Torque-speed relationship of the flagellar rotary motor of Escherichia coli

The output of a rotary motor is characterized by its torque and speed. We measured the torque-speed relationship of the flagellar rotary motor of Escherichia coli by a new method. Small latex spheres were attached to flagellar stubs on cells fixed to the surface of a glass slide. The angular speeds of the spheres were monitored in a weak optical trap by back-focal-plane interferometry in solutions containing different concentrations of the viscous agent Ficoll. Plots of relative torque (viscosity x speed) versus speed were obtained over a wide dynamic range (up to speeds of approximately 300 Hz) at three different temperatures, 22.7, 17.7, and 15.8 degrees C. Results obtained earlier by electrorotation (, Biophys. J. 65:2201-2216) were confirmed. The motor operates in two dynamic regimes. At 23 degrees C, the torque is approximately constant up to a knee speed of nearly 200 Hz, and then it falls rapidly with speed to a zero-torque speed of approximately 350 Hz. In the low-speed regime, torque is insensitive to changes in temperature. In the high-speed regime, it decreases markedly at lower temperature. These results are consistent with models in which torque is generated by a powerstroke mechanism (, Biophys. J. 76:580-587).     

Study of the torque of the bacterial flagellar motor using a rotating electric field.

Bacterial flagella are driven by a rotary motor that is energized by an electrochemical ion gradient across the cell membrane. In this study the torque generated by the flagellar motor was measured in tethered cells of a smooth-swimming Escherichia coli strain by using rotating electric fields to determine the relationship between the torque and speed over a wide range. By measuring the electric current applied to the sample cell and combining the data obtained at different viscosities, the torque of the flagellar motor was estimated up to 55 Hz, and also at negative rotation rates. By this method we have found that the torque of the flagellar motor linearly decreases with rotation rate from negative through positive rate of rotation. In addition, the dependence of torque upon temperature was also investigated. We showed that torque at the high speeds encountered in swimming cells had a much steeper dependence on temperature that at the low speeds encountered in tethered cells. From these results, the activation energy of the proton transfer reaction in the torque-generating unit was calculated to be about 7.0 x 10(-20) J.     

Control of speed modulation (chemokinesis) in the unidirectional rotary motor of Sinorhizobium meliloti

Swimming cells of Sinorhizobium meliloti are driven by flagella that rotate only clockwise. They can modulate rotary speed (achieve chemokinesis) and reorient the swimming path by slowing flagellar rotation. The flagellar motor is energized by proton motive force, and torque is generated by electrostatic interactions at the rotor/stator (FliG/MotA-MotB) interface. Like the Escherichia coli flagellar motor that switches between counterclockwise and clockwise rotation, the S. meliloti rotary motor depends on electrostatic interactions between conserved charged residues, namely, Arg294 and Glu302 (FliG) and Arg90, Glu98 and Glu150 (MotA). Unlike in E. coli, however, Glu150 is essential for torque generation, whereas residues Arg90 and Glu98 are crucial for the chemotaxis-controlled variation of rotary speed. Substitutions of either Arg90 or Glu98 by charge-neutralizing residues or even by their smaller, charge-maintaining isologues, lysine and aspartate, resulted in top-speed flagellar rotation and decreased potential to slow down in response to tactic signalling (chemokinesis-defective mutants). The data infer a novel mechanism of flagellar speed control by electrostatic forces acting at the rotor/stator interface. These features have been integrated into a working model of the speed-modulating rotary motor.     

The stall torque of the bacterial flagellar motor.

The bacterial flagellar motor couples the flow of protons across the cytoplasmic membrane to the rotation of a helical flagellar filament. Using tethered cells, we have measured the stall torque required to block this rotation and compared it with the torque of the running motor over a wide range of values of proton-motive force and pH. The stall torque and the running torque vary identically: both appear to saturate at large values of the proton-motive force and both decrease at low or high pH. This suggests that up to speeds of approximately 5 Hz the operation of the motor is not limited by the mobility of its internal components or the rates of proton transfer reactions coupled to flagellar rotation.     

Constraints on models for the flagellar rotary motor

Most bacteria that swim are propelled by flagellar filaments, each driven at its base by a rotary motor embedded in the cell wall and cytoplasmic membrane. A motor is about 45 nm in diameter and made up of about 20 different kinds of parts. It is assembled from the inside out. It is powered by a proton (or in some species, a sodium-ion) flux. It steps at least 400 times per revolution. At low speeds and high torques, about 1000 protons are required per revolution, speed is proportional to protonmotive force, and torque varies little with temperature or hydrogen isotope. At high speeds and low torques, torque increases with temperature and is sensitive to hydrogen isotope. At room temperature, torque varies remarkably little with speed from about -100 Hz (the present limit of measurement) to about 200 Hz, and then it declines rapidly reaching zero at about 300 Hz. These are facts that motor models should explain. None of the existing models for the flagellar rotary motor completely do so.     

Effect of Intracellular pH on the Torque-Speed Relationship of Bacterial Proton-Driven Flagellar Motor

Bacterial flagella responsible for motility are driven by rotary motors powered by the electrochemical potential difference of specific ions across the cytoplasmic membrane. The stator of proton-driven flagellar motor converts proton influx into mechanical work. However, the energy conversion mechanism remains unclear. Here, we show that the motor is sensitive to intracellular proton concentration for high-speed rotation at low load, which was considerably impaired by lowering intracellular pH, while zero-speed torque was not affected. The change in extracellular pH did not show any effect. These results suggest that a high intracellular proton concentration decreases the rate of proton translocation and therefore that of the mechanochemical reaction cycle of the motor but not the actual torque generation step within the cycle by the stator-rotor interactions.     

Quantitative measurements of proton motive force and motility in Bacillus subtilis.

The protein motive force of metabolizing Bacillus subtilis cells was only slightly affected by changes in the external pH between 5 and 8, although the electrical component and the chemical component of the proton motive force contributed differently at different external pH. The electrical component of the proton motive force was very small at pH 5, and the chemical component was almost negligible at pH 7.5. At external pH values between 6 and 7.7, swimming speed of the cells stayed constant. Thus, either the electrical component or the chemical component of the proton motive force could drive the flagellar motor. When the proton motive force of valinomycin-treated cells was quantitatively decreased by increasing the external K+ concentration, the swimming speed of the cells changed in a unique way: the swimming speed was not affected until about--100 mV, then decreased linearly with further decrease in the proton motive force, and was almost zero at about--30 mV. The rotation rate of a flagellum, measured by a tethered cell, showed essentially the same characteristics. Thus, there are a threshold proton motive force and a saturating proton motive force for the rotation of the B. subtilis flagellar motor.     

Viscoelastic dynamics of actin filaments coupled to rotary F-ATPase: angular torque profile of the enzyme

ATP synthase (F(O)F(1)) operates as two rotary motor/generators coupled by a common shaft. Both portions, F(1) and F(O), are rotary steppers. Their symmetries are mismatched (C(3) versus C(10-14)). We used the curvature of fluorescent actin filaments, attached to the rotating c-ring, as a spring balance (flexural rigidity of 8. 10(-26) Nm(2)) to gauge the angular profile of the output torque at F(O) during ATP hydrolysis by F(1) (see theoretical companion article (. Biophys. J. 81:1234-1244.)). The large average output torque (50 +/- 6 pN. nm) proved the absence of any slip. Variations of the torque were small, and the output free energy of the loaded enzyme decayed almost linearly over the angular reaction coordinate. Considering the threefold stepping and high activation barrier of the driving motor proper, the rather constant output torque implied a soft elastic power transmission between F(1) and F(O). It is considered as essential, not only for the robust operation of this ubiquitous enzyme under symmetry mismatch, but also for a high turnover rate of the two counteracting and stepping motor/generators.     

Dynamics of a tightly coupled mechanism for flagellar rotation. Bacterial motility, chemiosmotic coupling, protonmotive force.

The bacterial flagellar motor is a molecular engine that couples the flow of protons across the cytoplasmic membrane to rotation of the flagellar filament. We analyze the steady-state behavior of an explicit mechanical model in which a fixed number of protons carries the filament through one revolution. Predictions of this model are compared with experimentally determined relationships between protonmotive force, proton flux, torque, and speed. All such tightly coupled mechanisms produce the same torque when the motor is stalled but vary greatly in their behavior at high speed. The speed at zero load predicted by our model is limited by the rates of association and dissociation of protons at binding sites on the rotor and by the mobility of force generators containing transmembrane channels that interact with these sites. Our analysis suggests that more could be learned about the motor if it were driven by an externally applied torque backwards (at negative speed) or forwards at speeds greater than the zero-load speed.     

Effect of intracellular pH on rotational speed of bacterial flagellar motors

Weak acids such as acetate and benzoate, which partially collapse the transmembrane proton gradient, not only mediate pH taxis but also impair the motility of Escherichia coli and Salmonella at an external pH of 5.5. In this study, we examined in more detail the effect of weak acids on motility at various external pH values. A change of external pH over the range 5.0 to 7.8 hardly affected the swimming speed of E. coli cells in the absence of 34 mM potassium acetate. In contrast, the cells decreased their swimming speed significantly as external pH was shifted from pH 7.0 to 5.0 in the presence of 34 mM acetate. The total proton motive force of E. coli cells was not changed greatly by the presence of acetate. We measured the rotational rate of tethered E. coli cells as a function of external pH. Rotational speed decreased rapidly as the external pH was decreased, and at pH 5.0, the motor stopped completely. When the external pH was returned to 7.0, the motor restarted rotating at almost its original level, indicating that high intracellular proton (H+) concentration does not irreversibly abolish flagellar motor function. Both the swimming speeds and rotation rates of tethered cells of Salmonella also decreased considerably when the external pH was shifted from pH 7.0 to 5.5 in the presence of 20 mM benzoate. We propose that the increase in the intracellular proton concentration interferes with the release of protons from the torque-generating units, resulting in slowing or stopping of the motors.     

Theories of rotary motors

The bacterial flagellar motor and the ATP-hydrolysing F1 portion of the F1Fo-ATPase are known to be rotary motors, and it seems highly probable that the H+-translocating Fo portion rotates too. The energy source in the case of Fo and the flagellar motor is the flow of ions, either H+ (protons) or Na+, down an electrochemical gradient across a membrane. The fact that ions flow in a particular direction through a well-defined structure in these motors invites the possibility of a type of mechanism based on geometric constraints between the rotor position and the paths of ions flowing through the motor. The two best-studied examples of such a mechanism are the ''turnstile'' model of Khan and Berg and the ''proton turbine'' model of Läuger or Berry. Models such as these are typically represented by a small number of kinetic states and certain allowed transitions between them. This allows the calculation of predictions of motor behaviour and establishes a dialogue between models and experimental results. In the near future structural data and observations of single-molecule events should help to determine the nature of the mechanism of rotary motors, while motor models must be developed that can adequately explain the measured relationships between torque and speed in the flagellar motor.     

Quantification of flagellar motor stator dynamics through in vivo proton‐motive force control

The bacterial flagellar motor, one of the few rotary motors in nature, produces torque to drive the flagellar filament by ion translocation through membrane‐bound stator complexes. We used the light‐driven proton pump proteorhodopsin (pR) to control the proton‐motive force (PMF) in vivo by illumination. pR excitation was shown to be sufficient to replace native PMF generation, and when excited in cells with intact native PMF generation systems increased motor speed beyond the physiological norm. We characterized the effects of rapid in vivo PMF changes on the flagellar motor. Transient PMF disruption events from loss of illumination caused motors to stop, with rapid recovery of their previous rotation rate after return of illumination. However, extended periods of PMF loss led to stepwise increases in rotation rate upon PMF return as stators returned to the motor. The rate constant for stator binding to a putative single binding site on the motor was calculated to be 0.06 s^?1. Using GFP‐tagged MotB stator proteins, we found that transient PMF disruption leads to reversible stator diffusion away from the flagellar motor, showing that PMF presence is necessary for continued motor integrity, and calculated a stator dissociation rate of 0.038 s^?1.     

Function of Proline Residues of MotA in Torque Generation by the Flagellar Motor of Escherichia coli

Bacterial flagellar motors obtain energy for rotation from the membrane gradient of protons or, in some species, sodium ions. The molecular mechanism of flagellar rotation is not understood. MotA and MotB are integral membrane proteins that function in proton conduction and are believed to form the stator of the motor. Previous mutational studies identified two conserved proline residues in MotA (Pro 173 and Pro 222 in the protein from Escherichia coli) and a conserved aspartic acid residue in MotB (Asp 32) that are important for function. Asp 32 of MotB probably forms part of the proton path through the motor. To learn more about the roles of the conserved proline residues of MotA, we examined motor function in Pro 173 and Pro 222 mutants, making measurements of torque at high load, speed at low and intermediate loads, and solvent-isotope effects (D2O versus H2O). Proton conduction by wild-type and mutant MotA-MotB channels was also assayed, by a growth defect that occurs upon overexpression. Several different mutations of Pro 173 reduced the torque of the motor under high load, and a few prevented motor rotation but still allowed proton flow through the MotA-MotB channels. These and other properties of the mutants suggest that Pro 173 has a pivotal role in coupling proton flow to motor rotation and is positioned in the channel near Asp 32 of MotB. Replacements of Pro 222 abolished function in all assays and were strongly dominant. Certain Pro 222 mutant proteins prevented swimming almost completely when expressed at moderate levels in wild-type cells. This dominance might be caused by rotor-stator jamming, because it was weaker when FliG carried a mutation believed to increase rotor-stator clearance. We propose a mechanism for torque generation, in which specific functions are suggested for the proline residues of MotA and Asp32 of MotB.     

Thermal and Solvent-Isotope Effects on the Flagellar Rotary Motor near Zero Load

In Escherichia coli, the behavior of the flagellar rotary motor near zero load can be studied by scattering light from nanogold spheres attached to proximal hooks of cells lacking flagellar filaments. We used this method to monitor changes in speed when cells were subjected to changes in temperature or shifted from a medium made with H₂O to one made with D₂O. In H₂O, the speed increased with temperature in a near-exponential manner, with an activation enthalpy of 52 ± 4 kJ/mol (12.0 ± 1.0 kcal/mol). In D₂O, the speed increased in a similar manner, with an activation enthalpy of 50 ± 4 kJ/mol. The speed in H₂O was higher than that in D₂O by a factor of 1.53 ± 0.14. We performed comparison studies of variations in temperature and solvent isotope, using motors operating at high loads. The variations were small, consistent with previous observations. The implications of these results for proton translocation are discussed.     

Isotope and thermal effects in chemiosmotic coupling to the flagellar motor of Streptococcus

The torque generated by the flagellar motor of Streptococcus strain V4051 has been determined from rates of rotation of cells tethered by a single flagellum in media of different isotopic composition and temperature. Starved cells were energized artificially with either a potassium diffusion potential or a pH gradient. The torque increased linearly with protonmotive force. Identical results were obtained in media made with D2O or H2O; there was no solvent isotope effect. At a fixed protonmotive force, the torque was approximately constant over a temperature range of 4 degrees -38 degrees C. In cells chemotactically inert to changes in cytoplasmic pH, the motor turned counterclockwise when protons moved inward and clockwise when they moved outward. We conclude that the motor is a reversible engine driven by simple acid-base dissociation. A detailed model is discussed.     

A model for bacterial flagellar motor: free energy transduction and self-organization of rotational motion

A bacterial flagellar motor is an energy transducing molecular machine which shows some attractive characteristics. First, this motor is driven by a protonmotive force (PMF) across the membrane, two components of which, electric potential delta psi and chemical potential -(2.3RT/F)delta pH, are equivalently transduced to the mechanical work of the motor rotation. Second, a PMF threshold for rotation is observed. Third, this motor can rotate reversibly either counterclockwise (CCW) or clockwise (CW) at almost the same speed. To clarify the osmomechanical coupling of this motor, these characteristics must be explained consistently at the molecular level. In this paper, in order to allow quantitative analyses of the above characteristics, a theoretical model of a bacterial flagellar motor is constructed assuming that the torque generating sites are electrodes which can be charged by protons and that the electrostatic interaction between the electrodes generates the rotation torque. Electrode reaction reasonably derives the equivalence of delta psi and -(2.3RT/F)delta pH. In this model, rates of charging and discharging of protons are influenced by the motor rotation rate, so that the torque generating sites co-operatively work through the motor rotation. We named this kind of co-operativity among them "dynamic co-operativity" in torque generation. This co-operativity causes autocatalytic generation of motor torque and the existence of the rotation threshold. In this model, the appearance of the stable rotational states can be described by phase transition caused by the dynamic co-operativity among torque generating sites. According to this model, the flagellar motor has two stable rotational states corresponding to CCW and CW, which show the same torques. The motor selects one direction from them to rotate, and that is self-organization of rotational motion. Interpretation of the transition between the two stable rotational states as the chemotactic reversals of the flagellar motor is also discussed.     

Rapid changes in flagellar rotation induced by external electric pulses.

The bacterial flagellar motor is the only molecular rotary machine found in living organisms, converting the protonmotive force, i.e., the membrane voltage and proton gradients across the cell membrane, into the mechanical force of rotation (torque). We have developed a method for holding a bacterial cell at the tip of a glass micropipette and applying electric pulses through the micropipette. This method has enabled us to observe the dynamical responses of flagellar rotation to electric pulses that change the membrane voltage transiently and repeatedly. We have observed that acceleration and deceleration of motor rotation are induced by application of these electric pulses. The change in the rotation rate occurred within 5 ms after pulse application.     

Steps in the Bacterial Flagellar Motor

The bacterial flagellar motor is a highly efficient rotary machine used by many bacteria to propel themselves. It has recently been shown that at low speeds its rotation proceeds in steps. Here we propose a simple physical model, based on the storage of energy in protein springs, that accounts for this stepping behavior as a random walk in a tilted corrugated potential that combines torque and contact forces. We argue that the absolute angular position of the rotor is crucial for understanding step properties and show this hypothesis to be consistent with the available data, in particular the observation that backward steps are smaller on average than forward steps. We also predict a sublinear speed versus torque relationship for fixed load at low torque, and a peak in rotor diffusion as a function of torque. Our model provides a comprehensive framework for understanding and analyzing stepping behavior in the bacterial flagellar motor and proposes novel, testable predictions. More broadly, the storage of energy in protein springs by the flagellar motor may provide useful general insights into the design of highly efficient molecular machines.     

Rotary protein motors

Three protein motors have been unambiguously identified as rotary engines: the bacterial flagellar motor and the two motors that constitute ATP synthase (F(0)F(1) ATPase). Of these, the bacterial flagellar motor and F(0) motors derive their energy from a transmembrane ion-motive force, whereas the F(1) motor is driven by ATP hydrolysis. Here, we review the current understanding of how these protein motors convert their energy supply into a rotary torque.     

Analysis of bacterial flagellar rotation

Bacterial flagella have rotary motors at their base; embedded in the cytoplasmic membrane and powered by transmembrane ion gradients instead of ATP. Assays have been developed to measure the torque output of individual motors over a wide regime of load, to correlate the energizing proton flux with rotation speed and relate through genetic analysis motor structure to function. These assays promise substantial advances in understanding mechanochemical coupling in these motors. Here, I summarize the present status of our understanding of energy transduction in bacterial flagella and compare this with the case for muscle.