1. Qualitative Modeling and Simulation of Genetic Regulatory Networks
Hans Geiselmann1 and Hidde de Jong2
1Université Joseph Fourier, Grenoble
2INRIA Rhône-Alpes

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
These are the three parts of the presentation.

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
Start with a schematic overview of the life cycle of B. subtilis. Use this slide to draw attention to the question mark. This is the important developmental decision.

4. Regulatory interactions
Different types of interactions between genes, proteins, and small molecules are involved in the regulation of sporulation in B. subtilis
SinR~SinI
SinI inactivates SinR
AbrB -
AbrB represses sin operon
sinR  A H
sinI
SinR
SinI
sin operon
Spo0A˜P +
Spo0A~P activates 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 - +  H
KinA
phospho-
relay
Spo0A˜P
Spo0A A
spo0A
sinR
sinI
SinI
SinR
SinR/SinI
sigF
hpr (scoR)
abrB
Hpr
AbrB
spo0E
sigH
(spo0H)
Spo0E F
Signal
The assembled data from many laboratories yield a qualitative scheme of the molecular interactions. A prediction of the global behavior of this system is no longer possible. We therefore have to develop conceptual and computer tools to estimate the behavior of such regulation networks.

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
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 b - B a A
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
a1
maxa
maxb
a2
 b1
 b2 .

9. Analysis in regulatory domains
In every regulatory domain D, system monotonically tends towards target equilibrium set  (D)
maxb
model in D1 : D1
xa  a– a xa .
xb  b – b xb D3
xa  a– a xa .
xb  – b xb
model in D3 :
 (D1)  {(a /a , b /b )}
 (D1)
 (D3)
 (D3)  {(a /a , 0 )} xb
 b2
 b1
a1
a2
maxa xa
xa  a s-(xa , a2) s-(xb , b1 ) – a xa .
xb  b s-(xa , a1) s-(xb , b2 ) – b xb

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
 (D1)
 (D3) D1 D3 D2 xb xa D5
 (D5) D3
 (D3) D4
 (D4)
Filippov generalization of PL differential equations

11. Qualitative state and state transition
maxa
maxb
a2
 b1
 b2
 (D1) D2 D3
QS3
QS2
QS1 D1
QS1   D1, {(1,1)} 
Qualitative state consists 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'

12. State transition graphD2 D3 D4 D7 D5 D6 D1 D8 D9
D10
D11
D12
D13
D14
D15
D16
D17
D18
D24
D20
D21
D22
D23
D19
D25
QS3
QS2
QS1
QS4
QS5
QS10
QS15
QS20
QS25
QS24
QS23
QS22
QS21
QS16
QS11
QS6
QS7
QS12
QS17
QS18
QS19
QS13
QS14
QS8
QS9
a1
maxa
maxb
a6
 b1
 b2
Closure of qualitative states and transitions between qualitative states results in state transition graph
Transition graph contains qualitative equilibrium states and/or cycles

13. Qualitative PL model
No exact numerical values for parameters , , and  , but two types of inequality constraints:
Ordering of threshold concentrations of proteins
0 < a1 < a2 < maxa xb xa
a1
maxa
maxb
a2
 b1
 b2
0 < b1 < b2 < maxb
Ordering of target equilibrium values w.r.t. threshold concentrations
a2 < ka / ga < maxa
b2 < kb / gb < maxb
maxa xb xa
a1
maxb
a2
 b1
 b2
a /ga
kb /gb

14. Qualitative simulation
All quantitative PL models subsumed by qualitative PL model have same state transition graph
State transition graph derived from qualitative constraints on parameters
a1
maxa
maxb
a6
 b1
 b2
QS1 D1
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 1.3: 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 - +  H
KinA
phospho-
relay
Spo0A˜P
Spo0A A
spo0A
sinR
sinI
SinI
SinR
SinR/SinI
sigF
hpr (scoR)
abrB
Hpr
AbrB
spo0E
sigH
(spo0H)
Spo0E 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
initial state
division state
Incorporated in the slides before and after this one.
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
s12 s6 s7 s1 s2 s3 s4 s5 s8 s9
s10
s11
s13
ka1
ka3
maxka
KinA
se1
se3
maxse
Spo0E
ab1
maxab
AbrB s1 s2 s3 s4 s5 s6 s7 s8 s9
s10
s11
s12
s13
initial state
division state
initial state
division state
A more rigorous test of the network investigates the dynamics of the transitions between states. Sporulation is induced at time zero.

20. Sporulation vs. division behaviors
ka1
ka3
maxka
KinA
s24
s25 s1 s2
s21
s22
s23 s8
ka1
ka3
maxka
KinA
s12 s6 s7 s1 s2 s3 s4 s5 s8 s9
s10
s11
s13
se1
se3
maxse
Spo0E
se1
se3
maxse
s24
s25 s1 s2
s21
s22
s23 s8
Spo0E
s12 s6 s7 s1 s2 s3 s4 s5 s8 s9
s10
s11
s13
ab1
maxab
AbrB s1
s24
s25 s2
s21
s22
s23 s8
AbrB
ab1
maxab
s12 s6 s7 s4 s8 s9
s10
s11
s13 s1 s2 s3 s5
maxf
SigF s1
s24
s25 s2
s21
s22
s23 s8
SigF
maxf
maxsi
s12 s6 s7 s1 s2 s3 s4 s5 s8 s9
s10
s11
s13
si1
SinI s1
s24
s25 s2
s21
s22
s23 s8
SinI
si1
maxsi

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
KinA
kinA  H + + +
phospho-
relay
Spo0A˜P +
Spo0A - +  F
Using only the known interactions leads to an inconsistency. The concentration of Spo0E has to be kept low in order to maintain a stable sporulation state. We therefore need an additional interaction that negatively regulates Spo0E after sporulation has been initiated.  H
sigF
Spo0E
spo0E  A

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- topA- wt
fis-
topA- k2
k2 topA+
k20

24. Stress adaptation network
Preliminary working model of the regulatory network controlling stress adaptation
ATP
¬Nut + − − + −
Crp
cAMP −
Fis
fis
crp
Crp•cAMP
GyrAB − −
super-
coiling + + + −
gyrAB
GyrAB•SmbC
topA − +
SmbC
ClpXP
rpoS
smbC
RssB•p +
RpoS
Nut p
rssB
RssB

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. References
de Jong, H. (2002), Modeling and simulation of genetic regulatory systems: A literature review, J. Comp. Biol., 9(1):
de Jong, H., J. Geiselmann & D. Thieffry (2003), Qualitative modelling and simulation of developmental regulatory networks, On Growth, Form, and Computers, Academic Press,
de Jong, H., J. Geiselmann, C. Hernandez & M. Page (2003), Genetic Network Analyzer: Qualitative simulation of genetic regulatory networks, Bioinformatics,19(3):
Gouzé, J.-L. & T. Sari (2003), A class of piecewise-linear differential equations arising in biological models, Dyn. Syst., 17(4):
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:

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
QS8
a1
maxa
maxb
a2
 b1
 b2
QS7
QS5
QS6
QS1
QS2
QS3
QS4