Detailed comparison between StochSim and SSA

Morton-Firth and Bray's stochastic simulator (StochSim) and Gillespie's stochastic simulation algorithm (SSA) are two important methods for stochastic modelling and simulation of biochemical systems. They have been widely applied to different biological problems. A key question is discussed here: Are these two methods equivalent? These two methods are compared using fundamental probability analysis. The analysis clearly shows that, when the time step in the StochSim is chosen very small, the StochSim can be viewed as a first-order approximation to the SSA. The analysis also explains why the SSA is usually much more efficient than the StochSim for biochemical systems. However, when multistate species present in a system, the StochSim clearly shows its advantage. The Complexity analysis is used to explain this advantage. The hybrid SSA (HSSA) is proposed to combine the advantages of both the StochSim and SSA. When the populations of the multistate species are small, the HSSA is very efficient. Numerical experiments are presented to verify the analysis.