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A reaction's stoichiometric vector outer product multiplied by its reaction propensity function is its contribution to the infinitesimal state-increment covariance. Summing over channels gives $B$. Its normalized form $D$ divides by the two strictly positive mean copy numbers. The Fokker-Planck equation writes the associated second-derivative term with $B/2$; the covariance evolution and normalized stationary Lyapunov equation instead use $B$ and $D$ themselves. A simultaneous unit loss of two species contributes positive cross noise even when their long-time stationary covariance is negative.

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.

Daniel T. Gillespie developed stochastic simulation methods for reacting molecular populations. In the Gillespie algorithm, a reaction propensity function specifies an event rate, and the algorithm samples both the next reaction time and the reaction channel rather than evolving only mean concentrations.^[ref]

If the total propensity is $a_0>0$, the waiting time is exponentially distributed with rate $a_0$, and channel $j$ is selected with probability $a_j/a_0$ at the current state. His tau-leaping approach groups reaction events to accelerate simulation, introducing a controlled approximation rather than retaining the exact event-by-event law.^[ref]

The Gillespie algorithm samples statistically exact trajectories of a well-mixed chemical reaction network modeled by a continuous-time Markov chain. At copy-number state $x$, each reaction propensity function $a_r(x)$ is a nonnegative event rate and each stoichiometric vector ${\nu }_r$ specifies the update. Assume the rates have no explicit time dependence. Put $a_0(x)={\sum }_r{a}_r(x)$. If $a_0(x)=0$, no further chemical reaction can occur: the state is an absorbing state.

For $a_0(x)>0$, sample a waiting time $\tau$ from the exponential distribution with rate $a_0(x)$, independently choose channel $r$ with probability $a_r(x)/a_0(x)$, and update $(t,x)\leftarrow (t+\tau ,x+{\nu }_r)$. Recompute the reaction propensity functions and repeat. The no-event probability is $S(\tau )=e^{-a_0\tau }$ because $S^{\prime }=-a_{0}S$; multiplying by $a_r$ gives the joint waiting-time/channel probability density function $a_r{e}^{-a_0\tau }$. This factorization explains the two draws.^[ref]

For $A\to B$ with reaction propensity function $a(x)=k{x}_A$, the waiting time has mean $1/(k{x}_A)$, and $x_A=0$ is an absorbing state. Exactness concerns the specified chemical master equation; spatial mixing errors and explicitly time-varying rates require a different model or sampling rule.

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.

The normalized stationary fluctuation-dissipation relation for a reaction network isHere $M$ is the negative drift Jacobian matrix expressed in normalized population coordinates, and $D$ is the normalized noise matrix. More explicitly, with $S=\mathrm{diag}({\mu }_i)$, drift Jacobian $A$, stoichiometric vectors ${\nu }_r$, and stationary reaction propensity functions $a_r$, these matrices are $M=-S^{-1}AS$ and $D=S^{-1}B{S}^{-1}$, where $B={\sum }_r{a}_r{\nu }_r{\nu }_r^{\mathsf{T}}$. The linear noise approximation gives a covariance evolution $\dot{\eta }=-M\eta -\eta M^{\mathsf{T}}+D$; stationarity is the displayed Continuous Lyapunov equation.

In one dimension, the normalized restoring coefficient is $M=H/\tau =1/(2\tau )$. Squared chemical jumps determine the noise:The factor nine from a birth burst is essential; its three molecules arise in one correlated reaction event. The scalar Continuous Lyapunov equation is $2M\eta =D$, givingEquivalently, the normalized variance is $⟨s⟩/(H\mu )=2/((1/2)\mu )$. The Fano factor is approximately four. This calculation uses the prescribed mean closure and the linear noise approximation, so its small-relative-fluctuation regime is $\mu \gg 1$; it is not an exact assertion for arbitrarily small molecular copy numbers.

For the Gillespie algorithm, start from $t=0$ and an integer molecular copy number $x\ge 0$. At each step use the current state, without the mean approximation in parts (a)–(b). The reaction propensity functions are $a_{+}(x)=\lambda$ and $a_{-}(x)=\beta \sqrt{x}$, and their sum is $a_0(x)=\lambda +\beta \sqrt{x}$. Draw two independent uniform random variables $U_1,U_2$ on $(0,1)$ and setThe waiting time has an exponential distribution with rate $a_0(x)$. Keep the sample path constant on $[t,t+\Delta t)$, advance $t\leftarrow t+\Delta t$, and choose the reaction byRecord the post-jump state, recompute both reaction propensity functions, and repeat with fresh independent uniforms. At $x=0$ the death propensity is zero, so only a birth is selected and the trajectory remains nonnegative. For a prescribed terminal time $T$, if the next event would occur after $T$, keep the current state to $T$ and stop. This produces exact piecewise constant sample paths of the specified continuous-time Markov chain.

Let ${\mu }_i=⟨X_i⟩$. Applying the Markov jump-process generator to either coordinate gives the exact moment equationsIn each equation the joint reaction removes one molecule, at reaction propensity function $C{X}_1{X}_2$. Because the two reactants are different chemical species, the propensity uses their product; no same-species factorial correction is needed. The exact identity $⟨X_1{X}_2⟩={\mu }_1{\mu }_2+\mathrm{Cov}(X_1,X_2)$ shows explicitly where the equations fail to close. Neglecting this covariance relative to the product of means is the small-fluctuation moment closure. It gives the mean-field joint removal of two chemical species:The difference equation $d({\mu }_1-{\mu }_2)/dt=-\beta ({\mu }_1-{\mu }_2)$ actually holds exactly as well as after closure, because both the constant production and the joint removal terms cancel. The nonlinear moment closure will be used for the subsequent phase plane and stationary linear noise approximation calculations, as requested.

Write $n=n_1$ and $m=n_2$. The three stoichiometric vectors are $(1,0)$, $(0,1)$ and $(-1,-1)$. The law of mass action gives the respective reaction propensity functionsIn particular, producing the second chemical species does not consume the first chemical species. Since the two reactants in the third chemical reaction are different chemical species, the number of possible pairs is $nm$.

For each Markov chain state, the chemical master equation adds incoming probability flux and subtracts outgoing probability flux:Here $p(n,m,t)=0$ if either index is negative, and the initial condition is $p(n,m,0)=1_{\{n=m=0\}}$. These conventions also give the correct chemical master equation on the coordinate axes: for example, the third chemical reaction has zero outgoing reaction propensity function when either chemical species is absent.

Conditioning on the observed time. The native PDF defines ${\tau }_2$ as the first production time and then conditions on that production occurring at ${\tau }_2$. If the intended information is its numerical value ${\tau }_2=t$, the answer depends on $t$. We first compute this conditional probability using a conditional probability density; conditioning on an exact value of a continuous random variable means a density ratio, since $\mathbb{P}({\tau }_2=t)=0$.

Before ${\tau }_2$, no molecules of the second chemical species are present, so the joint-removal chemical reaction cannot occur. Molecules of the first chemical species arrive as a rate-$2$ Poisson process, and each independently starts a clock with a rate-$1/2$ exponential distribution for producing the first molecule of the second chemical species. Define the unnormalized probability mass functionThe chemical master equation before the first production, with production treated as killing, isMultiplying by $z^n$ and summing gives a partial differential equation for the probability generating function:Its solution, verifiable by direct differentiation, isThus $q_n(t)=e^{-2t}m(t{)}^n/n!$, the survival function is $S(t)=F(1,t)=e^{-2t+m(t)}$, and the conditional distribution of $A_1(t)$ given survival to $t$ is a Poisson distribution of mean $m(t)$.

Production in $(t,t+dt)$ from Markov chain state $n$ has joint probability $(n/2)q_n(t)dt+o(dt)$. Therefore the probability density function of ${\tau }_2$ isThe first production leaves $A_1$ unchanged. Taking the density ratio gives a size-biased distribution of the surviving count:In particular, under this reading the requested conditional probability isThe shift by one matters: an observation made exactly when production occurs preferentially selects larger counts, since their production reaction propensity function is larger.

Without observing the numerical time. If the conditioning merely repeats the definition of ${\tau }_2$, the requested probability is instead the unconditional count at that stopping time. Before production, $A_1$ increases by one at rate $2$, while production occurs at rate $n/2$ in Markov chain state $n$. The competing exponential clocks implyThe first factor requires a second arrival before production from count one; the second requires production before a third arrival. The first arrival itself is certain. Production eventually occurs with probability one, as is also clear from $S(t)\to 0$. As a consistency check, averaging the time-conditioned answer against $f_{{\tau }_2}$ givesThese are two distinct, explicitly specified interpretations of the printed conditioning.

An invariant distribution of a continuous-time Markov chain whose coordinates are independent Poisson distributions. This property needs verification from the reaction propensity functions; it does not follow merely from having an equilibrium for the deterministic reaction-rate equations.

For example, consider the cyclic chemical reaction network with stoichiometric vectors $(1,0)$, $(0,1)$ and $(-1,-1)$ and respective reaction propensity functions $\lambda$, $\beta n$, $\gamma nm$, where all three constants are positive. Put $a=\lambda /\beta$ and $b=\beta /\gamma$. For $\pi (n,m)=e^{-a-b}{a}^n{b}^m/(n!m!)$, the three incoming terms in the chemical master equation are respectively $\beta n\pi (n,m)$, $\gamma nm\pi (n,m)$ and $\lambda \pi (n,m)$. They exactly cancel the outgoing terms. Boundary terms vanish by the zero negative-index convention, and normalization follows from the two Poisson distributions. Establishing a limiting probability distribution additionally requires nonexplosion, an irreducible Markov chain and positive recurrence.

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.

Tau-leaping accelerates the Gillespie algorithm by approximating reaction rates as constant over an interval $\tau$, then drawing all reaction counts for that interval at once. At state $x$, with reaction propensity functions $a_r(x)$ and stoichiometric vectors ${\nu }_r$, the explicit step isThe leap condition requires the propensities to change little during the interval. Unlike event-by-event exact simulation, this introduces an approximation error controlled through the choice of $\tau$.^[ref]

For $A\to B$ with rate $k{x}_A$, the approximate number converted is Poisson with mean $k{x}_A\tau$. The exact number converted over the interval is instead binomial with parameters $x_A$ and $1-e^{-k\tau }$. The two agree to first order for small $k\tau$, but the Poisson draw can exceed $x_A$ and produce a negative count. Smaller or adaptive leaps and modified sampling rules address this limitation; freezing the rate is not automatically safe at low copy number.