Stoichiometric matrix analysis is a fundamental tool in the study of chemical reaction systems, particularly within systems biology and chemical engineering. It provides a structured way to represent and analyze the relationships between the components of a reaction network.
Consider the reaction system:
$$ \begin{gathered} A+B \xrightarrow{k_1} C \\ C \xrightarrow{k_2} D \end{gathered} $$
where:
Step 1: Define the Species and Reactions The chemical species involved:
$$ S=\{A, B, C, D\} $$
The reaction rates ( $r_1, r_2$ ) using mass-action kinetics:
Step 2: Construct the Stoichiometric Matrix The stoichiometric matrix ($N$) captures how each species changes due to the reactions.
$$ \begin{array}{|l|l|l|} \hline \text { Species } & \text { Reaction } 1(A+B \rightarrow C) & \text { Reaction } 2(C \rightarrow D) \\ \hline A & -1 & 0 \\ \hline B & -1 & 0 \\ \hline C & +1 & -1 \\ \hline D & 0 & +1 \\ \hline \end{array} $$
So, the stoichiometric matrix is:
$$ N=\left[\begin{array}{cc} -1 & 0 \\ -1 & 0 \\ +1 & -1 \\ 0 & +1 \end{array}\right] $$
Step 3: Find Conservation Laws To find conserved quantities, we determine the null space of $N$, meaning we solve:
$$ N^T c =0 $$
where $c =\left(c_1, c_2, c_3, c_4\right)^T$ represents the conserved quantities. Solving:
$$ \left[\begin{array}{cccc} -1 & -1 & +1 & 0 \\ 0 & 0 & -1 & +1 \end{array}\right]\left[\begin{array}{l} c_1 \\ c_2 \\ c_3 \\ c_4 \end{array}\right]=\left[\begin{array}{l} 0 \\ 0 \end{array}\right] $$