Positive equilibria of power law kinetics on networks with independent linkage classes

TL;DR

This paper investigates the structure of positive equilibria in power law kinetic systems on networks with independent linkage classes, extending known results from mass action systems to a broader class with applications in complex balancing and robustness.

Contribution

It characterizes positive equilibria in cycle terminal PL-FSK systems with ILC as poly-PLP sets and extends structural analysis of complex balanced systems to these kinetic models.

Findings

Positive equilibria form disjoint log-parametrized sets in certain PL systems.

Non-emptiness of positive equilibria in linkage classes is characterized for PL-RDK systems.

New criteria for absolute concentration robustness (ACR) in poly-PLP systems.

Abstract

Studies about the set of positive equilibria ($E_+$) of kinetic systems have been focused on mass action, and not that much on power law kinetic (PLK) systems, even for PL-RDK systems (PLK systems where two reactions with identical reactant complexes have the same kinetic order vectors). For mass action, reactions with different reactants have different kinetic order rows. A PL-RDK system satisfying this property is called factor span surjective (PL-FSK). In this work, we show that a cycle terminal PL-FSK system with $E_+\ne \varnothing$ and has independent linkage classes (ILC) is a poly-PLP system, i.e., $E_+$ is the disjoint union of log-parametrized sets. The key insight for the extension is that factor span surjectivity induces an isomorphic digraph structure on the kinetic complexes. The result also completes, for ILC networks, the structural analysis of the original complex balanced generalized mass action systems (GMAS) by M\"uller and Regensburger. We also identify a large set of PL-RDK systems where non-emptiness of $E_+$ is a necessary and sufficient condition for non-emptiness of each set of positive equilibria for each linkage class. These results extend those of Boros on mass action systems with ILC. We conclude this paper with two applications of our results. Firstly, we consider absolute complex balancing (ACB), i.e., the property that each positive equilibrium is complex balanced, in poly-PLP systems. Finally, we use the new results to study absolute concentration robustness (ACR) in these systems. In particular, we obtain a species hyperplane containment criterion to determine ACR in the system species.