Stochastic enzyme kinetics and the quasi-steady-state reductions: Application of the slow scale linear noise approximation `a la Fenichel

The linear noise approximation models the random fluctuations from the mean field model of a chemical reaction that unfolds near the thermodynamic limit. Specifically, the fluctuations obey a linear Langevin equation up to order $Omega^{-1/2}$, where $Omega$ is the size of the chemical system (usually the volume). Under the presence of disparate timescales, the linear noise approximation admits a quasi-steady-state reduction referred to as the slow scale linear noise approximation. However, the slow scale linear approximation has only been derived for fast/slow systems that are in Tikhonov standard form. In this work, we derive the slow scale linear noise approximation directly from Fenichel theory, without the need for a priori scaling and dimensional analysis. In so doing, we can apply for the first time the slow scale linear noise approximation to fast/slow systems that are not of standard form. This is important, because often times algorithms are only computationally expensive in parameter ranges where the system is singularly perturbed, but not in standard form. We also comment on the breakdown of the slow scale linear noise approximation near dynamic bifurcation points -- a topic that has remained absent in the chemical kinetics literature, despite the presence of bifurcations in simple biochemical reactions, such the Michaelis--Menten reaction mechanism.