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The stoichiometric vector of reaction $r$ records its net change in each chemical species. If the copy-number state is $x$, firing the reaction changes it to $x+{\nu }_r$.

Stoichiometry relates the quantities consumed and formed in a chemical reaction. Its coefficients are ratios of entity counts, or equivalently of amount of substance, rather than ratios of masses. For the water chemical equation, consuming two molecular hydrogen molecules per molecular oxygen molecule gives the same $2:1$ ratio as consuming two moles of molecular hydrogen per mole of molecular oxygen. Multiplying every entity count by the same factor preserves the ratio. See Flowers et al., Chemistry 2e, reaction stoichiometry.

Coefficients and subscripts do different jobs. The coefficient in $2{\mathrm{H}}_2\mathrm{O}$ counts two water molecules; the subscript says that each contains two hydrogen chemical atoms. Changing coefficients scales quantities. Changing a subscript changes the specified chemical substance.

This distinction catches a common counting error: one mole of water molecules contains two moles of hydrogen chemical atoms and one mole of oxygen chemical atoms. Always specify what is counted.

Stoichiometry records reactant and product multiplicities in a chemical reaction or a chemical reaction network. The change caused by one reaction is its stoichiometric vector; multiplying by the reaction rate gives its contribution to the concentration equations. For two identical reactants the rate-constant convention absorbs any combinatorial factor.

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.