Bulletin of Mathematical Biology

, Volume 75, Issue 6, pp 967–987 | Cite as

Probabilistic Approach for Predicting Periodic Orbits in Piecewise Affine Differential Models

  • Madalena Chaves
  • Etienne Farcot
  • Jean-Luc Gouzé
Original Article


Piecewise affine models provide a qualitative description of the dynamics of a system, and are often used to study genetic regulatory networks. The state space of a piecewise affine system is partitioned into hyperrectangles, which can be represented as nodes in a directed graph, so that the system’s trajectories follow a path in a transition graph.

This paper proposes and compares two definitions of probability of transition between two nodes A and B of the graph, based on the volume of the initial conditions on the hyperrectangle A whose trajectories cross to B. The parameters of the system can thus be compared to the observed transitions between two hyperrectangles. This property may become useful to identify sets of parameters for which the system yields a desired periodic orbit with a high probability, or to predict the most likely periodic orbit given a set of parameters, as illustrated by a gene regulatory system composed of two intertwined negative loops.


Piecewise affine models Genetic networks 



This work was supported in part by the INRIA-INSERM project ColAge and by ANR project GeMCo (ANR 2010 BLAN 0201 01).


  1. Belta, C., Habets, L., & Kumar, V. (2002). Control of multi-affine systems on rectangles with applications to hybrid biomolecular networks. In Proceedings of the 41st IEEE conference decision and control (CDC02) (pp. 534–539). Google Scholar
  2. Büeler, B., Enge, A., & Fukuda, K. (2000). Exact volume computation for convex polytopes: a practical study. In: G. Ziegler (ed.) Polytopes—combinatorics and computation. Basel: Birkhäuser. Google Scholar
  3. Casey, R., de Jong, H. & Gouzé, J. (2006). Piecewise-linear models of genetic regulatory networks: equilibria and their stability. J. Math. Biol., 52, 27–56. MathSciNetMATHCrossRefGoogle Scholar
  4. Chaves, M. & Gouzé, J. (2011). Exact control of genetic networks in a qualitative framework: the bistable switch example. Automatica, 47, 1105–1112. MATHCrossRefGoogle Scholar
  5. Chaves, M., Tournier, L., & Gouzé, J. L. (2010). Comparing Boolean and piecewise affine differential models for genetic networks. Acta Biotheor., 58(2), 217–232. CrossRefGoogle Scholar
  6. Edwards, R. (2000). Analysis of continuous-time switching networks. Physica D, 146, 165–199. MathSciNetMATHCrossRefGoogle Scholar
  7. Farcot, E. (2006). Geometric properties of a class of piecewise affine biological network models. J. Math. Biol., 52(3), 373–418. MathSciNetMATHCrossRefGoogle Scholar
  8. Farcot, E., & Gouzé, J. L. (2008). A mathematical framework for the control of piecewise-affine models of gene networks. Automatica, 44(9), 2326–2332. MathSciNetMATHCrossRefGoogle Scholar
  9. Farcot, E., & Gouzé, J. L. (2009). Periodic solutions of piecewise affine gene network models with non uniform decay rates: the case of a negative feedback loop. Acta Biotheor., 57, 429–455. CrossRefGoogle Scholar
  10. Friedman, N. (2004). Inferring cellular networks using probabilistic graphical models. Science, 303(5659), 799–805. CrossRefGoogle Scholar
  11. Giannakopoulos, F., & Pliete, K. (2001). Planar systems of piecewise linear differential equations with a line of discontinuity. Nonlinearity, 14, 1611–1632. MathSciNetMATHCrossRefGoogle Scholar
  12. Glass, L. (1975). Combinatorial and topological methods in nonlinear chemical kinetics. J. Chem. Phys., 63, 1325–1335. CrossRefGoogle Scholar
  13. Glass, L., & Kauffman, S. (1973). The logical analysis of continuous, nonlinear biochemical control networks. J. Theor. Biol., 39, 103–129. CrossRefGoogle Scholar
  14. Glass, L., & Pasternak, J. (1978). Stable oscillations in mathematical models of biological control systems. J. Math. Biol., 6, 207–223. MathSciNetMATHCrossRefGoogle Scholar
  15. Gouzé, J., & Chaves, M. (2010). Piecewise affine models of regulatory genetic networks: review and probabilistic interpretation. In: J. Lévine & P. Müllhaupt (Eds.), Lecture notes in control and information sciences: Vol470. Advances in the theory of control, signals and systems, with physical modelling (pp. 241–253). Berlin: Springer. Google Scholar
  16. Gouzé, J., & Sari, T. (2002). A class of piecewise linear differential equations arising in biological models. Dyn. Syst., 17(4), 299–316. MathSciNetMATHCrossRefGoogle Scholar
  17. Habets, L., & Schuppen, J. V. (2004). A control problem for affine dynamical systems on a full-dimensional polytope. Automatica, 40, 21–35. MATHCrossRefGoogle Scholar
  18. Hoffmann, A., Levchenko, A., Scott, M., & Baltimore, D. (2002). The IkB-NFkB signaling module: temporal control and selective gene activation. Science, 298, 1241–1245. CrossRefGoogle Scholar
  19. Lee, I., Date, S., Adai, A., & Marcotte, E. (2004). A probabilistic functional network of yeast genes. Science, 306(5701), 1555–1558. CrossRefGoogle Scholar
  20. Mestl, T., Lemay, C., & Glass, L. (1996). Chaos in high-dimensional neural and gene networks. Physica D, 98, 33–52. MathSciNetMATHCrossRefGoogle Scholar
  21. Shmulevich, I., Dougherty, E., Kim, S., & Zhang, W. (2002). Probabilistic Boolean networks: a rule-based uncertainty model for gene regulatory networks. Bioinformatics, 18(2), 261–274. CrossRefGoogle Scholar
  22. Snoussi, E. (1989). Qualitative dynamics of piecewise-linear differential equations: a discrete mapping approach. Dyn. Stab. Syst., 4(3–4), 189–207. MathSciNetMATHGoogle Scholar
  23. Thomas, R., & D’Ari, R. (1990). Biological feedback. Boca Raton: CRC-Press. MATHGoogle Scholar
  24. Tournier, L., & Chaves, M. (2009). Uncovering operational interactions in genetic networks using asynchronous boolean dynamics. J. Theor. Biol., 260(2), 196–209. MathSciNetCrossRefGoogle Scholar
  25. Wiback, S., Famili, I., Greenberg, H., & Palsson, B. (2004). Monte Carlo sampling can be used to determine the size and shape of the steady-state flux space. J. Theor. Biol., 228(4), 437–447. MathSciNetCrossRefGoogle Scholar

Copyright information

© Society for Mathematical Biology 2012

Authors and Affiliations

  • Madalena Chaves
    • 1
  • Etienne Farcot
    • 2
  • Jean-Luc Gouzé
    • 1
  1. 1.BIOCOREINRIASophia AntipolisFrance
  2. 2.Montpellier cedex 5France

Personalised recommendations