Flow-force relationships for a six-state proton pump model: intrinsic uncoupling, kinetic equivalence of input and output forces, and domain of approximate linearity.

General flow-force relations have been determined, by the Hill diagram method, for a six-state proton pump model with and without intrinsic uncoupling (molecular slipping). A computer-aided analysis of the resulting sigmoidal flow-force curves has been performed by using a set of physically meaningful rate constants. It is shown that gating effects and apparent irreversibility can arise from sigmoidicity. The regions of approximate linearity in the vicinity of inflection points, which may be far from equilibrium, have been examined with a view to characterization in terms of linear phenomenological equations, with due regard to the problems of kinetic equivalence of the forces and symmetry. The determination of thermodynamic parameters such as the degree of coupling, the phenomenological stoichiometry, and the efficiency in these regions is discussed, and their meaning is analyzed in relation to the parameters characterizing the Onsager domain close to equilibrium. The application of the phenomenological equations of near-equilibrium nonequilibrium thermodynamics to such regions is at best a simplification to be treated with great caution. A knowledge of the distance from equilibrium of the flow-controlling ranges of the forces (i.e., the ranges of approximate linearity) turns out to be crucial for the interpretation of thermodynamic parameters determined by manipulating one of the forces while the other remains constant, as well as for the interpretation of measurements of force ratios at static head. The latter approaches can give good estimates of the magnitude of the mechanistic stoichiometry and of the constant force if the pumps are highly coupled and are operating not far from equilibrium. The force-flow relationships are shown to be modified by intrinsic uncoupling, reflecting the regulatory influence of the forces on the extent and nature of the slip. Thus reaction slip increases, for example, as the force against which the proton pump operates increases. The possible physiological significance of regulated intrinsic uncoupling is discussed.