Nanoparticle Endocytosis: The Membrane-Wrapping Model

Nanoparticle & Membrane

Particle radius R 100 nm

Receptor density ρ 1.00×

Membrane bending rigidity κ 20 k_BT

Neck line tension γ 1.00 k_BT/nm

Wrapping State

f = 0%ΔG = 0 k_BT

No nanoparticle in flight. Press Spawn nanoparticle.

Attempts

Uptake success

— %

How it works

A nanoparticle sticking to receptors on a membrane only gets internalized if wrapping the membrane around it lowers the total free energy. For wrapping fraction f (0 = untouched, 1 = fully engulfed), three terms compete:

ΔG(f) = 8πκf                         bending cost (Helfrich, size-independent!)
      + 4πγR·√(f(1−f))              neck line-tension barrier (peaks at f=0.5)
      − w·4πR²·f                    receptor–ligand adhesion gain (w ∝ ρ)

df/dt = −Γ·ρ·dΔG/df                overdamped wrapping dynamics

Bending energy to fully wrap any sphere is 8πκ — independent of R. Adhesion energy scales with contact area (∝R²). This is exactly why very small nanoparticles (below ~20–30 nm) often fail to trigger receptor-mediated uptake: there isn't enough membrane area in contact to out-bid the fixed bending cost, no matter how sticky the ligand is.
The line-tension term is a barrier concentrated at the closing neck (largest halfway through wrapping) — it models the real energy dynamin-family GTPases must overcome to pinch a vesicle free.
Once f reaches 1 the neck is assumed to scission and the vesicle is released into the cytoplasm — a genuine uptake event.
Pathway label is a size-based overlay on top of the physics result, following the accepted diameter ranges for the three canonical routes: caveolae-mediated (~50–80 nm pits), clathrin-mediated (~80–200 nm coated pits), and macropinocytosis (>200 nm, receptor-independent membrane ruffling). If ΔG never turns favorable, the particle stalls and desorbs — modeling opsonization/clearance instead of uptake.

This is a simplified, illustrative energy-competition model (after Gao, Shi & Boal-style wrapping theory) — not calibrated to a specific real receptor system, but every term is a genuine physical quantity with the right size- and receptor-density scaling.

Nanoparticle Cell membrane Internalized vesicle

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