Stability analysis of a phase-separating droplet with active reactions reveals rich dynamics

Biomolecular condensates like the nucleolus, transcriptional condensates, and nuclear speckles spatially concentrate proteins and carry out irreversible reactions such as RNA transcription and splicing. To study the different qualitative dynamics that can arise in such active droplets, we consider a minimal two-species model where a conserved protein species phase-separates according to model B dynamics, while actively producing and passively interacting with a non-conserved molecular species (RNA) that slowly decays. A linear stability analysis reveals that the system relaxes to a stable pattern for most parameter settings, which can then be characterized by a Lyapunov functional. However, sustained oscillations can arise when proteins and RNA mutually repel each other, the former diffuses much faster than the latter, and the production of RNA is sufficiently fast. Our analysis correctly predicts the qualitative long-time behavior in numerical simulations of the full PDE model. In the case of sustained oscillations, our simulations reveal a spontaneous symmetry breaking of patterns due to shape instability. Our model provides a framework to map out the different qualitative dynamics of active droplets and can enable the engineering and control of such biomolecular systems.