Selection for Folding Stability Predicts Observed Covariation Between Protein Positions in the PDB

Statistical couplings between protein sites have been enormously useful for protein structure prediction, but there is still debate on the selective forces that generate them. Here we explicitly predict the couplings considering selection for protein folding stability, both against unfolding and misfolding, which we model through the Stability constrained model of protein evolution (SCPE). The SCPE is based on contact interactions, and it predicts stability against misfolding through the Random Energy Model in the space of contact matrices. We adopt the minimum selection principle, assuming that the multivariate amino acid distribution has minimum Kullback-Leibler divergence with respect to a site-unspecific distribution that is not influenced by protein folding stability. We predict the couplings at first order, assuming that they are small (small coupling approximation, SCA), and obtain an explicit formula that relates them with the contact interaction matrix and with the structural properties of the pairs of sites: presence of native contact, distance along the sequence and buriedness of the sites. We test these predictions on a representative set of the Protein Data Bank. For pairs of sites in contact, both predicted and observed couplings are inversely related with the contact energies of the amino acid pairs, while for pairs not in contact this relation is positive, i.e. negative design destabilizes pairs of amino acids for short-range pairs of sites that can form wrong contacts with high probability. Our results suggest that the strongest couplings correspond to native contacts, but an even larger number of couplings are influenced by negative design. Intriguingly, the most informative pairs are formed by surface sites, suggesting a role for protein-protein interactions and functional motions. We interpret the site-unspecific couplings, which are not influenced by selection for protein folding stability, as the result of variations of global mutational or selective forces across the lineages where the proteins evolve. Consistently, the strongest global couplings are self-couplings between equal amino acids. Cys-Cys, which form disulfide bridges, show the strongest global coupling, followed by pairs of metabolically costly amino acids (Trp-Trp, Tyr-Tyr, Trp-Tyr) and positively charged amino acids that tend to interact with nucleic acids (Lys-Lys, Arg-Arg). Ser, Thr, Asn and Gln form a cluster of globally correlated polar amino acids.