Designability and cooperative folding in a four-letter hydrophobic-polar model of proteins

The two-letter hydrophobic-polar (HP) model of Lau and Dill [Macromolecules 22, 3986 (1989)] has been widely used in theoretical studies of protein folding due to its conceptual and computational simplicity. Despite its success in elucidating various aspects of the sequence-structure relationship, thermodynamic behavior of the model is not in agreement with a sharp two-state folding transition of many single-domain proteins. To gain a better understanding of this discrepancy, we consider an extension of the HP model by including an "antiferromagnetic" (AF) interaction in the contact potential that favors amino acid residues with complementary attributes. With an enlarged four-letter alphabet, the density of states on the low energy side can be significantly decreased. Computational studies of the four-letter HP model are performed on 36-mer sequences on a square lattice. It is found that the designability of folded structures in the extended model exhibits strong correlation with that of the two-letter HP model, while the AF interaction alone selects a very different class of structures that resembles the Greek key motif for beta sheets. A procedure is introduced to select sequences which have the largest energy gap to the native state. Based on density of states and specific heat calculations in the full configuration space, we show that the optimized sequence is able to fold nearly as cooperatively as a corresponding Gō model.