Multi-equilibrium property of metabolic networks: MMN module

This paper studies the multi-equilibrium property of multiple substrate or multiple product modules with no inhibition (MMN modules). The rates of the metabolic reactions are characterized by Michaelis-Menten kinetics. A unified model is given for MMN modules by using a set of nonlinear ordinary differential equations (ODEs) with multi-variables. It is shown that the injectivity of the ODE is equivalent to the non-singularity of its Jacobian matrix, and an easy-to-verify sufficient and necessary condition is obtained. For a general class of MMN modules named α-MMN modules, a criterion of injectivity based on topology structure is given, with which it is shown that a specific class of α-MMN modules cannot produce multiple equilibria, and the equilibrium if existent is asymptotically stable.