Entropy-Driven Initiation and Cytoskeletal Viscoelasticity in Endocytosis: An Onsager Variational Framework.

Receptor-mediated endocytosis requires a particle to approach the cell membrane to within a few nanometers before ligand-receptor binding can occur. Existing continuum models often start from an already established contact and do not explicitly describe how crowding particles on the extracellular side influence the distribution of the particle near the membrane. We examine entropic depletion forces as one possible nonspecific contribution to this initial approach. For ideal depletants, the Asakura-Oosawa excluded-volume construction gives an exact depletion potential for the planar geometry before contact. The potential and force vanish continuously at the onset of excluded-volume overlap. This interaction provides a possible contribution to membrane proximity before specific binding, while its extension to curved wrapping geometries requires additional approximation. Within a reduced continuum model, we combine depletion attraction, ligand-receptor binding, membrane deformation, and cytoskeletal viscoelastic dissipation. The viscoelastic contact is formulated through a hereditary integral and a standard linear solid. The kinetic model gives a conditional minimum ligand density for complete engulfment, a finite particle-size window, and a stiffness-dependent upper limit. When the stationary radius lies inside the domain of finite positive wrapping times, the estimated wrapping time has a minimum at a radius that decreases with increasing binding energy density. At fixed viscosity and other independent parameters, the same time approximation predicts slower wrapping as cell stiffness increases. The two positive roots defining the size window merge at a limiting parameter value, which characterizes closure of the admissible size interval. Depletion attraction is interpreted as one possible contribution to particle-membrane association, alongside electrostatic interactions, steric effects, and membrane fluctuations. The present analysis identifies how nonspecific attraction, specific adhesion, and mechanical resistance can contribute to different stages of membrane wrapping.