Inter-state switching in stochastic gene expression: Exact solution, an adiabatic limit and oscillations in molecular distributions
We consider the stochastic gene expression process with inter-state flip-flops. An exact steady-state solution to the master equation is calculated. One of the main goals in this paper is to investigate whether the probability distribution of gene copies contains even-odd number oscillations. A master equation previously derived in the adiabatic limit of fast switching by Kepler and Elston \cite{kepler} suggests that the oscillations should be present. However our analysis demonstrates that the oscillations do not happen not only in the adiabatic case but they are entirely absent. We discuss the adiabatic approximation in detail. The other goal is to establish the master equation that takes into account external fluctuations that is similar to the master equation in the adiabatic approximation. The equation allows even-odd oscillations. The reason the behaviour occurs is an underlying interference of Poisson and Gaussian processes. The master equation contains an extra term that describes the gene copy number unconventional diffusion and is responsible for the oscillations. We also point out to a similar phenomenon in quantum physics.
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