Qualitative Modeling and Simulation of Genetic Regulatory Networks Hans Geiselmann1 and Hidde de Jong2 1Université Joseph Fourier, Grenoble 2INRIA Rhône-Alpes Hans.Geiselmann@ujf-grenoble.fr Hidde.de-Jong@inrialpes.fr

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.

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.

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

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.

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

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)

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 .

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

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

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'

State transition graph D2 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

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

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

Genetic Network Analyzer (GNA) Qualitative simulation method implemented in Java 1.3: Genetic Network Analyzer (GNA) Graphical interface to control simulation and analyze results

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

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

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

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.

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

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

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

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

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

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

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

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

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

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. de Jong, H., J. Geiselmann, C. Hernandez & M. Page (2003), Genetic Network Analyzer: Qualitative simulation of genetic regulatory networks, Bioinformatics,19(3):336-344. Gouzé, J.-L. & T. Sari (2003), 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 (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: http://www-helix.inrialpes.fr/gna

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