Stochastic Chemical Kinetics

Abstract

The time evolution of a well-stirred chemically reacting system is traditionally described by a set of coupled, first-order, ordinary differential equations. Obtained through heuristic, phenomenological reasoning, these equations characterize the evolution of the molecular populations as a continuous, deterministic process. But a little reflection reveals that the system actually possesses neither of those attributes: Molecular populations are whole numbers, and when they change they always do so by discrete, integer amounts. Furthermore, in excusing ourselves from the arduous task of tracking the positions and velocities of all the molecules in the system, which we hope to justify on the grounds that the system is “well-stirred”, we preclude a deterministic description of the system’s evolution; because, a knowledge of the system’s current molecular populations is not by itself sufficient to predict with certainty the future molecular populations. Just as rolled dice are essentially random or “stochastic” when we do not precisely track their positions and velocities and all the forces acting on them, so is the time evolution of a well-stirred chemically reacting system for all practical purposes stochastic. That said, discreteness and stochasticity are usually not noticeable in chemical systems of “test-tube” size or larger, and for most such systems the traditional continuous deterministic description seems to be adequate. But if the molecular populations of some reactant species are very small, as is often the case for instance in cellular systems in biology, discreteness and stochasticity can sometimes play an important role.