  The self-assembly of proteins into $\beta$-sheet-rich amyloid fibrils has
been observed to occur with sigmoidal kinetics, indicating that the system
initially is trapped in a metastable state. Here, we use a minimal
lattice-based model to explore the thermodynamic forces driving amyloid
formation in a finite canonical ($NVT$) system. By means of
generalized-ensemble Monte Carlo techniques and a semi-analytical method, the
thermodynamic properties of this model are investigated for different sets of
intersheet interaction parameters. When the interactions support lateral growth
into multi-layered fibrillar structures, an evaporation/condensation transition
is observed, between a supersaturated solution state and a thermodynamically
distinct state where small and large fibril-like species exist in equilibrium.
Intermediate-size aggregates are statistically suppressed. These properties do
not hold if aggregate growth is one-dimensional.
