hydrodynamics of the double-wave structure of insect spermatozoa
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doi: 10.1098/rsif.2011.0841, 1908-1924 first published online 1 February 20129 2012 J. R. Soc. Interface
On Shun Pak, Saverio E. Spagnolie and Eric Lauga spermatozoa flagellaHydrodynamics of the double-wave structure of insect
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J. R. Soc. Interface (2012) 9, 1908–1924
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*Author for c
doi:10.1098/rsif.2011.0841Published online 1 February 2012
Received 1 DAccepted 4 Ja
Hydrodynamics of the double-wavestructure of insect spermatozoa flagella
On Shun Pak, Saverio E. Spagnolie and Eric Lauga*
Department of Mechanical and Aerospace Engineering, University of California San Diego,9500 Gilman Drive, La Jolla, CA 92093-0411, USA
In addition to conventional planar and helical flagellar waves, insect sperm flagella have alsobeen observed to display a double-wave structure characterized by the presence of two super-imposed helical waves. In this paper, we present a hydrodynamic investigation of thelocomotion of insect spermatozoa exhibiting the double-wave structure, idealized here assuperhelical waves. Resolving the hydrodynamic interactions with a non-local slender bodytheory, we predict the swimming kinematics of these superhelical swimmers based on exper-imentally collected geometric and kinematic data. Our consideration provides insight into therelative contributions of the major and minor helical waves to swimming; namely, propulsionis owing primarily to the minor wave, with negligible contribution from the major wave. Wealso explore the dependence of the propulsion speed on geometric and kinematic parameters,revealing counterintuitive results, particularly for the case when the minor and major helicalstructures are of opposite chirality.
Keywords: swimming micro-organisms; flagellar hydrodynamics;insect sperm motility
1. INTRODUCTION
Locomotion in fluids is ubiquitous in nature, withexamples spanning a wide range in size from bacterialmotility to the swimming of whales. It plays fundamen-tal roles throughout the lives of animals in suchendeavours as predation and finding a mate for repro-duction [1]. Lying along the interface between biologyand fluid dynamics, biological locomotion at smallscales has received substantial attention from biologistsand engineers in recent years [2,3].
The physics governing locomotion in fluids is verydifferent for microscopic organisms (e.g. bacteria, sper-matozoa) and macroscopic organisms (e.g. fishes,humans). The dramatic difference is because of the com-petition between inertial and viscous effects in the fluidmedium. The Reynolds number, Re ¼ UL/n (with Uand L characteristic velocity and length scales, and n
the kinematic viscosity) is a dimensionless parameterthat measures the relative importance of the inertialforces to viscous forces in a fluid. Locomotion of largeranimals in fluids takes place at moderate to largeReynolds numbers, where inertial forces dominate. Atthis scale, swimming and flying are generally accom-plished by imparting momentum into the fluid oppositeto the direction of locomotion. Micro-organisms mean-while inhabit in a world of low Reynolds numbers,where inertia plays a negligible role and viscous damp-ing is paramount. The Reynolds number ranges from1026 for bacteria to 1022 for spermatozoa [4]. The
orrespondence ([email protected]).
ecember 2011nuary 2012 1908
absence of inertia imposes stringent constraints on amicro-organism’s locomotive capabilities.
Many micro-organisms propel themselves by propa-gating travelling waves along one or many slenderflagella [4]. The motility features of these flagelladepend on the cell type, either prokaryotic (cells with-out a nucleus) or eukaryotic (cells with nuclei). Theflagella of prokaryotic bacteria, such as those used byEscherichia coli, are helical in shape and are passivelyrotated at their base by a motor embedded in the cellwall. The rotation propagates an apparent helicalwave from the sperm head to the distal end of the flagel-lum, propelling the cell in the opposite direction.Eukaryotic flagella exhibit a different internal struc-ture, called an axoneme, which is composed ofmicrotubules, proteins and protein complexes such asdynein molecular motors. The dynein arms convertchemical energy contained in adenosine triphosphateinto mechanical energy, inducing active relative slidingbetween the microtubules, which in turn leads to bend-ing deformations that propagate along the flagellum.A common structure of the axoneme has a ring ofnine microtubule doublets spaced around the circum-ference and two additional central microtubules (theso-called 9 þ 2 axoneme). Other variations of theaxonemal structure have also been observed [5].
Generally, three levels of complexity in undulatorybeat patterns are observed in eukaryotic flagella [5,6],following a hierarchy in the structure of the axoneme:(i) the lowest in the hierarchy is a simple planar beatingpattern, as in human and sea-urchin spermatozoa fla-gella (figure 1a), with the common 9 þ 2 axonemestructure; (ii) a more complicated three-dimensional
This journal is q 2012 The Royal Society
(a)
(b)
(c)
Figure 1. A hierarchy of the complexity of flagellar beat-ing pattern observed in eukaryotic cells. (a) Planar-wavepattern in sea-urchin spermatozoa flagella [7]; (b) helical-wave pattern in Gryllotalpa gryllotalpa [6]; and (c) double-wave pattern in Haematopinus suis [6]. All images werereproduced with permission: (a) from Rikmenspoel & Isles[7]. Copyright 1985 Elsevier; (b,c) from Baccetti [6].Copyright 1972 Elsevier.
Hydrodynamics of insect spermatozoa O. S. Pak et al. 1909
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helical beating pattern is exhibited by some insect sper-matozoa with a 9 þ 9 þ 2 axoneme, as in Gryllotalpagryllotalpa (figure 1b); and (iii) the highest level of com-plexity is a double-wave pattern observed in some insectspermatozoa with a 9 þ 9 þ 2 axoneme and accessorybodies endowed with ATPase activity, as in Haematopi-nus suis (figure 1c). A vast diversity in sperm structureis found in insects [8], and the hierarchy described isalso observed even just within the realm of insect sper-matozoa flagella [5,6]. Figure 2 shows a schematic of apterygote insect flagellosperm and its ultrastructure(reproduced with permission from Werner & Simmons[5]). Similar to the 9 þ 2 axoneme observed in ciliaand flagella of many plant and animal cells, the centralcore of the insect sperm axoneme is composed of twocentral microtubules surrounded by a ring of ninemicrotubule doublets. However, the ring of nine micro-tubule doublets is surrounded by another nine accessorytubules, forming the characteristic 9 þ 9 þ 2 arrange-ment of the insect sperm axoneme. In addition to themore complicated microtubule arrangement, two pro-minent features of inset spermatozoa flagella are themitochondrial derivatives and accessory bodies runningalong the axoneme (see the study of Werner & Simmons[5] for a thorough review of insect sperm structure).
J. R. Soc. Interface (2012)
Although the structure of many different spermatozoahas been examined, the rapid and divergent evolution insperm morphology is not well understood [5,9]. Hydro-dynamic considerations of the relationship betweenflagellar morphology and functional parameters suchas the swimming speed may provide useful informationfor explaining the evolutionary divergence. Because ofits intricate nature, the double-wave structure is lesswell-explored than wave types sitting lower in the hierar-chy. Relevant studies on the planar and helical wavestructures are abundant and well-developed (see the clas-sical and recent reviews [2–4]), but we are not aware ofany hydrodynamic studies dedicated to the double-wave structure. Here, we present a hydrodynamic studyon the motility of insect spermatozoa exhibiting adouble-wave beat pattern.
The double-wave pattern is characterized by the sim-ultaneous presence of two kinds of waves—a minorwave with small amplitude and high frequency superim-posed on a major helical wave of large amplitude andlow frequency. The minor wave has also been observedto be approximately helical [10], and the combinedactivity of the two is described as a double-helical beat-ing pattern [5]. The double-wave structure was firstobserved in Tenebrio molitor and Bacillus rossius byBaccetti et al. [11,12], and was also later found inLygaeus [13], Culicoides melleus [14], Aedes notoscrip-tus [15], Ceratitis capitata [16], Drosophila obscura[17], Megaselia scalaris [18], and more recently in Aleo-chara curtula [10] and Drusilla canaliculata [19].Figure 3 compiles a collection of images of spermatozoaexhibiting the double-wave structure. Werner &Simmons [5] have presented a thorough review of thiscomplex structure in insect spermatozoa.
Most studies on insect spermatozoa focus on thesperm ultrastructure, there are very few studies oninsect sperm motility [5]. Many important geometricand kinematic data required for hydrodynamic model-ling of the double-wave structure are unavailable. Inparticular, we are not aware of any information aboutthe chirality of the minor helical structure relative tothe major helical structure. The generation and propa-gation mechanism of the double-wave is also not yetfully understood. Baccetti et al. [11,12] have suggestedthat the accessory bodies and the axoneme are responsiblefor the major and minor waves, respectively, whereasSwan [15] has stated that the major wave may be owingto the sliding of the accessory tubule against the axonemaldoublets. More recently, Werner et al. [10] have proposeda completely different line of thought, suggesting that themajor wave is not in act a real wave but a static helicalstructure formed owing to the coupling of static forces ofthe axoneme, mitochondrial derivatives and plasmamembrane. The apparent propagation of the majorwave could be owing to the passive rolling of the entirecell and might in fact be mistaken for an active, propagat-ing wave under the microscope. It has therefore beensuggested that the sperm motility is caused solely bythe minor wave. The relative extent of the contributionof the major and minor waves to propulsion is thus stillan open question [14]. With the hydrodynamic study pre-sented in this paper, we hope to provide physical insightson these unresolved problems.
acrosome(a) (b)
(g)
(h)
(c)
(d)
(e)
( f )
elav
ar
n
axrs
da
B A
at
cs
ct
oa
iars
d
itm
ca
md
md1 md2
ab
pc
nucleus
flagellum
tail end
centrioleadjunct
Figure 2. Schematic of the ground plan of a pterygote insect flagellosperm and its ultrastructure. (a) A typical filiform insectspermatozoon. Although not easily visible from the outside, it can be divided into five distinct parts: acrosome, nucleus, centrioleadjunct, flagellum, and tail end. (b) Cross section of the acrosome showing its trilayered arrangement of an inner acrosomal rod(ar), an acrosomal vesicle (av) and an outer extra acrosomal layer (el). (c) Cross section of the nucleus (n) showing condensedchromatin. (d) Cross section of the posterior centriole adjunct region. This part of the spermatozoon is characterized by theelectron-dense centriole adjunct material (ca), often surrounding the anterior part of the axoneme (ax) and the tip of one ofthe mitochondrial derivatives (md). (e) Cross section through a representative segment of the flagellum. In addition to the axo-neme, two accessory bodies (ab) and the two mitochondrial derivatives of often different size (md1, md2) can be seen. Themitochondrial derivatives typically bear paracrystalline inclusions (pc). ( f ) Cross section through the tail end showing disso-ciated axonemal tubules. (g) Cross-sectional representation of a typical 9 þ 9 þ 2 insect sperm axoneme. Nine microtubuledoublets (d) with associated dynein arms (da) and radial spokes (rs) are connected to two central microtubules (ct) via the cen-tral sheath (cs). The doublets are in turn surrounded by nine accessory tubules (at). Accessory tubules and doublets are linkedtogether by intertubular material (itm). (h) Schematic cross-sectional drawing of an axonemal doublet showing the protofilamentarrangement of the A and B subtubules. The radial spoke (rs), the inner dynein arm (ia) and the outer dynein arm (oa) areattached to the A subtubule. This figure and caption are reproduced with permission from Werner & Simmons [5]. Copyright2008 John Wiley and Sons.
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The structure of this paper is as follows. We idealizethe double-wave structure as the propagation of super-helical waves and model the hydrodynamics usingnon-local slender body theory (SBT) in §2. In §3, wepresent the computed hydrodynamic performance ofspermatozoa of different species and compare the pre-dictions with the available experimental data (§3.1).The features of superhelical swimming are illustratedby a specific model organism, namely the spermatozoaof Cu. melleus (§3.2). We then investigate the effectsof kinematic and geometric parameters on the propul-sion performance of a superhelical swimmer (§3.3).Finally, the limitations of the present study and direc-tions for future work are discussed in §4.
2. MATERIAL AND METHODS
2.1. Idealized double-wave structure: superhelicalswimmers
The experimentally observed double-wave structureof insect spermatozoa is mathematically idealized inthis paper as a superhelix (a small helix itself coiledinto a larger helix); we refer to the helical structurewith the larger wavelength as the major helix and
J. R. Soc. Interface (2012)
the other as the minor helix. To mathematicallydescribe a superhelix, we first construct the positionvector of a regular axial helix (the major helix)to be Hðs0Þ ¼ ½AM cosðkMas0Þ;AM sinðkMas0Þ;as0�, witha ¼ 1=
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ A2
M k2M
p. Here kM is the wavenumber, AM
is the amplitude, and s0 [ ½0;L0� and L0 are the arc-length parameter and the length of the major helix,respectively. From this basic helix, the local unit tangenttA, unit normal nA and unit binormal bA ¼ tA� nA vec-tors are determined, and we take them to define a localcoordinate system upon which the minor helix is con-structed [20]. The position vector of the combinedsuperhelical shape is then given by:
Xðs0Þ ¼ ½xðs0Þ; yðs0Þ; zðs0Þ�¼ Hðs0Þ þ Am cosðkms0Þ nAðs0Þ
ð2:1Þ
+ Am sinðkms0Þ bAðs0Þ; ð2:2Þ
where Am and km are the amplitude and wavenumber ofthe minor helix, respectively. Two different configur-ations will be considered: the ‘ þ ’ (plus) sign leads to asuperhelical structure where both the major and minorhelices have the same chirality, whereas the ‘ 2 ’(minus) sign represents the case of opposite chirality.
(a) ( f )
(b)
(g)
(h)
(c)
(d)
(e)
Figure 3. The double-wave beating pattern observed in insect spermatozoa flagella of different species: (a) Megaselia scalaris [18],(b) Haematopinus suis [6], (c) Culicoides melleus [14], (d) Tenebrio molitor [11], (e) Drosophila obscura [17], ( f ) T. molitor [13],(g) Aedes notoscriptus [15] and (h) Bacillus rossius [12]. All images were reproduced with permission: (a) from Curtis & Benner[18]. Copyright q 1991 John Wiley and Sons; (b) from Baccetti [6]. Copyright q 1972 Elsevier; (c) from Linley [14]. Copyright q
1979 John Wiley and Sons; (d) from Baccetti et al. [11]. Copyright q 1973 Plenum Publishing Corporation, with kind permissionfrom Springer Science þ Business Media B.V; (e) from Bressac et al. [17]. Copyright q 1991 John Wiley and Sons; ( f ) from Philips[13]. Copyright q 1974 John Wiley and Sons; (g) from Swan [15]. Copyright q 1981 John Wiley and Sons; (h) from Baccetti et al. [12].Copyright q 1973 Elsevier.
Hydrodynamics of insect spermatozoa O. S. Pak et al. 1911
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Note that s0 is no longer the natural arc-length par-ameter, but merely a regular parameter for describingthe swimmer’s geometry. The arc-length of the completesuperhelix, denoted by s, as a function of the parameter s0
is determined by numerical integration, and the totallength of the superhelix is denoted by L.
The major and minor helices are free to propagatewaves at different wave speeds. Denoting cM and cm
as the major and minor wave speeds, respectively, theposition vector at time t, X(s0,t), may be written incomponent form as:
xðs0; tÞ ¼ AM cos½kMðas0 � cMtÞ�� Am cos½kmðs0 � cmtÞ� cos½kMðas0 � cMtÞ�+ aAm sin½kmðs0 � cmtÞ� sin½kMðas0 � cMtÞ�;
ð2:3Þyðs0; tÞ ¼ AM sin½kMðas0 � cMtÞ�
� Am cos½kmðs0 � cmtÞ� sin½kMðas0 � cMtÞ�+ aAm sin½kmðs0 � cmtÞ� cos½kMðas0 � cMtÞ�,
ð2:4Þ
J. R. Soc. Interface (2012)
and
zðs0; tÞ ¼+aAmAMkM sin½kmðs0 � cmtÞ� þ as0: ð2:5Þ
In addition to the long flagellum, insect spermatozoahave cell bodies that are very slender and short when com-pared with the flagellum size. We expect that thehydrodynamic influence of the sperm head is negligibleand do not include such a body in our consideration.A typical superhelical swimmer is shown in figure 4 usingthe dimensionless parameters of Cu. melleus spermatozoa.
2.2. Hydrodynamic modelling
Flagellar swimming is a result of the interaction betweenthe actuating bodyand the surrounding fluid. A tractableand accurate approach to studying such hydrodynamicinteractions exploit the slenderness of the flagellum, inwhich the velocity along the flagellar centreline is relatedto the fluid forces along the same curve. In previousstudies, the fluid–body interaction has been modelledusing a resistive force theory (RFT) [21–23], in whichlocal forces acting on the flagellum at any station along
0.4
0.2
0.6
0.8
(a) (b)
1.0
0.8
0.6
0.4
0.2
0.020.02
–0.02 –0.02
0
00
1.0
0.8
0.6
0.4
0.2
0.020.02
–0.02 –0.02
0
00
0.02–0.02
0.02–0.02
top view
side view
0.25
0.15
0.05
–0.02 0.02
rescaled
0.02–0.02
0.02–0.02
0.4
0.2
0.6
0.8
rescaled
top view
0.25
0.15
0.05
–0.02 0.02
side view
Figure 4. Idealization of the double-wave structure as superhelices for the (a) same-chirality, and (b) opposite-chirality configurations,using the dimensionless parameters of Culicoides melleus spermatozoa (table 2). (Online version in colour.)
1912 Hydrodynamics of insect spermatozoa O. S. Pak et al.
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the filament are expressed in terms of the local velocity atthe same location. RFT takes only local effects intoaccount and neglects any hydrodynamic interactionsbetween different parts of the deforming body. Thelocal theory works well for simple geometries. However,owing to the complexity of the superhelical structure inthe problem under consideration here, the local theoryis inadequate (see §3.4 for details), and we employ insteadthe full non-local SBT [24] to study the hydrodynamics ofthe superhelical swimmers. The non-local theory cap-tures the hydrodynamic interactions between distantparts of a curved filament, while still taking advantageof the slenderness of flagellum to simplify the analysis.
The flagellum is modelled as a slender filament oflength L and circular cross section of radius eLrðsÞ,where e� 1 is the small aspect ratio of the flagellum(the maximum radius along the flagellum ar dividedby its total length L) and r(s) is the dimensionlessradius. The non-local SBT is algebraically accurate inratio of the slenderness e; namely, by setting theradius profile to rðsÞ ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4sðL� sÞ
p=L, the computed
fluid velocity is accurate to O(e2) [24].For a given velocity distribution v(s,t) along the fila-
ment at time t, the corresponding fluid force per unitlength f(s,t) is given implicitly by the non-local relation
8pmvðs; tÞ ¼ �L½fðs; tÞ� �K½fðs; tÞ�; ð2:6Þ
where
L½f �ðs; tÞ ¼ ½c0ðIþ ttÞ þ 2ðI� ttÞ�fðs; tÞ, ð2:7Þ
and
K½f �ðs; tÞ ¼ ðIþ ttÞðL
0
fð~s; tÞ � fðs; tÞj~s � sj d~s
þðL
0
Iþ RRjRð~s; s; tÞj �
Iþ ttj~s � sj
!fð~s; tÞd~s;
ð2:8Þ
J. R. Soc. Interface (2012)
are the local and non-local operators, respectively, m isthe shear viscosity of the fluid, c0 ¼ �lnð12eÞ . 0,Rð~s; s; tÞ ¼ Xð~s; tÞ �Xðs; tÞ, R ¼ R=jRj and t is thelocal unit tangent vector at the point s.
The swimmers of interest deform their shapes in a pre-scribed, time-varying fashion (the superhelical wavepattern, equations (2.3)–(2.5)), and the velocity createdon its surface by this deformation is given byvdeformðs; tÞ ¼ @Xðs; tÞ=@t. At every time instant t, theswimmer can be seen as a solid body with unknown trans-lational velocity U(t) and rotation rate V(t). Thevelocity created on the swimmer’s surface owing to swim-ming is then vswimðs; tÞ ¼ UþV� ½Xðs; tÞ�X0�, whereX0 is an arbitrary reference point (taken as the origin herefor simplicity). Therefore, the local velocity relative to thefluid v(s,t) is given by the sum of the deformation andswimming velocities: vðs; tÞ ¼ vdeform þ vswim ¼ UþV� ½Xðs; tÞ �X0� þ @Xðs; tÞ=@t. In this work, thewave propagation is towards the positive z-direction.Therefore, a negative swimming velocity (U) meansthat the propulsion occurs in a direction opposite to thewave propagation, while a positive swimming velocitymeans that both the propulsion and wave propagationoccur in the same direction.
Using a Galerkin method [25], we express the localforce f(s,t) as a finite sum of Legendre polynomials andsolve equation (2.6) for f(s) by requiring the equationto hold under inner products against the same basis func-tions. The first integral in the non-local operator K[f ] isdiagonalized in this space [26,27]. The system is closedby requiring the entire swimmer to be force-free andtorque-free, ðL
0fðsÞds ¼ 0; ð2:9Þ
and ðL
0½XðsÞ �X0� � fðsÞds ¼ 0; ð2:10Þ
Hydrodynamics of insect spermatozoa O. S. Pak et al. 1913
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providing at each moment in time a systemof six equations to solve for the six unknowns U(t)and V(t).
The swimming velocities determined in the mannerdescribed above represent velocities in a referenceframe fixed on the swimmer. In order to study the fullthree-dimensional swimming kinematics in the labora-tory frame (in which the body moves with velocityUðtÞ and rotation rate VðtÞ), we must include a trans-formation between the two. We denote the Cartesiancoordinate system moving with the swimmer, thebody frame, as ½ex ; ey; ez � and the Cartesian coordinatesystem in the laboratory frame as ½e1; e2; e3�. The evol-ution of the body frame, with respect to the laboratoryframe, is then governed by:
dEdt¼ VðtÞ �E; ð2:11Þ
where E ¼ ½ex ; ey; ez �T, along with the initial condition½ex ; ey; ez �ðt ¼ 0Þ ¼ ½e1; e2; e3�.
2.3. Non-dimensionalization
The process of non-dimensionalization is very useful inscience and engineering to identify the relevant dimen-sionless parameters governing the physics of theproblem. In theoretical studies, it always allows amore concise description of the system, while in exper-imental studies, it reduces the number of independentexperiments required to fully explore the problem.The present system is made dimensionless by scalinglengths of the order of 1/km, velocities by cm and timeby their ratio, 1/cmkm. The dimensionless positionvector describing the kinematics of the superhelix isthus given by:
x ¼ Rfcos½Kðas0 � ctÞ� � r cosðs0 � tÞ cos½Kðas0 � ctÞ�þ ar sinðs0 � tÞ sin½Kðas0 � ctÞ�g; ð2:12Þ
y ¼ Rfsin½Kðas0 � ctÞ� � r cosðs0 � tÞ sin½Kðas0 � ctÞ�� ar sinðs0 � tÞ cos½Kðas0 � ctÞ�g, ð2:13Þ
and z ¼ as0 þ arR2K sinðs0 � tÞ; ð2:14Þ
where all variables are now understood to be dimension-less. Four dimensionless parameters characterizing thekinematics are identified above: the dimensionlessamplitude of the major helix, R ¼ AMkm, the ratio ofthe wavenumbers characterizing the major and minorhelices, K ¼ kM/km, the ratio of the minor helix ampli-tude to the major helix amplitude, r ¼ Am/AM and theratio of the major wave speed to the minor wave speedc ¼ cM/cm. In the double-wave pattern observed ininsect spermatozoa, the major wave amplitude isalways larger than the minor wave amplitude (r , 1;although this need not be true for a general superhelix).In addition, it is also observed that the minorwave speed is always greater than the major wave speedin the double-wave structure of insect spermatozoa(c , 1).
J. R. Soc. Interface (2012)
2.4. Kinematic and geometric data
From the non-dimensionalization above, we haveidentified four dimensionless parameters (r, R, K, c)required to fully characterize the centreline motion ofa superhelical flagellum. Werner & Simmons [5] havecompiled a very useful table containing kinematic andgeometric data of the double-wave structure observedin insect spermatozoa in previous studies. The tablereveals that experimental measurements of the necess-ary quantities for hydrodynamic modelling are verylimited owing to the difficulties involved in inter-preting three-dimensional data from two-dimensionalimages [14].
Here, we follow the table compiled by Werner andSimmons and estimate the missing information basedon images of insect spermatozoa reported in the litera-ture. Table 1 contains reported and estimated dataon the double-wave structure, including the wave-lengths, amplitudes, frequencies, major and minorwave speeds, flagellum thickness, and the swimmingspeed of different insect spermatozoa. Our estimatedquantities are marked with asterisks to distinguishthem from reported quantities by the original papers.In some studies (e.g. for Lygaeus [13]), only relativelengths can be given because of the lack of scale barsin the reported images; these relative quantities aresquare-bracketed in tables 1 and 2. Geometric andkinematic data can display large variations evenwithin a species (T. molitor [11]). When the distributionof the quantities are not given, arithmetic means of theavailable measurements are used whenever appropriatein the present study.
The measurements are presented in the correspondingdimensionless quantities in table 2, using the scalingsdefined in §2.3. We have not found any information inthe referenced literature about the chirality of themajor and minor helical structures. We therefore presentresults below for both the same-chirality and opposite-chirality configurations. In our measurements (collectedin table 1), the wavelength of the minor wave is takento be the two-dimensional distance between adjacentminor wave peaks. A small correction factor is requiredto convert these two-dimensional quantities to the appro-priate wavelengths in describing the three-dimensionalsuperhelical structure. The correction factor dependson whether the superhelical structure is in the same-chirality (1=ð1� alm=lMÞ) or in the opposite-chirality(1=ð1þ alm=lMÞ) configuration.
3. RESULTS
In this section, we first use the framework descri-bed above to predict the swimming performance ofspermatozoa of different species, and compare ourhydrodynamic results with the available experimentalmeasurements. We then illustrate the basic featuresof superhelical swimming by focusing on a modelorganism, namely Cu. melleus, and proceed to performa parametric study investigating the effects of cer-tain kinematic parameters (minor and major wavespeeds) and geometric parameters (minor and majorwave amplitudes).
Tab
le2.
Dim
ensi
onle
sspa
ram
eter
sof
supe
rhel
ical
flage
lla:
K¼
k M/k m
;R¼
AM
k m;
r¼
Am
/A
M;
c¼
c M/c
m;
V¼
Vsp
erm
/cm;
a¼
a rk m
;n
refe
rsto
num
ber
ofla
rge
wav
elen
gths
.ð~ Uþ
and
~ U�
are
the
pred
icte
dav
erag
esw
imm
ing
velo
citi
esfo
rth
esa
me-
chiral
ity
and
oppo
site
-chi
ralit
ys
confi
gura
tion
s,re
spec
tive
ly.)
spec
ies
KR
rc
Na
V~ Uþð~ U�Þ
hþðh�Þð
%Þ
j~ Uþ�
Vj=
Vðj
~ U��
Vj=
VÞð%Þ
Aed
esno
tosc
ript
us0.
24*
3.4*
0.5*
0.41
*1
0.25
*0.
21(0
.16)
1.2
(0.3
4)B
acill
usro
ssiu
s0.
45*
4.2*
0.09
20.
67*
1.5
0.13
*0.
031
(0.0
33)
0.30
(0.1
4)C
erat
itis
capi
tata
0.25
16.8
0.07
50.
8*4–5a
0.28
*0.
11*
0.15
(0.0
39)c
0.40
(0.0
21)
36(6
5)C
ulic
oide
sm
elle
us0.
161.
50.
383
0.28
*0.
100.
051
(0.1
5)0.
30(1
.5)
49(5
0)D
roso
phila
obsc
ura
0.30
*7.
6*0.
03*
10.
18*
0.01
2(0
.009
6)0.
10(0
.036
)Lyg
aeus
0.21
*3*
0.14
*1–2b
0.34
*0.
033
(0.0
81)
0.24
(0.7
1)M
egas
elia
scal
aris
0.10
*8.
3*0.
05*
10.
35*
0.03
5(0
.065
)0.
16(0
.42)
Ten
ebri
om
olitor
0.36
8.4
0.29
0.32
40.
28*
0.34
0.40
(0.4
1)d
1.2
(0.8
6)18
(21)
a Ava
lue
of4
isad
opte
din
the
sim
ulat
ion.
bA
valu
eof
1is
adop
ted
inth
esi
mul
atio
n.c V
eloc
ity
inth
ez-
dire
ctio
noc
curs
inth
esa
me
dire
ctio
nas
the
wav
epr
opag
atio
n.dV
eloc
ity
inth
ez-
dire
ctio
noc
curs
inth
esa
me
dire
ctio
nas
the
wav
epr
opag
atio
n.
Tab
le1.
Kin
emat
ican
dge
omet
ric
data
ofth
edo
uble
-wav
est
ruct
ure.
(We
follo
wth
eta
ble
byW
erne
r&
Sim
mon
s[5
]w
ith
our
esti
mat
ions
(mar
ked
wit
has
teri
sks)
ofth
em
issi
ngqu
anti
ties
base
don
the
imag
esof
the
sper
mce
llre
port
edin
the
refe
renc
es;a r
refe
rsto
the
radi
usof
the
flage
llum
.)
maj
orw
ave
min
orw
ave
spec
ies
lM
(mm
)A
M(
m)
f M(H
z)c M
(mm
/S)
lb
(mm
)A
m(m
m)
f m(H
z)c m
(mm
/s)
2ar(m
m)
Vsp
erm
(mm
)re
fere
nces
Aed
esno
tosc
ript
us28
*3.
67*
3.4
95.2
*6.
8*1.
83*
3423
1.2*
0.55
*Sw
anet
al.[1
5]B
acill
usR
ossi
us40
*12
1248
0*17
.9*
1.1
4071
6*0.
74*
Bac
cett
iet
al.[
12]
Cer
atitis
capi
tata
a30
204
120*
7–8
1–2
2015
0*0.
68*
16B
acce
ttiet
al.[
16]
Cul
icoi
des
mel
leus
54.1
2.1
8.7
0.8
8.2
800.
77*
8.3
Lin
ley
[14]
Dro
soph
ilaob
scur
ab45
*16
13.3
*0.
5*20
.427
10.
76*
Bre
ssac
etal
.[17
]Lyg
aeus
c[1
]*[0
.1]*
[0.2
1]*
[0.0
14]*
130
[0.0
24]*
Phi
llips
[13]
Meg
asel
iasc
alar
is68
*9.
3*3.
121
0.8*
7*0.
5*0.
78*
12.7
Cur
tis
&B
enne
r[1
8]T
eneb
rio
mol
itor
d20
–30
(25)
9–15
(12)
0.9–2.
8(1
.85)
20–90
(55)
6–12
(9)
3–4
(3.5
)7–28
(17.
5)40
–30
0(1
70)
0.79
*16
–10
0(5
8)B
acce
ttiet
al.[
11]
a Onl
yfo
rwar
dm
ode
isco
nsid
ered
[16]
.bO
nly
mal
elo
ngsp
erm
isco
nsid
ered
here
[17]
.c S
cale
bars
wer
eno
tpr
esen
ted
wit
hth
eim
ages
inth
ere
fere
nce
and
only
rela
tive
leng
ths
(squ
are-
brac
kete
d)ca
nbe
esti
mat
ed.
dR
ound
brac
kets
deno
tear
ithm
etic
mea
ns.
1914 Hydrodynamics of insect spermatozoa O. S. Pak et al.
J. R. Soc. Interface (2012)
on August 30, 2012rsif.royalsocietypublishing.orgDownloaded from
0
0.1
0.2
0.3
0.4
(a) (b)
Aed
es
Bac
illu
s
Cer
atit
is
Cul
icoi
des
Dro
soph
ila
Lyg
aeus
Meg
asel
ia
Ten
ebri
o
U±
η±
(%)
0
0.4
0.8
1.2
1.6
Aed
es
Bac
illu
s
Cer
atit
is
Cul
icoi
des
Dro
soph
ila
Lyg
aeus
Meg
asel
ia
Ten
ebri
o
Figure 5. Predicted swimming performance: (a) average swimming speed, U+, of different species; (b) hydrodynamic efficiency,h+, of different species. Open symbols (blue circles) represent the same-chirality configuration (þ); filled symbols (red squaresand diamonds) represent the opposite-chirality configuration (2); for blue open circles and red squares, swimming occurs in theopposite direction as the wave propagation; on the contrary, red diamonds represent the cases of opposite-chirality configurationwhere the velocity in the z-direction occurs in the same direction as the wave propagation; black crosses represent experimentalmeasurements of V ¼ Vsperm/cm (see tables 1 and 2). (Online version in colour.)
Hydrodynamics of insect spermatozoa O. S. Pak et al. 1915
on August 30, 2012rsif.royalsocietypublishing.orgDownloaded from
3.1. Hydrodynamic performance
3.1.1. Propulsion speedThe propulsion speed is an important functional para-meter characterizing the motility of a sperm cell.There exist only very few measurements of the swim-ming speed of insect spermatozoa exhibiting thedouble-wave structure: Ce. capitata was observed toswim at a speed, Vsperm ¼ 16 mm s�1 [16] and theminor wave speed is estimated to be 150 mm s�1 (hence,V ¼ Vsperm=cm ¼ 0:11); Cu. melleus [14] was observedto swim at a speed of 8:3 mm s�1 and minor wavespeed of 80 mm s�1 (V ¼ 0.10). For the case of T. moli-tor, a wide range of sperm speeds were reported (from16 to 100 mm s�1) as well as minor wave speeds (40–300 mm s�1) [11]. The distributions were not reportedhowever; so using the arithmetic means we obtain aswimming speed to minor wave speed ratio of V ¼ 0.34.
For a superhelical swimmer in ourmodel, the swimmingkinematics are three-dimensional and unsteady in time,1
and the most relevant quantification of the propulsionspeed is an average swimming velocity in the laboratoryframe, denoted by ~U+, where the þ (plus) and 2
(minus) signs represent the same- and opposite-chiral-ity configurations, respectively. The average swimmingspeed is defined as ~U+ ¼ j½k ~Ux l; k ~Uy l; k ~Uz l�j, where j. . .jdenotes the magnitude of a vector, and k. . .l denotesa time average.2 Since the chirality configurationremains unknown, we present predictions for bothcases in figure 5a. Only the three sets of experimentalmeasurements of the swimming performance (V ¼
1The deformations (equations (2.3)–(2.5)) are higher frequencyoscillations modulated by lower frequency oscillation (a beat). Theredoes not exist a time period T, such that the deformation vectorrepeats itself: R(s,t þ T ) = R(s,t), 8T.2The time averaging is defined for a function f(t) as
Ð0t0 f(t) dt/t0,
where t0 is a sufficiently large time, such that the peak-to-peakfluctuation in time is less than 1 per cent of the final average value.
J. R. Soc. Interface (2012)
Vsperm/cm) mentioned above are available for compari-son and are superimposed on the same figure (figure 5).Predictions as ratios of the swimming speed relative tothe minor wave speed (dimensionless speed) for specieswith no measurement of V are also provided infigure 5a.
In most of the cases considered, the propulsion in thelongitudinal direction (z-direction) of these superhelicalswimmers is opposite to the direction of wave propa-gation (the major wave propagates in the positivez-direction, and the minor wave propagates along thecurved major helix and distally towards the positivez-direction), regardless of the chirality configuration.This is not unlike the behaviour of swimmers propagat-ing a planar sinusoidal or a regular helical wave, forwhich the swimming direction is also opposite to thedirection of the wave propagation. In superhelicalswimming, however, we also find cases of theopposite-chirality configuration, where the body swimsin the same direction as the wave propagation. Specifi-cally, we note a qualitative difference between thestudy of the same-chirality and opposite-chiralityconfigurations for Ce. capitata and T. molitor.In these simulations, a superhelical wave is set topropagate in the positive z-direction, and both theopposite-chirality configurations of Ce. capitata andT. molitor generate a positive Uz (swimming is in thesame direction as the wave propagation), while theircorresponding same-chirality configurations generate anegative Uz (swimming is in the opposite direction asthe wave propagation). The speed and efficiency ofthese peculiar swimmers are distinguished from othercases by red diamonds as shown in figure 5. It remainsa question whether this phenomenon may be observedin nature, as the chirality configuration and the swim-ming direction (relative to the wave propagation) ofactual insect spermatozoa displaying the double-wave
1916 Hydrodynamics of insect spermatozoa O. S. Pak et al.
on August 30, 2012rsif.royalsocietypublishing.orgDownloaded from
structure are still unclear. Nevertheless, this phenom-enon by itself is intriguing and will be furtherexplored in §3.3.1.
The swimming speed predictions lie at least withinthe same order of magnitude of the experimentalmeasurements for both chirality configurations. Thesame-chiralty results provide slightly better agreementthan the opposite-chiralty results (without taking theswimming direction into account). For the same-chirality configuration, the discrepancies, j ~Uþ �V j=V ,between the predictions and the experimental measure-ments read 36 per cent for Ce. capitata, 49 per cent forCu. melleus and 18 per cent for T. molitor. For theopposite-chirality configuration, the discrepancybetween our predictions and the experimental measure-ments, j ~U� �V j=V , are 65 per cent for Ce. capitata, 50per cent for Cu. melleus and 21 per cent for T. molitor.Given the primitive nature of the data employed (see§2.4), we consider the agreements here to be reasonable.However, we cannot draw definite conclusions on theissue of chirality configuration; further experimentalobservations are necessary.
It shall be remarked that the present study is largelyconstrained by the unavailability of experimentalmeasurement data. Critical kinematic information,such as the major and minor wave speeds, are oftennot reported in the literature and are impossible to esti-mate from the images available. The speed ratio c ¼cM/cm is not available for simulations for most species.The computations here are still possible because of theindependence of the swimming kinematics on the para-meter c (verified numerically and will be explained in§3.3.1). Therefore, the specific value of c is unimportantfor all cases considered here; we adopt c ¼ 0 in allsimulations hereafter unless otherwise stated.
3.1.2 Hydrodynamic efficiencyAnother important functional parameter is the hydro-dynamic efficiency of the swimmer. In the microscopicworld, the hydrodynamic efficiency is typically verylow. For a rigid helix, Lighthill [28] calculated theoreti-cally the maximum efficiency attainable to be about 8.5per cent, while the typical efficiency of biological cells isaround 1–2% [3,28–32]. For low Reynolds numberswimming, a common measure of the hydrodynamicefficiency, h, is the ratio of the rate of work requiredto drag the straightened flagellum through the fluid tothe rate of work done on the fluid by the flagellumduring swimming [28],
h+ ¼jkL ~U
2+
kÐ L0 v � f dsl
; ð3:1Þ
where jk ¼ 4p=c0 is the drag coefficient for a straight,slender rod, and the brackets indicate a time average.We calculate the hydrodynamic efficiencies for boththe same- and opposite-chirality configurations(figure 5b) for spermatozoa of different species. Forthe same-chirality configuration, the efficiency rangesfrom 0.16 per cent (M. scalaris) to 1.2 per cent(Ae. notoscriptus and T. molitor); for the opposite-chir-ality configuration, the efficiency ranges from 0.036 percent (Dro. obscura) to 1.5 per cent (Cu. melleus). The
J. R. Soc. Interface (2012)
efficiencies of these swimmers are comparable with typi-cal biological cells.
3.2. Model organism: Cu. melleus
In this section, we illustrate the features of superhelicalswimming by singling out a superhelical swimmer definedusing the geometric data of the sperm cell of Cu. melleus(table 2). See figure 4 for the swimmer geometry.
3.2.1. Swimming kinematicsThe swimming velocities computed for motion in thebody frame [Ux, Uy, Uz] and in the laboratory frame½ ~Ux ; ~Uy; ~Uz � of the superhelical swimmer with datafrom the sperm cell of Cu. melleus are plotted infigure 6 for the same-chirality (figure 6a) and opposite-chirality (figure 6b) configurations. We include in thesefigures the results for the ratio of the major and minorwave speeds c ¼ 0 (solid lines) and c ¼ 0.4 (dottedlines). When c ¼ 0, the deformation (equations (2.12)–(2.14) is 2p-periodic, therefore the swimming kinematicsin the body frame are also 2p-periodic. However, whenobserved in the laboratory frame, the coupling betweenthe translational and rotational kinematics renders theswimming velocities no longer 2p-periodic, and themotion is unsteady in time (figure 6c,d). We observe apattern consisting of higher frequency oscillations modu-lated by a lower frequency envelope. For the case of c ¼0.4, when observed in the body frame, we see modulatedwaveforms. However, when transformed to the labora-tory frame, the cases of c ¼ 0 and c ¼ 0.4 have identicalswimming kinematics (see the overlapped solid anddotted lines in figure 6c,d), implying that the swimmingkinematics in the laboratory frame are independent of themajor propagating wave speed. The value of c affects onlythe kinematics in the body frame; why the relative wavespeed is unimportant is described in greater detail in§3.3.1.
The three-dimensional swimming velocities give riseto a doubly-helicated trajectory (the presence of aminor structure on top of a major helical structure).However, this could be difficult to observe experimen-tally as the major amplitude of the doubly-helicatedtrajectory is usually much smaller than that of thesuperhelical swimmer; the swimmer would apparentlymove with a straight trajectory (with very small oscil-lations in the transverse direction). Recall that forregular helical swimming (r ¼ 0), the trajectory of thehelical swimmer reduces to a regular helix.
3.2.2. Head-less swimmingThe shape and size of the sperm head vary amongspermatozoa of different species: human and bullspermatozoa have relatively large, paddle-shaped heads,whereas insect spermatozoa have elongated heads whichare almost indistinguishable from the mid-piece. Theadditional hydrodynamic resistance from the presence ofa head would seemingly degrade the swimming perform-ance of the sperm cell. However, Chwang & Wu [33]showed that a sperm head is actually necessary for helicalswimming; without one, the motion is that of a rotatingrigid body, which cannot be realized in the absence ofan external force or torque [34]. To satisfy the zero
UzUyUx
UxUyUz
UzUyUx (c = 0.4)
(c = 0.4)(c = 0.4)
Ux (c = 0)Uy (c = 0)
(c = 0)
(c = 0.4)(c = 0.4)(c = 0.4)
(c = 0)(c = 0)(c = 0)
Uz
–0.20
–0.15
–0.10
–0.05
–0.20
–0.15
–0.10
–0.05
0
0.05
0.10
0.15(a) (b)
(c) (d)
0 20 40 60 80t t
100 120 140 160
0
0.05
0.10
0.15
0 20 40 60 80 100 120 140 160
Figure 6. Three-dimensional swimming velocities in the body frame U ¼ (Ux, Uy, Uz) for (a) the same-chirality and (b) the opposite-chirality configurations, and in the laboratory frame U ¼ ð ~Ux ; ~Uy ; ~UzÞ for (c) the same-chirality and (d) the opposite-chirality con-figurations, for Cu. melleus spermatozoa. (Online version in colour.)
Hydrodynamics of insect spermatozoa O. S. Pak et al. 1917
on August 30, 2012rsif.royalsocietypublishing.orgDownloaded from
net-torque condition, a sperm head is required to balancethe reaction torque acting on the flagellum. This con-straint does not apply for planar, sinusoidal wavemotion, which can swim without an anchor or load.
We pause to point out a subtle but important differ-ence between a rotating prokaryotic helical tail and aeukaryotic tail propagating a bending helical wave.For a eukaryotic tail propagating a bending wave, thefluid forces act to rotate the flagellum opposite to thedirection of the apparent helical rotation. The rotationowing to the fluid reaction creates torques owing tolocal spinning (rotation of the flagellum about its cen-treline), which balance the opposing torque generatedby the helical wave propagation. Therefore, a eukary-otic cell could theoretically swim without a spermhead, albeit very slowly because the flagellum is veryslender and the local torques are correspondinglysmall. Quantitatively, using a local drag model [33], itcan be shown that the head-less swimming speedscales as U=c � 2mk2=jkð1þ k2A2Þb2 þ Oðb4Þ, where bis the radius of a cross section of the flagellum, and kand A are the wavenumber and helical radius, respect-ively. Using the geometrical data of flagella ofEuglena viridis summarized by Brennen & Winet [4],and assuming a flagellar diameter of 2b � 0.25 mm[35,36], we find U/c � 1023. For a prokaryotic tail, how-ever, the helix rotates as a rigid body. In this case, thelocal spinning torques generated by the active helicalrotation and the passive rotation owing to the fluidreaction are identical in magnitude but opposite insign. The torque-free condition therefore requires thatthe fluid reaction counter-rotates the helix at preciselythe rotation rate of the helical wave propagation. Hence,there can be no effective helical rotation, and thebody cannot swim. Although head-less swimming is
J. R. Soc. Interface (2012)
theoretically possible for eukaryotic tails, the swimmingspeed would be exceptionally small as shown by the esti-mation above. The subtle difference between the twotypes of helical waves just described is often therefore neg-lected, and it is generally reasonable to state that head-less swimming is not possible using helical wavepropagation.
In superhelical swimming, a sperm head is not requiredfor self-propulsion (indeed, all cases reported in this paperwere studied with the absence of a sperm head) as a super-helical wave motion is in general not a rigid body motion.Furthermore, as actual insect spermatozoa heads are veryslender and short when compared with the entire lengthof the flagellum, the contribution of its hydrodynamicresistance and the hydrodynamic interactions withthe flagellum should be negligible. Therefore, we do notexpect the presence of a slender and short sperm headto introduce qualitative differences in the results.
3.3. Parametric study
Next, we explore the effects of certain kinematic (wavespeeds) and geometric parameters (wave amplitudes)on the swimming velocities of a superhelical swimmer.
3.3.1. The effect of the major/minor wave speed ratioIn this problem we have scaled the velocities upon theminor wave speed cm, which implies that the dimen-sionless swimming velocities scale linearly with theminor wave speed. Here, we examine the effect ofthe major wave speed on the swimming performanceof a superhelical swimmer and answer the question: towhat relative extent do the two waves contributeto the propulsion of the superhelical swimmer[14]? Specifically, we study the effect of the parameter
1918 Hydrodynamics of insect spermatozoa O. S. Pak et al.
on August 30, 2012rsif.royalsocietypublishing.orgDownloaded from
c ¼ cM/cm, which is the ratio of the major wave speed tothe minor wave speed. For every species shown intable 2, we fix all parameters but vary the value of cfrom zero to unity (c , 1, as cM , cm). It is foundthat the swimming velocities in the laboratory frameis independent of the value of c. We have alreadyshown in figure 6 the results for two values of c for illus-tration. While the resulting swimming motions in thesetwo cases differ significantly in the body frame, theyare identical when observed in a laboratory frame ofreference. The numerical results imply that the majorwave speed does not contribute to propulsion.
These findings may be understood by noting anambiguity in definition. First consider a single helicalfilament (without a sperm head) placed in a fluid.Such a body cannot swim on its own, for there is noload or cell to counterbalance the torque it wouldexert on the surrounding fluid during rotation, so itmust be motionless as seen in a fixed, laboratoryframe. However, this body may be represented as anactive helical body propagating a wave with velocity cplus a rigid body rotation that contributes a wavewith speed 2c. That the body frame may be chosenarbitrarily allows for such an ambiguity in the defi-nition of the swimming speed, but in the laboratoryframe this ambiguity disappears.
A similar argument can be used to show that c hasno bearing on the swimming speed of a superhelical flagel-lum in the laboratory frame. For a superhelical flagellum,the propagation of the major wave can be defined as arigid body rotation of the entire superhelical structureabout the longitudinal axis (z-axis) in the body frame.The apparent rotation rate is the sum of the rotationcaused by the active propagation of the major wave andthe rotation caused by fluid reaction to maintain theforce-free and torque-free conditions. For swimming with-out a head, only the apparent rotation rate is importantbecause the fluid forces and torques are functions of theapparent rotation rate alone (not the absolute value ofthe rotation rate owing to the active major wave propa-gation). Any alternation of the major wave speed isaccompanied by a corresponding change in the rotationrate caused by the fluid reaction, resulting in the sameapparent rotation rate and the same force and torque bal-ances. Therefore, the kinematics and dynamics do notdepend on the absolute value of the major wave speed c(or cm). In other words, the major wave speed does notcontribute to swimming.
For the case of swimming with a head, the fluid reac-tion will not only rotate the flagellum but also thesperm head, which creates extra torques and perturbsthe original force and torque balances. Altering theabsolute value of the major wave speed in this casetheoretically will affect the swimming velocities evenin the laboratory frame, because it changes the relativeportion of the rotation rate caused by the fluid reactionand hence the value of the extra torque from the spermhead. However, as the sperm head is slender and shortwhen compared with the overall length of the flagellumof actual insect spermatozoa, the extra-resistant forcesand torques created should be insignificant. Therefore,we suggest that the propagation of the major wave con-tributes very little to the propulsion (none in the case of
J. R. Soc. Interface (2012)
head-less swimming), and that it is the minor wavespeed that is primarily responsible for the propulsion.Note also that the hydrodynamic efficiencies in §3.1.2are independent of the value of c. Therefore, from ahydrodynamic efficiency point of view, there is noadvantage or disadvantage to actively propagate amajor wave. However, there are other energy costs nottaken into account here (for example, work done to pro-duce the sliding of microtubules within the flagellum) inactively propagating a major wave. Therefore, wespeculate that it might be energetically more favourablefor a doubly helicated organism to propagate only aminor wave.
3.3.2. The effects of the parameters R and rWe now examine the effect of geometrical dimensionlessparameters R and r, keeping all other parameters ofCu. melleus spermatozoa fixed. R is the dimensionlessmajor wave amplitude and r is the ratio of minor tomajor wave amplitude, hence the minor wave amplitudeis given by rR (see the schematic in figure 7). In general,one expects the propulsion speed to increase withthe wave amplitude for simple geometries. However,for a superhelical structure, the dependence of theaverage swimming speed on the minor wave amplitudedisplays interesting behaviour. Physically, both thepropulsive force and the bulkiness of the structure arevaried upon changing the major or minor wave ampli-tudes. The competition between these factors and thecoupling between kinematics in different directionscreate interesting geometric dependencies of theaverage swimming speed. We illustrate this by observ-ing the average swimming speed under differentframes of reference.
First, we look at the time-averaged swimming velocityin the body frame Ub ¼ j½kUxðtÞl; kUyðtÞl; kUzðtÞl�j.As shown in figure 7a(i), for the same-chirality con-figuration, the average swimming velocity growsmonotonically with R and r. Note that, keeping otherparameters fixed, increasing the value of R ¼ AMkm fora fixed r ¼ Am/AM geometrically means that both themajor AM and minor Am amplitudes are increased simul-taneously by the same proportion. There is a competitionbetween the increase in the overall hydrodynamic resist-ance owing to the increased bulkiness (correlated withincreases in AM and Am) and an enhanced propulsiveforce (correlated with an increase in Am). In the case ofthe same-chirality configuration, the latter effect domi-nates. However, for the opposite-chirality configuration,as shown in figure 7a(ii), non-monotonic variations inthe swimming speed with the major amplitude R areobserved for certain values of r, the ratio of the minorto major wave amplitudes.
Since the swimming kinematics are three-dimensional,variations of the geometric parameters affect swimmingvelocities and rotational rates in all directions. In parti-cular, a large propulsion speed in the body frame doesnot necessarily imply a large net propulsion speed inthe laboratory frame. The mean swimming speedsin the laboratory frame for both chirality configura-tions are shown in figure 7b(i–ii). The couplingbetween the swimming kinematics in all directions
1
0 0.1 0.2 0.3r
0.4 0.5 0 0.1 0.2 0.3r
0.4 0.5
2
3
4
5
6
7
8
9
10(a)
(b)
(c)
R
R
R
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
0
0.5
1.0
1.5
0
0
–0.2
–0.4
–0.6
–0.8
–1.0
–1.2
–1.2
–1.4
0.2
0.4
0.6
0.8
1.0
1.2
1.40.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0.60.5
0.4
0.3
0.2
0.1
–0.1
–0.2
0
0
0.2
0.4
0.6
0.8
1
RrR
(i) (ii)
(i) (ii)
10(i) (ii)
Figure 7. Parametric study of the dependence of the average swimming speed in the body-fixed frame Ub (a(i),(ii)) and in thelaboratory frame ~U+ (b(i),(ii)) as a function of the dimensionless parameters R and r (see the schematic for geometrical illus-tration). (c(i),(ii)) show the average swimming velocity in the z-direction in the body frame, kUzl, as a function of R and r.(a–c) on the left (right) refer to the same- (opposite-) chirality configuration. The dotted line in panel (c(ii)) represent the con-tour of kUzl ¼ 0. Geometric data of Cu. melleus spermatozoa are used for other fixed parameters. (Online version in colour.)
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produces more complicated variations in the propul-sion speed as a function of R and r. Non-monotonicbehaviours are observed in both cases.
We have already noted that some superhelical swim-mers propel themselves surprisingly in the samedirection as the wave propagation, unlike for planar orsingle helical wave propulsion. For the range of para-meters explored in this paper, this direction reversal isfound to occur only in the opposite-chirality configur-ation for sufficiently large R and r (figure 7c(ii)),while the average propagation velocity kUzl is always
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negative for the same-chirality configuration (i.e. theswimming direction is always opposite to the directionof the propagating wave; figure 7c(i)). In figure 8, weshow the detailed swimming velocities of two oppo-site-chirality superhelical swimmers, where one ofthem has its longitudinal propulsion in the oppositedirection relative to the wave propagation (R ¼ 7, r ¼0.4; figure 8a), and the other in the same direction asthe wave propagation (R ¼ 9, r ¼ 0.4; figure 8b). It isintriguing that a small change in the geometry (seethe corresponding superhelices in figure 8a(ii),b(ii))
Ux Uy Uz Ux Uy Uz
(a) (i) (ii) (i) (ii)(b)0.8 0.8
0.4
0
–0.4
–0.8
0.250.16
0.12
0.08
0.04
0
0.20
0.15
0.10
0.05
00.02 0.02
0–0.020
–0.02 –0.02
0.020 –0.02
0
0.4
0
–0.4
–0.80 20 40 60 80 100 120 140 160 0 20 40 60 80 100 120 140 160
Figure 8. Three-dimensional swimming velocities in the laboratory frame (a(i), b(i)) of two opposite-chirality superhelicalswimmers, and their corresponding geometries (a(ii), b(ii)). (a) R ¼ 7, r ¼ 0.4 and Uz is negative, while the wave propagationis towards the positive z-direction; (b) R ¼ 9, r ¼ 0.4 and Uz is positive, while the wave propagation is towards the positivez-direction. Geometric data of Cu. melleus spermatozoa are used for other parameters. (Online version in colour.)
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can lead to a swimming direction reversal. We argue inthe next section (§3.4) that this transition is relatedto the hydrodynamic interaction between distinctparts of the superhelical flagellum.
3.4. Comparison between slender body theoryand resistive force theory
In order to consider the relative importance of non-localhydrodynamic interactions in the swimming of super-helices, we now compare our results with thoseobtained using the more commonly used local dragmodel (equation (2.6), but neglecting the non-localterm K). The local drag model (so-called RFT) ignoreshydrodynamic interactions between distinct parts of thecurved flagellum, and is expected to work well [21–23]for simple geometries, where different parts of thebody are sufficiently well separated. However, formore complicated geometries, the local drag modelmay not capture even the correct qualitative features.Recently, Jung et al. [20] studied the rotationaldynamics of opposite-chirality superhelices and founda qualitative discrepancy (the rotational direction ofthe superhelix when being towed in a viscous flow)between the experimental results and the predictionsfrom the RFT. Other recent works have also shownthe inadequacy of the local drag model [32,37].
Figure 9 shows the swimming speeds computed withboth non-local SBT and RFT using two sets of geo-metrical data: Cu. melleus (figure 9a (same-chirality)and figure 9b (opposite-chirality)), and T. molitor(figure 9c (same-chirality) and figure 9d (opposite-chir-ality)). In these two cases, all other parameters are fixedbut the amplitude ratio r is varied from 0 to 0.5.Increasing r complicates the swimmer geometry, andthe hydrodynamic interactions are expected to bemore significant for large r. For the case of Cu. melleus(figure 9a,b), we see good (even quantitative) agreementbetween the results form the SBT (solid lines) and theRFT (dashed lines). However, for the case of T. molitor(figure 9c,d), which has larger values of R and K, thedeviation between the two models becomes significantfor the same-chirality configuration (figure 9c) as r
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increases. There are even qualitative discrepancies: inthe opposite-chirality configuration (figure 9c) for largevalues of r, the RFT fails to capture the transition inthe sign of kUzl predicted by the SBT. As no such tran-sition is found when the hydrodynamic interactions areignored, this transition may be attributed to non-localhydrodynamic interactions of the body with itself. In gen-eral, and perhaps unsurprisingly, we have shown that thelocal drag model breaks down when the geometry of thestructure is sufficiently intricate.
3.5. Asymptotic analysis
In the locomotion of some species of spermatozoa, theminor wave amplitude is much smaller than the majorwave amplitude (tables 1 and 2), which motivates usto perform an asymptotic analysis for r� 1. Such anasymptotic analysis linearizes the problem geometri-cally and allows the nonlinear effects to be taken intoaccount order by order, making the problem moreamenable to mathematical analysis. However, even inthe asymptotic consideration, three-dimensional forceand torque balances do not yield tractable analyticalresults. Hence, in the spirit of Chwang & Wu [33], weperform the force and torque balances only in the longi-tudinal (z-) direction using the RFT, which is expectedto be at least qualitatively correct in the asymptoticlimit r� 1. The local force fz and torque mz may beexpressed as:
fz ¼ cUzFz ðt; sÞUz þ cVzFz ðt; sÞVz þ cFz ðt; sÞ, ð3:2Þ
and
mz ¼ cUzMz ðt; sÞUz þ cVzMz ðt; sÞVz þ cMz ðt; sÞ; ð3:3Þ
where the coefficients are determined analytically fromequation (2.6) (without the non-local operator) and thegeometry of the swimmer (equations (2.12)–(2.14)).We consider regular perturbation expansions in r forevery term in equations (3.2) and (3.3), and enforcethe force-free and torque-free conditions order byorder. A non-zero time-averaged swimming velocityenters at O(r2). The leading order mean swimming vel-ocities for the cases of same- and opposite-chirality
r r
–0.20
–0.15
–0.10
–0.05
0
0.05
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0.15(a) (b)
(c) (d)
–0.20
–0.15
–0.10
–0.05
0
0.05
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0.15
0 0.1 0.2 0.3 0.4 0.5–0.7
–0.6
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0
0.1
0.2
0 0.1 0.2 0.3 0.4 0.5–0.2
–0.1
0
0.1
0.2
0.3
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0.5
0.6
Uz (SBT)Uy (SBT)
Ux (SBT)
Ux (RFT)
Uy (RFT)Uz (RFT)
Uz (SBT)Uy (SBT)
Ux (SBT)
Ux (RFT)
Uy (RFT)Uz (RFT)
Uz (SBT)Uy (SBT)
Ux (SBT)
Ux (RFT)
Uy (RFT)Uz (RFT)
Uz (SBT)Uy (SBT)
Ux (SBT)
Ux (RFT)
Uy (RFT)Uz (RFT)
Figure 9. Comparison between the RFT (dashed lines) and the non-local SBT (solid lines) using geometric data of Cu. melleusspermatozoa (a,b), and T. molitor spermatozoa (c,d). The panels on the left (right) refer to the same- (opposite-) chiralityconfiguration. (Online version in colour.)
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configurations read
~U+ � R2r2
2L2ð1þ R2K 2Þ3=2� fð2� 2L2Þðj� 1Þ
þ R2K 2½2ðj� 1 + KÞ � L2ð2j� 2 + KÞ�
� 2½j� 1þ R2K2ðj� 1 + KÞ� cos Lg;ð3:4Þ
where j ¼ j?=jk is the ratio of the drag coefficientsin the normal direction to the longitudinal direction(jk ¼ 4p=c0; j? ¼ 8p=ð2þ c0Þ). For very slender fila-ments, j � 2. We note that the substitution K! 2Kin equation (3.4) converts the same-chirality speed ~Uþto the opposite-chirality speed ~U�. The propulsionspeed has a quadratic dependence on the minor waveamplitude for small r, which is also found in planar[38] and helical [39] geometries.
These asymptotic results (equation (3.4)) are com-pared with the finite-amplitude simulations (both theRFT and the SBT predictions) in figure 10, for a verysmall amplitude ratio of r ¼ 0.01 and a slendernessratio of e ¼ 1/1000. There is excellent agreementbetween the asymptotic results and the finite amplitudeRFT simulations. The discrepancy between the non-local SBT and the local drag model highlights theimportance of non-local hydrodynamic interactions forsuch organisms.
We conclude by pointing out an intriguing theoreti-cal curiosity, that drag anisotropy ( j = 1) is not
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required for superhelical swimming: setting j ¼ 1 inequation (3.4), non-zero mean propulsion velocities (ofequal magnitude but opposite signs) are obtained forboth chirality configurations,
~U+ðj ¼ 1Þ ¼+R4K 3ðL2 � 2þ 2 cos LÞ
2Lð1þ R2K2Þ3=2r2: ð3:5Þ
Note that Becker et al.’s [40] argument of the require-ment of drag anisotropy for locomotion is only true forinextensible swimmers and does not apply here. Thesuperhelical kinematics described (equations (2.12)–(2.14)) are possible only when extensibility is allowed;the minor helix is built upon another curved structure(the major helix) and local extension and contractionis implied in the wave kinematics. When extensibilityis permitted, the relaxation of the drag anisotropyrequirement has been recently shown [41]. That theswimming speed is non-zero is owing to intrinsic vari-ations in length (and hence drag) embedded in thecurved geometry of the superhelices. A minor helixbuilt upon a major helix has relatively shorter lengthsin the regions closer to the longitudinal axis, creatingan overall imbalance of hydrodynamic drag even inthe isotropic drag case ( j ¼ 1). A similar example is atoroidal helix (a helix built upon a circle), which is anidealized model studied recently for dinoflagellates[42]. We expect that the propagation of a wave alonga toroidal helix should also require extensibility, andthat propulsion is still possible even without draganisotropy [41].
0.1 0.2 0.3 0.4 0.51
2
3
4
5
6
7
8
9
10(a) (b)
R
K0.1 0.2 0.3 0.4 0.5
K
0.0005
0.0003
0.001
0.00150.0020
0.002
5
0.0025
0.0020
0.0015
0.001
0.0005
0.0003
Figure 10. Comparison of the asymptotic results (dotted lines) with the finite-amplitude simulations using the RFT (dashedlines) and the non-local SBT (solid lines). Shown are the contour lines of the average swimming velocity U+ for the(a) same-chirality, and (b) opposite-chirality configurations for a small minor to major wave amplitude ratio of r ¼ 0.01.(Online version in colour.)
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4. DISCUSSION
In this paper, we have studied a morphologically interest-ing double-wave structure exhibited by various insectspermatozoa. The construction of such spermatozoa isconsiderably more complex than those for flagellawhich exhibit simpler planar or helical waves: the flagel-lum does not only have a more complicated 9 þ 9 þ 2arrangement of microtubules but also mitochondrialderivatives and accessory bodies running along the axo-neme [5]. We have mathematically idealized thedouble-wave structure as a superhelical structure andpresented a hydrodynamic study on superhelical swim-ming. The available data are primitive and sparse;nevertheless, we consider the agreement between exper-imental measurements and the theory explored hereinto be quite reasonable. Through numerical experiments,we have found that the major wave speed has little contri-bution to propulsion when the sperm head is small, as isthe case for insect spermatozoa. When there is no spermhead, the propulsion speed is independent of the majorwave speed and depends entirely upon the minor wavespeed. We have also explored the dependence of thepropulsion speed on the dimensionless major wave ampli-tude R and the ratio of the minor to major waveamplitudes r (figure 7), and counterintuitive behaviourshave been found for the opposite-chirality configuration.In particular, we have found that propulsion andwave propagation can occur in the same direction forsuperhelices in the opposite-chirality configuration.
The present study suggests that the major wave hasnegligible influence on the motility of a superhelicalswimmer. This finding favours the recent hypothesisby Werner et al. [10] that the major helical wave is astatic (non-propagating) structure; the minor wavestructure is solely responsible for the motility, and theapparent major wave propagation is simply owing tothe passive rotation of the entire geometry (see §1).However, in the study by Baccetti et al. [16] on themotility of Ce. capitata, they have adopted the same
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experimental techniques as in Gibbons et al. [43],which distinguished the rolling frequency from theapparent beat frequency. In their work, the rolling fre-quency was measured by stroboscopic observation ofthe eccentrically attached sperm head, and the flagellarbeat frequency was measured by the same means withthe sperm head adhered to the bottom of the obser-vation dish. Using this method, the major wave speedmeasured should be taken as an active propagationspeed. We are not in a position to provide a definiteanswer on whether or not the major wave propagatesactively in actual insect spermatozoa. However, accord-ing to the present study, we suspect that there might besome biological reasons, other than motility, for themajor wave to propagate actively. We also do not knowif the propagation of the major wave is a biological prere-quisite for the propagation of the minor wave. Furtherbiological studies are required to answer these questions.
It is illuminating to compare the superhelical swim-ming studied here with regular helical swimming. Theswimming trajectories are qualitatively different in thetwo cases: in regular helical swimming, the trajectoryis a regular helix, whereas in superhelical swimmingthe trajectory is doubly helicated. In addition, a regularhelical flagellum cannot swim on its own; a sperm headis required to swim (as the absence of a sperm head ren-ders the deformation of the regular helix a rigid bodymotion). By contrast, the propagation of superhelicalwaves along a flagellum is in general not equivalent torigid body motion, and hence ‘head-less’ swimming ispossible. This might help explain the presence of onlya small slender sperm head in insect spermatozoa. Inother words, superhelical swimming can be viewed asan alternative mechanism to the regular helicalswimming when only a small sperm head is available.
Finally, a non-local SBT was used in this work andcompared with a simpler and widely used local dragmodel. We showed that the RFT failed to captureeven the qualitative features of the swimmer when the
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geometry becomes complicated. The results suggestthat further hydrodynamic studies on superhelicalstructures require more advanced models than thelocal RFT, such as the SBT employed here. Optimiz-ation with respect to the efficiency of the swimmertaking into account the viscous dissipation, and otherenergy costs owing to bending and internal sliding offilaments [31], which have been neglected in thiswork, will be interesting for future work.
This work was funded in part by the US National ScienceFoundation (Grant CBET-0746285 to E.L.) and theCroucher Foundation (through a scholarship to O.S.P.).Useful conversations with Dr Michael Werner and Dr AnneSwan are gratefully acknowledged.
REFERENCES
1 Biewener, A. A. 2003 Animal locomotion. Oxford, UK:Oxford University Press.
2 Fauci, L. J. & Dillon, R. 2006 Biofluid mechanics of repro-duction. Annu. Rev. Fluid Mech. 38, 371–394. (doi:10.1146/annurev.fluid.37.061903.175725)
3 Lauga, E. & Powers, T. 2009 The hydrodynamics of swim-ming microorganisms. Rep. Prog. Phys. 72, 096601.(doi:10.1088/0034-4885/72/9/096601)
4 Brennen, C. & Winet, H. 1977 Fluid mechanics of pro-pulsion by cilia and flagella. Annu. Rev. Fluid Mech. 9,339–398. (doi:10.1146/annurev.fl.09.010177.002011)
5 Werner, M. & Simmons, L. W. 2008 Insect sperm motility.Biol. Rev. 83, 191–208. (doi:10.1111/j.1469-185X.2008.00039.x)
6 Baccetti, B. 1972 Insect sperm cells. Adv. Insect Physiol. 9,315–397. (doi:10.1016/S0065-2806(08)60279-9)
7 Rikmenspoel, R. & Isles, C. A. 1985 Digitized precisionmeasurements of the movements of sea urchin spermflagella. Biophys. J. 47, 395–410. (doi:10.1016/S0006-3495(85)83931-X)
8 Jamieson, B. G. M., Dallai, R. & Afzelius, B. A. 1999Insects: their spermatozoa and phylogeny. Enfield, NH:Science Publishers Inc.
9 Morrow, E. H. 2004 How the sperm lost its tail: the evol-ution of aflagellate sperm. Biol. Rev. 79, 795–814.(doi:10.1017/S1464793104006451)
10 Werner, M., Tscheulin, T., Speck, T., Zissler, D. &Peschke, K. 2002 Ultrastructure and motility pattern ofthe spermatozoa of Aleochara curtula (Coleoptera,Staphylinidae). Arth. Struct. Dev. 31, 243–254. (doi:10.1016/S1467-8039(02)00046-4)
11 Baccetti, B., Burrini, A. G., Dallai, R., Giusti, F., Mazzini,M., Renieri, T., Rosati, F. & Selmi, G. 1973 Structure andfunction in the spermatozoon of Tenebrio molitor (thespermatozoon of Arthropoda. XX). J. Mechanochem.Cell Motil. 2, 149–161.
12 Baccetti, B., Burrini, A. G., Dallai, R., Pallini, V., Periti, P.,Piantelli, F., Rosati, F. & Selmi, G. 1973 Structure andfunction in the spermatozoon of Bacilus rossius: the sperma-tozoon of Arthropoda. XIX. J. Ultrastruct. Res. 44, 1–73.
13 Phillips, D. M. 1974 Structural variants in invertebratesperm flagella and their relationship to motility. In Ciliaand flagella (ed. M. A. Sleigh), pp. 379–402, London,UK: Academic Press.
14 Linley, J. R. 1979 Activity and motility of spermatozoa ofCulicoides melleus (Diptera: Ceratopogonidae). Entomol.Exp. Appl. 26, 85–96. (doi:10.1111/j.1570-7458.1979.tb02902.x)
J. R. Soc. Interface (2012)
15 Swan, M. A. 1981 The generation and propagation of doublewaves in mosquito (Aedes notoscriptus) sperm-tails.Gamete Res. 4, 241–250. (doi:10.1002/mrd.1120040308)
16 Baccetti, B., Gibbons, B. H. & Gibbons, I. R. 1989Bidirectional swimming in spermatozoa of Tephritidflies. J Submicrosc. Cytol. Pathol. 21, 619–625.
17 Bressac, C., Joly, D., Devaux, J., Serres, C., Feneus, D. &Lachaise, D. 1991 Comparative kinetics of short and longsperm in sperm dimorphic Drosophila species. Cell Motil.Cytoskel. 19, 269–274. (doi:10.1002/cm.970190405)
18 Curtis, S. K. & Benner, D. B. 1991 Movement of sperma-tozoa of Megaselia scalaris (Diptera: Brachycera:Cyclorrhapha: Phoridae) in artificial and natural fluids.J. Morphol. 210, 85–99. (doi:10.1002/jmor.1052100108)
19 Werner, M., Gack, C., Speck, T. & Peschke, K. 2007Queue up, please! Spermathecal filling in the rove beetleDrusilla canaliculata (Coleoptera, Staphylinidae). Natur-wissenschaften 94, 837–841. (doi:10.1007/s00114-007-0257-8)
20 Jung, S., Mareck, K., Fauci, L. & Shelley, M. 2007Rotational dynamics of a superhelix towed in a Stokesfluid. Phys. Fluids 19, 103105. (doi:10.1063/1.2800287)
21 Wiggins, C. H., Riveline, D., Ott, A. & Goldstein, R. E.1998 Trapping and wiggling: elastohydrodynamics ofdriven microfilaments. Biophys. J. 74, 1043–1060.(doi:10.1016/S0006-3495(98)74029-9)
22 Yu, T. S., Lauga, E. & Hosoi, A. E. 2006 Experimentalinvestigations of elastic tail propulsion at low Reynoldsnumber. Phys. Fluids 18, 091701. (doi:10.1063/1.2349585)
23 Riedel-Kruse, I. H., Hilfinger, A., Howard, J. & Julicher,F. 2007 How molecular motors shape the flagellar beat.HFSP J. 1, 192–208. (doi:10.2976/1.2773861)
24 Johnson, R. E. 1980 An improved slender-body theory forstokes flow. J. Fluid Mech. 99, 411–431. (doi:10.1017/S0022112080000687)
25 Atkinson, K. E. 1997 The numerical solution of integralequations of the second kind. Cambridge, UK: CambridgeUniversity Press.
26 Gotz, T. 2000 Interactions of fibers and flow: asymp-totics, theory and numerics. PhD thesis, University ofKaiserslautern, Germany.
27 Tornberg, A. K. & Shelley, M. J. 2004 Simulating thedynamics and interactions of flexible fibers in Stokes flows.J. Comput. Phys. 196, 8–40. (doi:10.1016/j.jcp.2003.10.017)
28 Lighthill, J. 1975 Mathematical biofluid dynamics.Philadelphia, PA: SIAM.
29 Chattopadhyay, S., Moldovan, R., Yeung, C. & Wu, X. L.2006 Swimming efficiency of bacterium Escherichia coli.Proc. Natl Acad. Sci. USA 103, 13 712–13 717. (doi:10.1073/pnas.0602043103)
30 Childress, S. 1981 Mechanics of swimming and flying.Cambridge, UK: Cambridge University Press.
31 Spagnolie, S. E. & Lauga, E. 2010 The optimal elastic fla-gellum. Phys. Fluids 22, 031901. (doi:10.1063/1.3318497)
32 Spagnolie, S. E. & Lauga, E. 2011 Comparative hydrodyn-amics of bacterial polymorphism. Phys. Rev. Lett. 106,058103. (doi:10.1103/PhysRevLett.106.058103)
33 Chwang, A. T. & Wu, T. Y. 1971 A note on the helicalmovement of micro-organisms. Proc. R. Soc. Lond. B178, 327–346. (doi:10.1098/rspb.1971.0068)
34 Keller, J. B. & Rubinow, S. I. 1976 Swimming of flagel-lated microorganisms. Biophys. J. 16, 151–170. (doi:10.1016/S0006-3495(76)85672-X)
35 Margulis, L., Chapman, M., Guerrero, R. & Hall, J. 2006The last eukaryotic common ancestor (leca): acquisitionof cytoskeletal motility from aerotolerant spirochetesin the proterozoic eon. Proc. Natl Acad. Sci. USA 103,13 080–13 085. (doi:10.1073/pnas.0604985103)
1924 Hydrodynamics of insect spermatozoa O. S. Pak et al.
on August 30, 2012rsif.royalsocietypublishing.orgDownloaded from
36 Nicastro, D., McIntosh, J. R. & Baumeister, W. 2005 3Dstructure of eukaryotic flagella in a quiescent state revealedby cryo-electron tomography. Proc. Natl Acad. Sci. USA102, 15 889–15 894. (doi:10.1073/pnas.0508274102)
37 Chattopadhyay, S. & Wu, X. L. 2009 The effect of long-range hydrodynamic interaction on the swimming of asingle bacterium. Biophys. J. 96, 2023–2028. (doi:10.1016/j.bpj.2008.11.046)
38 Taylor, G. 1951 Analysis of the swimming of microscopicorganisms. Proc. R. Soc. Lond. A 209, 447–461. (doi:10.1098/rspa.1951.0218)
39 Taylor, G. 1952 The action of waving cylindrical tails inpropelling microscopic organisms. Proc. R. Soc. Lond. A211, 225–239. (doi:10.1098/rspa.1952.0035)
J. R. Soc. Interface (2012)
40 Becker, L. E., Koehler, S. A. & Stone, H. A. 2003 On self-propulsion of micro-machines at low Reynolds number:Purcell’s three-link swimmer. J. Fluid Mech. 490, 15–35.(doi:10.1017/S0022112003005184)
41 Pak, O. S. & Lauga, E. 2011 Extensibility enables loco-motion under isotropic drag. Phys. Fluids 23, 081702.(doi:10.1063/1.3624790)
42 Nguyen, H., Ortiz, R., Cortez, R. & Fauci, L. 2011 The actionof waving cylindrical rings in a viscous fluid. J. Fluid Mech.671, 574–586. (doi:10.1017/S0022112010006075)
43 Gibbons, B. H., Baccetti, B. & Gibbons, I. R. 1985 Liveand reactivated motility in the 9þ0 flagellum of Anguillasperm. Cell Motil. 5, 333–350. (doi:10.1002/cm.970050406)