Accurate Determination of Rate Constants of Very Slow, Tight-Binding Competitive Inhibitors by Numerical Solution of Differential Equations, Independently of Precise Knowledge of the Enzyme Concentration

This paper is concerned with the determination of rate constants characterizing the binding and release of a slow binding inhibitor to and from an enzyme, here almond β-glucosidase. We demonstrate the inability of the conventional method to yield reliable rate constants when one or more of these is less than 1 × 10−4 per second. Instead one must use the much more accurate fitting of rate constants of the set of simultaneous differential equations characterizing the kinetic model. This procedure has the added advantage, when properly used, that the rate constants found pertaining to the inhibitor are largely insensitive to the particular value used for the enzyme concentration; i.e., the same data set may be fitted using a range of enzyme concentrations with no change in the resulting parameters. Hence the method can be used when little is known about the enzyme, except for the value of Km, which is readily determined. Also, we report the somewhat unexpected finding that the association rate constant for the substrate (4-nitrophenyl-β-d-glucopyranoside) is about one-third of the value of the corresponding rate constant for the inhibitor. The method is used to determine rate constants at several temperatures for the strong, slow binding inhibitor 2-phenethylglucoimidazole 1, enabling us to compute standard thermodynamic functions. The identity of these functions with those of isofagomine (2) reported earlier leads us to argue that the two compounds share a common binding mechanism, involving the same groups, whereas the different stabilities of the enzyme-inhibitor complexes must reside in those parts of the molecules that are not identical.