  We consider a general incompressible finite model protein of size M in its
environment, which we represent by a semiflexible copolymer consisting of amino
acid residues classified into only two species (H and P, see text) following
Lau and Dill. We allow various interactions between chemically unbonded
residues in a given sequence and the solvent (water), and exactly enumerate the
number of conformations W(E) as a function of the energy E on an infinite
lattice under two different conditions: (i) we allow conformations that are
restricted to be compact (known as Hamilton walk conformations), and (ii) we
allow unrestricted conformations that can also be non-compact. It is easily
demonstrated using plausible arguments that our model does not possess any
energy gap even though it is supposed to exhibit a sharp folding transition in
the thermodynamic limit. The enumeration allows us to investigate exactly the
effects of energetics on the native state(s), and the effect of small size on
protein thermodynamics and, in particular, on the differences between the
microcanonical and canonical ensembles. We find that the canonical entropy is
much larger than the microcanonical entropy for finite systems. We investigate
the property of self-averaging and conclude that small proteins do not
self-average. We also present results that (i) provide some understanding of
the energy landscape, and (ii) shed light on the free energy landscape at
different temperatures.
