Critical Particle Sizes for the Engulfment of Nanoparticles by Membranes and Vesicles with Bilayer Asymmetry

The adhesion and engulfment of nanoparticles by biomembranes is essential for many processes such as biomedical imaging, drug delivery, nanotoxicity, and viral infection. Many studies have shown that both surface chemistry, which determines the adhesive strength of the membrane-particle interactions, and particle size represent key parameters for these processes. Here, we show that the asymmetry between the two leaflets of a bilayer membrane provides another key parameter for the engulfment of nanoparticles. The asymmetric membrane prefers to curve in a certain manner as quantitatively described by its spontaneous curvature. We derive two general relationships between particle size, adhesive strength, and spontaneous curvature that determine the instabilities of (i) the nonadhering or free state and (ii) the completely engulfed state of the particle. For model membranes such as lipid or polymer bilayers with a uniform composition, the two relationships lead to two critical particle sizes that determine four distinct engulfment regimes, both for the endocytic and for the exocytic engulfment process. For strong adhesion, the critical particle sizes are on the order of 10 nm, while they are on the order of 1000 nnn for weak or ultraweak adhesion. Our theoretical results are therefore accessible to both experimental studies and computer simulations of model membranes. In order to address the more complex process of receptor-mediated endocytosis, we take the adhesion-induced segregation of membrane components into account and consider bound and unbound membrane segments that differ in their spontaneous curvatures. To model protein coats as formed during clathrin-dependent endocytosis, we focus on the case in which the bound membrane segments have a large spontaneous curvature compared to the unbound ones. We derive explicit expressions for the engulfment rate and the uptake of nanoparticles, which both depend on the particle size in a nonmonotonic manner, and provide a quantitative fit to experimental data for clathrin-dependent endocytosis of gold nanoparticles.