Journal of Mathematical Biology

, Volume 52, Issue 1, pp 27–56 | Cite as

Piecewise-linear Models of Genetic Regulatory Networks: Equilibria and their Stability

Article

Abstract

A formalism based on piecewise-linear (PL) differential equations, originally due to Glass and Kauffman, has been shown to be well-suited to modelling genetic regulatory networks. However, the discontinuous vector field inherent in the PL models raises some mathematical problems in defining solutions on the surfaces of discontinuity. To overcome these difficulties we use the approach of Filippov, which extends the vector field to a differential inclusion. We study the stability of equilibria (called singular equilibrium sets) that lie on the surfaces of discontinuity. We prove several theorems that characterize the stability of these singular equilibria directly from the state transition graph, which is a qualitative representation of the dynamics of the system. We also formulate a stronger conjecture on the stability of these singular equilibrium sets.

Key words or phrases

Piecewise-linear systems Filippov solutions stability genetic regulatory networks 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alur, R., Belta, C., Ivančíc, F., Kumar, V., Mintz, M., Pappas, G.J., Rubin, H., Schlug, J.: Hybrid modeling and simulation of biomolecular networks. In: Di Benedetto, M.D., Sangiovanni-Vincentelli, A. (eds.) Hybrid Systems: Computation and Control (HSCC 2001), vol 2034 of LNCS, Springer-Verlag, 2001, pp. 19–32Google Scholar
  2. 2.
    Belta, C., Finin, P., Habets, L.C.G.J.M., Halasz, A., Imielinksi, M., Kumar, V., Rubin, H.: Understanding the bacterial stringent response using reachability analysis of hybrid systems. In: Alur, R., Pappas, G. (eds.) Hybrid Systems: Computation and Control (HSCC 2004), vol 2993 of LNCS. Springer-Verlag, Berlin, 2004Google Scholar
  3. 3.
    Boukal, D., Křivan, V.: Lyapunov functions for Lotka-Volterra predator-prey models with optimal foraging behavior. J. Math. Biol. 39, 493–517 (1999)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Branicky., M.: Multiple Lyapunov functions and other analysis tools for switching and hybrid systems. IEEE Trans. Automatic Control 43 (4), 175–482 (1998)Google Scholar
  5. 5.
    Bhatia, N.P., Szegö, G.P.: Dynamical systems: stability theory and applications. Number 35 in Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1967Google Scholar
  6. 6.
    Clarke, F.: optimization and nonsmooth analysis. Wiley, New York, 1983Google Scholar
  7. 7.
    Demongeot, J., Aracena, J., Thuderoz, F., Baum, T., Cohen., O.: Genetic regulation networks: circuits, regulons and attractors. C.R. Biologies 326, 171–188 (2003)CrossRefGoogle Scholar
  8. 8.
    di Bernardo, M., Budd, C.J., Champneys., A.R.: Grazing, skipping and sliding: Analysis of the nonsmooth dynamics of the DC/DC buck converter. Nonlinearity 11(4), 858–890 (1998)Google Scholar
  9. 9.
    Decarlo, R., Branicky, M., Pettersson, S., Lennartson., B.: Perspectives and results on the stability and stabilizability of hybrid systems. Proc. IEEE 88 (7), 1069–1083 (2000)CrossRefGoogle Scholar
  10. 10.
    Devloo, V., Hansen, P., Labbé., M.: Identification of all steady states in large networks by logical analysis. Bull. Math. Biol. 65, 1025–1051 (2003)CrossRefGoogle Scholar
  11. 11.
    de Jong., H.: Modeling and simulation of genetic regulatory systems: A literature review. J. Comput. Biol. 9 (1), 69–105 (2002)Google Scholar
  12. 12.
    de Jong, H., Geiselmann, J., Batt, G., Hernandez, C., Page., M.: Qualitative simulation of the initiation of sporulation in Bacillus subtilis. Bull. Math. Biol. 6 (2), 261–299 (2004)Google Scholar
  13. 13.
    de Jong, H., J-L.Gouzé, Hernandez, C., Page, M., Sari, T., Geiselmann., J.: Qualitative simulation of genetic regulatory networks using piecewise-linear models. Bull. Math. Biol. 6 (2), 301–340 (2004)Google Scholar
  14. 14.
    de Jong, H., Geiselmann, J., Hernandez, C., Page., M.: Genetic network analyzer: qualitative simulation of genetic regulatory networks. Bioinformatics 19 (3), 336–344 (2003)CrossRefGoogle Scholar
  15. 15.
    Edwards., R.: Analysis of continuous-time switching networks. Physica D 146, 165–199 (2000)CrossRefMathSciNetGoogle Scholar
  16. 16.
    Edwards, C., Spurgeon, S.K.: Sliding mode control: theory and applications. Taylor & Francis, 1998Google Scholar
  17. 17.
    Edwards, R., Siegelmann, H.T., Aziza, K., Glass., L.: Symbolic dynamics and computation in model gene networks. Chaos 11 (1), 160–169 (2001)CrossRefGoogle Scholar
  18. 18.
    Feigin., M.I.: The increasingly complex structure of the bifurcation tree of a piecewise-smooth system. Journal of Applied Maths and Mechanics 59, 853–863 (1995)CrossRefGoogle Scholar
  19. 19.
    Filippov, A.F.: Differential equations with discontinuous righthand sides. Kluwer Academic Publishers, Dordrecht, 1988Google Scholar
  20. 20.
    Glass, L., Kauffman., S.A.: The logical analysis of continuous non-linear biochemical control networks. J. Theor. Biol. 39, 103–129 (1973)CrossRefGoogle Scholar
  21. 21.
    Glass., L.: Classification of biological networks by their qualitative dynamics. J. Theor. Biol. 54 (1), 85–107 (1975)Google Scholar
  22. 22.
    Glass, L., Pasternack., J.S.: Stable oscillations in mathematical models of biological control systems. J. Math. Biol. 6, 207–223 (1978)MATHMathSciNetGoogle Scholar
  23. 23.
    Giannakopoulos, F., Pliete., K.: Planar systems of piecewise linear differential equations with a line of discontinuity. Nonlinearity 14, 1611–1632 (2001)CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    Gouzé, J.L., Sari., T.: A class of piecewise linear differential equations arising in biological models. Dyn. Syst 17 (4), 299–316 (2002)CrossRefGoogle Scholar
  25. 25.
    Ghosh, R., Tomlin, C.J.: Lateral inhibition through Delta-Notch signaling: A piecewise affine hybrid model. In: Di Benedetto, M.D., Sangiovanni-Vincentelli, A. (eds.) Hybrid Systems: Computation and Control (HSCC 2001), vol 2034 of LNCS, Springer-Verlag, Berlin, 2001, pp. 232–246Google Scholar
  26. 26.
    Hirsch, M.W., Smale, S.: Differential equations, dynamical systems, and linear algebra. Number 60 in Pure and Applied Mathematics. Academic Press, San Diego, 1974Google Scholar
  27. 27.
    Johansson, M., Rantzer., A.: Computation of piecewise quadratic Lyapunov functions for hybrid systems. IEEE Trans. Automatic Control 43 (4), 555–559 (1998)CrossRefMathSciNetGoogle Scholar
  28. 28.
    Kauffman, S.A.: The Origins of Order: Self-Organization and Selection in Evolution. Oxford University Press, 1993Google Scholar
  29. 29.
    Kohn., K.W.: Molecular interaction maps as information organizers and simulation guides. Chaos 11 (1), 1–14 (2001)CrossRefGoogle Scholar
  30. 30.
    Leine, R., Nijmeijer, H.: Dynamics and bifurcations in non-smooth mechanical systems. Number 18 in Lecture Notes in Applied and Computational Mechanics. Springer-Verlag, Berlin, 2004Google Scholar
  31. 31.
    Mestl, T., Plahte, E., Omholt., S.W.: A mathematical framework for describing and analysing gene regulatory networks. J. Theor. Biol. 176 (2), 291–300 (1995)CrossRefGoogle Scholar
  32. 32.
    Mestl, T., Plahte, E., Omholt., S.W.: Periodic solutions in systems of piecewise-linear differential equations. Dyn. Stabil. Syst. 10 (2), 179–193 (1995)MathSciNetGoogle Scholar
  33. 33.
    Plahte, E., Mestl, T., Omholt., S.W.: Global analysis of steady points for systems of differential equations with sigmoid interactions. Dyn. Stabil. Syst. 9 (4), 275–291 (1994)MathSciNetGoogle Scholar
  34. 34.
    Plahte, E., Mestl, T., Omholt., S.W.: A methodological basis for description and analysis of systems with complex switch-like interactions. J. Math. Biol. 36 (4), 321–348 (1998)CrossRefMathSciNetGoogle Scholar
  35. 35.
    Plahte, E., Kjóglum, S.: Analysis and generic properties of gene regulatory networks with graded response functions. Physica D, 201 (1), 150–176 (2005)CrossRefMathSciNetGoogle Scholar
  36. 36.
    Padden, B., Sastry., S.S.: A calculus for computing Filippov's differential inclusion with application to the variable structure control of robot manipulators. IEEE Trans. Circuits Systems 34, 73–82 (1987)CrossRefGoogle Scholar
  37. 37.
    Ptashne, M.: A genetic switch: phage λ and higher organisms. Cell Press & Blackwell Science, Cambridge, MA, 2nd edition, 1992Google Scholar
  38. 38.
    Ropers, D., de Jong, H., Page, M., Schneider, D., Geiselmann, H.: Qualitative simulation of the carbon starvationo response in Escherichia coli. Biosystems, 2005, to appearGoogle Scholar
  39. 39.
    Snoussi., E.H.: Qualitative dynamics of piecewise-linear differential equations: A discrete mapping approach. Dyn. Stabil. Syst., 4 ( 3 (4), 189–207 (1989)Google Scholar
  40. 40.
    Shevitz, D., Padden., B.: Lyapunov stability theory of nonsmooth systems. IEEE Trans. Automatic Control 39 (9), 1910–1914 (1994)CrossRefMathSciNetGoogle Scholar
  41. 41.
    Snoussi, E.H., Thomas., R.: Logical identification of all steady states: the concept of feedback loop characteristic states. Bull. Math. Biol. 55, 973–991 (1993)CrossRefMATHGoogle Scholar
  42. 42.
    Thomas, R., d'Ari, R.: Biological feedback. CRC Press, 1990Google Scholar
  43. 43.
    Thomas, R., Thieffry, D., Kaufman., M.: Dynamical behaviour of biological regulatory networks: I. Biological role of feedback loops and practical use of the concept of the loop-characteristic state. Bull. Math. Biol. 57 (2), 247–276 (1995)Google Scholar
  44. 44.
    Utkin, V.I.: Sliding modes in control and optimization. Communications and Control Engineering. Springer-Verlag, Berlin, 1992Google Scholar
  45. 45.
    Viretta, A.U., Fussenegger., M.: Modeling the quorum sensing regulatory network of human-pathogenic Pseudomonas aeruginosa. Biotech. Prog. 20, 670–678 (2004)CrossRefGoogle Scholar
  46. 46.
    Yagil, G., Yagil., E.: On the relation between effector concentration and the rate of induced enzyme synthesis. Biophys. J. 11 (1), 11–27 (1971)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  1. 1.COMORE INRIAUnité de recherche Sophia AntipolisSophia AntipolisFrance
  2. 2.HELIX INRIAUnité de recherche Rhône-AlpesSaint Ismier CedexFrance

Personalised recommendations