Positive equilibria in mass action networks: geometry and bounds

TL;DR

This paper introduces alternative polynomial systems linked to mass action networks, enabling easier analysis of positive equilibria, their bounds, and bifurcations, especially for quadratic networks.

Contribution

It develops a novel approach to study positive equilibria via alternative systems, simplifying analysis and deriving bounds and properties, with a focus on quadratic networks.

Findings

Alternative systems capture equilibria and degeneracy properties.

Bounds on the number of positive nondegenerate equilibria are established.

Techniques for quadratic networks lead to strengthened results.

Abstract

Any mass action network gives rise to a parameterised family of polynomial equations whose positive solutions are the positive equilibria of the network. Here, we consider alternative systems of equations, whose solutions are in smooth, one-to-one correspondence with positive equilibria of the network, and capture degeneracy or nondegeneracy of the corresponding equilibria. The construction leads us to consider partitions of networks in a natural sense, and we explore the implications of choosing different partitions. The alternative systems are in some situations simpler than the original mass action equations, which allows us to rapidly identify various algebraic and geometric properties of the positive equilibrium set. This includes the characterisation of toricity and local toricity, bounds on the number of positive nondegenerate equilibria on stoichiometric classes, semialgebraic descriptions of the parameter regions for multistationarity, and the study of bifurcations. After discussing the construction of the alternative systems, various consequences for particular classes of networks and numerous examples are presented. We also develop additional techniques specifically for quadratic networks, the most common class of networks in applications, and use these techniques to derive strengthened results for quadratic networks.