Exact Product Formation Rates for Stochastic Enzyme Kinetics

The rate of product formation is an important measure of the speed of enzyme reactions. Classical studies of enzyme reactions have been conducted in dilute solutions and under conditions that justified the substrate abundance assumption. However, such assumption is well-known to break down in the context of cellular biochemistry. Instead, the concentration of available substrate can become rate limiting. Here we use the chemical master equation to obtain expressions for the instantaneous and time averaged rate of product formation without invoking the conventional substrate abundance assumption. The expressions are derived for a broad range of enzyme reaction mechanisms, including those that involve one or many enzyme molecules, require multiple substrates, and exhibit cooperativity and substrate inhibition. Novel results include: (i) the relationship between the average rate of product formation (calculated over the time it takes for the reaction to finish) and the substrate concentration, for a Michaelis-Menten (MM) reaction with one enzyme molecule, is approximately given by a logarithmically corrected MM form; (ii) intrinsic noise decreases the sharpness of cooperative switches but enhances the filtering response of substrate inhibition; (iii) the relationship between the initial average rate of product formation and the initial substrate concentration for a MM reaction with no reversible reaction and with any number of enzyme and substrate molecules is a sum of Michaelis-Menten equations.