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We consider a general incompressible finite model protein of size M in itsenvironment, which we represent by a semiflexible copolymer consisting of aminoacid residues classified into only two species (H and P, see text) followingLau and Dill. We allow various interactions between chemically unbondedresidues in a given sequence and the solvent (water), and exactly enumerate thenumber of conformations W(E) as a function of the energy E on an infinitelattice under two different conditions: (i) we allow conformations that arerestricted to be compact (known as Hamilton walk conformations), and (ii) weallow unrestricted conformations that can also be non-compact. It is easilydemonstrated using plausible arguments that our model does not possess anyenergy gap even though it is supposed to exhibit a sharp folding transition inthe thermodynamic limit. The enumeration allows us to investigate exactly theeffects of energetics on the native state(s), and the effect of small size onprotein thermodynamics and, in particular, on the differences between themicrocanonical and canonical ensembles. We find that the canonical entropy ismuch larger than the microcanonical entropy for finite systems. We investigatethe property of self-averaging and conclude that small proteins do notself-average. We also present results that (i) provide some understanding ofthe energy landscape, and (ii) shed light on the free energy landscape atdifferent temperatures.

Exact Statistical Mechanical Investigation of a Finite Model Protein in its environment: A Small System Paradigm