The general modifier ("allosteric") unireactant enzyme mechanism: redundant conditions for reduction of the steady state velocity equation to one that is first degree in substrate and effector.

The general unireactant modifier mechanism in the absence of product can be described by the following linked reactions: E + S k1 in equilibrium k-1 ES k3----E + P; E + I k5 in equilibrium k-5 EI; EI + S k2 in equilibrium k-2 ESI k4----EI + P; and ES + I k6 in equilibrium k-6 ESI where S is a substrate and I is an effector. A full steady state treatment yields a velocity equation that is second degree in both [S] and [I]. Two different conditions (or assumptions) permit reduction of the velocity equation to one that is first degree in [S] and [I]. These are (a) that k-2k3 = k-1k4 (Frieden, C., J. Biol. Chem. 239, pp. 3522-3531, (1964)) and (b) that the I-binding reactions are at equilibrium (Reinhart, G. D., Arch. Biochem. Biophys. 224, pp. 389-401 (1983)). It is shown that each condition gives rise to the other (i.e., if the I-binding reactions are at equilibrium, then k-2k3 must equal k-1k4 and vice-versa). If one assumes equilibrium for the I-binding steps, the velocity equation derived by the method of Cha (J. Biol. Chem. 243, pp. 820-825 (1968)) is apparently second degree in [I] (Segel, I. H., Enzyme Kinetics, p. 838, Wiley-Interscience (1975)), but reduces to a first degree equation when the relationship derived by Frieden is inserted. If one starts by assuming a single equilibrium condition for I binding, e.g., k-5[EI] = k5[E][I] or k-6[ESI] = k6[ES][I], then a traditional algebraic manipulation of the remaining steady state equations provides first degree expressions for the concentrations of all enzyme species and also discloses the Frieden relationship.