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Amyloid fibrils are self-assembled protein aggregates implicated in a number of human diseases. Fragmentation-dominated models for the self-assembly of amyloid fibrils have had important successes in explaining the kinetics of amyloid fibril formation but predict fibril length distributions that do not match experiments. Here we resolve this inconsistency using a combination of experimental kinetic measurements and computer simulations. We provide evidence for a structural transition that occurs at a critical fibril mass concentration, or CFC, above which fragmentation of fibrils is suppressed. Our simulations predict the formation of distinct fibril length distributions above and below the CFC, which we confirm by electron microscopy. These results point to a new picture of amyloid fibril growth in which structural transitions that occur during self-assembly have strong effects on the final population of aggregate species with small, and potentially cytotoxic, oligomers dominating for long periods of time at protein concentrations below the CFC.