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We consider the stochastic gene expression process with inter-stateflip-flops. An exact steady-state solution to the master equation iscalculated. One of the main goals in this paper is to investigate whether theprobability distribution of gene copies contains even-odd number oscillations.A master equation previously derived in the adiabatic limit of fast switchingby Kepler and Elston \cite{kepler} suggests that the oscillations should bepresent. However our analysis demonstrates that the oscillations do not happennot only in the adiabatic case but they are entirely absent. We discuss theadiabatic approximation in detail. The other goal is to establish the masterequation that takes into account external fluctuations that is similar to themaster equation in the adiabatic approximation. The equation allows even-oddoscillations. The reason the behaviour occurs is an underlying interference ofPoisson and Gaussian processes. The master equation contains an extra term thatdescribes the gene copy number unconventional diffusion and is responsible forthe oscillations. We also point out to a similar phenomenon in quantum physics.

Inter-state switching in stochastic gene expression: Exact solution, an adiabatic limit and oscillations in molecular distributions