Global injectivity and multiple equilibria in uni- and bi-molecular reaction networks

Casian Pantea; Heinz Koeppl; Gheorghe Craciun

doi:10.3934/dcdsb.2012.17.2153

Article Contents

2012, Volume 17, Issue 6: 2153-2170. Doi: 10.3934/dcdsb.2012.17.2153

This issue Previous Article A discrete dynamical system arising in molecular biology Next Article Dynamics of a two-receptor binding model: How affinities and capacities translate into long and short time behaviour and physiological corollaries

Global injectivity and multiple equilibria in uni- and bi-molecular reaction networks

1.
Department of Electrical and Electronic Engineering, Imperial College London, London, SW7 2AZ
2.
Automatic Control Laboratory, Department of Information Technology and Electrical Engineering, Swiss Federal Institute of Technology (ETH) Zürich, Zürich, ETL I26 8092
3.
Department of Mathematics and Department of Biomolecular Chemistry, University of Wisconsin-Madison, Madison, WI 53706

Received: May 31, 2011

Revised: August 31, 2011

Published: August 2012

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Abstract

Dynamical system models of complex biochemical reaction networks are high-dimensional, nonlinear, and contain many unknown parameters. The capacity for multiple equilibria in such systems plays a key role in important biochemical processes. Examples show that there is a very delicate relationship between the structure of a reaction network and its capacity to give rise to several positive equilibria. In this paper we focus on networks of reactions governed by mass-action kinetics. As is almost always the case in practice, we assume that no reaction involves the collision of three or more molecules at the same place and time, which implies that the associated mass-action differential equations contain only linear and quadratic terms. We describe a general injectivity criterion for quadratic functions of several variables, and relate this criterion to a network's capacity for multiple equilibria. In order to take advantage of this criterion we look for explicit general conditions that imply non-vanishing of polynomial functions on the positive orthant. In particular, we investigate in detail the case of polynomials with only one negative monomial, and we fully characterize the case of affinely independent exponents. We describe several examples, including an example that shows how these methods may be used for designing multistable chemical systems in synthetic biology.

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Mathematics Subject Classification: 80A30, 92C45, 37C25, 65H10.

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