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Epithelial tricellular junctions act as interphase cell shape sensors to orient mitosis

The orientation of cell division along the interphase cell long-axis, the century old Hertwig’s rule, has profound roles in tissue proliferation, morphogenesis, architecture and mechanics1,2. In epithelial tissues, the shape of the interphase cell is influenced by cell adhesion, mechanical stress, neighbour topology, and planar polarity pathways3–12. At mitosis, epithelial cells usually round up to ensure faithful chromosome segregation and to promote morphogenesis1. The mechanisms underlying interphase cell shape sensing in tissues are therefore unknown. We found that in Drosophila epithelia, tricellular junctions (TCJ) localize microtubule force generators, orienting cell division via the Dynein associated protein Mud independently of the classical Pins/Gαi pathway. Moreover, as cells round up during mitosis, TCJs serve as spatial landmarks, encoding information about interphase cell shape anisotropy to orient division in the rounded mitotic cell. Finally, experimental and simulation data show that shape and mechanical strain sensing by the TCJ emerge from a general geometric property of TCJ distributions in epithelial tissues. Thus, in addition to their function as epithelial barrier structures, TCJs serve as polarity cues promoting geometry and mechanical sensing in epithelial tissues.

pupal notum epithelium. Within this tissue, more than ten thousand cells divide13, and, as in many epithelial tissues, the division of rounded mitotic cells takes place in the plane of the tissue and is influenced by their interphase cell shape (Extended Data Fig. 1a,b). One possibility is that Pins (vertebrate LGN) or Gαi polarization orients division as found in single cells in culture or during asymmetric divisions14. However, Pins and Gαi were homogenous around the cortex (Extended Data Fig. 1c,d). In contrast, the distribution of the Dynein associated protein Mud (vertebrate NuMa) suggested a role in orienting the spindle according to the interphase cell shape. GFP:Mudwas localized at the spindle poles and unexpectedly was also enriched at tricellular junctions (TCJs) where at least three cells meet (Fig. 1a, Extended Data Fig. 1e,h and Supplementary Video 1). Accordingly, in this tissue and other pupal or larval epithelial tissue GFP:Mud or endogenous Mud co-localized with Gliotactin (Gli), a septate TCJ marker15 ( Fig. 1b and Extended Data Fig. 1i-o). Furthermore, we established that in G2 phase GFP:Mud localizes at TCJs where it persists through mitosis (Extended Data Fig. 2). The TCJ localization of Mud was independent of Pins and Gαi in both interphase and mitotic cells (Fig. 1c-e and not shown). Accordingly, GFP:Mud lacking the Pins binding domain (GFP:MudΔPINS) localizes at TCJs (Fig. 1c). Whereas Mud loss of function did not affect Gli localization, loss of Gli led to a reduction of GFP:Mud localisation at the TCJs (Fig. 1d,e and Extended Data Fig. 3a). Likewise, loss of function of the Discs-large (Dlg) septate protein, which is necessary for Gli localization15 caused the disappearance of both Gli and GFP:Mud from the TCJs (Fig. 1d,e and Extended Data Fig. 3b-d). Collectively, our results show that independently of the Pins/Gαi pathway, epithelial mitotic cells harbour a cortical TCJ Mud distribution inherited from interphase.
Since astral microtubules (MT) contacted ChFP:Mud patches at TCJ (Extended Data Fig. 4a and Supplementary Video 2), we asked whether TCJs recruit or activate force generators to orient the spindle. Following experiments in C. elegans zygote16, we developed a laser ablation assay to estimate the relative magnitude and the direction of mechanical forces exerted by astral MTs on the centrosome within tissue (Extended Data Fig. 4b and 5). Astral MT ablation in wild-type (wt) cells caused the centrosomes to recoil away from the ablation site, suggesting that MTs predominantly exert pulling forces on spindle poles ( Fig. 2a-b and Supplementary Video 3). The loss of Mud or Dynein minus-end directed motor activity led to a reduction in centrosome recoil upon MT ablation (Fig. 2b). In agreement with the role of Gli and Dlg in promoting TCJ Mud localization, centrosome recoil velocities upon MT ablation were also reduced in gli and dlg mutant cells (Fig. 2b). Together, these results indicate that TCJs control the pulling forces exerted by astral MTs on the spindle via Mud and Dynein activities.
We then investigated whether the Mud distribution at TCJs accounts for the torque exerted by MTs on the spindle to dictate its orientation. To this end, we adapted a mechanical model predicting the spindle orientation according to cell shape17, 18. In this model, developed to describe isolated and non-epithelial cells which do not round up at mitosis, the pulling forces exerted by astral MTs scale with MT length and, as a consequence, the model predicts the preferred spindle orientation along the cell long-axis (Fig. 2c). To account for the contribution of Mud to MT pulling forces in epithelia, we modified the model to assume that astral MTs instead pull with a force proportional to the cortical GFP:Mud intensity and independent of MT length (Fig. 2d). We then measured the metaphase distribution of cortical GFP:Mud and cell shapes to compare the predictions based on cell shape and GFP:Mud intensity models for cells in metaphase (Fig. 2e). Strikingly, the model based on GFP:Mud distribution along the cortex predicted spindle orientation and its predictions were significantly better than the ones based on the metaphase cell shape (Fig. 2f- Table 1). In agreement with the fact that Pins does not regulate Mud localization at TCJs, spindle orientation predictions were similar in wt and pins tissues (Extended Data Fig. 7). To further test the contribution of Mud-dependent MT pulling forces to spindle orientation, we characterized a GFP:Mud mutant deleted of its coiled-coil domain (GFP:MudΔCC, Extended Data Fig. 8). GFP:MudΔCC co-localizes with Gli in wt or mud tissues, but GFP:MudΔCC cannot restore astral pulling forces in mud tissue ( Fig. 2i-k). Whereas the GFP:MudΔCC cortical localization predicted spindle orientation in wt tissue (Extended Data Fig. 8d), planar mitotic spindles were not oriented according to the distribution of GFP:MudΔCC in mud tissue (Fig. 2l). Collectively, these findings indicate that TCJ via Mud define the distribution of MT pulling forces specifying the spindle orientation in epithelial tissue.
Our finding that in metaphase the Mud distribution at TCJs is a better predictor of spindle orientation than is cell shape argues against a model where incomplete cell rounding ensures interphase cell shape "memorization". We therefore hypothesized that the interphase TCJ distribution might explain the Hertwig's rule in tissues. Since our theoretical analysis indicated that spindle orientation is mainly dictated by the anisotropy of the TCJ distribution (Extended Data Fig. 6h,i), we introduced a TCJ bipolarity quantity characterized by an anisotropy (η TCJ ) and orientation (θ TCJ ) to describe the TCJ angular distribution in a given cell ( Fig. 3a and Extended Data Fig. 6j). The TCJ bipolarity anisotropy and orientation can be similar to or distinct from the cell shape anisotropy (or elongation, η shape ) and the longaxis orientation (θ shape (Fig. 3a). We found that the anisotropy of TCJ bipolarity decreases much less than cell shape anisotropy during mitotic cell rounding (Fig. 3b). Also, division orientation predictions based on the TCJ distribution, unlike those based on cell shape, were similar in interphase and mitosis (Fig. 3c-c'). These findings support the notion that TCJ bipolarity is a persistent marker of the interphase cell elongation axis during mitotic rounding. We then measured each cell's average shape (θ shape ) and TCJ bipolarity (θ TCJ ) from 60 to 30 minutes prior to mitosis (from late G2 interphase to prior to mitotic rounding) as well as its division orientation (θ division ). Apart from cases where TCJ and shape orientation are aligned (|θ TCJ − θ shape |<10°), TCJ gives better division orientation predictions than cell shape does, and this improvement increases as the difference between shape and TCJ orientation increases (Fig. 3d-e',g and Extended Data Fig. 9a). This finding applies for both rounded cells (low η shape ) and elongated cells (high η shape ) (Extended Data Fig. 9b-c). This raises the question of why cells tend to divide according to their interphase cell long-axis. The distribution of the angular difference between TCJ and shape orientation is broad in rounded cells (low η shape ) but narrow in elongated cells (high η shape ) (Fig. 3h). Accordingly, cell shape does not predict cell division axis in rounded cells, and as cell shape anisotropy increases, the predictions based on cell shape agree better and better with the predictions based on TCJ bipolarity ( Fig. 3f-g and Extended Data Fig. 9b-c). Hence, in rounded cells TCJ bipolarity and cell shape orientations may be misaligned and division orientation follows TCJ bipolarity, whereas in elongated cells TCJ bipolarity and cell shape orientations are aligned in most cases, and the TCJ distribution ensures that cell division occurs along the former interphase cell shape long-axis. Lastly, cell division orientation along the interphase cell long-axis and TCJ distribution was strongly reduced in mud tissue ( Fig. 3i and Extended Data Fig. 5f-g). Altogether, we propose that TCJ, via Mud, constitute the dominant mechanism of division orientation along the interphase cell long-axis.
Why are the orientations of cell long-axes and TCJ distributions aligned? This can be understood by picturing regular hexagonal cells, which are then pulled. The cell elongation leads to the alignments of cell shape and TCJ bipolarity orientations with the pulling direction (Fig. 4a). Computer simulation can then be used to model the disordered case of epithelial cells whose shapes depend on adhesion and cortical tension19. The simulations reproduce the alignment between cell shape long-axis and TCJ bipolarity orientations as cell shape anisotropy increases as well as the average alignment of the TCJ bipolarity and mechanical strain orientation (Fig. 4b,c Extended data Fig. 10d and Supplementary Video 4). Therefore generic properties of epithelial cells, adhesion and cortical tension, are sufficient to reproduce the alignment of TCJ bipolarity and cell shape as their anisotropy increases. Furthermore, in agreement with the fact that global mechanical stress tends to elongate cells8,12, the alignment of TCJ bipolarity with mechanical stress increases as tissue stress anisotropy increases, thus accounting for orientation of divisions along the global mechanical stress orientation ( Fig. 4d and Extended Data Fig. 10c). Our findings hence indicate that the alignment of TCJ distribution with cell elongation and mechanical stress axis is a core geometric property of epithelial tissues and accounts for a role of TCJ as spatial landmarks providing the information needed for cell shape and mechanical strain orientation sensing.
Altogether our findings provide evidence that TCJ can serve as built-in interphase shape sensors to orient division when the interphase cell shape is well defined. This mechanism is distinct from the ones involving extracellular matrix retraction fibres or signalling17,20-23. It accounts for the integration of two properties of epithelial division: orientation along the interphase cell shape and mitotic rounding (Fig. 4e). The packing of tissue promotes contacts between three (or more) cells and the formation of TCJs. In addition to TCJ proteins implicated in epithelial barrier function24,25, several proteins are enriched at TCJs, including adhesion molecules, cytoskeleton regulators and Hippo pathway component25-29. The alignment of TCJ with cell shape or mechanical strain being a geometrical property of epithelia, TCJ might provide epithelial cells with an axial polarity (or bipolarity) to couple cell shape and tissue mechanics with adhesion, cytoskeleton organization and signalling.

Fly stocks and genetics
Drosophila melanogaster stocks and associated references are listed in Supplementary Table   2. Flies were crossed and experiments were performed at 25°C unless specified otherwise. Male or female pupae were used. Experiments using the temperature-sensitive allele of p150/Glued, D82glued30 were performed as previously described for temperaturecontrolled experiments in the Drosophila pupa13. Loss of function, gain of function and dual techniques31-33. Somatic clones were induced in the second instar larval stage by heatshock (20 min at 37°C for FRT19A and FRT40A, 1 h at 37 °C for FRT42D) and analyzed 3-4 days after clone induction in 12-20 hours after pupa formation (hAPF) pupae. The analyses of dlg m52 loss of function clones were performed in small clones (3 days after induction) to avoid AB polarity defects34.

Molecular biology
To create the GFP:Mud or ChFP:Mud transgenes under the control of Mud endogenous promoter, we used recombineering35,36 to introduce a GFP or ChFP tag at the ATG of the mud open reading frame in the CH322-147E14 BAC genomic clone (BACPAC Resources Center). This BAC clone contains ~19.5kb of X chromosome genomic region including ~3.3kb upstream of the mud mRNA 5' and 5.4kb downstream of the mud 3' mRNA and thus most of the coding regions of genes proximal and distal to mud locus. Fluorescent secondary antibodies were: alexa-488 goat-anti-rabbit IgG (1:500, Molecular Probes), Cy2, Cy3 and Cy5 donkey-anti-mouse IgGs (1:500, Jackson ImmunoResearch). Images were collected with confocal microscopes (LSM710NLO or LSM780, Carl Zeiss). All images are maximum projections of a z-stack unless otherwise indicated.

Live imaging microscopy
Live imaging-Pupae were prepared for live imaging as described previously46. Samples were imaged at 25°C or 29°C with either an inverted confocal spinning disk microscope from Nikon or Zeiss using either 40x NA1.3 OIL DIC H/N2 WD0.2 PL FLUOR, 60x NA1.4 OIL DIC N2 PL APO VC, 63x NA1.4 OIL DICII PL APO or 100x NA1.4 OIL DIC N2 PL APO VC objectives and either a CoolSNAP HQ2 (Photometrics), an EMCCD Evolve (Photometrics) or a CMOS (Hamamtsu) camera. Live images of FUCCI and GFP:Mud were acquired using a confocal microscope (LSM780, Carl Zeiss) and 63x NA 1.4 OIL DICII PL APO objective. To improve signal-to-noise ratio, videos and images for display and segmentation were either deconvolved using Huygens software (Scientific Volume Imaging b.v., NL) or denoised using Safir software47. Average projections of raw z-stack images were used for intensity measurements.
Unless specified otherwise, all experiments on dividing cells were performed during the first round of cell divisions in the notum tissue (12-20 hAPF). In the analyses (unless mentioned otherwise), the time (t) equals 0 was set at anaphase onset defined by the initial cell elongation and/or centrosome movements towards the cortex. GFP:Mud kymograph from interphase to mitosis-To generate the kymograph of the GFP:Mud signal around the cell contour from interphase to anaphase, the cell contour was manually segmented using the GFP:Mud signal. The GFP:Mud intensity of the fluorescence signal was recorded as a function of the angle θ of each contour pixel around the center of mass and then plotted as a line for each time point.

Laser ablations of astral MTs and estimation of cortical forces
Laser ablation of astral MTs were performed in cells labelled with Jupiter:GFP or α-Tub:GFP (MT markers) and Sas-4:RFP, Spd-2:RFP or YFP:Asl (centrosome markers) as well as Dlg:GFP or Nrg:GFP (septate junction markers). Images were captured using a twophoton laser-scanning microscope (LSM710 NLO, Carl Zeiss) equipped with a 63x NA 1.4 OIL DICII PL APO objective (digital zoom 3X) in single-photon bidirectional scan mode lasting δt=756 ms. The astral MTs of mitotic spindles, which were parallel to the plane of the epithelial tissue were severed (t 3 ) using the Ti:Sapphire laser (Mai Tai DeepSee, Spectra Physics) at 890 nm with < 100 fs pulses with a 80 MHz repetition rate typically set at 60% power.
To measure the recoil velocity (amplitude and orientation) of the centrosomes upon astral MT ablation, all spindles were registered horizontally with the ablated astral MTs at the top right and the centrosome positioned at the origin. Centrosome movements were then manually tracked. The velocity (amplitude and orientation) was measured between t 2 and t 20 .

Image quantifications and GFP:Mud and GFP:MudΔCC localization analyses
To measure and compare the TCJ accumulation of GFP:Mud in wt, pins, Gli and dlg cells (Fig. 1e), confocal z-stack average projections at the level of the septate junction (labelled by FasIII, Cora, PH:ChFP or mRFP) were generated using ImageJ from fixed (pins and dlg) or live (wt and Gli) tissues harbouring wt, Gli, dlg and pins mutant cells. Using FasIII, Cora, PH:ChFP or mRFP labelling, the positions of the TCJs in each cell were manually determined. The FasIII, Cora, PH:ChFP or mRFP labelling were used to draw a mask (5 pixels wide) delineating the cell outline at the level of the mitotic spindle (as determined by GFP:Mud localization at the spindle poles) of (pro)metaphase cells. Following background subtraction, the mean intensity at TCJs (10° over each TCJ) was divided by the mean intensity along the rest of the cell outline to obtain the accumulation at TCJs. Significance was determined using the Student's t-test.
To compare the GFP:Mud or GFP:MudΔCC distributions at TCJ in wt, pins and mud tissues  Fig 7b), their co-localization with Gli were quantitatively compared as follows. Confocal z-stack average projections at the level of the septate junction of fixed (pro)metaphase cells (as determined by DAPI staining) expressing either GFP:Mud or GFP:MudΔCC and labelled with Gli and Coracle (Cora, a septate junction marker) were generated using ImageJ. Using the Cora staining, a mask of 5-pixels wide was manually drawn to measure the raw fluorescent intensity profiles of GFP:Mud or GFP:MudΔCC and Gli in (pro)metaphase cells (as determined by DAPI staining) which were normalised by their total fluorescence intensity upon subtraction of the background intensity. The colocalization factor (C, vertical axis Fig. 2j and Extended Data Fig. 7b) between GFP:Mud or GFP:MudΔCC and Gli was then determined by calculating the area between the GFP:Mud or GFP:MudΔCC and Gli normalized intensity curves. C equals 1 if the two proteins perfectly co-localize and equals 0 if the two proteins do not colocalize. The comparisons of the distribution of GFP:Mud, GFP:MudΔCC in wt and mud tissues were performed blind (Fig 2j). Significance was determined using the Student's ttest.

Quantitative analyses of epithelial cell divisions orientation, cell shape and TCJ bipolarity
Measurement of apical-basal (AB) angle of the spindle α AB -The AB orientations of the mitotic spindle (α AB ) in the different experimental conditions were determined by measuring the orientation of the centrosomes (marked by Spd-2:RFP, Sas-4:RFP or Sas-4:GFP) relative to the plane of the epithelial tissue (labelled by α-Tub:GFP, Jupiter:GFP or RFP:α-Tub) using a custom ImageJ plugin. Statistical significance was assessed using the Kolmogorov-Smirnov test.
Analyses of cell rounding and mitotic spindle prediction from interphase cell shape and TCJ bipolarity-The analyses of cell rounding from t=-60 min to cytokinesis was performed using Dlg:GFP. The mitotic spindle being positioned at the level of the septate junction (Extended Data Fig. 1g- To compare the prediction based on cell shape long-axis versus TCJ bipolarity in wt (6 videos, n=4860 cells) and mud (8 videos, n=7770 cells) mutant tissue predictions were performed in a specific region of the tissue were spindle misorientation along the AB is very weak (Extended Data Fig. 5f-g).
Upon Dlg:GFP signal or E-Cad:GFP signal segmentation and cell tracking, the following measurements were determined using Matlab: (i) The experimental cell division orientation (θ division ) was determined as the orthogonal of the interface between the two daughter cells upon cytokinesis. θ division , which correlates very well with cell division orientation measured by the positions of the two centrosomes at metaphase (R=0.91, n=127 cells, data not shown) as established using time-lapse videos of E:Cad:GFP and Spd-2:mRFP (centrosome marker) labelled epithelial tissue.
(ii) To characterize cell shape elongation (η shape ) and cell shape long-axis orientation (θ shape ), each cell region was used to construct its inertia matrix.
where n pix is the number of pixels in the cell and are the vectors pointing from the barycenter of the cell to each pixel of the cell, p. Its eigenvalues λ S , S , with 0 < λ S < S , have the dimensions of a squared distance. Its eigenvector associated to S defines the direction of the cell's long-axis (θ shape ). The cell shape anisotropy was defined as a dimensionless number, which ranged from 0 for a cell perfectly circular, to 1 for an infinitely stretched cell: η shape = 1 − λ S / S .
(iii) To characterize the anisotropy (η TCJ ) and orientation of the TCJ (θ TCJ ) angular distribution, we built the "TCJ bipolarity" matrix V: where n TCJ is the number of TCJs in the cell and the are the unit vectors pointing from the barycenter of the cell to each cell TCJ, (Extended Data Fig. 6j). Its eigenvalues λ TCJ , TCJ , with 0 < λ TCJ < TCJ , are dimensionless numbers. Its eigenvector associated to defines the direction of the long-axis of the TCJ bipolarity (θ TCJ ). The TCJ distribution anisotropy was defined as a dimensionless number, ranging from 0 for TCJ uniformly distributed around the cell, to 1 for the theoretical case of TCJ split in two groups diametrically opposed: η TCJ = 1 − λ TCJ / TCJ .
Note that unlike the cell inertia S that is calculated using all the pixels making up the cell, the TCJ bipolarity V solely uses the unit vectors pointing from the cell center to each cell TCJ. By doing so, the TCJ bipolarity disentangle the characterization of TCJ distribution from cell shape measurement, and any correlation observed between the two quantities is not due a shape bias in the TCJ bipolarity measurement. In the example shown in Extended Data Fig. 6j, although the two cells have different shape anisotropies, they share the same set of vectors and have therefore the same TCJ bipolarity.
Both cell shape anisotropy and TCJ anisotropy were normalized to their respective averages over all the cells in the tissue. The cell shape and TCJ distribution anisotropies are represented with bars whose directions give the direction of their respective anisotropies and whose length is proportional to the magnitude of the normalized anisotropy.
To compare the orientations of the cell shape long-axis or the TCJ bipolarity axis with the cell division orientation, the cell shape and TCJ distribution tensors S and V were averaged during late interphase from 60 to 30 minutes prior to the end of cytokinesis. The orientations of the resulting averaged tensors were then compared to the experimental cell division orientation, θ divison .
The improvement of spindle orientation prediction is calculated as N S is the number of cells for which and N TCJ is the number of cells for which The mean improvement and its standard deviation were calculated using the improvement values from three different videos. Similar improvement values are found when determining N S and N TCJ for below 5°, 10°, 20° or 25°.
The correlation coefficient A between TCJ bipolarity and cell long-axis orientation is calculated as where N is the total number of cells analyzed. For each cell, θ i equals θ TCJ − θ shape and η i is the cell shape anisotropy. The correlation coefficient A ranged from -1 for complete anticorrelation to 1 for complete correlation. In between, a homogeneous distribution indicating an absence of correlation led to A = 0. The correlation coefficient was calculated over all 4504 simulated cells or over an equal number of experimental cells randomly picked (n=4505) among the 29388 cells analyzed in Fig. 3e-g and Extended Data Fig. 9.
As cell division is symmetric in size in the Drosophila notum we have focused on the anisotropy and the orientation of the distribution of the TCJ (bipolarity). The analysis of the asymmetry of TCJ distribution in epithelial tissue where epithelial cells undergo unequal size distribution might provide insights on how unequal daughter cells are generated in epithelial tissue.

Numerical simulations
We used numerical simulations based on the cellular Potts model, which is particularly relevant in biology to describe variable cell shape, size, packing and irregular fluctuating interfaces of cells48-50. We consider a 2D square lattice. Each pixel i has an integer index i The m-th cell is defined as the domain consisting of all pixels with the same index value i = m. The number of pixels that cell has defines its cell area. A cell shape changes when one of its pixels is attributed to another cell. Here, the evolution is driven by the minimization of a total energy E, which has three physical ingredients: interfacial energy, area constraints and an external force applied to the patch of cells. Since the calculations are performed on a lattice, we have The first term represents the contribution of the energy of the interfaces between the cells. Minimizing this term leads to perimeter minimization ( is the Kronecker symbol and Λ is interfacial energy). The second term keeps each cell area A m close to its predefined target value A 0 ( is the compressibility). The balance between this term and the preceding one simulates a tissue relaxing towards mechanical equilibrium. The third term describes an energy gradient51,52, i.e. an elastic force field, which pulls on the tissue in opposite directions (k is an elastic constant and is the x-position of the center of the simulated field).
The algorithm to minimize E uses Monte Carlo sampling and the Metropolis algorithm, as follows. We randomly draw (without replacement) a lattice pixel and one of its eight neighboring pixels. If both pixels belong to different cells, we try to copy the state of the neighboring pixel to the first one. If the copying decreases E, we accept it, and if it increases E, we accept it with probability P = e −∆E/T , where ∆E = E after − E before . The prefactor T is a fluctuation (random copying) allowance. Because all energy parameters are scalable with T, we can fix it without loss of generality; for numerical convenience, we choose numbers on the order of 10 3 . We define one Monte Carlo time step (MCS) as the number of random drawings equal to the number of lattice pixels. We ran simulations during 600 MCS to reach a tissue shape that no longer evolves (initial image). We then applied a constant bulk force that stretched a tissue and run the simulations for 600 MCS (final image). To obtain a large range of cell anisotropies in the simulations we used simulations with an elastic constant, k=[0;0.5;1;2;3;4;5;10]. For each value of k, 5 simulations (with about 170 cells in each simulation) were run. Using the initial and final images, the tissue elongation along the direction of stretching was calculated for each simulation as the relative increase in distances between landmarks. On the final image, the segmented cell contours were used to determine the tensors V (TCJ bipolarity) and S (inertia matrix) for each cell as well as their averages over all cells in the simulation.

Mechanical stress estimation and TCJ bipolarity orientation
To compare the mechanical stress and TCJ bipolarity orientations as a function of the normal mechanical stress difference ( yy − xx ), we used yy − xx experimental values of the estimated mechanical stress obtained up to a prefactor by Bonnet et al.53 measured from 12hAPF to 28hAPF in the medial region of the scutellum, where the mechanical stress is oriented along the medial-lateral axis ( xy = 0 and θ stress = 0). The experimental orientation of mechanical stress for each ablation was compared to the average TCJ bipolarity orientation determined using the segmented cell outlines of the E-Cad:GFP cells within the rim of ablated cells (Extended Data Fig. 10a,b).

Models for predicting spindle orientation
Predictions based on experimental Mud distribution and cell shape-For predictions of the mitotic spindle orientation based on GFP:Mud or GFP:MudΔCC signals (thereafter referred to as Mud fluorescence signal) in wt and mud tissues, average intensity (2-4 μm) projections centred around the plane of the centrosomes were generated using a custom ImageJ plugin for the t=-2 min and t=-1 min frames (t=0 corresponding to the anaphase onset). A 5 pixel mask that does not overlap with the GFP:Mud or GFP:MudΔCC centrosome signal was drawn around the cortex to determine using a Matlab script the shape of the dividing cell as well as the cortical Mud signal profiles. The experimental orientation of the mitotic spindle and the positions of the two centrosomes were manually determined using the GFP:Mud or GFP:MudΔCC accumulation at the spindle poles (Fig. 2e,i).
The model predicting spindle orientation from cell shape is similar to the one used in8,18. The model based on Mud fluorescence signal is adapted from this previous model and inputs the distribution of cortical intensity of GFP:Mud or GFP:MudΔCC obtained from a fluorescence image in a mitotic cell17,18. This intensity computed around the cell contour is renormalized, so that the sum of intensities around each treated cell is the same. For each cell, we aimed to compute the global torque T generated as a function of the spindle orientation angle θ (Fig. 2d,g). For each possible spindle orientation θ, (θ varying from 0 to π) we generate two asters of N MTs nucleated at a constant angular density ρ from centrosomes placed at a distance ±L/2 from the spindle center along the axis θ. Both L, which represents the spindle length, and the spindle center are computed from the experimental position of the two centrosomes (Fig. 2e). An MT projecting at an angle ϕ with respect to the spindle contacts the cortex at a given location with a GFP:Mud or GFP:MudΔCC concentration C mud ; and is assumed to pull on the spindle pole it is connected to with a force f[] that scales with C mud : f[C mud (θ,ϕ)]~(C mud ) α with the exponent α representing putative non-linearity in how Mud may influence astral MT pulling forces. This yields a torque, τ(θ,ϕ), at the spindle centre projected along the z-axis: (Eq. 1) The resultant total torque T(θ) generated by the two asters is then obtained by summing the projected torques over all MTs: where Φ is the total angular width of the aster. Initial tests of the model showed that, above a certain threshold, the number of MTs N (or equivalently the angular density: ρ = N/Φ), does not significantly impact axis definition (Extended Data Fig. 6c-f). Thus, in the model, we keep N as a silent parameter, by normalizing the total torque with N. The stable theoretical axis orientation θ theory , can be identified from the minima of the potential U(θ) computed as a primitive of T(θ), and compared with the experimental division axis θ division (Fig. 2f-h and Extended Data Fig. 6g).
The quality of the prediction was computed based on the magnitude of the angular deviation between the model and the experiments, |θ theory − θ division |. Overall, the model based on the Mud distribution accounts for observed spindle orientation, with a mean angular deviation Bosveld 15.6°. Although we cannot fully preclude the existence of Mud-independent secondary systems that contribute to spindle orientation, it is important to outline the structural differences in models and biological systems that could explain these differences. One first difference is that the Mud model infers a fluorescence signal distribution which could be in part affected by the imaging itself, yielding variations in Mud signal peak heights or widths which do not reflect the actual force field. In agreement with this, the same model run with cells where the neighbours are not photobleached makes predictions within 32.4°(n=241 cells, data not shown). Another probably more important difference is that in Drosophila epithelial cells the spindles jiggle and move with a timescale close to mitosis duration (data not shown). In contrast, in large cells like zygotes and blastomeres, division axes are stably set for tens of minutes with negligible movement and rotation of the spindles18,54. Although the lack of a standardized Mud distribution precludes us from computing the effective temperature of the system, the difference in spindle movements suggests that the noise in the Drosophila epithelial system studied is much larger than in these other cell types previously analysed.
Predictions at the tissue scale based on TCJ anisotropy-In order to study spindle orientation across the entire tissue, we used the bipolarity axis of the TCJ distribution in each cell as a proxy for a prediction based on the full Mud protein distribution (see Fig. 3 and the discussion of the tensor earlier in this Supplementary Information). This quantity has the advantages that it relies only on a marker (E-Cad:GFP or Dlg:GFP) that can be reliably imaged over the necessary length and time scales and that the predicted spindle orientations can be computed in a reasonable time, even for tens of thousands of cells. In this section, we show how the bipolarity axis arises naturally as an approximation to a more detailed description that explicitly calculates forces and torques.
We begin by examining in general terms how a cortical force distribution translates into a potential U(θ) governing the spindle orientation. In the context of this analytic formulation, we assume that the rounded, mitotic cells are approximately circles with radius R and center

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Here, we have suppressed the explicit dependence of the force f on C mud , and the kernel ũ( ) is a 2π-periodic function given, for −π < ≤ π, by where Ψ is related to Φ as cot(Φ/2) = cot(Ψ) − /sin(Ψ).
Importantly, ũ( ) is hence a symmetric function whose magnitude peaks at = 0. That is, within our model the net effect of the MTs is to act as a linear filter that smooths out the cortical Mud distribution without otherwise altering it. We therefore expect that the spindle should generally prefer to orient itself towards the direction of highest Mud concentration but that it is more sensitive to relatively wide, broad peaks of Mud than to the fine details of a rapidly varying Mud distribution. This idea can be expressed more formally by Fourier transforming the periodic functions U, f, and ũ. With U(θ) = ∑ n U n exp(inθ), and similarly for f and ũ, we have U n = ρLf n ũ n for n even and U n = 0 for n odd. The magnitudes of the coefficients ũ n are plotted in Extended Data Fig. 6i. As expected, the filter decreases the importance of components with higher n relative to the dominant n = 2 mode. Thus, the orientation of the n = 2 mode of the force distribution f( ) gives the natural, leading approximation to the location of the minimum of U(θ). For forces centered on the TCJs (where the Mud concentration is highest), the n = 2 mode of the TCJ distribution similarly gives the natural proxy for f 2 and thus for the orientation of the mitotic spindle. Moreover, the orientation of this bipolar mode can be computed far more efficiently than can the location of the global minimum of a potential including all Fourier modes. Indeed, it is a standard result that this orientation corresponds with the anisotropy axis of the second rank tensor V defined previously. This makes it especially appropriate for use in our tissue-scale calculations.

Statistics
Sample sizes vary in each experiment. Every experiment was done at least three independent times. Statistical significances of protein distribution and velocity amplitude were assessed using Student t-tests: the distribution normalities were checked using Kolmogorov-Smirnov test. In cases where the variances were different, significance was assessed using the unequal variance t-test. The angular distribution of velocity was assessed using Watson's U 2 test for circular data. GFP:Mud localization at TCJ and GFP:Mud or GFP:MudΔCC co-localization with Gli in the different mutant backgrounds were analysed blindly. Kolmogorov-Smirnov tests were used to analyse differences in α AB spindle orientations and differences in division orientation. P values superior to 0.05 are indicated as not significant (n.s.) in figure legends or graphs. Predictions of division orientation based on GFP:Mud or GFP:MudΔCC in mud or wt tissues were performed blindly. Experiments were not randomized and every experiment was repeated at least three independent times.

Code availability
Matlab code used to segment and track cells has been previously published13. Matlab code used to determine division orientation, cell shape and TCJ bipolarity upon cell segmentation and tracking are available upon request.  Europe PMC Funders Author Manuscripts in Fig. 3b.). Prior to mitosis (-30 min) the cell (marked by asterisk) is clearly elongated and divides according to its interphasic cell shape (5 min tissue (a-e). Since a large proportion of the spindles remain within the plane of the tissue, all analyses reported in the manuscript were performed on cell divisions that occurred within the plane of the tissue. Furthermore, in a central region of the tissue (box in f and f'), 88% of the divisions in mud tissue occur with an α AB in the range of the wt tissue (g). This region was analysed to compare TCJ bipolarity and cell shape based predictions of division orientation in wt and mud tissue (Fig. 3h-h'). (c-f) Cumulative plots of the differences between the theoretical spindle orientation (θ theory ) and the experimental spindle orientation (θ division ) angles in GFP:Mud expressing cells (same cells as in Fig. 2h) for different spindle lengths (c), MT number (d), angular extension of astral MTs (e) and different scaling factor between the GFP:Mud intensity and mechanical pulling force (f). The GFP:Mud model predictions are mostly independent of spindle length, the number of astral MTs, the angle covered by the astral MTs or the scaling factor between GFP:Mud intensity and MT pulling force. The magnitudes |ũ n | drop off substantially with increasing n, indicating that for many purposes it should be sufficient to approximate the function U by its lowest, n = 2 mode. To calculate numerical values for the Fourier coefficients, we took the average of the normalized spindle length ϵ = L/2R for the 140 cells (n) analysed in this paper, obtaining ϵ ≈ 0.76 ± 0.03; because it is difficult to precisely estimate Φ from the available data, coefficients are shown for Φ = 180° and 270° in agreement with the astral MT distribution observed in a. (j) Schematic illustrating the difference between cell shape and cell TCJ bipolarity measurements. An elongated cell and a rounded cell are overlaid (left panels) and shown side-by-side (middle and right panels). In this example, although the two cells have distinct shapes, they have the same TCJ bipolarity. The upper panels illustrate the measurement of cell shape, which uses all the pixels making up the cell (blue bars). The lower panels illustrate the measurement of TCJ bipolarity (red bars), which is solely based on the angular distribution of the TCJs (red dots), only using the unit vectors pointing from the cell center (black dot) to each cell TCJ. The TCJ bipolarity therefore characterizes TCJ distribution independently of cell shape, and a correlation observed between the two quantities is not due to a shape bias in the TCJ bipolarity measurement.   (a-a") Rose plots of the magnitude of the difference between experimental (θ division ) and predicted division orientations by the average (60-30 min prior to mitosis) interphase TCJ bipolarity (θ TCJ ) or cell long-axis (θ shape ) in cells for the indicated |θ TCJ -θ shape | intervals.
To facilitate the comparison between the left and the right rose plots, the data are duplicated relative to 0° line (light blue and light red). Kolmogorov-Smirnov test (p values), percentage of total cells (n=29388). Panels b and b" are identical to panels e and e' in Fig. 3. (b-b") Rose plots of the magnitude of the difference between experimental (θ division ) and predicted division orientations by the average (60-30min prior to mitosis) interphase TCJ bipolarity (θ TCJ ) or cell long-axis (θ shape ) for the indicated η shape intervals. To facilitate the comparison between the left and the right rose plots, the data are duplicated relative to 0° line (light blue and light red). Kolmogorov-Smirnov test (p values), percentage of total cells (n=29388). Panels c and c" are identical to panels f and f' in Fig. 3. (c) Plot of the spindle orientation prediction improvements (color-coded from dark blue to red) based on TCJ bipolarity over those based on cell shape versus the magnitude of their angular difference (|θ TCJ -θ shape |) and the cell shape anisotropy (η shape ). The plot height is the normalized cell number in each domain of the plot (29883 cell were analysed in total). As |θ TCJ -θ shape | increases, the TCJ bipolarity predictions improve over cell shape prediction for both rounded (low η shape ) and elongated cells (high η shape ). Whereas the rounded cells are characterized by an even distribution along the |θ TCJ -θ shape | axis, the elongated cells are mainly characterized by a strongly skewed distribution towards low |θ TCJ θ shape |.
Extended Data Figure 10. TCJ bipolarity aligns with mechanical stress.
(a,b) Images of the scutellum tissue before and after ablation (ablated region in yellow) in early and late pupa characterized by small isotropic stress (a) and high anisotropic stress (b). Tissue stress was estimated by determining the initial recoiled velocity upon circular ablation in the x and y directions53. First and last images of two time-lapse movies out of the 18 quantified in c are shown. Scale bars: 10μm (a, b). (c) Plot of the difference between the orientation of TCJ bipolarity (θ TCJ ) and principal strain axis (θ strain ) as a function of normal stress differences (σ yy -σ xx , note that σ xy = 0) as estimated up to a prefactor by circular laser ablation. Number of ablations (n) analysed is indicated. The same plot is shown in Fig. 4D. (d) Plot of the difference between the orientation of TCJ bipolarity (θ TCJ ) and the orientation of strain (θ strain ) as a function of the percentage of cell elongation applied to a simulated cell lattice. When cell elongation increases TCJ bipolarity orientation becomes aligned with the direction of cell elongation. Number of simulations (n) analysed is indicated.

Supplementary Material
Refer to Web version on PubMed Central for supplementary material.  (c,d) Cell shape (c) and Mud intensity (d) models: pulling forces scale with MT length (blue arrows) or Mud cortical intensity (red arrows) to exert a torque (T, arrows). (e-g) Experimental spindle orientation (green cross) and predictions based on cell shape (blue circles, f) or GFP:Mud intensity (red circles, g) potentials at t=-1min for cell in e (n=121 cells).  (e-f') Difference between experimental (θ division ) and predicted division orientations based on interphase TCJ bipolarity (θ TCJ ) or cell long-axis (θ shape ) for |θ TCJ -θ shape | intervals (ee') and indicated η shape intervals (f-f'  (a-b) Regular hexagonal cells (a) and simulated cell lattice (b) before and after elongation. (c) Differences (mean±s.d.) between θ shape and θ TCJ versus η shape for experimental (Fig. 3g) and simulated cells. A stimulation = 0.94. n: cell numbers. (d) Difference between θ TCJ and principal strain axis (θ strain ) versus normal stress differences. n: ablation number.
(e) Upon mitotic rounding, Mud interphase localization is maintained at TCJs orienting the spindle along the interphase cell long-axis.