Unbiased τ -leap methods for stochastic simulation of chemically reacting systems

The τ -leap method first developed by Gillespie [D. T. Gillespie, J. Chem. Phys. 115, 1716 (2001)] can significantly speed up stochastic simulation of certain chemically reacting systems with acceptable losses in accuracy. Recently, several improved τ -leap methods, including the binomial, multinomial, and modified τ -leap methods, have been developed. However, in all these τ -leap methods, the mean of the number of times, Km, that the mth reaction channel fires during a leap is not equal to the true mean. Therefore, all existing τ -leap methods produce biased simulation results, which limit the simulation accuracy and speed. In this paper, we analyze the mean of Km based on the chemical master equation. Using this analytical result, we develop unbiased Poisson and binomial τ -leap methods. Moreover, we analyze the variance of Km, and then develop an unbiased Poisson/Gaussian/binomial τ -leap method to correct the errors in both the mean and variance of Km. Simulation results demonstrate that our unbiased τ -leap method can significantly improve simulation accuracy without sacrificing speed.