Phase Separation on Deformable Membranes: interplay of mechanical coupling and dynamic surface geometry

Protein self-organization and pattern formation are critical processes at the cellular level, often driven by liquid-liquid phase separation. When proteins undergoing phase separation are membrane-bound, they can locally deform the membrane, leading to long-range membrane-mediated interactions between the proteins. We investigate how these membrane-mediated interactions affect the steady-state protein distribution. Through numerical simulations, linear stability analysis, and free energy arguments, we show that these interactions qualitatively alter the system's behavior. In contrast to typical demixing and complete coarsening, we observe arrested coarsening and the emergence of characteristic length scales. Our findings demonstrate that long-range membrane interactions drive pattern formation in systems that otherwise would not exhibit such behavior.