Inhibition and activation of enzymes. The effect of a modifier on the reaction rate and on kinetic parameters

A combined analysis of enzyme inhibition and activation is presented, based on a rapid equilibrium model assumption in which one molecule of enzyme bi nds one molecule of substrate (S) and/or one molecule of a modifier X. The modifier acts as activator (essential or non-essential), as inhibitor (tota l or partial), or has no effect on the reaction rate (nu), depending on the values of the equilibrium constants, the rate constants of the limiting ve locity steps, and the concentration of substrate ([S]). Different possibili ties have been analyzed from an equation written to emphasize that nu = f([ X]) is, in general and at a fixed [S], a hyperbolic function. Formulas for S-u (the value of [S], different from zero, at which nu is unaffected by th e modifier) and nu(su) (nu at that particular [S]) were deduced. In Linewea ver-Burk plots, the straight lines related to different M generally cross i n a point (P) with coordinates (S-u, nu(su)). In certain cases, point P is located in the first quadrant which implies that X acts as activator, as in hibitor, or has no effect, depending on [S]. Furthermore, we discuss: (1) t he apparent V-max and K-m displayed by the enzyme in different situations; (2) the degree of effect (inhibition or activation) observed at different c oncentrations of substrate and modifier; (3) the concept of K-e, a paramete r that depends on the concentration of substrate and helps to evaluate the effect of the modifier: it equals the value of [X] at which the increase or decrease in the reaction rate is half of that achieved at saturating [S]. Equations were deduced for the general case and for particular situations, and used to obtain computer-drawn graphs that are presented and discussed. Formulas for apparent V-max, K-m and K-e have been written in a way making it evident that these parameters can be expressed as pondered means.