Exact power spectrum in a minimal hybrid model of stochastic gene expression oscillations. (English) Zbl 07864252
Summary: Stochastic oscillations in individual cells are usually characterized by a nonmonotonic power spectrum with an oscillatory autocorrelation function. Here we develop an analytical approach to 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. 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’s [in: Numerical methods in the study of critical phenomena, Proc. Colloq., Carry-le-Rouet/Fr. 1980, Springer Ser. Synerg. 9, 180–193 (1981; Zbl 0489.92025)] two conjectures for single-cell gene expression kinetics.
MSC:
| 92C40 | Biochemistry, molecular biology |
| 34A38 | Hybrid systems of ordinary differential equations |
| 37H20 | Bifurcation theory for random and stochastic dynamical systems |
| 60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |
Keywords:
stochastic oscillation; bursting; autocorrelation; power spectrum; stochastic bifurcation; hysteresisCitations:
Zbl 0489.92025References:
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