Interaction between cell membranes and protein inclusions in the large-deformation regime

Biological membranes are dynamic surfaces whose shape and function are critically influenced by protein inclusions (PIs). While membrane deformations induced by PIs have been extensively studied in the small-deformation regime, a variety of processes involve strong membrane deformations. We investigate the interaction between lipid membranes and PIs in the large-deformation (LD) regime, with the finite-element method. We develop an approximate analytical solution that captures key features of the LD regime. We show that the force exerted by the membrane on a PI displays nonmonotonic behavior with respect to the PI vertical displacement. The qualitative features of this force appear to be independent of the protein geometry. For two interacting PIs, the membrane-mediated potential exhibits sub-power-law decay with interprotein distance, reflecting the complex nature of the elastic medium. The interaction potential shows that conical PIs with identical and opposite orientations repel and attract, respectively, confirming the analogy between PI orientation and electric charge, in the LD regime. In the presence of membrane flows, we identify a characteristic velocity that separates two regimes in which bending rigidity and viscous effects dominate, respectively, implying the onset of flow-induced deformations above such velocity threshold. Overall, our results provide quantitative predictions for membrane-protein systems in biologically relevant scenarios involving LDs, with implications for protein sorting, clustering, and membrane trafficking.