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A theory for the locomotion of spirochetes
J. theor. Biol. (1972) Xi, 53-60
A Theory for the Locomotion of Spirochetes CHANGYI
WANG AND THEODORE L. JAI-IN
Department of Mathematics, Michigan State University, East Lansing, Michigan, 48823, U.S.A. Department of Zoology, University of California at Los Angeles Cal& U.S.A. (Received 25 March 197 1, and in revised form 28 June 197 1) A hydrodynamic theory for the locomotion of spirochetes is presented. The theory is based on a possible arrangement of internal fibrils such that self rotation about a local body axis is possible. This self rotation is responsible for cancelling the torque produced by the traveling helical waves of the body. The hydrodynamic theory follows the method used by Taylor, but with different boundary conditions. A relationship between the radius of body cross section, amplitude and wavelength of the helical wave is obtained for a spirochete traveling with no apparent slippage of water. The angular velocity of rotation about a local body axis is also determined. The theoretical results compare favorably with direct measurements of body geometry.
1. IntrodRctioll It is generally accepted that the flagellum attached to an inert “head” functions as a locomotory organ for micro-organisms. The organism propels itself by sending traveling waves, either planar or helical, down the flagellum. These waves are created by the contraction of internal fibrils inside the flagellum (Jahn & Bovee, 1965; Holwill, 1966). A hydrodynamic theory of propulsion was fifst proposed by Taylor (1951, 1952), who showed that if a circular cylinder of finite radius is undulating with small amplitude traveling waves, then its propulsive speed is proportional to the wave speed, the square of the amplitude and the inverse square of the wavelength. Using a completely different approach, Hancock (1953) approximated the total flagellum effect by a series of elements of infinitesimal longitudinal
and Holwill
radius.
By following
this approach
and assuming
lateral and
drag coefficients for each element, Gray & Hancock (1955), & Burge (1963) arrived essentially at the same result: save for 53
54
C.-Y.
WANG
AND
T.
L.
JAHN
mastigonemes, the propulsive speed could never be greater than one half the wave speed. All existing hydrodynamic theories concluded that in order to send helical waves down a flagellum, an external torque must be supplied by the inert head. None of the above theories could explain the locomotion of spirochetes, which travel by means of helical waves and do not have torque-resisting “heads”. Furthermore, it is reported that Cristispira balbianii and Spirochaeta plicatilis travel in helical waves with no apparent slippage of water (Jahn & Landman, 1965; Jahn & Bovee, 1970). This means propulsive speed is equal to wave speed, a fact that could not be explained by Hancock’s approach. A possible model and corresponding hydrodynamic theory for spirochete locomotion is presented in this paper.
2. Rotation about Local Body Axis
It is necessary to first establish that the spirochete can rotate aboutitslocal body axis. Figure 1 shows a section of the spirochete body of radius a, wound around in a helix of radius b and wavelength 2n/k where k is the wave number.
FIG.
1. Geometric characteristicsof a helical body.
The spirochete is able to bend and produce helical waves by the contraction of internal fibrils (Gray, 1953). It is not known whether the internal fibrils are parallel to, or wound helically about the body axis. Contracted in sequence, both models could produce the same helical waves (Gray, 1953; Astbury, Beighton & Weibull, 1955). There is reason to believe that the spirochete body could only bend to a certain minimum radius of curvature. This is supported by the fact that some free living spirochetes are observed to curl into circles of radius no smaller than four to five diameters of the body.
A
THEORY
FOR
THE
LOCOMOTION
OF
SPIROCHETES
55
A hypothetical toroid shaped body at its minimum radius of curvature (Fig. 2) is used to demonstrate self rotation about a local body axis. As the
FIG. 2. Self rotation of a torus.
circular fibril A-A is contracted a rotation about its local body axis will be produced, since the torus could not shrink any further. As A-A reaches the location of C-C, B-B would be in the position to contract. For a helical body the radius of curvature of the body axis is (b2kZ + l)/bk2, a constant. In Fig. 3, if the fibril A-A-A-A-A is contracted, a rotation about its local body axis is produced. For a continuous rotation, the fibrils must be more than three in number. These fibrils are wound helically
FIG. 3. Schematic figure showing the mechanism for the self rotation about a local body axis.
around the body axis with the same pitch as the body, but should be distinguished from those discussed by Gray (1953) which maintain the helical shape or helical wave. 3. Hydrodynamics Since Hancock’s (1953) approach could not explain the lack of an apparent slippage of water produced by the spirochete, the approach used by Taylor (1952), with different boundary conditions, will be followed, In
56
C.-Y.
WANG
AND
T.
L.
JAHN
Taylor’s work, rotation is prevented by a large inert head. This condition should be relaxed since the spirochetes do not have a head. The self rotation creates a torque which just cancels the torque produced by the traveling helical wave. With the overall torque being zero, the hydrodynamic paradox of spirochete locomotion can be explained. Zero apparent slippage of water is also assumed, which yields a relationship between the geometric parameters a, b and k. For very low Reynolds numbers (10e3 to 10m6) the hydrodynamics are governed by the Stoke’s equations, which, written in cylindrical co-ordinates, become 1 1 2 -=u,,+~u,-~+~ue8-r3ve+u,,, cl ar 1 2 1 ap --= v,, + - v, - Jt t‘ee + - u + v,,, r2 + -i2 1dP
rp
ae
r
r2 ’
(1) (2)
1 1 1 ap --= w, + ; w, + r2 - w,+w,,. (3) P ax u, v and w are the velocity components in the direction of r, 0 and x respectively, p is the pressure and P is the viscosity coefficient and the subscripts denote partial differentiation. The continuity equation is u,+u/r+(l/r)v,+w,
Following
= 0.
(4)
Taylor, let the helical body be represented by r=a+bcoscp=a+bcos[8-k(x+Ur)],
(5)
where U is the wave speed and b/a = E is assumed to be small. The boundary conditions on the body are u = kUb sin cp- bR sin cp, (6) v = kUb cos rp-aQ,
(7)
w = w.
(8)
The first terms on the right-hand side of equations (6) and (7) result from the motion of the helical wave. The last terms in the same equations, neglected by Taylor, represent rotation about the local body axis with angular velocity rZ. The constant W is the propulsive velocity to be determined. At r = co, we require fmite pressure, and all the velocities decay to zero. Normalizing the lengths by a, time by a/U, the velocities by kbU and the pressure by pki% and assuming apriori that Q/(kU) = E~/I and /?is a constant of order unity, the boundary conditions equations (6), (7) and (8) become:
au U~,=l+ECOScp + O(s2) = sin ‘p + e2/? sin cp, dr,=1
A THEORY
FOR THE
a0 VI,&-& coscpz
LOCOMOTION
OF SPIROCHETES
+ O(s2)= cosfp- e/3,
57
(10)
r=1
wl,=,+e coscpaw -a'r-1+o(e2) = w E W,+eW,+O(2).
(11)
Expanding the pressure and velocities in a power seriesin E produces u = uo+eu~+O(e2), (12) v = vo+ El)1+0(&z), (13) w = wo+ ew1+ 0(&Z), (14) p = po + ep, + O(E2). (15) Substituting into the Stokes equations and the boundary conditions, the xeroth-order solution is found to be uo = F(z) sin fp, (16) 00= G(z)cosp, (17) wo = H(z) cos cp, (18) po = kaP(z) sin cp,
(19)
w, = 0,
(20)
where z = kar and 2F(z) = AzK,(z) +N,(z) + CK,(z), 2G(z) = - BK,(z) + CKo(d, 2H(z) = BKl(z)+CK,(z)+AzKo(z)-AKl(z). K,, K,, K2 are the modified Besselfunctions of the secondkind; A, B, constants defined by A= -
(21) (22)
(23) C are (24)
4W4K2(aY~,
B = 2aK:(a)/M,
G5.l
C = WKoWCl +fi%OK2C4/W ;
(26)
and a = ka, M = 2aKi(a)K2(a)-
2Ko(a)Kl(a)K2(a) - aKf(a)[K,(a)
+ K,(a)].
(27)
The rotation parameter fl does not affect the xeroth-order solution which is essentiallythat obtained by Taylor. The first-order problem may be separated into the sum of a steady part and a time-dependentpart harmonic in 2q. The steadypart which yields both the mean torque and the propulsive velocity is the point of interest. The governing equations for the steadypart are pi/a = u’; +(1/z)& -(l/z2)ul, (28)
58
C.-Y.
WANG
AND
T.
L.
JAHN
0 = u;+(l/z)o;-(1/22)u,, 0 = wY+(l/z)wi, u;+(l/z)u, = 0. The boundary conditions are
(29) (30) (31)
u&4 = 0,
(32)
u1tcO = - CB+WW(419
(33)
WI(a) = w, -(a/2)H’(a).
(34)
p1 is finite, and u i, ai and wi vanish at infinity. u1= 0,
The solution is
u1 = - a[T?+ (a/2)G’(a)]/z, WI = w, -(a/2)zf’(a) = 0.
(35) (36) (37)
Equation (36) implies the existence of a torque about the helix axis. In order to make this torque zero, the rotation fi about the local body axis must be equal to -cc/2G’(a), a constant, or Q = $kU[-(a/2)G’(a)]
Equation
= )(U/a)(kb)z[BK;(a)-CKb(a)].
(38)
(37) implies that W = kbU&W,
= U[(kb)‘/2]H’(a).
(39)
The propulsive velocity IV, is found to be the same as that obtained by Taylor. Therefore, given body geometry k, a, b and self rotation R, we can find the wave speed U from equation (38) and subsequently the forward speed W from equation (39). Slip occurs when W # U. For a spirochete to travel without apparent slippage of water, we must equate the propulsive speed to the wave speed. Letting W = U, equations (38) and (39) become (h/W)
= [(kb)‘/4][BK;(ka) kB = J2/[H’(ka)].
- CK&(ka)],
tw (41)
In summary, according to the theories of Taylor (1952) and Hancock (1953), an organism without an inert head capable of resisting torque cannot swim by means of traveling helical waves in the body. However, Jahn BELandman (1965) found that spirochetes can swim in this manner. This created a dilemma in that there was a direct conflict between fact and theory. The present theory removes that dilemma by permitting the torque to be balanced by self-rotation of the body. Therefore, it is now theoretically possible for spirochetes to swim the way they do.
A THEORY
FOR
THE
LOCOMOTION
OF
SPIROCHETES
59
4. Discussion
Jahn dc Landman (1965) assumed, in accordance with the theory of Taylor (1952), that the helical wave was composed of the algebraic summation of two sine waves, 90” to each other and 90” out of phase, and therefore was irrotational. This left the torque unbalanced. The present theory,as described above, assumes that rotation does occur, and that it cancels the torque. If the spirochete has a right-handed helical body, then the rotation about a local body axis is in the direction shown in Fig. 1. If the helical body is lefthanded, then the rotation would be in the opposite direction. Perhaps rotation about a local body axis can be detected by staining a point on the side of the spirochete and observing the frequency of reappearance of the dot while the organism is in motion. However, this has not been done. It is interesting to note that Gray (1953) did some experiments with grass snakes in sinuous tubes. A snake glides easily through both horizontal and vertical planar tubes, but is unable to negotiate a helical tube. Perhaps this is caused by the fact that grass snakes do not possess helical muscle fibers which would be responsible for rotation around a local body axis,
3-
FIG.
4. Comparison of theory
with
observation.
A, Spirochaetaplicatiiis (Stanier, Doudoroff & Adelberg, 1957); 0, Cristispira (Stanier, Doudoroff & Adelberg, 1957); 0, Cristispira (Jahn & Landman, 1965).
60
C.-Y.
WANG
AND
T.
L.
JAHN
or do not possess the neurocoordination system capable of producing a helical wave with the muscles it does have. If the spirochete is to travel without apparent slippage of water, the relation between the geometric parameters ka and kb is given by equation (41). Figure 4 shows the theoretical results compared to rough measurements of ka and kb from the microphotographs of Cristispira and Spirochaetaplicatilis. The comparison is encouraging, especially since the theoretical analysis is based on small amplitude helical waves (i.e. ka % kb) and in reality the amplitude is large. Also shown in the figure is the theoretical rotation rate (C&z/W) plotted against ka. If the forward speed W is measured from the cinemicrophotographs, the corresponding angular velocity R can then be obtained from the figure or from equation (40). We see that the larger the body radius a, the less self-rotation S2 is required to achieve the same propulsive speed. REFERENCES ASTBURY, W. T., BEIGI-IT~N, E. 8c WEIBULL, C. (1955). Symp. Sot. exp. Biol. 9,282. GRAY, J. (1953). Quart. Jl microsc. Sci. 94, 551. GRAY, J. & HANCOCK, G. J. (1955). J. exp. Biol. 32, 802. HANCOCK, G. J. (1953). Proc. R. Sot. A 217, 96. HOLWILL, M. E. J. (1966). Physiol. Rev. 46, 696. HOLWILL, M. E. J. & BURGE, R. E. (1963). Arch. Biochem. Biophys. 101, 249. JAI-IN, T. L. & BOVEE, E. C. (1965). A. Rev. Microbial. 19, 21. JAHN, T. L. & BOVEE, E. C. (1970). Proc. 2nd Znt. Congr. Parasitology, p. 168. JAI-IN, T. L. & LANDMAN, M. D. (1965). Trans. Am. microsc. Sot. 84, 395. STANIER, R. Y., DOUDOROFF, M. & ADELBERG, E. A. (1957). The Microbial World, p. 317. Englewood Cliffs, N.J. : Prentice-Hall. TAYUIR, G. I. (1961). Proc. R. Sot. A 209, 447. TAYLOR, G. I. (1952). Proc. R. Sot. A 211, 225.
A Theory for the Locomotion of Spirochetes CHANGYI
WANG AND THEODORE L. JAI-IN
Department of Mathematics, Michigan State University, East Lansing, Michigan, 48823, U.S.A. Department of Zoology, University of California at Los Angeles Cal& U.S.A. (Received 25 March 197 1, and in revised form 28 June 197 1) A hydrodynamic theory for the locomotion of spirochetes is presented. The theory is based on a possible arrangement of internal fibrils such that self rotation about a local body axis is possible. This self rotation is responsible for cancelling the torque produced by the traveling helical waves of the body. The hydrodynamic theory follows the method used by Taylor, but with different boundary conditions. A relationship between the radius of body cross section, amplitude and wavelength of the helical wave is obtained for a spirochete traveling with no apparent slippage of water. The angular velocity of rotation about a local body axis is also determined. The theoretical results compare favorably with direct measurements of body geometry.
1. IntrodRctioll It is generally accepted that the flagellum attached to an inert “head” functions as a locomotory organ for micro-organisms. The organism propels itself by sending traveling waves, either planar or helical, down the flagellum. These waves are created by the contraction of internal fibrils inside the flagellum (Jahn & Bovee, 1965; Holwill, 1966). A hydrodynamic theory of propulsion was fifst proposed by Taylor (1951, 1952), who showed that if a circular cylinder of finite radius is undulating with small amplitude traveling waves, then its propulsive speed is proportional to the wave speed, the square of the amplitude and the inverse square of the wavelength. Using a completely different approach, Hancock (1953) approximated the total flagellum effect by a series of elements of infinitesimal longitudinal
and Holwill
radius.
By following
this approach
and assuming
lateral and
drag coefficients for each element, Gray & Hancock (1955), & Burge (1963) arrived essentially at the same result: save for 53
54
C.-Y.
WANG
AND
T.
L.
JAHN
mastigonemes, the propulsive speed could never be greater than one half the wave speed. All existing hydrodynamic theories concluded that in order to send helical waves down a flagellum, an external torque must be supplied by the inert head. None of the above theories could explain the locomotion of spirochetes, which travel by means of helical waves and do not have torque-resisting “heads”. Furthermore, it is reported that Cristispira balbianii and Spirochaeta plicatilis travel in helical waves with no apparent slippage of water (Jahn & Landman, 1965; Jahn & Bovee, 1970). This means propulsive speed is equal to wave speed, a fact that could not be explained by Hancock’s approach. A possible model and corresponding hydrodynamic theory for spirochete locomotion is presented in this paper.
2. Rotation about Local Body Axis
It is necessary to first establish that the spirochete can rotate aboutitslocal body axis. Figure 1 shows a section of the spirochete body of radius a, wound around in a helix of radius b and wavelength 2n/k where k is the wave number.
FIG.
1. Geometric characteristicsof a helical body.
The spirochete is able to bend and produce helical waves by the contraction of internal fibrils (Gray, 1953). It is not known whether the internal fibrils are parallel to, or wound helically about the body axis. Contracted in sequence, both models could produce the same helical waves (Gray, 1953; Astbury, Beighton & Weibull, 1955). There is reason to believe that the spirochete body could only bend to a certain minimum radius of curvature. This is supported by the fact that some free living spirochetes are observed to curl into circles of radius no smaller than four to five diameters of the body.
A
THEORY
FOR
THE
LOCOMOTION
OF
SPIROCHETES
55
A hypothetical toroid shaped body at its minimum radius of curvature (Fig. 2) is used to demonstrate self rotation about a local body axis. As the
FIG. 2. Self rotation of a torus.
circular fibril A-A is contracted a rotation about its local body axis will be produced, since the torus could not shrink any further. As A-A reaches the location of C-C, B-B would be in the position to contract. For a helical body the radius of curvature of the body axis is (b2kZ + l)/bk2, a constant. In Fig. 3, if the fibril A-A-A-A-A is contracted, a rotation about its local body axis is produced. For a continuous rotation, the fibrils must be more than three in number. These fibrils are wound helically
FIG. 3. Schematic figure showing the mechanism for the self rotation about a local body axis.
around the body axis with the same pitch as the body, but should be distinguished from those discussed by Gray (1953) which maintain the helical shape or helical wave. 3. Hydrodynamics Since Hancock’s (1953) approach could not explain the lack of an apparent slippage of water produced by the spirochete, the approach used by Taylor (1952), with different boundary conditions, will be followed, In
56
C.-Y.
WANG
AND
T.
L.
JAHN
Taylor’s work, rotation is prevented by a large inert head. This condition should be relaxed since the spirochetes do not have a head. The self rotation creates a torque which just cancels the torque produced by the traveling helical wave. With the overall torque being zero, the hydrodynamic paradox of spirochete locomotion can be explained. Zero apparent slippage of water is also assumed, which yields a relationship between the geometric parameters a, b and k. For very low Reynolds numbers (10e3 to 10m6) the hydrodynamics are governed by the Stoke’s equations, which, written in cylindrical co-ordinates, become 1 1 2 -=u,,+~u,-~+~ue8-r3ve+u,,, cl ar 1 2 1 ap --= v,, + - v, - Jt t‘ee + - u + v,,, r2 + -i2 1dP
rp
ae
r
r2 ’
(1) (2)
1 1 1 ap --= w, + ; w, + r2 - w,+w,,. (3) P ax u, v and w are the velocity components in the direction of r, 0 and x respectively, p is the pressure and P is the viscosity coefficient and the subscripts denote partial differentiation. The continuity equation is u,+u/r+(l/r)v,+w,
Following
= 0.
(4)
Taylor, let the helical body be represented by r=a+bcoscp=a+bcos[8-k(x+Ur)],
(5)
where U is the wave speed and b/a = E is assumed to be small. The boundary conditions on the body are u = kUb sin cp- bR sin cp, (6) v = kUb cos rp-aQ,
(7)
w = w.
(8)
The first terms on the right-hand side of equations (6) and (7) result from the motion of the helical wave. The last terms in the same equations, neglected by Taylor, represent rotation about the local body axis with angular velocity rZ. The constant W is the propulsive velocity to be determined. At r = co, we require fmite pressure, and all the velocities decay to zero. Normalizing the lengths by a, time by a/U, the velocities by kbU and the pressure by pki% and assuming apriori that Q/(kU) = E~/I and /?is a constant of order unity, the boundary conditions equations (6), (7) and (8) become:
au U~,=l+ECOScp + O(s2) = sin ‘p + e2/? sin cp, dr,=1
A THEORY
FOR THE
a0 VI,&-& coscpz
LOCOMOTION
OF SPIROCHETES
+ O(s2)= cosfp- e/3,
57
(10)
r=1
wl,=,+e coscpaw -a'r-1+o(e2) = w E W,+eW,+O(2).
(11)
Expanding the pressure and velocities in a power seriesin E produces u = uo+eu~+O(e2), (12) v = vo+ El)1+0(&z), (13) w = wo+ ew1+ 0(&Z), (14) p = po + ep, + O(E2). (15) Substituting into the Stokes equations and the boundary conditions, the xeroth-order solution is found to be uo = F(z) sin fp, (16) 00= G(z)cosp, (17) wo = H(z) cos cp, (18) po = kaP(z) sin cp,
(19)
w, = 0,
(20)
where z = kar and 2F(z) = AzK,(z) +N,(z) + CK,(z), 2G(z) = - BK,(z) + CKo(d, 2H(z) = BKl(z)+CK,(z)+AzKo(z)-AKl(z). K,, K,, K2 are the modified Besselfunctions of the secondkind; A, B, constants defined by A= -
(21) (22)
(23) C are (24)
4W4K2(aY~,
B = 2aK:(a)/M,
G5.l
C = WKoWCl +fi%OK2C4/W ;
(26)
and a = ka, M = 2aKi(a)K2(a)-
2Ko(a)Kl(a)K2(a) - aKf(a)[K,(a)
+ K,(a)].
(27)
The rotation parameter fl does not affect the xeroth-order solution which is essentiallythat obtained by Taylor. The first-order problem may be separated into the sum of a steady part and a time-dependentpart harmonic in 2q. The steadypart which yields both the mean torque and the propulsive velocity is the point of interest. The governing equations for the steadypart are pi/a = u’; +(1/z)& -(l/z2)ul, (28)
58
C.-Y.
WANG
AND
T.
L.
JAHN
0 = u;+(l/z)o;-(1/22)u,, 0 = wY+(l/z)wi, u;+(l/z)u, = 0. The boundary conditions are
(29) (30) (31)
u&4 = 0,
(32)
u1tcO = - CB+WW(419
(33)
WI(a) = w, -(a/2)H’(a).
(34)
p1 is finite, and u i, ai and wi vanish at infinity. u1= 0,
The solution is
u1 = - a[T?+ (a/2)G’(a)]/z, WI = w, -(a/2)zf’(a) = 0.
(35) (36) (37)
Equation (36) implies the existence of a torque about the helix axis. In order to make this torque zero, the rotation fi about the local body axis must be equal to -cc/2G’(a), a constant, or Q = $kU[-(a/2)G’(a)]
Equation
= )(U/a)(kb)z[BK;(a)-CKb(a)].
(38)
(37) implies that W = kbU&W,
= U[(kb)‘/2]H’(a).
(39)
The propulsive velocity IV, is found to be the same as that obtained by Taylor. Therefore, given body geometry k, a, b and self rotation R, we can find the wave speed U from equation (38) and subsequently the forward speed W from equation (39). Slip occurs when W # U. For a spirochete to travel without apparent slippage of water, we must equate the propulsive speed to the wave speed. Letting W = U, equations (38) and (39) become (h/W)
= [(kb)‘/4][BK;(ka) kB = J2/[H’(ka)].
- CK&(ka)],
tw (41)
In summary, according to the theories of Taylor (1952) and Hancock (1953), an organism without an inert head capable of resisting torque cannot swim by means of traveling helical waves in the body. However, Jahn BELandman (1965) found that spirochetes can swim in this manner. This created a dilemma in that there was a direct conflict between fact and theory. The present theory removes that dilemma by permitting the torque to be balanced by self-rotation of the body. Therefore, it is now theoretically possible for spirochetes to swim the way they do.
A THEORY
FOR
THE
LOCOMOTION
OF
SPIROCHETES
59
4. Discussion
Jahn dc Landman (1965) assumed, in accordance with the theory of Taylor (1952), that the helical wave was composed of the algebraic summation of two sine waves, 90” to each other and 90” out of phase, and therefore was irrotational. This left the torque unbalanced. The present theory,as described above, assumes that rotation does occur, and that it cancels the torque. If the spirochete has a right-handed helical body, then the rotation about a local body axis is in the direction shown in Fig. 1. If the helical body is lefthanded, then the rotation would be in the opposite direction. Perhaps rotation about a local body axis can be detected by staining a point on the side of the spirochete and observing the frequency of reappearance of the dot while the organism is in motion. However, this has not been done. It is interesting to note that Gray (1953) did some experiments with grass snakes in sinuous tubes. A snake glides easily through both horizontal and vertical planar tubes, but is unable to negotiate a helical tube. Perhaps this is caused by the fact that grass snakes do not possess helical muscle fibers which would be responsible for rotation around a local body axis,
3-
FIG.
4. Comparison of theory
with
observation.
A, Spirochaetaplicatiiis (Stanier, Doudoroff & Adelberg, 1957); 0, Cristispira (Stanier, Doudoroff & Adelberg, 1957); 0, Cristispira (Jahn & Landman, 1965).
60
C.-Y.
WANG
AND
T.
L.
JAHN
or do not possess the neurocoordination system capable of producing a helical wave with the muscles it does have. If the spirochete is to travel without apparent slippage of water, the relation between the geometric parameters ka and kb is given by equation (41). Figure 4 shows the theoretical results compared to rough measurements of ka and kb from the microphotographs of Cristispira and Spirochaetaplicatilis. The comparison is encouraging, especially since the theoretical analysis is based on small amplitude helical waves (i.e. ka % kb) and in reality the amplitude is large. Also shown in the figure is the theoretical rotation rate (C&z/W) plotted against ka. If the forward speed W is measured from the cinemicrophotographs, the corresponding angular velocity R can then be obtained from the figure or from equation (40). We see that the larger the body radius a, the less self-rotation S2 is required to achieve the same propulsive speed. REFERENCES ASTBURY, W. T., BEIGI-IT~N, E. 8c WEIBULL, C. (1955). Symp. Sot. exp. Biol. 9,282. GRAY, J. (1953). Quart. Jl microsc. Sci. 94, 551. GRAY, J. & HANCOCK, G. J. (1955). J. exp. Biol. 32, 802. HANCOCK, G. J. (1953). Proc. R. Sot. A 217, 96. HOLWILL, M. E. J. (1966). Physiol. Rev. 46, 696. HOLWILL, M. E. J. & BURGE, R. E. (1963). Arch. Biochem. Biophys. 101, 249. JAI-IN, T. L. & BOVEE, E. C. (1965). A. Rev. Microbial. 19, 21. JAHN, T. L. & BOVEE, E. C. (1970). Proc. 2nd Znt. Congr. Parasitology, p. 168. JAI-IN, T. L. & LANDMAN, M. D. (1965). Trans. Am. microsc. Sot. 84, 395. STANIER, R. Y., DOUDOROFF, M. & ADELBERG, E. A. (1957). The Microbial World, p. 317. Englewood Cliffs, N.J. : Prentice-Hall. TAYUIR, G. I. (1961). Proc. R. Sot. A 209, 447. TAYLOR, G. I. (1952). Proc. R. Sot. A 211, 225.