Proposed model for the flagellar rotary motor with shear stress transmission.

Most bacteria that swim are propelled by flagellar filaments, which are driven by a rotary motor powered by proton flux. The motor consists of the rotor and the stator. The stator consists of about 8 MotA-Mot B complex. There seems to be no definite information about the structure between the rotor and the stator, and it is examined whether the experimental data can be explained based upon the following assumptions. (a) There is viscoelastic medium between the rotor and the stator. (b) MotA-MotB complex has an electric dipole moment and produces shear stress in the electric field by a proton in the channel. Calculation results based upon these assumptions are in good agreement with the following experimental observations. (1) One revolution of the flagellar rotation consists of a constant number of steps. (2) The rotation velocity of the rotor is proportional to the trans-membrane potential difference. (3) When the rotational velocity of a flagellum is changed by adjusting the viscosity of the outer fluid, the torque for the cell to rotate a flagellum is practically constant but sharply decreases when the rotational velocity increases over a critical value. (4) The rotation direction remains the same when the sign of the electrochemical potential gradient is reversed. (5) The cell produces constant torque to rotate the flagellum even when the cell is rotated by externally applied torque. (6) A simple switch mechanism is proposed for chemotaxis.

Flagellar motors of most bacteria are powered by protons moving down an electrochemical gradient in channels in the membrane1. The motor is a complex system as illustrated, e.g., in refs. 2, 3. In the following, however, a simplified system consisting of the stator, rotor and flagellum is considered to discuss the flagellar rotation mechanism. The rotor consists of C ring, S ring and M ring (cf. Fig. 1 of ref. 2). The stator consists of MotA and MotB complexes (cf. Fig. 1 of ref. 3).

In 2000, Berg4 summarized the constraints that any models for the flagellar rotary motor should satisfy (as summarized later in Sect. 1), and noted that none of the existing models completely satisfied the constraints. To discuss the problem, it seems worthwhile to note that the characteristics of the flagellar motor are quite different from those of FoF1-ATP synthase motor (for more details, cf. Appendix 1). The diameter of the flagellar motor is roughly 45 nm compared to about 10 nm of the ATP synthase motor. About 1200 protons are needed for one revolution of the flagellar motor4, while about 10 protons can cause one revolution of the ATP synthase motor. The ATP synthase motor is reversible in the sense that reversed rotation of the rotor causes proton pumping, while there is no evidence of proton pumping for the flagellar rotor. The ATP synthase motor seems a compact, chemically specific protein machine while there seems to be no definite evidence that tight connection of protein molecules exists between the rotor and stator in the flagellar motor. Around 2008, the present authors felt it worthwhile to examine whether there is any possibility that the stator can induce the rotor rotation when there is viscoelastic medium between the rotor and the stator and MotA-MotB complex has an electric dipole moment. The model proposed in the paper5 was constructed by putting weight on the possibility. Calculation results based upon the model well explained experimental observations, satisfying Berg constraints4. Some readers of the paper5, however, commented that it was very difficult to understand the model since there were many mathematical formulae whose physical meanings were only briefly explained. Also the present authors have found several immature discussions in the paper5. In the present review, the way of discussion in the paper5 is largely revised and a few illustrations are added to answer the readers’ comments. Also some of the mathematical symbols are altered.

Experimental data to be explained by the model are summarized in Sect. 1. Basic ideas of the model are explained in Sect. 2. The experimental data are explained by the model in Sect. 3. Summary of the study and discussion are given in Sect. 4. In Appendix 1, characteristics of the flagellar motor are compared with those of the FoF1-ATP synthase motor. Appendix 2 explains the way to calculate the electric field which is produced in the stator by a proton in the channel. In Appendix 3, discussion is done on the requirement on the piezoelectric property of the stator to rotate the rotor.

Experimental data to be explained by the model

Referring to the constraints by Berg4, any model for the flagellar rotary motor should explain the following experimental observations.

One revolution of the flagellar rotation consists of a constant number of steps irrespective of the transmembrane potential difference.

The rotation velocity of the rotor is proportional to the transmembrane potential difference.

When the rotational velocity of a flagellum is changed by adjusting the viscosity of the outer fluid, the torque for the cell to rotate the flagellum is practically constant independent of the velocity, but sharply decreases when the velocity increases over a critical value.

There are observations that rotation direction remains the same when the sign of the electrochemical potential gradient is reversed.

The cell produces constant torque to rotate the flagellum even when the cell is rotated by externally applied torque.

The cell has a switch that reverses the sense of the flagellar rotation for chemotaxis.

The model

Basic assumptions and illustration of the model

According to Braun & Blair6, MotA has two proton channels, and two protons practically simultaneously pass through the channels. For simplicity, however, the following discussion is done as if only one channel exists in the Mot complex and the two protons separately pass the channel. This simplification does not seem to cause any serious error in conclusion since the electric field is additive.

Discussion is done based upon the following two assumptions.

Assumption (i) MotA-MotB complex has a permanent electric dipole moment as a vector sum of dipole moments of the constituent protein molecules.

Assumption (ii) There is viscoelastic medium between the rotor and stator.

Concerning the assumption (i), it should be noted that protein molecules are structurally polar and have permanent electric dipole moments7,8. For instance, the dipole moment of flagellin is 800 Debye8.

Figure 1 is an illustration of the system to be discussed with large modification of sizes and forms of the rotor constituents to make the idea of the model clear. Figure 1(a) shows the system projected on the membrane plane. The rotor is shown by the red circle. The viscoelastic medium is indicated in blue. The part of the MotA-MotB complex between the proton channel and the rotor is important in the following discussion, and is denoted as Mot*, which is represented by the white parallelogram.

Figure 1

Schematic illustration of the system to be discussed. Sizes are not in scale. (a) A part of the flagellar rotary motor projected on the membrane plane with the coordinate system. Mot* is the part of MotA-MotB complex between the channel and the viscoelastic medium. (b) Mot* with the x component of electric field E and the electric dipole moment P and its y component P.

The channel is assumed to be perpendicular to the cytoplasmic membrane. The origin of the coordinate system is set at the center of the channel on the outer surface of the membrane. The z axis is set parallel to the proton channel along the proton movement and thus perpendicular to the sheet. The x axis is directed to the rotation axis of the rotor. The y axis is perpendicular to the x and z axes. The angle between the edge of the parallelogram of Mot* and the x axis is denoted as α which is a parameter to characterize the shape of Mot*. Figure 1(b) shows the inside of Mot*. E is the x component of electric field produced by a proton in the channel and P is the y component of dipole moment P.

A proton at rest in the channel

Figure 2 is to give an idea about the nature of the electric field in Mot* produced by a proton in the channel. The actual structure of Mot* is complex and calculation is done for a membrane consisting of uniform dielectric material by the method outlined in Appendix 2. The ordinate is E, which is the x component of electric field averaged in a volume regarded as the volume of Mot*. The abscissa is zp/dch where zp is the proton position on the z axis in the channel and dch is the channel length.

Figure 2

E as a function of zp /dch. E: volume-averaged x component of the electric field produced in Mot* by a proton resting at zp. dch: the channel length. Calculation is done for a simplified system and the values of E should be understood as approximate values. See Appendix 2 for the way of calculation.

The order of magnitude of E is 108 V/m, which is very strong field. Note that the breakdown field strength of bulk paraffin is (0.8∼1.2) × 107 V/m9. In the following we call the actual x component of the mean electric field produced by proton in Mot* as E for simplicity.

Suppose that the polarization P of Mot* has y component (P) as shown in Figure 1(b), then P tends to be parallel to E and a shear stress X appears in Mot*. The shear stress induces the strain x, which corresponds to the change of angle α in Figure 1(a). The shear stress X is proportional to the vector product of E and P and thus proportional to E and P and is given by where a is a constant.

In the present model, it is assumed that the piezoelectric activity of Mot* is the origin to cause the rotor rotation. Then a question arises how large the piezoelectric activity of Mot* should be to realize the rotor rotation. Discussion on this problem is done in Appendix 3. The conclusion there is that there is a good possibility that Mot* is designed as a shear stress transmitter to rotate the rotor.

A proton moving in the channel and shear stress flow

The transmembrane electrochemical potential ΔΨ is defined by where Ψout and Ψin are the electrochemical potentials in the outer and inner liquids, respectively. It is assumed that the proton moves in the channel against constant frictional resistance, and mean velocity of all protons in channels is denoted as vp. Then where A is a constant. The proton position zp is given as a function of time t:

The field E produced by a proton is 0 before the proton gets into the channel and increases as the proton moves to the halfway of the channel, then decreases and becomes 0 after the proton gets out of the channel, as seen in Figure 2. During this process the energy of eΔΨ is liberated in Mot* and disappears in Mot*, where e is the elementary electric charge. Exact treatment of variation of E in such a dynamic case should be done referring to Maxwell’s equations, but the proton does not seem to move very fast and the dynamics is discussed by the following approximation.

Mot* is in an elastically tensioned state with the shear stress X(zp) (Eq. 1) or with the associated elastic energy UXy(zp), and tends to release the tension by stress flow or energy flow out of Mot*. Parts of the flows get into the viscoelastic medium. The flows of X(zp(t)) and UXy(zp(t)) toward the rotor during dt are denoted as JXy(t)dt and JU(t)dt, respectively, where t is time. Expressions for X(zp(t)), UXy(zp(t)), JXy(t) and JU(t) are given below.

As discussed above, the stress X(zp) induces the strain x(zp). Generally, however, there is a time lag in appearing x. The viscoelastic medium is elastically soft and it is assumed that JXy(t) flows out of Mot* within a shorter duration than the time lag of x and thus x is negligibly small. Then, in Figure 1, the angle α practically does not change during the stress flow and P(zp) remains almost constant at the initial value at E = 0: P0(zp). Thus Eq. 1 can be approximated as As usually done in the theory of elasticity, the elastic energy UXy(zp(t)) is given by where cXy is an elastic constant.

The stress flow JXy(t) seems to be proportional to the stress Xy(zp(t)). The proportional coefficient seems to be a function of vp and is denoted as f(vp), whose expression is determined below. Now JXy(t) is expressed by The energy flow JU(t) is given by where cU is an elastic constant. The total energy output from Mot* to the rotor per one proton passage is denoted as IU: Here the integration is done from 0 to dch/vp. Combining Eqs. 5, 6, 7, 8 and 9 gives Here ∫ E(zp)2dzp is independent of vp since E(zp) is the electric field component when the proton is sitting at zp. Hence IU depends upon vp through the factor f(vp)2/vp. According to Eq. 9,IU is the integration of energy input to the rotor during one proton passage in the channel, and should be proportional to the transmembrane electrochemical potential ΔΨ, which is proportional to vp by Eq. 3. Hence, f(vp)2/vp should be proportional to vp, i.e., Here g is a constant. Then Eq. 7 becomes By the relation X(zp) = aP0(zp) (Eq. 5), JXy(t) becomes This expression of JXy(t) is very useful. By Eq. 13, the energy flow JU(t) = (cU/2) JX(t)2 (Eq. 8) becomes

Now let us consider about implication of the assumption (ii) that the blue part in Figure 1 is viscoelastic. The assumption implies that chemical, tight coupling by protein molecules does not exist in the blue part. However, it seems reasonable to expect that the blue part mainly consists of protein molecules. Viscoelasticity of polymers are discussed in detail in textbooks by Ferry10 and Saito11. Viscoelasticity of monolayers is discussed by Langevin12. By the assumption (ii), the blue part behaves like an elastic body as well as like a viscous liquid. Thus shear stress flow proceeds into the viscoelastic medium due to the elasticity and viscous flow occurs in response to the shear stress due to the liquid-like property. As a simplified molecular model, let us imagine that the viscoelastic medium is an ensemble of small cylinders perpendicular to the membrane surface. The cylinders are in thermal motion in potential wells. When the shear stress exists in the medium, some cylinders change their position within wells and contribute to the elastic response of the medium. On the other hand, some cylinders move across the potential barriers by the shear stress and viscous flow takes place corresponding to the liquid-like property. After the shear stress flow passes through the medium, the elastic response within wells leaves no hysteresis, but the viscous flow over the potential barrier leaves hysteresis, causing the step motion of the rotor. Note that there is the boundary condition that material movement should be continuous at the boundaries. Accordingly, there is no slippage at the boundaries between Mot* and the viscoelastic medium and between the viscoelastic medium and the rotor. Hence the rotor rotation corresponds to the relative motion between the two boundaries.

Although the involved process is complex, we regard the shear stress flow JXy(t) = (gaP0)vpE(vpt) (Eq. 13) as an effective force to rotate the rotor in the following discussion. Then the parameter g is related with the nature of the viscoelastic medium.

Explanation of experimental observations

In this section, the subsection number and the number in parenthesis are the same as the item number of the experimental result in Sect. 1.

Experimental result (1)

(1) One revolution of the flagellar rotation consists of a constant number of steps irrespective of the transmembrane potential difference4.

When the shear stress flow JX(t) exists at the rotor surface, the torque to rotate the rotor is rRJX(t), where rR is the radius of rotor. The rotation angle caused by the torque is denoted as θ. Then the friction force against the rotation is (bη)dθ(t)/dt, where η is the viscosity of the outer fluid and b is a constant determined by the shape and surface condition of the flagellum. Thus we have The step rotation of the rotor due to one proton passage is indicated as Δθ. Then, by integration of Eq. 15, we have where the integration is done from 0 to dch/vp. If we define IXy by Δθ becomes By the relations JXy(t) (Eq. 13) and vpdt = dzp from Eq. 4, Eq. 17 becomes Since ∫ E(zp)dzp is the area under the curve in Figure 2, IXy is constant. Hence Δθ given by Eq. 18 does not depend upon ΔΨ. Accordingly, one revolution of the flagellar rotation should consist of a constant number of steps irrespective of the transmembrane electrochemical potential, as experimentally observed.

Experimental result (2)

(2) The rotation velocity of the rotor is proportional to the transmembrane potential.

The number of protons passing through one rotary motor per unit time is denoted as n. Then n is proportional to the transmembrane electrochemical potential ΔΨ. That is, where B is a constant. The rotational velocity of the rotor is denoted as ω. Then ω is related with the step rotation Δθ by Combining Eqs. 20 and 21 gives

Since Δθ = (rR/bη)IXy by Eq. 18, Here IXy does not depend upon ΔΨ as proved with Eq. 19. Hence ω is proportional to ΔΨ. In Figure 3, the experimental data reported by Fung and Berg13 are cited by black circles. The abscissa is the electric potential difference Δφ which is equal to ΔΨ since chemical compositions in the outer and inner solutions are not changed in their study13 . The straight line is drawn with the formula,

Figure 3

Flagellar rotation velocity ω/2π as a function of the trans-membrane electric potential Δϕ, after Figure 7 of ref. 5. The data points are cited from Figure 4 of Fuang and Berg13.

Experimental result (3)

(3) When the rotational velocity of a flagellum is changed by adjusting the viscosity of the outer fluid, the torque for the cell to rotate the flagellum is practically constant independent of the velocity, but sharply decreases when the velocity increases over a critical value.

Figure 4 shows distribution of the data points (small circles) cited from ref. 4. The abscissa is the rotational velocity ω/2π of the flagellum and the ordinate is the relative torque. The critical velocity is indicated as ωcr. Similar data were reported for the Na+-driven flagellar motor in Figure 4(b) of the paper by Sowa et al.14, although bead sizes were changed instead of viscosity in their experiment. Change of the bead size means change of b instead of η in Eq. 15.

Figure 4

Relative value of torque as a function of the rotation velocity ω/2π, after Figure 8 of ref. 5. Data points (small circles) are cited from Figure 3 of Berg4. (a) 23°C and (b) 16°C. The horizontal line (ω) represents the relative torque according to the conclusion by Eq. 26 that the torque is constant for ω < ωcr. The curve (ω*) is the relative torque ωcr/ω (Eq. 44), which gives the upper limit of the relative torque for ω > ωcr.

The following discussion is done in three parts: (1) the torque for ω< ωcr, (2) the critical velocity ωcr, (3) the torque for ω > ωcr

The torque for ω < ωcr

The torque to rotate the rotor is denoted as T. The observed torque is obtained by multiplying the viscous resistance bη and the rotation velocity ω: By Δθ = (rR/bη)IXy (Eq. 18) and ω = BΔΨΔθ (Eq. 22), we haves

Note that the viscosity-dependent term bη disappears in Eq. 26. Since IXy is a constant as proved with Eq. 19, the torque T remains constant irrespective of the viscosity η for given ΔΨ. In Figure 4, the horizontal line (ω) indicates the relative value of torque T(ω)/T(ωcr), which is constant according to this conclusion. The experimental data are distributed around the straight line (ω) for ω < ωcr.

The critical velocity ωcr

The rotation of the flagellum in the viscous solution causes energy dissipation. Let the energy liberation during dt be D dt in the step rotation Δθ. Then D is given by the frictional torque bηdθ/dt multiplied by the rotation velocity dθ/dt: Since (bη)dθ(t)/dt = rR JXy(t) (Eq. 15) at the stationary rotation, D(t) becomes Since JXy(t) is independent of η, Eq. 28 implies that the energy dissipation D(t) becomes infinitely large as η approaches to 0. This is impossible and the necessary condition for the stationary flagellar rotation is given by where JU(t) is the energy input during dt and given by JU(t) = (cU/2) JX(t)2 (Eq. 8). Combining Eqs. 8, 28 and inequality in Eq. 29 gives Now ηcr is defined by Then the condition for the stationary rotation of the flagellum is given by In the stationary rotation, ω is given by ω = (BrR/bη)IXyΔΨ (Eq. 23). Hence the critical velocity ωcr corresponding to ηcr is given by Or by Eq. 31, The range of constant torque is limited in the range Accordingly, the straight line (ω) is terminated at ωcr in Figure 4.

The torque for ω > ωcr

When η < ηcr, all the input energy flow JU(t) will be absorbed by the flagellar rotation, although there is no balance of forces expressed by (bη)dθ(t)/dt = rRJXy(t) (Eq. 15). Presumably the flagellum rotates at random, but here let us suppose a fictional case that the rotation is smooth with a rotation velocity θ*. Then the energy liberation rate is given by multiplication of the torque bηdθ*/dt and the rotation velocity dθ*/dt. If this energy dissipation rate is set equal to the energy input JU(t) given by JU(t) = (cU/2) JX(t)2 (Eq. 8), we have Thus By using IXy in Eq. 17, integration of Eq. 37 gives The corresponding rotation velocity ω* is given by The parameters other than η are independent of η. If the parameters other than η are combined and expressed by a constant C*, then The torque T* is defined by bηω*. Then Combining Eqs. 40 and 41 gives Hence we have To see the nature of Eq. 43, ω* is replaced by ω as This relation is plotted as the curves (ω*) in Figure 4(a) and (b). The curve T*(ω)/T*(ωcr) = ωcr /ω is continuous with T(ω)/T(ωcr) = 1 at ω = ωcr. The flagellar rotation seems to be random for ω > ωcr and the data points are expected to distribute under the curve (ω*) which is derived on the assumption of fictional smooth rotation. In the calculation, the parameter values of ωcr are determined based upon this expectation.

Curve (ω*) more steeply declines in Figure 4(b) than in (a). The distribution of data points shows similar tendency.

Figure 4(a) for 23°C and (b) for 16°C indicate that ωcr sensitively depends upon temperature. It is a possibility that the viscoelastic medium becomes less liquid-like at lower temperature and reduces the shear stress input to the rotor (IXy), and causes a decrease of ωcr which is given by ωcr = {BcU/(2rR)}IXyΔΨ (Eq. 34).

Figure 5 is to help understanding the discussion in this section. Figure 5(a) shows the case that the viscosity of the outer fluid η is infinitely large. The rotor does not rotate and all the shear stress flow JXy propagates into the inside of the rotor without loss. In (b) η has a finite value larger than ηcr. The rotor rotation takes place with the force balance expressed by (bη)dθ(t)/dt = rRJXy(t) (Eq. 15) using a portion of JXy. The remaining part of JXy proceeds into the inside of the rotor. In (c) η < ηcr, and all JXy(t) is used for random rotation of the rotor.

Figure 5

Transmission of the shear stress flow JXy. (a) The viscosity η of the outer liquid is infinitely large. The rotor does not rotate and all JXy transmits into the inside of the rotor. (b) η has a finite value larger than ηcr. The rotor smoothly rotates by using a portion of the input energy. The remaining part transmits into the inside of the rotor. (c) η is smaller than ηcr. All input energy is used for random rotation of the rotor.

Let us consider about the energy efficiency of the flagellar motor, eU Since the energy dissipation rate in the rotor rotation is given by D(t) = rR2JXy(t)2/(bη) (Eq. 28) and all the energy dissipation per one proton passage is given by (eΔΨ), the energy efficiency eU is given by Here k is a constant. In this equation, parameters other than η do not depend upon η, and hence we have By Eq. 23, Eq. 46 becomes Thus the efficiency eU increases with increasing ω for ω < ωcr, and then decreases with increasing ω since randomness in the rotation reduces eU for ω < ωcr.

Effect of reversal of the sign of the electrochemical potential gradient

(4) There are observations that the rotation direction remains the same when the sign of the electrochemical potential gradient is reversed.

In 1980, Manson, et al.15 used Streptococcus bacteria and examined how the flagellar rotation velocity varies when the sign of the transmembrane electrochemical potential is reversed by changing ionic concentration in the outer solution. They observed that in some specimens, the direction of the rotation remained the same but in other specimens the direction was reversed when the proton movement was reversed. In 1982, Berg et al.16 carried out more detailed studies of the effects. They examined how the rotation velocity changes when pH changes from 7 to 8 in the outer solution. The rotation velocity became 0 around pH = 7.5. Experimental results were not very reproducible when pH was changed in the outer solution but there were cases that the rotation occurred in the same direction in both sides of pH = 7.5. An example of their results is cited by black circles in Figure 6. As pH increases from pH = 7.0, velocity ω decreases, becomes 0 around pH = 7.5, and then increases. The direction of the proton passage seems to be reversed at pH = 7.5, but the rotation takes place in the same direction in both sides of pH = 7.5.

Figure 6

Flagellar rotation velocity ω/2π as a function of pH of the outer solution, after Figure 9 of ref. 5. Data points are cited from Fig. 5b of Berg et al.16. The direction of the proton is inward for pH < 7.5 and outward for pH > 7.5, but the direction of the rotation remains the same in both sides of pH = 7.5.

Our model supports the experimental results in Figure 6 as discussed below. Hitherto we have discussed the case that the electrochemical potential Ψout of the outer solution is higher than Ψin of the inner potential. The transmembrane electrochemical potential is defined by ΔΨ = Ψout – Ψin (Eq. 2) and the proton velocity vp is expressed by vp = AΔΨ (Eq. 3). To discuss the case that ΔΨ changes its sign at pH = 7.5, we define vp by The proton positions in the channel zp(t) are given by zp(t) = vpt when Ψout > Ψin, while zp(t) = dch – vpt when Ψout < Ψin. In both cases, the electric field E(t) is produced by the proton as shown in Figure 2: E(t) is 0 at t = 0, becomes maximum around t = dch/(2vp) and returns to 0 at t = dch/vp. Thus the change of E(t) is similar although the direction of proton motion is reversed. Accordingly, we can use JXy(t) = (gaP0)vpE(zp(t)) (Eq. 13) and JU(t) = {(cU/2)(gaP0)2}vp2Ex2(zp(t)) (Eq. 14) in both cases of Ψout >Ψin, and Ψout < Ψin. As a result, the rotor rotates in the same direction with the same velocity irrespective of the sign of ΔΨ in accordance with the experimental results shown in Figure 6.

The rotation velocity ω is proportional to the potential difference |ΔΨ| when |ΔΨ| is relatively large as shown in Figure 3. If this relation holds for small |ΔΨ|, the ω vs. pH relation becomes as shown by the straight lines in Figure 6. The experimental data, however, indicate the tendency that tangent of the ω vs. pH relation becomes 0 in the vicinity of pH = 7.5, as indicated by the quadratic curve. The values of ω plotted in Figure 6 are smaller than those in Figure 3, and hence proton velocities should be smaller in the case of Figure 6. Therefore, the results in Figure 6 mean that slowly moving proton does not effectively produce the shear stress flow. Two causes can be considered to explain it. In deriving Eqs. 5 and 7, it is assumed that the shear stress flow JXy(t) occurs within a shorter duration than the time lag of x and x is negligibly small. It is, however, plausible that x becomes large so that effectiveness of JXy(t) is reduced when the proton moves very slowly. Also, the viscoelastic medium generally loses its ability to transmit the shear stress when the shear stress slowly changes.

Experimental result (5)

(5) The cell produces constant torque to rotate the flagellum even when the cell is rotated by externally applied torque.

Berg and his colleagues16–18 used the technique of electrorotation to apply the torque to cells of bacteria tethered to glass coverslips by a single flagellum. Cells were driven to rotate either forward or backward. Here forward means the direction of rotation driven by the flagellar motor itself. They used the intact cells which could normally rotate the flagellum (called motor intact) and the broken cells which lost the ability to rotate the flagellum (called motor broken). Berg and Turner17 observed a barrier to backward rotation for the motor intact but later Berry and Berg18,19 found that it was an artifact. According to Berry and Berg18,19, the relation between the rotation velocity of motor intact and the torque is approximately linear from −100 Hz to +100 Hz and parallel to the relation of motor broken.

To make the experimental results clear, let us denote the externally applied torque as Tapp and the rotation velocities of the broken and intact cells as ωbrk(Tapp) and ωint(Tapp), respectively. Then the experimental results by Berry and Berg18,19 are expressed by the two formulae. Here q is a constant and ωint(0) is the rotation velocity produced by the intact cell in the absence of Tapp. Figure 7 shows lines ωint and ωbrk corresponding to Eqs. 49 and 50.

Figure 7

Illustration of the effect of the externally applied torque on the flagellar rotation velocity, after Figure 10 of ref. 5. The cell is tethered by a single flagellum and the cell body is rotated electrically18,19. Tapp: externally applied torque on the cell. ωint: measured rotation velocity of the intact cell when the torque is applied externally. ωbrk: measured rotation velocity of the broken cell when the torque is applied externally.

As mentioned above, Berg and his colleagues17–19 used the technique of electrorotation to apply the torque to cells tethered to glass coverslips by a single flagellum. Theoretically, the relative motion between the cell and a flagellum is important, and the following discussion is done as if the cell is fixed and a flagellum is rotated by an external torque. The externally applied torque Tapp to the flagellum rotates the rotor. Then the rotor produces shear stress in the viscoelastic medium and in Mot*. Mot* deformed and an electric field is induced by piezoelectricity in Mot*. The transmembrane elecrochemical potential ΔΨ is, however, kept constant by the outer and inner liquids and the variation of the electric field inside of Mot* should satisfy this boundary condition. Accordingly, the mean electric field in the channel due to ΔΨ remains constant and the cell produces almost the same torque as in the absence of externally applied torque, resulting in the relation ωint(Tapp) = qTapp + ωint(0) (Eq. 50). Naturally the reversed rotation of the rotor does not induce proton pumping.

Experimental observation (6)

(6) The cell has a switch which reverses the sense of the flagellar rotation for chemotaxis.

One possible mechanism for the switch is that Mot* has two stable structures having positive and negative P0 as illustrated in Figure 8. Information flow from the receptor to the flagellar motor in the chemotaxis is illustrated, e.g., in Figure 2 of ref. 20. If there is a device to interchange the two structures in Figure 8 simultaneously in all Mot*s in response to the information flow, the shear forces JXy(t) = (gaP0)vpE(vpt) (Eq. 13) change their signs in all Mot*s and the direction of flagellar rotation is reversed for the chemotaxis. Presumably the proposed mechanism is simpler than in the case of chemically tight coupling models.

Figure 8

Proposed mechanism for reversal of the rotation direction in the chemotaxis. (a) The dipole moment P has the downward component and causes anti-clockwise rotor rotation. (b) The dipole moment P has the upward component and causes clockwise rotor rotation.

Summary and discussion

The study described above started from the question of whether there is any possibility that the stator can induce the rotor rotation in the absence of chemically tight connection by protein molecules between them. We assume that there is only viscoelastic medium between the rotor and stator and that Mot* has a permanent electric dipole moment. Then Mot* acts as a shear stress generator through the interaction between its dipole moment and the electric field produced by a proton passing in the channel. The shear stress transmits through the viscoelastic medium and leaves hysteresis which causes the step rotation of the rotor. Calculation results based upon the model well explain experimental observations as described in Sect. 3, and suggest that the proposed model is not very far from reality. For further discussion, however, detailed experimental data are needed on the nature of the viscoelastic medium and electric and piezoelectric properties of the Mot assembly.

One of the characteristic features of our model is that discussion is done on the basis of the force balance expressed by (bη)dθ(t)/dt = rRJXy(t) (Eq. 15) instead of the energy balance. Related shear stress flow is illustrated in Figure 5.

As noted in our previous review on muscle contraction mechanism21, protein molecules are structurally polar and biological systems should be considered as a system of four variables (electric field, polarization, stress, and strain) from the physical view point. The present study is based upon this idea. Systematic discussion on four variable systems can be found, e.g., in ref. 22, although this reference is for solid state materials. Recently Furuike et al.23 studied axleless F1-ATPase and found that neither fixed pivot nor rigid axle is needed for rotation of F1-ATPase. Their observation seems to suggest that some physical factors are involved in the mechanism of F1-ATPase rotary motor in addition to chemically specific factors.