adapt: analysis of dynamic adaptations in parameter trajectories

Data Integration in the Life Sciences

Feb. 5, 2015, Lorentz Center, Leiden

Natal van Riel

Systems Biology and Metabolic Diseases

[email protected], GEM-Z 3.109, tel. 040 247 5506

Objectives

• Follow-up on parameter estimation

• Propagation of Uncertainty

• ADAPT

/ biomedical engineering PAGE 22/5/2015

SlideShare

http://www.slideshare.net/natalvanriel

measuringmodelling




Today’s team

• Karen van Eunen (UMCG)

• Yared Paalvast (UMCG)

• Bert Groen (UMCG)

• Yvonne Rozendaal (TU/e)

• Natal van Riel (TU/e)

/ biomedical engineering PAGE 35-2-2015

Longitudinal - Treatment in time


Preclinical study of pharmaceutical

intervention

• data: control, treated for 1, 2, 4, 7, 14, and 21 days


0 10 200

100

200Hepatic TG

Time [days]

[um

ol/g]

0 10 200

1

2

3Hepatic CE

Time [days]

[um

ol/g]

0 10 200

2

4

6Hepatic FC

Time [days]

[um

ol/g]

0 10 200

50

100Hepatic TG

Time [days]

[um

ol]

0 10 200

0.5

1

1.5Hepatic CE

Time [days]

[um

ol]

0 10 200

2

4Hepatic FC

Time [days]

[um

ol]

0 10 200

1000

2000

3000Plasma CE

Time [days]

[um

ol/L]

0 10 200

1000

2000

3000HDL-CE

Time [days]

[um

ol/L]

0 10 200

500

1000

1500Plasma TG

Time [days]

[um

ol/L]

0 10 206

8

10

12VLDL clearance

Time [days]

[-]

0 10 20100

200

300

400ratio TG/CE

Time [days]

[-]

0 10 200

5

10

15VLDL diameter

Time [days]

[nm

]

0 10 200

1

2

3VLDL-TG production

Time [days]

[um

ol/h]

0 10 201

2

3Hepatic mass

Time [days]

[gra

m]

0 10 200

0.2

0.4DNL

Time [days]

[-]

Grefhorst et al. Atherosclerosis, 2012, 222: 382– 389

Modelling


Understanding (modeling) progressive diseases and effect of

treatment-in-time

Challenges:

• Many factors involved

• Different biological levels, many details unknown

• Dynamic interactions of molecular species, cells,

tissues/organs

• Multiple time scales (orders of magnitude different) - molecular

mechanisms governing cell behaviour versus gradual

(patho)physiological changes induced by a progressive disease

or therapeutic intervention

• In vivo values of parameters unknown


ADAPT

Analysis of Dynamic Adaptations in Parameter Trajectories


? ? ?


Data integration via dynamic

network models

System identification


M.C. Escher

Mechanism-based models for data integration

• Physical / biological interpretation of model variables and

parameters

• Structure based on known

physics and biology

• Parameter values estimated

from experimental data

(parameter identification)


biology physics

model model

scheme equations

‘Fitting’ of model to data

• Known from linear regression

• Which ‘estimator’?

• Which algorithm?

• What are the underlying principles?

• What is the effect of the uncertainty (‘noise’) in the data

• Can we get more out of this than a line through some

datapoints?

• Can we generalize this? (nonlinear, dynamic)


uu y

y u

Parameter Estimation

• Minimize the sum of squared model errors by varying model

parameters

• The parameter value for which criterion is minimal is the best

(most likely) estimate for the parameters


parameters

+

-

MODEL ERROR

input

MODEL OUTPUT

MODEL

( ) ( | ) ( )d k y k k

Dynamic systems and models

• Dynamic system (state-space representation)

• outputs:

• initial conditions:

• Stoichiometry matrix N


u2

u1 1 S1

S3S2S4

3

4 5

2

1 2 3 4 5v v v v v

1

2

3

4

1 0 1 1 0

1 1 0 0 0

1 1 0 0 0

0 0 0 1 1

S

S

S

S

N


Dynamic systems and models

• Network structure and stoichiometry are fixed

• Variables: concentrations S (in x)

reaction rates v (in f)

• Parameters Vmax, Km, …

• In general, output y(t) cannot be calculated analytically, but

results from numerical simulation

• Matlab ODE suite, e.g. ode45, ode15s

• Mathematical model: continuous time

• Computational model: discrete time

( , , )x f x u ty(t)u(tk)u(t) u(k)~

interpolatey(tk)

1 21

2

( ) ( )( )

( )max

m

u t S tv t V

K S t

A ‘driving’ / ‘forcing’ functionmeasured data is interpolated and used as input

Cubic spline

interpolation

Data interpolation

Matlab

• Linear interpolation

interp1

• Cubic Spline interpolation

csaps


0 30 60 90 120 150 1805

5.5

6

6.5

7

7.5

8

8.5

time [min]

G [m

mol/L]

raw data

spline interpolation

0 30 60 90 120 150 1805

5.5

6

6.5

7

7.5

8

8.5

time [min]

G [m

mol/L]

raw data

linear interpolation

Parameter estimation for Dynamic models

• Error model

• Maximum Likelihood Estimation


2

2

1 1

( ) ( | )( )

n Ni i

i k ik

d k y k

( ) ( | )i id k y k

( | ) ( )i iy k k

2

ˆ 0

ˆ arg min ( )


Unknowns to be estimated

• Initial conditions of dynamic models x0 often not known for

biological / biomedical systems

• If measured → uncertainty / error

• So typically

• But potentially not all parameters/initial conditions need to be

estimated

0[ , ]p x

0[ ', ']p x 0 0' 'p p x x


Parameter estimation for Dynamic models

• Parameter estimation: nesting of 2 numerical schemes


Examples

A theoretical example

• A metabolic system with

metabolite controlled,

negative transcriptional

feedback

• A progressive

perturbation acting on

the gene/protein circuit

encoding the repressor

• Time scales relevant to this phenotype:

• Metabolic network – seconds

• Gene regulatory circuit – minutes/hours

• Progressive adaptation to the perturbation – days…


R1

u2

u1 1 S1

S3S2S4

3

4 5

2

7

6

Van Riel et al. (2013) Interface Focus, 3(2): 20120084

A theoretical example

Experimental data:

• metabolic profile (S1, S2, S3, S4)

• 5 stages / 5 ‘snapshots’

(time 1, 2, 3, 4, 5)


R1

u2

u1 1 S1

S3S2S4

3

4 5

2

7

6

R1

u2

u1 1 S1

S3S2S4

3

4 5

2

7

6

Case 1: one model for each stage

• Transcription:

• Simulate steady-state xss

• Infer values for from the data for stage 2, 3, 4, 5

• Stoichiometry matrix

• ODE model


1 21 max

1i

u Sv V

K R

( )

( ), , ( )d t

f t tdt

x

N x p u

6 6 4

6 0.01

v k S

k

6k̂

1 0 1 1 0

1 1 0 0 0

1 1 0 0 0

0 0 0 1 1

N

Estimate transcription rate k6 for the time

points after the perturbation


R1

u2

u1 1 S1

S3S2S4

3

4 5

2

7

6

• Statistically acceptable fits and

accurate parameter estimates

0 1 2 3 40

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

days

S1

S2

S3

S4

1 2 3 4 5

Results case 1

• Case 1:

• Metabolic level: topology and interaction kinetics known

• Gene / protein level: topology known, kinetic parameters

unknown (changing)

• Kinetic parameters of the gene/protein circuit estimated from

experimental observations at the metabolic level (metabolic

profiling) during the different stages of progression

• Resulting in 5 separate simulation models (one for each stage)


stage 1 stage 5

Case 2: Lacking information at gene/protein

level

• Next, a more challenging but common scenario is explored:

• Metabolic level: topology known, uncertainty in interaction

kinetics (kinetic parameters)

• Gene / protein level: from functional genomics studies we

know that the intervention affects a gene/protein controlling

reaction 1 (but molecular details are lacking)

• Same experimental observations, reflecting progressive

metabolic adaptations after an intervention at time 0 (stage 1)


u2

u1 1 S1

S3S2S4

3

4 5

2

Analyze the data as individual ‘snapshots’

• Metabolic network without feedback

• The unknown adaptation at gene/protein level is translated into

an unknown, but inferable value for the metabolic rate constant

• However, like in the approach with case 1, this ignores the fact

that the snapshots are linked


1 1 1 2ˆv k u S

max1 1 2

4( )m

Vv u S

K f S

( )

( ), , ( )d t

f t tdt

x

N x p u

u2

u1 1 S1

S3S2S4

3

4 5

2

phenomenological parameter

k1 (‘undermodeling’)


Identifiability and Uncertainty


The Elephant in the Room,

Banksy exhibition, 2006

Bootstrapping

• Sampling based method


Vanlier et al. Math Biosci. 2013 Mar 25

Example cont’d – case 2

• Monte Carlo (drawing samples from the data distribution)

• MLE (weighting with the data variance)


u2

u1 1 S1

S3S2S4

3

4 5

2

Simulation of

the five

models,

with the

mean value

of the

ensemble of

parameter k1

for the

different

stages.

k1


ADAPT

Time-continuous description of the data

• ADAPT accounts for uncertainty in the data

• ADAPT accounts for potential differences in dynamic behavior


Gaussian distribution

Sampling replicates from error model

( , )d d N

Modelling phenotype transition (1)

34

treatment

disease progression

longitudinal discrete data: different phenotypes

Introducing time-dependent parameters

35

steady state model

Parameter trajectory estimation

36

steady state model

iteratively calibrate model to data: estimate parameters over time

minimize difference between data and model simulation


37

steady state model



38

steady state model


Modelling phenotype transition

longitudinal discrete data: different phenotypes

estimate continuous data: ensemble of cubic smooth spline

incorporate uncertainty in data: multiple describing functions


Estimated parameter trajectories


Results with ADAPT

• Using the model of the metabolic network to integrate and

connect metabolomic data obtained at different stages of

progressive adaptations after an intervention


u2

u1 1 S1

S3S2S4

3

4 5

2

Van Riel et al. (2013) Interface Focus, 3(2): 20120084

ADAPT of lipoprotein and lipid metabolism

• Connecting the longitudinal data

• Taking into account uncertainties


• Calculating unobserved quantities

Tiemann et al. (2013) PLoS

Comput Biol. 9: e1003166

Literature

• Hijmans BS, Tiemann CA, Grefhorst A, Boesjes M, van Dijk TH, Tietge UJ, Kuipers F,

van Riel NA, Groen AK, Oosterveer MH. A systems biology approach reveals the

physiological origin of hepatic steatosis induced by liver X receptor activation. FASEB

Journal, 2014 Dec 4. [Epub ahead of print]

• Tiemann CA, Vanlier J, Hilbers PA, and van Riel NA. Parameter adaptations during

phenotype transitions in progressive diseases. BMC Syst Biol. 5:174, 2011.

• Tiemann CA, Vanlier J, Oosterveer MH, Groen AK, Hilbers PAJ, and van Riel NAW.

Parameter trajectory analysis to identify treatment effects of pharmacological

interventions. PLoS computational biology 9: e1003166, 2013.

• van Riel NA, Tiemann CA, Vanlier J, and Hilbers PA. Applications of analysis of

dynamic adaptations in parameter trajectories. Interface Focus 3(2): 20120084, 2013.


Systems Biology of Disease Progression

http://www.youtube.com/watch?v=x54ysJDS7i8




adapt: analysis of dynamic adaptations in parameter trajectories

Science

fitting of model

dynamic systems

ode15s mathematical

parameter estimationminimize

dynamic network models

sum of squared model

parameters structure

functionmeasured data