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.

This noise-relevant event size weights absolute stoichiometric jump sizes by their molecular turnover flux. At stationarity the denominator is twice the balanced production or removal flux $v_i$, so the normalized diffusion diagonal is $D_{ii}=2⟨s_i⟩/({\tau }_i{\mu }_i)$. Fixed birth bursts of size $b$ and unit deaths give $⟨s⟩=(b+1)/2$. This quantity is not the event-count-weighted average absolute jump size.

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At stationarity, the mean number ${\mu }_i$ divided by the molecular removal flux $v_i$ is the mean time a molecule remains in the system. Birth or death event rates must be multiplied by the numbers of molecules changed in each event before computing this lifetime.

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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 population jumps from $n$ to $n+2$ at a constant immigration rate $\lambda$ and from $n$ to $n-1$ at rate $\beta n$. Applying its Markov jump-process generator to $n$ and $n^2$ gives the displayed mean and variance equations. For $\beta >0$ and finite initial second moment, the stationary mean and variance are $2\lambda /\beta$ and $3\lambda /\beta$. Interpreting the pair-arrival rate as proportional to population instead would produce a different branching process.

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.

For a Markov jump-process generator $L$, differentiating an integrable observable gives $d\mathbb{E}[f(X_t)]/dt=\mathbb{E}[Lf(X_t)]$, when the generator identity and interchange are justified. Choosing powers gives equations for moments. These need not form a closed finite system.

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