Multiple equilibria in SSN metabolic module

In this paper, a novel modeling framework based on network modularization decomposition is proposed to analyze the multi-stability of metabolic networks. We first decompose a metabolic network into four types of elementary modules according to the special structures of metabolic systems, and then focus on one key type of these modules-the single substrate and single product with no inhibition (SSN) module, by deriving a nonlinear ordinary differential equation (ODE) model based on the Hill kinetics. We show that the injectivity of the vector field of the ODE model is equivalent to the nonsingularity of the Jacobian matrix of this vector field, which enables us equivalently to convert an unverifiable sufficient condition on the absence of multiple equilibria of an SSN module into a verifiable one. Moreover, we prove that this sufficient condition is held when the SSN module has both input and output nodes. Such a theoretical result not only provides a general framework for modeling an SSN metabolic module, but also implies that if a biological system can be exactly modeled by an SSN module, then it cannot admit multiple equilibria (or steady states).