Liquid-liquid phase separation induces stochastic oscillations in gene regulation

Rhythmic gene expression underlies core physiological processes across organisms, from circadian timekeeping to stress responses. Recent experiments suggest that the regulation of such rhythmic dynamics involves protein compartmentalization mediated by liquid-liquid phase separation (LLPS), yet the mechanisms by which LLPS feeds back onto oscillatory behaviour remain unclear. Here we develop a minimal two-phase gene-expression model in which proteins are synthesized in the dilute phase, reversibly partition into a protein-dense droplet phase, and repress their own production via condensate-mediated regulation. In the deterministic limit, LLPS does not generate limit cycles; instead, nonlinear partitioning and timescale separation between phase separation and protein turnover convert purely relaxational dynamics into damped oscillatory transients, altering the approach to equilibrium without producing sustained oscillations. In the stochastic regime, intrinsic noise interacting with this near-focus dynamics is amplified into noise-sustained, near-periodic fluctuations with a characteristic timescale, as revealed by the power spectral density and autocorrelation functions. These results show how LLPS reshapes oscillatory signatures by encoding and filtering temporal signals in phase-specific ways, providing a quantitative framework for interpreting LLPS-rhythm coupling and for engineering biomolecular systems with tunable dynamic behaviour.