= Feedback attenuation of gene-expression noise
{title2=$\eta_{22}\simeq\frac1{Hm_2}+\frac{\beta_2}{H(\beta_1+\beta_2H)m_1}$}
For RNA birth/death rates $\lambda_1,\beta_1x_1$ and protein rates $\lambda_2x_1f(x_2),\beta_2x_2$, 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 $\begin{pmatrix}-\beta_1&0\\\beta_2&-\beta_2H\end{pmatrix}$ and the diffusion matrix is $\operatorname{diag}(2\beta_1/m_1,2\beta_2/m_2)$. The stationary covariance equation gives the displayed normalized protein variance. Its extrinsic part factors as $H^{-2}[\beta_2H/(\beta_1+\beta_2H)]m_1^{-1}$: 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.
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