A Stronger Necessary Condition for the Multistationarity of Chemical Reaction Networks

Abstract

To cite this version:Sylvain Soliman. A stronger necessary condition for the multistationarity of chemical reaction networks. Bulletin of Mathematical Biology, Springer Verlag, 2013, 75 (11) Abstract 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.

Inferring reaction systems from ordinary differential equations

“…for oscillations, e.g. for homeostasis) in different formalisms, and in particular for ODE systems in [43,39,40,41,42] and recently in [38] for the ODE semantics of non-linear reaction systems.…”

“…However, Thomas's original condition provides no information in presence of reactions with two reactants, since a reaction like for instance A + B −→ C immediately creates a positive circuit of negative influences between A and B in the associated SIG and DIG for any reasonable kinetics. This counter-example has been recently rule out in [38], where it is shown that Thomas's conditions can be made stronger for reactions models, by labeling the influence edges by the reactions they come from, and by restricting the analysis of circuits to circuits labeled by different reactions. With this stronger condition for multi-stationarity, the analysis of labeled circuits in the DIG of a reaction system does provide information on its capabilities of exhibiting multi-stationarity.…”

“…This result generalizes a previous result in [36,37] to reactions with inhibitors. It shows that the influence graph of a well-formed reaction system with inhibitors is essentially independent of the kinetics, can be computed in linear time in the number of reactions when the number of species per reaction is bounded, and can thus advantageously be used to perform multi-stationarity analyses by circuit analysis à la Thomas [38,39,40,41,42,43,5,4,3].…”

On the multistationarity of chemical reaction networks

“…In this paper we are looking for a common generalization of [16] and [22]. We state this as a conjecture about chemical reaction networks (Conjecture 1).…”

“…In Section 3, following [22], [6] and [24], we introduce the reaction labelled influence graph of a chemical reaction network, and we use it to formulate the main result of [22], with three important corollaries. Next, we state the main result of [16] (Theorem 2).…”