  We present the exact solution of a microscopic statistical mechanical model
for the transformation of a long polypeptide between an unstructured coil
conformation and an $\alpha$-helix conformation. The polypeptide is assumed to
be adsorbed to the interface between a polar and a non-polar environment such
as realized by water and the lipid bilayer of a membrane. The interfacial
coil-helix transformation is the first stage in the folding process of helical
membrane proteins. Depending on the values of model parameters, the
conformation changes as a crossover, a discontinuous transition, or a
continuous transition with helicity in the role of order parameter. Our model
is constructed as a system of statistically interacting quasiparticles that are
activated from the helix pseudo-vacuum. The particles represent links between
adjacent residues in coil conformation that form a self-avoiding random walk in
two dimensions. Explicit results are presented for helicity, entropy, heat
capacity, and the average numbers and sizes of both coil and helix segments.
