Positive equilibria of a class of power-law kinetics

Abstract

This paper studies a class of power-law kinetics, PL-ILK, for whose subset, PL-TIK, analogues of the Deficiency Zero Theorem and the Deficiency One Theorem (DOT) for mass action systems are valid. The DOT also includes the necessary and sufficient condition of Boros for uniqueness in the non-weakly reversible case. To our knowledge, this is the first set of kinetics beyond mass action kinetics (MAK) for which the DOT has been shown to be valid. A further interesting property of PL-TIK is a certain “robustness” relative to dependence of linkage classes: existence of a positive equilibrium for each linkage class implies the existence of a positive equilibrium for the whole network. For MAK systems, the PL-ILK property is equivalent to the reactant deficiency of the linkage class containing the zero complex being one, and zero for all other linkage classes. As shown in the Supplementary Materials, an initial survey of MAK and BST systems already reveals numerous examples with PL-ILK kinetics.