  We develop a theory of aggregation using statistical mechanical methods. An
example of a complicated aggregation system with several levels of structures
is peptide/protein self-assembly. The problem of protein aggregation is
important for the understanding and treatment of neurodegenerative diseases and
also for the development of bio-macromolecules as new materials. We write the
effective Hamiltonian in terms of interaction energies between protein
monomers, protein and solvent, as well as between protein filaments. The grand
partition function can be expressed in terms of a Zimm-Bragg-like transfer
matrix, which is calculated exactly and all thermodynamic properties can be
obtained. We start with two-state and three-state descriptions of protein
monomers using Potts models that can be generalized to include q-states, for
which the exactly solvable feature of the model remains. We focus on n X N
lattice systems, corresponding to the ordered structures observed in some real
fibrils. We have obtained results on nucleation processes and phase diagrams,
in which a protein property such as the sheet content of aggregates is
expressed as a function of the number of proteins on the lattice and
inter-protein or interfacial interaction energies. We have applied our methods
to A{\beta}(1-40) and Curli fibrils and obtained results in good agreement with
experiments.
