Fig. 5.

Theoretical studies. (A) Schematic of membrane wrapping around an elastic thin-shelled spherical nanocapsule in the adopted cylindrical coordinate (r, φ, z). Arclengths s and s_m of the nanocapsule and membrane are measured from the bottom pole, respectively. (B) Left: Elastic energy E_el versus the wrapping degree f at the normalized membrane tension σR²/κ = 3 and adhesion energy γ = 0. Inset, energy derivative d[E_el/(πκ)]/df versus f. Right: Selected wrapping configurations at σR²/κ = 3 and γR²/κ = 7 for hard and soft (B/κ = 20.45) nanocapsules. (C) Schematic of receptor-mediated endocytosis of a spherical hard nanocapsule of radius R (Left). During the wrapping process, the receptor density distribution is nonuniform with receptors depleted near the binding region. The resulting gradient of receptor density ξ in turn induces global receptor diffusion toward the binding site (Right). (D) Wrapping degree f versus the normalized time tξ₀D for different values of the nanocapsule radius R and ligand density ξ_L at the binding energy per receptor–ligand bond of 15 k_BT, D = 0.1 μm²/s, κ = 20 k_BT, σR²/κ = 3 and ξ₀ = 50/μm², 100/μm², and 500/μm². The parameters used in the theoretical model are also summarized in SI Appendix, Table S1.