# Positive Equilibria of Hill-Type Kinetic Systems

Source: [Positive Equilibria of Hill-Type Kinetic Systems](<https://www.sota2.com/research/paper/68fa574d1e209131ce9a45d2>)

Collection index: [llms.txt](<https://www.sota2.com/research/llms.txt>)

Data updated: 2025-10-25T14:39:36.307Z

Authors: Bryan S. Hernandez, Eduardo R. Mendoza

Published: 2020-09-05T07:47:49.000Z

## 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 positive linear combination of power-law kinetic systems called poly-PL kinetic (PYK) system $\\left( {\\mathscr{N},{K\_\\text{PY}}} \\right)$ to the given HTK system. The associated system has the key property that its equilibria sets coincide with those of the Hill-type system, i.e., ${E\_ + }\\left( {\\mathscr{N},K} \\right) = {E\_ + }\\left( {\\mathscr{N},{K\_\\text{PY}}} \\right)$ and ${Z\_ + }\\left( {\\mathscr{N},K} \\right) = {Z\_ + }\\left( {\\mathscr{N},{K\_\\text{PY}}} \\right)$. This allows us to identify two novel subsets of the Hill-type kinetics, 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 power-law (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 M\\"uller and Regensburger on generalized mass action systems to PL-complex balanced HT-RDK systems. In addition, we derive the theory of balanced concentration robustness (BCR) 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 includes quotients of poly-PL functions.

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## Original sources and implementation

[Original paper](<https://arxiv.org/abs/2009.02487>)

[PDF](<https://arxiv.org/pdf/2009.02487.pdf>)