Swain-Schaad exponent

For harmonic oscillators recall that the ZPE s, (ZPE = (l/2)hc(//p,)1/2), and ZPE differences scale proportionally to (1/p-h) and (1/ jid), respectively. The q s are oscillator reduced masses and / is the isotope independent force constant. Thus, writing equations analogous to Equation 10.19 for tritium substitution, and taking the ratio, we obtain kH/kT = (kH/kD)x where x, the Swain-Schaad exponent in the harmonic approximation is expressed... 

This apparently naive expectation has been confirmed theoretically and experimentally. The exponent 1.44 is known as the Swain-Schaad exponent or coefficient, and is used in various forms of which the most common are ... 

Ah/At = 7.4 and A /Ax = 1.8 and isotopic activation energy differences that are within the experimental error of zero. The values of the two A-ratios correspond to a Swain-Schaad exponent of 3.4, not much different from the semiclassical expectation of 3.3. The a-secondary isotope effects are 1.19 (H/T), 1.13 (H/D), and 1.05 (D/T), which are exactly at the limiting semiclassical value of the equilibrium isotope effect. The secondary isotope effects generate a Swain-Schaad exponent of 3.5, again close to the semiclassical expectation. At the same time that the isotope effects are temperature-independent, the kinetic parameter shows... 

Bahnson et al. extended the series of mutations to include ones in which reductions occurred in the second-order rate constant / cat/ M by as much as a factor of 100. No substantial changes were observed in the primary isotope effects or their Swain-Schaad exponent. However, the precisely measured secondary isotope effects changed systematically as the rate constant decreased, such that the Swain-Schaad exponent decreased monotonically with decreasing fecat/ M from exponent of 8.5 for the L57F mutant (reactivity equivalent to the wild-type enzyme) to an exponent of 3.3 for the V203G mutant, slower by 100-fold. 

Fig. 6 Illustration from Chin and Klinman. Increased catalytic activity of horse-liver alcohol dehydrogenase in the oxidation of benzyl alcohol to benzaldehyde by NAD, measured by cat/ M (ordinate), correlates with the Swain-Schaad exponent for the -secondary isotope effect (abscissa), for which values above about four are indicators of tunneling. This is a direct test of the hypothesis that tunneling in the action of this enzyme contributes to catalysis. As the rate increases by over two orders of magnitude and then levels off, the anomalous Swain-Schaad exponents also increase and then level off. Reproduced from Ref. 28 with the permission of the American Chemical Society.

The observation of a primary tritium isotope effect (H/T) that is substantially larger than the value predicted on the basis of the semiclassical Swain-Schaad relation (Chart 3) from a heavy-hydrogen (DAT) isotope effect. The same information can be expressed in terms of a Swain-Schaad exponent required to relate the two isotope effects that is substantially larger than the semiclassical value of 3.26. 

The resulting generalized Swain-Schaad exponent varies from 1.44 for pH = 0 to 1.77 for pH = 1. A more realistic upper limit that takes into account coupling to promoting modes vill be 1.65 corresponding to pH-0.75. Such generalized Swain-Schaad exponents can be used to estimate tunneling contributions to proton transfer [21]. 

The generalized Swain-Schaad exponent (29.52) is directly applicable to most two-proton transfers obviously, this holds true for each step of a stepwise process, but it also applies to concerted processes in which the two proton isotopes are the same, since the effective mass and harmonic frequency of the relevant symmetric or antisymmetric XH-stretch modes are essentially the same as those of their onedimensional components. However, if the two proton isotopes are different, this argument no longer suffices because the normal mode that represents the frequency and effective mass of the transfer coordinate in the transition state correlates with two distinct normal modes in the equilibrium configuration. Hence there is no unambiguous Swain-Schaad type exponent relating HD to HH and DD transfer. 

It is noteworthy that the Swain-Schaad exponents are temperature independent for both primary and secondary isotope effects at pH 6.1. Two scenarios can be considered. The first is that there is a significant commitment to catalysis which is obscuring the full value of the isotope effect on k st/Km- It is anticipated that, because the primary and secondary exponents are temperature independent, this commitment would be temperature independent. Jonsson et al. have used North-rop s expression [60] for correcting observed isotope effects based on the assumption of a temperature-independent commitment (Eq. (10.22)). A commitment of 0.6 for the oxidation of benzylamine brings the secondary exponent to about 3.3,... 

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