Journal of Mathematical Biology

, Volume 52, Issue 4, pp 524–570 | Cite as

Discrete time piecewise affine models of genetic regulatory networks



We introduce simple models of genetic regulatory networks and we proceed to the mathematical analysis of their dynamics. The models are discrete time dynamical systems generated by piecewise affine contracting mappings whose variables represent gene expression levels. These models reduce to boolean networks in one limiting case of a parameter, and their asymptotic dynamics approaches that of a differential equation in another limiting case of this parameter. For intermediate values, the model present an original phenomenology which is argued to be due to delay effects. This phenomenology is not limited to piecewise affine model but extends to smooth nonlinear discrete time models of regulatory networks.

In a first step, our analysis concerns general properties of networks on arbitrary graphs (characterisation of the attractor, symbolic dynamics, Lyapunov stability, structural stability, symmetries, etc). In a second step, focus is made on simple circuits for which the attractor and its changes with parameters are described. In the negative circuit of 2 genes, a thorough study is presented which concern stable (quasi-)periodic oscillations governed by rotations on the unit circle – with a rotation number depending continuously and monotonically on threshold parameters. These regular oscillations exist in negative circuits with arbitrary number of genes where they are most likely to be observed in genetic systems with non-negligible delay effects.


Discrete Time Unit Circle Periodic Oscillation Contracting Mapping Genetic System 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Buescu, J.: Exotic attractors, Birkhäuser, 1997Google Scholar
  2. 2.
    Bugeaud, Y., Conze, J.-P.: Dynamics of some contracting linear functions modulo 1. Noise, oscillators and algebraic randomness, Lect. Notes Phys. 550, Springer, 2000Google Scholar
  3. 3.
    Coutinho, R.: Dinâmica simbólica linear. PhD Thesis, Technical University of Lisbon, 1999Google Scholar
  4. 4.
    Coutinho, R., Fernandez, B.: Extended symbolic dynamics in bistable CML: Existence and stability of fronts. Physica D 108, 60–80 (1997)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    de Jong, H.: Modeling and simulation of genetic regulatory systems: a literature review. J. Compt. Biol. 9, 69–105 (2002)Google Scholar
  6. 6.
    de Jong, H., Gouzé, J.-L., Hernandez, C., Page, M., Sari, T., Geiselmann, J.: Qualitative simulation of genetic regulatory networks using piecewise-linear models. Bul. Math. Bio. 66, 301–340 (2004)CrossRefGoogle Scholar
  7. 7.
    Edwards, R.: Analysis of continuous-time switching networks. Physica D 146, 165–199 (2000)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Edwards, R., Glass, L.: Combinatorial explosion in model gene networks. Chaos 10, 691–704 (2000)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Edwards, R., Siegelmann, H.T., Aziza, K., Glass, L.: Symbolic dynamics and computation in model gene networks. Chaos 11, 160–169 (2001)CrossRefGoogle Scholar
  10. 10.
    Elaydi, S.: An introduction to difference equations. Springer, 1996Google Scholar
  11. 11.
    Ford, A., Cavana, R.: Special issue: Environmental and Resource Systems. Systems Dynamics Review 20, 2 (2004)Google Scholar
  12. 12.
    Gardner, T.: Design and construction of synthetic gene regulatory networks. PhD Thesis, Boston University, 1997Google Scholar
  13. 13.
    Glass, L.: Combinatorial and topological methods in nonlinear chemical kinetic. J. Chem. Phys. 63, 1325–1335 (1975)CrossRefGoogle Scholar
  14. 14.
    Glass, L., Pasternack, J.S.: Prediction of limit cycles in mathematical models of biological oscillations. Bull. Math. Biol. 40, 27–44 (1978)CrossRefMathSciNetGoogle Scholar
  15. 15.
    Gambaudo, J-M., Tresser, C.: On the dynamics of quasi-contractions. Bol. Soc. Bras. Mat. 19, 61–114 (1988)MATHMathSciNetGoogle Scholar
  16. 16.
    Hoppensteadt, F., Izhikevich, E.: Weakly connected neural networks. Springer, 1997Google Scholar
  17. 17.
    Keener, J.P.: Chaotic behavior in piecewise continuous difference equations. Trans. Amer. Math. Soc. 261, 589–604 (1980)CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Kocic, V., Ladas, G.: Global behaviour of nonlinear difference equations of higher order with applications. Kluwer, 1993Google Scholar
  19. 19.
    Lima, R., Ugalde, E.: Dynamical complexity of discrete time regulatory networks. Nonlinearity 19, 237–259 (2006)CrossRefMathSciNetGoogle Scholar
  20. 20.
    Mallet-Paret, J., Nussbaum, R.: A bifurcation gap for a singularly perturbed delay equation. In: Chaotic dynamics and fractals, Academic Press, pp. 263–286 1986Google Scholar
  21. 21.
    Monk, N.: Oscillatory expression of Hes1, p53, and NF-kB driven by transcriptational time delays. Current Bio. 13, 1409–1413 (2003)CrossRefGoogle Scholar
  22. 22.
    Meir, E., von Dassow, G., Munro, E., Odell, G.: Robustness, flexibility and the role of lateral inhibition in the neurogenic network. Curr. Biol. 12, 778–786 (2002)CrossRefGoogle Scholar
  23. 23.
    Mendoza, L., Thieffry, D., Alvarez-Bullya, E.: Genetic control of flower morphogenesis in Arabidopsis thaliana: a logical analysis. Bioinformatics 15, 593–606 1999Google Scholar
  24. 24.
    Novak, B., Tyson, J.: Modeling the control of DNA replication in fission yeast. Proc. Natl. Acad. Sci. 94, 9147–9152 (1997)CrossRefGoogle Scholar
  25. 25.
    Robinson, C.: Dynamical systems: Stability, symbolic dynamics and chaos. CRC Press, 1999Google Scholar
  26. 26.
    Rémy, E., Mossé, B., Chaouiya, C., Thieffry, D.: A description of dynamical graphs associated with elementary regulatory circuits. Bioinformatics 19 (Suppl. 2), ii172–ii178 (2003)Google Scholar
  27. 27.
    Sanchez, L., van Helden J., Thieffry, D.: Establishment of the dorso-ventral pattern during embryonic development of Drosophila melanogaster: a logical analysis. J. Theo. Biol. 189, 377–389 (1997)CrossRefGoogle Scholar
  28. 28.
    Soulé, C.: Graphic requirements for multistationarity. Complexus 1, 123–133 (2003)CrossRefGoogle Scholar
  29. 29.
    Thomas, R.: Boolean formalization of genetic control circuits. J. Theor. Biol. 42, 563–585 (1973)CrossRefGoogle Scholar
  30. 30.
    Thomas, R., D'Ari, R.: Biological feedbacks. CRC Press (1990)Google Scholar
  31. 31.
    Tyson, J., Chen, K., Novak, B.: Sniffers, buzzers, toggles and blinkers: dynamics of regulatory and signaling pathways in the cell. Current Opinion in Cell Biology 15, 221–231 (2003)CrossRefGoogle Scholar
  32. 32.
    Thieffry, D., Romero, D.: The modularity of biological regulatory networks. BioSystems 50, 49–59 (1999)CrossRefGoogle Scholar
  33. 33.
    Zeidler, E.: Nonlinear functional analysis and its applications: I Fixed-points theorems. Springer, 1996Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • R. Coutinho
    • 1
  • B. Fernandez
    • 2
  • R. Lima
    • 2
  • A. Meyroneinc
    • 2
  1. 1.Departamento de MatemáticaInstituto Superior TécnicoLisboa CodexPortugal
  2. 2.Centre de Physique Théorique CNRSUniversités de Marseille I et IIMarseille CEDEX 09France

Personalised recommendations