= Stoichiometric matrix
{title2=$S$}
A <stoichiometric matrix> records how each reaction changes the amounts of the species in a <chemical reaction network>. Rows correspond to species and columns to reactions; an entry is positive for production and negative for consumption. For a well-mixed system of fixed volume, with concentration vector $\mathbf{x}$ and reaction-rate vector $\mathbf{v}$, the mass-balance model is $\dot{\mathbf{x}}=S\mathbf{v}$.
Consider input to $A$, conversion $A\to B$, and removal of $B$, at rates $u,v,w$. Counting only the two internal species gives
$$
S=\begin{pmatrix}1&-1&0\\0&1&-1\end{pmatrix},\qquad
\frac{d}{dt}\begin{pmatrix}A\\B\end{pmatrix}=S\begin{pmatrix}u\\v\\w\end{pmatrix}=\begin{pmatrix}u-v\\v-w\end{pmatrix}.
$$
At a <steady state>, $S\mathbf{v}=0$, so $u=v=w$. All three rates can be equal at many different magnitudes: <stoichiometry> constrains the allowable balance without supplying the <chemical kinetics> that sets the actual rate. Genome-scale <metabolic networks> use this same bookkeeping for many reactions.<The Escherichia coli MG1655 in silico metabolic genotype: its definition, characteristics, and capabilities by J. S. Edwards and B. O. Palsson>[2000, metabolic reconstruction and capability analysis; the two-species calculation here is illustrative]{ref}
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