Qualitative Modeling and Simulation of Genetic Regulatory Networks

Hans Geiselmann1 and Hidde de Jong2

1Université Joseph Fourier, Grenoble 2INRIA Rhône-Alpes

Hans.Geiselmann@ujf-grenoble.fr

Hidde.de-Jong@inrialpes.fr

2

Overview

1. Introduction

2. Initiation of sporulation in Bacillus subtilis

3. Modeling and simulation of genetic regulatory networks

4. Stress adaptation in Escherichia coli

5. Conclusions and work in progress

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 analysis of hybrid systems

Method supported by computer tool GNA

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

Differential equation models of regulatory networks are piecewise-linear (PL)

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

Filippov generalization of PL differential equations

xb

xa

0

D5

(D5)

D3

(D3)

D4

(D4)

11

Qualitative state and state transition

Qualitative state consists 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

a6

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

No exact numerical values for parameters , , and , but two types of inequality constraints:

Qualitative PL model

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

14

Qualitative simulation

Qualitative simulation determines all qualitative states that are reachable from initial state through successive transitions

a1 maxa0

maxb

a6

b1

b2

QS1

D1

All quantitative PL models subsumed by qualitative PL model have same state transition graph

State transition graph derived from qualitative constraints on parameters

15

Genetic Network Analyzer (GNA) Qualitative simulation method implemented in Java 1.3:

Genetic Network Analyzer (GNA)

Graphical interface to control simulation and analyze results

16

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

-

-

17

Model of sporulation network

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

11 differential equations, with 59 parameter inequalities

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

With few exceptions, parameter inequalities are uniquely determined by biological data

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

18

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

19

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

20

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

21

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

Grossman, 1995; Hoch, 1993

Sporulation mutants disable positive or negative feedback loops

+ phospho- relay

Spo0A˜P

Spo0A-

Spo0Espo0E A

+

KinAkinA H +

+ sigFHF

+

22

Simulation of stress adaptation in E. coli Adaptation of E. coli to 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

Project inter-EPST « Validation de modèles de réseaux… »

ENS, Paris ; INRIA RA ; UJF, Grenoble

23

Data on stress adaptation

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

24

Stress adaptation network

Preliminary working model of the regulatory network controlling stress adaptation

fis

+Fis

crp

Crp

gyrAB

GyrAB

rpoS

RpoS

smbC

topA

rssB

RssB•p

SmbC

super-coiling

+ +

ATP

cAMP

¬Nut

− −

+

+p

RssB

Nut

Crp•cAMP

+−−

−−

−

+

−

ClpXP

GyrAB•SmbC

25

Evolution of stress adaptation network

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

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

wt crp Suppressor

26

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

27

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

Model editor, 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

28

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

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

Tewfik Sari Université de Haute Alsace, Mulhouse

Dominique Schneider Université Joseph Fourier, Grenoble

29

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.

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

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

Gouzé, J.-L. & T. Sari (2003), 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 (2002), Qualitative simulation of

genetic regulatory networks using piecewise-linear models, RR-4407, INRIA.

de Jong, H., J. Geiselmann, G. Batt, C. Hernandez & M. Page (2002), Qualitative simulation of the initiation

of sporulation in B. subtilis, RR-4527, INRIA.

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

30

Soundness of qualitative simulation Qualitative simulation is sound:

Given qualitative PL model and initial domain D0

Let X be set of solutions x(t) on [0, ] of quantitative PL models

corresponding to qualitative model, such that x(0) D0

Every x X covered by some path in transition graph

a1 maxa0

maxb

a2

b1

b2

QS1 QS2 QS3 QS4

QS5QS6

QS7

QS8