Two-stage linear gene-expression covariance ($\mathrm{var}P=q_P/(2{\gamma }_P)+k_P^2{q}_R/[2{\gamma }_R{\gamma }_P({\gamma }_R+{\gamma }_P)]$)

For linear RNA removal and protein production driven by independent additive white noises, the stationary RNA variance is $q_R/(2{\gamma }_R)$ and the protein variance is the displayed expression. Steady count birth-and-death intensities $q_R=2{\gamma }_R⟨R⟩$, $q_P=2{\gamma }_P⟨P⟩$ give the protein Fano factor $1+k_P/({\gamma }_R+{\gamma }_P)$. This specializes arbitrary noise strengths; it is not an identity valid for every additive-noise model.