Nematic Materials on Curved Surfaces with Immersed Inclusions 

Biological interfaces—such as the nuclear envelope, cell membranes, and epithelial tissues—play central roles in regulating structure and dynamics in living systems. Owing to their microstructure, these interfaces often exhibit internal degrees of freedom that manifest as in-plane order. The nuclear envelope, a double-membrane composite reinforced by lamin fibers and populated with nuclear pore complexes (NPCs), protects the genome and mediates nucleo-cytoplasmic transport in eukaryotic cells. Motivated by such biological interfaces that host embedded structures like NPCs, we investigate nematic ordering on spherical shells perforated by multiple circular inclusions. Using a Landau–de Gennes Q-tensor framework on spherical surfaces, we model systems with both stationary and motile inclusions. For N stationary inclusions, we explore how tangential anchoring at inclusion rims shapes the nematic texture and determines the interior defect charge, which follows a strict topological budget of 2-N. We further extend the model to account for inclusion motility driven by elastic and active stresses. By coupling Q-tensor dynamics with overdamped motion of inclusions on the sphere, we capture the feedback between the nematic field and inclusion trajectories. This approach provides insight into how geometry, nematic order, and active stresses influence self-organization on active, curved surfaces.