Zero-Eigenvalue Turing Instability in General Chemical Reaction Networks

Abstract

Biochemical reaction networks with diffusion are usually modeled by reaction-diffusion systems of equations and are studied in connection with pattern formation in biology. We say thatВ В  Turing instability occurs if a spatially homogeneous equilibrium is asymptotically stable as a solution of the ordinary differential equation system and unstable as a solution ofВ В  the correspondingВ  reaction-diffusion systems of equations. We describe aВ  necessary condition for zero-eigenvalue Turing instability, i.e., Turing instability arising from a real eigenvalue changing sign from negative to positive,В  for generalВ В  chemical reaction networksВ  with any number of species,В  modeled with mass-action kinetics. The reaction mechanisms are represented by the species-reaction graph (SR graph) which is an undirected bipartite graph. If the SR graph satisfies certain conditions, similar to the conditions for ruling outВ  multiple equilibria in spatially homogeneous differential equations systems, then the correspondingВ  mass-action reaction-diffusion system cannot exhibit zero-eigenvalue Turing instability for anyВ  parameter values, rate constants and diffusion coefficients.