GLOSSARY

TOP ABSTRACT GLOSSARY INTRODUCTION THEORETICAL PARAMETERS OF THE MODEL SIMULATION OF EXPERIMENTAL... DISCUSSION APPENDIX A APPENDIX B REFERENCES

dc
distance between the channel axis and the circumference of the rotor	dm
thickness of rotor (membrane)	d,,
distance between charge Q,, and the position (xo, 0, z) on the channel axis	eo
elementary charge	f
frequency of rotation; subscripts br, bt, fl, lf, and mot for body roll, body tethered cell, flagellum, level flow, and motor, respectively	f'mot
dimensionless motor frequency defined in Eq. 33	G°
standard free energy	Gc
free energy due to electrostatic interaction between protons in different channels	Gp
free energy due to electrostatic interaction between protons in the same channel	Gr
free energy due to electrostatic interaction of a proton with the rotor charges	h
Planck's constant	Ji,j
flow of protons between states i and j of a channel	Jex
flow of protons from the periplasmic space into a channel	Jex,tot
total proton flow from the periplasmic space into all channels	Jin
flow of protons from the channel into the cytoplasm	Jin,tot
total proton flow from all channels into the cytoplasm	JH
proton flow through motor	Ki,i+1
equilibrium constant of transition between states i and i + 1	k
Boltzmann's constant	ksubscript
proportionality constant (subscripts ext, tor, mot, , v relate to Eqs. 37, 39, 39, 41, 44, respectively)	kbt, kbr, kfl
frictional drag coefficients for tethered cell, body roll, flagellum	n
number of channels	P
/ (pitch)	pi
probability of the ith state	pHin, pHex
pH in the cytoplasm, suspending medium	Q,,
th charge on a row in th repeat with charge number	q
number of charges in a row	R
radius of the rotor	r
number of repeats	s
number of proton binding sites in a channel	T
absolute temperature	Tref
reference temperature	Ti
torque generated by the ith state	Tch
torque generated by a single channel	Ttot
total torque, generated by all channels	Tmot
torque generated by the motor	Ttor
torque due to torsion in the hook/filament complex	Text
external torque exerted on the motor	Text,r
relative external torque exerted on the motor	t
time	Uex
externally applied voltage	x
x-coordinate	xo
distance between axes of rotor and channel	y
y-coordinate	Zi
charge number of the ith proton binding site in a channel	z
z-coordinate	zi
z-coordinate of ith proton binding site in a channel	zi#
z-coordinate of the ith transition state between sites i and i + 1 in a channel	i,i+1
transition probability per unit time (intrinsic rate constant) for the transition from state i to state i + 1	*i,i+1
first order rate constant including the effect of electrostatic interaction, membrane potential, and proton concentration	i
relative rate constant of ith transition
tilt angle
horizontal angle corresponding to tilt	o
permittivity in vacuo
relative permittivity	f,i, b,i
dimensionless distances defined in Eqs. B3 and B4
viscosity of the medium
rotation angle
transmission coefficient of transition state theory; running index for charges Q on a row
running index for repeats on the rotor	µH
chemical potential of protons	H
electrochemical potential of protons
charge number of charges Q on rotor circumference (±1)	H
proton stoichiometry (number of protons transferred per revolution)
rotation angle as a fraction of repeat angle	j
phase shift of the jth channel
angle of repeat	,,, ',,
angles defined in Fig. 1	m
membrane potential	m,0
resting membrane potential

	INTRODUCTION

TOP ABSTRACT GLOSSARY INTRODUCTION THEORETICAL PARAMETERS OF THE MODEL SIMULATION OF EXPERIMENTAL... DISCUSSION APPENDIX A APPENDIX B REFERENCES

The rotary motor responsible for the spinning of a bacterial flagellum is one of the most intriguing of microbiological systems, and it presents a major challenge from the viewpoint of bioenergetics. A flagellum is a complex macromolecular machine that can be divided into three parts: 1) the filament that protrudes from the cell body and has a helical shape, 2) the basal body that is anchored to both the outer cell wall and the cell membrane, and 3) the hook that connects the filament to the basal body (reviewed in Caplan and Kara-Ivanov, 1993; Macnab, 1996). The basal body consists of a central rod and 5 ring-shaped structures. Two rings (L and P) are adjacent to the cell wall and are thought to act as a bushing for the rod. Two rings (M and S) are adjacent to the cell membrane and are thought to constitute the rotor of the motor. The fifth ring (C) is bell shaped, protrudes into the cytoplasm, and is probably built from the components that are responsible for switching (see below). A ring of particles (also called studs or force-generating units) are embedded in the cell membrane around the M ring and are thought to act as the stator of the motor. The MS ring consists of the proteins called FliF and FliG, whereas the particles are formed by the proteins called MotA and MotB. The proteins FliM and FliN, together with a part of FliG, form the C ring.

The flagellar motor is a mechanochemical energy converter. Its driving force is the difference in electrochemical potential of protons, _H, between the periplasmic space (the space between cell wall and cell membrane) and the cytoplasm. The output force is the torque exerted on the filament, and the conjugate flow is the frequency of rotation of the filament with respect to the cell body. Flagella can rotate counterclockwise (CCW) or clockwise (CW) as seen looking from the tip toward the cell body, without a reversal of the driving force. All flagella of a cell rotate in the same direction at any given time. Switching between rotation directions occurs spontaneously, and the switching frequency is modulated by chemotactic agents (reviewed in Eisenbach, 1996). When rotating in CCW mode, the filaments of all flagella are bundled together, and the cell swims linearly with an approximately constant speed. After switching to CW rotation, the flagella fly apart and the cell tumbles. The steady state of rotation after switching is reached within milliseconds, i.e., the inertia of the rotating parts is negligibly small. At steady state, the motor rotates more or less smoothly, although a stepping of the motor can be deduced from the analysis of fluctuations of its frequency (Samuel and Berg, 1995, 1996). These features form the basis for any model of the flagellar motor.

The mechanism of coupling of the transmembrane flow of protons to the rotation of a flagellum is not as yet understood. A wide variety of models of the flagellar motor have been developed in recent years (for a review see, e.g., Caplan and Kara-Ivanov, 1993; Berg and Turner, 1993; Schuster and Khan, 1994). Among these, the two types of mechanism that have been analyzed quantitatively, in terms of what was known about the structure and function of the motor at the time, are those based on fixed elastic elements analogous to muscle cross-bridges (Berg and Khan, 1983; Läuger, 1988; Meister et al., 1989), and those based on electrostatic interaction (Berry, 1993; Doering et al., 1995; Elston and Oster, 1997). Coupling between the linear motion of protons and rotation can be accomplished by means of a helical array of rotor elements interacting with a linear array of stator elements. Helical arrays or tilted rows were suggested both by Läuger (1977) and Macnab (1979). Berry (1993) was the first to consider a purely electrostatic model in which no structural complementarity is required between the rotor and the force-generating units. He assumed the presence of alternating tilted rows of positive and negative charges around the rotor, and showed that torque can be developed in such a system. Unfortunately, his analysis was over-simplified and led to a number of incorrect and misleading conclusions. However, because Berry's concept is both electrostatically sound and physicochemically convincing, we have used it as the basis of a new model that takes explicit account of the presently known structural and functional aspects of the flagellar motor.

	THEORETICAL

TOP ABSTRACT GLOSSARY INTRODUCTION THEORETICAL PARAMETERS OF THE MODEL SIMULATION OF EXPERIMENTAL... DISCUSSION APPENDIX A APPENDIX B REFERENCES

Torque balances and sign convention

It has become customary to assign a positive sense of rotation to a flagellum that rotates CCW when viewed from the outside of the cell. Similarly, by convention the externally applied torque T_ext causing such a rotation is considered positive. Hence, in the steady state of rotation of a tethered cell

(1)

where T_mot denotes the torque generated by the motor, and f_bt is the rotational frequency of the cell body measured in Hz. The frictional drag coefficient of the cell body rotating about a tether is given by k_bt, and  denotes the viscosity of the medium. Because T_mot represents a torque exerted by the stator on the cell body and by the rotor on the flagellar filament, the torque balance for a freely swimming cell at steady state reads

(2)

Here, f_br and f_fl denote the body-roll and flagellar (bundle) frequencies, respectively, whereas k_br and k_fl are the corresponding coefficients relating frictional drag to viscosity. T_tor represents torque due to torsion in the hook/filament complex which, as an approximation, may be assumed to be proportional to f_fl. The motor frequency f_mot, i.e., the rotational frequency of the rotor with respect to the cell body, is related to the different frequencies by

(3)

under tethered or swimming conditions, respectively (Caplan and Kara-Ivanov, 1993).

We define positive proton flow to be directed from the extracellular space to the cytoplasm. Accordingly, the positive driving force for protons, i.e., the electrochemical potential difference _H, must also be directed inward. Hence _H = _H,ex  H,in = µ_H + e_om, where the chemical potential difference µ_H = µ_ex  µ_in = kT ln 10 (pH_in  pH_ex), and the membrane potential _m = _ex  in. Here e_o, k, and  denote the elementary charge, Boltz-mann's constant, and the electrical potential, respectively, whereas the subscripts ex and in refer to the suspending medium and the cytoplasm, respectively. Note that pH and potential are assumed to be equal in the periplasmic space and in the suspending medium.

Electrostatics

Figure 1 shows a sketch of the rotor, i.e., the MS ring, which is approximated as a disk with radius R and thickness d_m. Two adjacent tilted rows of q charges, one a negative set ( = 1) and the other a positive set ( = 1), form a repeat unit. The number of repeats is r, and hence the angle of the repeat unit is

(4)

The pitch of the rows, P, is defined for our purposes as /, where  is the angular offset between the top and bottom surfaces of the rotor. The parameter  measures the rotation angle  as the fractional remainder of the quotient /,

(5)

and varies between 0 and 1. The free energy G_r for a proton at the position (x_o, 0, z) that arises from the electrostatic interaction with all charges Q_,, on the rotor can be calculated by a linear superposition using Coulomb's law (see Appendix A),

(6)

For reasons to be discussed below, the relative permittivity  is given the argument d to indicate that it is distance dependent.

View larger version (21K): [in this window] [in a new window]   FIGURE 1   Schematic diagrams of the rotor showing the principle geometric parameters, including the alternating tilted rows of positive and negative fixed charges. (A) The rotor (radius R, thickness dm), and one representative force-generating unit containing of a single channel (at a distance dc from the perimeter of the rotor), are situated within a rectangular Cartesian coordinate system as shown. The rod and hook (not shown), to which the flagellar filament is attached, project up from the center of the rotor in the z direction, toward the exterior of the bacterium. The coordinates of a particular position in the channel occupied by a proton are depicted, as well as those of a particular fixed charge Q,, on the rotor (the th charge on a row in the th repeat with charge number ). (B) Eight representative repeat units are illustrated, with the angular position of the selected fixed charge relative to its repeat unit indicated. The angle of repeat is , the tilt and corresponding horizontal offset angles are  and , respectively, and  is the angle of rotation.

Kinetics

Most of the data available to test our model were obtained in experiments performed with motors having a full complement of force-generating units, for which average frequencies were determined. Under these conditions, a smooth rotation of the motor is normally seen on all time scales except when measuring fractions of a rotation (Kara-Ivanov et al., 1995). Hence, we can safely ignore Brownian motion and use a deterministic instead of a statistical approach to the kinetics of the system. Because d/dt = 2f, it follows from Eqs. 4 and 5 that

(7)

Figure 2 shows a kinetic scheme for the transport of protons through the channel of a force-generating unit. The circles represent binding sites for protons, and are filled if a proton is bound. The number of binding sites s includes two mandatory outer binding sites adjacent to the aqueous phases and any additional inner binding sites that may be present. Let p_i denote the probability of the ith state, then, differentiating the probabilities with respect to time t, we have

(8)

(9)

(10)

(11)

(12)

(13)

As shown in Appendix B, the forward and backward rate constants, ^*_i,i+1 and ^*_i+1,i, include intrinsic transition probabilities per unit time, and several factors depending on the electrostatic interaction energies G_r, G_c, and, where appropriate, G_p (cf., Eqs. 6, 26, B16, and B17), the membrane potential _m, and pH_ex and pH_in. The scheme used here includes s + 2 states, although, in principal, it could include many more states. However, as will be discussed below, these states can be omitted here without loss of precision.

View larger version (23K): [in this window] [in a new window]   FIGURE 2   Channel kinetics. (A) Diagram showing the possible states of the channel following proton binding from, or release to, the periplasmic space or the cytoplasm, and internal proton transitions. Each line represents a forward and backward reaction to which a rate constant is assigned. The channel has s binding sites numbered in sequence from the cell exterior to the cell interior, and each state is labeled by the number of the site occupied. The empty state is designated s + 1, and the only state considered having more than one bound proton is designated s + 2. (B) Schematic cycle diagram showing the two constituent cycles designated a and b, and the sign convention adopted for the cycle fluxes. (C) The positions (along the z coordinate) of the ith proton-binding site zi and the ith transition state zi# in the channel.

Torque generation and proton flow in a single channel

The torque generated by a charge with charge number Z on the channel axis is given by Z[G_r/]_,z. The torque T_i generated by the ith state of a channel then becomes

(14a)

(14b)

where Z_j denotes the charge number of the jth binding site. Note that the partial derivatives are weighted by the probabilities p_i of the channel states. The partial derivative of G_r with respect to  is, in view of the sign convention chosen,

(15)

The torque associated with a given channel is

(16)

The flow J_i,j between states i and j is given by

(17)

The flows of protons from the periplasmic space into the given channel, J_ex, and out of the channel into the cytoplasm, J_in, then become

(18)

(19)

In general, J_ex() and J_in() are not equal at any given value of . However, the integral of these flows over  (from 0 to 1), i.e., the number of protons entering and leaving the channel during a rotation equal to a repeat unit, must be equal.

Channel ensemble

In general, the motor includes n channels, which we arbitrarily number from 1 to n. As is evident from Fig. 3, the value of  depends on the geometrical arrangement of the channels and hence, in general, is different for each channel. This is taken into account by assigning a phase shift _j to the jth channel. The total torque of the motor T_tot() and the total proton flows J_ex,tot() and J_in,tot() generated by the channels then become

(20)

(21)

(22)

For a symmetric arrangement of channels, we find

(23)

The averages of T_tot and J_ex,tot or J_in,tot are obtained by integration and yield the motor torque and the proton flow through the motor, respectively,

(24)

(25)

Another aspect arising from the channel ensemble pertains to the electrostatic interaction between protons in the channels. The free energy G_c for a proton at the position {x_o, 0, z} in the channel shown in Fig. 1 A that arises from the electrostatic interaction with all protons in the other channels can be calculated by a linear superposition using Coulomb's law (see Appendix A),

(26)

Note that the interaction terms are weighted by the probabilities p_i of the channel states, because the positions of the protons in a channel are only known for a given state.

View larger version (11K): [in this window] [in a new window]   FIGURE 3   Schematic diagram showing a situation in which the number of repeat units constituting the rotor is unequal to an integral multiple of the number of channels, and hence the two are not in register. The channel indicators indicate the positions of a symmetrical array of channels around the rotor.

Numerical computations

It is convenient to scale all rate constants relative to a particular rate constant, and, for this purpose, we choose _2,1. Accordingly, making use of the equilibrium constants in Eqs. B14-B17, we write

(27)

(28)

(29)

(30)

(31)

Note that ₁  1. When inserting Eqs. 27-31 into Eqs. 8-12 and dividing the resulting differential equations by _2,1, one obtains, by Eq. 7,

(32)

where f'_mot denotes a dimensionless frequency defined as

(33)

At static head, i.e., f'_mot = 0, we get the variables p_i() as the solution of the system of linear equations derived from Eqs. 8-13 with dp_i/dt set to 0. In the case f'_mot  0, the differential equations require numerical integration (taking into account Eq. 13), which was performed by the network simulation technique (Walz et al., 1995a). The integration was carried out over two repeats, and only the results from the second repeat were used. Moreover, the results were checked for compliance with the equality on the right hand side of Eq. 25. From the p_i() values, all other variables can be calculated by means of Eqs. 14-25.

Because the free energy G_c from the interaction of protons in the channels depends on the probabilities p_i of the states (Eq. 26), which, in turn, are also determined by G_c via the rate constants ^*_i,j (see Appendix B), an iterative procedure is necessary. The p_i values obtained in a given iteration step were used to calculate G_c for the next step, and the procedure was finished when both T_mot and J_H changed less than 1% in the subsequent iteration step.

Adjustment of parameter values was done either manually or, whenever possible, by means of the nonlinear fitting program MODFIT (McIntosh and McIntosh, 1980).

	PARAMETERS OF THE MODEL

TOP ABSTRACT GLOSSARY INTRODUCTION THEORETICAL PARAMETERS OF THE MODEL SIMULATION OF EXPERIMENTAL... DISCUSSION APPENDIX A APPENDIX B REFERENCES

Electron micrographs prepared by freeze-fracture (Khan et al., 1988) show disk-like structures that are surrounded by a ring of particles exhibiting a diameter of about 5 nm. Both the diameter of the disk and the number of particles vary with species as demonstrated by the values obtained for Escherichia coli and Streptococcus (Table 1). The diameter of the disk for Salmonella typhimurium was deduced from image reconstructions of isolated motor complexes (Sosinsky et al., 1992). The proteins FliF and FliG were found to be present in equimolar ratio and 26 copies of FliF were estimated for the M ring of S. typhimurium (see also Macnab, 1996; a somewhat higher but not inconsistent value for FliG was reported by Zhao et al., 1996). Based on these data, we have chosen the values of R listed in Table 1 and have calculated the missing values for the number of particles and FliF, FliG by assuming their proportionality to R.

Because FliG carries clusters of charged residues (Kihara et al., 1989), which were found essential for torque generation (Lloyd and Blair, 1997) it is reasonable to assume that the lines of charges are associated with this protein. Moreover, the particles were shown to be the MotA/MotB complexes (Khan et al., 1988). It would seem reasonable to set r and n equal to the number of FliG subunits and the number of particles, respectively. However, the simulations can reproduce the experimental data only if two repeats per FliG and two channels per MotA/MotB complex are assumed, thus yielding the values for r and n listed in Table 1. For q and P, the smallest values were chosen (Table 2) which yield a sufficiently large torque.

No detailed structural information about the channels in the MotA/MotB particles is yet available. However, motility-deficient mutants indicate that the important part of these channels is the domain formed by -helices adjacent to the cell membrane (Schuster and Khan, 1994). Hence, we have chosen d_m, which is also the length of the channels, equal to 5 nm, i.e., the average thickness of a biological membrane. Sharp et al. (1995), using tryptophan-scanning mutagenesis of MotA, could not find any typical proton binding sites within the channel and concluded that these sites are most likely provided by water molecules interspersed between the -helices. Hence, we set the charge number Z_i of all binding sites to 0. We assume the channel axis with the sites to be located close to the boundary of the particles (d_c = 0.65 nm) such that an equal spacing of all channels in the particle ring exists. Moreover, we assume an equal spacing of the proton binding sites in a channel, i.e., (Fig. 2 C)

(34)

and a symmetrical barrier for the transition state (Walz and Caplan, 1995),

(35)

which also yields (Eqs. B3, B4, and 34)

(36)

The parameters s, pK_1,ex, pK_1,in, K_i,i+1, G_p, and _i (i = 5-8) are adjusted to yield the best fit to the experimental data. The values for pK_2,ex, and pK_2,in are calculated by means of Eqs. B16 and B17. The equilibrium constants K_i,i+1 of the inner transitions have to comply with Eq. B18, so only s  2 values for the equilibrium constants need to be adjusted. The relative rate constants _i for the inner transitions are all set to unity, whereas those for the outer transitions are assumed to be equal but are adjusted. Values for the rate constant _2,1 were found to vary considerably even for different cells of the same species under identical conditions. Hence, this parameter is not included in Table 2, but its value will be given in the context of each system to be discussed. Because the geometry of the motor is found to be similar in the investigated species and because there is no evidence for differences in the structure of the MotA/MotB particles of different species, the parameter values listed in Table 2 are taken to be valid for all species.

A possible dependence of pK_1,ex, pK_1,in, and the equilibrium constants K_i,i+1 on temperature is not taken into account. Because there is no information available on the effect of temperature on proton binding in the MotA/MotB proteins, any assumption about such a dependence would be purely speculative. Note, however, that pK_2,ex and pK_2,in are slightly temperature dependent as is evident from Eqs. B16 and B17. In contrast, the rate constants _i,j strongly depend on temperature. This will be treated below, together with the arguments for a temperature independence of the _i.

Strictly speaking, the use of a relative permittivity (or dielectric constant)  in calculations of electrostatic interactions is legitimate only for macroscopic phases, and its validity on the microscopic level is at least questionable. In fact, the polarizability in macromolecular complexes depends on atoms and bonds, and the electric field in such complexes is influenced by the considerable change in polarizability at the boundaries of the macromolecules that are exposed to water. These effects can be taken into account providing the structure of the complexes is known at the atomic level (Sharp and Honig, 1990). However, experience has shown that, for systems where this information is lacking, a reasonable approximation can be found by means of a relative permittivity whose value depends on the distance between the charges. Both a proportionality to this distance (Harvey, 1989) and an exponential dependence (Elston and Oster, 1997) have been used; we have chosen the former case with a proportionality constant as given in Table 2.

Two additional parameters not pertaining to the model but essential for the simulations are _m and pH_in. Data for E. coli at room temperature (Felle et al., 1980), at 30°C (Hirota et al., 1981), and 28°C (Kashket, 1982) are presented in Fig. 4 A. It seems that _m and pH_in vary with temperature, but the data are not sufficient to derive a temperature dependence, which therefore was neglected. Khan et al. (1990) reported data for Streptococcus (Fig. 4 B), whereas Shioi et al. (1980) determined _m and pH_in for Bacillus subtilis at 30°C (Fig. 4 C). The only data for S. typhimurium that could be found in the literature are _m  145 mV for pH_ex = 7 (Shioi et al., 1982), and _m = 162 ± 13 mV, pH_in  pH_ex = 7.5 (Shioi and Taylor, 1984), both at 30°C. These values are close to those shown in Fig. 4 A, so the latter were also used for this species. If not indicated otherwise, values according to the curves drawn through the points in Fig. 4 were used in the simulations.

View larger version (16K): [in this window] [in a new window]   FIGURE 4   Dependence of membrane potential, m, and pH in the cytoplasm, pHin, on pH in the suspending medium, pHex, for different species. Experimental data for m and pHin are represented by closed circles and triangles, respectively. The solid and long broken lines were drawn through the points and served to determine values for the simulations. (A) E. coli at room temperature (closed symbols, data taken from Felle et al., 1980); short broken line, m at 30°C (Hirota et al., 1981); open circles and dotted line, m and pHin at 28°C (Kashket, 1982). (B) Streptococcus at room temperature, data taken from Khan et al. (1990). (C) B. subtilis at 30°C, data taken from Shioi et al. (1980).

	SIMULATION OF EXPERIMENTAL RESULTS

TOP ABSTRACT GLOSSARY INTRODUCTION THEORETICAL PARAMETERS OF THE MODEL SIMULATION OF EXPERIMENTAL... DISCUSSION APPENDIX A APPENDIX B REFERENCES

The analysis of the model yields the torque T_mot generated by the motor as a function of the dimensionless frequency f'_mot. In experiments, however, only frequencies at the steady state of rotation can be measured. Under this condition, Eqs. 1-3, and 33 can be used to convert the simulated torque-frequency relationship into the form of the experimental data. The values for k_bt, k_br, k_fl, and _2,1 are thereby adjusted.

Torque-frequency relationship for tethered cells

When all filaments except one are removed from a bacterial cell and the remaining filament is glued to a support, one obtains what is known as a tethered cell. The rotation frequency of the cell body can then be measured, and an external torque T_ext can be applied by means of a rotational electric field. Typical results obtained by Berg and Turner (1993) in such a setup with an E. coli cell deficient in switching are shown in Fig. 5. Because it is difficult to calculate the torque exerted by the rotational electric field, T_ext is expressed by means of a relative external torque T_ext,r, such that

(37)

If too large a T_ext (in either direction) is applied, the motor is irreversibly broken. For such a cell, frequency should be proportional to T_ext (cf., Eq. 1 with T_mot = 0), as shown by the straight line through the origin in Fig. 5 A. The slight deviations of the data points from this line indicate that the motor was not fully broken in this case. A fit of these data with a model that includes a factor accounting for the active fraction of the unbroken motor yielded the result shown by the broken line in Fig. 5 A. This fraction turned out to be 14%, which may suggest that three channels remained active. The heavy line represents the result of the simulation for an intact motor. It fits the experimental data except for the two points indicated by open symbols. However, these points are most likely not real but artifacts caused by an inhomogeneity in the rotational field (Berry and Berg, 1996). In a case where this artifact is absent, the experimental points are well represented by the simulated curve (Fig. 5 B). Moreover, the point on the abscissa of Fig. 5 A appears to indicate that the motor opposes clockwise rotation by means of a barrier. Using optical tweezers, however, Berry and Berg (1997) showed that this is not the case.

View larger version (19K): [in this window] [in a new window]   FIGURE 5   Dependence of body rotation frequency, fbt, of tethered E. coli cells on externally applied torque. A rotational electric field with variable strength was used to subject the cell body to different torques, which, however, can be given only as a relative quantity Text,r in arbitrary units. Circles and triangles represent an intact and an almost fully broken motor, respectively; open circles indicate artifactual data. (A) Data taken from Fig. 10 of Berg and Turner (1993); temperature 32°C, pHex = 7.2, hence pHin = 7.43 and m = 134 mV (cf. Fig. 4 A). (B) Data taken from Fig. 6 b of Berry and Berg (1996); room temperature, pHex = 7, hence pHin = 7.43 and m = 125 mV (cf. Fig. 4 A). In both cases, switching-deficient mutants were used. The curves represent results of simulations with the model using the parameter values listed in Tables 1 and 2. The solid straight lines through the origin indicate the behavior of a fully broken motor. Parameter fitting yielded the following values: 2,1 × 105 s = 1.1 ± 0.2 and 4.4 ± 0.7, kbt/(nN·nm·rad1·Hz1) = 0.112 ± 0.004 and 0.075 (manually adjusted because of overparametrization), kext/(nN·nm·rad1) = 0.213 ± 0.008 and 2.79 ± 0.02 for the data in (A) and (B), respectively. Thus, one arbitrary unit of Text,r corresponds to 0.21 and 2.8 nN·nm·rad1, respectively (Eq. 37). The value of the fraction of unbroken motor used when simulating the broken motor data in (A) turned out to be 0.14 ± 0.02.

This behavior of tethered cells in a rotating field appears to be independent of the measuring technique and the species used. Whereas the frequency of the rotating field in the experiments presented in Fig. 5 was set at 2.25 MHz and the field strength was varied by means of the voltage applied to the electrodes, Iwazawa et al. (1993) used frequencies comparable to those of rotating cells and determined the minimal voltage necessary to synchronize cell and field rotation. Their results, obtained with 9 small and 11 large cells, are shown in Fig. 6, together with the simulated curves. The relatively large scatter of the points arises from a considerable variation between cells (see below), which could not have been taken into account because individual cells are not identifiable in the data set. Unfortunately, these authors did not inactivate the motors, and the behavior of the cells under this condition (thin lines in Fig. 6) cannot be checked against experimental data. Washizu et al. (1993) applied the same procedure as Berg and Turner (1993) but with six instead of four electrodes and a field frequency of 0.5 MHz. They used cells of a switching-deficient mutant of S. typhimurium and inactivated the motor by ultraviolet (UV) irradiation. The results thus obtained are depicted in Fig. 7, and seem to indicate a slight hysteresis, which, however, was not observed by Berg and Turner (1993) and, most likely, arises from limited experimental accuracy. In this case, too, simulations w