Two step aggregation kinetics of Amyloid-β proteins from fractal analysis

Self-aggregation in proteins has long been studied and modeled due to its ubiquity and importance in many biological contexts. Several models propose a two step aggregation mechanism, consisting of linear growth of fibrils and branch formation. Single molecule imaging techniques such as total internal reflection fluorescence (TIRF) microscopy can provide direct evidence of such mechanisms, however, analyzing such large datasets is challenging. In this paper, we analyze for the first time, images of growing amyloid fibrils obtained from TIRF microscopy using the techniques of fractal geometry, which provides a natural framework to disentangle the two types of growth mechanisms at play. We find that after an initial linear growth phase, identified by a plateau in the average fractal dimension with time, the occurrence of branching events leads to a further increase in the fractal dimension with a final saturation value {approx} 2. We also simulate the aggregation process using the identified linear growth and secondary nucleation mechanisms, using an event driven algorithm. We theoretically model this system using a set of coupled nonlinear differential equations describing a mean field model for branching and linear growth, which we use to characterize the growth process observed in simulations as well as experiments. Finally, we provide estimates for the parameter regimes that govern the two step aggregation process observed in experiments.