NBCR Summer Institute 2006:Multi-Scale Cardiac Modeling with

Continuity 6.3

Tuesday:Modeling Cardiac Myocyte Excitation-

Contraction Coupling and its Regulation

Andrew McCulloch, Anushka Michailova and Stuart Campbell

Ionic currents

Calcium handling

Crossbridge interactions

Coronary artery flow

Myofilament activation

Wall stress and strain

Action potential propagation

Whole ventricular electromechanics

Circulatory system dynamics Neurohumoral

regulation

Cellsignaling

Torso bioelectricfields

Tissue perfusion

Purine metabolism

Mitochondrial metabolism

Ventricular systolic

pressures and cardiac output

adenosine

substrates

Ventricular filling pressures and output impedance

Epicardial potential fields

Transmembrane potentials

Total transmembrane ionic current

Myofilament tension

Intramyocardial pressure and

volume

Coronary artery flows

Regional wall stresses, strains,

displacements

Coronary ostia

pressure

Regulation of peripheral

resistances, fluid volumes, HR

Autonomic mediatorsS

ys

tem

s

Ph

ys

iolo

gy

M

od

els

Cel

l S

yste

ms

mo

del

sC

on

stit

uti

ve

Mo

del

s

Fin

ite

ele

men

t a

nd

bo

un

dar

y el

em

ent

mo

de

ls

Circu

latio

nV

entric

les

Myo

card

ium

Myo

cyte

To

rso

Multi-scale Modeling

crossbridge lattice

multicellularventriclescirculation

Scale Class of Model Mechanics Example Electrophysiology Example

Lumped parameter model Arterial circuit equivalent Equivalent dipole EKG External boundary conditions

Hemodynamic loads No flux condition

Continuum PDE model Galerkin FE stress analysis Collocation FE model Constitutive model Constitutive law for stress Anisotropic diffusion Multicellular network model

Tissue micromechanics model

Resistively coupled network

Cell-cell/cell-matrix coupling

Matrix micromechanics model

Gap junction model

Whole cell systems model Myocyte 3-D stiffness and contractile mechanics

Myocyte ionic current and flux model

Subcellular compartment model

Sarcomere dynamics model Intracellular calcium fluxes

Stochastic state-transition model

Crossbridge model of actin-myosin interaction

Single channel Markov model

Weighted ensemble Brownian dynamics

Single cross-bridge cycle Ion transport through single channel

Hierarchical collective motions

Actin, myosin, tropomyosin Na+, K+ and Ca+ channels

Organ system Whole organ Tissue Multi-cellular Single cell Subcellular Macromolecular Molecular Atomic Molecular dynamics

simulation PDB coordinates PDB coordinates

Models at each physical scale and the bridges between them

Classes of Model

Class of Model Biomechanics Electrophysiology Biotransport

Continuum PDEs Equations of motion

Monodomain equation

Reaction-diffusion equation

Constitutive Model

Strain energy formulation

Anisotropic conductivities

Diffusion model

Systems Model Myofilament activation and interactions

Ionic currents and action potential

Reaction network model

Tuesday: Modeling Myocyte Excitation-Contraction Coupling

and its RegulationCardiac mechanics: myofilament models

Cardiac electrophysiology: ionic models

Regulation of excitation-contraction coupling: biochemical models

Intracellular Diffusion of Second Messengers – cAMP, Ca

Getting set up with Continuity 6.3

A simple reaction diffusion model with Continuity 6.3

Wednesday: Finite Element Discretization and Mesh Fitting

Finite element approximation and discretization

Finite element interpolation – Lagrange and Hermite basis functions with Continuity 6.3

Tensor-product interpolation for two and three dimensions

Curvilinear world coordinates and coordinate conversion

Fibers and fields in Continuity 6.3

Least squares fitting of anatomic meshes and fiber angles with Continuity 6.3

Non-homogeneous strain analysis

Thursday: Monodomain Modeling in Cardiac Electrophysiology

Cardiac myocyte ionic models

Modeling cardiac action potential propagation

Collocation FEM for monodomain problems with Continuity 6.3

Friday: Cardiac Mechanics and Electromechanics

Modeling Ventricular Wall Mechanics

Determinants of ventricular wall stress

Galerkin FEM for ventricular stress analysis

Systolic wall strains and anisotropy

Ventricular Electromechanics

Cardiac resynchronization therapy

Ventricular-Vascular Coupling

Cardiac Mechanics: Myofilament Models

Cardiac Myocytes

• Rod-shaped• Striated• 80-100 m long• 15-25 m diameter

Myocyte Connections

• Myocytes connect to an average of 11 other cells (half end-to-end and half side-to-side)

• Functional syncytium

• Myocytes branch (about 12-15º)

• Intercalated disks– gap junctions

– connexons– connexins

Fiber-Sheet Structure

x510

endocardium

midwall

epicardium

Myocyte Ultrastructure

• Sarcolemma• Mitochondria (M)

~30%• Nucleus (N)• Myofibrils (MF)• Sarcoplasmic

Reticulum and T-tubule network

Striated Muscle Ultrastructure

Electron micrograph of longitudinal section of freeze-substituted, relaxed rabbit psoas muscle. Sarcomere shows A band, I band, H band, M line, and Z line. Scale bar, 100 nm. From Millman BM, Physiol. Rev. 78: 359-391, 1998

The Sarcomere

The Sarcomere

Anisotropic Isotropic to polarized light~ 2.0 m

Hexagonal Arrangement of Myofilaments in Cross-Section

Crossbridge Cycle

Excitation-Contraction

Coupling

• Calcium-induced

calcium release• Calcium current• Na+/Ca2+ exchange• Sarcolemmal Ca2+

pump• SR Ca2+ ATP-

dependent pump

Click image to view animation of calcium cycling

http://www.meddean.luc.edu/lumen/DeptWebs/physio/bers.html

Isometric Tension in Skeletal Muscle:Sliding Filament Theory

(a) Tension-length curves for frog sartorius muscle at 0ºC

(b) Developed tension versus length for a single fiber of frog semitendinosus muscle

Isometric Testing

2.1

2.0

1.9

Sarcomerelength, m

2.0

Tension, mN

time, msec200100 300 500 700600400

1.0

Muscle isometric

Sarcomere isometric

Isometric Length-Tension Curve

Peak developed isometric twitch tension (total-passive)

High calcium

Low calcium

muscle isometric

sarcomere isometric

Passive

Length-Dependent Activation

Isometric peak twitch tension in cardiac muscle continues to rise at sarcomere lengths >2 m due to sarcomere-length dependent increase in myofilament calcium sensitivity

Isotonic Testing

Isovelocity release experiment conducting during a twitch

Cardiac muscle force-velocity relation corrected for viscous forces of passive cardiac muscle which reduce shortening velocity

Modeling Myofilament Force Production

• Ca2+ binding to TnC causes tropomyosin to change to a permissive state

• Force development occurs as actin-myosin crossbridges form

• Crossbridges can ‘hold’ tropomyosin in the permissive state even after Ca2+ has dissociated

Roff Roff

RonRon

A1 A1

0

0

0

*

*

*

koff kon koff

fg g f

Ca2+

Ca2+

Ca2+

Ca2+

Ca2+

Ca2+

Myofilament Activation/Crossbridge Cycling Kinetics

Non-permissive Tropomyosin

Permissive Tropomyosin

Permissive Tropomyosin, 1-3 crossbridges attached (force generating states)

Ca2+

boundto TnC

Ca2+

notboundto TnC

*

kb

kn

This scheme is used to find A(t), the time-course of attached crossbridges for a given input of [Ca2](t)

Myofilament Model Equations• Total force is the product of the total number of attached

crossbridges, average crossbridge distortion, and crossbridge stiffness:

txtAF

SLxx

A

Agx 0

• Average crossbridge distortion is obtained by the solution to the following differential equation:

Uniaxial Resting Mechanics (Contribution of Collagen)

0

2

4

6

8

10

0 0.05 0.1 0.15 0.2 0.25

oim

WT

Str

ess (m

N/m

m2)

Strain

n=9 WT, n=7 oimP<0.05

Passive Biaxial Properties

0.250.200.150.100.050.000

2

4

6

8

10

Equibiaxial Strain

Str

ess

(kP

a)

Fiber stress

Cross-fiber stress

Strain Energy

( )

( )

ij ij

ijij

d d d d

. . where d d d

W T I Q

Wi e T W I Q

e r

¶r

¶e

= = -

= = -

• W is the strain-energy function; its derivative with respect to the strain is the stress.

• This is equivalent to saying that the stress in a hyperelastic material is independent of the path or history of deformation.

• Similarly, when a force vector field is the gradient of a scalar energy function, the forces are said to be conservative; they work they do around a closed path is zero.

• The strain energy in an elastic material is stored as internal energy or free energy (related to entropy) …

Cauchy Stress Tensor is Eulerian

R

Sn

tf(n)

lim

a a0

a

e1

e2

e3

Tij = ti•ej

T11

T12T22

T33

T13

T31 T23

T21

T32

Tij is the component in the xj direction of the traction

vector t(n) acting on the face normal to the xi axis in the

deformed state of the body. The "true" stress.

Cauchy’s formula: t(n) = n•TIn index notation: ( )

j i ijnt nT=

The (half) Lagrangian Nominal stress tensor S

t N SNR Rj

( )j

Lagrangian Stress Tensors

The symmetric (fully) Lagrangian Second Piola-Kirchhoff stress tensor P S F F F T F P

F

T T T

RS ij SR

(det )

det

1

PX

x

X

xT PR

i

s

j

b g

• Useful mathematically but no direct physical interpretation• For small strains differences between T, P, S disappear

SRj is the component in the xj direction of the traction

measured per unit reference area acting on the surface normal to the (undeformed) XR axis. Useful experimentally

S = detF.F-1.T ST

Example: Uniaxial Stress

undeformed length = Lundeformed area = A

deformed length = ldeformed area = a

TF

a

SF

A

la

LA

L

l

F

a

L

lT

T= = = =

0 0

PL

lS

F

A= =

1

Cauchy Stress

Nominal Stress

Second Piola-Kirchhoff Stress

aFF

l

L

A

Hyperelastic Constitutive Law for Finite Deformations

Second Piola-Kirchhoff Stress

Cauchy Stress

RSSRRSSRRSRS 2

1

E

W

E

W

E

W

C

W

C

WP

RSS

j

R

iij E

W

X

x

X

xT

o

2-D Example:Exponential Strain-Energy Function

221132222

2111 2 where

2

1EEbEbEbQCeW Q

PW

E

PW

EC bE b E e

PW

EC b E b E e

Q

Q

RSRS

1111

2222

1 11 3 22

2 22 3 11

b g

b g

Stress components have interactions

3-D Orthotropic Exponential Strain-Energy Function

From: Choung CJ, Fung YC. On residual stress in arteries. J Biomech Eng 1986;108:189-192

29

28

27

654

23

22

21

222

where2

1

RZRZ

RRRRZZZZ

RRZZ

Q

EbEbEb

EEbEEbEEb

EbEbEbQ

CeW

Strain Energy Functions

W 0.21e

9.4 I1 3

1

0.35 e

66F 1

2

1

W =0.6 ( )eQ – 1 ,where, in the dog

Q = 26.7E211 2.0( )E

222E

233E

223E

232 14.7( )E

212E

221E

213E

231 ,

and, in the rat

Q = 9.2E211 2.0( )E

222E

233E

223E

232 3.7( )E

212E

221E

213E

231 .

For the outer third, they obtained

W = 4.8(F–1)2 + 3.4(F–1)3 + 0.77(I1–3) – 6.1(I1–3)(F–1) + 6.2(I1–3)2

for the midwall region

W = 5.3(F–1)2 + 7.5(F–1)3 + 0.43(I1–3) – 7.7(I1–3)(F–1) + 5.6(I1–3)2,and for the inner layer of the wallW = 0.51(F–1)2 + 27.6(F–1)3 + 0.74(I1–3) – 7.3(I1–3)(F–1) + 7.0(I1–3)2.

W 0.36

132

32 2

30

30 3

31

31 3

Transversely Isotropic (Isotropic + Fiber) Exponential

Transversely Isotropic Exponential

Transversely Isotropic Polynomial

Orthotropic Power Law

Incompressible MaterialsStress is not completely determined by the strain because a hydrostatic pressure can be added to Tij without changing CRS. The extra condition is the kinematic incompressibility constraint

I F3

2

1 2 3

21 detb g b g

To avoid derivative of W tending to

W W I I p I 1 2 3

1

21,b g b g

p is a Lagrange multiplier (a negative stress)

PW

I

I

C

W

C

I

Cp

X

x

X

x

C

RSRS RS RS

R S

i

RS

2 21

1 2

1

i

Ionic Models of Cardiac Myocyte Electrophysiology

Intracellular IonConcentrations

Extracellular IonConcentrations

Resting Membrane Potential

5 - 15 mM

140 mM

0.5 mM

10-4 mM

7x10-5 mM

5 - 15 mM

Na+ 145 mM

K+ 5 mM

Mg2+ 1-2 mM

Ca2+ 1-2 mM

H+ 4x10-5 mM

Cl- 110 mM

• An imbalance of total ionic charge leads to a potential difference across a cell membrane:

Resting Membrane Potential

Vm = Vo - Vi

• Only a slight imbalance is needed to result in a potential difference

• If 1/100,000th of available cytosolic K+ ions crossed the membrane of a spherical cell 10µm DØ, the membrane potential changes by 100 mV

Ventricular Action Potential

1. Through a stimulus such as current injection, Vm reaches a threshold voltage

2. Voltage-gated Na+ ion channels change to open state, Na+ ions enter depolarizing the cell

3. A peak is reached near the equilibrium potential for Na+ ions

4. Other ion channels react, namely K+ and Ca2+ channels repolarizing the cell

5. Over time the cell returns to a resting equilibrium state

1

2

3

4

5P

oten

tial (

mV

)

Time (msec)

threshold voltage

Ion transporters, including ligand- and voltage-gated channels, exchangers and ATP-dependent pumps

io

C

C

zF

RTV ln

Ion Motion

V Potential Co Concentration of ion outside cellCi Concentration of ion inside cellR Gas Constantz ValenceF Faraday’s constant

At 37°C, RT/F = 26 mV

Nernst Equation

• Ions cross the membrane by two methods: Active Transport and Diffusion

ATP

3Na2+

2K+

Na+

Closed Open Inactivated

Electrochemical Equilibrium

• Ions in a resting cell are in electrochemical equilibrium

Goldman-Hodgkin-Katz Equation

ioni

ionion

ionionoion

m CP

CP

zF

RTV

ln

Vm Membrane PotentialPion Permeability of membrane to particular ion[C]ion Concentration of a particular ion

Requires the assumption of a constant electric field in the membrane

Voltage Clamping• Whole cell and patch clamping

techniques reveal ionic current activity

• Allows control over membrane voltage

• And ionic concentrations on both sides of the membrane

• Hodgkin and Huxley observed sodium and potassium currents in nerve cells

• Hodgkin, A.L. and A.F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve. J Physiol, 1952. 117: p. 500-544.

Hodgkin-Huxley Ionic Currents

ClmClCl

NamNaNa

KmKK

EVgI

EVgI

EVgI

ClCl

NaNa

KK

gg

hmgg

ngg

3

4

i

oCl

i

oNa

i

oK

Cl

Cl

F

RTE

Na

Na

F

RTE

K

K

F

RTE

ln

ln

lnwhere and

Nernst potentialsConductances = 1/resistivity

Ohm’s Law:

n, m, and h are gating variables

Fitting Conductance Parameters

Potassium Gating (1)

• n is the gating variable for the K+ current

• n = #open/(#open+#closed) = fraction of open channels = probability a channel is

open

• αn = rate of channel opening• βn = rate of channel closing• αn and βn are f(Vm) found by

fitting

4ngg KK Potassium conductance was found empirically to have behavior:Corresponds to four protein subunits with equal open probability.

nndt

dnnn 1

Exponent represents # of gates per channel

channels conducting offraction the4

1then5.0 2 nn

Potassium Gating (2)

where

Solution of (1) neglecting αn and βn is:

and n∞ comes from steady state:

nndt

dnnn 1

nn

nn

nn nn 10

nto ennnn

nn

n

1

(1)

no is an initial condition,

So (1) can be written: n

nn

dt

dn

Fitting Opening/Closing Rates

Membrane Capacitance

• The cell membrane acts as a capacitor • Membrane capacitance, Cm, is a material property defined per area

Conductive Cytoplasm

Conductive Extracellular Space

Non-conductive, dielectric membrane

• Charge is related to potential by the definition of capacitance:

q = CmVm

q Charge on one side of the capacitor (coulombs/cm2)Cm Membrane capacitance ( farad/cm2 [farad = coulomb/volt])Vm Membrane potential (mV)

Currents that Cross the Cell Membrane

• By definition the change of charge over time is current• Note that current units are per area -- µA/cm2

• Capacitive Current – Changes in capacitive charge of the membrane over time

• Ionic currents – Charges on ions crossing the membrane through ion channels – Calculated by channel gating models

dt

dVC

dt

dqI m

mc

– – – – –

+ +

+ +

+ +

+ +

Mismatch in the amount of charge on either side of the capacitor results in the recruitment of additional charges to the area near the membrane.

– – –

CaKNaion IIII

ClKNam

m

ioncm

IIIdt

dVC

III

Hodgkin-Huxley Nerve Cell Analog

ClmClNamNaKmKm

m EVgEVhmgEVngdt

dVC 34

4 ODEs: membrane potential and 3 gating variables

hhdt

dh

mmdt

dm

nndt

dn

hh

nm

nn

1

1

1

αi and βi are functions of Vm determined by experimental curve fit

Functional Integration: Ionic Currents

FitzHugh R (1960) J Gen Physiol 43: 867-96Noble D (1960) Nature 188: 495-7

Cable Theory• A cell might be modeled as a cable

– A cable consists of a conductive interior (cytoplasm) – with the material property conductivity, 3-D tensor D (mS/cm)– Surrounded by an insulator (cell membrane)

Physically, current flux in a cable occurs in the direction of greatest potential drop

mV J G

kx

jx

ix

where ˆˆˆ321

E

Φi represents potential inside

the cable (intracellular potential)

• Electric field vector, E (mV/cm), is defined as a potential drop maintained spatially in a material

GEJ

Ohm’s Law:• Flux vector, J, is proportional to the electric field vector • Flux is the current density inside the cable (µA/cm2)

- - - ++++++++ - - - - -

- - - - - - - -++++++++

+++++ - - -- - - - - +++

++++++++- - - - - - - -

Propagation

Repolarization Depolarization

Initial Activation Site

Inactivation

Conservation of Charge (1)Current entering a section of cable (dvolume) must equal current that leaves the section of cable.

Inward currents are positiveOutward currents are negative

area area surfc iond d I I d J J J

Flux in through darea

Flux out through darea

Sum of currents through membrane dsurf– =

J J + dJ

IcIionEnd Area = darea

Membrane surface Area = dsurf

dvolume

areaarea dddd JJ)(JJ Total current traveling into and out of the cable through neighboring conductive volume, i.e. the ends (µA/cm2):

• Change in flux in the x1 direction [ (1/cm)(µA/cm2)(cm3) = µA ]:

• Total change in flux for general 3D cable (µA): volumevolume

JJJ

321

321 ddxxx

d xxx JJ

volumeJJ

1321

1

11 dx

dxdxdxx

xx

darea

Conservation of Charge (2)• Total current leaving through the membrane

surface dS [ (µA/cm2)(cm2) = µA ]:

surf surf surfmm c ion m ion

dVI d I I d C I d

dt

J J + dJ

IcIion

Membrane surface Area = dsurfNote that Iion is calculated by the system of ODE’s described earlier and depends on Vm

• Conservation of charge results in: volume surfmm ion

dVd C I d

dt

J

• With dsurf/dvolume set equal to the surface to volume ratio of a cell (Sv) and flux expressed in terms of potential, we have:

mm V m ion

dVV S C I

dt

G

1 1mm ion

V m m

dVV I

S C dt C G

3D Cable Equation or Monodomain Equation

1mm ion

m

dVV I

dt C D

D has units of diffusion (cm2/msec), by combining G with Sv (1/cm) and Cm (µF/cm2)

A Note on Units

• Unit of conductivity (mS/cm), Siemens are inverse of resistivity:

secμ1000μμ1

2222

2m

cm

mF

cm

F

cm

F

cmmS

cmF

cm

cmmS

CS mV

G

• Therefore the units of flux are:

• From conductivity to diffusion:

V

A1S

22 cm

μA

cm1,000,000

A

mV1000

V

cm

mV

cm1000

1

cm

mV

cm

mS

GEJ

• All terms in the cable equation have units of (mV/msec) including:

secsecμ

μ

11

m

mVV

F

V

F

AA

FI

C ionm

Saucerman, JJ et al., J Biol Chem 278: 47997 (2003)

-Adrenergic Regulation of Excitation-Contraction Coupling

Modeling Cell Signaling Pathways

Biochemical Processes

Binding• Complex formation• InhibitionTransformation• Enzymatic conversion• DegradationCompartmentation• Organelles and

subspaces• Scaffolds and complexesTranslocation• Transport between

compartments

Modeling Biochemical Networks

1. Build a network diagram of all components

2. Revise network diagram to include detail of each protein state and reaction, removing any detail that will not be modeled

3. Translate the mechanistic diagram into a system of equations

4. Parameterize the model

5. Validate model predictions with experimental observations/data

6. Apply model with analysis

References/Additional Info

• Saucerman JJ, McCulloch AD. Mechanistic systems models of cell signaling networks: a case study of myocyte adrenergic regulation. Prog Biophys Mol Biol. 2004 Jun-Jul;85(2-3):261-78.

• Practical Kinetic Modeling of Large Scale Biological Systems, Robert Phair www.bioinformaticsservices.com

• Irwin Segel, Enzyme Kinetics• CellML website www.cellml.org• Alliance for Cellular Signaling www.afcs.org

1 & 2. Mechanistic Network Diagram

40 differential equations, 20 algebraic equations, 78 parameters from 57 papers

Saucerman, JJ et al., J Biol Chem 278: 47997 (2003)

Deterministic ODEs

• Each protein, complex or distinguishable chemical species is considered a state variable

• Write an ODE for each state variable

Ex: dX/dt = Rbind + Rconv + Rtrans

Binding Reactions

f

b

k

f bk

b f

d LRL R LR k L R k LR

dtd L d R

k LR k L Rdt dt

Rf = kf[L][R]; Rb = kb[C]Dissociation constant kd = kb/kf, units M

•Receptor-ligand binding–Cytokines, neurotransmitters, catecholamines

•Small molecules binding to proteins–IP3 binding to IP3 regulated Ca2+ channels

•Protein-protein binding–Complex formation, dimerization

•Mass action kinetics:

1 2

1

1 1 2

1

1 2

tot

Michaelis-Menten Enzyme Reaction Scheme:

>> :

Fractional formation of complex

0

:

k k

k

m

m

E S ES E P

d ESk E S k k ES

dt

d ES k E S E SES

dt k k k

ES ES S

E E ES k

S

S

S E E

Quasi Steady - State Assumption

2 2 tot max

Rate of product formation:

m m

d P S S d SV k ES k E V

dt k S k S dt

Transformation Reactions

1 2

1

1

1

1

1 2

m

1 1 2

ax

Michaelis-Menten Enzyme Reaction Scheme:

, :

Example: Competitive Inhibitioni

d ca

k k

k

d

d

d

k

k k

k k

E S ES E P

kk

k

k E S

k

E SES

k k k

SV V

k S

I E EI

E S ES

Quasi - Equilibrium Assumption

1

t

totcat

d i

E P

E SV k

k I k S

General Derivation of Rate Lawswith the QEA

1. Draw explicit diagram, labeled with constants

2. Write expression for equilibrium constants

3. Write fractional ES complex eqn

4. Manipulate eqn to cancel out [ES]

5. Rate eqn as V = kcat*[Etot]*([ES]/[Etot])

Quasi-Equilibrium Pools

Model Formulation

_ __ _

tot 1 1 bg

[ ]([ : 1 : ] [ 1 : ]) [ ]

Gs = Gs + L: AR:Gs + AR:Gs +

ODEs, e.g.

Algebraic Equations, e.g.

Gs

GTP totgact hyd GTP tot

d Gk L AR Gs AR Gs k G

dt

• Time scale of interest between 0.1 seconds and 10 minutes

• Conservation relations• Results in differential-algebraic equations , e.g G-

protein activation module:• Need a DAE solver (e.g. Radau, Daspk) or

iterative root finder.

Parameterization

Several options:

1. Using “approximate” values without any explicit experimental basis

2. Using parameters obtained or derived from the literature

3. Curve fitting (brute force, GA’s, etc.)

4. Hybrids of above approaches where appropriate, possibly guided by sensitivity analysis

Model Validation

Need to find independent experimental data for comparison

Validate individual modules before piecing together

Try to find both time course and dose/response data

Analysis

Methods from control theory:• Sensitivity analysis, dX(t)/dP, dX/dP*(P/X)• Stability analysis (oscillations,chaos)• Bifurcation analysis

In-silico experiments:• Inhibition with drugs

• Overexpression (Xtot = c*Xtot_old)

• Knockouts (Xtot ~ 0)

Imaging PKA-mediated phosphorylation gradients with local cAMP uncaging

UV

0 µm

65 µm

Michailova A, DelPrincipe F, Egger M, Niggli E. Biophys J, 1999, 76:A459

A: Fluo-3 fluorescence (yellow line).

B: Ca2+ signal recorded in the cell center (blue) and periphery (red).

C: Spatially averaged Ca2+ signal across the entire cell.

D: L-type Ca2+ current and voltage-clamp protocol.

E: A line-scan image recorded from the yellow line in (A).

F: Spatial profile of Ca2+ during the rising phase of Ca2+.

G: Ca2+ signal as a surface plot, computed from the line-scan image in (E).

Ca2+ Signaling, Buffering and Diffusion In Atrial Myocytes

• Cylindrical cell geometry is assumed. Model cell has diameter and volume corresponding to the real atrial myocyte.

• The cell has two spaces - restricted subsarcolemmal (RSP) and myofibrillar (MYOF).• Ca2+ and mobile buffers fluo-3 and calmodulin diffuse throughout the cell.• SR Ca2+ release and uptake are not included• Ca2+ enters via L-type Ca2+ current. The activity of the Na+/ Ca2+ exchanger at rest is

compensated by a Ca2+-leak.• Ca2+ binds to fluo-3, calmodulin, troponin-C and phospholipids without cooperativity.• Initial total concentrations of the mobile buffers (fluo-3 or calmodulin) are spatially uniform.• Diffusion constants for Ca2+ bound to fluo-3 or calmodulin are approximately equal to the

diffusion constants for fluo-3 or calmodulin.

Model Formulation

22 2 2 2

2 2

2 2

2

[ ][ ] [ ][ ] [ ] [ ][ ] [ ]

[ ][ ] [ ][ ] [ ]

[ ][ ] [ ][ ] [ ]

[ ][ ][ ] [ ]

m

m

Ca s s s s m m m m i

mB m m m m m

mCaB m m m m m

ss s s s

CaD Ca k Ca B k CaB k Ca B k CaB J

tB

D B k Ca B k CaBt

CaBD CaB k Ca B k CaB

tCaB

k Ca B k CaBt

Michailova A, DelPrincipe F, Egger M, Niggli E. Biophys J, 1999, 76:A459

A: Model cell has a diameter (15.6 m) and capacitance (41 pF) corresponding to the real cell.

B: Ca2+ signal calculated for the cell center (blue), periphery (red) and fuzzy space (green).

C: Time-course of spatially averaged Ca2+ signal.

D: Simulated Ca2+ current.

E: Calculated diffusion of Ca2+ in space and time as a line-scan image.

F: Spatial profile of Ca2+ during the rising phase of Ca2+ (at 100 ms).

G: Ca2+ signal as a surface plot.

Model Results

GENERAL EQUATIONS

sin

sin

[ ].( [ ])

( , , , , )

( , , , , )

[ ]([ ] [ ])[ ] [ ]

source kC i j m

i j m

sourcei i

kj j

m m m

CD C J J R

t

J f C t x y z

J f C t x y z

mR k B m C k m

tB C m

where: [C] intracellular specie concentration; DC diffusion constant for [C]; Ji

source – i source for [C]; Jjsink – j sink for [C]; km

+, km- - kinetic constants

Example reaction-diffusion model of cell with Continuity: Example reaction-diffusion model of cell with Continuity: Intracellular CaIntracellular Ca2+2+ transport transport

22 2

2

[ ] [ ][ ]

[ ]([ ] [ ])[ ] [ ]

Caii LCC NCX leak

Tn i Tn

Ca CaTnD Ca J J J

t tCaTn

k Tn CaTn Ca k CaTnt

where: [Ca2+]i free intracellular Ca2+; Tn stationary buffer troponin C; DCa diffusion constant for free Ca2+; kTn

+, kTn- kinetic rate constants; JLCC L-type Ca2+ current; JNCX

Na/Ca exchange current; Jleak Ca2+ leak. Sources and sinks for Ca2+ were modeled as in Michailova et al., Biophys J 2002.

2

0

on region, R 0

,0

, on boundary, S 0

, on boundary, S 0

uD u g t

tu u

u u t t

uq t t

n

S S R

x

x x

x

x

Transient Heat Equation

• Evolved first from the matrix methods of structural

analysis in the early 1960’s• Uses the algorithms of linear algebra• Later found to have a more fundamental foundation• The essential features are in the formulation• There are two alternative formulations that are broadly

equivalent in most circumstances– Variational formulations, e.g. the Rayleigh-Ritz method– Weak or weighted residual formulations, e.g.the

Galerkin method• Both approaches lead to integral equations instead of

differential equations (the strong form)

The Finite Element Method

The Finite Element Method• Solution is discretized using a finite number of functions

– Piecewise polynomials (elements)– Continuity across element boundaries ensured by

defining element parameters at nodes with associated basis functions,

12 13

14 15

21 22

23 24

• FE equations are derived from the weak form of the governing equations

R = 0Finite differences:Finite elements:

R = 0

The Finite Element Method• Integrate governing equations in each element

• Assemble global system of equations by adding contributions from each element

1 2

5 6

7 8

3 4

Element equationsk k k k k k k k

k k k k k k k k

k k k k k k k k

k k k k k k k k

k k k k k k k k

k k k k k k k k

k k k k k k k k

k k k k k k k k

u

u

u

u

u

u

u

u

11 12 13 14 15 16 17 18

21 22 23 24 25 26 27 28

31 32 33 34 35 36 37 38

41 42 43 44 45 46 47 48

51 52 53 54 55 56 57 58

61 62 63 64 65 66 67 68

71 72 73 74 75 76 77 78

81 82 83 84 85 86 87 88

1

2

3

4

5

6

7

8

L

N

MMMMMMMMMMM

O

Q

PPPPPPPPPPP

L

N

MMMMMMMMMMM

O

Q

PPPPPPPPPPP

L

N

MMMMMMMMMMM

O

Q

PPPPPPPPPPP

f

f

f

f

f

f

f

f

1

2

3

4

5

6

7

8

12 13

14 15

21 22

23 24

Global equations

L

N

MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM

O

Q

PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP

L

N

MMMMMMMMMMMM

MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM

O

Q

PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP

L

N

MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM

O

Q

PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP

PPPP

Consider the strong form of a linear partial differential equation, e.g. 3-D Poisson’s equation with zero boundary conditions:

0

),,(2

2

2

2

2

2

u

zyxfz

u

y

u

x

uOn region R

on boundary S

Strong Form Lu = f

Variational Principle, e.g. minimum potential energy

Rv

vfLvu Vd)2(min

Weighted Residual (weak) form, e.g. virtual work

0Vd)( R

wfLu

Integral Formulations

0

),(2

2

2

2

u

yxfy

u

x

uOn region S

on boundary C

Weak form

SS

yxwfyxwy

u

x

udddd

2

2

2

2

Integrate by parts

d d d d d dS C C S

u w u w u ux y w x w y f w x y

x x y y y x

Where, u and w vanish at the boundary

0 0

Weak Form for 2-D Poisson’s Equation

• Choose a finite set of approximating (trial) functions, i(x,y), i = 1, 2, …, N

• Allow approximations to u in the formU(x,y) = U11 + U22 + U33 + … + UNN

(that can also satisfy the essential boundary conditions)

• Solve N discrete equations for U1, U2, U3, …, UN

ij

jij

si

S

iNN

iNN

FUK

yxf

yxyy

Uy

Uxx

Ux

U

dd

dd...... 11

11

Galerkin’s Method for 2-D Poisson’s Equation

yxfF

Kyxyyxx

K

Sii

jiS

jijiij

dd

dd

[K]U = F

[K] is the stiffness matrix and F is the load (RHS) vector

[K] is symmetric and positive definite

Galerkin’s Method for 2-D Poisson’s Equation

• Galerkin is more general than Rayleigh-Ritz. If we add u/x, symmetry & the variational principle are lost, but Galerkin still works

• If w is chosen as Dirac delta functions at N points, weighted residuals reduces to the collocation method

• If w is chosen as the residual functions Lu-f, weighted residuals reduces to the least squares method

• By choosing w to be the approximating functions, Galerkin’s method requires the error (residual) in the solution to be orthogonal to the approximating space.

• The integration by parts (Green-Gauss theorem) automatically introduces the Neumann (natural) boundary conditions

• The Dirichlet (essential) boundary conditions must be satisifed explicitly when solving [K]U=F

• Since discretized integrals are sums, contributions from many elements are assembled into the global stiffness matrix by addition.

• The Ritz-Galerkin FEM finds the approximate solution that minimizes the error in the energy

Comments on Galerkin’s Method

2 on region, R 0u

D u g tt

x

Weak form

2

2

d d

d d d

d d d d

d d d d

d d d

R R

R R R

R R R R

R R R S

R R R

uD u w V g w V

t

uw V D u w V g w V

t

uw V D u w V wD u A g w V

t

u uw V D u w V g w V w D A

t n

uw V D u w V g w V

t

n

dS

w D A q

Transient Heat Equation

Strong form

d d d d

Let

Let

d d d . d

or [ ] [ ] [ ]

R R R S

i i i ii

j

ii j i i j j j

R R R S

ij i ij i ji i

uw V D u w V g w V w D q A

t

U U U

w

UV D U V g V D q A

t

M U t K U t f

M U K U f

x ξ ξ ξ

Galerkin FE Equations

A system of linear ordinary differential equations, that could be solved with one of the many sophisticated time integration methods for solving ODEs.[M] is called the mass matrix

1 2 3

[0,1]

or [ ] [ ] [ ]

where d

d d d

d

where is the determinant of the Jacobian,

and d d

and d

ij i ij i ji i

ij i j

R

ki j

l

i j

e

k

l

ij i j i j

R e

j j

R

M U K U f

M V

x

J

xJ

K D V D J

f g V D q

M U K U f

ξ

ξ

dj

S

A

Finite Element Equations

Consider the Finite Difference Discretization

2 32 3 41 1

1 2 62 3

2 32 3 41 1

1 2 62 3

22

1 1

Taylor's expan

( , ) (

2

sion

, ) ki

k kkk ki i

i i i

k kkk ki i

i i i

k k ki i i

u x t u i x k t u

u u uu u x x x O x

x x x

u u uu u x x x O x

x x x

u u u x

42

21 1

2 2

1 2

1

central difference approximation2

k

i

k k k ki i i

i

kk ki i

i

k k ki i

i

uO x

x

u u uu

x x

uu u t O t

t

u uu

t t

Finite Difference Method for 1-D Heat Equation

2

2

12 41 1

2

11 12

Applying the finite difference approximations to the 1-D heat equation:

This (backward ) scheme

0

2,

2

is

k k k k ki i i i i

k k k k ki i i i i

u uD

t x

u u u u uD O t x

t xt

u u D u u

explicit Eule

u

r

x

2

1 1 1 11 11 1 1 12 22

stable if:

Accuracy and stability can be improved with an scheme:

This is called the - scheme.

1

2

2 2k k k k k k k ki i i i i i i i

implicit

Crank Nicho

tD

x

tu u D

lso

u u u u u ux

n

Time Discretization: The -Rule

1

1

Approximate

by 1

backward Euler scheme (explicit) 0

forward Euler scheme (implicit) 1

1Crank-Nicholson scheme (implicit)

2

k kk k

uG

t

u uG G

t

Applying the -Rule to the Heat Equation

2

12 1 2

11

11

on region, R 0

1 on region, R

d 1 d

d d

Weighted residual formula

d d

d

tion

k kk k

k kk k

R R

R S

kk

R R

R

uu g t

t

u uu u g

t

u uw V u u w V

t

g w V w q A

uw V u w V

t

g w V

x

x

d d 1 dk

k

S R R

uw q A w V u w V

t

Crank-Nicholson Galerkin Equations

11

11

d d

d d d 1 d

Let and

d d

d . d d 1 d

kk

R R

kk

R S R R

i i i i ji

kki

i j i i j

R R

kki

j j i j i i j

R S R R

uw V u w V

t

ug w V w q A w V u w V

t

U U U w

UV U V

t

Ug V q A V U V

t

M

x ξ ξ ξ

1

1 1

where [ ], [ ], and [ ] are the same as defined before

k kk ki i

ij ij i j ij ij ii i i i

U UK U f M K U

t t

M K f