ENERGETICS OF CILIARY MOVEMENT IN SABELLARIA AND MYTILUS

Transcription

1 J. Exp. Biol. (1969), 50, With 3 text-figures 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

2 734 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 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

3 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^ \ 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 , 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

4 736 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 p.p.s., and briefly by Gosselin (1966) and Sleigh (1968) from films taken at 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

5 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, c T, 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 df = C N arydy (2) where y is the distance of the element from the fixed end of the cylinder and C N 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. dp = C N a>ydy. (3)

6 738 M. A. SLEIGH AND M. E. J. HOLWILL The rate at which work is done by the whole cylinder (length I) is 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, W E, done during the effective stroke is W E = ^C N w 2 Pt E, (5) where t E 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 = C T 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 = C T V(L-x)dx, (7) where L is the initial length of the straight region. The work, W s, needed to move the entire straight region in the recovery stroke is dw = W s = \C T VL\ (8) (4) (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 = C N Vsin dds (9) while the tangential force, ct, is dt = C T V(i + cos 0)ds. (10) The rate at which work is done on the element ds is therefore dp = C T V*r[zsm*O + (i+cm0)*]d0 (11) where rdd has been substituted for ds.

7 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( )- (sin20 i! -sin20 1 ) + 2(sin0 2 -sin0 1 )], (12) where d lt 6 2 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 W c = C T Vht R [2-5(6 t -d 1 )-l(sin2d i -sm26 1 ) + 2(sin6 2 -s\nd 1 )], (13) where t R 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 Ak 2 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 4x1 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 qak z. 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 cm 2, 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,

8 740 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 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 Sabellana c. 0 c. 180 Mytxlus 50 O-2 I O Variable from 8 to o-oi 28 c. 0 c 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.

9 Energetics of ciliary movement in Sabellaria and Mytilus 741 (b) Work done against elastic forces. For the purposes of calculation the magnitude of qak 2 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 Effective stroke: Viscous Elastic Total Recovery stroke: Viscous Elastic Total Total work for complete cycle Sabellaria Mytilus o I 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 qak 2 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 qak 2 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

10 742 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 qak 2 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

11 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, BROKAW, C. J. (1968). Mechanisms of sperm movement. Symp. Soc exp. Btol. 22, 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, GIBBONS, I. R. & ROWE, A. J. (1965). Dynein: A protein with adenosine tnphosphatase activity from cilia. Science, N. Y. 149, GOSSELIN, R. E. (1966). Physiologic regulators of ciliary motion. Am. Rev. resp. Dts. 93, Suppl GRAY, J. (1930). The mechanism of ciliary movement. VI. Photographic and stroboscopic analysis of ciliary movement. Proc. Roy. Soc. B 107, GRAY, J. & HANCOCK, G. J. (1955). The propulsion of sea-urchin spermatozoa. J. exp. Btol. 32, GRIMSTONE, A. V. & KLUC, A. (1966). Observations on the substructure offlagellarfibres.j. Cell Set. 1, HARRIS, J. E. (1961) The mechanics of ciliary movement. In The Cell and the Organism (eds. J. A. Ramsay and V. B. Wigglesworth), pp Cambridge University Press. HOLWILL, M. E. J. (1965). The motion of Strigouwnas oncopelti. J. exp. Biol. 43, HOLWIIX, M. E. J. (1966). Physical aspects of flagellar movement. Physiol. Rev. 46, HOLWILL, M. E. J. & BURGE, R. E. (1963). A hydrodynamic study of the motihty of flagellated bacteria. Arch. Btochem. Biophys. 101, Z HOLWILL, M. E. J. & SLEIGH, M. A. (1967). Propulsion by hispid flagella. J. exp. Biol. 47, KmosiTA, H. & KAMADA, T. (1939). Movement of abfrontal cilia of Mytilus. Jap. J. Zool. 8, MACHIN, K. E. (1958). Wave propagation along flagella. J. exp. Biol. 35, RIKMENSPOEL, R. (1965). The tail movement of bull spermatozoa. Observations and model calculations. Biophys. J. 5, RIKMENSPOEL, R. (1966). Elastic properties of the sea urchin sperm flagellum. Biophys. J. 6, 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 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, SLEIGH, M. A. (1969). Coordination of the rhythm of beat in some ciliary systems. Int. Rev. Cytol. as TAYLOR, G I. (1952). The action of waving cylindrical tails in propelling microscopic organisms. Proc. Roy. Soc. A an, YONEDA, M. (i960). Force exerted by a single cihum of Mytilus edulis. I. J. exp. Biol. 37, YONEDA, M. (1962). Force exerted by a single cihum of Mytilus edulis. II. J. exp. Biol. 39,

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