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.Ion-motive force (IMF) comprises electrical and chemical transmembrane potentials and is defined aswhere
Vm is the membrane voltage and
Δμ = kBT ln(
Cin/
Cout) and
q are the transmembrane chemical potential difference and charge of the ions, respectively, with
Cin and
Cout 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
1F
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 () 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).
Open in a separate windowBacterial flagellar motor speed measurement. (
A) Schematic of the bacterial flagellar motor and the polystyrene bead rotation assay. Stator proteins couple ion flux to rotation of the rotor. The hook is a universal joint that connects external flagellar filaments to the rotor. Motor rotation is observed via back-focal-plane interferometry of a polystyrene bead attached to the truncated filament. (
B) A typical 10-ms
X-Y trace of a 200-nm polystyrene bead attached to a fully energized motor, sampled at 10 kHz. (
C) The power spectrum of (
X +
iY) from
B shows a clear peak at 660 Hz. (
D) The stable rotational speed of the same motor for 10 s.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. Fully energized motors with a full complement of stators, driven by H
+ (
10) or Na
+ (
11,
12), show the same characteristic torque–speed curve. There is a plateau of nearly constant torque from stall up to a certain limiting speed, and at higher speeds torque falls more rapidly to zero. The junction between these two domains is relatively sharp and has been called the “knee”. In the wild-type
E. coli motor, the knee is at ∼200 Hz and the zero-torque speed is ∼300 Hz (
17). Na
+-driven motors are faster: In
V. alginolyticus and the chimeric motor in
E. coli the knee speeds are ∼450 Hz and the zero-torque speeds are ∼700 Hz and ∼900 Hz, respectively (
11,
12). Torque–speed curves have also been measured for fully energized motors with reduced numbers of stators. The torque generated by H
+-driven motors with
N stators appears to be simply
N times the torque generated by a single stator at the same speed (
18), a result that has recently been confirmed by measurements close to the zero-torque limit (
17,
19). Previously measured torque–speed curves for single-stator Na
+-driven motors do not extend beyond the knee due to a lack of data points at extremely low load (
12). Various models of the mechanism of torque generation in the BFM have been proposed and tested by their ability to predict the torque–speed curve (
16,
20–
24). The torque plateau is believed to represent a regime where motion of the attached load rather than internal processes in the motor’s mechano-electrochemical cycle is rate limiting, and the system operates with high efficiency close to thermodynamic equilibrium. Thus, the plateau torque is close to the stall torque, and the work done against viscous drag is nearly equal to the free energy of driving ions that is consumed by the motor. As the load is reduced and the motor speeds up, the knee represents the point at which internal processes become rate limiting, and the zero-torque speed is a measure of the limiting rate of the cycle in the absence of load.The recent development of a Na
+-driven chimeric flagellar motor in
E. coli (
25) opened new possibilities for investigation of the mechano-electrochemical cycle of the BFM (
26). The PMF that drives the wild-type motor of
E. coli is the primary form of free energy in the cell and is consequently tightly regulated and relatively difficult to manipulate. Furthermore, extreme changes in H
+ concentration equate to extremes of pH, which are expected to affect the stability of motor proteins. On the other hand, the SMF in
E. coli is secondary and relatively easy to manipulate, and extremes of [Na
+] are tolerated by the cell. We have recently demonstrated that the electrical and chemical potential components of the SMF in
E. coli can be varied independently via the pH and [Na
+] of the surrounding medium (
27,
28) (
Table S1).In this paper, we measured the torque–speed curves of single-stator Na
+-driven chimeric motors for each of 25 different combinations of membrane voltage and sodium gradient (
27,
28) (
Table S1). We chose single-stator motors, to isolate the properties of the fundamental mechanochemical cycle from the complication of interactions between different stators in the same motor. Our set of torque–speed curves is an order of magnitude bigger than any previous torque–speed dataset and allowed a systematic exploration of the parameter space of a minimal four-state kinetic model of the BFM mechanochemical cycle. The four-state kinetic model derives from the simplest possible representation of a carrier-like ion transport mechanism (
29,
30) and has been extensively used to model the BFM (
31). Rather than choosing a single set of model parameters by educated guesswork, as in all previous attempts to model the flagellar motor, we performed a comprehensive search for sets of values of the 25 model parameters that were consistent with our measured torque–speed curves.
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