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