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

Authors

  • Hong-Bo Lei,

    1. Key Laboratory of Systems and Control, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, People's Republic of China
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  • Ji-Feng Zhang,

    Corresponding author
    1. Key Laboratory of Systems and Control, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, People's Republic of China
    • Key Laboratory of Systems and Control, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, People's Republic of China
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  • Luonan Chen

    1. Key Laboratory of Systems Biology, Shanghai Institutes for Biological Sciences, Chinese Academy of Sciences, Shanghai 200031, People's Republic of China
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SUMMARY

It 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 blocks—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 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. Copyright © 2011 John Wiley & Sons, Ltd.

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