Vibrational-analysis modeling techniques have been applied to the hydride transfer step in dehydrogenase-catalyzed reactions which exhibit exalted α-secondary deuterium isotope effects, leading to the conclusion that non-resonant tunneling occurs in these reactions. Details of the calculations are presented. The calculations also indicate that tunneling is responsible for the violations of the Rule of the Geometric Mean observed in dehydrogenase reactions ( i.e., the magnitude of the α-secondary isotope effect for hydride transfer from NADH is smaller when deuterium is transferred than when protium is transferred), and that such violations are diagnostic of tunneling. The observation of Hermes et al. [J. Am. Chem. Soc., 106 (1984) 5479] (Ref. 1 in the list) that the violations of the Rule disappear as the cofactor is altered m structure and k cal / K m drops by 300-fold can be interpreted as indicating that, for formate dehydrogenase, non-resonant tunneling is associated with 2–3 orders of magnitude of the enzymic acceleration. The temperature dependence of the solvent isotope effect is employed as a probe for tunneling phenomena in reactions of the serine proteases. For chymotrypsin with Suc-Phe-NHNp, trypsin and thrombin with Bz-Arg-OEt, elastase and subtilisin with DFP (irreversible inhibition with catalytic recruitment), with all enzymes in homogeneous buffered solution, the solvent isotope effects exhibit the temperature dependence expected either for simple hydrogen bridging in a semiclassical transition state or for non-resonant transition-state tunneling. These results contrast with the findings of Khoshtariya et al. [Bioorg. Khim., 5 (1979) 1243, and preceding papers] (Ref. 17 in the list) with the same enzymes studied under different conditions, in which solvent isotope effects were temperature-independent, indicative of the importance of resonant tunneling. The evidence for tunneling in the acid-base component of serine-protease catalysis is therefore ambiguous, and specification of its role must await further study.