Journal of Mathematical Biology

, Volume 65, Issue 2, pp 375–402 | Cite as

The interaction graph structure of mass-action reaction networks



Behaviour of chemical networks that are described by systems of ordinary differential equations can be analysed via the associated graph structures. This paper deals with observations based on the interaction graph which is defined by the signs of the Jacobian matrix entries. Some of the important graph structures linked to network dynamics are signed circuits and the nucleus (or Hamiltonian hooping). We use mass-action chemical reaction networks as an example to showcase interesting observations about the aforementioned interaction graph structures. We show that positive circuits and specific nucleus structures (associated to multistationarity) are always present in a great generic class of mass-action chemical and biological networks. The theory of negative circuits remains poorly understood, but there is some evidence that they are indicators of stable periodicity. Here we introduce the concept of non-isolated circuits which indicate the presence of a negative circuit.


Multistationarity Oscillations Feedback circuits Interaction graph Jacobian matrix 

Mathematics Subject Classification (2000)

34C10 92C42 


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Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Warwick Systems Biology Centre, Coventry HouseUniversity of WarwickCoventryUK
  2. 2.Laboratoire Jean-Alexandre Dieudonné, U.M.R. C.N.R.S. 6621Université de Nice Sophia-Antipolis Parc ValroseNice Cedex 02France

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