The bacterial flagellar motor is a large rotary molecular machine that propels swimming bacteria, powered by a transmembrane electrochemical potential difference. It consists of an ∼50-nm rotor and up to ∼10 independent stators anchored to the cell wall. We measured torque-speed relationships of single-stator motors under 25 different combinations of electrical and chemical potential. All 25 torque-speed curves had the same concave-down shape as fully energized wild-type motors, and each stator passes at least 37 ± 2 ions per revolution. We used the results to explore the 25-dimensional parameter space of generalized kinetic models for the motor mechanism, finding 830 parameter sets consistent with the data. Analysis of these sets showed that the motor mechanism has a "powerstroke" in either ion binding or transit; ion transit is channel-like rather than carrier-like; and the rate-limiting step in the motor cycle is ion binding at low concentration, ion transit, or release at high concentration. where V m is the membrane voltage and Δμ = k B T ln(C in /C out ) and q are the transmembrane chemical potential difference and charge of the ions, respectively, with C in and C out the internal and external ion concentrations. In most species the primary form of biological free energy is the proton-motive force (PMF), the IMF for H + ions (1). Physiological PMF is typically in the range −150 mV to −200 mV, with the inside electrically negative and slightly alkaline relative to the outside. Some organisms use sodium-motive force (SMF) to drive numerous cellular processes, such as bacterial motility (2), ATP synthesis (3), and active membrane transport (4). Arguably the most important process driven by IMF is ATP synthesis, which generates cellular ATP by forced rotation of the F 1 part of F 1 F O ATP-synthase. F 1 is mechanically coupled to and rotated by F O , which like the bacterial flagellar motor (BFM) is an ion-driven rotary motor. Understanding the mechanism of these and other ion-driven molecular machines is a fundamental challenge in cellular energetics and biophysics. The BFM (Fig. 1A) is a rotary molecular machine that propels many species of swimming bacteria. It couples ion flow, for example protons (H + ) in Escherichia coli or sodium ions (Na + ) in Vibrio alginolyticus, to the rotation of extracellular helical flagellar filaments at hundreds of revolutions per second (Hz) (2, 5, 6). Torque is generated by interactions between stator complexes (containing the proteins MotA and MotB in E. coli and PomA and PomB in V. alginolyticus) and the rotor protein FliG (7). In E. coli, each motor can be powered by any number between 1 and at least 11 functionally independent stators (8), which exchange with a membrane-bound pool of "spare" stators on a timescale of minutes (9).The most important biophysical method for studying the torque-generating mechanism of the BFM has been to measure its torque-speed curves. This has been done using varying viscous load (10-14) or external torque (15, 16) to control the speed. Fu...