Generalized Equations for the Analysis of Inhibitions of Michaelis-Menten and Higher-Order Kinetic Systems with Two or More Mutually Exclusive and Nonexclusive Inhibitors

The following simple and generalized equation not involving kinetic constants for substrates (Km) or inhibitors (Km) describes the summation of the effects of two mutually exclusive and reversible inhibitors on enzyme systems obeying Michaelis-Menten kinetics: where fv and fi(= 1 −fv) are the fractional velocity and the fractional inhibition of the reaction, respectively, and I50 is the concentration of the inhibitor (I) that is required to produce 50% inhibition (i.e. the median effect). More specifically, (fv)1 and (fv)2 are the fractional velocities in the presence of inhibitors I1, and I2, respectively, and (fv)1,2 the fractional velocity in the simultaneous presence of both inhibitors. For mutually nonexclusive reversible inhibitors that follow Michaelis-Menten kinetics, the relationship becomes: Similar relationships apply to situations involving more than two inhibitors, for which generalized equations are given in the text. The above relationships express summation of inhibitory effects, irrespective of the number of substrates, the number of inhibitors, the type of reversibie inhibition (competitive, noncompetitive, or uncompetitive), or the kinetic mechanisms (sequential or ping-pong) of the enzyme reaction under consideration. These concepts have been extended to higher-order (Hill-type) systems in which each reversible inhibitor has more than one binding site. If such inhibitors are mutually exclusive: where m is a Hill-type coefficient, assumed to be the same for each inhibitor. If the inhibitors, I1, and I2, are mutually nonexclusive, this relationship becomes: It may be seen that (for higher-order mutually exclusive and nonexclusive inhibitors), when (fi)1,2= 0.5, the above relationships are equal to unity, and hence require no knowledge of the magnitude of the values of m for each inhibitor. The above relations at (fi)1,2= 0.5 describe I50-isobolograms (curves for isoeffective combinations of inhibitors). The above-described relationships lend themselves to simple graphical representations, directly applicable to experimental results. Thus, plots of log (= 0.5 describe I50-isobolograms (curves for isoeffective combinations of inhibitors). The above-described relationships lend themselves to simple graphical representations, directly applicable to experimental results. Thus, plots of log [(fi)−1– 1−1 versus log I for each inhibitor individually lead to their corresponding m values (i.e. slope) and log I50, values (i.e. intercept at y= 0). Linearity of such plots is diagnostic of Michaelis-Menten (where m= 1) or Hill (where m > 1) kinetics. Furthermore, if the two inhibitors are mutually exclusive and have the same m values, the plot of log (fi)−1– 1−1 with respect to log I1+I2 will be linear and parallel to the plots for each component inhibitor. For mutually nonexclusive inhibitors, if the plot of log (fi)−1– 1−1 with respect to log I are linear for each individual inhibitor, then the plot of log (fi)−1– 1−1 with respect to log I1+I2 will be concave upward, and will intersect ihe plot for the morc potent of the two inhibitors.