Entropy-Driven Initiation and Cellular Uptake Mediated by Viscoelastic Cytoskeleton: A Kinetic Phase Diagram from Onsager Variational Principle

A fundamental question in receptor-mediated endocytosis remains unanswered: what initial driving force brings ligands and receptors into close proximity?

While previous models assume pre-existing contact and overlook this initiation problem, we propose that entropic forces from nanoscale biomolecules in crowded cellular environments provide the essential driving mechanism.

We develop a unified continuum model rooted in the Onsager variational principle, where engulfment depth serves as the generalized coordinate and the driving force derives from a free energy landscape of entropic, binding, membrane, and cytoskeleton contributions.

The framework naturally incorporates: (i) entropy-driven adhesion as initiation; (ii) ligand-receptor binding as the sustaining force; (iii) membrane deformation via the Helfrich-Canham Hamiltonian; and (iv) cytoskeleton viscoelasticity through the elastic-viscoelastic correspondence principle.

The kinetic phase diagram predicts a critical biomolecule concentration for initiation, a lower bound of ligand density for complete engulfment, a finite size window for engulfable particles, and an optimal virus radius of 30--60 nm that decreases with increasing binding energy.

The Onsager solubility condition naturally yields the phase boundaries.

The model exhibits asymptotic consistency with the classic Asakura-Oosawa result in the large-particle flat-surface limit.

Stiffer cells lead to longer engulfment times and narrower size windows.

Strikingly, the optimal size matches HIV-1 dimensions under physiologically realistic parameters.

This work provides a variational foundation for cellular uptake with implications for virology, nanotechnology, and drug delivery.