  Protein structures in nature often exhibit a high degree of regularity
(secondary structures, tertiary symmetries, etc.) absent in random compact
conformations. We demonstrate in a simple lattice model of protein folding that
structural regularities are related to high designability and evolutionary
stability. We measure the designability of each compact structure by the number
of sequences which can design the structure, i.e., which possess the structure
as their nondegenerate ground state. We find that compact structures are
drastically different in terms of their designability; highly designable
structures emerge with a number of associated sequences much larger than the
average. These structures are found to have ``protein like'' secondary
structure and even tertiary symmetries. In addition, they are also
thermodynamically more stable than ordinary structures. These results suggest
that protein structures are selected because they are easy to design and stable
against mutations, and that such a selection simutaneously leads to
thermodynamic stability.
