  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.
