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Multi‐equilibrium property of metabolic networks: MMN module
Author(s) -
Guo Jin,
Zhang JiFeng,
Zhao Yanlong
Publication year - 2012
Publication title -
international journal of robust and nonlinear control
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.361
H-Index - 106
eISSN - 1099-1239
pISSN - 1049-8923
DOI - 10.1002/rnc.2945
Subject(s) - mismatch negativity , injective function , jacobian matrix and determinant , mathematics , property (philosophy) , set (abstract data type) , pure mathematics , computer science , arithmetic , topology (electrical circuits) , combinatorics , psychology , philosophy , electroencephalography , epistemology , psychiatry , programming language
SUMMARY This paper studies the multi‐equilibrium property of the multiple substrates and multiple products with no inhibition (MMN) module. On the basis of the topological structure, a model for such module is established in the form of a set of nonlinear ordinary differential equations. It is shown that the injectivity of the MMN module is equivalent to the nonsingularity of Jacobian matrix of its rate function, and a necessary and sufficient condition for the injectivity is obtained by using the Hadamard product. For non‐injective MMN module, a sufficient condition for existence of multiple positive equilibria is provided by introducing the concept of input‐matrix. For a type of commonly encountered MMN module— A ‐MMN module—a structure‐oriented criterion for judging its injectivity is given. For A ‐MMN modules with some special structure, it is shown that there does not exist multiply equilibria and the equilibrium (if exists) is asymptotically stable. Examples and simulations are given to illustrate the results obtained. Copyright © 2012 John Wiley & Sons, Ltd.