Protein Aggregation — Nucleation & Growth (Amyloid Fibrils)

Primary nucleation rate k_n 1e-4

Elongation rate k_+ 0.50

Secondary nucleation k_2 1e-4

Initial monomer m₀ 1.00

Monomer remaining m(t)—

Aggregate mass M(t)—

Fibril number P(t)—

Half-time t½—

Physics (Knowles-Waudby-Dobson)

Amyloid aggregation follows:

dP/dt = k_n·mⁿ + k_2·M·mⁿ²

dM/dt = 2k_+·m·P

dm/dt = −dM/dt

Primary nucleation creates new fibril ends (rate ∝ mⁿ). Secondary nucleation (rate ∝ M·mⁿ²) is catalysed by existing fibrils — this autocatalysis produces the characteristic sigmoidal curve with lag phase.

Linked to Alzheimer's (Aβ) and Parkinson's (α-synuclein). The half-time t½ decreases sharply as seeding increases secondary nucleation.