A deficiency-based approach to parametrizing positive equilibria of biochemical reaction systems

TL;DR

This paper establishes conditions under which the positive equilibria of biochemical reaction systems can be parametrized, linking network deficiency and reversibility to equilibrium structure, with applications to signaling pathways.

Contribution

It introduces a deficiency-based framework for parametrizing positive equilibria in generalized mass-action systems, connecting network properties to equilibrium sets.

Findings

Zero deficiency networks have complex-balanced equilibria that can be parametrized.

Weakly reversible, zero kinetic deficiency networks have nonempty, rational parametrizations.

Application to biochemical pathways demonstrates practical utility for studying multistationarity and ACR.

Abstract

We present conditions which guarantee a parametrization of the set of positive equilibria of a generalized mass-action system. Our main results state that (i) if the underlying generalized chemical reaction network has an effective deficiency of zero, then the set of positive equilibria coincides with the parametrized set of complex-balanced equilibria and (ii) if the network is weakly reversible and has a kinetic deficiency of zero, then the equilibrium set is nonempty and has a positive, typically rational, parametrization. Via the method of network translation, we apply our results to classical mass-action systems studied in the biochemical literature, including the EnvZ-OmpR and shuttled WNT signaling pathways. A parametrization of the set of positive equilibria of a (generalized) mass-action system is often a prerequisite for the study of multistationarity and allows an easy check for the occurrence of absolute concentration robustness (ACR), as we demonstrate for the EnvZ-OmpR pathway.