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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 xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" article-type="research-article" dtd-version="1.1d1"><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">04165</article-id><article-id pub-id-type="doi">10.7554/eLife.04165</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></article-categories><title-group><article-title>Active torque generation by the actomyosin cell cortex drives left–right symmetry breaking</article-title></title-group><contrib-group><contrib contrib-type="author" id="author-16407" equal-contrib="yes"><name><surname>Naganathan</surname><given-names>Sundar Ram</given-names></name><xref ref-type="aff" rid="aff1">1</xref><xref ref-type="aff" rid="aff2">2</xref><xref ref-type="aff" rid="aff3">3</xref><xref ref-type="fn" rid="equal-contrib">†</xref><xref ref-type="other" rid="par-1"/><xref ref-type="fn" rid="con1"/><xref ref-type="fn" rid="conf2"/></contrib><contrib contrib-type="author" id="author-17040" equal-contrib="yes"><name><surname>Fürthauer</surname><given-names>Sebastian</given-names></name><xref ref-type="aff" rid="aff2">2</xref><xref ref-type="aff" rid="aff3">3</xref><xref ref-type="fn" rid="equal-contrib">†</xref><xref ref-type="fn" rid="pa1">‡</xref><xref ref-type="other" rid="par-2"/><xref ref-type="other" rid="par-3"/><xref ref-type="fn" rid="con2"/><xref ref-type="fn" rid="conf2"/></contrib><contrib contrib-type="author" id="author-17041"><name><surname>Nishikawa</surname><given-names>Masatoshi</given-names></name><xref ref-type="aff" rid="aff1">1</xref><xref ref-type="aff" rid="aff2">2</xref><xref ref-type="aff" rid="aff3">3</xref><xref ref-type="other" rid="par-1"/><xref ref-type="fn" rid="con4"/><xref ref-type="fn" rid="conf2"/></contrib><contrib contrib-type="author" id="author-11130"><name><surname>Jülicher</surname><given-names>Frank</given-names></name><xref ref-type="aff" rid="aff2">2</xref><xref ref-type="fn" rid="con5"/><xref ref-type="fn" rid="conf1"/></contrib><contrib contrib-type="author" corresp="yes" id="author-11621"><name><surname>Grill</surname><given-names>Stephan W</given-names></name><xref ref-type="aff" rid="aff1">1</xref><xref ref-type="aff" rid="aff2">2</xref><xref ref-type="aff" rid="aff3">3</xref><xref ref-type="corresp" rid="cor1">*</xref><xref ref-type="fn" rid="con3"/><xref ref-type="fn" rid="conf2"/></contrib><aff id="aff1"><label>1</label><institution content-type="dept">Biotechnology Center</institution>, <institution>Technical University Dresden</institution>, <addr-line><named-content content-type="city">Dresden</named-content></addr-line>, <country>Germany</country></aff><aff id="aff2"><label>2</label><institution>Max Planck Institute for the Physics of Complex Systems</institution>, <addr-line><named-content content-type="city">Dresden</named-content></addr-line>, <country>Germany</country></aff><aff id="aff3"><label>3</label><institution>Max Planck Institute of Molecular Cell Biology and Genetics</institution>, <addr-line><named-content content-type="city">Dresden</named-content></addr-line>, <country>Germany</country></aff></contrib-group><contrib-group content-type="section"><contrib contrib-type="editor"><name><surname>Ferrell</surname><given-names>James</given-names></name><role>Reviewing editor</role><aff><institution>Stanford University</institution>, <country>United States</country></aff></contrib></contrib-group><author-notes><corresp id="cor1"><label>*</label>For correspondence: <email>stephan.grill@biotec.tu-dresden.de</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>Courant Institute of Mathematical Sciences, New York University, New York, United States</p></fn></author-notes><pub-date publication-format="electronic" date-type="pub"><day>17</day><month>12</month><year>2014</year></pub-date><pub-date pub-type="collection"><year>2014</year></pub-date><volume>3</volume><elocation-id>e04165</elocation-id><history><date date-type="received"><day>28</day><month>07</month><year>2014</year></date><date date-type="accepted"><day>12</day><month>11</month><year>2014</year></date></history><permissions><copyright-statement>© 2014, Naganathan et al</copyright-statement><copyright-year>2014</copyright-year><copyright-holder>Naganathan et al</copyright-holder><license xlink:href="http://creativecommons.org/licenses/by/4.0/"><license-p>This article is distributed under the terms of the <ext-link ext-link-type="uri" xlink:href="http://creativecommons.org/licenses/by/4.0/">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="elife04165.pdf"/><abstract><object-id pub-id-type="doi">10.7554/eLife.04165.001</object-id><p>Many developmental processes break left–right (LR) symmetry with a consistent handedness. LR asymmetry emerges early in development, and in many species the primary determinant of this asymmetry has been linked to the cytoskeleton. However, the nature of the underlying chirally asymmetric cytoskeletal processes has remained elusive. In this study, we combine thin-film active chiral fluid theory with experimental analysis of the <italic>C. elegans</italic> embryo to show that the actomyosin cortex generates active chiral torques to facilitate chiral symmetry breaking. Active torques drive chiral counter-rotating cortical flow in the zygote, depend on myosin activity, and can be altered through mild changes in Rho signaling. Notably, they also execute the chiral skew event at the 4-cell stage to establish the <italic>C. elegans</italic> LR body axis. Taken together, our results uncover a novel, large-scale physical activity of the actomyosin cytoskeleton that provides a fundamental mechanism for chiral morphogenesis in development.</p><p><bold>DOI:</bold> <ext-link ext-link-type="doi" xlink:href="10.7554/eLife.04165.001">http://dx.doi.org/10.7554/eLife.04165.001</ext-link></p></abstract><abstract abstract-type="executive-summary"><object-id pub-id-type="doi">10.7554/eLife.04165.002</object-id><title>eLife digest</title><p>Most living things have left and right sides that are not identical. A well-known example of this ‘left–right asymmetry’ is the position of the human heart within the human body. While the human heart is always on the left, in other situations it is possible for either the left side or the right side to be preferred: for example, some people prefer to write with their right hand, while others prefer to write with their left hand.</p><p>In animals, left–right asymmetry starts early in the development of the embryo. A structure in cells called the cytoskeleton is known to be responsible for generating the asymmetry in many species. The cytoskeleton is mostly made of two types of proteins—rod-like proteins called microtubules and filaments of a protein called actin—but it is not clear how it is involved in establishing left–right asymmetry.</p><p>The cytoskeleton has many functions in the cell: for example, it maintains the shape of the cell, it splits the contents of the cell during cell division, and it transports various things around inside the cell. The cytoskeleton is constantly moving and changing shape: all this activity involves another protein called myosin that binds to the actin filaments and moves along them to generate pulling forces.</p><p>Naganathan et al. studied newly fertilized embryos of the nematode worm <italic>Caenorhabditis elegans</italic> when they contained just one cell. The experiments showed that myosin can generate turning forces that twist the actin cortical layer, leading to local rotations in the cytoskeleton that make the cell asymmetrical. This is controlled by a group of proteins called Rho proteins.</p><p>Next, Naganathan et al. studied embryos that contained four cells. Again, myosin generates local rotations in the cytoskeleton, which are involved in setting up left–right body direction in this stage of development. These experiments show that changes in the cytoskeleton of individual cells can drive asymmetry in the whole embryo. The next challenge will be to understand how myosin is controlled so that rotations only occur during specific cell divisions.</p><p><bold>DOI:</bold> <ext-link ext-link-type="doi" xlink:href="10.7554/eLife.04165.002">http://dx.doi.org/10.7554/eLife.04165.002</ext-link></p></abstract><kwd-group kwd-group-type="author-keywords"><title>Author keywords</title><kwd>actomyosin cortex</kwd><kwd>active torques</kwd><kwd>L/R symmetry breaking</kwd><kwd>hydrodynamics</kwd><kwd>Wnt signaling</kwd></kwd-group><kwd-group kwd-group-type="research-organism"><title>Research organism</title><kwd><italic>C. elegans</italic></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>281903</award-id><principal-award-recipient><name><surname>Naganathan</surname><given-names>Sundar Ram</given-names></name><name><surname>Nishikawa</surname><given-names>Masatoshi</given-names></name></principal-award-recipient></award-group><award-group id="par-2"><funding-source><institution-wrap><institution-id institution-id-type="FundRef">http://dx.doi.org/10.13039/501100001659</institution-id><institution>Deutsche Forschungsgemeinschaft</institution></institution-wrap></funding-source><award-id>FU-961/1-1</award-id><principal-award-recipient><name><surname>Fürthauer</surname><given-names>Sebastian</given-names></name></principal-award-recipient></award-group><award-group id="par-3"><funding-source><institution-wrap><institution-id institution-id-type="FundRef">http://dx.doi.org/10.13039/501100000854</institution-id><institution>Human Frontier Science Program</institution></institution-wrap></funding-source><award-id>Cross disciplinary fellowship LT000871/2014</award-id><principal-award-recipient><name><surname>Fürthauer</surname><given-names>Sebastian</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-cml-version</meta-name><meta-value>2.0</meta-value></custom-meta><custom-meta specific-use="meta-only"><meta-name>Author impact statement</meta-name><meta-value>The actomyosin cytoskeleton generates active chiral torques that lead to left-right asymmetry in <italic>C. elegans</italic> embryos.</meta-value></custom-meta></custom-meta-group></article-meta></front><body><sec sec-type="intro" id="s1"><title>Introduction</title><p>Most organisms are bilaterally asymmetric with morphologically distinct left and right hand sides. Bilateral asymmetry of organisms, organs, and tissues emerges early in development and is dependent on chiral symmetry breaking of cells and subcellular structures (<xref ref-type="bibr" rid="bib13">Hayashi and Murakami, 2001</xref>; <xref ref-type="bibr" rid="bib34">Shibazaki et al., 2004</xref>; <xref ref-type="bibr" rid="bib6">Danilchik et al., 2006</xref>; <xref ref-type="bibr" rid="bib46">Xu et al., 2007</xref>; <xref ref-type="bibr" rid="bib14">Hejnol, 2010</xref>; <xref ref-type="bibr" rid="bib38">Tamada et al., 2010</xref>; <xref ref-type="bibr" rid="bib42">Vandenberg and Levin, 2010</xref>; <xref ref-type="bibr" rid="bib31">Savin et al., 2011</xref>; <xref ref-type="bibr" rid="bib39">Taniguchi et al., 2011</xref>; <xref ref-type="bibr" rid="bib44">Wan et al., 2011</xref>; <xref ref-type="bibr" rid="bib17">Huang et al., 2012</xref>). In many species the primary determinant of chirality has been linked to the cytoskeleton with both the microtubule (<xref ref-type="bibr" rid="bib26">Nonaka et al., 1998</xref>; <xref ref-type="bibr" rid="bib18">Ishida et al., 2007</xref>) and the actomyosin cytoskeleton (<xref ref-type="bibr" rid="bib6">Danilchik et al., 2006</xref>; <xref ref-type="bibr" rid="bib16">Hozumi et al., 2006</xref>; <xref ref-type="bibr" rid="bib36">Spéder et al., 2006</xref>) (AD Bershadsky, personal communication, November 2013) playing prominent roles. Generally, how chiral molecules and chiral molecular interactions generate chiral morphologies on larger scales remains to be a fundamental problem (<xref ref-type="bibr" rid="bib41">Turing, 1952</xref>; <xref ref-type="bibr" rid="bib4">Brown and Wolpert, 1990</xref>; <xref ref-type="bibr" rid="bib15">Henley, 2012</xref>). For example, it has been observed that myosin motors can rotate actin filaments in motility assays (<xref ref-type="bibr" rid="bib30">Sase et al., 1997</xref>; <xref ref-type="bibr" rid="bib1">Beausang et al., 2008</xref>). Yet, it remains unknown which types of large-scale mechanical activities arise from such types of chiral molecular interactions. In this study, we describe that the actomyosin cytoskeleton can generate active torques at cellular scales, and that the cell uses active torques to break chiral symmetry.</p></sec><sec sec-type="results|discussion" id="s2"><title>Results and discussion</title><p>We investigated chiral behaviours of the actomyosin cell cortex in the context of polarizing cortical flow in the 1-cell <italic>Caenorhabditis elegans</italic> embryo (<xref ref-type="bibr" rid="bib25">Munro et al., 2004</xref>; <xref ref-type="bibr" rid="bib23">Mayer et al., 2010</xref>). The cell cortex, sandwiched between the membrane and cytoplasm, is a thin actin gel containing myosin motors and actin binding proteins (<xref ref-type="bibr" rid="bib28">Pollard and Cooper, 1986</xref>; <xref ref-type="bibr" rid="bib5">Clark et al., 2013</xref>). Given the chirality of cortical constituents, we first asked if cortical flow breaks chiral symmetry. We quantified the cortical flow velocity field <italic>v</italic> using particle image velocimetry in <italic>C. elegans</italic> zygotes containing GFP-tagged non-muscle myosin II (NMY-2) (<xref ref-type="bibr" rid="bib23">Mayer et al., 2010</xref>). Flow proceeds primarily along the anteroposterior (AP) axis (<italic>x</italic>-direction), however, we also observed flow vectors to have a small component in the direction orthogonal to the AP axis (<italic>y</italic>-direction). Notably, the posterior and anterior halves of the cortex counter-rotate relative to each other (<xref ref-type="fig" rid="fig1">Figure 1A,B</xref>, <xref ref-type="fig" rid="fig1s1">Figure 1—figure supplement 1</xref>, <xref ref-type="other" rid="video1">Video 1</xref>), with <italic>y</italic>-velocities of ∼–2.5 μm/min and ∼1 μm/min respectively (<xref ref-type="fig" rid="fig1">Figure 1D</xref>). We define the chiral counter-rotation velocity <italic>v</italic><sub><italic>c</italic></sub> as the difference between spatially averaged <italic>y</italic>-velocities in the posterior and the anterior region (<xref ref-type="fig" rid="fig1">Figure 1B</xref>) and measured <italic>v</italic><sub><italic>c</italic></sub> at 858 time points during flow in 25 embryos. We find that the distribution of <italic>v</italic><sub><italic>c</italic></sub> is shifted towards negative values, with a mean of −2.9 ± 0.3 μm/min (mean ± error of mean at 99% confidence unless stated otherwise, <xref ref-type="fig" rid="fig1">Figure 1C</xref>). Thus, counter-rotating cortical flow breaks chiral symmetry at the 1-cell stage, with the posterior half undergoing a counterclockwise rotation when viewed from the posterior pole (<xref ref-type="fig" rid="fig1">Figure 1A</xref>). Notably, chiral counter-rotating flow precedes the previously reported chiral whole-cell rotation of the zygote during cell division (<xref ref-type="bibr" rid="bib33">Schonegg et al., 2014</xref>).<fig-group><fig id="fig1" position="float"><object-id pub-id-type="doi">10.7554/eLife.04165.003</object-id><label>Figure 1.</label><caption><title>Chiral flow depends on myosin activity.</title><p>(<bold>A</bold>) Sketch of a <italic>C. elegans</italic> embryo. Curved arrows illustrate chiral counter-rotating flow in the anterior (A, red) and posterior (P, green) half of the embryo, respectively. (<bold>B</bold>) Time-averaged cortical flow field (arrows) at the bottom surface of a representative <italic>C. elegans</italic> embryo viewed from the outside of the embryo in this and all other images. Arrow colors indicate <italic>y</italic>-velocity. Scale bar, 5 μm. Velocity scale arrow, 20 μm/min. (<bold>C</bold>) Histogram of instantaneous chiral counter-rotation velocity <inline-formula><mml:math id="inf1"><mml:mrow><mml:mrow><mml:msub><mml:mi>v</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msub><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:mi>P</mml:mi></mml:msub></mml:mrow><mml:mo>−</mml:mo><mml:msub><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:mi>A</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula>, where <inline-formula><mml:math id="inf2"><mml:mrow><mml:msub><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:mi>A</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> <inline-formula><mml:math id="inf3"><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><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:mi>P</mml:mi></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math></inline-formula> is the average of the <italic>y</italic>-component of the velocity <italic>v</italic> over the left (right) shaded area in (<bold>B</bold>), for non-RNAi (858 frames from 25 embryos; gray) and <italic>mlc-4 (RNAi)</italic> (8 hrs; 223 frames from 7 embryos; beige). Dashed vertical lines indicate mean <italic>v</italic><sub><italic>c</italic></sub>. (<bold>D</bold>) <italic>y</italic>-velocity <italic>v</italic><sub><italic>y</italic></sub> along the AP axis averaged over 18 vertical stripes as indicated, for non-RNAi (black, averaged over 25 embryos) and <italic>mlc-4 (RNAi)</italic> (beige, averaged over 7 embryos). Error bars, SEM.</p><p><bold>DOI:</bold> <ext-link ext-link-type="doi" xlink:href="10.7554/eLife.04165.003">http://dx.doi.org/10.7554/eLife.04165.003</ext-link></p></caption><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="elife04165f001"/></fig><fig id="fig1s1" position="float" specific-use="child-fig"><object-id pub-id-type="doi">10.7554/eLife.04165.004</object-id><label>Figure 1—figure supplement 1.</label><caption><title>Similar flow fields were obtained when imaging actin as compared to imaging myosin <xref ref-type="other" rid="video8">Video 8</xref>.</title><p>Quantification of chiral flow for myosin and actin using a dual-colored transgenic line (SWG003). (<bold>A</bold>) Histogram of instantaneous chiral counter-rotation velocity <inline-formula ><mml:math id="inf13"><mml:mrow><mml:msub><mml:mi>v</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msub><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:mi>P</mml:mi></mml:msub><mml:mo>−</mml:mo><mml:msub><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:mi>A</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula>, where <inline-formula ><mml:math id="inf14"><mml:mrow><mml:msub><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:mi>A</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula><inline-formula><mml:math id="inf15"><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><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:mi>P</mml:mi></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math></inline-formula>is the average of the <italic>y</italic>-component of the velocity <italic>v</italic> over the left (right) shaded area in <xref ref-type="fig" rid="fig1">Figure 1B</xref>, for NMY-2::GFP (245 frames from 6 embryos; gray) and Lifeact::tagRFP-T (245 frames from 6 embryos; beige). Dashed vertical lines indicate mean <italic>v</italic><sub><italic>c</italic></sub>. (<bold>B</bold>) <italic>y</italic>-velocity <italic>v</italic><sub><italic>y</italic></sub> along the AP axis averaged over 18 vertical stripes, as indicated in <xref ref-type="fig" rid="fig1">Figure 1B</xref>, for NMY-2::GFP (black, averaged over 6 embryos) and Lifeact::tagRFP-T (beige, averaged over 6 embryos). Error bars, SEM.</p><p><bold>DOI:</bold> <ext-link ext-link-type="doi" xlink:href="10.7554/eLife.04165.004">http://dx.doi.org/10.7554/eLife.04165.004</ext-link></p></caption><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="elife04165fs001"/></fig></fig-group><media id="video1" content-type="glencoe play-in-place height-250 width-310" xlink:href="elife04165v001.mov" mimetype="video" mime-subtype="mov"><object-id pub-id-type="doi">10.7554/eLife.04165.005</object-id><label>Video 1.</label><caption><title>Cortical flow breaks chiral symmetry.</title><p>Cortical flow during AP polarization of the <italic>C. elegans</italic> zygote exhibits chiral behaviors with the posterior and the anterior halves of the cortex counter-rotating relative to each other.</p><p><bold>DOI:</bold> <ext-link ext-link-type="doi" xlink:href="10.7554/eLife.04165.005">http://dx.doi.org/10.7554/eLife.04165.005</ext-link></p></caption></media></p><p>Since AP flow depends on myosin activity (<xref ref-type="bibr" rid="bib23">Mayer et al., 2010</xref>), we asked if chiral flow does so as well. We tested if reducing myosin activity through RNAi of the myosin regulatory light-chain <italic>mlc-4</italic> reduces the chiral counter-rotation velocity <italic>v</italic><sub><italic>c</italic></sub>. We found that 8 hrs of <italic>mlc-4 (RNAi)</italic> not only reduces the AP flow velocity (<xref ref-type="bibr" rid="bib25">Munro et al., 2004</xref>) but also significantly reduces <italic>v</italic><sub><italic>c</italic></sub> (Wilcoxon rank sum test at 99% confidence; mean: −1.1 ± 0.4 μm/min, <xref ref-type="fig" rid="fig1">Figure 1C</xref>, <xref ref-type="other" rid="video2">Video 2</xref>) when compared to non-RNAi embryos. We conclude that chiral flow depends on myosin activity.<media id="video2" content-type="glencoe play-in-place height-250 width-310" xlink:href="elife04165v002.mov" mimetype="video" mime-subtype="mov"><object-id pub-id-type="doi">10.7554/eLife.04165.006</object-id><label>Video 2.</label><caption><title>Chiral flow depends on myosin activity.</title><p>8 hrs of <italic>mlc-4 (RNAi)</italic> leads to a substantial reduction of both AP and chiral flow.</p><p><bold>DOI:</bold> <ext-link ext-link-type="doi" xlink:href="10.7554/eLife.04165.006">http://dx.doi.org/10.7554/eLife.04165.006</ext-link></p></caption></media></p><p>We next sought to understand how myosin activity can drive both AP and chiral flow. We pursue the idea that molecular-scale torque generation (<xref ref-type="bibr" rid="bib30">Sase et al., 1997</xref>; <xref ref-type="bibr" rid="bib1">Beausang et al., 2008</xref>) leads to the emergence of active torques on larger scales and make use of a physical description of the cell cortex as a thin film of an active chiral fluid (<xref ref-type="bibr" rid="bib9">Fürthauer et al., 2012</xref>, <xref ref-type="bibr" rid="bib10">2013</xref>). In our description, force and torque generation at the molecular scale give rise to both an active contractile tension <italic>T</italic> and an active torque density <italic>τ</italic> (<xref ref-type="fig" rid="fig2">Figure 2A</xref>) to drive cortical flow. Under conditions of azimuthal symmetry (<xref ref-type="fig" rid="fig2s1">Figure 2—figure supplement 1</xref>, appendix), the AP flow velocity (<italic>v</italic><sub><italic>x</italic></sub>) and the <italic>y</italic>-velocity (<italic>v</italic><sub><italic>y</italic></sub>) obey the equations of motion,<disp-formula id="equ1"><mml:math id="m1"><mml:mrow><mml:msub><mml:mo>∂</mml:mo><mml:mi>x</mml:mi></mml:msub><mml:mi>T</mml:mi><mml:mo>=</mml:mo><mml:mi>η</mml:mi><mml:msubsup><mml:mo>∂</mml:mo><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msubsup><mml:msub><mml:mi>v</mml:mi><mml:mi>x</mml:mi></mml:msub><mml:mo>−</mml:mo><mml:mi>γ</mml:mi><mml:msub><mml:mi>v</mml:mi><mml:mi>x</mml:mi></mml:msub></mml:mrow></mml:math></disp-formula><disp-formula id="equ2"><label>(1)</label><mml:math id="m2"><mml:mrow><mml:msub><mml:mo>∂</mml:mo><mml:mi>x</mml:mi></mml:msub><mml:mi>τ</mml:mi><mml:mo>=</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac><mml:mi>η</mml:mi><mml:msubsup><mml:mo>∂</mml:mo><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msubsup><mml:msub><mml:mi>v</mml:mi><mml:mi>y</mml:mi></mml:msub><mml:mo>−</mml:mo><mml:mi>γ</mml:mi><mml:msub><mml:mi>v</mml:mi><mml:mi>y</mml:mi></mml:msub><mml:mtext> </mml:mtext><mml:mo>,</mml:mo></mml:mrow></mml:math></disp-formula>where <italic>η</italic> is the 2D viscosity of the cortical layer and <italic>γ</italic> quantifies friction with membrane and cytoplasm (<xref ref-type="bibr" rid="bib23">Mayer et al., 2010</xref>). From the structure of <xref ref-type="disp-formula" rid="equ2">Equation 1</xref>, we see that gradients in active tension <italic>T</italic> along the AP axis drive AP flow, while gradients in active torque density <italic>τ</italic> along the AP axis drive chiral flow orthogonal to the AP axis (<xref ref-type="fig" rid="fig2">Figure 2B</xref>, bottom sketch). We introduce the chirality index <italic>c</italic> = <italic>τ</italic>/<italic>T</italic>, which quantifies their relative magnitude. We assume that both active tension <italic>T</italic> and active torque density <italic>τ</italic> are proportional to the local myosin concentration, leading to a single value of the chirality index <italic>c</italic> that is constant over the embryo. This remains a useful approximation even for cases where <italic>T</italic> and <italic>τ</italic> exhibit more complex dependencies on myosin concentration or where they are independently regulated (see below). In such cases the single value of <italic>c</italic> we determine corresponds to an average, capturing the overall chirality index of the embryo (see appendix). Accordingly, we calculated the theoretical AP and chiral flow profiles from the experimentally determined myosin distribution and found a best match with the experimental profiles for a hydrodynamic length of <inline-formula><mml:math id="inf4"><mml:mrow><mml:mi>ℓ</mml:mi><mml:mo>=</mml:mo><mml:msqrt><mml:mrow><mml:mi>η</mml:mi><mml:mo>/</mml:mo><mml:mi>γ</mml:mi></mml:mrow></mml:msqrt><mml:mo>=</mml:mo><mml:mn>16</mml:mn><mml:mo>±</mml:mo><mml:mn>0.6</mml:mn></mml:mrow></mml:math></inline-formula> μm (<xref ref-type="bibr" rid="bib23">Mayer et al., 2010</xref>) and an overall chirality index of <italic>c</italic> = 0.58 ± 0.09 (<xref ref-type="fig" rid="fig2">Figure 2B</xref>; see appendix). We conclude that a significant part of myosin activity is utilized for generating active torques. The handedness of active torques is clockwise when viewed from the outside of the embryo as indicated by the positive sign of the chirality index <italic>c</italic>. When considering the observed AP myosin gradients, active torques of this handedness give rise to counterclockwise flow in the posterior domain when viewed from the posterior tip, see <xref ref-type="fig" rid="fig2">Figure 2B</xref> for an illustration.<fig-group><fig id="fig2" position="float"><object-id pub-id-type="doi">10.7554/eLife.04165.007</object-id><label>Figure 2.</label><caption><title>The cortex actively generates torques.</title><p>(<bold>A</bold>) Left, myosin heads consume ATP to pull (<xref ref-type="bibr" rid="bib20">Kron and Spudich, 1986</xref>) and twist (<xref ref-type="bibr" rid="bib30">Sase et al., 1997</xref>; <xref ref-type="bibr" rid="bib1">Beausang et al., 2008</xref>) actin filaments, leading to the generation of a force dipole (top, magenta) and a torque dipole (bottom, beige). Right, these can generate an active tension and an active torque density at larger scales, causing an isolated piece of cortex to contract (top) and rotate (bottom). Gray surface, membrane; cube with wire frames, non-contracted (non-rotated) piece of cortex; magenta (beige) cubes, contracted (rotated) piece of cortex. The gray arrow points from the outside to the inside of the cell and the rotation is clockwise when viewed from the outside. (<bold>B</bold>) Top, myosin intensity (blue markers) and velocity profiles (magenta markers, AP flow velocity <italic>v</italic><sub><italic>x</italic></sub>; beige markers, <italic>y</italic>-velocity <italic>v</italic><sub><italic>y</italic></sub>) along the AP axis (<xref ref-type="fig" rid="fig1">Figure 1B,D</xref>) for the non-RNAi condition (averaged over 25 embryos). Error bars, SEM. Magenta and beige curves, respective theoretical velocity profiles (<italic>c</italic> = 0.58 ± 0.09). Bottom, sketch of a <italic>C. elegans</italic> embryo with clockwise active torques in beige (as viewed from the outside of the embryo). A gradient in myosin concentration along the AP axis (see plot above) leads to a gradient in active torques (shown here with varying sizes of the clockwise torques), resulting in a chiral flow (red and green arrows) orthogonal to the gradient.</p><p><bold>DOI:</bold> <ext-link ext-link-type="doi" xlink:href="10.7554/eLife.04165.007">http://dx.doi.org/10.7554/eLife.04165.007</ext-link></p></caption><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="elife04165f002"/></fig><fig id="fig2s1" position="float" specific-use="child-fig"><object-id pub-id-type="doi">10.7554/eLife.04165.008</object-id><label>Figure 2—figure supplement 1.</label><caption><title>Myosin distribution is azimuthally symmetric.</title><p>(<bold>A</bold>) Left, representative image of the actomyosin cortex labeled with GFP-tagged NMY-2 (gray, myosin) at 30% cortical retraction. Scale bar, 5 μm. Right, histogram of the difference in spatially averaged myosin fluorescence intensity between the posterior and anterior halves of the embryo, quantified from the respective shaded regions for non-RNAi embryos (N = 250 frames from 25 embryos; gray). Dashed line, mean of the difference in myosin intensity. Only the last 50 s of cortical flow was utilized from each video for generating this histogram. (<bold>B</bold>) Left, representative image of the actomyosin cortex labeled with GFP-tagged NMY-2 (gray, myosin). Right, histogram of the difference in spatially averaged myosin fluorescence intensity between the top and bottom halves of the embryo in the posterior from the respective shaded regions for non-RNAi embryos (N = 858 frames from 25 embryos; gray). Dashed line, mean of the difference in myosin intensity. The entire cortical flow period was utilized from each video for generating this histogram.</p><p><bold>DOI:</bold> <ext-link ext-link-type="doi" xlink:href="10.7554/eLife.04165.008">http://dx.doi.org/10.7554/eLife.04165.008</ext-link></p></caption><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="elife04165fs002"/></fig></fig-group></p><p>We next sought to investigate if changing myosin activity affects the overall chirality index. To this end, we performed a series of mild-to-stronger (<xref ref-type="bibr" rid="bib1a">Baggs et al., 2009</xref>) <italic>mlc-4 (RNAi)</italic> experiments with feeding times of 4, 6, and 8 hrs, respectively, and determined <italic>c</italic> for each condition. We refer to this as weak perturbation RNAi experiments as we aim to identify principle phenotypical alterations upon a mild deviation from non-RNAi conditions, similar to determining the linear response to a small perturbation. While AP flow velocity <italic>v</italic><sub><italic>x</italic></sub> and the chiral flow velocity <italic>v</italic><sub><italic>c</italic></sub> were generally reduced at 4 and 6 hrs of <italic>mlc-4 (RNAi)</italic> (<xref ref-type="fig" rid="fig3">Figure 3A</xref>, <xref ref-type="fig" rid="fig3s1 fig3s2 fig3s3">Figure 3—figure supplement 1–3</xref>, <xref ref-type="other" rid="video3">Video 3</xref>), <italic>c</italic> remained unchanged from non-RNAi conditions (<italic>c</italic>, 0.61 ± 0.07 at 4 hrs and 0.52 ± 0.06 at 6 hrs of RNAi, compared to 0.58 ± 0.09 for non-RNAi; <xref ref-type="fig" rid="fig3">Figure 3A</xref>, <xref ref-type="fig" rid="fig3s3">Figure 3—figure supplement 3</xref>). However, 8 hrs of <italic>mlc-4 (RNAi)</italic> not only resulted in a large reduction of both AP and chiral flow velocities but also led to a significant reduction of <italic>c</italic> (0.14 ± 0.04, <xref ref-type="fig" rid="fig3">Figure 3A</xref>, <xref ref-type="fig" rid="fig3s3">Figure 3—figure supplement 3</xref>). This indicates that the overall ratio of active torque density to active tension is not changed by weak reduction of <italic>mlc-4</italic> activity but is altered at stronger RNAi conditions when cortical structure is affected (<xref ref-type="fig" rid="fig3s4">Figure 3—figure supplement 4A</xref> and <xref ref-type="other" rid="video2">Video 2</xref>).<fig-group><fig id="fig3" position="float"><object-id pub-id-type="doi">10.7554/eLife.04165.009</object-id><label>Figure 3.</label><caption><title>Ratio of active torque to active tension is modulated by Rho.</title><p>(<bold>A</bold>) Chiral counter-rotation velocity <italic>v</italic><sub><italic>c</italic></sub> (top), AP velocity <italic>v</italic><sub><italic>x</italic></sub> (middle), and chirality index <italic>c</italic> (bottom) for non-RNAi (gray), <italic>mlc-4</italic> (4, 6, 8 hrs RNAi), <italic>ect-2</italic> (4, 6, 8 hrs RNAi), and <italic>rga-3</italic> (3, 5, 40 hrs RNAi). Error bars, error of the mean with 99% confidence. Yellow bars, significant difference to non-RNAi condition; brown bars, no significant difference. (<bold>B</bold>) Histogram of instantaneous chiral counter-rotation velocity <italic>v</italic><sub><italic>c</italic></sub> for <italic>mlc-4</italic> (left; 6 hrs; 235 frames from 7 embryos), <italic>ect-2</italic> (middle; 6 hrs; 338 frames from 9 embryos), and <italic>rga-3</italic> (right; 5 hrs; 402 frames from 10 embryos) RNAi. Gray histograms, non-RNAi condition. Dashed lines, mean <italic>v</italic><sub><italic>c</italic></sub>. (<bold>C</bold>) Respective time-averaged cortical flow field (arrows) of representative embryos (gray, myosin). Arrow colors indicate <italic>y</italic>-velocity <italic>v</italic><sub><italic>y</italic></sub>. Scale bar, 5 μm. Velocity scale arrow, 20 μm/min. (<bold>D</bold>) Respective average myosin intensity (blue markers) and velocity profiles (magenta markers, AP flow velocity <italic>v</italic><sub><italic>x</italic></sub>; beige markers, <italic>y</italic>-velocity <italic>v</italic><sub><italic>y</italic></sub>) along the AP axis for each RNAi condition. Error bars, SEM. Magenta and beige curves, respective theoretical velocity profiles. Dashed lines, non-RNAi theoretical velocity profiles.</p><p><bold>DOI:</bold> <ext-link ext-link-type="doi" xlink:href="10.7554/eLife.04165.009">http://dx.doi.org/10.7554/eLife.04165.009</ext-link></p></caption><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="elife04165f003"/></fig><fig id="fig3s1" position="float" specific-use="child-fig"><object-id pub-id-type="doi">10.7554/eLife.04165.010</object-id><label>Figure 3—figure supplement 1.</label><caption><title>Chiral counter-rotation velocity <italic>v</italic><sub><italic>c</italic></sub> for <italic>mlc-4</italic>, <italic>ect-2</italic>, and <italic>rga-3</italic> RNAi.</title><p>Each graph presents the instantaneous chiral counter-rotation velocity <italic>v</italic><sub><italic>c</italic></sub> histogram for the RNAi condition specified (beige). The histogram from the non-RNAi condition is shown in gray. Downward arrows indicate a significant decrease and upward arrows indicate a significant increase in <italic>v</italic><sub><italic>c</italic></sub> compared to the non-RNAi condition (Wilcoxon rank sum test with 99% confidence). The number of hours of RNAi is as indicated.</p><p><bold>DOI:</bold> <ext-link ext-link-type="doi" xlink:href="10.7554/eLife.04165.010">http://dx.doi.org/10.7554/eLife.04165.010</ext-link></p></caption><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="elife04165fs003"/></fig><fig id="fig3s2" position="float" specific-use="child-fig"><object-id pub-id-type="doi">10.7554/eLife.04165.011</object-id><label>Figure 3—figure supplement 2.</label><caption><title>AP velocity <italic>v</italic><sub><italic>x</italic></sub> for <italic>mlc-4</italic>, <italic>ect-2</italic>, and <italic>rga-3</italic> RNAi.</title><p>Each graph presents the instantaneous AP velocity <italic>v</italic><sub><italic>x</italic></sub> histogram for the RNAi condition specified (magenta). The histogram from the non-RNAi condition is shown in gray. Downward arrows indicate a significant decrease and upward arrows indicate a significant increase in <italic>v</italic><sub><italic>x</italic></sub> compared to the non-RNAi condition (Wilcoxon rank sum test with 99% confidence). The number of hours of RNAi is as indicated.</p><p><bold>DOI:</bold> <ext-link ext-link-type="doi" xlink:href="10.7554/eLife.04165.011">http://dx.doi.org/10.7554/eLife.04165.011</ext-link></p></caption><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="elife04165fs004"/></fig><fig id="fig3s3" position="float" specific-use="child-fig"><object-id pub-id-type="doi">10.7554/eLife.04165.012</object-id><label>Figure 3—figure supplement 3.</label><caption><title>Theoretical velocity profiles for <italic>mlc-4</italic>, <italic>ect-2 ,</italic>and <italic>rga-3</italic> RNAi.</title><p>Each graph presents the respective average myosin intensity (blue markers) and velocity profiles (magenta markers, AP flow velocity <italic>v</italic><sub><italic>x</italic></sub>; beige markers, <italic>y</italic>-velocity <italic>v</italic><sub><italic>y</italic></sub>) along the AP axis for each RNAi condition specified. Error bars, SEM. Magenta and beige curves, respective theoretical velocity profiles. Dashed lines, non-RNAi theoretical velocity profiles. The number of hours of RNAi is as indicated. See respective <xref ref-type="other" rid="video1 video2 video3">Videos 2–5</xref>.</p><p><bold>DOI:</bold> <ext-link ext-link-type="doi" xlink:href="10.7554/eLife.04165.012">http://dx.doi.org/10.7554/eLife.04165.012</ext-link></p></caption><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="elife04165fs005"/></fig><fig id="fig3s4" position="float" specific-use="child-fig"><object-id pub-id-type="doi">10.7554/eLife.04165.013</object-id><label>Figure 3—figure supplement 4.</label><caption><title>Comparison of cortical structure through imaging GFP-tagged NMY-2.</title><p>(<bold>A</bold>) Representative images for non-RNAi, 4, 6, 8, and 11 hrs of <italic>mlc-4 (RNAi)</italic> are shown. Cortical structure is disrupted by 8 hrs of <italic>mlc-4 (RNAi)</italic> (magnified view with characteristic foci size obtained from spatial myosin fluorescence intensity–intensity correlation, compare to non-RNAi). (<bold>B</bold>) Representative images for non-RNAi, 4, 6, 8, and 11 hrs of <italic>ect-2 (RNAi)</italic>. Cortical structure is disrupted by 8 hrs of <italic>ect-2 (RNAi)</italic> (magnified view with characteristic foci size, compare to non-RNAi). (<bold>C</bold>) Representative images for non-RNAi, 3, 5, and 40 hrs of <italic>rga-3 (RNAi)</italic>. Cortical structure is disrupted by 5 hrs of <italic>rga-3 (RNAi)</italic> (magnified view with characteristic foci size, compare to non-RNAi). Scale bars, 10 μm. Error bars, error of the mean with 99% confidence.</p><p><bold>DOI:</bold> <ext-link ext-link-type="doi" xlink:href="10.7554/eLife.04165.013">http://dx.doi.org/10.7554/eLife.04165.013</ext-link></p></caption><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="elife04165fs006"/></fig></fig-group><media id="video3" content-type="glencoe play-in-place height-250 width-310" xlink:href="elife04165v003.mov" mimetype="video" mime-subtype="mov"><object-id pub-id-type="doi">10.7554/eLife.04165.014</object-id><label>Video 3.</label><caption><title>Chirality of the cortex is unaffected under weak perturbation of myosin activity.</title><p>4 and 6 hrs of <italic>mlc-4 (RNAi)</italic> leads to a proportional change of AP and chiral flow, with the chirality index remaining unchanged under these conditions.</p><p><bold>DOI:</bold> <ext-link ext-link-type="doi" xlink:href="10.7554/eLife.04165.014">http://dx.doi.org/10.7554/eLife.04165.014</ext-link></p></caption></media></p><p>We next asked whether there are conditions that modify active torque generation without affecting active tension. To this end, we tested if small changes in Rho signaling, which regulates myosin activity as well as actin dynamics (<xref ref-type="bibr" rid="bib22">Maekawa et al., 1999</xref>), have a different impact on AP and chiral flow and thus change <italic>c</italic>. We performed a series of mild-to-stronger RNAi of the Rho GEF <italic>ect-2</italic> and the Rho GAP <italic>rga-3</italic>. We found that weak perturbation RNAi of <italic>ect-2</italic> led to a substantial decrease in chiral but not AP flow and thus a decrease in the overall chirality index <italic>c</italic> when compared to non-RNAi conditions (<xref ref-type="fig" rid="fig3">Figure 3A,D</xref>; see also <xref ref-type="fig" rid="fig3s4">Figure 3—figure supplement 4B</xref>, <xref ref-type="other" rid="video4">Video 4</xref>). Conversely, weak perturbation RNAi of <italic>rga-3</italic> led to a substantial increase in chiral but not AP flow and thus an increase in the overall chirality index <italic>c</italic> (<xref ref-type="fig" rid="fig3">Figure 3A,D</xref>; see also <xref ref-type="fig" rid="fig3s4">Figure 3—figure supplement 4C</xref>, <xref ref-type="other" rid="video5">Video 5</xref>). Thus, a weak perturbation of <italic>ect-2</italic> and <italic>rga-3</italic> affects chiral but not AP flow, unlike a weak perturbation of <italic>mlc-4</italic> which affects both. We conclude that a principle phenotypical alteration upon mild modifications of Rho pathway activity is a change of the chirality index, or, in other words, mild modifications of Rho pathway activity change active torque generation without affecting active tension.<media id="video4" content-type="glencoe play-in-place height-250 width-310" xlink:href="elife04165v004.mov" mimetype="video" mime-subtype="mov"><object-id pub-id-type="doi">10.7554/eLife.04165.015</object-id><label>Video 4.</label><caption><title>Chiral flow decreases with decreasing Rho activity.</title><p><italic>ect-2 (RNAi)</italic> leads to a substantial reduction in chiral flow with a minimal change in AP flow.</p><p><bold>DOI:</bold> <ext-link ext-link-type="doi" xlink:href="10.7554/eLife.04165.015">http://dx.doi.org/10.7554/eLife.04165.015</ext-link></p></caption></media><media id="video5" content-type="glencoe play-in-place height-250 width-310" xlink:href="elife04165v005.mov" mimetype="video" mime-subtype="mov"><object-id pub-id-type="doi">10.7554/eLife.04165.016</object-id><label>Video 5.</label><caption><title>Chiral flow increases with increasing Rho activity.</title><p><italic>rga-3 (RNAi)</italic> leads to a substantial increase in chiral flow with a minimal change in AP flow.</p><p><bold>DOI:</bold> <ext-link ext-link-type="doi" xlink:href="10.7554/eLife.04165.016">http://dx.doi.org/10.7554/eLife.04165.016</ext-link></p></caption></media></p><p>We next asked whether actomyosin active torques participate in bilateral symmetry breaking, since this requires a chiral process. In <italic>C. elegans</italic>, embryonic handedness is determined at the 4-cell stage when the ABa and ABp cells skew clockwise by ∼20° (as viewed dorsally in the AP–LR plane, <xref ref-type="fig" rid="fig4">Figure 4A</xref>) (<xref ref-type="bibr" rid="bib45">Wood, 1991</xref>; <xref ref-type="bibr" rid="bib2">Bergmann et al., 2003</xref>). We first tested if the clockwise skew in ABa is accompanied by chiral cortical flow. Strikingly, we observed chiral cortical flow in ABa, with the cortex in both future daughter cells counter-rotating (<italic>v</italic><sub><italic>c</italic></sub> = −5.2 ± 1.1 μm/min, mean ± error of mean at 95% confidence, <xref ref-type="other" rid="video6">Video 6</xref>). The handedness of chiral flow is identical to that at the 1-cell stage, indicative of a presence of active torques with the same sign of <italic>c</italic>. If these chiral counter-rotating flows participate in the clockwise skew of both daughter cells, we would expect that changing active torque generation should affect the chiral skew at the 4-cell stage. To this end, we performed weak perturbation RNAi of the Rho pathway members, <italic>ect-2</italic> and <italic>rga-3</italic>, to specifically modify active torques. We first tested whether chiral flows are affected at the 4-cell stage under these conditions. We found that 4.5 hrs of <italic>ect-2 (RNAi)</italic> led to a significant decrease in chiral flow velocity, <italic>v</italic><sub><italic>c</italic></sub> (−3.4 ± 1.4 μm/min), while 4.5 hrs of <italic>rga-3 (RNAi)</italic> led to a significant increase in <italic>v</italic><sub><italic>c</italic></sub> in the ABa cell (−6.7 ± 0.7 μm/min, <xref ref-type="fig" rid="fig4">Figure 4B</xref>, <xref ref-type="other" rid="video6">Video 6</xref>), similar to our observations at the 1-cell stage. We next tested whether changing chiral flow velocity at the 4-cell stage is concomitant with a change in the degree of clockwise skew. Indeed, we found that 4.5 hrs of <italic>ect-2 (RNAi)</italic> led to a significant decrease in skew (15.8° ± 4.9°) in the ABa cell measured in the AP–LR plane, while 4.5 hrs of <italic>rga-3 (RNAi)</italic> led to a significant increase in skew (37.8° ± 6.1°) when compared to non-RNAi conditions (23.6° ± 3.7°; <xref ref-type="fig" rid="fig4">Figure 4A</xref> and <xref ref-type="fig" rid="fig4s1">Figure 4—figure supplement 1</xref>). Similar results were obtained in ABp (<xref ref-type="fig" rid="fig4s1">Figure 4—figure supplement 1</xref>). Thus, changing counter-rotating chiral flow velocity in these cells by weak perturbation of the Rho pathway leads to a change in the degree of skew. This suggests that active torque generation and chiral counter-rotating flow participate in the execution of the LR symmetry breaking chiral skew event at the 4-cell stage.<fig-group><fig id="fig4" position="float"><object-id pub-id-type="doi">10.7554/eLife.04165.017</object-id><label>Figure 4.</label><caption><title>Active torques participate in <italic>L</italic>/<italic>R</italic> body axis establishment.</title><p>(<bold>A</bold>) A schematic of the skew angle measurement in the AP–LR plane. Gray dashed line, initial nuclei position; black dashed line, skewed nuclei position; beige arrows, direction of cortical flow on the dorsal surface (<xref ref-type="other" rid="video6">Video 6</xref>). To the right are the chiral skew angles of ABa for non-RNAi (gray), <italic>ect-2 (RNAi)</italic> (4.5 hrs) and <italic>rga-3 (RNAi)</italic> (4.5 hrs) in the AP–LR plane. Gray circles, skew angle in individual videos; shaded areas, SEM; green horizontal lines, mean skew angle; red horizontal lines, median skew angle; yellow shaded areas, knockdown conditions with a significant difference (95% confidence with the Wilcoxon rank sum test) from the non-RNAi condition. (<bold>B</bold>) Chiral counter-rotation velocity <italic>v</italic><sub><italic>c</italic></sub> for non-RNAi (gray), <italic>ect-2 (RNAi)</italic> (4.5 hrs) and <italic>rga-3 (RNAi)</italic> (4.5 hrs) quantified at the 4-cell stage during ABa cytokinesis. Note that one outlier was removed for computing mean <italic>v</italic><sub><italic>c</italic></sub> for <italic>rga-3 (RNAi)</italic>. The expected flow profiles from our theoretical description, given a stripe of high myosin activity (corresponding to the cleavage plane), is shown in <xref ref-type="fig" rid="fig4s5">Figure 4—figure supplement 5</xref>. (<bold>C</bold>) Overall chirality index <italic>c</italic>, for non-RNAi (gray) and for Wnt signaling genes (40 hrs RNAi) that impact the establishment of the <italic>L</italic>/<italic>R</italic> body axis. Interestingly, <italic>gsk-3</italic> not only results in a reduced chiral counter-rotation velocity but also in an increased AP velocity (<xref ref-type="fig" rid="fig4s2 fig4s3 fig4s4">Figure 4—figure supplements 2–4</xref>). Error bars, error of the mean with 99% confidence. Yellow bars, significant difference to non-RNAi condition; brown bars, no significant difference.</p><p><bold>DOI:</bold> <ext-link ext-link-type="doi" xlink:href="10.7554/eLife.04165.017">http://dx.doi.org/10.7554/eLife.04165.017</ext-link></p></caption><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="elife04165f004"/></fig><fig id="fig4s1" position="float" specific-use="child-fig"><object-id pub-id-type="doi">10.7554/eLife.04165.018</object-id><label>Figure 4—figure supplement 1.</label><caption><title>Chiral skew quantifications during bilateral symmetry breaking of the organism.</title><p>(<bold>A</bold>) A schematic of the skew angle measurement in the AP–LR plane. Gray dashed line, initial nuclei position; black dashed line, skewed nuclei position; beige arrows, direction of cortical flow on the dorsal surface (ABa counter-rotating flow shown in <xref ref-type="other" rid="video6">Video 6</xref>). Below, skew angles of ABa and ABp for non-RNAi (gray), <italic>ect-2 (RNAi)</italic> (4.5 hrs) and <italic>rga-3 (RNAi)</italic> (4.5 hrs) in the AP–LR plane. Gray circles, skew angle in individual videos; shaded areas, SEM; green horizontal lines, mean skew angle; red horizontal lines, median skew angle; yellow shaded areas, knockdown conditions with a significant difference (95% confidence with the Wilcoxon rank sum test) from the non-RNAi condition. (<bold>B</bold>) Same as (<bold>A</bold>) with the skew determined in the DV–LR plane. (<bold>C</bold>) Skew calculated without projections on to a particular plane.</p><p><bold>DOI:</bold> <ext-link ext-link-type="doi" xlink:href="10.7554/eLife.04165.018">http://dx.doi.org/10.7554/eLife.04165.018</ext-link></p></caption><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="elife04165fs007"/></fig><fig id="fig4s2" position="float" specific-use="child-fig"><object-id pub-id-type="doi">10.7554/eLife.04165.019</object-id><label>Figure 4—figure supplement 2.</label><caption><title>Chiral counter-rotation velocity <italic>v</italic><sub><italic>c</italic></sub> for RNAi of Wnt signaling genes.</title><p>Each graph presents the instantaneous chiral counter-rotation velocity <italic>v</italic><sub><italic>c</italic></sub> histogram for the RNAi condition specified (beige). The histogram from the non-RNAi condition is shown in gray. Downward arrows indicate a significant decrease and upward arrows indicate a significant increase in <italic>v</italic><sub><italic>c</italic></sub> compared to the non-RNAi condition (Wilcoxon rank sum test with 99% confidence). The number of hours of RNAi is as indicated.</p><p><bold>DOI:</bold> <ext-link ext-link-type="doi" xlink:href="10.7554/eLife.04165.019">http://dx.doi.org/10.7554/eLife.04165.019</ext-link></p></caption><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="elife04165fs008"/></fig><fig id="fig4s3" position="float" specific-use="child-fig"><object-id pub-id-type="doi">10.7554/eLife.04165.020</object-id><label>Figure 4—figure supplement 3.</label><caption><title>AP velocity <italic>v</italic><sub><italic>x</italic></sub> for RNAi of Wnt signaling genes.</title><p>Each graph presents the instantaneous AP velocity <italic>v</italic><sub><italic>x</italic></sub> histogram for the RNAi condition specified (magenta). The histogram from the non-RNAi condition is shown in gray. Downward arrows indicate a significant decrease and upward arrows indicate a significant increase in <italic>v</italic><sub><italic>x</italic></sub> compared to the non-RNAi condition (Wilcoxon rank sum test with 99% confidence). The number of hours of RNAi is as indicated.</p><p><bold>DOI:</bold> <ext-link ext-link-type="doi" xlink:href="10.7554/eLife.04165.020">http://dx.doi.org/10.7554/eLife.04165.020</ext-link></p></caption><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="elife04165fs009"/></fig><fig id="fig4s4" position="float" specific-use="child-fig"><object-id pub-id-type="doi">10.7554/eLife.04165.021</object-id><label>Figure 4—figure supplement 4.</label><caption><title>Theoretical velocity profiles for RNAi of Wnt signaling genes.</title><p>Each graph presents the respective average myosin intensity (blue markers) and velocity profiles (magenta markers, AP flow velocity <italic>v</italic><sub><italic>x</italic></sub>; beige markers, <italic>y</italic>-velocity <italic>v</italic><sub><italic>y</italic></sub>) along the AP axis for each RNAi condition specified. Error bars, SEM. Magenta and beige curves, respective theoretical velocity profiles. Dashed lines, non-RNAi theoretical velocity profiles. The number of hours of RNAi is as indicated. See <xref ref-type="other" rid="video7">Video 7</xref>.</p><p><bold>DOI:</bold> <ext-link ext-link-type="doi" xlink:href="10.7554/eLife.04165.021">http://dx.doi.org/10.7554/eLife.04165.021</ext-link></p></caption><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="elife04165fs010"/></fig><fig id="fig4s5" position="float" specific-use="child-fig"><object-id pub-id-type="doi">10.7554/eLife.04165.022</object-id><label>Figure 4—figure supplement 5.</label><caption><title>Theoretical velocity profiles for a stripe of high myosin activity.</title><p>The graph presents theoretical axial velocity <italic>v</italic><sub><italic>x</italic></sub> (magenta) and chiral velocity <italic>v</italic><sub><italic>y</italic></sub> (beige) profiles given a stripe of high myosin activity (blue). The gradient in myosin activity will then lead to axial flows along the gradient and directed towards the stripe as well as chiral flows orthogonal to the gradient leading to counter-rotations.</p><p><bold>DOI:</bold> <ext-link ext-link-type="doi" xlink:href="10.7554/eLife.04165.022">http://dx.doi.org/10.7554/eLife.04165.022</ext-link></p></caption><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="elife04165fs011"/></fig></fig-group><media id="video6" content-type="glencoe play-in-place height-250 width-310" xlink:href="elife04165v006.mov" mimetype="video" mime-subtype="mov"><object-id pub-id-type="doi">10.7554/eLife.04165.023</object-id><label>Video 6.</label><caption><title>Chiral flow accompanies the LR symmetry breaking skew event at the 4-cell stage.</title><p>Dorsal view of a representative 4-cell stage embryo, with ABa and ABp cells exhibiting counter-rotating cortical flow during cytokinesis, for 4.5 hrs of <italic>ect-2 (RNAi)</italic>, non-RNAi and 4.5 hrs of <italic>rga-3 (RNAi)</italic>. Anterior view of these Videos is shown at the bottom for visualizing counter-rotation in ABa cell. Flashing cyan arrows indicate the direction of counter-rotating cortical flow. Note that counter-rotation of AB cells is significantly reduced in <italic>ect-2 (RNAi)</italic> and significantly increased in <italic>rga-3 (RNAi)</italic> compared to the non-RNAi condition. Quantification of chiral flow velocities (<xref ref-type="fig" rid="fig4">Figure 4B</xref>) was performed in the ABa cell (marked in red).</p><p><bold>DOI:</bold> <ext-link ext-link-type="doi" xlink:href="10.7554/eLife.04165.023">http://dx.doi.org/10.7554/eLife.04165.023</ext-link></p></caption></media></p><p>Finally, we tested whether genes that affect establishment of the L/R body axis impacts chiral flow. To investigate this, we quantified chiral flow velocities and the overall chirality index <italic>c</italic> at the 1-cell stage under conditions of RNAi of the Wnt signaling genes <italic>dsh-2, gsk-3, mig-5, mom-2</italic>, and <italic>mom-5</italic>, which are known to regulate aspects of bilateral symmetry breaking (<xref ref-type="bibr" rid="bib43">Walston et al., 2004</xref>; <xref ref-type="bibr" rid="bib27">Pohl and Bao, 2010</xref>). Strikingly, we found that all these conditions (except <italic>mom-5</italic>) led to reduced chiral flow and a significant reduction of the overall chirality index <italic>c</italic> at the 1-cell stage (<xref ref-type="fig" rid="fig4">Figure 4C</xref>, <xref ref-type="fig" rid="fig4s2 fig4s3 fig4s4">Figure 4—figure supplement 2–4</xref>, <xref ref-type="other" rid="video7">Video 7</xref>). These results are indicative of a fundamental link between genes that affect LR symmetry breaking and chiral counter-rotating flow. Since Wnt-induced signals in many systems propagate through Rho GTPases to promote morphological changes (<xref ref-type="bibr" rid="bib32">Schlessinger et al., 2009</xref>), we speculate that these effects are propagated through Rho signaling (<xref ref-type="fig" rid="fig3">Figure 3</xref>). Taken together, our results indicate that active torque generation and chiral counter-rotating flows participate in the establishment of the L/R body axis of <italic>C. elegans</italic>.<media id="video7" content-type="glencoe play-in-place height-250 width-310" xlink:href="elife04165v007.mov" mimetype="video" mime-subtype="mov"><object-id pub-id-type="doi">10.7554/eLife.04165.024</object-id><label>Video 7.</label><caption><title>Wnt signaling genes regulate chiral flow.</title><p>RNAi of Wnt signaling genes leads to a substantial reduction in chiral flow.</p><p><bold>DOI:</bold> <ext-link ext-link-type="doi" xlink:href="10.7554/eLife.04165.024">http://dx.doi.org/10.7554/eLife.04165.024</ext-link></p></caption></media><media id="video8" content-type="glencoe play-in-place height-250 width-310" xlink:href="elife04165v008.mov" mimetype="video" mime-subtype="mov"><object-id pub-id-type="doi">10.7554/eLife.04165.025</object-id><label>Video 8.</label><caption><title>Chiral flow observed with an actin probe.</title><p>Cortical flow visualized through Lifeact::tagRFP-T exhibits similar chiral behaviors.</p><p><bold>DOI:</bold> <ext-link ext-link-type="doi" xlink:href="10.7554/eLife.04165.025">http://dx.doi.org/10.7554/eLife.04165.025</ext-link></p></caption></media></p><p>To conclude, the actomyosin cytoskeleton in <italic>C. elegans</italic> generates active chiral torques with clockwise handedness when viewed from the outside of the cell. They drive a specific pattern of chiral flows which can be understood quantitatively based on the physics of active gels with chiral asymmetries (<xref ref-type="bibr" rid="bib21">Kruse et al., 2005</xref>; <xref ref-type="bibr" rid="bib9">Fürthauer et al., 2012</xref>, <xref ref-type="bibr" rid="bib10">2013</xref>). Furthermore, our weak perturbation RNAi experiments indicate that Rho activity affects cortical chirality in a way that does not depend on its role in activating myosin. On the one hand, this raises an interesting question whether active chiral torques arise directly from chiral interactions between actin and myosin (<xref ref-type="fig" rid="fig2">Figure 2A</xref>) or whether they rather emerge through myosin molecular force generation and non-trivial tension–torque coupling (<xref ref-type="bibr" rid="bib12">Gore et al., 2006</xref>; <xref ref-type="bibr" rid="bib7">De La Cruz et al., 2010</xref>) in the actomyosin network. On the other hand, through these weak perturbation RNAi experiments we have identified specific conditions under which cortical chirality and active torques can be selectively modified. Bilateral symmetry breaking requires a chiral process, and we used these specific conditions to demonstrate that in <italic>C. elegans</italic>, this chiral process could be provided by active chiral torque generation of the actomyosin cortical layer for driving the spindle skew at the 4-cell stage. We note that a plausible scenario for driving spindle skew by counter-rotating flows is similar to the rotation that a crawler excavator or a digger can execute on the spot. This is done by such a machine rotating its two chains in opposite directions. In our context, the chain rotations correspond to the counter-rotating flows and the rotation of the machine corresponds to the spindle rotation giving rise to the skew.</p><p>Our results imply that active torques are generated at multiple stages during development, in the zygote during polarity establishment, without immediate consequences with respect to LR symmetry breaking, and again at the 4-cell stage, but here as an instructional and mechanistic event that helps to break left/right symmetry. Chiral morphogenetic rearrangements have been observed at other stages in <italic>C. elegans</italic> development (<xref ref-type="bibr" rid="bib27">Pohl and Bao, 2010</xref>) and during the first cleavage (<xref ref-type="bibr" rid="bib33">Schonegg et al., 2014</xref>; <xref ref-type="bibr" rid="bib35">Singh and Pohl, 2014</xref>), as well as in other systems (<xref ref-type="bibr" rid="bib34">Shibazaki et al., 2004</xref>; <xref ref-type="bibr" rid="bib6">Danilchik et al., 2006</xref>; <xref ref-type="bibr" rid="bib11">Géminard et al., 2014</xref>). It is interesting to speculate that all these events might be driven by active torque generation in the actomyosin layer. As such, our work paves the way for a mechanistic understanding of chiral morphogenesis of cells, tissues, and organisms.</p></sec><sec sec-type="materials|methods" id="s3"><title>Materials and methods</title><sec id="s3-1"><title><italic>C. elegans</italic> strains</title><p>The following transgenic lines were used in this study: TH455 <italic>(unc-119(ed3) III; zuIs45[nmy-2::NMY-2::GFP + unc-119(+)] V; ddIs249[TH0566(pie1::Lifeact::mCherry:pie1)])</italic> for imaging cortical flow, LP133 <italic>(nmy-2(cp8[NMY-2::GFP + unc-119(+)]) I; unc-119(ed3) III)</italic> (<xref ref-type="bibr" rid="bib8">Dickinson et al., 2013</xref>) for imaging counter-rotation of AB cells, and SWG003 <italic>(nmy-2(cp8[NMY-2::GFP + unc-119(+)]) I; unc-119(ed3) III; gesIs002[unc-119(ed3) III; (pie-1::Lifeact::tagRFP-T::pie-1 + unc-119(+))])</italic> for quantifying chiral flow fields with an actin probe. For imaging the chiral skew event at the 4-cell stage, a mCherry::Histone; mCherry::PH-PLC1<italic>δ</italic>1 transgenic line was generated by crossing OD70 (<xref ref-type="bibr" rid="bib19">Kachur et al., 2008</xref>) to a line expressing Moesin::GFP and mCherry::Histone obtained from the Piano lab (New York University, New York, USA). <italic>C. elegans</italic> worms were cultured on OP50-seeeded NGM agar plates as described (<xref ref-type="bibr" rid="bib3">Brenner, 1974</xref>).</p></sec><sec id="s3-2"><title>RNA interference</title><p>RNAi experiments were performed by feeding (<xref ref-type="bibr" rid="bib40">Timmons et al., 2001</xref>). Worms were placed on feeding plates (NGM agar containing 1 mM isopropyl-<italic>β</italic>-D-thiogalactoside and 50 μg ml<sup>−1</sup> ampicillin) and incubated for the specified number of hours at 25°C. We defined feeding time (number of hours of RNAi) as the time between transfer of worms to the feeding plate and putative fertilization of the egg. Worms were dissected in M9 buffer and the embryos were mounted on 2% agarose pads for image acquisition. <italic>rga-3</italic> feeding clone was obtained from Ahringer lab (Gurdon institute, Cambridge, United Kingdom), <italic>ect-2</italic> and <italic>mlc-4</italic> from Hyman lab (MPI-CBG, Dresden, Germany). Feeding clones <italic>dsh-2, gsk-3, mig-5, mom-2</italic>, and <italic>mom-5</italic> were obtained from Source Bioscience (Nottingham, United Kingdom).</p><p>For performing weak perturbation RNAi experiments (from 3–12 hrs of RNAi), L4 staged worms were first incubated overnight on OP50 plates at 25°C. Young adults were then transferred to respective RNAi feeding plates. For performing 40 hr RNAi experiments, early L4 staged worms were directly transferred to respective RNAi feeding plates and incubated at 25°C.</p></sec><sec id="s3-3"><title>Image acquisition</title><p>All videos were acquired at 23–24°C, with a spinning disc confocal microscope using a Zeiss C-Apochromat 63X/1.2 NA objective lens and a Yokogawa CSU-X1 scan head. The following emission filter was used for all acquisitions unless specifically stated: 525/50 nm bandpass filter from Semrock (Rochester, New York). Micromanager software (Vale lab, UCSF) was used to acquire videos using the Hamamatsu ORCA-flash camera.</p><p>Confocal videos of cortical NMY-2::GFP for non-RNAi, <italic>mlc-4, ect-2,</italic> and <italic>rga-3 (RNAi)</italic> were acquired using an Andor iXon EMCCD camera (512 by 512 pixels). A stack consisting of three z-planes (0.5 μm apart) with a 488 nm laser and an exposure of 150 ms was acquired at an interval of 5 s from the onset of cortical flow until the first cell division. The maximum intensity projection of the stack at each time point was then subjected for further analysis.</p><p>Confocal videos of cortical NMY-2::GFP for <italic>dsh-2, gsk-3, mig-5, mom-2</italic>, and <italic>mom-5 (RNAi)</italic> were acquired using an Andor Neo sCMOS camera (2560 by 2160 pixels). A stack consisting of two z-planes (0.5 μm apart) with a 488 nm laser and an exposure of 150 ms was acquired at an interval of 5 s from the onset of cortical flow until the first cell division. The maximum intensity projection of the stack at each time point was then subjected for further analysis.</p><p>Chiral skew at the 4-cell stage was imaged by using mCherry::Histone; mCherry::PH-PLC1<italic>δ</italic>1 dual transgenic line. Confocal videos were acquired using a Hamamatsu ORCA-flash 4.0 camera (2048 by 2048 pixels). A stack consisting of 25–30 z-planes (1 μm apart) with a 561 nm laser and an exposure of 300 ms (emission filter – 641/75 nm bandpass filter from Semrock) was acquired at an interval of 30 s from metaphase of the AB lineage at the 4-cell stage until telophase of the AB lineage at the 8-cell stage.</p><p>Counter-rotation of the AB cells was imaged using the LP133 strain. Embryos at the 4-cell stage were first identified and an eye-lash tool was then used to rotate the embryo to obtain a dorsal view. Confocal videos from the dorsal side of the embryo were then acquired using a Hamamatsu ORCA-flash 4.0 camera (2048 by 2048 pixels). A stack consisting of 25 z-planes (0.5 μm apart) with a 488 nm laser and an exposure of 100 ms was acquired at an interval of 5 s from the start of telophase of the AB lineage at the 4-cell stage until cytokinesis.</p></sec><sec id="s3-4"><title>Flow velocity analysis</title><p>2D cortical flow velocity fields were obtained by performing Particle Image Velocimetry (PIV) (<xref ref-type="bibr" rid="bib29">Raffel et al., 2007</xref>) using the freely available PIVlab MATLAB algorithm (pivlab.blogspot.de). PIVlab was employed by performing a 3-step multi pass (with linear window deformation), where the final interrogation area was 16 pixels with a step of 8 pixels.</p><p>To obtain the flow profiles, 2D velocity fields were projected to the AP axis by dividing the embryo into 18 bins along the AP axis (<xref ref-type="fig" rid="fig1">Figure 1B</xref>), and by spatially averaging the x-component or the y-component of velocity along each bin in a single frame. The average velocity in each bin was then averaged over time across the entire flow period (from start of the flow till pseudocleavage). These time-averaged flow profiles were then averaged across all embryos for one experimental condition. Bin extent in the <italic>y</italic> direction was restricted to a stripe of about 13 μm (<xref ref-type="fig" rid="fig1">Figure 1B</xref>). An accurate quantification of chiral flow fields is only possible when the flow axis is approximately aligned with the long axis of the embryo, and we removed from our analysis a small number of embryos (3/28 embryos for the non-RNAi condition) which in the bottom plane analysis appeared to clearly polarize from the side.</p><p>Chiral counter-rotation velocity <italic>v</italic><sub><italic>c</italic></sub> was quantified in each frame by subtracting the y-component of velocity in the anterior (spatially averaged across bins 3 to 6, <xref ref-type="fig" rid="fig1">Figure 1B</xref>) from the y-component of velocity in the posterior (spatially averaged across bins 13 to 16, <xref ref-type="fig" rid="fig1">Figure 1B</xref>). <italic>v</italic><sub><italic>c</italic></sub> from each frame was computed across the entire flow period from all embryos of one experimental condition and a histogram was plotted.</p><p>For quantification of chiral counter-rotation velocity <italic>v</italic><sub><italic>c</italic></sub> in the ABa cell, the cleavage plane as viewed from the dorsal side of the embryo was first manually identified using FIJI. PIV was then performed and the component of velocity vectors parallel to the cleavage plane was calculated. <italic>v</italic><sub><italic>c</italic></sub> was then computed by subtracting velocity components in the right daughter cell (spatially averaged across a box of width 5 μm close to the cleavage plane) from the velocity components in the left daughter cell (spatially averaged across a box of size 5 μm close to the cleavage plane). <italic>v</italic><sub><italic>c</italic></sub> from each frame was computed across the counter-rotation flow period and a time-averaged <italic>v</italic><sub><italic>c</italic></sub> was reported. We removed from our analysis one video each from non-RNAi and <italic>ect-2 (RNAi)</italic> conditions and 2 videos from <italic>rga-3 (RNAi)</italic> condition that resulted in a marked whole body rotation along the DV-LR plane.</p></sec><sec id="s3-5"><title>Chiral skew analysis</title><p>Chiral skew analysis was performed by analyzing the multi-stack videos using Imaris v7.6.5 and v7.7 software (Bitplane, Zurich, Switzerland). The anterior and posterior pole positions were first manually identified and used to define the AP-vector. The DV-vector was then obtained utilizing the nuclear position of EMS and the AP-vector. The initial and skewed nuclei vectors of ABa and ABp cells at the beginning and end of telophase, respectively, were determined by identifying the corresponding nuclei positions. We used MATLAB to determine the angle between the initial and skewed nuclei vectors of ABa and ABp. We first determined the skew angle by projecting all vectors on to the AP–LR plane (dorsal view, <xref ref-type="fig" rid="fig4s1">Figure 4—figure supplement 1A</xref>). This is the plane in which the skew has been reported previously (<xref ref-type="bibr" rid="bib45">Wood, 1991</xref>; <xref ref-type="bibr" rid="bib2">Bergmann et al., 2003</xref>). Next, we determined the skew angle in the DV–LR plane (anterior view, <xref ref-type="fig" rid="fig4s1">Figure 4—figure supplement 1B</xref>) by projecting all vectors on to the DV–LR plane. Finally, we also determined the full (3D) angle between the respective initial and skewed vectors.</p></sec><sec id="s3-6"><title>Additional information on the chiral skew event at the 4-cell stage</title><p>As described in the main text (<xref ref-type="fig" rid="fig4">Figure 4A</xref>), in the ABa cell, <italic>ect-2 (RNAi)</italic> (4.5 hrs) led to a significantly reduced skew in the AP–LR plane, whereas <italic>rga-3 (RNAi)</italic> (4.5 hrs) led to a significantly increased skew. Similar to the ABa cell, the ABp skew was also significantly reduced under <italic>ect-2 (RNAi)</italic> condition (<xref ref-type="fig" rid="fig4s1">Figure 4—figure supplement 1A</xref>). However, the skew was unchanged in the ABp cell for <italic>rga-3 (RNAi)</italic>. We next determined the unprojected full (3D) skew angle for each condition (<xref ref-type="fig" rid="fig4s1">Figure 4—figure supplement 1C</xref>). Intriguingly, we observed that the unprojected full ABp cell skew was marginally increased under <italic>rga-3 (RNAi)</italic> condition (<xref ref-type="fig" rid="fig4s1">Figure 4—figure supplement 1C</xref>) even though the projected AP–LR plane skew remained unchanged (<xref ref-type="fig" rid="fig4s1">Figure 4—figure supplement 1A</xref>). Similarly, we observed that the unprojected full ABa cell skew was not significantly different for <italic>ect-2 (RNAi)</italic> compared to the non-RNAi condition (<xref ref-type="fig" rid="fig4s1">Figure 4—figure supplement 1C</xref>). This difference in skew between the AP–LR projected and the unprojected angles for the same conditions indicated that there could be an additional skew in a different plane. To this end, we determined the skew in the DV–LR plane. Interestingly, we detected an ∼20° skew in this plane even for the non-RNAi condition (<xref ref-type="fig" rid="fig4s1">Figure 4—figure supplement 1B</xref>). Generally, the difference between the full (3D) skew angles and the AP–LR projected angles is due to this additional skew in the DV–LR plane. This illustrates that the chiral skew at the 4-cell stage is more complex than previously reported, with a chiral rotation in the DV–LR plane in addition to the AP–LR plane.</p></sec><sec id="s3-7"><title>Foci size analysis</title><p>A characteristic myosin foci size was determined by performing spatial myosin fluorescence intensity autocorrelation in MATLAB. The autocorrelation was performed in a stripe of about 27 μm wide and 13 μm high in the anterior of the embryo and the analysis was carried out in each frame during the first 75 s of cortical flow. The spatial autocorrelation decay determined in each analysis frame was then fitted with a single exponential function and we define the decay length of this fit as the characteristic foci size. The foci size thus determined in each frame was then averaged over all analysis frames for a single embryo and an ensemble average was reported.</p></sec><sec id="s3-8"><title>Appendix</title><sec id="s3-8-1"><title>Cortical distribution of myosin is azimuthally symmetric</title><p>Through laser ablation experiments, we have shown that cortical flow is driven by gradients in active tension along the AP axis. We consider that active tension is proportional to myosin fluorescence intensity (<xref ref-type="bibr" rid="bib23">Mayer et al., 2010</xref>). Accordingly, we observed that a histogram (<xref ref-type="fig" rid="fig2s1">Figure 2—figure supplement 1A</xref>) of the difference in spatially averaged myosin fluorescence intensities between the posterior and anterior halves of the embryo (see shaded areas in <xref ref-type="fig" rid="fig2s1">Figure 2—figure supplement 1A</xref>) is shifted towards negative values, indicating an asymmetric myosin distribution along the AP axis. Similarly, it is plausible that chiral flows could originate from a gradient in myosin fluorescence intensity (i.e., myosin activity) along the azimuthal direction. To test for such a gradient, we analyzed the difference in spatially averaged myosin fluorescence intensities between the top and bottom halves of the embryo in the posterior (see shaded areas in <xref ref-type="fig" rid="fig2s1">Figure 2—figure supplement 1B</xref>). This analysis was performed in each frame and compiled across the entire flow period from all embryos of the non-RNAi condition. The resulting histogram (<xref ref-type="fig" rid="fig2s1">Figure 2—figure supplement 1B</xref>) shows that there is no detectable gradient in myosin fluorescence intensity along the azimuthal direction. Thus, we conclude that the embryo does not appear to break azimuthal symmetry with respect to myosin distribution and that chiral flow does not originate from myosin activity gradients along the azimuthal direction.</p></sec><sec id="s3-8-2"><title>Cortical equation of motion</title><p>We derive equations of motion for the cell cortex, describing it as a thin film of active chiral fluid (<xref ref-type="bibr" rid="bib9">Fürthauer et al., 2012</xref>, <xref ref-type="bibr" rid="bib10">2013</xref>). We start by stating the conservation of momentum and angular momentum. Momentum conservation in the absence of external forces is expressed by,<disp-formula id="equ3"><label>(2)</label><mml:math id="m3"><mml:mrow><mml:msub><mml:mo>∂</mml:mo><mml:mi>t</mml:mi></mml:msub><mml:mo>(</mml:mo><mml:mi>ρ</mml:mi><mml:msub><mml:mi>v</mml:mi><mml:mi>α</mml:mi></mml:msub><mml:mo>)</mml:mo><mml:mo>=</mml:mo><mml:msub><mml:mo>∂</mml:mo><mml:mi>β</mml:mi></mml:msub><mml:msubsup><mml:mi>σ</mml:mi><mml:mrow><mml:mi>α</mml:mi><mml:mi>β</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mi>o</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:mtext> </mml:mtext><mml:mo>,</mml:mo></mml:mrow></mml:math></disp-formula>where <italic>ρv</italic> is the momentum density, with the mass density <italic>ρ</italic> and the center of mass velocity <italic>v</italic>, and <inline-formula><mml:math id="inf5"><mml:mrow><mml:msubsup><mml:mi>σ</mml:mi><mml:mrow><mml:mi>α</mml:mi><mml:mi>β</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mi>o</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:msubsup></mml:mrow></mml:math></inline-formula> is the stress tensor. A summation convention over repeated indices is implied. In general, the stress tensor <inline-formula><mml:math id="inf6"><mml:mrow><mml:msubsup><mml:mi>σ</mml:mi><mml:mrow><mml:mi>α</mml:mi><mml:mi>β</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mi>o</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:msub><mml:mi>σ</mml:mi><mml:mrow><mml:mi>α</mml:mi><mml:mi>β</mml:mi></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msubsup><mml:mi>σ</mml:mi><mml:mrow><mml:mi>α</mml:mi><mml:mi>β</mml:mi></mml:mrow><mml:mi>a</mml:mi></mml:msubsup><mml:mo>−</mml:mo><mml:mi>P</mml:mi><mml:msub><mml:mi>δ</mml:mi><mml:mrow><mml:mi>α</mml:mi><mml:mi>β</mml:mi></mml:mrow></mml:msub><mml:mo>/</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:math></inline-formula> can be decomposed into a symmetric traceless part, an antisymmetric part, and a trace part corresponding to the hydrostatic pressure, which we denote as <italic>σ</italic><sub><italic>αβ</italic></sub>, <inline-formula><mml:math id="inf7"><mml:mrow><mml:msubsup><mml:mi>σ</mml:mi><mml:mrow><mml:mi>α</mml:mi><mml:mi>β</mml:mi></mml:mrow><mml:mi>a</mml:mi></mml:msubsup></mml:mrow></mml:math></inline-formula>, and <italic>P</italic>, respectively.</p><p>Conservation of angular momentum in the absence of external torques reads,<disp-formula id="equ4"><label>(3)</label><mml:math id="m4"><mml:mrow><mml:msub><mml:mo>∂</mml:mo><mml:mi>t</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>I</mml:mi><mml:msub><mml:mtext>Ω</mml:mtext><mml:mrow><mml:mi>α</mml:mi><mml:mi>β</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mo>∂</mml:mo><mml:mi>γ</mml:mi></mml:msub><mml:msub><mml:mi>M</mml:mi><mml:mrow><mml:mi>α</mml:mi><mml:mi>β</mml:mi><mml:mi>γ</mml:mi></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:mn>2</mml:mn><mml:msubsup><mml:mi>σ</mml:mi><mml:mrow><mml:mi>α</mml:mi><mml:mi>β</mml:mi></mml:mrow><mml:mi>a</mml:mi></mml:msubsup><mml:mtext> </mml:mtext><mml:mo>.</mml:mo></mml:mrow></mml:math></disp-formula></p><p>In this study, we introduced the intrinsic or ‘spin’ angular momentum density <italic>I</italic>Ω<sub><italic>αβ</italic></sub>, with Ω<sub><italic>αβ</italic></sub> and <italic>I</italic> being the spin rotation rate and the moment of inertia density, respectively. Moreover, we introduced the spin angular momentum flux tensor <italic>M</italic><sub><italic>αβγ</italic></sub>. Note that in general the moment of inertia tensor <italic>I</italic> is a symmetric fourth rank tensor (<xref ref-type="bibr" rid="bib37">Stark and Lubensky, 2005</xref>; <xref ref-type="bibr" rid="bib9">Fürthauer et al., 2012</xref>). Here, we choose it to be diagonal for simplicity.</p><p>We describe the cell cortex as an active chiral fluid with the constitutive equations,<disp-formula id="equ5"><label>(4)</label><mml:math id="m5"><mml:mrow><mml:msub><mml:mi>σ</mml:mi><mml:mrow><mml:mi>α</mml:mi><mml:mi>β</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mi>η</mml:mi><mml:msub><mml:mi>u</mml:mi><mml:mrow><mml:mi>α</mml:mi><mml:mi>β</mml:mi></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mi>ζ</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>p</mml:mi><mml:mi>α</mml:mi></mml:msub><mml:msub><mml:mi>p</mml:mi><mml:mi>β</mml:mi></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mi>δ</mml:mi><mml:mrow><mml:mi>α</mml:mi><mml:mi>β</mml:mi></mml:mrow></mml:msub><mml:mo>/</mml:mo><mml:mn>3</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mtext> </mml:mtext><mml:mo>,</mml:mo></mml:mrow></mml:math></disp-formula><disp-formula id="equ6"><label>(5)</label><mml:math id="m6"><mml:mrow><mml:msubsup><mml:mi>σ</mml:mi><mml:mrow><mml:mi>α</mml:mi><mml:mi>β</mml:mi></mml:mrow><mml:mi>a</mml:mi></mml:msubsup><mml:mo>=</mml:mo><mml:mi>η</mml:mi><mml:mo>′</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mtext>Ω</mml:mtext><mml:mrow><mml:mi>α</mml:mi><mml:mi>β</mml:mi></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mi mathvariant="italic">ω</mml:mi><mml:mrow><mml:mi>α</mml:mi><mml:mi>β</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mtext> </mml:mtext><mml:mo>,</mml:mo></mml:mrow></mml:math></disp-formula><disp-formula id="equ7"><label>(6)</label><mml:math id="m7"><mml:mrow><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac><mml:msub><mml:mi>M</mml:mi><mml:mrow><mml:mi>α</mml:mi><mml:mi>β</mml:mi><mml:mi>γ</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mi>κ</mml:mi><mml:msub><mml:mo>∂</mml:mo><mml:mi>γ</mml:mi></mml:msub><mml:msub><mml:mtext>Ω</mml:mtext><mml:mrow><mml:mi>α</mml:mi><mml:mi>β</mml:mi></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi>ζ</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:msub><mml:mi mathvariant="italic">ϵ</mml:mi><mml:mrow><mml:mi>α</mml:mi><mml:mi>β</mml:mi><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mi>p</mml:mi><mml:mi>ν</mml:mi></mml:msub><mml:msub><mml:mi>p</mml:mi><mml:mi>γ</mml:mi></mml:msub><mml:mtext> </mml:mtext><mml:mo>,</mml:mo></mml:mrow></mml:math></disp-formula>where we introduced the strain rate <italic>u</italic><sub><italic>αβ</italic></sub> = (∂<sub><italic>α</italic></sub><italic>v</italic><sub><italic>β</italic></sub> + ∂<sub><italic>β</italic></sub><italic>v</italic><sub><italic>α</italic></sub>)/2 and the vorticity <italic>ω</italic><sub><italic>αβ</italic></sub> = (∂<sub><italic>α</italic></sub><italic>v</italic><sub><italic>β</italic></sub> − ∂<sub><italic>β</italic></sub><italic>v</italic><sub><italic>α</italic></sub>)/2. The phenomenological coefficients <italic>η</italic>, <italic>η</italic>′, and <italic>κ</italic> describe the passive response of the material. The coefficients <italic>ζ</italic> and <italic>ζ</italic><sub>1</sub> quantify active processes. Notably, <italic>ζ</italic> quantifies the tendency of molecular motors to generate active shear in the material, whereas <italic>ζ</italic><sub>1</sub> quantifies their tendency to produce active angular momentum fluxes. In general, additional couplings to the constitutive equations (<xref ref-type="disp-formula" rid="equ5">Equations 4</xref>–<xref ref-type="disp-formula" rid="equ6">6</xref>) exist, see (<xref ref-type="bibr" rid="bib9">Fürthauer et al., 2012</xref>). In this study, we constrain ourselves to the simplest set of equations which reproduces the phenomenology observed in the <italic>C. elegans</italic> cell cortex. The vector <italic>p</italic> denotes the broken symmetry of the cortical material in the thin direction and reflects that inhomogeneous protein distributions lead to different boundary conditions with the cytosol and the cell membrane (<xref ref-type="bibr" rid="bib10">Fürthauer et al., 2013</xref>). Thus in the following we consider <italic>p</italic><sub><italic>x</italic></sub> = <italic>p</italic><sub><italic>y</italic></sub> = 0 and <italic>p</italic><sub><italic>z</italic></sub> = 1, thus <italic>p</italic><sup>2</sup> = 1.</p><p>To obtain equations of motion for the cortical material, we use the constitutive <xref ref-type="disp-formula" rid="equ5">Equations 4</xref>–<xref ref-type="disp-formula" rid="equ6">6</xref> together with the force balance <xref ref-type="disp-formula" rid="equ3">Equation 2</xref> and the torque balance <xref ref-type="disp-formula" rid="equ4">Equation 3</xref> at low Reynolds number. The 3D torque and force balances for the cortical material read,<disp-formula id="equ8"><label>(7)</label><mml:math id="m8"><mml:mrow><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>η</mml:mi><mml:mo>+</mml:mo><mml:mi>η</mml:mi><mml:mo>′</mml:mo></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:msubsup><mml:mo>∂</mml:mo><mml:mi>γ</mml:mi><mml:mn>2</mml:mn></mml:msubsup><mml:msub><mml:mi>v</mml:mi><mml:mi>α</mml:mi></mml:msub><mml:mo>+</mml:mo><mml:mi>η</mml:mi><mml:mo>′</mml:mo><mml:msub><mml:mo>∂</mml:mo><mml:mi>β</mml:mi></mml:msub><mml:msub><mml:mtext>Ω</mml:mtext><mml:mrow><mml:mi>α</mml:mi><mml:mi>β</mml:mi></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mo>∂</mml:mo><mml:mi>α</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>ζ</mml:mi><mml:mo>−</mml:mo><mml:mi>P</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>/</mml:mo><mml:mn>3</mml:mn><mml:mo>+</mml:mo><mml:msub><mml:mo>∂</mml:mo><mml:mi>β</mml:mi></mml:msub><mml:mi>ζ</mml:mi><mml:msub><mml:mi>p</mml:mi><mml:mi>α</mml:mi></mml:msub><mml:msub><mml:mi>p</mml:mi><mml:mi>β</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mtext> </mml:mtext><mml:mo>,</mml:mo></mml:mrow></mml:math></disp-formula><disp-formula id="equ9"><label>(8)</label><mml:math id="m9"><mml:mrow><mml:mi>κ</mml:mi><mml:msubsup><mml:mo>∂</mml:mo><mml:mi>γ</mml:mi><mml:mn>2</mml:mn></mml:msubsup><mml:msub><mml:mtext>Ω</mml:mtext><mml:mrow><mml:mi>α</mml:mi><mml:mi>β</mml:mi></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mo>∂</mml:mo><mml:mi>γ</mml:mi></mml:msub><mml:msub><mml:mi>ζ</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:msub><mml:mi mathvariant="italic">ϵ</mml:mi><mml:mrow><mml:mi>α</mml:mi><mml:mi>β</mml:mi><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mi>p</mml:mi><mml:mi>ν</mml:mi></mml:msub><mml:msub><mml:mi>p</mml:mi><mml:mi>γ</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mi>η</mml:mi><mml:mo>′</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mtext>Ω</mml:mtext><mml:mrow><mml:mi>α</mml:mi><mml:mi>β</mml:mi></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mi>ω</mml:mi><mml:mrow><mml:mi>α</mml:mi><mml:mi>β</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mtext> </mml:mtext><mml:mo>,</mml:mo></mml:mrow></mml:math></disp-formula>where we used the incompressibility condition <italic>∂</italic><sub>γ</sub><italic>v</italic><sub>γ</sub> = 0. After some algebra we obtain a fourth order differential equation for the velocity field,<disp-formula id="equ10"><label>(9)</label><mml:math id="m10"><mml:mtable columnalign="left"><mml:mtr><mml:mtd><mml:mn>0</mml:mn><mml:mo>=</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac><mml:mi>η</mml:mi><mml:msubsup><mml:mo>∂</mml:mo><mml:mi>γ</mml:mi><mml:mn>2</mml:mn></mml:msubsup><mml:msub><mml:mi>v</mml:mi><mml:mi>α</mml:mi></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mo>∂</mml:mo><mml:mi>α</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>ζ</mml:mi><mml:mo>−</mml:mo><mml:mi>P</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>/</mml:mo><mml:mn>3</mml:mn><mml:mo>+</mml:mo><mml:msub><mml:mo>∂</mml:mo><mml:mi>β</mml:mi></mml:msub><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:mi>ζ</mml:mi><mml:msub><mml:mi>p</mml:mi><mml:mi>α</mml:mi></mml:msub><mml:msub><mml:mi>p</mml:mi><mml:mi>β</mml:mi></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mo>∂</mml:mo><mml:mi>γ</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>ζ</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:msub><mml:mi mathvariant="italic">ϵ</mml:mi><mml:mrow><mml:mi>α</mml:mi><mml:mi>β</mml:mi><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mi>p</mml:mi><mml:mi>ν</mml:mi></mml:msub><mml:msub><mml:mi>p</mml:mi><mml:mi>γ</mml:mi></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>]</mml:mo></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mtext>    </mml:mtext><mml:mo>−</mml:mo><mml:mfrac><mml:mi>κ</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>η</mml:mi><mml:mo>′</mml:mo></mml:mrow></mml:mfrac><mml:msubsup><mml:mo>∂</mml:mo><mml:mi>ν</mml:mi><mml:mn>2</mml:mn></mml:msubsup><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>η</mml:mi><mml:mo>+</mml:mo><mml:mi>η</mml:mi><mml:mo>′</mml:mo></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:msubsup><mml:mo>∂</mml:mo><mml:mi>γ</mml:mi><mml:mn>2</mml:mn></mml:msubsup><mml:msub><mml:mi>v</mml:mi><mml:mi>α</mml:mi></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mo>∂</mml:mo><mml:mi>α</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>ζ</mml:mi><mml:mo>−</mml:mo><mml:mi>P</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>/</mml:mo><mml:mn>3</mml:mn><mml:mo>+</mml:mo><mml:msub><mml:mo>∂</mml:mo><mml:mi>β</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>ζ</mml:mi><mml:msub><mml:mi>p</mml:mi><mml:mi>α</mml:mi></mml:msub><mml:msub><mml:mi>p</mml:mi><mml:mi>β</mml:mi></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>]</mml:mo></mml:mrow><mml:mtext> </mml:mtext><mml:mo>.</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula></p><p>The length scale <inline-formula><mml:math id="inf8"><mml:mrow><mml:msqrt><mml:mrow><mml:mi>κ</mml:mi><mml:mo>/</mml:mo><mml:mi>η</mml:mi><mml:mo>′</mml:mo></mml:mrow></mml:msqrt></mml:mrow></mml:math></inline-formula> is the length on which the intrinsic rotation rate Ω<sub><italic>αβ</italic></sub> decays to the vorticity <italic>ω</italic><sub><italic>αβ</italic></sub> and is set by some molecular lengths in the system (<xref ref-type="bibr" rid="bib37">Stark and Lubensky, 2005</xref>; <xref ref-type="bibr" rid="bib9">Fürthauer et al., 2012</xref>, <xref ref-type="bibr" rid="bib10">2013</xref>). An upper bound for <inline-formula><mml:math id="inf9"><mml:mrow><mml:msqrt><mml:mrow><mml:mi>κ</mml:mi><mml:mo>/</mml:mo><mml:mi>η</mml:mi><mml:mo>′</mml:mo></mml:mrow></mml:msqrt></mml:mrow></mml:math></inline-formula> is the thickness <italic>d</italic> of the cell cortex, which is smaller than 1 μm (<xref ref-type="bibr" rid="bib5">Clark et al., 2013</xref>). Thus to describe long ranged flows in the cell cortex, we can consider the limit of <inline-formula><mml:math id="inf10"><mml:mrow><mml:msqrt><mml:mrow><mml:mi>κ</mml:mi><mml:mo>/</mml:mo><mml:mi>η</mml:mi><mml:mo>′</mml:mo></mml:mrow></mml:msqrt><mml:mo>→</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math></inline-formula> and obtain,<disp-formula id="equ11"><label>(10)</label><mml:math id="m11"><mml:mrow><mml:mn>0</mml:mn><mml:mo>=</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac><mml:mi>η</mml:mi><mml:msubsup><mml:mo>∂</mml:mo><mml:mi>γ</mml:mi><mml:mn>2</mml:mn></mml:msubsup><mml:msub><mml:mi>v</mml:mi><mml:mi>α</mml:mi></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mo>∂</mml:mo><mml:mi>α</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>ζ</mml:mi><mml:mo>−</mml:mo><mml:mi>P</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>/</mml:mo><mml:mn>3</mml:mn><mml:mo>+</mml:mo><mml:msub><mml:mo>∂</mml:mo><mml:mi>β</mml:mi></mml:msub><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:mi>ζ</mml:mi><mml:msub><mml:mi>p</mml:mi><mml:mi>α</mml:mi></mml:msub><mml:msub><mml:mi>p</mml:mi><mml:mi>β</mml:mi></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mo>∂</mml:mo><mml:mi>γ</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>ζ</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:msub><mml:mi mathvariant="italic">ϵ</mml:mi><mml:mrow><mml:mi>α</mml:mi><mml:mi>β</mml:mi><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mi>p</mml:mi><mml:mi>ν</mml:mi></mml:msub><mml:msub><mml:mi>p</mml:mi><mml:mi>γ</mml:mi></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>]</mml:mo></mml:mrow><mml:mtext> </mml:mtext><mml:mo>.</mml:mo></mml:mrow></mml:math></disp-formula></p><p>Using the approximation <inline-formula><mml:math id="inf11"><mml:mrow><mml:msubsup><mml:mi>σ</mml:mi><mml:mrow><mml:mi>z</mml:mi><mml:mi>z</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mi>o</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:mo>≈</mml:mo><mml:msup><mml:mi>P</mml:mi><mml:mrow><mml:mi>e</mml:mi><mml:mi>x</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:msup><mml:mo>/</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:math></inline-formula>, valid in thin films where <italic>P</italic><sup><italic>ext</italic></sup> is the constant external pressure, as well as the incompressibility condition ∂<sub><italic>γ</italic></sub><italic>v</italic><sub><italic>γ</italic></sub> = 0, we can rewrite this expression as,<disp-formula id="equ12"><label>(11)</label><mml:math id="m12"><mml:mrow><mml:mn>0</mml:mn><mml:mo>=</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac><mml:mi>η</mml:mi><mml:msubsup><mml:mo>∂</mml:mo><mml:mi>γ</mml:mi><mml:mn>2</mml:mn></mml:msubsup><mml:msub><mml:mi>v</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>+</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac><mml:mi>η</mml:mi><mml:msub><mml:mo>∂</mml:mo><mml:mi>i</mml:mi></mml:msub><mml:msub><mml:mo>∂</mml:mo><mml:mi>j</mml:mi></mml:msub><mml:msub><mml:mi>v</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mo>∂</mml:mo><mml:mi>i</mml:mi></mml:msub><mml:msubsup><mml:mi>p</mml:mi><mml:mi>z</mml:mi><mml:mn>2</mml:mn></mml:msubsup><mml:mi>ζ</mml:mi><mml:mo>+</mml:mo><mml:msub><mml:mo>∂</mml:mo><mml:mi>j</mml:mi></mml:msub><mml:msub><mml:mo>∂</mml:mo><mml:mi>z</mml:mi></mml:msub><mml:msub><mml:mi>ζ</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:msub><mml:mi mathvariant="italic">ϵ</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi><mml:mi>z</mml:mi></mml:mrow></mml:msub><mml:mtext> </mml:mtext><mml:mo>.</mml:mo></mml:mrow></mml:math></disp-formula></p><p>Here, the roman indices <italic>i</italic> and <italic>j</italic> denote the in film directions <italic>x</italic> and <italic>y</italic>. Finally, we integrate <xref ref-type="disp-formula" rid="equ12">Equation 11</xref> over the film thickness <italic>d</italic> and obtain.<disp-formula id="equ13"><label>(12)</label><mml:math id="m13"><mml:mrow><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac><mml:mi>η</mml:mi><mml:msubsup><mml:mo>∂</mml:mo><mml:mi>j</mml:mi><mml:mn>2</mml:mn></mml:msubsup><mml:msub><mml:mrow><mml:mover accent="true"><mml:mi>v</mml:mi><mml:mo>¯</mml:mo></mml:mover></mml:mrow><mml:mi>i</mml:mi></mml:msub><mml:mo>+</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac><mml:mi>η</mml:mi><mml:msub><mml:mo>∂</mml:mo><mml:mi>i</mml:mi></mml:msub><mml:msub><mml:mo>∂</mml:mo><mml:mi>j</mml:mi></mml:msub><mml:msub><mml:mrow><mml:mover accent="true"><mml:mi>v</mml:mi><mml:mo>¯</mml:mo></mml:mover></mml:mrow><mml:mi>j</mml:mi></mml:msub><mml:mo>−</mml:mo><mml:mi>γ</mml:mi><mml:msub><mml:mrow><mml:mover accent="true"><mml:mi>v</mml:mi><mml:mo>¯</mml:mo></mml:mover></mml:mrow><mml:mi>i</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mo>∂</mml:mo><mml:mi>i</mml:mi></mml:msub><mml:mi>T</mml:mi><mml:mo>+</mml:mo><mml:msub><mml:mi mathvariant="italic">ϵ</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mo>∂</mml:mo><mml:mi>j</mml:mi></mml:msub><mml:mi>τ</mml:mi><mml:mtext> </mml:mtext><mml:mo>,</mml:mo></mml:mrow></mml:math></disp-formula>where we introduced the averaged velocity<disp-formula id="equ14"><label>(13)</label><mml:math id="m14"><mml:mrow><mml:msub><mml:mrow><mml:mover accent="true"><mml:mi>v</mml:mi><mml:mo>¯</mml:mo></mml:mover></mml:mrow><mml:mi>i</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mi>d</mml:mi></mml:mfrac><mml:munderover><mml:mstyle displaystyle="true"><mml:mo>∫</mml:mo></mml:mstyle><mml:mn>0</mml:mn><mml:mi>d</mml:mi></mml:munderover><mml:mi>d</mml:mi><mml:mi>z</mml:mi><mml:msub><mml:mi>v</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mtext> </mml:mtext><mml:mo>,</mml:mo></mml:mrow></mml:math></disp-formula>the active cortical tension<disp-formula id="equ15"><label>(14)</label><mml:math id="m15"><mml:mrow><mml:mi>T</mml:mi><mml:mo>=</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mi>d</mml:mi></mml:mfrac><mml:munderover><mml:mstyle displaystyle="true"><mml:mo>∫</mml:mo></mml:mstyle><mml:mn>0</mml:mn><mml:mi>d</mml:mi></mml:munderover><mml:mi>d</mml:mi><mml:mi>z</mml:mi><mml:mi>ζ</mml:mi><mml:mtext> </mml:mtext><mml:mo>,</mml:mo></mml:mrow></mml:math></disp-formula>the active cortical torque density<disp-formula id="equ16"><label>(15)</label><mml:math id="m16"><mml:mrow><mml:mi>τ</mml:mi><mml:mo>=</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mi>d</mml:mi></mml:mfrac><mml:munderover><mml:mstyle displaystyle="true"><mml:mo>∫</mml:mo></mml:mstyle><mml:mn>0</mml:mn><mml:mi>d</mml:mi></mml:munderover><mml:mi>d</mml:mi><mml:mi>z</mml:mi><mml:msub><mml:mo>∂</mml:mo><mml:mi>z</mml:mi></mml:msub><mml:msub><mml:mi>ζ</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mtext> </mml:mtext><mml:mo>,</mml:mo></mml:mrow></mml:math></disp-formula>and the friction coefficient <italic>γ</italic><disp-formula id="equ17"><label>(16)</label><mml:math id="m17"><mml:mrow><mml:mi mathvariant="italic">γ</mml:mi><mml:msub><mml:mrow><mml:mover accent="true"><mml:mi>v</mml:mi><mml:mo>¯</mml:mo></mml:mover></mml:mrow><mml:mi>i</mml:mi></mml:msub><mml:mi mathvariant="normal">≃</mml:mi><mml:mfrac><mml:mn>1</mml:mn><mml:mi>d</mml:mi></mml:mfrac><mml:mi>η</mml:mi><mml:msub><mml:mo>∂</mml:mo><mml:mi>z</mml:mi></mml:msub><mml:msub><mml:mi>v</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:msubsup><mml:mo>|</mml:mo><mml:mn>0</mml:mn><mml:mi>d</mml:mi></mml:msubsup><mml:mo>=</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mi>d</mml:mi></mml:mfrac><mml:munderover><mml:mstyle displaystyle="true"><mml:mo>∫</mml:mo></mml:mstyle><mml:mn>0</mml:mn><mml:mi>d</mml:mi></mml:munderover><mml:mi>d</mml:mi><mml:mi>z</mml:mi><mml:mi>η</mml:mi><mml:msubsup><mml:mo>∂</mml:mo><mml:mi>z</mml:mi><mml:mn>2</mml:mn></mml:msubsup><mml:msub><mml:mi>v</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mtext> </mml:mtext><mml:mo>.</mml:mo></mml:mrow></mml:math></disp-formula></p><p>Since the cell is azimuthally symmetric, the <italic>x</italic> and <italic>y</italic> components of the <xref ref-type="disp-formula" rid="equ13">Equation 12</xref> decouple such that,<disp-formula id="equ18"><label>(17)</label><mml:math id="m18"><mml:mrow><mml:mi>ℓ</mml:mi><mml:msubsup><mml:mo>∂</mml:mo><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msubsup><mml:msub><mml:mrow><mml:mover accent="true"><mml:mi>v</mml:mi><mml:mo>¯</mml:mo></mml:mover></mml:mrow><mml:mi>x</mml:mi></mml:msub><mml:mo>−</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mi>ℓ</mml:mi></mml:mfrac><mml:msub><mml:mrow><mml:mover accent="true"><mml:mi>v</mml:mi><mml:mo>¯</mml:mo></mml:mover></mml:mrow><mml:mi>x</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mo>∂</mml:mo><mml:mi>x</mml:mi></mml:msub><mml:mi>T</mml:mi><mml:mtext> and </mml:mtext></mml:mrow></mml:math></disp-formula><disp-formula id="equ19"><label>(18)</label><mml:math id="m19"><mml:mrow><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac><mml:mi>ℓ</mml:mi><mml:msubsup><mml:mo>∂</mml:mo><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msubsup><mml:msub><mml:mrow><mml:mover accent="true"><mml:mi>v</mml:mi><mml:mo>¯</mml:mo></mml:mover></mml:mrow><mml:mi>y</mml:mi></mml:msub><mml:mo>−</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mi>ℓ</mml:mi></mml:mfrac><mml:msub><mml:mrow><mml:mover accent="true"><mml:mi>v</mml:mi><mml:mo>¯</mml:mo></mml:mover></mml:mrow><mml:mi>y</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mo>∂</mml:mo><mml:mi>x</mml:mi></mml:msub><mml:mi>τ</mml:mi><mml:mtext> </mml:mtext><mml:mo>.</mml:mo></mml:mrow></mml:math></disp-formula></p><p>Here, we have introduced the hydrodynamic length <inline-formula><mml:math id="inf12"><mml:mrow><mml:mi>ℓ</mml:mi><mml:mo>=</mml:mo><mml:msqrt><mml:mrow><mml:mi>η</mml:mi><mml:mo>/</mml:mo><mml:mi>γ</mml:mi></mml:mrow></mml:msqrt></mml:mrow></mml:math></inline-formula>.</p><p>In summary, molecular force generation by actin and myosin is represented by active force and torque dipoles. These give rise to an active tension and an active torque density in the material and can generate flows. Spatial gradients in active tension are generated by a spatially inhomogeneous distribution of myosin motors. These give rise to a flow along the gradient of myosin density (<xref ref-type="bibr" rid="bib23">Mayer et al., 2010</xref>), see <xref ref-type="disp-formula" rid="equ18">Equation 17</xref>. Spatial gradients in active torque density on the other hand give rise to a flow orthogonal to the gradient of myosin density (<xref ref-type="bibr" rid="bib10">Fürthauer et al., 2013</xref>) (see <xref ref-type="disp-formula" rid="equ19">Equation 18</xref>) in a direction that is set by the chirality of the torque dipoles in the cell cortex. The ratio of active torque densities to active tension thus generated is quantified by the chirality index,<disp-formula id="equ20"><label>(19)</label><mml:math id="m20"><mml:mrow><mml:mi>c</mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:mrow><mml:munderover><mml:mstyle displaystyle="true"><mml:mo>∫</mml:mo></mml:mstyle><mml:mn>0</mml:mn><mml:mi>d</mml:mi></mml:munderover><mml:mi>d</mml:mi><mml:mi>z</mml:mi><mml:msub><mml:mo>∂</mml:mo><mml:mi>z</mml:mi></mml:msub><mml:msub><mml:mi>ζ</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow><mml:mo>/</mml:mo><mml:mrow><mml:munderover><mml:mstyle displaystyle="true"><mml:mo>∫</mml:mo></mml:mstyle><mml:mn>0</mml:mn><mml:mi>d</mml:mi></mml:munderover><mml:mi>d</mml:mi><mml:mi>z</mml:mi><mml:mi>ζ</mml:mi></mml:mrow></mml:mrow><mml:mtext> </mml:mtext><mml:mo>.</mml:mo></mml:mrow></mml:math></disp-formula></p><p>A sketch of this mechanism is provided in <xref ref-type="fig" rid="fig2">Figure 2B</xref>, bottom sketch.</p></sec><sec id="s3-8-3"><title>Non-linear dependency on myosin levels</title><p>In general, the active torque density <italic>τ</italic> and the active tension <italic>T</italic> are functions of the myosin density, which in turn is proportional to NMY-2::GFP fluorescence intensity <italic>I</italic>(<italic>x</italic>). In the absence of myosin, <italic>T</italic> and <italic>τ</italic> vanish. For increasing <italic>I</italic>, we express active tension and active torque density in a Taylor expansion,<disp-formula id="equ21"><mml:math id="m21"><mml:mtext>T</mml:mtext><mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mi>α</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mi>I</mml:mi><mml:mo>+</mml:mo><mml:msub><mml:mi>α</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:msup><mml:mi>I</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:mi>O</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msup><mml:mi>I</mml:mi><mml:mn>3</mml:mn></mml:msup></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mtext> </mml:mtext><mml:mo>,</mml:mo></mml:mrow></mml:math></disp-formula><disp-formula id="equ22"><label>(20)</label><mml:math id="m22"><mml:mi>τ</mml:mi><mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mi>β</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mi>I</mml:mi><mml:mo>+</mml:mo><mml:msub><mml:mi>β</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:msup><mml:mi>I</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:mi>O</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msup><mml:mi>I</mml:mi><mml:mn>3</mml:mn></mml:msup></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mtext> </mml:mtext><mml:mo>,</mml:mo></mml:mrow></mml:math></disp-formula>with coefficients <italic>α</italic><sub>0</sub>, <italic>α</italic><sub>1</sub>, <italic>β</italic><sub>0</sub>, and <italic>β</italic><sub>1</sub>. The chirality index <italic>c</italic> = <italic>τ</italic>/<italic>T</italic> then shows a myosin dependence of the form,<disp-formula id="equ23"><label>(21)</label><mml:math id="m23"><mml:mrow><mml:mi>τ</mml:mi><mml:mo>/</mml:mo><mml:mi>T</mml:mi><mml:mo>≃</mml:mo><mml:mfrac><mml:mrow><mml:msub><mml:mi>β</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi>β</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mi>α</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi>α</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mi>I</mml:mi></mml:mrow></mml:mfrac><mml:mo>≃</mml:mo><mml:mfrac><mml:mrow><mml:msub><mml:mi>β</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mi>α</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mrow></mml:mfrac><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mfrac><mml:mrow><mml:msub><mml:mi>β</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mi>β</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mrow></mml:mfrac><mml:mo>−</mml:mo><mml:mfrac><mml:mrow><mml:msub><mml:mi>α</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mi>α</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mrow></mml:mfrac></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mtext> </mml:mtext><mml:mo>,</mml:mo></mml:mrow></mml:math></disp-formula>where we have neglected higher order terms. Rho signalling may directly influence the levels of myosin and thereby <italic>I</italic>. Also, Rho signaling affects the structure of the actin cytoskeleton (<xref ref-type="fig" rid="fig3s4">Figure 3—figure supplement 4</xref>), and could alter the fraction of active myosin, possibly even in a position-dependent manner. Rho exhibits a spatial profile similar to that of NMY-2 (<xref ref-type="bibr" rid="bib24">Motegi and Sugimoto, 2006</xref>), which is captured by the function <italic>I</italic>(<italic>x</italic>). The non-linearities induced by Rho can thus be accounted for by non-linearities in <italic>I</italic>(<italic>x</italic>). We can therefore approximate such additional effects of Rho signaling by modifications of the position-independent parameters <italic>α</italic><sub>0</sub>, <italic>α</italic><sub>1</sub>, <italic>β</italic><sub>0</sub>, and <italic>β</italic><sub>1</sub>.</p><p>However, our experimental data do not allow us to faithfully estimate the values of <italic>α</italic><sub>1</sub> and <italic>β</italic><sub>1</sub>. Thus, we consider only linear dependences in our fitting procedure using <italic>T</italic> = <italic>αI</italic> and <italic>τ</italic> = <italic>βI</italic>. The values for <italic>α</italic> and <italic>β</italic> and thus the chirality index <italic>c</italic> = <italic>α</italic>/<italic>β</italic> that our fitting routine determines correspond to averages of <italic>α</italic><sub>0</sub> + <italic>α</italic><sub>1</sub><italic>I</italic> and <italic>β</italic><sub>0</sub> + <italic>β</italic><sub>1</sub><italic>I</italic> along the embryo.</p></sec><sec id="s3-8-4"><title>Fitting phenomenological coefficients</title><p>To compare our theory to experiment, we numerically solve <xref ref-type="disp-formula" rid="equ18">Equations 17</xref> and <xref ref-type="disp-formula" rid="equ18">18</xref> using a finite difference scheme, using the two extreme points of the measured velocity profile as boundary conditions. <xref ref-type="disp-formula" rid="equ18">Equations 17</xref> and <xref ref-type="disp-formula" rid="equ18">18</xref> depend on three free parameters, the hydrodynamic length <italic>ℓ</italic>, and the constants <italic>α</italic> and <italic>β</italic> which relate the measured fluorescence intensity <italic>I</italic> to the active tension and the active torque density such that <italic>T</italic> = <italic>αI</italic> and <italic>τ</italic> = <italic>βI</italic>, respectively. We adjust these parameters by performing a least squares fit of the solutions to <xref ref-type="disp-formula" rid="equ18">Equations 17</xref> and <xref ref-type="disp-formula" rid="equ18">18</xref> to the measured velocity profile. In this way we obtain values for the hydrodynamic length <italic>ℓ</italic> and the chirality index <italic>c</italic> = <italic>τ</italic>/<italic>T</italic> numerically. We obtain error estimates for <italic>ℓ</italic>, <italic>α</italic>, and <italic>β</italic> from the Hessian of the residual of the least square fit with respect to the parameters.</p></sec></sec></sec></body><back><ack id="ack"><title>Acknowledgements</title><p>We are grateful to S Schonegg, A A Hyman, and W B Wood for their observation and analysis of the chiral whole-cell rotation of the dividing <italic>C. elegans</italic> zygote, without which we would not have recognized the importance of observing chiral cortical flow at an earlier stage. We are also grateful to B Fievet, J Rodriguez, and J Ahringer for observations of chiral flow in a PAR suppressor screen. We would also like to thank J S Bois, M Labouesse, J Prost, S B Reber, K Vijay Kumar, A-C Reymann, P Gross, P Chugh, and D Needleman for valuable discussions and critical comments. This work was supported by grant no. 281903 from the European Research Council (ERC) and SF acknowledges DFG and Human Frontier Science Program (HFSP) for funding.</p></ack><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>FJ: Reviewing editor, <italic>eLife</italic>.</p></fn><fn fn-type="conflict" id="conf2"><p>The other 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>SRN, 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>SF, Conception and design, Analysis and interpretation of data, Drafting or revising the article</p></fn><fn fn-type="con" id="con3"><p>SWG, Conception and design, Analysis and interpretation of 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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 “Active torque generation by the actomyosin cell cortex drives chiral symmetry breaking” for consideration at <italic>eLife</italic>. Your article has been favorably evaluated by Janet Rossant (Senior editor), a Reviewing editor, and 2 reviewers.</p><p>These results are interesting and timely as they provide strong experimental and theoretical support to a role of the actin cytoskeleton in breaking L/R symmetry in early stage embryos. Although this idea has been supported by some data on a few model organisms (including a recent paper by Schonegg et al., Genesis 2014 describing the chiral flow at the 1-cell stage and showing its dependence on nmy-2 activity) and has been discussed in recent reviews, the findings and concepts reported here provide strong advances to the field with a focus on the quantitative characterization of the chiral flows. The results point towards a significant connection between actomyosin-based chiral flows and LR symmetry breaking at embryo/tissue scales, with implications for LR symmetry-breaking and chiral morphogenesis in many other contexts.</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. We have one major point and a number of minor points for you to consider when revising your manuscript.</p><p>Major point:</p><p>1) The authors are inferring a connection between chiral flows in the zygote, those that occur at the 4-cell stage in ABa and ABp, and the skew of ABa and ABp that represents the first observable manifestation of embryonic handedness. But the links between observations made in the zygote and those at the 4-cell stage, remain tenuous. What would greatly strengthen this connection in my opinion would be to document a correlation between effects on chiral flow and skew - both measured at the 4-cell stage, for a few key perturbations (e.g. weak ect-2(RNAi), weak rga-3./4(RNAi, gsk-3(RNA) and possibly one of (dsh-2, mig-5, mom-2). My sense is that this could be done with relatively little additional effort.</p><p>Minor points:</p><p>2) The quantitative analysis of chiral flow and its interpretation in terms of a physical model for active chiral fluids is nicely done, but we think the authors need to tone down the claim that they have “discovered that the actomyosin cytoskeleton generates active torques...” What they have in fact shown is that a physical model assuming active torques can reproduce the observed kinematics.</p><p>3) It would be nice if the authors could comment on two related points:</p><p>(a) In the physical theory, counter-rotating flow depends on a gradient of active torque density. In the zygote this gradient is presumed to arise from a gradient of Myosin II activity, accounting for the simultaneous existence of AP and counterrotating flow. What about at the 4-cell stage? Are there axial flows (and possibly gradients of Myosin density) that coincide with the reported counter-rotation? Is the overall pattern of chiral + axial flow consistent with the physical theory?</p><p>(b) A basic assumption of the physical model is that actomyosin interactions have an intrinsic handedness that determines the handedness of the active local torque, which in turn determines the handedness of counter-rotating flow. But what happens when LR asymmetry is reversed as in cold-treated embryos or certain mutants? Is there a change in the handedness of the flow? Or (a more likely scenario) does this imply that there are other processes that can completely override any contribution of chiral flow to dictate handedness. We are not asking that the authors resolve this issue experimentally, but it would be nice if they could discuss briefly the implications.</p><p>4) When referring to Drosophila work, the authors should cite the 2 original Nature papers characterizing the myosinID gene: <xref ref-type="bibr" rid="bib16">Hozumi et al. 2006</xref> and Speder et al. 2006. In their reference list, the journal for Hozumi et al. is not spelled out correctly (should be Nature instead of Nat Cell Biol).</p><p>5) The chiral flows are intracellular by nature suggesting a cell autonomous effect. However the text is confusing at it is stated that the origin of the skew comes from 1-cell stage while concluding that it originates from chiral torque during 4-cell stage. The specific contribution of each flow (1-cell vs 4-cell) on the establishment and/or maintenance of handedness is thus unclear. Uncoupling the contribution of each flow would clarify the model and more clearly address the origin and duration of the effect of the flow on later events. Of note, results from Schonegg et al. suggest an independent origin of embryo rotation and handedness (see Table 2 in Schonegg et al.).</p><p>6) The authors mention that they could not find any defect in early embryos upon perturbing the flow at the 1-cell stage. We presume they have looked at embryo cortex rotation (and spindle asymmetry) and if so, their findings seem to contradict results from Schonegg et al. showing that defective acto-myosin network blocks early rotation of the 1-cell zygote (see Table 1 in Schonegg et al.). Comments?</p><p>7) Although they raise the question, the authors do not discuss what may control flow directionality, which is key to understanding symmetry breaking? Could it be asymmetry of the actin network, intrinsic molecular chirality of myosin and actin, etc? I think that expanding the discussion on this point would help clarify and complement the model.</p><p>8) Could the authors explain what in their view could be the basis of the independence of contractile tension (along A/P axis) and torque density (along L/R axis)?</p><p>9) Could the authors be more clear on how could a directional flow at the cortex induce asymmetry, for example of the spindle?</p></body></sub-article><sub-article article-type="reply" id="SA2"><front-stub><article-id pub-id-type="doi">10.7554/eLife.04165.027</article-id><title-group><article-title>Author response</article-title></title-group></front-stub><body><p><italic>1) The authors are inferring a connection between chiral flows in the zygote, those that occur at the 4-cell stage in ABa and ABp, and the skew of ABa and ABp that represents the first observable manifestation of embryonic handedness. But the links between observations made in the zygote and those at the 4-cell stage, remain tenuous. What would greatly strengthen this connection in my opinion would be to document a correlation between effects on chiral flow and skew - both measured at the 4-cell stage, for a few key perturbations (e.g. weak ect-2(RNAi), weak rga-3./4(RNAi, gsk-3(RNA) and possibly one of (dsh-2, mig-5, mom-2). My sense is that this could be done with relatively little additional effort</italic>.</p><p>This is a very good suggestion. As requested, we have now quantified chiral flow velocities for wild type, <italic>ect-2</italic> and <italic>rga-3</italic> RNAi conditions in the ABa cell. We find that changes in skew angle are indeed concomitant with changes in chiral counterrotation velocity vc in ABa, with <italic>ect-2</italic> (vc: -3.4 ± 1.4 μm/min, n=8, skew: 15.8 ± 4.9°, n=14) RNAi leading to reduced chiral flows and a reduced skew, and <italic>rga-3</italic> (vc: -6.7 ± 0.7 μm/min, n=8, skew: 37.8 ± 6.1°, n=13) leading to increased chiral flows and an increased skew compared to wild type (vc: -5.2 ± 1.1 μm/min, n=10, skew: 23.6 ± 3.7°, n=12).</p><p>We have included a new plot, <xref ref-type="fig" rid="fig4">Figure 4B</xref>, with wild type, <italic>ect-2</italic> and <italic>rga-3</italic> chiral velocity quantifications and have updated <xref ref-type="other" rid="video6">Video 6</xref> that displays chiral flow under <italic>ect-2</italic> and <italic>rga-3</italic> RNAi conditions in addition to the non-RNAi condition. An in-depth analysis of all Wnt pathway members and their control of chiral flow, together with a mechanistic picture of how chiral flows execute the clockwise skew event warrants an independent analysis, which is beyond the scope of our manuscript.</p><p>However, the new <xref ref-type="fig" rid="fig4">Figure 4B</xref> allows us to conclude that active torque generation and chiral counterrotatory flows participate in execution of left-right (LR) symmetry breaking of the embryo.</p><p><italic>2) The quantitative analysis of chiral flow and its interpretation in terms of a physical model for active chiral fluids is nicely done, but we think the authors need to tone down the claim that they have “discovered that the actomyosin cytoskeleton generates active torques...” What they have in fact shown is that a physical model assuming active torques can reproduce the observed kinematics</italic>.</p><p>We agree with this suggestion and have toned down the wording accordingly.</p><p><italic>3) It would be nice if the authors could comment on two related points:</italic></p><p><italic>(a) In the physical theory, counter-rotating flow depends on a gradient of active torque density. In the zygote this gradient is presumed to arise from a gradient of Myosin II activity, accounting for the simultaneous existence of AP and counterrotating flow. What about at the 4-cell stage? Are there axial flows (and possibly gradients of Myosin density) that coincide with the reported counter-rotation? Is the overall pattern of chiral + axial flow consistent with the physical theory?</italic></p><p>This is a very good point, we do observe axial flows at the 4-cell stage. The cleavage plane comprises a stripe of high myosin activity leading to a gradient on either side of the stripe (<xref ref-type="other" rid="video6">Video 6</xref>). This will lead to both chiral flows and axial flows as observed in our movies at the 4-cell stage. However, quantification of axial flows is non-trivial because of an additional rotational movement (<xref ref-type="fig" rid="fig4s1">Figure 4–figure supplement 1B</xref>), which also enters our quantification as an axial flow (when viewed from the dorsal side of the embryo).</p><p>A full quantification of the different flows generated at the 4-cell stage is beyond the scope of the paper. Please note that this would require a precise analysis of cell-cell interactions with neighbors. However, on a qualitative level, the myosin distributions in ABa and ABp (<xref ref-type="other" rid="video6">Video 6</xref>) are such that the overall pattern of chiral and axial flow is consistent with the predictions from our physical theory.</p><p>The expected flow profiles from our theoretical description given a stripe of high myosin activity (corresponding to the cleavage plane) is shown in newly added <xref ref-type="fig" rid="fig4s5">Figure 4–figure supplement 5</xref>.</p><p><italic>(b) A basic assumption of the physical model is that actomyosin interactions have an intrinsic handedness that determines the handedness of the active local torque, which in turn determines the handedness of counter-rotating flow. But what happens when LR asymmetry is reversed as in cold-treated embryos or certain mutants? Is there a change in the handedness of the flow? Or (a more likely scenario) does this imply that there are other processes that can completely override any contribution of chiral flow to dictate handedness. We are not asking that the authors resolve this issue experimentally, but it would be nice if they could discuss briefly the implications</italic>.</p><p>This is a very interesting question. In the case of cold treatment, a 0.5% increase of sinistral handed embryos was observed by Wood WB et al. (1996) when the worms were maintained at 10°C. It has to be noted that the myosin ATPase activity and actin filament sliding velocity also reduce by 4-fold at low temperatures (Yanagida T et al., Nature, 1996). Therefore, we would expect a marked reduction in myosin driven activities and significantly reduced chiral flows in cold-treated embryos. Thus, in the case of cold-treated embryos, because chiral flows are likely to be significantly reduced, additional unknown processes could effect a marginal increase in sinistral handed embryos.</p><p>Similarly, in the case of certain mutants such as <italic>gpa-16(it43)</italic> reported by <xref ref-type="bibr" rid="bib2">Bergmann et al. (2003)</xref>, it is likely that the handedness of flow is unaffected.</p><p>Given that <italic>gpa-16</italic> genetically interacts with a few <italic>par</italic> genes (<xref ref-type="bibr" rid="bib2">Bergmann et al., 2003</xref>), it is possible that cortical localization of myosin is impaired, which could affect chiral flow. Also, the skew event facilitated by chiral flows is likely to be dependent on neighboring cell-cell interactions as well (see response to point 9). Given the aberrant behavior of mitotic spindles in the first three cleavages prior to LR symmetry breaking (<xref ref-type="bibr" rid="bib2">Bergmann et al., 2003</xref>), the <italic>it143</italic> mutant could have slightly misplaced neighbors that could as well result in impaired LR symmetry breaking.</p><p>It remains to be seen which processes act in concert with chiral flows to dictate embryonic handedness and as the reviewers have indicated we also expect chiral flow handedness to be unaffected in cold-treated embryos as well as the different mutants.</p><p><italic>4) When referring to Drosophila work, the authors should cite the 2 original Nature papers characterizing the myosinID gene:</italic> <xref ref-type="bibr" rid="bib16"><italic>Hozumi et al. 2006</italic></xref> <italic>and</italic> <italic>Speder et al. 2006</italic><italic>. In their reference list, the journal for Hozumi et al. is not spelled out correctly (should be Nature instead of Nat Cell Biol)</italic>.</p><p>We have included the suggested paper in the reference section and corrected the Hozumi et al., reference.</p><p><italic>5) The chiral flows are intracellular by nature suggesting a cell autonomous effect. However the text is confusing at it is stated that the origin of the skew comes from 1-cell stage while concluding that it originates from chiral torque during 4-cell stage. The specific contribution of each flow (1-cell vs 4-cell) on the establishment and/or maintenance of handedness is thus unclear. Uncoupling the contribution of each flow would clarify the model and more clearly address the origin and duration of the effect of the flow on later events. Of note, results from Schonegg et al. suggest an independent origin of embryo rotation and handedness (see Table 2 in Schonegg et al.)</italic>.</p><p>We are sorry for the misunderstanding and we have amended the conclusions in the manuscript to read better. We did not intend to mean that the origin of the skew comes from the 1-cell stage, but rather wanted to state that there is a general link between LR symmetry breaking genes and chiral flows. We completely agree with Schonegg et al., in the fact that cortical chirality independently regulates early chiral events at the 1-cell stage as well as the chiral event at the 4-cell stage.</p><p>Chiral flow generation is an intrinsic property of the actomyosin cortex and is generated independently at multiple stages during <italic>C. elegans</italic> development.</p><p><italic>6) The authors mention that they could not find any defect in early embryos upon perturbing the flow at the 1-cell stage. We presume they have looked at embryo cortex rotation (and spindle asymmetry) and if so, their findings seem to contradict results from Schonegg et al. showing that defective acto-myosin network blocks early rotation of the 1-cell zygote (see Table 1 in Schonegg et al.)</italic>. <italic>Comments?</italic></p><p>We apologize for the poor wording in this part of the main text. We intended to comment that counterrotatory flows during polarity establishment at the 1-cell stage cannot provide information about LR symmetry breaking of the embryo because of the presence of just a single established body axis.</p><p>We have corrected the main text accordingly. It is important to understand that the chiral counterrotatory flows described in this paper observed before pronuclei meeting at the 1-cell stage and during the 4-cell stage, are mechanistically different from the chiral flows that lead to a whole body rotation during cytokinesis of the 1-cell stage described in Schonegg et al.</p><p>The latter event is also likely to require the generation of clockwise torques, however, the physical basis of this whole body rotation during cytokinesis is not understood.</p><p><italic>7) Although they raise the question, the authors do not discuss what may control flow directionality, which is key to understanding symmetry breaking? Could it be asymmetry of the actin network, intrinsic molecular chirality of myosin and actin, etc? I think that expanding the discussion on this point would help clarify and complement the model</italic>.</p><p>At the 1-cell stage, when viewed from the posterior pole, we observe the flow direction to be counterclockwise in the posterior and clockwise in the anterior. This directionality is due to the generation of clockwise torques (when viewed from the outside) in the cortical layer.</p><p>A gradient in these torques then leads to either counterclockwise or clockwise flows depending on the direction of the gradient as illustrated in <xref ref-type="fig" rid="fig2">Figure 2b</xref>, bottom sketch. It will now be interesting to investigate the molecular basis of clockwise torques and as the reviewers have indicated, the observed chirality is likely due to a combination of an intrinsic chirality of the actin filament, a mismatch in the step size of myosin and the helical pitch of actin and a symmetry broken orthogonal to the cortical plane with the membrane on one side and the cytoplasm on the other side. A clarification of this is beyond the scope of our current work.</p><p><italic>8) Could the authors explain what in their view could be the basis of the independence of contractile tension (along A/P axis) and torque density (along L/R axis)?</italic></p><p>This is a very good point that was not properly discussed. We have demonstrated that myosin molecular motors are the origin of both active tension and active torque in the cortex. There are two possibilities wherein (i) myosin either directly produces torque (dipoles) or (ii) myosin generated force (dipoles) are converted to torque dipoles via the structure of the actin network.</p><p>Our finding that active torques can be regulated independently from active tension hints at the latter possibility (ii). A definitive answer is however, beyond the scope of the present study.</p><p><italic>9) Could the authors be more clear on how could a directional flow at the cortex induce asymmetry, for example of the spindle?</italic></p><p>We propose that the mechanism by which chiral flows execute the skew is similar to a crawler excavator or a digger with both chains rotating in opposite directions, allowing the whole machine to turn on the spot.</p><p>Chiral counterrotatory flows in the ABa and ABp daughter cells with the help of adhesion to the substrate below (EMS cell) would then cause the daughter cells to roll past each other pushing the left daughters more anterior than the right. A corresponding and clarifying sentence has been added to the manuscript.</p></body></sub-article></article> |