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We consider stochastic descriptions of chemical reaction networks in whichthere are both fast and slow reactions, and for which the time scales arewidely separated. We develop a computational algorithm that produces thegenerator of the full chemical master equation for arbitrary systems, and showhow to obtain a reduced equation that governs the evolution on the slow timescale. This is done by applying a state space decomposition to the fullequation that leads to the reduced dynamics in terms of certain projections andthe invariant distributions of the fast system. The rates or propensities ofthe reduced system are shown to be the rates of the slow reactions conditionedon the expectations of fast steps. We also show that the generator of thereduced system is a Markov generator, and we present an efficient stochasticsimulation algorithm for the slow time scale dynamics. We illustrate thenumerical accuracy of the approximation by simulating several examples.Graph-theoretic techniques are used throughout to describe the structure of thereaction network and the state-space transitions accessible under the dynamics.

A Multi-Time-Scale Analysis of Chemical Reaction Networks : II. Stochastic Systems