Free nanoparticles diffuse through the extracellular fluid by Brownian motion. Whenever one drifts close to a cell, it may be internalised — the probability per frame depends on the particle's size (endocytosis peaks around ~50 nm; particles far smaller or larger are taken up more slowly) and its surface charge (cell membranes carry a net negative charge, so cationic particles adhere and are taken up faster than anionic ones). Each uptake event adds mass to that cell's accumulated intracellular dose, weighted by particle volume (∝ size³), and the population's viability follows a Hill-equation dose-response curve.
P(uptake) = k · size_factor(d) · charge_factor(q) · dt
size_factor(d) = exp(−(d−50)²/(2·35²))
charge_factor(q) = clamp(0.5 + q/100, 0.15, 1.4)
response(D) = Dⁿ / (Dⁿ + EC50ⁿ), n=2
viability = 100·(1 − response)
- Particle size — sets the endocytosis bell curve; mid-size (~50 nm) particles are internalised fastest, but each successful uptake of a larger particle delivers far more mass (dose ∝ size³).
- Surface charge (zeta potential) — a positive zeta potential attracts particles to the anionic cell membrane and speeds uptake; a strongly negative one slows it.
- Exposure concentration — how many free particles are maintained in the fluid; higher concentration raises the collision rate with every cell.
- Dose-response curve — the inset plot shows the theoretical Hill curve for the current EC50; the marker tracks the population's live average dose against it.
Real-world relevance: this is the core dose-response logic behind in-vitro nanotoxicology assays — cellular uptake efficiency (governed by size and surface chemistry) determines the internal dose, and the internal dose, not the applied concentration, is what actually predicts cytotoxicity.