Kinetic scheme

A kinetic scheme is a network (a directed graph) of distinct states (although repetition of states may occur and this depends on the system), where each pair of states i and j are associated with directional rates, ${\displaystyle A_{ij}}$ (and ${\displaystyle A_{ji}}$). It is described with a master equation: a first-order differential equation for the probability ${\displaystyle {\vec {P}}}$ of a system to occupy each one its states at time t (element i represents state i). Written in a matrix form, this states: ${\displaystyle {\frac {d{\vec {P}}}{dt}}=\mathbf {A} {\vec {P}}}$, where ${\displaystyle \mathbf {A} }$ is the matrix of connections (rates) ${\displaystyle A_{ij}}$.

In a Markovian kinetic scheme the connections are constant with respect to time (and any jumping time probability density function for state i is an exponential, with a rate equal the value of all the exiting connections).

When detailed balance exists in a system, the relation ${\displaystyle A_{ji}P_{i}(t\rightarrow \infty )=A_{ij}P_{j}(t\rightarrow \infty )}$ holds for every connected states i and j. The result represents the fact that any closed loop in a Markovian network in equilibrium does not have a net flow.

Matrix ${\displaystyle \mathbf {A} }$ can also represent birth and death, meaning that probability is injected (birth) or taken from (death) the system, where then, the process is not in equilibrium. These terms are different than a birth–death process, where there is simply a linear kinetic scheme.

• A birth–death process is a linear one-dimensional Markovian kinetic scheme.
• Michaelis–Menten kinetics are a type of a Markovian kinetic scheme when solved with the steady state assumption for the creation of intermediates in the reaction pathway.