= Solution
If RNA concentration does not fluctuate, protein synthesis is an immigration-death process, so its intrinsic normalized <variance> is $\boxed{1/m_2}$. If protein birth/death noise is suppressed and protein responds deterministically to the fluctuating RNA input, the remaining normalized <variance> is
$$
\boxed{\eta_{22}^{\rm ext}=\frac{\beta_2}{\beta_1+\beta_2}\eta_{11}.}
$$
Writing $\tau_i=1/\beta_i$, its factor is $\tau_1/(\tau_1+\tau_2)$. Slow environmental changes are followed almost fully; rapid environmental fluctuations are averaged over the longer protein response time. Thus extrinsic noise is the magnitude of relative RNA fluctuations multiplied by a dynamical averaging factor.
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