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The disguised toric locus and affine equivalence of reaction networks

Sabina J. Haque, Matthew Satriano, Miruna-Stefana Sorea, Polly Y. Yu

Abstract

Under the assumption of mass-action kinetics, a dynamical system may be induced by several different reaction networks and/or parameters. It is therefore possible for a mass-action system to exhibit complex-balancing dynamics without being weakly reversible or satisfying toric constraints on the rate constants; such systems are called disguised toric dynamical systems. We show that the parameters that give rise to such systems are preserved under invertible affine transformations of the network. We also consider the dynamics of arbitrary mass-action systems under affine transformations, and show that there is a bijection between their sets of positive steady states, although their qualitative dynamics can differ substantially.

The disguised toric locus and affine equivalence of reaction networks

Abstract

Under the assumption of mass-action kinetics, a dynamical system may be induced by several different reaction networks and/or parameters. It is therefore possible for a mass-action system to exhibit complex-balancing dynamics without being weakly reversible or satisfying toric constraints on the rate constants; such systems are called disguised toric dynamical systems. We show that the parameters that give rise to such systems are preserved under invertible affine transformations of the network. We also consider the dynamics of arbitrary mass-action systems under affine transformations, and show that there is a bijection between their sets of positive steady states, although their qualitative dynamics can differ substantially.
Paper Structure (10 sections, 7 theorems, 44 equations, 7 figures)

This paper contains 10 sections, 7 theorems, 44 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Reaction networks and mass-action systems
  4. Dynamical equivalence
  5. Complex-balanced and disguised toric dynamical systems
  6. Affine equivalence of networks
  7. Main result
  8. Extension to detailed-balanced systems
  9. Discussion
  10. The disguised toric locus and projective transformations

Key Result

Lemma 3.4

The net reaction vector of the vertex $\mathbf{A}({\boldsymbol{y}}_i)\in \mathbf{A}(V_s)$ in $(\mathbf{A}(G), {\boldsymbol{\kappa}})$ is $\mathbf{M} {\boldsymbol{w}}_i$, where ${\boldsymbol{w}}_i$ is the net reaction vector of the vertex ${\boldsymbol{y}}_i \in V_s$ in $(G,{\boldsymbol{\kappa}})$.

Figures (7)

  • Figure 1: A reaction network in $\mathbb{R}^2$. See \ref{['ex:intro-mas']} for its associated dynamics.
  • Figure 2: Three networks with vertices as labelled in (a). For rate constants satisfying some linear constraints as shown in \ref{['ex:DE']}, the corresponding mass-action systems can be made dynamically equivalent.
  • Figure 3: (a) A network that is dynamically equivalent to the reversible pair ${\boldsymbol{y}}_1 \rightleftharpoons {\boldsymbol{y}}_3$. (b) Its image under the map $(x,y) \mapsto (y, xy)$ can never be dynamically equivalent to a reversible pair.
  • Figure 4: (a) The image of the network in \ref{['fig:intro-mas']} under an invertible affine transformation (see \ref{['ex:doubletargets']}), and (b) its phase portrait. For comparison, (c) the phase portrait of the network in \ref{['fig:intro-mas']}. The rate constants are taken to be $\kappa_1 = \cdots = \kappa_4 = 1$ for simplicity.
  • Figure 5: (a) A multistationary mass-action system ($\kappa_1 = \kappa_2 = 1$) and (c) its phase portrait. (b) Its image under an affine map as in \ref{['ex:MS-not']}, and (d) its phase portrait, showing the system is not multistationary. In (c) and (d), particular stoichiometric compatibility classes are highlighted in yellow.
  • ...and 2 more figures

Theorems & Definitions (34)

  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • Definition 2.4
  • Definition 2.5
  • Example 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 3.1
  • Remark 3.2
  • ...and 24 more