Stochastic chemical kinetics and the quasi-steady-state assumption: Application to the Gillespie algorithm

Abstract

Biochemical dynamics are often determined by series of single molecule events such as gene expression and reactions involving protein concentrations at nanomolar concentrations. Molecular fluctuations, consequently, may be of biological significance. For example, heterogeneity in clonal populations is believed to arise from molecular fluctuations in gene expression. A realistic description, therefore, requires a probabilistic description of the biochemical dynamics as deterministic descriptions cannot capture the inherent molecular fluctuations. The Gillespie algorithm [D. T. Gillespie, J. Phys. Chem. 81, 2350 (1977)] is a stochastic procedure for simulating chemical systems at low concentrations. A limitation of stochastic kinetic models is that they require detailed information about the chemical kinetics often unavailable in biological systems. Furthermore, the Gillespie algorithm is computationally intensive when there are many molecules and reaction events. In this article, we explore one approximation technique, well known in deterministic kinetics, for simplifying the stochastic model: the quasi-steady-state assumption (QSSA). We illustrate how the QSSA can be applied to the Gillespie algorithm. Using the QSSA, we derive stochastic Michaelis–Menten rate expressions for simple enzymatic reactions and illustrate how the QSSA is applied when modeling and simulating a simple genetic circuit.