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| <?xml version="1.0" encoding="UTF-8"?><!DOCTYPE article PUBLIC "-//NLM//DTD JATS (Z39.96) Journal Archiving and Interchange DTD v1.1d1 20130915//EN" "JATS-archivearticle1.dtd"><article article-type="research-article" dtd-version="1.1d1" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><front><journal-meta><journal-id journal-id-type="nlm-ta">elife</journal-id><journal-id journal-id-type="hwp">eLife</journal-id><journal-id journal-id-type="publisher-id">eLife</journal-id><journal-title-group><journal-title>eLife</journal-title></journal-title-group><issn publication-format="electronic">2050-084X</issn><publisher><publisher-name>eLife Sciences Publications, Ltd</publisher-name></publisher></journal-meta><article-meta><article-id pub-id-type="publisher-id">02403</article-id><article-id pub-id-type="doi">10.7554/eLife.02403</article-id><article-categories><subj-group subj-group-type="display-channel"><subject>Research article</subject></subj-group><subj-group subj-group-type="heading"><subject>Biophysics and structural biology</subject></subj-group><subj-group subj-group-type="heading"><subject>Human biology and medicine</subject></subj-group></article-categories><title-group><article-title>Rheotaxis facilitates upstream navigation of mammalian sperm cells</article-title></title-group><contrib-group><contrib contrib-type="author" equal-contrib="yes" id="author-11030"><name><surname>Kantsler</surname><given-names>Vasily</given-names></name><xref ref-type="aff" rid="aff1"/><xref ref-type="fn" rid="equal-contrib">†</xref><xref ref-type="fn" rid="pa1">‡</xref><xref ref-type="fn" rid="pa2">§</xref><xref ref-type="other" rid="par-1"/><xref ref-type="fn" rid="con1"/><xref ref-type="fn" rid="conf1"/></contrib><contrib contrib-type="author" equal-contrib="yes" id="author-11031"><name><surname>Dunkel</surname><given-names>Jörn</given-names></name><xref ref-type="aff" rid="aff1"/><xref ref-type="fn" rid="equal-contrib">†</xref><xref ref-type="fn" rid="pa3">¶</xref><xref ref-type="other" rid="par-1"/><xref ref-type="fn" rid="con2"/><xref ref-type="fn" rid="conf1"/></contrib><contrib contrib-type="author" id="author-11032"><name><surname>Blayney</surname><given-names>Martyn</given-names></name><xref ref-type="aff" rid="aff2"/><xref ref-type="fn" rid="con4"/><xref ref-type="fn" rid="conf1"/></contrib><contrib contrib-type="author" corresp="yes" id="author-3082"><name><surname>Goldstein</surname><given-names>Raymond E</given-names></name><xref ref-type="aff" rid="aff1"/><xref ref-type="corresp" rid="cor1">*</xref><xref ref-type="other" rid="par-1"/><xref ref-type="fn" rid="con3"/><xref ref-type="fn" rid="conf1"/></contrib><aff id="aff1"><institution content-type="dept">Department of Applied Mathematics and Theoretical Physics</institution>, <institution>University of Cambridge</institution>, <addr-line><named-content content-type="city">Cambridge</named-content></addr-line>, <country>United Kingdom</country></aff><aff id="aff2"><institution content-type="dept">Science</institution>, <institution>Bourn Hall Clinic</institution>, <addr-line><named-content content-type="city">Cambridge</named-content></addr-line>, <country>United Kingdom</country></aff></contrib-group><contrib-group content-type="section"><contrib contrib-type="editor"><name><surname>Hyman</surname><given-names>Anthony A</given-names></name><role>Reviewing editor</role><aff><institution>Max Planck Institute of Molecular Cell Biology and Genetics</institution>, <country>Germany</country></aff></contrib></contrib-group><author-notes><corresp id="cor1"><label>*</label>For correspondence: <email>R.E.Goldstein@damtp.cam.ac.uk</email></corresp><fn fn-type="con" id="equal-contrib"><label>†</label><p>These authors contributed equally to this work</p></fn><fn fn-type="present-address" id="pa1"><label>‡</label><p>Department of Physics, University of Warwick, Coventry, United Kingdom</p></fn><fn fn-type="present-address" id="pa2"><label>§</label><p>Skolkovo Institute of Science and Technology, Skolkovo, Russia</p></fn><fn fn-type="present-address" id="pa3"><label>¶</label><p>Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, United States</p></fn></author-notes><pub-date date-type="pub" publication-format="electronic"><day>27</day><month>05</month><year>2014</year></pub-date><pub-date pub-type="collection"><year>2014</year></pub-date><volume>3</volume><elocation-id>e02403</elocation-id><history><date date-type="received"><day>26</day><month>01</month><year>2014</year></date><date date-type="accepted"><day>29</day><month>04</month><year>2014</year></date></history><permissions><copyright-statement>© 2014, Kantsler et al</copyright-statement><copyright-year>2014</copyright-year><copyright-holder>Kantsler et al</copyright-holder><license xlink:href="http://creativecommons.org/licenses/by/3.0/"><license-p>This article is distributed under the terms of the <ext-link ext-link-type="uri" xlink:href="http://creativecommons.org/licenses/by/3.0/">Creative Commons Attribution License</ext-link>, which permits unrestricted use and redistribution provided that the original author and source are credited.</license-p></license></permissions><self-uri content-type="pdf" xlink:href="elife02403.pdf"/><abstract><object-id pub-id-type="doi">10.7554/eLife.02403.001</object-id><p>A major puzzle in biology is how mammalian sperm maintain the correct swimming direction during various phases of the sexual reproduction process. Whilst chemotaxis may dominate near the ovum, it is unclear which cues guide spermatozoa on their long journey towards the egg. Hypothesized mechanisms range from peristaltic pumping to temperature sensing and response to fluid flow variations (rheotaxis), but little is known quantitatively about them. We report the first quantitative study of mammalian sperm rheotaxis, using microfluidic devices to investigate systematically swimming of human and bull sperm over a range of physiologically relevant shear rates and viscosities. Our measurements show that the interplay of fluid shear, steric surface-interactions, and chirality of the flagellar beat leads to stable upstream spiralling motion of sperm cells, thus providing a generic and robust rectification mechanism to support mammalian fertilisation. A minimal mathematical model is presented that accounts quantitatively for the experimental observations.</p><p><bold>DOI:</bold> <ext-link ext-link-type="doi" xlink:href="10.7554/eLife.02403.001">http://dx.doi.org/10.7554/eLife.02403.001</ext-link></p></abstract><abstract abstract-type="executive-summary"><object-id pub-id-type="doi">10.7554/eLife.02403.002</object-id><title>eLife digest</title><p>A sperm cell must complete a long and taxing journey to stand a chance of fertilising an egg cell. This quest covers a distance that is thousands of times longer than the length of a sperm cell. It also passes through the diverse environments of the cervix, the uterus and, finally, the oviduct, where there might be an egg to fertilise. How the sperm cells manage to stay on course over this distance is a mystery, although it has been suggested that many different factors, including chemical signals and fluid flow, are involved.</p><p>The fluids that the sperm cells travel through are not static. Evidence suggests that contractions of the cervix and uterus help to pump sperm cells along the first part of their journey. However, mucus flows out of the oviduct in the opposite direction to way the sperm cells need to go.</p><p>Sperm cells mostly move along the walls of the cervix, uterus, and oviduct. This means that sperm cells must contend with two properties of the fluids they travel through—the viscosity (or ‘thickness’) of the fluid, and the fact that different parts of the fluid will flow at different speeds, depending on how close it is to the wall (‘shear flow’).</p><p>Kantsler et al. have now used a technique called microfluidics—which involves forcing tiny amounts of liquid to flow through very narrow channels—to study how the movement of human and bull sperm cells along a surface is affected by the viscosity and flow rate of the fluid they are swimming through. The sperm cells were found to swim upstream, moving along the walls of the channels in a spiral movement. This is likely to help the sperm cells to find the egg, because spiralling around the oviduct will increase the chances of meeting the egg.</p><p>Kantsler et al. also built a mathematical model that describes how the sperm cells move. Although further work is needed to better understand the role played by chemical signals, understanding how fluid flow and viscosity influence sperm cells could lead to more effective artificial insemination techniques.</p><p><bold>DOI:</bold> <ext-link ext-link-type="doi" xlink:href="10.7554/eLife.02403.002">http://dx.doi.org/10.7554/eLife.02403.002</ext-link></p></abstract><kwd-group kwd-group-type="author-keywords"><title>Author keywords</title><kwd>sperm</kwd><kwd>rheotaxis</kwd><kwd>fertilization</kwd></kwd-group><kwd-group kwd-group-type="research-organism"><title>Research organism</title><kwd>human</kwd><kwd>other</kwd></kwd-group><funding-group><award-group id="par-1"><funding-source><institution-wrap><institution-id institution-id-type="FundRef">http://dx.doi.org/10.13039/501100000781</institution-id><institution>European Research Council</institution></institution-wrap></funding-source><award-id>247333</award-id><principal-award-recipient><name><surname>Kantsler</surname><given-names>Vasily</given-names></name><name><surname>Dunkel</surname><given-names>Jörn</given-names></name><name><surname>Goldstein</surname><given-names>Raymond E</given-names></name></principal-award-recipient></award-group><funding-statement>The funders had no role in study design, data collection and interpretation, or the decision to submit the work for publication.</funding-statement></funding-group><custom-meta-group><custom-meta><meta-name>elife-xml-version</meta-name><meta-value>2</meta-value></custom-meta><custom-meta specific-use="meta-only"><meta-name>Author impact statement</meta-name><meta-value>Mammalian sperm subject to shear flow swim in upstream spirals along the walls bounding such flows, thereby demonstrating a robust mechanism for upstream navigation to the ovum.</meta-value></custom-meta></custom-meta-group></article-meta></front><body><sec id="s1" sec-type="intro"><title>Introduction</title><p>During their journey from ejaculation to fertilisation, human spermatozoa have to find and maintain the right swimming direction over distances that may exceed their head-to-tail length (∼100 μm) by a 1000-fold. On their path to the egg cell, mammalian sperm encounter varied physiological environments and are exposed to a variety of chemical gradients and must overcome counterflows. Whilst chemotactic sensing (<xref ref-type="bibr" rid="bib18">Kaupp et al., 2008</xref>) is assumed to provide important guidance in the immediate vicinity of the ovum (<xref ref-type="bibr" rid="bib31">Spehr et al., 2003</xref>), it is not known which biochemical (<xref ref-type="bibr" rid="bib3">Brenker et al., 2012</xref>) or physical mechanisms (<xref ref-type="bibr" rid="bib36">Winet et al., 1984</xref>) keep the sperm cells on track as they pass through the rugged landscapes of cervix, uterus, and oviduct (<xref ref-type="bibr" rid="bib17">Katz et al., 2005</xref>; <xref ref-type="bibr" rid="bib6">Eisenbach and Giojalas, 2006</xref>; <xref ref-type="bibr" rid="bib33">Suarez and Pacey, 2006</xref>). The complexity of the mammalian reproduction process and not least the lack of quantitative data make it very difficult to assess the relative importance of the various proposed long-distance navigation mechanisms (<xref ref-type="bibr" rid="bib6">Eisenbach and Giojalas, 2006</xref>; <xref ref-type="bibr" rid="bib9">Fauci and Dillon, 2006</xref>), ranging from cervix contractions (<xref ref-type="bibr" rid="bib9">Fauci and Dillon, 2006</xref>; <xref ref-type="bibr" rid="bib33">Suarez and Pacey, 2006</xref>) and chemotaxis (<xref ref-type="bibr" rid="bib18">Kaupp et al., 2008</xref>) to thermotaxis (<xref ref-type="bibr" rid="bib2">Bahat et al., 2003</xref>) and rheotaxis (<xref ref-type="bibr" rid="bib23">Miki and Clapham, 2013</xref>). Aiming to understand not only qualitatively but also quantitatively how fluid-mechanical effects may help steer mammalian spermatozoa over large distances, we report here a combined experimental and theoretical study of sperm swimming in microfluidic channels, probing a wide range of physiologically relevant conditions of shear and viscosity. For both human and bull spermatozoa, we find that their physical response to shear flow, combined with an effective shape-regulated surface attraction (<xref ref-type="bibr" rid="bib16">Kantsler et al., 2013</xref>) and head–tail counter-precession, favours an upstream spiralling motion along channel walls. The robustness of this fluid-mechanical rectification mechanism suggests that it is likely to play a key role in the long-distance navigation of mammalian sperm cells. Thus, the detailed analysis reported below not only yields new quantitative insights into the role of biophysical processes during mammalian reproduction but could also lead to new diagnostic tools and improved artificial insemination techniques (<xref ref-type="bibr" rid="bib22">Merviel et al., 2010</xref>).</p><p>Recent experiments on red abalone (<xref ref-type="bibr" rid="bib26">Riffell and Zimmer, 2007</xref>; <xref ref-type="bibr" rid="bib38">Zimmer and Riffell, 2011</xref>), a large marine snail that fertilises externally, showed that weak fluid flows can be beneficial to the reproduction of these organisms, suggesting that shear flows could have acted as a selective pressure in gamete evolution. In higher organisms, which typically fertilise internally, sperm transport is much more complex and the importance of shear flows relative to chemotaxis, peristaltic pumping, or thermotaxis still poses an open problem as it is difficult to perform well-controlled in vivo studies. The complex uterine and oviduct topography (<xref ref-type="bibr" rid="bib33">Suarez and Pacey, 2006</xref>) and large travelling distances render it unlikely that local chemotactic gradients steer sperm cells during the initial and intermediate stages of the sexual reproduction process. Experimental evidence (<xref ref-type="bibr" rid="bib19">Kunz et al., 1996</xref>) suggests that rapid sperm transport right after insemination is supported by peristaltic pumping driven by muscular contractions of the uterus, but it is not known how sperm navigate in the oviduct. Thermotaxis, the directed response of sperm to local temperature differences, was proposed as a possible long-range rectification mechanism of sperm swimming in rabbits (<xref ref-type="bibr" rid="bib2">Bahat et al., 2003</xref>), but recently questioned (<xref ref-type="bibr" rid="bib23">Miki and Clapham, 2013</xref>) as it is likely to be inhibited by convective currents that form in the presence of temperature gradients. On the other hand, it has long been known that, similar to bacteria (<xref ref-type="bibr" rid="bib21">Marcos et al., 2012</xref>) and algae (<xref ref-type="bibr" rid="bib4">Chengala et al., 2013</xref>), mammalian sperm (<xref ref-type="bibr" rid="bib1">Adolphi, 1905</xref>; <xref ref-type="bibr" rid="bib27">Roberts, 1970</xref>) are capable of performing rheotaxis, by aligning against a surrounding flow (<xref ref-type="bibr" rid="bib21">Marcos et al., 2012</xref>; <xref ref-type="bibr" rid="bib23">Miki and Clapham, 2013</xref>), but this effect has yet to be systematically quantified in experiments (<xref ref-type="bibr" rid="bib9">Fauci and Dillon, 2006</xref>; <xref ref-type="bibr" rid="bib33">Suarez and Pacey, 2006</xref>). Specifically, it is not known at present how sperm cells respond to variations in shear rate and viscosity, and how long they need to adapt to temporal changes in the flow direction. Answering these questions is essential for understanding which physical effects may be important at different stages of the mammalian fertilisation process.</p><p>To quantify the swimming strategies of sperm cells under well-controlled flow conditions, we performed a series of microfluidic experiments in cylindrical and planar channels (<xref ref-type="fig" rid="fig1">Figure 1</xref>), varying systematically shear rates <inline-formula><mml:math id="inf1"><mml:mrow><mml:mover accent="true"><mml:mi>γ</mml:mi><mml:mo>˙</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula> and viscosities <italic>μ</italic> through the physiologically and rheotactically relevant regime, up to <italic>μ</italic> = 20 mPa·s which is roughly 10× the viscosity of natural seminal fluid (<xref ref-type="bibr" rid="bib25">Owen and Katz, 2005</xref>). These measurements revealed the interesting result that both human and bull spermatozoa do not simply align against the flow, but instead swim upstream on spiral-shaped trajectories along the walls of a cylindrical channel (<xref ref-type="fig" rid="fig1">Figure 1A</xref>; and <xref ref-type="other" rid="video1">Video 1</xref>). The previously unrecognised transversal velocity component can be attributed to the chirality of the flagellar beat. The resulting helical swimming patterns enable the spermatozoa to explore collectively the full surface of a cylindrical channel, suggesting that rheotaxis can help sperm to navigate their way through the oviduct and find the egg cell (<xref ref-type="bibr" rid="bib23">Miki and Clapham, 2013</xref>). Using high-speed imaging, we also determined the dynamical response of human and bull spermatozoa to flow reversal at different viscosities, which is essential for understanding how active swimming, rheotaxis, and uterine peristalsis can combine to facilitate optimal sperm transport. To rationalise the experimental observations, we identify below a simple mathematical model that reproduces the main results of our measurements.<fig id="fig1" position="float"><object-id pub-id-type="doi">10.7554/eLife.02403.003</object-id><label>Figure 1.</label><caption><title>Sperm swim on upstream spirals against shear flow.</title><p>(<bold>A</bold>) Background-subtracted micrograph showing the track of a bull sperm in a cylindrical channel (viscosity <italic>μ</italic> = 3 mPa·s shear rate <inline-formula><mml:math id="inf2"><mml:mrow><mml:mrow><mml:mover accent="true"><mml:mi>γ</mml:mi><mml:mo>˙</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:mn>2.1</mml:mn><mml:msup><mml:mi>s</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math></inline-formula> ), channel boundary false-coloured with black, see <xref ref-type="other" rid="video1">Video 1</xref> for raw data. (<bold>B</bold>) Schematic representation not drawn to scale. The conical envelope of the flagellar beat holds the sperm close to the surface (<xref ref-type="bibr" rid="bib16">Kantsler et al., 2013</xref>). The vertical flow gradient exerts a torque that turns the sperm against the flow, but is counteracted by a torque from the chirality of the flagellar wave, resulting in a mean diagonal upstream motion. (<bold>C</bold>) Tracks of bull sperm near a flat channel surface. (<bold>D</bold>) Upstream and transverse mean velocities <inline-formula><mml:math id="inf3"><mml:mrow><mml:mo>〈</mml:mo><mml:msub><mml:mi>v</mml:mi><mml:mrow><mml:mi>y</mml:mi><mml:mo>,</mml:mo><mml:mi>x</mml:mi></mml:mrow></mml:msub><mml:mo>〉</mml:mo></mml:mrow></mml:math></inline-formula> vs shear flow speed <italic>u</italic><sub>20</sub> at 20 μm from the surface for different viscosities. All velocities are normalised by the sample mean speed <italic>v</italic><sub>0<italic>μ</italic></sub> at <inline-formula><mml:math id="inf4"><mml:mrow><mml:mrow><mml:mover accent="true"><mml:mi>γ</mml:mi><mml:mo>˙</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math></inline-formula>. For human sperm, in order of increasing viscosity <italic>v</italic><sub>0<italic>μ</italic></sub> = 53.5 ± 3.0, 46.8 ± 3.7, 36.8 ± 3.3, 29.7 ± 3.9 μm/s, and for bull sperm <italic>v</italic><sub>0<italic>μ</italic></sub> = 70.4 ± 11.8, 45.6 ± 4.7, 32.4 ± 4.8, 29.6 ± 4.1 μm/s, where uncertainties are standard deviations of mean values from different experiments. Each data point is an average over >1000 sperms. (<bold>E</bold>) Histograms for selected points in (<bold>D</bold>).</p><p><bold>DOI:</bold> <ext-link ext-link-type="doi" xlink:href="10.7554/eLife.02403.003">http://dx.doi.org/10.7554/eLife.02403.003</ext-link></p></caption><graphic xlink:href="elife02403f001"/></fig><media content-type="glencoe play-in-place height-250 width-310" id="video1" mime-subtype="mov" mimetype="video" xlink:href="elife02403v001.mov"><object-id pub-id-type="doi">10.7554/eLife.02403.004</object-id><label>Video 1.</label><caption><title>Human sperm cell swimming on a spiral trajectory (green) against a shear flow in a cylindrical channel (fluid viscosity 3 mPa·s; channel diameter 300 μm; channel boundaries marked in red).</title><p>Scale bar 100 μm.</p><p><bold>DOI:</bold> <ext-link ext-link-type="doi" xlink:href="10.7554/eLife.02403.004">http://dx.doi.org/10.7554/eLife.02403.004</ext-link></p></caption></media></p></sec><sec id="s2" sec-type="results"><title>Results</title><sec id="s2-1"><title>Shear and viscosity dependence</title><p>In the experiments, samples of human and bull spermatozoa were injected into microfluidic channels of spherical or rectangular cross-section (‘Materials and methods’). The cells were then exposed to well-defined Poiseuille shear flows, corresponding to parabolic flow profiles (<xref ref-type="fig" rid="fig1">Figure 1B</xref>). Even in the absence of flow, sperm cells tend to accummulate at surfaces (<xref ref-type="bibr" rid="bib28">Rothschild, 1963</xref>; <xref ref-type="bibr" rid="bib5">Denissenko et al., 2012</xref>) due to a combination of steric repulsion (<xref ref-type="bibr" rid="bib16">Kantsler et al., 2013</xref>) and hydrodynamic forces (<xref ref-type="bibr" rid="bib10">Fauci and McDonald, 1995</xref>; <xref ref-type="bibr" rid="bib7">Elgeti et al., 2010</xref>; <xref ref-type="bibr" rid="bib11">Friedrich et al., 2010</xref>; <xref ref-type="bibr" rid="bib12">Gaffney et al., 2011</xref>; <xref ref-type="bibr" rid="bib24">Montenegro-Johnson et al., 2012</xref>). This can be explained by the fact that, in essence, the flagellar beat traces out a cone which, upon collision, aligns with a solid surface, so that the sperm's propulsion vector points into the boundary and the cells become effectively trapped at the surface (<xref ref-type="bibr" rid="bib16">Kantsler et al., 2013</xref>). In the presence of a Poiseuille shear flow, cells close to the channel boundaries experience an approximately linear vertical flow profile, whose slope is given by the shear rate <inline-formula><mml:math id="inf5"><mml:mrow><mml:mover accent="true"><mml:mi>γ</mml:mi><mml:mo>˙</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula> (<xref ref-type="fig" rid="fig1">Figure 1B</xref>). To quantify the effects of shear rate and viscosity on sperm swimming, we tracked a large number of individual cells (typically <italic>N</italic> >10,000) in planar microfluidic channels (<xref ref-type="fig" rid="fig1">Figure 1C</xref>) at different shear rates <inline-formula><mml:math id="inf6"><mml:mrow><mml:mover accent="true"><mml:mi>γ</mml:mi><mml:mo>˙</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>, ranging from 0.2 s<sup>−1</sup> to 9 s<sup>−1</sup>, and different dynamic viscosities <italic>μ</italic>, ranging from 1 mPa·s (that of water) to 20 mPa·s (‘Materials and methods’). The cell tracks were then used to reconstruct the velocities of sperm swimming close to the boundary. Mean values and histograms of the upstream and transverse velocity components from those measurements are summarised in <xref ref-type="fig" rid="fig1">Figure 1D,E</xref>. Since sperm motility depends on viscosity and may vary among different samples, it is advisable to normalise sperm velocities <bold><italic>v</italic></bold> = (<italic>v</italic><sub><italic>x</italic></sub>,<italic>v</italic><sub><italic>y</italic></sub>), that have been measured at different values of <inline-formula><mml:math id="inf7"><mml:mrow><mml:mover accent="true"><mml:mi>γ</mml:mi><mml:mo>˙</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula> and <italic>μ</italic>, by the mean sample speed <inline-formula><mml:math id="inf8"><mml:mrow><mml:msub><mml:mi>v</mml:mi><mml:mrow><mml:mn>0</mml:mn><mml:mi>μ</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mo>〈</mml:mo><mml:mo>|</mml:mo><mml:mi mathvariant="bold-italic">v</mml:mi><mml:mo>|</mml:mo><mml:mo>〉</mml:mo></mml:mrow><mml:mi>μ</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> at zero shear <inline-formula><mml:math id="inf9"><mml:mrow><mml:mrow><mml:mover accent="true"><mml:mi>γ</mml:mi><mml:mo>˙</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math></inline-formula>, and also to rescale the flow velocity accordingly. <xref ref-type="fig" rid="fig1">Figure 1D</xref> shows the thus-normalised mean upstream and mean transverse swimming velocities <inline-formula><mml:math id="inf10"><mml:mrow><mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mo>〈</mml:mo><mml:msub><mml:mi>v</mml:mi><mml:mrow><mml:mi>y</mml:mi><mml:mo>,</mml:mo><mml:mi>x</mml:mi></mml:mrow></mml:msub><mml:mo>〉</mml:mo></mml:mrow><mml:mi>μ</mml:mi></mml:msub></mml:mrow><mml:mo>/</mml:mo><mml:mrow><mml:msub><mml:mi>v</mml:mi><mml:mrow><mml:mn>0</mml:mn><mml:mi>μ</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mrow></mml:mrow></mml:math></inline-formula> for bull and human spermatozoa as a function of the dimensionless rescaled shear flow speed <inline-formula><mml:math id="inf11"><mml:mrow><mml:msub><mml:mi>u</mml:mi><mml:mrow><mml:mn>20</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mover accent="true"><mml:mi>γ</mml:mi><mml:mo>˙</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mi>A</mml:mi></mml:mrow><mml:mo>/</mml:mo><mml:mrow><mml:msub><mml:mi>v</mml:mi><mml:mrow><mml:mn>0</mml:mn><mml:mi>μ</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mrow></mml:math></inline-formula>, where <italic>A</italic> = 10 μm is the approximate amplitude of a typical flagellar beat (i.e., the maximum distance of the flagellar tip from the surface during a beat is 2<italic>A</italic>). The results for the upstream velocity reveal that both human and bull sperm exhibit optimal upstream swimming at rescaled flow speeds <italic>u</italic><sub>20</sub> ∼ 1, implying that there is an optimal shear regime for the rectification of sperm swimming. Remarkably, however, we also find that, at low viscosities (<italic>μ</italic> ≪10 mPa·s), human spermatozoa exhibit a substantial shear-induced transverse velocity component that becomes suppressed at very high viscosities. By contrast, for bull spermatozoa, the mean transverse component is generally weaker and less sensitive to viscosity variations. These statements are also corroborated by the corresponding velocity histograms in <xref ref-type="fig" rid="fig1">Figure 1E</xref>.</p><p>Qualitatively, the above observations for stationary shear flows can be explained as follows. Once a sperm cell has become trapped at a surface, its tail explores, on average, regions of higher flow velocity than the head, resulting in a net torque that turns the head against the flow (<xref ref-type="fig" rid="fig1">Figure 1B</xref>). This shear-induced rectification is counter-acted by variability in the cells swimming direction. If the shear velocity is too low the orientational ‘noise’, which is caused by a combination of intrinsic fluctuations in the cells' swimming apparatus, thermal fluctuations, and elastohydrodynamic effects, inhibits upstream swimming, whereas if the shear velocity becomes too large the sperm will simply be advected downstream by the flow, implying that there exists an optimal intermediate shear rate for upstream swimming. Interestingly, we find that the maximum of the upstream velocity decreases more strongly with viscosity for bull sperm than for human sperm (<xref ref-type="fig" rid="fig1">Figure 1D</xref>). This could be due to differences in cell morphology, as previous numerical studies (<xref ref-type="bibr" rid="bib30">Smith et al., 2011</xref>) for bacterial cells suggest that differences in head shape can substantially alter swimming behavior. Bull sperm have a flatter head than human sperm, which likely suppresses the rotational motion of the cell at high viscosities thus leading to an effectively smaller vertical beat amplitude <italic>A</italic>. This could explain why, at high values of <italic>μ</italic>, the tail beat of bull sperm becomes essentially two-dimensional and constricted to the vicinity of the surface, so that alignment against the flow becomes less efficient.</p><p>To understand the unexpectedly strong transverse velocity component of human sperm at <italic>μ</italic> ≲5 mPa·s, as typical of the seminal fluid (<xref ref-type="bibr" rid="bib25">Owen and Katz, 2005</xref>), it is important to recall that sperm of invertebrae and mammals are known to exhibit different chiral beat patterns depending on environmental conditions (<xref ref-type="bibr" rid="bib13">Gibbons, 1982</xref>; <xref ref-type="bibr" rid="bib15">Ishijima and Hamaguchi, 1993</xref>; <xref ref-type="bibr" rid="bib37">Woolley and Vernon, 2001</xref>; <xref ref-type="bibr" rid="bib29">Smith et al., 2009</xref>), and that shear flows are capable of separating particles along the transverse direction according to their chirality (<xref ref-type="bibr" rid="bib20">Marcos et al., 2009</xref>; <xref ref-type="bibr" rid="bib34">Talkner et al., 2012</xref>). Human sperm exhibit a strong helical beat component at low-to-moderate values of <italic>μ</italic> (<xref ref-type="other" rid="video2">Video 2</xref>), but this chirality becomes suppressed at high viscosities (<xref ref-type="other" rid="video3">Video 3</xref>) resulting in more planar wave forms (<xref ref-type="bibr" rid="bib29">Smith et al., 2009</xref>). For comparison, the beat of a bull sperm flagellum is more similar to a rigidly rotating planar wave even at low viscosities (<xref ref-type="other" rid="video4">Video 4</xref>), thus exhibiting a weaker chirality and leading to smaller transverse velocities (<xref ref-type="fig" rid="fig1">Figure 1D</xref>). Since the flagellar beating pattern can be controlled not only by viscosity but also by changes in calcium concentration (<xref ref-type="bibr" rid="bib15">Ishijima and Hamaguchi, 1993</xref>), higher organisms appear to possess several means for tuning transverse and upstream swimming of sperm.<media content-type="glencoe play-in-place height-250 width-310" id="video2" mime-subtype="mov" mimetype="video" xlink:href="elife02403v002.mov"><object-id pub-id-type="doi">10.7554/eLife.02403.005</object-id><label>Video 2.</label><caption><title>Human sperm cells swimming in a low-viscosity fluid (3 mPa·s) near the wall of a planar channel.</title><p>The video shows that, at low viscosity, the flagellar beat of a human sperm cell typically exhibits a considerable chiral component. This follows from the fact that the flagellum never appears as a straight line (in contrast to bull sperms at same viscosity, compare <xref ref-type="other" rid="video4">Video 4</xref>). Scale bar 20 μm.</p><p><bold>DOI:</bold> <ext-link ext-link-type="doi" xlink:href="10.7554/eLife.02403.005">http://dx.doi.org/10.7554/eLife.02403.005</ext-link></p></caption></media><media content-type="glencoe play-in-place height-250 width-310" id="video3" mime-subtype="mov" mimetype="video" xlink:href="elife02403v003.mov"><object-id pub-id-type="doi">10.7554/eLife.02403.006</object-id><label>Video 3.</label><caption><title>Human sperm cells swimming in a high-viscosity fluid (20 mPa·s) near the wall of a planar channel.</title><p>The video shows that, at very high viscosity, the chiral beat component becomes considerably weaker for there now exist instances where the flagellum appears as an almost straight line, indicating that the beat pattern approaches the shape of a planar rotating wave. Scale bar 20 μm.</p><p><bold>DOI:</bold> <ext-link ext-link-type="doi" xlink:href="10.7554/eLife.02403.006">http://dx.doi.org/10.7554/eLife.02403.006</ext-link></p></caption></media><media content-type="glencoe play-in-place height-250 width-310" id="video4" mime-subtype="mov" mimetype="video" xlink:href="elife02403v004.mov"><object-id pub-id-type="doi">10.7554/eLife.02403.007</object-id><label>Video 4.</label><caption><title>Bull sperm cells swimming in a low-viscosity fluid (3 mPa·s) near the wall of a planar channel.</title><p>The video shows that, even at low viscosity, the flagellar beat of bull sperm is approximately planar. This follows from the fact that at certain instances the flagellum appears as a line (in contrast to human sperms at same viscosity, compare <xref ref-type="other" rid="video2">Video 2</xref>). Scale bar 20 μm.</p><p><bold>DOI:</bold> <ext-link ext-link-type="doi" xlink:href="10.7554/eLife.02403.007">http://dx.doi.org/10.7554/eLife.02403.007</ext-link></p></caption></media></p></sec><sec id="s2-2"><title>Dynamical response</title><p>In addition to typically outward directed mucus flow in the oviduct, sperm cells are also exposed to temporally varying flows driven by uterine contractions (<xref ref-type="bibr" rid="bib9">Fauci and Dillon, 2006</xref>; <xref ref-type="bibr" rid="bib33">Suarez and Pacey, 2006</xref>). To probe the dynamical response of sperm to changes in the flow direction, we performed additional experiments, where we tracked the motion of bull and human spermatozoa after a sudden flow reversal at two different viscosities (<xref ref-type="fig" rid="fig2">Figure 2</xref>; <xref ref-type="other" rid="video5 video6">Videos 5, 6</xref>). In those experiment, sperm were first given time to align against a stationary shear flow, then the flow direction was reversed, <italic>u</italic><sub><italic>y</italic></sub>→−<italic>u</italic><sub><italic>y</italic></sub>, with a switching time <1 s. Upon flow reversal, a sperm cell typically performs a U-turn (<xref ref-type="fig" rid="fig2">Figure 2A,B</xref>). The characteristic radius of curvature of the trajectory and the typical turning time <italic>τ</italic> were found to increase strongly with viscosity. At low viscosity, <italic>μ</italic> ∼ 1 mPa·s, sperm realign rapidly against the new flow direction with a typical response time of <italic>τ</italic> ∼ 5 s to 10 s, and the curvature radius is of the order of one or two sperm lengths <italic>ℓ</italic> ∼ 60 μm (<xref ref-type="other" rid="video5">Video 5</xref>). By contrast, at a larger viscosity of <italic>μ</italic> ∼ 12 mPa·s, which is roughly 4× higher than the natural viscosity of the ejaculate, both curvature radius and response time increase by approximately a factor of 5 (<xref ref-type="other" rid="video6">Video 6</xref>). Interestingly, these response times are of the order of typical cervical contractions (<xref ref-type="bibr" rid="bib19">Kunz et al., 1996</xref>), suggesting a possible fine-tuning between muscular activity of the uterus and turning behavior of sperm cells. In particular, immediately after the flow reversal, sperm orientation and flow direction point for a short period of time in approximately the same direction, leading to a momentarily increased transport velocity (see velocity peaks in <xref ref-type="fig" rid="fig2">Figure 2C</xref>). Thus, by switching flow directions back and forth at an optimal rate, the transport efficiency of an initially rectified sperm population can be enhanced.<fig id="fig2" position="float"><object-id pub-id-type="doi">10.7554/eLife.02403.008</object-id><label>Figure 2.</label><caption><title>Temporal response of sperm cells to a reversal of the flow direction depends sensitively on viscosity.</title><p>(<bold>A</bold>) At low viscosity, sperm perform sharp U-turns, see also <xref ref-type="other" rid="video2">Video 2</xref>. (<bold>B</bold>) At high viscosity, the typical radius of the U-turns increases substantially (<xref ref-type="other" rid="video3">Video 3</xref>). White/black arrows show orientations of several cells before/after turning. (<bold>C</bold>) Flow velocity at distance 5 μm from the channel surface (blue, ‘Flow’), mean upstream velocity <inline-formula><mml:math id="inf12"><mml:mrow><mml:mo>〈</mml:mo><mml:msub><mml:mi>v</mml:mi><mml:mi>y</mml:mi></mml:msub><mml:mo>〉</mml:mo></mml:mrow></mml:math></inline-formula> (red, ‘Up’) and mean transverse velocity <inline-formula><mml:math id="inf13"><mml:mrow><mml:mo>〈</mml:mo><mml:msub><mml:mi>v</mml:mi><mml:mi>x</mml:mi></mml:msub><mml:mo>〉</mml:mo></mml:mrow></mml:math></inline-formula> (green, ‘Trans’) as function of time. The typical response time of sperm cells after flow reversal increases with viscosity. Peaks reflect a short period when mean swimming direction and flow direction are aligned. The time series for human sperm also signal a suppression of the beat chirality at high viscosity, consistent with <xref ref-type="fig" rid="fig1">Figure 1D</xref>.</p><p><bold>DOI:</bold> <ext-link ext-link-type="doi" xlink:href="10.7554/eLife.02403.008">http://dx.doi.org/10.7554/eLife.02403.008</ext-link></p></caption><graphic xlink:href="elife02403f002"/></fig><media content-type="glencoe play-in-place height-250 width-310" id="video5" mime-subtype="mov" mimetype="video" xlink:href="elife02403v005.mov"><object-id pub-id-type="doi">10.7554/eLife.02403.009</object-id><label>Video 5.</label><caption><title>Reorientation of a human sperm cell swimming in a low-viscosity fluid (1 mPa·s) in a planar channel, after a sudden reversal of the flow direction at time <italic>t</italic> = 0.</title><p><bold>DOI:</bold> <ext-link ext-link-type="doi" xlink:href="10.7554/eLife.02403.009">http://dx.doi.org/10.7554/eLife.02403.009</ext-link></p></caption></media><media content-type="glencoe play-in-place height-250 width-310" id="video6" mime-subtype="mov" mimetype="video" xlink:href="elife02403v006.mov"><object-id pub-id-type="doi">10.7554/eLife.02403.010</object-id><label>Video 6.</label><caption><title>Reorientation of two human sperm cells, swimming in a high-viscosity fluid in a planar channel, after a sudden reversal of the flow direction at time <italic>t</italic> = 0.</title><p><bold>DOI:</bold> <ext-link ext-link-type="doi" xlink:href="10.7554/eLife.02403.010">http://dx.doi.org/10.7554/eLife.02403.010</ext-link></p></caption></media></p></sec><sec id="s2-3"><title>Minimal model</title><p>To test whether our understanding of the experimental observations is correct and to provide a basis for future theoretical studies, we used resistive force theory to infer a minimal mathematical model that incorporates the main physical mechanisms discussed above (details are provided in the <xref ref-type="supplementary-material" rid="SD1-data">Supplementary file 1</xref>). The model assumes that the effective two-dimensional motion of a sperm cell, that swims close to a surface in the presence of a shear flow can be described in terms of its position vector <bold><italic>R</italic></bold>(<italic>t</italic>) = (<italic>X</italic>(<italic>t</italic>),<italic>Y</italic>(<italic>t</italic>)) and its orientation unit vector <bold><italic>N</italic></bold>(<italic>t</italic>) = (<italic>N</italic><sub><italic>x</italic></sub>(<italic>t</italic>),<italic>N</italic><sub><italic>y</italic></sub>(<italic>t</italic>)). Focussing on an effective description of the main physical effects and assuming that the flow is in <italic>y</italic>-direction (<xref ref-type="fig" rid="fig1">Figure 1B</xref>), the equations of motions for <bold><italic>R</italic></bold> and <bold><italic>N</italic></bold> read<disp-formula id="equ1"><label>(1)</label><mml:math id="m1"><mml:mrow><mml:mrow><mml:mover accent="true"><mml:mi mathvariant="bold-italic">R</mml:mi><mml:mo>˙</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:mi>V</mml:mi><mml:mi mathvariant="bold-italic">N</mml:mi><mml:mo>+</mml:mo><mml:mi>σ</mml:mi><mml:mrow><mml:mover accent="true"><mml:mi>U</mml:mi><mml:mo>¯</mml:mo></mml:mover></mml:mrow><mml:msub><mml:mi mathvariant="bold-italic">e</mml:mi><mml:mi>y</mml:mi></mml:msub><mml:mo>,</mml:mo></mml:mrow></mml:math></disp-formula><disp-formula id="equ2"><label>(2)</label><mml:math id="m2"><mml:mrow><mml:mrow><mml:mover accent="true"><mml:mi mathvariant="bold-italic">N</mml:mi><mml:mo>˙</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:mi>σ</mml:mi><mml:mrow><mml:mover accent="true"><mml:mi>γ</mml:mi><mml:mo>˙</mml:mo></mml:mover></mml:mrow><mml:mi>α</mml:mi><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:msub><mml:mi>N</mml:mi><mml:mi>x</mml:mi></mml:msub><mml:msub><mml:mi>N</mml:mi><mml:mi>y</mml:mi></mml:msub></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:msubsup><mml:mi>N</mml:mi><mml:mi>y</mml:mi><mml:mn>2</mml:mn></mml:msubsup><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>+</mml:mo><mml:mi>σ</mml:mi><mml:mrow><mml:mover accent="true"><mml:mi>γ</mml:mi><mml:mo>˙</mml:mo></mml:mover></mml:mrow><mml:mi>χ</mml:mi><mml:mi>β</mml:mi><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:msubsup><mml:mi>N</mml:mi><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msubsup><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:msub><mml:mi>N</mml:mi><mml:mi>x</mml:mi></mml:msub><mml:msub><mml:mi>N</mml:mi><mml:mi>y</mml:mi></mml:msub></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>+</mml:mo><mml:msup><mml:mrow><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>2</mml:mn><mml:mi>D</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi mathvariant="bold-italic">I</mml:mi><mml:mo>−</mml:mo><mml:mi mathvariant="bold-italic">NN</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>·</mml:mo><mml:mi mathvariant="bold-italic">ξ</mml:mi><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>.</mml:mo></mml:mrow></mml:math></disp-formula></p><p><xref ref-type="disp-formula" rid="equ1">Equation 1</xref> states that the net in-plane velocity <inline-formula><mml:math id="inf14"><mml:mrow><mml:mrow><mml:mover accent="true"><mml:mi mathvariant="bold-italic">R</mml:mi><mml:mo>˙</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mrow></mml:math></inline-formula> of a cell arises from two main contributions: self-swimming at typical speed <italic>V</italic> in the direction of the cell orientation <bold><italic>N</italic></bold>, and advection by the flow, where <italic>σ</italic> = ±1 defines the flow direction and <inline-formula><mml:math id="inf15"><mml:mrow><mml:mrow><mml:mover accent="true"><mml:mi>U</mml:mi><mml:mo>¯</mml:mo></mml:mover></mml:mrow><mml:mo>></mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math></inline-formula> the mean flow speed experienced by the cell. As explained in detail in the <xref ref-type="supplementary-material" rid="SD1-data">Supplementary file 1</xref>, the nonlinear structure of <xref ref-type="disp-formula" rid="equ2">Equation 2</xref> ensures that the length of the orientation vector <bold><italic>N</italic></bold> remains conserved, assuming that the change in orientation, <inline-formula><mml:math id="inf16"><mml:mrow><mml:mrow><mml:mover accent="true"><mml:mi mathvariant="bold-italic">N</mml:mi><mml:mo>˙</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mrow></mml:math></inline-formula>, is caused by three effects: shear-induced alignment against the flow with rate <inline-formula><mml:math id="inf17"><mml:mrow><mml:mrow><mml:mover accent="true"><mml:mi>γ</mml:mi><mml:mo>˙</mml:mo></mml:mover></mml:mrow><mml:mi>α</mml:mi></mml:mrow></mml:math></inline-formula>, where <italic>α</italic> >0 is numerical factor that encodes geometry of the flagellar beat, shear-and-chirality-induced turning at rate <inline-formula><mml:math id="inf18"><mml:mrow><mml:mrow><mml:mover accent="true"><mml:mi>γ</mml:mi><mml:mo>˙</mml:mo></mml:mover></mml:mrow><mml:mi>β</mml:mi></mml:mrow></mml:math></inline-formula> with <italic>χ</italic>∈{−1, 0, +1} and <italic>β</italic> >0 encoding chirality and shape of the flagellar beat, and variability (<xref ref-type="bibr" rid="bib32">Su et al., 2012</xref>) in the swimming direction, modeled as a Stratonovich-type two-dimensional Gaussian white noise <bold><italic>ξ</italic></bold> with amplitude <italic>D</italic> (<xref ref-type="bibr" rid="bib14">Han et al., 2006</xref>). <xref ref-type="disp-formula" rid="equ1">Equations 1</xref> and <xref ref-type="disp-formula" rid="equ2">2</xref> were obtained by approximating the flagellum by a rigid conical helix, with the polar geometry of the enveloping cone dictating the mathematical structure of the deterministic turning terms (see <xref ref-type="supplementary-material" rid="SD1-data">Supplementary file 1</xref> for details of the calculation). The simplifying assumptions underlying <xref ref-type="disp-formula" rid="equ1">Equations 1</xref> and <xref ref-type="disp-formula" rid="equ2">2</xref> imply that this minimal model does not accurately capture the dynamics of individual cells at zero shear, as the deterministic terms in <xref ref-type="disp-formula" rid="equ2">Equation 2</xref> neglect the intrinsic curvature of cell trajectories. However, when analyzing the in-plane curvature for a large number of human sperm trajectories (>100,000 sample points from more than 1200 cells) at zero shear, we found a broad distribution of curvatures with a small positive mean curvature of (5.6 ± 1.3)·10<sup>−4</sup> μm at low viscosity (1 mPa·s) and a small negative mean curvature (−1.9 ± 0.1)·10<sup>−3</sup> μm at high viscosity (12 mPa·s), where the different signs are consistent with the observed change in the transverse velocity for human sperm at high viscosity (<xref ref-type="fig" rid="fig1">Figure 1D</xref>). To account at least partially for these curvature variations, we include in <xref ref-type="disp-formula" rid="equ2">Equation 2</xref> the Gaussian white noise term. Compared with more accurate models that resolve the details of the flagellar dynamics (<xref ref-type="bibr" rid="bib7">Elgeti et al., 2010</xref>; <xref ref-type="bibr" rid="bib12">Gaffney et al., 2011</xref>), <xref ref-type="disp-formula" rid="equ1">Equations 1</xref> and <xref ref-type="disp-formula" rid="equ2">2</xref> provide a strongly reduced description which, however, turns out be sufficient for rationalising our experimental observations (<xref ref-type="fig" rid="fig3">Figure 3</xref>). Values for <italic>V</italic> and <inline-formula><mml:math id="inf19"><mml:mrow><mml:mover accent="true"><mml:mi>U</mml:mi><mml:mo>¯</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula> can be directly estimated from experiments, and sign conventions in <xref ref-type="disp-formula" rid="equ2">Equation 2</xref> have been chosen such that <italic>χ</italic> = +1 for human sperm at low viscosity (for weakly chiral bull sperm one can use <italic>χ</italic> = 0 in a first approximation). The model parameters (<italic>α</italic>, <italic>β</italic>, <italic>D</italic>) can be inferred from the experimental data (<xref ref-type="supplementary-material" rid="SD1-data">Supplementary file 1</xref>). By performing systematic parameter scans, we found that values <italic>α</italic>∈[0.2, 0.4], <italic>β</italic>∈[0.05, 0.1] and <italic>D</italic>∈[0.2, 0.3] rad/s yield good quantitative agreement with the experimental results for both stationary flow (<xref ref-type="fig" rid="fig3">Figure 3A</xref>) and flow reversal (<xref ref-type="fig" rid="fig3">Figure 3B</xref>), suggesting that the coupling between shear flow and beat chirality dominates the transverse velocity dynamics. We may therefore conclude that, despite some strong simplifications, the effects included in the model capture indeed the main physical mechanisms relevant for understanding sperm motion in shear flow near a surface.<fig id="fig3" position="float"><object-id pub-id-type="doi">10.7554/eLife.02403.011</object-id><label>Figure 3.</label><caption><title>Model simulations reproduce main experimental observations.</title><p>(<bold>A</bold>) Upstream and transverse velocity for different values of the variability (effective noise) parameter <italic>D</italic> in units rad/s and dimensionless shape factors (<italic>α</italic>, <italic>β</italic>). (<bold>B</bold>) Time response of a chiral swimmer with <italic>χ</italic> = +1 (‘Human’) and a non-chiral swimmer with <italic>χ</italic> = 0 (‘Bull’) to a reversal of the flow direction at time <italic>t</italic> = 0. Blue dashed line shows fluid flow <italic>u</italic><sub><italic>y</italic></sub> at 5 <italic>μ</italic>m from the boundary. Simulation parameters (<italic>N</italic> = 1000 trajectories, <italic>A</italic> = 10 μm, <italic>ℓ</italic> = 60 μm, <italic>V</italic> = 50 μm/s) were chosen to match approximately those for viscosity 1 mPa·s in <xref ref-type="fig" rid="fig2">Figure 2C</xref>.</p><p><bold>DOI:</bold> <ext-link ext-link-type="doi" xlink:href="10.7554/eLife.02403.011">http://dx.doi.org/10.7554/eLife.02403.011</ext-link></p></caption><graphic xlink:href="elife02403f003"/></fig></p></sec></sec><sec id="s3" sec-type="discussion"><title>Discussion</title><p>In conclusion, we have reported detailed quantitative measurements of sperm motion in shear flow. Our experimental results show that upstream swimming of mammalian sperm due to rheotaxis is more complex than previously thought. Human sperm cells were found to exhibit a significant transverse velocity component that could be of relevance in the fertilisation process, as the ensuing spiralling motion enables spermatozoa to explore collectively a larger surface area of the oviducts, thereby increasing the probability of locating egg cells. Our theoretical analysis implies that the transverse velocity component arises from a preferred handedness in the flagella beat in the presence of shear flow, in contrast to recent findings for male microgametes of the malaria parasite <italic>Plasmodium berghei</italic> (<xref ref-type="bibr" rid="bib35">Wilson et al., 2013</xref>). Due to the large sample size, our data provide substantial statistical evidence for the hypothesis that mammalian sperm have evolved to achieve optimal upstream swimming near surfaces, possibly exploiting the enhanced fluid production in the female reproductive system during the fertile phase (<xref ref-type="bibr" rid="bib8">Eschenbach et al., 2000</xref>) and after intercourse (<xref ref-type="bibr" rid="bib23">Miki and Clapham, 2013</xref>). The improved quantitative knowledge derived from this data may help to design more efficient artificial insemination strategies, for example, by optimising the viscosity and chemical composition of fertilisation media and adjusting injection techniques to maximise upstream swimming of sperm cells. Combined with recent measurements (<xref ref-type="bibr" rid="bib16">Kantsler et al., 2013</xref>), which clarified the importance of flagella-mediated contact interactions for the accumulation of sperm cells at surfaces, the results presented here yield a cohesive picture of the mechanistic and fluid-mechanical (<xref ref-type="bibr" rid="bib11">Friedrich et al., 2010</xref>) aspects of long-distance sperm navigation. Future work should focus on merging these insights with quantitative studies of chemotaxis (<xref ref-type="bibr" rid="bib31">Spehr et al., 2003</xref>; <xref ref-type="bibr" rid="bib18">Kaupp et al., 2008</xref>; <xref ref-type="bibr" rid="bib38">Zimmer and Riffell, 2011</xref>; <xref ref-type="bibr" rid="bib3">Brenker et al., 2012</xref>) to obtain a differentiated understanding of the interplay between physical and chemical factors during various stages of the mammalian reproduction process.</p></sec><sec id="s4" sec-type="materials|methods"><title>Materials and methods</title><sec id="s4-1"><title>Sperm sample preparation</title><p>Cryogenically frozen bull spermatozoa were purchased from Genus Breeding. For each experiment, a bull sperm sample of 250 μl was thawed in a water bath at 37°C for 15 s. Human samples from healthy undisclosed normozoospermic donors were obtained from Bourn Hall Clinic. Donors provided informed consent in accordance with the regulations of The University of Cambridge Human Biology Research Ethics Committee. For each experiment, bull and human samples were washed three times by centrifugation at 500 rcf for 5 min with the appropriate medium. The medium for bull spermatozoa contained 72 mM KCl, 160 mM sucrose, 2 mM Na-pyruvate, and 2 mM Na-phosphate buffer at pH 7.4 (<xref ref-type="bibr" rid="bib37">Woolley and Vernon, 2001</xref>). Human sperm medium was based on a standard Earle's Balanced Salt Solution, containing 66.4 mM NaCl, 5.4 mM KCl, 1.6 mM CaCl<sub>2</sub>, 0.8 mM MgSO<sub>4</sub>, N<sub>2</sub>H<sub>2</sub>PO<sub>4</sub> 1 mM, NaHCO<sub>3</sub> 26 mM, D-Glucose 5.5 mM supplemented with 2.5 mM Na pyruvate and 19 mM Na-lactate pH adjusted to 7.2 by bubbling the medium with CO<sub>2</sub>. Viscosity of the medium was modified by adding methylcellulose (M0512; Sigma-Aldrich; St. Louis, MO; approximate molecular weight 88,000) at concentrations 0%, 0.2%, 0.4%, 0.5% wt/vol. The absence of circular trajectories at zero-shear implies that the sperm are capacitated (<xref ref-type="bibr" rid="bib23">Miki and Clapham, 2013</xref>).</p></sec><sec id="s4-2"><title>Microfluidics</title><p>Microfluidic channels were manufactured using standard soft-lithography techniques. The master mould was produced from SU8 2075 (MicroChem Corp.; Newton, MA) spun to a 340 microns thickness layer and exposed to UV light through a high resolution mask to obtain the desired structures. The microfluidic chip containing the channels cast from PDMS (Sylgard 184; Dow Corning; Midland, MI) and bonded to covered glass. The channel has rectangular cross-section of 0.34 × 3 mm. We treated PDMS surfaces of the channels prior the experiment with 10% (wt/vol) Polyethylene glycol (m.w. 8000; Sigma) solution in water for 30 min to avoid adhesion of sperm cells to the walls. Sperm suspension was introduced through inlets with a micro-syringe pump (Harvard Apparatus; Kent, UK) at controlled flow rates of 0.1–40 μl/min. The concentration of the sperm cells in the experiments was kept below 1% volume fraction.</p></sec><sec id="s4-3"><title>Microscopy</title><p>To identify the swimming characteristics of individual sperm cells, the trajectories were reconstructed by applying a custom-made particle-tracking-velocimetry (PTV) algorithm to image data taken with a Zeiss Axio Observer inverted microscope (20x or 10x objective, 25 fps). The flagella dynamics was captured with a Fastcam SA-3 Photron camera (San Diego, CA; 125 fps, 40x/NA 0.6 objective). Calibration of the velocity profile in the channel was performed by measuring trajectories of fluorescent beads for different distances from the coverslip via PTV. The measured velocity profile is found identical to the calculated values from solving the Stokes equations for the given geometry. Values of the shear rate <inline-formula><mml:math id="inf20"><mml:mrow><mml:mover accent="true"><mml:mi>γ</mml:mi><mml:mo>˙</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula> in the different experiments were reconstructed from the flow velocity at distance 20 μm from the wall (see below).</p></sec><sec id="s4-4"><title>Additional experimental information</title><p>Effects of viscosity variation and shear-rate variation were studied in experiments that were performed in a rectangular channel with a cross-section 0.34 × 3 mm, by observing sperm motion at lower and upper channel walls. The field of view (normally 800 × 800 μm) was chosen at the middle of the channel (in <italic>x</italic>-direction), where the in-plane velocity gradient is negligible due to the high aspect ratio of the channel (<xref ref-type="fig" rid="fig4">Figure 4B</xref>). The <italic>v</italic><sub><italic>y</italic></sub>-velocity profiles, measured along the <italic>z</italic>-coordinate, were found to be in perfect agreement with the theoretically expected parabolic flow profile for this geometry (<xref ref-type="fig" rid="fig4">Figure 4B</xref>). The shear rate <inline-formula><mml:math id="inf21"><mml:mrow><mml:mover accent="true"><mml:mi>γ</mml:mi><mml:mo>˙</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula> at a given flow rate was determined from the flow velocity at distance 20 μm from the wall. The depth of field of the objective was <5 μm to ensure that we only observed cells that swam close to the surface. Trajectories of individual sperm cells were analysed in MATLAB. The sample size in a single experiment exceeds 100,000 velocity vectors, each measurement for a given viscosity and a shear rate was repeated a few times with different sperm samples. Supplemental data tables that summarise the statistical information for each experiment are given in <xref ref-type="supplementary-material" rid="SD2-data">Supplementary file 2</xref>.<fig id="fig4" position="float"><object-id pub-id-type="doi">10.7554/eLife.02403.012</object-id><label>Figure 4.</label><caption><p>(<bold>A</bold>) Schematic of the microfluidic channel and field of view (turquoise region) in the sperm motility measurements. (<bold>B</bold>) Velocity profile at the center of the channel. Red symbols are values of the vertical velocity profile <italic>v</italic><sub><italic>y</italic></sub>(<italic>z</italic>) measured by PTV for the flow rate 0.1 μl/s. The solid line shows the theoretically calculated flow profile for the same flow rate. In motility experiments, values for the velocity gradient near the boundary (pink region) were obtained by measuring the flow velocity at 20 μm from the boundary. (<bold>C</bold>) Theoretical 2D flow speed profile in (<italic>x</italic>,<italic>z</italic>)-plane at flow rate 0.1 μl/s.</p><p><bold>DOI:</bold> <ext-link ext-link-type="doi" xlink:href="10.7554/eLife.02403.012">http://dx.doi.org/10.7554/eLife.02403.012</ext-link></p></caption><graphic xlink:href="elife02403f004"/></fig></p></sec></sec></body><back><sec sec-type="additional-information"><title>Additional information</title><fn-group content-type="competing-interest"><title>Competing interests</title><fn fn-type="conflict" id="conf1"><p>The authors declare that no competing interests exist.</p></fn></fn-group><fn-group content-type="author-contribution"><title>Author contributions</title><fn fn-type="con" id="con1"><p>VK, Conception and design, Acquisition of data, Analysis and interpretation of data, Drafting or revising the article</p></fn><fn fn-type="con" id="con2"><p>JD, Conception and design, Analysis and interpretation of data, Drafting or revising the article</p></fn><fn fn-type="con" id="con3"><p>REG, Conception and design, Analysis and interpretation of data, Drafting or revising the article</p></fn><fn fn-type="con" id="con4"><p>MB, Conception and design, Acquisition of data, Drafting or revising the article</p></fn></fn-group><fn-group content-type="ethics-information"><title>Ethics</title><fn fn-type="other"><p>Human subjects: Human samples from healthy undisclosed normozoospermicdonors were obtained from Bourn Hall Clinic. Donors provided informed consent in accordance with the regulations of The University of Cambridge Human Biology Research Ethics Committee, which granted approval to this research under application number HBREC.2013.15.</p></fn></fn-group></sec><sec sec-type="supplementary-material"><title>Additional files</title><supplementary-material id="SD1-data"><object-id pub-id-type="doi">10.7554/eLife.02403.013</object-id><label>Supplementary file 1.</label><caption><p>This file contains a detailed mathematical derivation of the minimal model in <xref ref-type="disp-formula" rid="equ1">Equations 1</xref> and <xref ref-type="disp-formula" rid="equ2">2</xref> of the main text, a description of the parameter estimation procedure and a brief summary of numerical methods.</p><p><bold>DOI:</bold> <ext-link ext-link-type="doi" xlink:href="10.7554/eLife.02403.013">http://dx.doi.org/10.7554/eLife.02403.013</ext-link></p></caption><media mime-subtype="pdf" mimetype="application" xlink:href="elife02403s001.pdf"/></supplementary-material><supplementary-material id="SD2-data"><object-id pub-id-type="doi">10.7554/eLife.02403.014</object-id><label>Supplementary file 2.</label><caption><p>This file contains data tables that summarise the statistical information for each experiment.</p><p><bold>DOI:</bold> <ext-link ext-link-type="doi" xlink:href="10.7554/eLife.02403.014">http://dx.doi.org/10.7554/eLife.02403.014</ext-link></p></caption><media mime-subtype="xlsx" mimetype="application" xlink:href="elife02403s002.xlsx"/></supplementary-material></sec><ref-list><title>References</title><ref id="bib1"><element-citation publication-type="journal"><person-group person-group-type="author"><name><surname>Adolphi</surname><given-names>H</given-names></name></person-group><year>1905</year><article-title>Die Spermatozoen der Säugetiere schwimmen gegen den Strom</article-title><source>Anatomischer 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contrib-type="editor"><name><surname>Hyman</surname><given-names>Anthony A</given-names></name><role>Reviewing editor</role><aff><institution>Max Planck Institute of Molecular Cell Biology and Genetics</institution>, <country>Germany</country></aff></contrib></contrib-group></front-stub><body><boxed-text><p>eLife posts the editorial decision letter and author response on a selection of the published articles (subject to the approval of the authors). An edited version of the letter sent to the authors after peer review is shown, indicating the substantive concerns or comments; minor concerns are not usually shown. Reviewers have the opportunity to discuss the decision before the letter is sent (see <ext-link ext-link-type="uri" xlink:href="http://elifesciences.org/review-process">review process</ext-link>). Similarly, the author response typically shows only responses to the major concerns raised by the reviewers.</p></boxed-text><p>Thank you for sending your work entitled “Rheotaxis facilitates upstream navigation of mammalian sperm cells” for consideration at <italic>eLife</italic>. Your article has been favorably evaluated by a Senior editor and 3 reviewers, one of whom, Tony Hyman, is a member of our Board of Reviewing Editors.</p><p>The Reviewing editor and the other reviewers discussed their comments before we reached this decision, and the Reviewing editor has assembled the following comments to help you prepare a revised submission.</p><p>It has been known for a long time that spermatozoa are not just dragged along by a fluid flow, but they tend to swim upstream of the flow, under certain circumstances. A recent paper by Miki et al. in Curr. Biol. (2013) showed that there are fluid flows in the oviduct of mammals for some time after coitus that guide sperm swimming upstream towards the egg. Your paper adds to our knowledge by providing the first quantitative analysis of how mammal spermatozoa swimming is guided by surrounding fluid flows and provides a substantial contribution to the understanding of how sperm are transported to the egg before fertilization. The theory is elegant and the arguments presented are quite clear.</p><p>Our main criticism is that in many places you make strong statements without sufficient justification. We would like you to go through your manuscript and consider making adjustments at these places.</p><p>For instance in the fifth paragraph of the Introduction section, you state “Human sperm exhibit a strongly helical beat component at low-to-moderate values of μ, but this chirality becomes suppressed at high viscosities due to increased friction (<xref ref-type="fig" rid="fig1">Figure 1d</xref>). For comparison, the beat of a bull sperm flagellum is more similar to a rigidly rotating planar wave, thus exhibiting a weaker chirality and leading to smaller transverse velocities at physiologically relevant values of μ.” This is a crucial point of the paper and needs to be validated by citations.</p><p>There are several statements in the Discussion section that interpret the observations, but it is unclear how strong the arguments presented are. You need to either tone down your statements, or provide better explanations. For example the statement “This is a consequence of the fact that bull sperm have a flatter head which suppresses the rotational motion of the tail ...” is presented as a clear fact. However, I do not see direct evidence for the claim that the head shape is the precise cause for the observation. There are several such problems in the Discussion section that leave the reader with strong statements that remain unclear as to the precise supporting evidence. A second example is “... but the chirality becomes suppressed at high viscosities due to increased friction (<xref ref-type="fig" rid="fig1">Figure 1d</xref>)”. <xref ref-type="fig" rid="fig1">Figure 1d</xref> does not however provide any information about the beat shape, only about the swimming path. Also the “due to increased friction” referring to a changed chirality of beat shape seems to be just a hypothesis.</p><p>You do not comment that their swimming paths are straight at zero shear rates. Chiral asymmetry of the beat would allow for a bend in the swimming path that has been observed experimentally (probably for non-capacitated sperm). Why is this possibility not appearing in the simple model? Is there not a measurable radius of curvature of the trajectories observed in the experiments?</p><p>There remains some confusion by the symmetries of the terms in <xref ref-type="disp-formula" rid="equ2">Equation (2)</xref> that are derived in the theory section. Both alignment terms in <xref ref-type="disp-formula" rid="equ2">Equation (2)</xref> have the same structure and thus should have similar symmetries (one aligns in y-direction, one in x-direction). However, the term proportional to beta was introduced as a chiral term, while the term proportional to alpha is a shear-aligning term that exits for nonchiral elongated objects. This is somewhat puzzling. Also, the shear alignment term usually has a nematic symmetry as shear only provides an axis but no direction. However, the terms in <xref ref-type="disp-formula" rid="equ2">Equation (2)</xref> describe alignment with a stable direction (not axis), which is a vectorial symmetry. The theory would be more compelling if the underlying symmetries would become clear and their relation to chiral, vector or tensor asymmetries would be transparent.</p></body></sub-article><sub-article article-type="reply" id="SA2"><front-stub><article-id pub-id-type="doi">10.7554/eLife.02403.016</article-id><title-group><article-title>Author response</article-title></title-group></front-stub><body><p>The main criticism expressed in the decision letter was that we made several strong statements without sufficient justification. To address these concerns, we have carefully reformulated the corresponding text parts, added several references that provide support for our conclusions and also provide some additional data and videos.</p><p><italic>1) For instance in the fifth paragraph of the Introduction section, you state “Human sperm exhibit a strongly helical beat component at low-to-moderate values of μ, but this chirality becomes suppressed at high viscosities due to increased friction (</italic><xref ref-type="fig" rid="fig1"><italic>Figure 1d</italic></xref><italic>). For comparison, the beat of a bull sperm flagellum is more similar to a rigidly rotating planar wave, thus exhibiting a “weaker chirality and leading to smaller transverse velocities at physiologically relevant values of μ.” This is a crucial point of the paper and needs to be validated by citations</italic>.</p><p>We now include three additional Videos (now listed as <xref ref-type="other" rid="video2 video3 video4">Videos 2, 3, 4</xref>) from our experiments that illustrate representative beat patterns of human and bull sperm cells at different viscosities. <xref ref-type="other" rid="video2 video3">Videos 2 and 3</xref> show human sperm at low (3 mPa·s) and high (20 mPa·s) viscosity. At low viscosity (<xref ref-type="other" rid="video2">Video 2</xref>), the projection of the rotating flagellum never appears as a straight line, implying that the beat pattern has a strong chiral component. By contrast, at high viscosity (<xref ref-type="other" rid="video3">Video 3</xref>), one can observe instances where the projected flagellum appears as an almost straight line, suggesting that the beat pattern approaches the shape of a rotating planar wave. For comparison, the new <xref ref-type="other" rid="video4">Video 4</xref> shows bull sperm at low (3 mPa·s) viscosity. Here the projected beat patterns are qualitatively similar to those of human sperm at high viscosity. We reformulated the corresponding manuscript parts accordingly and also added a reference to an earlier recent experimental study that reports qualitatively similar findings for human sperm cells [DJ Smith et al, Cell Motil. Cytoskeleton 66, 220 (2009)].</p><p><italic>2) … the discussion section is quite unclear. Here are several statements that interpret the observations but it is not clear how strong the arguments presented are. You need to either tone down your statements, or do a better job of explaining them. For example the statement “This is a consequence of the fact that bull sperm have a flatter head which suppresses the rotational motion of the tail ...” is presented as a clear fact. However, I do not see direct evidence for the claim that the head</italic> <italic>shape is the precise cause for the observation.”</italic></p><p>We have now reformulated this text part, and we also added a reference that emphasizes the potential importance of cell-morphology (in particular, head-shape) for swimming behavior near surfaces [DJ Smith et al, Biophys. J. 100, 2318 (2011)].</p><p><italic>3) There are several such problems in the Discussion section that leave the reader with strong statements that remain unclear as to the precise supporting evidence. A second example is “... but the chirality becomes suppressed at high viscosities due to increased friction (</italic><xref ref-type="fig" rid="fig1"><italic>Figure 1d</italic></xref><italic>)”.</italic> <xref ref-type="fig" rid="fig1"><italic>Figure 1d</italic></xref> <italic>does not however provide any information about the beat shape, only about the swimming path and the “due to increased friction” referring to a changed chirality of beat shape seems just a hypothesis</italic>.</p><p>As stated in our response to point 1), the new <xref ref-type="other" rid="video2 video3 video4">Videos 2, 3, 4</xref> provide some additional visual evidence for the suppression of chirality in the beat patterns of human sperms at higher viscosity. During the revision process we have reformulated this text part and removed the phrase “due to increased friction”<italic>.</italic></p><p><italic>4) You do not comment that their swimming paths are straight at zero shear rate. Chiral asymmetry of the beat would allow for a bend in the swimming path that has been observed experimentally (probably for non-capacitated sperm). Why is this possibility not appearing in the simple model? Is there not a measurable radius of curvature of the trajectories</italic> <italic>observed in the experiments?</italic></p><p>We have reanalyzed cell trajectories (>100,000 data points from >1200 cells) and found a broad distribution of curvatures at zero shear for human sperm with a small positive mean curvature of 5.6e-04 +/-1.3e-04 per micron at low viscosity (1mPa·s) and a small negative mean curvature - 1.9e-03 +/- 1e-04 per micron at high viscosity (12mPa·s). The difference in the sign of the mean value is consistent with the observed change of the transverse velocity in <xref ref-type="fig" rid="fig1">Figure 1D</xref>. This is evident from the derivation in the Supplementary file 1, our simple model cannot account for this effect because it is based on the analytically tractable approximation of a rigid conical helix. To account at least partially for this variability in trajectory curvature, we included in <xref ref-type="disp-formula" rid="equ2">Equation (2)</xref> the Gaussian white noise term that neglects the small bias due to the shifted mean of the curvature distribution. The reasonable agreement between the simulation results and the experimental data suggests that, even with this simplification, <xref ref-type="disp-formula" rid="equ1">Equations (1)</xref> and <xref ref-type="disp-formula" rid="equ2">(2)</xref> can provide a useful minimal description of the upstream/transverse swimming behaviour under shear. We have modified the discussion in the model section of the main text to state the above more clearly.</p><p><italic>5) There remains some confusion by the symmetries of the terms in Equation. (2) that are derived in the theory section. Both alignment terms in</italic> <xref ref-type="disp-formula" rid="equ2"><italic>Equation (2)</italic></xref> <italic>have the same structure and thus should have similar symmetries (one aligns in y-direction, one in x-direction). However, the term proportional to beta was introduced as a chiral term, while the term proportional to alpha is a shear-aligning term that exits for nonchiral elongated objects.” This is somewhat puzzling. Also, the shear alignment term usually has a nematic symmetry as shear only provides an axis but no direction. However, the terms in</italic> <xref ref-type="disp-formula" rid="equ2"><italic>Equation (2)</italic></xref> <italic>describe alignment with a stable direction (not axis), which is a vectorial symmetry. The theory would be more compelling if the underlying symmetries would become clear and their relation to chiral, vector or tensor asymmetries would be transparent</italic>.</p><p>Perhaps the most important difference compared with earlier studies is that we model the flagellum by a conical helix (and not by a cylindrical helix). There is a dominant geometric polarity that determines the leading-order contributions in our model. This is more clearly stated in the substantially extended discussion of the model in the main text.</p></body></sub-article></article> |