2020/09/05 by Bryan S. Hernandez, Hernandez, Bryan S., Eduardo R. Mendoza +1 · 1 citation
Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #math.DS
paper · pdf · doi:10.48550/arxiv.2009.02487
arxiv created 2020/09/05 · arxiv updated 2020/09/08
This work introduces a novel approach to study properties of positive equilibria of a chemical reaction network \mathscrN endowed with Hill-type kinetics K, called a Hill-type kinetic (HTK) system (\mathscrN,K), 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 ( \mathscrN,KPY ) 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_ + ( \mathscrN,K ) = E_ + ( \mathscrN,KPY ) and Z_ + ( \mathscrN,K ) = Z_ + ( \mathscrN,KPY ). 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 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 (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.