Multistationarity in the space of total concentrations for systems that admit a monomial parametrization

TL;DR

This paper uses real algebraic geometry to analyze multistationarity in chemical reaction networks with monomial parametrizations, revealing scale invariance and conditions for multistationarity based on total concentrations.

Contribution

It introduces a method to determine multistationarity independent of rate constants using semi-algebraic conditions and applies chamber decomposition to characterize multistationarity regions.

Findings

Multistationarity is scale invariant in the space of total concentrations.

Multistationarity depends on total concentrations exceeding certain thresholds.

The approach extends to include total concentrations, providing new insights into multistationarity regions.

Abstract

We apply tools from real algebraic geometry to the problem of multistationarity of chemical reaction networks. A particular focus is on the case of reaction networks whose steady states admit a monomial parametrization. For such systems we show that in the space of total concentrations multistationarity is scale invariant: if there is multistationarity for some value of the total concentrations, then there is multistationarity on the entire ray containing this value (possibly for different rate constants) -- and vice versa. Moreover, for these networks it is possible to decide about multistationarity independent of the rate constants by formulating semi-algebraic conditions that involve only concentration variables. These conditions can easily be extended to include total concentrations. Hence quantifier elimination may give new insights into multistationarity regions in the space of total concentrations. To demonstrate this, we show that for the distributive phosphorylation of a protein at two binding sites multistationarity is only possible if the total concentration of the substrate is larger than either the total concentration of the kinase or the total concentration of the phosphatase. This result is enabled by the chamber decomposition of the space of total concentrations from polyhedral geometry. Together with the corresponding sufficiency result of Bihan et al. this yields a characterization of multistationarity up to lower dimensional regions.