  Stochastic oscillations in individual cells are usually characterized by a
non-monotonic power spectrum with an oscillatory autocorrelation function. Here
we develop an analytical approach of stochastic oscillations in a minimal
hybrid model of stochastic gene expression including promoter state switching,
protein synthesis and degradation, as well as a genetic feedback loop. The
oscillations observed in our model are noise-induced since the deterministic
theory predicts stable fixed points. The autocorrelated function, power
spectrum, and steady-state distribution of protein concentration fluctuations
are computed in closed form without making any approximations. Using the
exactly solvable model, we illustrate sustained oscillations as a circular
motion along a stochastic hysteresis loop induced by gene state switching. A
triphasic stochastic bifurcation upon the increasing strength of negative
feedback is observed, which reveals how stochastic bursts evolve into
stochastic oscillations. In our model, oscillations tend to occur when the
protein is relatively stable and when gene switching is relatively slow.
Translational bursting is found to enhance the robustness and broaden the
region of stochastic oscillations. These results provide deeper insights into
R. Thomas' two conjectures for single-cell gene expression kinetics.
