Axis convergence in C. elegans embryos

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In brief Bhatnagar et al. show that convergence of the anteroposterior axis with the long axis in one-cell C. elegans embryo is primarily driven by the pseudocleavage furrow-a contractile ring that forms as a result of active forces in the actomyosin cortex.This study demonstrates the role of cell mechanics in establishment of body axes.

INTRODUCTION
The development of a single-cell embryo into an adult animal features the establishment of up to three anatomical body axes-anterior-posterior (AP), dorsal-ventral, and left-right.In general, these axes are intrinsic to the embryo 1 yet consistently orient relative to geometric cues.In the nematode worm C. elegans, the AP axis coincides with the geometric long axis of the ellipsoidal egg. 2 However, during the establishment of the AP axis, the two axes may not always coincide.In this case, the AP axis actively converges toward the long axis of the egg. 2 In this study, we seek to understand how the ellipsoidal geometry of the C. elegans egg directs this convergence of the AP axis with the long axis.
4][5][6][7] These domains form as a result of cell polarity establishment about 30 min after fertilization.2][13][14][15] Polarization proceeds via guided self-organized mechanochemical feedback between the PAR network 5 and surface stresses and flows in the actomyosin cortex [16][17][18][19] : centrosomal microtubules load cytoplasmic pPARs onto the cortex. 12,13,2021 Cortical flows circulate away from the male pronucleus and toward the future anterior end, [5][6][7]21 further redistributing PAR proteins by advective transport.[21][22][23] Altogether, mutual antagonism between pPARs and aPARs and their transport by cortical flows leads to the formation of mutually exclusive domains of aPARs and pPARs: aPARs at the anterior and pPARs at the posterior, 20 giving rise to a polarized cell and defining the embryonic AP body axis. 2 The ellipsoidal-like geometry of the C. elegans zygote is imposed by the eggshell surrounding the zygote, which forms shortly after fertilization.24 The zygote has one long axis about 50 mm in length and two short axes each about 30 mm in length.25,26 How, then, is the AP axis oriented along the geometric long axis of the egg?As described above, the position of the male pronucleus imparts positional information to the cell polarity machinery.The sperm tends to enter the oocyte at the tip, 2,27 possibly due to how the oocyte enters the spermatheca. 28,29As the male pronucleus is typically located near this tip of the embryo, 30 the orientation of the AP axis is along the long axis of the egg ellipsoid.However, occasionally the male pronucleus is positioned away from the tip, 2,30 yet the AP axis eventually converges with the geometric long axis. 2 In these cases, the male pronucleus ''posteriorizes''; that is, it moves toward the closest tip of the embryo.2,30 This dynamically reorients the AP axis toward the long axis of the embryo, until the two axes converge. Howeve, the physical mechanisms that drive this process of axis convergence remain poorly understood.
How does the AP axis converge toward the long axis?2][33][34][35] Such cytoplasmic flows, as observed in the one-cell embryo, 36 have previously been proposed to drive posteriorization 2,27 and thus AP axis convergence.These flows are created by anterior-directed cortical flows at the surface. 36Cytoplasmic flows, in turn, could transport the male pronucleus 30,37 toward the closest tip.We refer to this scenario as the cytoplasmic-flow-dependent mechanism.Another possibility is that cortical flows drive axis convergence in C. elegans embryos via the pseudocleavage furrow, 38 a contractile ring-like structure that forms at the boundary between the aPAR and pPAR domains. 8,9The ring arises by compressive-flow-driven alignment of actin filaments. 39The pseudocleavage furrow is dispensable for establishing the AP axis 40 but might modulate the dynamics of AP axis formation. 41n particular, if the furrow does not lie in a plane orthogonal to the long axis of the egg, constriction of the furrow 39 would reposition it to minimize its circumference, reorienting the furrow to be in a plane orthogonal to the long axis.We refer to this scenario as the pseudocleavage-furrow-dependent mechanism.
To evaluate these two mechanisms, we measure cortical flows and quantify axis convergence during AP axis establishment.We develop a mathematical model of axis convergence that recapitulates the measured flows-both cortical and cytoplasmic.An experimental reduction of cortical flows demonstrates that they are required for axis convergence.Experiments and numerical simulations that diminish the pseudocleavage furrow demonstrate that the pseudocleavage-furrow-dependent mechanism is the primary driver of axis convergence, but the cytoplasmicflow-dependent mechanism also contributes.Experiments and numerical simulations that change the shape of embryos demonstrate the contribution of geometry to the rate of axis convergence: rounder embryos exhibit slower axis convergence.This reduction in rate is consistent between numerical simulations and experimental observations.This effect can also be described by a minimal model focusing on the action of the pseudocleavage furrow.Taken together, our results demonstrate how actomyosin-dependent active surface mechanics and flows translate the ellipsoidal geometry of the embryo into the orientation of the AP axis.

Cortical flows drive AP axis convergence
We first set out to understand how the AP axis converges with the long axis of the embryo.To this end we observed the behavior of PAR domains during AP axis establishment via time-lapse imaging of the one-cell C. elegans embryo during this phase.We used a C. elegans line expressing PAR-6::mCherry (marking the aPAR domain) together with PAR-2::GFP (marking the pPAR domain) to visualize the PAR domains (STAR Methods).In agreement with previous observations, 2 we found that in 33 out of 57 embryos where the AP axis is established off-axis (with a measured AP axis that started more than 10 deg out of convergence with the geometric long axis), the PAR domains moved such that within 7 min the AP axis converged with the long axis of the embryo (Figure 1A).We also find that AP axis convergence occurs concomitantly with the posteriorization of the male pronucleus (Figures 1B and 1C, male pronucleus visualized as a dark circle in the cytoplasm in NMY-2::GFP-labeled embryos; Figure S1).
Furthermore, the angular velocity of the male pronucleus is highest when the angular position of the male pronucleus is largest; that is, the further the male pronucleus is from the posterior tip, the faster it moves toward the tip.Here, angular position refers to the angle between the long axis of the embryo and the line connecting the center of the male pronucleus to the center of the embryo.Specifically, we measured mean angular velocities of À2.1 ± 8.3 deg/s for angles between 0 and 10 deg compared with À18.9 ± 12.9 deg/s for angular positions between 30 and 40 deg (Figure 1C, negative velocities denote movements toward the posterior tip).Thus, for the range of angular positions measured (below approximately 45 deg), the rate of axis convergence is greatest when the axis is furthest displaced from the long axis.
Given the importance of cortical flows for AP axis establishment, 20,22,23 we next investigate their role in axis convergence.We used RNAi to impair function of mlc-4, an activator of myosin activity that regulates cortical flows. 42In mlc-4 RNAi embryos (Figures 1D and S1B), flows were indeed reduced (Figure S2), the rate of axis convergence greatly diminished, and the AP axis failed to converge to the long axis (Figure 1D).Specifically, we measured mean angular velocities of À0.02 ± 0.04 deg/s for angular positions between 0 and 10 deg and À0.01 ± 0.03 deg/s for angular positions between 30 and 40 deg, with 10/18 embryos failing to re-orient their PAR domains (Figures 1E and  1F).[45] A mathematical model accounts for AP axis convergence To understand how cortical flows could drive axis convergence, we developed a mathematical model.The model considers both active surface flows in the cortex itself 21 and bulk cytoplasmic flows that are a consequence of cortical flows 36 (Figure 2; STAR Methods).This mathematical model should allow us to answer two questions: (1) do both mechanisms, the cytoplasmic-flow-dependent mechanism and the pseudocleavagefurrow-dependent mechanism (Figure 2A), contribute to axis convergence and, if so, to what extent?(2) Are the two mechanisms together sufficient to capture the spatiotemporal dynamics of the entire process?
In the mathematical model (STAR Methods), we approximate the geometry of the C. elegans embryo by a fixed ellipsoidal shape.0][51][52] Active stresses in this surface fluid layer arise through myosin activity 20,21,47 and generate cortical flows.The cytoplasm itself is considered as an incompressible Newtonian fluid present in the interior of the ellipsoid, with flows in it approximated as Stokes flow.At the interface between the cortex and the cytoplasm, the cytoplasm flows at the same velocity as the cortex; that is, cortical flows at the surface set a no-slip boundary condition for the bulk cytoplasm within the ellipsoid. 36The effect of bulk flows onto the surface fluid is captured by an effective frictional coefficient g in the surface fluid layer 21,47 (Equation 14 in STAR Methods).We assume that the friction between the embryo and the eggshell is negligible and therefore consider only one friction term.With this coupling of surface and bulk flows, the cytoplasmic flow-field in the bulk is fully determined by the cortical flow-field on the surface 36 (Equation 15 in STAR Methods).The model considers the male pronucleus to be the organizer of active surface flows.The pronucleus is advected by the bulk cytoplasmic flows as a fluid particle with a large viscosity. 53In addition, the male pronucleus is both in direct physical contact with the cortical layer and also attached to it via microtubules that radiate from the two centrosomes, which are adjacent to the male pronucleus, toward the cell cortex. 12,27,30We capture the impact of these interactions in the model by a drag coefficient between the male pronucleus and the cortical layer (d; Equation 18 in STAR Methods).The position of the male pronucleus determines the instantaneous distribution of myosin in the cortex and also defines the instantaneous AP axis (Equation 9 in STAR Methods).The resulting gradients of myosin are responsible for the active stress distribution in the cortical layer, which in turn drives cortical flows on the surface.The resulting cytoplasmic flows and their advection of the male pronucleus capture the cytoplasmic-flow-dependent mechanisms of axis convergence.This mechanism arises due to embryo geometry (Figure S3A).The pseudocleavagefurrow-dependent mechanism arises due to flow-induced local shear in the cortical layer, which causes actin filaments to align into a ring. 39Active stresses depend on the state of local filament alignment.Anisotropic active stresses along the ring of actin filaments generate the pseudocleavage furrow and drive rotation of the whole-cell surface 39,54 (Figure 2A; Equation 12 in STAR Methods).Thus, the effects of actin alignment are embodied in the pseudocleavage-furrow-dependent mechanism.Consequently, the distinction between the two mechanisms is whether they take actin alignment into account (Figure 2A).Altogether, for a given position of the male pronucleus, the mathematical model calculates cortical flows, cytoplasmic flows, and flow-derived male pronucleus velocity.
We implemented the model using the adaptive finite element toolbox AMDiS. 55,56We solved for cortical flows on the ellipsoid surface using a surface finite element method for tangential vector and tensor quantities. 57We solved the Stokes problem for cytoplasmic flows in bulk using standard finite element methods within a diffuse domain. 58We solved the coupled surface (cortex) and bulk (cytoplasm) flows concurrently.We selected numerical parameters such that the numerical solutions are stable and independent of these choices (STAR Methods).Thus, the (B and E) Angular position of the male pronucleus (on y axis, in deg), plotted as a function of time relative to end of posteriorization (on x axis, in s) for unperturbed control embryos (B; N = 57; same data as Figure 1C) and nop-1;mel-11 RNAi embryos (E; N = 69).Blue thin lines in (B) and pink thin lines in (E) represent the experimentally observed trajectories of the male pronucleus in unperturbed embryos (B) and nop-1;mel-11 RNAi embryos (E), respectively; each line is an embryo.Blue thick line in (B) represents the predicted trajectory of the male pronucleus using the full mathematical model calibrated for unperturbed embryos.Pink dashed thick line in (E) represents the predicted trajectory using the mathematical model without the pseudocleavage furrow and calibrated for nop-1;mel-11 RNAi embryos (STAR Methods; Figure S3; model parameters are listed in Table S1).
(legend continued on next page) calculation of cortical flows, cytoplasmic flows, and flow-derived pronucleus velocity depends only on the four model parameters: hydrodynamic length l H , active force relaxation l A , nematic stress relaxation l N , and drag coefficient between the cortex and male pronucleus d (Equation 22 in STAR Methods; Table S1), together with the two axes lengths that describe the geometry of the egg: semi-major axis a and semi-minor axis b.
We sought model parameters that recapitulate the cortical and cytoplasmic flows we observed experimentally in unperturbed embryos.For this, we set the axes of the embryo ellipsoid to match the average axes lengths (with semi-major axis length a = 28.7 mm and semi-minor axis length b = 16.4 mm; Figure S5A).Because cortical flows are the basis for cytoplasmic flows, we sought parameters that result in calculated cortical flows that best match the experimental cortical flows (STAR Methods).Cortical flows arise from myosin depletion near the male pronucleus; based on previous work, 20 the model depletes myosin within 15.9 mm of the male pronucleus.With the ellipsoid dimensions and depletion distance set, we numerically solved the model for cortical flows over a range of angular positions of the male pronucleus (0-19 deg; Figure S3).Our calculated cortical flows best match our experimentally measured cortical flows with the following parameter values: hydrodynamic length l H = 10 mm, active force relaxation l A = 11.5 mm 2 s À1 , and nematic stress relaxation l N = 152.5 mm 2 s À1 (Table S1; see STAR Methods for calibration details).Bulk cytoplasmic flows are determined uniquely from cortical flows via the no-slip boundary condition.Calculated cytoplasmic flows show good agreement with experimental cytoplasmic flows (Figure 2B).Thus, for an optimal set of model parameters, our model faithfully recapitulates experimental cortical and cytoplasmic flows observed in unperturbed embryos.
Given that the mathematical model captures surface and bulk flow, we next evaluated how well the model predicts the movement of the male pronucleus toward the posterior tip.Specifically, we quantified (1) angular position, taken as the angle between the line connecting the male pronucleus and embryo center to the long axis (Figure 3B), and (2) posteriorization velocity, taken as the component of the male pronucleus velocity tangential to the cortex, at the point on the cortex closest to the pronucleus (Figure 3C).The remaining model parameter that is needed to calculate male pronucleus dynamics is the drag coefficient d, which reflects direct interactions between the male pronucleus and the cortex (STAR Methods).Using model parameters that give the best match to experiment for cortical and cytoplasmic flows, we determined that a value of d = 0.61 gives the best match to experimentally observed posteriorization velocities (for angular positions between 0 and 20 deg; Figure 3C).By integrating these best-match velocities, we obtain a predicted trajectory of the male pronucleus that agrees well with our experimental observations (Figure 3B; STAR Methods).Because numerical simulations closely match experimental measurements, we conclude that our model faithfully recapitulates the dynamics of the male pronucleus.
With this description in place, we are now in a position to evaluate to which degree the two mechanisms-the cytoplasmicflow-dependent mechanism and the pseudocleavage-furrowdependent mechanism-contribute to axis convergence.
The pseudocleavage furrow is a major contributor to AP axis convergence In the mathematical model, the cytoplasmic-flow-dependent mechanism and the pseudocleavage-furrow-dependent mechanism are additive; furrow formation requires cortical flows, but the flows-although influenced by the furrow-do not require the furrow.Recall that the pseudocleavage-furrow-dependent mechanism embodies the overall effect of actin alignment, of which the major effect is the formation of the furrow.Regardless, because the two mechanisms are separable, the model enables us to probe the function of the pseudocleavage furrow in AP axis convergence.Specifically, we can assess the importance of the pseudocleavage-furrow-dependent mechanism by modulating the l N parameter, which controls the actin-alignment-dependent active stresses.We observe that reducing the l N parameter leads to slower posteriorization velocities, but the AP axis still converges to the long axis (Figure S3).Thus, the model predicts that the pseudocleavage-furrow-dependent mechanism is essential to achieve normal rates of AP axis convergence.
To verify this prediction experimentally, we generated embryos lacking a pseudocleavage furrow but with normal cortical flows.To generate embryos with unaltered cortical flows, but without a pseudocleavage furrow, we performed double RNAi of nop-1 and mel-11.NOP-1 modulates activity of the small GTPase RHO-1, a major regulator of actomyosin in the early embryo. 45Embryos from mothers that are mutant for NOP-1 lack a pseudocleavage furrow. 40MEL-11 is a myosin phosphatase 59 that suppresses the activity of myosin in the cortex. 60RNAi of nop-1 eliminates the pseudocleavage furrow in the embryo, 45 but cortical flows are also reduced (Figure S2; STAR Methods).Under double nop-1; mel-11 RNAi conditions, every embryo that we observed lacked a pseudocleavage furrow (69/69 embryos).Cortical flow speeds in the double-RNAi embryos were also comparable to unperturbed controls (average cortical flow speed: 3.34 ± 0.52 mm/min in double RNAi of nop-1 and mel-11, as compared with 4.12 ± 0.59 mm/min for unperturbed control embryos; STAR Methods; Figure S2).Thus, nop-1; mel-11 RNAi experimentally generates embryos lacking a pseudocleavage furrow, but with normal cortical flows.
To evaluate the prediction of the mathematical model, we measured the rate of convergence in nop-1; mel-11 RNAi (C and F) Posteriorization velocity of the male pronucleus (velocity component of the male pronucleus along the cell cortex, on y axis, in mm/s) as a function of its angular position (on x axis, in deg) in unperturbed embryos (C; N = 57; Figures S4A and S4E) and in nop-1;mel-11 RNAi embryos (F; N = 69; Figures S4B and S4F).See inset in (C) for visual definition.Blue circles in (C) and pink squares in (F) denote the average posteriorization velocity for each angular position bin (bin size: 3 deg), with error bars representing the 95% confidence interval of the average, measured in unperturbed embryos (C) and nop-1;mel-11 RNAi embryos (F), respectively.Blue curve in (C) and pink dashed curve in (F) denote the predicted posteriorization velocities using the full mathematical model (C) and mathematical model without pseudocleavage furrow (F), respectively (STAR Methods; Figure S3; model parameters are listed in Table S1).Pink dotted curve and light blue dash-dot curve in (C) denote the contribution of the cytoplasmic-flow-dependent mechanism and pseudocleavage-furrow-dependent mechanism, respectively, to the predicted posteriorization velocities using the full model (blue curve, C). embryos.Qualitatively, the rate was slower (compare Figures 3E  and 3F and Figures 3B and 3C).We calibrated our model based on the experimentally observed cortical flows in nop-1; mel-11 RNAi embryos.Because we see no pseudocleavage furrow in these embryos, we set l N = 0 mm 2 s À1 , thereby excluding the pseudocleavage-furrow-dependent mechanism.Model parameters that yield the best match to experimental cortical flows for the l N = 0 mm 2 s À1 condition are l H = 11 mm and l A = 7 mm 2 s À1 (Table S1; Figure S3), which indicates that cortical mechanics are slightly altered in this condition.We leave the drag coefficient d = 0.61 unchanged and find that the model prediction for the l N = 0 mm 2 s À1 condition broadly matches both of the experimental readouts of AP axis convergence-posteriorization velocities and the male pronucleus trajectories-in nop-1; mel-11 RNAi embryos (Figures 3E and 3F).Thus, the cytoplasmicflow-dependent mechanism accounts for the slow AP axis convergence in nop-1; mel-11 perturbed embryos (Figures 3C  and 3F; average posteriorization velocities, calculated over angular positions 0-20 deg: À0.01 ± 0.06 mm/s in nop-1; mel-11 RNAi compared with À0.03 ± 0.06 mm/s for unperturbed embryos).We conclude that the pseudocleavage furrow is a major contributor to the dynamics of AP axis convergence, as predicted by the mathematical model.

Embryo geometry directs the rate of AP axis convergence
Our mathematical model considers active surface stresses driving dynamics that depend on local curvature and, therefore, embryo geometry.The embryo takes the form of an ellipsoid, with one long axis a and two equal short axes b.We focus on the aspect ratio a=b.To investigate how AP axis convergence depends on geometry, we calculated posteriorization velocities for embryos with different aspect ratios while holding the volume constant, setting all other model parameters to the values for unperturbed embryos (Table S1; STAR Methods).Embryos with smaller aspect ratios show slower posteriorization velocities (Figure S6A).Thus, the mathematical model predicts that rounder embryos have a slower rate of AP axis convergence.
To verify this prediction experimentally, we used RNAi to knock down ima-3 to generate embryos with aspect ratios smaller than unperturbed embryos. 61IMA-3 is a member of the importin a family of nuclear transport factors. 61,62Reducing ima-3 levels by RNAi results in smaller, rounder embryos 61 (average a=b = 1.48 ± 0.18 for ima-3 RNAi embryos compared with 1.78 ± 0.15 for unperturbed embryos; Figures 4A, 4B, S5B, and S5C).To obtain the model prediction for the rounder ima-3 embryos, we used the same model parameters as the unperturbed embryos, but changed the aspect ratio of the ellipsoid to match the average aspect ratio of the rounder ima-3 RNAi embryos (Table S1).Using the same model parameters is valid because both cortical flows and myosin concentrations in the cortex and cytoplasm are comparable between ima-3 RNAi and unperturbed embryos (STAR Methods; see Figure S2 for cortical flows and Figure S5D for myosin concentrations).
The model prediction for this updated geometry matches both of the experimental readouts of AP axis convergence-male pronucleus trajectories and posteriorization velocities-as observed in ima-3 RNAi embryos (Figures 4C and 4D).Additionally, the experimentally observed posteriorization velocities are faster in unperturbed embryos compared with ima-3 embryos (Figure 4D; average posteriorization velocity over angular positions 0-20 deg: À0.03 ± 0.06 mm/s for unperturbed compared with À0.02 ± 0.06 mm/s in ima-3 RNAi embryos).Consequently-as predicted by the model-rounder embryos have a slower rate of AP axis convergence.
Conversely, our mathematical model predicts that embryos with aspect ratios larger than those observed in unperturbed embryos should exhibit faster axis convergence.To test this prediction, we use RNAi to knock down C27D9.1 to generate more elliptical embryos 63,64 (Figure S1F).C27D9.1 is an uncharacterized gene, which is an ortholog of human fucosyltransferase and is important for body morphogenesis, embryo development, and reproduction. 61,65,66RNAi-mediated knockdown of C27D9.1 results in more elliptical embryos-that is, embryos with higher aspect ratio compared with unperturbed embryos 63,64 (average a=b = 1.98 ± 0.16, N = 37 for C27D9.1 RNAi embryos compared with 1.78 ± 0.15 for unperturbed embryos; Figures S5B and S5C).However, we observe a much larger overlap between the distribution of aspect ratios observed in C27D9.1 embryos and unperturbed embryos compared with that observed between ima-3 RNAi embryos and unperturbed embryos (STAR Methods; Figure S5A).Because we observe that both cortical flows and myosin concentrations in the cortex and cytoplasm are comparable between C27D9.1 RNAi and unperturbed embryos (STAR Methods; see Figure S2 for cortical flows and Figure S5D for myosin concentrations), the same as when comparing ima-3 RNAi embryos with unperturbed embryos, we combined the three datasets into a single combined dataset-thus considering aspect ratios between a=b = 1.1-2.3.We partition the combined dataset into three subsets based on aspect ratios-''round'' (with aspect ratio a=b< 1:53, average aspect ratio a=b = 1:37 ± 0:12, N = 19), ''mid'' (with aspect ratio 1:53 <a=b< 2:01, average aspect ratio a=b = 1:77 ± 0:14, N = 91) and ''elliptical'' (with aspect ratio a=b> 2:01, average aspect ratio a=b = 2:14 ± 0:09, N = 19), as shown in Figure 5A.Cutoffs for the aspect ratios are selected to be at the 15th and 85th percentile of the aspect ratio distribution of the combined dataset (approximately one standard deviation above and below the mean aspect ratio observed in the full combined dataset, a=b = 1:76 ± 0:24, N = 129).Experimentally observed posteriorization velocities in the three subsets show that more elliptical embryos show faster axis convergence: for each angular position bin, we observe fastest posteriorization velocity in the elliptical subset, followed by the mid subset, and slowest in the round subset (Figure 5B).Model predictions using the mathematical model for the three subsets, using the same model parameters as those determined for the unperturbed embryos but with average aspect ratio observed in the corresponding subset, predict the same qualitative trend-faster axis convergence in more elliptical embryos (Figure S6B).However, while the model prediction captures the posteriorization velocities observed in round and mid subsets, we observe that the model prediction for the elliptical subset is slower at higher angular positions compared with experimentally observed posteriorization velocities (Figure S6B).This discrepancy at higher aspect ratios reveals limitations of our mathematical model that might arise because we consider a fixed ellipsoidal cell shape.We expect that a model explicitly considering cell shape changes driven by the pseudocleavage furrow 39 would, due to dependencies on local curvature, give rise to altered dynamics and faster axis convergence, in particular at larger aspect ratios.
Our results thus far lead us to conclude that there is a relation between the rate of axis convergence and embryo geometrythe more elliptical the embryo, the faster the rate of AP axis convergence and the more spherical the embryo, the slower the rate of AP axis convergence.

A minimal model explains how geometry directs AP axis convergence
Why does the rate of AP axis convergence depend on embryo geometry?Our mathematical model and experimental results show that the pseudocleavage-furrow-dependent mechanism is a major contributor to the dynamics of AP axis convergence.In the mathematical model, the pseudocleavage-furrow-dependent mechanism embodies the overall effect of actin alignment and associated active stresses, of which the major effect is the formation of the pseudocleavage furrow.In the embryo, the pseudocleavage furrow is a contractile ring.Thus, a simple, minimal model of AP axis convergence would be a contractile ring rotating on an ellipsoid.Is this minimal model sufficient to capture the relation between geometry and AP axis convergence?
In this minimal model (STAR Methods), the ellipsoid has a long axis of length 2a and short axis of length 2b.The centers of the (C) Angular position of the male pronucleus (on y axis, in deg), plotted as a function of time relative to end of posteriorization (on x axis, in s) for ima-3 RNAi embryos.Orange thin lines represent the experimentally observed trajectories; each line is an embryo.Orange thick line represents the predicted trajectory of the male pronucleus using the full mathematical model, using the parameters from the unperturbed embryos (Table S1), for ima-3 embryos (aspect ratio: 1.48) (STAR Methods).(D) Posteriorization velocity of the male pronucleus (on y axis, in mm/s) as a function of its angular position (on x axis, in deg) in unperturbed embryos (blue, N = 57, reproduced from Figure 3C) and in ima-3 RNAi embryos (orange, N = 35; Figures S4C and S4G).See inset in Figure 3C for visual definition.Blue circles and orange diamonds denote the average posteriorization velocity for each angular position bin (bin size: 3 deg), with error bars representing the 95% confidence interval of the average, measured in unperturbed embryos (blue circles) and ima-3 RNAi embryos (orange diamonds), respectively.Curves denote the predicted posteriorization velocities using the full mathematical model, using the parameters from the unperturbed embryos (Table S1)-blue for unperturbed embryos (aspect ratio: 1.78, reproduced from Figure 3C), orange for ima-3 embryos (aspect ratio: 1.48) (STAR Methods).
ring and the ellipsoid are coincident.The angle between the normal to the plane of the ring and the long axis of the ellipsoid is a.The sole degree of freedom in the model is a; thus, changes in a encapsulate all the dynamics of the model.Changes in a are opposed by a coefficient of friction (Figure 5C).In addition, we assume that the male pronucleus moves with the same angular The histogram plots the number of embryos (along y axis) in a given aspect ratio bin (along x axis, bins of width 0.1).For (B), the combined dataset is split into three subsets: round (aspect ratio < 1.53, average aspect ratio = 1.37 ± 0.12), mid (1.53 < aspect ratio < 2.01, average aspect ratio = 1.77 ± 0.14), and elliptical (aspect ratio > 2.01, average aspect ratio = 2.14 ± 0.09).Aspect ratio cutoffs are depicted in dashed black line.Cutoffs are selected to be at the 15th and the 85th percentile of the aspect ratio distribution of the combined dataset (see Figure S5A for overlap between aspect ratios of unperturbed, ima-3 RNAi, and C27D9.1 RNAi embryos).For details on C27D9.1 RNAi embryos, see Figures S1F, S2, S4D, S4H, and S5.(B) Posteriorization velocity of the male pronucleus (on y axis, in mm/s) as a function of its angular position (on x axis, in deg) for the three aspect ratio subsets of the combined dataset, as depicted in Figure 5A.Dark blue upward triangles, green diamonds, and black downward triangles denote the average observed posteriorization velocity in the round (aspect ratio < 1.53) subset, mid (1.53 < aspect ratio < 2.01) subset, and elliptical (aspect ratio > 2.01) subset, respectively.Error bars indicate the 95% confidence interval of the average.Predicted posteriorization velocity for the average aspect ratio in each subset using the minimal model are depicted as solid lines (aspect ratio = 1.37 for round in dark blue, aspect ratio = 1.77 for mid in green, and aspect ratio = 2.14 for elliptical in black).See Figures S6A and S6B for predictions using the mathematical model.See Figures S6C and S6D and STAR Methods for parameters used for minimal model.(C) Schematic depicting the effective model of a contractile ring (red ellipse) with fixed line tension (T) moving on an ellipsoid (black thick line, long axis of length 2a along x axis and short axes of length 2b each along y and z axes).The ring is free to rotate around the z axis (red block arrows).Its position is defined by the angle a between the long axis and normal to the plane of the ring and is the same as the angular position of the male pronucleus.v is the posteriorization velocity of the male pronucleus.(D) Ratio of the posteriorization velocity of the male pronucleus with the angular position of the male pronucleus (on y axis, in mm s À1 deg À1 ) as a function of embryo aspect ratios (on x axis).Black circles with error bars denote the observations from the combined dataset, considering angular positions within 3-21 deg.Black circles denote the average posteriorization velocity in each aspect ratio bin (bin size: 0.2).Error bars represent the 95% confidence interval in each bin.For each aspect ratio bin, the ratio of posteriorization velocity with the angular position is averaged over angular positions 3-21 deg of the combined dataset-we exclude the angular position bin 0-3 deg as v=a is not defined for a = 0 deg.Curves denote the predictions from full mathematical model (blue), mathematical model using cytoplasmic-flow-dependent mechanism only (red), and minimal model (gray, v=a The predictions using the mathematical model, with (blue) or without (red) the pseudocleavage-furrow-dependent mechanism, are obtained by averaging predicted v=a over the same range of angular positions for different aspect ratios (see Figure S6A for the predictions using the full mathematical model and Table S1 for parameter values).For minimal model, C = À2.64 3 10 À3 mm s À1 deg À1 (STAR Methods).See also Figures S6C-S6F.velocity as the ring.For small angles a, we obtain the following relation between the posteriorization velocity v of the male pronucleus and a (Equation 30 in STAR Methods): Note that this minimal model only considers the dynamics of the pseudocleavage furrow and thus only captures the pseudocleavage-furrow-dependent mechanism of axis convergence.We find that predicted posteriorization velocities obtained using this simple minimal model for the average axes lengths observed in round, mid, and elliptical subsets of the combined dataset capture reasonably well the observed posteriorization velocities for different angular position bins in the three subsets (Figure 5B).The expression for the posteriorization velocity v in the minimal model indicates that the posteriorization velocity is proportional to the angular position a, with the proportionality constant determined by the embryo axes lengths a and b.We thus next considered the ratio of posteriorization velocity with the angular position v=a (Figure 5D).We find that the measured v= a as a function of embryo aspect ratio is well captured by both the minimal model and the full mathematical model, but not the cytoplasmic-flow-dependent mechanism alone (Figures 5D and  S6E).We thus conclude that the pseudocleavage-furrowdependent mechanism is a major driver of AP axis convergence.

DISCUSSION
Here, we show that in C. elegans two additive mechanisms drive the convergence of the AP axis with the geometric long axis of the egg.Both are driven by flows in the actomyosin cortex, which behaves as an active nematic fluid layer.Flows in the cortical layer are organized by the centrosomes associated with the male pronucleus. 12,20Cortical flows generate cytoplasmic flows and the pseudocleavage furrow.Both features translate the geometry of the ellipsoidal embryo to orient the AP axis toward the long axis of the embryo.Typically, both mechanisms are engaged, but the pseudocleavage-furrow-dependent mechanism dominates.
Previous work has shown that the pseudocleavage furrow is largely dispensable for proper AP axis formation and orientation. 9,40We argue that in embryos without a pseudocleavage furrow, AP axis convergence takes place because the cytoplasmic flows advect the male pronucleus toward the closest tip.In embryos with a pseudocleavage furrow, AP axis convergence takes place rapidly because the male pronucleus migrates quickly due to action of the furrow.Furthermore, in pseudocleavage-furrow-deficient embryos, incomplete AP axis convergence early in development could be corrected later by slow movement of PAR domains. 37,67Such a correction may also be geometry-dependent due to the tendency of PAR domains to form in regions of high curvature. 15Thus, multiple redundant mechanisms ensure proper positioning of the AP axis even if the pseudocleavage furrow fails to form.
To conclude, the AP axis converges to the long axis because cortical flows translate the geometry of the embryo into directed movement.The cortical layer is sensitive to geometry due to the nematic nature of the cortex: the alignment of actin filaments in the cortical layer depends on local curvature.Myosin motors acting on aligned filaments generate active anisotropic stresses physically embodied as the pseudocleavage furrow. 39The nematic ordering of actin filaments thus gives rise to curvaturesensitive stresses that lead to the rotation of the pseudocleavage furrow, a directional movement.9][70][71][72][73][74] Here, our mathematical model shows how the nematic behaviors of the actomyosin cortex, in the context of the ellipsoidal shape of the embryo, give rise to the orientation of an anatomical feature-the AP axis of the C. elegans embryo.

STAR+METHODS
Detailed methods are provided in the online version of this paper and include the following:

Lead contact
Further information and requests for resources and reagents should be directed to and will be fulfilled by the Lead Contact, Stephan W. Grill (grill@mpi-cbg.de)

Materials availability
C. elegans strains used in this study are available upon reasonable request from the authors.

Data and code availability
All analysis results, such as male pronucleus trajectories and cortical flows measured for all movies, along with all original code used to analyze movies has been deposited and publicly available at https://doi.org/

EXPERIMENTAL MODEL AND STUDY PARTICIPANT DETAILS
C. elegans strains C. elegans strains were maintained according to standard procedures. 78C. elegans strains were cultured on plates containing nematode growth medium seeded with OP50 Escherichia coli bacteria.All strains were maintained at 20 C. All strains used in this study are listed in the key resources table.

Bacterial strains
The OP50 strain was used for maintenance of worm strains.IPTG-containing RNAi plates were seeded with cultures of HT115(DE3) carrying the appropriate vector for RNAi studies, as derived from Ahringer RNAi library. 65

METHOD DETAILS
Feeding RNAi RNAi was performed by feeding bacteria expressing dsRNA to C. elegans worms.All RNAi clones used in this study were derived from the Ahringer RNAi library. 65RNAi induction was performed via feeding, as previously described. 79Briefly, RNAi feeding plates were created by seeding NGM agar plates, containing 1 mM isopropyl-b-D-thiogalactoside (IPTG) and 50 mg mL À1 ampicillin, with bacteria expressing dsRNA targeting the gene of interest.The nop-1, mel-11, ima-3 and C27D9.1 RNAi clones were obtained from the Ahringer RNAi library (Source Bioscience). 65The mlc-4 RNAi clone was obtained from the Hyman lab. 80For single RNAi experiments, only the RNAi clone of interest was grown on the RNAi feeding plates.For double RNAi of nop-1 and mel-11, both RNAi clones were grown simultaneously on the RNAi feeding plates.RNAi interference was then performed by transferring young L4 larvae onto these RNAi feeding plates before imaging.Indicated hours of RNAi treatment reflect the time that the worms spent on the feeding plate.
Table above displays feeding times for RNAi conditions.Feeding time for each RNAi described in this manuscript.For each RNAi, young L4 larvae were transferred onto RNAi feeding plates before imaging.Indicated hours of RNAi treatment reflect the time that the worms spent on the feeding plate.
Live imaging C. elegans embryos were dissected from adult worms and mounted in M9 buffer (22 mM KH 2 PO, 42 mM Na 2 HPO 4 , 86 mM NaCl) to provide osmotic support.20 mm polystyrene beads were added to the M9 buffer as spacers for imaging.All movies of C. elegans embryos were obtained at room temperature, using spinning-disk confocal microscopy.Three different microscope setups were used -System 1: Zeiss Axio Observer Z1 equipped with Yokogawa CSU-X1 scan head, a PlanApochromat 63X/1.2NA Water objective, an Andor iXon emCCD camera (512 pixel by 512 pixel), operated using Andor iQ software; System 2: Zeiss Axio Observer Z1 equipped with Yokogawa CSU-X1 scan head, a C-Apochromat 63X/1.2NA Water objective, a Hamamatsu ORCA-Flash4.0V2 CMOS camera (2048 pixel by 2048 pixel), operated using Micro-Manager 81 ; and System 3: Nikon Ti2-E microscope equipped with Yokogawa CSU-X1 scan head, a SR PlanApochromat 60X/1.27NA Water objective, a Hamamatsu ORCA-FusionBT CMOS camera (2304 pixel by 2304 pixel), operated using NIS-Elements software.System 1 was used for all movies obtained in strain TH120.System 2 was used for all movies obtained in strain SWG070, except for C27D9.1 RNAi embryos.System 3 was used for all movies obtained for C27D9.1 RNAi embryos.System 1 uses excitation laser wavelengths 488 nm for GFP, and 594 nm for mCherry.System 2 and System 3 both use excitation laser wavelengths 488 nm for GFP, and 561 nm for mCherry.
C. elegans embryos were imaged starting from the beginning of polarizing flows until pronuclear meeting.T=0 s was selected at the end of flow phase -the time-point after which the male pronucleus moves away from the cortex and towards the female pronucleus.All movies were synchronized using this time-point.mlc-4 RNAi embryos were synchronized using the pronuclear meeting (observed at T=180 s in unperturbed embryos) instead, due to a lack of cortical flows in these embryos -see Figure S1.
Movies of PAR domains in strain TH120 for unperturbed control and mlc-4 RNAi embryos were acquired using System 1.At each time-point in a movie, two images were acquired at the mid-plane of the embryo -one for each channel using the corresponding excitation lasers: 488 nm for GFP and 594 nm for mCherry.For unperturbed control embryos, movies were acquired at 5 s intervals between time-points, with 80 ms exposure time for GFP and 900 ms exposure time for mCherry.For mlc-4 RNAi embryos, movies were acquired at 10 s intervals between time-points, with 80 ms exposure time for GFP and 300 ms exposure time for mCherry.Pixel size = 0.211 mm.Movies of myosin in strain SWG070 for unperturbed control and all RNAi embryos except C27D9.1 RNAi embryos were acquired using System 2. At each time-point in a movie, three images were acquired at the mid-plane of the embryo -one for each channel.One image is acquired using transmitted light (data not shown), and the rest two are acquired using the corresponding excitation lasers: 488 nm for GFP and 561 nm for mCherry.All movies were acquired at 3 s intervals between time-points, with 50 ms exposure time for transmitted light, 200 ms exposure time for GFP and 150 ms exposure time for mCherry.Pixel size = 0.105 mm.
Movies of myosin in strain SWG070 of C27D9.1 RNAi embryos were acquired using System 3, using the same procedure as for System 2. All movies were acquired at 3 s intervals between time-points, with 200 ms exposure time for transmitted light, 200 ms exposure time for GFP and 200 ms exposure time for mCherry.Pixel size = 0.108 mm.

QUANTIFICATION AND STATISTICAL ANALYSIS
In brief, the time-lapse movies of C. elegans embryos thus acquired were analyzed to quantify the following: embryo geometry, posteriorization of the male pronucleus and cortical flows.Embryo geometry was quantified by fitting an ellipse to the boundary of the embryo mid-plane observed in the movies.Posteriorization was quantified by tracking the male pronucleus, identified as a dark circle in the cytoplasm in fluorescent microscopy images.Cortical flows were quantified similarly to Gross et al. 20 using particle image velocimetry on the cortical layer of the embryo.Only NMY-2::GFP labelled embryos were used for measuring cortical flows.All softwares used in this study are listed in the key resources table.

Image Analysis
Each movie was analyzed in the following way: 1 Each movie was cropped to ensure that only a single embryo was present in each frame.Anterior and posterior end of the embryo were manually annotated.For strain TH120, the tip of the embryo closest to the initial position of the posterior PAR domain (labeled by PAR-2::GFP) is identified as the posterior end.For strain SWG070, the tip of the embryo closest to the initial site of NMY-2 depletion (pPARs dependent) is identified as the posterior end.In both cases, the opposite end is designated the anterior end.For embryos that show complete AP axis convergence, it was verified that the above identification matches the position of the PAR domains (in case of strain TH120) or NMY-2 anterior cap (in case of strain SWG070) after end of axis convergence process.The first frame in the movie at which the male pronucleus appears and the last frame before the male pronucleus moves away from the cortex were manually selected.The male pronucleus was identified as a dark circle in the cytoplasm in the fluorescently labelled channels (PAR-2::GFP in TH120, NMY-2::GFP in SWG070), located near to the posterior end.Any embryos where both pronuclei are located near to the same end were discarded.Only frames between the first and last frame selected above were analyzed.We computationally rearranged embryos such that the anterior tip of egg faces the left and the posterior tip faces the right, and that off-axis posterorization events occurred from above the AP axis.To achieve this orientation each movie was rotated, and if necessary, mirrored.Manual annotation and rotation were done in Fiji, 76,82 using bilinear interpolation. 2 To track the position of the male pronucleus as it posteriorizes, rotated movies for strain SWG070 were analyzed using a custom Python 77 script.First, the cell membrane of the embryo in each frame was segmented by thresholding, using the phDomain:: mCherry marker for the cell membrane, and then fit to an ellipse.Semi-major and semi-minor axes lengths and directions of the fitted ellipses are averaged over all frames to obtain the semi-major and semi-minor axes for the embryo as a whole.Next, NMY-2::GFP channel image was denoised using non-local means denoising, 83 with the following parameters: Filter strength = 3 pixel, template window size = 4 pixel, search window size = 12 pixel.Next, the male pronucleus was segmented in these denoised NMY-2::GFP frames using successive thresholding -the male pronucleus was identified as a dark circle in the cytoplasm, present in the right half of the embryo.These segmentations were verified manually.Finally, the trajectory of the male pronucleus was obtained by noting the position of the male pronucleus, as segmented, as a function of number of frames, and therefore time.Posteriorization velocities were calculated as the component of male pronucleus velocity perpendicular to the line connecting the male pronucleus center to the closest point on the cell membrane.Following additional packages were used in this Python script: openCV, 84 tifffile, 85 scipy, 86 numpy, 87 pandas 88 and matplotlib. 89For movies with strain TH120, no boundary segmentations were done.Male pronucleus was identified manually using a custom Fiji 76 macro, in a similar fashion to the automatic Python script.3 Cortical flow fields were only determined in strain SWG070 embryos using a custom MATLAB 75 script, as previously described. 20Briefly, cortical flows were determined by calculating the displacement field of the yolk granules near the cell membrane in NMY-2::GFP denoised frame, using PIVlab algorithm in MATLAB. 90The displacement field was calculated for each frame, in a region extending 30 pixel deep in the embryo and perpendicular to the cell membrane.These 2D velocity fields were binned and projected onto the cell membrane, thus yielding the measured cortical flows for each frame in the movie.A multi-pass (4 passes) PIV was utilized in all cases, with initial template size of 24 pixel and step size of 4 pixel.
For unperturbed control, ima-3 RNAi and C27D9.1 RNAi embryos (strain SWG070), myosin concentrations in the cytosol and cortex were also calculated, in intensity per pixel units.Intensity is measured in arbitrary units, corresponding to the camera readings in the movies.Cortex is identified as a region 15 pixel deep from the cell membrane.Cytosol is identified as the interior region of the embryo, left after the cortical region is removed.Myosin concentration in each region is estimated as the average intensity per pixel in the corresponding region.

Data analysis
The above image analysis pipeline provides the angular position of the male pronucleus, posteriorization velocity and cortical flows for each frame in each movie.This data is then analyzed, using MATLAB, 75 in the following way:

Posteriorization velocity vs angular position
To smooth short-term fluctuations in posteriorization velocity and angular position, a sliding average with a window of 7 frame is used for each movie.Then, all posteriorization velocities and angular positions from all movies in the condition of interest are combined together.Angular positions are binned in 3 deg bins.Mean posteriorization velocity for each angular position bin is calculated by averaging over all posteriorization velocities corresponding to the angular positions included in the bin.95% confidence interval for the mean posteriorization velocities are calculated using a two-sided t-test, using the statistics toolbox in MATLAB. 75

Cortical flows
Cortical flows for all movies in the condition of interest are combined together, and binned by the angular position of the male pronucleus, with 3 deg bin width.The mean cortical flows for an angular position bin are calculated by averaging all cortical flows within an angular position bin.Cortical flows are averaged spatially -i.e., average flow velocity at a certain point on the cortex is calculated by averaging over all observed flow velocities at that point.

Mathematical model
To obtain a theoretical description for the convergence of body axes to embryo shape we define a feedback loop, connecting mechanical processes in cortex and cytoplasm.First contribution in this loop are flows and actin alignment in the cortex.These intertwined effects are driven by gradients in myosin distribution due to a localized depletion in cortical layer triggered by the centrosome.Second contribution comes from these cortical flows driving cytoplasmic flows, yielding an advective transport and drag of the male pronucleus and thereby the attached centrosome.With the centrosomal trigger displaced the myosin distribution rearranges, cortical activity and flow profiles are displaced as well and the loop is closed.
In our modelling we neglect temporal minor fluctuations in the cell shape and considered the shape as a fixed spherical ellipsoid with typical in-vivo dimensions ½a;b;c = ½28:7mm;16:4mm;16:4mm.The cortical layer is assumed as a thin film enveloping the cytoplasmic bulk with constant thickness h ( a.We denote E as surface of the ellipsoid and b E the enclosed bulk.For these two domains, cortical layer and enclosed cytoplasmic bulk, we apply a coarse-grained/continuous modeling framework laid out in Ju ¨licher et al. 51 to obtain a suitable model for each domain.Due to the intricate physics of the cortex and its curved thin film geometry we will focus the modeling discussion on this part of the feedback loop.Furthermore, to separate the modeling assumptions and the impacts of confined geometries and its curvature we derive the cortex model in three steps.As initial step we discuss two established models of Gross et al. 20 and Reymann et al. 39 as well as their experimentally determined phenomenological parameters for an a priori estimate of dominant effects.In the second step we will derive a model for the cortex physics in an unbounded, volume like bulk domain, following the approach of Ju ¨licher et al. 51 As third step we transfer the volume model to a bounded thin film domain (thickness h), discuss boundary conditions and perform the thin film limit 70,91 h/0 to obtain the final cortex model.For the remaining feedback loop contributions, cytoplasmic streaming and male pronucleus transport, we will straight forward apply established models of Niwayama et al., 36 Tanaka and Araki. 53For sake of numerical feasibility, we consider the coupling between cortex and cytoplasm as ''one way'', enabling a sequential evaluation of the associated models.

Mechanics of the cortex
The cortical layer and its governing physics are subject of an extended research providing a wide range of experimental observations and theoretical descriptions.For our modeling we will use two recent models as building blocks as well as their experimental observations as guiding concepts.
As first building block we use the results of Gross et al., 20 which provides a description of the interplay of myosin depletion, flows due to myosin gradients and forming of PAR domains as a bistable system.We will neglect the formation of PAR domains and take from this model the connection between myosin depletion and cortical force generation.Reviewing the determined phenomenological parameters we observe that myosin concentration is dominated by un-binding dynamics between cortex and cytoplasmic bulk.Furthermore the observed cortical flow field is of compressible nature driven only by forces due to myosin gradients.Please note due to the limitation to one dimensional domain and absence of curvature the model 20 is invariant under translation such that a displacement of the centrosome yields a quasi instantaneous, equivalent displacement in myosin and flow fields.Second building block in modeling cortical dynamics are the results of Reymann et al. 39 There it has been established how compression of actin filaments yield a localized, ribbon like domain with aligned filaments.Such local aligned filaments, in combination with myosin motors, exert forces normal to the cortical layer yielding the ingression of the pseudocleavage furrow.Key mechanism in the model, driving the localized alignment, is the reactive coupling between deviatoric stresses and filament alignment.The results of Reymann et al. 39 consider this coupling as one way connection, neglecting any effects of alignment on cortical flow fields.Furthermore a cylinder like domain and rotational symmetric flow field and alignment are considered such that only normal (w.r.t cylinder domain) forces are yielded.Here we will use the modelling approach but focus on the tangential parts of the force generation and neglect the normal parts.Furthermore the considered ellipsoidal domain yields a significant different curvature distribution compared to a cylinder.
For the subsequent modeling we will combine these two approaches, along the concepts of Ju ¨licher et al., 51 to obtain a cortex model coupling momentum balance, with flows driven by forces due to myosin gradients as well as due to localized actin alignment, and alignment induced by flow compression.To estimate the impact of possible coupling mechanisms we review the experimental observations and calibrated phenomenological parameters of Reymann et al. 39 There the actin alignment is described by the uniaxial nematic order parameter 92 Q with flux D c Q=Dt and adjoint force H.The nematic state Equation 3 is rescaled along a characteristic nematic relaxation time t = G=A (nematic dissipation rate versus nematic energy density) and a nematic distortion length l D = ffiffiffiffiffiffiffiffi ffi L=A p (nematic elastic coefficient versus nematic energy density).Observations in Reymann et al. 39 indicate a relaxation time below measurement resolution, such that we will neglect flux contributions to nematic state equations.Furthermore, we use the observed values of l D = 4:72 and k Qk = Oð0:1Þ to estimate the order of magnitude for possible nematic stress contributions to the momentum balance.We yield k Hk = AOð0:1Þ for alignment stress and k QH À HQk = AOð0:01Þ; kA l 2 D VQ: VQk = AOð0:01Þ for nematic anti symmetric and equilibrium stress.Assuming a weak nematic energy density A% Oð1Þ we will neglect the anti symmetric and equilibrium stress contributions in further modeling.
Given the parameter regime specified by Gross et al., 20 Reymann et al. 39 for C. elegans zygote during pseudocleavage, the relaxation of cortical dynamics is considered fast compared to the timescale of axis convergence.
We first describe a bulk description of the cortex, which is transformed using a thin film limit into a relevant surface model.We consider the cortical layer as a fluid constituted by a mixture of rodlike particles (actin) and spherical shaped particles (myosin), with particle number densities A and M. 1) Actin is assumed to be uniformly distributed (Azconst) in the domain varying only in the degree of local ordering. 39Myosin motor distribution is assumed to exhibit temporal and spatial patterns mainly driven by local depletion near centrosome position x N and un/ rebinding bulk. 20Possible gradients of M in the neighboring cytoplasmic bulk, as discussed in Geßele et al., 67 as well as advective transport are neglected.Following Gross et al., 20 the particle number density balance reads therefore 2) where total amount of myosin is conserved by Þ models the depletion near P E ½x N with radius d M .P E ½x N refers to the point in the domain U (i.e.cortex) closest to the male pronucleus while j denotes the volume to neighboring bulk ratio to balance the myosin reservoir.Following Reymann et al., 39 Ju ¨licher et al. 51 the nematic alignment of actin is described by hydrodynamic variables Q and H along the state equation 3) with L the dimensionless parameter determining the strength of reactive flow-alignment coupling, the deviatoric (symmetric and trace free) deformation tensor of cortical flows D = ½VV + VV T =2 À½V $V=3I and F U a Landau-de Gennes like free energy describing the actin molecules tendency towards isotropic ordering and parallel alignment in case of ordering.In the considered parameter regime of fast cortical dynamics we yield 4) For momentum balance we use the approach of Gross et al. 20 for velocity V with friction g, as representative for momentum transfer to neighboring bulk, and active force creation due to gradients in M À V$s + gV = xVM (Equation 5) Due to the modulations in M we can not assume constant density, such that V$Vs0.The viscous stress therefore contains shear and bulk contributions, with associated viscosity h (shear), h B (bulk) and secondary viscosity h S = h B + 2=3h.6) We expand this approach by incorporating the rodlike nature of actin filaments, its local alignment and active tension in alignment direction in the stress modeling, following Ju ¨licher et al. 51 and the discussion above.We yield for total stress 7) with additional phenomenological parameter j for active tension strength.Furthermore we assume a shear-bulk viscosity relation of h B = À 2=3h and linear isentropic pressure pzC p r. Therefore the momentum balance reads 8) with the two force generating mechanisms of isotropic force generation due to myosin activity (A), rescaled by x = x + C p m M , and stresses due to actin alignment (N).
To derive now the relevant surface model from the bulk formulation of Equations 2, 4, and 8 we follow the thin film limit approach described in detail at Nitschke et al. 69 There, the basic idea is to consider a tubular extension E h of surface E with constant thickness h.In this volume E h we use the previously derived bulk model endowed with physical meaningful boundary conditions on vE h and perform the limit h/0.By using the thin film metric of E h as a bridge between bulk and surface domain we yield a compatible and covariant surface model.We highlight that in the process of thin film limit the tensorial degree of the considered physical field as well as the boundary conditions are decisive for the resulting coupling of geometry and physics in the surface model.See Nestler et al. 93 for a detailed discussion for nematic ordering.
Given the results of Nitschke et al., 70 Arroyo and DeSimone 91 we choose boundary conditions for the bulk model.With n the outward pointing normal of E, n h the constant extension (in normal direction) of n to E h , P½: the tangential projection to the tangent space of E and P h ½: for vE h , we impose d VM$n h = 0, no additional flux across vE h beside the contributions by binding dynamics d V$n h = 0, no flow across vE h and V n V = 0 no variations of velocity in normal direction.d P h ½s $n h = 0, normal components of stress do not contain tangential parts.Under this condition momentum balance of tangential and normal components decouple.d n h $Q$n h = 0, no nematic ordering in normal direction which is suitable for mono layers of filaments, see Nestler et al. 93 Furthermore we require VQ$n h = 0 which ensures continuous values of Q in h/0.
Denoting the covariant differential operators on E by div and grad the thin film limit yields for myosin particle number balance 9) where Þ with a geodesic distance measure s on E applied and P E ½x N denotes the closest point on the cortex w.r.t to male pronucleus center.We denote the tangential part by lower case letters, e.g.P½Vj E = v.Under the chosen boundary conditions, we yield for h/0 for velocity divergence V $V/div v. Please note that in the transition from volume to surface the notion of trace changes, so the surface deviatoric deformation tensor is given by d = ½gradv + gradv T =2 À ½div v=2g, where g denotes the metric of E. Therefore the thin film limit of deviatoric deformation includes an additional term.We express this term by using the augmenting surface metric (the so called thin film metric) G = g + nn ðdiv vÞg for h/0 (Equation 10) Due to the boundary condition chosen for nematics such additional term does not occur for nematic ordering and molecular field.Therefore we define the tangential parts of nematic order tensor P½Qj E = q and molecular field P½Hj E = h.With K the Gaussian curvature, B = À P½Vn the shape operator of E we obtain the thin film limit of nematic free energy, see Nitschke et al., 69 via 1 =h F Eh ðQÞ/F E ðqÞ.11) and nematic state equation on the surface The total stress for h/0 reads therefore 13) Using the previous isentropic pressure approximation and the results of Arroyo and DeSimone, 91 the tangential part of momentum balance on E reads 14) Cytoplasmic bulk and male pronucleus transport Following the results of Niwayama et al. 36 we model the cytoplasmic bulk as an Newtonian fluid in Stokes regime yielding a parameter free model.We neglect all cell organelles except the male pronucleus, which is described by a smooth density function fðxÞ = ½1 À tanhð3ðk x À x N k Àr N Þ =eÞ=2 with r N the male pronucleus radius and e the density transition width.By Tanaka and Araki 53 we include the male pronucleus in the Stokes model as a region of significantly increased viscosity (u N = 100) enforcing a homogeneous flow field inside the male pronucleus.The drag induced by cortical activity is modelled by a velocity boundary condition.Therefore the cytoplasmic bulk model for bulk velocity U and pressure P reads 15) The male pronucleus velocity U N is then calculated from two contributions.The first contribution is due to advective transport of the male pronucleus via cytoplasmic flows. 36,53Here we evaluate the tangential part, w.r.t.ellipsoid normal n N at closest point on the cortex P E ½x N , of the average cytoplasmic velocity in male pronucleus domain 16) The second contribution is due to the drag of the male pronucleus due to local movement of the cortical layer.Such a component arises due to the male pronucleus being considered to be in direct contact with the cortical layer and coupled to the cortex via microtubules. 12,27,30Note that such a coupling between the cortex and male pronucleus is not considered to be tight.To capture these interactions between the cortex and the male pronucleus, we introduce a phenomenological drag coefficient d ˛½0; 1.Such modelling could be understood as basic approximation of the interaction of cortex and cytoplasmic bulk in normal direction.For this purpose we evaluate an average cortical velocity across a domain matching the projection of the male pronucleus on the cortex, considering only tangential contribution 17) The total velocity of the male pronucleus thus reads: 18) with U 1 N ðx N Þ due to advection by cytoplasmic flows and dU 2 N ðx N Þ due to drag with the cortex.

Parameter calibration for Mathematical model
Before starting parameter calibration we will review the cortex model regarding the included phenomenological parameters and available experimental data to arrive at an feasible calibration problem.Furthermore we remove units from the complete feedback cycle by dedimensionalizing the state Equations 9, 14, and 15 along characteristic time 1 s and 1 mm.
In the given parameter regime we can rewrite the nematic state Equation 12and yield h = À Ltd resulting in a strong similarity in force fields of associated viscous and flow alignment stresses.Furthermore, we lack the experimental data to effectively discriminate the contributions of viscous and alignment forces.We therefore define and calibrate an effective viscosity h accounting for the combined effect.
(Equation 20) Reviewing the active nematic force contribution jdiv ð6m qÞ we observe the presence of contributions parallel to viscous forces.To reduce the ''overlap'' in these force contributions we follow Reymann et al. 39 and consider only active stresses along the nematic ribbon.21) Finally we highlight that Gross et al. 20 applies a slightly different approach in modeling isotropic active forces by x Ã grad 6m (instead of xgradM) where 6m = M =½M + M Ã .Applying such modeling indeed improved the reproduction of cortical flow field data.Therefore we adapt this modeling for the chemical potential and apply it also in the prefactor of active stress along the nematic ribbon.
Given this refined modeling, we follow Gross et al. 20 and Reymann et al. 39 and cancel friction by introducing cortical hydrodynamic length l H = ffiffiffiffiffiffiffiffi h=g p as well as a nematic stress relaxation l N = j=g and active force relaxation l A = x Ã =g.The cortical particle numbers for Myosin, momentum balance and nematic state equation read therefore 22) Please note, that ðv; qÞ scale with l A , e.g.Cl A /ð Cv; CqÞ, enabling us to fit l A separately by requiring to match kinetic energy in model cortex flow fields with experimental data.
Using parameter values for the Myosin and nematic state equation as specified in Gross et al., 20 Reymann et al. 39 only the two parameters l H and l N need to be calibrated.

Calibration for unperturbed control embryos
The calibration procedure for the cortex model uses experimental data of cortical flow profiles on a planar slice of E. Such profiles v d ða i Þ are available for a set of male pronucleus positions x N ða i Þ with displacement angles a i , see Figure S3B (blue lines) for unperturbed control embryos data or Figure S3F (blue lines) for nop-1; mel-11 RNAi embryos data.The plane is determined by AP axis and male pronucleus center, we label the slice of the cortex by P. The cortical velocities are described w.r.t a counter clockwise tangential direction t and are plotted along an oriented geodesic distance s on the ellipse with the male pronucleus position as origin, s = 0. Please note, such coordinate system corresponds to shift of the fixed coordinate system of experimental data where s = 0 is always aligned with the posterior pole.We use this data to calibrate the parameters l H , l N , l A of cortical model in a two step process.
As first step we solve Equation 22 and obtain ðM; v Ã ; q Ã Þ for each a i and for an array of values ðl H ; l N Þ while keeping l Ã A fixed.The value of l A is then determined by fitting the kinetic energy E P ðvÞ = R P v 2 dP across all velocity profiles min 23) As second step we evaluate a mixed calibration cost measure E = L + QR combining a similarity measure L and an imbalance measure R with weighting factor Q. Both measures are relative w.r.t to the model results.Basic motivation for the imbalance measure is the observation of a distinct correlation of male pronucleus displacement a i and the ratio of velocity magnitudes pointing towards to and away from the posterior pole.While this ratio is 1 for aligned male pronucleus (a = 0) the imbalance increases with further male pronucleus displacement.Therefore, the imbalance measure penalizes deviations in the magnitude ratios between experimental and model data across all displacement angles.We formalize this approach by 24) 25) A variation of the calibration parameter Q demonstrates the relevance of imbalance feature.For Q = 0 the calibration results in a model failing to capture the imbalances and yield weak male pronucleus velocities U 1 N and U 2 N .Yet choosing a high value, e.g.Q = 50, we yield model cortical flow profiles with weak similarity to experimental data.For the final error measure we use a value Q = 5 to weight the error measures.See Figure S3C for the resulting calibration cost landscape for unperturbed control embryos cortical flow data.In Figure S3B and Figure S3F (red lines) we plot the velocities of best fit models and observe a good reproduction of the experimental velocities.We summarize the obtained parameter values in Table S1.
After calibrating the cortex model parameters we evaluate the cytoplasmic flows and resulting male pronucleus velocities due to advective transport U 1 N as well as cortical drag U 2 N for a range of displacement angles a j ˛½0; 30 + (10 points).We compare the model male pronucleus velocity contributions to the in-vivo observed male pronucleus velocities, see Figure S3D.From this comparison we determine the effective drag factor d = 0:61 as optimal to reproduce male pronucleus velocities for a ˛½0; 20 + .For stronger displacements deviations in male pronucleus velocities remain, regardless of the value of d.
To complement the parameter calibration we investigate the sensitivities of model parameters w.r.t. the impact on male pronucleus velocity.Due to linearity of U to boundary velocities v in Equation 15we yield equivalent change in male pronucleus velocity for changes in l A .For the remaining model parameters l H and l N we evaluate a set of models with ±50% variation in the parameters, see Figure S3E for the comparison of resulting male pronucleus velocities.Here we observe that even for such strong parameter variations the qualitative behavior of the model remains unchanged underlining the robustness of the calibrated parameter values.Model for nop-1; mel-11 RNAi Here we use the parameters of the unperturbed model and enforce the absence of nematic effects by setting l N = 0.Only the active force relaxation is calibrated by fitting the kinetic energy as described in Equation 23.The Myosin state equation parameters are again used as given by Gross et al. 20 See Figure S3F for a comparison of experimental data (blue lines) and model results (red lines) for fitted parameters.Again we observe a good reproduction of experimental data for cortex velocities as well as male pronucleus velocities (effective drag remains unchanged d = 0:61), see Figure S3F.

Male pronucleus trajectory evaluation
From the evaluated male pronucleus velocities we can evaluate male pronucleus trajectories by integrating Equation 18for a given initial male pronucleus displacement x N ða = 45 + Þ for an arbitrary time domain t ˛½0; 1000.We describe the obtained male pronucleus trajectory by aðtÞ.To compare them to experimentally observed trajectories we introduce a temporal reference point b t.This defined for each set of experimental trajectories (unperturbed control, nop-1; mel-11 and ima-3) by choosing b t such that the average distance to experimental trajectories is minimized.In practice we obtain an average experimental trajectory a d ðt i Þ by temporal binning (100 bins) and perform least square error fitting to determine b t min t˛R

Shape variation
To investigate the geometric sensitivity in the mechanical feedback loop we variate the shape of the shape of the ellipsoid.Each shape is characterized along the ratio of long/short body axis a=b.To avoid a change in characteristic length scales we perform such shape variations while preserving volume of the enclosed cytoplasmic bulk.Please note the ima-3 RNAi embryos exhibit an average shape with ellipsoidal axes ½a; b; c = ½21mm; 14mm; 14mm.This shape is on the one hand side closer to a sphere compared to the unperturbed control embryos and on the other hand side significantly smaller wrt.volume V IMA = V WT = 0:57.

Numerical methods
Given the rotational symmetry of E along long body axis, the ''one way'' coupling of cortex and cytoplasmic flows, the instantaneous relaxation of momentum as well as the assumption of a male pronucleus close to the cortex we apply a catalogue approach for parameter calibration and evaluation of male pronucleus trajectories.
Assuming a constant male pronucleus cortex distance k x N À P E ½x N k = 3:5 and male pronucleus radius r N = 3 we can identify the male pronucleus position by the polar angle a = arctanðx =zÞ ˛½0; p=2.We discretize this domain equidistant with 30 points.For each a i we evaluate cortical and cytoplasmic flows as well as male pronucleus velocities.
All finite element methods are implemented and solved by AMDiS toolbox. 55,56

Catalogue evaluation
We use a regular triangulation of E (10k vertices) to formulate a surface FEM (linear Lagrange elements) 94 and solve the geodesic equations by the heat method of Crane et al. 95 with source at P E ½x N ðf i Þ. Thereby we obtain the geodesic distances dðxÞ on E.
The same triangulation and surface FEM approach is used to solve Equation 9 and obtain M for a depletion centered at P E ½x N ða i Þ.The non-linearity is handled by a relaxation scheme (100 steps).
For the coupled vector valued PDE (Equation 14) and Q tensor valued PDE (Equation 12) we use a embedded component wise FEM (linear Lagrange elements) 57 on the previous used surface grid, using the penalty specified in Nestler et al. 57 with prefactor u v = u q = 1000 to ensure tangentiality.Geometric quantities are evaluated from analytical results for E. The coupled system contains nonlinear terms in the active nematic stress.We apply an iterative solution procedure, treating the nonlinear terms explicitly.Details and convergence properties are described in Reuther and Voigt, 96 Reuther et al., 97 Nestler and Voigt, 19 Brandner et al. 98 To discretize the volume b E we use an adaptive, parametric and regular tetrahedral grid where refinement adaption is used at the interface 4 ˛½0:05; 0:95 with e = 0:25.The resulting mesh contains 300k vertices.On this mesh we use Taylor-Hood elements to solve the variational formulation of Equation 15.Pressure is fixed by imposing zero mean.Boundary conditions are imposed by using the diffuse domain approach. 58rom the catalogue of flow fields for the set of a i we evaluate the associated male pronucleus velocities.To evaluate the male pronucleus trajectory, starting at a 0 , we perform explicit time integration and linear interpolation of male pronucleus velocities from the catalogue.

Minimal model
To provide a physically intuitive understanding of axis convergence, we consider a simpler model of a contractile active ring rotating on an ellipsoid -representing the embryo.As before, we neglect any temporal changes in the shape of the embryo, considering a fixed axisymmetric ellipsoid of axis lengths a; b; b; a > b for the semi-major and semi-minor axes respectively.The long axis (length 2a) defines the x-axis, while the two short axes (length 2b each) define the y-and z-axes, respectively.The contractile ring has only one degree of freedom (it can only rotate about the z-axis).The plane of the contractile ring makes an angle a with the yz plane (see Figure 5D).
Following assumptions are made: d The ring has a constant line tension T -independent of both the orientation of the ring (i.e.a) and the shape of the ellipsoid (i.e.a; b).In our construction, T has units of force.d The frictional torque acting on the ring is given by g$ _ a d No inertial terms: Torque generated by the ring is perfectly balanced by the torque generated by friction.
Under the above assumptions, we want to calculate _ a as a function of a; b; a for small angles a.Note that a; b are free to vary -we do not assume that our ellipsoid is close to a sphere.Since the minimal model is only concerned with the rotation of the pseudocleavage furrow, and does not explicitly consider cytoplasmic or cortical flows in the embryo, the minimal model only captures the pseudocleavage furrow-dependent mechanism.
We first consider the mechanical energy stored in the contractile ring, which can be given by: E = TCðaÞ (Equation 26) where CðaÞ is the total circumference of the ring.The ring can be described as an intersection of ellipsoid with the plane in which the ring resides.The ring plane makes an angle of a with the y-z plane, and passes through the origin.Thus, it can be described by the equation: y = À x cot a.The ellipsoid itself can be described as x 2 a 2 + y 2 +z 2 b 2 = 1.On taking the intersection of the ring plane with the ellipsoid, we get the equation describing the ring: Plane:y = Àx cot a Ellipsoid: . Then, the equation in the intersection part above can be written as: Note that the ring is an ellipse in its plane -since the intersection of an ellipsoid and a plane must be an ellipse.This can also be seen by the form by k r !k 2 : We can therefore calculate semi-major a ring and semi-minor b ring axes of the ellipse formed by the ring.We directly read them off the expression of k r !k 2 : r where e ring is the eccentricity of the ellipse formed by the ring.Note that since the ring rotates near the yz plane (i.e. the plane with the two short axes), the short axis of the ring is just b.
The circumference of the ring is then given by the circumference of this ellipse.The formula for the circumference of an ellipse is: q d4 (Equation 27) Using Equations 26 and 27, we can write down the energy as: The torque can then be expressed as: Using the Taylor expansion around a = 0, we can expand t to linear order in a as t = tj a = 0 + dt da j a = 0 a + Oða 2 Þ.Note that on a = 0, the torque also goes to zero.To obtain the linear order expansion, calculate the derivative of the torque at a = 0. dt da Thus, by Taylor expansion, the torque to linear order in a is: 28) Per our assumption of negligible inertial terms and the form of friction experienced by the ring, the torque balance of the ring can be written as: Putting in the expression for t we just obtained, we get 29) To get the posteriorization velocity of the male pronucleus v, we assume that the change in the angular position of the male pronucleus from the long axis is the same as the angle a itself.Effectively, the male pronucleus acts as if it is rigidly attached to the normal vector to the ring (Figure 5D).Then, the position of the male pronucleus is given by: On ellipsoid: The posteriorization velocity of the male pronucleus is then given by (for small angles a, velocity is almost parallel to the cortex, hence we can take the full magnitude) For small angle a, r nucl za À a 3 b 2 a 2 + Oða 3 Þ and _ Therefore, using Equation 29, we get: ! a (Equation 30) where we select the negative sign to ensure correspondence with observed behavior.Note that in the minimal model, in contrast to the mathematical model, the velocity of the male pronucleus is calculated solely as a tight coupling between the contractile ring (pseudocleavage furrow) and the male pronucleus: the male pronucleus is assumed to be rigidly attached perpendicular to the pseudocleavage furrow.Therefore the change in angle between the plane of the contractile ring and the long axis of the ellipsoid is considered to be the same as the change in the angular position of the male pronucleus in the minimal model.This constraint in the minimal model implicitly captures the movement of the male pronucleus as a consequence of the (additional) cytoplasmic and cortical flow modes which are induced in the pseudocleavage furrow-dependent mechanism (see Figure 2A) and link the rotation of the pseudocleavage furrow to the movement of the male pronucleus.

Estimating relation between v and R = a=b
To obtain a relation between the posteriorization velocity v and aspect ratio R = a=b for the combined dataset, we consider the following procedure: To then transform Equation 30 into a relation between v=a and R = a=b, we look at the distribution of semi-major a and semi-minor b axes length in the combined dataset (Figure S6C).We note that the combined dataset has a wide range of a, but b are mostly concentrated around the average value of 15.9 ± 1.3 mm (see red line in Figure S6D bottom).Therefore, we consider b to be fairly constant throughout the combined dataset, allowing us to rewrite Equation 30 as: 33) where C = Àkb 2 is treated as a constant when considering the relation between v=a and R = a=b alone.Therefore, C can be calculated as: C = Àkb 2 = À À 1:05 3 10 À 5 s À 1 deg À 1 mm À 1 Á ð15:9mmÞ 2 = À 2:64 3 10 À 3 mms À 1 deg À 1 (Equation 34) Comparison between experimental results and the prediction from Equation 33 can be observed in Figures 5D and S6E, where the experimental results are black circles with error bars and the prediction from Equation 33 in gray, using the value of C obtained in Equation 34.We also compare the experimentally observed average posteriorization velocities and minimal model prediction over all aspect ratios for a range of angular positions -see Figure S6F.For these angular position ranges, a is set to the midpoint of each angular position range for the corresponding prediction.

Additional experimental details
Reduced cortical flows in nop-1 RNAi embryos NOP-1 is essential for the formation of a pseudocleavage furrow. 45It acts upstream of RHO-1, one of the main regulators of actomyosin in C. elegans.We find that 24 hours RNAi of nop-1 leads to an absence of the pseudocleavage furrow (8 out of 10 embryos), mid) normalized by the area under the frequency histogram for the aspect ratios in unperturbed embryos.Specifically, we calculate the overlap D between aspect ratio distributions in ima-3 (or C27D9.1)RNAi embryos and unperturbed embryos as: minðP cond ðRÞ; P unperturbed ðRÞÞ where the sum is taken over aspect ratio bin centers R, P cond ðRÞ denotes the frequency of embryos observed in that aspect ratio bin in the specified condition (ima-3 RNAi, C27D9.1 RNAi or unperturbed).By definition, D unperturbed = 1 -that is, the unperturbed embryos' aspect ratio distribution overlaps completely with itself.For ima-3 RNAi embryos, we calculate D imaÀ 3 = 0:36 ± 0:07.For C27D9.1 RNAi embryos, we calculate D C27D9:1 = 0:55 ± 0:07.We report here the standard error of the mean for the overlap, calculated using bootstrapping (1000 resamples).Thus, we observe a much larger overlap in the distribution of aspect ratios observed in C27D9.1 RNAi embryos and unperturbed embryos, compared to ima-3 RNAi embryos and unperturbed embryos.

Figure 1 .
Figure 1.AP axis convergence and male pronucleus posteriorization depend on myosin activity (A) AP axis convergence in unperturbed control embryos is visualized by labeling anterior PAR (PAR-6::mCherry, in magenta) and posterior PAR (PAR-2::GFP, in cyan) domain.The male pronucleus (green dashed circle) moves with the posterior PAR domain, aligning the AP axis (cyan to magenta arrow) to the geometric long axis (white arrow).T = 0 s is set at the end of posteriorization.See also Video S1.Scale bars, 10 mm (for A and D).(B) Male pronucleus positions (each point represents the center) during posteriorization in NMY-2::GFP-labeled (Figure S1A) unperturbed embryos, color coded by time relative to end of posteriorization (see color bar).Scale bars, 5 mm.Ellipse has semi-major axis a = 28.4mm and semi-minor axis b = 17.5 mm lengths (for B and E).(C) Angular position of the male pronucleus (on y axis) as a function of time relative to end of posteriorization (on x axis) in NMY-2::GFP-labeled (light blue; Figure S1A) and PAR-2::GFP, PAR-6::mCherry (dark blue) unperturbed embryos.Each line represents an individual embryo.(D) No AP axis convergence is observed in mlc-4 RNAi embryos, noted by no re-orientation of the posterior (cyan) and anterior (magenta) PAR domains.See (A) for details.See also Video S2.See Figure S2 for cortical flow measurements in mlc-4 RNAi embryos.(E) Male pronucleus positions in mlc-4 RNAi embryos labeled with NMY-2::GFP (Figure S1B) or PAR-6::mCherry and PAR-2::GFP, indicating lack of posteriorization in mlc-4 RNAi embryos.See (B) for details.(F) Angular position of the male pronucleus (on y axis) as a function of time relative to end of posteriorization (on x axis) in NMY-2::GFP-labeled (gray; Figure S1B) and PAR-2::GFP, PAR-6::mCherry (black) mlc-4 RNAi embryos.Each line represents an individual embryo.

Figure 2 .Figure 3 .
Figure 2. Numerical simulations capture the experimentally observed cortical and cytoplasmic flows during AP axis establishment in unperturbed embryos (A) Schematic detailing the two proposed mechanisms-cytoplasmic-flow-dependent mechanism (left) and pseudocleavage-furrow-dependent mechanism (right).3D schematic depicts the cortical forces induced by off-axis location of male pronucleus (green filled sphere) in the cortex: color represents magnitude (transparent to red, increasing magnitude); green arrows represent direction.In the 2D schematic, green circle represents the male pronucleus and outline around the embryo represents the cortex (transparent to red, increasing force magnitude), with black arrows representing cortical flow fields (magnitude, length of arrow).Streamlines inside the embryo represent cytoplasmic flows (blue to red, increasing speed).Black line indicates the 2D schematic boundary.Each schematic represents the forces and flows that result from the corresponding mechanism alone-the full description considers both together, combined linearly.(B) Cortical and cytoplasmic flow speeds, experimentally observed (left) compared with numerical simulations (right), for different male pronucleus positions (dashed green circle) in unperturbed embryos.Numerical simulations consider both cytoplasmic-flow-dependent mechanism and pseudocleavage-furrowdependent mechanisms (TableS1; FigureS3).In each panel, ellipse interior represents cytoplasmic flows and outer ellipse represents cortical flows.See color bar for cortical flow velocities.

Figure 4 .
Figure 4. Rounder embryos generated by ima-3 RNAi show slower AP axis convergence, consistent with predictions from mathematical model (A) AP axis convergence observed in rounder ima-3 RNAi embryos via time-lapse microscopy of embryos labeled with NMY-2::GFP (in gray).T = 0 s denotes end of posteriorization of the male pronucleus.Scale bars, 10 mm.See also Figure S1E.(B) Comparing aspect ratio (length/width) between unperturbed and ima-3 RNAi embryos.Histogram depicts the distribution of average aspect ratio of an embryo in unperturbed embryos (blue, N = 57, average aspect ratio = 1.78 ± 0.15) and ima-3 RNAi embryos (orange, N = 35, average aspect ratio = 1.48 ± 0.18) plotted as the frequency of embryos (fraction of embryos, along y axis) observed in a given aspect ratio bin (along x axis, bins of width 0.1).See also Figures S2 (for cortical flow comparison) and S5 and Video S3.(C) Angular position of the male pronucleus (on y axis, in deg), plotted as a function of time relative to end of posteriorization (on x axis, in s) for ima-3 RNAi embryos.Orange thin lines represent the experimentally observed trajectories; each line is an embryo.Orange thick line represents the predicted trajectory of the male pronucleus using the full mathematical model, using the parameters from the unperturbed embryos (TableS1), for ima-3 embryos (aspect ratio: 1.48) (STAR Methods).(D) Posteriorization velocity of the male pronucleus (on y axis, in mm/s) as a function of its angular position (on x axis, in deg) in unperturbed embryos (blue, N = 57, reproduced from Figure3C) and in ima-3 RNAi embryos (orange, N = 35; FiguresS4C and S4G).See inset in Figure3Cfor visual definition.Blue circles and orange diamonds denote the average posteriorization velocity for each angular position bin (bin size: 3 deg), with error bars representing the 95% confidence interval of the average, measured in unperturbed embryos (blue circles) and ima-3 RNAi embryos (orange diamonds), respectively.Curves denote the predicted posteriorization velocities using the full mathematical model, using the parameters from the unperturbed embryos (TableS1)-blue for unperturbed embryos (aspect ratio: 1.78, reproduced from Figure3C), orange for ima-3 embryos (aspect ratio: 1.48) (STAR Methods).

Figure 5 .
Figure 5.A minimal effective model can capture the relation between embryo geometry and rate of AP axis convergence (A) Histogram depicts the distribution of aspect ratio (length/width) in the combined dataset (N = 129) comprising unperturbed embryos, ima-3 RNAi embryos, and C27D9.1 RNAi embryos.The histogram plots the number of embryos (along y axis) in a given aspect ratio bin (along x axis, bins of width 0.1).For (B), the combined dataset is split into three subsets: round (aspect ratio < 1.53, average aspect ratio = 1.37 ± 0.12), mid (1.53 < aspect ratio < 2.01, average aspect ratio = 1.77 ± 0.14), and elliptical (aspect ratio > 2.01, average aspect ratio = 2.14 ± 0.09).Aspect ratio cutoffs are depicted in dashed black line.Cutoffs are selected to be at the 15th and the 85th percentile of the aspect ratio distribution of the combined dataset (see FigureS5Afor overlap between aspect ratios of unperturbed, ima-3 RNAi, and C27D9.1 RNAi embryos).For details on C27D9.1 RNAi embryos, see FiguresS1F, S2, S4D, S4H, and S5.(B) Posteriorization velocity of the male pronucleus (on y axis, in mm/s) as a function of its angular position (on x axis, in deg) for the three aspect ratio subsets of the combined dataset, as depicted in Figure5A.Dark blue upward triangles, green diamonds, and black downward triangles denote the average observed posteriorization velocity in the round (aspect ratio < 1.53) subset, mid (1.53 < aspect ratio < 2.01) subset, and elliptical (aspect ratio > 2.01) subset, respectively.Error bars indicate the 95% confidence interval of the average.Predicted posteriorization velocity for the average aspect ratio in each subset using the minimal model are depicted as solid lines (aspect ratio = 1.37 for round in dark blue, aspect ratio = 1.77 for mid in green, and aspect ratio = 2.14 for elliptical in black).See FiguresS6A and S6Bfor predictions using the mathematical model.See FiguresS6C and S6Dand STAR Methods for parameters used for minimal model.(C) Schematic depicting the effective model of a contractile ring (red ellipse) with fixed line tension (T) moving on an ellipsoid (black thick line, long axis of length 2a along x axis and short axes of length 2b each along y and z axes).The ring is free to rotate around the z axis (red block arrows).Its position is defined by the angle a between the long axis and normal to the plane of the ring and is the same as the angular position of the male pronucleus.v is the posteriorization velocity of the male pronucleus.(D) Ratio of the posteriorization velocity of the male pronucleus with the angular position of the male pronucleus (on y axis, in mm s À1 deg À1 ) as a function of embryo aspect ratios (on x axis).Black circles with error bars denote the observations from the combined dataset, considering angular positions within 3-21 deg.Black circles denote the average posteriorization velocity in each aspect ratio bin (bin size: 0.2).Error bars represent the 95% confidence interval in each bin.For each aspect ratio bin, the ratio of posteriorization velocity with the angular position is averaged over angular positions 3-21 deg of the combined dataset-we exclude the angular position bin 0-3 deg as v=a is not defined for a = 0 deg.Curves denote the predictions from full mathematical model (blue), mathematical model using cytoplasmic-flow-dependent mechanism only (red), and minimal model (gray, v=a = C À R À 1 a form similar to the canonical form for an ellipse.Calling the parametric angle for this ellipse 4, we then can write the position vector describing the ring r !as: r != ðA cos 4; ÀA cot a cos 4; b sin 4Þ; A = ab ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi a 2 cot 2 a+b 2 p where 4˛½ À p; pÞ.

x 2 nucl a 2 + y 2 nucl b 2 = 1 r 2 nucl cos 2 a a 2 + r 2 nucl sin 2 a b 2 = 1
Angle with x-axis (long axis): x nucl = r nucl cos a; y nucl = r nucl sin a 0 0r nucl = ab ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a 2 sin 2 a+b 2 cos 2 a p straight line to the posteriorization velocity versus angular position graph for the unperturbed embryos (FigureS6C) 2 Use the average semi-minor axis length in the combined dataset to transform Equation 30 into a relation between v and R = a= b (FigureS6D).3 Compare result from transformed Equation 30 and result from full mathematical model with experimental results (Figures5C and S6E).To estimate k = h pT g i , we will use the dataset comprised only of unperturbed embryos.As shown in Figure S6C, we plot the average posteriorization velocity observed in these embryos as a function of the angular position of the male pronucleus, and choose a set of angular positions where the posteriorization velocity looks mostly linear with respect to the angular position.We chose the angular position bins: 3-15 deg.From the slope m = À3.37310À 3 mm s À1 deg À1 of the linear fit in the selected angular position range (3-15 deg), we can obtain k using Equation 30 as: To obtain the predictions using the minimal model for the three subsets of the combined dataset (Round, Mid and Elliptical -see Figure S5A) in Figure 5B, we use Equation 30 directly.Specifically, for each subset we calculate the average axes lengths -obtaining a = 21.3 ± 2.8 mm, b = 15.5 ± 1.4 mm for the Round subset (a=b< 1:53), a = 28.4 ± 3.2 mm, b = 16.1 ± 1.3 mm for the Mid subset (1:53 <a=b< 2:01) and a = 33.0 ± 1.8 mm, b = 15.5 ± 0.9 mm for the Elliptical subset (a=b> 2:01).Inputting these in Equation 30, along with k = pT g as obtained in Equation 31, directly yields us a relation between posteriorization velocity v and angular position a -as plotted in Figure 5B.Note that the minimal model predictions in Figure 5B are thus a partial fit -since we obtain k in Figure S6C by fitting a line to the posteriorization velocity vs angular position plot for the unperturbed embryos.

TABLE
and performed all the simulations.The results were analyzed by A.B., M.K., P.G., and M.N.The mathematical model was conceived by M.N., P.G., A.V., and S.W.G., while the minimal model was conceived by A.B., M.L., and S.W.G.The main text was written by A.B., M.N., M.L., A.V., and S.W.G.
d RESOURCE AVAILABILITY B Lead contact B Materials availability B Data and code availability d EXPERIMENTAL MODEL AND STUDY PARTICIPANT DE-TAILS B C. elegans strains B Bacterial strains d METHOD DETAILS B Feeding RNAi B Live imaging d QUANTIFICATION AND STATISTICAL ANALYSIS B Image Analysis B Data analysis B Mathematical model B Parameter calibration for Mathematical model B Numerical methods B Minimal model B Additional experimental details designed 10.5281/zenodo.8276100.Original code for numerical simulations of the mathematical model are also deposited and publicly available at https://doi.org/10.5281/zenodo.8276100.DOIs are also listed in the key resources table.Any additional information required to reanalyze the data reported in this paper is available from the lead contact upon request.