MOTIVATION: The constraints under which sequence, structure and function coevolve are not fully understood. Bringing this mutual relationship to light can reveal the molecular basis of binding, catalysis and allostery, thereby identifying function and rationally guiding protein redesign. Underlying these relationships are the epistatic interactions that occur when the consequences of a mutation to a protein are determined by the genetic background in which it occurs. Based on prior data, we hypothesize that epistatic forces operate most strongly between residues nearby in the structure, resulting in smooth evolutionary importance across the structure. METHODS AND RESULTS: We find that when residue scores of evolutionary importance are distributed smoothly between nearby residues, functional site prediction accuracy improves. Accordingly, we designed a novel measure of evolutionary importance that focuses on the interaction between pairs of structurally neighboring residues. This measure that we term pair-interaction Evolutionary Trace yields greater functional site overlap and better structure-based proteome-wide functional predictions. CONCLUSIONS: Our data show that the structural smoothness of evolutionary importance is a fundamental feature of the coevolution of sequence, structure and function. Mutations operate on individual residues, but selective pressure depends in part on the extent to which a mutation perturbs interactions with neighboring residues. In practice, this principle led us to redefine the importance of a residue in terms of the importance of its epistatic interactions with neighbors, yielding better annotation of functional residues, motivating experimental validation of a novel functional site in LexA and refining protein function prediction. CONTACT: [email protected]. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.