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.

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