Nucleation of amyloid fibrils

We consider nucleation of amyloid fibrils in the case when the process occurs by the mechanism of direct polymerization of practically fully extended protein segments, i.e., beta-strands, into beta-sheets. Applying the classical nucleation theory, we derive a general expression for the work to form a nanosized amyloid fibril (protofilament) constituted of successively layered beta-sheets. Analysis of this expression reveals that with increasing its size, the fibril transforms from one-dimensional to two-dimensional aggregate in order to preserve the equilibrium shape corresponding to minimal formation work. We determine the size of the fibril nucleus, the fibril nucleation work, and the fibril nucleation rate as explicit functions of the concentration and temperature of the protein solution. The results obtained are applicable to homogeneous nucleation, which occurs when the solution is sufficiently pure and/or strongly supersaturated.