Grégory Batt Ph.D. defense -- under supervision of Hidde de Jong, Helix research group

Validation of Qualitative Models of Genetic Regulatory Networks

A Method Based on Formal Verification Techniques

Grégory BattPh.D. defense

--

under supervision of Hidde de Jong,

Helix research group

INRIA Rhône-Alpes

--

Ecole doctorale

Mathématiques, Sciences et technologies de l’information, Informatique

Université Joseph Fourier

Stress response in Escherichia coli

Bacteria capable of adapting to a variety of changing environmental conditions

Stress response in E. coli has been much studied

Model for understanding adaptation of pathogenic bacteria to their host

Nutritional stress

Osmotic stress

Heat shock

Cold shock

…

Nutritional stress response in E. coli

Response of E. coli to nutritional stress conditions: transition from exponential phase to stationary phase

Important developmental decision: profound changes of morphology,

metabolism, gene expression,...

log (pop. size)

time

> 4 h

Network controlling stress response Response of E. coli to nutritional stress conditions controlled by

genetic regulatory networkDespite abundant knowledge on network components, no global view of

functioning of network available

rrnP1 P2

CRP

crp

cya

CYA

cAMP•CRP

FIS

TopA

topA

GyrAB

P1-P4P1 P2

P2P1-P’1

P

gyrABP

Signal (carbon starvation)

DNA supercoiling

fis

stable RNAs

protein

gene

promoter

Modeling and simulation

Genetic regulatory network controlling E. coli stress response is large and complex

Modeling and simulation indispensable for dynamical analysis of genetic regulatory networks

Systematic prediction of possible network behaviors

Current constraints on modeling and simulation: knowledge on molecular mechanisms rare quantitative information on kinetic parameters and molecular

concentrations absent

Qualitative methods developed for analysis of genetic networks using coarse-grained models

Model validation

Available information on structure of network controlling E. coli stress response is incomplete

Model is working hypothesis and needs to be tested

Model validation is prerequisite for use of model as predictive and explanatory tool

Check consistency between model predictions and experimental

data

consistency?experimental datanetwork predictions

x = f (x) .

model

Model validation

Available information on structure of network controlling E. coli stress response is incomplete

Model is working hypothesis and needs to be tested

Model validation is prerequisite for use of model as predictive and explanatory tool

Check consistency between model predictions and experimental

data

Current constraints on model validation:

available experimental data essentially qualitative in nature

model validation must be automatic and efficient

Objectives and approach of thesis

Objective of thesis:

Development of automated and efficient method for testing whether

predictions from qualitative models of genetic regulatory networks are

consistent with experimental data on dynamics of system

Approach based on formal verification of hybrid systems qualitative analysis of piecewise-linear models of genetic networks

model checking for testing consistency between predictions and data

Expected contributions: scalable method with sound theoretical basis

implementation of method in user-friendly computer tool

applications to validation of models of networks of biological interest

Overview

I. Introduction

II. Method for model validation

1. Piecewise-linear (PL) differential equation models

2. Symbolic analysis using qualitative abstraction

3. Verification of properties by means model-checking techniques

III. Genetic Network Analyzer 6.0

IV. Validation of model of nutritional stress response in E. coli

V. Discussion and conclusions

Overview

I. Introduction








PL differential equation models

Genetic networks modeled by class of differential equations using step functions to describe switch-like regulatory interactions

xa a s-(xa , a2) s-(xb , b ) – a xa .

xb b s-(xa , a1) – b xb .

x : protein concentration

, : rate constants : threshold concentration

x

s-(x, θ)

0

1

Hybrid, piecewise-linear (PL) models of genetic regulatory networks Glass and Kauffman, J. Theor. Biol., 73

b

B

a

A

Analysis of the dynamics in phase space:

Partition of phase space into mode domains

a10

maxb

a2

b

maxa

Qualitative analysis of network dynamics

a10

maxb

a2

b

maxa a10

maxb

a2

b

maxaa10

maxb

a2

b

maxa

M1 M2 M3 M4 M5

M10

M15M14M13M12M11

M6 M7 M8 M9

x = h (x), x \ .


a10

maxb

a2

b

maxa


xa a s-(xa , a2) s-(xb , b ) – a xa.

xb b s-(xa , a1) – b xb .

a10

maxb

a2

b

maxa

xa a – a xa .

xb b – b xb .

aa

bb

0 < a1 < a2 < a/a < maxa

0 < b < b/b < maxb

0 a – a xa 0 b – b xb

.

M1

x = h (x), x \


a10

maxb

a2

b

maxa


a10

maxb

a2

b

maxa

xa – a xa .

xb b – b xb .

M11

0 < a1 < a2 < a/a < maxa

0 < b < b/b < maxb

. x = h (x), x \

bb


a10

maxb

a2

b

maxa


a10

maxb

a2

b

maxa

M2

aa

bb

M3M1

. x = h (x), x \


Extension of PL differential equations to differential inclusions using Filippov approach:

a10

maxb

a2

b

maxa


a10

maxb

a2

b

maxaaa

bb

M3

a10

maxb

a2

b

maxaa10

maxb

a2

b

maxaaa

M3M1M5

Gouzé and Sari, Dyn. Syst., 02

.

M2 M4

. x = h (x), x \

x H (x), x


In every mode domain M, the system either converges monotonically towards focal set, or instantaneously traverses M

a10

maxb

a2

b

maxa


a10

maxb

a2

b

maxa a10

maxb

a2

b

maxaa10

maxb

a2

b

maxa

M1 M2 M3 M4 M5

M10

M15M14M13M12M11

M6 M7 M8 M9

. x H (x), x

de Jong et al., Bull. Math. Biol., 04Gouzé and Sari, Dyn. Syst., 02

Partition does not preserve unicity of derivative sign

Predictions not adapted to comparison with available experimental data:

temporal evolution of direction of change of protein concentrations

Problem for model validation

a10

maxb

a2

b

maxaa10

maxb

a2

b

maxa a10

maxb

a2

b

maxaa10

maxb

a2

b

maxa

M1 M2 M3 M4 M5

M10

M15M14M13M12M11

M6 M7 M8 M9

xa < 0, xb ?x M11:. .

Finer partition of phase space: flow domains

In every domain D, the system either converges monotonically towards focal set, or instantaneously traverses D

In every domain D, derivative signs are identical everywhere

a10

maxb

a2

b

maxa


a10

maxb

a2

b

maxa a10

maxb

a2

b

maxaa10

maxb

a2

b

maxa

bb

D12 D22 D23 D24

D17 D18

D21 D20

D1 D3 D5 D7 D9

D15

D27 D26 D25

D11 D13 D14

D2 D4 D6 D8

D10 D16

D19

xa < 0, xb > 0x D17:. .

Continuous transition system

PL system, = (,,H), associated with continuous PL transition system, -TS = (,→,╞), where

continuous phase space




→ transition relation

and x and x’ in same or in adjacent domain

: transition from x to x’ iff a solution reaches x’ from x

maxamaxa

a10

maxb

a2

b

a10

maxb

a2

b

bb x1 → x2, x1 → x3,

x3 → x4x2 → x3,x1 x2

x3x4

x5




→ transition relation ╞ satisfaction relation

and -TS have equivalent reachability properties

: describes derivative sign of solutions at x

maxamaxa

a10

maxb

a2

b

a10

maxb

a2

b

bb

x1 x2

x3x4

. x1╞ xa > 0,x5 . x1╞ xb

> 0,

. x4╞ xa < 0, . x4╞ xb

> 0,

Discrete abstraction

Qualitative PL transition system, -QTS = (D, →,╞), where

D finite set of domains :D = {D1, …, D27}

D1 D ;

maxamaxa

a10

maxb

a2

b

a10

maxb

a2

b

bb

D12 D22 D23 D24

D17 D18

D21 D20

D1 D3 D5 D7 D9

D15

D27 D26 D25

D11 D13 D14

D2 D4 D6 D8

D10 D16

D19



D finite set of domains → quotient transition relation : transition from D to D’ iff there exist

xD, x’D’ such that x → x’

D1 D ; D1 →~ D1, D1 →~ D11, D11 →~ D17,

maxamaxa

a10

maxb

a2

b

a10

maxb

a2

b

bb

x1

D17

D1

D11

x1 x2

x3x4

x5

D1 D1

D11

D17



D finite set of domains → quotient transition relation

╞ quotient satisfaction relation: D╞ p iff there exists xD such that x╞ p

maxamaxa

a10

maxb

a2

b

a10

maxb

a2

b

bb

x1

D17

D1

D11

x1 x2

x3x4

x5

D1 D ; D1 →~ D1, D1 →~ D11, D11 →~ D17, D1╞ xa>0, D1╞ xb>0, D4╞ xa < 0. . .



D finite set of domains → quotient transition relation

╞ quotient satisfaction relation

Quotient transition system -QTS is a simulation of -TS (but not a bisimulation)

D1 D3 D5 D7 D9

D15

D27D26D25

D11 D12 D13 D14

D2 D4 D6

D8

D10

D16D17

D18

D20

D19

D21

D22

D23

D24

maxamaxa

a10

maxb

a2

b

a10

maxb

a2

b

bb

D12 D22 D23 D24

D17 D18

D21 D20

D1 D3 D5 D7 D9

D15

D27 D26 D25

D11 D13 D14

D2 D4 D6 D8

D10 D16

D19

D1

D11

D17

D18

Alur et al., Proc. IEEE, 00

Discrete abstraction Important properties of -QTS :

-QTS provides finite and qualitative description of the dynamics of

system in phase space

-QTS is a conservative approximation of : every solution of

corresponds to a path in -QTS

-QTS is invariant for all parameters , , and satisfying a set of

inequality constraints

-QTS can be computed symbolically using parameter inequality

constraints: qualitative simulation

Use of-QTS to verify dynamical properties of original system Need for automatic and efficient method to verify properties of -QTS

Batt et al., HSCC, 05

Model-checking approach

Model checking is automated technique for verifying that discrete transition system satisfies certain temporal properties

Computation tree logic model-checking framework: set of atomic propositions AP

discrete transition system is Kripke structure KS = S, R, L ,

where S set of states, R transition relation, L labeling function over AP

temporal properties expressed in Computation Tree Logic (CTL)

p, ¬f1, f1f2, f1f2, f1→f2, EXf1, AXf1, EFf1, AFf1, EGf1, AGf1, Ef1Uf2, Af1Uf2,

where pAP and f1, f2 CTL formulas

Computer tools are available to perform efficient and reliable model checking (e.g., NuSMV, SPIN, CADP)

Validation using model checking

Atomic propositionsAP = {xa = 0, xa < a

1, ... , xb < maxb, xa < 0, xa= 0, ... , xb > 0}

Observed property expressed in CTL

There Exists a Future state where xa > 0 and xb > 0

and from that state, there Exists a Future state where xa < 0 and xb > 0

. .

. .

EF(xa > 0 xb > 0 EF(xa < 0 xb > 0) ). . . .

. . .

0

xb

time

time0

xa

xa > 0.xb > 0.

xb > 0.xa < 0.

Validation using model checking

Discrete transition system computed using qualitative simulation

Use of model checkers to check consistency between experimental data and predictions

Fairness constraints used to exclude spurious behaviors

Yes

Consistency?0

xb

time

time0

xa

xa > 0.xb > 0.

xb > 0.xa < 0.

EF(xa > 0 xb > 0 EF(xa < 0 xb > 0) ). . . .

D1 D3 D5 D7 D9

D15

D27D26D25

D11 D12 D13 D14

D2 D4 D6

D10

D16D17

D18

D20

D19

D21

D22

D23

D24

D8

Batt et al., IJCAI, 05

Overview

I. Introduction








Genetic Network Analyzer

Model validation approach implemented in version 6.0 of GNA, freely available for academic research Batt et al., Bioinformatics, 05

Integration into environmentfor explorative genomics atGenostar SA

structure into packages

class diagram of kernel


GNA implemented in Java 1.4

> 17000 lines of code in 6 packages

35% of lines modified with respect to version 5.5

(up to 60% in kernel)


Rules for symbolic computation of refined partition and corresponding transition relation and domain properties

Tailored algorithms and implementation favor upscalability

Export functionalities to model checkers (NuSMV, CADP)

Overview

I. Introduction








Nutritional stress response in E. coli Entry into stationary phase is an important developmental

decision

?exponential phase

stationary phase

signal of nutritional

deprivation

How does lack of nutrients induce decision to stop growth?

Model of nutritional stress response Carbon starvation network modeled by PL model

7 PL differential equations, 40 parameters and 54 inequality constraints

Ropers et al., Biosystems, in press

Signal (carbon starvation)

CRP

crp

cya

CYA

cAMP•CRP

fis

Fis

Supercoiling

TopA

topA

GyrAB

P1-P4 P1 P2

P2P1-P’1

rrnP1 P2

stable RNAs

PgyrABP

How does response emerge from network of interactions?

Validation of stress response model

Qualitative simulation of carbon starvation:

66 reachable domains (< 1s.)

single attractor domain (asymptotically stable equilibrium point)

Experimental data on Fis:

CTL formulation:

Model checking with NuSMV: property true (< 1s.)

“Fis concentration decreases and becomes steady in stationary phase”

Ali Azam et al., J. Bacteriol., 99

EF(xfis < 0 EF(xfis = 0 xrrn < rrn) ). .

Validation of stress response model

Other properties: “cya transcription is negatively regulated by the complex cAMP-CRP”

“DNA supercoiling decreases during transition to stationary phase”

Inconsistency between observation and prediction calls for model revision or model extension

Nutritional stress response model extended with global regulator RpoS

True (<1s)

Kawamukai et al., J. Bacteriol., 85

Balke and Gralla, J. Bacteriol., 87

False (<1s)

AG(xcrp > 3crp xcya > 3

cya xs > s → EF xcya < 0).

EF( (xgyrAB < 0 xtopA > 0) xrrn < rrn). .

Novel prediction of stress response model

Qualitative simulation of carbon upshift response: 1143 reachable domains (< 2s)

several strongly connected components

Are some strongly connected components attractors?

Attractor corresponds to damped oscillations towards stable equilibrium point: unexpected prediction

Experimental verification of model predictions

Time-series measurements of protein concentrations in parallel and at

high sampling rate using gene reporter system

AG(statesInSCCi → AG statesInSCCi) True (<1s, i=3)

Grognard et al., in preparation

Overview

I. Introduction








Summary

Development of automated and efficient method for testing whether predictions from qualitative models of genetic regulatory networks are consistent with experimental data on system dynamics

Use of discrete abstraction that yields predictions well-adapted to comparison with available experimental data

Combination of tailored symbolic analysis and model checking for verification of dynamical properties of hybrid models of large and complex networks

Biological relevance demonstrated on validation of models of networks of biological interest Batt et al., Bioinformatics, 05

Batt et al., IJCAI, 05

Batt et al., HSCC, 05

Discussion

Discrete abstractions used for analysis of continuous and hybrid models

symbolic reachability analysis of hybrid automata models more precise analysis of system dynamics need for complex decision procedures no treatment of discontinuities in vector field

qualitative simulation using qualitative differential equations more general class of model methods are not scalable

Model checking used for analysis of discrete models verification of properties of logical models

intuitive connection between underlying continuous dynamics and discrete representation

no explicit representation of dynamical phenomena at threshold concentrations

Ghosh and Tomlin,

Systems Biology, 04

Heidtke and Schulze-Kremer,

Bioinformatics, 98

Bernot et al.,

J. Theor. Biol., 04

Perspectives

Further integration of model-checking task into GNA

Property specification, verification, interpretation of diagnostics

Exploitation of advanced model-checking techniques

Partial order reduction, graph minimization, modular model checking, ...

Extensions of model validation

model inference: complete partially-specified models

model revision: modify inconsistent models

network design: find model satisfying set of design constraints

Thanks for your attention!

Documents

Grégory Batt Ph.D. defense -- under supervision of Hidde de Jong, Helix research group