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Decomposable and essentially univariate mass-action systems: Extensions of the deficiency one theorem

Abhishek Deshpande, Stefan Müller

TL;DR

The work generalizes the deficiency one theorem by introducing dependency-one concepts based on monomial dependency and a coefficient-polytope framework, enabling modular analysis of mass-action networks decomposed into independent subnetworks. By recasting positive equilibria as parametrized generalized polynomial equations and applying Birch’s theorem within a decomposable structure, the authors obtain a dependency-one theorem for essentially univariate subsystems and extend it to mass-action systems with arbitrary deficiency and multiple terminal components. This yields a concise, modular proof of the extended deficiency-one theorem and recovers classical results as special cases, while providing practical criteria for the existence and uniqueness of positive equilibria in each stoichiometric and kinetic compatibility class. The framework is validated through concrete examples where traditional deficiency-one conditions fail, yet the dependency-one criteria guarantee a unique positive equilibrium.

Abstract

The classical and extended deficiency one theorems by Feinberg apply to reaction networks with mass-action kinetics that have independent linkage classes or subnetworks, each with a deficiency of at most one and exactly one terminal strong component. The theorems assume the existence of a positive equilibrium and guarantee the existence of a unique positive equilibrium in every stoichiometric compatibility class. In our work, we use the $\textit{monomial dependency}$ which extends the concept of deficiency. First, we provide a dependency one theorem for parametrized systems of polynomial equations that are essentially univariate and decomposable. As our main result, we present a corresponding theorem for mass-action systems, which permits subnetworks with arbitrary deficiency and arbitrary number of terminal strong components. Finally, to complete the picture, we derive the extended deficiency one theorem as a special case of our more general dependency one theorem.

Decomposable and essentially univariate mass-action systems: Extensions of the deficiency one theorem

TL;DR

The work generalizes the deficiency one theorem by introducing dependency-one concepts based on monomial dependency and a coefficient-polytope framework, enabling modular analysis of mass-action networks decomposed into independent subnetworks. By recasting positive equilibria as parametrized generalized polynomial equations and applying Birch’s theorem within a decomposable structure, the authors obtain a dependency-one theorem for essentially univariate subsystems and extend it to mass-action systems with arbitrary deficiency and multiple terminal components. This yields a concise, modular proof of the extended deficiency-one theorem and recovers classical results as special cases, while providing practical criteria for the existence and uniqueness of positive equilibria in each stoichiometric and kinetic compatibility class. The framework is validated through concrete examples where traditional deficiency-one conditions fail, yet the dependency-one criteria guarantee a unique positive equilibrium.

Abstract

The classical and extended deficiency one theorems by Feinberg apply to reaction networks with mass-action kinetics that have independent linkage classes or subnetworks, each with a deficiency of at most one and exactly one terminal strong component. The theorems assume the existence of a positive equilibrium and guarantee the existence of a unique positive equilibrium in every stoichiometric compatibility class. In our work, we use the which extends the concept of deficiency. First, we provide a dependency one theorem for parametrized systems of polynomial equations that are essentially univariate and decomposable. As our main result, we present a corresponding theorem for mass-action systems, which permits subnetworks with arbitrary deficiency and arbitrary number of terminal strong components. Finally, to complete the picture, we derive the extended deficiency one theorem as a special case of our more general dependency one theorem.

Paper Structure

This paper contains 17 sections, 15 theorems, 61 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Previous results on parametrized systems of polynomial equations
  3. Dependency one systems
  4. One class
  5. Decomposable systems
  6. Reaction networks
  7. Network decomposition
  8. Positive equilibria
  9. "Independent subnetworks" and "decomposability"
  10. Dependency one mass-action systems
  11. Deficiency one
  12. Examples
  13. Proof of Theorem \ref{['thm:existence']}
  14. Reaction networks
  15. Index notation
  16. ...and 2 more sections

Key Result

Theorem 1

Consider the parametrized system of generalized polynomial equations $A \, (c \circ x^B) = 0$. The solution set $Z_c = \{ x \in \mathbb{R}^n_> \mid A \, (c \circ x^B) = 0 \}$ can be written as where is the solution set on the coefficient polytope $P$.

Figures (3)

  • Figure 1: Mass-action system $(G_k,y)$ with 5 vertices and 4 source vertices and hence $\delta = 5-1-2 = 2$, but $d = 4-1-2 = 1$.
  • Figure 2: Mass-action system $(G_k,y)$ with 5 vertices and 4 source vertices and hence $\delta = 5-1-2 = 2$, but $d = 4-1-2 = 1$.
  • Figure 3: Mass-action system $(G_k,y)$ with 5 vertices and 3 source vertices and hence $\delta = 5-1-2 = 2$, but $d = 3-1-2 = 0$.

Theorems & Definitions (35)

  • Theorem 1: MuellerRegensburger2023b, Theorem 1
  • Theorem 2: Birch's theorem
  • Definition 3
  • Theorem 4: $d=1$, one class
  • proof
  • Theorem 5
  • proof
  • Definition 6
  • Theorem 7: $d \le 1$
  • proof
  • ...and 25 more