Graph-theoretic conditions for  zero-eigenvalue Turing instability in general  chemical reaction networks

Maya Mincheva; Gheorghe Craciun

doi:10.3934/mbe.2013.10.1207

Article Contents

2013, Volume 10, Issue 4: 1207-1226. Doi: 10.3934/mbe.2013.10.1207

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Graph-theoretic conditions for zero-eigenvalue Turing instability in general chemical reaction networks

Maya Mincheva^1, and
Gheorghe Craciun^2,

1.
Department of Mathematical Sciences, Northern Illinois University, Dekalb, IL 60115
2.
Department of Mathematics and Department of Biomolecular Chemistry, University of Wisconsin-Madison, Madison, WI 53706

Received: June 30, 2012

Revised: February 28, 2013

Published: May 2013

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Abstract

We describe a necessary condition for zero-eigenvalue Turing instability, i.e., Turing instability arising from a real eigenvalue changing sign from negative to positive, for general chemical reaction networks modeled with mass-action kinetics. The reaction mechanisms are represented by the species-reaction graph (SR graph), which is a bipartite graph with different nodes representing species and reactions. If the SR graph satisfies certain conditions, similar to the conditions for ruling out multiple equilibria in spatially homogeneous differential equations systems, then the corresponding mass-action reaction-diffusion system cannot exhibit zero-eigenvalue Turing instability for any parameter values. On the other hand, if the graph-theoretic condition for ruling out zero-eigenvalue Turing instability is not satisfied, then the corresponding model may display zero-eigenvalue Turing instability for some parameter values. The technique is illustrated with a model of a bifunctional enzyme.

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Mathematics Subject Classification: Primary: 92C15; Secondary: 80A30.

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