Validity of rate equation results for reaction rates in reaction networks with fluctuations

Systems of reacting particles, which are well mixed or distributed homogeneously in space, are commonly modeled by deterministic rate equations. These equations, which are based on the mean-field approximation, are valid for macroscopic systems. However, they neglect fluctuations in the particle populations. As a result, under conditions of strong fluctuations, the reaction rates obtained from the rate equations are highly inaccurate. To account for the fluctuations, stochastic methods are required. However, these methods are computationally intensive and may become infeasible for complex reaction networks. Therefore, it is useful to identify the conditions under which the rate equations provide accurate results. Naively, one expects strong fluctuations when the average population sizes of some of the reactants are of order one or lower. Here we present a systematic approach, for testing the validity of the rate equations, in which we define characteristic scales in terms of the rate constants of the network. We show that the rate equations fail to accurately reproduce the reaction rates when the system size is reduced below these scales. Surprisingly, the rate equations are found to be applicable in a wider range than expected. Their validity depends not only on the population sizes of the reactive species but also on the kinetic properties of the reaction network.