Domijan, Mirela and Kirkilionis, Markus (2009) Bistability and oscillations in chemical reaction networks. Journal of Mathematical Biology, Vol.59 (No.4). pp. 467-501. doi:10.1007/s00285-008-0234-7 ISSN 0303-6812.

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Abstract

Bifurcation theory is one of the most widely used approaches for analysis of dynamical behaviour of chemical and biochemical reaction networks. Some of the interesting qualitative behaviour that are analyzed are oscillations and bistability (a situation where a system has at least two coexisting stable equilibria). Both phenomena have been identified as central features of many biological and biochemical systems. This paper, using the theory of stoichiometric network analysis (SNA) and notions from algebraic geometry, presents sufficient conditions for a reaction network to display bifurcations associated with these phenomena. The advantage of these conditions is that they impose fewer algebraic conditions on model parameters than conditions associated with standard bifurcation theorems. To derive the new conditions, a coordinate transformation will be made that will guarantee the existence of branches of positive equilibria in the system. This is particularly useful in mathematical biology, where only positive variable values are considered to be meaningful. The first part of the paper will be an extended introduction to SNA and algebraic geometry-related methods which are used in the coordinate transformation and set up of the theorems. In the second part of the paper we will focus on the derivation of bifurcation conditions using SNA and algebraic geometry. Conditions will be derived for three bifurcations: the saddle-node bifurcation, a simple branching point, both linked to bistability, and a simple Hopf bifurcation. The latter is linked to oscillatory behaviour. The conditions derived are sufficient and they extend earlier results from stoichiometric network analysis as can be found in (Aguda and Clarke in J Chem Phys 87:3461-3470, 1987; Clarke and Jiang in J Chem Phys 99:4464-4476, 1993; Gatermann et al. in J Symb Comput 40:1361-1382, 2005). In these papers some necessary conditions for two of these bifurcations were given. A set of examples will illustrate that algebraic conditions arising from given sufficient bifurcation conditions are not more difficult to interpret nor harder to calculate than those arising from necessary bifurcation conditions. Hence an increasing amount of information is gained at no extra computational cost. The theory can also be used in a second step for a systematic bifurcation analysis of larger reaction networks.

Item Type:	Journal Article
Subjects:	Q Science > QH Natural history > QH301 Biology
Divisions:	Faculty of Science, Engineering and Medicine > Science > Mathematics
Journal or Publication Title:	Journal of Mathematical Biology
Publisher:	Springer
ISSN:	0303-6812
Official Date:	October 2009
Dates:	Date Event October 2009 Published
Volume:	Vol.59
Number:	No.4
Number of Pages:	35
Page Range:	pp. 467-501
DOI:	10.1007/s00285-008-0234-7
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Restricted or Subscription Access
Funder:	Commonwealth Scholarship, European Commission
Grant number:	12990

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