Translational burst size ($b=r/{\alpha }_R$)

The expected number of proteins produced by one messenger RNA before its removal. With translation rate $r$ and exponential removal rate ${\alpha }_R$, $b=r/{\alpha }_R$. Independent Poisson translation during an exponential lifetime yields a geometric burst count on nonnegative integers, with variance $b(1+b)$. In the short-message-lifetime limit with independent unit protein removal, the protein Fano factor tends to $1+b$.

For independent Poisson process arrivals of geometric translational bursts with mean $b$, arrival rate $\lambda$, and unit protein removal rate $\alpha N$, the stationary mean is $\bar{N}=\lambda b/\alpha$. Starting with zero molecules, the expected value is $m(t)=\bar{N}(1-e^{-\alpha t})$. The variance satisfies $v^{\prime }=\lambda b(1+2b)+\alpha m-2\alpha v$. An integrating factor gives the displayed expression, and its stationary limit is $\bar{N}(1+b)$. Concentration fluctuations require a specified cell volume to convert concentrations into molecule counts.