The mean frequency of transcriptional bursting and its variation in single cells.

The recent in vivo RNA detection technique has allowed real-time monitoring of gene transcription in individual living cells, revealing that genes can be transcribed randomly in a bursting fashion that short periods of rapid production of multiple transcripts are interspersed with relatively long periods of no production. In this work, we utilize the three state model to study how environmental signals and the intrinsic cellular contexts are combined to regulate stochastic gene transcription. We introduce a system of three master equations to model the stochastic occurrence of transcriptional bursting. As this system cannot be solved analytically, we introduce a linear operator, called the master operator. It is of significant mathematical interests of its own and transforms the mean frequency of transcriptional bursting mu(t) and the second moment mu2(t) into the unique solutions of the respective operator equations. Following this novel approach, we have found the exact forms of mu(t) and the variance sigma2(t). Our analysis shows that the three state transition process produces less noisy transcription than a single Poisson process does, and more transition steps average out rather than propagate fluctuations of transcripts among individual cells. The noise strength phi(t) = sigma2(t)/mu(t) displays highly non-trivial dynamics during the first two to three transcription cycles. It declines steeply from the beginning until reaching the absolute minimum value, and then bounces back suddenly to a flat level close to the steady-state. Our numerical simulations further demonstrate that the cellular signals that produce the least noisy population at steady-state may not generate the least noisy population in a finite time, and suggest that measurements at steady-state may not necessarily capture most essential features of transcription noise.