Quasiequilibrium approximation of fast reaction kinetics in stochastic biochemical systems

We address the problem of eliminating fast reaction kinetics in stochastic biochemical systems by employing a quasiequilibrium approximation. We build on two previous methodologies developed by [Haseltine and Rawlings, J. Chem. Phys. 117, 6959 (2002)] and by [Rao and Arkin, J. Chem. Phys. 118, 4999 (2003)]. By following Haseltine and Rawlings, we use the numbers of occurrences of the underlying reactions to characterize the state of a biochemical system. We consider systems that can be effectively partitioned into two distinct subsystems, one that comprises "slow" reactions and one that comprises "fast" reactions. We show that when the probabilities of occurrence of the slow reactions depend at most linearly on the states of the fast reactions, we can effectively eliminate the fast reactions by modifying the probabilities of occurrence of the slow reactions. This modification requires computation of the mean states of the fast reactions, conditioned on the states of the slow reactions. By assuming that within consecutive occurrences of slow reactions, the fast reactions rapidly reach equilibrium, we show that the conditional state means of the fast reactions satisfy a system of at most quadratic equations, subject to linear inequality constraints. We present three examples which allow analytical calculations that clearly illustrate the mathematical steps underlying the proposed approximation and demonstrate the accuracy and effectiveness of our method.