Quasi-steady state and singular perturbation reduction for reaction networks with non-interacting species

TL;DR

This paper rigorously analyzes the conditions under which quasi-steady state (QSS) reduction aligns with singular perturbation reduction in chemical reaction networks, especially for non-interacting species, providing practical criteria and examples.

Contribution

It establishes necessary and sufficient conditions for QSS and singular perturbation reductions to coincide in linear cases and applies these to reaction networks with non-interacting species.

Findings

Derived graphical conditions for parameter selection.

Identified parameter choices where QSS and singular perturbation reductions agree.

Extended previous results to broader classes of reaction networks.

Abstract

Quasi-steady state (QSS) reduction is a commonly used method to lower the dimension of a differential equation model of a chemical reaction network. From a mathematical perspective, QSS reduction is generally interpreted as a special type of singular perturbation reduction, but in many instances the correspondence is not worked out rigorously, and the QSS reduction may yield incorrect results. The present paper contains a thorough discussion of QSS reduction and its relation to singular perturbation reduction for the special, but important, case when the right hand side of the differential equation is linear in the variables to be eliminated. For this class we give necessary and sufficient conditions for a singular perturbation reduction (in the sense of Tikhonov and Fenichel) to exist, and to agree with QSS reduction. We then apply the general results to chemical reaction networks with non-interacting species, generalizing earlier results and methods for steady states to quasi-steady state scenarios. We provide easy-to-check graphical conditions to select parameter values yielding to singular perturbation reductions and additionally, we identify a choice of parameters for which the corresponding singular perturbation reduction agrees with the QSS reduction. Finally we consider a number of examples.