K+ appears to decrease the affinity of the (Na+ + K+)-dependent ATPase (ATP phosphohydrolase, EC 3.6.1.3) for its substrate, Mg2+ - ATP, and Mg2+ - ATP, in turn, appears to decrease the affinity of the enzyme for K+. These antagonisms have been investigated in terms of a quantitative model defining the magnitude of the effects as well as identifying the class of K+ sites on the enzyme involved. K+ increased the apparent Km for Mg2+ - ATP, an effect that was antagonized competitively by Na+. The data can be fitted to a model in which Mg2+ - ATP binding is prevented by occupancy of alpha-sites on the enzyme by K+ (i.e. sites of moderate affinity for K+ accessible on the "free" non-phosphorylated enzyme, in situ on the external membrane surface). By contrast, occupancy of these alpha-sites by Na+ has no effect on Mg2+ - ATP binding to the enzyme. On the other hand, Mg2+ - ATP decreased the apparent affinity of the enzyme for K+ at the alpha-sites, in terms of (i) the KD for K+ measured by K+-accelerated inactivation of the enzyme by F-, and (ii) the concentration of K+ for half-maximal activation of the K+-dependent phosphatase reaction (which reflects the terminal hydrolytic steps of the overall ATPase reaction). These data fit the same quantitative model. Although this formulation does not support schemes in which ATP binding effects the release of transported K+ from discharge sites, it is consistent with observations that K+ can inhibit the enzyme at low substrate concentrations, and that Li+, which has poor efficacy when occupying these alpha-sites, can stimulate enzymatic activity at high K+ concentrations by displacing the inhibitory K+.