The Levinthal Problem in Amyloid Aggregation: Identification of Good Coordinates in a Flat Reaction Space

Abstract

Successful protein folding in physiological timescales requires a biased free energy landscape to restrict the otherwise prohibitive search space. The existence of this bias is a necessary outcome of evolution since sequences that fold slowly do not contribute to the fitness of the organism. In contrast, the evolutionary pressure for efficient folding pathways is not present in the formation of pathological aggregates. Accordingly, the measured growth rates for amyloid fibrils are much slower than might be expected for the formation of beta sheets. Analytic theory shows that fibril growth rates are consistent with a random search over the alignments of intermolecular H-bonds and that solution conditions that accelerate this search (i.e. weakening bonds) can increase aggregation rates. This theory identifies two reaction coordinates, the alignment between molecules and the number of formed H-bonds, which we use to devise a novel Markov State Model to simulate fibril growth in atomistic detail. This model is used to simulate the growth of beta amyloid (16-22) and three mutants. The simulations qualitatively capture the non-additive effects of the mutations, but interestingly there is no obvious trend to the mutation effects in the lifetimes of the molecular alignments or individual H-bonds. Instead, the changes in the growth rate emerge from the accumulation of many small perturbations over a large ensemble of trajectories. We conclude with a discussion of theory development with an eye towards the simulation of very slow processes like fibril nucleation.