Stochastic chemical kinetics and the total quasi-steady-state assumption: application to the stochastic simulation algorithm and chemical master equation

Recently the application of the quasi-steady-state approximation _QSSA_ to the stochastic simulation algorithm _SSA_ was suggested for the purpose of speeding up stochastic simulations of chemical systems that involve both relatively fast and slow chemical reactions _Rao and Arkin, J. Chem. Phys. 118, 4999 _2003__ and further work has led to the nested and slow-scale SSA. Improved numerical efficiency is obtained by respecting the vastly different time scales characterizing the system and then by advancing only the slow reactions exactly, based on a suitable approximation to the fast reactions. We considerably extend these works by applying the QSSA to numerical methods for the direct solution of the chemical master equation _CME_ and, in particular, to the finite state projection algorithm _Munsky and Khammash, J. Chem. Phys. 124, 044104 _2006__, in conjunction with Krylov methods. In addition, we point out some important connections to the literature on the _deterministic_ total QSSA _tQSSA_ and place the stochastic analogue of the QSSA within the more general framework of aggregation of Markov processes. We demonstrate the new methods on four examples: Michaelis–Menten enzyme kinetics, double phosphorylation, the Goldbeter–Koshland switch, and the mitogen activated protein kinase cascade. Overall, we report dramatic improvements by applying the tQSSA to the CME solver.