The Dimension of the Disguised Toric Locus of a Reaction Network

TL;DR

This paper investigates the set of rate constants that make a reaction network's dynamics equivalent to complex-balanced systems, extending the understanding of toric systems to those not inherently toric, with applications to biological models.

Contribution

It provides a method to compute the exact dimension of the disguised toric locus for reaction networks, broadening the analysis of complex-balanced systems.

Findings

Derived the dimension formula for the disguised toric locus.

Applied the results to biological models like circadian clocks.

Enhanced understanding of reaction network dynamics.

Abstract

Under mass-action kinetics, complex-balanced systems emerge from biochemical reaction networks and exhibit stable and predictable dynamics. For a reaction network $G$, the associated dynamical system is called $\textit{disguised toric}$ if it can yield a complex-balanced realization on a possibly different network $G_1$. This concept extends the robust properties of toric systems to those that are not inherently toric. In this work, we study the $\textit{disguised toric locus}$ of a reaction network - i.e., the set of positive rate constants that make the corresponding mass-action system disguised toric. Our primary focus is to compute the exact dimension of this locus. We subsequently apply our results to Thomas-type and circadian clock models.