A two-state model of an enzyme with an allosteric regulatory site capable of metabolizing the regulatory ligand. Simplified mathematical treatments of transient and steady state kinetics of an activator and its competitive inhibition as applied to adenylyl cyclases.

The simplest model that can account for the existence of a system in which allosteric activation of an enzyme results in modification of its V,,, (V-system) is a twostate model where the enzyme exists in either of two preferred conformations: one termed zero (O) state that predominates in the absence of ligand and the other termed prime (‘) state that predominates in the presence of ligand. Analytical solutions for the time-dependent changes occurring after addition of one allosteric activator or of two competing allosteric activators with differing efficacies in stimulating the system were calculated by assuming that all but the active, ligand-occupied species of the enzyme are in rapid equilibrium. An analytical solution to an example in which rapid equilibrium was assumed only to apply to the equilibrium between the two states of the ligand-free forms of the enzyme is also given. The solutions presented include the option that the allosteric ligand binding site be capable of also metabolizing the ligand in a manner independent of microscopic reversibility principles and thermodynamic equilibrium laws that govern all other reaction rates of the two-state model. Simulations are presented that indicate that the presence of a ligand-metabolizing site results in both a lowering in the fractional activation induced by the ligand that cannot be overcome by excess ligand and in a lowering of the apparent K, with which the ligand activates the two-state enzyme. Simulations that analyze time transients in the absence of the ligand-metabolizing activity with an ineffective ligand (having high apparent K, and giving low fractional activation at steady state) were not different from those obtained with an effective ligand acting on a system showing at steady state an equally low fractional activity and equally high apparent K, due to the presence of a ligand-metabolizing activity on the enzyme. We discuss limitations and consequences of external regulation of equilibria of the two-state model that take into consideration so-called microscopic reversibility constraints.