An Accurate Numerical Solution to the Kinetics of Breakable Filament Assembly

Proteinaceous aggregation occurs through self-assembly—a process not entirely understood. In a recent article, Knowles and colleagues (2009) presented an analytical theory for amyloid fibril growth via secondary rather than primary nucleation. Remarkably, with only a single kinetic parameter, the authors were able to unify growth characteristics for a variety of experimental data. In essence, they seem to have uncovered the underlying allometric law governing the evolution of filament elongation simply from two coupled nonlinear ordinary differential equations originally obtained from a master equation. While this work adds significantly to our understanding of filament self-assembly, it required an “approximate” analytical solution representation for the moments of the chain length distribution. If this were always true, the discovery of such scaling laws would be infrequent. Here, we show that the same results are found by purely numerical means. In addition, the numerical method used features a highly accurate solution strategy for the coupled Ordinary Differential Equations (ODEs) based only on a fundamental finite difference scheme and convergence acceleration. Once a reliable numerical solution has been established, a dimensional analysis then provides the scaling laws.