Three component phase separation along a deformable surface with applications to viral budding

The interplay between phase separation and membrane deformation is a common feature of many biological processes, from lipid raft organization to the assembly and budding of enveloped viruses. These systems often exhibit heterogeneous structures composed of multiple, functionally distinct components. To understand how surface deformations and multiple components influence phase separation, we develop a continuum model that describes the spatial distribution of three interacting components along a deformable surface. Each component induces a characteristic curvature and interacts pairwise with the others, jointly determining the coupled evolution of composition and surface shape. By comparing three-component systems to the classical two-component case, we demonstrate that introducing an additional phase can enhance phase separation even without curvature effects. Upon introducing curvature, surface deformations can act as a feedback mechanism that promotes either mixing or demixing depending on component characteristic curvatures. The resulting curvature patterns give rise to diverse surface morphologies, including ring-like domains of one phase surrounding elevated regions of another phase. In the context of viral assembly and budding, such mechanisms could underlie the coordinated dynamics of membrane remodeling and protein organization leading to budding, scission, and virion formation.