A cilium can be stopped by applying the tip of a glass microneedle during the effective stroke. The force required to stop the cilium has been measured by the author by the displacement of a flexible needle on the basis of its compliance. This force was expressed in terms of a torque, at the base of the cilium, which was 4× 10−7 dyne. cm. The second step would be to know the magnitude of the torque when the cilium is beating freely without constraint. Such a study would require a hydrodynamical treatment, as was given in the case of swimming spermatozoa (Taylor, 1951, 1952; Hancock, 1953; Gray & Hancock, 1955; Machin, 1958; Carlson, 1959).
In Part I of the present paper the form and velocity of the cilia in motion in media of various viscosities are reported, in Part II an equation for the torque at the base of the cilium is given, and in Part III the amount of the torque is calculated from the equation and the physiological significance is discussed.
PART I. ANGULAR VELOCITY AND PATTERN OF BEAT OF A CILIUM IN MEDIA OF VARIOUS VISCOSITIES
Material and preparation
Large abfrontal gill cilia of Mytilus edulis were used as described in the previous paper. The mean length of the cilia is about 55 μ, although individual length varies between 40 and 90 μ. In order to orient the abfrontal cilia so as to make them beat in the horizontal plane a gill filament was stretched across a trough, keeping about 0·2 mm. distant from the surfaces of a slide and a cover-glass. Thoroughness of exchange of media was assured, judging from the fact that the cilia were forced to bend down-stream during the changing of the medium.
Preparation of viscous media
3 g. of sodium carboxymethyl cellulose was dissolved in 100 c.c. of sea water on water-bath and filtered through cheese cloth. To remove undissolved particles the filtrate was centrifuged for 30 min. at 26,500 g. Any effect of the remaining particles on viscosimetry and on ciliary beating is regarded as negligible, since the particles remaining in the supernatant were about 2 μ or less in diameter, and the concentration was of the order of 1 particle/8000 μ3. This supernatant was used as stock solution. Employing an Ostwald viscometer, the viscosity of the stock solution was found to be 0·40 poise at 20°C. Examination by the capillary flow method revealed that the solution behaved as a Newtonian fluid at the maximum rate of shear of between 3 and 60 sec.−1. Less viscous solutions were prepared by diluting the stock solution with sea water. The colloid osmotic pressure of sodium carboxymethyl cellulose is negligible, the stock solution being practically isosmotic with sea water using Mactra egg as an osmometer.
Photographic recording of the cilium
The track of the beat of the cilium was recorded photographically with exposure of sec. with 40 × objective and 10 × ocular. By employing phase optics the track of the cilium during sec. was recorded as a bright area against the dark background on the print. Such exposure was just enough for the cilium to cover 10–50° excursion during the effective stroke, from which the angular velocity of the stroke may be obtained, provided the shutter speed is reliably accurate. In the case of very small velocities sec. exposure was employed. Calibration of the speed of the Copal Shutter no. 00 here employed revealed that standard error of the mean in ten measurements was 2 and 3 % respectively for and for sec. exposure.*
Pattern of the effective stroke
Typical examples of records of the effective stroke are shown in Pl. 1. The general patterns were quite similar to the ciné-photographs published by Gray (1930, 1931) and by Kinosita & Kamada (1939). It would, however, be appropriate to remind the reader of a few points of importance, in reference to the diagram of Kinosita & Kamada reproduced in Text-fig. 1.
Gray defined the effective stroke as from phase 6 to 10 of the diagram in Text-fig. 1, when the cilium remains straight (phases 1–5 are recovery stroke). At the transition from the recovery to the effective stroke (phases 5–6), when the cilium becomes completely straight, the motion becomes momentarily distinctly slower. In fact, even at sec. exposure, the cilium often appears to be motionless at the transition. Thereafter the cilium moves rapidly with an almost uniform velocity to the end of the effective stroke (phase 10 in Text-fig. x). In the paragraphs to follow, this rapid and uniform phase of the effective stroke will be dealt with exclusively, for the sake of simplicity in analysis. Force exerted or work done by the cilia which will be calculated later should therefore be considered as the highest value within the beating cycle of the cilium.
From measurements made on the photographs the mean angular velocity in sea water for thirty-five individual cilia was found to be 11·4 rad./sec., with a standard error of 3·1 rad./sec.
Effect of viscosity on pattern of beat and on angular velocity
Angular velocities between phases 6 and 10 in media of various viscosities are shown in Text-fig. 2. Gradual decrease in the angular velocity is evident as the viscosity increases to a relative viscosity of 23·5, i.e.
On the other hand, the curvature of the cilium, which in normal sea water is very small or non-existent, increases slightly as the viscosity rises. The critical inclination of the cilium when the effective stroke begins is not affected by change in viscosity. When the relative viscosity increases as far as 40, damping of the distal part of the cilium is so great that only the proximal part oscillates. The pattern of beating then becomes too complicated and unstable to be analysed.
PART II. HYDRODYNAMICAL CONSIDERATIONS AND MODEL EXPERIMENT
The main object of this paper is to obtain the force exerted during free motion of the cilium. As far as the large abfrontal cilia of Mytilus are concerned the only part which bends actively is the extreme base (Yoneda, 1960). In the following sections the force will therefore be expressed in terms of a torque around the base of the cilium.
Both size and velocity of the cilia are so small (i.e. low Reynolds number) that inertial force is considered to be negligible as compared with viscous force. Thus the problem is reduced to calculation of the amount of viscous reactive force of the medium when a cilium of a given size is beating with a given angular velocity in a medium of a given viscosity.
For the purposes of calculation some simplifying assumptions are needed, which are as follows: (a) the cilium remains straight during the effective stroke; (b) the cilium beats in a stationary medium. (The two conditions do not always hold true. An estimate of the error arising from these simplifications will be given in the Appendix.)
The average thickness of the cilia at various levels was determined from photographs (Table 1 and Text-fig. 4). Assuming that the shape of the cilium is a solid of revolution of such optical section, a model of the cilium can be constructed with considerable fidelity as to shape. The arrangement for the model experiment is shown in Text-fig. 5. A symmetrical pair of model cilia (M) continuous at the base, each being 4 cm. in length and i-6 mm. in diameter at the base, was carved from a bamboo rod. This was pierced through with a needle (N) at the centre (base). The needle with the model was hung by a piano wire (W) which would serve as a torsion balance. Torque can be calculated from the torsion of the wire, as recorded by lampmirror-scale system (m), the factor being 0·344 dyne.cm per degree. A cylindrical vessel (C) of diameter 12 cm., filled with glycerol, was brought beneath the assembly and centered visually. The model was then submerged 1·5 cm. below the surface of glycerol and 2·5 cm. above the bottom of the vessel. The vessel could be rotated simple belt-drive at speeds of from 0·002 to 0·5 rad./sec.
Experiment and result
PART III. CALCULATION AND DISCUSSION
The torque generated by a single cilium in sea water was calculated by equation (8). From Text-fig. 7 it can be seen that larger cilia will exert stronger torques. On the average, for a cilium of 55μ in length and mean angular velocity of 11·4 rad./sec. the calculated torque is 2·2 × 10−8 dyne.cm.* In the previous paper the torque exerted by the cilium when it was arrested was found to be × 10−7 dyne.cm., a higher value than that in the normal environment; that is, in free motion a force of only one-twentieth of the maximum is developed.
The effect of viscosity on the torque developed is summarized in Table 2 and illustrated in Text-fig. 8(a). A tendency for the torque to increase with increase in viscosity, up to a maximum when the cilium is arrested, suggests that some sort of machinery analogous to that of muscle must be involved in ciliary movement. According to Sleigh (1956) peristomial cilia of Stentor resemble the abfrontal cilia of Mytilus in this respect. Despite viscosity increase of up to 3·5 times, the frequency of beat of the Stentor cilia maintained the level of 70% of its initial value with no change in amplitude. In Sleigh’s data, since the frequency is proportional to the mean velocity of the stroke, increase of the torque of the peristomial cilia by 2·5 times ( = 3·5× 0·7) is obtained for 3·5-fold shift in viscosity according to equation (8). Papers by Gosselin (1958) and Aiello (1960), both working on the influence of viscosity on the motion of molluscan cilia, cannot be used for comparison owing to lack of information necessary for calculation.
Results by Pigoh & Szarski (1955) on the swimming speed of Paramecia, however, suggest the contrary. The speed was found to be inversely proportional to the viscosity within the range of relative values from 1 to 15. Since Stokes’s law implies that the force required to move a Paramecium is proportional to the product of velocity and viscosity, it can be concluded that the net force exerted by the cilia of Paramecium is unchanged, regardless of change in viscosity. Such discrepancy cannot be explained at present.
Using Gray’s (1955) data Carlson (1959) calculated the power developed by a swimming spermatozoon of the sea urchin, Psammechinus miliaris, which was 2·92 ×10−7 erg/sec., being roughly comparable to, but definitely higher than, the value (6× 10−8 erg/sec.) for one subunit of the abfrontal cilium of Mytilus. Gray (1955) and Machin (1958) concluded that any part of the sperm tail can bend actively, while in the case of the abfrontal cilium, only the proximal part bends actively (Kinosita & Kamada, 1939; Yoneda, 1960). The difference in power developed by these two structures might be explained on this same basis.
Angular velocities of the large abfrontal cihum of Mytilus beating in media of various viscosities were measured.
- An equation for the force produced by a single cihum in free motion was derived by hydrodynamical methods. Expressing the force in terms of torque referred to the base of the cihum :
The torque exerted by the cihum in normal sea water was 2 × 10−8 dyne.cm. Increase of viscosity resulted in increase in the torque.
The maximum power developed by the cihum was calculated to be 1·2×10−6 erg/sec. Assuming that a single cihum is made up of 20 subunits, the power developed by a single subunit is 6 ×10−8 erg/sec., which is of the same order of magnitude as the power developed by a single spermatozoon of a sea urchin (Carlson, 1959).
The author expresses his thanks to Prof. Katsuma Dan for his valuable advice and help in preparing the manuscript. He also thanks Prof. Tsunetaka Sasaki of the Department of Chemistry of the University for suggesting the design of the model experiment, and Mr Shozo Ishizaka of the Department of Physics of St Paul’s University for discussion of the hydrodynamical problem.
In the application of hydrodynamics to the problem of ciliary activity, two simplifications have been made; the extent of error introduced by these assumptions will be estimated.
(1) Correction for bending of the cilium during the effective stroke
As seen in Pl. 1, the cilia are sometimes bent backwards, which will reduce the viscous resistance. Assuming the shape of the curved cilium to be expressed by a logarithmic spiral (Text-fig. 9) the decreasing curvature passing distally along the cilium can be expressed by the following relation (in polar co-ordinates) :
(2) Correction for the movement of the medium caused by small cilia
Besides the large abfrontal cilia, which are sparsely distributed, the abfrontal gill surface bears numerous minute cilia. Such small cilia are sometimes so vigorous that a continuous current of water is generated near the gill surface, the direction coinciding with that of the effective stroke of the large abfrontal cilia. One should therefore subtract the velocity of this current from the observed velocity of the large cilium, in order to obtain the actual value of the torque exerted by the large cilium.
By suspending minute chalk granules in the medium, and by measuring the tracks of the granules photographically, the velocity profile of the current as a function of the distance from the gill surface was obtained as shown in Text-fig. 10. To take account of the effect of the current, equation (2) should be modified, by substituting v – vm for v as follows :
EXPLANATION OF PLATE
The author is indebted to Mr Toshio Nakatsubo of The Olympus Optical Co. for the calibration.
This figure may become smaller (at the most by 30 %) if due corrections are made for bending of the cilium and for the current caused by small cilia (see Appendix).