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

Abstract

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). In the presence of disparate timescales, the linear noise approximation admits a quasi-steady-state 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 non-classical singularly perturbed problem. In so doing, we disprove a commonly-accepted qualifier for the validity of stochastic quasi-steady-state approximation of the Michaelis--Menten reaction mechanism.

arXiv comment25 pages, 4 figures

References in corpus (5)

• The validity of quasi steady-state approximations in discrete stochastic simulations
• Limitations of the stochastic quasi-steady-state approximation in open biochemical reaction networks
• Rigorous elimination of fast stochastic variables from the linear noise approximation using projection operators
• Classical quasi-steady state reduction -- A mathematical characterization
• The Quasi-Steady-State Approximations revisited: Timescales, small parameters, singularities, and normal forms in enzyme kinetic

Cited by in corpus (1)

• Model Reduction for the Chemical Master Equation: an Information-Theoretic Approach