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The helix as propeller of microorganisms
THE
HELIX AS PROPELLER MICROORGANISMS’” KKISTIAK
Department of Mathematics.
OF
E. SCHREINER University of Oslo, 040
3. Norua!
Abstract- A phenomenological approach is employed in determining the forces and moments on a rotating helix by the resistance of the surrounding liquid a~ low Reynolds numbers. and it i\ shown how they may be balanced by the forces and moments on a spherical head. \Vithin the restrictions of the analysis it is found that if the head rotates it can only have one helical propeller. But if the number of flagella is In. half of them rotating in each direction. the head will not rotate. The r;LIe of energy transmission to the surrounding liquid is evaluated in each case: it is found IO be lower in the latter case. and in the former case it increases with the square of the ratio between helix and head radii. The efficiency of the propulsion system is found to be practically independent of the viscosity of the surrounding liquid. INTRODUCTION
were able to treat the more realistic case when a sinusoidal wave of finite amplitude progresses backwards in the flagellum. A similar approach was employed by Holwill and Burge (1963) for the case of a rotating helical flagellum. Both Taylor and Holwill 6: Burge showed that the rotation of the helical flagellum produces an axial thrust that can compensate for the drag on the head of the microorganism. Further. Taylor showed that the rotation of the helix produces a torque around the axis that must be balanced. In the present paper we will employ a phenomenological approach to study the feasibility of rotating helical flagella as propellers of microorganisms. and we will consider some ways the axial torque can be balanced by the presence of the head. It turns out that roration claims a major. part of the rate of work needed to propel the organism.
THE PROPULSION of microorganisms by flagellar motion is easily observed in the microscope. and it has therefore been studied in the hope that it might throw some light on the life functions of the cell. (For a review of the literature see for instance Holwill ( 19661.) The success of this approach depends on our knowledge about the mechanics of flagellar propulsion. But this again depends on the understanding of the hydrodynamics of the translation of bodies at low Reynolds numbers. This is. however, a complicated problem. and analytical solutions to such motions have only been found for bodies of the simplest geometries. Thus Kaplun (1957) has found the solution for the flow past a circular cylinder, and Proudman and Pearson (1957) the solution for the flow past a sphere. Whatever understanding we have about the swimming of microorganisms has therefore been reached using approximate methods. Thus Taylor (I 952) was able to treat only the cases of a long circular cylinder bent into a sinusoidal wave or a helix. when the amplitude of the wave was in the order of the diameter of the cylinder. By employing a phenomenological approach, Gray and Hancock ( 19_(j)
FORCESAND MOMEh’J’SON THE ROTATING HELIX We consider a circular cylindrical helix of radius R and pitch angle 8. Fig. I(a). moving in axial direction with Velocity U. The radius of the flagellum is r, and its curve length is f. R, 8, and r are all assumed constant along the helix. The helix rotates about the axis with an
*Received 5 May 1970. 73
K. E. SCHKEINER
are denoted cT and c.~ respectively, and we put cT = yc..,,. Hancock ( 1953) has determined cr and c,~ for infinite straight cylinders in unbounded fluid, and he found y = 4 in this case. The presence of external boundaries to the fluid will influence the drug on cylinders, however,see Happel and Brenner ( 1965). Chap. 7. In the present cahe the cylinder. i.e. the flagellum - is coiled into a helix of finite length. The other parts of the helix will then act as external boundaries to the fluid aroundeach element, as will the head and the eventual other flagella of the organism. No attempt to determine the proper values of cs, cT. and y for the present problem will be made here: we only note that the values determined by Hancock should represent lower bounds. To simplify the analysis we assume that cs and cr are constant along the helix. The force on the element of length 61 from its translation (I) through the surrounding liquid is then element
Fig. I. Sketch of the helix. with notutions used in evaluating forces and moments on the element M. u. und on the helix. b.
6F=-
angular velocity wt. and the flagellum has a ‘speedometer wire’ rotation of angular velocity 0s. The ve\ocity of an element of the filament is then v=o,Rj+Uk=
o’j’+,c*‘k’.
- cr( o,Rsin
B-6 Usin 8)j’ 8-
Ucos 8) k’)
= - c,~N{ [w,R(sin2B+ycos28) - Usin Bcos 6(, I - y)]j
(I)
where i, j, k are unit vectors in radial. peripheral and axial directions respectively. relative to the cylinder of the helix, and j’ and k’ are unit vectors in a coordinate system rotated an angle 8 about the i-axis. j’ is thus along the element, and k’ normal to it. Further, the element has a rotation that is the sum of the rigid body rotation and the ‘speedometer wire’ rotation,
o=c++a_=w,k+c+j' = (wlsin 8 + u2) j’ + wlcos Ok’.
+ c,VllS’k’)
=-61{cr(~,Rcos
(~,Rcos8+UsinB)j’
-(w,RsinO-Ucos$)k’=
Sl(c+‘j’
(2)
The frictional coefficiiznts per unit length for motion along and normal to the axis of the
- [w,Rsin
8~0s B( 1 -y)
- U(cosz8+ysin28)]k} = - c,,3l{P, j - P,k}.
(3
The forces due to the ‘Magnus effect’ on the rotating element are of the order of the Reynolds number times the forces due to the translation, and they are therefore neglected since the Reynolds number is expected to bc small. The torque on the element due to the rotation (2) is SM,=-
61{4(o,sin
8+w.,)j’+przo,cos
Ok’} (4
THE
HELlX
AS PROPELLER
where the j’-component is due to the rotation about the axis of the cylindrical element. the k’-component is due to the nonuniformity of the longitudinal velocity of the surrounding liquid relative to the element. and cuand p are the frictional coefficients per unit length for the two motions. The element of force (3) gives an element of torque at a point on the helical axis )6&I,; = /d x 6F].
OF
75
MICROORGANISMS
where we have used 16) in the last equality. The element of torque at 0 from this element offorce is by (5). (7) and (8) dM, = -s
dl!,[-- cos t/A, i sin t/lj, A t!r/y~k~] [P,sin rljil + P&OS Jjj, - P,k,]
x
cvR’
= ~0s
dlb{ [P,r,@bcos nili P&n c//Ii,
- [P,fg&lr sin II/ - P,cos tL]j, + P,k,}.
(5)
(9) where IdI is the distance between the element and the point on the axis. Consider the part of the helix between the rear end and an element at curve length I, from this end, Fig. I(b). The station 1, may equivalently be characterized by the angular distance from the rear end
CD,=
Since R. 0. y. and c,, are.all assumed constant along the helix, (8) and (9) may be integrated directly, giving the force on the rear part of the helix. and its contribution to the torque at 0. Thus F, = 2
a,,++?
{-
[ 1 - cos t1j,,]P,i, - sin (JOP,j, + clr,,P,k,}
Similarily the angular distance to the element at station I is given by @=
q,+!y.
( 10)
and M, = - 5
{ [ P,r.@&,sin I/J,, 1 -cosJ,,)]i,
(P,fp9-fp,)(
and the angular distance between I and I, is thus (6) The radius vector of station 1 relative to the point 0 on the helical axis on the normal from station I, is then d =-R[-cosIG,+~~~I_IJ~,+IIJ~~~~,],
(7)
where [il, j,, k,] is a coordinate system on the axis parallel to the [i, j. k] system at station I,. 1n this system the force (3) at station 1 is
1,cos 8 where we have put = ch.
R
In the coordinate system [i,. j,. k,] at 0 the torque (4) due to the rotation (2) is 6M, = -Gl{crr2(w,sin + /3r%,cos
ok’}
= -61([(cY-_)r%,sin + ~*f+cos
ecos 8
81j
+ [r2w1 (asin’6,+/3cos’8~ + c&.&n
dF = - cNdl{ P,sin llri, -t P,cos $jI - P 2kl}
19+o?)j’
01k}
= - r%1{ Tj + T,k} = s
dJI{f ,sin Ilti, + f ,cos
CLj, - f2kl},
(8)
= -rW{
T,sin +i, + T,COS $j, + T,k,}.
K. E. SCHREINEK
76
Using (6) we integrate this over the helix behind station I,. to find the torque at 0 from the ‘speedometer wire’ rotation of this part of the helix
on the helix lb,. = L cos B/R = X-w, and ( 13) and ( 14) take the particularly simple form F = c.yLP,k,.
(15’)
and &I = - +RLP, (12)
-r2f,T2k,.
I, cos
whereagain&,=
R
e.
Putting JI,. = L cos B/R we then have from ( IO) that the total force on the helix is F =
csL,
-
*
-r
-y
‘L pliL
PJ,
-i- I, P,kL
(13) where [i,.j,, kL] is the [i,. jr, k,] system at the front end. From ( I I) and ( 12) the torque at the front end of the axis is rvl = -
P,rRBsinJI,.-(P,t.~e-Pi). l-cos&, JIL
-sin
tiLt +).‘L&IL I
- {csRLP,
GLIL I
1
+r’L1-cos~I.
7
i
sin “LL TI jL I
GL
-I- rZLT2}kl.
(14)
We recall that PI = o,R(sin28+ycos20) - I/sin 19cos61(1 - y),
- L/(cos2e+ysinZe), ec0s e + LY~COS 8,
and Tz = wl(crsin28+pcos^9) When there is an integer
+af+sin
number
- (c,yRLP, +r’LT.?)k,,
8.
of windings
(16)
For simplicity the further respectively. analysis will be restricted to helices with integer number of windings. The sum of the external forces and the sum of the external moments must both be zero on a body moving freely at constant velocity. Due to the presence of the jL-component in the expression ( 16). ( IS) and ( 16) can both be zero only if P,. Pz. and T2 are all zero. But this is impossible since P, = 0 and P2 = 0 implies y = 0. while we have seen that y 2 &. We thus confirm Taylors ( 1952) conclusion that the organism must have a head to balance the torque on the helix. In that case thr: 3rganism may ot course have several flagella. Since the torque from the ‘speedometer wire’ rotation is unable to balance the torque from the rigid body rotation. we will simplify the analysis by assuming r2 % R’. in which case the contribution to the torque from the ‘speedometer wire’ rotation may be neglected Further, since the force and moment on the head induced by its motion through the liquir may be evaluated only for the simplesl geometries, we will consider heads of spheri Cal shape. PROPULSION
P, = w,RsinfIcos9(1--y)
T, = ( (Y- p)w,sin
Mj,.
BY SEVERAL
HELICES
We consider a microorganism with spherical head and n helical flagella. The above analysis of forces and torques is only valid fo helices in axial translation. and we therefore assume the flagella to be parallel. If they are constructed in the same way. we expect then to be equal. except that they may be either righthanded or lefthanded screws, with corres ponding direction of rotation. This is denotes
THE
HELIX
AS PROPELLER
by putting Wri = 2 WI,
7;
MICROORGANISXlS
From (3). using ( 19).
( -)
6i=T& $-f.
p+
U-L
I--Y
At low Keynolds numbers the translational drag on the head is approximately DU = 67r@ U. where A is the radius of the head and p is the coefficient of viscosity of the liquid. and the forces balance if c F-DUkt=O or,from(JS)and(3). 4
OF
tB+y)~$-+y(B+
1 I-$
=F 0. I
(20)
and thus n = h, with 111flagella rotating in each direction. Further. since j,. rotate5 with the helix, the transverse component in ( 16) can only be balanced by the similar component on another helix in the same rotation. with a phase difference t between the two helices. Thus m must also be an even number. m = Sp. and consequently II = 4p. The total rate of work by the organism on the surrounding liquid is
[c.vL{w,rRsin Hicos H,( J -y)
- U(coszBi+ysin’Oi)}] = nc,vL
[w,Rsin
2 =($)H,ad+ n($),,,,i,. (2’ )
-DU 6cos
0(
1 - y)
-U(cos2B+ysin’6)]-DU = nc,,rLP, - DU = 0.
When the head does not rotate (17)
= Du?
‘91)
( __
Head
(18)
Further. from (I ) and (3). With -&
01
.I
= B we then get (19)
=&J--[(B+~)~~B+(B+J)--&].
(%)H.,ir
= -1
VW l.
=ch.
The axial torque on the organism may balance either because the sum of the axial torques on the helices is zero, or because the head rotates. In the former case
(w,Rj;Uk).(P,j-P,k)6/ I 1. = c.,~L&, RP, - UP,) = csLRw,P,
-1
n
DU?.
123)
where we have used (18) in the last equality. Introducing (22) and (23) in (2 1) we get
If the number of lefthanded may be written
or, substituting (20).
c,vRL[(n-m)(w,R(sin’8+ycos~8) - Usin &OS e( 1 -y)} + ycosz8)
dr = nc,,,LRw, P, ,
helices is m, this
for w1 and P, from (19) and
- m{o,R(sinzB
- Usin Bcos @(I -?)}I
or
= 0 +(1+y)(B+I)fB+y)+y(B+1)2~
(n-2m)P,
= 0.
fk??@I ‘(24)
K. E. SCHREINEK
78
For a given rate of energy input the velocity will be maximal when 1 dw --3:-DU’ dr
+(l+y,(E+
operate at the optimal pitch angle given by (26). and the head does not rotate. E-’ is expressed by (27). which has been plotted in Fig. 2 as a function of B = D/ncvL for some values of y = crlc.~
1(25)
lJ(B+y)+y(B+l)‘$j
E”
is minimal. The optimal value of the pitch angle is then found, by putting
to be
e. = arc,,[ (z
\/;;)I”]-
(26)
With this (15) takes the form
(27) The optimal value of B = D/nc,,,L is found from (17). by putting
1
to be
value
of the pitch
x 800= 2. Since DU2 is the rate of ‘work needed by external forces to propel the head at the velocity U, E-
3
Fig. 2. Dimensionless nte of work E-l = ( I/D L:‘~(dw/dc for an organism with several halicnl propellers as a func tion of R = Dltq L for some virlues of y = cT/cv.
B,=$l with the corresponding angle
2
(&-$y’
is a measure of the efficiency of the helices in propelling the microorganism. If the helices
PROPULSION
BY ONE HELIX
If the sum of the axial torques on th helices does not cancel. the head must rotat if the total moment shall balance. But if th head rotates the several flagella must. if the are long enough, be wound up into one tail. I they are too short to be wound up. the a: sumptions of our analysis breaks down. bot because the flagella are no longer paralle and because of the proximity of the heat whose effect on the flow around the heiice has been ignored. In addition to the case of th organism wj.L several flagella and a nonrota
THE HELIX
AS PROPELLER
ing head. we are left with the case of the spher-
ical head driven by one helical tail. if the velocity and rotation of the head are vII and o,,. respectively. and we again neglect the small contribution from the Magnus effect, the force on the head is
OF XllCROORGANlShlS
Rotating joints are unknown in organisms. and the ‘speedometer wire’ rotation of the head and the tail must be numerically the same. Therefore. if k,‘, is the direction of the ‘speedometer wire’ rotation of the head. WH =
FH= - Dv,,
7’)
wI
k,, + W2k;,.
(28)
and the torque is
T,, = - TO,/.
(‘9)
If k;, makes an angle A with k,,. and the plane Of k;,, and k,, makes an angle K with the plane of j,, and kL, then the torque on the head may’ be written
D = 6npA and 7 = @pA;’ = $DA’, and A is again the radius of the head. If iI,. j,. k,, are unit vectors in radial. periwhere
pheral,
and
axial
direction
relative
= w,sj,, + Uk,, = - wIs sin &,iL -t o1s
cos
= Tw,sin
to the
cylinder of the helix in a coordinate system with origo at the centre of the head, and the centre is at a distance s from the helical axis, then vH
TH = - Tw,, = - T(o,k,, i o,k;,J
- T(w, +C+COS h)k,,.
M lotal=
Sin &iL
-
hsin
c.vLP2= DU.
(3 1)
which is equivalent with ( 181, with n = 1. o1 and P, are therefore also in the present
case given by the expressions where now B = DlcNL.
(19) and (201,
T(w, +w,cos A) ]kL = 0.
c,vRLP,tge* csRLP,+
Tqsin
A= 0
T(o, +C+COSA) = 0.
(33)
1%)
The configuration is then as indicated in Fig. 3. From (33) and (34) we get *=-
(c.vLP2 - DU)kL = 0.
and it follows that s = 0, i.e. the centre of the head is on the axis of the helix and thus r& is irrelevant, and
hcos K]j,_
It then follows that K = 0 or n. and
Do,s COS c&j,. +
AL
- [c,VRLP,+
(30)
F ,0(a,= Dw,s
T-sin
- [c,+.RLP,rg6,+ Tysin
FH = Dqs sin &iL - Dqs cos r#+,j,- DUkL. Since the total force on the organism must be zero we then have from (I 5) and (30)
(32)
Since the total torque must be zero. we then have from (16) and (32), with rz + R’.
(bHjL+ Ukr.
where & is the angle between iL at the front end of the helix and i,,. 1n the system [iL, jL,kL1 the force on the head is then
hsin f& - Tw,sin hcos Kj,_
c,RLP, + Tw, Tcos h
(35)
and tgA = +.
(36)
where the positive sign is used when K = 0. The rate of work done by the organism on the surrounding liquid is again given by
(37)
K. E. SCHREINER
while = -F,J!-T,,w,, Heiltl
= DU’
where we have used (IS) and (3 I), and (32). Further from (32). since K = O.or r. Wf = elk,, + wzk;, = 2 w,sin AjL + (w, + ~2cos A) k,., and thus drv = DU’+T(~,‘+2w,~cosh+~~~). ( clr >Heud Introducing(38) d,v dr=
(39)
and 09) in (37) we then have Tr+(yfw,
cash).
(40)
Using (35) and (36) this may, after some manipulation, be written dH’= 1 yvRLP, dt
Introducing then have
[ ( I+ rg2B)c,vRLP, + Tw,]. (41) w, and P, from ( 19) and (SO) we
Fig. 3. Sketch of the organism with spherical head and onehelical propeller.
From (23) and (34) we have = -DU=+
c,~RLw~P,
=-DL/z-
T(w,+o~&osh),
+2y(B+l)iB+y)+y”(B+I)~~.
HCllX
(38)
I (42)
The first term on the right side is equal to (24). with n= 1, and the second term thus repre-
THE
HELIX
AS PROPELLER
OF MICROORGANISMS
81
sents the additional rate of work because the head rotates. When the head is spherical T = $DA2, and the efficiency E of the single helix in propelling a spherical head is then expressed by 200 -
1 -=--
E
1
drta
DU? dt
150 -
+ (l+y)(B+I)(B+y)+y(B+l)~&
I
100
X (B+y)9gz8+2y(B+ [ +y’(B+
l$
tg2e
l)(B+y)
1.
(43)
As in the former case, with a given rate of energy input, the velocity is maximal when (43) is minimal. In the present case it is more complicated to use this to determine the optimal value of the pitch angle directly. Instead (42) was plotted as a function of B for some values of R/A and y, and for a range of values of 8 (the plots were drawn by a Control Data 3 300 computer), and from these sets the curves giving the lowest values of E-l were selected. The selected diagrams are shown in Figs. 4-6, where the corresponding optimal values of the pitch angle are indicated. DISCUSSION
AND CONCLUSIONS
Figures 2, 4, 5 and 6 show that E-l varies slowly with B = DlncNL in a wide neighbourhood of the optimal value of B. The total curvelength IL of the helix or helices and the frictional coefficient c,~are therefore not very critical parameters. The frictional coefficient may be written C‘V = k/E, where
BMVd4No.
l
depends on the geometry of the prob-
1-F
-
so -
F Fig. 4. Dimensionless rate of work E-’ = (l/DU%d~~/dr) for an organism with one helical propeller as a function of B = Dlc,J.. for some values of R/A. and the corresponding optimal values of 8. y = 0.50.
lem. For an infinite cylinder with external boundaries to the liquid parallel to the cylinder e is a function of the logaritm of the ratio between cylinder radius and the distance to the external wails, and for a finite cylinder in otherwise unbounded liquid Qis a function of the logaritm of the length to width ratio of the cylinder. And when an infinite cylinder moves in an unbounded liquid, Lamb (1932) has shown that 4 is a function of the logaritm of the cylinder Reynolds number, which in the present case is approximately 2rq Rplp, where p is the density of the liquid. Eis then at the most a function of the logaritm of the viscosity of the surrounding liquid. and since E-’ varies only slowly with B, the efficiency of the
K. E. SCHREINER
82
I.I.SA
0.w
lOO-
W-
I-
Fig. 5. Similar io Fig. 4. with y - 0-B.
propulsion system is effectively independent of the viscosity of the liquid. From (27) and (43) U - d/[ (E/D) (dwldr) 1, and the velocity of the microorganism is. with a constant energy input, inversely proportional to the square root of the coefficient of viscosity of the liquid it moves through. The variation of E with y is more pronounced. This indicates a defect in the analysis, since, as already mentioned, y will depend on the distance to ‘exterior’ boundaries to the liquid around each element on the helix. Thus small values of R, and possibly the several flagella, should give higher values of y. with correspondingly higher values of E-l and lower efficiency of the propulsion system. With only one helical propeller the efficiency
Fig. 6. Similarto Fig. 4. with y - O-60.
decreases rapidly with increasing ratio between helix and head radii R/A. This inincrease should be less pronounced if the head- as is often observed-is elongated in the direction of progression and flattened parallel to this direction. For then the ratio D/T should be lower, and by (42) this gives a lower value of ( l/DU*) (dwldr). For small values of the ratio R/A the efficiency will be lower, and E-l higher, than indicated in the diagrams, since the claim on the available energy input from the ‘speedometer wire’ rotation of the helix then can no longer be ignored. We have not considered how the front end of the helix is connected to the head. It is obvious, however, that also the motion of this part of the tail will make a claim on the energ)
THE
HELIX
AS PROPELLER
budget, probably without a comparable contribution to the propulsion of the system. Saire for the assumption that the head is spherical, all our assumptions and simplifications, including the selection of optimal values of the pitch angle, thus tend to give too favourable values for the efficiency of the propulsion system. If the radius of the head of the microorganism is I pm, and it moves through water, say, where I_L- 10e2, with a velocity U= 100 pmlsec - values that are in the observed range -the head Reynolds number is Re Head= 2A
P
-
2.
10-4,
while, with I - iR, (r2 Q R*) the Reynolds number of the helix is ZrRo,p ReHe,ix - I-(
From(l9),withB
2 R’o,p - - 3 cc
-
- 1 andy=f,
OF MICROORGANiSMS
83
this interior deformation should change with time, to support the helical motion and the resulting forces and moments from the exterior. This will give restrictions on the parameters of the helix. Thus the strains in the flagellum are proportional to the ratio between the radius of the flagellum and the radius of curvature in the helix, and there is an obvious relation between this radius of curvature, and the radius and the pitch angle of the helix. Further, the necessary change of deformation in the flagellum cannot be performed without energy losses. The efficiency of the helix as propeller for microorganisms can only be determined when both the external and the internal mechanics are taken into account. The external mechanics of the system seem now to be fairly well understood. It may be hoped that the results of the present study can be used in establishing some of the boundary conditions needed to solve the problems of the internal mechanics of the system, and thus lead to a more complete understanding of the mechanics and physics of the helical propulsion system.
RUl - 7u, and taking R - A - 1 pm, then Re
Helix
‘-
5.10-d.
The Reynolds numbers are thus small, as expected. With E-l = 50, say, the rate at which energy is transmitted to the liquid in the above motion is in the order of dw dr = ikr@ L/? X SO - IO-’ erg/set, while, according to Rotschild (1961). the available energy input rate is an order of magnitude higher. In the ‘present study we have, however, only considered the exterior mechanics of the system. Actually, this cannot be separated from the interior mechanics, that is how the flagellum should be constructed, how it should deform in the interior, and how
REFERENCES Gray, J. and Hancock, C. J. (1955) The propulsion of sea-urchin spermatozoa. 1. erp. Biol. 32.802-8 14. Hancock. G. J. (1953) The self-propulsion of microscopic organisms through liquids. Proc. R. Sot. Land. A217. 96-121. Happel,J. and Brenner, H. (I 965) Lobt*Reynolds number Hydro-dynamics. Prentice-Hall, Englewood Cliffs. NJ. Holwill. M. E. J. and Burge. R. E. (1963) A hydrodynamic study of the motility of flagellated bacteria, Archs Biochem. Biophys. 101.249-260. Holwill. M. E. J. (1966) Physical aspects of flagellar movement. Physiol. Reo. 46.696-785. Kaplun, S. (1957) Low Reynolds number flow past a circular cylinder. 1. Marh. Mech. 6.595-603. Lamb, H. (1932) Hydrodynamics. 6th Edn. p. 614. Cambridge Universities Press, Cambridge. Proudman. I. and Pearson. J. R. A. (1957) Expansion at small Reynolds numbers for the flow past a sphere and a circular cylinder. J. Flnid Mech. 2.237-262. Rotschild. Lord. (1961) Sperm energetics. In The C&l and the Organism. (Edited by J. A. Ramsay and V. 8. Wigglesworth). p. 9. Cambridge Universities Press, Cambridge. Taylor, G. 1. (1952) ihe action of waving cylindrical tails in propelling microscopic organisms. Proc. R. Sac. Land. A211.225-239.
HELIX AS PROPELLER MICROORGANISMS’” KKISTIAK
Department of Mathematics.
OF
E. SCHREINER University of Oslo, 040
3. Norua!
Abstract- A phenomenological approach is employed in determining the forces and moments on a rotating helix by the resistance of the surrounding liquid a~ low Reynolds numbers. and it i\ shown how they may be balanced by the forces and moments on a spherical head. \Vithin the restrictions of the analysis it is found that if the head rotates it can only have one helical propeller. But if the number of flagella is In. half of them rotating in each direction. the head will not rotate. The r;LIe of energy transmission to the surrounding liquid is evaluated in each case: it is found IO be lower in the latter case. and in the former case it increases with the square of the ratio between helix and head radii. The efficiency of the propulsion system is found to be practically independent of the viscosity of the surrounding liquid. INTRODUCTION
were able to treat the more realistic case when a sinusoidal wave of finite amplitude progresses backwards in the flagellum. A similar approach was employed by Holwill and Burge (1963) for the case of a rotating helical flagellum. Both Taylor and Holwill 6: Burge showed that the rotation of the helical flagellum produces an axial thrust that can compensate for the drag on the head of the microorganism. Further. Taylor showed that the rotation of the helix produces a torque around the axis that must be balanced. In the present paper we will employ a phenomenological approach to study the feasibility of rotating helical flagella as propellers of microorganisms. and we will consider some ways the axial torque can be balanced by the presence of the head. It turns out that roration claims a major. part of the rate of work needed to propel the organism.
THE PROPULSION of microorganisms by flagellar motion is easily observed in the microscope. and it has therefore been studied in the hope that it might throw some light on the life functions of the cell. (For a review of the literature see for instance Holwill ( 19661.) The success of this approach depends on our knowledge about the mechanics of flagellar propulsion. But this again depends on the understanding of the hydrodynamics of the translation of bodies at low Reynolds numbers. This is. however, a complicated problem. and analytical solutions to such motions have only been found for bodies of the simplest geometries. Thus Kaplun (1957) has found the solution for the flow past a circular cylinder, and Proudman and Pearson (1957) the solution for the flow past a sphere. Whatever understanding we have about the swimming of microorganisms has therefore been reached using approximate methods. Thus Taylor (I 952) was able to treat only the cases of a long circular cylinder bent into a sinusoidal wave or a helix. when the amplitude of the wave was in the order of the diameter of the cylinder. By employing a phenomenological approach, Gray and Hancock ( 19_(j)
FORCESAND MOMEh’J’SON THE ROTATING HELIX We consider a circular cylindrical helix of radius R and pitch angle 8. Fig. I(a). moving in axial direction with Velocity U. The radius of the flagellum is r, and its curve length is f. R, 8, and r are all assumed constant along the helix. The helix rotates about the axis with an
*Received 5 May 1970. 73
K. E. SCHKEINER
are denoted cT and c.~ respectively, and we put cT = yc..,,. Hancock ( 1953) has determined cr and c,~ for infinite straight cylinders in unbounded fluid, and he found y = 4 in this case. The presence of external boundaries to the fluid will influence the drug on cylinders, however,see Happel and Brenner ( 1965). Chap. 7. In the present cahe the cylinder. i.e. the flagellum - is coiled into a helix of finite length. The other parts of the helix will then act as external boundaries to the fluid aroundeach element, as will the head and the eventual other flagella of the organism. No attempt to determine the proper values of cs, cT. and y for the present problem will be made here: we only note that the values determined by Hancock should represent lower bounds. To simplify the analysis we assume that cs and cr are constant along the helix. The force on the element of length 61 from its translation (I) through the surrounding liquid is then element
Fig. I. Sketch of the helix. with notutions used in evaluating forces and moments on the element M. u. und on the helix. b.
6F=-
angular velocity wt. and the flagellum has a ‘speedometer wire’ rotation of angular velocity 0s. The ve\ocity of an element of the filament is then v=o,Rj+Uk=
o’j’+,c*‘k’.
- cr( o,Rsin
B-6 Usin 8)j’ 8-
Ucos 8) k’)
= - c,~N{ [w,R(sin2B+ycos28) - Usin Bcos 6(, I - y)]j
(I)
where i, j, k are unit vectors in radial. peripheral and axial directions respectively. relative to the cylinder of the helix, and j’ and k’ are unit vectors in a coordinate system rotated an angle 8 about the i-axis. j’ is thus along the element, and k’ normal to it. Further, the element has a rotation that is the sum of the rigid body rotation and the ‘speedometer wire’ rotation,
o=c++a_=w,k+c+j' = (wlsin 8 + u2) j’ + wlcos Ok’.
+ c,VllS’k’)
=-61{cr(~,Rcos
(~,Rcos8+UsinB)j’
-(w,RsinO-Ucos$)k’=
Sl(c+‘j’
(2)
The frictional coefficiiznts per unit length for motion along and normal to the axis of the
- [w,Rsin
8~0s B( 1 -y)
- U(cosz8+ysin28)]k} = - c,,3l{P, j - P,k}.
(3
The forces due to the ‘Magnus effect’ on the rotating element are of the order of the Reynolds number times the forces due to the translation, and they are therefore neglected since the Reynolds number is expected to bc small. The torque on the element due to the rotation (2) is SM,=-
61{4(o,sin
8+w.,)j’+przo,cos
Ok’} (4
THE
HELlX
AS PROPELLER
where the j’-component is due to the rotation about the axis of the cylindrical element. the k’-component is due to the nonuniformity of the longitudinal velocity of the surrounding liquid relative to the element. and cuand p are the frictional coefficients per unit length for the two motions. The element of force (3) gives an element of torque at a point on the helical axis )6&I,; = /d x 6F].
OF
75
MICROORGANISMS
where we have used 16) in the last equality. The element of torque at 0 from this element offorce is by (5). (7) and (8) dM, = -s
dl!,[-- cos t/A, i sin t/lj, A t!r/y~k~] [P,sin rljil + P&OS Jjj, - P,k,]
x
cvR’
= ~0s
dlb{ [P,r,@bcos nili P&n c//Ii,
- [P,fg&lr sin II/ - P,cos tL]j, + P,k,}.
(5)
(9) where IdI is the distance between the element and the point on the axis. Consider the part of the helix between the rear end and an element at curve length I, from this end, Fig. I(b). The station 1, may equivalently be characterized by the angular distance from the rear end
CD,=
Since R. 0. y. and c,, are.all assumed constant along the helix, (8) and (9) may be integrated directly, giving the force on the rear part of the helix. and its contribution to the torque at 0. Thus F, = 2
a,,++?
{-
[ 1 - cos t1j,,]P,i, - sin (JOP,j, + clr,,P,k,}
Similarily the angular distance to the element at station I is given by @=
q,+!y.
( 10)
and M, = - 5
{ [ P,r.@&,sin I/J,, 1 -cosJ,,)]i,
(P,fp9-fp,)(
and the angular distance between I and I, is thus (6) The radius vector of station 1 relative to the point 0 on the helical axis on the normal from station I, is then d =-R[-cosIG,+~~~I_IJ~,+IIJ~~~~,],
(7)
where [il, j,, k,] is a coordinate system on the axis parallel to the [i, j. k] system at station I,. 1n this system the force (3) at station 1 is
1,cos 8 where we have put = ch.
R
In the coordinate system [i,. j,. k,] at 0 the torque (4) due to the rotation (2) is 6M, = -Gl{crr2(w,sin + /3r%,cos
ok’}
= -61([(cY-_)r%,sin + ~*f+cos
ecos 8
81j
+ [r2w1 (asin’6,+/3cos’8~ + c&.&n
dF = - cNdl{ P,sin llri, -t P,cos $jI - P 2kl}
19+o?)j’
01k}
= - r%1{ Tj + T,k} = s
dJI{f ,sin Ilti, + f ,cos
CLj, - f2kl},
(8)
= -rW{
T,sin +i, + T,COS $j, + T,k,}.
K. E. SCHREINEK
76
Using (6) we integrate this over the helix behind station I,. to find the torque at 0 from the ‘speedometer wire’ rotation of this part of the helix
on the helix lb,. = L cos B/R = X-w, and ( 13) and ( 14) take the particularly simple form F = c.yLP,k,.
(15’)
and &I = - +RLP, (12)
-r2f,T2k,.
I, cos
whereagain&,=
R
e.
Putting JI,. = L cos B/R we then have from ( IO) that the total force on the helix is F =
csL,
-
*
-r
-y
‘L pliL
PJ,
-i- I, P,kL
(13) where [i,.j,, kL] is the [i,. jr, k,] system at the front end. From ( I I) and ( 12) the torque at the front end of the axis is rvl = -
P,rRBsinJI,.-(P,t.~e-Pi). l-cos&, JIL
-sin
tiLt +).‘L&IL I
- {csRLP,
GLIL I
1
+r’L1-cos~I.
7
i
sin “LL TI jL I
GL
-I- rZLT2}kl.
(14)
We recall that PI = o,R(sin28+ycos20) - I/sin 19cos61(1 - y),
- L/(cos2e+ysinZe), ec0s e + LY~COS 8,
and Tz = wl(crsin28+pcos^9) When there is an integer
+af+sin
number
- (c,yRLP, +r’LT.?)k,,
8.
of windings
(16)
For simplicity the further respectively. analysis will be restricted to helices with integer number of windings. The sum of the external forces and the sum of the external moments must both be zero on a body moving freely at constant velocity. Due to the presence of the jL-component in the expression ( 16). ( IS) and ( 16) can both be zero only if P,. Pz. and T2 are all zero. But this is impossible since P, = 0 and P2 = 0 implies y = 0. while we have seen that y 2 &. We thus confirm Taylors ( 1952) conclusion that the organism must have a head to balance the torque on the helix. In that case thr: 3rganism may ot course have several flagella. Since the torque from the ‘speedometer wire’ rotation is unable to balance the torque from the rigid body rotation. we will simplify the analysis by assuming r2 % R’. in which case the contribution to the torque from the ‘speedometer wire’ rotation may be neglected Further, since the force and moment on the head induced by its motion through the liquir may be evaluated only for the simplesl geometries, we will consider heads of spheri Cal shape. PROPULSION
P, = w,RsinfIcos9(1--y)
T, = ( (Y- p)w,sin
Mj,.
BY SEVERAL
HELICES
We consider a microorganism with spherical head and n helical flagella. The above analysis of forces and torques is only valid fo helices in axial translation. and we therefore assume the flagella to be parallel. If they are constructed in the same way. we expect then to be equal. except that they may be either righthanded or lefthanded screws, with corres ponding direction of rotation. This is denotes
THE
HELIX
AS PROPELLER
by putting Wri = 2 WI,
7;
MICROORGANISXlS
From (3). using ( 19).
( -)
6i=T& $-f.
p+
U-L
I--Y
At low Keynolds numbers the translational drag on the head is approximately DU = 67r@ U. where A is the radius of the head and p is the coefficient of viscosity of the liquid. and the forces balance if c F-DUkt=O or,from(JS)and(3). 4
OF
tB+y)~$-+y(B+
1 I-$
=F 0. I
(20)
and thus n = h, with 111flagella rotating in each direction. Further. since j,. rotate5 with the helix, the transverse component in ( 16) can only be balanced by the similar component on another helix in the same rotation. with a phase difference t between the two helices. Thus m must also be an even number. m = Sp. and consequently II = 4p. The total rate of work by the organism on the surrounding liquid is
[c.vL{w,rRsin Hicos H,( J -y)
- U(coszBi+ysin’Oi)}] = nc,vL
[w,Rsin
2 =($)H,ad+ n($),,,,i,. (2’ )
-DU 6cos
0(
1 - y)
-U(cos2B+ysin’6)]-DU = nc,,rLP, - DU = 0.
When the head does not rotate (17)
= Du?
‘91)
( __
Head
(18)
Further. from (I ) and (3). With -&
01
.I
= B we then get (19)
=&J--[(B+~)~~B+(B+J)--&].
(%)H.,ir
= -1
VW l.
=ch.
The axial torque on the organism may balance either because the sum of the axial torques on the helices is zero, or because the head rotates. In the former case
(w,Rj;Uk).(P,j-P,k)6/ I 1. = c.,~L&, RP, - UP,) = csLRw,P,
-1
n
DU?.
123)
where we have used (18) in the last equality. Introducing (22) and (23) in (2 1) we get
If the number of lefthanded may be written
or, substituting (20).
c,vRL[(n-m)(w,R(sin’8+ycos~8) - Usin &OS e( 1 -y)} + ycosz8)
dr = nc,,,LRw, P, ,
helices is m, this
for w1 and P, from (19) and
- m{o,R(sinzB
- Usin Bcos @(I -?)}I
or
= 0 +(1+y)(B+I)fB+y)+y(B+1)2~
(n-2m)P,
= 0.
fk??@I ‘(24)
K. E. SCHREINEK
78
For a given rate of energy input the velocity will be maximal when 1 dw --3:-DU’ dr
+(l+y,(E+
operate at the optimal pitch angle given by (26). and the head does not rotate. E-’ is expressed by (27). which has been plotted in Fig. 2 as a function of B = D/ncvL for some values of y = crlc.~
1(25)
lJ(B+y)+y(B+l)‘$j
E”
is minimal. The optimal value of the pitch angle is then found, by putting
to be
e. = arc,,[ (z
\/;;)I”]-
(26)
With this (15) takes the form
(27) The optimal value of B = D/nc,,,L is found from (17). by putting
1
to be
value
of the pitch
x 800= 2. Since DU2 is the rate of ‘work needed by external forces to propel the head at the velocity U, E-
3
Fig. 2. Dimensionless nte of work E-l = ( I/D L:‘~(dw/dc for an organism with several halicnl propellers as a func tion of R = Dltq L for some virlues of y = cT/cv.
B,=$l with the corresponding angle
2
(&-$y’
is a measure of the efficiency of the helices in propelling the microorganism. If the helices
PROPULSION
BY ONE HELIX
If the sum of the axial torques on th helices does not cancel. the head must rotat if the total moment shall balance. But if th head rotates the several flagella must. if the are long enough, be wound up into one tail. I they are too short to be wound up. the a: sumptions of our analysis breaks down. bot because the flagella are no longer paralle and because of the proximity of the heat whose effect on the flow around the heiice has been ignored. In addition to the case of th organism wj.L several flagella and a nonrota
THE HELIX
AS PROPELLER
ing head. we are left with the case of the spher-
ical head driven by one helical tail. if the velocity and rotation of the head are vII and o,,. respectively. and we again neglect the small contribution from the Magnus effect, the force on the head is
OF XllCROORGANlShlS
Rotating joints are unknown in organisms. and the ‘speedometer wire’ rotation of the head and the tail must be numerically the same. Therefore. if k,‘, is the direction of the ‘speedometer wire’ rotation of the head. WH =
FH= - Dv,,
7’)
wI
k,, + W2k;,.
(28)
and the torque is
T,, = - TO,/.
(‘9)
If k;, makes an angle A with k,,. and the plane Of k;,, and k,, makes an angle K with the plane of j,, and kL, then the torque on the head may’ be written
D = 6npA and 7 = @pA;’ = $DA’, and A is again the radius of the head. If iI,. j,. k,, are unit vectors in radial. periwhere
pheral,
and
axial
direction
relative
= w,sj,, + Uk,, = - wIs sin &,iL -t o1s
cos
= Tw,sin
to the
cylinder of the helix in a coordinate system with origo at the centre of the head, and the centre is at a distance s from the helical axis, then vH
TH = - Tw,, = - T(o,k,, i o,k;,J
- T(w, +C+COS h)k,,.
M lotal=
Sin &iL
-
hsin
c.vLP2= DU.
(3 1)
which is equivalent with ( 181, with n = 1. o1 and P, are therefore also in the present
case given by the expressions where now B = DlcNL.
(19) and (201,
T(w, +w,cos A) ]kL = 0.
c,vRLP,tge* csRLP,+
Tqsin
A= 0
T(o, +C+COSA) = 0.
(33)
1%)
The configuration is then as indicated in Fig. 3. From (33) and (34) we get *=-
(c.vLP2 - DU)kL = 0.
and it follows that s = 0, i.e. the centre of the head is on the axis of the helix and thus r& is irrelevant, and
hcos K]j,_
It then follows that K = 0 or n. and
Do,s COS c&j,. +
AL
- [c,VRLP,+
(30)
F ,0(a,= Dw,s
T-sin
- [c,+.RLP,rg6,+ Tysin
FH = Dqs sin &iL - Dqs cos r#+,j,- DUkL. Since the total force on the organism must be zero we then have from (I 5) and (30)
(32)
Since the total torque must be zero. we then have from (16) and (32), with rz + R’.
(bHjL+ Ukr.
where & is the angle between iL at the front end of the helix and i,,. 1n the system [iL, jL,kL1 the force on the head is then
hsin f& - Tw,sin hcos Kj,_
c,RLP, + Tw, Tcos h
(35)
and tgA = +.
(36)
where the positive sign is used when K = 0. The rate of work done by the organism on the surrounding liquid is again given by
(37)
K. E. SCHREINER
while = -F,J!-T,,w,, Heiltl
= DU’
where we have used (IS) and (3 I), and (32). Further from (32). since K = O.or r. Wf = elk,, + wzk;, = 2 w,sin AjL + (w, + ~2cos A) k,., and thus drv = DU’+T(~,‘+2w,~cosh+~~~). ( clr >Heud Introducing(38) d,v dr=
(39)
and 09) in (37) we then have Tr+(yfw,
cash).
(40)
Using (35) and (36) this may, after some manipulation, be written dH’= 1 yvRLP, dt
Introducing then have
[ ( I+ rg2B)c,vRLP, + Tw,]. (41) w, and P, from ( 19) and (SO) we
Fig. 3. Sketch of the organism with spherical head and onehelical propeller.
From (23) and (34) we have = -DU=+
c,~RLw~P,
=-DL/z-
T(w,+o~&osh),
+2y(B+l)iB+y)+y”(B+I)~~.
HCllX
(38)
I (42)
The first term on the right side is equal to (24). with n= 1, and the second term thus repre-
THE
HELIX
AS PROPELLER
OF MICROORGANISMS
81
sents the additional rate of work because the head rotates. When the head is spherical T = $DA2, and the efficiency E of the single helix in propelling a spherical head is then expressed by 200 -
1 -=--
E
1
drta
DU? dt
150 -
+ (l+y)(B+I)(B+y)+y(B+l)~&
I
100
X (B+y)9gz8+2y(B+ [ +y’(B+
l$
tg2e
l)(B+y)
1.
(43)
As in the former case, with a given rate of energy input, the velocity is maximal when (43) is minimal. In the present case it is more complicated to use this to determine the optimal value of the pitch angle directly. Instead (42) was plotted as a function of B for some values of R/A and y, and for a range of values of 8 (the plots were drawn by a Control Data 3 300 computer), and from these sets the curves giving the lowest values of E-l were selected. The selected diagrams are shown in Figs. 4-6, where the corresponding optimal values of the pitch angle are indicated. DISCUSSION
AND CONCLUSIONS
Figures 2, 4, 5 and 6 show that E-l varies slowly with B = DlncNL in a wide neighbourhood of the optimal value of B. The total curvelength IL of the helix or helices and the frictional coefficient c,~are therefore not very critical parameters. The frictional coefficient may be written C‘V = k/E, where
BMVd4No.
l
depends on the geometry of the prob-
1-F
-
so -
F Fig. 4. Dimensionless rate of work E-’ = (l/DU%d~~/dr) for an organism with one helical propeller as a function of B = Dlc,J.. for some values of R/A. and the corresponding optimal values of 8. y = 0.50.
lem. For an infinite cylinder with external boundaries to the liquid parallel to the cylinder e is a function of the logaritm of the ratio between cylinder radius and the distance to the external wails, and for a finite cylinder in otherwise unbounded liquid Qis a function of the logaritm of the length to width ratio of the cylinder. And when an infinite cylinder moves in an unbounded liquid, Lamb (1932) has shown that 4 is a function of the logaritm of the cylinder Reynolds number, which in the present case is approximately 2rq Rplp, where p is the density of the liquid. Eis then at the most a function of the logaritm of the viscosity of the surrounding liquid. and since E-’ varies only slowly with B, the efficiency of the
K. E. SCHREINER
82
I.I.SA
0.w
lOO-
W-
I-
Fig. 5. Similar io Fig. 4. with y - 0-B.
propulsion system is effectively independent of the viscosity of the liquid. From (27) and (43) U - d/[ (E/D) (dwldr) 1, and the velocity of the microorganism is. with a constant energy input, inversely proportional to the square root of the coefficient of viscosity of the liquid it moves through. The variation of E with y is more pronounced. This indicates a defect in the analysis, since, as already mentioned, y will depend on the distance to ‘exterior’ boundaries to the liquid around each element on the helix. Thus small values of R, and possibly the several flagella, should give higher values of y. with correspondingly higher values of E-l and lower efficiency of the propulsion system. With only one helical propeller the efficiency
Fig. 6. Similarto Fig. 4. with y - O-60.
decreases rapidly with increasing ratio between helix and head radii R/A. This inincrease should be less pronounced if the head- as is often observed-is elongated in the direction of progression and flattened parallel to this direction. For then the ratio D/T should be lower, and by (42) this gives a lower value of ( l/DU*) (dwldr). For small values of the ratio R/A the efficiency will be lower, and E-l higher, than indicated in the diagrams, since the claim on the available energy input from the ‘speedometer wire’ rotation of the helix then can no longer be ignored. We have not considered how the front end of the helix is connected to the head. It is obvious, however, that also the motion of this part of the tail will make a claim on the energ)
THE
HELIX
AS PROPELLER
budget, probably without a comparable contribution to the propulsion of the system. Saire for the assumption that the head is spherical, all our assumptions and simplifications, including the selection of optimal values of the pitch angle, thus tend to give too favourable values for the efficiency of the propulsion system. If the radius of the head of the microorganism is I pm, and it moves through water, say, where I_L- 10e2, with a velocity U= 100 pmlsec - values that are in the observed range -the head Reynolds number is Re Head= 2A
P
-
2.
10-4,
while, with I - iR, (r2 Q R*) the Reynolds number of the helix is ZrRo,p ReHe,ix - I-(
From(l9),withB
2 R’o,p - - 3 cc
-
- 1 andy=f,
OF MICROORGANiSMS
83
this interior deformation should change with time, to support the helical motion and the resulting forces and moments from the exterior. This will give restrictions on the parameters of the helix. Thus the strains in the flagellum are proportional to the ratio between the radius of the flagellum and the radius of curvature in the helix, and there is an obvious relation between this radius of curvature, and the radius and the pitch angle of the helix. Further, the necessary change of deformation in the flagellum cannot be performed without energy losses. The efficiency of the helix as propeller for microorganisms can only be determined when both the external and the internal mechanics are taken into account. The external mechanics of the system seem now to be fairly well understood. It may be hoped that the results of the present study can be used in establishing some of the boundary conditions needed to solve the problems of the internal mechanics of the system, and thus lead to a more complete understanding of the mechanics and physics of the helical propulsion system.
RUl - 7u, and taking R - A - 1 pm, then Re
Helix
‘-
5.10-d.
The Reynolds numbers are thus small, as expected. With E-l = 50, say, the rate at which energy is transmitted to the liquid in the above motion is in the order of dw dr = ikr@ L/? X SO - IO-’ erg/set, while, according to Rotschild (1961). the available energy input rate is an order of magnitude higher. In the ‘present study we have, however, only considered the exterior mechanics of the system. Actually, this cannot be separated from the interior mechanics, that is how the flagellum should be constructed, how it should deform in the interior, and how
REFERENCES Gray, J. and Hancock, C. J. (1955) The propulsion of sea-urchin spermatozoa. 1. erp. Biol. 32.802-8 14. Hancock. G. J. (1953) The self-propulsion of microscopic organisms through liquids. Proc. R. Sot. Land. A217. 96-121. Happel,J. and Brenner, H. (I 965) Lobt*Reynolds number Hydro-dynamics. Prentice-Hall, Englewood Cliffs. NJ. Holwill. M. E. J. and Burge. R. E. (1963) A hydrodynamic study of the motility of flagellated bacteria, Archs Biochem. Biophys. 101.249-260. Holwill. M. E. J. (1966) Physical aspects of flagellar movement. Physiol. Reo. 46.696-785. Kaplun, S. (1957) Low Reynolds number flow past a circular cylinder. 1. Marh. Mech. 6.595-603. Lamb, H. (1932) Hydrodynamics. 6th Edn. p. 614. Cambridge Universities Press, Cambridge. Proudman. I. and Pearson. J. R. A. (1957) Expansion at small Reynolds numbers for the flow past a sphere and a circular cylinder. J. Flnid Mech. 2.237-262. Rotschild. Lord. (1961) Sperm energetics. In The C&l and the Organism. (Edited by J. A. Ramsay and V. 8. Wigglesworth). p. 9. Cambridge Universities Press, Cambridge. Taylor, G. 1. (1952) ihe action of waving cylindrical tails in propelling microscopic organisms. Proc. R. Sac. Land. A211.225-239.