Qualitative Modeling and Simulation of
Genetic Regulatory Networks
Hidde de Jong
Projet HELIX
Institut National de Recherche en Informatique et en Automatique
Unité de Recherche Rhône-Alpes
655, avenue de l’Europe
Montbonnot, 38334 Saint Ismier CEDEX
Email: Hidde.de-Jong@inrialpes.fr
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
2
Life cycle of Bacillus subtilis
 B. subtilis can sporulate when the environmental conditions
become unfavorable
division
cycle
?
sporulationgermination
cycle
metabolic and
environmental signals
3
Regulatory interactions
 Different types of interactions between genes, proteins, and
small molecules are involved in the regulation of sporulation
in B. subtilis
AbrB
SinR~SinI
SinR
Spo0A~P activates sin operon
SinI
sinR
A
sinI
sin operon
H
+
AbrB represses sin operon
Spo0A˜P
SinI inactivates SinR
 Quantitative information on kinetic parameters and
molecular concentrations is usually not available
4
Genetic regulatory network of B. subtilis
 Reasonably complete genetic regulatory network controlling
the initiation of sporulation in B. subtilis
SinR/SinI
H A
spo0A
H
+
+
kinA
SinI
-
Spo0A
Signal
+
SinR
KinA
phosphorelay
-
protein
F
Spo0A˜P
-
spo0E
promoter
H
A
sigH A
(spo0H)
sigF
abrB
-
A A
+
-
-
+
-
AbrB
-
H
sinI
+
Spo0E
gene
A
sinR
+
H
Hpr
A hpr (scoR)
 Genetic regulatory network is large and complex
5
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
de Jong, Gouzé et al. (2004), Bull. Math. Biol., 66(2):301-340
 Method exploits related work in a variety of domains:



mathematical and theoretical biology
qualitative reasoning about physical systems
control theory and hybrid systems
6
PL models of genetic regulatory networks
 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 , b1 ) – a xa
b

b s-(xa ,
a1) s-(xb ,
1
b2 ) – 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, Kauffman (1973), J. Theor. Biol., 39(1):103-129
7
Domains in phase space
 Phase space divided into domains by threshold planes
maxb
xb
 b2
 b1
0
.
.
.
.
a1
a2
maxa
xa
 Different types of domains: regulatory and switching domains
Switching domains located on threshold plane(s)
8
Analysis in regulatory domains
 In every regulatory domain D, system monotonically tends
towards target equilibrium set  (D)
maxb
xb
 (D1)
 b2
b
 b1
D3
 (D3)
D1
0
.
.
x
model in D31 : xa  a– a xa
a1
a2

–b–b xb xb
 (D31)  {(a /a , 0b)}/b )}
maxa
xa
.
.
x
xa  a s-(xa , a2) s-(xb , b1 ) – a xa
b
 b s-(xa , a1) s-(xb , b2 ) – b xb
9
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
 (D1)
xb
D1
D2
xb
D3
D3
D4
D5
 (D5)
0
 (D3)
0
 (D4)  (D3)
xa
xa
 Filippov generalization of PL differential equations
Gouzé, Sari (2002), Dyn. Syst., 17(4):299-316
10
Qualitative state and state transition
maxb
 (D1)
 b2

D2
b1
D1
0
a1
D3
a2
2
1 1  DQS
1, {(1,1)}
QS3
QSQS
maxa
 Qualitative state is discrete abstraction, consisting of domain
D and relative position of target equilibrium set  (D)
 Transition between qualitative states associated with D
and D', if trajectory starting in D reaches D'
11
State transition graph
QS21
maxb
 b2

QS17
D21 D22 D23 D24 D25
D16 D17 D18 D19 D20
QS16
D15
QS11
D11
D12
D13
D14
0
a1
a2
QS7
QS6
maxa
QS2
QS20
QS19
QS14
QS15
QS13
QS8
QS1
QS25
QS18
QS12
D6 D7 D8 D9 D10
D1 D2 D3 D4 D5
b1
QS22 QS23 QS24
QS3
QS9
QS4
QS10
QS5
 Closure of qualitative states and transitions between qualitative
states results in state transition graph
Transition graph contains qualitative equilibrium states and/or cycles
12
Robustness of state transition graph
 State transition graph, and hence qualitative dynamics, is
dependent on parameter values
maxb
 (D1)
maxb  (D1)
 b2
 b2

D6 D7

D6 D7
b1
D2
b1
D2
D1
0
a1
a2
QS6
QS7
QS1
QS2
D1
maxa
0
a1
a2
maxa
QS6
QS1
13
Inequality constraints
 Same state transition graph obtained for two types of
inequality constraints on parameters , , and  :

Ordering of threshold concentrations of proteins
0 < a1 < a2 < maxa

0 < b1 < b2 < maxb
Ordering of target equilibrium values w.r.t. threshold concentrations
a2 < a / a < maxa
b2 < b / b < maxb
maxb
maxb
b /b
 b2
xb
 b1
0
xb
a1
a2
 b2
 b1
0
maxa
xa
a1
a2
maxa
a /a
xa
14
Qualitative simulation
 PL model supplemented with inequality constraints results in
qualitative PL model
max
b
b2
b1
maxb
 b2
QS1 QS2
maxa

a2
a1
D1
b1
0
xb
a1
a6
maxa
QS1
QS2
QS3
QS 4
QS3 QS 4
xa
QS1 QS2
QS3 QS 4
 Given qualitative PL model, qualitative simulation
determines all qualitative states that are reachable from initial
state through successive transitions
15
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
16
Simulation of sporulation in B. subtilis
 Simulation method applied to analysis of regulatory network
controlling the initiation of sporulation in B. subtilis
SinR/SinI
H A
spo0A
H
+
+
kinA
SinI
-
Spo0A
Signal
+
SinR
KinA
phosphorelay
-
F
Spo0A˜P
-
spo0E
H
sigH A
(spo0H)
sigF
abrB
-
A A
+
-
-
+
-
AbrB
-
H
sinI
+
Spo0E
A
A
sinR
+
H
Hpr
A hpr (scoR)
17
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
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
division state
initial state
82 states
19
Simulation of sporulation network
 Behavior can be studied in detail by looking at transitions
between qualitative states
Predicted qualitative temporal evolution of protein concentrations
maxka
ka3
ka1
s3 se
max
initial state
s1
se3
s2
s5
s6
s7
s
8
KinA
s4
s 1 s 2 s 3 s 4 s 5 s 6 s 7 s8
s9
s9 s10 s11 s12s10
s13
s11 division state
s12
Spo0E
s13
se1
s1 s2 s3 s4 s 5 s6 s7 s8
maxab
ab1
s9 s10 s11 s12 s13
AbrB
s1 s2 s3 s4 s 5 s6 s7 s8
s9 s10 s11 s12 s13
20
Sporulation vs. division behaviors
maxka
maxka
ka3
KinA
ka1
maxse
se3
s1 s2 s 3 s4 s5 s6 s7 s8
s9 s10 s11 s12 s13
Spo0E
maxf
maxsi
si1
maxse
s1 s2 s21 s22 s23 s24 s25 s8
se3
Spo0E
se1
s1 s2 s 3 s4 s5 s6 s7 s8
ab1
KinA
ka1
se1
maxab
ka3
s1 s2 s21 s22 s23 s24 s25 s8
s9 s10 s11 s12 s13
maxab
AbrB
AbrB
ab1
s 1 s 2 s 3 s 4 s 5 s 6 s 7 s8
s1 s2 s21 s22 s23 s24 s25 s8
s9 s10 s11 s12 s13
maxf
SigF
s 1 s 2 s 3 s4 s 5 s 6 s 7 s 8
s9 s10 s11 s12 s13
si1
SinI
s 1 s 2 s 3 s 4 s5 s 6 s 7 s 8
maxsi
s9 s10 s11 s12 s13
SigF
s1 s2 s21 s22 s23 s24 s25 s8
SinI
s1 s2 s21 s22 s23 s24 s25 s8
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), Ann. Rev. Genet., 29:477-508
Hoch (1993), Ann. Rev. Microbiol., 47:441-465
KinA
+
+
Spo0A
Spo0E
phosphorelay
kinA
H
+
Spo0A˜P
-
+
+
spo0E
H
sigF
F
A
 Sporulation mutants disable positive or negative feedback loops
22
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
23
Data on stress response
 Gene transcription changes dramatically
when the network is perturbed by a
mutation
 The superhelical density of DNA
modulates the activity of many bacterial
promoters
wt
fistopAk2
k2 topA+
fis- topAk20
 Small signaling molecules participate in
global regulation mechanisms (cAMP,
ppGpp, …)
24
Draft of stress response network
CRP
Fis
P
fis
crp
P1 P2
Activation
Supercoiling
P1/P1’ P3
TopA
GyrAB
P
Cya
cya
Stress
signal
gyrAB
topA P1 Px1
RssB
GyrI
gyrI
P
P
σS
Stable RNAs
nlpD1
nlpD2
P1 P2
rrn
P1
ClpXP
rssB
P
rpoS
Laget et al. (2004)
25
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
26
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
Clarke et al. (1999), Model Checking, MIT Press
 Computer tools are available to perform automated, efficient
and reliable model checking (NuSMV)
27
Model checking
Use of model-checking techniques


QS8
transition graph transformed into Kripke structure
properties expressed in temporal logic
.
.
.
x.a<0
There Exists a Future state where xa>0 and xb>0 and
.
.
xb=0
starting from that state, there Exists a Future state where x =0 and x <0
a
QS7
.x <0
.xa>0
.
.
.
b
.
EF(xa>0 Λ xb>0 Λ EF(xa=0 Λ xb<0))
b
.
x.a=0
xb=0
QS6
QS5
.
.xxa>0
>0
.
.xxa>0
<0
b
.x =0
.a
Yes!
xb<0
b
QS1
QS2
QS3
QS4
28
Summary of approach
 Test validity of B. subtilis sporulation models
.
.
EF(xhpr>0 Λ EF EG(xhpr=0))
Signal
Kripke
structure
temporal logic
“ [The expression of the gene hpr] increase in
-
model
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)
checking
Batt et al. (2004), SPIN-04, LNCS
29
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
30
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
31
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
32
References
de 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
33