Depletion Capture and Rayleigh--Onsager Dynamics in Receptor-Mediated Endocytosis

Abstract

Receptor-mediated endocytosis requires a particle to approach a membrane in a crowded extracellular environment, form ligand-receptor bonds, and wrap the membrane against bending, tension, and cytoskeletal viscoelastic resistance. We develop a two-stage theory. The capture stage uses the exact ideal hard-sphere Asakura--Oosawa sphere-plane excluded-volume geometry. The wrapping stage combines a spherical-cap coordinate, a quadratic single-mode free energy, a standard linear solid with one internal strain variable, and a two-variable Rayleigh--Onsager (RO) gradient flow. The creep compliance is mapped exactly onto a long-time spring in parallel with a Maxwell branch, preserving both the creep retardation and stress relaxation times. The free energy is strictly convex by its Hessian convexity, and the linear wrapping system has two real negative eigenvalues and an explicit double-exponential solution. Two independent arguments establish strict monotonicity for positive initial drive: the irreducible Metzler structure with its positive semigroup and the interlacing of Laplace poles with positive velocity residues. The first-passage equation has a unique finite root below the unconstrained long-time limit. Normal-cone conditions specify the projected dynamics on $0\le x\le 2$, and the projected system retains the Lyapunov inequality of the unconstrained gradient flow. From the same free energy we derive long-time adhesion thresholds, the threshold-minimizing radius, the finite-size window, and the critical cell modulus. The $95\%$ spherical-area criterion is treated as an operational near-complete threshold, with $90\%$, $95\%$, and $99\%$ compared. The capture analysis includes the exact overlap volume, depletion force, Carnahan--Starling pressure rescaling, Stokes approach time, and Brenner wall-drag correction.

arXiv comment22 pages, 14 figures, 4 tables