Tau-leaping

Tau-leaping accelerates the Gillespie algorithm by approximating reaction rates as constant over an interval $\tau$, then drawing all reaction counts for that interval at once. At state $x$, with reaction propensity functions $a_r(x)$ and stoichiometric vectors ${\nu }_r$, the explicit step isThe leap condition requires the propensities to change little during the interval. Unlike event-by-event exact simulation, this introduces an approximation error controlled through the choice of $\tau$.^[ref]

For $A\to B$ with rate $k{x}_A$, the approximate number converted is Poisson with mean $k{x}_A\tau$. The exact number converted over the interval is instead binomial with parameters $x_A$ and $1-e^{-k\tau }$. The two agree to first order for small $k\tau$, but the Poisson draw can exceed $x_A$ and produce a negative count. Smaller or adaptive leaps and modified sampling rules address this limitation; freezing the rate is not automatically safe at low copy number.