Qualitative Simulation of the Carbon
Starvation Response in Escherichia coli
Delphine Ropers
INRIA Rhône-Alpes
655 avenue de l’Europe
Montbonnot, 38334 Saint Ismier CEDEX, France
Email: Delphine.Ropers@inrialpes.fr
Web: http://www-helix.inrialpes.fr/article593.html
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
2
Stress response in Escherichia coli
 Bacteria able to adapt to a variety of changing environmental
conditions
Heat shock
Nutritional stress
Osmotic stress
Cold shock
…
 Stress response in E. coli has been much studied
Model for understanding adaptation of pathogenic bacteria to their host
3
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)
>4h
time
4
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
5
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
protein
P
gene
fis
gyrAB
P
P1-P’1
P2
cya
promoter
FIS
GyrAB
CYA
DNA
supercoiling
cAMP•CRP
TopA
CRP
tRNA
rRNA
P1-P4
Signal (lack of carbon source)
topA
P1
P2
rrn
P1
P2
crp
Ropers et al. (2006),
Biosystems, in press
6
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
7
PL differential equation models
 Genetic networks modeled by class of differential equations
using step functions to describe regulatory interactions
.
.
x
s-(x, θ)
xa  a s-(xa , a2) s-(xb , b ) – a xa
b

b s-(xa ,
1
a1) – b xb
0
A
B
a

x
x : protein concentration
 : threshold concentration
 ,  : rate constants
b
 Differential equation models of regulatory networks are
piecewise-linear (PL)
Glass and Kauffman (1973), J. Theor. Biol., 39(1): 103-129
8
Qualitative analysis of network dynamics
 Analysis of the dynamics in phase space
 Phase space partition: unique derivative sign pattern in domains
 Qualitative abstraction yields state transition graph
maxb
.
D20
b/b D18
D16
b D10
D5: x. a > 0
xb < 0
D21
D19
D17
D11
D1
0
D21
D20
.
D15
D3
D5 D7
D9
D2
D4
D8
a1
D6
D27
a2 a/a maxa
D25
D27
D26
D19 D23
D
s-17(xaaxD,a22 a2) s-(xb , b ) – a xa
a–
s-(xbax,b Da112) – D13b xb D14 D15
Db11–
.
xDa16   

.
xDb10   

0 < a1 < a2 < a/a < maxa
D18
D24
D23 D25 D26
D22
D12 D13 D14
D24
D5
0 < b <D1 b/Db 3<2 max
4b
D
D
.
x >0
D1: .a
xb > 0
.
x >0
D5: .a
xb < 0

D9
D7
D6
D8
.
D7:
xa = 0
.
xb < 0
 Abstraction preserves unicity of derivative sign pattern
9
Validation of qualitative models
 Predictions well adapted to comparison with available
experimental data: changes of derivative sign patterns
D21
D24
D19
D23
D17
D22
D20
xa
D18
D16
0
D25
D26
D27
time
0
D11
D10
xb
D12
D13
D15
D14
Concistency?
.
.
x >0
xa > 0
b
.
.
x >0
xa < 0
b
time
Yes
D1
.
D1: x.a > 0
xb > 0
D5 D7
D4
D6
D3
D2
.
x <0
D17: . a
xb > 0
D9
.
D18:
D8
xa = 0
.
xb = 0
 Model validation: comparison of derivative sign patterns in
observed and predicted behavior
10
Genetic Network Analyzer (GNA)
 Qualitative simulation method implemented in Java: Genetic
Network Analyzer (GNA)
Integration into environment
for explorative genomics by
Genostar Technologies SA
de Jong et al. (2003) Bioinformatics
Batt et al. (2005), Bioinformatics
Page et al. (2006)
http://www-helix.inrialpes.fr/gna
11
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
?
sporulationgermination
cycle
metabolic and
environmental signals
12
Model of sporulation network
 Piecewise-linear model of network controlling sporulation
11 differential equations, with 59 inequality constraints
SinR/SinI
H A
spo0A
H
+
+
kinA
SinI
-
Spo0A
Signal
+
SinR
KinA
F
Spo0A˜P
-
-
H
A
sigH A
(spo0H)
sigF
abrB
-
A
A
+
-
-
+
-
AbrB
-
H
sinI
+
phosphorelay
Spo0E
spo0E
A
sinR
+
H
Hpr
A hpr (scoR)
de Jong, Geiselmann et al. (2004), Bull. Math. Biol., 66(2): 261-300
13
Model of carbon starvation network E. coli
 Carbon starvation network modeled by PL model
7 differential equations, with 36 inequality constraints
P
fis
gyrAB
P
P1-P’1
P2
cya
FIS
GyrAB
CYA
Supercoiling
Activation
Superhelical
density of DNA
CRP•cAMP
TopA
Signal (lack of carbon source)
CRP
tRNA
rRNA
P1-P4
topA
P1
P2
rrn
P1
P2
crp
Ropers et al. (2006),
BioSystems, in press
14
Modeling of rrn module
 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
Schneider et al. (2003), Curr. Opin. Microbiol., 6:151-156
Hill rate law:
FIS
frrnP1( xFIS ) =
P1 P2
rrn
( xFIS )n
( xFIS )n + Kon
Step-function approximation:
frrnP1 ( xFIS )  s+( xFIS , FIS )
stable RNAs
FIS
.
xrrn  rrn1 s+( xFIS , FIS ) + rrn2 – rrn xrrn
15
Modeling of CRP activation
 CRP activation in presence of carbon starvation signal
ATP + CYA*
cAMP + CRP
K1
K4
CYA*•ATP
k2
CYA* + cAMP
k3
CYA
degradation/export
CRP•cAMP
Activation
Signal
CRP•cAMP
CRP
P1
P2
crp
 Modeling of CRP activation using mass-action law
Quasi steady-state assumption simplifies model
xCRP•cAMP =
k2 xCYA xCRP
k2 xCYA + k3 K4
16
Modeling of crp activation by CRP·cAMP
 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
Barnard et al. (2004), Curr. Opin. Microbiol., 7:102-108
Rate law:
CYA
CRP•cAMP
Activation
( xCRP•cAMP )n
fcrpP2( xCRP•cAMP ) =
( xCRP•cAMP )n + Kon
Signal
CRP
P1
P2
crp
xCRP•cAMP =
CYA concentration (M)
CRP concentration (M)
k2 xCYA xCRP
k2 xCYA + k3 K4
Step-function approximation:
fcrpP2  s+(xCYA , CYA) s+(xCRP , CRP) s+(xSIGNAL , SIGNAL)
.
xcrp  crp1 + crp2 s+(xCYA , CYA1) s+(xCRP , CRP1) s+(xSIGNAL , SIGNAL) – crp xcrp
17
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
18
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
CRP
GyrAB
TopA
CYA
rrn
FIS
Signal
19
Insight into carbon starvation response
 Sequence of qualitative events leading to adjustment of
growth of cell after carbon starvation signal
Role of the mutual inhibition of FIS and CRP•cAMP
P
fis
gyrAB
P
P1-P’1
P2
cya
FIS
GyrAB
CYA
Supercoiling
Activation
Superhelical
density of DNA
CRP•cAMP
TopA
Signal (lack of carbon source)
CRP
tRNA
rRNA
P1-P4
topA
P1
P2
rrn
P1
P2
crp
20
Extension of carbon starvation network
 Model does not reproduce observed downregulation of negative
supercoiling
Missing component in the network?
P
P
fis
P1 P2nlpD
gyrAB
P1-P’1 P2
GyrI
cya
P
σS
FIS
GyrAB
CYA
Supercoiling
gyrI
Activation
TopA
P5 P1-P4
Ropers et al. (2006)
rpoS
Stress
signal
RssB
CRP
tRNA
rRNA
topA
P1 P2
P1 P2
rrn
crp
PA
rssA PB rssB
21
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
equilibrium
state
equilibrium
state
CRP
GyrAB
CYA
TopA
rrn
FIS
Signal
22
Insight into response to carbon upshift
 Sequence of qualitative events leading to adjustment of cell
growth after a carbon upshift
Role of the negative feedback loop involving Fis and DNA supercoiling
P
fis
gyrAB
P
P1-P’1
P2
cya
FIS
GyrAB
CYA
DNA
supercoiling
Activation
TopA
Signal (lack of carbon)
CRP
tRNA
rRNA
P1-P4
topA
P1
P2
rrn
P1
P2
crp
23
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
gene reporter
system on
plasmid
bla
gfp or lux
reporter
gene
ori
• Reporter gene expression reflects expression of gene of interest
24
Monitoring gene expression: population
 Integration of the gene reporter system into bacterial cell
Time-series measurement of
fluorescence or luminescence
E. coli
genome
Global
regulator
Reporter
gene
rrn GFP
GFP or
Luciferase
 Real-time measurement of reporter-gene expression in
bacterial population
25
Monitoring gene expression: single cell
 Integration of the gene reporter system into bacterial cell
Phase contrast
E. coli
genome
Fluorescence
Global
regulator
Reporter
gene
GFP or
Luciferase
Cts/cell
 Real-time measurement of reporter-gene expression in
gyrA GFP
individual bacteria
Mihalcescu et al. (2004), Nature, 430(6995):81-85
Time (min)
26
Work in progress
 Model predictions verified?
CRP
GyrAB
TopA
CYA
rrn
FIS
Signal
 We will know soon!
27
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 welladapted 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
28
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
29
30
Automated verification of properties
Use of model-checking techniques to verify
(observed) properties of dynamics of network
transition graph transformed into Kripke
structure
 properties expressed in temporal logic
.
.
There Exists a Future state where xa>0 and xb>0 and
.
.
starting from that state, there Exists a Future state where x =0 and x <0

.
x.a<0
x =0
QS8
QS7
b
.x <0
.xa>0
b
a
.
x.a=0
xb=0
QS6
QS5
.
.xxa>0
>0
.
.xxa>0
<0
b
.
.
.
b
.
EF(xa>0 Λ xb>0 Λ EF(xa=0 Λ xb<0))
.x =0
.a
Yes!
xb<0
b
QS1
QS2
QS3
QS4
Batt et al. (2005), Bioinformatics, 21(supp. 1): i19-i28
31
Analysis of attractors of PL systems
 Search of attractors of PL systems in phase space
Combinatorial, but efficient algorithms
maxb
 b2

b1
0
a1
a2
maxa
 Analysis of stability of attractors, using properties of state
Casey et al. (2005), J. Math. Biol., in press
transition graph
Definition of stability of equilibrium points on surfaces of discontinuity
32