Interpretation of V/K isotope effects for enzymatic reactions exhibiting multiple isotopically sensitive steps

Formulations of an enzyme mechanism where only a single step is presumed to be isotopically sensitive can be written in terms of forward and reverse commitments to catalysis. These commitments provide a natural and intuitive way of interpreting the observed isotope effects. Unfortunately, when multiple isotopically sensitive steps are present in the mechanism, including effects associated with pre-equilibria of the unbound substrate, the observed V/K kinetic isotope effect is expressed as a complicated expression of the intrinsic rate constants for each step, the interpretation of which is not always immediately obvious. We show here that V/K isotope effects from unbranched or rapid-equilibrium random Michaelis-Menten systems containing multiple isotopically sensitive steps can be written as a weighted average of the intrinsic isotope effects on each step, where this intrinsic isotope effect from each step is the product of the equilibrium isotope effect on the formation of the reacting intermediate for that step and the intrinsic kinetic effect on the forward rate constant for that step, and the weighting factors are simply the reciprocal sum of the forward and reverse commitments for each step i plus unity, 1/(C(fi)+C(ri)+1), equivalent to the sensitivity index [Ray, W.J., 1983.