Periodic Solutions of Piecewise Affine Gene Network Models with Non Uniform Decay Rates: The Case of a Negative Feedback Loop

Etienne Farcot (1, 2) , Jean-Luc Gouzé (3)

1 UMR AGAP - Amélioration génétique et adaptation des plantes méditerranéennes et tropicales
2 VIRTUAL PLANTS - Modeling plant morphogenesis at different scales, from genes to phenotype
3 COMORE - Modeling and control of renewable resources

This paper concerns periodic solutions of a class of equations that model gene regulatory networks. Unlike the vast majority of previous studies, it is not assumed that all decay rates are identical. To handle this more general situation, we rely on monotonicity properties of these systems. Under an alternative assumption, it is shown that a classical fixed point theorem for monotone, concave operators can be applied to these systems. The required assumption is expressed in geometrical terms as an alignment condition on so-called focal points. As an application, we show the existence and uniqueness of a stable periodic orbit for negative feedback loop systems in dimension 3 or more, and of a unique stable equilibrium point in dimension 2. This extends a theorem of Snoussi, which showed the existence of these orbits only.

Domaines: Modélisation et simulation

Etienne Farcot, Jean-Luc Gouzé. Periodic Solutions of Piecewise Affine Gene Network Models with Non Uniform Decay Rates: The Case of a Negative Feedback Loop. Acta Biotheoretica, 2009, 57 (4), pp.429-455. ⟨10.1007/s10441-009-9086-9⟩. ⟨hal-00831799⟩