Mixtures of tight-binding enzyme inhibitors. Kinetic analysis by a recursive rate equation

When two or more tight-binding inhibitors are present in an enzyme assay, the equation that relates the initial velocity v to the concentration of reactants cannot be written in an algebraically explicit form. Rather, for n inhibitors it is an implicit polynomial equation of degree n + 1 with respect to v. The complexity of the polynomial coefficients dramatically increases with each added inhibitor. Solving the transcendental rate equation by traditional methods of numerical mathematics has proven tedious because of the sensitivity of these methods to initial estimates and because of the existence of multiple roots. However, the equation can be rearranged into a convenient recursive form, one in which the velocity appears on both sides and the solution is found iteratively. The algebraic form of the recursive rate equation is remarkably simple and differs from the rate equation for classical rather than tight-binding inhibition only by an added term. The numerical stability and the speed of convergence were tested on the case of two competitive inhibitors. Initial estimates of velocity that spanned 12 orders of magnitude converged within five iterations. The velocities computed with the recursive method for a single tight-binding inhibitor were identical with the values predicted by the Morrison equation. The method is used to analyze experimental data for the inhibition of rat liver dihydrofolate reductase by mixtures of the anticancer drug methotrexate and its metabolic precursor form, methotrexate-alpha-aspartate (a prodrug).