Figure 1 shows a simplified structure of the flagellar rotary motor. The motor has a diameter of approximately 45 nm [1] and is surrounded by a membrane with an averaged thickness of approximately 7 nm. The motor consists of Rotor, Stator and the intermediate layer between them, which is hereafter called the RS layer. The stator consists of complexes of Mot A and Mot B proteins. The complex is symbolized as Mot* and the Stator is assumed to have eight Mot*s [5], as illustrated in Figure 1(a). Each Mot* has two proton channels [6]. Two protons seem to pass practically simultaneously through the two channels [6]. For simplicity, we suppose that two protons simultaneously pass through the combined single channel shown in Figure 1. The proton pair is called 2e, where e represents the charge of proton.

Under normal conditions, the proton channels conduct protons from the lumen to the cytoplasm. The electrolytes in the lumen and the cytoplasm are referred to as the outer liquid and the inner liquid, respectively. A flagellum has its root in the center of the rotor, and rotates at the speed of, for example, 10 or 100 Hz, depending upon the viscous resistance of the outer liquid. The Mot* shown at the extreme right in Figure 1 will be discussed as a representative of Mot*s. The origin of the coordinate system is set at the outer surface of the single channel situated at its center, the x axis is directed toward the axis of the rotor, the z axis is perpendicular to the membrane, directing inward to the cytoplasm, and the y axis is orthogonal to the x and z axes, as shown in Figure 1.

Experimental observations on the flagellar rotary motor to be explained by the model are summarized as follows:

Our basic assumption is that the electric field produced by protons plays an important role in the flagellar rotation. As the first trial model, the proton channel was assumed to be tilted to the z-axis on the y–z plane. Then the proton passage in the channel will give Rotor a torque if the surface of Rotor has a suitable charge distribution. This model, however, did not explain the experimental data and was discarded. Then, it is assumed that the proton channel is perpendicular to the membrane as shown in Figure 1(b) and Mot* has an electric polarization along the y-axis so that Mot* acts as a shear force generator. This model successfully explained experimental observations.

In order to obtain rough idea on the strength of the field produced during the proton passage in Mot*, a dielectric membrane is considered as described in Appendix, where Coulomb field is calculated by the method of images as a function of z_p, the proton-pair position on the z axis (the normal of the dielectric membrane). Since the distance between the channel and the Rotor surface is uncertain, a cylinder of radius 2.5 nm is considered as an example. The x component of the field is averaged over the half the cylinder of x > 0 and the average is denoted as E_x. The mean z component of the field is zero since the membrane is sandwiched by conductors. Also the volume-averaged E_y is zero due to the symmetry of the dielectric membrane. Hence, only the x component of the field is considered. Figure 2 shows E_x as a function of z_p/d_ch, where d_ch is the channel length. Thus z_p/d_ch stands for the relative position of the proton pair in the channel. The proton pair sits at the outside end of the channel when z_p/d_ch = 0 and at the inside end when z_p/d_ch = 1. The calculation was made by setting d_ch equal to 7 nm. The order of magnitude of E_x(z_p) around the middle of z_p (3.5 nm) is 3×10⁸ V/m. Note that the breakdown field strength of bulk paraffin is about 10⁷ V/m, but the field can be stronger than 5×10⁷ V/m in a synthetic polymer film of thickness 50 mm (for instance, cf. [7]). It seems possible that the breakdown fields are stronger than 5×10⁷ V/m in microscopic systems. Since the field of 3×10⁸ V/m is very strong, it is plausible that it can produce significant effects on Mot* structures if Mot* has charged parts in it. Note that Figure 2 is to give an idea on the strength of the field and actual field in Mot* should have more complex characteristics, e. g., it is asymmetric since the membrane structure is polar. For simplicity, however, the same notation E_x will be used for the x component of the actual field in Mot*in the following discussion.

Protein molecules are structurally polar and generally they have large permanent electric dipole moments [8, 9]. For instance, G-actin has the dipole moment of 600 Debye and flagellin about 800 Debye [9]. As shown in Figure 2 of [10], the MotA complex consists of 4 copies and MotB consists of 2 copies in Mot*. All the copies are expected to have permanent electric dipoles. We call the polarization corresponding to the vector sum of their dipoles as Mot* polarization, and denote its y component as P_0y. Interaction between P_0y and E_x will cause a shear stress and strain in Mot*. Figure 3(a1) indicates Mot* by the rectangle arrow and electric charges due to P_0y by + and −. The arrow in (a2) indicates the field E_x. The positive charges of Mot* will move to left and the negative charges move to right if Mot* is free to deform. Thus the rectangle will deform as indicated in (a2) when the lower side of the Mot is fixed. The deformation will take place as shown in (a3) when the right-hand side of Mot* is fixed, and will tend to rotate Rotor through RS layer as shown in Figure 3(b). The angle indicated in Figure 3(a3) is the shear strain x_y. The shear stress which produces x_y is denoted as X_y, which is approximately proportional to P_0y and E_x(z_p), i.e. to P_0yE_x(z_p) when the proton pair 2e sit at z_p. The stress X_y(z_p) is written by

where a is a constant determined by the mechanical properties and shape of Mot*. Here, Mot* is embedded in the membrane as a part of the flagellar motor system, and the stress X_y(z_p) tends to transmit into its surroundings (cf. for instance [11]). As discussed in the next section, RS layer is viscoelastic and a portion of the stress X_y(z_p) will be transmitted into RS layer. The way of the transmission will be discussed in Section 3.3.

Let us quantitatively examine the possibility that Rotor rotates by the strain x_y. According to Figure 1 of [1], the diameter of Mot A is comparable to the radius of Rotor. Hence the strain x_y will be of the same order of magnitude as Rotor rotation angle. As an example for large piezoelectric strain, let us consider the ferroelectric crystal, KH₂PO₄ [12]. In this crystal, x_y = d₃₆ E_z and its maximum d₃₆ is 1.4×10⁻⁹ m/V at −150 °C. If E_z is 10⁸ V/m, x_y becomes 0.14 radian. If this value is adopted as x_y of Mot*, 2π /0.14 = 45 proton pairs are required for one revolution. If d₃₆ is one tenth of 1.4×10⁻⁹ m/V, 2π/0.014 = 450 proton pairs are required. It is known that about 1000 protons are required per revolution of Rotor [1]. The cited d₃₆ is the value at the temperature where KH₂PO₄ does not have the permanent polarization. In the presence of the permanent polarization, the shear strain may become larger. If Mot* is designed as a shear force generator having the permanent polarization, it will be relatively easy to produce shear stress that is large enough to produce the flagellar rotation.

The flagellar motor is embedded in liquid membranes. It is assumed that the RS layer consists of lipid membranes. According to [13], lipid membranes are generally viscoelastic, that is, they behave like a viscous liquid as well as an elastic medium in which shear stress can propagate. Here, by the terminology of viscoelasticity, we expect the characteristics of RS layer that the shear stress can transmit in it and that the layer acts as lubricant or (cylindrical) ball bearings for the rotation of Rotor. It seems permissible to assume that RS layer has a finite thickness though it is thin, and thus can have such characteristics.

Concerning the nature of the lipid membrane, an observation by Nakayama et al. [14] seems interesting. They extracted lipids from the cytoplasmic membrane of E. coli cells and studied a relation between the growth temperature and the solid-liquid phase transition in the lipid membrane with X-ray diffraction method. They found that cytoplasmic membranes of the cells grown at 17, 27 and 37 °C exhibited the lipid solid-liquid phase transitions at 10, 15 and 28.5 °C, respectively. That is, the transition temperatures were below the growth temperatures by 7, 12 and 8.5 °C, respectively, and thus the lipid parts of the cytoplasmic membranes were always in the liquid phase. Nakayama et al. [14], who carried out also chemical assays, and found that the molar ratio of the saturated to unsaturated fatty acids increases in phospholipids extracted from cytoplasmic membranes as the growth temperature increases. It seems plausible that E. coli cells adjust the molar ratio of the saturated to unsaturated fatty acids in order that the viscoelastic properties of lipid layer remain constant irrespective of the growth temperature.

Figure 4 illustrates responses of the system against the shear force transmission, when RS layer is simply viscous (a), and when the system is viscoelastic (b). A part of Rotor, RS layer and Mot* are represented by rectangles (or parallelograms after deformation). Figures (a1) and (b1) show the states before the shear stress takes place. (a2) and (b2) show the states when the shear stress is passing. The shift of the boundary between Rotor and RS layer corresponds to the step rotation of Rotor and the deviation of the Rotor shape from rectangle (b2) indicates the propagation of the shear stress in Rotor. (a3) and (b3) show the states after the shear stress has passed. In Figure (a2), the arrows ν_M indicates the viscous stream following the movement of the edge of Mot*. As usually occurs in a viscous fluid (cf. for instance [15]), ν_M causes a stream ν_R near Rotor, causing movement S_v(t) of Rotor. Positive ν_M will induce positive S_v(t), while negative ν_M negative S_v(t), and Rotor will returns to the original position when the shear stress passes as indicated in (a3). In the case of (b), the shear stresses transmitted into RS layer due to the viscoelastic property of RS layer. In addition to the direct effect of viscosity as illustrated in (a2), two effects of the transmitted stress are expected as illustrated in (b2). One is the effect of the elastic stress itself that causes movement S_e(t) and elastic deformation of Rotor (b2), but these changes return to zero after the shear stress passes (b3). The second is the effect that the shear stress causes viscoelastic movement S_ve(t) and thus rotation of Rotor (b2). The time derivative of S_ve(t) is proportional to the shear stress F_y(t) as usually treated as Maxwell’s model of viscoelastic systems [16, 17], and leaves the step rotation ΔS_ve after an impulsive stress has passed (b3), as discussed mathematically in Section 4-1a.

The transmembrane electrochemical potential difference ΔΨ is defined by

where Ψ_in and Ψ_out are the electrochemical potentials in the inner and outer liquids, respectively. The absolute value of the proton velocity is denoted as ν_p, The frictional resistance against the proton motion in the channel is assumed constant as generally done. Then ν_p is proportional to |ΔΨ| :

where A is a constant. In this section the case of ΔΨ > 0 is considered (the case of ΔΨ < 0 is discussed in Section 4.2.). Then the proton position z_p is given as a function of time t by

where t changes from 0 to d_ch/ν_p. The strength of the stress flow transmitted into RS layer is denoted as F_y(t). It seems reasonable to assume that F_y(t) is proportional to the stress X_y(ν_pt) produced in Mot* given by Eq. 1 so that F_y(t) is a function of ν_p, expressed by

where f(ν_p) is a function of ν_p to be determined. The energy input to the RS layer during dt is denoted as Wdt. Then, W is given by

where c_RS is an elastic constant of RS layer. The total energy input associated with the proton pair passage is denoted as I_W:

From Eqs, 5 and 6, I_W becomes

where integration is done from 0 to d_ch/ν_p. Since the proton velocity ν_p is constant and dt is given by dz_p/ν_p, Eq. 8 can be rewritten as

The energy I_W should be proportional to the work done by the membrane during the proton pair passage and thus proportional to |ΔΨ| and to ν_p by Eq. 3. The integral ∫E_x(z_p)²dz_p is a constant independent of ν_p. Hence, f(ν_p)²/ν_p should be proportional to ν_p:

where γ is a constant. The value of γ is assumed to be independent of other parameters, e. g., c_RS. This assumption means that Mot* is a constant shear force transmitter. The model based upon this postulation explains various experimental observations as seen below. Now Eq. 5 becomes

Integration of the force of the force F_y(t) is denoted as I_y:

where the integration is made from 0 to d_ch/ν_p. Combining Eqs. 4, 11 and 12 gives

Accordingly, I_y does not depend upon ν_p nor ΔΨ.

As will be discussed in Section 4.2, Eq. 11 will be modified in the case of ΔΨ < 0 (cf. Eq. 59), but the value of I_y will remain unchanged.

In this section, discussion is continued for the case of ΔΨ > 0.

Several workers examined how the flagellar rotation changes when the direction of the proton movement is reversed by changing the sense of the transmembrane electrochemical potential difference, Ψ, as described in detail in Section 5.2. In this section, the effect of reversal of the proton passage direction is discussed based on the model.

The electric field E_x due to the proton pair is determined by the proton position, z_p, as given in Figure 2, irrespective of the direction of their motion. Hence, the shear force X_y(z_p) = aP_0yE_x(z_p) (Eq. 1) does not depend upon the direction of proton passage. Therefore, the rotation direction of the Rotor is the same irrespective of the direction of the proton motion, as evident also in Figure 3.

Some readers, however, may feel puzzled about the above conclusion because, if ν_p is replaced by - ν_p in F_y(t) = aγP_0yE_x(z_p)ν_p (Eq. 11), then F_y(t) becomes negative so that the impulse I_y (Eq. 12) and Δθ (Eq. 21) become negative, and thus the rotation direction will be reversed. This conjecture, however, is not correct since the above expression of F_y(t) is derived for the case of Δ Ψ> 0 Let us derive F_y(t) in the case of ΔΨ<0 The position of a proton pair for the reverse passage, is given by

Then E_x, which is expressed by E_x(ν_pt) as a function of t for the normal direction, becomes E_x(d_ch – ν_pt) for the reverse passage. Then dz_p = − ν_pdt, and Eq. 8 becomes

where the integration by z_p is done from d_ch to 0. By changing the integration range from (0 to d_ch) to (d_ch to 0),

This equation is the same as Eq. 9, and hence we have f(ν_p)=γν_p, as in Eq, 10, where ν_p is the absolute value defined by Eq. 3. Therefore, F_y(t) becomes

The integration I_y = ∫F_y (t)dt is the same for the normal and reversed proton passages since the area under the curve E_x(t) and that for E_x(d_ch – ν_pt) are equal. Hence, the rotation velocity ω given by Eq. 25 has the same value and the same sense irrespective of the sign of ΔΨ. That is,

where C is the same constant as defined by Eq. 26. Eq. 60 has the same form as Eq. 25, but now it is certain that C has the same value for positive and negative. This matter will be discussed in comparison with experimental data in Section 5.2.

Berg and his colleagues [18–20] used the technique of electrorotation to apply torque to E. coli bacterial cells tethered to glass coverslips by a single flagellum, as described in more detail in Section 5.3.

Theoretically, the relative motion between the cell and a flagellum is important, and the following discussion is made as if the cell is fixed and a flagellum is rotated by an external force. The externally applied torque is denoted as T_app, which transmits from the flagellum into Mot* through the elasticity of the RS layer, deforming Mot* as shown in Figure 3(b). This deformation will induce an electric field, E_x’ by the inverse piezoelectric effect. The x component of the electric field, E_x’, however, does not affect the proton motion along the z axis. Although T_app might produce other components of the field, E_y’ and E_z’ in Mot*, E_y’ does not affect the motion of the proton along the z axis and E_z’ will be compensated as an average by the ions in the outer and inner liquids, since the transmembrane potential tends to keep E_z constant. Accordingly, T_app does not change the velocity ν_p of the protons in the channel nor the number n of proton pairs passing the membrane per unit time:

The torque produced by the cell under the external torque is denoted as T_cell(T_app) and the torque at T_app= 0 as T_cell(0). Since T_app does not change the way of proton motion, T_app does not affect the torque produced by the cell:

The rotation velocity produced by the cell itself under the external torque is denoted as ω_cell (T_app) and the torque at T_app= 0 as ω_cell(0). Eq. 63 means that the deriving force for the cell to produce the flagellar rotation is not affected by T_app. Accordingly, ω_cell does not depend upon T_cell:

Effects of T_cell will be discussed in comparison with experimental observations in Section 5.3.

Bacteria have a switch to change the direction of flagellar rotation for chemotaxis as described, for instance, in [21]. Presumably a large-scale structural modification is needed to explain the chemotaxis if we assume a direct contact between Mot* and the Rotor. In our model, however, the coupling is intermediated by P_0y and is effected by the shear force F_y(t) = aγP_0yE_x(z_p)ν_p (Eq. 11). Therefore, the rotation direction can be reversed simply by changing the sign of P_0y. More detailed discussion on this matter is given in Section 5.4.

In the experiment [22] mentioned in Section 5-1b, Fung and Berg also tried to examine how the flagellar rotation changes when the direction of the proton movement is reversed by changing the sense of the transmembrane electric potential difference Δφ. It was, however, difficult to reach definite conclusion. They observed that, when the sense of Δφ was reversed, the motors rotated in the same direction for a short time in three of 17 trials, the motor rotated in the reversed direction for a few seconds in five of the trials, the motors stopped immediately in the remaining nine trials. On the other hand, Manson, et al. [24] used Steptococcus bacteria and examined how the flagellar rotation velocity varies when the direction of proton motion 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. Berg et al. [25] carried out more detailed studies of the effects. An example of their results are cited by black circles in Figure 9. They changed pH in the outer solution from 7 to 8, and observed that the rotation velocity decreases, becoming zero around pH = 7.5 and then increases. The direction of the proton passage seems to be reversed at pH = 7.5, but the direction of the rotation remains the same. This feature is consistent with our conclusion ω = C|ΔΨ| (Eq. 60) where C is the same constant for positive and negative ΔΨ. In the experiment cited in Figure 9, the electrochemical potential difference ΔΨ is proportional to (pH – 7.5). The solid lines in Figure 9 show results calculated by

These lines illustrate the theoretical prediction that ω is positive irrespectively of the sign of ΔΨ, but the lines do not well represent the distribution of the experimental data. Especially, the rotation velocity is close to zero in the vicinity of pH= 7.5 where the proton pair moves slowly. In the case of large electrical potential difference, however, experimentally observed ω are well reproduced by the linear relation ω = CΔφ as seen in Figure 7. Therefore, Figure 9 indicates that there is a deviation from the linear relation when the proton pair moves slowly, and suggests that some correction is necessary in the postulation that Mot* is a constant shear force transmitter. A possible model would be that the deformation x_y of Mot* has two components, one of which can immediately follow the stress X_y and the other responses against X_y with significant delay. In [2] a model was proposed assuming such a response delay. It was postulated that there is a large delay in the deformation x_y against the stress X_y in Mot* for not-very-slow proton passage, but x_y can quasi-stationarily follow X_y for slow proton passage. Then the shear force integral I_y (Eq. 12) becomes practically constant for usual proton passage but approach to zero for slow proton passage. The calculated values of ω/2π, which are cited from Figure 11 of [2], are given by the dashed curve in Figure 9, which well reproduces the general tendency of the experimental data.

Invariance of the rotation direction similar to the data shown in Figure 9 was observed in some bacterial strains, but the rotation direction was reversed in some other strains when the direction of the proton passage is reversed [25]. These indefinite results were attributed to the sensitivity of the switch mechanism to ionic circumstances in [25]. More experimental studies seem desired. Sometimes the invariance of the rotation direction was doubted in relation with a possibility of perpetual flagellar rotation caused by Brownian motion of protons in the channel. A comment is given about this problem in Sect. 6.1.

As mentioned in Section 4.3, Berg and his colleagues [18–20] used the technique of electrorotation to apply torque to cells of the bacterium E. coli tethered to glass coverslips by a single flagellum. Cells were driven to rotate either forward or backward. Here forward means the rotation driven by the flagellar motor itself. They used the intact cells which can normally rotate the flagellum (called motor intact) and those which lost the ability to rotate actively the flagellum (called motor broken). Berg and Turner [18] observed a barrier to backward rotation for the motor intact but later Berry and Berg [19, 20] found that it was an artifact and showed that the relation between rotation velocity and torque is approximately linear in the range over 100 Hz in either direction and parallel to the relation of the cell broken. Let us denote the rotation velocities of the intact and broken cells as ω_int and ω_brk, respectively. The externally applied torque is expressed by T_app as in Section 4.3. With these symbols, the experimental results by Berry and Berg [19, 20] are schematically illustrated by the two lines ω_int and ω_brk in Figure 10.

The rotation velocity ω_brk is proportional to T_app:

where q is a constant. In Section 4.3, the rotation velocity produced by the cell itself under the torque is denoted as ω_brk(T_app), and the relation (Eq. 64) is derived as

This means that ability of the cell to produce the flagellar rotation is not affected by T_app and hence it is reasonable to expect the additive relation between ω_brk(T_app) and ω_cell(0) for the intact cell. Thus, considering Eq. 71, we have

That is, the model gives parallel relation between ω_int(T_app) and ω_brk(T_app), in agreement with experimental observation illustrated in Figure 10. As mentioned above, experimentally the parallel relations in Figure 10 holds from −100 Hz to +100 Hz. This implies that ω_cell(T_app) = ω_cell(0) (Eq. 72) holds and the proton motion is not affected even for negative T_app. Therefore, negative T_app does not pump out protons unlike the case of F₀F₂-ATP synthase (cf. Section 6.2).

Bacteria have a switch to change the direction of flagellar rotation for chemotaxis. For details of the switch, readers may refer, for instance, to [21].

As described in Section 3.1, the MotA complex consists of 4 copies and MotB complex consists of 2 copies in Mot*. All the copies are expected to have permanent electric dipoles. We defined the polarization corresponding to the vector sum of their dipoles as Mot* polarization, and denoted its y component as P_0y. As mentioned in Section 4.4, the flagellar rotation direction can be switched by changing the sign of the polarization P_0y in our model. If the dipoles of copy molecules have a tendency to be approximately antiparallel to each other, their vector sum, and thus Mot* polarization and P_0y, will be sensitive to change in structure and orientation of the copy molecules. Then, relatively small structural or orientation changes in the copy molecules can cause change of the sign of P_0y. It is a possibility that Stator can assume two stable structures having positive and negative P_0y and the chemotaxis system determines which of the two is realized through the chemical information flow as illustrated in Figure 2 of [21]. Then the shear forces F_y(t) = aγP_0yE_x(z_p)ν_p (Eq. 11) simultaneously change their sign in all Mot*s of Stator and the flagellar rotation is reversed.

As mentioned in Section 1, protein molecules are structurally polar and thus piezoelectric, and there should be interaction between an electric field and their mechanical deformation. Hence theoretically the flagellar rotary motor should be treated as a four-variable system, i. e., as a system in which strain, stress, electric field and polarization are interacting with each other. Our model is constructed based on this viewpoint and succeeds in explaining various experimental data. The viewpoint of four-variable system was applied to discuss the muscle contraction mechanism and successful to explain experimental observations [26]. It is assumed that Mot* has a permanent polarization and acts as a shear force generator. The shear force is transmitted to the Rotor through the viscoelastic RS layer. Experimental observations listed as items (1) ~ (6) in Section 2.2 are explained by the model. The results are summarized below with the same item numbers.

As mentioned in Section 2.1, each Mot* has two proton channels [6], but two protons seem to pass practically simultaneously through the two channels [6]. For simplicity, we have supposed that the two protons simultaneously pass through the combined single channel as shown in Figure 1. It seems plausible that Coulomb field produced by one proton in one channel is very strong at the other channel as suggested by Figure 2, and largely deforms the potential barrier for the proton in the other channel so as to stimulate simultaneous passage of the proton pair.

Sowa et al. [27] observed 26 steps per one flagellar revolution. It is an interesting result. As mentioned in Section 1, however, many (about 1000) protons are needed for one flagellar revolution, and our present viewpoint is that the flagellar rotation is a more physical rather than chemically specific process. From this viewpoint, the observed 26 steps seem to relate with rugged potential distribution for the flagellar rotation which is caused by the structural periodicity of the ring of FliG proteins.

A question arose whether perpetual flagellar rotation occurs by Brownian motion of the protons in channel if the reversal of proton passage causes the flagellar rotation in the same direction. An answer can be found in the experimental data in Figure 9, where the three data points indicate that ω is almost zero when pH is close to 7.5, illustrating that the flagella rotation does not occur when proton-motive force is small and protons move slowly. Theoretically, there seem to be two origins causing this observation. (1) Generally the viscoelastic medium loses its ability to transmit the shear stress when the shear changes very slowly with time, as described in text books (For instance, [17]). (2) When Mot* deformation is faster than rotor reaction, then the shear force X_y is released mainly within Mot* and RS layer interact with Mot* through strain rather than stress. Case (2) was treated mathematically with a finite time constant of Mot* deformation in [2]. The time constant was set so that almost all shear stress is released by deformation of Mot* for slow proton passage. The calculation results in [2] are cited by the dotted curve in Figure 9, which has zero tangent at pH = 7.5. Thermal motion of protons in channels seems slow in channels so that it is decoupled from the motion of the flagellar rotor. Accordingly, thermal fluctuation of the proton does not cause the perpetual flagellar rotation.

Another example of biological rotary motors is F₀F₂-ATP synthase. Protons pass through the channels due to the transmembrane electrochemical potential difference and rotates the rotor which consists of F₀c ring and γ protein. The reversed rotation of the rotor by ATP hydrolysis pumps out protons through the channels (for instance, cf. [28]). On the other hand, in the case of the bacterial flagellar motors there is no observation of proton pumping.

From the structural viewpoint, the motor of F₀F₂-ATP synthase is quite different from the bacterial flagellar motors. The diameter of the flagellar motor is about 45 nm while that of F₀F₂-ATP synthase is about 10 nm [29]. The diameter of a γ protein molecule which rotates in F₁ is about 2 nm, and has binding sites at every 120°. Thus there seems to be a device to prevent backward motion like a microscopic ratchet. The γ protein molecule is surrounded by about 10 F₀c-subunits [28]. About 10 protons can cause 360° rotation of the rotor. This means that one proton passage per F₀c-subunit is large enough to cause one revolution of the rotor. Presumably each hopping step of the proton motion is connected with each sep of chemical relation, and such models as proposed by [30, 31] seem close to reality. In these models, reversibility of the motor is possible: flow-in protons cause ATP synthesis and ATP dehydration causes the proton pumping.

In the case of the flagellar rotary motor, the stator consists of about eight pairs of Mot* (Figure 1) and about 1000 protons (500 proton pairs) are needed for one revolution of the rotor [1]. Hence 500/8 = 63 relative positions exist for the proton pair per Mot*. It seems difficult to suppose 63 chemical reaction sites for proton pair per Mot*, and it seem plausible that the flagellum rotates by a chemically non-specific force such as proposed in our model.