Multi‐equilibrium property of metabolic networks: Exclusion of multi‐stability for SSN metabolic modules

Abstract

SUMMARYIt is a fundamental and important problem whether or not a metabolic network can admit multiple equilibria in a living organism. 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 viewpoints. In this paper, a structure-oriented modularization research framework is proposed to analyze the multi-stability of metabolic networks. We first decompose a metabolic network into four types of basic building blocks (called metabolic modules) according to the particularity of its structure, and then focus on one type of these basic building blocksthe 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 its Jacobian matrix, which enables us equivalently to convert an unverifiable sufficient condition for the absence of multiple equilibria of an SSN module into a verifiable one. Moreover, we prove that this sufficient condition holds for the SSN module in a living organism. Such a theoretical result not only provides a general framework for modeling metabolic networks, but also shows that the SSN module in a living organism cannot be multi-stable.

Analysis of complex balanced chemical reaction networks with fixed boundary concentrations

“…, n1 , k i and k´i denote the forward and the reverse reaction constants respectively of the i th reaction. As shown in (1), some of the species in the network can have constant influx and/or proportional efflux, where for i " 1, . .…”

“…The SS networks are defined (see also [1,Definition 2.1]) as reaction networks in which each reaction has one species as a reactant and another species as a product. Furthermore, we assume that our reaction networks are connected.…”

Multi‐equilibrium property of metabolic networks: MMN module

“…Multi‐equilibrium of metabolic networks is a fundamental and significant property in systems biology , including the capacity for admitting multiple equilibria and the dynamic behaviors of the equilibria. The former considers the existence of multiple equilibria and the latter focuses mainly on their stabilities.…”

“…On the basis of the topological structure of networks, some works related to the multi‐equilibrium property have been carried out in , and a summary on these works can be found in . For chemical reaction networks, Craciun and Feinberg developed a very rich theory to injectivity tests by means of the Jacobian matrix of the rate function based on the law of mass action.…”

“… proposed a ‘modularization’ idea: regarding a metabolic network as an assembly of some basic building blocks (called metabolic modules) with specific structures and functions, then to investigate the entire network by analyzing the functional characteristic and interactions of these basic modules. According to the numbers of the substrates and products and the existence of inhibition, decomposed a metabolic network into four classes of basic modules, called the Single substrate and Single product with No inhibition (SSN), the Single substrate and Single product with Inhibition (SSI), the Multiple substrate and Multiple product with No inhibition (MMN) and the Multiple substrate and Multiple product with Inhibition (MMI), respectively. Here, a feasible method of realizing such decomposition may be to take initially each metabolic reaction as a module and then extend it to the maximum through adding its neighbor nodes according to the definition of basic module.…”