Uniqueness of weakly reversible and deficiency zero realizations of dynamical systems

TL;DR

This paper proves that the underlying weakly reversible deficiency zero network of a dynamical system is uniquely identifiable when it exists, and such systems are notably stable and simple compared to general mass-action systems.

Contribution

It establishes the well-posedness of identifying weakly reversible deficiency zero networks and highlights their stability and simplicity in dynamical behavior.

Findings

Uniqueness of WR0 network realization when it exists

WR0 systems have a unique positive steady state

WR0 systems cannot exhibit oscillations or chaos

Abstract

A reaction network together with a choice of rate constants uniquely gives rise to a system of differential equations, according to the law of mass-action kinetics. On the other hand, different networks can generate the same dynamical system under mass-action kinetics. Therefore, the problem of identifying "the" underlying network of a dynamical system is not well-posed, in general. Here we show that the problem of identifying an underlying weakly reversible deficiency zero network is well-posed, in the sense that the solution is unique whenever it exists. This can be very useful in applications because from the perspective of both dynamics and network structure, a weakly reversibly deficiency zero ($\textit{WR}_\textit{0}$) realization is the simplest possible one. Moreover, while mass-action systems can exhibit practically any dynamical behavior, including multistability, oscillations, and chaos, $WR_0$ systems are remarkably stable for any choice of rate constants: they have a unique positive steady state within each invariant polyhedron, and cannot give rise to oscillations or chaotic dynamics. We also prove that both of our hypotheses (i.e., weak reversibility and deficiency zero) are necessary for uniqueness.