Page 1
Biophysical Journal
Volume 67
October 1994
17671774
The Movement of Spermatozoa with Helical Head:
Theoretical Analysis and Experimental Results
F. Andrietti and G. Bernardini
Dipartimento di Biologia, Universit& di Milano, Milano, Italy
ABSTRACT
head and a helical pattern of flagellar beating, such as Xenopus sperms. The theoretical approach is similar to that taken by
Chang and Wu (1971) in the study of helical flagellar movement. The model used in the present study allows us to determine
the velocity of propulsion (U) and the frequency of rotation of the sperm head (fh) as a function of the frequency of the wave
of motion (ft) traveling along the tail. The results relative to the case of helical and planar flagellar waves are compared. Our
main finding is that the helical shape of the head seems to increase the efficiency of propulsion of the spermatozoon when
compared with the more commonly shaped spherical head. Experimentally measured values of fh versus U may be fitted by
a linear plot whose slope is much higher than that corresponding to the case of a planar flagellar beating. This fact is consistent
with an effectively threedimensional (nonplanar) movement of the flagellar tail. However, the results do not fit those predicted
from a circular helix, suggesting that a different shape of the flagellar beating should be considered.
The present work is concerned with the study of the swimming of flagellated microscopic organisms with a helical
INTRODUCTION
Helical shape is a feature largely scattered among unrelated
microscopic organisms; fern spermatozoids, spirilla, some
Phytoflagellata, and Xenopus sperms are a few examples.
The analysis of their movement has been performed by
means of hydrodynamic (Lowndes, 1943; Brown, 1945;
Chwang et al., 1972) and mathematical models (Hollwill,
1966; Mysercough and Swan, 1989). One of the points
stressed by such studies concerns the contribution ofthebody
shape to themovement ofthe organisms. However, theworks
mentioned were concerned either with the case of spirilla
(Chwang et al., 1972; Mysercough and Swan, 1989) or
Euglena (Holwill, 1966). A spirillar locomotory apparatus
consists of one or more bundles of flagella at the end of the
body, beating on a conical surface of revolution. Euglena is
not comparable with Xenopus sperms either in locomotory
apparatus or in body shape. Because both cases are different
from that presented by Xenopus sperms, this requires a quite
different analysis. In fact these organisms are very thin cells
with a corkscrew shape of the head and a helical, three
dimensional flagellar beating. They move following a path
which is, at least approximately, straight.
The aim ofthe presentwork is to investigate the movement
of Xenopus spermatozoa by making use of a simple theory
and to relate the results of the theoretical study with the
experimental data obtained by the video recording of the
swimming of the sperms reported in a previous paper
(Bernardini et al., 1988), in which a description of the ex
perimental method used in this study appears.
THEORY
Helical flagellar wave
We will consider the case of a sperm moving uniformly with
a translational movement U and rotating around its axis with
an angular velocityQh. Dealing with helical structures, it is
suitable to use cylindrical coordinates. We will consider a
system ofsuch coordinates, (r, 0, x), fixed to the sperm head,
with the origin in the point of attachment of the tail and the
xaxis coincident with that of the head and the tail, pointing
toward the end of the tail (Fig. 1).
In these coordinates the head is described as
r = hh
O= khx,
(1)
where hh is the head radius, kh = 2ir/Sh is the head wave
number and ah its wave length.
In the same coordinate system the flagellar tail movement
is described as
r=ht
0 = kx +(fltQh)t = ktx +(ktc,fh)t
wherehtis the radial amplitude ofthe helical wave ofthe tail,
St its wavelength, k,=2'ni8,the tail wavenumber, fl,the
velocity ofrotation ofthe flagellar tail with respect to a fixed
frame of reference, Qt fh the apparent tail angular ve
locity, and ct =Qit/ktthe phase velocity ofthewave ofmotion
traveling along the tail.
By applying the Gray and Hancock theory (1955) to the
case of helical structures we have for the tail (Chwang and
Wu, 1971)
Ft,n= Ct,n(ktCt Qh)ht Uktht]dx
and
dFt s=CI,J(k,C, flh)ktht' + U] dx
whereFt,n Ft,s Ct,n9Ct are the tail normal (n) and tangential
(s) forces and coefficients of resistance, respectively.
(1')
Receivedfor publication 21 April 1994 and in finalform 26 July 1994.
Address reprint requests to Dr. F. Andrietti, Dipartimento di Biologia, Uni
versitA Degli Studi di Milano, Via Celoria 26, 20133 Milano, Italy. Tel.: 39
226604460; Fax: 39 22361070.
X 1994 by the Biophysical Society
00063495/94/10/1767/08
$2.00
(3)
(4)
1767
Page 2
Volume 67
October 1994
FLAGELLAR
TAIL
SPERM
HEAD
sperm forward velocity U
//
1/
//
'I
I,
I,
I,,
J
head angular velocity ALh
FIGURE 1
tozoon with a helical head and a helicalflagellar
tail. For explanation of symbols and values of
parameters see Table 1.
Schematic diagram of a sperma
/
Lt,x
Mbt
z
2bh Vl
Thh $\
B3
Lh,x
y1
I'
7
NK
.
N
ntSt
/1\
nhah
(3')
By analogy, we obtain for the head
dFh,n=Ch,nfQhhh Ukhhh]dx
and
dFh,S= Ch,j Qfhkhhh + U]dX
with an obvious meaning of symbols.
The propulsive forcedFt,,originated by an element ds of
the tail in the direction of the xaxis isgiven by Chwangand
Wu (1971) as
dFx=(dFt,nsin ,tdFt,sCOs t)
whereAtis the angle between the tail and the xaxisgiven
by
(5)
Taking into account the relationships
COS pt = (1 + K2)1/2,
whereK,=htlc,,we findby Eqs. 3, 4, 5, 7, 8,
rnt 8
Ft,X= J
x
sin t=Kt(l + Kt2)1
dFt,xdx
(4')
(10)
=Ct,sn8[(kct
)h K42U(2K12 + 1)](1+Kt2)1t2
rnt8
Mtx=
dM4
dx
=Ct,snt8tht[UKt+fhht(2+Ktt)K1(2+Ktt)ct]
X(1+K2)12
tan
=(2r1cT/)h,=ktht,
(6)
whereas the force in the e direction is given by
dFx=(dFt,ncos P3t+dFt,ssin 1t)
which, in turn, gives a moment of force about the xaxis
dMt,X=htdFt,
The total propulsive force and torque isgiven by integra
tion of Eqs. S and 7 between x = 0 and x =ntst,wherent
is the number of flagellar wavelengths.
According to Gray and Hancock (1955), the force coef
ficients are given by
=ht(dFtncos igt+dFtssin Pt)
(7)
C;n=2Ci,s
i = t, h
Ci's=2i,m/[ln(28j/bj)0.5]
(8)
(9)
(11)
By similar reasoning we find
rnhah
Fh,x =J
dh,
d
(10')
Ch,Snhh [fhhhKh U(2Kh2
rnhah
dMh,xdx
1)](1
Kh2)
1/2
Mhx=
Kh2)](1+K2
(11')
=ChSnhkhh[UKh+f4hh(2 + Kb2)] (1 + Kh2)112
with obvious meaning of symbols not yet defined. Observe
thatFh,xandMh,xhave the meaning of head viscousdragand
torque, because of the head translational and rotational
movement, respectively.
h. x
I> 0
I
I
I'll
\, ...
'\\\
I
%
e0
3k *c
Biophysical Joumal
1768
r
X
i=t,h
Page 3
Swimming of Sperm with Helical Head
Finally,we have to consider that, as long as the spermbody
rotates around the xaxis with the angular velocity fh, every
crosssection of the helical head also rotates around the
tangent of its central curve (i.e., a line passing through the
center of each crosssection), giving rise to an additional
viscous torque Lh. The same is true for the tail, which gives
rise to an additional viscous torqueL,(see Chwang and Wu
(1971) and their Fig. 2). These additional torques are small
compared withMh,xandMt,x and they are given here only for
sake of completeness. The xcomponent dLtxof the infini
tesimal torque of a cylindrical tail element of radiusb,is
given by Chwang and Wu (1971) and Lamb (1932).
dLt X=4'ujbt2fl Cosftdx
where
,u is the viscosity coefficient, so that
4 x
f
dLt x=4,r(3bt2hnt8tCOS At
(12)
Analogously we find
4hX= 4'nUbh2fhnh6hcos 3h
The balance of total forces and moments in the xdirection
requires
(12')
Ftx+Fh,x= 0
(13)
Mtx+Mh,X+Lx+Lhx= 0
(14)
By Eqs. 13, 14, 1012, and 10'12' we obtain
A1,5U +Al1,2(h=Ctsnt8tKt2Ct(l+Kt2)112
(15)
A2,1U +A2,2Qh=Ct,sntt
+K2)h1t2 Kt(2+ 1t2)ct
where
AC,1
=C,ntt6(2Kt2+ 1) (1 + K 2)1/2
+Ch,nh8h(2Kh2+ 1) (1 + Kh2)1/2
Ais=Ctsnt8thtKt(1+Kt2)
A2,1=Ctsnt8thtKt(1+Kt2)112+Ch,snh8hhhKh(l+Kt2)1/2
112 +Ch,snh&hhhKh(l+ Kh2)1/2
A2,2=Ct,snt8tht2[2+ 1t2 + C
Ch,Snh[2 + Kh +Ct, s
(h
(1 +Kt2)1/2
L
ht,J
+Ch,sn,hhhhL2+Kh2
Chj(s
2
+hKh2
Resolution ofEq. 15 allows determinatii
values lh and U.
A different expression for the force coef
by Lighthill (1975, p. 52) is thefollowing
on Eqs. 8 and 9.
81Tg
Ci,n
ln(0.0324 i2/bi2) + 1
Ci,S
ln(0.032482/bi2)1
The use offorce coefficients givenby Eqs. 8'9' gives rise
to formulae not reported here, which are only slightly more
complicated than the previous ones.
Planar flagellar wave
In the case of a planar wave the flagellar movement does not
induce a torque. However, if the head has a helical shape, an
indirect torque is produced given by Eq. 11' or, more gen
erally, leaving coefficientsCh,.,andCh,Sfree to assume in
dependent values
Mh,X= nh hh2(1 +Kh2)1/2
X [(Ch,nCh,S)khU+(Ch,n+Ch,sKh2)fh]
Because of the rotation induced by the head, the tail
will rotate around the xaxis with the angular velocity fl
=
the tail generates a force perpendicular to each tail ele
ment ds given by
(16)
h' producinga viscoustorqueM' . The rotation of
dFt,n=Cts,nVt,0dS
(17)
whereVt,,,the velocity of rotation of each element ds, is
given byy,f yt(x)=htsin[k,(x+ctt)]is the form of the
wave generated by the tail conforming to a sine curve, and
ds = [1 +(dytIdx)2]112dx
From Eqs. 17 and 18 we have
(18)
m
=fdM't dx= T&Ytctnvt41+ (dx)21l2
=Ctnht2fl
sin[k(x+ctt)]
t
X (1 +h2k2Cos2[lk(x+ctt)])1' dX
The above integral is in general timedependent, because
ntis anonintegernumber. However, if weaveragethe dif
ferent values of the moment during a given period of time,
phase differences will cancel and the resulting average value
of the moment(M'tx)will be given by
rS
(M',x)=ntCt,ht2f J
sin2ktx[1+ht2kt2Cos2(ktx)]112 dcx
xuhprp n,jvv
WllMlC llUW t11U Hilotgral is calculinc UL '11U anu is mnue
pendent of time.
The evaluation of the above integral cannot be given in
closed form, but it may be expressed by means ofelliptic
integrals. However, anapproximate valuemay be obtained
taking into account that cos2ktx=1/2[1 + cos(4'r/5)x]. The
second term in the brackets may be ignored in a first ap
proximation, given that its contribution to the integral will be
less than the first term.Substitutingcos2krx
root of the integral, find
(Mrx) (nt2)Ct,nh fl8t[l+ K12/2]1/2
thL1
iC
natlLrlaLlu at L
= n anA ie;i1i
on ofthe unknown
ficients developed
;, which improves
i=
, h
(8')
1/2in thesquare
i= t,h
(9')
(19)
Andriefti and Bernardini
1769
Page 4
Volume 67October 1994
A second viscous torque to be taken into account is again
that generated by the rotation of each tail and head cross
sections around the tangent of its central curve, given by Eqs.
12 (whencosp,B= 0) and 12', respectively.
The balance of all moments requires
Mh,X+(Mtf,x)+Lt,x+Lh,x= 0
In a planar flagellar beating the presence of an additional
torque perpendicular to the plane of the flagellar movement
should be considered. In the simplified model of Gray and
Hancock (1955) that we are following here, this additional
component is disregarded. It has been taken into account,
instead, in more elaborate treatments such as that given by
Higdon (1979). Because of this additional moment, the tra
jectory of the organism during one cycle of flagellar beating
is represented by a yawing motion rather than by a straight
line. However, for any symmetrical wave offlagellar beating
such as that considered here, the organism has no net rotation
over a whole cycle of flagellar beating, and the trajectory is
symmetrical with respect to the direction of movement.
Moreover, it has been shown (Higdon, 1979) that, at least in
the case of organisms with a spherical head, when the ratio
between the radius of the head and the length of the flagellar
tail is higher than 20, the use ofthe force coefficients ofGray
and Hancock (1955) gives results accurate to within 10%. In
our casewe are dealing with helical heads. However, the ratio
between the flagellar length and the "equivalent" spherical
head radius (see Table 1) is higher than 20. Moreover, the
yawing motion should be less in the case of a more elongated
helical head than in that of a spherical one. For all these
reasons we will disregard the torque perpendicular to the
plane of the flagellar beating, because this error is certainly
not greater than and is probably even less than those inherent
to any model based on the resistiveforce theory.
(20)
By substituting Eqs. 16, 19, 12, and 12' in Eq. 20, we
may easily derive the following linear relationship be
tween U and fQ
fl _(Ch,sCh,nf)nh6hKhhh
U
{I}
(21)
where
{ } = Ct
(nt/2)5t[l + (Kt2/2)]112(1 +Kh2)1/2
+ nh6h[4TiLbh + hh2(Ch,n+ ChSKh2)]
+ 4,uib12n6t1(1 +Kh2)1/2
To evaluate fl and U, the balance of the propulsive forces
and resistances is also required. From Gray and Hancock
(1955) we find that the total propulsive force along the xaxis
is given by
It Slt[
A2
1
ct(Ct,n Ct,s)
[(1 + A2)/2J dX
u(uf
L
A2)1/2dx=] =I
UL
where A =dyt/dx=Ktcos[k,(x+ctt)].The integrals I and
L, when averaged with respect to time, may be calculated as
(= nt (Ct,n Cts)
(1+ Kt2Cos2ktx,12]
J=
Ct,s +Ct,nKt2cOs2kdx
()
(1+Kt2cos2kx)1
2
(22)
(23)
TABLE 1
Meaning of symbols and values of experimental data
Wi
Fj,.;Fis~; Fj,.; Fi,9
MC,.
Ci,.; Ci,t
Pi
kik
Ki
Li,.
I, (flagellar length)
at (flagellar wavelength)
n,(number ofwavelengths along theflagellum)
ht (wave amplitude of the flagellar wave centralcurve)
b,(flagellar crosssectional radius)
'h (helical head length)
6h (helical head wavelength)
nh (number ofwavelengths along the helicahead)
hh (wave amplitude of the helical head central curve)
bh (helical head crosssectional radius)
fh (frequency ofgyration of the helical head)
f (frequencyofflagellar movement)
f (apparent frequency of gyration of the tail)
a (radius of the equivalent spherical head)
*Bernardini et al. (1988); all other data are from unpublished results.
= index related to the flagellar tail (t) or the sperm head (h)
= angular velocity
= normal, tangential, xdirected and 0 directed forces
= torque component in the xdirection
= normal and tangential force coefficients
= angle between helical tail or head and xdirection
= wave number = 2ir/S
= h.k.
= torque induced by the rotation of each cross section of tail
or head sections around its central curve
= 40 ,um*
= 19.2,Am
= 1.22
= (lt  nt2)"'2/2rnt= 4.3,Am,for helical waves
=(,/w/<)[Rt/(nt,8t 1)1/2= 3.6,Am,for planar waves
= 0.1 ,um
=22 ,m*
= 9.2 ,um
= 1.5*
= (l nh 8^)'/21mb = 1.82 ,um
= 0.4 ,um*
= 0.5 Hz
= 1.0 Hz
= 0.5 Hz
=[¾3/4bh2lih]3= 1.38 ,im
1770
Biophysical Journal
Page 5
Swimming of Sperm with Helical Head
The balance of forces in the xdirection requires (I) 
U(L) +Fh,X= 0. When FhX is given by Eq. 10' generalized
to the case whenCh,flandCh,sare independent values, we
obtain from the balance equation
[(ChfKh2+ Ch,S) (1 +Kh2) 1f2nh8h+(L)]U
(25)
+[Khhh (1+Kh2) Y2nh8h](Ch,nCh,s)f= (
where (1) and (L) are given by Eqs. 22 and 23, respectively.
Eqs. 25 and 21 allow determination of the unknown vari
ables U and Q. The evaluation of integrals from Eqs. 22 and
23 may be given in terms of elliptic integrals. However, an
approximate value ofthese integrals may be obtained as seen
above. Ifwe substitutecos2ktx
integrals from Eqs. 22 and 23, in a first approximation they
reduce to
Ctntt(Ct Ct,s)(Kt2/2)[1+(Kt2/2)]1/2
(L)
1/2in the denominator of the
(1)
(26)
ntIt[Ct,s+Ct,n (Kt2/2)]11+(Kt2/2)]i12
(27)
RESULTS
As a result implicit in the use of resistive force theory, both
Qhand Udepend linearlyon thefrequencyftof rotation of
the flagellum about the xaxis. This is also shown by the fact
that the parameterctis present only in the right side ofEq. 15,
andf =ctkt/27r. Giventhatft=ft/21r,also the "apparent"
angular velocity (Q of rotation of the tail, fl =
2mf, 2Tfh, depends linearlyonft.The ratiosfQ/Qh, Q/Qt
fQ/U andf1t/U, instead, are independent fromft.
In Fig. 2 (solid line, left axis) is plotted the value of
f/fh=Q/Qhversus different values of theflagellar length lt,
when the flagellar beating is as depicted in Fig. 1, i.e., shaped
in the form of a circular helix reversed with respect to that
represented by the helical head. The values of the other geo
metrical parameters, corresponding to the flagellar tail and
head of Xenopus spermatozoon, are given in Table 1. Co
Qh =
Pm
2.5
.4
2.0
\3
f/fh
1.5
ht
2
1.0

0.51
25
*
30
35
40 pm
it
FIGURE 2
1, plot (dashed line, right ordinate axis). The *indicates the flagellar length
giving the samne value off/fh found experimentally (see Table 1). Helical
flagellar pattern as in Fig. 1, Gray and Hancock (1955) force coefficients.
See text for further explanations.
f/fh versus It plots (solid line, left ordinate axis) andh,versus
efficients 8 and 9 from Gray and Hancock (1955) have been
used. We see that at the experimental measured value of the
flagellum, 40 ,um (Table 1), the "apparent" angular ve
locity of the tail is 20% of"h. From measures based on
photographs taken from video recordings of Xenopus sper
matozoa, the two velocities showed, on the contrary, similar
values (Table 1). According to Fig. 2, this result may be
obtained with a reduced tail length of about 28 ,um (* in Fig.
2, solid line). A possible reason for the discrepancy between
the observed and calculated tail length will be discussed fur
ther on. Observe that at the reduced tail length the radial
amplitude of a helical wave calculated according to the for
mula given in Table 1, still maintaining the same value ofSt
andnt, is reduced from more than 4 ,um to 2 ,um (dashed
line, right axis).
Fig. 2 also allows us to determine Qt/Qh= Q/Qh + 1,
or flt/fl; in this way the frequencyftof rotation of the
flagellum around its xaxis, a quantity indirectly observ
able, may at least in principle be estimated (see Discus
sion section).
In Fig. 3, A and B is plotted the ratiofh/U versus different
values ofIt, in the case of a helical (A) and a planar (B)
flagellar wave. The values of the other geometrical param
eters are those of Table 1. Curves (GH) are calculated using
Gray and Hancock (1955) coefficients 8 and 9. The other
curves are calculated according to different force coeffi
cients. In curves Li and L2 the coefficientsCt,nandCt,Shave
been determined according to Lighthill's (1975) equations
(Eqs. 8', 9'), which give a ratioC1,Xt,n= 0.66. However,
both Gray and Hancock's (1955) and Lighthill's (1975) for
mulae hold only whenkihi<< 1. This may be true for the
flagellar tail, not for the helical head with the parameter val
ues given in Table 1. For arbitrary values of khhh, Ch,S/Ch,n
may have a value somewhat higher than 0.5 (Brokaw, 1970).
For this reason we have used values that are both larger than
0.66, i.e.,ChS/Chn= 0.7 (Li) andChS/Chfn= 0.8 (L2). These
values, even if arbitrary, may give an idea of the effect of
increasing the ratio of the force coefficients. For the GH
curve of Fig. 3 A, we see that for the measured tail length of
40 ,u the value offh/U is about 0.4gm, against the value
of 0.21 pum1 observed experimentally. However, for the
abovementioned value ofIt= 28 ,um (*) the theoretical value
decreases to 0.267 ,um/s, not too far from the experimental
data. In the case of curves Li and L2 of Fig. 3 A, the fit to
the experimental values is still worse.
In Fig. 4 is a plot offhIU versus Itwhen the direction of
rotation of the flagellum around the xaxis is inverted with
respect to the situation illustrated in Fig. 1, i.e., when the
flagellum has the same sense ofrotation ofthe head. Observe
the great increase ofthe ratiofh/Uwith respect to the previous
case represented in Fig. 3 A (curve GH).
In Fig. 5 different linear plots offh versus U are shown,
corresponding to a tail length of40 ,tm, calculated in the case
of helical (hw, hw') and planar (pw) flagellar waves, com
pared with the curve fitted to the experimental data taken
from Fig. 6 ofBernardini et al. (1988). The ratio of the force
coefficients has been determined according to the Gray and
Hancock (1955) theory forhw and pw, and with an increased
Andrietti and Bernardini
1771
Page 6
Volume 67
October 1994
pm1
3
2.0
fh/u
1.0
lPm
It
35
25
30
35
it
FIGURE 4 fh/U versus It plot for spermatozoa with an inverted helical
pattern ofthe flagellar beating. Gray and Hancock (1955) force coefficients.
Geometrical parameter values as in Fig. 2.
si
8
6
4
40pm
It
FIGURE 3
helical flagellar wave (A) and planar flagellar wave (B). GH, Gray and
Hancock (1955) force coefficients; *, flagellar length giving afhUvalue
close to that found experimentally. Lighthill (1975) force coefficients for the
flagellar tail (Li, L2):Ch,S/Ch,n= 0.7 (Li), Ch,S/Chn= 0.8 (L2). Helical
pattern of flagellar beating in (A) as in Fig. 1. Other explanations in the text.
fhlU versus l, plots for spermatozoa with helical head and
value of the force coefficient ratios, as in L2 of Fig. 3, for
hw'. This same increased ratio for the case of a planar flagel
lar wave gives rise to a curve below pw, not represented
in the figure.
In Fig. 6 different plots off/U versus lt are shown. A first
one (hh), corresponding to a sperm with a helical flagellar
beating and a helical head, is calculated according to Eq. 15.
Observe that the curve reaches a minimum at about 33gm.
Ifwe assume that the valueoff,is proportional to the energy
required to move the flagellum, this flagellar length should
be that which requires a minimum power for propelling the
spermatozoon. A second plot (hs), for the case of a sper
matozoon with a spherical head with the same volume of the
helical one, is calculated according to Eq. 35 of Chwang and
Wu (1971).
The reason for the increase in the propulsion efficiency of
spermatozoa with helical heads, with respect to those pro
vided with spherical ones, should be related to the fact that
the first kind ofheads not only exert a resistive effect against
2
0
10
20
30
40
50
U
.
1
FIGURE 5
helical flagellar wave; pw, planar flagellar wave; ec, curve fitted to ex
perimental data (0). The force coefficients are calculated according to
Gray and Hancock (1955) for hw, pw; as in L2 of Fig. 3 for hw'. The
values of the geometrical parameters of the flagellar tail and head are those
of Table 1.
fh versus U plots for spermatozoa with It = 40 ,um: hw, hw',
the fluid but also a propulsive one, owing to the rotation
induced by the helical movement ofthe flagellum, caused by
the head spinning in the fluid. In fact the experimental curve
(ec) of Fig. 5 matches very closely that of an "ideal" screw,
with the same pitch of the spermatozoon head, advancing in
the medium without slipping. In fact, this coincidence is only
casual, and the experimental curve should be the result of a
compromise between the situation of a spermatozoon with
helical flagellar waves (hw in Fig. 5), which spinsmuch more
rapidly than the ideal screw, and that of a spermatozoon with
a planar flagellar wave (pw in Fig. 5), which spins much more
slowly.
In Fig. 6 are also shownf/U versusItplots in the case of
planar waves. One of them corresponds to a spermatozoon
with a helical head (ph), and is calculated according to Eqs.
pmr
1.2
0.8
fh/u
0.4
0
2!
pm1l
2
X10
2.0
1.5
fh/U
1.0
0.5
25
40Pm
LI
30 E5
1772
Biophysical Journal
Page 7
Swimming of Sperm with Helical Head
pm1i
2.5
2.0
ft/U15 1h.
1.0
0.5

~~~~hh
Ps
25
30
35
40 pm
it
FIGURE 6
and helical head (hh, hi), helical flagellar wave and spherical head (hs),
planar flagellar wave and helical head (ph), and planar flagellar wave and
spherical head (ps). Helical pattern of the flagellar beating: hh, hs, ph, ps,
as in Fig. 1; hi, reversed. Gray and Hancock (1955) force coefficients. The
values of the geometrical parameters of the flagellar tail and head are those
of Table 1.
f,/Uversus1, plots for spermatozoa with: helical flagellar wave
21 and 25, making allowance for Eqs. 26 and 27. A second
plot (ps), concerning a spermatozoon with a spherical head
with the same volume of the helical one, has been calculated
according to approximate formulas given previously by Gray
and Hancock (1955). For planar waves the propulsion effi
ciency of the tail does not reach any maximum in the range
of flagellar length considered, because it increases mono
tonically withIt. Contrary to what has been observed in the
case of helical flagellar waves, in the case of planar waves
the spermatozoa with helical heads are shown to be far less
efficient than those provided with spherical ones. This is
because, in such a situation, owing to the low speed of ro
tation, the contribution of a helical head shape to the ad
vancing of the organism is very low and it is not able to
counterbalance its increased resistance to the propulsive
force.
A last plot of Fig. 6 (hi) represents the case when the
direction of rotation of the flagellar beating is reversed as in
Fig. 4.
DISCUSSION
A rigorous analysis of the kinematics of Xenopus sperms'
motion should probably reveal that they move following a
helical path, like many other similar organisms. Their tra
jectory should be characterized by their own pitch, radius,
and angular frequency and analyzed according to these pa
rameters (see, e.g., Crenshaw (1989)). In fact, in a first ap
proximation the trajectories followed byXenopus sperms ap
pear to be quite straight, and the present analysis will account
only for the two more important parameters that characterize
the sperms' movement, i.e., their translational and rotational
velocities (U andflh, respectively).
By inspection ofphotographs taken from video recordings
ofXenopus spermatozoawe have observed that theampli
tude of the flagellum lateral displacement seems to decrease
or to increase slightly during rotation. This is not what we
would expect for aplanar sinusoidal or circular helical flagel
lar beating. However, it could be consistent with a flattened
helical pattern with an elliptical or oval yz projection. Ob
serve that to have a circular helix with a reduced value ofht,
by still maintaining the same value of at andnt,we have to
reduce the value ofItas is shown in Fig. 2 (dashed line, right
axis). On the other hand, video recording or direct obser
vation may not discriminate unequivocally between a planar
or a threedimensional flagellar beating. In fact, because of
the rotation of the whole sperm induced by the head, even
a planar wave should "look" threedimensionally shaped. An
indirect evidence of an effective threedimensional flagellar
beating (Bernardini et al., 1988) should be confirmed on the
basis of the present theoretical analysis.
In general the rapid sperm motion prevents accurate and
quick measurements of (apparent) flagellar beatfrequencyf.
However, when a lower mobility in a highosmolarity me
dium is induced (Bernardini et al., 1988), more reliable mea
surements may be performed. From a series of video re
cordings in such a situation we have observed that the
spermatozoon head shows a yawing motion. A similar os
cillation has been observed in other cases in flagellate or
ganisms (Holwill, 1965), and it could originate from a trans
versal component because of asymmetries arising in the
course of the flagellar beating. We have already said that
such a yawing should be expected for a planar flagellar beat
ing. In the threedimensional case, the transversal component
should be less relevant, at least in the case ofa circular helical
pattern offlagellar beating. It couldbe quite relevant, instead,
for a flattened or elliptical helix. In our case of induced slow
mobility, pitches showed a frequency of about 1 Hz, corre
lated to a value of both f and fh of about 0.5 Hz. In this
condition the theory predictsf,= f +fh= 1 Hz. This fact
seems to indicate what we would expect, i.e., that the head
pitching is correlated to the frequency of rotation of the fla
gellum around the xaxis. Because the evaluation ofthe yaw
ing frequency is much easier to perform than the direct meas
ure of the flagellar beating frequency, it may give a more
reliable method to determinef.
Microphotographs seem to show a direction of rotation of
the flagellum contrary to that observed in the head. This is
also what we would expect on a theoretical ground, to in
crease the efficiency of the propulsion (Fig. 6, compare
curves hi and hh) and by comparing Figs. 3 A, and 4 and the
experimental curve (ec) of Fig. 5.
Experimental measured values offh versus U are fitted by
a linear plot that is intermediate to those relative to helical
and planar flagellar waves (Fig. 5), but nowhere close to
either of them. The difference between theoretical and ex
perimental curves may not be attributed to the use of non
suitable force coefficients, because according to other co
efficient ratios thefh/U slope is still higher in case of helical
flagellar beating (compare GH, Li, and L2 of Fig. 3 A, and
hw and hw' of Fig. 5) and still smaller for planar flagellar
beating (compare GH, Li, and L2 of Fig. 3 B). The result of
Andrietti and Bernardini
1773
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1774
Biophysical Journal
Volume 67
October 1994
the present analysis is consistent with the fact that the flagel
lar beating of Xenopus spermatozoa is effectively three
dimensional and nonplanar. However, it is not consistent
with the assumption ofa circular helical wave. Given that this
difference should not depend on the use of nonsuitable force
coefficient ratios, it could be attributed to a different geom
etry of the (threedimensional) flagellar shape. We suggest
here, as a possibility thatwe hope to explore in a future work,
a shape conforming to a flattened helix, with an elliptical yz
projection.
A last point to stress is that, in the case of a (circular)
helical flagellar beating, the helical shape of the head seems
to increase considerably the velocity of propulsion of the
spermatozoon, with respect to the energy expenditure, com
pared with the more common spherical shape, as we see in
Fig. 6 (curves hs and hh). On the contrary, in the case of
planar flagellar waves, a helical head decreases, instead of
increases, movement efficiency (Fig. 6, curves ps and ph).
Regarding helical and planar flagellar waves, the first ones
seem to be far less efficient than the second ones, in the case
of spherical heads (Fig. 6, curves hs and ps). This result is
the opposite ofthat shownby Chwang andWu (1971) in their
Fig. 10. However, the conditions are not comparable, be
cause the parameter values ofTable 1 are very different from
those giving optimal results for helical waves. Instead, in the
case of helical heads, there is no definite advantage, in terms
ofefficiency, in having planar or helical flagellarwaves (Fig.
6, curves hh and ph); for higher values of tail length there is
a greater efficiency with a planar wave, for smaller values,
on the contrary, with a helical one. This result does not
change, at least qualitatively, when using different force co
efficients (plots not presented here).
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