J. exp. Biol. (1977), 71, 157-170
157
~'ith 12 figures
&ii
inted in Great Britain
ASYMMETRIC WAVEFORMS IN ECHINODERM
SPERM FLAGELLA
BY STUART F. GOLDSTEIN
Department of Genetics and Cell Biology, University 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 analysed.
2. Bends develop at the base in pairs, as they do in flagella with symmetrical 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 distally. This sliding implies that (a) microtubules are not rigidly cross-Linked within straight regions, and (b) bends
can propagate normally in spite of appreciable extrinsic microtubular sliding.
3. Both the sperm head and the asymmetry of the waveform appear to
affect the swim path of a spermatozoon.
INTRODUCTION
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. id), 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
Jaase (Goldstein, 1969). This means that patterns of microtubular sliding can be
Pferred from measurements of angles in photographs of actively beating flagella.
6
KXB 71
158
STUART F . GOLDSTEIN
Symmetrical waveforms of tritonated sea urchin sperm flagella have been describ^
in previous studies (Goldstein, 1975, 1976a). In the study presented here, the waveforms offlagellaof 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).
MATERIALS AND METHODS
Spermatozoa studied included those of the California sea urchins Strongylocmtrotw
purpuratus (Light et al. 1967; Mortensen, 1940) and Lyteckinus pictus (Light et al.
1967; Mortensen, 1940), the Bermuda sea urchins Tripneustes csculentus (Clark, 1942;
Mortensen, 1940), Echinometra hicunter (Clark, 1942; Mortensen, 1940), and Lytechinus variegatui (Clark, 1942; Mortensen, 1940), and a Bermuda brittle star, Ophiocoma
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, 19766). 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 x 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 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 typically obscures about 0-5-1-0 /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. 20-0 Hz. (d) L. variegatus. Beat freq. appror. 40 Hz. Flash freq. 21-6 Hz.
(e) T. esculentus. Beat freq. approx. 48 Hz. Flash freq. 15-1 Hz. (/) O. echinata. Beat freq.
approx. 40 Hz. Flash freq. 137 Hz. (g) S. purpuratus. Beat freq. approx. 35 Hz. Flash freq.
18-3 Hz. (h) E. hicunter. Beat freq. approx. 40 Hz. Flash freq. 13-2 Hz.
Journal of Experimental Biology, Vol. 71
Of),
STUART F. GOLDSTEIN
Fie. 1
(/I)
{Facing p. 158)
fnurnal nf Experimental Biolopv. Vol. 71
Fig. 2
Fig. 2. Marker is 40 /<m for all photographs except z(c) (10/im). (a) E. lucunter. Beat freq.
approx. 40 Hz. Flash freq. 21-6 Hz. Taken with film moving, for analysis of waveform.
(6) Same spermatozoon as Fig. 2 (a). Taken with film stationary, 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 analysis 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 stationary, to show swim
path. (/) L. pictus. Headless. Flash freq. 125 Hz.
STUART F. GOLDSTEIN
Asymmetric waveforms in echinoderm sperm flagella
E. lucunter
S. purpuralus
L. piclus
40
40
40
20
20
20
• g o
20
40
0
60
o
20
40
60
;
0
A
20
40
60
Swim radius Om)
T. esculentus
L. variegatus
20
20
10
10
n .-n
0
40
80
il
> 100
J 11 1 1 in fl / /
0
40
80
120
>150
Fig. 3. Distributions of radii of swim paths of sea urchin spermatozoa.
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. 2/).
RESULTS
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. z(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 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 6-13.
Swim path
Free-swimming spermatozoa swim in helical paths; when swimming against a
teverglass the path is usually at least approximately circular (Gray, 1955). Examples
m the species used in this study, and of the variety of swim path radii observed, are
6-3
i6o
STUART F. GOLDSTEIN
2-4 -
2-2
•
20
-
X
^ T \
1-8
A
1-6
4
1
1
1-4
2
i
/
/
/
I
ngl
"u 1-2
<
10
i
1
/
i
If
I
1
0-6
y
04 •
0-2
0
y
t
t
\
1
J
1
0-8
vi
/i /
*
*
i
\
fI
/f
\
A/
i
i
10
1
15
20
1
25
30
Exposure
1
1
1
I
35
40
45
50
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 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 whici|
the following bend began to form varied somewhat among individual spermatozo™
Asymmetric waveforms in echinoderm sperm flagella
161
2-4
2-2
20
T3
e
1-8
1-6
1-4
10
1-2
1-4
1-6
1-8
20
2-2
2-4
2-6
2-8
30
# principal (rad)
Fig. 5. Relationship between peak angles of principal bends and reverse bends. Broken line is
locus of points for perfectly symmetrical waveforms. Solid line is regression line.
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
asymmetricflagellatend to have both larger principal bends and smaller reverse bends
than less asymmetric ones: the regression line for Fig. 5 is y = 3-12 — 0-582*; the
correlation coefficient is -0-453.
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 newly forming bends were too small to be measured
in these photographs. They could be measured by the time they had reached about
2 /im; 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 /im, depending
on the degree of asymmetry; those of reverse bends were typically 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
162
STUART F. GOLDSTEIN
10
9
8
7
I
I3
VA
/
i
4
f
0
2
10
5
15
20
25
30
35
40
45
50
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 generally differed by 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 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.
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.
Asymmetric waveforms in echmoderm sperm flagella
163
5>
10
15
20
25
30
Exposure
35
40
45
50
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 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, i/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 i/R
and the asymmetry in bend angles. The relationship between i/R and the difference
between the average angles of the principal and reverse bends, dp — dT, is shown in
Fig. 10. The relationship between i/i? and the relative difference in average bend
angles, (Bp — 5r)/5f, 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
164
STUART F. GOLDSTEIN
!
3
a
15
20
25
30
Exposure
35
40
45
50
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 y — — 0-00294 + o-ooi 55* and^ = 0-00422
+ 0-0984*, 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 n are y = 0-0284 + 0-00119*
and y = 0-0390 + 0-0682*, 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 fim, compared
to about 4-7/im, 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, 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 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
Asymmetric waveforms in echinoderm sperm flagella
165
40
36
32
•
28
124
o 20
I .6
Q
12
10
15
20
25
30
Exposure
35
40
45
50
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 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.
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 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.
166
STUART F. GOLDSTEIN
y
y
/
010
y
009 -
(
V
y
)6
y
007 •
0
o
O
O
/
/
/
y
ooe.^
ure
turf)
I
006
0-05
•
004
•
003
•
002
° X° °?T < / X
y
y
/
/
y
A
?o
/
/
y
008
U
/
y
y
y
y
/
y
y''
/ / X• /
/ • / / • •/ -^
> »^ • • /
V
001
t
0
01
0-2
0-3
0-4
0-5
0-6
A«(rad)
0-7
0-8
0-9
10
11
12
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 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 fiagella. It has beenj
argued that straight regions must be quite stiff to resist being curved by hydrodynami™
Asymmetric waveforms in echinoderm sperm
flagella
167
010
009
^ ^
0 / 0
008
•
O
o oo
007
o./
1/ °
'OO
0
A\
•
Cy
g 005
/
/
o
006
/
/ /
•
y/
/
004
•f
/
003
002
001
/
..
/
• .y•«"•*•
,
01
/
/ ^
/
=
•
/
/
/x
0-2
0-3
0-4
05
1
I
1
1
1
0-6
0-7
0-8
0-9
10
Fig. 11. Radius of swim path as a function of relative difference between average bend angles
of principal and reverse bends. Symbols same a« in Fig. IO.
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 crossconnexions 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. z(d).
As a fully 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 any point within a bend associated with its propagation is:
d$_dd
dt~&
dsv
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)
between exposures 5 and 9 is 1-64 fim/exp, as shown in Fig. 9; that for bend 2 between
:posures 14 and 21 is i-34/im/exp. The corresponding values of r for bends 1 and 2
e 7-0 /*m and 4-5 fim, respectively. The rates of change of angle within bends 1 and
e
168
STUART F. GOLDSTEIN
0-3
20
0-2
19
01
0
•o
2
|
-o-i
i
I
-0-3
1-5
|
D.
-0-4
s
S
X
I'3
-0-5
o
o.
OJJ
-0-6 <
11
-0-7
10
-0-8
-0-9
-10
10
12
14 16
Exposure
18
20
22
22
24
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 0-235 rad/exp and
0-298 rad/exp, respectively. 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%,
respectively, of those associated with their propagation. As Fig. 12 shows, this extrinsic sliding is not constant, and can change direction. Earlier analysis of flagella
with symmetrical 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 analysis in the present study indicates that a bend can travel in spite of appreciable extrinsic sliding imposed upon it; any effects of this extrinsic sliding appear to
be subtle. This apparent insensitivity to microtubular sliding is complemented by a
sensitivity to artificially imposed bending (Lindemann & Rikmenspoel, 1972^
Shingyoji, Murakami & Takahashi, 1977). These observations suggest that bending'
Asymmetric waveforms in echinoderm sperm
flagella
169
is controlled primarily by bend curvature (Machin, 1963; Brokaw, 1971, 1972) rather
than by shear velocity (Brokaw, 1975).
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.
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BROKAW, C. J., JofS'-iN, R. & BOBROW, L. (1974). Calcium ion regulation of flagellar beat symmetry
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170
STUART F. GOLDSTEIN
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