  Protein oligomers have been implicated as toxic agents in a wide range of
amyloid-related diseases. Yet it has remained unsolved whether the oligomers
are a necessary step in the formation of amyloid fibrils, or just a dangerous
by-product. Analogously, it has not been resolved if the amyloid nucleation
process is a classical one-step nucleation process, or a two-step process
involving pre-nucleation clusters. We use coarse-grained computer simulations
to study the effect of non-specific attractions between peptides on the primary
nucleation process underlying amyloid fibrillization. We find that for peptides
that do not attract, the classical one-step nucleation mechanism is possible,
but only at non-physiologically high peptide concentrations. At low peptide
concentrations, which mimic the physiologically relevant regime, attractive
inter-peptide interactions are essential for fibril formation. Nucleation then
inevitably takes place through a two-step mechanism involving prefibrillar
oligomers. We show that oligomers not only help peptides meet each other, but
create an environment that facilitates the conversion of monomers into the
$\beta$-sheet rich form characteristic of fibrils. Nucleation typically does
not proceed via the most prevalent oligomers, but via an oligomer size that is
only observed in rare fluctuations, which is why such aggregates might be hard
to capture experimentally. Finally, we find that the nucleation of amyloid
fibrils cannot be described by classical nucleation theory: in the two-step
mechanism the critical nucleus size increases both with an increase in
concentration and in the inter-peptide interactions, in direct contrast with
predictions from classical nucleation theory.
