  Protein structures are a very special class among all possible structures. It
was suggested that a ``designability principle'' plays a crucial role in
nature's selection of protein sequences and structures. Here we provide a
theoretical base for such a selection principle, using a novel formulation of
the protein folding problem based on hydrophobic interactions. A structure is
reduced to a string of 0's and 1's which represent the surface and core sites,
respectively, as the backbone is traced. Each structure is therefore associated
with one point in a high dimensional space. Sequences are represented by
strings of their hydrophobicities and thus can be mapped into the same space. A
sequence which lies closer to a particular structure in this space than to any
other structures will have that structure as its ground state. Atypical
structures, namely those far away from other structures in the high dimensional
space, have more sequences which fold into them, and are thermodynamically more
stable. We argue that the most common folds of proteins are the most atypical
in the space of possible structures.
