Qualitative Simulation of the Carbon Starvation Response in Escherichia coli

Delphine Ropers

INRIA Rhône-Alpes655 avenue de l’Europe

Montbonnot, 38334 Saint Ismier CEDEX, France

Email: Delphine.Ropers@inrialpes.fr Web: http://www-helix.inrialpes.fr/article593.html

2

Overview

1. Introduction: nutritional stress response in E. coli

2. Qualitative modeling and simulation of genetic regulatory

networks

3. Modeling of carbon starvation response in E. coli

4. Experimental validation of model predictions

5. Work in progress

3

Stress response in Escherichia coli

Bacteria able to adapt to a variety of changing environmental conditions

Nutritional stress

Osmotic stress

Heat shock

Cold shock

…

Stress response in E. coli has been much studied

Model for understanding adaptation of pathogenic bacteria to their host

4

Nutritional stress response in E. coli

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

Changes in morphology, metabolism, gene expression, …

log (pop. size)

time

> 4 h

5

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

large and complex genetic regulatory network

Cases et de Lorenzo (2005),

Nat. Microbiol. Rev., 3(2):105-118

No global view of functioning of network available, despite abundant knowledge on network components

6

Analysis of carbon starvation response Objective: modeling and experimental studies directed at

understanding how network controls nutritional stress response

First step: analysis of the carbon starvation response in E. coli

rrnP1 P2

CRP

crp

cya

CYA

cAMP•CRP

FIS

TopA

topA

GyrAB

P1-P4P1 P2

P2P1-P’1

P

gyrABP

Signal (lack of carbon source)

DNA supercoiling

fis

tRNArRNA Ropers et al. (2006),

Biosystems, in press

protein

gene

promoter

7

Qualitative modeling and simulation

Current constraints on modeling and simulation: Knowledge on molecular mechanisms rare

Quantitative information on kinetic parameters and molecular

concentrations absent

Method for qualitative simulation of large and complex genetic regulatory networks using coarse-grained models

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

Batt G. et al. (2005), Hybrid Systems: Computation and Control, LNCS 3414, 134-150.

Method used to simulate initiation of sporulation in Bacillus subtilis and quorum sensing of Pseudomonas aeruginosa

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

Viretta and Fussenegger (2004), Biotechnol. Prog., 20(3):670-8

8

PL differential equation models

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

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

Glass and Kauffman (1973), J. Theor. Biol., 39(1): 103-129

b

B

a

A

9

Analysis of the dynamics in phase space

Phase space partition: unique derivative sign pattern in domains

Qualitative abstraction yields state transition graph

Abstraction preserves unicity of derivative sign pattern

a10

maxb

a2

b

maxa

Qualitative analysis of network dynamics

xa a s-(xa , a2) s-(xb , b ) – a xa.

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

xb b – b xb .

.

.xa > 0xb < 0D5:

0 < a1 < a2 < a/a < maxa

0 < b < b/b < maxb

.

. ...

.xa > 0xb > 0

xa > 0xb < 0

xa = 0xb < 0D1: D5: D7:

a10

maxb

a2

b

maxaaa

bbbb

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 D3 D5 D7 D9

D15

D27D26D25

D11 D12 D13 D14

D2 D4 D6

D8

D10

D16D17

D18

D20

D19

D21

D22

D23

D24

10

Predictions well adapted to comparison with available experimental data: changes of derivative sign patterns

Model validation: comparison of derivative sign patterns in observed and predicted behavior

D1 D3 D5 D7 D9

D15

D27D26D25

D11 D12 D13 D14

D2 D4 D6

D8

D10

D16D17

D18

D20

D19

D21

D22

D23

D24

Validation of qualitative models

. .xa < 0xb > 0

xa > 0xb > 0

xa= 0xb= 0

.

. ..D1: D17: D18:

Concistency?

Yes0

xb

time

time0

xa

xa > 0.xb > 0.

xb > 0.xa < 0.

11

Genetic Network Analyzer (GNA)

de Jong et al. (2003) Bioinformatics

Batt et al. (2005), Bioinformatics

Page et al. (2006)

http://www-helix.inrialpes.fr/gna

Integration into environmentfor explorative genomics by Genostar Technologies SA

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

12

Initiation of sporulation in Bacillus subtilis

Validation of method by analysis of well-understood network

Control of initiation of sporulation in Bacillus subtilis

?division cycle

sporulation-germination

cycle

metabolic and environmental signals

13

Model of sporulation network Piecewise-linear model of network controlling sporulation

11 differential equations, with 59 inequality constraints

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

-

-

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

14

Model of carbon starvation network E. coli Carbon starvation network modeled by PL model

7 differential equations, with 36 inequality constraints

Superhelical density of DNA

rrnP1 P2

Activation

CRP

crp

cya

CYA

CRP•cAMP

FIS

TopA

topA

GyrAB

P1-P4P1 P2

P2P1-P’1

P

gyrABP

Signal (lack of carbon source)Supercoiling

fis

tRNArRNA

Ropers et al. (2006),BioSystems, in press

15

( xFIS )n + Kon

( xFIS )n

frrnP1( xFIS ) =

Hill rate law:

FIS

frrnP1 ( xFIS ) s+( xFIS , FIS )

Step-function approximation:

Modeling of rrn module

FIS

rrnP1 P2

stable RNAs

Regulatory mechanism of control by FIS at promoter rrn P1• FIS binds to multiple sites in promoter region• FIS forms a cooperative complex with RNA polymerase

.xrrn rrn1 s+( xFIS , FIS ) + rrn

2 – rrn xrrn

Schneider et al. (2003), Curr. Opin. Microbiol., 6:151-156

16

ATP + CYA*K1

CYA*•ATP CYA* + cAMP

cAMP + CRPK4

k2

CRP•cAMP

k3degradation/export

Modeling of CRP activation

CRP•cAMP Activation

CRP

CYA

Signal

crpP1 P2

CRP activation in presence of carbon starvation signal

Modeling of CRP activation using mass-action law

Quasi steady-state assumption simplifies model

k2 xCYA + k3 K4

k2 xCYA xCRP

xCRP•cAMP =

17

Regulatory mechanism of control by CRP•cAMP at crp P2 • CRP•cAMP binds to a single site• CRP•cAMP forms a cooperative complex with RNA polymerase

Modeling of crp activation by CRP·cAMP

Barnard et al. (2004), Curr. Opin. Microbiol., 7:102-108

CRP•cAMP Activation

CRP

CYA

Signal

crpP1 P2 CYA concentration (M) CRP concentration (M)

( xCRP•cAMP )n + Kon

( xCRP•cAMP )n

fcrpP2( xCRP•cAMP ) =

Rate law:

k2 xCYA + k3 K4

k2 xCYA xCRP

xCRP•cAMP =

Step-function approximation:fcrpP2 s+(xCYA , CYA) s+(xCRP , CRP) s+(xSIGNAL , SIGNAL)

xcrp crp1 + crp

2 s+(xCYA , CYA1) s+(xCRP , CRP

1) s+(xSIGNAL , SIGNAL) – crp xcrp.

18

Simulation of stress response network

Qualitative analysis of attractors: two equilibrium states

• Stable state, corresponding to exponential-phase conditions

• Stable state, corresponding to stationary-phase conditions

19

Simulation of stress response network

Simulation of transition from exponential to stationary phase

State transition graph with 27 states generated in < 1 s, 1 stable equilibrium state

CYA

FIS

GyrAB

Signal

TopA

rrn

CRP

20

Insight into carbon starvation response

Sequence of qualitative events leading to adjustment of growth of cell after carbon starvation signal

Superhelical density of DNA

rrnP1 P2

Activation

CRP

crp

cya

CYA

CRP•cAMP

FIS

TopA

topA

GyrAB

P1-P4P1 P2

P2P1-P’1

P

gyrABP

Signal (lack of carbon source)Supercoiling

fis

tRNArRNA

Role of the mutual inhibition of FIS and CRP•cAMP

21

Extension of carbon starvation network

Ropers et al. (2006)

Missing component in the network?

Model does not reproduce observed downregulation of negative supercoiling

Activation Stress signal

CRP

crp

cya

CYA

fis

FIS

Supercoiling

TopA

topA

GyrAB

P1-P4P1 P2

P2P1-P’1

rrnP1 P2

P

gyrABP

tRNArRNA

GyrI

gyrIP

rpoSP1 P2nlpD

σS

RssB

rssAPA PB rssB

P5

22

Simulation of response to carbon upshift Simulation of transition from stationary to exponential phase after carbon upshift

State transition graph with 300 states generated in < 1 s, qualitative cycle

CYA

FIS

CRP

GyrAB

Signal

TopA

rrn

equilibrium state

equilibrium state

23

Insight into response to carbon upshift

Sequence of qualitative events leading to adjustment of cell growth after a carbon upshift

rrnP1 P2

CRP

crp

cya

CYA

Activation

FIS

TopA

topA

GyrAB

P1-P4P1 P2

P2P1-P’1

P

gyrABP

Signal (lack of carbon)

DNA supercoiling

fis

tRNArRNA

Role of the negative feedback loop involving Fis and DNA supercoiling

24

Experimental validation of model predictions Simulations yield novel predictions that call for experimental verification

Comparison with observed qualitative evolution of protein concentrations

Monitoring gene expression by means of gene reporter system

• Reporter gene under control of promoter region of gene of interest

promoter region

bla

ori

gfp or luxreporter

gene

gene reporter system on

plasmid

• Reporter gene expression reflects expression of gene of interest

25

Global regulator

GFP or Luciferase

E. coli genome

Reporter gene

Integration of the gene reporter system into bacterial cell

Monitoring gene expression: population

Real-time measurement of reporter-gene expression in bacterial population

Time-series measurement of fluorescence or luminescence

rrn GFP

26

Integration of the gene reporter system into bacterial cell

Real-time measurement of reporter-gene expression in individual bacteria

Monitoring gene expression: single cell

Phase contrast FluorescenceGlobal regulator

GFP or Luciferase

E. coli genome

Reporter gene

Mihalcescu et al. (2004), Nature, 430(6995):81-85

Cts

/ce

llTime (min)

gyrA GFP

27

Work in progress

Model predictions verified?

We will know soon!

CYA

FIS

GyrAB

Signal

TopA

rrn

CRP

28

Conclusions

Understanding of functioning and development of living organisms requires analysis of genetic regulatory networks

From structure to behavior of networks

Need for mathematical methods and computer tools well-adapted to available experimental data

Coarse-grained models and qualitative analysis of dynamics

Biological relevance attained through integration of modeling and experiments

Models guide experiments, and experiments stimulate models

29

Contributors

Grégory Batt, INRIA Rhône-Alpes, France

Danielle Bonaccio, Université Joseph Fourier, Grenoble, France

Hidde de Jong, INRIA Rhône-Alpes, France

Hans Geiselmann, Université Joseph Fourier, Grenoble, France

Jean-Luc Gouzé, INRIA Sophia-Antipolis, France

Irina Mihalcescu, Université Joseph Fourier, Grenoble, France

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

Corinne Pinel, Université Joseph Fourier, Grenoble, France

Delphine Ropers, INRIA Rhône-Alpes, France

Tewfik Sari Université de Haute Alsace, Mulhouse, France

Dominique Schneider Université Joseph Fourier, Grenoble, France

30

31

Automated verification of properties

Use of model-checking techniques to verify (observed) properties of dynamics of network

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!

transition graph transformed into Kripke structure

Batt et al. (2005), Bioinformatics, 21(supp. 1): i19-i28

properties expressed in temporal logic

32

Casey et al. (2005), J. Math. Biol., in press

Analysis of stability of attractors, using properties of state transition graph

Definition of stability of equilibrium points on surfaces of discontinuity

Analysis of attractors of PL systems

Search of attractors of PL systems in phase space

Combinatorial, but efficient algorithms

a1 maxa0

maxb

a2

b1

b2