Avoiding negative populations in explicit Poisson tau-leaping

Abstract

The explicit tau-leaping procedure attempts to speed up the stochastic simulation of a chemically reacting system by approximating the number of firings of each reaction channel during a chosen time increment tau as a Poisson random variable. Since the Poisson random variable can have arbitrarily large sample values, there is always the possibility that this procedure will cause one or more reaction channels to fire so many times during tau that the population of some reactant species will be driven negative. Two recent papers have shown how that unacceptable occurrence can be avoided by replacing the Poisson random variables with binomial random variables, whose values are naturally bounded. This paper describes a modified Poisson tau-leaping procedure that also avoids negative populations, but is easier to implement than the binomial procedure. The new Poisson procedure also introduces a second control parameter, whose value essentially dials the procedure from the original Poisson tau-leaping at one extreme to the exact stochastic simulation algorithm at the other; therefore, the modified Poisson procedure will generally be more accurate than the original Poisson procedure.

Hybrid modeling and simulation of stochastic effects on progression through the eukaryotic cell cycle

“…19,29,31 Cao et al 31 proposed to partition the system, based simply on the species population numbers. For species whose population numbers are less than a threshold, all related reactions are simulated by the SSA, while other reactions are simulated by the tau-leaping method.…”

Numerical methods in the context of compartmental models in epidemiology

“…The explicit τ -leaping algorithm above has been shown to provide accurate simulation results; furthermore, it is substantially faster than the SSA for many "not-too-stiff" systems 12 (see [22]; see also [24] for theoretical evidence for this). We illustrate the gain of computation time by numerical experiments in sections 4.4 and 4.5 below for a modification of this algorithm (cf.…”

“…That makes it impossible for a critical Poisson process to drive any component of Z N to a negative value. The theoretical justification for each step in this modified τ -leaping procedure is given in [22].…”

“…Unfortunately, this algorithm is rather slow in some circumstances, typically when there are too many jumps of the Poisson processes on a fixed time interval. Therefore, we discuss faster approximate simulation methods in the subsequent sections: in section 4.2, we introduce the explicit τ -leaping algorithm (see [16]) in the form of [21]; in section 4.3, we discuss modifications of the τ -leaping algorithm: implicit and mid-point τ -leaping (section 4.3.1) and the algorithm from [22] which ensures that the simulated process remains in its domain, e.g., by preventing the components of the process from becoming negative (section 4.3.2). We apply the SSA and the modified τ -leaping algorithm from section 4.3.1 to the specific models from Examples 1.1 and 1.2 in sections 4.4 and 4.5, respectively.…”

An exact accelerated stochastic simulation algorithm

“…The simplest versions of both -leap and R-leap have difficulties with the potential of producing negative numbers of reactants, which can be fixed by modifications such as binomial tau-leap 3 and modified tau-leap. 4 Since ERleap is exact, it intrinsically avoids this potential pitfall; stochastic moves to negative reactant states have zero probability and will be rejected. We demonstrate by computational experiments that ER-leap can execute in time sublinear in the number of reaction events to be simulated, while remaining exact.…”