Modeling and Analysis of Gene Regulatory Networks

  • Gilles Bernot
  • Jean-Paul Comet
  • Adrien Richard
  • Madalena Chaves
  • Jean-Luc Gouzé
  • Frédéric Dayan


This chapter describes basic principles for modeling genetic regulatory networks, using three different classes of formalisms: discrete, hybrid, and continuous differential systems. A short review of the mathematical tools for each formalism is presented. Based on several simple examples, which are worked out in detail, this chapter illustrates the study and analysis of the networks’ dynamics, their temporal evolution and asymptotic behaviors.


Gene Regulation Network State Graph Interaction Graph Stable Steady State Regular Domain 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



It is a pleasure for GB, JPC and AR to thank the biologist Janine Guespin-Michel, who has actively participated to the definition of our formal logic methodology in such a way that our techniques from computer science and the SMBioNet software become truly useful for biologists. She has also been at the origin of the Pseudomonas aeruginosa hypothesis. The authors would also like to thank F. Cazals for his remarks and careful reading of the chapter.


  1. 1.
  2. 2. Scholar
  3. 3.
    U. Alon. An Introduction to Systems Biology: Design Principles of Biological Circuits. Chapman & Hall/CRC, Boca Raton, 2006.Google Scholar
  4. 4.
    J. Aracena. On the number of fixed points in regulatory boolean networks. Bulletin of Mathematical Biology, 70(5):1398–1409, 2008.Google Scholar
  5. 5.
    J. Aracena, J. Demongeot, and E. Goles. Positive and negative circuits in discrete neural networks. IEEE Transactions of Neural Networks, 15:77–83, 2004.Google Scholar
  6. 6.
    J. Barnat, L. Brim, I. Černá, S. Dražan, J. Fabrikozá, and D. Šafránek. On algorithmic analysis of transcriptional regulation by ltl model checking. Theoretical Computer Science, 2009.Google Scholar
  7. 7.
    G. Batt, M. Page, I. Cantone, G. Goessler, P. Monteiro, and H. de Jong. Efficient parameter search for qualitative models of regulatory networks using symbolic model checking. Bioinformatics, 26(18):i603–i610, 2010.Google Scholar
  8. 8.
    G. Bernot, J.-P. Comet, A. Richard, and J. Guespin. A fruitful application of formal methods to biological regulatory networks: Extending Thomas’ asynchronous logical approach with temporal logic. J. Theor. Biol., 229(3):339–347, 2004.Google Scholar
  9. 9.
    R. Casey, H. de Jong, and J.L. Gouzé. Piecewise-linear models of genetic regulatory networks: equilibria and their stability. J. Math. Biol., 52:27–56, 2006.Google Scholar
  10. 10.
    M. Chaves and J.L. Gouzé. Exact control of genetic networks in a qualitative framework: the bistable switch example. Automat., 47:1105–1112, 2011.Google Scholar
  11. 11.
    A. Cimatti, E. Clarke, E. Giunchiglia, F. Giunchiglia, M. Pistore, and M. Roven. NuSMV2: An Open Source Tool for Symbolic Model Checking. In International Conference on Computer-Aided Verification (CAV 2002), 2002.Google Scholar
  12. 12.
    O. Cinquin and J. Demongeot. Roles of positive and negative feedback in biological systems. C.R.Biol., 325(11):1085–1095, 2002.Google Scholar
  13. 13.
    F. Corblin, E. Fanchon, and L. Trilling. Applications of a formal approach to decipher discrete genetic networks. BMC Bioinformatics, 11(385), 2010.Google Scholar
  14. 14.
    F. Corblin, S. Tripodi, E. Fanchon, D. Ropers, and L. Trilling. A declarative constraint-based method for analyzing discrete genetic regulatory networks. Biosystems, 98(2):91–104, 2009.Google Scholar
  15. 15.
    F. Dardel and F. Képès. Bioinformatics: genomics and post-genomics. Wiley, Chichester, 2005.Google Scholar
  16. 16.
    H. De Jong. Modeling and simulation of genetic regulatory systems: a literature review. Journal of computational biology, 9(1):67–103, 2002.Google Scholar
  17. 17.
    H. de Jong, J.L. Gouzé, C. Hernandez, M. Page, T. Sari, and J. Geiselmann. Qualitative simulation of genetic regulatory networks using piecewise linear models. Bull. Math. Biol., 66:301–340, 2004.Google Scholar
  18. 18.
    L. Edelstein-Keshet. Mathematical models in Biology. SIAM classics in applied mathematics, Philadelphia, 2005.Google Scholar
  19. 19.
    E.A. Emerson. Handbook of theoretical computer science, Volume B : formal models and semantics, chapter Temporal and modal logic, pages 995–1072. MIT Press, 1990.Google Scholar
  20. 20.
    S. Gama-Castro, H. Salgado, M. Peralta-Gil, A. Santos-Zavaleta, L. Muniz-Rascado, H. Solano-Lira, V. Jimenez-Jacinto, V. Weiss, J. S. Garcia-Sotelo, A. Lopez-Fuentes, L. Porron-Sotelo, S. Alquicira-Hernandez, A. Medina-Rivera, I. Martinez-Flores, K. Alquicira-Hernandez, R. Martinez-Adame, C. Bonavides-Martinez, J. Miranda-Rios, A. M. Huerta, A. Mendoza-Vargas, L. Collado-Torres, B. Taboada, L. Vega-Alvarado, M. Olvera, L. Olvera, R. Grande, E. Morett, and J. Collado-Vides. RegulonDB version 7.0: transcriptional regulation of Escherichia coli K-12 integrated within genetic sensory response units (Gensor Units). Nucleic Acids Research, 2010.Google Scholar
  21. 21.
    T.S. Gardner, C.R. Cantor, and J.J. Collins. Construction of a genetic toggle switch in Escherichia coli. Nature, 403:339–342, 2000.Google Scholar
  22. 22.
    L. Glass and S.A. Kauffman. The logical analysis of continuous, nonlinear biochemical control networks. J. Theor. Biol., 39:103–129, 1973.Google Scholar
  23. 23.
    J.L. Gouzé. Positive and negative circuits in dynamical systems. Journal of Biological Systems, 6:11–15, 1998.Google Scholar
  24. 24.
    F. Grognard, J.-L. Gouzé, and H. de Jong. Piecewise-linear models of genetic regulatory networks: theory and example. In I. Queinnec, S. Tarbouriech, G. Garcia, and S. Niculescu, editors, Biology and control theory: current challenges, Lecture Notes in Control and Information Sciences (LNCIS) 357, pages 137–159. Springer-Verlag, 2007.Google Scholar
  25. 25.
    J. Guespin-Michel and M. Kaufman. Positive feedback circuits and adaptive regulations in bacteria. Acta Biotheor., 49(4):207–18, 2001.Google Scholar
  26. 26.
    M. Kaufman, C. Soulé, and R. Thomas. A new necessary condition on interaction graphs for multistationarity. Journal of Theoretical Biology, 248:675–685, 2007.Google Scholar
  27. 27.
    H.K. Khalil. Nonlinear systems. Prentice Hall, New Jersey, 2002.Google Scholar
  28. 28.
    E. Klipp, R. Herwig, A. Howald, C. Wierling, and H. Lehrach. Systems Biology in Practice. Wiley-VCH, Weinheim, 2005.Google Scholar
  29. 29.
    E. Plahte, T. Mestl, and S.W. Omholt. Feedback loops, stability and multistationarity in dynamical systems. Journal of Biological Systems, 3:569–577, 1995.Google Scholar
  30. 30.
    E. Remy, P. Ruet, and D. Thieffry. Graphic requirement for multistability and attractive cycles in a boolean dynamical framework. Advances in Applied Mathematics, 41(3):335–350, 2008.Google Scholar
  31. 31.
    A. Richard. Positive circuits and maximal number of fixed points in discrete dynamical systems. Discrete Applied Mathematics, 157(15):3281–3288, 2009.Google Scholar
  32. 32.
    A. Richard. Negative circuits and sustained oscillations in asynchronous automata networks. Advances in Applied Mathematics, 44(4):378–392, 2010.Google Scholar
  33. 33.
    A. Richard and J.-P. Comet. Necessary conditions for multistationarity in discrete dynamical systems. Discrete Applied Mathematics, 155(18):2403–2413, 2007.Google Scholar
  34. 34.
    D. Ropers, H. de Jong, M. Page, D. Schneider, and J. Geiselmann. Qualitative simulation of the carbon starvation response in Escherichia coli. Biosystems, 84(2):124–152, 2006.Google Scholar
  35. 35.
    E.H. Snoussi. Qualitative dynamics of a piecewise-linear differential equations : a discrete mapping approach. Dynamics and stability of Systems, 4:189–207, 1989.Google Scholar
  36. 36.
    E.H. Snoussi. Necessary conditions for multistationarity and stable periodicity. Journal of Biological Systems, 6:3–9, 1998.Google Scholar
  37. 37.
    C. Soulé. Graphical requirements for multistationarity. ComPlexUs, 1:123–133, 2003.Google Scholar
  38. 38.
    C. Soulé. Mathematical approaches to differentiation and gene regulation. C.R. Biologies, 329:13–20, 2006.Google Scholar
  39. 39.
    R. Thomas. On the relation between the logical structure of systems and their ability to generate multiple steady states and sustained oscillations. In Series in Synergetics, volume 9, pages 180–193. Springer, 1981.Google Scholar
  40. 40.
    G. Yagil and E. Yagil. On the relation between effector concentration and the rate of induced enzyme synthesis. Biophys. J., 11:11–27, 1971.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Gilles Bernot
    • 1
  • Jean-Paul Comet
    • 1
  • Adrien Richard
    • 1
  • Madalena Chaves
    • 2
  • Jean-Luc Gouzé
    • 2
  • Frédéric Dayan
    • 3
  1. 1.I3S – UMR 6070 CNRS/UNSA, Algorithmes-Euclide-BSophia AntipolisFrance
  2. 2.Inria Sophia Antipolis MéditerranéeBiocore project-teamSophia AntipolisFrance
  3. 3.SOBIOS SAValbonne Sophia AntipolisFrance

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