Kinetic matrix

Consider a chemical reaction network $\mathcal{N} = ( \mathcal{S}, \mathcal{C}, \mathcal{R})$ endowed with mass-action kinetics and using complex-centered indexing of the reactions and rate constants $k(i,j)$ , for $i, j = 1, \ldots, m$ . Then the kinetic or Kirchhoff matrix $A_k \in \mathbb{R}_{\geq 0}^{m \times m}$ of the system is defined as

$A_k = \left[ \begin{array}{cccc} - \sum_{i=1}^m k(1,i) & k(2,1) & \cdots & k(m,i) \\ k(1,2) & - \sum_{i=1}^m k(2,i) & \cdots & k(m,2) \\ \vdots & \vdots & \ddots & \vdots \\ k(1,m) & k(2,m) & \cdots & - \sum_{i=1}^m k(m,i) \end{array} \right]$

where $k(i,j) > 0$ is the rate constant corresponding to the reaction $(\mathcal{C}_i,\mathcal{C}_j) \in \mathcal{R}$ (and $k(i,j) = 0$ if $(\mathcal{C}_i,\mathcal{C}_j) \not\in \mathcal{R}$ ).

The kinetic matrix of a system keeps track of the connections in the reaction graph of a network, as well as the corresponding reaction weights associated with mass-action kinetics. For example, for off-diagonal entries $i \not= j$ , it is clear that $[A_k]_{ji} \geq 0$ for all $i,j = 1, \ldots, m,$ and $[A_k]_{ji} > 0$ if and only if $(\mathcal{C}_i, \mathcal{C}_j) \in \mathcal{R}$ . In this sense, a kinetics matrix can be said to indicate the structure of a chemical reaction network underlying a mass action system. It is also clear that $\vec{1}^T A_k = \vec{0}$ where $\vec{1} \in \mathbb{R}^m$ and $\vec{0} \in \mathbb{R}^m$ are the vectors of all ones and zeroes, respectively. This implies, among other things, that a kinetics matrix has a non-trivial kernel.