A flux-based approach for analyzing the disguised toric locus of reaction networks

TL;DR

The paper develops a flux-based framework to study the disguised toric locus $\mathcal{K}^{\operatorname{dt}}(G)$ of reaction networks, proving that it is a contractible manifold with boundary and that its interior is captured by a maximal weakly reversible realization graph $G^{\max}$. By introducing the disguised toric flux cone $\mathcal{F}^{\operatorname{dt}}(G)$ and establishing a homeomorphism between $({\boldsymbol{x}}_0+\mathcal{S})_{>0}\times\mathcal{F}^{\operatorname{dt}}(G)$ and $\mathcal{K}^{\operatorname{dt}}(G)$, the authors reduce the computation to linear programming (for flux cones) followed by a quantifier-elimination step (to recover rate constants). The method is demonstrated on several biologically relevant networks (square, Lotka–Volterra autocatalator, clock, tetrahedron, and higher-dimensional examples), showing that $\mathcal{K}^{\operatorname{dt}}(G)$ often has positive measure and is substantially larger than the toric locus $\mathcal{K}^{\operatorname{t}}(G)$. This framework enables efficient, scalable analysis of stability-friendly regimes across complex networks and provides a practical route to certify global stability via disguised toricity. The work thus advances the toolkit for dynamical systems analysis in biochemistry and ecology by turning a hard semi-algebraic problem into a tractable combination of linear and nonlinear steps.

Abstract

Dynamical systems with polynomial right-hand sides are very important in various applications, e.g., in biochemistry and population dynamics. The mathematical study of these dynamical systems is challenging due to the possibility of multistability, oscillations, and chaotic dynamics. One important tool for this study is the concept of reaction systems, which are dynamical systems generated by reaction networks for some choices of parameter values. Among these, disguised toric systems are remarkably stable: they have a unique attracting fixed point, and cannot give rise to oscillations or chaotic dynamics. The computation of the set of parameter values for which a network gives rise to disguised toric systems (i.e., the disguised toric locus of the network) is an important but difficult task. We introduce new ideas based on network fluxes for studying the disguised toric locus. We prove that the disguised toric locus of any network $G$ is a contractible manifold with boundary, and introduce an associated graph $G^{\max}$ that characterizes its interior. These theoretical tools allow us, for the first time, to compute the full disguised toric locus for many networks of interest.

Figures (2)

• Figure 1: Some slices of the toric flux cone $\mathcal{F}^{\operatorname{t}}(G)$ and the disguised toric flux cone $\mathcal{F}^{\operatorname{dt}}(G)$ (left), as well as the toric locus $\mathcal{K}^{\operatorname{t}}(G)$ and the disguised toric locus $\mathcal{K}^{\operatorname{dt}}(G)$ (right) for the partly reversible square $G$ from \ref{['ex:running_example']}.
• Figure 2: Illustration of the idea of the proof of \ref{['lem:rank-1']}.

Theorems & Definitions (50)

• Theorem A: \ref{['thm:homeo_Psi']} and \ref{['cor:Kdt_G']}
• Example 1.1
• Theorem B: \ref{['thm:G_max_enough_K']} and \ref{['cor:Kdt_G']}
• Example 1.2
• Definition 2.1
• Definition 2.2
• Remark 2.3
• Definition 2.4
• Definition 2.5
• Definition 2.6