We set out to quantify chiral rotatory flows in the actomyosin cell cortex of the first nine cell divisions in early C. elegans development. In order to image C. elegans embryogenesis, various mounting techniques have been described that either mildly compress the embryo or mount the embryo uncompressed. We first compared the degree of embryo compression using two common mounting methods (Figure 1—figure supplement 1): (1) Attaching the embryos to an agarose pad (Bargmann and Avery, 1995) and (2) embedding the embryos in low-melt agarose (Naganathan et al., 2014). As reported before (Walston and Hardin, 2010), we found that the first method indeed compressed the embryos and leads to an aspect ratio of 1.29 ± 0.14. Embedding embryos in low-melt agarose resulted in an aspect ratio of 1.02 ± 0.3 and hence a reduced compression. Measurements of early embryos in the adult uterus revealed an aspect ratio of 1.16 ± 0.7. (Figure 1—figure supplement 1), indicating that in-utero embryo compression strength is intermediate. Throughout this study, we used the classic mounting method using an agarose pad, unless stated otherwise.

We started with investigating the first two divisions, the divisions of the P₀ zygote and the AB cell, and quantified flows from embryos containing endogenously tagged non-muscle myosin-II (NMY-2)::GFP using spinning disc microscopy (Videos 1 and 2). We used particle image velocimetry (PIV) to determine cortical flow velocities in two rectangular regions of interest (ROIs) placed on opposite sides of the cytokinetic ring (Figure 1A, Figure 1—figure supplement 2). In particular, we were interested in chiral cortical flows that yield velocity contributions parallel to the cytokinetic ring, referred to as ‘y-direction flows’ (Figure 1—figure supplement 2). Beginning at the onset of furrow ingression, y-direction flows were averaged in each ROI over a time period of 21 s. This narrow time window provides a characterization of cortical flows restricted to mid-cytokinesis, and it ensures that shape changes of the dividing cell are minor and do not affect the velocity measurements. Note that, over the full time course from early to late cytokinesis, cortical flow velocities vary (Figure 1—figure supplement 2), while we did not observe major differences in the myosin distribution (Figure 1—figure supplement 3).

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Strikingly, while y-direction flows in both ROIs of the P₀ cell occur with the same orientation (Figure 1A, left), they have opposite orientations in AB cells (Figure 1A, right), such that the two dividing cell halves of the AB cell effectively spin in opposite directions. We refer to such flows as chiral counter-rotating flows (Figure 1—figure supplement 2, Videos 1 and 2). To quantify the speed and the handedness of counter-rotating flows more generally, we use the measured velocities in each ROI, denoted as $\underset{¯}{v_1}$ and $\underset{¯}{v_2}$, and define a chiral counter-rotation flow velocity $v_c=\underset{¯}{e_z}\cdot (\underset{¯}{e_x}\times \underset{¯}{v_2}-\underset{¯}{e_x}\times \underset{¯}{v_1})$, where $\underset{¯}{e_x}$ is a unit length base vector pointing from the cytokinetic ring toward the pole, $\underset{¯}{e_y}$ is an orthogonal unit length vector parallel to the cytokinetic plane (Figure 1A), and $\underset{¯}{e_z}=\underset{¯}{e_x}\times \underset{¯}{e_y}$. With this definition, $|v_c|$ quantifies the speed of counter-rotating flows and the sign of $v_c$ denotes their handedness (see Materials and methods and Figure 1—figure supplement 2). For the AB cell division, we then find $v_c$ = −7.01 ± 0.34 µm/min (mean ± error of the mean at 95% confidence, unless otherwise stated), indeed indicating the presence of counter-rotating flows with consistent handedness. Furthermore, for the division of the P₀ cell, we find that counter-rotating flows are essentially absent ($v_c$ = 0.01 ± 0.51 µm/min). Instead, the cortical flows of dividing P₀ cells, where both cell halves move in the same direction, correspond to net-rotating flows that resemble whole embryo rotations (Schonegg et al., 2014). We conclude that out of the first two cell divisions in the developing nematode, only one is accompanied by chiral counter-rotating actomyosin flows.

This raises the question which of the later cell divisions display chiral counter-rotating flows, and which do not. To answer this, we quantified chiral counter-rotating flows for the next seven cell divisions as described above. We found that cells of the AB lineage (ABa, ABp, ABar and ABpr) exhibit chiral counter-rotating flows with $v_c$ = −3.46 ± 0.33 µm/min, −3.68 ± 0.41 µm/min, −1.24 ± 0.19 µm/min and −1.16 ± 0.22 µm/min respectively (Figure 1B). In contrast, average chiral counter-rotating flow velocities of cells of the P/EMS lineage (P₁, EMS and P₂) are small: 0.15 ±0.38 µm/min, 0.03 ± 0.14 µm/min and 0.25 ± 0.19 µm/min, respectively (Figure 1B). Like P₀, cells of the P/EMS lineage, instead exhibit net-rotating flows, where the y-direction flows have the same orientation in both cell halves (Figure 1—figure supplement 4, Videos 1–5). To conclude, early cell divisions of the AB lineage, but not of the P/EMS lineage, undergo chiral counter-rotating flows. We note that due to a decrease in the overall cortical flow speed, also the chiral counter-rotating flow velocity $v_c$ in the AB lineage decreases as development progresses and cells become smaller (Figure 2B, Figure 1—figure supplement 5). This is accompanied by a decrease in the myosin concentration in the cytokinetic ring (Figure 1—figure supplement 5) and is consistent with overall contractility being reduced when development progresses. Together, these findings show that both the presence and the strength of chiral counter-rotating flows are related to the fate of the dividing cell.

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We have shown previously that chiral counter-rotating cortical flows are driven by active torque generation that depends on the myosin distribution (Naganathan et al., 2014). Given that chiral counter-rotating flows arise only during AB and not P/EMS cell divisions, we hypothesize that myosin distributions during division of the AB and the P/EMS-lineage cells differ in specific features. To investigate this, we determined the NMY-2 distribution along the long axis of dividing cells for the first six cell divisions (Figure 1C), and extracted two features (see Materials and methods): (1) a myosin ratio that characterizes the difference in myosin intensity between the two cell halves (Figure 1D, Figure 1—figure supplement 6) and (2) the relative myosin peak position along the long axis, which characterizes the cytokinetic ring position and the degree of asymmetry in the cell division (Figure 1E). We note that these quantities could be coupled, either mechanically (Sedzinski et al., 2011) or via par polarity complexes that facilitate cellular myosin asymmetry (Munro et al., 2004; Gross et al., 2019) and asymmetric positioning of the cleavage plane (Galli and van den Heuvel, 2008; Colombo et al., 2003). We find that myosin ratios in dividing AB, ABa and ABp cells are 1.0 ± 0.13, 0.92 ± 0.29, and 1.00 ± 0.24, respectively, indicating that there is no significant difference in cortical myosin concentration between the two halves of dividing cells from the AP lineage. In contrast, the myosin ratios in P₀, P₁ and EMS are 1.67 ± 0.3, 2.07 ± 0.61, 1.68 ± 0.33, respectively (Figure 1E). Furthermore, we find that the relative position of myosin peaks along the long axis of dividing AB, ABa and ABp cells are 0.49 ± 0.01, 0.49 ± 0.01 and 0.51 ± 0.02, respectively. In contrast, myosin peaks in dividing P₀, P₁ and EMS cells are positioned at 0.58 ± 0.01, 0.57 ± 0.02 and 0.59 ± 0.01 relative to cell length, respectively (Figure 1D). To conclude, the early cell divisions of the AB lineage are symmetric in terms of myosin intensity ratio and cytokinetic ring position, while early cell divisions of the P/EMS lineage are asymmetric in these two features.

We next asked if the presence and absence of chiral counter-rotating flows in AB and P/EMS cells can be accounted for by the observed difference in myosin ratio and cytokinetic ring position. To this end, we utilized a thin film active chiral fluid theory (Fürthauer et al., 2013; Naganathan et al., 2014) that describes active chiral processes in a fluid-like material and the flows these processes are expected to give rise to. We then considered a minimal model that allows to compute $y$-direction flows, and hence chiral counter-rotating flow velocities $v_c$, from a given myosin ratio and cytokinetic ring position (Appendix, Videos 6–8). We find that a symmetric myosin distribution and cytokinetic ring position (myosin ratio $\approx 1$, relative myosin peak position $\approx 0.5$) result in a maximum value of $v_c$. Accordingly, myosin ratios different from one and an asymmetric ring position result in smaller values of $v_c$, where the model suggests that $v_c$ is more sensitive to changes in the myosin ratio than to changes in the ring position (Figure 1—figure supplement 7).

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Our results indicate that the switch between chiral and non-chiral flows between the two lineages could in principle be attributed to the observed difference in myosin ratio, cytokinetic ring position or both (Appendix 1—table 1). We note, however, that while the thin film active chiral fluid theory can quantitatively relate y-direction flows of dividing AB, ABa, and ABp cells with the myosin profiles along their long axis (Figure 1—figure supplement 8, Appendix 1—table 1), this is not possible for the P/EMS lineage. We suspect that this is due to specific features of the overall dynamics that are not included in our theoretical description, such as movements of the midbody remnant particular to the P/EMS linage (Singh and Pohl, 2014), as well as more general aspects that we have neglected for simplicity, such as the shape dynamics of the dividing cell (Mietke et al., 2019) and possible inhomogeneities in friction forces due to mechanical interactions with the surrounding. This indicates that our understanding of the mechanisms that lead to the cortical flows in P/EMS linage divisions is still incomplete. Taken together, our analysis implies a mechanism whereby the occurrence of chiral counter-rotating actomyosin flows depends on the amount of cortical myosin in the cell halves and the position of the cytokinetic ring.

Next, we asked whether symmetric cell division is indeed required for chiral counter-rotating flows to emerge. The first cell division in wild-type embryos is asymmetric and does not display chiral counter-rotating flows. The asymmetry of the P₀ division is controlled by Par polarity complexes (Kemphues et al., 1988) and depletion of the posterior Par protein, PAR-2, led to a symmetric myosin distribution with a myosin ratio of one and centered ring position (Figure 1—figure supplement 9). Strikingly, par-2 depletion also triggered the emergence of chiral counter-rotating flow during P₀ cytokinesis ($v_c$ = −5.71 ± 0.89 µm/min; Figure 1—figure supplement 9) demonstrating that inducing symmetric cell division via par-2 (RNAi) is sufficient to trigger chiral counter-rotating flows in P₀. Moreover, our hydrodynamic theory predicts that a centered ring position should result in chiral counter-rotating flows regardless of myosin asymmetry (Figure 1—figure supplement 7). To test this, we performed lin-5 (RNAi) treatment and analyzed cortical actomyosin flows. LIN-5 is involved in generating asymmetric pulling forces that displace the spindle from the center (Galli and van den Heuvel, 2008; Lorson et al., 2000). We first confirmed that indeed the myosin distribution in P₀ is still asymmetric in these embryos (albeit reduced when compared with control), while the ring position is at the center (Figure 1—figure supplement 10). Quantifying cortical actomyosin flows revealed that lin-5 (RNAi) triggered the emergence of chiral counter-rotating flows ($v_c$ = −1.73 ± 0.74 µm/min; Figure 1—figure supplement 10). Given that cortical myosin levels upon lin-5 (RNAi) are comparable to those observed in control, the emergence of chiral counter-rotating flows is unlikely to be the result of higher overall actomyosin activity (Figure 1—figure supplement 10). Although we cannot fully uncouple the effect of myosin asymmetry and ring position, these experimental results are qualitatively consistent with our theoretical predictions and thus suggest that both, asymmetries in myosin distribution and ring position, counteract the emergence of chiral counter-rotating flows. We conclude that symmetric cell division is required for chiral counter-rotating actomyosin flows to emerge during cytokinesis.

We next focus our attention on the mechanism by which dividing cells assume their correct position inside the developing embryo. Specific rotations of the spindle (termed spindle skews) are thought to be a key mechanism by which dividing cells reposition themselves in the embryo (Naganathan et al., 2014; Singh and Pohl, 2014; Liro and Rose, 2016; Bergmann et al., 2003). One hypothesis is that spindle skews are driven by spindle elongation (Hennig et al., 1992). In this scenario, spindle elongation causes a skew because the spindle rearranges within an asymmetric cell shape, thus responding to the constraints provided by neighboring cells and the eggshell (Figure 3A). An alternative hypothesis is that spindle skews are driven by chiral counter-rotating flows of the dividing cell halves (Naganathan et al., 2014). In this case, the mechanism is akin to a bulldozer rotating on the spot by spinning its chains in opposite directions. If the cortical surface experiences a spatially inhomogeneous friction, e.g. when the friction with the eggshell is different from the friction with a neighboring cell, chiral counter-rotating cortical flows of the two cell halves will lead to a torque that acts on the cell and can lead to a cell skew (Figure 3A). In this scenario, the cell orientation changes over time, and the spindle simply aligns accordingly.

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To discriminate between these two hypotheses, we first asked if those cells that undergo spindle skews also exhibit chiral counter-rotating flows. We quantified spindle skews by determining the positions of spindle poles at the beginning and at the end of cytokinesis in embryos expressing TBB-2::mCherry for the first 11 cell divisions (Video 9). The angle between the lines that join the two spindle poles at the onset and after cell division defines the spindle skew angle (Figure 2A). We find that cells of the early AB lineage undergo an average skew angle of 21.17 ± 3.22° (AB: 37.51 ± 2.51°, ABa: 27.64 ± 2.29°, ABp: 23.98 ± 1.19°, ABal: 17.73 ± 1.98°, ABar: 15.51 ± 3.51°, ABpl: 10.43 ± 1.92° and ABpr: 15.42 ± 2.11°). In comparison, P/EMS lineage cells undergo significantly reduced spindle skews with an average skew angle of of 2.32 ± 0.43° (P₀: 2.31 ± 1.67°, P₁: 2.25 ± 0.66°, EMS: 2.66 ± 1.25°, P₂: 4.36 ± 1.67°). Note also, that spindle skew angles in the AB lineage decrease as development progresses, while the P/EMS lineage spindle skew angles remain small and approximately constant (Figure 2B). Given that AB lineage cells, but not P/EMS lineage cells, undergo chiral counter-rotating flows, we conclude that cells that exhibit significant chiral counter-rotating flows also undergo significant spindle skews during early development (Figure 2B).

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In our earlier work, we showed that the skew of the ABa and ABp cells is in fact a three-dimensional process, tilting the spindle in both the anteroposterior-left-right (AP-LR) plane and in the orthogonal left-right-dorsoventral (LR-DV) plane (Naganathan et al., 2014). Therefore, in addition to the well known skew of the AB cell in the anteroposterior-dorsoventral (AP-DV) plane that sets up the dorsoventral axis, we hypothesized that there might be an additional skew in the orthogonal LR-DV plane (Figure 3E, Figure 3—figure supplement 1). In order to test this hypothesis, we examined spindle orientation throughout AB cell cytokinesis from an anterior view down the anteroposterior axis, but did initially not observe any obvious skews in the orthogonal LR-DV plane (Figure 3—figure supplement 1). However, as discussed above, the embryo mounting method used in this study slightly compresses the embryos along their left-right axis, which could constrain spindle movements along this direction. In order to determine whether the spindle skew in the LR-DV plane is concealed by embryo compression, embryos were embedded in low-melt agarose to prevent compression and subsequently analyzed. Chiral counter-rotating flows in the AB cell could still be observed and the spindle skew in the AP-DV plane was similar in compressed and uncompressed conditions (37.51 ± 2.51° and 36.67 ± 9.24°, respectively). However, the absence of embryo compression revealed a significant clockwise skew in the LR-DV plane (when viewed from the anterior) with a skew angle of 31.85 ± 11.11° (Figure 3G, Figure 3—figure supplement 1, Video 10). We did not observe any significant skew in the P₁ cell in the LR-DV plane (Figure 3—figure supplement 1). These findings show that skews of the AB cell division axis are in fact also a three-dimensional process, where the skew occurs in both the AP-DV and LR-DV planes. Note that, under the assumption that the friction forces at the interface between AB and P₁ cell are high compared to those between AB cell and the surrounding vitelline layer and egg shell, the handedness of the chiral counter-rotating flows is consistent with the observed orientation of the spindle skew in the LR-DV plane. Using the same argument, the skew in the AP-DV plane could in principle result from friction forces between the AB cell and the egg shell that are different on the left and on the right side (see section 'Left-right asymmetric friction forces can account for the skew of the AB cell’ for a further analysis of this hypothesis).

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The results obtained so far are consistent with a scenario in which cell fate determines the presence of chiral counter-rotating flows, and these chiral counter-rotating flows then drive spindle skews. We test for these two causal relationships separately. We first asked if cell fate determines the presence of chiral counter-rotating flows. In this case, we expect that on the one hand, anteriorizing the worm leads to equal and significant chiral counter-rotating flows for the second and third division. On the other hand, we expect that posteriorizing the worm would lead to an absence of chiral counter-rotating flows. Consistent with the first expectation, anteriorizing the embryo via par-2; chin-1; lgl-1 (RNAi) (Beatty et al., 2013) results in the second and the third cell division exhibiting chiral counter-rotating flows with $v_c$ values that are not significantly different from one another (−6.53 ± 1.13 µm/min and −6.67 ± 0.8 µm/min, respectively; Wilcoxon rank sum test at 95% confidence; Figure 3B,C). Consistent with the second expectation, posteriorizing the embryo via par-6 (RNAi) (Hung and Kemphues, 1999) results in both the second and the third cell division not exhibiting counter-rotating flows ($v_c$ = 0.07 ± 0.71 µm/min and −0.85 ± 1.13 µm/min, respectively; Figure 3B,C, Videos 11–13). Hence, cell fate determines the presence or absence of chiral counter-rotating flows. Furthermore, switching the cell fate results in concomitant changes in the spindle skew angle in the AP-DV plane for both the second and third cell division in compressed as well as non-compressed embryos (Figure 3D,H). Strikingly, we observed that spindles of the two AB-like cells in uncompressed anteriorized embryos undergo clockwise skew in the LR-DV plane when viewed from the closest pole (skew angles of each AB-like cell = 26.93 ± 6.78° and 27.44 ± 8.66°) (Figure 3H, Figure 3—figure supplement 2, Videos 14–16). Again, assuming that the friction forces at the cell-cell contact surface are high, the orientation of these skews is consistent with the handedness of the observed chiral counter-rotating flows (Figure 3—figure supplement 1). We conclude that cell fate determines both the presence of chiral counter-rotating flows and the degree as well as the direction of spindle skew.

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We next tested the second causal relationship, and asked if chiral counter-rotating flows in the cortex drive spindle skews. Borrowing from the analogy of a bulldozer, for which the speed of the chiral counter-rotating chains sets the speed at which the entire bulldozer rotates, we investigated if increasing or decreasing the chiral counter-rotating flow velocities results in corresponding changes of the spindle skew rate. We first evaluated if we can increase and decrease chiral counter-rotating flow velocities in the AB cell, by RNAi of RhoA regulators. Weak perturbation RNAi of ect-2 results in $v_c$ = −3.57 ± 0.57 µm/min, which is reduced in comparison to the unperturbed embryo ($v_c$ = −6.6 ± 0.37 µm/min, Wilcoxon rank sum test at 95% confidence, Figure 4A,B). Conversely, weak perturbation RNAi of rga-3 increases the chiral counter-rotating flow velocity to $v_c$ = −8.71 ± 0.58 µm/min (Figure 4A,B). Note that these perturbations impact only chiral counter-rotating flows, and do not significantly change overall contractility, as illustrated by the on-axis flow into the contractile ring, which is not significantly altered (Wilcoxon rank sum test at 95% confidence, Figure 4C). We now use these perturbations to test whether chiral counter-rotating flows determine the rate of spindle skew. We indeed find that decreasing the chiral counter-rotating flow velocity by mild ect-2 (RNAi) leads to concomitant reduction of the rate of spindle skew (average peak rate: 0.59 ± 0.05°/min, Video 17) as compared to 1.01 ± 0.1°/min in control embryos (Figure 4D, Video 18). Conversely, increasing RhoA signaling by mild rga-3 (RNAi) treatment leads to an increase of both the chiral counter-rotating flow velocity and the rate of spindle skew (average peak rate: 1.29 ± 0.12°/min (Video 19, Figure 4D). Note that the peak rates of spindle elongation remain unchanged as compared to the control for both perturbations (Figure 4D, Figure 4—figure supplement 1). We conclude that changing chiral counter-rotating flow velocities results in concomitant changes of the rates of spindle skews without impacting on the dynamics of spindle elongation. In addition, given that the contractility-driven flows remain unaltered upon rga-3 (RNAi) and ect-2 (RNAi), overall actomyosin contractility is unlikely to contribute significantly to cell division skews. These results instead support our second hypothesis, that chiral counter-rotating flows mechanically drive cell skews and consequently also spindle skews.

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The position of the mitotic spindle is determined by an evolutionary conserved protein complex, containing GPR-1/2 and its interacting partner LIN-5/NuMa, that facilitates pulling forces on astral microtubules at the cell cortex (Srinivasan et al., 2003; Galli et al., 2011; Colombo et al., 2003). Therefore, we next asked whether spindle skews during anaphase of dividing AB lineage cells are dependent on this known spindle orientation pathway. To address this, we analyzed actomyosin flows and the cell division skew of AB upon depletion of lin-5 and gpr-1/2. Both perturbations resulted in defects that have been previously described, including symmetric cell division of P₀ (Figure 1—figure supplement 10 and data not shown) and misaligned spindles in the P₁ and AB cells (Figure 4—figure supplement 2) indicating that RNAi treatment was effective. Moreover, the AB cell exhibited chiral counter-rotating flows during division ($v_c$ = −5.05 ± 1.12 µm/min) and, although the AB spindle was often misaligned, it skewed substantially during anaphase (spindle skew angle = 23.14 ± 7.46°, Figure 4—figure supplement 2). These results argue against an important role for the known spindle orientation pathway in driving spindle skews. Alternatively, given that the LIN-5-dependent spindle orientation pathway is known to reorient the metaphase spindle in the P₁ cell, this pathway might inhibit actomyosin dependent spindle skews in the P/EMS lineage during anaphase. Analyzing spindle skews in P₁ upon lin-5 (RNAi) and gpr-1/2 (RNAi) revealed no significant difference when compared to control embryos, which suggests that the spindle orientation pathway does not overrule actomyosin-dependent spindle skews (Figure 4—figure supplement 2). Altogether, we conclude that the LIN-5/GPR-1/2-dependent spindle orientation pathway determines the initial spindle orientation, while chiral actomyosin flows drive the spindle skews observed during anaphase.

If chiral counter-rotating flows indeed drive spindle skews, we predict that a complete absence of chiral flows leads to a complete absence of spindle skews. We test this prediction in the AB cell, by inactivating cortical flows entirely through a fast acting temperature sensitive nmy-2(ts) mutant (Liu et al., 2010). Worms were dissected to obtain one-cell embryos and mounted at the permissive temperature at 15°C that allows them to develop normally. Approximately 60 s after the completion of the first (P₀) cell division, the temperature was rapidly shifted to the restrictive temperature of 25°C using a CherryTemp system. Imaging was started as soon as the AB cell entered anaphase. In all nmy-2(ts) mutant embryos (12 out of 12), an ingressing cytokinetic ring was absent and cytokinesis failed after temperature shifting, indicating that NMY-2 was effectively inactivated (Davies et al., 2014; Videos 20 and 21). Interestingly, while the dynamics of spindle elongation, including the peak rate of spindle elongation and the final spindle length, are essentially unchanged as compared to control embryos at restrictive temperature, the average peak rate of spindle skew was significantly reduced (0.33 ± 0.09°/min as compared to 1.43 ± 0.13°/min in control embryos, Wilcoxon rank sum test at 95% confidence, Figure 5A,B, Figure 5—figure supplement 1, Videos 20 and 21). This demonstrates that a functional actomyosin cortex with chiral cortical flows is required for spindle skews of the AB cell. In light of our experiments above, we conclude that spindle skews in the AB cell are independent of spindle elongation, but are instead a consequence of the cell skews that are driven by chiral counter-rotating flows. Finally, we performed temperature shift experiments for the first 11 divisions in the nematode, and found that all cells belonging to the AB cell lineage show a significantly reduced skew in the nmy-2(ts) embryos at restrictive temperatures as compared to the control (Figure 5C). We conclude that a functional actomyosin cortex with chiral cortical flows is required for spindle skews of early AB lineage cell divisions during worm development.

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We propose that chiral counter-rotating flows, together with inhomogeneous friction forces with the surroundings, drive cell division skews in a manner similar to a bulldozer rotating on the spot by rotating its chains in opposite directions. Following this analogy, the direction of the cell division skew is determined by (1) the handedness of the chiral counter-rotating flow and (2) the distribution of friction forces across the cellular surface. As described above, the AB cell simultaneously skews in two orthogonal planes, AP-DV and LR-DV. Assuming that there are high friction forces at the AB-P₁ interface, the clockwise skew (when viewed from the anterior down the anteroposterior axis) in the LR-DV plane is consistent with the handedness of the chiral counter-rotating cortical flows (Figure 3—figure supplement 1). Similarly, for the observed clockwise skew in the AP-DV plane (when viewed from the left down the left-right axis) to be consistent with the handedness of chiral counter-rotating flows of AB, we expect that the total friction forces between the AB cell and vitelline layer and/or the egg shell are elevated on the right side of the embryo, as compared to the left side (Figure 5—figure supplement 3).

In order to test this hypothesis, we first set out to identify signatures of left-right asymmetric friction forces. Because such forces are expected to reduce cortical flow speed, we argued that cortical flow speeds may be different on the future left and right sides of the dividing AB cell. Note that in our analysis, we have so far only reported values of the chiral counter-rotating flow velocity $v_c$ for flows measured on the right side of the embryo. Measuring the chiral counter-rotating flow velocities of the AB cell also on the left side, we indeed find that y-direction flows in each cell half, and consequently $v_c$, are significantly increased on the left side, as compared to the right side (Figure 5—figure supplement 3, velocity values are given in Appendix 1—table 3). This observation is consistent with the hypothesis that the friction forces experienced by the AB cell are higher on the right side than on the left side.

Based on these qualitative insights, we investigated if the observed spindle skew rates can be quantitatively predicted from measured chiral counter-rotating flow velocities and basic geometric features of the dividing AB cell. This was done in two steps: First, we tested if the cortical flow velocities are consistent with the geometric predictions that follow from the bulldozer analogy. In the second step, we investigated if physical interactions with the egg shell are sufficient to explain the skewing rate and direction.

For the first step, we note that the velocities of the bulldozer chains on top and bottom (corresponding to the left and right side of the embryo) can be measured (A) while standing next to the spinning bulldozer (measurement in the ’lab frame’) or (B) while sitting in the co-rotating drivers cab, from which one always expects to observe two perfectly counter-rotating chains. The differences in the velocities measured in A) and B) are due to the overall rotation of the system, that is, the spinning of the bulldozer or, in our case, the skewing of the dividing AB cell in the AP-DV plane. Formalizing this geometric description, we derived a parameter-free relation between the skew rate of the dividing AB cell and the cortical y-direction flows on the left and right side (see Appendix). We measured y-direction flows (measurement in the 'lab frame') in wild type, ect-2 (RNAi) and rga-3 (RNAi) on the left and right side of the embryo and found for all three conditions that the calculated angular skew velocities agree well with the experimentally measured average AP-DV spindle skew rates of the dividing AB cell (Appendix 1—table 3). This provides a first, direct quantitative relationship between chiral actomyosin flows and cell division skews. Additionally, it shows that the skewing motion and chiral flows across the AB cell cortex are compatible with a simplified geometry that is analogous to a spinning bulldozer.

In the second step, we used the geometric model to derive the torque balance that connects the AB cell skews to the mechanical interactions with the surrounding. For the skew in the AP-DV plane, we consider a scenario where the relevant torques arise from friction forces between the counter-rotating cell halves and the egg-shell on the left and right side, and then tested, if this is sufficient to explain the observed amplitude and handedness of the spindle skews. Neglecting contributions from the $AB$-$P_1$ cell contact and extra-embryonic fluid that may counteract the AP-DV skew, the torque balance of the system implies an angular skew velocity given by (see Appendix)

In this expression, ${\gamma }_L$ and ${\gamma }_R$ denote local friction coefficients, and $A_L$ and $A_R$ denote the corresponding contact surface areas between the AB cell and the egg shell on the left and right side. The velocity $v$, which is positive for all conditions, is related to the measured chiral counter-rotating flow velocity $v_c$, and $R$ is the distance between the cytokinetic ring and the ROIs where y-direction flows are measured (see Appendix). $\omega <0$ corresponds to a clockwise skew when the embryo is viewed from the left side. Equation 1 implies that, if chiral counter-rotating flows are present ($v\ne 0$), any left-right asymmetry in total friction forces ${\gamma }_L{A}_{L}v$ and ${\gamma }_R{A}_{R}v$ leads to an angular skew in the AP-DV plane, while their relative amplitude defines the skew handedness. In order to test this minimal physical model, we estimated the contact surface areas on the left and right side by measuring the area of the cortex that is in focus in our confocal movies ($A_R/A_L\approx 1.3\pm 0.2$, Figure 5—figure supplement 3, Appendix 1—table 4). The local friction coefficients were estimated by fitting the thin film active chiral fluid theory to measured flow profiles on the left and right side, where we assumed that the cortical viscosity does not significantly vary across the cell surface and hence that differences in the hydrodynamic length arise from differences in friction coefficients (${\gamma }_R/{\gamma }_L\approx 1.7$, Figure 5—figure supplement 3, see Appendix). Together with ${\displaystyle R=4.5\mu m}$ and $v$ determined from the measured y-direction flows (see Appendix), Equation 1 allows for each condition to estimate the angular skew velocity that is expected from the torque balance of the system. The resulting angular skew velocities ${\displaystyle \omega =-0.42\pm 0.15{}^{\circ }}$/min; ${\displaystyle -0.28\pm 0.04{}^{\circ }}$/min; ${\displaystyle -0.49\pm 0.19{}^{\circ }}$/min (wt; ect-2 (RNAi); rga-3 (RNAi)) correctly predict the experimentally observed skew handedness and faithfully recapitulate the behavior of the average spindle skew rates in all three conditions (see Appendix). Together, these results lend credence to our conclusion that chiral actomyosin flows drive the three-dimensional cell division skew of the AB cell. Furthermore, they suggest that inhomogeneous mechanical interactions with the surrounding are sufficient to explain the handedness and amplitude of these skews.

C. elegans worms were cultured on NGM agar plates seeded with OP50 as previously described (Brenner, 1974). The following C. elegans strains were used in this study: LP133 : nmy-2(cp8[nmy-2::GFP + unc-119(+)]) I; unc-119(ed3) III for imaging cortical flows (Dickinson et al., 2013). SWG063 : nmy-2(cp8[nmy-2::GFP + LoxP])I; unc-119(ed3)III; weIs21[Ppie-1::mCherry::beta-tubulin::pie-1 3’UTR] (cross between (LP133 and JA1559 (Gift from J. Ahringer lab)) for measuring spindle skews in the AB cell. SWG204 : nmy-2(ne3409ts) I; unc-119(ed3)III; weIs21[pJA138(Ppie-1::mCherry::beta-tubulin::pie-1 3’UTR)] (cross between WM179 and JA1559) for temperature-sensitive experiments. TH155 : unc-119(ed3)III; weIs21[pJA138(Ppie-1::mCherry::beta-tubulin::pie-1 3’UTR)]; PH::GFP (cross between JA1559 and OD70 [Kachur and Pilgrim, 2008]) for measuring spindle skews with the membrane marker and to measure relative cytokinetic ring position along the cell division axis.

All RNAi experiments in this study were performed at 20°C (Sijen et al., 2001). For performing weak perturbation RNAi experiments, L4 staged worms were first incubated overnight on OP50 plates. Young adults were then transferred to respective RNAi feeding plates (NGM agar containing 1 mM isopropyl-$\beta$-D-thiogalactoside and 50 µg ml-1 ampicillin seeded with RNAi bacteria). Feeding times for RNAi experiments were 4–5 hr for ect-2 (RNAi) and 6–8 hr for rga-3 (RNAi). For par-2, chin-1, lgl-1 (RNAi), par-6 (RNAi), par-2 (RNAi), lin-5 (RNAi) and gpr-1/2 (RNAi) conditions, young L4 were transferred to feeding plates 24 hr before imaging. Embryos used as controls for all RNAi experiments were grown on plates seeded with bacteria containing a L4440 empty vector, and exposed for the same number of hrs as the corresponding experimental RNAi perturbed worms.

The indicated hours of RNAi treatment reflects the time that worm spent on the feeding plate. All dissections were done in M9 buffer to obtain early embryos. Embryos were mounted on 2% agarose pads for image acquisition for the all cell divisions except 4–6 cell stage (Bargmann and Avery, 1995). Four cell embryos and uncompressed embryos were mounted using low-melt agarose (Naganathan et al., 2014). The rga-3 feeding clone was obtained from the Ahringer lab (Gurdon institute, Cambridge, United Kingdom) and the ect-2 feeding clone from the Hyman lab (MPI-CBG, Dresden, Germany). RNAi feeding clones for par-2, par-6, chin-1 and lgl-1 were obtained from Source Bioscience (Nottingham, United Kingdom).

All the imagining done in this study was performed at room temperature (22–23°C) with a spinning disk confocal microscope. Two different spinning disk microscope systems were used for the study. System 1: Axio Observer Z1 - ZEISS spinning disk confocal microscope; Apochromat 63X/1.2 NA objective lens; Yokogawa CSU-X1 scan head; Andor iXon EMCCD camera (512 by 512 pixels) , Andor iQ software. System 2: Axio Observer Z1 - ZEISS spinning disk confocal microscope; Apochromat 63X/1.2 NA objective lens; Yokogawa CSU-X1 scan head ; Hamamatsu ORCA-flash 4.0 camera (2048 by 2048 pixels); micromanager software (Edelstein et al., 2014).

Confocal videos of cortical NMY-2::GFP for the first three cell divisions for control and RNAi conditions were acquired using system (1) A stack was acquired using 488 nm laser and an exposure of 150 ms was acquired at an interval of 3 s between image acquisition consisting of three z-planes (0.5 µm apart). The maximum intensity projection of the stack at each time point was then used for further analysis. 21 s of time frames were analyzed for each embryo starting from the time frame when the cytokinetic ring has ingressed ~10%. Time intervals between consecutive frames were increased at later cell stage to avoid photo-toxicity and bleaching. From the 4–6 cell stage onwards, imaging was performed using system (2) A time period of 20 s was analyzed for measuring chiral counter-rotation flow velocity. Image stacks with a spacing of 0.5 µm for 20 z-slices from the bottom surface toward the inside of the embryo were taken with 488 nm laser with exposure of 150 ms at an interval of 5 s during the time of cytokinetic ring ingression and projected with maximum intensity projection.

Imaging of the spindle poles (TBB-2::mCherry) was done using SWG63 (Figures 2A–B, 3D, G, H, 4A and 5A–C). TH155 strain was used to image TBB-2::mCherry together with PH::GFP to determine cytokinetic ring position and spindle skew angle in Figure 2B for 11-cell stages. SWG204 strain was used for temperature-sensitive experiments (Figure 4A–C). Spindle pole imaging for AB cell (TBB-2::mCherry) was performed by acquiring nine-z-planes (1 µm apart) with a 594 nm laser and exposure of 150 ms on the second system at an interval of 3 s. For long-term imaging using TH155 strain (Figure 2B), TBB-2::mCherry together with PH::GFP was imaged in 24 z-slices (1 µm apart) with 594 nm laser and exposure of 150 ms at an interval of 30 s (Video 9). For uncompressed embryos, 21 and 25–31 µm z-stacks with z-spacing of 0.5 and 1 µm apart were recorded at time interval between 5 and 10 s using system two for measuring cortical flows and spindle poles during division, respectively. For a faster temporal resolution, we restricted to imaging either spindle skews or cortical flows for individual embryos for all perturbations.

Cortical flow velocity fields in the 2D $xy$-plane were obtained using a MATLAB code based on the freely available Particle Image Velocimetry (PIVlab) MATLAB algorithm (Raffel, 2007). Throughout the study, we used the three-step multi-pass with linear window deformation, a step size of 8 pixels and a final interrogation area of 16 pixels.

To determine the average flow fields that were fitted by the chiral thin film theory (Figure 1—figure supplement 8), we first tiled the flow field determined by PIV into nine sections along the long-axis of cells. Within each section, we averaged the flow field over the entire duration during which flows occur. Since we were only interested in flows that are generated by the cytokinetic ring, we excluded embryos in which we could not differentiate whole body rotation from cytokinesis (Schonegg et al., 2014). Note that cortical flow velocities change over the duration of cytokinesis and are likely to be influenced by changes in the AB cell geometry due to asymmetric ring ingression (Schonegg et al., 2014). Hence, for quantitative comparisons, we restricted the analysis of flows on the future right side (except for Par polarity perturbations where axis establishment was perturbed) and to the narrow time frame as described in the main text. Flow velocities observed from the future left side have been reported in Figure 5—figure supplement 3 and in Appendix 1—table 3. From all the imaged cell divisions, a small subset of measurements, in which the optical focus on the cortical plane was lost during imaging, was discarded.

From the measured cortical flows, we extracted properties that are later used to quantify their chiral counter-rotating nature in the following way. First, we identified the cleavage plane or the cytokinetic ring by eye. We then defined a region of interest (ROI) on each side of the cytokinetic ring (Figure 1A). In order to ensure that PIV measurements were not affected by the saturating fluorescence signal from the cytokinetic ring, we placed the inner boundary of each ROI at a distance of 1 µm from the ring. The position of the outer boundary of each ROI (along the long axis) was scaled with the size of the analyzed cells (10 µm, 8.5 µm, 6 µm, 6.5 µm, 6.5 µm, 6.5 µm for P₀, AB, P₁, ABa, ABp and EMS, respectively, measured from the position of the cytokinetic ring). For P₂, ABal and ABpr cells the distance of the outer ROI boundary to the cytokinetic ring varied around 4 µm, depending on the cell surface area visible in the imaging focal plane. For every cell, the width of ROIs was set equal to the length of the cytokinetic ring visible in the focal plane. In each ROI, we averaged the observed flow field over a period of 21 s. Consistent with earlier findings, we observed a whole cell rotation prior to P₀ cytokinesis (Schonegg et al., 2014; Singh et al., 2019). We discarded a small subset of embryos (4 out of 17), where this whole cell rotation coincided with the time frames in which we performed the flow analysis.

In order to quantify chiral flows, we defined a chiral counter-rotating flow velocity $v_c$ together with a handedness as follows. First, we introduce a unit vector that is orthogonal to the contractile ring and points toward the cell pole that is in the direction of the anteroposterior-axis of the overall embryo ($\underset{¯}{e_x}$, see Figure 1—figure supplement 2). A second unit vector $\underset{¯}{e_y}$ is used, which is parallel to the cytokinetic ring and orthogonal to $\underset{¯}{e_x}$. We denote the average velocities determined in the two halves of the cell (measured within the ROIs described above) as $\underset{¯}{v_1}$ and $\underset{¯}{v_2}$. With this, we define the chiral counter-rotating flow velocity $v_c$ by

where $\underset{¯}{e_z}=\underset{¯}{e_x}\times \underset{¯}{e_y}$ is the third base vector of a right-handed orthonormal coordinate system. With this definition, $|v_c|$ represents the speed of chiral counter-rotating flows, while the sign of $v_c$ determines the flow’s handedness: If cortical flows appear clockwise when viewed from the cytokinetic plane toward each pole, one has $v_c>0$ and the counter-rotating flows are left-handed. If cortical flows appear counter-clockwise, one has $v_c<0$ and counter-rotating flows are right-handed. In order to quantify mean chiral counter-rotating velocity, we averaged $v_c$ over multiple embryos for every timepoint that was analyzed. From the analysis of the first nine cell divisions, we found that intensities in the P/EMS lineage are lower and hence the PIV flow analysis is less coherent in the P/EMS lineage as compared to the AB lineage, leading to increased uncertainty in PIV measurements.

To characterize net-rotating flows in a similar fashion, we use the net-rotating flow velocity $v_r$ defined by (Figure 1—figure supplement 4)

If average flows in both cell halves are equal and in the same direction, that is, $\underset{¯}{v_1}=\underset{¯}{v_2}$, there are no counter-rotating flows ($v_c=0$) and the corresponding net-rotating flow is quantified by $v_r$.

Finally, in order to determine a flow measure that captures cortical flows along the cell’s long axis and into the contractile ring, we define the contractile flow velocity ($v_{\text{contr}}$) depicted in Figure 3B by

For contractile cortical flows into the cytokinetic ring, we have $\underset{¯}{e_x}\mathbf{\cdot }{\underset{¯}{v}}_1>0$ and $\underset{¯}{e_x}\mathbf{\cdot }{\underset{¯}{v}}_2<0$, and therefore $v_{\text{contr}}<0$.

Flow speed during cell division was calculated by adding the magnitude of mean absolute values of flow vectors in the two boxes from either side of the cytokinetic ring in the dividing cells.

In order to measure spindle skew angles, z-stacks (nine planes, each 1 µm apart recorded in 3 s time intervals) of the spindle poles were first projected on the plane perpendicular to the cytokinetic ring. For uncompressed embryos mounted using low-melt agarose method, z-stacks (25–31 z-planes; 1 µm apart with maximum possible temporal resolution between 5 and 10 seconds time intervals) were captured. For P₀, AB and P₁ the imaging plane was already perpendicular to the plane of cytokinesis while for the remaining cell divisions the embryos were rotated accordingly using the clear volume plugin (Royer et al., 2015) in FIJI. Subsequently, the spindle skew angle was defined as the angle between the line that joins the spindle poles at the onset of spindle elongation (anaphase-B) (initial spindle angle) and at completion of cytokinesis (final spindle angle) (Figure 2A). Spindle elongation length was defined as the distance between the two spindle poles during the same time-frames. To measure the dynamics of spindle elongation and skew angle in the AB cell (Figure 4—figure supplement 1, Figure 5—figure supplement 1) the spindle poles were automatically tracked and the spindle skew angle was plotted using custom-written MATLAB code (refer to spindle skew analysis MATLAB code). In order to compare the dynamics of spindle elongation and spindle skew, we first synchronized all the AB cells in each condition at the onset of anaphase-B and, subsequently, averaged over multiple embryos. The difference beween the spindle angle at each timepoint and the spindle angle at anaphase-B onset timepoint was measured and plotted over time. The maximum value of the slope for spindle elongation and spindle skew angle curves for each embryo between time window of 60–120 s (Figure 4—figure supplement 1, Figure 5—figure supplement 1) was identified as peak rate of spindle elongation and peak rate of spindle skew. In order to reduce noise in calculating slope, a moving average was performed to smooth the spindle skew angle and elongation profiles using moving slope function in MATLAB. Average peak rates were calculated by averaging the maximum values over multiple embryos for each condition.

In order to obtain cortical myosin distributions (Figure 1C, Source data 1), we normalized the axis of cell division and calculated the mean myosin intensity in a 20 pixel-wide stripe along this axis for the same timepoints when $v_c$ was calculated. The myosin ratio (Figure 1E) was determined by dividing the mean intensity at the anterior (0–20% of the normalized cell division axis) by the mean intensity at the posterior (80–100% of the normalized cell division axis) for P₀, AB, P₁ and EMS. For ABa and ABp the myosin intensity was calculated similarly but along the L/R axis. We have restricted myosin intensity measurements to the 20% outer segments of the cell division axis in order to prevent the strong fluorescent signal from the cytokinetic ring from affecting this analysis (Figure 1—figure supplement 6). Similar to myosin ratio, myosin concentration at the anterior was normalized with mean posterior myosin concentration for multiple embryos and plotted (Figure 1—figure supplements 9 and 10). To determine the position of the cytokinetic ring, we performed midplane imaging using a strain expressing a membrane marker (PH::GFP) and a tubulin marker to label spindle poles (TBB-2::mCherry). The cell division axis was defined as the line through the two opposing spindle poles and was normalized between cell boundaries. The relative cytokinetic ring position was defined as the position where the ingressing ring intersects the normalized cell division axis.

For visualization purposes (Figure 1C), we fitted the myosin distributions using the fitting function

The errorfunction erf and the Gaussian represent contributions from the anteroposterior myosin asymmetry and from the contractile ring, respectively. The fitting parameters $I_P$, $I_A$ , $I_R$, $w$ and $x_r$, respectively, describe myosin intensities at the posterior pole, the anterior pole and in the cytokinetic ring, as well as the width and position of the ring.

We used the cherry temp temperature control stage from Cherry Biotech for all temperature sensitive experiments in the study. L4 worms carrying tbb-2::mCherry (SWG63, control) and nmy-2(ts); tbb-2::mCherry (SWG204) were grown at 15°C (permissive temperature for nmy-2(ts) mutants) overnight. During image acquisition, embryos were mounted using agar pad mounting method and temperature was maintained using the cherry temp temperature control stage from Cherry Biotech. For imaging the dynamics of spindle elongation and skew angle in the AB cell for nmy-2(ts) and control embryos, 60 s after the successful completion of the first cell division (P₀ cell division), temperature was raised to 25°C. Embryos were imaged approximately 5 min after the temperature shift to minimize photo toxicity. Similarly, for later cell stages, the temperature shift was carried out right after completion of cytokinesis of the mother cell. A small subset of embryos in which spindle poles of daughter cells could not be visualized were discarded.

In order to determine embryo thickness, we collected an ensemble of up to nine two-cell embryos from SWG59, SWG70, OD70 strains mounted using the agar pad method or low-melt agarose method. To measure embryo thickness in utero, young adult worms (24 hrs post L4) were paralyzed in 0.1% tretramisole (Sigma-Aldrich T1512) for 3 min on a cover slip and mounted on 2% agarose pads. The AB cell in the two-cell embryos was projected on a view down the anterior end along the anteroposterior axis. Thickness of the embryo was determined by measuring the end to end distance between the opposing membrane signal (denoted by arrows in Figure 1—figure supplement 1). For each embryo, aspect ratio was calculated as the ratio of thickness values of the AB cell when projected along the AP-DV plane and the LR-DV plane as shown in Figure 1—figure supplement 1. All embryo rotations for post-processing and time-lapse along various axis were captured using clearvolume as described in the spindle skew analysis section.

Cortical flows in Caenorhabditis elegans embryos have been successfully described using the thin film theory of active chiral fluids (Naganathan et al., 2014; Naganathan et al., 2016). We use the fact that cells are approximately axisymmetric, and we assume that the active forces during cell division vary mostly along their symmetry axis, in the following referred to as long axis. In this case, the hydrodynamic equations of a chiral thin film can be formulated as an effectively one-dimensional system of equations, capturing flows parallel ($v_x$) and orthogonal ($v_y$) to the long axis:

Here, $\eta$ is the viscosity of the cortex and $\gamma$ is the friction with surrounding material. Note that the assumption of a homogeneous friction $\gamma$ is a strong simplification, where we neglect details of the potentially inhomogeneous mechanical interactions between cells, as well as with the extra-embryonic fluid and the egg shell. Gradients of active tension $T$ and active torques $\tau$ lead to contractile and chiral flows, respectively. It has been demonstrated previously that active tension and active torques both dependent on the local myosin concentration in the cortex (Mayer et al., 2010; Naganathan et al., 2014; Naganathan et al., 2016). Using the fluorescent myosin intensity $I(x)$ as a proxy for the myosin concentration, we can write in the simplest case $T=T_{0}I(x)$ and $T={\tau }_{0}I(x)$.

Denoting the length of the measurement domain by $L$ (approximately the length of the cell’s long axis), we can identify three independent parameters in the model Equations S1 and S2: The characteristic contractile and chiral velocities $v_T=L{T}_0/\eta$ and $v_{\tau }=L{\tau }_0/\eta$, respectively, as well as the hydrodynamic length $\mathrm{\ell }=\sqrt{\eta /\gamma }$.

For the fitting of cortical flows for given myosin profiles, we closely follow our previous work (Naganathan et al., 2014). Briefly, we use experimentally determined myosin profiles $I(x)$ in Equations S1 and S2 to calculate flows $v_x$ and $v_y$. We then determine the parameters $v_T$, $v_{\tau }$ and $\mathrm{\ell }$ for which the predicted flows best match the experimental data. To extract unique solutions from Equations S1 and S2, we use the experimentally measured flow velocities at the boundaries of the measurement domain as boundary conditions. Note that due to imaging limitations for smaller cells (after the eight-cell stage) as development progresses, it was not possible to extract spatially resolved velocity line profiles and fits for the cortical flows in ABal cell and ABpr cell. The final best fit parameters for AB, ABa and ABp cells are shown in Appendix 1—table 1. The corresponding contractile flow profiles $v_x$ and chiral flow profiles $v_y$ are shown in Figure 1—figure supplement 8. 

We also considered asymmetrically dividing cells (P₀, P₁, EMS) in which, instead of counter-rotating flows ($|v_c|>0$, $v_r\approx 0$), mainly net-rotating flows occur ($v_c\approx 0$, $|v_r|>0$) (Figure 1—figure supplement 4). Following the same fitting procedure as described in the previous section, we noticed that the chiral thin film theory Equations S1 and S2 generally could not account quantitatively for the observed cortical flow profiles. However, using the theory it is still possible to rationalize qualitative properties of the observed cortical flows, as we discuss in the following.

While the chiral thin film theory does not recapitulate all of the experimentally observed flows quantitatively, the theory allows linking qualitative predictions to key properties of observed myosin distributions and chiral flows. In particular, we noticed that the myosin profiles in asymmetrically diving cells consistently featured a rather asymmetrically positioned contractile ring (Figure 1D). Furthermore, the myosin profiles of asymmetrically dividing cells are asymmetric with respect to the cytokinetic ring and exhibit plateaus toward the anterior cell poles (Figure 1C and Figure 1E). An overview of these qualitative properties for the P/EMS lineage and the AB-lineage is given in Appendix 1—table 2.

To establish how a contractile ring and an overall myosin asymmetry generally affect flows predicted by the chiral thin film theory, we generalize the chiral thin film equations (Naganathan et al., 2014) to curved surfaces and solve them on an ellipsoid using corresponding synthetic torque profiles (Figure 1—figure supplement 7). We find that a symmetrically placed ring pattern yields perfectly counter-rotating flows ($v_r=0$, $|v_c|>0$, Figure 1—figure supplement 7, left), while anterior-posterior myosin asymmetry ratio results in net-rotating flow ($|v_r|>0$, $v_c=0$, Figure 1—figure supplement 7, middle) while an anterior-posterior myosin asymmetry along with a symmetrically placed ring pattern yields net-rotating flows such as observed in P/EMS cells (Figure 1—figure supplement 7, right). To study the combined effect of a varying position of the contractile ring and a myosin asymmetry, we develop in the following section a simple minimalistic model of the system.

In the following, we develop a minimal description of the occurrence of counter- and net-rotating flows based on key characteristics of the myosin profiles discussed in the previous section. We consider a simplified myosin profile in the form

with

Here, $\delta (x)$ is the delta function, $\mathrm{\Theta }(x)$ the Heaviside step function and $x_r$ is the position of an idealized cytokinetic ring. The intensity profile $I_r(x)$ captures the presence of a myosin peak (the later contractile ring) of width $w$ and average intensity $I_R$ at $x_r$. The profile $I_{ap}(x)$ represents the large-scale asymmetry in myosin between anterior pole ($I_A$) and posterior pole ($I_P$). We solve Equation S2 on a domain of length $L$ and for boundary conditions $v_y(0)=v_y(L)=0$ using the simplified myosin intensities given in Equations S4 and S5, which yields

Here, $\alpha =\sqrt{2\gamma L^2/\eta }$ is an inverse dimensionless hydrodynamic length and $v_{\tau }=L{\tau }_0/\eta$ the characteristic velocity associated with active torques. The coefficients in Equation S6 are given by

With these coefficients, Equation S6 describes flows $v_y^r$ resulting from the contractile ring given in Equation S4 and flows $v_y^{ap}$ resulting from the asymmetric myosin profile given in Equation S5. Note that, as expected, flows due to the presence of the contractile ring vanish for $I_R=0$ and flows due to the presence of a myosin asymmetry vanish if $I_A=I_P$. General flow profiles are given as superposition of the two contributions: $v_y=v_y^r+v_y^{ap}$.

Finally, to evaluate these solutions, we define a dimensionless counter-rotating velocity ${\tilde{v}}_c$ and the net-rotating velocity ${\tilde{v}}_r$ in analogy to the quantities introduced in the main text and methods, Equations 2 and 3, as

Here, ${\tilde{v}}_y=v_y/v_{\tau }$ and we consider a window of 15 % ($\mathrm{\Delta }/L=0.15$) of the total length toward either side of the idealized contractile ring over which the mean flow velocity is determined (Figure 1A, Figure 1—figure supplements 2 and 4). Using this minimal model, we can now investigate how the combined contributions of a large-scale myosin asymmetry ($I_{ap}$) and a varying contractile ring position $(I_r)$ affect the counter-rotating velocity ${\tilde{v}}_c$ and net-rotating velocity ${\tilde{v}}_r$. In particular, we fix for the discussion $\alpha =1$ and $w/L=0.1$, as well as $I_R/I_P=2$. In this case, the counter-rotating flow velocity ${\tilde{v}}_c$ and the net-rotating flow velocity ${\tilde{v}}_r$ are only functions of the relative ring position $x_r/L$ and the anterior-posterior myosin ratio $I_A/I_P$ with properties shown in Figure 1—figure supplement 7 and described in the following.

For an anterior-posterior symmetric myosin profile, counter-rotating flows $|{\tilde{v}}_c|$ have a weak maximum in their amplitude if the contractile ring is at a centered position $x_r/L=0.5$ (Figure 1—figure supplement 7). Furthermore, a centered contractile ring has no effect on the counter-rotating flow measure ${\tilde{v}}_c$ if a large-scale myosin asymmetry $I_A/I_P>1$ is introduced (Figure 1—figure supplement 7). However, for an off-centered ring $x_r/L>0.5$ the presence of a myosin asymmetry will contribute flows that reduce ${\tilde{v}}_c$ and can even lead to vanishing counter-rotating flows ${\tilde{v}}_c=0$ if $I_A/I_P$ is sufficiently large. While this matches the qualitative experimental observations listed in Appendix 1—table 2, the required asymmetry predicted by the theory is significantly larger than the experimentally measured one.

Also, the behavior of ${\tilde{v}}_r$ in our minimal model is compatible with the qualitative properties listed in Appendix 1—table 2. In particular, for a large-scale myosin asymmetry $I_A/I_P>1$, the net-rotation velocity ${\tilde{v}}_r$ is negative for any ring position, while it can becomes positive for $x_r/L>0.5$ if $I_A=I_P$ (Figure 1—figure supplement 7). Furthermore, net-rotating flows vanish for a symmetric myosin profile $I_A=I_P$ and a centered contractile ring $x_r/L=0.5$ (Figure 1—figure supplement 7), which corresponds to the observations made in AB-lineage cells. Finally, the presence of large-scale myosin asymmetries $I_A/I_P\ne 1$ is generally expected to contribute to net rotating flows (Figure 1—figure supplement 7). This holds true for essentially arbitrary positions of the contractile ring and indicates that such asymmetries could play an important role for developing a better understanding of net-rotating flows in the P/EMS lineage cells in the future.

In the following, we introduce a minimal model that can quantitatively link cell skews to the chiral cortical flows and mechanical interactions with the surrounding. Note that while the model describes the general scenario of two confined, counter-rotating surfaces, we focus its application to the case of the dividing AB cell and its skew in the AP-DV plane, for which all the required experimental information is available to make quantitative predictions.

We consider an idealized configuration of two connected incompressible surfaces that represent the cortices of the future ABa and ABp cell during AB cell division and reorientation-skew. Chiral counter-rotating cortical flows correspond to rotations of these surfaces in opposite directions around a common axis. For simplicity, we neglect more complex aspects of the surface deformations that occur during the cell division and skewing inside the constraining egg shell. This system is analogous to the two chains under a bulldozer, where the motion of the chains represent chiral cortical actomyosin flows and the operators cab, with a fixed orientation relative to the two chains, indicates the orientation of the dividing AB cell. The two connected surfaces are additionally confined by two opposing rigid surfaces that are parallel to the AP-DV-midplane and represent the egg shell (Figure 5—figure supplement 3).

We first describe the surface motion in a reference frame that is co-rotating with the angular skew occurring in the AP-DV-plane. We assume that active chirality leads to a counter-rotating surface motion, and the local velocities of the ABa and ABp cell on the the embryo’s future left side at a distance $R$ from the cytokinetic ring are given by $-v_a$ and $v_p$, respectively. Observing the two counter-rotating surfaces in a common co-rotating reference frame, the local velocities of each cell on the future right side are then given by $v_a$ and $-v_p$, respectively (Figure 5—figure supplement 3). In experiments, cortical flows ($v_L^{(a)}$, $v_L^{(p)}$, $v_R^{(a)}$, $v_R^{(p)}$) of the ABa and ABp cell on each surface are measured in the lab frame, which is defined as the reference frame in which the egg shell is at rest. These measurements are related to the parameters of the simplified geometric model by

Here, ${\omega }_g$ is the angular skew velocity around an axis that is orthogonal to the AP-DV plane and $\delta$ describes a possible displacement of the rotation axis from the center (Figure 5—figure supplement 3). If $\delta >0$ ($\delta <0$) the axis of rotation is shifted toward the ABa (ABp) cell. The sign in front of ${\omega }_g$ is chosen such that ${\omega }_g>0$ (${\omega }_g<0$) corresponds to anti-clockwise (clockwise) skews when viewed from the left side down the left-right axis (Figure 5—figure supplement 3). From Equations S9–S12, we then find

where $\alpha =(v_L^{(a)}+v_R^{(a)})/2$ and $\beta =(v_L^{(p)}+v_R^{(p)})/2$. Note that Equations S9–S16 essentially represent the transformation of velocities between two reference frames that rotate relative to each other.

To test if this simplified geometric representation of two counter-rotating and skewing surfaces is consistent with experiments, we measured $v_L^{(a)}$, $v_L^{(p)}$, $v_R^{(a)}$ and $v_R^{(p)}$ during the late state of the AB cell skew and computed the cell skew parameters using Equations S13–S16 for the wt, ect-2 and rga-3 conditions (Appendix 1—table 3). For all three conditions, the calculated angular skew velocities ${\omega }_g$ agree well with the average angular skewing velocities of the spindle measured in experiments (Appendix 1—table 4, right column). This strongly suggests that chiral cortical flows across the cortex are not only qualitatively, but even quantitatively linked to the corresponding skewing motion of the spindle. Furthermore, we find for all three conditions $\delta /R\approx 1$ (Appendix 1—table 3), which implies that the axis around which the AB cell skews is shifted toward the ABa cell. During later stages of the skew, the time point at which the velocities $v_L^{(a)}$, $v_L^{(p)}$, $v_R^{(a)}$ and $v_R^{(p)}$ given in Appendix 1—table 3 were measured, this can indeed be observed as a feature of the spindle skew (Figure 5—figure supplement 3): Both spindle poles of the AB cell move nearly parallel to the AP axis, but the ABp cell’s spindle pole does so faster. As a result, the rotation axis is located closer to the ABa spindle pole than to the ABp spindle pole.

Chiral surface flows and the motion of the AB cell relative to the surrounding material can in general give rise to friction forces and external torques that act on the cell. For the angular cell skew in the AP-DV-plane, the relevant frictions forces would have to occur dominantly through interactions with the egg shell on the future left and right side of the embryo (Figure 5—figure supplement 3). The corresponding flows are described by the velocities $v_L^{(a)}$, $v_L^{(p)}$, $v_R^{(a)}$ and $v_R^{(p)}$ measured in the lab frame, such that the external torques on future left and right side are given, respectively, by

Here, $F_L^{(a)}=-{\gamma }_L{A}_L^{(a)}{v}_L^{(a)}$ depicts the total external friction force on the future left side of the ABa cell (similarly for the other forces), $A_L=A_L^{(a)}+A_L^{(p)}$ and $A_R=A_R^{(a)}+A_R^{(p)}$ are the total contact surface areas between the AB cell and egg shell on the left and right side (Figure 5—figure supplement 3), respectively, and we have used Equations S9–S12 for simplicity with $v_a=v_p=v$. The external torques in Equation S17 and S18 transfer a total torque

onto the AB cell. This total torque on the other hand is balanced by a dissipative torque ${\tau }_{\omega }$ that arises through the angular skew of the dividing AB cell inside the egg shell, that is,

For simplicity, we consider ${\tau }_{\omega }=-\overline{\eta }{R}^3\omega$, where $\overline{\eta }$ denotes an effective rotational viscosity that captures interactions of the skewing AB cell with the extra-embryonic fluid inside the egg shell and potentially also interactions with the $P_1$ cell. Combining Equations S17–S20, we find

with

where $\mathrm{\Delta }{A}_L=A_L^{(p)}-A_L^{(a)}$, $\mathrm{\Delta }{A}_R=A_R^{(p)}-A_R^{(a)}$. Equation S21 with $\mathrm{\Gamma }$ given in Equation S22 provides a minimal model that can be used to estimate the expected angular cell skew velocity $\omega$ from the measured cortical velocities ($v_L^{(a)},v_L^{(p)},v_R^{(a)},v_R^{(p)}$) that define $v$, basic known measures of the cell geometry ($A_L,A_R,R$) and the friction parameters (${\gamma }_L$, ${\gamma }_R$, $\overline{\eta }$).

In the following, we want to use Equation S21 together with the experimentally measured parameters listed in Appendix 1—table 3 to calculate the expected angular skew frequency $\omega$. Note that the agreement of ${\omega }_g$ calculated from Equation S13 with the experimental values validates the simplifying geometric assumptions of our model, while Equation S21) indicates how specific mechanical interactions with the surrounding can lead to the handedness and amplitude of the angular skew.

To compute the required coefficient $\mathrm{\Gamma }$ given in Equation S22, we use that $\delta /R\approx 1$ (Appendix 1—table 3) for all conditions, and we consider approximately similar contact surface areas of the ABa and ABp cell halves on each side ($A_L^{(a)}\approx A_L^{(p)}$, $A_R^{(a)}\approx A_R^{(p)}$). In this case, $\mathrm{\Gamma }$ only depends on the total contact surface areas $A_L$ and $A_R$ that we estimate from confocal images ($A_R/A_L\approx 1.3$, Figure 5—figure supplement 3). Furthermore, we can express the friction coefficients in Equation S22 as ${\gamma }_L={\eta }_L/{\mathrm{\ell }}_L^2$ and ${\gamma }_R={\eta }_R/{\mathrm{\ell }}_R^2$, where ${\mathrm{\ell }}_L$ and ${\mathrm{\ell }}_R$ are the hydrodynamic lengths. The latter are determined by fitting the chiral thin film theory Equations S1 and S2 to measured flow profiles on the left and right side of dividing AB cells (Figure 5—figure supplement 3), from which we find ${({\mathrm{\ell }}_L/{\mathrm{\ell }}_R)}^2={\eta }_L{\gamma }_R/({\eta }_R{\gamma }_L)\approx 1.7$. Note, that we can in general not exclude spatial variations of the cortical viscosity that would lead to ${\eta }_L\ne {\eta }_R$ and accordingly affect the estimated local friction parameters. However, even if viscosity variations would fully account for the observed difference in hydrodynamic length such that ${\gamma }_L={\gamma }_R$, the contact area asymmetry with $A_R>A_L$ would still be sufficient to explain the handedness of the observed skews. Finally, to allow for quantitative predictions, we make the simplifying assumption ${\eta }_L\approx {\eta }_R=\eta$, such that Equation S22 is for all three conditions given by

In the second step, we have assumed that the cortical viscosity $\eta$ is large compared to the effective viscosity $\overline{\eta }$ that describes interactions with the extra-embryonic fluid ($\overline{\eta }R/\eta \approx 0$). Equation S21 with $\mathrm{\Gamma }$ given in Equation S23 is equivalent to Equation 1 and shows that contributions to the hydrodynamic length and to the contact surface area that are different on the future left and right side can generate an angular cell skew, where their relative values define the handedness of the skew. With Equation S23, the angular skew velocity $\omega$ given in Equation S21 is fully determined by parameters known from experiments and the corresponding values for all three conditions are shown in Appendix 1—table 4. Despite the geometric simplifications, our minimal model can predict the experimentally measured cell skew velocities for all three conditions within a relative error margin of approximately 5–25%.

The funders had no role in study design, data collection and interpretation, or the decision to submit the work for publication.

We thank Masatoshi Nishikawa for critical inputs on the theory, A Honigmann and P Gönczy for critical reading of the manuscript. We also thank T Hyman and B Bowerman for sharing C. elegans strains. We thank T Hyman, Kwabena Badu-Nkansah and all the members of Grill lab for critical input and discussions. We thank Julia Hubatsch for providing movies of C. elegans gonads. Some strains were provided by the CGC, which is funded by NIH Office of Research Infrastructure Programs (P40 OD010440). Furthermore, we thank the light microscopy facility of CMCB of TU Dresden, and the light microscopy facility of MPI-CBG for their support. We would like to acknowledge a long standing collaboration with Frank Jülicher on active chiral matter, and thank him for many discussions that were essential for the work here. LP acknowledges funding from the European Union’s Horizon 2020 research and innovation program under the Marie Sklodowska-Curie grant agreement No 641639. TCM was supported by the European Molecular Biology Organization (EMBO) long-term fellowship ALTF 1033–2015, and by the Dutch Research Council (NWO) Rubicon fellowship 825.15.010. AM thanks the ELBE Phd fellowship from the Center for Systems Biology, Dresden. SWG was supported by the DFG (SPP 1782, GSC 97, GR 3271/2, GR 3271/3, GR 3271/4), the European Research Council (grants 281903 and 742712).

© 2020, Pimpale et al.

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