(untitled) (CSIAM Transactions on Applied Mathematics, 2024)
The fluctuation of mRNA molecule numbers within an isogenic cell population is primarily attributed to randomly switching between active (ON) and inactive (OFF) periods of gene transcription. In most studies the waiting-times for ON or OFF states are modeled as exponential distributions. However, increasing data suggest that the residence durations at ON or OFF are non-exponential distributed for which the traditional master equations cannot be presented. By combining Kolmogorov forward equations with alternating renewal processes, we present a novel method to compute the average transcription level and its noise by circumventing the bottleneck of master equations under gene ON and OFF switch. As an application, we consider lifetimes of OFF and ON states having Erlang distributions. We show that: (i) multiple steps from OFF to ON force the oscillating transcription while multiple steps from ON to OFF accelerate the transcription, (ii) the increase of steps between ON and OFF rapidly reduces the transcription noise to approach its minimum value. This suggests that a large number of steps between ON and OFF are not needed in the model to capture the stochastic transcription data. Our computation approach can be further used to treat a series of transcription cycles which are non-lattice distributed.