Stability of Equilibria for Hybrid Models of Genetic Regulatory Networks

Richard Casey1 — Hidde de Jong2 — Jean-Luc Gouzé1

1INRIA Comore, BP 93, 06902 Sophia-Antipolis, France and 2INRIA Helix, 655 avenue de l’Europe, Montbonnot, 38334 Saint Ismier Cedex, France,

ABSTRACT. A formalism based on piecewise-linear (PL) differential equations has been shown to be well-suited to modelling genetic regulatory networks. The discontinuous vector field inherent in the PL models leads to 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.

KEYWORDS: Genetic regulatory networks, piecewise-linear differential models, hybrid systems, stability

1. Introduction

A class of piecewise-linear (PL) models, originally proposed by Glass and Kauffman [GLA 73], has been widely used in modelling genetic regulatory networks. The properties of these PL models have been well-studied in the mathematical biology literature, by for example Glass and Pasternack [GLA 78], Snoussi [SNO 89], Plahte et al [PLA 94], Mestl et al [MES 95b], Thomas et al [THO 95], Edwards [EDW 00], Gouzé and Sari [GOU 02], and more recently in the hybrid systems literature ...

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