DERIVATION AND PROPERTIES OF MICHAELIS-MENTEN TYPE AND HILL TYPE EQUATIONS FOR REFERENCE LIGANDS

The dose-effect relationships of the reference ligands (e.g. inhibitors) in the presence of the primary ligands (e.g., substrates) in enzyme kinetic models were analyzed systematically. Enzyme reactions with different number of substrates, different reaction mechanisms and different types and mechanisms of inhibition yield equation (A), fa/fu = D/Dm, for the first-order interactions and equation (B), fa/fu = (D/Dm)m, for the m-order interactions; D is dose, Dm is the D required for the median-effect, and fa and fu are the fractions of the system that are affected and unaffected by D, respectively. Conversion of equation (A) gives fa = 1/[1 + (Dm/D)] which has the same form as the Michaelis-Menten equation, v/Vmax = 1/]1 + (Km/S)]. The logarithmic form of equation (B) gives the Hill type equation. In equations (A) and (B), the fractional velocity is expressed with respect to the control velocity rather than the maximal velocity; D can be the reference ligands or environmental factors rather than the restriction to the primary ligands. A plot of log (D) vs. log[(fu)-1 - 1] or log[fa)-1 - 1]-1 will be a useful procedure for transforming dose-effect relationships and for analyzing mass action characteristics in various systems. The plot will give the slope m, and the intercepts of the dose-effect lines with the median-effect axis (i.e., log[(fu)-1 - 1] = 0) will give Dm values. Any cause-consequence relationships that give straight lines in the plot will give m and Dm values, and thus give empirical equation for the system. Both equations (A) and (B) were compared with relevant existing equations in terms of limitations and applicabilities. Mechanism-dependent equations involving kinetic or equilibrium constants can be transformed to a mechanism-independent general equation involving the median-effect value.