Stochastic chemical kinetics

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.

Compartment-based stochastic diffusion represents molecular motion by nearest-neighbour jumps on a spatial lattice. Jump rates of order $D/h^2\pm v/(2h)$ converge to an advection--diffusion equation as compartment width $h$ tends to zero.

The asymmetric simple exclusion process permits at most one particle per lattice site and biases jumps in one direction. A mean-field hydrodynamic limit has advective flux proportional to $c(1-c)$ because both an occupied departure site and vacant arrival site are required.

An immigration--death process has constant birth rate $b$ and linear death rate $dy$ in state $y$. Its stationary distribution is Poisson with mean $b/d$.

A stochastic quasi-steady-state approximation replaces fast species by their stationary conditional distribution given the slow species. Averaging slow propensities over that distribution yields a reduced Markov jump process.

A slow-scale stochastic simulation algorithm applies the Gillespie algorithm to propensities averaged over fast conditional equilibrium, avoiding explicit simulation of every fast reaction.

The molecular copy number of a chemical species is the nonnegative integer number of its molecules in a specified reactor or compartment.

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 $a_r(x)$.

A power-law propensity uses the same reactant monomial as a deterministic law of mass action, scaled by reactor volume. For example, a bimolecular channel may be assigned $a(x)=\alpha x^2/V$. This convention differs at small copy number from the combinatorial propensity $\alpha x(x-1)/V$ and must therefore be stated explicitly.

For reactions with stoichiometric vectors ${\nu }_r$ and reaction propensity functions $a_r$, the probability mass function $p(x,t)$ obeysIt is the forward operator of a Markov jump process equation for the copy-number continuous-time Markov chain.

The Gillespie algorithm samples an exact path of a well-mixed stochastic reaction network. At state $x$, let $a_0(x)={\sum }_r{a}_r(x)$; draw a waiting time with exponential distribution of rate $a_0$, choose reaction $r$ with probability $a_r/a_0$, and update $x\leftarrow x+{\nu }_r$.

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.

A moment closure replaces higher moments in a moment hierarchy by functions of retained lower moments. It converts an infinite hierarchy into a finite approximate system.

A central-moment closure sets selected high-order central moments to zero. For example, setting the third central moment of $X$ to zero gives