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

Periodic Solutions of Piecewise Affine Gene Network Models with Non Uniform Decay Rates: The Case... 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Biotheoretica Springer Journals

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

Acta Biotheoretica , Volume 57 (4) – Oct 8, 2009

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References (18)

Publisher
Springer Journals
Copyright
Copyright © 2009 by Springer Science+Business Media B.V.
Subject
Philosophy; Evolutionary Biology; Philosophy of Biology
ISSN
0001-5342
eISSN
1572-8358
DOI
10.1007/s10441-009-9086-9
pmid
19838808
Publisher site
See Article on Publisher Site

Abstract

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.

Journal

Acta BiotheoreticaSpringer Journals

Published: Oct 8, 2009

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