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The Computation of the Disguised Toric Locus of Reaction Networks

Gheorghe Craciun, Abhishek Deshpande, Jiaxin Jin

TL;DR

This work advances the computational understanding of disguised toric dynamics in reaction networks by deriving and implementing an efficient method to compute the dimension of the disguised toric locus. It proves that the complex-balanced (toric) restrictions and the $\mathbb{R}$-realizability constraints act independently on the dimension, enabling a straightforward dimension formula. An explicit algorithm computes $\dim(\mathfrak{J}_{\mathbb{R}}(G',G))$ via vertex-wise kernel dimensions, which, together with classical graph-theoretic quantities, yields the dimension of the overall disguised toric locus. The approach is demonstrated on Brusselator-type, Thomas-type, and circadian clock models, where the disguised toric locus attains full dimension within the positive parameter space, underscoring its potential to reveal toric-like robustness in broader biological networks.

Abstract

Mathematical models of reaction networks can exhibit very complex dynamics, including multistability, oscillations, and chaotic dynamics. On the other hand, under some additional assumptions on the network or on parameter values, these models may actually be toric dynamical systems, which have remarkably stable dynamics. The concept of disguised toric dynamical system" was introduced in order to describe the phenomenon where a reaction network generates toric dynamics without actually being toric; such systems enjoy all the stability properties of toric dynamical systems but with much fewer restrictions on the networks and parameter values. The \emph{disguised toric locus} is the set of parameter values for which the corresponding dynamical system is a disguised toric system. Here we focus on providing a generic and efficient method for computing the dimension of the disguised toric locus of reaction networks. Additionally, we illustrate our approach by applying it to some specific models of biological interaction networks, including Brusselator-type networks, Thomas-type networks, and circadian clock networks.

The Computation of the Disguised Toric Locus of Reaction Networks

TL;DR

This work advances the computational understanding of disguised toric dynamics in reaction networks by deriving and implementing an efficient method to compute the dimension of the disguised toric locus. It proves that the complex-balanced (toric) restrictions and the -realizability constraints act independently on the dimension, enabling a straightforward dimension formula. An explicit algorithm computes via vertex-wise kernel dimensions, which, together with classical graph-theoretic quantities, yields the dimension of the overall disguised toric locus. The approach is demonstrated on Brusselator-type, Thomas-type, and circadian clock models, where the disguised toric locus attains full dimension within the positive parameter space, underscoring its potential to reveal toric-like robustness in broader biological networks.

Abstract

Mathematical models of reaction networks can exhibit very complex dynamics, including multistability, oscillations, and chaotic dynamics. On the other hand, under some additional assumptions on the network or on parameter values, these models may actually be toric dynamical systems, which have remarkably stable dynamics. The concept of disguised toric dynamical system" was introduced in order to describe the phenomenon where a reaction network generates toric dynamics without actually being toric; such systems enjoy all the stability properties of toric dynamical systems but with much fewer restrictions on the networks and parameter values. The \emph{disguised toric locus} is the set of parameter values for which the corresponding dynamical system is a disguised toric system. Here we focus on providing a generic and efficient method for computing the dimension of the disguised toric locus of reaction networks. Additionally, we illustrate our approach by applying it to some specific models of biological interaction networks, including Brusselator-type networks, Thomas-type networks, and circadian clock networks.

Paper Structure

This paper contains 8 sections, 10 theorems, 97 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Reaction Networks and Dynamical Equivalence
  4. Flux Systems and Flux Equivalence
  5. Disguised Toric Locus and R-disguised Toric Locus
  6. Computing the Dimension of the Disguised Toric Locus
  7. Applications
  8. Discussion

Key Result

Lemma 2.9

Let $(G, \boldsymbol{k})$ and $(G, \boldsymbol{k}')$ be two mass-action systems. Then $(G, \boldsymbol{k}) \sim (G, \boldsymbol{k}')$ if and only if $\boldsymbol{k}' - \boldsymbol{k} \in \mathcal{D}_{\textbf{0}} (G)$.

Figures (4)

  • Figure 1: Two E-graphs $G = (V, E)$ and $G' = (V', E')$.
  • Figure 2: (a) A version of the Selkov model, denoted by $G = (V, E)$. (b) The network $G' = (V', E')$, a weakly reversible subgraph of the complete graph associated with the source vertices of $G$.
  • Figure 3: (a) The Thomas-type model, denoted by $G = (V, E)$. (b) The network $G' = (V', E')$, a weakly reversible subgraph of the complete graph associated with the source vertices of $G$.
  • Figure 4: (a) The circadian clock model, denoted by $G = (V, E)$. (b) The network $G' = (V', E')$, a weakly reversible subgraph of the complete graph associated with the source vertices of $G$.

Theorems & Definitions (42)

  • Definition 2.1: craciun2015toriccraciun2019polynomialcraciun2020endotactic
  • Definition 2.2
  • Definition 2.3: adleman2014mathematicsguldberg1864studiesvoit2015150gunawardena2003chemicalyu2018mathematicalfeinberg1979lectures
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • Definition 2.7
  • Remark 2.8
  • Lemma 2.9: disg_2
  • Example 2.10
  • ...and 32 more