Accelerated stochastic simulation of the stiff enzyme-substrate reaction

The enzyme-catalyzed conversion of a substrate into a product is a common reaction motif in cellular chemical systems. In the three reactions that comprise this process, the intermediate enzyme-substrate complex is usually much more likely to decay into its original constituents than to produce a product molecule. This condition makes the reaction set mathematically stiff. We show here how the simulation of this stiff reaction set can be dramatically speeded up relative to the standard stochastic simulation algorithm (SSA) by using a recently introduced procedure called the slow-scale SSA. The speedup occurs because the slow-scale SSA explicitly simulates only the relatively rare conversion reactions, skipping over occurrences of the other two less interesting but much more frequent reactions. We describe, explain, and illustrate this simulation procedure for the isolated enzyme-substrate reaction set, and then we show how the procedure extends to the more typical case in which the enzyme-substrate reactions occur together with other reactions and species. Finally, we explain the connection between this slow-scale SSA approach and the Michaelis-Menten [Biochem. Z. 49, 333 (1913)] formula, which has long been used in deterministic chemical kinetics to describe the enzyme-substrate reaction.