Algebraic expressions for the waveforms of sea urchin sperm flagella

J. theor. Biol. (1985) 116, 127-147

Algebraic Expressions for the Waveforms of Sea Urchin Sperm Flagella ROBERT RIKMENSPOEL

Department of Biological Sciences, State University of New York, Albany, New York 12222, U.S.A. (Received 25 January 1985, and in revised form 29 April 1985) The waveforms of live sea urchin sperm flagella were digitized with videodigitizing apparatus. The flagellar waveforms were expressed by the coordinates of 20 points spaced 2 txm apart along the flagella. The waveforms were condensed into simple algebraic expression with four parameters. Each of these parameters showed a systematic variation with the flagellar frequency. These average trends of the parameters made it possible to define an average, idealized, waveshape as a function of the flagella frequency, in the range 8-80 Hz. Introduction

The molecular basis o f the system that produces the forces necessary to power the motility of flagella is now well understood. Summers & Gibbons (1971) have conclusively shown that a sliding filament mechanism operates in the axoneme with the dynein molecules attaching as cross bridges between the longitudinal tubulin fibers. This mechanism is analogous to that of the myosin-actin system in striated muscle (Huxley, 1969). The clarification of the force producing mechanisms in flagella has stimulated the development of theoretical models of flagellar motility. In some of the theoretical models a purely analytical approach, in small amplitude approximations, was used (Rikmenspoel, 1971, 1978a; Lubliner & Blum, 1971 ; Brokaw & Rintala, 1975). In general, these models are sufficient only to obtain gross properties, such as the wavelength, and the wave amplitude, of the flagellar motion. The simple structure o f the axoneme, with nine longitudinal fibers between which the dynein cross bridges are formed, is attractive for the formulation of detailed models in which the attachment and detachment o f every cross bridge is followed as a function of time. Such detailed models, when making use o f exact, large amplitude, algebra can lead to the calculation o f the shape o f waves in flagella. Several groups o f investigators have developed detailed models to obtain flagellar waveforms that simulated those observed 127 0022-5193/85/170127 + 21 $03.00/0

(~) 1985 Academic Press Inc. (London) Ltd

128

R. RIKMENSPOEL

in live flagella (Brokaw, 1972; Rikmenspoel, 1976, 1982; Rikmenspoel & Rudd, 1973; Hines & Blum, 1978, 1979). To compare waveforms computed for model flagella with those of live sperm, it is necessary to have quantitative data on observed waveforms. In this laboratory apparatus has recently been developed to scan and digitize cinemicrographic images of sperm flagella (Rikmenspoel & Isles, 1985). In this paper the analysis of the waveform data obtained with this equipment is described. It has been possible to summarize the waveforms into simple algebraic expressions, and to derive the average, or idealized, waveforms of sea urchin sperm in the range of flagellar frequencies of 8-80 Hz.

Experimental Data The data analysed in this paper were all taken on sea urchin sperm of the species Arbacia. The sperm had been filmed at a framing rate of 400 or 200 frames/see, at a normal viscosity of 1.4 cpoise, and at temperatures varying from 22°C to 6°C. Of the sperm selected for detailed analysis a sequence of up to 60 consecutive flagellar images, at a time interval of 2-5, 5-0 or 10.0 msec depending on the flagellar frequency, was rephotographed on 35 mm film. For each sperm the sequence spanned from 4 to 6 flagellar periods. The rephotographed images were scanned by a video-digitizing apparatus described previously (Rikmenspoel & Isles, 1985). In the output of the digitizing equipment, a flagellar waveform was represented by the coordinates, relative to a fixed coordinate system, of 20 points along the flagellum, spaced 2 g.m (measured along the flagellum) apart. The first point was at 2 txm from the proximal junction, the final point was therefore at 40 ~m from the proximal junction. The accuracy of the measurement of the coordinates of the flagellar points was approximately +0.1 Izm (S.D.). For each of the 20 points along the flagella a value for the local curvature of the flagellum was derived by the digitizing equipment. This determination of the curvature did not involve the fitting of a circle through adjacent points, and it should be considered an independent measurement. An accuracy of approximately +50 cm -I in the curvature, representing 2 to 3% of the maximal flagellar curvatures, was obtained. In total 21 sperm were subjected to the algebraic analysis below. The flagellar frequency of the sperm varied from 8.8 to 81.5 Hz. Since the gross waveproperties of wavelength, wave amplitude and frequency of the sperm varied smoothly over the temperature range of 22°C to 6°C (Rikmenspoel & Isles, 1985), the sperm were treated as one continuous sample over the frequency range. A detailed description of the experimental procedures and

W A V E F O R M S O F SEA U R C H I N

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the equipment employed in the digitizing process has been given recently (Rikmenspoel & Isles, 1985). For the algebraic analysis below s will always represent a longitudinal coordinate, measured along the flagellum. The proximal, head-flagellar, junction is at s = 0. The Cartesian coordinates x and y refer to the coordinates of points on the flagellum relative to an external coordinate system. The symbol U will be used to denote the amplitude of a flagellar waveform when reduced to an algebraic expression. Mathematical Procedures OUTLINE F r e e l y m o v i n g s e a u r c h i n s p e r m d o n o t p r o g r e s s in a s t r a i g h t l i n e , b u t

follow a closed, circular path (Gray, 1955; Brokaw, Goldstein & Miller, 1970; Rikmenspoel, 1978b). The overall shape of the sperm appears to be slightly bent, as can be perceived in Fig. 1 below. Evidence has been presented that the flagellar waveforms are basically symmetrical shapes, which have become asymmetrical by bending (Rikmenspoel & Isles, 1985). The first step in a mathematical analysis of the waveforms must thus be to straighten the overall bend in the sperm, and to remove the forward progression of the sperm. This should result in symmetric, stationary flagellar waveforms. Once straight, symmetric, waveforms have been obtained, algebraic expressions are derived, which represent the waveshapes with good accuracy. This means that the waveshapes are essentially reduced to a number of parameters (four in the present case), and an algebraic prescription which together make it possible to compute waveshapes.

40[ FloQellum

),..,.

[ "" 20 ~L

~-~" . . . . . ~. . . . . . . . . 20

, .........

30 x (M.m)

Center I_ine. . . . . . . 40

50

FIG. I. Flage|lar waveform o f a sea urchin sperm as defined by the digitized data. The dots

were plotted from the coordinates put out by the digitizing equipment. The solid line was drawn by eye through the points. The position of the head of the sperm was sketched in by hand (dotted). The dashed curve represents the "centerline" of the sperm as explained in the text.

130

R. R I K M E N S P O E L

Average, or idealized, waveforms are subsequently derived, by searching for trends in the parameters as a function of the flagellar frequency. Finally it will be considered how well the average waveforms do represent the local curvatures in live flagella. As was noted in the section Experimental Data, independent measurements of the local curvatures in the sperm flagella have been obtained with the digitizing apparatus. Each of the above points will be considered in detail below. REMOVAL OF THE OVERALL CURVATURE A N D F O R W A R D P R O G R E S S I O N

For each of the sperm to be analyzed one period of motion was selected during which the flagellum was positioned approximately parallel to the x-axis, as illustrated in Fig. 1. This reduced the projection effects in the course of the analysis. The various sperm were represented by 7-15 flagellar positions, evenly spaced over the period. It has been shown for the sperm in the present sample, that the transverse motion of a fixed point on a flagellum was purely sinusoidal in time, to within an accuracy of approximately 2% (Rikmenspoel & Isles, 1985). 35 ~'"*"X

/

y = at + b + A sin (cu t + cz',

• ot*b

30

25 ,

,

I

. . . .

15

I

,

,

i

75

. . . .

20 P h o t o . . . .

i

100 Time,

I

. . . .

30

25 n u m b e r . . . .

i

. . . .

125

i

150

t(msec)

FIG. 2. Transverse motion (in the y-direction) of a point on a sperm flagellum at s = 10 ixm from the head, during one period of motion. The dots represent the output of the digitizing equipment. The lines were computed according to the equations shown, with the values of a, b, A and ~ determined by least squares fitting.

Figure 2 illustrates the time course of the ("transverse") y-coordinate of a sperm, at s = 10 I~m from the head, over a period of motion. It can be perceived in Fig. 2 that, due to the forward motion of the sperm which is not exactly parallel to the x-axis, a slant is present. The time course of y has therefore been approximated by y = at+b+Asin

(tot+a)

where t is the time and to = 2~-xflagellar frequency.

(1)

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For each location s on the flagella, the values of a, b, A and ot from equation (1) were determined by least squares fitting, as described previously (Rikmenspoel, 1978b). This yielded for every sperm analysed, a n array o f 20 values, as a function o f s, for a = a(s), b = b(s), A = A ( s ) and a = a(s). For the 21 sperm of the present sample, with 20 flagellar locations on each sperm, a total of 420 time series, as illustrated in Fig. 2, were analysed. The standard deviation, 8, o f the measured values o f y from the fitted lines of equation (1), over all data points from all sperm, a m o u n t e d to 8 = 0.23 ~m. The sperm at a temperature of 6°C were noticeably less regular in their motion than those at the higher temperatures. The 15 sperm at a temperature above 6°C showed an overall 8 = 0.20 ~m, the sperm at 6°C had 8 = 0.30 ~m. O f the 300 time series for the sperm above 6°C, 136 could be fitted to a standard deviation 8 <- 0-1 I.Lm,o f the 120 time series for sperm at 6°C only 16 had 8-<0-1 p.m. For a given sperm, at location s, the value of U

U(s) = A ( s ) sin [tot + a ( s ) ]

(2)

represents the " s m o o t h e d " , algebraic, a p p r o x i m a t i o n of the symmetric transverse motion relative to a centerpoint. The y-coordinate of this centerpoint, yo is not stationary in time, but it shifts due to the overall progression of the sperm yc(s) = a(s)t + b(s). (3) The p r o c e d u r e outlined above can also be applied to the time series for the x-coordinate at the various locations on the flagella by writing

x(s) = p ( s ) t + q(s) + R ( s ) sin [tot + O(s)]

(4)

Since the forward m o v e m e n t of a sea urchin sperm is not particularly regular, the fit for the time series for x for the sperm is less good than that for y. The overall standard deviation o f the x value to equation (4) was 6 = 0.35 ~.m. For one flagellar position at a time t = T the set o f coordinates

yc(s) = a(s) T + b(s)

(5a)

xc(s) = p(s) T + q(s)

(Sb)

defines a center line for that flageilar position. In Fig. 1 the thus derived center line is shown as the dashed curve. The curvature o f a segment o f the centerline, shown in Fig. 1, can be determined by fitting a circle through the coordinate points for that segment, as given in equations (5a) and (5b). Mainly due to the a p p a r e n t irregularity o f the original data for x, from which the values for xe are derived, the calculated curvatures o f segments

132

R. R I K M E N S P O E L

of the centerline showed appreciable scatter. Meaningful values for the curvature could only be obtained as the average over a section of at least 8 ~m long. For the various flagellar positions of a sperm, the average curvature o f an 8 ~m long section of the centerline varied by a p p r o x i m a t e l y 12% (s.d.), but it did not a p p e a r to show a trend as a function of time. The centerline drawn in Fig. 1 illustrates that its curvature is almost uniform over the entire length. When the centerline curvature in the section s = 2-10 ~ m o f a sperm was taken as p(2-10) = 1, that in the section s = 12-20 t~m was found as p(12-20) = 0 . 8 8 + 0 . 1 8 and that at s = 2 4 - 3 2 l~m as p(24-30) = 0 . 7 8 ± 0 . 1 7 (average and standard deviation over the 21 sperm in the sample). The overall curvature of a sperm can apparently meaningfully be taken as the average o f the above values for p(2-10), p(12-20) and p(24-30). It was noted above that sea urchin sperm progress in a curved, circular path. In Fig. 3 are plotted the curvature of the circular path o f the sperm "6 - ~ - 400 F

3oo u ~ °" u

": .:

200 100

0

~ 100 200 3 0 0 4 0 0 500 600 Curvoture of pofh (cm-l)

FIG. 3. Relation between the curvature of the path along which the sea urchin sperm progressed, and the average curvature of the centerline o f the sperm. The line was drawn by eye through the points.

in the present sample, determined as described previously (Rikmenspoel & Isles, 1985), versus the average curvature of the centerline of the sperm. It can be seen in Fig. 3 that the overall curvature of the sperm is only weakly correlated with the curvature of the path, and that the regression does not pass through the origin of the graph. This suggests that the overall curvature of the sperm is not the m a j o r cause of the fact that the sperm swim in a circular path. This point will be elaborated further in the Discussion. Above it has been shown that it is possible to derive a (curved) centerline for the sperm around which a point at a fixed position on the flagellum, s, executes a symmetric, sinusoidal, motion in time. A reduced transverse motion at s, relative to the center line, Yr~d can be taken as

Yreo(s) = y ( s ) - yc(s)

(6)

where y ( s ) represents the original data points as shown in Fig. 1, and yc(s)

WAVEFORMS OF SEA U R C H I N SPERM FLAGELLA

133

the y c o o r d i n a t e o f the c e n t e r l i n e , as d e f i n e d in e q u a t i o n (5a). F o r each flagellar p o s i t i o n , a string o f 20 r e d u c e d c o o r d i n a t e s , Yr~d, was thus o b t a i n e d . F l a g e l l a r w a v e f o r m s were o b t a i n e d b y m a k i n g use o f the fact t h a t the d i s t a n c e 8s, m e a s u r e d a l o n g the flagellum, b e t w e e n successive p o i n t s o n the flagellum was c o n s t a n t at 2 ixm. T h e v a l u e for the r e d u c e d x c o o r d i n a t e , x~ed, was t h u s d e r i v e d from 8X~d=(SS

2

o~y. 2, . d j ~1/2 .

-

(7)

T h e initial flagellar p o i n t at s = 2 l~m was o b t a i n e d , with e q u a t i o n (4), as

X,od(S =

2) =

R(s

= 2) sin [~0F+ @(s = 2)].

(8)

This r e m o v e d the f o r w a r d p r o g r e s s i o n o f the sperm.

(a)

4°I,o

F

20,-.,................................ - i........... ?;:5~.................. 20

50

30

40

"~ 20[-

40

50

0

40c

20

~0

o

~o

zo

30

0

10

20

30

_~

"--'~'~. so

10

60 ~

....

=L

~-*-~-x,,

20 2%....... ~0' ...... 4'0 ....... -~ ....

4O (d)

[

. . . . .

~ . . . . . . . .

t . . . . . . . .

i . . . . . . . .

1

--

~t

~...h,..~.

o

x(,u.m)

t.

Io

t h.,

~ i.,+,

20

h,

H

h,ll

i

5o

x(H.m)

FIG. 4. Flagellar waveforms of four sea urchin sperm at flagellar frequencies of 56-3 Hz (a), 46.8 Hz (b), 26.4 Hz (c) and 13-3 Hz (d). On the left side are shown the waveforms, plotted from the rough data put out by the digitizing equipment. On the right side of the figure the corresponding waveforms are shown, after removal of the average curvature of the sperm and of the forward progression. All lines were drawn by eye through the points. In order not to disrupt the clarity of the figure, only three fiagellar positions are shown for each sperm. The head flagellar junction of all sperm is to the left.

134

R. RIKMENSPOEL

The reduced, straightened, waveforms obtained with equations (6), (7) and (8) were reasonably smooth. Figure 4 shows four examples of waveforms plotted from the rough data, and the corresponding reduced waveshapes. It can be seen that even when the sperm is positioned at a sizeable angle with the x-axis, as in Fig. 4(c), the procedure works well. It should be noted that to obtain the waveforms as illustrated in Fig. 4, only the values o f the y-coordinates in the original data were used. The reduced waveforms are inherently symmetrical and straight. Furthermore, the transformation of the bend waveshapes, on the left in Fig. 4, into the straightened waveshapes, was carried out such that the flagellar length was conserved. As is explained in the Discussion, for a comparison of waveforms derived from theoretical models with those from live sperm, it is important that the flagellar length is a constant. A L G E B R A I C A P P R O X I M A T I O N S OF T H E R E D U C E D W A V E F O R M S

In the previous section it was shown that the transverse motion o f a point, s, on a flagellum, around the centerline can be approximated as U = A(s) sin [tot + a(s)]. The approximation was good to approximately 0.2 ~m (s.d.). The amplitude function, A(s), is shown for a typical sperm in Fig. 5. The figure illustrates that the amplitude reaches a maximum between s = 20 p.m and s = 30 p.m, and decreases toward both the proximal and the distal ends of the flagellum. This is in agreement with earlier observations (Rikmenspoel, 1978b; Hiramoto & Baba, 1978). The function A(s) could for every sperm be fitted with a single polynomial expression:

A(s) = Al(1 + A2s2+ A3s4).

(9)

The line drawn in Fig. 5 represents the one given by equation (9) with the values A~ = 3.22 p.m, A2 = 1.98 x 10 -3 ~ m -2, A3 = - 1 . 5 2 x 10 -6 ~zm-4, deter-

\ I

0

FIG. 5. A m p l i t u d e function,

10 20 30 Distance from head, s(Fcm)

A(s),

40

o f the flagellar wave in a typical sea urchin sperm, as a

function o f the location on the flagellum. The line was c o m p u t e d according to equation (9)

in the text.

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mined by least squares fitting. For several sperm of the sample the values for A(s) at the extreme distal end of the flagella were not conforming well to the expression in equation (9). For these sperm, the distal portion of the flagellum was omitted from the least square fitting procedure for A~, A2 and A3. Figure 6 shows the values for A~, A2 and m 3 for all 21 sperm in the sample. The graphs were constructed so that individual sperm can be identified, and the values of A ~ , . . . , A 3 read with reasonable precision by the reader. The parameters A2 and A3 show only a weak trend as a function of flagellar frequency. This reflects the fact that the form of the function A(s) did not vary much with the flagellar frequency.

~

_e L o_:_.,~__..o _.z_.__---o-

-~ ° 0

4

o o

~o

"

°

l

10 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0 "

• x

•

o •o

I o

I

10 2 0 30 4 0 5 0 6 0 7 0 8 0 9 0 5

oo

0

1~ 2'0 3'o 4'o sb 6'o zb 8'0 90 Flclgellor frequency (Hz)

FIG. 6. Value of the parameters A~, A2 and A3, from equation (9) in the text, for the various sperm in the present sample, plotted as a function of the flagellar frequency. The different symbols refer to sperm at different temperatures: O, 6°C; x, 12°C; O, 16°C; Z], 22°C. The solid lines were drawn, as explained in the text, to represent the average trend of the parameters as a function of the flagellar frequency. The precision of the fit of the data points for A(s) to the expression in equation (9) was to a standard deviation, 6, averaged over all sperm, of 6 =0-18 txm. The sperm with the best fit showed 6 =0.11 ~.m, the one with the worst fit: 6 = 0.34 p.m. The phase function a ( s ) defines the travelling of the flagellar wave toward the distal end (Rikmenspoel, 1965, 1978b). Figure 7 shows the values for a ( s ) obtained for the same sperm as represented in Fig. 5. The shape of ~(s) is a simple monotonic function, with the strongest curvature near the

136

R.

RIKMENSPOEL

//7

~'-8

2_o

-11 -~0 -9 -8

,-~

-6 ~"o

-~-o

-5 - ~ -4s~

t ~/J" -2 -I

o "~6zO

3o 40°

Distance from head, s (,urn)

FIG. 7. Phase function, ~(s), for the flagetlar wave of a sea urchin sperm, as a function of the location on the flagellum. The line was computed according to equation (10) in the text. The vertical scale on the right represents the values of a(s) obtained from the data. The vertical scale on the left was shifted such that a(0)= 0. proximal end o f the flagellum. In the extreme distal part o f the flagellum a (s) b e n d s sharply u p w a r d . This shape was f o u n d in all sperm o f the sample. For all sperm a(s) c o u l d be fitted to a function o f the f o r m a ( s ) = a o + m s °75.

(10)

ao and a , were d e t e r m i n e d by least squares fitting, while omitting the distal points where a ( s ) b e n d s u p w a r d . A v e r a g e d over all sperm in the s a m p l e the s t a n d a r d deviation o f the data points for a ( s ) f r o m the fitted curves was ~ = 0.056 rad ( a p p r o x i m a t e l y 3°). The s p e r m with the best fit s h o w e d = 0 . 0 3 1 rad, the one with the worst fit: ~ = 0 . 1 1 3 rad. In e q u a t i o n (10), ao represents the, accidental, p h a s e shift between the c i n e m i c r o g r a p h y c a m e r a and the flagellar period selected for analysis. The only p a r a m e t e r with significance in e q u a t i o n (10) is a,. For m o d e l calculations the function a(s) can be normalized to o r ( 0 ) = 0 , as s h o w n in Fig. 7. Figure 8 shows the values for a~ for the sperm a n a l y s e d in the present paper. The spread in at is small, and only a very w e a k trend as a f u n c t i o n o f f r e q u e n c y is a p p a r e n t in oq. This is in a g r e e m e n t with earlier observations that the wavelength o f the flagellar waves varies but little in different sea

~-" -07,

o -o.6~

o

'E-o.5~ ^~__~__.~.".--.-~----#

"~ -0.1 F

0

I0

20 50 40 50 60 70 80 90 Flogellar frequency (Hz)

FIG. 8. Value of a, from equation (10) in the text for the sperm analysed in this paper, as a function of the flagellar frequency. The symbols refer to sperm at different temperatures: O, 6°C; x, 12~C; 0, 16'~C;I-q, 22°C. The line was drawn by eye through the points.

WAVEFORMS

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URCHIN

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FLAGELLA

137

urchin sperm and over a wide range o f frequencies (Brokaw & Josslin, 1973; Rikmenspoel & Isles, 1985). With the values of A~, A2, A 3 and a0 and a~, algebraic approximations for the waveforms for the sperm can be computed with equations (2), (9) and (10), making use of (in analogy with the manner the reduced waveforms of Fig. 4 were constructed)

8x = (Ss 2 - 8U2) '/2.

(11)

Figure 9 shows the algebraically approximated waveforms for the sperm shown in Fig. 4 above. The standard deviation in the vertical, U, direction o f the data points from the computed waveforms, averaged over all sperm in the sample was 8 = 0.30 ~m. If the standard deviation was taken from the normal distance of the data points to the computed curves, a 8 ~ 0.2 ~m is found. It can be seen in the examples shown in Fig. 9 that in the distal parts of the flagella the fit is noticeably less good than in the more proximal parts.

(o)

0

(b)

10

20

50

.~ (c)

O

tO

20

50

10

20

50

(d)

0

10

2.0

30

0

x(~m) FIG. 9. Algebraic representation of the reduced waveforms of Fig. 4 for four sea urchin sperm. The data points are as for Fig. 4. The lines were c o m p u t e d with (2), (9), (10) a n d (11) from the text. The flagellar frequencies of the sperm are indicated. THE AVERAGE FLAGELLAR WAVEFORMS

The values A1, A2, A3 and al shown in Figs 6 and 8 above can be used to obtain the "average" waveforms of the sea urchin sperm as a function of frequency. The lines drawn in the graphs in Fig. 6 represent the average trends adopted for this purpose for A~, A2 and A3. In the graphs for A2 and A3 in Fig. 6 it can be seen that several points indicate " h i g h " values for the parameters. These points represented sperm in which either the fit to the curve for A(s) was rather poor, or sperm for which a distal part of

138

R. R I K M E N S P O E L

the flagellum was omitted from the fitting procedure. The points were therefore given a low weight. The lines shown in Fig. 6 were, with some trial and error, adopted such that the maximum amplitude o f the computed average waveshapes, as a function of frequency, conformed well to the maximum amplitude in the live sperm of the present sample, as shown in Fig. 10. The average trend for the phase parameter a~ is shown by the line in Fig. 8. Since the scattering in ch is small, the line was drawn by eye.

o °

. t~

o

0

o

e o~"'~'-"--.-~

~b 2'o ~'o 4b 5'o 6'o 7b 8'o 9o Ftogellar

frequency

(Hz}

FIG. 10. Maximal amplitude of the average, idealized waveforms of sea urchin sperm, as a function of the ftagellar frequencies. The points indicate the measured values for the live sperm analysed in this paper. C), 6°C: x, 12°C; 0 , 16°C; Eli, 22°C.

Table 1 shows the numerical values for the average A,, A2, A3 and ct, for the flagellar frequency range 10-80 Hz. Figure 11 shows the average, computed, waveforms, at 10, 20, 40 and 60 Hz, for the flageltar segment s = 2-40 ~xm. The waveforms were plotted without forward progression. It should be kept in mind that the algebraic approximations of equations (9) and (10) for A(s) and a(s) are valid only for s->21xm. In the most TABLE 1

Values for the parameters A~, A2 and A 3 of the amplitude function, equation (9), and for a~, of the phase function, equation (10), for the average, idealized waveforms of sea urchin sperm flagella Flagellar frequency (Hz)

Al (~m)

A2 (10 -2 ~m -2)

A3 (10 -s t~m -4)

ott (rad p.m -0"75)

10 20 30 40 50 60 70 80

3"5 3"7 3-7 3"4 3-2 2-9 2'7 2-5

0"15 0" 16 0-18 0" 19 0"20 0"21 0"23 0-24

-0'10 -0" 11 -0-12 - 0 ' 13 -0" 15 -0" 16 -0'18 -0"19

-0"430 -0"443 -0-456 -0'469 -0'482 -0-495 -0"508 -0-521

WAVEFORMS

OF SEA URCHIN

(o)

FLAGELLA

139

(b)

"g o ~-5 :;

SPERM

0

10

20

30

(c)

0

10

20

30

0

10

20

30

(d)

....

0

, . . . . . . .

10

H,

. . . . . .

20

*L,

,,

30

x(~m) FIG. l |. Idealized, average waveforms for sea urchin sperm flagella at frequencies of (a) 10 Hz, (b) 20 Hz, (c) 40 Hz (d) 60 Hz. The head of the sperm is supposed to be on the left. The waveforms shown cover the section of the flagella corresponding to s = 2-40 o.m.

proximal segment s = 0-2 Ixm no data were available on the flagellar coordinates (Rikmenspoel & Isles, 1985). Detailed micrographs o f invertebrate sperm (Goldstein, 1975; Hiramoto & Baba, 1978; Rikmenspoel & Isles, 1985) show that in the 2 lxm section closest to the head, very sharp bending of the flagella occurs, which is not represented in the data and the calculations in the present paper. The consequences of this for model calculations will be elaborated on in the Discussion. FLAGELLAR CURVATURES

The digitized data on flagellar motion presented by Rikmenspoel & Isles (1985) included measurements of the curvature of the flagella as a function of time and position. These measurements gave the true, local, curvature of the flagella at the 20 points o f which the x and y coordinates were obtained. The question arises of how well the average waveforms derived in the previous section can represent the curvatures of the live flagella. The computed waveforms, as those in Fig. 11, made use o f equation (11) to arrive at values for the x-coordinate. Since in general the conversion of the length o f a line segment to a projected length involves elliptic integrals, a closed form algebraic expression for the x-coordinate of points on the computed flagella cannot be obtained. A simple direct calculation o f the local curvatures of the computed waveforms is therefore not possible. Instead, curvatures have been calculated by fitting a circle through 4 ~m long segments of the waveforms. The values for the curvature thus obtained showed a strong dependency on the location o f the flagellum. At a given

140

R. R I K M E N S P O E L

location, only a weak trend of the curvatures with the flagellar frequency was apparent. This is in agreement with the measurements of the curvatures of live sperm flagella (Rikmenspoel, 1978b; Rikmenspoel & Isles, 1985). Figure 12 shows the m a x i m u m curvature (taken as one-half of the peak to peak value during one flagellar cycle) of the idealized c o m p u t e d waveforms, as a function of the distance along the flagellum, s. The line drawn in Fig. 12 represents the average of the m a x i m u m curvatures of computed flagella in the range 10-80 Hz. The standard deviation of approximately 10% over the different computed flagella is shown in the figure. 4000[

~o 2000" ~{~ [ "%-.-,-~ "'~l'v'~t'-' "~, looo I i .... 0

~ .... L .... t .... 10 20 30 40 Distance f r o m heod, s(/.zrn)

FIG. 12. Maximal curvature, taken as one-half of the peak to peak value in the time series, as a function of the location of the flagellum. The solid line represents the average for the idealized flagellar waveforms over the frequency range 10-80 Hz. The vertical bars represent typical standard deviations over the idealized waveforms at different frequencies. The symbols represent measurements on live sperm, taken from Rikmenspoel & Isles (1985). Measured values for the m a x i m u m (local) curvatures of the live flagella are inserted in Fig. 12. It can be seen that, except near the distal tip, the computed values are close to the experimental ones. This shows that the maximal curvatures occurring during the flagellar cycle can be meaningfully obtained by fitting a circle through a short (e.g. 41~m) segment o f the flagellar waveform. In a recent model calculation (Rikmenspoel, 1982) it was shown that the time course within a cycle of motion of the curvature of flagella is an accurate reflection of the time course of the active, contractile m o m e n t in the flagella. The measurements on live flagella (Rikmenspoel & Isles, 1985) have shown that the time course of the curvature has to be considered to be purely sinusoidal in time for live flagella. Figure 13(a) shows the time course of the curvature for the idealized c o m p u t e d waveforms for a flagellum at 40 Hz. As noted above the curvatures were calculated by fitting a circle through a 4 txm long segment of the flagellum. It can be seen that, compared to the purely sinusoidal time course of the amplitude in Fig. 13(c), the time course of the calculated curvature is flattened at the peaks of the curve.

WAVEFORMS

OF

SEA

URCHIN

SPERM

vo

FLAGELLA

Curvalure

141

(b)

-

c,l ~I .... ~. .... I'

o

om

0 5

Eo u

0

30

5070

nenl

100 200 300

}-5 <

T i m e --~

0

'

'

'

'

50 50 7 0 1 0 0 2 0 0 5 0 0 Frequency~ v ( H z )

FIG. 13. Time series (a) and Fourier spectrum (b) for the curvature, at s = 20 o.m from the head, for the idealized waveform at a flagellar frequency of 40 Hz. Purely sinusoidal time series (c) and featureless Fourier spectrum (d) for the amplitude, at s = 20 Izm,for the idealized sperm at a flagellar frequency of 40 Hz. The waveforms and the curvatures for the computed waveforms were calculated for a time interval covering five periods of the motion. The Fourier transform F ( v ) of the time series for the curvatures and amplitudes

F(v) = c I

g(t) e -"°' dt

(12)

were then calculated. In equation (12), v is the frequency, to =2~-v, g(t) represents the time series for the curvature or the amplitude, and c is a normalization constant such that F(vo) = 1 with vo = flagellar frequency. In Fig. 13(b) the Fourier spectrum for the curvature time course in Fig. 13(a) is shown. The flattening at the peaks in the time course manifests itself as a third harmonic c o m p o n e n t of approximately 10% of the height of the fundamental peak. The Fourier spectrum with the absence o f a third harmonic is given for comparison in Fig. 13(d). For all locations on the idealized flagella at all flagellar frequencies between 10 and 80 Hz, the time course o f the curvature and its Fourier spectrum were very close to those shown in Figs 13(a) and (b). As mentioned above, the true local curvature in live flagella should be considered to have a purely sinusoidal time course and a featureless Fourier spectrum. The results shown in Figs 13(a) and (b) therefore indicate that for a detailed consideration of the flagellar curvature and its time course, a determination of the curvature by fitting a circle through a flagellar segment is probably not satisfactory.

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RIKMENSPOEL

Discussion

The results in the previous section show that it is possible to find a very simple algebraic expression for the waveshapes in sea urchin sperm fagella. The complete waveform can be described with only four numbers A~, A2, A3 and a~. The average waveshape changes rather little over a wide range of frequencies, as illustrated in Fig. 11. This suggests that the waveshape is basically determined by the structure of the flagellum. The wide range of frequencies ( ~ 1 0 to 80 Hz) in the sperm studied was obtained by varying the temperature of the sperm. With the aid of the results of model calculations done previously (Rikmenspoel, 1978a, 1982) it was concluded that for spermatozoa at a temperature above 10°C, the internal, active, moments did not change much with temperature (Rikmenspoel & Isles, 1985). In the present study this applies to the sperm with a flagellar frequency o f 20 Hz and higher, as can be observed from Figs 6, 8, and 10. For spermatozoa at a temperature of at least 10°C, the waveforms defined by the parameter values in Table 1, and illustrated in Fig. 11 should, in the range 20-80 Hz, probably be considered to depend only on the frequency. The sperm of the present sample at a temperature o f 6°C all showed flagellar frequencies around 10 Hz. Rikmenspoel & Isles (1985) concluded that the active moments in these sperm was reduced compared to those at temperature above 10°C, leading to a lower wave amplitude. The line through the data points for the parameter A~ in Fig. 6 appears to show a break for the sperm at 6°C. The waveform for the sperm at 10 Hz, defined by the values in Table 1, should therefore be considered valid only for sperm at 6°C. Occasionally, sperm have been observed at higher temperatures (above 10°C) with a low frequency of 8-10 Hz (Rikmenspoel, 1978b). A representative waveform for these sperm can probably be obtained by a straight extrapolation of the line shown for A~ in Fig. 6, towards frequencies <20 Hz. This would lead to an increased amplitude for these slow sperm, in agreement with previous observations (Rikmenspoel, 1978b). For none of the sperm analysed in the present paper, were data available for the most proximal flagellar section, corresponding to s = 0-2 ~m. Due to the presence of the mitochondrion at the head flagellar junction (s = 0), and the frequent occurrence of some glare from the sperm head on the photographic images, precise data on the flagellar position and curvature are difficult to obtain in the extreme proximal section. Extrapolation of the data for the flagellar curvature in Fig. 12, suggests that near the head, the maximal curvature is very large, of the order of 104 cm -~. Visual inspection of photographs of sperm flagella appears to confirm this.

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143

Curvature calculations were performed for the average waveforms, as in Fig. 11, in the section near the head. For this purpose, the curves for A(s) and a(s) were varied at will from the form prescribed by equations (9) and (10), in the flagellar segment s = 0-4 ~m. From these calculations it was found that variations in A(s) had a small influence on the flagellar curvature. A large curvature was directly related to the steepness of the slope of c~(s) in the proximal segment, however. The slope of a ( s ) , its derivative, is to be interpreted as the wave number at that location. A steep slope of a(s) near s = 0, thus corresponds to a large wave number, or a small wavelength in the flagellum near the head. Visual inspection of photographs of sperm flagella indeed shows that the curvatures develop near the head as a sharp bending over a short section (Gibbons & Gibbons, 1980; Goldstein, 1979; Rikmenspoel, 1978b). Detailed models for sperm flagella should reproduce these qualitative features of a very high curvature and a steeply sloping a(s) in the region near the head-flagellar junction. With the methods used to reduce the flagellar waveforms, the length of the flagella, l, was conserved throughout. General theoretical treatments of the deformation of long thin structures (Landau & Lifshitz, 1970), and in particular applications to the case of sperm flagella (Machin, 1958; Rikmenspoel, 1978a), have all shown the occurrence of terms proportional to 14 in the equations of motion. This indicates a sharp dependency o f wave properties on the flagellar length. Flagellar models, intended to yield quantitative values for flagellar wave properties should therefore be formulated such that the flagellar length is a constant, and does not depend on e.g. the wave amplitude. The use of small amplitude approximation should probably be avoided when detailed and quantitative wave properties are to be studied. For the practical usage of the results in this paper, averaged, idealized, waveforms can be calculated for any frequency with the values o f A~, A2, A3 and a t taken from Table 1, or read from Figs 6 and 8. If the theoretical model, the results of which are to be compared with the average waveforms, uses s, the distance from the head measured along the flagellum, as the independent variable, a convenient and direct comparison with the results in this paper is available at the selected s-values. For a model using Cartesian, x and y, coordinates this is not possible since a closed form expression for the x coordinate of points on the flagella is not obtained in the algebraic procedures used in the present paper. In this event, it will probably be necessary to compute the average, idealized, waveforms for a large number of s-values (e.g. 1000) on the flagellum. The wave amplitude U, at the desired values o f the Cartesian running coordinate x, can then be obtained by interpolation.

144

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In general, theoretical models for sperm flagella, such as those of Brokaw (t972), Hines & Blum (t978, 1979) or Rikmenspoel (1982), will probably be directed toward reproducing the average, idealized, waveforms illustratted in Fig. 11. In live sperm flagella variations of the waveshapes occur, however which are reflected in the scattering of the values for A,, A2, A3 and a~ around the average trends shown in Figs 6 and 8. A completely satisfactory theoretical model for sperm flagella should be able to reproduce the variations of the observed waveforms by a variation of the essential model quantities within reasonable bounds. In other words, a theoretical model should not be critically locked in the values for its essential model quantities, such as e.g. the flagellar stiffness, the (local) density of dynein molecules or the kinetic constants of the dynein cross bridge attachment and detachment. Live sea urchin sperm almost always swim in a circular path. This indicates that an asymmetry is present in these sperm, which causes the deviation from the progression in a straight line. Rikmenspoel & Isles (1985) have shown that in the proximal part ( s = 0 - 1 0 ~m) of the sea urchin sperm flagella an asymmetry in the local flagellar curvature occurs. The asymmetry manifested itself in that toward one side, the flagellar curvature was bound to an upper limit. The time series of the curvature at one location as a function o f time then shows flattened, truncated peaks, but only on one side (compare Fig. 13 above). This one sided truncation o f the peaks in the time series gave rise to a second harmonic component in the Fourier spectrum of the time series. The amount of asymmetry in the curvatures of the proximal section ( s - - 0 - 1 0 ~m) of the live sea urchin sperm showed a good correlation with the curvature ( - - l / r a d i u s ) o f the path of the sperm (Rikmenspoel & Isles, 1985). It has been previously shown (Rikmenspoel, 1982) that the time course o f the curvature o f sea urchin sperm flagella is a direct reflection o f the time course of the internal, active, moment in the flagella. It was therefore concluded by Rikmenspoel & Isles (1985) that the active moment in the sea urchin sperm flagella was limited to an upper bound, on one side of the axoneme only, and that the curved path was caused by this one-sided limitation. The above asymmetry in the curvature of the sperm flagella was present only in the proximal part of the flagella (s = 0-10 p~m). In the present analysis it was found that the sea urchin sperm should be considered to have an overall bending which is approximately constant over the length o f the flagellum, as illustrated in Fig. 1. This suggests that a rather uniformly distributed asymmetry should be present in the sperm flagella. The original data on the curvature of the sperm were therefore reanalysed.

WAVEFORMS

OF

SEA

URCHIN

SPERM

FLAGELLA

145

Each of the time series for the curvature at all flagellar locations on the 21 sperm of the present sample was fitted to a periodic curve p( t) = p~ + RE sin (tot + ¢)

(13)

where p ( t ) is the time series for the curvature, and to=2zrxflagellar frequency, p~ represents the average displacement of the time series # ( t ) from zero (the "off-set" of the curvature). The values of p~, equation (13), were found to be almost uniformly distributed over the flagellar length. Figure 14 shows the off-set of the curvature time series as a function of the location on the flagellum. The values shown in Fig. 14 were obtained by averaging the off-set at a given location over the 21 sperm in the sample.

® ~3OOr ~.7 200 ~ ~ 100 f "" . .•. . . . . . . .

~g

(a) *•.eo

~° .... ;d" 2b"' '3~ '(4bb)

o ~ ,o~

Oil

e e

....... •

40

0

20

•

•

40 s(/~m)

50

Oisfence from head,

FIG. 14. Off-set of the flagellar curvature (as explained in the text) of sea urchin sperm, as a function of the location on the flagellum. The points represent the average over the 21 sperm in the sample. In (a) the off-set is expressed in absolute units, in (b) as a percentage of the maximal curvatures shown in Fig. 12.

The data in Fig. 14 indicate that an average asymmetry of the local curvatures of a sperm flagellum can be taken as the average of the curvature off-set at the various locations on that flagellum. Figure 15 shows that the thus defined average •it-set of the curvatures of a sperm flagellum is strongly correlated with the curvature of the path of the sperm. /

600

..c 500 B. 400 "6 ® 300 200 I00

jr.

~

°o • •

I

100

I

200

O f f - s e t of flaget~r curvature

I

300

(cm-~)

FIG, 15. Correlation between the off-set of the flagellar curvature and the curvature of the path of the sperm. Each dot represents one sperm of the present sample.

146

R. RIKMENSPOEL

The off-set of the curvature is probably related to the active moment developed in the flagella being larger on one side compared to the other side. The results shown in Figs 14 and 15 therefore suggest that the primary cause of the curved path of the sea urchin sperm is that, all over the length of the flagella, the active moment toward one side is larger than toward the other side. Near the proximal junction of the sperm flagella the magnitude of the flagellar curvature is much larger than in the more distal parts, as illustrated in Fig. 12 above. From model calculations (Rikmenspoel, 1982) this author has concluded that the magnitude of the internal, active, moments in sperm flagella is maximal in the proximal part of the flagellum. Since larger active moments would lead to larger flagellar curvatures (Rikmenspoel, 1982) these two results appear to support each other. That a saturation of the active moments only occurs in the proximal section, as manifested by an upper limit to the observed curvatures, would accordingly be caused by the fact that the magnitude of the active moments is largest in the proximal section. One question left unanswered is that the average curvature of the centerline of the sea urchin sperm seems not to be directly correlated to the curvature o f the path of the sperm (compare Figs 1 and 3 above). The forward progression of the sperm, and also the deviation of the progression from a straight line, involves the hydrodynamic interaction of the moving flagellum and the surrounding fluid (Taylor, 1952; Hancock, 1953). It would appear unlikely that the problem can be resolved on the basis o f the data used in this paper alone, without recourse to hydrodynamic theory. This investigation was supported in part by the NSF through grant PCM 80-3700. Robert Rikmenspoel died on 9 December, 1984. The finished manuscript was submitted posthumously.

REFERENCES BROKAW, C. J. (1972), Biophys. J. 12, 564. BROKAW, C. J., GOLDSTEIN, S. F. & MILLER, R. L. (1970). In Comparative Spermatology. (Baccetti, B. ed.). pp. 475-486. New York: Academic Press. BROKAW, C. J. & JOSSLIN, R. (1973). J. exp. Biol. 59, 617. BROKAW, C. J. & RIN'rALA, D. R. (1975). J. Mechano chem. Cell Mot. 3, 77. GIBBONS, I. R. & GIBBONS, B. H. (1980). J. Muscle Res. Cell Motil. 1, 31. GOLDSTEIN, S. F. (1979). J. Cell Biol. 80, 61. GOLDSTEIN, S. F. (1975). In Swimming and Flying in Nature. (Wu, T. Y., Brokaw, C. J. & Brennen, C. eds.), pp. 127-132. New York: Plenum Press. GRAY, J. (1955). J. exp. Biol. 32, 775. HANCOCK, G. J. (1953). Proc. R. Soc. (Lond.) A, 209, 447. HINES, M. & BLUM, J. J. (1978). Biophys. J. 23, 41.

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HINES, M. & BLUM, J. J. (1979). Biophys. Z 25, 421. HIRAMOTO, Y. & BABA, S. A. (1978). J. exp. Biol. 76, 85. HUXLEY, H. E. (1969). Science (Washington, DC) 164, 1356. LANDAU, L. D. & LIFSHITZ, E. M. (1970). Theory of Elasticity (2nd edition). Reading, Massachusetts: Addison-Wesley. LUBLINER, J. & BEUM, J. J. (1971). J. theor. Biol. 35, 796. MACHIN, K. E. (1958). J. exp. Biol. 35, 796. RIKMENSPOEL, R. (1965). Biophys. J. 5, 365. RIKMENSr'OEL, R. (1971). Biophys. J. II, 446. RIKMENSPOEL, R. (1976). Biophys..I. 16, 445. RIKMENSPOEL, R. (1978a). Biophys. J. 23, 177. RIKMENSPOEL, R. (1978b). J. Cell Biol. 76, 310. RIKMENSPOEL, R. (1982). J. theor. Biol. 96, 617. RIKMENSPOEE, R. & RUDD, W. G. (1973). Biophys..It 1~, 955. RIKMENSPOEL, R. & ][SEES, C. A. (1985). Biophys. J. (in press). SUMMERS, K. E. ~,~ GIBBONS, I. R. (1971). Proc. hath. Acad. Sci. U.S.A. 68, 3092. TAYLOR, G. I. (1952). Proc. R. Soc. (Lond.) A, 211, 225.