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

Journal of Mathematical Biology volume 52, pages 27–56 (2006)Cite this 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.

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Affiliations

  1. COMORE INRIA, Unité de recherche Sophia Antipolis, 2004 route des Lucioles, BP 93, 06902, Sophia Antipolis, France

    Richard Casey & Jean-Luc Gouzé

  2. HELIX INRIA, Unité de recherche Rhône-Alpes, 655 avenue de l'Europe, Montbonnot, 38334, Saint Ismier Cedex, France

    Hidde de Jong

Authors
  1. Richard Casey
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  2. Hidde de Jong
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  3. Jean-Luc Gouzé
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Correspondence to Jean-Luc Gouzé.

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Casey, R., Jong, H. & Gouzé, JL. Piecewise-linear Models of Genetic Regulatory Networks: Equilibria and their Stability. J. Math. Biol. 52, 27–56 (2006). https://doi.org/10.1007/s00285-005-0338-2

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  • DOI: https://doi.org/10.1007/s00285-005-0338-2

Key words or phrases

  • Piecewise-linear systems
  • Filippov solutions
  • stability
  • genetic regulatory networks