= Transient protein-count variance in the instantaneous-burst model
{title2=$v(t)=\bar N[(1+b)-e^{-\alpha t}-be^{-2\alpha t}]$}
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 $v'=\lambda b(1+2b)+\alpha m-2\alpha v$. 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.
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