A Computational Approach to Multistationarity in Poly-PL Kinetic Systems

TL;DR

This paper introduces a computational method combining the Multistationarity Algorithm with a transformation technique to analyze multistationarity in reaction networks with poly-PL kinetics, extending previous methods to more general systems.

Contribution

It extends existing algorithms to determine multistationarity in reaction networks with poly-PL kinetics using a transformation to dynamically equivalent PL systems.

Findings

The method can determine multistationarity in a broader class of kinetic systems.

Dynamically equivalent PL systems reflect the multistationarity properties of original PYK systems.

The approach simplifies analyzing complex kinetic systems by transforming them into more tractable forms.

Abstract

One important question that interests those who work in chemical reaction network theory (CRNT) is this: Does the system obtained from a reaction network admit a positive equilibrium and if it does, can there be more than one within a stoichiometric class? The higher deficiency algorithm (HDA) of Ji and Feinberg provided a method of determining the multistationarity capacity of a CRN with mass action kinetics (MAK). An extension of this, called Multistationarity Algorithm (MSA), recently came into the scene tackling CRNs with power law kinetics (PLK), a kinetic system which is more general (having MAK systems as a special case). For this paper, we provide a computational approach to study the multistationarity feature of reaction networks endowed with kinetics which are non-negative linear combinations of power law functions called poly-PL kinetics (PYK). The idea is to use MSA and combine it with a transformation called STAR-MSC (i.e., $S$-invariant Termwise Addition of Reactions via Maximal Stoichiometric Coefficients) producing PLKs that are dynamically equivalent to PYKs. This leads us to being able to determinine the multistationarity capacity of a much larger class of kinetic systems. We show that if the transformed dynamically equivalent PLK system is multistationary for a stoichiometric class for a set of particular rate constants, then so is its original corresponding PYK system. Moreover, the monostationarity property of the transformed PLK system also implies the monostationarity property of the original PYK system.