This is the 2D companion to the 3D lysosomal-permeabilization simulator, computed independently rather than a flattened render of the 3D scene. The 3D version tracks a single whole-membrane damage number D(t). Real membranes don't fail uniformly — damage accumulates wherever nanoparticles happen to collide most, so this version splits the lysosomal membrane cross-section into 48 angular sectors, each with its own independent damage state Dᵢ(t), driven by contacts landing specifically in that sector:
dDᵢ/dt = k0 · R · S(r) · fᵢ(t) · [1 + β·diss] − k_repair · (1 − Dᵢ)
fᵢ(t) = contacts in sector i this frame / (N particles / 48 sectors)
S(r) = 1/√(r / r0)
λᵢ(Dᵢ) = λ_max · exp(6·(Dᵢ − 1)) (per-sector Poisson pore nucleation, Dᵢ > 0.55)
Free nanoparticles undergo 2D Brownian motion inside the disk, displaced each frame by a random step scaled to the Stokes–Einstein diffusion coefficient D = k_BT / (6πηr) — identical physics to the 3D version, since that relation doesn't depend on how many spatial dimensions you draw the particle in. When a particle strikes the boundary its angle sets which sector absorbs the contact. As any one sector's Dᵢ climbs past 0.55, that sector's local pore-nucleation rate rises sharply; the first sector to nucleate a pore releases its own burst of cathepsin markers (red) outward from that exact position, not from a random point on the shell.
The strip beneath the disk unrolls all 48 sectors' damage into a single bar chart against angle — a genuinely 2D-native diagnostic with no 3D equivalent (you'd need a full spherical-surface unwrap to get the analogous view in 3D). It makes the localization of failure visible directly: watch one or two bars climb and rupture while the rest of the membrane stays intact, exactly the patchy failure pattern reported for lipid bilayers under point-source stress.
- Particle count / size — dose and geometry: more particles and smaller radii raise both contact frequency and the S(r) curvature term, in every sector they happen to hit.
- Surface reactivity — stand-in for surface charge/redox activity; scales the local damage rate k0·R uniformly, but which sector rises fastest is still set by where contacts land.
- Intralysosomal dissolution — ion release in the acidic lumen, adding to local membrane stress everywhere a particle has been in contact.
This is a simplified, illustrative rate model — real LMP kinetics depend on lipid composition, particle coating chemistry and cell type — but the qualitative behaviour (dose- and reactivity-dependent damage accumulation, a rupture threshold, and localized stochastic pore nucleation) matches the mechanism reported across the nanotoxicology literature for cationic polymers, metal-oxide nanoparticles and some carbon nanomaterials.