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For RNA birth/death rates ${\lambda }_1,{\beta }_1{x}_1$ and protein rates ${\lambda }_2{x}_{1}f(x_2),{\beta }_2{x}_2$, linearize at stationary means $m_1,m_2$. Put $H=1-\partial \log f/\partial \log x_2=1+h$. The relative-fluctuation drift is $\left(\begin{array}{cc}-{\beta }_1& 0\\ {\beta }_2& -{\beta }_{2}H\end{array}\right)$ and the diffusion matrix is $\mathrm{diag}(2{\beta }_1/m_1,2{\beta }_2/m_2)$. The stationary covariance equation gives the displayed normalized protein variance. Its extrinsic part factors as $H^{-2}[{\beta }_{2}H/({\beta }_1+{\beta }_{2}H)]m_1^{-1}$: static susceptibility squared times a time-averaging factor times environmental noise. Faster response alone weakens averaging; feedback reduces susceptibility and thereby lowers noise. The formula is exact for affine rates when $H=1$, and is a linear noise approximation for nonlinear feedback.

For a stable one-species linear noise approximation, the normalized stationary fluctuation-dissipation relation for a reaction network becomesTo see the factors explicitly, the linearized restoring rate is $a=H/\tau =4\beta m$, while the chemical noise intensity is the rate-weighted squared step sizeStationarity of the variance gives $2a\mathrm{Var}x=B$. Using $\lambda =2\beta m^2$ therefore yieldsThis is the stationary fluctuation result within the stated negligible-fluctuation/linear-noise approximation, not an exact identity for the boundary-corrected discrete chemistry.

For the constitutive system, every propensity is affine in the copy numbers. The exact first- and second-moment equations therefore close: expectations of drift are determined by the means, and covariance drift by the covariance. The stationary fluctuation-dissipation calculation is consequently exact, even though the full joint stationary distribution need not be Gaussian.

For feedback, expectations such as $E[x_{1}f(x_2)]$ involve nonlinear moments and correlations. Replacing them by a linear expansion about the mean is a linear noise approximation, so the covariance formula is approximate. It works well around a stable single equilibrium when relative fluctuations are small and the rates vary smoothly over their typical range, generally at sufficiently large molecule numbers. Low counts, strong nonlinearities, multiple attractors and proximity to instability can invalidate it.

Let $\mu =⟨X⟩$. The Markov jump-process generator applied to $X$ gives the exact moment equation $\dot{\mu }=3\lambda -\beta ⟨\sqrt{X}⟩$. With the prescribed moment closure, $⟨\sqrt{X}⟩\approx \sqrt{\mu }$, stationarity givesThe production flux counts molecules, not reaction events: it is $v^{+}=3\lambda$, while the removal flux is $v^{-}=\beta \sqrt{\mu }=3\lambda$. The reaction-rate elasticity of removal relative to constant production isThe mean molecular lifetime is population divided by its stationary molecular removal flux. ThereforeThe noise-relevant average chemical event size is the molecular-flux-weighted reaction event size. For the stoichiometric vectors $+3$ and $-1$ it isThis weights each absolute jump by the molecular turnover it contributes. Equivalently, the balanced production and removal fluxes give equal weight to their sizes $3$ and $1$, yielding $(3+1)/2$. An event-count-weighted average would be $3/2$, since three quarters of firing events are deaths at stationarity; that different average does not determine the diffusion term in the linear noise approximation.

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.

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.

Let $\xi =x-⟨x⟩$ and $C_{ij}=⟨{\xi }_i{\xi }_j⟩$. In the linear noise approximation, $A_i=a_{ik}{\xi }_k$ and $B_{ij}=b_{ij}$. At the stationary mean, the preceding moment identity gives ${\dot{C}}_{ij}=a_{ik}{C}_{kj}+a_{jk}{C}_{ik}+b_{ij}$. Hence the stationary covariance matrix solves the Continuous Lyapunov equationwhich is precisely the requested component formula because $C$ is symmetric.

The deterministic drift vanishes at the positive stable pointWhen $n_{\ast }$ is large, fluctuations sample only a narrow range around it, so the linear noise approximation replacesThe resulting Ornstein-Uhlenbeck stationary density is the stated normal distribution withand

This is a birth-death process with birth and death ratesIts transition diagram has the two arrowsThe gain into state $n$ comes from a death in state $n+1$ or a birth in state $n-1$, while the loss is the sum of both rates out of state $n$. The birth-death master equation is consequently

Multiplying by $n$, summing over the nonnegative integers, and shifting the summation indices shows that each birth contributes $+1$ and each death contributes $-1$. Thus the first-moment equation of a birth-death process givesAt a stationary state, use the variance identity $⟨n^2⟩=⟨n{⟩}^2+(\Delta n{)}^2$. Solving the resulting quadratic equation givesThe minus root is inadmissible whenever it is negative, namely when $(\Delta n{)}^2<\gamma /\alpha$. Equality would give zero mean, which is also incompatible with a positive immigration rate $\gamma$, so for $\gamma >0$ the minus branch requires $(\Delta n{)}^2>\gamma /\alpha$ even to be a possible mean.

For a continuum approximation, write $b(n)=\gamma +\beta n$ and $d(n)=\alpha n(n-1)$. Applying the Kramers-Moyal expansion to the two gain terms and retaining derivatives through second order gives the Fokker-Planck equationwhere the negative drift and infinitesimal jump variance areThis truncation requires the typical population and the scale on which $P$ varies to be much larger than the unit jump size, with the rates varying smoothly across that scale.

The positive zero of $g$ isso in particular $n_{\ast }\sim \beta /\alpha$ as $\alpha /\beta \to 0$. Put $x=n-n_{\ast }$. The linear noise approximation useswhereand, because $b(n_{\ast })=d(n_{\ast })$,In a stationary state with zero Fokker-Planck probability current,Its normalized solution is the normal distributionIt follows that

These estimates agree with the exact stationary first-moment relation on its plus branch to leading order: inserting $(\Delta n{)}^2\sim \beta /\alpha$ gives $⟨n⟩\sim \beta /\alpha$. Moreover,Thus the stationary mass lies far from the boundary $n=0$, is narrow relative to its mean, yet changes across many lattice sites. These are precisely the large-population and slow-variation conditions needed for the diffusion approximation.