Coarse-graining protein energetics in sequence variables

We show that cluster expansions (CE), previously used to model solid-state materials with binary or ternary configurational disorder can be extended to the protein design problem. We present a generalized CE framework, in which properties such as energy can be unambiguously expanded in the amino-acid sequence space. The CE coarse grains over non-sequence degrees of freedom (e.g., side-chain conformations) and thereby simplifies the problem of designing proteins, or predicting the compatibility of a sequence with a given structure, by many orders of magnitude. The CE is physically transparent, and can be evaluated through linear regression on the energies of training sequences. [PRL 95, 148103 (2005)]. We show, as example, that good prediction accuracy is obtained with up to pairwise interactions for a coiled-coil backbone, and that triplet and/or quadruplet interactions are important in the energetics of the more globular zinc-finger and WW domain backbones. In the coiled-coil system, where experimental data is available, the calculated pair interaction parameters compare favorably with measured coupling energies. The clear advantage of a CE driven optimization over a direct one is demonstrated by searching for low-energy sequences on the zinc-finger backbone. Other possible applications of our approach are also discussed.