Absolute Concentration Robustness and Multistationarity in Reaction Networks: Conditions for Coexistence

TL;DR

This paper investigates the conditions under which reaction networks can simultaneously exhibit multistationarity and absolute concentration robustness (ACR), providing bounds on network size and showing their typical incompatibility in small bimolecular networks.

Contribution

It establishes necessary conditions for coexistence of multistationarity and ACR in reaction networks, including bounds on species, complexes, and reactions, and relates ACR to non-multistationarity in small networks.

Findings

Coexistence requires at least 3 species, 5 complexes, and 3 reactions.

In small bimolecular networks, ACR generally implies the absence of multistationarity.

Bounds on reactions depend on the number of conservation laws.

Abstract

Many reaction networks arising in applications are multistationary, that is, they have the capacity for more than one steady state; while some networks exhibit absolute concentration robustness (ACR), which means that some species concentration is the same at all steady states. Both multistationarity and ACR are significant in biological settings, but only recently has attention focused on the possibility for these properties to coexist. Our main result states that such coexistence in at-most-bimolecular networks (which encompass most networks arising in biology) requires at least $3$ species, $5$ complexes, and $3$ reactions. We prove additional bounds on the number of reactions for general networks based on the number of linear conservation laws. Finally, we prove that, outside of a few exceptional cases, ACR is equivalent to non-multistationarity for bimolecular networks that are small (more precisely, one-dimensional or up to two species). Our proofs involve analyses of systems of sparse polynomials, and we also use classical results from chemical reaction network theory.