  Self-assembly of proteins is a biological phenomenon which gives rise to
spontaneous formation of amyloid fibrils or polymers. The starting point of
this phase, called nucleation exhibits an important variability among
replicated experiments.To analyse the stochastic nature of this phenomenon, one
of the simplest models considers two populations of chemical components:
monomers and polymerised monomers. Initially there are only monomers. There are
two reactions for the polymerization of a monomer: either two monomers collide
to combine into two polymerised monomers or a monomer is polymerised after the
encounter of a polymerised monomer. It turns out that this simple model does
not explain completely the variability observed in the experiments. This paper
investigates extensions of this model to take into account other mechanisms of
the polymerization process that may have impact an impact on fluctuations.The
first variant consists in introducing a preliminary conformation step to take
into account the biological fact that, before being polymerised, a monomer has
two states, regular or misfolded. Only misfolded monomers can be polymerised so
that the fluctuations of the number of misfolded monomers can be also a source
of variability of the number of polymerised monomers. The second variant
represents the reaction rate $\alpha$ of spontaneous formation of a polymer as
of the order of $N^{-\nu}$, with $\nu$ some positive constant. First and second
order results for the starting instant of nucleation are derived from these
limit theorems. The proofs of the results rely on a study of a stochastic
averaging principle for a model related to an Ehrenfest urn model, and also on
a scaling analysis of a population model.
