This is a genuinely 2D-native solve of the same Finke-Watzky nucleation-autocatalysis mechanism as the 3D version, but computed and rendered by different methods entirely:
A → B nucleation, propensity a1 = k1·A
A + B → 2B autocatalytic, propensity a2 = (k2/N)·A·B
Gillespie direct method (exact stochastic simulation, not Euler-integrated ODE):
a0 = a1 + a2
wait time τ ~ Exponential(a0) [drawn fresh from current a0 — valid by
the memoryless property even when a
slider changes mid-wait]
event type chosen ∝ {a1, a2}; each event moves exactly one monomer A→B
The 3D version integrates the mean-field ODE with Euler steps and interpolates particle positions toward a target fraction. This version instead runs the exact chemical master equation for discrete monomer counts (A_count + B_count = N always, exactly — never approximated), so the lag-phase length itself is genuinely stochastic run to run, exactly as real single-aggregate ThT traces show run-to-run lag variability that a smooth ODE curve cannot.
Converted monomers don't teleport to a shrinking sphere: each one performs its own Brownian random walk across the 2D soma cross-section until it physically touches the existing cluster, then sticks — a diffusion-limited aggregation (DLA) process, the same growth mechanism believed to underlie the branched, radiating fibril morphology of real Lewy bodies (rather than the isotropic icosahedron scaling used for the 3D core/halo).
The bottom panel is a phase-plane (rate-law) diagram: because this is a closed conversion (A = N − B exactly, no separate mean-field approximation needed), the instantaneous rate is a pure function of state, dB/dt = (1−B)(k1 + k2·B) — a "hump" that is zero at B=0 and B=1 and peaks at B* = (k2−k1)/(2k2). The live dot rides this curve as the simulation evolves — a state-space view no time-series chart can show directly.
- k1 (nucleation) — spontaneous A→B propensity; raising it shortens the (now stochastic) lag phase.
- k2 (autocatalytic/elongation) — how strongly existing cluster mass recruits new monomer; raising it steepens the hump and the growth phase.
- Seeding — starts with a fraction of B already on the cluster, skipping nucleation, exactly as templated seed-amplification assays (RT-QuIC) do.