Stochastic enzyme kinetics and the quasisteadystate reductions: Application of the slow scale linear noise approximation à la Fenichel
Abstract
The linear noise approximation models the random fluctuations from the meanfield 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). In the presence of disparate timescales, the linear noise approximation admits a quasisteadystate reduction referred to as the \textit{slow scale} linear noise approximation (ssLNA). Curiously, the ssLNAs reported in the literature are slightly different. The differences in the reported ssLNAs lie at the mathematical heart of the derivation. In this work, we derive the ssLNA directly from geometric singular perturbation theory and explain the origin of the different ssLNAs in the literature. Moreover, we discuss the loss of normal hyperbolicity and we extend the ssLNA derived from geometric singular perturbation theory to a nonclassical singularly perturbed problem. In so doing, we disprove a commonlyaccepted qualifier for the validity of stochastic quasisteadystate approximation of the MichaelisMenten reaction mechanism.
 Publication:

arXiv eprints
 Pub Date:
 January 2021
 DOI:
 10.48550/arXiv.2101.04814
 arXiv:
 arXiv:2101.04814
 Bibcode:
 2021arXiv210104814E
 Keywords:

 Physics  Chemical Physics;
 Condensed Matter  Statistical Mechanics;
 Quantitative Biology  Quantitative Methods;
 37M10 (Primary);
 92E20;
 60J22;
 60H10;
 60H35;
 8210 (Secondary)
 EPrint:
 25 pages, 4 figures