In the past one hundred years, deterministic rate equations have been successfully used to infer enzyme-catalysed reaction mechanisms and to estimate rate constants from reaction kinetics experiments conducted in vitro. In recent years, sophisticated
experimental techniques have been developed that allow the measurement of enzyme- catalysed and other biopolymer-mediated reactions inside single cells at the single molecule level. Time course data obtained by these methods are considerably noisy because molecule numbers within cells are typically quite small. As a consequence, the interpretation and analysis of single cell data requires stochastic methods, rather than deterministic rate equations. Here we concisely review both experimental and theoretical techniques which enable single molecule analysis with particular emphasis on the major developments in the field of theoretical stochastic enzyme kinetics, from its inception in the mid-twentieth century to its modern day status. We discuss the differences between stochastic and deterministic rate equation models, how these depend on enzyme molecule numbers and substrate inflow into the reaction compartment and how estimation of rate constants from single cell data is possible using recently developed stochastic approaches.
Genetic feedback loops in cells break detailed balance and involve bimolecular reactions; hence exact solutions revealing the nature of the stochastic fluctuations in these loops are lacking. We here consider the master equation for a gene regulatory
feedback loop: a gene produces protein which then binds to the promoter of the same gene and regulates its expression. The protein degrades in its free and bound forms. This network breaks detailed balance and involves a single bimolecular reaction step. We provide an exact solution of the steady-state master equation for arbitrary values of the parameters, and present simplified solutions for a number of special cases. The full parametric dependence of the analytical non-equilibrium steady-state probability distribution is verified by direct numerical solution of the master equations. For the case where the degradation rate of bound and free protein is the same, our solution is at variance with a previous claim of an exact solution (Hornos et al, Phys. Rev. E {bf 72}, 051907 (2005) and subsequent studies). We show explicitly that this is due to an unphysical formulation of the underlying master equation in those studies.
