This 2D view unrolls the 13-protofilament microtubule cylinder into a flat lattice strip (top band) — the same way a cartographer unrolls a globe into a map. Each column is one protofilament, each row one tubulin repeat along the axon. Tau bound to a lattice site sits directly on this grid; once hyperphosphorylation knocks it loose, it drops into the cytosol strip below and undergoes a genuine 2D random walk, exactly as in the 3D cylindrical model but projected onto its own 2D plane rather than a camera view of 3D space.
Free tau aggregates by nucleation-dependent polymerization (Oosawa–Kasai kinetics, as characterized for tau by Friedhoff et al. and Barghorn & Mandelkow): a slow, high-order nucleation step forms a small seed, followed by fast monomer-by-monomer elongation at the fibril ends into paired helical filaments (PHFs), drawn here as growing 2D polylines in the cytosol strip.
Detachment: P(off) = 1 - exp(-k_off · φ · dt)
Diffusion: Δx,Δy = N(0,1) · sqrt(2·D·dt) (2D isotropic random walk)
Nucleation: P(nuc) = k_n · dt · [M_free]^nc (nc ≈ 2, critical nucleus)
Elongation: P(add) = k_e · dt for M_free within capture radius r_c of a fibril end
- φ (phosphorylation) — sets the detachment rate; higher φ empties the microtubule lattice faster, raising the free-monomer pool that fuels aggregation.
- Tau pool N — total tau molecules simulated; more tau means more nucleation attempts and denser tangles at the same φ.
- Inject Seed Fibril — drops a preformed nucleus directly into the cytosol strip, modeling prion-like templated seeding, which bypasses the slow nucleation step and accelerates tangle growth.
- The strip chart under the cytosol band plots the three population fractions live — the mean-field signature of the same stochastic process running above it.
Color key: teal dots are tau still bound to (and stabilizing) the unrolled microtubule lattice; amber dots are free, hyperphosphorylated monomers random-walking in the cytosol strip; red dots/lines are tau locked into a growing paired-helical-filament tangle.