The disguised toric locus and affine equivalence of reaction networks

TL;DR

This paper investigates disguised toric dynamical systems in mass-action kinetics, revealing how affine transformations preserve certain parameters and steady states, thus deepening understanding of network equivalences and dynamics.

Contribution

It introduces the concept of disguised toric systems, showing affine invariance of parameters and steady states, and explores how dynamics can differ despite these invariances.

Findings

Parameters for disguised toric systems are preserved under invertible affine transformations.

There is a bijection between positive steady states of affine-transformed systems.

Qualitative dynamics can differ significantly despite invariant steady states.

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.