Editing Stoichiometry (section)

== Stoichiometry matrix ==
{{Main|Chemical reaction network theory}}

In complex reactions, stoichiometries are often represented in a more compact form called the stoichiometry matrix. The stoichiometry matrix is denoted by the symbol '''N'''.<ref>{{cite journal |last1=Ghaderi |first1=Susan |last2=Haraldsdóttir |first2=Hulda S. |last3=Ahookhosh |first3=Masoud |last4=Arreckx |first4=Sylvain |last5=Fleming |first5=Ronan M.T. |title=Structural conserved moiety splitting of a stoichiometric matrix |journal=Journal of Theoretical Biology |date=August 2020 |volume=499 |pages=110276 |doi=10.1016/j.jtbi.2020.110276|pmid=32333975 |bibcode=2020JThBi.49910276G |doi-access=free |hdl=1887/3134882 |hdl-access=free }}</ref><ref>{{cite journal |last1=Hofmeyr |first1=Jan-hendrik S. |title=Metabolic control analysis in a nutshell |journal=In Proceedings of the 2 Nd International Conference on Systems Biology |date=2001 |pages=291–300 |citeseerx=10.1.1.324.922 }}</ref><ref>{{cite journal |last1=Reder |first1=Christine |title=Metabolic control theory: A structural approach |journal=Journal of Theoretical Biology |date=21 November 1988 |volume=135 |issue=2 |pages=175–201 |doi=10.1016/s0022-5193(88)80073-0|pmid=3267767 |bibcode=1988JThBi.135..175R }}</ref>

If a reaction network has ''n'' reactions and ''m'' participating molecular species, then the stoichiometry matrix will have correspondingly ''m'' rows and ''n'' columns.

For example, consider the system of reactions shown below:
: {{chem2|S1 → S2}}
: {{chem2|5 S3 + S2 → 4 S3 + 2 S2}}
: {{chem2|S3 → S4}}
: {{chem2|S4 → S5}}

This system comprises four reactions and five different molecular species. The stoichiometry matrix for this system can be written as:
: <math>
\mathbf{N} = \begin{bmatrix}
  -1 &  0 &  0 & 0 \\
   1 &  1 &  0 & 0 \\
   0 & -1 & -1 & 0 \\
   0 &  0 &  1 & -1 \\
   0 &  0 &  0 & 1 \\
\end{bmatrix}
</math>

where the rows correspond to {{chem2|S1, S2, S3, S4 and S5,}} respectively. The process of converting a reaction scheme into a stoichiometry matrix can be a lossy transformation: for example, the stoichiometries in the second reaction simplify when included in the matrix. This means that it is not always possible to recover the original reaction scheme from a stoichiometry matrix.

Often the stoichiometry matrix is combined with the rate vector, '''v''', and the species vector, '''x''' to form a compact equation, the [[biochemical systems equation]], describing the rates of change of the molecular species:
: <math>
\frac{d\mathbf{x}}{dt} = \mathbf{N} \cdot \mathbf{v}.
</math>