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For reactions with <stoichiometric vectors> $\nu_r$ and <reaction propensity functions> $a_r$, the probability mass function $p(x,t)$ obeys
$$
\partial_t p(x,t)=\sum_r\left[a_r(x-\nu_r)p(x-\nu_r,t)-a_r(x)p(x,t)\right].
$$
It is the <forward operator of a Markov jump process> equation for the copy-number <continuous-time Markov chain>.


Chemical master equation

A moment hierarchy is the coupled family of equations obtained by applying a <Markov jump-process generator> to powers of the state. Nonlinear <reaction propensity functions> make a moment of one order depend on higher-order moments, so the hierarchy generally does not close after finitely many equations.


Moment hierarchy

Stochastic chemical kinetics models molecular copy numbers as a <continuous-time Markov chain>. Each reaction has a state-dependent <reaction propensity function> and changes the state by its <stoichiometric vector> when it fires.


Stochastic chemical kinetics

{title2=$a_r(x)$}

The reaction propensity $a_r(x)$ is the instantaneous rate at which reaction $r$ fires when the copy-number state is $x$. Conditional on the present state, its reaction clock has an <exponential distribution> of rate $a_r(x)$.


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 Incoming links: Reaction propensity function