Biochemical activity and core stability are essential properties of proteins, maintained usually by conserved amino acids. Structural dynamics emerged in recent years as another essential aspect of protein functionality. Structural dynamics enable the adaptation of the protein to binding substrates and to undergo allosteric transitions, while maintaining the native fold. Key residues that mediate structural dynamics would thus be expected to be conserved or exhibit coevolutionary patterns at least. Yet, the correlation between sequence evolution and structural dynamics is yet to be established. With recent advances in efficient characterization of structural dynamics, we are now in a position to perform a systematic analysis. In the present study, a set of 34 enzymes representing various folds and functional classes is analyzed using information theory and elastic network models. Our analysis shows that the structural regions distinguished by their coevolution propensity as well as high mobility are predisposed to serve as substrate recognition sites, whereas residues acting as global hinges during collective dynamics are often supported by conserved residues. We propose a mobility scale for different types of amino acids, which tends to vary inversely with amino acid conservation. Our findings suggest the balance between physical adaptability (enabled by structure-encoded motions) and chemical specificity (conferred by correlated amino acid substitutions) underlies the selection of a relatively small set of versatile folds by proteins.