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We present an effective procedure for selecting the stepsize in the binomial $\tau$-leap method, which is an efficient technique for the discrete simulation of biochemical reaction systems. We use the difference of the propensity functions to approximate their derivatives, thus giving a derivative-free implementation. We compare the difference between the stepsizes obtained by existing procedures and the new procedure, and compare their relative efficiencies when simulating biochemical reaction systems. Numerical results indicate that the new procedure is very efficient and robust. More importantly, this new procedure is easy to implement and leads one step further towards a general purpose computer program for the efficient simulation of stochastic biochemical reaction systems.   References R. E. Bank and C. C. Douglas, Sparse matrix multiplication package (SMMP), Adv. Comput. Math. 1 (1993) 127--137. K. Burrage and T. Tian, Poisson Runge-Kutta methods for chemical reaction systems, in Advances in Scientific Computing and Applications , Y. Lu et al., eds, Science Press, Beijing/New York, 82--96, 2004. K. Burrage, T. Tian and P. Burrage, A Multi-scaled Approach for Simulating Chemical Reaction Systems, Prog. Biophys. Mol. Bio. 85 (2004) 217--234. D. T. Gillespie, Exact stochastic simulation of coupled chemical reactions, J. Phys. Chem. 81 (1997) 2340--2361. D. T. Gillespie, Approximate accelerated stochastic simulation of chemical reaction systems, J. Chem. Phys. 115 (2001) 1716--1733. D. T. Gillespie and L. R. Petzold, Improved leap-size selection for accelerated stochastic simulation, J. Chem. Phys. 119 (2003) 8229--8234. E. L. Haseltine and J. B. Rawlings, Approximate simulation of coupled fast and slow reactions for stochastic chemical kinetics, J. Chem. Phys. 117 (2002) 6959--6969. A. M. Kierzek, STOCKS: stochastic kinetic simulations of biochemical systems with Gillespie algorithm, Bioinformatics 18 (2002) 470--481. J. Puchalka and A. M. Kierzek, Bridging the gap between stochastic and deterministic regimes in the kinetic simulations of the biochemical reaction networks, Biophys. J. 86 (2004) 1357--1372. C. Rao and A. Arkin, Stochastic chemical kinetics and the quasi-steady-state assumption: application to the Gillespie algorithm, J. Chem. Phys. 118 (2003) 4999--5010. M. Rathinam, L. R. Petzold, Y. Cao and D. T. Gillespie, Stiffness in stochastic chemically reacting systems: The implicit tau-leaping method, J. Chem. Phys. 119 (2003) 12784--12794. T. Tian and K. Burrage, Binomial leap methods for simulating chemical kinetics, J. Chem. Phys. 121 (2004) 10356--10364.

An effective stepsize selection procedure for discrete simulation of biochemical reaction systems