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 binds one molecule of substrate (S) and/or one molecule of a modifier X. The modifier acts as activator (essential or non-essential), as inhibitor (total 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 velocity steps, and the concentration of substrate ([S]). Different possibilities 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 the modifier) and nu(su) (nu at that particular [S]) were deduced. In Lineweaver-Burk plots, the straight lines related to different M generally cross in 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 inhibitor, or has no effect, depending on [S]. Furthermore, we discuss: (1) the apparent V-max and K-m displayed by the enzyme in different situations; (2) the degree of effect (inhibition or activation) observed at different concentrations of substrate and modifier; (3) the concept of K-e, a parameter 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.