J. Exp. Biol. (1969), 50, 733-743
With 3 text-figures
733
Printed m Great Britain
ENERGETICS OF CILIARY MOVEMENT IN
SABELLARIA AND MYTILUS
BY M. A. SLEIGH
Department of Zoology, The University, Bristol
AND M. E. J. HOLWILL
Department of Physics, Queen Elizabeth College,
Campden Hill Road, London W.8
(Received 11 September 1968)
INTRODUCTION
The movement of many flagella is symmetrical and because of this symmetry it is
possible to calculate the amount of work done in each cycle of a sinusoidal or helical
beat (e.g. Taylor, 1952; Holwill & Burge, 1963). In the case of bull and sea-urchin
spermatozoa the work done against viscosity appears to be less than the energy
available from the breakdown of ATP by the whole organism (Rothschild, 1962;
Brokaw, 1965), although for bull sperm the external work expended has not yet been
accurately computed (Rikmenspoel, 1965). It is not unreasonable to suppose, as
suggested by several authors, that only a portion of the ATP broken down by the
entire spermatozoon is used to provide energy for mechanical deformation of the
flagellum.
The unilateral ciliary beat is usually considered to be made up of an effective stroke
during which the cilium remains approximately straight and swings as if hinged near
the base, and a recovery stroke during which a wave of flexure passes up the cilium to
return the whole organelle to its starting position. Equations have been derived for the
calculation of the torque required to move the cilium through fluid at the observed
rate during the effective stroke (Harris, 1961; Yoneda, 1962; Holwill, 1966), and
Yoneda (i960) was able to show, by arresting the movement of a compound abfrontal
cilium of Mytilus with aflexibleglass microneedle, that the force exerted represented
a torque at the ciliary base comparable with the values obtained later from theoretical
equations.
Recent improvements in the amount of quantitative information about ciliary beat
cycles, which have resulted from the application of high-speed cinematography to
selected ciliary organelles (Sleigh, 1968), suggest that it should be possible to calculate
the viscous work done by certain cilia during the recovery stroke as well as during the
effective stroke. Also, some estimates of the elastic work can be made using information
about the changes in shape of the cilium during the cycle. Two cilia showing rather
different patterns of beat have been chosen for this study, and it is interesting to compare the rate of working at different parts of the cycle in these two examples.
47
Exp. Biol. 50, 3
734
M. A. SLEIGH AND M. E. J. HOLWILL
MATERIALS AND METHODS
The cilia selected for study were from ciliary rows on the dorsal segmental gills of
the polychaete Sabellaria, and solitary abfrontal cilia from the ctenidial filaments of
Mytilus. In both cases the gill structure was cut from the animal and mounted in
seawater on a microscope slide for filming at 300 p.p.s. with a Vinten H.S. 450 cine
camera. These observations were made at room temperature (c. 200 C.)
OBSERVATIONS
The conical segmental gills (= dorsal cirri) of Sabellaria bear ciliated cells in
oblique uniseriate rows which form incomplete loops around the gill. A 30 /i length
of the ciliary row includes, on average, eleven compound cilia and four ciliated cells.
Each cell carries between sixty and ninety cilia whose bases are evenly distributed
over the cell in rather irregular rows. Although the shafts of the twenty to thirty
component organelles of the three compound cilia borne by each cell adhere together,
there is no trace of aggregation of basal bodies beneath each compound structure.
The basal bodies are 0-2 to 0-5 /i apart and have striated roots which run down into
the cell, but no other regular root structures which could interconnect the basal bodies
have been seen. The synchronous beat of the component cilia of each compound
structure must be assumed to result from some adhesion, or at least close mechanical
interference, between the units of the bundle. No evidence was found of any material
that might cause adhesion of the ciliary shafts, and, since the arrays of basal bodies are
apparently continuous across the cell boundaries, it is quite possible that a compound
cilium may contain units from two different cells. The length of the cilia on different
gills varies, but is usually between 30 and 50 fi. The cilia beat in sequence along the
row, with dexioplectic metachronism, and the effective stroke is directed towards the
tip of the gill (Sleigh, 1969).
The movements of such a cihum throughout the beat cycle are shown in Fig. 1,
which is more complete than the outlines previously published (Sleigh, 1962, 1968).
The resting cilium lies fairly close to the gill surface on the right—the surface of the
ciliated cell is set a little below that of the other cells. During the effective stroke
(0-24 msec.) the ciliary shaft swings around the basal region, straightening to a vertical
position and bending to the left. Before the full swing of the effective stroke is complete, the extreme basal region of the ciliary shaft has begun to move back to the right
(15 msec), so beginning the recovery stroke. In the final part of the beat the tip of
the cilium trails in the water as the bent region of the cilium is propagated up the
shaft (24-60 msec), maintaining, in this case, a fairly constant radius of curvature.
The paths traced out by various points along the ciliary shaft are also shown in Fig. 1;
while the basal regions move to and fro along much the same line, the tip moves
quickly through a wide arc in the effective stroke, but follows the same path as more
proximal regions during the recovery stroke. In the cycle shown here the cilium
completes one recovery stroke before the next effective stroke commences, but an
overlap of adjacent beats is frequently observed, and in these cases there is still a bent
region at the tip of the cilium at the start of the effective stroke.
The solitary compound cilia which occur on the abfrontal surface of ctenidial
Energetics of ciliary movement in Sabellaria and Mytilus
735
filaments of Mytilus vary considerably in size; they all have a similar beat, although the
example shown in Fig. 2 is of one of the larger ones. These compound structures are
believed to be built up of about twenty-five cilia which normally beat in unison, but
occasionally fray into two or more groups which beat independently. The beat of these
cilia is only slightly oblique to the long axis of the gill filament, and is easily seen if
filaments are laid on their sides.
v
42
48
^
6
\
54
\
Fig. 1. The position of a compound cilium from a dorsal cirrus of Sabdlana at intervals
(indicated in msec.) during a single beat. The interrupted lines show the movements of points
on the cihum 8 /*, 16 /t, 24 fi and 32 ft from its base.
150
120
180
210,
240, 0
Fig. 2. The position of a compound abfrontal cilium of Mytilus at intervals (indicated in
msec.) during a single beat. The interrupted lines show the movements of points on the cilium
•7 /*. 33 /* aT>d 5° /* from its base.
47-2
736
M. A. SLEIGH AND M. E. J. HOLWILL
Movements of Mytilus abfrontal cilia have been described by Gray (1930) and
Kinosita & Kamada (1939) from firms taken at 24-64 p.p.s., and briefly by Gosselin
(1966) and Sleigh (1968) from films taken at 300-400 p.p.s. Cilia of this type also rest
with the shaft close to the gill surface, but in this case the rest is taken at the end of the
effective stroke, rather than at the end of the recovery stroke (Fig. 2). In the resting
position the cilium is bent at the base, and the beat commences with a movement of
the basal region of the shaft to the right (0-60 msec.) and the rapid propagation of the
ciliary flexure to the tip of the shaft. Before the bend has reached the tip, the whole
(a)
Fig. 3. Selected positions showing the effective stroke (a, b) and the recovery stroke (c) of the
compound cilium of Sabellaria shown in Fig. ib. d, e, f are idealized forms of a, b and c
respectively from which mathematical analysis of the movement can be performed.
shaft swings towards the left to begin the effective stroke (75 msec.). After moving
quickly at the start of the effective stroke the cilium usually slows up near the vertical
position, and then moves steadily back to the gill surface to complete the beat.
The two cilia chosen as examples differ in the duration of their beat cycles, and in
the relative durations of the effective and recovery strokes, as well as in the position
of rest between cycles. These differences are reflected in the calculations of work done
by the cilia.
THEORY
In this section equations will be derived to permit the estimation of the work done
by a cilium during a cycle of its movement. For this purpose it is necessary to make
some approximations to the actual motion of the organelle. Figure id, e,/shows the
Energetics of ciliary movement in Sabellaria and Mytilus
737
idealized motion of the compound cilium of Sabellaria alongside the tracings taken
directly from a cinematographic film of the ciliary movement. Thus, for the purposes
of analysis, the effective stroke will be considered as the rotation of a rigid cylindrical
rod about one of its ends (Fig. ^d, e). During the recovery stroke the cilium is regarded
as a cylinder, one end of which corresponds to the basal end which is bent into a
circular arc (Fig. 3/). As the recovery stroke proceeds, the circular arc progresses to
the other end of the cylinder in such a way that the motion of the centre of the circle
(of which the arc is a part) is a straight line parallel to the straight portion of the
cylinder. The arc is assumed to remain constant in length and radius during the
recovery period, so that the initial straight section becomes shorter and a new,
lengthening straight region is formed adjacent to the gill surface.
The viscous forces acting during the motion will be evaluated by the use of surface
coefficients of resistance first employed in connexion with flagellar motility by Gray
& Hancock (1955) and later successfully applied to the movement of a variety of
flagellated micro-organisms (e.g. Holwill, 1965; Holwill & Burge, 1963; Holwill &
Sleigh, 1967). Two coefficients of resistance will be considered, one normal and the
other tangential to the surface of a cylinder. The tangential surface coefficient is
defined as the force acting per unit length of the cylinder when the velocity of the
cylinder in the direction of its axis is unity. A similar definition holds for the normal
coefficient and, for thin cylinders, it can be shown that the normal coefficient is twice
the tangential one (Gray & Hancock, 1955). For a straight cylinder of length / and
radius r the tangential coefficient, cT, is given by
-
where ji is the viscosity of the fluid surrounding the cylinder (Gray & Hancock,
IQ
55)Work is also necessary to overcome the elasticity of the cilium and this will be
estimated later by using the theory of bending beams. Since the elastic constants of
the cilium are not well established, the estimates of the elastic work done will not be
so reliable as those relating to the work done in overcoming external viscous forces.
The calculation of the work against viscous forces performed during the ciliary
cycle will be split into two parts, the first dealing with the effective stroke and the
second with the recovery stroke, while a third section will deal with equations from
which the work done against elastic forces may be estimated.
The effective stroke
The force acting on an element dy (Fig. 3 d) of a cylinder rotating about one end
- with angular velocity co is
(2)
dF = CNarydy
where y is the distance of the element from the fixed end of the cylinder and CN is
the normal coefficient of resistance. Since the velocity is everywhere normal to the
axis of the cylinder no tangential forces arise from viscous interactions during the
effective stroke. The rate (dP) at which work is done by the element is the force
multiplied by the velocity, i.e.
(3)
dP = CNa>ydy.
738
M. A. SLEIGH AND M. E. J. HOLWILL
The rate at which work is done by the whole cylinder (length I) is
(4)
Harris (1961) and Yoneda (1962) obtained expressions for the torque at the base of a
cilium from which expressions for the rate of working can be obtained. Although these
expressions are a little different algebraically from equation (4), the numerical values
obtained from them are of the same order of magnitude.
If it is assumed that the angular velocity remains constant throughout the effective
stroke, the work, WE, done during the effective stroke is
WE = ^CNw2PtE,
(5)
where tE is the time taken to execute the effective stroke.
The recovery stroke
It is convenient to consider the viscous work done during the recovery stroke in
two parts: (a) the work necessary to move the straight region and (b) the work needed
to move the circular arc.
(a) Movement of the straight region
If the straight region moves with velocity V parallel to its axis, then at any instant
the force on an element dx at a distance x from the tip of the cylinder (Fig. 3/) is
dF = CT Vdx.
(6)
The work, dw, done on the element dx during recovery is equal to this force multiplied
by the distance through which the element moves. Thus
dw = CTV(L-x)dx,
(7)
where L is the initial length of the straight region. The work, Ws, needed to move
the entire straight region in the recovery stroke is
dw = Ws = \CTVL\
(8)
(b) Movement of the curved region
Consider the element ds in the curved region of the cylinder (Fig. 3/). The velocity
of the element ds may be considered in two components, a velocity V parallel to the
axis of motion of the centre of the circle of which the arc is part and a velocity V
tangential to the arc at ds. The angle 6 is that between the radius to ds and a reference
axis drawn perpendicular to the straight region of the cylinder.
The force, dN, acting normal to ds is
dN = CNVsin dds
(9)
dT = CTV(i + cos 0)ds.
(10)
while the tangential force, cT, is
The rate at which work is done on the element ds is therefore
dP = CTV*r[zsm*O + (i+cm0)*]d0
where rdd has been substituted for ds.
(11)
Energetics of ciliary movement in Sabellaria and Mytilus
739
The rate of working, P, for the entire arc is thus
dp = P = C r ^r[2-5(0 2 -0 1 )-£(sin20 i ! -sin20 1 ) + 2(sin0 2 -sin0 1 )], (12)
where dlt 62 are the angles between the reference axis and the radii to the ends of the
arc.
The work done in moving the curved region during the recovery stroke is thus
Wc = CTVhtR[2-5(6t-d1)-l(sin2di-sm261)
+ 2(sin62-s\nd1)],
(13)
where tR is the time occupied by the recovery stroke.
Work done to overcome elastic forces
When a beam is bent into an arc of radius p it can be shown that the energy per
unit length, E, stored in the beam is
E
=
where q is Young's modulus of the material of the beam and Ak2 is the second moment
of area of the cross-section of the beam (see e.g. Champion & Davy, 1952). This
energy must be equal to the work done to bend the beam against elastic forces.
PRACTICAL APPLICATION OF THE EQUATIONS
(1) Gill cilia of Sabellaria
(a) Work done to overcome viscous forces. In the movement to be considered about
twenty-five cilia move as a unit in the manner shown by Fig. 3 a, b, c. The estimation
of the work done in the effective stroke is evidently best performed by two separate
calculations relevant to Fig. 3 d and e, each of which occupies one-half of the total time
taken for the effective stroke. The length of cilium involved in the pendulous beat of
Fig. 1 e is about 23/i. On this basis, using equation (5) and the figures given in Table 1,
the work done against viscous resistances by the group of cilia during the effective
stroke is about 9 x io" 8 ergs. The work done by each component cilium is thus about
4 x 1 o~9 ergs if it is assumed that the load is shared equally by all the cilia.
From equations (8) and (13) the work done by the group of cilia to overcome the
viscous resistance during the recovery stroke is about 2-5 x io~® ergs. The average
work done by each component cilium is thus about io~9 ergs.
(b) Work done against elastic forces. In evaluating the work done in the elastic
deformation of a cilium it is necessary to assume a value for the quantity qAkz. The
value will depend on which structures within the cilium provide most of the resistance
to bending. Holwill (1965) has estimated the magnitude of this product for the
flagellum of Crithidia oncopelti (formerly Strigomonas oncopelti) under conditions
where the membrane, the nine peripheral fibrils, the two central fibrils or the matrix
of the flagellum were each separately assumed to constitute the compressive elements
within the organelle. Of these four structures, the membrane yields the highest value
of 2 x io~12 dyne cm2, although in point of fact the value could be lower by a factor
of one or two orders of magnitude if other features within the flagellum were considered. In the flagellum of the sea urchin spermatozoon, for example, Rikmenspoel
(1966) has calculated that the value of this product is 6 x io~ u dyne cm.2. However,
740
M. A. SLEIGH AND M. E. J. HOLWILL
to obtain an upper limit for the value of the work done against the elastic forces, the
value of 2 x io~12 dyne cm.2 will be used in the present study.
During the effective stroke the tip of the group of cilia straightens while the base
of the cilium produces a bend that is first convex towards the leading edge of the
cilium, later becoming concave in this direction. Using equation (14) and assuming
that none of the energy expended in overcoming the elastic forces is recoverable, the
work done by each cilium in the effective stroke to overcome its natural rigidity is
about 1-9 x io" 8 ergs.
During the recovery stroke the entire cilium is effectively bent in an arc of radius
4 n and about 20 fi is unbent again. Assuming once more that none of the energy
can be recovered, the work done by a single cilium against elastic forces during the
recovery stroke is about 3-2 x io" 8 ergs.
Thus, in a complete cycle the work done by a single cilium is about 5-6 x io" 8 ergs.
The work done during the various parts of the cycle is summarized in Table 2.
(2) Abfrontal cilia of Mytilus
(a) Work done to overcome viscous forces. About twenty-five component cilia move
as a single structure in the manner depicted in Fig. 2. The movement contains the
same features as those illustrated in the idealised beat of Fig. 3 d, e, f, and the relevant
dimensions are given in Table 1.
Table 1. Parameters involved in the movement of cilia
Sabellana
Length of cibum (JL)
Diameter of individual cilium (ji)
Diameter of compound cilium (ji)
Angular velocity of effective
stroke (sec."1)
Time to complete effective stroke
(msec.)
Tune to complete recovery stroke (msec.)
Viscosity of medium (poise)
Length of straight region in recovery
stroke (jt)
Radius of arc during recovery stroke (ji)
Velocity of straight region during
recovery stroke (cm. sec."1)
0* n
32
0 2
10
Mytxlus
50
O-2
I O
130
Variable from
8 to 18
24
165
36
75
0 01
22
4
013
c. 0
c. 180
o-oi
28
8
0093
c. 0
c. 180
The angular velocity of the cilium is not constant throughout the effective stroke.
To calculate the work done, therefore, the stroke was divided into three parts, during
each of which the angular velocity remains essentially constant. Thus, for the first
75 msec, of the effective stroke the angular velocity was taken to be 8-2 sec."1, for
the next 60 msec, 17-5 sec."1 and for the final 30 msec, 8-7 sec."1. Using equation
(5) the work done during each effective stroke by the group of cilia is found to be
about 2-7 x io" 8 ergs and by a single component cilium about io~9 ergs. From equations (8) and (13) it is found that the work done during the recovery stroke by the
compound structure is about 5 x io" 8 ergs; the work done by a single cilium is thus
about 2 x io"9 ergs.
Energetics of ciliary movement in Sabellaria and Mytilus
741
(b) Work done against elastic forces. For the purposes of calculation the magnitude
of qAk2 will again be taken as 2 x io~12 dyne cm.2. During the effective stroke, a region
of length about 3-5 /i at the base of the cilium is bent into an arc of radius 2-2 fi. The
work done to overcome rigidity is thus about 7 x icr 9 ergs. During the recovery stroke
the entire cilium is effectively bent into an arc of radius 8 /i while some 25 fi of the
cilium is unbent. The work done by a single cilium against elastic forces during the
recovery stroke is thus about I-I x io" 8 ergs.
A summary of the work done at various parts of the cycle is given in Table 2.
Table 2. Work done (in ergs x io^8) by a single cilium
Sabellaria
Effective stroke: Viscous
Elastic
Total
Recovery stroke: Viscous
Elastic
Total
Total work for complete cycle
04
19
23
Mytilus
01
33
0-7
o-8
0-2
1-I
1-3
5-6
2-1
0 1
32
DISCUSSION
The work done against viscous forces by the cilia described in this account is of the
same order of magnitude as that calculated for various flagella (see Holwill, 1966, for
references; Holwill & Sleigh, 1967). Good agreement is found between the work
calculated here for the effective stroke of Mytilus cilia and that found by Yoneda (1962)
using a slightly different form of analysis (the values are respectively io~9 and about
7 x io~9 ergs/cilium/effective stroke).
The viscous work done by Sabellaria cilia in the recovery stroke is less than that
done during the effective stroke, a result which is to be expected since the resultant
movement of water as a consequence of ciliary movement is in the direction of the
effective stroke. In the case of Mytilus, on the other hand, the cilium performs more
viscous work in the recovery phase than in the effective stroke, and in this case there
is little resultant water movement; the cilium probably has some function other than
the propulsion of water.
Estimates of the elastic work done given in Table 2 are believed to be less accurate
than those for viscous work, as noted earlier. The work done to overcome ridigity in
the recovery stroke is some 50% greater than that in the effective stroke and the total
elastic work done is very much greater than the total of viscous work. It is possible
that some of the energy used in the elastic deformation of the cilium may be stored and
used to assist the active bending forces, so that not all of the elastic work is wasted,
and the active forces may not need to develop as much power as indicated in Table 2.
Brokaw (1965) has suggested that the stiffness (of which qAk2 is a measure) of certain
sperm flagella decreases in the region of active bending (where energy is dissipated
in overcoming elastic forces) remaining at a high value elsewhere. Further, the value
adopted for qAk2 may be too large. Machin (1958) has shown that for a system operating under optimum conditions for wave propagation, the energy dissipated elastically
is one-third of that used to overcome viscous forces. To meet this requirement for the
742
M. A. SLEIGH AND M. E. J. HOLWILL
cilia studied here, the elastic work done during the effective stroke should be about
i-3 x io~fl ergs for Sabellaria cilia and 3 x io~10 ergs for Mytilus cilia, although it is
possible that the ratio of elastic to viscous work done in cilia may be greater than that
inflagellabecause of the different (typical) forms of beating of the two organelles. The
magnitude of the quantity qAk2 may therefore need to be reduced to a value about
one-tenth of that assumed earlier. The evidence at present available does not permit
us to decide whether Young's modulus or the second moment of area should be
reduced, so that further discussion of this topic at this stage will not be fruitful. From
the above considerations it seems likely that the work done in elastic deformation is
less than that calculated, so that the relative magnitudes of the total work done for
cilia from Mytilus and Sabellaria correspond more nearly to those relating to the viscous
work in each case.
It is of interest to compare the calculated work done with the energy available from
ATP which is believed to be responsible for supplying the energy necessary for
flagellar and ciliary activity. The protein dynein, which constitutes the arms on the
peripheral fibrils of a cilium (Gibbons, 1965), appears to be the enzyme which
liberates the energy from ATP within the cilium. Assuming that the pairs of arms are
spaced at 170 A intervals along each peripheral fibril (Gibbons & Rowe, 1965;
Grimstone & Klug, 1966), then a single cilium from Sabellaria will contain about
3-4x10* arms, while one from Mytilus will have about 5-4x10* arms. If each arm is a
single molecule of dynein, then the molar contents of dynein in Sabellaria and
Mytilus cilia are about 5-7 x io" 20 and 9 x io" 20 respectively.
If each molecule of dynein de-phosphorylates one molecule of ATP per ciliary beat,
and if the amount of available energy from this deformation is 10 kcal/mole of ATP,
then in each beat of the Sabellaria cilium the energy that can be used in movement
would be 2-4 x icr 8 ergs/beat and in Mytilus it would be 3-8 x io^ 8 ergs/beat. This is
comfortably in excess of the total viscous work in each case by an amount comparable
with that found by Brokaw (1968) for the flagellum of a sea-urchin spermatozoon.
If the elastic work is as great as the figures in Table 2 suggest, it would be necessary
for more than one ATP molecule to be broken down by each dynein molecule per
beat cycle—perhaps one in the effective stroke and one in the recovery stroke.
SUMMARY
1. High-speed cinephotography has been used to study the movements performed
by compound cilia from the segmental gills of Sabellaria and from the abfrontal face
of the gill filaments of Mytilus.
2. The two types of cilium have distinctly different beat patterns.
3. Equations are derived which allow the calculation of the energy necessary to
overcome viscous resistance during the effective and recovery strokes of a cilium in
terms of its dimensions and angular frequency.
4. In Sabellaria cilia the energy needed to overcome viscous forces is greater for the
effective stroke than for the recovery stroke, but the reverse is true for Mytilus
abfrontal cilia.
5. Estimates of the work done to overcome elastic forces are probably too high, but
it appears that the elastic work done in the recovery stroke is greater than that in the
Energetics of ciliary movement in Sabellaria and Mytilus
743
effective stroke for cilia of both types if the stiffness remains constant throughout
the beat.
6. The energy released if each fibrillar arm causes the breakdown of one ATP
molecule per beat cycle is greater than that required to overcome viscous resistance
to ciliary motion.
It is a pleasure to acknowledge the technical assistance of Miss Sheila Manning;
this assistance and the cine equipment were provided by grants from the Science
Research Council.
REFERENCES
BROKAW, C. J. (1965). Non-sinusoidal bending waves of spermflagella.J. exp. Biol. 43, 155—69.
BROKAW, C. J. (1968). Mechanisms of sperm movement. Symp. Soc exp. Btol. 22, 101—16.
CHAMPION, F. C. & DAVY, N. (1952) Properties of Matter. London: Blackie and Son.
GIBBONS, I. R. (1965). Chemical dissection of cilia. Arch. Biol. Liege 76, 317-52.
GIBBONS, I. R. & ROWE, A. J. (1965). Dynein: A protein with adenosine tnphosphatase activity from
cilia. Science, N. Y. 149, 424-26.
GOSSELIN, R. E. (1966). Physiologic regulators of ciliary motion. Am. Rev. resp. Dts. 93, Suppl. 41-59.
GRAY, J. (1930). The mechanism of ciliary movement. VI. Photographic and stroboscopic analysis of
ciliary movement. Proc. Roy. Soc. B 107, 313-32.
GRAY, J. & HANCOCK, G. J. (1955). The propulsion of sea-urchin spermatozoa. J. exp. Btol. 32, 802-14.
GRIMSTONE, A. V. & KLUC, A. (1966). Observations on the substructure offlagellarfibres.J. Cell Set. 1,
351-62.
HARRIS, J. E. (1961) The mechanics of ciliary movement. In The Cell and the Organism (eds. J. A.
Ramsay and V. B. Wigglesworth), pp. 22-36. Cambridge University Press.
HOLWILL, M. E. J. (1965). The motion of Strigouwnas oncopelti. J. exp. Biol. 43, 125-37.
HOLWIIX, M. E. J. (1966). Physical aspects of flagellar movement. Physiol. Rev. 46, 696—785.
HOLWILL, M. E. J. & BURGE, R. E. (1963). A hydrodynamic study of the motihty of flagellated bacteria.
Arch. Btochem. Biophys. 101, Z49-60.
HOLWILL, M. E. J. & SLEIGH, M. A. (1967). Propulsion by hispid flagella. J. exp. Biol. 47, 267-76.
KmosiTA, H. & KAMADA, T. (1939). Movement of abfrontal cilia of Mytilus. Jap. J. Zool. 8, 291-310.
MACHIN, K. E. (1958). Wave propagation along flagella. J. exp. Biol. 35, 796-801.
RIKMENSPOEL, R. (1965). The tail movement of bull spermatozoa. Observations and model calculations.
Biophys. J. 5, 365-92.
RIKMENSPOEL, R. (1966). Elastic properties of the sea urchin sperm flagellum. Biophys. J. 6, 471-9.
ROTHSCHILD, Lord (1962) Sperm energetics: An account of work in progress. In The Cell and
the Organism (eds. J. A. Ramsay and V. B. Wigglesworth), pp. 9—21. Cambridge University Press.
SLEIGH, M. A. (1962). The Biology of Cilia and Flagella. Oxford: Pergamon.
SLEIGH, M. A. (1968). Patterns of ciliary beating. Symp. Soc. exp. Biol. 23, 131-50.
SLEIGH, M. A. (1969). Coordination of the rhythm of beat in some ciliary systems. Int. Rev. Cytol.
as- 31-54.
TAYLOR, G I. (1952). The action of waving cylindrical tails in propelling microscopic organisms.
Proc. Roy. Soc. A a n , 225-39.
YONEDA, M. (i960). Force exerted by a single cihum of Mytilus edulis. I. J. exp. Biol. 37, 461-8.
YONEDA, M. (1962). Force exerted by a single cihum of Mytilus edulis. II. J. exp. Biol. 39, 307-17.