Qualitative Simulation of the Carbon Starvation Response in Escherichia coli

Qualitative Simulation of the Carbon Starvation Response in Escherichia coli

Hidde de Jong1 Delphine Ropers1 Johannes Geiselmann1,2

1INRIA Grenoble-Rhône-Alpes2Laboratoire Adaptation Pathogénie des Microorganismes

CNRS UMR 5163Université Joseph Fourier

Email: [email protected]

2

Overview

1. Carbon starvation response of Escherichia coli

2. Modeling and simulation: objective and constraints

3. Qualitative modeling and simulation of carbon starvation

4. Experimental validation of carbon starvation model

5. Conclusions

3

Escherichia coli

Rocky Mountain Laboratories, NIAID, NIH2 µm

1 µm

4

Escherichia coli stress responses E. coli is able to adapt and respond to a variety of stresses in its environment

Model organism for understanding adaptation of pathogenic bacteria to their host

Nutritional stress

Osmotic stress

Heat shock

Cold shock

…

Storz and Hengge-Aronis (2000), Bacterial Stress Responses, ASM Press

5

Nutritional stress response in E. coli Response of E. coli to nutritional stress conditions: transition

from exponential phase to stationary phaseChanges in morphology, metabolism, gene expression, …

log (pop. size)

time

> 4 h

6

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

7

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, 84(2):124-52

protein

gene

promoter

8

Constraints on modeling and simulation Current constraints on modeling and simulation:

Knowledge on molecular mechanisms rare Quantitative information on kinetic parameters and molecular

concentrations absent

Possible strategies to overcome the constraints Parameter estimation from experimental data Parameter sensitivity analysis Model simplifications

Intuition: essential properties of system dynamics robust against moderate changes in kinetic parameters and rate laws

Stelling et al. (2004), Cell, 118(6):675-86

9

Qualitative modeling and simulation Qualitative modeling and simulation of large and complex

genetic regulatory networks using simplified DE models

Applications of qualitative simulation: initiation of sporulation in Bacillus subtilis

quorum sensing in Pseudomonas aeruginosa

onset of virulence in Erwinia chrysanthemi

Relation with discrete, logical models of gene regulation

de Jong, Gouzé et al. (2004), Bull. Math. Biol., 66(2):301-40Batt et al. (2007), Automatica, in press

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

Viretta and Fussenegger, Biotechnol. Prog., 2004, 20(3):670-678

Sepulchre et al., J. Theor. Biol., 2007, 244(2):239-57

Thomas and d’Ari (1990), Biological Feedback, CRC Press

10

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-affine (PA)

b

B

a

A

Glass and Kauffman (1973), J. Theor. Biol., 39(1):103-29

11

Analysis of dynamics of PA models in phase space

a10

maxb

a2

b

maxa

Mathematical analysis of PA models

xa a s-(xa , a2) s-(xb , b ) – a xa.xb b s-(xa , a1) – b xb .

a10

maxb

a2

b

maxaaa

bb xa a – a xa .xb b – b xb .

D1

12


a10

maxb

a2

b

maxa



xa a – a xa .xb – b xb .

a10

maxb

a2

b

maxaaa

D5

13


Extension of PA differential equations to differential inclusions using Filippov approach

a10

maxb

a2

b

maxa



a10

maxb

a2

b

maxa

D3

Gouzé, Sari (2002), Dyn. Syst., 17(4):299-316

14


Extension of PA differential equations to differential inclusions using Filippov approach

a10

maxb

a2

b

maxa



a10

maxb

a2

b

maxa

D7

Gouzé, Sari (2002), Dyn. Syst., 17(4):299-316

15

Phase space partition: unique derivative sign pattern in regions Qualitative abstraction yields state transition graph

Shift from continuous to discrete picture of network dynamics

a10

maxb

a2

b

maxa

Qualitative analysis of network dynamics

a10

maxb

a2

b

maxa .. .

.

..xa > 0

xb > 0xa > 0xb < 0

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

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

16

State transition graph invariant for parameter constraints


D1 D3

D11 D12

a10

maxb

a2

b

maxaa10

maxb

a2

b

maxaaa

bb

D1

D11 D12

D3

0 < a1 < a2 < a/a < maxa

0 < b < b/b < maxb

17



D1 D3

D11 D12 0 < a1 < a2 < a/a < maxa

0 < b < b/b < maxb

a10

maxb

a2

b

maxaa10

maxb

a2

b

maxaaa

bb

D1

D11 D12

D3

18



D1 D3

D11 D12 0 < a1 < a2 < a/a < maxa

0 < b < b/b < maxb

a10

maxb

a2

b

maxaa10

maxb

a2

b

maxaaa

bb

D1

D11 D12

D3

D1

D11

a10

maxb

a2

b

maxaa10

maxb

a2

b

maxaaa

bb

D1

D11 D12

D3

0 < a/a < a1 < a2 < maxa

0 < b < b/b < maxb

19

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 behaviors

Need for automated and efficient tools for model validation

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.

20

Compute state transition graph and express dynamic properties in temporal logic

Use of model checkers to verify whether experimental data and model predictions are consistent

Verification using model checking

Yes

Concistency?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

D8

D10

D16D17

D18

D20

D19

D21

D22

D23

D24

Batt et al. (2005), Bioinformatics, 21(supp. 1): i19-28

21

Analysis of stability of attractors, using properties of state transition graphDefinition of stability of equilibrium points on surfaces of discontinuity

D1 D3 D5 D7 D9

D15

D27D26D25

D11 D12 D13 D14

D2 D4 D6

D10

D16D17

D18

D20

D19

D21

D22

D23

D24

D8 a1

0

maxb

a2

b

maxaa10

maxb

a2

b

maxa

bb

Analysis of attractors of PA systems Search of attractors of PA systems in phase space

Casey et al. (2006), J. Math Biol., 52(1):27-56

22

Genetic Network Analyzer (GNA)

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

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

de Jong et al. (2003), Bioinformatics, 19(3):336-44

Distribution by Genostar SA

Batt et al. (2005), Bioinformatics, 21(supp. 1): i19-28

23

Analysis of carbon starvation response Modeling and experimental studies directed at understanding

how network controls carbon starvation response

Bottom-up strategy: 1) Initial model of carbon starvation response

rrnP1 P2

CRP

crp

cya

CYA

cAMP•CRP

FIS

TopA

topA

GyrAB

P1-P4P1 P2

P2P1-P’1

P

gyrABP


supercoiling

fis

tRNArRNA


protein

gene

promoter

24

Analysis of carbon starvation response

Bottom-up strategy: 1) Initial model of the carbon starvation response

Search and curate data available in the literature and databases

2) Experimental verification of model predictions

3) Extension of model to take into account wrong predictions

Additional global regulators: IHF, HNS, ppGpp, FNR, LRP, ArcA, …

Modeling and experimental studies directed at understanding how network controls carbon starvation response

25

Modular structure of carbon starvation network

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


Modeling of carbon starvation network

26

Modeling of carbon starvation network Can the initial model explain the carbon starvation response of E. coli cells?

Translation of biological data into a mathematical model

rrnP1 P2

CRP

crp

cya

CYA

cAMP•CRP

FIS

TopA

topA

GyrAB

P1-P4P1 P2

P2P1-P’1

P

gyrABP


supercoiling

fis

tRNArRNA

27

From nonlinear kinetic model to PA model Modeling process consists of reducing classical nonlinear

kinetic model to PA model

Nonlinear kinetic model

Nonlinear reduced kinetic model

Piecewise-linear model

Quasi-steady state approximation

Piecewise-linear approximation

28

Nonlinear kinetic model Nonlinear kinetic ODE model of 12 variables and 46 parameters

Regulation of gene expression (Hill)

FIS

rrnP1 P2

stable RNAs

29


Regulation of gene expression (Hill) Enzymatic reactions (Michaelis-Menten)

ATP + CYA*K1

CYA*•ATP CYA* + cAMPk2 k3

degradation/export

30


Regulation of gene expression (Hill) Enzymatic reactions (Michaelis-Menten) Formation of biochemical complexes (mass action)

cAMP + CRPK4

CRP•cAMP

31

Modeling of carbon starvation network

32

Identification of slow and fast processes in network

Quasi steady state approximation

rrnP1 P2

CRP

crp

cya

CYA

cAMP•CRP

FIS

TopA

topA

GyrAB

P1-P4P1 P2

P2P1-P’1

P

gyrABP


supercoiling

fis

tRNArRNA

33

Quasi steady state approximation

rrnP1 P2

CRP

crp

cya

CYA

cAMP•CRP

FIS

TopA

topA

GyrAB

P1-P4P1 P2

P2P1-P’1

P

gyrABP


supercoiling

fis

tRNArRNA

Heinrich and Schuster (1996), The Regulation of Cellular Systems, Chapman & Hall

34

Nonlinear reduced model QSSA model of 7 variables and 46 parameters

35

Piecewise-linear approximation Approximation of Hill function with step function Approximation of sigmoidal surfaces with product of step

functions

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

1) s+(xSIGNAL , SIGNAL)

CYA concentration (M) CRP concentration (M)

CRP• cAMP Activation

CRP

CYA

Signal

crpP1 P2

36

Model of carbon starvation network PADE model of 7 variables and 36 parameter inequalities

37

Attractors of stress response network Analysis of attractors of PA model: two steady states

• Stable steady state, corresponding to exponential-phase conditions

• Stable steady state, corresponding to stationary-phase conditions

38

Simulation of stress response network Simulation of transition from exponential to stationary phase

State transition graph with 27 states, 1 stable steady state

CYA

FIS

GyrAB

Signal

TopA

rrn

CRP

39

Insight into nutritional stress response Sequence of qualitative events leading to adjustment of

growth of cell after nutritional stress signal


rrnP1 P2

Activation

CRP

crp

cya

CYA

CRP•cAMP

FIS

TopA

topA

GyrAB

P1-P4P1 P2

P2P1-P’1

P

gyrABP

Signal (lack of nutrients)Supercoiling

fis

tRNArRNA

Role of the mutual inhibition of Fis and CRP•cAMP

40

Reporter gene systems

Use of reporter gene systems to monitor gene expression

promoter region

bla

ori

gfp or luxreporter

gene

cloning promoter regions on plasmid

Simulations yield predictions that cannot be verified with currently avaliable experimental data

rrnB

fis

crp

rpoS

topA

gyrB

gyrA

nlpD

41

Global regulator

GFP

E. coli genome

Reporter gene

Integration of fluorescent or luminescent reporter gene systems into bacterial cell

Monitoring of gene expression

excitation

emission

Expression of reporter gene reflects expression of host gene of interest

Global regulator

Luciferase

E. coli genome

Reporter operon

emission

42

Real-time monitoring: microplate reader Use of automated microplate reader to monitor in parallel in

single experiment expression of different reporter genes fluorescence/luminescent intensity absorbance (OD) of bacterial culture

Upshift experiments in M9/glucose medium

96-well microplate

Well withbacterial culture

Different gene reporter system in wells

43

Analysis of reporter gene expression data WellReader: program for analysis of reporter gene expression

data

44

Validation of carbon starvation response model

Translate expression profiles into temporal logic and verify properties by means of model checking

“Fis concentration decreases and becomes steady in stationary phase”

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

Geiselmann et al (2007), in preparation

45


Translate expression profiles into temporal logic and verify properties by means of model checking

“GyrA concentration decreases at the onset of stationary phase”

EF (xgyrA < 0 xrrn < rrn). False

Geiselmann et al (2007), in preparation

46

Suggestion of missing interaction Model does not reproduce observed downregulation of negative

supercoiling


rrnP1 P2

Activation

CRP

crp

cya

CYA

CRP•cAMP

FIS

TopA

topA

GyrAB

P1-P4P1 P2

P2P1-P’1

P

gyrABP

Signal (lack of nutrients)Supercoiling

fis

tRNArRNA

Missing interaction in the network?

47

Extension of stress response network

Activation Stress signal

CRP

crp

cya

CYA

fis

FIS

Supercoiling

TopA

topA

GyrAB

P1-P4 P1 P2

P2P1-P’1

rrnP1 P2

P

gyrABP

tRNArRNA

Ropers et al. (2006)

GyrI

gyrIP

rpoSP1 P2nlpD

σS

RssB

rssAPA PB rssB

P5

Missing component in the network?

Model does not reproduce observed downregulation of negative supercoiling

48

Perspectives Refining model validation by monitoring gene expression in

single cells Inference of regulatory networks from gene expression data

Use hybrid system identification methods adapted to PL models

Composite models of E. coli stress response on genetic and metabolic level

Integrated tools for model checking and qualitative simulation using high-level specification languagesPrerequisite for further upscaling

Collaboration with Irina Mihalcescu, Université Joseph Fourier

Drulhe et al. (2007), IEEE Trans. Autom. Control/Circ. Syst. I, in press

Collaboration with Daniel Kahn, INRA, and Jean-Luc Gouzé, INRIA

Pedro T. Monteiro, PhD thesis, IST and UCBL

49

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 dataCoarse-grained models and qualitative analysis of dynamics

Biological relevance attained through integration of modeling and experimentsModels guide experiments, and experiments stimulate models

50

Contributors and sponsorsGrégory Batt, Université Joseph Fourier, Grenoble, France

Bruno Besson, INRIA Rhône-Alpes, France

Hidde de Jong, INRIA Rhône-Alpes, France

Hans Geiselmann, Université Joseph Fourier, Grenoble, France

Jean-Luc Gouzé, INRIA Sophia-Antipolis, France

Radu Mateescu, INRIA Rhône-Alpes, 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

Ministère de la Recherche,

IMPBIO program European Commission,

FP6, NEST program INRIA, ARC program Agence Nationale de la

Recherche, BioSys program

51


Validation of model using model checking “Fis concentration decreases and becomes steady in stationary phase”

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

“DNA supercoiling decreases during transition to stationary phase”

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

TrueAG(xcrp > 3crp xcya > 3

cya xs > s → EF xcya < 0).

True

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

Ali Azam et al. (1999), J. Bacteriol., 181(20):6361-6370

Kawamukai et al. (1985), J. Bacteriol., 164(2):872-877

Balke, Gralla (1987), J. Bacteriol., 169(10):4499-4506

Documents

Qualitative Simulation of the Carbon Starvation Response in Escherichia coli