Asymmetric Waveforms in Echinoderm Sperm Flagella

The symmetrical, planar waveforms of echinoderm sperm flagella have been variously described as sine waves (Gray, 1955), meander-like waves (Brokaw, Goldstein & Miller, 1970; Rikmenspoel, 1971; Silvester & Holwill, 1972), or circular arcs connected by straight lines (Brokaw & Wright, 1963; Brokaw, 1965).

Tritonated sea urchin spermatozoa can beat quite symmetrically in reactivating solutions containing low concentrations of calcium (Brokaw, Josslin & Bobrow, 1974). Live echinoderm spermatozoa which are attached to a surface by their heads can exhibit symmetrical waveforms (Fig. 1a), but they can also exhibit asymmetrical waveforms (Fig. 1 b). When swimming freely next to a surface, they typically exhibit asymmetrical planar waveforms: they 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 asymmetrically beating flagella.

The sliding microtubule model of flagellar motility (Satir, 1965, 1974) predicts that the amount of sliding occurring between two microtubules at any point along a flagellum during any portion of a beat cycle is directly proportional to the change in the angle between a tangent to the flagellum at that point and a tangent to the flagellar base (Goldstein, 1969). This means that patterns of microtubular sliding can be referred from measurements of angles in photographs of actively beating flagella.

Symmetrical waveforms of tritonated sea urchin sperm flagella have been described in previous studies (Goldstein, 1975, 1976a). In the study presented here, the waveforms of flagella of live spermatozoa of a number of echinoderms have been analysed, with attention paid to asymmetries 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).

Spermatozoa studied included those of the California sea urchins Strongylocentrotus purpuratus (Light et al. 1967; Mortensen, 1940) and Lytechinus pictus (Light et al. 1967; Mortensen, 1940), the Bermuda sea urchins Tripneustes esculentus (Clark, 1942; Mortensen, 1940), Echinometra lucunter (Clark, 1942; Mortensen, 1940), and Lytechinus variegatus (Clark, 1942; Mortensen, 1940), and a Bermuda brittle star, Ophiocama echinata (Clark, 1942). They were observed in artificial sea water containing 0·25% bovine serum albumin (BSA) (Sigma Chemical Co., St Louis, MO 63178). The BSA improved the uniformity and longevity of beating and reduced the tendency 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 previously (Goldstein, 1976 a).

Headless spermatozoa were produced by passage through a pipette (Brokaw, 1970) in BSA sea water at pH 5-3, in which they were immotile (Goldstein, 1976b). They 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 stationary (Brokaw, 1970) or moving (Goldstein, 1976a), as previously described, usually at a magnification on the film of × 160·200.

Measurements were made on prints as previously described (Goldstein, 1976a). Bends and straight regions were followed through a set of exposures from their start at the flagellar base until they 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 reliably. The graphs of parameters were approximated by 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 typically increased to a maximum value and then decreased slightly. The‘peak’ value of a parameter that was followed through N images is the value averaged over the N/4 consecutive images which gave the highest total value.

The measurements of angles depend on reliable estimates of tangents to the flagellar base. The midpiece typically obscures about 0·5–1·0 μm of the flagellum.

In this study the base of a flagellum has usually 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 sharply 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. 2f).

The waveforms of these flagella could usually be approximated well as circular arcs connected by straight segments. The most common deviation from this idealization was a temporary departure of reverse bends from circular as the following bend formed.

Except for differences in the degree of asymmetry discussed below, no essential differences in waveform were noted among the species observed. The spermatozoon of Figs. 2(d) and 2(e), in which the asymmetry is relatively pronounced, is used to illustrate the general findings. For analysis of Fig. 2(d), in which there are 19 exposures to a beat ‘cycle’, a single principal bend travelling from base to tip is constructed from bend 4 in images 10–24 Plus bend 2 in images 6–24 ; a reverse bend is constructed from bend 3 in images 1–24 plus bend 1 in images 6–13.

Free-swimming spermatozoa swim in helical paths; when swimming against a overglass the path is usually at least approximately circular (Gray, 1955). Examples w the species used in this study, and of the variety of swim path radii observed, are 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.

The development of the angles subtended by 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 by principal bends were apparent from the start of their formation. A bend usually 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 which the following bend began to form varied somewhat among individual spermatozed The maximum value was typically reached by about two-thirds of a beat cycle. 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 asymmetry in bend angle, as described below. The variations in swim paths shown in Fig. 3 therefore reflect variations in the asymmetries of bend angles, both among and within species. Fig. 5 shows the peak angles attained by the principal and reverse bends of the spermatozoa whose swim path radii are shown in Figs. 10 and 11. More asymmetric flagella tend to have both larger principal bends and smaller reverse bends than less asymmetric ones: the regression line for Fig. 5 is y = 3·12 −0·582x; the correlation coefficient is −0·453.

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 newly forming bends were too small to be measured in these photographs. They could be measured by the time they had reached about 2 μm ; these values were typically obtained by about 30 % of a beat cycle after a bend angle had begun to develop.

The radii of reverse bends were always larger than those of principal bends. Peak values of the radii of principal bends were typically between 3 and 6 μm, depending on the degree of asymmetry; those of reverse bends were typically between 5 and 6 μm. 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 differences between their angles, so that the peak lengths of the principal and reverse bends generally differed by not more than 10 %, even when the peak value of the angle of a principal bend was twice that of the reverse bend.

The development of straight regions of the spermatozoon of Fig. 2 (d) is shown in Fig. 8. Straight regions began to form as newly developing bends travelled away from the base. This usually 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 patently odd waveforms.

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.

The relationship between the asymmetry of the waveform of a spermatozoon and the curvature of its trajectory through the water was studied by plotting the curvature (the inverse of the radius, 1/R) of the swim path against various functions of the angles and radii of the principal and reverse bends. The curvature tended to increase with asymmetry in bend radii. However, better correlations were found between 1/R and the asymmetry in bend angles. The relationship between 1 /R and the difference between the average angles of the principal and reverse bends, is shown in Fig. 10. The relationship between1/R and the relative difference in average bend angles, is shown in Fig. 11. In Figs. 10 and 11 bend angles less than one radian 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 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 E. lucunter) in Figs, 10 and 11 are y = − 0·00294 + 0·00155x and y = 0-00422 + 0-0984x, respectively; the respective correlation coefficients are 0·927 and 0·923.

The values for the spermatozoa of E. lucunter have been indicated separately from those of the other species in Figs. 10 and 11, to illustrate that they swam in smaller circles than would be expected from the asymmetry in their waveforms. The regression lines for the spermatozoa of E. lucunter in Figs. 10 and 11 are y = 0·0284 + 0·00119x and y = 0·0390 + 0·0682x, respectively; the respective correlation coefficients are 0·774 and 0·771. 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 statistically highly significant. The spermatozoa of E. lucunter have an unusually long head (about 7·-8 μm, compared to about 4·7 μm, 5·6 μm, 4·7 μm, 4·7 μm, and 3·1 μm for S. purpuratus, L. pictus, T. esculentus, L. variegatus, and O. echinata, respectively). The axis of this head was tilted at an angle to the point on the flagellum emerging from the midpiece. This angle typically varied from about 0–0·1radian when a reverse bend formed at the base to about 1·0 radian when a principal bend formed, as shown in Fig. 2(c). The axis of the head was usually tangential to the swim path. The head may therefore act as a rudder, steering the spermatozoon into a tighter circle than would normally result from the asymmetry of its flagellar waveform.

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 completely cancel one another; the associated net sliding is transferred distally 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.

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, dθ/dt, of 0·065 rad/exposure, between exposures 5 and 9; the straight region just distal to reverse bend 3 has a maximum rate of 0·167 rad/exp, 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 ciliary straight regions (Satir, 1965, 1974). The occurrence of microtubular sliding within straight regions implies that the outer doublets are not rigidly 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 flagella. It has beem argued that straight regions must be quite stiff to resist being curved by hydrodynamic forces (Brokaw, 1965), suggesting that the doublets may not be completely free to slide within them. It is not known whether ATPase activity or transient cross-connexions occur within straight regions.

Sliding associated with any 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 0·065 rad/exp and 0·167 rad/exp are transferred to bends 1 and 2, respectively, in Fig. 2(d).

The curvature of the swim path of a spermatozoon increases with the asymmetry in bend angles, as shown in Figs. 10 and 11. The scatter in these figures is probably not due entirely 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 by some function of amplitude. A relationship between asymmetry of bend amplitude and curvature of swim path has been suggested for bull spermatozoa by Rikmenspoel, van Herpen & Eijkhout (1960), but the method of estimating asymmetry was not given. Theoretical work on the hydrodynamic effects of asymmetry has begun (Yundt, Shack & Lardner, 1975; Keller & Rubinow, 1976).

The spermatozoa of E. lucunter swim in smaller circles than would be expected from the asymmetry of their waveform. This suggests that their long, tilted head can produce an appreciable hydrodynamic effect and suggests a possible general function for the variety of shapes exhibited by sperm heads. It is possible that the anomalous swimming of these spermatozoa is due to an anomaly in their waveform, but the only unusual feature noted was a greater tendency 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 asymmetry in flagellar waveforms are not clear. They may simply reflect differences in the natures of the principal and reverse bends (Goldstein, 1976b). On the other hand, the asymmetry may be important in producing an optimum waveform, affecting the shape of the path of the free-swimming spermatozoon, which could in turn affect the probability of hitting an egg.

I greatly appreciate the generous help of the other members of the Bermuda Cell Motility and Development Group. I am also indebted to Dr C. J. Brokaw, in whose laboratory a number of the photographs were taken; and to Dr W. Schmid, of the International Institute for Hermonography, for statistical analyses. Support came from National Science Foundation grant no. BMS73-06710-A01 to me, National Science Foundation grant no. GB43627 to the Bermuda Cell Motility and Development Group, and National Institutes of Health grant no. GM-18711 to C. J. Brokaw.

Contribution no. 713 from the Bermuda Biological Station for Research.