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

Pantea, C. and Koeppl, H. and Craciun, G. (2012):
Global injectivity and multiple equilibria in uni- and bi-molecular reaction networks.
In: Discrete and Continuous Dynamical Systems - Series B, American Institute of Mathematical Sciences, pp. 2153-2170, 17, (6), [Online-Edition: https://aimsciences.org/journals/displayArticlesnew.jsp?pape...],
[Article]

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.

Item Type: Article
Erschienen: 2012
Creators: Pantea, C. and Koeppl, H. and Craciun, G.
Title: Global injectivity and multiple equilibria in uni- and bi-molecular reaction networks
Language: English
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.

Journal or Publication Title: Discrete and Continuous Dynamical Systems - Series B
Volume: 17
Number: 6
Publisher: American Institute of Mathematical Sciences
Divisions: 18 Department of Electrical Engineering and Information Technology > Institute for Telecommunications > Bioinspired Communication Systems
18 Department of Electrical Engineering and Information Technology
18 Department of Electrical Engineering and Information Technology > Institute for Telecommunications
Date Deposited: 04 Apr 2014 11:40
Official URL: https://aimsciences.org/journals/displayArticlesnew.jsp?pape...
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