 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 isWriting ${\tau }_i=1/{\beta }_i$, 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.