The probability that complex enzyme kinetic curves can be caused by activators of inhibitors

Numerous chemical compounds are known that alter the rate of conversion of substrates into products in enzyme-catalysed reactions by interacting with the enzyme rather than substrates. Where this takes place in such a way that the effect is reversible on removing the compound, say by dialysis, and where the compound is unchanged chemically by the enzyme system, we refer to such a compound as a modifier. So protons, inorganic salts, activators, inhibitors or even specific allosteric effectors would all be modifiers, and any chemically reasonable kinetic scheme that is proposed to account for such effects is referred to as modifier mechanism. Three versions of a modifier mechanism of enzyme action are studied. The implicit representation is 2:2 in [S] (with alpha(0)=0) and 2:2 in [M] (with alpha(0) not equal0), and this is a short-hand scheme for the minimum chemical formulation, the explicit one, involving discrete ES and EP species, which is 2:2 in [S] (with alpha(0)=0) and 3:3 in [M] (with alpha(0) not equal0). If m extra steps are allowed between interconversion of ES and EP species, the degree of the rate equation remains 2:2 in [S] (with alpha(0)=0), but increases to degree (m+3):(m+3) in modifier (with alpha(0) not equal0). It is proved that this increase in degree is genuine and that highly complex v([M]) (i.e. v-versus-[M]) curves can occur. Computation of the probabilities of the five possible double-reciprocal plots in 1/v versus 1/[S] show that all of these formulations of the modifier mechanism give similar probabilities, and these are characteristic for the mechanism and quite distinct from the intrinsic curve-shape probabilities. It is also established that the probabilities of alternative complex v([M]) plots are similar for the various formulations, and again the probabilities of the allowed complex curves for the mechanism are quite distinct from the instrinsic probabilities of the ten possible v([M]) curves for a 2:2 function (with alpha(0) not equal0). The computer studies reported lead to several conclusions about the probability of modifiers leading to inhibition or activation or causing changes in v([S]) curve shapes, and suggest that differentiation between model mechanisms may be facilitated by knowledge of the intrinsic curve-shape probabilities for the appropriate degree rational function and the characteristic way that this is altered by specific mechanisms. It is shown that, although in some instances new curve-shape complexities are possible when schemes are considered that allow for interconversion of ES and EP species, these are highly improbable and, for theoretical purposes, schemes formulated with node compression provide good approximations to the more complicated explicit schemes. By node compression we refer to the procedure whereby enzyme kinetic schemes are simplified by replacing sequences of steps such as ESright harpoon over left harpoonX(1)right harpoon over left harpoonX(2)right harpoon over left harpoon...right harpoon over left harpoonEP... by a single step... ES/EP... that does not formally recognize the existence of the intermediate species. We show that the modifier mechanism studied is one where this process alters the form of the rate equation.