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First published online March 2, 2006
Journal of Experimental Biology 209, 985-986 (2006)
Published by The Company of Biologists 2006
doi: 10.1242/jeb.02120
JEB Classics |
FLAGELLAR PROPULSION
California Institute of Technology
brokawc{at}its.caltech.edu
In this JEB Classics paper, Sir James Gray and G. J. Hancock explained how
spermatozoa are propelled by flagellar bending waves
(Gray and Hancock, 1955
). This
paper was a lasting success because it provided an easy-to-understand solution
to a complicated hydrodynamic problem, and because it provided a quantitative
prediction of the swimming speed that was almost identical to the swimming
speed measured in Gray's accompanying paper on the movement of sea urchin
spermatozoa (Gray, 1955).
Gray came to this work with a full understanding of undulatory propulsion
in long, thin animals such as snakes (Gray,
1953). Bending waves passing along the length of an elongated body
will propel an animal forward if the body pushes laterally against its
surroundings with a force that is greater than the force required to drag the
body parallel to its length (see Fig.
1). But the microscopic realm, where momentum is insignificant,
and the resistances are purely viscous, was uncharted territory. It was known
that the viscous resistance of a very long thin body moving perpendicular to
its length was twice the resistance moving parallel to its length (e.g.
Burgers, 1938), but it was not
known whether this fact had any relevance to an actively bending flagellum.
Sir Geoffrey Taylor was the first to prove that propagated bending waves on a
body in a viscous fluid would propel the body
(Taylor, 1951;
Taylor, 1952). Taylor's
analysis was limited to very small amplitude waves, and could not be applied
quantitatively to real situations, such as the sea urchin sperm flagellum,
where the peak-to-peak amplitude is about one third of the wavelength. Hancock
followed up on this by performing the difficult mathematical analysis required
for sinusoidal bending waves with realistic amplitudes
(Hancock, 1953). By discarding
terms containing the value of the radius of the flagellum, he obtained an
equation for the forward swimming velocity as a function of the frequency,
amplitude and wavelength of the bending waves at the theoretical limit of 0
radius. Although it seems counterintuitive that a flagellum with 0 radius can
propel a cell, this is a natural result of the `no slip condition' for
hydrodynamics in a viscous fluid, which requires that the velocity of fluid at
the surface of a moving object must be the same as the velocity of the object.
Hancock then used numerical calculations to explore the amount of reduction in
velocity that occurred when the radius was greater than 0. From his analysis
Hancock could see that, at least for a very thin filament, it was reasonable
to estimate the forces on each small element along the length by treating each
element as part of a straight filament moving at velocity V. By
considering the filament in this way it was possible to multiply the local
normal and longitudinal velocity components, VN and
VL, by resistance constants, CN and
CL, to calculate the force generated by each element (see
Fig. 1). Gray and Hancock used
this approximate method, now known as resistive force theory, to add up the
forces on each element resulting from bend propagation along the flagellum and
from forward movement at an unknown velocity. The summation of all of these
forces along the x axis must be 0, so the equation can be solved to
obtain the forward propulsion velocity. It turns out that the resistance
constants enter the forward velocity equation only as the ratio
CN/CL, which is 2.0 for an infinitesimally thin
filament. Gray's photographs of sea urchin spermatozoa revealed a waveform
that could be approximated by a sine wave, so it was possible to use a sine
wave as a model and integrate along the length. The use of resistive force
theory was validated by the result that it generated the same equation for
swimming velocity obtained from the more rigorous analysis in Hancock's 1953
paper. After adding the effect of the sperm head to the equations, the
predicted velocity was almost exactly the same as Gray's measured
velocity.
|
In effect, Gray and Hancock separated the analysis of movement in viscous
fluids into two parts. One part, the calculation of the effects of radius and
wavelength on the values of the resistance constants, remained the province of
hydrodynamicists, who have reexamined the problem repeatedly. Much of this
work is reviewed by Lighthill (Lighthill,
1976) and Dresdner et al.
(Dresdner et al., 1980). The
durability of the original conclusions results from the fact that the swimming
velocity depends only on the ratio of the resistance constants, and
hydrodynamic refinement usually changes CN and CL in the
same direction, with minimal change in their ratio. The second part of the
analysis, the use of resistance constants to analyze a particular movement
pattern, was made accessible for biologists by Gray and Hancock's publication.
Holwill and Burge used resistive force theory to analyze the swimming of
bacteria with helical flagella (Holwill
and Burge, 1963). I used it together with new descriptions of
sperm flagellar bending waves as functions of length along the curve of the
flagellum, rather than along a hypothetical x axis, to obtain simpler
equations for calculating sperm swimming speed and energy expenditure
(Brokaw, 1965;
Brokaw, 1975). Actually,
Carlson first used resistive force theory to calculate the energy expended
against viscous resistances by a swimming sperm cell, by integrating the
product of force and velocity along the length of the flagellum
(Carlson, 1959). However,
energy expenditure calculation requires absolute values of CL, not
just the ratio CN/CL. The value of CL
originally proposed by Hancock has been updated
(Lighthill, 1976;
Dresdner et al., 1980) leading
to energy expenditures 35% greater than the first calculations. These new
values still show that the energy output is well within the energy available
from sperm metabolism. Resistive force theory is only a first order
approximation to a very complex hydrodynamic problem. It is limited to
specific cases where the fluid movement near any element along the length is
perturbed in only a limited manner by the movement of nearby elements, and is
not perturbed by any other influences, such as the presence and/or movement of
other nearby objects in the fluid. More recent hydrodynamic work has attempted
to develop methods for these situations, where simple resistive force theory
cannot be used. For instance, it is inadequate for situations where a small
flagellum is propelling a large cell, or where multiple, closely spaced
flagella or cilia are producing movement. A new methodology developed to deal
with such problems has succeeded in demonstrating that metachronal
coordination of ciliary beating can be explained by hydrodynamic interactions
between cilia (Gueron and Levit-Gurevich,
2001).
Resistive force theory, like the Stokes equations on which it is based, is
a linear theory, which makes it easily invertible to calculate velocities
resulting from specified forces. This, in fact, is exactly what Gray and
Hancock did, to calculate the forward swimming velocity. More recently,
resistive force theory has been used for simulations of flagellar movement,
which use ideas about internal force generation to calculate the form and
propagation of flagellar bending waves as well as velocities of movement in a
viscous environment. These simulations require computations of rates of
flagellar bending at many points along a flagellum, given a specified
distribution of internal active forces. The result is a set of equations with
many unknown velocities, which can be solved by standard numerical methods as
long as the velocities are linear functions of all of the force specifications
(Brokaw, 1972;
Hines and Blum, 1978;
Brokaw, 2002; etc.). Since the
mechanisms actually used by flagella to initiate and propagate bending waves
are still unknown, resistive force theory is still being actively used for
these investigations, and more than 20% of the citations of this 1955 JEB
Classics paper have accumulated since January 2000. These include new and
unexpected interest from the field of nanotechnology, seeking to construct
microscopic artificial swimmers that mimic the movements of spermatozoa (e.g.
Dreyfus et al., 2005).
Footnotes
Charles J. Brokaw writes about Sir James Gray and G. J. Hancock's 1955 publication on the propulsion of sea urchin spermatozoa. Gray and Hancock's paper can be accessed free at the JEB archive.
References
Brokaw, C. J. (1965). Non-sinusoidal bending
waves of sperm flagella. J. Exp. Biol.
43,155
-169.
Brokaw, C. J. (1972). Computer simulation of
flagellar movement. I. Demonstration of stable bend propagation and bend
initiation by the sliding filament model. Biophys. J.
12,564
-586.
Brokaw, C. J. (1975). Spermatozoan motility: a biophysical survey. Biol. J. Linn. Soc. 7 (Suppl. 1),423 -439.
Brokaw, C. J. (2002). Computer simulation of flagellar movement. VIII: Coordination of dynein by local curvature control can generate helical bending waves. Cell Motil. Cytoskel. 53,103 -124.[CrossRef][Medline]
Burgers, J. M. (1938). On the motion of small particles of elongated form suspended in a viscous liquid. In Second Report on Viscosity and Plasticity, pp.113 -127. Amsterdam: North Holland Publ. Co.
Carlson, F. D. (1959). The motile power of a swimming spermatozoon. In Proc. First Natl. Biophysics Conf. (ed. H. Quastler and H. J. Morowitz), pp.443 -449. New Haven: Yale University Press.
Dresdner, R. D., Katz, D. F. and Berger, S. A. (1980). The propulsion by large amplitude waves of uniflagellar micro-organisms of finite length. J. Fluid Mech. 97,591 -621.[CrossRef]
Dreyfus, R., Baudry, J., Roper, M. L., Fermigier, M., Stone, H. A. and Bibette, J. (2005). Microscopic artificial swimmers. Nature 437,862 -865.[CrossRef][Medline]
Gray, J. (1953). Undulatory movement. Quart. J. Microsc. Sci., NS 94,551 -578.
Gray, J. (1955). The movement of sea urchin spermatozoa. J. Exp. Biol. 32,775 -801.[Abstract]
Gray, J. and Hancock, G. J. (1955). The propulsion of sea urchin spermatozoa. J. Exp. Biol. 32,802 -814.[Abstract]
Gueron, S. and Levit-Gurevich, K. (2001). The three-dimensional motion of slender filaments. Math. Meth. Applied Sci. 24,1577 -1603.[CrossRef]
Hancock, G. J. (1953). The self-propulsion of microscopic organisms through liquids. Proc. R. Soc. A 217,96 -121.
Hines, M. and Blum, J. J. (1978). Bend
propagation in flagella. I. Derivation of equations of motion and their
simulation. Biophys. J.
23, 41-57.
Holwill, M. E. J. and Burge, R. E. (1963). A hydrodynamic study of the motility of flagellated bacteria. Arch. Biochem. Biophys. 101,249 -260.[CrossRef][Medline]
Lighthill, M. J. (1976). Flagellar hydrodynamics. SIAM Rev. 18,161 -230.[CrossRef]
Taylor, G. I. (1951). Analysis of the swimming of microscopic organisms. Proc. R. Soc. A 209,447 -461.
Taylor, G. I. (1952). The action of waving cylindrical tails in propelling microscopic organisms. Proc. R. Soc. A 211,225 -239.
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