Boundary-layer analysis of the partial engulfment of a small particle by a lipid membrane

Abstract

We study the axisymmetric partial engulfment of a small rigid sphere by a fluid Helfrich membrane. When the size ratio $\eps$ between the particle and the membrane is small, the neck that joins the wrapped cap to the surrounding membrane is an elastic boundary layer, and we analyse it by matched asymptotic expansions. The leading inner surface is a catenoid, a minimal surface that stores no Helfrich energy, so that the energy of partial engulfment is carried by the first correction and appears only at order $\eps^{2}\ln(1/\eps)$. We obtain it by solving the inhomogeneous Jacobi equation of the catenoid, forced by the spontaneous curvature and by the ambient mean curvature that the neck must match, and we give its coefficient in closed form. The boundary layer can then be integrated out, and the neck replaced by a scalar self-energy carried at the pole, so that the outer field can be closed independently of the inner one. For a membrane coupled to a tension reservoir we derive the binding threshold, which turns out to be independent of both the tension and the spontaneous curvature, the complete-wrapping threshold, and the hysteresis of the envelopment transition. The neck self-energy law is confirmed against the full nonlinear shape equations.