Dielectric Motors—A new hypothesis for the bacterial flagella

J. theor. Biol. (1989) 139, 39-59

Dielectric Motors--A New Hypothesis for the Bacterial Flagella G. FUHRt AND R. HAGEDORN

Department of Biology, Humboldt- University of Berlin, Berlin, G.D.R. (Received 21 June 1988, and accepted in revised form 17 January 1989) A new hypothesis to explain bacterial flagellar rotation developed in its basic idea in a recent paper (Fuhr & Hagedorn, 1987, Stud. Biophys. 121, 25) is carried on and calculated in detail. The principle of motor action is related to dielectric motors driven by a constant electric field, called in the literature "Quincke-rotation". This model is based on polarization and charge relaxation processes on the interface of dielectrics (flagellar base) and is related to some intermediate molecular configurations which allowed the cell to determine the direction of flagellar rotation, as well as to start and stop the motor. A quantitative analysis of the flagellar rotational behavior related to different models of the flagellum is given, which shows good agreement with experimental observations described in the literature. Several motor variants are discussed and the principle is extended to macromolecules incorporated into the cytoplasmic membrane.

Introduction In contrast to eukaryotic flagella and cilia, bacterial flagella propel the cell by rotating. Flagellar rotation is driven by a m o t o r anchored in the cell m e m b r a n e , and this rotation is regulated in response to information transmitted by chemoreceptors on the surface of the cell. Proteins required for energy and sensory transduction have been identified and localized in the cytoplasm and cell m e m b r a n e (for review see Adler, 1975; Silverman, 1980). Flagellar function can be divided into several subfunctions, such as motor assembly, energy conversion and sensory processing. The m o t o r is the basal structure, and energization of the m o t o r probably requires mot proteins (products of mot A and mot B genes) around the basal structure in the inner m e m b r a n e (Szmelcman & Adler, 1976; Silverman, 1980). Sensory processing is accomplished by interacting proteins dispersed in the periplasm, cytoplasm and the m e m b r a n e system o f the cell. The motion of bacteria, flagellar m o r p h o l o g y and development, as well as the genetic mechanism were investigated using mutants defective in t h e synthesis or function of the flagellar system and in chemotaxis (Silverman, 1980; Silverman & Simon, 1972, 1974; Stocker, 1949; Suzuki et al., 1978). Mutants which fail to assemble a recognizable flagellar structure are calledfla-mutants. Mot- mutants assemble a morphologically complete flagellar structure, but the flagella of these cells are incapable of rotation in either direction. Che- mutants possess structurally complete flagella that rotate, but the tTo whom all correspondence should be addressed at Humboldt-Universit§t zu Berlin, Department of Biology, Invalidenstr. 42, Berlin 1040, G.D.R. 39 0022-5193/89/130039 + 20 $03.00/0 © Academic Press Limited

40

G. FUHR

AND

R. H A G E D O R N

cells are not capable of chemotaxis. Investigations on these distinctive phenotypes showed that flageilar function is thus the result of the interaction of at least three subfunctions: motor assembly, energy conversion and sensory processing. The structure of the flagellum is well known, and has been studied extensively by electron microscopy (De Pamphilis & Adler, 1971; Dimmitt & Simon, 1971). The flagella is composed of three morphologically distinct regions: the helical filament (assembled from a single protein subunit, flagellin), the hook and the basal structure (for dimensions and localization, see Fig. 1). The maximum length of the flagella is 20/~m, whilst the average diameter is 10 to 20 nm. The basal structure is the most complex part of the flagella. In case of Escherichia coli four rings are mounted on the rod, whereas flagella from Gram-positive bacteria have only the two inner rings (S, M). In E. coil the outer ring is attached to the outer membrane and the inner ring to the cytoplasmic membrane. The M- or the M- and S-ring are the flagellar motor in the strict sense. By the use of mutants defective in various steps in oxidative phosphorylation, it has been shown that the source of energy for flagellar rotation is an intermediate of oxidative phosphorylation, specifically a proton motive force or in particular the transmembrane potential (Thipayathasana & Valentine, 1974; Larsen et al., 1974). Rotation could be clockwise or counterclockwise, and it was suggested that the ability to modulate the direction of rotation was the basis for the mechanism of chemotaxis and the action of the MOT proteins located in the neighbourhood of the M- and S-ring (Silverman & Simon, 1974; Berg, 1974). The motion of the intact bacteria is of two types: predominantly straight movements, which are occasionally terminated by abrupt changes in the direction, called twiddles or tumbles. Straight movement is correlated with clockwise flagellar rotation whilst the tumbling is accompanied by conterclockwise rotation of the flagella or inverted, respectively (Silverman, 1980). Normal flagellar rotation speeds reaches values of more than 40 rev/sec. Several mechanisms have been offered to explain how the electrochemical gradient across the membrane might drive the flagellar motor (Berg, 1975; Lauger, 1977; Adam, 1977; Macnab, 1979; Glagolev, 1980; Oosawa & Masi, 1982; Khan & Berg, 1983; Berg & Khan, 1983; Jou et al., 1986). The spectrum of explanations extends from cytomembranous streaming (Adam, 1977) via protometers (Glagolev, 1980) to the concept of flagellar rotation as a nonequilibrium phase transition (Jou et al., 1986). All calculations, of course, are approximations in every model, but what needs to be shown is (a) that the derived torque is a driving one, and (b) that it is high enough to induce flagellar rotation of 40 rev/sec. This paper will restrict itself to the bacterial flagellar motor. A new hypothesis is developed, which contains only parameters which have well-defined physical meanings that may, in principle, be related to the molecular properties of various structural elements. In a recent paper (Fuhr & Hagedorn, 1987) we developed the idea of dielectric motors in relation to polarizable macromolecules and bacterial flagella. In this paper now the flagellar model is extended and detailed calculations are added. We aim to develop a better understanding of the mechanism of the flagellar rotation using a closed derivation based on

DIELECTRIC MOTORS

41

/

1

• ' d II . ~ I I ~ - -

#." #-o,

#..."

;',-'-.',' • -'.-',,..,*~-'_'"~:'-'~-~,;'I ~

1

1

1

1

1

1

1

1

1

~

i

i

-

,,

-

~

•

~,

8

~"..... ":":"~~22'":::~:::~" a ' ~ :I~~,x, ,~~. . . ~ .lOnnl '

'

10

g

FIG. !. Structure of the basal flagellar part of E. coli, and its incorporation of the four rings into the corresponding cell compartments in connection with Silverman (1980). l--filament 7--rod 2--hook 8--S-ring 3--L-ring 9--protein complex 4--cylinder 10--M-ring 5--P-ring 11--cytoplasmic membrane 6--peptidoglycan layer 12---outer membrane

classical electrodynamics. The article is divided into three sections dealing with rotation of small dielectric particles like bacterial flagella, as well as special membrane proteins: (i) characterization of the fundamental principle of dielectric motors common to all systems; (ii) calculation of bacterial flageilar rotation;

42

G. FUHR AND R. HAGEDORN

(iii) discussion of the theoretical results. Calculation o f the driving torque is given in the Appendix.

Mechanism of Dielectric Motors Small spherical and cylindrical particles would rotate when immersed in liquids and subjected to electromagnetic fields. This p h e n o m e n o n was first analysed by Hertz in 1881 and recorded by Quincke in 1896. Rotation occurs as a result of polarization processes and charge separation at the interfaces of dielectrics (such as the rotor and the surrounding medium). When a suitable potential difference is applied the rotor continues to move in whichever sense it is initially displaced (Fig. 2). I f the rotor is displaced two mechanisms which are effectively in opposition now become operative. Firstly, there is a net torque acting on the rotor due to non-alignment of the field and charge m a x i m a axes denoted with the dipole P. The other effect arises because the equilibrium charge distribution on the rotor has been

® ~,, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ "

,/

~ \ \ \ k \ \ k \ \\\N\\RXNXXXXX\\X\\\K\

F

F ~a

~ XXX\\\\\\\\\ \ \\ \\\X\\\\\\\\~

(a¿

gk \ X \ \ \ \ \ \ \ \ \ \

\\\\\\\

\ \\\\\\\'~

Q (hi

FIG. 2. Charge distribution around the rotor surface following polarization by an exogenous electrical field. A--rotor stationary, B--after rotor displacement or in steady motion, p is the effective vector dipole moment, e,,, e~, G~. and G~ are the external and internal permittivity and conductivity, respectively.

DIELECTRIC

MOTORS

43

disturbed. A current flows through the liquid in an attempt to re-establish the original charge configuration. Since the liquid is low conductive a significant time, characterized by the relaxation time ~', would be required for this latter condition to be attained so that with the rotor in motion a steady-state charge distribution is established instead [Fig. 2(b)]. Stable rotation is observed when the driving torque is exactly balanced by that due to viscous drag. The axis of rotation always lies in a plane perpendicular to the electric field vector (Hertz, 1881; Quincke, 1896; Jones, 1984). There are three different cases of particle rotation in dependence on the applied field form: (i) rotation in electrostatic fields, (ii) rotation in alternating fields, and (iii) rotation in rotating fields. Case (i) is called "Quincke rotation", case (iii) "Electrorotation" (Quincke, 1986; Jones, 1984; Glaser & Fuhr, 1986). Quincke rotation was observed on macroscopic bodies and theoretical analysed by (Seeker & Scialom, 1968; Coddington et al., 1970; Simpson & Taylor, 1971). Rotation of ceils (plant protoplasts, yeast cells) was observed in alternating electric fields by several authors (Teixeira-Pinto et al., 1960, Fiiredi & Ohad, 1964; Pohl & Crane, 1971). A fundamental description is given by Holzapfel et al. (1982). Finally, measurement of the spin of cells in rotating high-frequency electric fields (Electrorotation) is a new method in field of biological research (Arnold & Zimmermann, 1982). Particle rotation is the result of the same charge separation but induced by a rotating field vector. In difference to Quincke rotation the direction of rotation is determined by the electrical properties of both the particle, as well as the surrounding medium and the field frequency. This technique has recently been shown to be a valuable method for studying the electrical behavior of single cells and organelles. For theoretical descriptions of this phenomenon see Fuhr et al. (1986) and Fuhr & Kuzmin (1986). All cases (rotation in constant, alternating and rotating electrical fields) are possible mechanisms by which are might-seek to interpret bacterial flagellar motion, but the existence of a static transmembrane electrical potential differences indicates Quincke rotation to be the most reasonable or simple type of mechanism. Therefore, our calculations are restricted to the basic expressions relevant to Quincke rotation. The general equation of motion of a symmetrical particle around an axis of symmetry (z-axis) is given by 003,%= N - N F - N a (1) where O is the moment of inertia, 03, is the acceleration ez is the unit vector in z-directional, N is the driving torque, N F is the frictional torque and Na is the torque due to a possible repulsion. The quantities O, NF, N and Na depend on the geometry of the moving particle. In the case of flagella the driving part and the filament (see Figs 1 and 3) lead to different terms in eqn (1). In Table 1 these terms are summarized for flagella of type A and B according to Fig. 3. The difficulty in calculating the speed of flagellar rotation lies in the derivation of the driving torque.

44

G. F U H R

AND

R, H A G E D O R N

i.:Re (a)

dl

•----.~- .--

• ~ .

External

(c)

_1

S* Membrane

~)--

r

I

solution

FIG. 3. Denotions of the parameters used for calculation of flagellar motion. The flagellum is divided into a driving part (basal body) incorporated into the membrane and a driven part (filament) denoted by (*). A--flagellum with spherical basal part, B--flagellum with cylindrical basal part, C--flagellum with single-shell basal part.

TABLE 1

Moment of inertia 0 and frictional torque NF used to calculate flagellar motion according to flagella of types A and B (Fig. 3). M is the mass, ~7 is the coefficient of microvicosity around the driving part, 77* is the coefficient of viscosity around the cylindrical filament and KR is a constant related to the repulsing force (for the other denotions see Fig. 3). flagella type

O

A

MR2+M ~-*-R*24

B

MR2+2

Nv =

2 M__~*2R*:

Ctorez

Na

8~R3to"e~+4rr*g*2s*t°"e=

KRR2fa'dte~

41r~R2Sto,ez+4Tr*R*~-S*to,.ez

KnR2f to dtez

DIELECTRIC

MOTORS

45

A very simple solution to the problem is to use the effective dipole concept to calculate the torque N that produces particle rotation. As shown by Jones (1984), this results are identical to that derived with the Maxwell-stress integral (for details see Appendix). In the case of Quincke rotation, and also for Electrorotation, the dirving torque is N=PxE

(2)

where P is the effective dipole moment due to polarization and E is the field strength on the surface of the particle. The general equation common to all rotational phenomena is N = A. B. E 2

to,/to~ 1 + (Wr/Wc) 2

(3)

where to, is the relative velocity between the field vector and the particle, w~ = 1/~" is a constant describing the relaxation of the field induced charge separation and A and B are constants depending on the geometry and dielectric properties of the driving flagellar part (sphere or cylinder, respectively) and the surrounding medium (see Table 2; for derivation see Appendix). In general, w, is the relative rotational velocity between the field and the particle: (4)

Wr ~--"//O.)lield -- O.)particle//.

In the case of Quincke rotation Wne~d= 0. In the case of Electrorotation in good approximation tOpa~icte= 0. Under steady state conditions the driving torque is exactly balanced by the frictional torque, so that (5)

N = NF.

Following the acceleration oJ, = 0, N n = 0 and w, =-t%a~ticte = - % . The steady state angular velocity (%o0) is given by the expression O)poO ~--"

. tOc -

• E

(6)

(for A, B, C see Tables 1 and 2). TABLE

2

Characteristic relaxation frequency wc and values o f the constants A and B according to Fig. 3 and (3). flagella type

A, B

w,. = l / r

A = 12¢reoeeR 3

A

B

G~e,. - G,,e~

(el +

2 % ) ( G i + 2G,,)

G~+2G~ e0(e ~+ 2 % )

A = 8w%e,.R2S B=

G i e ,, - G,.el

G~ + G,.

(e~+ e,,)(G~+ G,.)

eo(e~ + t,.)

46

G. F U H R A N D R. H A G E D O R N

From eqn (6) it follows that rotation occurs only if -A.B - - . C

E 2 > oJc.

(7)

Therefore, rotation of particles in electrostatic fields is found to depend on a threshold value of the field Ecrit =

[c ],: oac

(8)

Additionally, the criterion for Quincke rotation of a homogeneous particle is

Geei>Giee,

(9)

otherwise the torque is not a driving but a breaking torque and no rotation occurs. This is also the criterion for a Maxwell-Wagner type of dielectric dispersion and indicates the close connection between both phenomena. The direction of rotation in electrostatic fields is determined by the initial displacement of the particle, e.g. by external forces or Brownian motion. Several authors observed "spontaneous" rotation of a particle in either direction, when wpl,=o)=0 and E > Ecrit , which is caused by field inhomogenities (Secker & Scialom, 1968). The electric power input given by the product of current I and the voltage U can be equated with the mechanical work done by the particle against viscous drag: Pel = U * I = N*6%.

(10)

Using eqn (3) the current flow corresponding to the motor action is given by t=

N . ~op_ A. B . U d2

u.

~o~,/co~ 1 + (oJp/o~c)2

(11)

where d is the interelectrode gap. The linear relation between the current and the driving voltage was observed by S e c k e r & Scialom (1968). From the sphere's frame of reference, the electric field appears as a circulary polarized vector, and therefore the current is also partly a displacement current, in analogy to the behaviour of capacitors or dielectrically dispersive media generally. The flagella models (type A and B) are strongly simplified. As is known from the literature (Adler, 1975; Silverman, 1980) a more detailed model of the flagellum is a single-shelled cylinder (in which the shell represents the flagellin helix, and the interior is bound water; possibly the driving part is a single-shelled sphere or cylinder too [Fig. 3(c)]. The quantities A and B are now voluminious terms and given by a = 4 rreoe,R3;

( C, B,'~ B = \-~, --~2 ]

B1 = - G s ( ee - ei) - es( G~ - Gi) dl R [ ( G , - G . ~ ) ( 2 e , . + e ~ ) + ( 2 . G.~+G~)(e~-e.~)]

DIELECTRIC

MOTORS

47

B2 = + G~(e~+2. e ~ ) + e . ~ ( G i + 2 G ~ ) (12)

+2dl

R [(e,-e,)(G~-G~)+(ee-e,)(G,-G~)]

dl

C~ = - G s ( G e - G,)----R ( G , - G~)(2G~ + G~)

C2 = Gs( G, + 2 .

G

e)

+2dl

---~ ( G, - Gs)( Ge - Gs)

where d~ denotes the thickness of the shell, and Gs and es are the shell conductivity and permittivity. For the complete torque calculation see Fuhr & Kuzmin (1986).

Application of this Mechanism to Flaggelar Rotation

It is necessary that a new hypothesis for the mechanism of flagellar motion considers the well-known phenomena described in the literature, and thus can account for the following observations: (i) flagellar rotation speed of more than 40 rev/sec; (ii) spin direction reversal and spin induction regulated by proteins ( m o t proteins?) localized in the vicinity of the basal structure; (iii) source of energy in the form of potential gradients across the membrane and the basal structure of the flagella; (iv) the driving part of the flagella is the M-ring, constituting only 5 nm of the rod; (v) the membrane potential as a source of energy should be of the order of 10 to 100 mV as measured on several bacteria; (vi) flagellar rotation exhibits a threshold value of the electric field below which rotation does not occur and the torque speed diagram is approximately linear (Lowe et al., 1987). Figure 1 shows the model of flagellar base used for calculations. Because its axis of rotation always lies in a plane perpendicular to the electric field vector, only the tangential component of the membrane potential can be used to interpret flagella rotation (Fig. 4). In other words, it is postulated that there exists a potential difference across the basal structure incorporated into the membrane. This indeed is the case since proteins, membrane pores in the neighbourhood of the flagellar ring or asymmetric water layers on the basal flagellar surface lead to field deformations or, for example, the flagellar rod crosses the membrane by an angle a (Fig. 4). Therefore an asymmetric charge distribution on the membrane interface around the flagellar basal structure is sufficient, for example by interaction of a pump and a pore [Fig. 4(a)], to produce the necessary potential difference (see Jou et ai., 1986). The permittivities and conductivities also, enter into several terms of our equation and therefore deserve special attention. Both quantities are defined for macroscopic molecular ensembles but not for single molecules. Most calculations at the molecular level were done by introducing effective permittivities and conductivities, more or

48

G. FUHR AND R. HAGEDORN

C_~'"~ H-I-

c,)

I

C

.~

(b)

2

C

(c)

(d)

f (f)

FIG. 4. Six examples describing configurations which lead to tangential field components and following potential differences across the basal structure (M-ring or parts of the rod) A--interacting pump (5) and pore (3) 1--S-ring; 2--M-ring; 4---cytoplasmic membrane, B--asymmetric charge distribution on the membrane surface around the flagellar base caused by a conductive membrane area, C--field deformation caused by two conductive membrane proteins, D--flagellar base with protein in the neighbourhood and asymmetric water layer, E--membrane deformation in the vicinity of the flagellar base, F--flagellar rod inclined to the axis across the membrane.

DIELECTRIC

49

MOTORS

less different to that measured in macroscopic media (for review see Honig et al., 1986). For the dielectrics in and around the lipid bilayers the following permittivities were assumed. Pure water has a permittivity of about 80 at 20°C. Chatiges due to temperature or ionic composition are generally negligible. Most oils and aliphatic hydrocarbons have permittivities in the range of 2-5. Permittivities in the range between 10 and 30 are reasonable for the region between bulk water and lipid hydrocarbons. Thus it is reasonable that the permittivity should vary with perpendicular distance from the interface of membranes and macromolecules between 2 and 80. It is frequently well known that proteins have permittivities between 4 and 20. Such values have been introduced to calculate molecular interactions with lipid bilayers and found useful for estimating the effects of the dipole potential in the interfacial region (Kell & Harris, 1985; Honig et al., 1986, and others). Conductivities inside the lipid bilayer and of bound water are very small. This is caused by the fact that mobile ions move to the membrane interfaces forced by charges on both membrane sides and image energy. The values discussed in the literature range between 10 -4 and 10 -1° S / m . We also used effective permittivities and conductivities in our calculations as described above. However, exact values of both the permittivities and conductivities around the basal flagellar structure and the M-ring are not available: therefore we varied all parameters as follows. At first, as the simplest model, we assume the driving structure to be a homogeneous sphere or cylinder, respectively. The set of parameters corresponding to Fig. 5 was: ee = 2 . . . 80; ei = 2 . . . 20; R = 5... 8

r/=0.1 ...0.001 Ns/m 2

Ge = 1 0 - 8 . . . 10 -1° S/m; Gi---- 1 0 - 9 . . .

* 10 -9

m;

R*

S* = 5 . . . 20

=

10 -li S/m;

77* = 0.001 N s / m 2

5

S=3...5.10-9m

* 10 -6

* 10 -9

m;

m;

d = 10 -8 m.

The threshold strongly depends on the electric properties of the driving part and the surrounding membrane (Fig. 5). The value of threshold is on the order of 10-50 mV across the flagellar part incorporated into the membrane and the rotation speed increases very fast with increasing potential difference. Variations of flagella parameters in dependence on the potential difference are shown in Fig. 5. The charge relaxation time r = 1/~oc is on the order of 1 0 - 3 . . . 1 s (see Table 2). The rotation speed can not, of course, increase without limit. Saturation of the polarization processes, as well as non-linear frictional forces on the filament (not the driving part) limit the velocity (Fig. 6). That means the flagellar rotation at high rotation speeds can lead to turbulent streaming in the external medium surrounding the filament. How non-linear frictional forces and saturation processes play a role quantitatively in the case of flagellar motility is difficult to estimate. However, the potential inducing the driving torque is high enough to propel the flagella. Note that in our calculation the frictional torque of the filament is underestimated since the flagellum is not of the sort pictured in Fig. 3 but a large spiral. Under consideration of a frictional torque ten times higher as used above we get a driving torque of several d y n . cm, as measured by several authors (Lowe et al.,

50

G. FUHR AND R. HAGEDORN 400

400 I (o)

/

(b)

3O0

/

300

2

I

/

3

20(

I00 ~ - . ~ . . . . . . . . . . . . . . . . .

IO( ,/Ill

I

J

20

J

J

~.-'-r""Y~t

t

80

Ic)

G

J

60

40

Ioo

20

0

2,/

t 40

J

t

60

,

d)

tOO

2

3I

>

,

80

J

4 200

200 5

"--

IOO

--I

3

I00

1 20

40

J

........................

60

80

I00

0

20

40

4

60

80

I00

U{mVI

FIG. 5. Theoretical rotation speed (v = %,/2~r) for bacterial flagella (type A and B, Fig. 3) as a function of the potential difference (U) across the basal flagellar part. A--variation in G, (l--10 -9 S/m, 2--10 -'° S/m) B--variation in G,. ( l - - 1 0 -v S/m, 2--10 -s S/m, 3--10 -9 S/m, 4--10 -~° S/re.) C--variation in e~ ( 1 - 2 , 2--5, 3--10, 4--20, 5--40.) D--variation in e,. (1-2, 2-10, 3-40, 4-80). 1987; M e i s t e r & B e r g , 1987) a n d 40 rev/sec.

necessary

to induce

a rotation

of more

than

B e f o r e d i s c u s s i n g t h i s r e s u l t in d e t a i l let u s c a l c u l a t e t h e r o t a t i o n s p e e d i n t h e case of a single-shelled driving body (that means for example a bound-water-filled protein sphere or cylinder, respectively) immersed in phospholipids or a thin water layer. According

to e q n s (6) a n d (12) w e u s e d t h e f o l l o w i n g s e t o f p a r a m e t e r s : el = 5 . . .

40;

G i = 10 -5 • • • 10 -7

e, = 2 . . .

20;

G,, -- 10 -7 . . . 10 -9

e,. = 2 . . .

80;

G,, = 10 -7 . . . 1 0 - ' °

51

DIELECTRIC MOTORS

2

0

~,,t

E

FIG. 6. Quantitative influence of saturation and non-linear frictional forces on the rotational behavior of dielectrics immersed in semiinsulating liquids and subjected to electrostatic fields. l--linear part; 2--range of non-linear behavior.

with a thickness of the protein wall of 3 * 10 -9 m: the other quantities are identical to the first set of parameters. As shown in Fig. 7 the situation is much more complex. In this case a driving torque always obtained provided that G~ or Ge and e~ or ee are different in comparison to Gs and es. The behavior of the rotational speed of the flagellum as a function of the potential difference across the flagellar base is demonstrated in Fig. 7 in detail. This model leads to a threshold for rotation between 0.5 and 30 mV, dependent on the dielectric properties and the frictional torque. In other words, only 5 to 10 mV potential difference across the driving flagellar base are necessary to produce more than 40 rev/sec. The length (S*) of the flagella and outside viscosity (7*) play a subordinate role up to 100 p~m and 0-1 N s / m 2, respectively, by reason of their square root dependence on the rotation speed. Note that kT corresponds to potential differences of nearly 20 mV and therefore the motor action can be influenced by termal effects. By reason of the functional arrangement of the flagellum there is no limitation that the flagellar motor can work below this value.

Discussion of the Calculated Results

As demonstrated, the basal M-ring causes the production of flagellar rotation as described in literature, even under consideration of higher frictional forces induced by a spiralic filament. For steady state motion a tangential component between 1/5 and 1/20 o f the transmembrane potential is necessary, if U,, is supposed with 50 mV. Up to now flagellar rotation occurs only if the rotation is induced by an initial displacement (for example Brownian molecular motion). Under these conditions

52

G. F U H R AND R. H A G E D O R N

1/ ooF//

,°: ,

I 0

--

200/-

20

40 40 60

80

//

0

I00

40

60

810

I00

2O

40

6O

8O

I00

(c)

3

200

100

= ~oo I 0

20

20

40

l

60

1

J

I00

80

2ooL I' ///

(f I

~//( • }

I00[ 0

I

20

40

60

BO

,J I00

0

T/.:-..-7.i,

b'(mY)

20

40

60

J

1 80

t I00

FIG. 7. Theoretical rotation speed (v) for flagella (type C, Fig. 3) as a function of the potential difference ( U ) A--variation in "0" 1--10-3 Ns/m2, 2--10-2 Ns/m2, 3- 1 0 - ~ Ns/m2, 4---1 N s / m 2. B--variation in S* (length of the filament) (1--1 I,J,m, 2--5 ixm, 3--10 ixm, 4----20 IJ.m, 5--50 ~m, 6--100 ~,m). C--variation in G, ( l - - l . 5 * 1 0 - 6 S / m , 2 - - 1 0 - 6 S / m , 3 - - 7 . 1 0 - 7 S / m , 4----6.3.10-7S/m, 5 - 6.1 * 10-7 S/m). D--variation in G, (1--10 -7 S / m , 2--10 -8 S/m. 3--10 -9 S / m , 4----5 * 10 -t° S/m}. E--variation in G,. ( 1--10 -~ S/m, 2--10 -7 S/m, 3--2 * 10 -7 S/m, 4---2.5 * 10 -7 S/m, 5--3 * 10-7 S/m). F--variation in e,. (1--80, 2--20, 3--10, 4---5, 5--2).

the direction of rotation is not determined and spontaneous rotation in either direction occurs. In the particular model discussed here, determined flagellar rotation and spin reversal, as well as starting and stopping of flagellar rotation, can be effected by protein complexes able to change their conformation or localization in comparison to the flagellar base. Such conformational changes, however, in our interpretation lead to a rapid field change across the flagellar basal structure and correspond to a field vector movement in one direction in analogy to the initial displacement of the rotor in Fig. 2. Additionally, it can be assumed that protein configurations exist, which decrease the tangential field strength below the critical value and thus rotation stops immediately [for example if the pore in Fig. 4(a) is

DIELECTRIC MOTORS

53

closed]• Therefore, all experimental findings on flagellar rotation may be interpreted in a very simple manner without additional requirements. In our opinion, and in correspondence with data in the literature, the mot proteins are the most favourable macromolecules to fulfil this regulatory function. Note that no close contact between protein and the flagellar structure is necessary• On the other hand the question arises to develop alternative constructions of the flagellar motor, since it is not obligatory that the M- and S-ring are mounted into the membrane as schematical shown in Fig. 1 and assumed in the literature (Adler, 1975; Silverman, 1980). At first, from a physical point of view, dielectric rotation occurs by charge separation and movement on the boundary layer between the dielectrics (flagellar base and surrounding medium). That means, in the case of Geei > Giee, that the net charge is localized on the outside of the boundary layer. Than the flagellum is driven by forces acting on this charge separation transformed via frictional effects to the flagellar base. A very probable construction of the basal part is shown in Fig. 8. Here the two basal rings are not incorporated into but attached to the membrane. The driven part in the strict sense is the rod between the rings ( M and S). The charge separation and resulting streaming around the rod is given by the

t

I 1

3

4

I

2

m

H-I 7 FIG. 8• Model of the flagellar motor. The M- and S-ring are bearing and isolating systems.The motor in the strict sense is the rod between the two rings surrounded by a water layer. l--pump 5--cytoplasmic membrane 2--rod (s--mot protein complex 3--S-ring 7--M-ring 4~water layer

54

G. F U H R

AND

R. H A G E D O R N

liquid layer, higher in its conductivity compared with that of the flagellar material, localized around the cylindrical part between both rings. The function of the rings using this interpretation are bearing and electric isolating systems. This configuration is a mechanical stable system and the torque high enough to induce flagellar rotation of more than 40 rev/sec. The calculated torques discussed in this paper and related to different flagellar models are the minimal values which can be assumed. Additional driving forces of the same nature as described induced by potential gradients across other parts of the flagellar base, such as the L- and P-ring, the rod, as well as the M- and S-ring can lead to an amplification of the acting forces. The most probable mechanism for generation of additional potential differences across the flagellar base are potential gradients generated in the double layer of the outer and inner membrane. Therefore, a large number of possible constructions of the flagellar motor can be developed. Detailed structural investigations of the bacterial flagella, especially the arrangement of the M- and S-ring in the membrane, are necessary before a final discussion of the driving forces is ingenious. Conclusion

In contrast to dielectrophoretic leviation, permanent dipole-field interaction and surface forces, Quincke rotation or rotation of spherical or cylindrical particles in alternating fields are largely unknown. Application of these rotation phenomena to particle treatment processes has been seriously considered in the case of Electrorotation, but only for the investigation of the electrical cell properties. Now the theoretical estimates given above on the velocity of flagellar rotation which yield the necessary driving torque lead indeed to values in the range of 20 to 100 rev/sec and can be used to interpret the mechanism of bacterial flagellar rotation. Our model corresponds with the findings described in the literature and is compatible with several models previously proposed for the motor mechanism (see Jou et al., 1986). The thresh01d-dependent Quincke rotation seems to us the simplest solution of the problem. A single or multilayer construction of the basal flagellar part (bound water, protein part, and surface layers) are probable too, but also homogeneous disks or cylinders fulfils the conditions to interpret flagellar rotation. At values higher than the threshold, however, the rotation speed increases in dependence on charge relaxation times relative rapidly (but linear) and reaches later nearly constant values due to non-linear frictional forces and saturation processes. Our calculations proof that quantitatively the observed bacterial movement and flagellar behavior can be explained. There are, of course, some effects described in the literature (Meister & Berg, 1987) which are not in agreement with our hypothesis. (1) The flagellar motor exerts maximum torque at stall, while the dielectric motor exerts zero torque at stall. (2) There are some references that the flagellar motor works too at zero transmembrane potential. The first point can be answered as follows:

DIELECTRIC MOTORS

55

The dielectric motor exerts maximum torque after displacement of less than 1/10 revolution. Therefore, it seems to be difficult to differentiate between torque at stall and the starting rotational behavior. The second point, however, is difficult to understand and must be checked detailed in future. We have suggested this mechanism on purely theoretical interest, but we think too that it should be helpful to develop a characteristic experiment controlling our hypothesis practically. On mature considerations we found that all characteristic parameter dependencies, as existence of a critical field strength, nearly linear increasing of the rotation speed with increasing potential difference, as well as spin direction reversal and the necessary field deformating protein complex near the flagellar base are even arguments used to prove the other models too. As a result of our model the rotation speed of the flagellum is dependent on the square root o f the external viscosity (7/*): this should be measurable, whereas the effect can be overshadowed by non-linear frictional forces. As indirect proof we have made measurements on a macroscopic dielectric rotor corresponding in its geometry to the basal flagellar structure, and have found a rotation under analogous conditions as described above. However, our calculations to flagellar rotation only take into account the M-ring or parts of the rod as the driving part. The torque also increases if both rings, Mand S-ring, as well as parts of the rod, are supported by a potential difference. It should be noted also that the mechanism of dielectric motion is a general principle and can therefore be extended to other macromolecules localized in or attached to the membrane as discussed in a previous paper (Fuhr & Hagedorn, 1987). Equation (3) shows that the total torque can be positive or negative and resulting driving or breaking forces are acting. That means, all polarizable macromolecules exposed to electric fields are influenced by these forces and therefore steady state motion is stimulated or supppressed, respectively. Electrostatic interactions with proteins and charged molecule interactions with lipid bilayers have been reviewed several times (Mclaughlin 1977; Warshel & Russei, 1985; Honig et al., 1986). The theoretical membrane potential profiles combining Born, image, dipole and neutral energy profiles indicate potential differences across macromolecules up to several 100 mV. Membrane proteins are thus exposed to intense electric fields, and many systems are known in which the conformations are functionally coupled to the electric field for the purpose of switching, transport and energy transduction. Voltage dependent conformational changes involve the motion of a charge or dipole along a direction normal to the membrane surfaces. Displacement currents associated with the opening of voltage-dependent ionic channels have been detected and referred to as gating currents. The mobility of membrane molecules such as protein complexes is more or less limited by frictional or restoring forces of different nature. Therefore, it seems that only a minor part of most molecules is able to rotate in the way described for bacterial flagella. The other molecules can move in one or more direction until the driving force is balanced by the restoring forces. In this case the complete equation

56

G. FUHR AND R. HAGEDORN

of motion [eqn (1)] holds. Depending on the coefficients and interacting forces in the equation of motion we obtain solutions describing molecule rotation, oscillation and damped oscillation, as well as molecule distortion. The frequency of rotation or oscillation depends on the dielectric properties of both the particle and the surrounding medium and varied in the range of realistic membrane potentials between several Hz up to more than 1 kHz. Molecular dipoles oscillating in an electrostatic field themselves generate alternating dipole fields and can stimulate other molecules localized in the neighbourhood to oscillate too. Therefore, the described phenomena of dielectric motion may not be only another way to interpret bacterial flagellar rotation but also the mechanism of energy transformation (constant electric fields into alternating fields), trigger processes, and constant oscillators. Additional, potential waves around the cell stimulating cytoplasmic and ciliar motion are imaginable, but of course, speculative. On the other hand, if Giee > Geei the polarization process leads to a breaking torque o f the same order of magnitude and every molecular motion is strongly suppressed. Therefore, both driving and breaking forces, respectively, have to be considered if macromolecular motion is calculated. Note that permanent molecule dipoles and their interactions with outside fields can overshadow the described polarization effect such that motion is suppressed completely. Therefore, it has to be expected that across the flagellar base no permanent dipole moment is measurable (which should be checked experimentally too). On the other hand it seems probable that the motor principle developed by bacteria is no particular case in cellular systems and the evolutionary process possibly started with rotating macromolecules incorporated into the membrane. We would like to thank Professor R. Glaser, Dr E. Donath, Dr R. Ehwald and Dr D. Kell for informative discussion and critical reading of the manuscript. REFERENCES ADAM, G. (1977). J. theor. Biol. 65, 713. ADLER, J. (1975). Ann. Reo. Biochem. 44, 341. ARNOLD, W. M. • ZIMMERMANN, U. (1982). Z. Naturforsch. 37, 908. BERG, H. C. (1974). Nature, Lond. 249, 77. BERG, H. C. (1975). Nature, Lond. 254, 389. BERG, H. C. & KHAN, S. (1983). Motility and Recognition in Cell Biology. Berlin: Walter de Gruyter. CODDINGTON, Z., POLLARD, A. F. & HOUSE, H. (1970). J. Phys. D. App. Phys. 3, 1212. DE PAMPHILIS, M. L. & ADLER, J. (1971). J. Bacteriol. 105, 384. DIMMZ3-r, K. & SIMON, M. I. (1971L J. Bacteriol. 108, 282. FUHR, G., (3LASER, R. & HAGEDORN, R. (1986). Biophys. J. 49, 395. FUHR, G. & HAGEDORN, R. (1987). Stud. Biophys. 121, 23. FUHR, G. & KUZMIN, P. (1986). Biophys. J. 50, 789. FOREDI, A. A. & OHAD,J. (1964). Biochim. Biophys. Acta 79, 1. GLAGOLEV, A. W. (1980). J. theor. Biol. 82, 171. GLASER, R. t~ FUHR, G. (1986). Electrical Double Layers in Biology. New York: Plenum Press. HERTZ, H. R. (1881). Wied. Ann. 13, 266. HYDWEILLER, A. (1899). Ann. d. Physik 69, 531. HOLZAPFEL, C., VIENKEN, J. • ZIMMERMANN, U. (1982). J. Membr. Biol. 67, 13. HONIG, B. H., HUBELL, W. L. & FLEWELLING, R. F. (1986). A. Rev. Biophys. 15, 163. JONES, T. B. (1984). Trans. Industry Applications IA-20, 845.

DIELECTRIC MOTORS

57

Jou, D., PEREZ-GARCIA,C. & LIEBOT, J. E. (1986). J. theor. Biol. KEEL, D. B. & HARRIS, Ch. M. (1985). Eur. Biophys. J. 12, 181. KHAN, S. & BERG, H. C. (1983). Cell 32, 913. LARSEN, S. H., ADLER, J., GARGUS, J. & HOGG, R. W. (1974). Proc. hath. Acad. Sci. U.S.A. "/1, 1239. LAUGER, P. (1977). Nature, Lond. 268, 360. LOWE, G., MEISTER, M. & BERG, H. C. (1987). Nature, Lond. 325, 637. MCLAUGHLIN, S. (1977). Curt. Top. Membr. Transp. 9, 71. MACNAB, R. (1979). Trends. Biochem. Sci. 4, 10. MEISTER, M. & BERG, H. C. (1987). Biophys. J. 52, 413. OOSAWA, F. & MASI, J. (1982). Z Phys. Soc. Japan 51, 631. POHL, H. A. & CRANE, J. S. (1971). Biophys. J. II, 711. QU1NCKE, G. (1986). Wied. Ann. 59, 417. SECKER, P. E. & SCIAEOM, I. N. (1968). J. App. Physics 39, 2957. SILVERMAN, M. (1980). Q. Rev. Biol. 55, 395. SILVERMAN, M. R. & SIMON, M. I. (1972). J. Bacteriol. ll2, 986. SILVERMAN, M. R. & SIMON, M. I. (1974). J. Bacteriol. 120, 1196. SIMPSON, P. & TAYLOR, R. J. (1971). J. Phys. D. Appl. Phys. 4, 1893. STOCKER, B. A. O. (1949). J. Hygiene 47, 398. SUZUKI, H., hNO, T., HORIGUCHI, T. & YAMAGUCHI,S. (1978). J. Bacteriol. 133, 904. SZMELCMAN, S. & ADLER, J. (1976). Proc. ham. Acad. Sci. U.S.A. 73, 4387. TEIXEIRA-PINTO, A. A., NEJELSKI, L. L., CUTLER, J. L. & HEELER, J. H. (1960). Exp. Cell Res. 20, 548. THIPAYATHASANA,P. & VALENTINE, R. C. (1974). Biochim. Biophys. Acta. WARSHEL, A. & RUSSEL, S. T. (1985). Q. Reo. Biophys. 18, 283. APPENDIX Torque Calculation in the Case of Quincke Rotation T h e easiest w a y to c a l c u l a t e the t o r q u e a c t i n g on a r o t a t i n g s p h e r e in an e l e c t r o s t a t i c field is to use the d i p o l e c o n c e p t (see J o n e s , 1984) a n d r o t a t i n g c o - o r d i n a t e systems. F r o m t h e s p h e r e s f r a m e o f r e f e r e n c e , the electric field s e e m s a c i r c u l a r l y p o l a r i z e d vector E = E e x p (joJt).

(A1)

T h e t o r q u e o n a p o l a r i z a b l e s p h e r e in a r o t a t i n g field is given b y N=PxE

(A2)

w h e r e P is t h e effective d i p o l e m o m e n t i n d u c e d b y the e x t e r n a l field. T o c a l c u l a t e t h e effective d i p o l e m o m e n t , w e m u s t s o l v e t h e L a p l a c e e q u a t i o n div g r a d ~b = 0.

(A3)

T h e f o l l o w i n g p o t e n t i a l e q u a t i o n s c a n be u s e d ~bi=Ei. r

(r< R)

~be = - E . r-~

p ._____.r_r 4 ,1reoeer3

(r> R)

(A4)

h e r e Ei is the field s t r e n g t h i n s i d e the s p h e r e , r d e n o t e s the p o l a r vector. A t the i n t e r f a c e s , t w o b o u n d a r y c o n d i t i o n s a l l o w us to solve t h e p o t e n t i a l e q u a t i o n system 6 , = 6~

and

e , - ~ - r = e, 0--7-

(A5)

58

G. F U H R A N D R. H A G E D O R N

The effective dipole moment is given by P = 4¢re°e~R3 (~:i + 2 ~ ) " E

where e~ = ee - j Ge

and

ei = e i - j - -

EOto

Gi

(A6)

~OtO

are complex permittivities. The total torque N = 12,a-eoeeRaE 2

( Glee - G~ei)

wr/toc

(2ee + el)(2Ge + G,)" 1 + (toJtoc) 2"

(A7)

In the case of Quincke rotation to~ = -top, the angular velocity of the sphere, in the case o f rotating field application, tar = o~aeld. Analogous to this the torque acting on a cylinder or multilayered particle can be calculated. For detailed torque derivation see Fuhr & Kuzmin (1986).