Stoichiometric matrix ($S$)

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 givesAt 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.^[ref]