Apparent Kinetic Isotope Effects for Multi-Step Steady-State Reactions

The apparent kinetic isotope effect (KIE) for a multistep steady-state reaction can be expressed simply as a sum of terms, one for each transition state (TS) in the serial sequence, each of which is the product of the KIE for an individual TS (with respect to a common reference state) and a weighting factor, which is the degree of kinetic significance for that TS. This requires knowledge of the relative Gibbs energies of the sequential TSs but not of any intermediates, and it involves a much simpler expression than the conventional method for analysis of KIEs for enzyme reactions. A numerical example is presented to illustrate how the same apparent KIE may result from numerous combinations of individual KIEs and weighting factors. It is proposed that computed apparent KIEs should be compared directly with experimentally observed KIEs rather than with derived intrinsic KIEs of possibly dubious validity. The results of DFT calculations for an S_N1 nucleophilic displacement are presented to show how the apparent KIE varies, as the relative concentration of the nucleophilic species ranges from 0.1 to 10, between limiting values corresponding to either the first or second step being completely rate limiting.