Normalized stationary fluctuation-dissipation relation for a reaction network ($M\eta +\eta M^{\mathsf{T}}=D$)

Let $A$ be the drift Jacobian at a stable mean-field equilibrium, $S=\mathrm{diag}({\mu }_i)$, and $B={\sum }_r{a}_r{\nu }_r{\nu }_r^{\mathsf{T}}$ the reaction noise matrix. For normalized fluctuations, put $M=-S^{-1}AS$, $D=S^{-1}B{S}^{-1}$, and ${\eta }_{ij}=\mathrm{Cov}(X_i,X_j)/({\mu }_i{\mu }_j)$. Stationarity of the linear-noise covariance gives the displayed Continuous Lyapunov equation. It applies to driven reaction networks without requiring thermal equilibrium. A mode with zero restoring eigenvalue and nonzero noise has no stationary covariance.

// Target: mathematical-biology.bigb

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.