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Thomas's necessary conditions for the existence of multiple steady states ingene networks have been proved by Soul\'e with high generality for dynamicalsystems defined by differential equations. When applied to (protein) reactionnetworks however, those conditions do not provide information since they aretrivially satisfied as soon as there is a bimolecular or a reversible reaction.Refined graphical requirements have been proposed to deal with such cases. Inthis paper, we present for the first time a graph rewriting algorithm forchecking the refined conditions given by Soliman, and evaluate its practicalperformance by applying it systematically to the curated branch of theBioModels repository. This algorithm analyzes all reaction networks (of size upto 430 species) in less than 0.05 second per network, and permits to concludeto the absence of multistationarity in 160 networks over 506. The shortcomputation times obtained in this graphical approach are in sharp contrast tothe Jacobian-based symbolic computation approach. We also discuss the case ofone extra graphical condition by arc rewiring that allows us to conclude on 20more networks of this benchmark but with a high computational cost. Finally, westudy with some details the case of phosphorylation cycles and MAPK signallingmodels which show the importance of modelling the intermediate complexationswith the enzymes in order to correctly analyze the multistationaritycapabilities of such biochemical reaction networks.

Graphical Requirements for Multistationarity in Reaction Networks and their Verification in BioModels