The tertiary structures of globular proteins have remarkable and complex symmetries. What forces cause them? We find that a very simple model reproduces some of those symmetries. Proteins are modeled as copolymers of specific sequences of hydrophobic (H) and polar (P) monomers (HP model) configured as self-avoiding flights on simple three-dimensional cubic lattices. The model has no parameters; we just seek the conformations that have the global maximum number of HH contacts for any given sequence. Finding global optima for chains in this model has not been computationally possible before for chains longer than 36-mers. We report here a procedure that can find all the globally optimal conformations, the number of which defines the degeneracy of a sequence, for chains up to 88 monomers long. It is about 37 orders of magnitude faster than previous exact methods. We find that degeneracy is an important aspect of sequence design. So far, we have found that four-helix bundles, alpha/beta-barrels, and parallel beta-helices are globally optimal conformations of polar/nonpolar sequences that have minimal degeneracy.

Yue, K., & Dill, K. A. (1995). Forces of tertiary structural organization in globular proteins. Proceedings of the National Academy of Sciences, 92(1), 146–150.