 Solution

The messenger RNA has steady state $X_{R,\ast }={\beta }_R/{\alpha }_R$. Since biological translation produces proteins at rate $r{X}_R$, setting the protein degradation rate equal to this production rate givesThe translational burst size is the expected number of proteins made during one messenger RNA lifetime, whose expected value is $1/{\alpha }_R$. It is dimensionless, rather than a production rate. With independent Poisson process translation and an exponential distribution for the lifetime, the resulting count has a geometric distribution on nonnegative integers with expected value $b$ and variance $b(1+b)$.