= 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 is
$$
K_r\sim\operatorname{Poisson}(a_r(x)\tau)\text{ independently},\qquad X(t+\tau)\approx x+\sum_r\nu_rK_r.
$$
The 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 $\tau$.<Approximate accelerated stochastic simulation of chemically reacting systems by Daniel T. Gillespie>[Section III, Poisson tau-leap construction]{ref}
For $A\to B$ with rate $kx_A$, the approximate number converted is Poisson with mean $kx_A\tau$. 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.
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