  Self-assembly of proteins into amyloid aggregates is an important biological
phenomenon associated with human diseases such as Alzheimer's disease. Amyloid
fibrils also have potential applications in nano-engineering of biomaterials.
The kinetics of amyloid assembly show an exponential growth phase preceded by a
lag phase, variable in duration as seen in bulk experiments and experiments
that mimic the small volumes of cells. Here, to investigate the origins and the
properties of the observed variability in the lag phase of amyloid assembly
currently not accounted for by deterministic nucleation dependent mechanisms,
we formulate a new stochastic minimal model that is capable of describing the
characteristics of amyloid growth curves despite its simplicity. We then solve
the stochastic differential equations of our model and give mathematical proof
of a central limit theorem for the sample growth trajectories of the nucleated
aggregation process. These results give an asymptotic description for our
simple model, from which closed form analytical results capable of describing
and predicting the variability of nucleated amyloid assembly were derived. We
also demonstrate the application of our results to inform experiments in a
conceptually friendly and clear fashion. Our model offers a new perspective and
paves the way for a new and efficient approach on extracting vital information
regarding the key initial events of amyloid formation.
