Graphtheoretic methods for the analysis of chemical and biochemical networks. II. Oscillations in networks with delays
 Maya Mincheva,
 Marc R. Roussel
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Delaydifferential 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 massaction systems, i.e. systems in which the rates of reaction are given by massaction kinetics, but where the appearance of products may be delayed. Necessary conditions for delayinduced instability are presented in terms both of a directed graph (digraph) constructed from the Jacobian matrix of the corresponding ODE model and of a speciesreaction bipartite graph which directly represents a chemical mechanism. Methods based on the bipartite graph are particularly convenient and powerful. The condition for a delayinduced 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.
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 Title
 Graphtheoretic methods for the analysis of chemical and biochemical networks. II. Oscillations in networks with delays
 Journal

Journal of Mathematical Biology
Volume 55, Issue 1 , pp 87104
 Cover Date
 20070701
 DOI
 10.1007/s0028500700982
 Print ISSN
 03036812
 Online ISSN
 14321416
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 Chemical reactions
 Graph
 Delayinduced instability
 Oscillations
 34K18
 Authors

 Maya Mincheva ^{(1)} ^{(2)}
 Marc R. Roussel ^{(1)}
 Author Affiliations

 1. Department of Chemistry and Biochemistry, University of Lethbridge, Lethbridge, AB T1K 3M4, Canada
 2. Department of Mathematics, University of WisconsinMadison, Madison, WI, 537061388, USA