Single enzyme pathways and substrate fluctuations

The ability to dynamically probe single enzymes allows the experimental investigation of enzyme kinetics with unprecedented resolution. In this paper we develop a simple theory which predicts that certain classes of enzyme pathways can be distinguished by studying the turnover rate, V, as a function of substrate concentration, [S]. In particular, we study the steady state of a single enzyme interacting with a bath of substrate molecules, and analyse it as a stochastic process. The V([S]) relation is found to depend sensitively on the manner in which substrate molecules in the bath are replenished. We focus on a gedanken experiment in which the average substrate concentration is kept fixed by allowing molecules to enter the bath at a constant rate. We derive the exact relationship between V and [S], which has a relatively simple form, though different to that of the Michaelis-Menten (MM) equation. Interestingly, the MM equation is exactly recovered if the substrate concentration is instantaneously maintained with molecular precision. We examine the new V([S]) relation for a number of enzyme pathways and find that it differentiates between enzyme reactions involving one or many intermediate enzyme-substrate complexes. This, in principle, allows one to probe the internal conformations of enzymes by careful measurement of V([S]) curves in appropriately designed single enzyme experiments.