Qualitative Modeling and Simulation of Genetic Regulatory Networks

Hidde de Jong

Projet HELIXInstitut National de Recherche en Informatique et en Automatique

Unité de Recherche Rhône-Alpes655, avenue de l’Europe

Montbonnot, 38334 Saint Ismier CEDEX

Email: Hidde.de-Jong@inrialpes.fr

2

Overview

1. Introduction

2. Modeling and simulation of genetic regulatory networks

3. Genetic Network Analyzer (GNA)

4. Applications

Initiation of sporulation in Bacillus subtilis

Nutritional stress response in Escherichia coli

5. Validation of models of genetic regulatory networks

6. Conclusions

3

Life cycle of Bacillus subtilis

B. subtilis can sporulate when the environmental conditions become unfavorable

?division cycle

sporulation-germination

cycle

metabolic and environmental signals

4

Regulatory interactions

AbrB

- AbrB represses sin operon

SinR~SinI

SinI inactivates SinR Spo0A˜P

+

Spo0A~P activates sin operon

sinR A HsinI

SinRSinI

sin operon

Quantitative information on kinetic parameters and molecular concentrations is usually not available

5

Genetic regulatory network of B. subtilis Reasonably complete genetic regulatory network controlling

the initiation of sporulation in B. subtilis

Genetic regulatory network is large and complex

protein

gene

promoter

kinA

-

+

HKinA

+ phospho- relay

Spo0A˜P

+

Spo0A

H A

A H

spo0A-

sinR sinI

SinISinR

SinR/SinI

-

sigF H

+

+

hpr (scoR)A

A AabrB

-

-

HprAbrB

spo0E A

sigH(spo0H)

A

-

-

-Spo0E

H

F

-

+

+Signal

-

-

6

Qualitative modeling and simulation

Computer support indispensable for dynamical analysis of genetic regulatory networks: modeling and simulation

precise and unambiguous description of network

systematic derivation of behavior predictions

Method for qualitative simulation of large and complex genetic regulatory networks

Method exploits related work in a variety of domains: mathematical and theoretical biology qualitative reasoning about physical systems control theory and hybrid systems

de Jong, Gouzé et al. (2004), Bull. Math. Biol., 66(2):301-340

7

PL models of genetic regulatory networks

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

b

-

B

a

-

A

- -

xa a s-(xa , a2) s-(xb , b1 ) – a xa .

xb b s-(xa , a1) s-(xb , b2 ) – b xb .

x : protein concentration

, : rate constants : threshold concentration

x

s-(x, θ)

0

1

Differential equation models of regulatory networks are piecewise-linear (PL) Glass, Kauffman (1973), J. Theor. Biol., 39(1):103-129

8

Domains in phase space

Phase space divided into domains by threshold planes

Different types of domains: regulatory and switching domains

Switching domains located on threshold plane(s)

xb

xa

a1maxa0

maxb

a2

b

1

b

2

.

.

.

.

9

In every regulatory domain D, system monotonically tends towards target equilibrium set (D)

Analysis in regulatory domains

xb

xa

a1maxa0

maxb

a2

b

1

b

2

xa a s-(xa , a2) s-(xb , b1 ) – a xa .

xb b s-(xa , a1) s-(xb , b2 ) – b xb .

D3

xa a– a xa .xb – b xb .

model in D3 :model in D1 :

D1

xa a– a xa .xb b – b xb .

(D1) {(a /a , b /b )}

(D1)

(D3)

(D3) {(a /a , 0 )}

10

Analysis in switching domains

In every switching domain D, system either instantaneously traverses D, or tends towards target equilibrium set (D)

D and (D) located in same threshold hyperplane

xb

xa

0

(D1)

(D3)

D1 D3D2

xb

xa

0

D5

(D5)

D3

(D3)

D4

(D4)

Filippov generalization of PL differential equationsGouzé, Sari (2002), Dyn. Syst., 17(4):299-316

11

Qualitative state and state transition

Qualitative state is discrete abstraction, consisting of domain D and relative position of target equilibrium set (D)

a1 maxa0

maxb

a2

b1

b2

QS1 D1, {(1,1)}

Transition between qualitative states associated with D and D', if trajectory starting in D reaches D'

(D1)

D1

D2 D3

QS3QS2QS1

12

State transition graph

Closure of qualitative states and transitions between qualitative states results in state transition graphTransition graph contains qualitative equilibrium states and/or cycles

a1 maxa0

maxb

a2

b1

b2

D2 D3 D4

D7

D5

D6

D1

D8 D9 D10

D11 D12 D13 D14 D15

D16 D17 D18

D24

D20

D21 D22 D23

D19

D25

QS3QS2QS1 QS4 QS5

QS10

QS15

QS20

QS25QS24QS23QS22QS21

QS16

QS11

QS6

QS7

QS12

QS17 QS18

QS19

QS13

QS14

QS8 QS9

13

a1 maxa0

maxb

a2

b1

b2

Robustness of state transition graph

State transition graph, and hence qualitative dynamics, is dependent on parameter values

(D1)

D2

D6 D7

QS6

QS2QS1

QS7

D1

a1 maxa0

maxb

a2

b1

b2

(D1)

D1

QS6

QS1

D2

D6 D7

14

Same state transition graph obtained for two types of inequality constraints on parameters , , and :

Inequality constraints

Ordering of threshold concentrations of proteins a1 a2 maxa

xb

xa

a1maxa0

maxb

a2

b

1

b

2

b1 b2 maxb

Ordering of target equilibrium values w.r.t. threshold concentrationsa2 a / a maxa b2 b / b maxb

maxa

xb

xa

a10

maxb

a2

b

1

b

2

a /a

b /b

15

Qualitative simulation

Given qualitative PL model, qualitative simulation determines all qualitative states that are reachable from initial state through successive transitions

QS1a1 maxa0

maxb

a6

b1

b2

D1

PL model supplemented with inequality constraints results in qualitative PL model

b1

maxbxb

QS1

b2

QS2 QS3 QS 4

a1

maxaxa

QS1

a2

QS2 QS3 QS 4QS2 QS3 QS 4

16

Genetic Network Analyzer (GNA) Qualitative simulation method implemented in Java 1.4:

Genetic Network Analyzer (GNA)

Graphical interface to control simulation and analyze results

de Jong et al. (2003), Bioinformatics, 19(3):336-344

17

Simulation of sporulation in B. subtilis Simulation method applied to analysis of regulatory network

controlling the initiation of sporulation in B. subtilis

kinA

-

+

HKinA

+ phospho- relay

Spo0A˜P

+

Spo0A

H A

A H

spo0A-

sinR sinI

SinISinR

SinR/SinI

-

sigF H

+

+

hpr (scoR)A

A AabrB

-

-

HprAbrB

spo0E A

sigH(spo0H)

A

-

-

-Spo0E

H

F

-

+

+Signal

-

-

18

Model of sporulation network

Essential part of sporulation network has been modeled by qualitative PL model:

11 differential equations, with 59 inequality constraints

Most interactions incorporated in model have been characterized on genetic and/or molecular level

With few exceptions, inequality constraints are uniquely determined by biological data

If several alternative constraints are consistent with biological data, every alternative considered

de Jong, Geiselmann et al. (2004), Bull. Math. Biol., 66(2):261-300

19

Simulation of sporulation network

Simulation of network under under various physiological conditions and genetic backgrounds gives results consistent with observations

Sequences of states in transition graphs correspond to sporulation (spo+)

or division (spo –) phenotypes

82 states

initial state

division state

20

s12s6 s7s1 s2 s3 s4 s5 s8 s9 s10 s11

s13

ka1

ka3

maxka

KinA

s12s6 s7s1 s2 s3 s4 s5 s8 s9 s10 s11

s13

se1

se3

maxse

Spo0E

s12s6 s7s1 s2 s3 s4 s5 s8 s9 s10 s11

s13

ab1

maxab AbrB

Simulation of sporulation network

Behavior can be studied in detail by looking at transitions between qualitative states

Predicted qualitative temporal evolution of protein concentrations

initial state

division state

s1 s2

s3

s4

s5s6

s7 s8 s9

s10

s11

s12s13

initial state

division state

21

Sporulation vs. division behaviors

s12s6 s7s1 s2 s3 s4 s5 s8 s9 s10 s11

s13

ka1

ka3

maxka

KinA

s12s6 s7s1 s2 s3 s4 s5 s8 s9 s10 s11

s13

se1

se3

maxse

Spo0E

s12s6 s7s1 s2 s3 s4 s5 s8 s9 s10 s11

s13

ab1

maxab AbrB

s12s6 s7s4 s8 s9 s10 s11s13

s1 s2 s3 s5

maxf SigF

maxsi

s12s6 s7s1 s2 s3 s4 s5 s8 s9 s10 s11

s13

si1 SinI

s24 s25s1 s2 s21 s22 s23 s8

ka1

ka3

maxka

KinA

se1

se3

maxse

s24 s25s1 s2 s21 s22 s23 s8

Spo0E

s1 s24 s25s2 s21 s22 s23 s8

SinIsi1

maxsi

s1 s24 s25s2 s21 s22 s23 s8

SigFmaxf

s1 s24 s25s2 s21 s22 s23 s8

AbrBab1

maxab

22

Analysis of simulation results

Qualitative simulation shows that initiation of sporulation is outcome of competing positive and negative feedback loops regulating accumulation of Spo0A~P

Sporulation mutants disable positive or negative feedback loops

+ phospho- relay

Spo0A˜P

Spo0A-

Spo0Espo0E A

+

KinAkinA H +

+ sigFHF

+

Grossman (1995), Ann. Rev. Genet., 29:477-508

Hoch (1993), Ann. Rev. Microbiol., 47:441-465

23

Nutritional stress response in E. coli Response of E. coli to nutritional stress conditions controlled by

network of global regulators of transcription

Fis, Crp, H-NS, Lrp, RpoS,…

Network only partially known and no global view of its functioning available

Computational and experimental study directed at understanding of:

How network controls gene expression to adapt cell to stress conditions

How network evolves over time to adapt to environment

Projects: inter-EPST, ARC INRIA, and ACI IMPBio

ENS, Paris ; INRIA ; UJF, Grenoble ; UHA, Mulhouse

24

Data on stress response

Gene transcription changes dramatically when the network is perturbed by a mutation

Small signaling molecules participate in global regulation mechanisms (cAMP, ppGpp, …)

The superhelical density of DNA modulates the activity of many bacterial promoters

fis-

to

pA

-

wt

fis-

top

A-

k2k2

to

pA

+

k20

25

Draft of stress response network

CRP

Stable RNAs

Supercoiling Activation

Stress signal

Fis fis P

rrnP1 P2

nlpD1nlpD2

σS

rpoSP1

RssB

rssBP

ClpXP P

crpP1 P2

topA Px1P1

cyaP1/P1’ P3

gyrAB

gyrI

Cya

P

P

GyrAB

GyrI

TopA

Laget et al. (2004)

26

Evolution of stress response network

Stress response network evolves rapidly towards optimal adaptation to a particular environment

Small changes of the regulatory network have large effects on gene expression

wt crp Suppressor

27

Validation of network models Bottleneck of qualitative simulation: visual inspection of large

state transition graphs

Goal: develop a method that can test if state transition graph satisfies certain properties

Is transition graph consistent with observed behavior?

Model checking is automated technique for verifying that finite state system satisfies certain properties

Computer tools are available to perform automated, efficient and reliable model checking (NuSMV)

Clarke et al. (1999), Model Checking, MIT Press

28

Model checking Use of model-checking techniques

transition graph transformed into Kripke structure properties expressed in temporal logic

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

.

xa<0xb=0

.

.

xa<0xb>0.

xa>0xb<0.

xa=0xb<0

.

.xa>0xb>0

.

.

xa=0xb=0

.

.

QS1 QS3 QS4

QS5

QS7

QS8

QS6

QS2

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

..

..

Yes!

29

Summary of approach

Test validity of B. subtilis sporulation models

EF(xhpr>0 Λ EF EG(xhpr=0)). .

“ [The expression of the gene hpr] increase in proportion of the growth curve, reached a

maximum level at the early stationary phase [(T1)] and remained at the same level during the

stationary phase” (Perego and Hoch, 1988)

-

-Signal

model checking

Kripke structure temporal logic

Batt et al. (2004), SPIN-04, LNCS

30

Conclusions

Implemented method for qualitative simulation of large and complex genetic regulatory networks

Method based on work in mathematical biology and qualitative reasoning

Method validated by analysis of regulatory network underlying initiation of sporulation in B. subtilis

Simulation results consistent with observations

Method currently applied to analysis of regulatory network controlling stress adaptation in E. coli

Simulation yields predictions that can be tested in the laboratory

31

Work in progress

Validation of models of regulatory networks using gene expression data

Model-checking techniques

Search of attractors in phase space and determination of their stability

Further development of computer tool GNA

Connection with biological knowledge bases, …

Study of bacterial regulatory networks

Sporulation in B. subtilis, phage Mu infection of E. coli, signal

transduction in Synechocystis, stress adaptation in E. coli

32

Contributors

Grégory Batt INRIA Rhône-Alpes

Hidde de Jong INRIA Rhône-Alpes

Hans Geiselmann Université Joseph Fourier, Grenoble

Jean-Luc Gouzé INRIA Sophia-Antipolis

Céline Hernandez INRIA Rhône-Alpes, now at SIB, Genève

Eva Laget INRIA Rhône-Alpes and INSA Lyon

Michel Page INRIA Rhône-Alpes and Université Pierre Mendès France, Grenoble

Delphine Ropers INRIA Rhône-Alpes

Tewfik Sari Université de Haute Alsace, Mulhouse

Dominique Schneider Université Joseph Fourier, Grenoble

33

Referencesde Jong, H. (2002), Modeling and simulation of genetic regulatory systems: A literature review, J. Comp.

Biol., 9(1):69-105.

de Jong, H., J. Geiselmann & D. Thieffry (2003), Qualitative modelling and simulation of developmental

regulatory networks, On Growth, Form, and Computers, Academic Press,109-134.

Gouzé, J.-L. & T. Sari (2002), A class of piecewise-linear differential equations arising in biological

models, Dyn. Syst., 17(4):299-316.

de Jong, H., J.-L. Gouzé, C. Hernandez, M. Page, T. Sari & J. Geiselmann (2004), Qualitative simulation of

genetic regulatory networks using piecewise-linear models, Bull. Math. Biol., 66(2):301-340.

de Jong, H., J. Geiselmann, C. Hernandez & M. Page (2003), Genetic Network Analyzer: Qualitative

simulation of genetic regulatory networks, Bioinformatics,19(3):336-344.

de Jong, H., J. Geiselmann, G. Batt, C. Hernandez & M. Page (2004), Qualitative simulation of the initiation

of sporulation in B. subtilis, Bull. Math. Biol., 66(2):261-340.

GNA web site: http://www-helix.inrialpes.fr/article122.html