Cluster expansion models for flexible-backbone protein energetics

Protein structure prediction and design often involve discrete modeling of side-chain conformations on structural templates. Introducing backbone flexibility into such models has proven important in many different applications. Backbone flexibility improves model accuracy and provides access to larger sequence spaces in computational design, although at a cost in complexity and time. Here, we show that the influence of backbone flexibility on protein conformational energetics can be treated implicitly, at the level of sequence, using the technique of cluster expansion. Cluster expansion provides a way to convert structure-based energies into functions of sequence alone. It leads to dramatic speed-ups in energy evaluation and provides a convenient functional form for the analysis and optimization of sequence-structure relationships. We show that it can be applied effectively to flexible-backbone structural models using four proteins: alpha-helical coiled-coil dimers and trimers, zinc fingers, and Bcl-xL/peptide complexes. For each of these, low errors for the sequence-based models when compared with structure-based evaluations show that this new way of treating backbone flexibility has considerable promise, particularly for protein design.