Accurate parameter optimization leads to predictive dynamical models for systems biology

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Hansen, Jérôme Azé, Romain Yvinec, Pascale Crepieux, Eric Reiter, Anne Poupon

To cite this version:

Thomas Bourquard, Mohammed Akli Ayoub, Domitille Heitzler, Nikolaus Hansen, Jérôme Azé, et al..

Accurate parameter optimization leads to predictive dynamical models for systems biology. Design,

Optimization and Control in Systems and Synthetic Biology DOC’15, Ecole Normale Supérieure -

Paris (ENS Paris). FRA., Nov 2015, Paris, France. �hal-02799275�

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Accurate parameter optimization leads to predictive dynamical models for systems biology

Angiotensin signaling model: Data Fitting, Convergence and Identifiability issues

T. Bourquard

¹

, M. A. Ayoub

¹

, D. Heitzler

¹

, N. Hansen

²

, J.

Az´ e

³

, R. Yvinec

¹

, P. Cr´ epieux

¹

, E. Reiter

¹

& A. Poupon

¹

1BIOS group, UMR7247, INRA Tours

2LRI, UMR8623, CNRS, Orsay

3LIRMM, UMR5506, CNRS, Montpellier

(3)

Optimization for parameter estimation

Results on the angiontensin signaling pathway

(4)

Outline

Signaling Biology

Modeling and data fitting framework Optimization for parameter estimation

Results on the angiontensin signaling pathway

(5)

surface and leads to a signal.

I

The bound receptor-ligand complex leads

to a cascade of reactions (enzymatic

catalysis, phosphorylation,...) up to some

effector molecule that leads to a cellular

response.

(6)

Signaling Modeling Optimization Angiontensin

General issues

I

A Ligand binds a receptor and signals.

I

The bound receptor-ligand complex leads to a cascade of reactions.

I

G Protein Coupled Receptor (GPRC) : Family of receptor, widely targeted by drugs.

I

Two main pathways : G protein pathway

and

β-arrestin

(signal vs internalization)

(7)

I

The bound receptor-ligand complex leads to a cascade of reactions.

I

G Protein Coupled Receptor (GPRC)

I

Two pathways : G protein and

β-arrestin.

I

More complex issues :

β-arrestin induced

pathway leads to a different signal on

the same effector.

(8)

General issues

I

A Ligand binds a receptor and signals.

I

The bound receptor-ligand complex leads to a cascade of reactions.

I

G Protein Coupled Receptor (GPRC)

I

Complex interactions between G protein and

β-arrestin pathways.

Drug discovery

I

Signaling through one pathway and

not another one : Bias (synthetic

hormone, mutant receptor, small

molecules...)

(9)

I

The bound receptor-ligand complex leads to a cascade of reactions.

I

G Protein Coupled Receptor (GPRC)

I

Complex interactions between G protein and

β-arrestin pathways.

Computational Modeling

I

Help deciphering the intricate effect of each pathway.

I

Quantify the precise effect of a

specific couple Ligand-Receptor.

(10)

GPCR signaling through ERK phosphorylation

The extracellular signal-regulated kinase ERK is activated both by the G protein and the

β-arrestin pathway but (Ahn et

al. J Biol Chem (2004)) :

I

The spatial distribution are distinct.

I

The kinetics are distinct.

(11)

ERK is activated both by the G protein and the

β-arrestin pathway but (Ahn et al. J Biol Chem (2004)) :

I

The spatial distribution are distinct.

I

The kinetics are distinct.

Transient and sustained ERK activa-

tion have been shown to regulate cell

fates such as growth and differen-

tiation.(Sasagawa et al. Nat Cell Biol

(2005))

(12)

GPCR signaling through ERK phosphorylation

The extracellular signal-regulated kinase ERK is activated both by the G protein and the

β-arrestin pathway but (Ahn et al. J Biol Chem (2004)) :

I

The spatial distribution are distinct.

I

The kinetics are distinct.

β-arrestin 2 dependent ERK pathway can

be activated independently of G pro- teins with a mutant receptor (Wei et al.

PNAS (2004)).

(13)

I

Angiotensin II type 1A receptor (AT1AR) transfected in cultured human embryonic kidney (HEK 293 cells).

I

ERK phosphorylation data : Phosphorylated ERK in immunoblots, quantified by densitometry (Kim et al. PNAS 2005 )

I

DAG accumulation and PKC activity data, measured in real time by FRET sensors.

I

Four perturbed conditions in addition to control :

I β-arrestin 2 siRNA

I G protein-coupled receptor kinases (GRK2/3 and GRK5/6) siRNA

I PKC inhibitor.

(14)

Outline

Signaling Biology

Modeling and data fitting framework Optimization for parameter estimation

Results on the angiontensin signaling pathway

(15)

I

Starting point : graph of interaction

of molecules (based on biological know-

ledge, literature)

(16)

Overview of the methodology

I

Graph of interaction of molecules

I

Law of mass-action : Ordinary Diffe- rential Equations (ODE) produce time- dependent trajectories, that depend on parameters (kinetic rate, initial condi- tion)

d [B ]

dt = k

0

[A]

−

k

1

[B]

.

d [B ]

dt = k

₀

[A][C ]

−

k

₁

[B]

.

(17)

I

Graph of interaction of molecules.

I

Law of mass-action : ODE.

I

Quality of the model based on the in-

troduction of a cost function (based on

statistical error model, or heuristic argu-

ments).

(18)

Overview of the methodology

I

Graph of interaction of molecules.

I

Law of mass-action : ODE.

I

Cost function.

I

Optimization of the cost function (Fre-

quentist / Bayesian approach). Numeri-

cal search.

(19)

I

Graph of interaction of molecules.

I

Law of mass-action : ODE.

I

Cost function.

I

Optimization.

I

Validation data, prediction and experi-

mental design...

(20)

Outline

Signaling Biology

Modeling and data fitting framework Optimization for parameter estimation

Results on the angiontensin signaling pathway

(21)

The major difficulties are due to

I

Large dimension of parameter space (10

−

100) and state space (> 10).

I

Few molecule concentrations measured, and not in absolute numbers.

I

Large ODE’s may be numerically costly to simulate if they are stiff.

I

Parameters can be non-identifiable (non-convexity, presence of many local minima)

I

Hybrid local and global method, based on heuristics

(HYPE, T. Bourquard & A. Poupon)

(22)

The major difficulties are due to

I

Large dimension of parameter space (10

−

100) and state space (> 10).

I

Few molecule concentrations measured, and not in absolute numbers.

I

Large ODE’s may be numerically costly to simulate if they are stiff.

I

Parameters can be non-identifiable (non-convexity, presence of many local minima)

A variety of methods can be employed : local/global methods and deterministic/stochastic methods, hybrid method.

I

Gradient descent methods with many random initial start (D2D, Raue A., et al. Bioinformatics (2015)).

I

Hybrid local and global method, based on heuristics

(HYPE, T. Bourquard & A. Poupon)

(23)

Random set of parameters Random set of parameters

Genetic algorithm Genetic algorithm CMA-ES CMA-ES

Final set of parameters

A random set of parameters is optimized using Genetic algorithm. The resulting parameter set is optimized using CMA-ES.

This operation is repeated until a sufficient number of parameter sets giving small errors is obtained.

1 individual = (param1, param2)

Parameter 1

Parameter 2

Mutation

Parameter 1

Parameter 2

Cross-over

Parameter 1

Parameter 2

New parents

Parameter 1

Parameter 2

+

-

Error

GENETIC ALGORITHM:

● A random set of n parameter sets is chosen, called parents.

● Mutation and cross-over produce new sets of parameters, as offsprings.

● The n best parameter sets giving the lowest errors are selected, and become the parents of next generation.

δ1,5 δ1,10

δ1,20

δ1,40

δ1,60

δ1,80 δ1,100

Error function

CMA-ES:

●Start with one set of parameter, called parent.

●Local mutations (gaussian) produce offsprings.

●The next parent is the average of the children weighted by their errors.

●Successive steps have memory, mutation are made along the direction of the previous steps. (Hansen et al. 2003)

Parameter 1

Parameter 2

Parameter 1

Parameter 2

New parent : weighted average + best direction

Parameter 1

Parameter 2

New generation

Parameter 1

Parameter 2

Mutation

(24)

Critical assessment of methods

How to judge different method ? How to asses the quality of a fit ?

I

Toy models with in silico simulated data / Benchmark models.

I

Absolute value of the error function.

I

Speed of the algorithm.

I

Convergence curve (number of best error function value over number of runs/function evaluation.

I

Robustness of the minima.

(25)

How to judge different method ? How to asses the quality of a fit ?

Why/How to deal with Non-Identifiability (NI) ?

I

It slows down the numerical search and leads to unreliable results.

I

Theoretical NI : reduction / algebraic relations.

I

Numerical NI : distinguish between structural and practical

NI. Calculate sensitivity and one-dimensional profile likelihood.

(26)

Toy models

(27)

-1 -0.9 -0.8 -0.7 -0.6 -0.5

Model1

D2D HYPE

-1 -0.5 0 0.5

1 1.5

2

Model2

lo g

10

( χ

2

)

-1 -0.5

0 0.5 1 1.5 2

Model3

-1.5 -1 -0.5 0

Model4

runs

0 20 40 60 80 100

-1 -0.9 -0.8 -0.7 -0.6 -0.5

Model5

runs

0 20 40 60 80 100

0 0.5

1 1.5

2

Model6

(28)

Outline

Signaling Biology

Modeling and data fitting framework Optimization for parameter estimation

Results on the angiontensin signaling pathway

(29)

kinases (GRK) in the balance of signals.

(30)

Full model (Heitzler el al. MSB 2012) : Control + 4 pert.

(31)

0 10 20 30 40 50 60 70 80 90 100

pERK (a.u.)

0 30 60 90

0 10 20 30 40 50 60 70 80 90 100

pERK (a.u.)

0 30 60 90

0 10 20 30 40 50 60 70 80 90 100

pERK (a.u.)

0 30 60 90

120 pERK control vs GRK2/3 siRNA

0 10 20 30 40 50 60 70 80 90 100

pERK (a.u.)

0 30 60 90

Time

0 10 20 30 40 50 60 70 80 90 100

DAG (a.u.)

0 0.05

0.1 0.15 0.2 0.25 0.3

DAG control

Time

0 10 20 30 40 50 60 70 80 90 100

PKC (a.u.)

-0.02 0 0.02 0.04

0.06 PKC control

In black : control experiments.

In Red : perturbated experiments.

(32)

The model correctly predicts some validation data...

Fitting error

Predictivity

Remark

Good correlation between error and prediction.

(33)

0 20 40 60 80 100 120 140 160 180 10⁰

10¹ 10² 10³ 10⁴

likelihood

converged fits initial value

Critical assessments :

I

Convergence curve. Identifiability of parameters.

I

Parsimonious use of parameters. Model selection.

(34)

Model reduction

From 50 parameters...

(35)

(36)

Model reduction

I

The reduced model still fit (reasonably) well...

0 10 20 30 40 50 60 70 80 90 100

pERK (a.u.)

0 30 60 90

120 pERK control vs Barr2 siRNA

0 10 20 30 40 50 60 70 80 90 100

pERK (a.u.)

0 30 60 90

120 pERK control vs PKC inhib

0 10 20 30 40 50 60 70 80 90 100

pERK (a.u.)

0 30 60 90

0 10 20 30 40 50 60 70 80 90 100

pERK (a.u.)

0 30 60 90

Time

0 10 20 30 40 50 60 70 80 90 100

DAG (a.u.)

0 0.05

0.1 0.15 0.2 0.25 0.3

DAG control

Time

0 10 20 30 40 50 60 70 80 90 100

PKC (a.u.)

-0.02 0 0.02 0.04

0.06 PKC control

(37)

I

and the convergence is better...

run index (sorted by likelihood)

0 100 200 300 400 500 600 700 800 900

10⁰ 10¹ 10² 10³

likelihood

(38)

Model reduction

I

The reduced model still fit well...

I

and the convergence is better...

I

but the identifiability is still poor !

0.20.4 0.6 0.8 1 1.2 1.4 1.6 log10(k08) parameter #7

−0.2 0 0.2 0.4 0.6 0.8 1 1.2 log10(k09) parameter #8

−0.9 −0.8 −0.7 −0.6 −0.5 log10(k07) parameter #6

1.5 2 2.5 3 3.5

log10(k05) parameter #4

0 1 2 3 4

21 22 23 24 25 26

log10(k06) χ2 PL

parameter #5

0 1 2 3 4

−3.5 −3 −2.5 −2 −1.5 log10(k04) parameter #3

3 4 5 6

21 22 23 24 25 26

95% (point-wise)

log10(k0) χ2 PL

parameter #1

(39)

I

and the convergence is better...

I

but the identifiability is still poor ! Let’s reduced further ?

0.20.4 0.6 0.8 1 1.2 1.4 1.6 log10(k08) parameter #7

−0.2 0 0.2 0.4 0.6 0.8 1 1.2 log10(k09) parameter #8

−0.9 −0.8 −0.7 −0.6 −0.5 log10(k07) parameter #6

1.5 2 2.5 3 3.5

0 1 2 3 4

21 22 23 24 25 26

log10(k06) χ2 PL

parameter #5

0 1 2 3 4

−3.5 −3 −2.5 −2 −1.5 log10(k04) parameter #3

3 4 5 6

21 22 23 24 25 26

95% (point-wise)

log10(k0) χ2 PL

parameter #1

(40)

Systematic reduction

(41)

(42)

Systematic reduction

(43)

(44)

Systematic reduction

(45)

(46)

Systematic reduction

(47)

(48)

Systematic reduction

(49)

I

a model with 19 parameters

I

good convergence properties (10% of runs reached the optima)

I

most parameters are identifiable

0 102030405060708090100

0 30 60 90 120

pERK (a.u.)

pERK control vs Barr2 siRNA

0102030405060708090100

0 30 60 90 120

pERK (a.u.)

pERK control vs PKC inhib

0 102030405060708090100

0 30 60 90 120

pERK (a.u.)

pERK control vs GRK2/3 siRNA

0102030405060708090100

0 30 60 90 120

pERK (a.u.)

pERK control vs GRK5/6 siRNA

0 102030405060708090100

0 0.05 0.1 0.15 0.2 0.25 0.3

Time

DAG (a.u.)

DAG control

0102030405060708090100

-0.02 0 0.02 0.04 0.06

Time

PKC (a.u.)

PKC control

run index (sorted by likelihood)

0 50 100 150 200 250 300 350 400 450 500

10⁰ 10¹ 10² 10³

likelihood

−0.2 0 0.2 0.4 log

10(k09) parameter #2

0.55 0.6 0.65 0.7

log10(r26) parameter #7

0.05 0.10.15 0.2 0.25

log10(r27) parameter #8

−2.2−2−1.8−1.6−1.4 log

0.20.30.40.50.60.7

15 16 17 18 19 20

95% (point-wise)

log10(k05) χ2 PL

parameter #1

−0.9−0.85−0.8−0.75−0.7−0.65−0.6 log

−0.6−0.55−0.5−0.45−0.4 15 16 17 18 19 20

log 10(k18) χ2 PL

parameter #5

−3.5 −3−2.5 −2 log

(50)

What do we learn (so far) ?

I

The three path-

ways are a neces-

sary condition to re-

produce the data.

(51)

necessary.

I

An internalization

pathway, inde-

pendent of the

β-arrestin

signa-

ling pathway, is

mandatory.

(52)

What do we learn (so far) ?

I

Three pathways are necessary.

I

An independent internalization path- way is mandatory.

I

A minimal model

with three reversible

pathway with 10 pa-

rameter is able to fit

the phospho ERK

data and its para-

meter are all iden-

tifiable.

(53)

necessary.

I

An independent internalization path- way is mandatory.

I

A minimal model can fit the

phospho ERK

data and is identifiable.

I

The best mo- del able to fit all data present non-identifiability

⇒

Experimental

design.

(54)

Conclusion

I

A full model able to fit the data (Heitzler el al. MSB 2012 ).

I

Accurate parameter estimation leads to accurate prediction.

I

Further improvements with model reduction/selection.

I

Parameter identifiability with a good fit can be achieved.

I

We have shed light on the importance of three pathways in

GPCR signaling, and its regulations.

(55)

I

A full model able to fit the data (Heitzler el al. MSB 2012 ).

I

Accurate parameter estimation leads to accurate prediction.

I

Further improvements with model reduction/selection.

I

Parameter identifiability with a good fit can be achieved.

I

We have shed light on the importance of three pathways in GPCR signaling, and its regulations.

Thanks for your attention !

(56)

validation data

0 10 20 30 40 50 60 70 80 90 100

pERK (a.u.)

0 30 60 90 120

pERK with GRK23,GRK56 overexpression and G inhibition

++GRK23 ++GRK56 --Ginhib