  Native protein folds often have a high degree of symmetry. We study the
relationship between the symmetries of native proteins, and their
designabilities -- how many different sequences encode a given native
structure. Using a two-dimensional lattice protein model based on
hydrophobicity, we find that those native structures that are encoded by the
largest number of different sequences have high symmetry. However only certain
symmetries are enhanced, e.g. x/y-mirror symmetry and $180^o$ rotation, while
others are suppressed. If it takes a large number of mutations to destabilize
the native state of a protein, then, by definition, the state is highly
designable. Hence, our findings imply that insensitivity to mutation implies
high symmetry. It appears that the relationship between designability and
symmetry results because protein substructures are also designable. Native
protein folds may therefore be symmetric because they are composed of repeated
designable substructures.
