# Kinetic matrix

(Redirected from Kirchhoff matrix)

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

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

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

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 ${\displaystyle i\not =j}$, it is clear that ${\displaystyle [A_{k}]_{ji}\geq 0}$ for all ${\displaystyle i,j=1,\ldots ,m,}$ and ${\displaystyle [A_{k}]_{ji}>0}$ if and only if ${\displaystyle ({\mathcal {C}}_{i},{\mathcal {C}}_{j})\in {\mathcal {R}}}$. 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 ${\displaystyle {\vec {1}}^{T}A_{k}={\vec {0}}}$ where ${\displaystyle {\vec {1}}\in \mathbb {R} ^{m}}$ and ${\displaystyle {\vec {0}}\in \mathbb {R} ^{m}}$ 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.