ASYMMETRIC WAVEFORMS IN ECHINODERM SPERM FLAGELLA

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1 J. exp. Biol. (1977), 71, &ii ~'ith 12 figures inted in Great Britain ASYMMETRIC WAVEFORMS IN ECHINODERM SPERM FLAGELLA BY STUART F. GOLDSTEIN Department of Genetics and Cell Biolog, Universit of Minnesota, and Bermuda Biological Station for Research (Received 27 April 1977) SUMMARY 1. Dark-field, multiple-exposure photographs of live spermatozoa of a number of echinoderms were analsed. 2. Bends develop at the base in pairs, as the do in flagella with smmetrical waveforms. However, the angles of these bends do not cancel, so that microtubular sliding - of up to over 50 % of that associated with bend propagation - is transferred distall. This sliding implies that (a) microtubules are not rigidl cross-linked within straight regions, and (b) bends can propagate normall in spite of appreciable extrinsic microtubular sliding. 3. Both the sperm head and the asmmetr of the waveform appear to affect the swim path of a spermatozoon. INTRODUCTION The smmetrical, planar waveforms of echinoderm sperm flagella have been variousl described as sine waves (Gra, 1955), meander-like waves (Brokaw, Goldstein & Miller, 1970; Rikmenspoel, 1971; Silvester & Holwill, 1972), or circular arcs connected b straight lines (Brokaw & Wright, 1963; Brokaw, 1965). Tritonated sea urchin spermatozoa can beat quite smmetricall in reactivating solutions containing low concentrations of calcium (Brokaw, Josslin & Bobrow, 1974). Live echinoderm spermatozoa which are attached to a surface b their heads can exhibit smmetrical waveforms (Fig. id), but the can also exhibit asmmetrical waveforms (Fig. 1 b). When swimming freel next to a surface, the tpicall exhibit asmmetrical planar waveforms: the travel in a curved path, with the bends whose convex sides face outward from the swim path (the 'principal bends') subtending a larger angle than those whose convex sides face inwards (the' reverse bends') (Gibbons & Gibbons, 1972). Brokaw (1970) has made some measurements on asmmetricall beating flagella. The sliding microtubule model of flagellar motilit (Satir, 1965, 1974) predicts that the amount of sliding occurring between two microtubules at an point along a flagellum during an portion of a beat ccle is directl proportional to the change in the angle between a tangent to the flagellum at that point and a tangent to the flagellar Jaase (Goldstein, 1969). This means that patterns of microtubular sliding can be Pferred from measurements of angles in photographs of activel beating flagella. 6 KXB 71

2 158 STUART F. GOLDSTEIN Smmetrical waveforms of tritonated sea urchin sperm flagella have been describ^ in previous studies (Goldstein, 1975, 1976a). In the stud presented here, the waveforms offlagellaof live spermatozoa of a number of echinoderms have been analsed, with attention paid to asmmetries and to the inferred patterns of microtubular sliding. Some of the results on angles of bends have been presented at a meeting (Goldstein & Pivonka, 1975). MATERIALS AND METHODS Spermatozoa studied included those of the California sea urchins Stronglocmtrotw purpuratus (Light et al. 1967; Mortensen, 1940) and Lteckinus pictus (Light et al. 1967; Mortensen, 1940), the Bermuda sea urchins Tripneustes csculentus (Clark, 1942; Mortensen, 1940), Echinometra hicunter (Clark, 1942; Mortensen, 1940), and Ltechinus variegatui (Clark, 1942; Mortensen, 1940), and a Bermuda brittle star, Ophiocoma echinata (Clark, 1942). The were observed in artificial sea water containing 0-25% bovine serum albumin (BSA) (Sigma Chemical Co., St Louis, MO 63178). The BSA improved the uniformit and longevit of beating and reduced the tendenc of spermatozoa to adhere to glass, without changing the qualitative appearance of beating from the best of that seen in sea water lacking BSA. Slides and coverglasses were treated as described previousl (Goldstein, 1976 a). Headless spermatozoa were produced b passage through a pipette (Brokaw, 1970) in BSA sea water at ph 5-3, in which the were immotile (Goldstein, 19766). The were photographed while swimming near the interface of this suspension and BSA sea water at ph 8-3. Results were recorded with dark-field, multiple-exposure photographs, taken with the film either stationar (Brokaw, 1970) or moving (Goldstein, 1976a), as previousl described, usuall at a magnification on the film of x Measurements were made on prints as previousl described (Goldstein, 1976a). Bends and straight regions were followed through a set of exposures from their start at the flagellar base until the began to travel off the tip. The value of a parameter was assumed to be equal to zero on the exposure just preceding the first image on which it was large enough to be measured reliabl. The graphs of parameters were approximated b straight line segments between the measured values, and the 'average' value of an angle, radius, or length is the value as averaged over these straight segments. These parameters tpicall increased to a maximum value and then decreased slightl. The ' peak' value of a parameter that was followed through N images is the value averaged over the Nj\ consecutive images which gave the highest total value. The measurements of angles depend on reliable estimates of tangents to the flagellar base. The midpiece tpicall obscures about im of the flagellum. Fig. 1. Marker of Fig. 2 is 40 im in all photographs except 1 (a) (16 *m). (a) L. pictus. Beat freq. approx. 19 Hx. Flash freq. 8-6 Hz. Taken with film moving. (b) T. uculentut. Beat freq. approx. 40 Hz. Flash freq. 158 Hz. Taken with film moving, (c) L. pictus. Beat freq. approx. 40 Hz. Flash freq Hz. (d) L. variegatus. Beat freq. appror. 40 Hz. Flash freq Hz. (e) T. esculentus. Beat freq. approx. 48 Hz. Flash freq Hz. () O. echinata. Beat freq. approx. 40 Hz. Flash freq. 137 Hz. (g) S. purpuratus. Beat freq. approx. 35 Hz. Flash freq Hz. (h) E. hicunter. Beat freq. approx. 40 Hz. Flash freq Hz.

3 Journal of Experimental Biolog, Vol. 71 Fie. 1 Of), (I) STUART F. GOLDSTEIN {Facing p. 158)

4 fnurnal nf Experimental Biolopv. Vol. 71 Fig. 2 Fig. 2. Marker is 40 <m for all photographs except z(c) (10im). (a) E. lucunter. Beat freq. approx. 40 Hz. Flash freq Hz. Taken with film moving, for analsis of waveform. (6) Same spermatozoon as Fig. 2 (a). Taken with film stationar, to show swim path, (c) E. lucunter. Basal region, (d) S. purpuratus. Beat freq. approx. 35 Hz. Flash freq. 203 Hz. Taken with film moving, for analsis of waveform. Exposure farthest to right is exposure 1 in Figs. 4, 6-9, and 12. («) Same spermatozoon a9 Fig. i(d). Taken with film stationar, to show swim path. () L. pictus. Headless. Flash freq. 125 Hz. STUART F. GOLDSTEIN

5 Asmmetric waveforms in echinoderm sperm flagella E. lucunter S. purpuralus L. piclus ; A go o 20 T. esculentus Swim radius Om) 20 L. variegatus n.-nil J in fl > >150 Fig. 3. Distributions of radii of swim paths of sea urchin spermatozoa. In this stud the base of a flagellum has usuall been assumed to subtend a constant angle to the axis of the head. This appeared to be more reliable than the apparent angle of a sharpl bending flagellum as it entered the midpiece. It also produced a more conservative estimate of the net sliding in the pair of bends forming at the base. Estimates of microtubular sliding made on this assumption were in agreement with those made from measurements on headless flagella (Fig. 2). RESULTS The waveforms of these flagella could usuall be approximated well as circular arcs connected b straight segments. The most common deviation from this idealization was a temporar departure of reverse bends from circular as the following bend formed. Except for differences in the degree of asmmetr discussed below, no essential differences in waveform were noted among the species observed. The spermatozoon of Figs. z(d) and 2(e), in which the asmmetr is relativel pronounced, is used to illustrate the general findings. For analsis of Fig. 2(d), in which there are 19 exposures to a beat 'ccle', a single principal bend travelling from base to tip is constructed from bend 4 in images P^U8 bend 2 in images 6-24; a reverse bend is constructed from bend 3 in images 1-24 plus bend 1 in images Swim path Free-swimming spermatozoa swim in helical paths; when swimming against a teverglass the path is usuall at least approximatel circular (Gra, 1955). Examples m the species used in this stud, and of the variet of swim path radii observed, are 6-3

6 i6o STUART F. GOLDSTEIN "u 1-2 ngl < i I J 1 I 1 i 1 * f * i \ t \ t A A I f fi 1 vi i i X T ^ \ \ 0 i i Exposure I Fig. 4. Bend angles of principal bend (solid line) and reverse bend (broken line) offlagellumof Fig. shown in Figs. 1 (c-h). Distributions of radii of the swim paths of various species are shown in Fig. 3. Each of the samples shown contained at least 100 spermatozoa. There was some variation between samples, and samples could change somewhat with age. Angles of bends The development of the angles subtended b the principal and reverse bends of the spermatozoon of Fig. 2 (d) is shown in Fig. 4. The faster rates of increase in the angles subtended b principal bends were apparent from the start of their formation. A bend usuall began to form when the angle of the previous bend had attained about half its maximum value, although the point in the development of one bend at whici the following bend began to form varied somewhat among individual spermatozo

7 Asmmetric waveforms in echinoderm sperm flagella T3 e # principal (rad) Fig. 5. Relationship between peak angles of principal bends and reverse bends. Broken line is locus of points for perfectl smmetrical waveforms. Solid line is regression line. The maximum value was tpicall reached b about two-thirds of a beat ccle. The angle of a bend often increased somewhat as the bend approached the tip. The radius of the swim path of a spermatozoon was related to the degree of asmmetr in bend angle, as described below. The variations in swim paths shown in Fig. 3 therefore reflect variations in the asmmetries of bend angles, both among and within species. Fig. 5 shows the peak angles attained b the principal and reverse bends of the spermatozoa whose swim path radii are shown in Figs. 10 and 11. More asmmetricflagellatend to have both larger principal bends and smaller reverse bends than less asmmetric ones: the regression line for Fig. 5 is = *; the correlation coefficient is Radii and lengths of bends The development of the radii of the bends of the spermatozoon of Fig. 2(d) is shown in Fig. 6. The development of the lengths of these bends is shown in Fig. 7. The initial radii and lengths of newl forming bends were too small to be measured in these photographs. The could be measured b the time the had reached about 2 im; these values were tpicall obtained b about 30 % of a beat ccle after a bend angle had begun to develop. The radii of reverse bends were alwas larger than those of principal bends. Peak values of the radii of principal bends were tpicall between 3 and 6 im, depending on the degree of asmmetr; those of reverse bends were tpicall between 5 and im. A bend radius often decreased somewhat as the bend approached the tip. Differences between the radii of the principal and reverse bends tended to cancel

8 162 STUART F. GOLDSTEIN I 3 7 I i VA 4 f Exposure Fig. 6. Radii of principal bend (solid line) and reverse bend (broken line) of flagellum of Fig. a{d). differences between their angles, so that the peak lengths of the principal and reverse bends generall differed b not more than io %, even when the peak value of the angle of a principal bend was twice that of the reverse bend. Straight regions The development of straight regions of the spermatozoon of Fig. 2 (d) is shown in Fig. 8. Straight regions began to form as newl developing bends travelled awa from the base. This usuall happened when the angle of a bend had reached about half its maximum value, just before the following bend began to form (see Fig. 9). There was no regular difference between the length of the straight region distal to a principal bend and that of the one proximal to it; occasional spermatozoa with appreciable differences between these straight regions exhibited patentl odd waveforms. Propagation of bends The positions of bends travelling along the flagellum of Fig. 2(d) are shown in Fig. 9. No regular differences were noted between the speeds of principal bends and those of reverse bends.

9 Asmmetric waveforms in echmoderm sperm flagella 163 5> Exposure Fig. 7. Lengths of principal bend (solid line) and reverse bend (broken line) of flagellum of Fig. 2(d). Curvature of swim path The relationship between the asmmetr of the waveform of a spermatozoon and the curvature of its trajector through the water was studied b plotting the curvature (the inverse of the radius, ir) of the swim path against various functions of the angles and radii of the principal and reverse bends. The curvature tended to increase with asmmetr in bend radii. However, better correlations were found between ir and the asmmetr in bend angles. The relationship between ir and the difference between the average angles of the principal and reverse bends, d p d T, is shown in Fig. 10. The relationship between ii? and the relative difference in average bend angles, (B p 5 r )5 f, is shown in Fig. 11. In Figs. 10 and 11 bend angles less than one fcadian have been neglected; i.e. the bend angle was assumed to be zero until the nrst image in which it was at least one radian. Scatter was reduced when these small

10 164 STUART F. GOLDSTEIN! 3 a Exposure Fig. 8. Lengths of straight regions proximal to principal bend (solid line) and reverse bend (broken line) of flagellum of Fig. a(d). angles near the base were neglected. Curvature of swim path increased with the difference in peak angles of bends, but the correlation was not as good as those in Figs. 10 and 11. The regression line for the main group of spermatozoa (all species except i?. lucunter) in Figs. 10 and 11 are o-ooi 55* and^ = *, respectivel; the respective correlation coefficients are and The values for the spermatozoa of E. lucunter have been indicated separatel from those of the other species in Figs. 10 and 11, to illustrate that the swam in smaller circles than would be expected from the asmmetr in their waveforms. The regression lines for the spermatozoa of E. lucunter in Figs. 10 and n are = * and = *, respectivel; the respective correlation coefficients are and Most of the values for E. lucunter lie outside of the 95% prediction intervals in Figs. 10 and 11 (Sokal & Rohlf, 1969); the differences in the distributions of E. lucunter and those of the other spermatozoa are statisticall highl significant. The spermatozoa of E. lucunter have an unusuall long head (about 7-8 fim, compared to about 4-7im, 5-6 fim, 4-7 tm, 4-7 im, and 3-1 fim for 5. purpuratus, L. pictus, T. esculentus, L. variegatus, and O. echinata, respectivel). The axis of this head was tilted at an angle to the point on the flagellum emerging from the midpiece. This angle tpicall varied from about o-o-i radian when a reverse bend formed at thj base to about i-o radian when a principal bend formed, as shown in Fig. 2{c). Thw

11 Asmmetric waveforms in echinoderm sperm flagella o 20 I.6 Q Exposure Fig. 9. Positions of principal bend (solid line) and reverse bend (broken line) of flagellum of Fig. z(d). Vertical lines indicate exposures in which bends leave base and basal straight regions begin to develop. axis of the head was usuall tangential to the swim path. The head ma therefore act as a rudder, steering the spermatozoon into a tighter circle than would normall result from the asmmetr of its flagellar waveform. Microtubular slidirig The difference in the rates of increase of the angles of the principal and reverse bends shown in Fig. 4 implies that there is a net microtubular sliding in the two bends developing nearest the base: these bends do not completel cancel one another; the associated net sliding is transferred distall along the flagellum. The angles between the flagellar base and the straight regions distal to pairs of bends developing near the base of the flagellum of Fig. 2(d) are shown in Fig. 12. The angles are shown for straight regions just proximal to both a principal and a reverse bend.

12 turf) 166 STUART F. GOLDSTEIN 010 I ure U V t ooe.^ o 0 O O X? <?o X '' T X -^ >»^ ( V)6 A A«(rad) Fig. io. Radius of swim path as a function of difference between average angles of principal and reverse bends. Circles represent spermatozoa of E. luamter; dots represent those of other species. Arrow indicates spermatozoon of Figs, a(rf-e). Solid line indicates regression line; broken lines indicate 95 % prediction interval. DISCUSSION Microtubular sliding distal to bends forming near the base, as indicated in Fig. 12, implies sliding within the straight regions distal to these bends. In the flagellum of Fig. 2 (d) the straight region just distal to principal bend 2 has a maximum rate of change of angle, ddjdt, of radexposure, between exposures 5 and 9; the straight region just distal to reverse bend 3 has a maximum rate of radexp, between exposures 14 and 21. There is also appreciable sliding in a straight region between the two bends developing near the base (Goldstein, 1975, 1976a). Flagellar straight regions cannot, therefore, be characterized as regions in which no sliding can occur; this is also true of ciliar straight regions (Satir, 1965, 1974). The occurrence of microtubular sliding within straight regions implies that the outer doublets are not rigidl cross-linked within them. In cilia, the spokes appear to be connected to the central sheath within bends but disconnected from it within straight regions (Warner & Satir, 1974); there is no reason to doubt that this is also true in fiagella. It has beenj argued that straight regions must be quite stiff to resist being curved b hdrodnami

13 Asmmetric waveforms in echinoderm sperm flagella ^ ^ g 005 = o 0 C...- «" * O o oo o. 1 'OO A\ ^ f, x 1 I Fig. 11. Radius of swim path as a function of relative difference between average bend angles of principal and reverse bends. Smbols same a«in Fig. IO. forces (Brokaw, 1965), suggesting that the doublets ma not be completel free to slide within them. It is not known whether ATPase activit or transient crossconnexions occur within straight regions. Sliding associated with an change in the angle between the base and a straight region just proximal to a bend is transferred to that bend, so that changes of radexp and radexp are transferred to bends 1 and 2, respectivel, in Fig. z(d). As a full formed bend travels along a flagellum, there is a continual change in the angle between the flagellar base and points within that bend. The rate of change of angle at an point within a bend associated with its propagation is: d$_dd dsv dt~& Jt~r~' where s = distance along the flagellum, v = speed of propagation of the bend along the flagellum, and r = radius of the bend. The value of v for bend 1 in Fig. z(d) e between exposures 5 and 9 is 1-64 fimexp, as shown in Fig. 9; that for bend 2 between :posures 14 and 21 is i-34imexp. The corresponding values of r for bends 1 and 2 e 7-0 *m and 4-5 fim, respectivel. The rates of change of angle within bends 1 and

14 168 STUART F. GOLDSTEIN o 2 i I 0 -o-i 1-5 s I' D. S -0-4 X o -0-5 o. OJJ -0-6 < Exposure Fig. i a. Total angles of pairs of developing bends, as measured between base and straight region just proximal to a principal bend (solid line) and a reverse bend (broken line) of nagellum of Fig. i{d). 2 associated with their propagation in these exposures is therefore radexp and radexp, respectivel. The speeds of extrinsic sliding imposed on bends 1 and 2 from the straight regions just proximal to them are, then, about 28% and 56%, respectivel, of those associated with their propagation. As Fig. 12 shows, this extrinsic sliding is not constant, and can change direction. Earlier analsis of flagella with smmetrical waveforms (Goldstein, 1975, 1976a) indicated that a bend can travel in the absence of sliding or viscous forces from other regions of a flagellum. The analsis in the present stud indicates that a bend can travel in spite of appreciable extrinsic sliding imposed upon it; an effects of this extrinsic sliding appear to be subtle. This apparent insensitivit to microtubular sliding is complemented b a sensitivit to artificiall imposed bending (Lindemann & Rikmenspoel, 1972^ Shingoji, Murakami & Takahashi, 1977). These observations suggest that bending'

15 Asmmetric waveforms in echinoderm sperm flagella 169 is controlled primaril b bend curvature (Machin, 1963; Brokaw, 1971, 1972) rather than b shear velocit (Brokaw, 1975). The curvature of the swim path of a spermatozoon increases with the asmmetr in bend angles, as shown in Figs. 10 and 11. The scatter in these figures is probabl not due entirel to experimental errors. The improvement which occurred when small bends near the base were neglected suggests that the angle of a bend should be weighted b some function of amplitude. A relationship between asmmetr of bend amplitude and curvature of swim path has been suggested for bull spermatozoa b Rikmenspoel, van Herpen & Eijkhout (1960), but the method of estimating asmmetr was not given. Theoretical work on the hdrodnamic effects of asmmetr has begun (Yundt, Shack & Lardner, 1975; Keller & Rubinow, 1976). The spermatozoa of E. lucunter swim in smaller circles than would be expected from the asmmetr of their waveform. This suggests that their long, tilted head can produce an appreciable hdrodnamic effect and suggests a possible general function for the variet of shapes exhibited b sperm heads. It is possible that the anomalous swimming of these spermatozoa is due to an anomal in their waveform, but the onl unusual feature noted was a greater tendenc than that in most spermatozoa for reverse bends to be somewhat non-circular during the formation of the principal bends which follow. The reasons for the existence of asmmetr in flagellar waveforms are not clear. The ma simpl reflect differences in the natures of the principal and reverse bends (Goldstein, 1976b). On the other hand, the asmmetr ma be important in producing an optimum waveform, affecting the shape of the path of the free-swimming spermatozoon, which could in turn affect the probabilit of hitting an egg. I greatl appreciate the generous help of the other members of the Bermuda Cell Motilit and Development Group. I am also indebted to Dr C. J. Brokaw, in whose laborator a number of the photographs were taken; and to Dr W. Schmid, of the International Institute for Hermonograph, for statistical analses. Support came from National Science Foundation grant no. BMS A01 to me, National Science Foundation grant no. GB43627 to the Bermuda Cell Motilit and Development Group, and National Institutes of Health grant no. GM to C. J. Brokaw. Contribution no. 713 from the Bermuda Biological Station for Research. REFERENCES BROKAW, C. J. (1965). Non-sinusoidal bending waves of sperm flagella. J. exp. Biol. 43, BROKAW, C. J. (1970). Bending moments in free-swimming flagella. J. exp. Biol. 53, BROKAW, C. J. (1971). Bend propagation b a sliding filament model for flagella. J. exp. Biol. 55, BROKAW, C. J. (1972). Computer simulation of flagellar movement. I. Demonstration of stable bend propagation and bend initiation b the sliding filament model. Biopks. J. 12, BROKAW, C. J. (1975). Molecular mechanism for oscillation in flagella and muscle. Proc. natn. Acad. Sci. U.S.A. 7a, BROKAW, C. J., GOLDSTEIN, S. F. & MILLER, R. L. (1970). Recent studies on the motilit of spermatozoa from some marine invertebrates. In Comparative Spermatolog (ed. B. Baccetti), pp London: Academic Press. BROKAW, C. J., JofS'-iN, R. & BOBROW, L. (1974). Calcium ion regulation of flagellar beat smmetr in reactivated sea urchin spermatozoa. Biochem. Biophs. Res. Commun. 58, BROKAW, C. J. & WRIGHT, L. (1963). Bending waves of the posterior flagellum of Ceratium. Science, N.Y. 14a,

16 170 STUART F. GOLDSTEIN CLARK, H. L. (1943). Echinoderm fauna of Bermuda. Bull. Museum comp. zoo!., Harvard 89, GIBBONS, B. H. & GIBBONS, I. R. (1972). Flagellar movement and adenosine triphosphatase activit in sea urchin sperm extracted with Triton X-100. J. Cell Biol, 54, GOLDSTEIN, S. F. (1969). Irradiation of sperm tails b laser microbeam. J. exp. Biol. 51, GOLDSTEIN, S. F. (1975). Morpholog of developing bends in sperm flagella. In Swimming and Fling in Nature (ed. T. Y. Wu, C. J. Brokaw and C. Brennan). New York: Plenum Publishing Co. GOLDSTEIN, S. F. (1976a). Form of developing bends in reactivated sperm flagella. J. exp. Biol. 64, GOLDSTEIN, S. F. (19766). Bend initiation in quiescent sperm flagella. J. Cell Biol. 70, 71A. GOLDSTEIN, S. F. & PIVONKA, P. R. (1975). Microtubular sliding in asmmetricall beating flagella. J. Cell Biol. 67, 136A. GRAY, J. (1955). The movement of sea urchin spermatozoa. J. exp. Biol. 33, KELLER, J. B. & RUBINOW, S. I. (1976). Swimming of flagellated microorganisms. Biopks. J LIGHT, S. F., SMITH, R. I., PITELKA, F. A., ABBOTT, D. P. & WEESNER, F. M. (1967). Inttrtidal Invertebrates of the Central California Coast, pp Berkele: Universit of California Press. LINDEMANN, C. B. & RIKMENSPOEL, R. (1972). Sperm flagella: autonomous oscillations of the contractile sstem. Science, N. Y. 175, MACHIN, K. E. (1963). The control and snchronisation of flagellar movement. Proc. R. Soc. B 158, MORTENSEN, TH. (1940). A Monograph of the EcUnoidea, vol. 3: 2, 3. Copenhagen: C. A. Reitzel. RIKMENSPOEL, R. (1971). Contractile mechanisms in flagella. Biophs. J. 11, RIKMENSPOEL, R., VAN HERPEN, G. & EIJKHOUT, P. (i960). Cinematographic observations of the movements of bull sperm cells. Pks. Med. Biol. 5, SATIR, P. (1965). Studies on cilia. II. Examination of the distal region of the ciliar shaft and the role of the filaments in motilit. J. Cell Biol. 36, SATIR, P. (1974). The present status of the sliding microtubule model of ciliar motion. In Cilia and Flagella (ed. M. A. Sleigh), pp London: Academic Press. SHINGYOJI, C, MURAKAMI, A. & TAKAHASHI, K. (1977). Local reactivation of Triton-extracted flagella b iontophoretic application of ATP. Nature, Lond. 365, SILVESTER, N. R. & HOLWILL, M. E. J. (1972). An analsis of hpothetical flagellar waveform*. J. theor. Biol. 35, SOKAL, R. R. & ROHLF, F. J. (1969). Biometr, p San Francisco: W. H. Freeman. WARNER, F. D. & SATIR, P. (1974). The structural basis of ciliar bend formation. J. Cell Biol. 63, YUNDT, A. P., SHACK, W. J. & LARDNER, T. J. (1975). Applicabilit of hdrodnamic analses of gpermatozoan motion. J. exp. Biol. 6a,