Multi‐equilibrium property of metabolic networks: MMN module

Abstract

SUMMARYThis paper studies the multi‐equilibrium property of the multiple substrates and multiple products with no inhibition (MMN) module. On the basis of the topological structure, a model for such module is established in the form of a set of nonlinear ordinary differential equations. It is shown that the injectivity of the MMN module is equivalent to the nonsingularity of Jacobian matrix of its rate function, and a necessary and sufficient condition for the injectivity is obtained by using the Hadamard product. For non‐injective MMN module, a sufficient condition for existence of multiple positive equilibria is provided by introducing the concept of input‐matrix. For a type of commonly encountered MMN module— scriptA‐MMN module—a structure‐oriented criterion for judging its injectivity is given. For scriptA‐MMN modules with some special structure, it is shown that there does not exist multiply equilibria and the equilibrium (if exists) is asymptotically stable. Examples and simulations are given to illustrate the results obtained. Copyright © 2012 John Wiley & Sons, Ltd.