Bulletin of Mathematical Biology

, Volume 73, Issue 10, pp 2277–2304

Oscillations in Biochemical Reaction Networks Arising from Pairs of Subnetworks

Original Article


Biochemical reaction models show a variety of dynamical behaviors, such as stable steady states, multistability, and oscillations. Biochemical reaction networks with generalized mass action kinetics are represented as directed bipartite graphs with nodes for species and reactions. The bipartite graph of a biochemical reaction network usually contains at least one cycle, i.e., a sequence of nodes and directed edges which starts and ends at the same species node. Cycles can be positive or negative, and it has been shown that oscillations can arise as a result of either a positive cycle or a negative cycle. In earlier work it was shown that oscillations associated with a positive cycle can arise from subnetworks with an odd number of positive cycles. In this article we formulate a similar graph-theoretic condition, which generalizes the negative cycle condition for oscillations. This new graph-theoretic condition for oscillations involves pairs of subnetworks with an even number of positive cycles. An example of a calcium reaction network with generalized mass action kinetics is discussed in detail.


Biochemical and chemical reaction networks Bipartite graph Generalized mass action kinetics Oscillations Negative feedback cycle 


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Copyright information

© Society for Mathematical Biology 2011

Authors and Affiliations

  1. 1.Department of Mathematical SciencesNorthern Illinois UniversityDeKalbUSA

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