Regulation of the membrane wrapping transition of a cylindrical target by cytoskeleton adhesion

The adsorption of external objects to the cell membrane often triggers cellular responses involving large deformations. In phagocytosis, upon contact with the target, the cell creates large extensions that wrap around the target and ultimately lead to its engulfment. Although active force generation, in particular by actin polymerization, is required for completion of this process, the elastic deformation of the cell membrane upon adhesion to an external object might play an important part in its initiation. In this paper, the elastic deformation of a bilayer owing to the binding of a cylindrical object is studied, taking into account the membrane bending rigidity and the surface tension, the membrane adhesion to both the external target and inner cytoskeleton. The problem is studied within the framework of the Helfrich-Hamiltonian and using force balance relations and the proper boundary conditions that are related to the adhesion energy coefficients. It is shown that membrane wrapping around the target may be a continuous or abrupt transition upon increasing the target binding energy, depending on the value of the parameter. The degree of wrapping and the shape of the membrane in the vicinity of the object are computed numerically, and analytical expressions are given for the boundaries separating the different wrapping regimes in the parameter space.