Multi-equilibrium property of metabolic networks: SSI module

BACKGROUND: Revealing the multi-equilibrium property of a metabolic network is a fundamental and important topic in systems biology. Due to the complexity of the metabolic network, it is generally a difficult task to study the problem as a whole from both analytical and numerical viewpoint. On the other hand, the structure-oriented modularization idea is a good choice to overcome such a difficulty, i.e. decomposing the network into several basic building blocks and then studying the whole network through investigating the dynamical characteristics of the basic building blocks and their interactions. Single substrate and single product with inhibition (SSI) metabolic module is one type of the basic building blocks of metabolic networks, and its multi-equilibrium property has important influence on that of the whole metabolic networks. RESULTS: In this paper, we describe what the SSI metabolic module is, characterize the rates of the metabolic reactions by Hill kinetics and give a unified model for SSI modules by using a set of nonlinear ordinary differential equations with multi-variables. Specifically, a sufficient and necessary condition is first given to describe the injectivity of a class of nonlinear systems, and then, the sufficient condition is used to study the multi-equilibrium property of SSI modules. As a main theoretical result, for the SSI modules in which each reaction has no more than one inhibitor, a sufficient condition is derived to rule out multiple equilibria, i.e. the Jacobian matrix of its rate function is nonsingular everywhere. CONCLUSIONS: In summary, we describe SSI modules and give a general modeling framework based on Hill kinetics, and provide a sufficient condition for ruling out multiple equilibria of a key type of SSI module.