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

TL;DR

This paper introduces a fast graph rewriting algorithm to verify refined graphical conditions for multistationarity in reaction networks, demonstrating efficiency and practical applicability on large biological models.

Contribution

First implementation of a graph rewriting algorithm for checking refined multistationarity conditions in reaction networks, with systematic evaluation on BioModels data.

Findings

Algorithm analyzes networks up to 430 species in under 0.05 seconds.

Identifies absence of multistationarity in 160 out of 506 networks.

Additional graphical condition detects 20 more networks at high computational cost.

Abstract

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.