Reversible-equivalent-monomolecular tau: A leaping method for “small number and stiff” stochastic chemical systems

Leaping methods provide for efficient and approximate time stepping of chemical reaction systems modeled by continuous time discrete state stochastic dynamics. We investigate the application of leaping methods for “small number and stiff” systems, i.e. systems whose dynamics involve different time scales and have some molecular species present in very small numbers, specifically in the range 0 to 10. We propose a new explicit leaping scheme, reversible-equivalent-monomolecular tau (REMM-τ), which shows considerable promise in the simulation of such systems. The REMM-τ scheme is based on the fact that the exact solution of the two prototypical monomolecular reversible reactions S1 ↔ S2 and S ↔ 0 as a function of time takes a simple form involving binomial and/or Poisson random variables. The REMM-τ method involves approximating bimolecular reversible reactions by suitable monomolecular reversible reactions as well as considering each reversible pair of reactions in the system to be operating in isolation during the time step τ. We illustrate the use of the REMM-τ method through a number of biologically motivated examples and compare its performance to those of the implicit-τ and trapezoidal implicit-τ algorithms. In most cases considered, REMM-τ appears to perform better than these two methods while having the important advantage of being computationally faster due to the explicit nature of the method. Furthermore when stepsize τ is increased the REMM-τ exhibits a more robust performance than the implicit-τ or the trapezoidal implicit-τ for small number stiff problems.