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Qualitative Simulation of the Carbon Starvation Response in Escherichia coli
Hidde de Jong1 Delphine Ropers Johannes Geiselmann1,2 1INRIA Grenoble-Rhône-Alpes 2Laboratoire Adaptation Pathogénie des Microorganismes CNRS UMR 5163 Université Joseph Fourier
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Overview Carbon starvation response of Escherichia coli
Modeling and simulation: objective and constraints Qualitative modeling and simulation of carbon starvation Experimental validation of carbon starvation model Conclusions
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Rocky Mountain Laboratories, NIAID, NIH
Escherichia coli The average human gut contains about 1 kg of bacteria Approximatively 0.1% are Escherichia coli E. coli, along with other enterobacteria, synthesize vitamins which are absorbed by our body (e.g., vitamin K, B-complex vitamins) Rocky Mountain Laboratories, NIAID, NIH 2 µm 1 µm
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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 Storz and Hengge-Aronis (2000), Bacterial Stress Responses, ASM Press Heat shock Nutritional stress Cold shock Osmotic stress …
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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) > 4 h time
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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): No global view of functioning of network available, despite abundant knowledge on network components
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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 rrn P1 P2 CRP crp cya CYA cAMP•CRP FIS TopA topA GyrAB P1-P4 P1-P’1 P gyrAB Signal (lack of carbon source) DNA supercoiling fis tRNA rRNA Ropers et al. (2006), BioSystems, 84(2):124-52 protein gene promoter
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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
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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):301-40 Batt et al. (2007), Automatica, in press de Jong, Geiselmann et al. (2004), Bull. Math. Biol., 66(2): Viretta and Fussenegger, Biotechnol. Prog., 2004, 20(3): Sepulchre et al., J. Theor. Biol., 2007, 244(2):239-57 Thomas and d’Ari (1990), Biological Feedback, CRC Press
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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) 1 b B a A Differential equation models of regulatory networks are piecewise-affine (PA) Glass and Kauffman (1973), J. Theor. Biol., 39(1):103-29
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Mathematical analysis of PA models
Analysis of dynamics of PA models in phase space maxb a1 maxb a2 b maxa ka/ga kb/gb xa a – a xa . xb b – b xb D1 b a1 a2 maxa xa a s-(xa , a2) s-(xb , b ) – a xa . xb b s-(xa , a1) – b xb
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Mathematical analysis of PA models
Analysis of dynamics of PA models in phase space maxb a1 maxb a2 b maxa xa a – a xa . xb – b xb b D5 a1 a2 ka/ga maxa xa a s-(xa , a2) s-(xb , b ) – a xa . xb b s-(xa , a1) – b xb
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Mathematical analysis of PA models
Analysis of dynamics of PA models in phase space Extension of PA differential equations to differential inclusions using Filippov approach maxb a1 maxb a2 b maxa b D3 a1 a2 maxa xa a s-(xa , a2) s-(xb , b ) – a xa . xb b s-(xa , a1) – b xb Gouzé, Sari (2002), Dyn. Syst., 17(4):
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Mathematical analysis of PA models
Analysis of dynamics of PA models in phase space Extension of PA differential equations to differential inclusions using Filippov approach maxb a1 maxb a2 b maxa b D7 a1 a2 maxa xa a s-(xa , a2) s-(xb , b ) – a xa . xb b s-(xa , a1) – b xb Gouzé, Sari (2002), Dyn. Syst., 17(4):
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Qualitative analysis of network dynamics
Phase space partition: unique derivative sign pattern in regions Qualitative abstraction yields state transition graph Shift from continuous to discrete picture of network dynamics . xa > 0 xb > 0 xb < 0 xa = 0 D1: 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 maxb a1 maxb a2 b maxa b a1 a2 maxa
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Qualitative analysis of network dynamics
State transition graph invariant for parameter constraints a1 maxb a2 b maxa ka/ga kb/gb D1 D11 D12 D3 D1 D3 D11 D12 0 < qa1 < qa2 < a/a < maxa 0 < qb < b/b < maxb
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Qualitative analysis of network dynamics
State transition graph invariant for parameter constraints a1 maxb a2 b maxa ka/ga kb/gb D1 D11 D12 D3 D1 D3 D11 D12 0 < qa1 < qa2 < a/a < maxa 0 < qb < b/b < maxb
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Qualitative analysis of network dynamics
State transition graph invariant for parameter constraints a1 maxb a2 b maxa ka/ga kb/gb D1 D11 D12 D3 D1 D3 D11 D12 0 < qa1 < qa2 < a/a < maxa 0 < qb < b/b < maxb a1 maxb a2 b maxa ka/ga kb/gb D1 D11 D12 D3 D11 0 < a/a < qa1 < qa2 < maxa D1 0 < qb < b/b < maxb
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Validation of qualitative models
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 D27 D26 D25 D11 D12 D13 D14 D2 D4 D6 D8 D10 D16 D17 D18 D20 D19 D21 D22 D23 D24 xb time xa xa > 0 . xb > 0 xa < 0 Concistency? Yes . xa < 0 xb > 0 xa > 0 xa= 0 xb= 0 D1: D17: D18: 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.
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Verification using model checking
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 D1 D3 D5 D7 D9 D15 D27 D26 D25 D11 D12 D13 D14 D2 D4 D6 D8 D10 D16 D17 D18 D20 D19 D21 D22 D23 D24 xb time xa xa > 0 . xb > 0 xa < 0 EF(xa > 0 xb > 0 EF(xa < 0 xb > 0) ) . Concistency? Yes Batt et al. (2005), Bioinformatics, 21(supp. 1): i19-28
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Analysis of attractors of PA systems
Search of attractors of PA systems in phase space D1 D3 D5 D7 D9 D15 D27 D26 D25 D11 D12 D13 D14 D2 D4 D6 D10 D16 D17 D18 D20 D19 D21 D22 D23 D24 D8 a1 maxb a2 b maxa maxb kb/gb b a1 a2 maxa Analysis of stability of attractors, using properties of state transition graph Definition of stability of equilibrium points on surfaces of discontinuity Casey et al. (2006), J. Math Biol., 52(1):27-56
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Genetic Network Analyzer (GNA)
Qualitative simulation method implemented in Java: Genetic Network Analyzer (GNA) Distribution by Genostar SA de Jong et al. (2003), Bioinformatics, 19(3):336-44 Genostar slide Batt et al. (2005), Bioinformatics, 21(supp. 1): i19-28
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Analysis of carbon starvation response
Modeling and experimental studies directed at understanding how network controls carbon starvation response Bottom-up strategy: Initial model of carbon starvation response rrn P1 P2 CRP crp cya CYA cAMP•CRP FIS TopA topA GyrAB P1-P4 P1-P’1 P gyrAB Signal (lack of carbon source) DNA supercoiling fis tRNA rRNA Ropers et al. (2006), BioSystems, 84(2): protein gene promoter
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Analysis of carbon starvation response
Modeling and experimental studies directed at understanding how network controls carbon starvation response Bottom-up strategy: Initial model of the carbon starvation response Search and curate data available in the literature and databases Experimental verification of model predictions Extension of model to take into account wrong predictions Additional global regulators: IHF, HNS, ppGpp, FNR, LRP, ArcA, …
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Modeling of carbon starvation network
Modular structure of carbon starvation network Superhelical density of DNA rrn P1 P2 Activation CRP crp cya CYA CRP•cAMP FIS TopA topA GyrAB P1-P4 P1-P’1 P gyrAB Signal (lack of carbon source) Supercoiling fis tRNA rRNA Ropers et al. (2006), BioSystems, 84(2):
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Modeling of carbon starvation network
Can the initial model explain the carbon starvation response of E. coli cells? rrn P1 P2 CRP crp cya CYA cAMP•CRP FIS TopA topA GyrAB P1-P4 P1-P’1 P gyrAB Signal (lack of carbon source) DNA supercoiling fis tRNA rRNA Translation of biological data into a mathematical model
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From nonlinear kinetic model to PA model
Modeling process consists of reducing classical nonlinear kinetic model to PA model Nonlinear kinetic model Quasi-steady state approximation Nonlinear reduced kinetic model Piecewise-linear approximation Piecewise-linear model
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Nonlinear kinetic model
Nonlinear kinetic ODE model of 12 variables and 46 parameters Regulation of gene expression (Hill) FIS rrn P1 P2 stable RNAs
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Nonlinear kinetic model
Nonlinear kinetic ODE model of 12 variables and 46 parameters Regulation of gene expression (Hill) Enzymatic reactions (Michaelis-Menten) ATP + CYA* K1 CYA*•ATP CYA* + cAMP k2 k3 degradation/export
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Nonlinear kinetic model
Nonlinear kinetic ODE model of 12 variables and 46 parameters Regulation of gene expression (Hill) Enzymatic reactions (Michaelis-Menten) Formation of biochemical complexes (mass action) cAMP + CRP K4 CRP•cAMP
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Modeling of carbon starvation network
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Quasi steady state approximation
Identification of slow and fast processes in network P fis P gyrAB P1-P’1 P2 cya FIS GyrAB CYA DNA supercoiling cAMP•CRP Signal (lack of carbon source) TopA CRP tRNA rRNA P1-P4 topA rrn P1 P2 crp P1 P2
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Quasi steady state approximation
Identification of slow and fast processes in network Change of variables and quasi steady-state approximation P fis P gyrAB P1-P’1 P2 cya FIS GyrAB CYA DNA supercoiling cAMP•CRP Signal (lack of carbon source) TopA CRP tRNA rRNA P1-P4 topA P1 P2 rrn P1 P2 crp Heinrich and Schuster (1996), The Regulation of Cellular Systems, Chapman & Hall
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Nonlinear reduced model
QSSA model of 7 variables and 46 parameters
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Piecewise-linear approximation
Approximation of Hill function with step function Approximation of sigmoidal surfaces with product of step functions s+(xCYA , CYA1) s+(xCRP , CRP1) s+(xSIGNAL , SIGNAL) CYA concentration (M) CRP concentration (M) CRP• cAMP Activation CRP CYA Signal crp P1 P2
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Model of carbon starvation network
PADE model of 7 variables and 36 parameter inequalities
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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
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Simulation of stress response network
Simulation of transition from exponential to stationary phase State transition graph with 27 states, 1 stable steady state CRP GyrAB TopA CYA rrn FIS Signal
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Insight into nutritional stress response
Sequence of qualitative events leading to adjustment of growth of cell after nutritional stress signal Role of the mutual inhibition of Fis and CRP•cAMP P fis P gyrAB P1-P’1 P2 cya FIS GyrAB CYA Supercoiling Activation Superhelical density of DNA Signal (lack of nutrients) CRP•cAMP TopA CRP tRNA rRNA P1-P4 topA P1 P2 crp P1 P2 rrn
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cloning promoter regions on plasmid
Reporter gene systems Simulations yield predictions that cannot be verified with currently avaliable experimental data Use of reporter gene systems to monitor gene expression rrnB promoter region bla ori gfp or lux reporter gene cloning promoter regions on plasmid fis crp nlpD rpoS topA gyrB gyrA
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Monitoring of gene expression
Integration of fluorescent or luminescent reporter gene systems into bacterial cell Global regulator Global regulator E. coli genome E. coli genome emission excitation Reporter gene Reporter operon GFP Luciferase emission Expression of reporter gene reflects expression of host gene of interest
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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 Well with bacterial culture Different gene reporter system in wells 96-well microplate Upshift experiments in M9/glucose medium
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Analysis of reporter gene expression data
WellReader: program for analysis of reporter gene expression data
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Validation of carbon starvation response model
Translate expression profiles into temporal logic and verify properties by means of model checking Geiselmann et al (2007), in preparation “Fis concentration decreases and becomes steady in stationary phase” EF(xfis < 0 EF(xfis = 0 xrrn < qrrn) ) . True
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Validation of carbon starvation response model
Translate expression profiles into temporal logic and verify properties by means of model checking Geiselmann et al (2007), in preparation “GyrA concentration decreases at the onset of stationary phase” . EF (xgyrA < 0 xrrn < qrrn) False
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Suggestion of missing interaction
Model does not reproduce observed downregulation of negative supercoiling Missing interaction in the network? P fis P gyrAB P1-P’1 P2 cya FIS GyrAB CYA Supercoiling Activation Superhelical density of DNA Signal (lack of nutrients) CRP•cAMP TopA CRP tRNA rRNA P1-P4 topA P1 P2 crp P1 P2 rrn
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Extension of stress response network
Model does not reproduce observed downregulation of negative supercoiling Activation Stress signal CRP crp cya CYA fis FIS Supercoiling TopA topA GyrAB P1-P4 P1 P2 P1-P’1 rrn P gyrAB tRNA rRNA Ropers et al. (2006) GyrI gyrI rpoS nlpD σS RssB rssA PA PB rssB P5 Missing component in the network?
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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 languages Prerequisite 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
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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 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
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Contributors and sponsors
Grégory 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
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Validation of carbon starvation response model
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” Ali Azam et al. (1999), J. Bacteriol., 181(20): EF(xfis < 0 EF(xfis = 0 xrrn < qrrn) ) . True Kawamukai et al. (1985), J. Bacteriol., 164(2): AG(xcrp > q3crp xcya > q3cya xs > qs → EF xcya < 0) . True Balke, Gralla (1987), J. Bacteriol., 169(10): EF( (xgyrAB < 0 xtopA > 0) xrrn < qrrn) . False
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