# Stoichiometric Matrix

### From bio-physics-wiki

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
\begin{equation}
\frac{dP}{dt}=k_+ \cdot S_1 \cdot S_2 \tag{1}
\end{equation}
or for reversible reactions
\begin{equation}
\frac{dP}{dt}=k_+ \cdot S_1 \cdot S_2-k_- \cdot P^2 \tag{2}
\end{equation}
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
\begin{equation}
\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}
\end{equation}
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
\begin{equation}
\frac{dS_i}{dt}=\sum_{j=1}^{r}n_{ij}v_j \hspace{1cm} for \hspace{1cm} i=1, \dots , m.
\end{equation}
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 $S_i$
\begin{equation}
\mathbf{N}=\left\lbrace n_{ij} \right\rbrace \text{ for } i=1, \dots , m \text{ and } j= 1, \dots , r
\end{equation}
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.

Further Reading:

- Martin Feinberg - Feinberg Lectures on Chemical Reaction Networks [1]
- Bernhard Palsson - Systems Biology: Properties of Reconstructed Networks
- Edda Klipp et al. - Systems Biology: A Textbook