Protein Turbines 1: The Bacterial Flagellar Motor

The bacterial flagellar motor is driven by a flux of ions between the cytoplasm and the periplasmic lumen. Here we show how an electrostatic mechanism can convert this ion flux into a rotary torque. We demonstrate that, with reasonable parameters, the model can reproduce many of the experimental measurements. INTRODUCTION The bacterial flagellar motor (BFM) is a rotary engine that derives its energy from the electrochemical gradient established between the cell cytoplasm and the periplasmic lumen. This gradient drives ion flow through the motor, which is transduced into a rotary torque. When the motor rotates counterclockwise, the helical flagella propagate a wave away from the cell body. This causes adjacent flagella to intertwine and form a propulsive corkscrew that drives the bacterium through the fluid medium at speeds of up to 25 p,rnls (Anderson, 1975; Childress, 1981). When the motors reverse their direction of rotation, the individual flagella fly apart, causing the bacterium to tumble (Macnab, 1977). When the flagella reverse again to their swimming mode, the bacterium's direction has been randomly reoriented. Normally, reversals occur spontaneously around 1/s. However, a bacterium can bias its random walk up concentration gradients of chemoattractants by adjusting its tumbling frequency (Berg, 1983). Many models have been proposed for this molecular engine (reviewed by Latiger, 1990). Here we propose a novel mechanism for transducing an electrochemical potential gradient into a mechanical torque, compare its performance with experimental observations, and contrast it with earlier models. PROPERTIES OF THE BACTERIAL FLAGELLAR MOTOR Although a complete molecular structure of the flagellar motor is not yet available, microscopic, biochemical, and genetic studies have sketched a rough geometrical picture of motor assembly (Katayama et al., 1996; Macnab, 1996; Schuster and Khan, 1994; Sharp et al., 1995a). Fig. 1 is a schematic diagram showing the major components and their relative sizes (Francis et al., 1994; Schuster and Khan, 1994). The energy-transducing elements consist of the CReceived for publication 5 February 1997 and in final form 28 Apri/1997. Address reprint requests to Dr. George Oster, Department of ESPN, University of California, 201 Wellman Hall, Berkeley, CA 94720-3112. Tel.: 510-642-5277; Fax: 510-642-5277; E-mail: [email protected]. © 1997 by the Biophysical Society 0006-3495/97 tosno3tl9 $2.oo ring, believed to constitute the rotor, and the 8-16 MotA/ MotB complexes, believed to constitute the torque-generating stator elements. The rotor radius is -20-25 nm and probably carries charges around its periphery (Macnab, 1996). The MotA and MotB proteins consist of four and one transmembrane a-helices, respectively, and are thought to be proton channels. The site of torque generation appears to be the cytoplasmic domain of MotA, which also contains a-helical domains (Sharp et al., 1995a, b; Tang et al., 1996). The motor is driven by the electrochemical potential gradient between the intermembrane space and the cytoplasm. In most bacteria (e.g., Escherichia coli) this gradient is set up by proton pumps; however, protonation may not play an essential role in the transduction, because other flagellar motors are driven by sodium. At low speeds, the motor torque appears to be roughly proportional to the "proton-motive force," defined as PMF = dt/J (2.3 RT/ F)dpH, where F is the Faraday constant, R the gas constant, T the absolute temperature, dpH the pH difference between the lumen and cytoplasm, and dt/J the transmembrane potential (50-200 m V). In terms of motor performance, a dpH of 2 is roughly equivalent to a membrane potential difference of 120 mV. Kinetic studies suggest that, at normal swimming conditions of -100 Hz, -1200 protons pass through the motor per revolution, or -10 protons/s. A variety of techniques have been applied to determine the mechanical behavior of the motor (Berg, 1995; Berg and Turner, 1993; Berry and Berg, 1996; Meister and Berg, 1987). These experiments have measured the rotor speed as a function of an externally applied torque. From the loadvelocity curve, the motor torque versus velocity curve can be directly computed. Fig. 2 describes the major features of these relationships. In the absence of an external torque, a tethered bacterium rotates at w0 = 10 Hz. The rotational drag coefficient is '= 5-10 pN-s-nm. (The rotational drag coefficient of a cylinder of length L and radius r spinning about its base in a fluid of viscosity TJ is ' = 47TTJL?. For a bacterium 3 p,m long with a 0.5-p,m radius in water, ' = 5-10 pN-nm-s.) Thus the torque developed by the motor is -300-600 pN-nm. For 10 stator elements, each stator develops -30-60 pN-nm, and the motor speed increases linearly with the number of active stators. In addition to measuring the mean velocity as a function of load, important information can be gleaned from studies 704 Biophysical Journal Volume 73 August 1997 .. Cytoplasmic Membrane Cytoplasm OO:::x::ol; FIGURE I Motor structure. The flagellar rod passes through a series of rings. The L (lipid) ring, P (peptidoglycan) ring, and MS (membranesupramembrane) rings are thought to act as structural bushings, anchoring the motor to the cell wall. The C (cytoplasmic) ring is thought to be the motor rotor, and consists of the three proteins FliG, FliM, and FliN. The stator elements consist of 8-16 MotA/MotB complexes. Each MotA consists of four a-helices that have a large cytoplasmic domain. Each MotB is a single a-helix that is anchored to the peptidoglycan layer. Together, the five a-helices of the MotA/MotB complex constitute a channel that conducts protons from the lumen to the cytoplasm. on the statistical variability of the motor's motion under different circumstances. In particular, first-passage time statistics (e.g., time to rotate a given number of revolutions) at low speeds suggest that the motor operates nearly like a "stepper"; that is, a fixed step size is taken at random times (Samuel and Berg, 1995). Like other statistical measures, these experiments provide information that is independent of mean value measurements and can be used to make independent estimates of motor characteristics (Peskin and Oster, 1995; Svoboda et al., 1994). THE ION TURBINE: FIXED STATOR MODEL The operating principle of the ion turbine is quite simple, and can be understood as follows. Consider two cylindrical surfaces in close apposition, and constrained so that their only degree of freedom is for the inner cylinder to rotate with respect to the fixed outer cylinder. On each surface place an arbitrary distribution of point charges. In general, the electrostatic potential field defined by the charge distributions will have many equilibria, and in the absence of any constraints the inner surface will rotate by some angle until mechanical equilibrium is achieved in some local minimum. Suppose a single positive ion is now placed on one of the negative charges, thus neutralizing it. Then the system will no longer be in mechanical equilibrium and the inner cylinder will rotate to a new local equilibrium position. To tum this system into a rotary motor, the fixed charges must be located in such a way that the successive equilibria cause the cylinders to rotate in one direction. There is no unique solution to this geometrical problem; however, we shall construct one plausible charge geometry based on certain structural features of the E. coli flagellar motor. In the bacterial flagellar motor, the proton path is somewhat ambiguous. It is generally thought to follow the a-helices of the MotA/MotB channel complex. But the torquegenerating regions appear to be the cytoplasmic domain of MotA, which abuts the FliG component of the rotor. Blair and co-workers have isolated acidic and basic residues on both rotor (FliG) and stator (MotA and MotB) that appear to be necessary for torque generation (Lloyd and Blair, 1997; Tang et al., 1996). Here we will illustrate the principle of torque generation by separating the acidic residues that constitute the protonation sites and the basic residues that constitute the gating sites on the stator and rotor, respectively. Fig. 3 shows a schematic of the model. We will assume that the rotor carries positive charges arranged in helical rows. Each row consists of four charges spaced in a 2 + 2 configuration, shown in Fig. 3. Each of the eight stators carries two negative charges located vertically between the two positive rotor charges. We emphasize that many charge configurations are possible, and it is not necessary to segregate positive charges on the rotor and negative charges on the stator, although there is some experimental support for this choice. What is necessary is the tilt of the rotor charges with respect to the stator charges, although it is equally effective to have the rotor charges vertical and the stator charges tilted. We have chosen the simple charge distribution shown in Fig. 3 for illustrative purposes, and we shall speak of the mobile ions as protons, but the same argument applies to the sodium-driven motor. Initially, we will assume that the stator is rigid and immovable; later we shall relax this assumption. When a positive rotor charge is not blocking their entrance, protons can enter the stator from the top (peri plasmic space) to associate with the top negative charge on the stator. When the middle rotor charges are not blocking the proton's path, it can jump to the lower stator charge, and when the bottom charge rotates out of the way, the proton can hop out to the cytoplasm. The ability of the proton to proceed through the stator depends on the rotor's angular position, e, for this determines the electrostatic landscape the proton must traverse. This landscape is specified by the charge distribution on the rotor and on the stator. Denote by V( e, z) the electrostatic potential field set up by the distribution of fixed charges on the rotor and stator. Fig. 5 a shows the electrostatic surface experienced by a proton as it passes through the stator. (In computing the electrostatic field, we have neglected the rotor curvature.) The force exerted on the rotor by a stator depends on the occupancy of the stator charged sites. We will model the stator sites as Coulomb potential wells, the depth of which we estimate from the pKa of glutamic and aspartic acid residues that could comprise the stator sites. The potential Elston and Oster Protein Turbines 705