Isotopic fractionation in proteins as a measure of hydrogen bond length

If a deuterated molecule containing strong intramolecular hydrogen bonds is placed in a hydrogenated solvent, it may preferentially exchange deuterium for hydrogen. This preference is due to the difference between the vibrational zero-point energy for hydrogen and deuterium. It is found that the associated fractionation factor Φ is correlated with the strength of the intramolecular hydrogen bonds. This correlation has been used to determine the length of the H-bonds (donor-acceptor separation) in a diverse range of enzymes and has been argued to support the existence of short low-barrier H-bonds. Starting with a potential energy surface based on a simple diabatic state model for H-bonds, we calculate Φ as a function of the proton donor-acceptor distance R. For numerical results, we use a parameterization of the model for symmetric O-H⋯O bonds [R. H. McKenzie, Chem. Phys. Lett. 535, 196 (2012)]. We consider the relative contributions of the O-H stretch vibration, O-H bend vibrations (both in plane and out of plane), tunneling splitting effects at finite temperature, and the secondary geometric isotope effect. We compare our total Φ as a function of R with NMR experimental results for enzymes, and in particular with an earlier model parametrization Φ(R), used previously to determine bond lengths.