueness positivity We have obtained global existence uniqueness positivity

and boundedness of solutionsand boundedness of solutionsndash We have proven global stability of a unique steady state We have proven global stability of a unique steady state

within our region of interestwithin our region of interestbull Full model analysisFull model analysis

ndash Using SIMULINK we have estimated unknown rate Using SIMULINK we have estimated unknown rate parameters to obtain best fits to multiple DAG traces IPparameters to obtain best fits to multiple DAG traces IP33 and Caand Ca2+2+ all in response to 25 all in response to 25M UDPM UDP

ndash We have performed sensitivity analysis using the Latin We have performed sensitivity analysis using the Latin Hypercube sampling technique in combination with Hypercube sampling technique in combination with standardized regression coefficients to determine which standardized regression coefficients to determine which model parameters are responsible for most of the model model parameters are responsible for most of the model output uncertaintyoutput uncertainty

Future DirectionsFuture DirectionsFuture DirectionsFuture Directions

bull We have conducted multiple experiments to We have conducted multiple experiments to pharmacologically inhibit several different enzymes we pharmacologically inhibit several different enzymes we believe to play a role in this signaling pathway as believe to play a role in this signaling pathway as suggested by current known metabolic pathways and by suggested by current known metabolic pathways and by modeling resultsmodeling results

bull The next step is to perform gene knockdowns on The next step is to perform gene knockdowns on specific enzymes to verify results of inhibitor dataspecific enzymes to verify results of inhibitor data

bull The model output also suggests a time delay from The model output also suggests a time delay from receptor activation to PIPreceptor activation to PIP22 hydrolysis An upcoming task hydrolysis An upcoming task is to investigate the outcomes of adding such a delay is to investigate the outcomes of adding such a delay termterm

bull The model could be greatly enhanced by incorporating The model could be greatly enhanced by incorporating spatial dynamics so this will also be a major focus for spatial dynamics so this will also be a major focus for future researchfuture research

AcknowledgementsAcknowledgementsAcknowledgementsAcknowledgements

Mathematics DepartmentMathematics Departmentbull Hannah L Callender Hannah L Callender

PhDPhD

CollaboratorsCollaboratorsbull Alliance for Cellular SignalingAlliance for Cellular Signalingbull UT SouthwesternUT Southwestern

ndash Paul Sternweis PhDPaul Sternweis PhDndash Dianne DeCamp PhDDianne DeCamp PhD

Brown LabBrown Labbull H Alex Brown PhDH Alex Brown PhDbull Jeffrey S Forrester PhDJeffrey S Forrester PhDbull Mark Byrne PhDMark Byrne PhDbull Anita Preininger PhDAnita Preininger PhDbull Michelle ArmstrongMichelle Armstrongbull Andrew GoodmanAndrew Goodmanbull Pavlina Ivanova PhDPavlina Ivanova PhDbull Steve Milne PhDSteve Milne PhD

Sensitivity AnalysisSensitivity AnalysisSensitivity AnalysisSensitivity Analysis

bull Sensitivity analysis techniques are valuable tools Sensitivity analysis techniques are valuable tools designed to answer questions regarding which of the designed to answer questions regarding which of the uncertain input variables is more important in uncertain input variables is more important in determining the uncertainty in our output determining the uncertainty in our output

bull Likewise sensitivity analysis can provide insight into Likewise sensitivity analysis can provide insight into which parameter should be studied in more detail in which parameter should be studied in more detail in order to reduce the most variance in the model output order to reduce the most variance in the model output

bull The ability to answer these types of questions could The ability to answer these types of questions could lead to important insight into the design of new lead to important insight into the design of new experiments and in determining which experiments experiments and in determining which experiments would give us the most valuable informationwould give us the most valuable information

Sampling MethodSampling MethodSampling MethodSampling Method

bull Generate a random sample of our space of input Generate a random sample of our space of input variables over a ten percent variation from each variables over a ten percent variation from each parameters nominal value using the parameters nominal value using the Latin Hypercube Latin Hypercube Sampling (LHS) methodSampling (LHS) method

bull Uses Uses Standardized Regression Coefficients (SRCs) Standardized Regression Coefficients (SRCs) obtained by performing multiple linear regression obtained by performing multiple linear regression analysisanalysisndash offers a measure of sensitivity that is multi-dimensionally offers a measure of sensitivity that is multi-dimensionally

averaged over the entire space of parameter values averaged over the entire space of parameter values ndash SRCs give insight into degree of nonlinearity in the model SRCs give insight into degree of nonlinearity in the model

Computing RComputing Ryy22 values valuesComputing RComputing Ryy22 values values

bull SRCs are only reliable measures of sensitivity when SRCs are only reliable measures of sensitivity when degree of nonlinearity is ldquosmallrdquodegree of nonlinearity is ldquosmallrdquo

bull Use model coefficients of determination RUse model coefficients of determination Ryy22 given by given by

bull where ywhere yii is the estimate of y is the estimate of yii obtained from the regression obtained from the regression modelmodel

bull RRyy22 ge 07 ensures SRCs are good sensitivity measures ge 07 ensures SRCs are good sensitivity measures

SRCs (as sensitivity measures) for SRCs (as sensitivity measures) for DAGDAG

SRCs (as sensitivity measures) for SRCs (as sensitivity measures) for DAGDAG

SRCs (as sensitivity measures) for IPSRCs (as sensitivity measures) for IP33SRCs (as sensitivity measures) for IPSRCs (as sensitivity measures) for IP33

Effects of 3 most sensitive DAG Effects of 3 most sensitive DAG parametersparameters

Effects of 3 most sensitive DAG Effects of 3 most sensitive DAG parametersparameters

A B

C

AA k kdp2dp2 degradation of pool 2 DAG degradation of pool 2 DAG

BB k kdp1dp1 degradation of pool 1 DAG degradation of pool 1 DAG

CC k kap2ap2 production of pool 2 DAG production of pool 2 DAG

ConclusionsConclusionsConclusionsConclusions

bull We have developed a model of the UDP signaling We have developed a model of the UDP signaling pathway in RAW 2647 macrophages which can predict pathway in RAW 2647 macrophages which can predict the responses of multiple species of DAG as well as the the responses of multiple species of DAG as well as the responses of IPresponses of IP33 Ca Ca2+2+ receptor dynamics G-protein receptor dynamics G-protein activation and PIPactivation and PIP22 hydrolysis hydrolysis

bull Simplified model resultsSimplified model resultsndash We have obtained global existence uniqueness positivity We have obtained global existence uniqueness positivity

and boundedness of solutionsand boundedness of solutionsndash We have proven global stability of a unique steady state We have proven global stability of a unique steady state

within our region of interestwithin our region of interestbull Full model analysisFull model analysis

ndash Using SIMULINK we have estimated unknown rate Using SIMULINK we have estimated unknown rate parameters to obtain best fits to multiple DAG traces IPparameters to obtain best fits to multiple DAG traces IP33 and Caand Ca2+2+ all in response to 25 all in response to 25M UDPM UDP

ndash We have performed sensitivity analysis using the Latin We have performed sensitivity analysis using the Latin Hypercube sampling technique in combination with Hypercube sampling technique in combination with standardized regression coefficients to determine which standardized regression coefficients to determine which model parameters are responsible for most of the model model parameters are responsible for most of the model output uncertaintyoutput uncertainty

Future DirectionsFuture DirectionsFuture DirectionsFuture Directions

bull We have conducted multiple experiments to We have conducted multiple experiments to pharmacologically inhibit several different enzymes we pharmacologically inhibit several different enzymes we believe to play a role in this signaling pathway as believe to play a role in this signaling pathway as suggested by current known metabolic pathways and by suggested by current known metabolic pathways and by modeling resultsmodeling results

bull The next step is to perform gene knockdowns on The next step is to perform gene knockdowns on specific enzymes to verify results of inhibitor dataspecific enzymes to verify results of inhibitor data

bull The model output also suggests a time delay from The model output also suggests a time delay from receptor activation to PIPreceptor activation to PIP22 hydrolysis An upcoming task hydrolysis An upcoming task is to investigate the outcomes of adding such a delay is to investigate the outcomes of adding such a delay termterm

bull The model could be greatly enhanced by incorporating The model could be greatly enhanced by incorporating spatial dynamics so this will also be a major focus for spatial dynamics so this will also be a major focus for future researchfuture research

AcknowledgementsAcknowledgementsAcknowledgementsAcknowledgements

Mathematics DepartmentMathematics Departmentbull Hannah L Callender Hannah L Callender

PhDPhD

CollaboratorsCollaboratorsbull Alliance for Cellular SignalingAlliance for Cellular Signalingbull UT SouthwesternUT Southwestern

ndash Paul Sternweis PhDPaul Sternweis PhDndash Dianne DeCamp PhDDianne DeCamp PhD

Brown LabBrown Labbull H Alex Brown PhDH Alex Brown PhDbull Jeffrey S Forrester PhDJeffrey S Forrester PhDbull Mark Byrne PhDMark Byrne PhDbull Anita Preininger PhDAnita Preininger PhDbull Michelle ArmstrongMichelle Armstrongbull Andrew GoodmanAndrew Goodmanbull Pavlina Ivanova PhDPavlina Ivanova PhDbull Steve Milne PhDSteve Milne PhD

Sampling MethodSampling MethodSampling MethodSampling Method

bull Generate a random sample of our space of input Generate a random sample of our space of input variables over a ten percent variation from each variables over a ten percent variation from each parameters nominal value using the parameters nominal value using the Latin Hypercube Latin Hypercube Sampling (LHS) methodSampling (LHS) method

bull Uses Uses Standardized Regression Coefficients (SRCs) Standardized Regression Coefficients (SRCs) obtained by performing multiple linear regression obtained by performing multiple linear regression analysisanalysisndash offers a measure of sensitivity that is multi-dimensionally offers a measure of sensitivity that is multi-dimensionally

averaged over the entire space of parameter values averaged over the entire space of parameter values ndash SRCs give insight into degree of nonlinearity in the model SRCs give insight into degree of nonlinearity in the model

Computing RComputing Ryy22 values valuesComputing RComputing Ryy22 values values

bull SRCs are only reliable measures of sensitivity when SRCs are only reliable measures of sensitivity when degree of nonlinearity is ldquosmallrdquodegree of nonlinearity is ldquosmallrdquo

bull Use model coefficients of determination RUse model coefficients of determination Ryy22 given by given by

bull where ywhere yii is the estimate of y is the estimate of yii obtained from the regression obtained from the regression modelmodel

bull RRyy22 ge 07 ensures SRCs are good sensitivity measures ge 07 ensures SRCs are good sensitivity measures

SRCs (as sensitivity measures) for SRCs (as sensitivity measures) for DAGDAG

SRCs (as sensitivity measures) for SRCs (as sensitivity measures) for DAGDAG

SRCs (as sensitivity measures) for IPSRCs (as sensitivity measures) for IP33SRCs (as sensitivity measures) for IPSRCs (as sensitivity measures) for IP33

Effects of 3 most sensitive DAG Effects of 3 most sensitive DAG parametersparameters

Effects of 3 most sensitive DAG Effects of 3 most sensitive DAG parametersparameters

A B

C

AA k kdp2dp2 degradation of pool 2 DAG degradation of pool 2 DAG

BB k kdp1dp1 degradation of pool 1 DAG degradation of pool 1 DAG

CC k kap2ap2 production of pool 2 DAG production of pool 2 DAG

ConclusionsConclusionsConclusionsConclusions

bull We have developed a model of the UDP signaling We have developed a model of the UDP signaling pathway in RAW 2647 macrophages which can predict pathway in RAW 2647 macrophages which can predict the responses of multiple species of DAG as well as the the responses of multiple species of DAG as well as the responses of IPresponses of IP33 Ca Ca2+2+ receptor dynamics G-protein receptor dynamics G-protein activation and PIPactivation and PIP22 hydrolysis hydrolysis

bull Simplified model resultsSimplified model resultsndash We have obtained global existence uniqueness positivity We have obtained global existence uniqueness positivity

and boundedness of solutionsand boundedness of solutionsndash We have proven global stability of a unique steady state We have proven global stability of a unique steady state

within our region of interestwithin our region of interestbull Full model analysisFull model analysis

ndash Using SIMULINK we have estimated unknown rate Using SIMULINK we have estimated unknown rate parameters to obtain best fits to multiple DAG traces IPparameters to obtain best fits to multiple DAG traces IP33 and Caand Ca2+2+ all in response to 25 all in response to 25M UDPM UDP

ndash We have performed sensitivity analysis using the Latin We have performed sensitivity analysis using the Latin Hypercube sampling technique in combination with Hypercube sampling technique in combination with standardized regression coefficients to determine which standardized regression coefficients to determine which model parameters are responsible for most of the model model parameters are responsible for most of the model output uncertaintyoutput uncertainty

Future DirectionsFuture DirectionsFuture DirectionsFuture Directions

bull We have conducted multiple experiments to We have conducted multiple experiments to pharmacologically inhibit several different enzymes we pharmacologically inhibit several different enzymes we believe to play a role in this signaling pathway as believe to play a role in this signaling pathway as suggested by current known metabolic pathways and by suggested by current known metabolic pathways and by modeling resultsmodeling results

bull The next step is to perform gene knockdowns on The next step is to perform gene knockdowns on specific enzymes to verify results of inhibitor dataspecific enzymes to verify results of inhibitor data

bull The model output also suggests a time delay from The model output also suggests a time delay from receptor activation to PIPreceptor activation to PIP22 hydrolysis An upcoming task hydrolysis An upcoming task is to investigate the outcomes of adding such a delay is to investigate the outcomes of adding such a delay termterm

bull The model could be greatly enhanced by incorporating The model could be greatly enhanced by incorporating spatial dynamics so this will also be a major focus for spatial dynamics so this will also be a major focus for future researchfuture research

AcknowledgementsAcknowledgementsAcknowledgementsAcknowledgements

Mathematics DepartmentMathematics Departmentbull Hannah L Callender Hannah L Callender

PhDPhD

CollaboratorsCollaboratorsbull Alliance for Cellular SignalingAlliance for Cellular Signalingbull UT SouthwesternUT Southwestern

ndash Paul Sternweis PhDPaul Sternweis PhDndash Dianne DeCamp PhDDianne DeCamp PhD

Brown LabBrown Labbull H Alex Brown PhDH Alex Brown PhDbull Jeffrey S Forrester PhDJeffrey S Forrester PhDbull Mark Byrne PhDMark Byrne PhDbull Anita Preininger PhDAnita Preininger PhDbull Michelle ArmstrongMichelle Armstrongbull Andrew GoodmanAndrew Goodmanbull Pavlina Ivanova PhDPavlina Ivanova PhDbull Steve Milne PhDSteve Milne PhD

Computing RComputing Ryy22 values valuesComputing RComputing Ryy22 values values

bull SRCs are only reliable measures of sensitivity when SRCs are only reliable measures of sensitivity when degree of nonlinearity is ldquosmallrdquodegree of nonlinearity is ldquosmallrdquo

bull Use model coefficients of determination RUse model coefficients of determination Ryy22 given by given by

bull where ywhere yii is the estimate of y is the estimate of yii obtained from the regression obtained from the regression modelmodel

bull RRyy22 ge 07 ensures SRCs are good sensitivity measures ge 07 ensures SRCs are good sensitivity measures

SRCs (as sensitivity measures) for SRCs (as sensitivity measures) for DAGDAG

SRCs (as sensitivity measures) for SRCs (as sensitivity measures) for DAGDAG

SRCs (as sensitivity measures) for IPSRCs (as sensitivity measures) for IP33SRCs (as sensitivity measures) for IPSRCs (as sensitivity measures) for IP33

Effects of 3 most sensitive DAG Effects of 3 most sensitive DAG parametersparameters

Effects of 3 most sensitive DAG Effects of 3 most sensitive DAG parametersparameters

A B

C

AA k kdp2dp2 degradation of pool 2 DAG degradation of pool 2 DAG

BB k kdp1dp1 degradation of pool 1 DAG degradation of pool 1 DAG

CC k kap2ap2 production of pool 2 DAG production of pool 2 DAG

ConclusionsConclusionsConclusionsConclusions

bull We have developed a model of the UDP signaling We have developed a model of the UDP signaling pathway in RAW 2647 macrophages which can predict pathway in RAW 2647 macrophages which can predict the responses of multiple species of DAG as well as the the responses of multiple species of DAG as well as the responses of IPresponses of IP33 Ca Ca2+2+ receptor dynamics G-protein receptor dynamics G-protein activation and PIPactivation and PIP22 hydrolysis hydrolysis

bull Simplified model resultsSimplified model resultsndash We have obtained global existence uniqueness positivity We have obtained global existence uniqueness positivity

and boundedness of solutionsand boundedness of solutionsndash We have proven global stability of a unique steady state We have proven global stability of a unique steady state

within our region of interestwithin our region of interestbull Full model analysisFull model analysis

ndash Using SIMULINK we have estimated unknown rate Using SIMULINK we have estimated unknown rate parameters to obtain best fits to multiple DAG traces IPparameters to obtain best fits to multiple DAG traces IP33 and Caand Ca2+2+ all in response to 25 all in response to 25M UDPM UDP

ndash We have performed sensitivity analysis using the Latin We have performed sensitivity analysis using the Latin Hypercube sampling technique in combination with Hypercube sampling technique in combination with standardized regression coefficients to determine which standardized regression coefficients to determine which model parameters are responsible for most of the model model parameters are responsible for most of the model output uncertaintyoutput uncertainty

Future DirectionsFuture DirectionsFuture DirectionsFuture Directions

bull We have conducted multiple experiments to We have conducted multiple experiments to pharmacologically inhibit several different enzymes we pharmacologically inhibit several different enzymes we believe to play a role in this signaling pathway as believe to play a role in this signaling pathway as suggested by current known metabolic pathways and by suggested by current known metabolic pathways and by modeling resultsmodeling results

bull The next step is to perform gene knockdowns on The next step is to perform gene knockdowns on specific enzymes to verify results of inhibitor dataspecific enzymes to verify results of inhibitor data

bull The model output also suggests a time delay from The model output also suggests a time delay from receptor activation to PIPreceptor activation to PIP22 hydrolysis An upcoming task hydrolysis An upcoming task is to investigate the outcomes of adding such a delay is to investigate the outcomes of adding such a delay termterm

bull The model could be greatly enhanced by incorporating The model could be greatly enhanced by incorporating spatial dynamics so this will also be a major focus for spatial dynamics so this will also be a major focus for future researchfuture research

AcknowledgementsAcknowledgementsAcknowledgementsAcknowledgements

Mathematics DepartmentMathematics Departmentbull Hannah L Callender Hannah L Callender

PhDPhD

CollaboratorsCollaboratorsbull Alliance for Cellular SignalingAlliance for Cellular Signalingbull UT SouthwesternUT Southwestern

ndash Paul Sternweis PhDPaul Sternweis PhDndash Dianne DeCamp PhDDianne DeCamp PhD

Brown LabBrown Labbull H Alex Brown PhDH Alex Brown PhDbull Jeffrey S Forrester PhDJeffrey S Forrester PhDbull Mark Byrne PhDMark Byrne PhDbull Anita Preininger PhDAnita Preininger PhDbull Michelle ArmstrongMichelle Armstrongbull Andrew GoodmanAndrew Goodmanbull Pavlina Ivanova PhDPavlina Ivanova PhDbull Steve Milne PhDSteve Milne PhD

SRCs (as sensitivity measures) for SRCs (as sensitivity measures) for DAGDAG

SRCs (as sensitivity measures) for SRCs (as sensitivity measures) for DAGDAG

SRCs (as sensitivity measures) for IPSRCs (as sensitivity measures) for IP33SRCs (as sensitivity measures) for IPSRCs (as sensitivity measures) for IP33

Effects of 3 most sensitive DAG Effects of 3 most sensitive DAG parametersparameters

Effects of 3 most sensitive DAG Effects of 3 most sensitive DAG parametersparameters

A B

C

AA k kdp2dp2 degradation of pool 2 DAG degradation of pool 2 DAG

BB k kdp1dp1 degradation of pool 1 DAG degradation of pool 1 DAG

CC k kap2ap2 production of pool 2 DAG production of pool 2 DAG

ConclusionsConclusionsConclusionsConclusions

bull We have developed a model of the UDP signaling We have developed a model of the UDP signaling pathway in RAW 2647 macrophages which can predict pathway in RAW 2647 macrophages which can predict the responses of multiple species of DAG as well as the the responses of multiple species of DAG as well as the responses of IPresponses of IP33 Ca Ca2+2+ receptor dynamics G-protein receptor dynamics G-protein activation and PIPactivation and PIP22 hydrolysis hydrolysis

bull Simplified model resultsSimplified model resultsndash We have obtained global existence uniqueness positivity We have obtained global existence uniqueness positivity

and boundedness of solutionsand boundedness of solutionsndash We have proven global stability of a unique steady state We have proven global stability of a unique steady state

within our region of interestwithin our region of interestbull Full model analysisFull model analysis

ndash Using SIMULINK we have estimated unknown rate Using SIMULINK we have estimated unknown rate parameters to obtain best fits to multiple DAG traces IPparameters to obtain best fits to multiple DAG traces IP33 and Caand Ca2+2+ all in response to 25 all in response to 25M UDPM UDP

ndash We have performed sensitivity analysis using the Latin We have performed sensitivity analysis using the Latin Hypercube sampling technique in combination with Hypercube sampling technique in combination with standardized regression coefficients to determine which standardized regression coefficients to determine which model parameters are responsible for most of the model model parameters are responsible for most of the model output uncertaintyoutput uncertainty

Future DirectionsFuture DirectionsFuture DirectionsFuture Directions

bull We have conducted multiple experiments to We have conducted multiple experiments to pharmacologically inhibit several different enzymes we pharmacologically inhibit several different enzymes we believe to play a role in this signaling pathway as believe to play a role in this signaling pathway as suggested by current known metabolic pathways and by suggested by current known metabolic pathways and by modeling resultsmodeling results

bull The next step is to perform gene knockdowns on The next step is to perform gene knockdowns on specific enzymes to verify results of inhibitor dataspecific enzymes to verify results of inhibitor data

bull The model output also suggests a time delay from The model output also suggests a time delay from receptor activation to PIPreceptor activation to PIP22 hydrolysis An upcoming task hydrolysis An upcoming task is to investigate the outcomes of adding such a delay is to investigate the outcomes of adding such a delay termterm

bull The model could be greatly enhanced by incorporating The model could be greatly enhanced by incorporating spatial dynamics so this will also be a major focus for spatial dynamics so this will also be a major focus for future researchfuture research

AcknowledgementsAcknowledgementsAcknowledgementsAcknowledgements

Mathematics DepartmentMathematics Departmentbull Hannah L Callender Hannah L Callender

PhDPhD

CollaboratorsCollaboratorsbull Alliance for Cellular SignalingAlliance for Cellular Signalingbull UT SouthwesternUT Southwestern

ndash Paul Sternweis PhDPaul Sternweis PhDndash Dianne DeCamp PhDDianne DeCamp PhD

Brown LabBrown Labbull H Alex Brown PhDH Alex Brown PhDbull Jeffrey S Forrester PhDJeffrey S Forrester PhDbull Mark Byrne PhDMark Byrne PhDbull Anita Preininger PhDAnita Preininger PhDbull Michelle ArmstrongMichelle Armstrongbull Andrew GoodmanAndrew Goodmanbull Pavlina Ivanova PhDPavlina Ivanova PhDbull Steve Milne PhDSteve Milne PhD

SRCs (as sensitivity measures) for IPSRCs (as sensitivity measures) for IP33SRCs (as sensitivity measures) for IPSRCs (as sensitivity measures) for IP33

Effects of 3 most sensitive DAG Effects of 3 most sensitive DAG parametersparameters

Effects of 3 most sensitive DAG Effects of 3 most sensitive DAG parametersparameters

A B

C

AA k kdp2dp2 degradation of pool 2 DAG degradation of pool 2 DAG

BB k kdp1dp1 degradation of pool 1 DAG degradation of pool 1 DAG

CC k kap2ap2 production of pool 2 DAG production of pool 2 DAG

ConclusionsConclusionsConclusionsConclusions

bull We have developed a model of the UDP signaling We have developed a model of the UDP signaling pathway in RAW 2647 macrophages which can predict pathway in RAW 2647 macrophages which can predict the responses of multiple species of DAG as well as the the responses of multiple species of DAG as well as the responses of IPresponses of IP33 Ca Ca2+2+ receptor dynamics G-protein receptor dynamics G-protein activation and PIPactivation and PIP22 hydrolysis hydrolysis

bull Simplified model resultsSimplified model resultsndash We have obtained global existence uniqueness positivity We have obtained global existence uniqueness positivity

and boundedness of solutionsand boundedness of solutionsndash We have proven global stability of a unique steady state We have proven global stability of a unique steady state

within our region of interestwithin our region of interestbull Full model analysisFull model analysis

ndash Using SIMULINK we have estimated unknown rate Using SIMULINK we have estimated unknown rate parameters to obtain best fits to multiple DAG traces IPparameters to obtain best fits to multiple DAG traces IP33 and Caand Ca2+2+ all in response to 25 all in response to 25M UDPM UDP

ndash We have performed sensitivity analysis using the Latin We have performed sensitivity analysis using the Latin Hypercube sampling technique in combination with Hypercube sampling technique in combination with standardized regression coefficients to determine which standardized regression coefficients to determine which model parameters are responsible for most of the model model parameters are responsible for most of the model output uncertaintyoutput uncertainty

Future DirectionsFuture DirectionsFuture DirectionsFuture Directions

bull We have conducted multiple experiments to We have conducted multiple experiments to pharmacologically inhibit several different enzymes we pharmacologically inhibit several different enzymes we believe to play a role in this signaling pathway as believe to play a role in this signaling pathway as suggested by current known metabolic pathways and by suggested by current known metabolic pathways and by modeling resultsmodeling results

bull The next step is to perform gene knockdowns on The next step is to perform gene knockdowns on specific enzymes to verify results of inhibitor dataspecific enzymes to verify results of inhibitor data

bull The model output also suggests a time delay from The model output also suggests a time delay from receptor activation to PIPreceptor activation to PIP22 hydrolysis An upcoming task hydrolysis An upcoming task is to investigate the outcomes of adding such a delay is to investigate the outcomes of adding such a delay termterm

bull The model could be greatly enhanced by incorporating The model could be greatly enhanced by incorporating spatial dynamics so this will also be a major focus for spatial dynamics so this will also be a major focus for future researchfuture research

AcknowledgementsAcknowledgementsAcknowledgementsAcknowledgements

Mathematics DepartmentMathematics Departmentbull Hannah L Callender Hannah L Callender

PhDPhD

CollaboratorsCollaboratorsbull Alliance for Cellular SignalingAlliance for Cellular Signalingbull UT SouthwesternUT Southwestern

ndash Paul Sternweis PhDPaul Sternweis PhDndash Dianne DeCamp PhDDianne DeCamp PhD

Brown LabBrown Labbull H Alex Brown PhDH Alex Brown PhDbull Jeffrey S Forrester PhDJeffrey S Forrester PhDbull Mark Byrne PhDMark Byrne PhDbull Anita Preininger PhDAnita Preininger PhDbull Michelle ArmstrongMichelle Armstrongbull Andrew GoodmanAndrew Goodmanbull Pavlina Ivanova PhDPavlina Ivanova PhDbull Steve Milne PhDSteve Milne PhD

Effects of 3 most sensitive DAG Effects of 3 most sensitive DAG parametersparameters

Effects of 3 most sensitive DAG Effects of 3 most sensitive DAG parametersparameters

A B

C

AA k kdp2dp2 degradation of pool 2 DAG degradation of pool 2 DAG

BB k kdp1dp1 degradation of pool 1 DAG degradation of pool 1 DAG

CC k kap2ap2 production of pool 2 DAG production of pool 2 DAG

ConclusionsConclusionsConclusionsConclusions

bull We have developed a model of the UDP signaling We have developed a model of the UDP signaling pathway in RAW 2647 macrophages which can predict pathway in RAW 2647 macrophages which can predict the responses of multiple species of DAG as well as the the responses of multiple species of DAG as well as the responses of IPresponses of IP33 Ca Ca2+2+ receptor dynamics G-protein receptor dynamics G-protein activation and PIPactivation and PIP22 hydrolysis hydrolysis

bull Simplified model resultsSimplified model resultsndash We have obtained global existence uniqueness positivity We have obtained global existence uniqueness positivity

and boundedness of solutionsand boundedness of solutionsndash We have proven global stability of a unique steady state We have proven global stability of a unique steady state

within our region of interestwithin our region of interestbull Full model analysisFull model analysis

ndash Using SIMULINK we have estimated unknown rate Using SIMULINK we have estimated unknown rate parameters to obtain best fits to multiple DAG traces IPparameters to obtain best fits to multiple DAG traces IP33 and Caand Ca2+2+ all in response to 25 all in response to 25M UDPM UDP

ndash We have performed sensitivity analysis using the Latin We have performed sensitivity analysis using the Latin Hypercube sampling technique in combination with Hypercube sampling technique in combination with standardized regression coefficients to determine which standardized regression coefficients to determine which model parameters are responsible for most of the model model parameters are responsible for most of the model output uncertaintyoutput uncertainty

Future DirectionsFuture DirectionsFuture DirectionsFuture Directions

bull We have conducted multiple experiments to We have conducted multiple experiments to pharmacologically inhibit several different enzymes we pharmacologically inhibit several different enzymes we believe to play a role in this signaling pathway as believe to play a role in this signaling pathway as suggested by current known metabolic pathways and by suggested by current known metabolic pathways and by modeling resultsmodeling results

bull The next step is to perform gene knockdowns on The next step is to perform gene knockdowns on specific enzymes to verify results of inhibitor dataspecific enzymes to verify results of inhibitor data

bull The model output also suggests a time delay from The model output also suggests a time delay from receptor activation to PIPreceptor activation to PIP22 hydrolysis An upcoming task hydrolysis An upcoming task is to investigate the outcomes of adding such a delay is to investigate the outcomes of adding such a delay termterm

bull The model could be greatly enhanced by incorporating The model could be greatly enhanced by incorporating spatial dynamics so this will also be a major focus for spatial dynamics so this will also be a major focus for future researchfuture research

AcknowledgementsAcknowledgementsAcknowledgementsAcknowledgements

Mathematics DepartmentMathematics Departmentbull Hannah L Callender Hannah L Callender

PhDPhD

CollaboratorsCollaboratorsbull Alliance for Cellular SignalingAlliance for Cellular Signalingbull UT SouthwesternUT Southwestern

ndash Paul Sternweis PhDPaul Sternweis PhDndash Dianne DeCamp PhDDianne DeCamp PhD

Brown LabBrown Labbull H Alex Brown PhDH Alex Brown PhDbull Jeffrey S Forrester PhDJeffrey S Forrester PhDbull Mark Byrne PhDMark Byrne PhDbull Anita Preininger PhDAnita Preininger PhDbull Michelle ArmstrongMichelle Armstrongbull Andrew GoodmanAndrew Goodmanbull Pavlina Ivanova PhDPavlina Ivanova PhDbull Steve Milne PhDSteve Milne PhD

ConclusionsConclusionsConclusionsConclusions

bull We have developed a model of the UDP signaling We have developed a model of the UDP signaling pathway in RAW 2647 macrophages which can predict pathway in RAW 2647 macrophages which can predict the responses of multiple species of DAG as well as the the responses of multiple species of DAG as well as the responses of IPresponses of IP33 Ca Ca2+2+ receptor dynamics G-protein receptor dynamics G-protein activation and PIPactivation and PIP22 hydrolysis hydrolysis

bull Simplified model resultsSimplified model resultsndash We have obtained global existence uniqueness positivity We have obtained global existence uniqueness positivity

and boundedness of solutionsand boundedness of solutionsndash We have proven global stability of a unique steady state We have proven global stability of a unique steady state

within our region of interestwithin our region of interestbull Full model analysisFull model analysis

ndash Using SIMULINK we have estimated unknown rate Using SIMULINK we have estimated unknown rate parameters to obtain best fits to multiple DAG traces IPparameters to obtain best fits to multiple DAG traces IP33 and Caand Ca2+2+ all in response to 25 all in response to 25M UDPM UDP

ndash We have performed sensitivity analysis using the Latin We have performed sensitivity analysis using the Latin Hypercube sampling technique in combination with Hypercube sampling technique in combination with standardized regression coefficients to determine which standardized regression coefficients to determine which model parameters are responsible for most of the model model parameters are responsible for most of the model output uncertaintyoutput uncertainty

Future DirectionsFuture DirectionsFuture DirectionsFuture Directions

bull We have conducted multiple experiments to We have conducted multiple experiments to pharmacologically inhibit several different enzymes we pharmacologically inhibit several different enzymes we believe to play a role in this signaling pathway as believe to play a role in this signaling pathway as suggested by current known metabolic pathways and by suggested by current known metabolic pathways and by modeling resultsmodeling results

bull The next step is to perform gene knockdowns on The next step is to perform gene knockdowns on specific enzymes to verify results of inhibitor dataspecific enzymes to verify results of inhibitor data

bull The model output also suggests a time delay from The model output also suggests a time delay from receptor activation to PIPreceptor activation to PIP22 hydrolysis An upcoming task hydrolysis An upcoming task is to investigate the outcomes of adding such a delay is to investigate the outcomes of adding such a delay termterm

bull The model could be greatly enhanced by incorporating The model could be greatly enhanced by incorporating spatial dynamics so this will also be a major focus for spatial dynamics so this will also be a major focus for future researchfuture research

AcknowledgementsAcknowledgementsAcknowledgementsAcknowledgements

Mathematics DepartmentMathematics Departmentbull Hannah L Callender Hannah L Callender

PhDPhD

CollaboratorsCollaboratorsbull Alliance for Cellular SignalingAlliance for Cellular Signalingbull UT SouthwesternUT Southwestern

ndash Paul Sternweis PhDPaul Sternweis PhDndash Dianne DeCamp PhDDianne DeCamp PhD

Brown LabBrown Labbull H Alex Brown PhDH Alex Brown PhDbull Jeffrey S Forrester PhDJeffrey S Forrester PhDbull Mark Byrne PhDMark Byrne PhDbull Anita Preininger PhDAnita Preininger PhDbull Michelle ArmstrongMichelle Armstrongbull Andrew GoodmanAndrew Goodmanbull Pavlina Ivanova PhDPavlina Ivanova PhDbull Steve Milne PhDSteve Milne PhD

Future DirectionsFuture DirectionsFuture DirectionsFuture Directions

bull We have conducted multiple experiments to We have conducted multiple experiments to pharmacologically inhibit several different enzymes we pharmacologically inhibit several different enzymes we believe to play a role in this signaling pathway as believe to play a role in this signaling pathway as suggested by current known metabolic pathways and by suggested by current known metabolic pathways and by modeling resultsmodeling results

bull The next step is to perform gene knockdowns on The next step is to perform gene knockdowns on specific enzymes to verify results of inhibitor dataspecific enzymes to verify results of inhibitor data

bull The model output also suggests a time delay from The model output also suggests a time delay from receptor activation to PIPreceptor activation to PIP22 hydrolysis An upcoming task hydrolysis An upcoming task is to investigate the outcomes of adding such a delay is to investigate the outcomes of adding such a delay termterm

bull The model could be greatly enhanced by incorporating The model could be greatly enhanced by incorporating spatial dynamics so this will also be a major focus for spatial dynamics so this will also be a major focus for future researchfuture research

AcknowledgementsAcknowledgementsAcknowledgementsAcknowledgements

Mathematics DepartmentMathematics Departmentbull Hannah L Callender Hannah L Callender

PhDPhD

CollaboratorsCollaboratorsbull Alliance for Cellular SignalingAlliance for Cellular Signalingbull UT SouthwesternUT Southwestern

ndash Paul Sternweis PhDPaul Sternweis PhDndash Dianne DeCamp PhDDianne DeCamp PhD

Brown LabBrown Labbull H Alex Brown PhDH Alex Brown PhDbull Jeffrey S Forrester PhDJeffrey S Forrester PhDbull Mark Byrne PhDMark Byrne PhDbull Anita Preininger PhDAnita Preininger PhDbull Michelle ArmstrongMichelle Armstrongbull Andrew GoodmanAndrew Goodmanbull Pavlina Ivanova PhDPavlina Ivanova PhDbull Steve Milne PhDSteve Milne PhD

AcknowledgementsAcknowledgementsAcknowledgementsAcknowledgements

Mathematics DepartmentMathematics Departmentbull Hannah L Callender Hannah L Callender

PhDPhD

CollaboratorsCollaboratorsbull Alliance for Cellular SignalingAlliance for Cellular Signalingbull UT SouthwesternUT Southwestern

ndash Paul Sternweis PhDPaul Sternweis PhDndash Dianne DeCamp PhDDianne DeCamp PhD

Brown LabBrown Labbull H Alex Brown PhDH Alex Brown PhDbull Jeffrey S Forrester PhDJeffrey S Forrester PhDbull Mark Byrne PhDMark Byrne PhDbull Anita Preininger PhDAnita Preininger PhDbull Michelle ArmstrongMichelle Armstrongbull Andrew GoodmanAndrew Goodmanbull Pavlina Ivanova PhDPavlina Ivanova PhDbull Steve Milne PhDSteve Milne PhD