Symbolic reachability analysis of genetic regulatory networks using discrete abstractions

We use hybrid-systems techniques for the analysis of reachability properties of a class of piecewise-affine (PA) differential equations that are particularly suitable for the modeling of genetic regulatory networks. More specifically, we introduce a hyperrectangular partition of the state space that forms the basis for a discrete abstraction preserving the sign of the derivatives of the state variables. The resulting discrete transition system provides a qualitative description of the network dynamics that is well-adapted to available experimental data and that can be efficiently computed in a symbolic manner from inequality constraints on the parameters.

[1]  R Thomas,et al.  Dynamical behaviour of biological regulatory networks--I. Biological role of feedback loops and practical use of the concept of the loop-characteristic state. , 1995, Bulletin of mathematical biology.

[2]  George J. Pappas,et al.  Discrete abstractions of hybrid systems , 2000, Proceedings of the IEEE.

[3]  Jean-Luc Gouzé,et al.  Hybrid Modeling and Simulation of Genetic Regulatory Networks: A Qualitative Approach , 2003, HSCC.

[4]  El Houssine Snoussi.  Qualitative dynamics of piecewise-linear differential equations: a discrete mapping approach , 1989 .

[5]  Benjamin Kuipers,et al.  Qualitative reasoning: Modeling and simulation with incomplete knowledge , 1994, Autom..

[6]  Johannes Geiselmann,et al.  Symbolic Reachability Analysis of Genetic Regulatory Networks using Qualitative Abstraction , 2004 .

[7]  Vijay Kumar,et al.  Computational Techniques for Analysis of Genetic Network Dynamics , 2005, Int. J. Robotics Res..

[8]  D. Schneider,et al.  Qualitative simulation of the carbon starvation response in Escherichia coli. , 2006, Bio Systems.

[9]  O Bernard,et al.  Transient behavior of biological loop models with application to the Droop model. , 1995, Mathematical biosciences.

[10]  H. Hong.  An improvement of the projection operator in cylindrical algebraic decomposition , 1990, ISSAC '90.

[11]  Hidde de Jong,et al.  Modeling and Simulation of Genetic Regulatory Systems: A Literature Review , 2002, J. Comput. Biol..

[12]  R. Coutinho,et al.  Discrete time piecewise affine models of genetic regulatory networks , 2005, Journal of mathematical biology.

[13]  H. D. Jong,et al.  Qualitative simulation of the initiation of sporulation in Bacillus subtilis , 2004, Bulletin of mathematical biology.

[14]  Jan Lunze,et al.  Qualitative modelling of linear dynamical systems with quantized state measurements , 1994, Autom..

[15]  Radu Mateescu,et al.  Validation of qualitative models of genetic regulatory networks by model checking: analysis of the nutritional stress response in Escherichia coli , 2005, ISMB.

[16]  Erik Plahte,et al.  Analysis and generic properties of gene regulatory networks with graded response functions , 2005 .

[17]  Rajeev Alur,et al.  A Theory of Timed Automata , 1994, Theor. Comput. Sci..

[18]  L. Glass,et al.  Symbolic dynamics and computation in model gene networks. , 2001, Chaos.

[19]  Dominique Schneider,et al.  Qualitative Analysis and Verification of Hybrid Models of Genetic Regulatory Networks: Nutritional Stress Response in , 2005, HSCC.

[20]  T. Mestl,et al.  A mathematical framework for describing and analysing gene regulatory networks. , 1995, Journal of theoretical biology.

[21]  Bruce H. Krogh,et al.  Verification of infinite-state dynamic systems using approximate quotient transition systems , 2001, IEEE Trans. Autom. Control..

[22]  R. Utsumi,et al.  Negative regulation of adenylate cyclase gene (cya) expression by cyclic AMP-cyclic AMP receptor protein in Escherichia coli: studies with cya-lac protein and operon fusion plasmids , 1985, Journal of bacteriology.

[23]  Thomas A. Henzinger,et al.  Hybrid Automata with Finite Bisimulatioins , 1995, ICALP.

[24]  Eduardo D. Sontag,et al.  Molecular Systems Biology and Control , 2005, Eur. J. Control.

[25]  Edmund M. Clarke,et al.  Model Checking , 1999, Handbook of Automated Reasoning.

[26]  H. D. Jong,et al.  Piecewise-linear Models of Genetic Regulatory Networks: Equilibria and their Stability , 2006, Journal of mathematical biology.

[27]  L. Glass,et al.  Stable oscillations in mathematical models of biological control systems , 1978 .

[28]  S. Ueda,et al.  Growth Phase-Dependent Variation in Protein Composition of the Escherichia coli Nucleoid , 1999, Journal of bacteriology.

[29]  Andreas Kremling,et al.  A Quantitative Approach to Catabolite Repression in Escherichia coli* , 2006, Journal of Biological Chemistry.

[30]  C. Tomlin,et al.  Symbolic reachable set computation of piecewise affine hybrid automata and its application to biological modelling: Delta-Notch protein signalling. , 2004, Systems biology.

[31]  Olivier Bernard,et al.  Global qualitative description of a class of nonlinear dynamical systems , 2002, Artif. Intell..

[32]  H. D. Jong,et al.  Qualitative simulation of genetic regulatory networks using piecewise-linear models , 2004, Bulletin of mathematical biology.

[33]  Rajeev Alur,et al.  Progress on Reachability Analysis of Hybrid Systems Using Predicate Abstraction , 2003, HSCC.

[34]  Etienne Farcot,et al.  Geometric properties of a class of piecewise affine biological network models , 2006, Journal of mathematical biology.

[35]  J. Gouzé,et al.  A class of piecewise linear differential equations arising in biological models , 2002 .

[36]  Gary J. Balas,et al.  Hybrid Systems: Review and Recent Progress , 2003 .

[37]  Aleksej F. Filippov,et al.  Differential Equations with Discontinuous Righthand Sides , 1988, Mathematics and Its Applications.

[38]  R. Edwards.  Analysis of continuous-time switching networks , 2000 .

[39]  L. Glass.  Classification of biological networks by their qualitative dynamics. , 1975, Journal of theoretical biology.

[40]  L. Glass,et al.  The logical analysis of continuous, non-linear biochemical control networks. , 1973, Journal of theoretical biology.