Kinetic matrix

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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.

The kinetics matrix differs from the weighted Laplacian matrix of a weighted directed graph only by the sign of its entries.

Kernel of A_k

Relationship to Markov processes