The Glansdorff–Prigogine stability criterion for biochemical reaction networks

Abstract

Local stability analysis of steady states in mathematical models of biochemical reaction networks is an important tool for systems biology. The second variation of the Gibbs energy around a steady state is a positive definite function and a candidate for a Lyapunov function. A sufficient condition for the local stability is the local negative definiteness of the time derivative of this function. This is expressed by the Glansdorff–Prigogine stability criterion. Previously, the criterion was criticized to be overly conservative and difficult to check. Here, we derive an easily testable form of the criterion for models of biochemical networks. The criterion can be evaluated with incomplete knowledge of the parameters. For ideal mass-action kinetics, it depends only on the steady state fluxes. For reaction systems in ideal solutions, the Glansdorff–Prigogine criterion is overly conservative and we give a tighter criterion that depends on the same subset of the parameters as the Glansdorff–Prigogine criterion. Whenever these criteria are indefinite, there exist parameter values such that the steady state is unstable. By means of simple example systems we explore these aspects and discuss the possible uses of the criteria for systems biology.