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

Richard Casey 1 Hidde de Jong 2 Jean-Luc Gouzé 1
1 COMORE - Modeling and control of renewable resources
LOV - Laboratoire d'océanographie de Villefranche, CRISAM - Inria Sophia Antipolis - Méditerranée
2 HELIX - Computer science and genomics
Inria Grenoble - Rhône-Alpes, LBBE - Laboratoire de Biométrie et Biologie Evolutive - UMR 5558
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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  • HAL Id : inria-00071250, version 1

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Richard Casey, Hidde de Jong, Jean-Luc Gouzé. Piecewise-linear Models of Genetic Regulatory Networks: Equilibria and their Stability. [Research Report] RR-5353, INRIA. 2004. ⟨inria-00071250⟩

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