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.
