Derivation of Steady State Parametrizations of Chemical Reaction Networks with n Independent and Identical Subnetworks

The long-term behavior of a chemical reaction network (CRN) is usually described by steady states. Recently, Hernandez et al. provided a method and a computational package for deriving positive steady states of CRNs via the concept of network decomposition. In particular, a given CRN is decomposed into stoichiometrically independent subnetworks; then, positive steady state parametrizations of these subnetworks are derived individually and merged to obtain a positive steady state parametrization of the given network. However, the framework applies to a fixed number of subnetworks. In this work, we establish a systematic approach to solving steady state parametrizations of CRNs that can be decomposed into n stoichiometrically independent and structurally identical subnetworks, where n >= 2 is any positive integer. Specifically, we apply the method to the n-site processive phosphorylation/dephosphorylation model. That is, we compute the positive steady state parametrization for the case when n = 2 via the concept of network decomposition using the result of parametrizing positive steady states of the network when n = 1. Then, we generalize the parametrization for any positive integer n >= 2 via the principle of mathematical induction.