Graph-theoretic methods for the analysis of chemical and biochemical networks. II. Oscillations in networks with delays Authors Maya Mincheva Department of Chemistry and Biochemistry University of Lethbridge Department of Mathematics University of Wisconsin-Madison Marc R. Roussel Department of Chemistry and Biochemistry University of Lethbridge Article

First Online: 31 May 2007 Received: 24 December 2005 Revised: 15 March 2007 DOI :
10.1007/s00285-007-0098-2

Cite this article as: Mincheva, M. & Roussel, M.R. J. Math. Biol. (2007) 55: 87. doi:10.1007/s00285-007-0098-2
Abstract Delay-differential equations are commonly used to model genetic regulatory systems with the delays representing transcription and translation times. Equations with delayed terms can also be used to represent other types of chemical processes. Here we analyze delayed mass-action systems, i.e. systems in which the rates of reaction are given by mass-action kinetics, but where the appearance of products may be delayed. Necessary conditions for delay-induced instability are presented in terms both of a directed graph (digraph) constructed from the Jacobian matrix of the corresponding ODE model and of a species-reaction bipartite graph which directly represents a chemical mechanism. Methods based on the bipartite graph are particularly convenient and powerful. The condition for a delay-induced instability in this case is the existence of a subgraph of the bipartite graph containing an odd number of cycles of which an odd number are negative.

Keywords Chemical reactions Graph Delay-induced instability Oscillations This work was funded by the Natural Sciences and Engineering Research Council of Canada.

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