Energetics of Ciliary Movement in Sabellaria and Mytilus

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 (1960) was able to show, by arresting the movement of a compound abfrontal cilium of Mytilus with a flexible glass 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.

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. 20° C.)

The conical segmental gills (= dorsal cirri) of Sabellaria bear ciliated cells in oblique uniseriate rows which form incomplete loops around the gill. A 30 μ 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 μ 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 μ. 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 cilium 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 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.

Movements of Mytilus abfrontal cilia have been described by Gray (1930) and Kinosita & Kamada (1939) from films 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 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.

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 3,d, e, f shows the 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. 3,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 f). 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.

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.

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.

Consider the element ds in the curved region of the cylinder (Fig. 3 f). 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 θ is that between the radius to ds and a reference axis drawn perpendicular to the straight region of the cylinder.

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. i e is about 23μ 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 9×10⁻⁸ ergs. The work done by each component cilium is thus about 4×10⁻⁹ 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 2·5×10⁻⁸ ergs. The average work done by each component cilium is thus about 10⁻⁹ ergs.

In evaluating the work done in the elastic deformation of a cilium it is necessary to assume a value for the quantity qAk². 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 2×10⁻¹² 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 6×10⁻¹⁴ dyne cm.². However, to obtain an upper limit for the value of the work done against the elastic forces, the value of 2×10⁻¹² 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 1·9×10⁻⁸ ergs.

During the recovery stroke the entire cilium is effectively bent in an arc of radius 4 μ and about 20 μ 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 3·2×10⁻⁸ ergs.

Thus, in a complete cycle the work done by a single cilium is about 5·6×10⁻⁸ ergs. The work done during the various parts of the cycle is summarized in Table 2.

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.

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 2·7×10⁻⁸ ergs and by a single component cilium about 10⁻⁹ ergs. From equations (8) and (13) it is found that the work done during the recovery stroke by the compound structure is about 5×10⁻⁸ ergs; the work done by a single cilium is thus about 2×10⁻⁹ ergs.

For the purposes of calculation the magnitude of qAk² will again be taken as 2×10⁻¹² dyne cm.². During the effective stroke, a region of length about 3·5 μ at the base of the cilium is bent into an arc of radius 2·2 μ. The work done to overcome rigidity is thus about 7×10⁻⁹ ergs. During the recovery stroke the entire cilium is effectively bent into an arc of radius 8 μ while some 25 μ, of the cilium is unbent. The work done by a single cilium against elastic forces during the recovery stroke is thus about 1.1×10⁻⁸ ergs.

A summary of the work done at various parts of the cycle is given in Table 2.

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 10⁻⁹ and about 7×10⁻⁹ 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 cilia studied here, the elastic work done during the effective stroke should be about 1·3×10⁻⁹ ergs for Sabellaria cilia and 3×10⁻¹⁰ ergs for Mytilus cilia, although it is possible that the ratio of elastic to viscous work done in cilia may be greater than that in flagella because 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 Å intervals along each peripheral fibril (Gibbons & Rowe, 1965; Grimstone & Klug, 1966), then a single cilium from Sabellaria will contain about 3·4×10⁴ arms, while one from Mytilus will have about 5·4×10⁴ arms. If each arm is a single molecule of dynein, then the molar contents of dynein in Sabellaria and Mytilus cilia are about 5·7×10⁻²⁰ and 9×10⁻²⁰ 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 2·4×10⁻⁸ ergs/beat and in Mytilus it would be 3·8×10⁻⁸ 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.

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.