# Stoichiometric Matrix

In another article we discussed the Michaelis-Menten enzyme kinetics, where the change in concentration basically obeys the law of mass action. This means that the reaction rate $dP/dt$ is proportional to the probability of a collision of the reactants and therefore proportional to the substrate concentration $S_1$ or the product of substrate concentrations $S_1 \cdot S_2$ leading for the reaction $S_1+S_2 \rightarrow 2P$ or $S_1+S_2 \rightleftharpoons 2P$ to equation $$\frac{dP}{dt}=k_+ \cdot S_1 \cdot S_2 \tag{1}$$ or for reversible reactions $$\frac{dP}{dt}=k_+ \cdot S_1 \cdot S_2-k_- \cdot P^2 \tag{2}$$ Since not all rate constants $k$ for the metabolic system of a whole organisms are available, other approaches to investigate biochemical systems are employed, for example the stoichiometric coefficients. Using stoichiometric coefficients we may write the equations (1) and (2) as $$\frac{dS_1}{dt}=-\mathbf{1}v \hspace{1cm} \frac{dS_2}{dt}=-\mathbf{1}v \hspace{1cm} \frac{dP}{dt}=\mathbf{2}v \tag{3}$$ with stoichiometric coefficients $-1,-1,2$

Reactions of metabolic networks can be described by the system or balance equations. The representation of a metabolic network consisting of $m$ substances and $r$ reactions is given by $$\frac{dS_i}{dt}=\sum_{j=1}^{r}n_{ij}v_j \hspace{1cm} for \hspace{1cm} i=1, \dots , m.$$ The equations are called balance equations since substrate and product have to be balanced, in our example (3), this means for each molecule $S_1$ and $S_2$ disappearing, one molecule $P$ has to be created.

The stoichiometric coefficients $n_{ij}$ could be represented in a stoichiometric matrix $\mathbf{N}$ with reactions $v_j$ and compounds $S_i$ $$\mathbf{N}=\left\lbrace n_{ij} \right\rbrace \text{ for } i=1, \dots , m \text{ and } j= 1, \dots , r$$ Examples for Network representations with the stoichiometric matrix

Network Stoichimetric Matrix
$\mathbf{N}=\begin{pmatrix} -1\\ -1\\ -1\\ 1\\ 2\\ \end{pmatrix}$
$\mathbf{N}=\begin{pmatrix} -1 & 0 & 0 & 0\\ 1 & -1 & 0 & 0\\ 0 & 1 & -1 & 0\\ 0 & 0 & 1 & -1 \\ 0 & 0 & 0 & 1 \\ \end{pmatrix}$
$\mathbf{N}=\begin{pmatrix} 1 & -1 & -1\\ \end{pmatrix}$
$\mathbf{N}=\begin{pmatrix} 1 & -1 & 0 & -1\\ 0 & 2 & -1 & 0\\ 0 & 0 & 0 & 1\\ \end{pmatrix}$
$\mathbf{N}=\begin{pmatrix} 1 & -1 & 0 & 0\\ 0 & 0 &- 1 & 1\\ 0 & 0 & 1 & -1\\ \end{pmatrix}$

In the first network there is only one reaction (column) and five species (rows). $S_1,S_2,S_3$ are consumed and have therefore a minus sign, $S_4$ and two molecules $S_5$ are produced in this reaction, therefore they have a positive sign. Try to derive the stoichiometric matrix from the other networks.