Bulletin of Mathematical Biology

, Volume 75, Issue 11, pp 2289–2303 | Cite as

A Stronger Necessary Condition for the Multistationarity of Chemical Reaction Networks

  • Sylvain SolimanEmail author
Original Article


Biochemical reaction networks grow bigger and bigger, fed by the high-throughput data provided by biologists and bred in open repositories of models allowing merging and evolution. Nevertheless, since the available data is still very far from permitting the identification of the increasing number of kinetic parameters of such models, the necessity of structural analyses for describing the dynamics of chemical networks appears stronger every day.

Using the structural information, notably from the stoichiometric matrix, of a biochemical reaction system, we state a more strict version of the famous Thomas’ necessary condition for multistationarity. In particular, the obvious cases where Thomas’ condition was trivially satisfied, mutual inhibition due to a multimolecular reaction and mutual activation due to a reversible reaction, can now easily be ruled out.

This more strict condition shall not be seen as some version of Thomas’ circuit functionality for the continuous case but rather as related and complementary to the whole domain of the structural analysis of (bio)chemical reaction systems, as pioneered by the chemical reaction network theory.


Jacobian matrix Influence graph Feedback circuit Multistationarity Chemical reaction network 



We would like to thank Steven Gay, a Ph.D. student in our team, who was the first to formulate the conjecture proven in this article, as well as its variant on oscillations. We also thank Christophe Soulé for very interesting discussions that have led to Sect. 5. Finally, we thank the whole Contraintes team for numerous and fruitful discussions on this topic.


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Copyright information

© Society for Mathematical Biology 2013

Authors and Affiliations

  1. 1.Equipe ContraintesInria Paris-RocquencourtLe Chesnay CedexFrance

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