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Positive equilibria in mass action networks: geometry and bounds

Murad Banaji, Elisenda Feliu

TL;DR

This work develops a partitioned, Gale-duality-based framework to study the geometry of positive equilibria in mass action networks. By recasting the equilibrium conditions into alternative, lower-dimensional solvability systems tied to partitions of the network, the authors derive explicit parameterisations of $\mathcal{E}_\kappa$, characterize degeneracy, and obtain bounds (Bézout, BKK, and combinatorial) on the number of positive equilibria on stoichiometric classes. They introduce the finest partition to maximise toricity and present a rich set of results, including toric/locally toric structures and strong bounds for quadratic networks, with direct implications for multistationarity regions and bifurcation analysis. The framework yields concrete, implementable criteria for when equilibria are finite, unique, or degenerate, and provides monomial parameterisations in many cases, offering practical tools for analyzing complex biochemical networks and their dynamical behavior.

Abstract

We present results on the geometry of the positive equilibrium set of a mass action network. 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 start by deriving 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 derivation leads us to consider partitions of networks in a natural sense, and we explore the implications of choosing different partitions. The alternative systems are often simpler than the original mass action equations, sometimes giving explicit parameterisations of positive equilibria, and allowing us to rapidly identify various algebraic and geometric properties of the positive equilibrium set, including toricity and local toricity. We can use the approaches we develop to bound the number of positive nondegenerate equilibria on stoichiometric classes; to derive semialgebraic descriptions of the parameter regions for multistationarity; and to study bifurcations. We present the main construction, various consequences for particular classes of networks, and numerous examples. 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.

Positive equilibria in mass action networks: geometry and bounds

TL;DR

This work develops a partitioned, Gale-duality-based framework to study the geometry of positive equilibria in mass action networks. By recasting the equilibrium conditions into alternative, lower-dimensional solvability systems tied to partitions of the network, the authors derive explicit parameterisations of , characterize degeneracy, and obtain bounds (Bézout, BKK, and combinatorial) on the number of positive equilibria on stoichiometric classes. They introduce the finest partition to maximise toricity and present a rich set of results, including toric/locally toric structures and strong bounds for quadratic networks, with direct implications for multistationarity regions and bifurcation analysis. The framework yields concrete, implementable criteria for when equilibria are finite, unique, or degenerate, and provides monomial parameterisations in many cases, offering practical tools for analyzing complex biochemical networks and their dynamical behavior.

Abstract

We present results on the geometry of the positive equilibrium set of a mass action network. 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 start by deriving 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 derivation leads us to consider partitions of networks in a natural sense, and we explore the implications of choosing different partitions. The alternative systems are often simpler than the original mass action equations, sometimes giving explicit parameterisations of positive equilibria, and allowing us to rapidly identify various algebraic and geometric properties of the positive equilibrium set, including toricity and local toricity. We can use the approaches we develop to bound the number of positive nondegenerate equilibria on stoichiometric classes; to derive semialgebraic descriptions of the parameter regions for multistationarity; and to study bifurcations. We present the main construction, various consequences for particular classes of networks, and numerous examples. 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.
Paper Structure (22 sections, 36 theorems, 109 equations, 1 figure)

This paper contains 22 sections, 36 theorems, 109 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries on chemical reaction networks
  3. Existence and number of positive equilibria: general theory
  4. Partition subnetworks
  5. The main equation for positive equilibria
  6. Four matrices associated with a general mass action network
  7. Parameterising the equilibrium set
  8. Positive equilibria in stoichiometric classes
  9. Alternative Bézout and BKK bounds
  10. The finest partition
  11. Special cases
  12. $\mathcal{P}$-overdetermined and source deficient networks
  13. $\mathcal{P}$-toric networks
  14. Networks of full rank with few sources
  15. Networks with $\mathcal{P}$-independent sources
  16. ...and 7 more sections

Key Result

Proposition 2.1

Given a reaction network, let $\mathcal{Z} \subseteq \mathbb{R}^m_+$ be the set of rate constants $\kappa$ for which $\mathcal{E}_\kappa\neq \emptyset$, and let $\mathcal{Z}_{sc} \subseteq \mathbb{R}^m_+ \times \mathbb{R}^{n-r}$ be the set of rate constants and total amounts $(\kappa,K)$ for which t Additionally:

Figures (1)

  • Figure 1: Good labellings in \ref{['lemcomb']}(iv)-(vi).

Theorems & Definitions (96)

  • Proposition 2.1
  • Proposition 2.2
  • Lemma 3.1
  • proof
  • Remark 3.2
  • Remark 3.3: The geometry of Equation \ref{['eqbasiclog1']}
  • Remark 3.4: Decompositions of Equation \ref{['eqbasiclog1']}
  • Lemma 3.5
  • proof
  • Remark 3.6: Empty matrices
  • ...and 86 more