Pursuing a materials science approach to understanding the deformability of enzymes, we introduce measurements of the phase of the mechanical response function within the nanorheology paradigm. Driven conformational motion of the enzyme is dissipative as characterized by the phase measurements. The dissipation originates both from the surface hydration layer and the interior of the molecule, probed by examining the effect of point mutations on the mechanics. We also document changes in the mechanics of the enzyme examined, guanylate kinase, upon binding its four substrates. GMP binding stiffens the molecule, ATP and ADP binding softens it, while there is no clear mechanical signature of GDP binding. A hyperactive two-Gly mutant is found to possibly trade specificity for speed. Global deformations of enzymes are shown to be dependent on both hydration layer and polypeptide chain dynamics.