The Swain-Schaad Relationship

Quantum tunneling in enzyme-catalyzed reactions early indications 35 The rule of the geometric mean ( no isotope effects on isotope effects ) The Swain-Schaad relationship 36... 

There are two ways in which an enzymic reaction can fail to satisfy the Swain-Schaad relationship, one of which is if tunneling occurs. In order to use violations of this rule to diagnose the presence of tunneling, it is necessary to eliminate the other possible reason for a violation, namely, limitation of the rate by more than one step. The derivation of the Swain-Schaad equation in Chart 3 assumes that the step that produces the isotope effect is fully rate-limiting, and if this should be untrue, then the relationship should fail without any significance for tunneling. 

Chart 7. Huskey s rules for violations of the rule of the geometric mean and the Swain-Schaad relationships... 

For further important work on this and related concepts, see Rucker, J. and Kliman, J.P. (1999). Computational study of tunneling and coupled motion in alcohol dehydrogenase-catalyzed reactions Implication for measured hydrogen and carbon isotope effects. J. Am. Chem. Soc. 121, 1997 -2006, and Kohen, A. and Jensen, J.H. (2002). Boundary conditions for the Swain-Schaad relationship as a criterion for hydrogen tunneling. J. Am. Chem. Soc. April 17, 124(15), 3858-3864. 

The disintegration of a substance in a first-order manner. 2. A breakdown of the Swain-Schaad relationship in kinetic isotope effect studies usually as a consequence of tunneling or kinetic complexity. 

It is important to note that these equations are based on the Swain-Schaad relationship, which assumes that there is no tunnelling in any of the isotopic reactions (the KIEs are semiclassical) and that the relationship between the KIEs is determined only by the masses of the hydrogen, deuterium and tritium atoms. The secondary and kfyko KIEs calculated both with and... 

Significant deviations from the Swain-Schaad relationship (Swain et al., 1958) in the direction ( H/ D) > (kKjkT)0 693. 

Small deviations from the Swain-Schaad relationship have been attributed to tunneling in a base-catalyzed elimination (Shiner and Martin, 1964). In view of the approximate nature of the relationship small deviations must be viewed skeptically. 

Experiments with the oxidation of benzyl alcohol by NAD+, catalyzed by yeast alcohol dehydrogenase, yielded (fccat/XM)D/(fccat/XM)x = 1.72 — 1.76 (standard deviations about 0.03-0.06). These experiments involved multiple labeling so an exact interpretation must take into account Huskey s rules for this situation (Chart 7 below). Application of the Swain-Schaad relationship predicts an H/T effect of 5.9-6.3 (propagated errors 0.2-0.6). The observed H/T effects are 7.0-7.2 (standard deviations about 0.1), providing a very strong indication of the importance of tunneling in this reaction. 

Finally the temperature dependence of the primary isotope effects was determined. Here the traditional expectations of Chart 3 were fully met the results translate into AH/AD = 1.1 0.1, aD — aH = 0.8 kcal/mol. Thus the amount of tunneling present, adequate to produce the observed exaltation of secondary isotope effects, violations of the Swain-Schaad relationship, and violations of the Rule of the Geometric Mean in the neighborhood of room temperature, does not lead to anomalies in either the ratio of isotopic pre-exponential factors nor the isotopic activation energy difference over the temperature range studied (approximately 0-40 °C). As will be seen later, the temperature dependence of isotope effects for reactions that include tunneling is in general a complex, unresolved issue. 

The Swain Schaad relationship which yields boundaries for the ratios of the rate constants for the primary kinetic isotope effects for H, D and T with deviations being interpreted as evidence of tunneling. 

Before proceeding with the KIE analysis for adiabatic PT, it is worth stressing, for comparison with the standard picture, that there are four common experimental observations which are consistent with the standard picture for nonturmeling PT KIEs, and which are thus viewed as supporting that picture (i) the Arrhenius temperature dependence of the KIE (as well as of the individual isotope rate constants) (ii) the KIE - AGrxn behavior described in Section 10.1 (i.e. maximal for the symmetric case) (iii) the KIE range is -2-10 and (iv) the wide applicability of the Swain-Schaad relationship [13, 46] connecting KIE ratios (e.g. kn/kx = These observations have done much to maintain the stan-... 

The Swain-Schaad relationship has been an important experimental probe for PT reaction KIEs [11, 24, 46]. We have used [4] one of its forms for illustration... 

There are two extremes for this rate law (i) when k i k2, then kg g = [ kj / k i ] k2, and (ii) when k2 k i, then kg = kj. For case (i) second order kinetics and near unity experimental PKIE values are expected. On the other hand for case (ii) second order kinetics are measured with normal experimental values of the PKIE that obey the Swain-Schaad relationship. 

A major contribution to the analysis of isotope effects associated with cases where there is no single rate-limiting step was made by the Streitwieser group in 1971 [4, 12]. Using single temperature rate constants for all three hydrogen isotopes and the Swain-Schaad relationship, they calculate an internal-return parameter, a = k i/k2, associated with the experimental rate constants for each of the isotopic exchange measurements. The rate constant for the actual hydron transfer step can now be calculated ... 

The exchange reactions of 9-phenylfluorenes have experimental PKIE values that are normal in magnitude, but do not obey the Swain-Schaad relationship. 

Several investigators examined this relationship under extreme temperatures (20-1000 K), and as a probe for tunneling [26-28]. This isotopic relationship was also used in experimental and theoretical studies to suggest coupled motion between primary and secondary hydrogens for hydride transfer reactions, such as elimination in the gas phase, and in organic solvents [29, 30]. The power of the Swain-Schaad relationship is that it appears independent of the details of the reaction s potential surface and thus can be used to relate unknown KIEs (see Section 12.3.2). 

Two common uses of the Swain-Schaad relationship in enzymology are described in the following sections. 

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