I've been wondering for a very long time what properties are know on what I will call "kinetic" matrices, for lack of a proper name. These matrices $k_{ij}$ have the following properties:
- $\forall j\neq i, k_{ij} \geq 0$
- $ \forall i, \sum_{j} k_{ij} = 0$
These kind of matrices are very common in enzyme kinetics. For instance the Michaelis-Menten equations, arising for instance from the chemical reaction:
$$\mathrm{E + S} \mathop{\rightleftharpoons}^{k_\mathrm{f}}_{k_\mathrm{b}} \mathrm{ES} \mathop{\to}^{k_\mathrm{cat}} \mathrm{E + P}$$
For this system, the evolution over time of the concentrations of $\mathrm{E}$ and $\mathrm{ES}$ can be deduced from the differential equation:
$$ \dot{\mathbf{x}} = k \mathbf{x} $$
With $\mathbf{x} = (E, ES)$ and $k$ the following matrix:
$$ k = \left( \begin{array}{cc} -k_f s & k_b + k_{cat}\\ k_f s & -k_b - k_{cat}\\ \end{array} \right) $$
There are a few (more or less) obvious properties, such as:
- they are not invertible;
- their eigenvalues have negative real parts;
- the eigenvalues that are not $0$ have eigenvectors with sums of coefficients equal to $0$.
I'm sure these matrices have been heavily studied before, I just don't seem to find how they are called, and where to learn more about them. What is known about them ?
