  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.
