  Nucleation processes are at the heart of a large number of phenomena, from
cloud formation to protein crystallization. A recently emerging area where
nucleation is highly relevant is the initiation of filamentous protein
self-assembly, a process that has broad implications from medicine to
nanotechnology. As such, spontaneous nucleation of protein fibrils has received
much attention in recent years with many theoretical and experimental studies
focussing on the underlying physical principles. In this paper we make a step
forward in this direction and explore the early time behaviour of filamentous
protein growth in the context of nucleation theory. We first provide an
overview of the thermodynamics and kinetics of spontaneous nucleation of
protein filaments in the presence of one relevant degree of freedom, namely the
cluster size. In this case, we review how key kinetic observables, such as the
reaction order of spontaneous nucleation, are directly related to the physical
size of the critical nucleus. We then focus on the increasingly prominent case
of filament nucleation that includes a conformational conversion of the
nucleating building-block as an additional slow step in the nucleation process.
Using computer simulations, we study the concentration dependence of the
nucleation rate. We find that, under these circumstances, the reaction order of
spontaneous nucleation with respect to the free monomer does no longer relate
to the overall physical size of the nucleating aggregate but rather to the
portion of the aggregate that actively participates in the conformational
conversion. Our results thus provide a novel interpretation of the common
kinetic descriptors of protein filament formation, including the reaction order
of the nucleation step or the scaling exponent of lag times, and put into
perspective current theoretical descriptions of protein aggregation.
