Positive equilibria of Hill-type kinetic systems

Abstract

This work introduces a novel approach to study properties of positive equilibria of a chemical reaction network \({\mathscr {N}}\) endowed with Hill-type kinetics K, called a Hill-type kinetic (HTK) system \(\left( {\mathscr {N}},K\right)\), including their multiplicity and concentration robustness in a species. We associate a unique poly-PL kinetic (PYK) system \(\left( {{\mathscr {N}},{K_\text {PY}}} \right)\) to the given HTK system, where PYK is a positive linear combination of PL functions. The associated system has the key property that its equilibria sets coincide with those of the Hill-type system. This allows us to identify two novel subsets of the (HTKs), called PL-equilibrated and PL-complex balanced kinetics, to which recent results on absolute concentration robustness (ACR) of species and complex balancing at positive equilibria of PL kinetic systems can be applied. Our main results also include the Shinar–Feinberg ACR Theorem for PL-equilibrated HT-RDK systems (i.e., subset of complex factorizable HTK systems), which establishes a foundation for the analysis of ACR in HTK systems, and the extension of the results of Müller and Regensburger on generalized mass action systems to PL-complex balanced HT-RDK systems. In addition, we derive the theory of balanced concentration robustness in an analogous manner to ACR for PL-equilibrated systems. Finally, we provide further extensions of our results to a more general class of kinetics, which include quotients of poly-PL functions.