Analyzing Multistationarity in Chemical Reaction Networks using the Determinant Optimization Method

TL;DR

This paper applies the determinant optimization method to identify multistationarity in a family of sequestration networks, explicitly constructing reaction rates and steady states, and proving the existence of nondegenerate steady states, thus advancing understanding in chemical reaction network theory.

Contribution

It provides a detailed construction of reaction rates and steady states for sequestration networks using the determinant optimization method, confirming multistationarity and resolving part of a conjecture.

Findings

Explicit reaction rates and steady states for certain sequestration networks

Proof of nondegenerate steady states in these networks

Validation of the determinant optimization method for multistationarity

Abstract

Multistationary chemical reaction networks are of interest to scientists and mathematicians alike. While some criteria for multistationarity exist, obtaining explicit reaction rates and steady states that exhibit multistationarity for a given network -- in order to check nondegeneracy or determine stability of the steady states, for instance -- is nontrivial. Nonetheless, we accomplish this task for a certain family of sequestration networks. Additionally, our results allow us to prove the existence of nondegenerate steady states for some of these sequestration networks, thereby resolving a subcase of a conjecture of Joshi and Shiu. Our work relies on the determinant optimization method, developed by Craciun and Feinberg, for asserting that certain networks are multistationary. More precisely, we implement the construction of reaction rates and multiple steady states which appears in the proofs that underlie their method. Furthermore, we describe in detail the steps of this construction so that other researchers can more easily obtain, as we did, multistationary rates and steady states.