Emeline Daniel

Marion Daudé

Irina Kolotuev

Kristi Charish

Vanessa Auld

Roland Le Borgne
email: roland.leborgne@univ-rennes1.fr

Coordination of Septate Junctions Assembly and Completion of Cytokinesis in Proliferative Epithelial Tissues

Keywords: Epithelial Cells, Cytokinesis, Paracellular Diffusion Barrier, Septate Junctions, Tricellular Junctions, Drosophila, Live Imaging, Scanning Face Block Electron Microscopy and Transmission Electron Microscopy

How permeability barrier function is maintained when epithelial cells divide is largely unknown.

Here, we have investigated how the bi (BSJ) and tricellular (TSJ) septate junctions are remodeled throughout completion of cytokinesis in Drosophila epithelia. We report that, following cytokinetic ring constriction, the midbody assembles, matures within SJ and is displaced basally in two phases. In a first slow phase, the neighboring cells remain connected to the dividing cells by means of SJ-containing membrane protrusions pointing to the maturing midbody.

Fluorescence recovery after photobleaching (FRAP) experiments revealed that SJ present within the membrane protrusions correspond to the old SJ present prior to cytokinesis. In contrast, new SJ are assembled below the adherens junctions and spread basally to build a new belt of SJ, in a manner analogous to a conveyor belt. Loss-of-function of a core BSJ component, the Na/K-ATPase pump Nervana2 subunit, revealed that the apical-to-basal spread of BSJ drives the basal displacement of the midbody. In contrast, loss of the TSJ protein Bark beetle indicated that remodeling of TSJ is rate limiting, and slowed down midbody migration. In the second phase, once the belt of SJ is assembled, the basal displacement of the midbody is accelerated and ultimately leads to abscission. This last step is temporally uncoupled from the remodeling of SJ.

We propose that cytokinesis in epithelia involves the coordinated polarized assembly and remodeling of SJ both in the dividing cell and its neighbors to ensure the maintenance of permeability barrier integrity in proliferative epithelia.

INTRODUCTION

Epithelia serve as mechanical and chemical barriers essential for the function of all organs.

Adherens junctions (AJ) form the mechanical barrier [START_REF] Harris | Adherens junctions: from molecules to morphogenesis[END_REF], while tight (TJ)/septate (SJ) junctions create the permeability barrier, in vertebrates and invertebrates respectively [START_REF] Tepass | Epithelial cell polarity and cell junctions in Drosophila[END_REF][START_REF] Shin | Tight junctions and cell polarity[END_REF][START_REF] Banerjee | Organization and function of septate junctions: an evolutionary perspective[END_REF][START_REF] Tsukita | Multifunctional strands in tight junctions[END_REF]. Throughout life, epithelia are continuously growing, undergoing morphogenesis or regenerating in part due to cell division. Common to each division is the generation of a new cell-membrane interface and two cell vertices. These cell shape changes imply junctional plasticity with major disassembly and de novo assembly of junctions. However, AJ and TJ/SJ are composed of protein complexes exhibiting different rates of assembly/disassembly ranging from time scales of minutes for AJ, to tens of minutes or hours for TJ/SJ [START_REF] Oshima | Analysis of protein dynamics within the septate junction reveals a highly stable core protein complex that does not include the basolateral polarity protein Discs large[END_REF][START_REF] Higashi | Maintenance of the Epithelial Barrier and Remodeling of Cell-Cell Junctions during Cytokinesis[END_REF][START_REF] Levayer | Spatial regulation of Dia and Myosin-II by RhoGEF2 controls initiation of E-cadherin endocytosis during epithelial morphogenesis[END_REF][START_REF] Shen | The tight junction protein complex undergoes rapid and continuous molecular remodeling at steady state[END_REF][START_REF] Yamada | Deconstructing the cadherin-catenin-actin complex[END_REF][START_REF] Huang | Differential regulation of adherens junction dynamics during apical-basal polarization[END_REF]. Previous work revealed that formation of the new adhesive interface is coordinated with cytokinesis and the strength of adhesion with neighbors impacts both the kinetics of AJ remodeling and the geometry of the new AJ [START_REF] Higashi | Maintenance of the Epithelial Barrier and Remodeling of Cell-Cell Junctions during Cytokinesis[END_REF][START_REF] Herszterg | Interplay between the dividing cell and its neighbors regulates adherens junction formation during cytokinesis in epithelial tissue[END_REF][START_REF] Founounou | Septins regulate the contractility of the actomyosin ring to enable adherens junction remodeling during cytokinesis of epithelial cells[END_REF][START_REF] Guillot | Adhesion disengagement uncouples intrinsic and extrinsic forces to drive cytokinesis in epithelial tissues[END_REF][START_REF] Morais-De-Sa | Adherens junctions determine the apical position of the midbody during follicular epithelial cell division[END_REF][START_REF] Lau | Anisotropic stress orients remodelling of mammalian limb bud ectoderm[END_REF][START_REF] Firmino | Cell Division Drives Epithelial Cell Rearrangements during Gastrulation in Chick[END_REF]. AJ determine the positioning of the midbody immediately basal to the forming AJ in invertebrates [START_REF] Herszterg | Interplay between the dividing cell and its neighbors regulates adherens junction formation during cytokinesis in epithelial tissue[END_REF][START_REF] Founounou | Septins regulate the contractility of the actomyosin ring to enable adherens junction remodeling during cytokinesis of epithelial cells[END_REF][START_REF] Guillot | Adhesion disengagement uncouples intrinsic and extrinsic forces to drive cytokinesis in epithelial tissues[END_REF][START_REF] Morais-De-Sa | Adherens junctions determine the apical position of the midbody during follicular epithelial cell division[END_REF], or apical to AJ in vertebrates [START_REF] Higashi | Maintenance of the Epithelial Barrier and Remodeling of Cell-Cell Junctions during Cytokinesis[END_REF][START_REF] Jinguji | Electron microscopic observations on the maintenance of the tight junction during cell division in the epithelium of the mouse small intestine[END_REF][START_REF] Dubreuil | Midbody and primary cilium of neural progenitors release extracellular membrane particles enriched in the stem cell marker prominin-1[END_REF]. Similar to the mechanical barrier, permeability barrier must be preserved throughout cytokinesis. How permeability barrier maintenance is coordinated with cytokinetic abscission during epithelial division remains largely unexplored.

Cytokinesis starts at the onset of anaphase when the spindle drives the assembly of an actomyosin ring that constricts and draws in the cell membrane [START_REF] Green | Cytokinesis in Animal Cells[END_REF][START_REF] Glotzer | The molecular requirements for cytokinesis[END_REF]. After constriction the spindle and the actomyosin ring form the midbody and the midbody ring, which eventually directs the physical separation of daughter cells, i.e. the abscission [START_REF] Fededa | Molecular control of animal cell cytokinesis[END_REF][START_REF] Agromayor | Knowing when to cut and run: mechanisms that control cytokinetic abscission[END_REF]. The midbody ring retains components of the actomyosin contractile ring including actin and septin, non-muscle Myosin II, and Anillin [START_REF] Kechad | Anillin acts as a bifunctional linker coordinating midbody ring biogenesis during cytokinesis[END_REF]. The midbody consists in a densely packed array of antiparallel microtubules containing among others the chromosomal passenger complex (CPC), and eventually recruits the endosomal sorting complex (ESCRT-III) to allow for membrane remodeling during membrane constriction and fission [START_REF] Agromayor | Knowing when to cut and run: mechanisms that control cytokinetic abscission[END_REF][START_REF] Guizetti | Cortical constriction during abscission involves helices of ESCRT-IIIdependent filaments[END_REF][START_REF] Eikenes | ALIX and ESCRT-III coordinately control cytokinetic abscission during germline stem cell division in vivo[END_REF][START_REF] Matias | Abscission is regulated by the ESCRT-III protein shrub in Drosophila germline stem cells[END_REF][START_REF] Mierzwa | Dynamic subunit turnover in ESCRT-III assemblies is regulated by Vps4 to mediate membrane remodelling during cytokinesis[END_REF]. Whether these fundamental mechanisms extensively described in isolated cells [START_REF] Green | Cytokinesis in Animal Cells[END_REF][START_REF] Glotzer | The molecular requirements for cytokinesis[END_REF] apply to epithelia remains unknown.

In addition, because epithelial cytokinesis is a multicellular process, whether junctional complexes exert a role during midbody maturation is also unknown. TJ/SJ seal the intercellular space between adjacent cells forming a physical barrier restricting paracellular flux and preventing pathogens from crossing epithelia [START_REF] Shin | Tight junctions and cell polarity[END_REF]. In Drosophila, SJ are characterized by septae, ladder-like rungs that span between adjacent cells [START_REF] Tepass | Epithelial cell polarity and cell junctions in Drosophila[END_REF]. The bicellular septate junctions (BSJ) are made of a large protein complex comprising more than thirteen proteins [START_REF] Banerjee | Organization and function of septate junctions: an evolutionary perspective[END_REF][START_REF] Batz | The transmembrane protein Macroglobulin complement-related is essential for septate junction formation and epithelial barrier function in Drosophila[END_REF][START_REF] Furuse | Claudins in occluding junctions of humans and flies[END_REF], with a core complex that includes Neurexin IV (Nrx-IV), Coracle (Cora), Neuroglian (Nrg), and the α-and β-subunits of the Na + /K + ATPase pump (ATPα and Nervana 2 (Nrv2)) [START_REF] Lamb | Drosophila coracle, a member of the protein 4.1 superfamily, has essential structural functions in the septate junctions and developmental functions in embryonic and adult epithelial cells[END_REF][START_REF] Genova | Neuroglian, Gliotactin, and the Na+/K+ ATPase are essential for septate junction function in Drosophila[END_REF][START_REF] Schwabe | GPCR signaling is required for blood-brain barrier formation in drosophila[END_REF][START_REF] Paul | The Na+/K+ ATPase is required for septate junction function and epithelial tube-size control in the Drosophila tracheal system[END_REF]. At three-cell vertices, BSJ are disjointed and specialized tricellular junctions (TSJ) mediate cell contacts [START_REF] Ikenouchi | Tricellulin constitutes a novel barrier at tricellular contacts of epithelial cells[END_REF][START_REF] Oda | Tricellulin regulates junctional tension of epithelial cells at tricellular contacts through Cdc42[END_REF][START_REF] Krug | Tricellulin forms a barrier to macromolecules in tricellular tight junctions without affecting ion permeability[END_REF][START_REF] Byri | The Triple-Repeat Protein Anakonda Controls Epithelial Tricellular Junction Formation in Drosophila[END_REF][START_REF] Schulte | Gliotactin, a novel marker of tricellular junctions, is necessary for septate junction development in Drosophila[END_REF]. Drosophila TSJ contains the Neuroligin-like transmembrane protein Gliotactin (Gli) [START_REF] Schulte | Gliotactin, a novel marker of tricellular junctions, is necessary for septate junction development in Drosophila[END_REF], which is recruited by the triple-extracellular repeat protein Bark beetle (Bark, a.k.a. Anakonda) [START_REF] Byri | The Triple-Repeat Protein Anakonda Controls Epithelial Tricellular Junction Formation in Drosophila[END_REF][START_REF] Hildebrandt | Bark beetle controls epithelial morphogenesis by septate junction maturation in Drosophila[END_REF]. It is currently unknown how, when and where SJ components are recruited and stabilized to assemble de novo BSJ and TSJ to ensure maintenance of permeability barrier function throughout cytokinesis.

Here, we analyzed the maturation of the midbody and the remodeling of SJ upon cell division in the Drosophila pupal notum and larval wing imaginal disc (WID) epithelia (Figure S1). They are composed of cuboidal and columnar epithelial cells respectively, undergo numerous cell divisions, can be subjected to in vivo quantitative live imaging and to genetic dissection. We identified how new BSJ and TSJ are remodeled as the midbody matures to ensure the de novo permeability barrier maintenance in proliferative epithelia.

RESULTS

Maturation and basal displacement of the midbody during epithelial cytokinesis

To explore cytokinesis, we imaged the actomyosin contractile ring using the non-muscle Myosin II light chain tagged with RFP or mCherry (thereafter referred to as MyoII::RFP) together with the AJ marker E-Cadherin (E-Cad)::GFP or the SJ marker ATPα::YFP using time-lapse confocal microscopy on the notum (Figures 1A-B).

Upon completion of constriction, the actomyosin ring was located below the newly formed adhesive interface (Figure 1A; Movie S1), as previously described [START_REF] Herszterg | Interplay between the dividing cell and its neighbors regulates adherens junction formation during cytokinesis in epithelial tissue[END_REF][START_REF] Founounou | Septins regulate the contractility of the actomyosin ring to enable adherens junction remodeling during cytokinesis of epithelial cells[END_REF][START_REF] Guillot | Adhesion disengagement uncouples intrinsic and extrinsic forces to drive cytokinesis in epithelial tissues[END_REF][START_REF] Morais-De-Sa | Adherens junctions determine the apical position of the midbody during follicular epithelial cell division[END_REF]. MyoII::RFP remained associated with and marked the midbody ring (Figure 1A-B). The midbody moved towards the basal pole over two main phases. In the first, slow phase, the midbody moved ~3 µm within the SJ domain (Figures 1B,C,andS2A). In the second phase, upon exit from the SJ domain, the midbody moved faster basally (from t=1h30min, Figure 1B,C,).

By analogy to isolated cells [START_REF] Green | Cytokinesis in Animal Cells[END_REF][START_REF] Glotzer | The molecular requirements for cytokinesis[END_REF], the composition and the morphology of the midbody should evolve as it matures into an intercellular bridge. This prediction was by imaging midbody components. For each marker, any drift was corrected to artificially immobilize the midbody, allowing us to represent the changes in composition in the form of kymographs. We found that MyoII, Septin 2 (Sep2) and Septin 7 (a.k.a. Peanut (Pnut)), remained localized to the midbody ring for more than 120 minutes after the onset of anaphase (Figure 1D). FRAP experiments revealed that MyoII::RFP and Pnut::mCherry remained stably associated within the midbody (Figures 1E-F, S2B-C, Movies S3 and S4). In contrast, actin visualized using LifeAct::mCherry and microtubules visualized with Jupiter::GFP were no longer detected in the midbody at t=26 ± 4 min and t=42 ± 6 min respectively after the onset of anaphase (Figure 1D; Movie S2).

We next confirmed these data, using transmission-(TEM) and serial block face scanning-(SFB-SEM) electron microscopy [START_REF] Denk | Serial block-face scanning electron microscopy to reconstruct three-dimensional tissue nanostructure[END_REF] in the notum (Figure 1G These data indicate that basal displacement and maturation of the midbody is conserved in the notum and WID, raising the question of when and where completion of cytokinesis occurs.

Epithelial cell abscission

TEM of the WID revealed that intercellular bridges were ultimately detected in intracellular compartments (Figures 2A andS3A), presumably resulting from internalization. Such bridges followed an asymmetric pattern with one side opened to the cytosol and the other side sealed by membrane as a result of abscission (Figure 2A). The internalized bridges were eventually degraded (Figures 2B andS3B-E) and most were localized in the basal side (Figure S3F).

To monitor the timing of abscission in the notum, we found that CPC components Aurora B (AurB) and Survivin (Svv) left the midbody concomitantly with microtubules (Figures 1D and2C; AurB::GFP: t=45 ± 7 min; Svv::GFP: t=40 ± 5 min). The ESCRT-III component Shrub::GFP (Shrb, the Chmp4B ortholog) was recruited at the midbody at t=22 ± 8 min following the onset of anaphase (Figure 2C), suggesting that cells are set for abscission ~40 minutes post-anaphase onset. As an independent measure of abscission, we analyzed cytoplasmic isolation of the two daughter cells [START_REF] Guizetti | Cortical constriction during abscission involves helices of ESCRT-IIIdependent filaments[END_REF][START_REF] Steigemann | Aurora B-mediated abscission checkpoint protects against tetraploidization[END_REF]43] using the photoconvertible KAEDE. Photoconverted KAEDE can freely diffuse between daughter cells through the intercellular bridge prior to cytoplasmic isolation, and diffusion will cease upon abscission (Figure S3G-H). When KAEDE was photoconverted in one daughter cell, we found that even 5 hours after the onset of anaphase, cytoplasmic isolation had not yet occurred (Figure 2D). These data demonstrate that recruitment of Shrb to the midbody and cytoplasmic isolation occur at distinct times.

Given the time lag between midbody maturation and abscission, and the two phases of midbody displacement, the question arises as to what forces are driving midbody basal displacement?

Because it occurred within the SJ, we asked how SJ are remodeled during cytokinesis and could contribute to midbody displacement.

Connection between dividing cells and their neighbors during cytokinesis

To monitor SJ remodeling, we imaged MyoII::RFP together with ATPα::YFP in the notum. We observed that the neighboring interphase cells make membrane protrusions that pointed towards the midbody, hereafter called finger-like protrusions (Figure 3A). Such protrusions labeled with Nrg::GFP were also observed in the WID (Figure S4A,B) suggesting this is a common feature of Drosophila epithelial cytokinesis.

To unambiguously establish that neighbors retain contact with dividing cells, we generated clones devoid of ATPα::YFP in a tissue expressing ATPα::YFP and MyoII::RFP. When a YFPnegative cell (Figure 3B) divided next to an ATPα::YFP-positive cell, the YFP signal exclusively originated from the neighboring cell. The latter remained connected with the mitotic cell at the level of the midbody where membrane deformation was maximal due to cytokinetic ingression (Figure 3C). At the time the new AJ were already assembled (Figure 3C, t=10 min), the fingerlike protrusions containing ATPα::YFP were already present and persisted for more than 2h30.

3D modeling of a SEM dataset further demonstrated the presence of finger-like protrusions at the level of the midbody in the notum as well as in WID (Figures 3D-H andS4C-D). TEM revealed that dividing cells form a tight, four-cell contact with their neighbors at the level of the midbody (Figure 3I-K). As the midbody matured, these finger-like protrusions were displaced towards the basal pole (Figure 3C). The maintenance of finger-like protrusions through the basal displacement suggests that these contacts have biological significance in dividing epithelia.

De novo assembly of SJ at the cleavage site

During cytokinesis, new BSJ will assemble along the interface between daughter cells. Meanwhile, at the cleavage site, the dividing cells need to disassemble and remodel the prior SJ to create a TSJ at the newly formed cell corners.

To monitor BSJ assembly, we imaged clones of dividing cells expressing ATPα::YFP surrounded by interphase cells devoid of ATPα::YFP (Figures 4A andS5A). The new BSJ, marked by the only source of ATPα::YFP in the dividing cell, were found to assemble apical to the midbody (Figure 4A). As the midbody was displaced basally, BSJ continued to assemble culminating in the formation of the characteristic belt of BSJ seen in interphase cells (Figure 4A, t=1h20). It was at this point the intercellular bridge moved toward the basal side with faster kinetics.

To monitor TSJ remodeling, we imaged Gli::YFP, which was detected apical to the midbody (Figure 4B). In contrast to ATPα::YFP, which was uniformly recruited along the new membrane interface, Gli::YFP appeared first as a single puncta (Figure 4B, t=23 min), which expanded to two (t=33 min), three (t=38 min) where the addition of subsequent punctae yielded the appearance of a pearl necklace (t=1h23) along the finger-like protrusions. Between t=1h23 and t=2h33, the pearl-like distribution was no longer detected, and Gli::YFP was exclusively detected at the three-cell corners. It was after this stage the intercellular bridge was displaced faster basally (Figure 4B, red arrows at t=2h23 to t=2h33). Bark::GFP, which recruits and stabilizes Gli at TSJ [START_REF] Byri | The Triple-Repeat Protein Anakonda Controls Epithelial Tricellular Junction Formation in Drosophila[END_REF], exhibited comparable spatio-temporal localization than Gli::YFP throughout cytokinesis (Figure S5B).

Overall our findings suggest that BSJ assembly during cytokinesis is an inherently slow process and the completion of SJ assembly is linked to the different phases of midbody displacement.

Slow and polarized SJ assembly along the apico-basal axis

BSJ are highly stable in Drosophila embryos [START_REF] Oshima | Analysis of protein dynamics within the septate junction reveals a highly stable core protein complex that does not include the basolateral polarity protein Discs large[END_REF] and thus the duration of SJ remodeling during cytokinesis could be limited by slow junctional turnover. We tested this possibility using FRAP and found that most of ATPα::YFP was found in an immobile fraction in mitotic cells. About 90 minutes were required to recover ~80% of ATPα::YFP fluorescence, confirming a stable protein complex (Figure 5A,D,G). The kinetics of fluorescence recovery of ATPα::YFP in interphase cells (Figure 5B,E,H) was only slightly faster (ANOVA, p-value=0.019 *, Figure 5A,D,G) indicating that SJ renewal is largely cell-cycle independent. Similar to ATPα::YFP, Bark was stable at the TSJ as ~80% of Bark::GFP signal was recovered in 90 minutes in interphase cells (Figure 5C,F,I).

FRAP experiments revealed that the recovery of fluorescence was polarized. The recovery of ATPα::YFP fluorescence occurred first apically, 1 µm below the AJ, in mitotic and interphase cells (Figure 5D-E, G-H, black frames). At 2 µm below the AJ, recovery occurred later (Figure 5E, H, blue frame) (ANOVA, p-value<2.2x10 -16 ***). Bark recovery was also polarized along the apical basal axis (Figure 5F,I).

Very little or no ATPα::YFP signal was recovered in the finger-like protrusions formed by the neighboring interphase cells (Figure 5D, G, red frame). These BSJ likely correspond to the prior SJ present before mitosis.

Collectively, these data show that BSJ and TSJ are assembled apically, and spread from the apical to basal side as they age, in a manner akin to a conveyor belt.

Coordination between assembly of SJ and basal displacement of the midbody

The above findings raise the possibility that assembly of BSJ and TSJ regulate the basal displacement of the finger-like protrusions connected to the midbody. To test this prediction, we analyzed the kinetics of the midbody basal displacement upon loss of BSJ or TSJ components.

Depletion of the core BSJ components Nrx-IV and Cora, disrupted cell polarity leading to cell delamination and prevented further analyses (data not shown). We found that loss of the core component Nrv2, impaired SJ integrity as judged by the strong reduction in ATPα::YFP, Fas3, Gli and Cora staining without affecting E-Cad localization (Figures 6A-B andS6A-C). FRAP analyses in nrv2 k13315 mutant cells indicated that the remaining ATPα::YFP resided mostly in a mobile fraction with ~80% recovery and t1/2=30 ± 6 sec (Figures 6C andS6H, K). This is in striking contrast to controls where ~19% of ATPα::YFP was in a mobile fraction (t1/2=28 ± 6 sec) (Figures 6C andS6G, J) and the bulk of ATPα::YFP in an immobile fraction (Figure 5H). In contrast to the percentage of recovery (t-test, p-value=1.26x10 -14 ***), the halftimes are not significantly different (t-test, p-value=0.28) between wild-type and nrv2 k13315 mutant cells. These data suggest that the assembly of ATPα::YFP-containing junctions were impaired upon loss of Nrv2. Only the mobile fraction remained, which displayed rapid recovery. Furthermore, the finger-like protrusions were no longer detected in nrv2 k13315 mutant cells (Figure 6B), even within wild-type cells dividing next to nrv2 k13315 mutant cells (Figure 6A). Finally, the midbody was not displaced basally in nrv2 k13315 mutant cells over time compared to wild-type cells (Figure 6D,E, ANCOVA, p-value<2.2x10 -16 ***). Collectively, these data suggest that polarized assembly of BSJ drives midbody basal displacement.

To test whether TSJ influence midbody basal displacement, we analyzed the consequence of loss of Bark [START_REF] Byri | The Triple-Repeat Protein Anakonda Controls Epithelial Tricellular Junction Formation in Drosophila[END_REF]. While dispensable for E-Cad localization, as expected, Bark was needed to recruit Gli [START_REF] Byri | The Triple-Repeat Protein Anakonda Controls Epithelial Tricellular Junction Formation in Drosophila[END_REF] (Figure S6E). Loss of Bark also disrupted the BSJ, where ATPα::YFP, Fas3 and Cora staining was reduced and spread basally (Figures 6F-G and S6D, F), and membrane deformations were observed (Figure S6F, white inset). Despite these morphological changes, loss of Bark did not modified the stability of ATPα::YFP as measured by FRAP (Figures 6H andS6I, L) (ANOVA, p-value=0.62). In contrast to Nrv2, loss of Bark did not modify the assembly or the stability of BSJ complexes, and loss of Bark did not impair the finger-like protrusions with neighboring cells at cytokinesis (Figure 6F-G). Also in contrast to Nrv2, the basal displacement of the midbody was accelerated in bark L200 mutant cells (Figure 6I-J; ANCOVA, p-value<2.2x10 - 16 ***), indicating that the remodeling of TSJ is rate-limiting, slowing down the midbody basal displacement.

Based on these data, we propose a model where the remodeling of SJ and TSJ between dividing cells and their neighbors regulates the kinetics of midbody basal displacement (Figure 7).

DISCUSSION

Coordination between SJ remodeling and basal displacement of the midbody

Epithelial growth requires the formation of cell-cell junctions and physical separation of daughters upon cell division. Here, in both cuboidal and columnar epithelia we have characterized a multicellular mechanism that coordinates completion of cytokinesis and de novo formation of SJ (Figure 7): 1) Following actomyosin ring constriction, the midbody forms just basal to the AJ within the SJ and matures into an intercellular bridge that is displaced basally. 2) Neighboring interphase cells maintain SJ contacts, the finger-like protrusions connecting to the maturing midbody of the dividing cells. At this position, the ménage à quatre formed contains preexisting SJ and is stable. 3) Novel BSJ are assembled below the AJ and above the finger-like protrusions and spread towards the basal side, thereby driving the basal displacement of the finger-like protrusion plus midbody. Once the new BSJ is complete the midbody is positioned outside the SJ belt and basal displacement accelerated. Overall, we propose that this multicellular process is to ensure the maintenance of the permeability barrier throughout cytokinesis.

Role of polarized delivery and assembly of BSJ in midbody basal displacement

Our data show that, as in embryos [START_REF] Oshima | Analysis of protein dynamics within the septate junction reveals a highly stable core protein complex that does not include the basolateral polarity protein Discs large[END_REF], newly synthesized SJ components assemble into stable protein complexes exhibiting a low rate of diffusion, endocytosis and recycling. These stable complexes assemble below the AJ and then propagate towards the basal pole, in a manner analogous to a conveyor belt, both in interphase and during cytokinesis.

Loss of the core SJ component Nrv2 prevents both assembly of stable BSJ complexes and basal displacement of the midbody. Thus, we propose that the apico-basal flux of newly assembled BSJ is the propelling force for the basal displacement of the finger-like protrusions connected to the midbody. The conveyor belt model first predicts that newly synthesized and/or slowly recycling SJ components are delivered apically in the vicinity of AJ. The model also predicts that SJ disassembly occurs at the basal rim of SJ belt. Future work will determine what regulates the polarized traffic of SJ components and the disassembly of SJ to control both the positioning and thickness of the SJ belt.

Site of TSJ assembly and role in midbody displacement

Our study sheds light on the remodeling of TSJ at cytokinesis. Bark and Gli are initially detected as puncta next to the midbody, prior to the pearl necklace distribution along the finger-like protrusions. These dotted structures could represent new TSJ, as at this location a new threeway contact is formed between the two daughter cells and a neighbor. However, FRAP analyses suggest that these punctae do not contain exchangeable components and are not moving laterally (our unpublished results). Moreover, the appearing of Gli and Bark punctae in mitotic cells is faster (20 min post-anaphase) than the de novo assembly of TSJ in interphase cells (~80% recovery in 90 min). Analyses of the punctae by TEM will be necessary to test if they are bona fide TSJ or components in the process of assembly. Alternatively, the old TSJ between dividing cells and neighbors prior to mitosis could undergo a change in distribution from a uniform distribution into clusters giving rise to the pearl necklace. Redistribution could represent an intermediate step in TSJ disassembly and may explain the fast kinetics of assembly of pearl necklace structure. Future work using for example photoconvertible probes would help address this issue.

Finally, loss of Bark resulted in acceleration of the midbody basal displacement. The de novo assembly of TSJ in the pearl necklace structure could be the rate-limiting step that imposes the slower speed of basal displacement. Alternatively, if pearl necklace represents the remodeling of prior TSJ connections with neighboring cells, TSJ disassembly maybe the rate-limiting step.

Regardless, our data argue that the TSJ components present in the pearl necklace slow down the midbody basal displacement driven by the BSJ mediated conveyor belt.

Completion of epithelial cytokinesis

Our study reveals differences in midbody maturation and organization of the intercellular bridge between isolated and epithelial cells. In epithelial cells, the densely packed microtubule array that normally sits in the center of the intercellular bridge and the Flemming body were not observed [START_REF] Mierzwa | Cytokinetic abscission: molecular mechanisms and temporal control[END_REF]. Instead, we found an intercellular bridge with a uniform density of microtubules that progressively disappear. The electron dense material detected along the plasma membrane of the intercellular bridge could act as a rigid scaffold preventing membrane constriction. The SJ core components, plus MyoII and Pnut at the interface made between the intercellular bridge and the finger-like protrusions might be part of the electron dense material and therefore contribute to membrane rigidity. Alternatively, this scaffold could prevent tension release within the intercellular bridge, therefore preventing premature abscission [START_REF] Lafaurie-Janvore | ESCRT-III assembly and cytokinetic abscission are induced by tension release in the intercellular bridge[END_REF].

We also found that abscission is asymmetric with the bridge remnant internalized and then degraded in one of the daughter cells in a mode similar to the internalization and autophagy described for isolated cells [START_REF] Ettinger | Proliferating versus differentiating stem and cancer cells exhibit distinct midbody-release behaviour[END_REF]. However, we cannot exclude that abscission could also occur on both side of the bridge leading to its release followed by its recapture by endocytosis in some epithelial cells [START_REF] Agromayor | Knowing when to cut and run: mechanisms that control cytokinetic abscission[END_REF][START_REF] Ettinger | Proliferating versus differentiating stem and cancer cells exhibit distinct midbody-release behaviour[END_REF][START_REF] Elia | Incisive imaging and computation for cellular mysteries: lessons from abscission[END_REF][START_REF] Chai | Apoptotic regulators promote cytokinetic midbody degradation in C. elegans[END_REF][START_REF] Crowell | Engulfment of the midbody remnant after cytokinesis in mammalian cells[END_REF].

Finally, while epithelial cells seem to be set for abscission about 40 minutes following the onset of anaphase, cytoplasmic isolation had not occurred 5 hours later. Photoconversion data furthermore suggests that cytoplasmic isolation takes place about one hour prior to one of the sister entering mitosis (Figure S3H), possibly at the mid-to late G2 phase as reported for germline stem cells [START_REF] Matias | Abscission is regulated by the ESCRT-III protein shrub in Drosophila germline stem cells[END_REF].

Conservation in vertebrates?

Despite the opposite apico-basal positioning of permeability barrier relative to the mechanical barrier, a number of similarities are encountered in vertebrates and invertebrates epithelial cells.

First, mechanical barrier transmission during epithelial cytokinesis is a multicellular process [START_REF] Higashi | Maintenance of the Epithelial Barrier and Remodeling of Cell-Cell Junctions during Cytokinesis[END_REF][START_REF] Herszterg | Interplay between the dividing cell and its neighbors regulates adherens junction formation during cytokinesis in epithelial tissue[END_REF][START_REF] Founounou | Septins regulate the contractility of the actomyosin ring to enable adherens junction remodeling during cytokinesis of epithelial cells[END_REF][START_REF] Guillot | Adhesion disengagement uncouples intrinsic and extrinsic forces to drive cytokinesis in epithelial tissues[END_REF][START_REF] Morais-De-Sa | Adherens junctions determine the apical position of the midbody during follicular epithelial cell division[END_REF][START_REF] Lau | Anisotropic stress orients remodelling of mammalian limb bud ectoderm[END_REF][START_REF] Firmino | Cell Division Drives Epithelial Cell Rearrangements during Gastrulation in Chick[END_REF]. Second, the permeability barrier is also maintained throughout epithelial cytokinesis [START_REF] Higashi | Maintenance of the Epithelial Barrier and Remodeling of Cell-Cell Junctions during Cytokinesis[END_REF].

Third, is the recruitment of tricellular junction components in close vicinity to the midbody, supporting the idea that polarized delivery, membrane composition and topology at midbody is engineered for tricellular junction remodeling. Fourth, epithelial cytokinesis is polarized along the apico-basal axis leading to the embedding of midbody en route to abscission within the permeability barrier both in vertebrates [START_REF] Higashi | Maintenance of the Epithelial Barrier and Remodeling of Cell-Cell Junctions during Cytokinesis[END_REF][START_REF] Jinguji | Electron microscopic observations on the maintenance of the tight junction during cell division in the epithelium of the mouse small intestine[END_REF][START_REF] Dubreuil | Midbody and primary cilium of neural progenitors release extracellular membrane particles enriched in the stem cell marker prominin-1[END_REF] and in Drosophila (this study). Based on our findings, we anticipate that the coordination between permeability barrier transmission and completion of cytokinesis in Drosophila is conserved in vertebrates for maintenance of tissue integrity in proliferative epithelia. After contractile ring closure, a long adhesive interface is assembled between daughter cells (cyan and blue) at AJ level. At SJ level, neighboring cells (yellow and orange) exhibit membrane protrusions that maintain cell-cell contacts with the dividing cell at the level of the midbody (magenta), forming a ménage à quatre. TSJ (light green), first detected in dotted structures just above the midbody, evolve as a line as the midbody descents basally and are ultimately found at the tricellular contacts contacts between the daughter and the neighbors. Concomitantly, BSJ (dark green) assembles below AJ and above the midbody and this apical to basal outspreading of BSJ thus drives the basal displacement of the midbody and membrane protrusions. Upon completion of SJ assembly, the midbody leaves the SJ belt and moves faster towards the basal pole.

STAR METHODS

CONTACT FOR REAGENT AND RESOURCE SHARING

Further information and requests for resources and reagents should be directed to and will be fulfilled by the Lead Contact, Roland Le Borgne (roland.leborgne@univ-rennes1.fr).

EXPERIMENTAL MODEL AND SUBJECT DETAILS

Drosophila stocks and genetics

Drosophila melanogaster stocks were maintained and crossed at 25°C. Somatic clones were induced using the FLP-FRT technique using the hs-FLP and by two heat shocks (60 min at 37°C) at second and third instar larvae.

The ap-GAL4 or pnr-GAL4 drivers were used to drive the expression of UAS-Pnut::mCherry, UAS-LifeAct::RFP or UAS-AurB::GFP. The expression of UAS-Shrb::GFP was driven by the ay-GAL4 driver. Finally, the sca-GAL4 driver was used to drive the expression of the photoconvertible probe UAS-KAEDE. MyoII::RFP (also called Sqh above) flies were designed and generated by inDroso functional genomics (Rennes, France) using CRISPR-mediated HR. RFP was inserted just before the stop codon of endogenous MyoII gene with a flexible GVG linker and the resulting flies was validated by sequencing.

Drosophila genotypes

All EM experiments have been performed on w 118 strain (Figures 1G-M, 2A-B, 3D-K).

METHOD DETAILS

Immunofluorescence

Pupae aged for 16h30 to 18h after puparium formation (APF) were dissected using Cannas microscissors in 1X Phosphate-Buffered Saline (1X PBS, pH 7.4) and fixed 15 min in 4% paraformaldehyde at room temperature [START_REF] Jinguji | Electron microscopic observations on the maintenance of the tight junction during cell division in the epithelium of the mouse small intestine[END_REF]. Following fixation, dissected nota were permabilized using 0.1% Triton X-100 in 1X PBS (PBT), incubated with primary antibodies diluted in PBT for 2 hours at room temperature. After 3 washes of 5 minutes in PBT, nota were incubated with secondary antibodies diluted in PBT, followed by 2 washes in PBT, and one wash in PBS, prior mounting in 0,5% N-propylgallate dissolved in 1X PBS + 90% glycerol.

Live-imaging and image analyses

Live imaging was performed on pupae aged for 16h30 APF at 20-25°C [START_REF] Jinguji | Electron microscopic observations on the maintenance of the tight junction during cell division in the epithelium of the mouse small intestine[END_REF]. Pupae were sticked on a glass slide with a double-sided tape, and the brown pupal case was removed over the head and dorsal thorax using microdissecting forceps. Pillars made of 4 to 5 glass coverslips were positioned at the anterior and posterior side of the pupae, respectively. A glass coverslip covered with a thin film of Voltalef 10S oil is then placed on top of the pillars such that a meniscus is formed between the dorsal thorax of the pupae and the glass coverslip. Images were acquired with a LSM Leica SP5 or SP8 equipped with a 63X N.A. 1.4. and controlled by LAS AF software. Confocal sections were taken every 0.5 μm. All images were processed and assembled using ImageJ/FIJI software and Adobe Illustrator.

Midbody tracking

The 3D coordinates of the midbody were manually tracked over time to be artificially realigned to generate the kymographs.

The apico-basal position of the midbody was calculated measuring the distance between the middle of the new AJ and the midbody (both labelled with MyoII::RFP) at each time. The x, y, z coordinates of AJ and midbody were manually tracked to record positions at each time, and then are used to calculate the distance using the Pythagorean Theorem.

KAEDE photoconversion

Photoconversion assays were performed in pupae expressing the green to red photoconvertible probe KAEDE. KAEDE was photoconverted (405 nm laser at 0.1% power, point bleach, 4 iterations of 100 ms each) using a LSM Leica SP8 equipped with a 63X N.A. 1.4 PlanApo objective. Confocal stacks were acquired every 5 min after photoconversion.

Fluorescence recovery after photobleaching

FRAP experiments were performed in pupae expressing MyoII::RFP or Pnut::mCherry together with Sep2::GFP used as a marker of midbody. MyoII::RFP and Pnut::mCherry were bleached (561 nm laser at 40% or 30% power, respectively, point bleach, one iteration of 500 ms) using a LSM Leica SP5 and a 63X N.A. 1.4 PlanApo objective. Confocal stacks were acquired every 2 min after photobleaching. FRAP experiments were performed in pupae expressing ATPα::YFP or Bark::GFP together with MyoII::RFP. Regions of interest corresponding to the entire junctions (interphase cells) or the finger-like protrusions pointing to the midbody (mitotic cells) were bleached (488 nm laser at 100% power, 1 iteration of 100 ms) using a LSM Leica SP5 or SP8 equipped with a 63X N.A. 1.4 PlanApo objective. Confocal stacks were acquired every 5 min after photobleaching.

TEM sample preparation

Drosophila wing discs (third instar larvae) and pupal notum (16h APF) were dissected in PBS at room temperature and immediately processed [83,84]. Briefly, the samples were fixed in 1% paraformaldehyde and 2.5% glutaraldehyde in 0.1M cacodylate buffer for 2 hours. Then, they were stained for 1 hour in 2% (wt/vol) osmium tetroxide and 1.5% (wt/vol) K4[Fe(CN)6] in cacodylate buffer followed by 1 hour in 1% (wt/vol) tannic acid in 100 mM cacodylate buffer. Finally, they were incubated 30 min in 2% (wt/vol) osmium tetroxide followed by 1% (wt/vol) uranyl acetate for 2 hours. After the dehydration cycles, samples were embedded in Epon-Araldite mix. To assure precise orientation and access to the samples 2-step flat embedding procedure was used [84].

TEM observation and analysis

Rigid flat blocks were trimmed using 90° diamond trim tool (Diatome, Biel, Switzerland) to delimit the zone of interest based on laser marks. For standard TEM observations, samples were directly processed to thin sectioning using 35° diamond knife (Diatome, Biel, Switzerland). For high-resolution analyses, sections were collected on formvar-coated slot grids (EMS) and analyzed with a JEOL JEM-1400 electron microscope operated at 80kV, equipped with a Gatan Orius SC 1000 camera, piloted by the Digital Micrograph program.

For 3-view analysis, flat blocks were glued onto an aluminum cryo specimen pin using superglue (cyanoacrylate, Permabond). For trimming, care was taken to orient the sample so that the imaging plane was perpendicular to the pin axis to form a trapezoid face approximately 500 x 500 x 150 μm before mounting on a Gatan 3-view within an FEI Quanta 250 FEG scanning electron microscope. For the purpose of this study the imaging settings were: accelerating voltage: 3.8kV; spot size: 3.5; final lens aperture: 30 μm; chamber pressure: 66 Pa; quadrant magnification: 3500x giving a horizontal field width of approximation 40 μm; image dimensions: 4096 x 4096; pixel dwell time: 10 μs; cut thickness: 100nm.

Raw data was converted to an MRC file stack using IMOD [83]. Imaging noise was removed by a standard Gaussian smoothing using a 3x3 kernel in order to aid manual segmentation. TEM data analysis was performed using FIJI, IMOD 3Dmod module for particular regions annotations and modelling. Photoshop was used for alignment of serial sections and pseudo-coloring of different zones in image.

QUANTIFICATION AND STATISTICAL ANALYSIS

Signal recovery upon photobleaching

On kymographs focused on junctions at -1 μm, -2 μm or at midbody level (level 0 corresponds to the AJ level), the mean of fluorescence intensity was measured on 13 pixels along the time.

In FRAP experiments on MyoII::RFP or Pnut::mCherry, this was performed for the photobleached midbody, control midbody (another non-FRAP cell at the same stage of division) and 3 backgrounds. In FRAP experiments on ATPα::YFP or Bark::GFP, this was performed for the photobleached junctions, control junctions (3 for interphase cell assay, 1 for mitotic cell assay) and 3 backgrounds. Note that in mitotic cells, the control junction is the finger-like protrusion opposite to the FRAPed finger-like protrusion.

The differences between averaged controls and averaged backgrounds were normalized with 3

STAR METHODS

KEY RESOURCES

  -M; Movie S5) and WID (FigureS2D-G).The maturing midbody was characterized by the presence of an array of microtubules (Figures1G-H and S2D; Movie S6) predominantly found in the apical side of dividing cells (Figures1L-Mand S2G, magenta). At an intermediate stage, most of microtubules disappeared and were replaced by internal membranes (Figures1I, L-M, green, and S2E; Movie S7), until the maturation into intercellular bridges that were found more basally (Figures 1J,L-M and S2F,G, Cyan; Movie S8).

Figure 1 :

 1 Figure 1: Midbody assembles and displaces toward the basal pole (A-B) Time-lapse of dividing cells expressing MyoII::RFP (magenta) with E-Cad::GFP (A, cyan, n>30; see also Movie S1) or ATPα::YFP (B, green, n>30). Cyan line highlights the level of AJ and red arrows point to the midbody. (C) Plot of the quantitation of the apical to basal movement of the midbody over time relative to AJ (cyan line: 0 to -0.5 µm) and SJ (green line: -0.5 to -3 µm) (n=20). (D) Kymographs and quantitation of the residence time of MyoII::RFP (n=66), Sep2::GFP (n=45), Pnut::mCherry (n=18), LifeAct::mCherry (n=27) or Jupiter::GFP (n=20; see also Movie S2) present in the midbody. (E) Kymographs of MyoII::RFP (magenta) imaged with Sep2::GFP (green) present in the midbody over time in control condition (left panels) or upon photobleaching (right panels). Photobleaching of MyoII::RFP was done 10 minutes following contractile ring constriction (yellow dashed line); see also Movies S3 and S4. (F) Plot of the quantitation of the fluorescence signal recovery of MyoII::RFP (n=7) after photobleaching over time. (G-J) TEM micrographs of dividing notum cells sectioned along the apico-basal axis and parallel to the midbody. H is a high magnification of the inset depicted in G. Three stages of midbody are shown: maturing midbody (H, n>20; see also Movie S5), intermediate midbody (I, n>20; see also Movie S6) and intercellular bridge (J, n>20; see also Movie S7). (K-M) Top view (K) or orthogonal sections (L-M) showing the distribution of maturating midbody (magenta), intermediate midbody (green) and intercellular bridges (cyan) identified using the SFB-SEM volume reconstitution. M is a maximal projection of orthogonal sections depicted in L. Two daughter cells (cyan and blue) and their two neighbors (yellow and orange) were represented; see also Movie S8. Time is in hour:minute (A-B) or in minutes (C-F), with t=0 min corresponding to the onset of anaphase. Scale bars represent 3 µm (A-B) or 200 nm (H-J). Ap: apical; B: basal. See also Figure S1 and Figure S2.

Figure 2 :

 2 Figure 2: Abscission is temporally uncoupled to Shrb recruitment.

Figure 3 :

 3 Figure 3: Dividing cells remain connected to their neighbors throughout cytokinesis. (A) Time-lapse of a dividing cell expressing MyoII::RFP (magenta) with ATPα::YFP (green) (n>30). Orange line highlights the contact of the neighbors and the dividing cell. (B-C) Time-lapse of a clone of cells expressing MyoII::RFP (magenta) and surrounded by cells expressing MyoII::RFP with ATPα::YFP (green, with nls-GFP) (n=18). Yellow line highlights a dividing cell devoid of ATPα::YFP, identified by the loss of nls-GFP. (C) is a high magnification of the red inset depicted in (B).(D-H) 3D model based on SFB-SEM dataset illustrating the membrane contact made between daughter cells (cyan and blue) and their neighbors (yellow and orange). At AJ level, the daughters have assembled a long adhesive interface (E); while at SJ level, they are connected by the midbody (green) (F). Orthogonal views of daughters (G) and of neighbors with the fingerlike protrusions (H), both connected to the midbody (green).(I-K) TEM micrograph of dividing notum epithelial cells sectioned parallel to the midbody (I) and ultrathin serial sections parallel (J) or orthogonal (K) to the midbody. Neighbor cells (yellow and orange) are tightly connected to the dividing cells.

Figure 4 :

 4 Figure 4: De novo assembly of SJ at the cleavage site. (A) Time-lapse of a dividing clonal cell expressing MyoII::RFP (magenta) with ATPα::YFP (green) (identified by the loss of nls-RFP, not shown) and surrounded by cells devoid of ATPα::YFP (n=13). (B) Time-lapse of a dividing cell expressing MyoII::RFP (magenta) with Gli::YFP (green) (n=11). High magnifications on right panels correspond to the red inset depicted at t=10 min. Yellow arrows show the first Gli::YFP puncta and white arrow shows the final Gli::YFP distribution at TSJ. Time is in hour:minute, with t=0 min corresponding to anaphase onset. Yellow line delineates the dividing cell and its daughters, cyan lines highlight the position of AJ and red arrows point to the midbody. Distances correspond to the position relative to the plane of AJ labeled with MyoII::RFP. Scale bars represent 3 μm. See also Figure S5.

Figure 5 :

 5 Figure 5: New SJ junctions assemble apically. (A-C) Time-lapse of dividing (A) or interphase (B-C) cells expressing MyoII::RFP (magenta) with ATPα::YFP (A-B, green) or Bark::GFP (C, green). The photobleached ROI correspond to the yellow insets. (D-F) Kymographs representing the photobleaching of ATPα::YFP (D-E, green) or Bark::GFP (F, green) imaged with MyoII::RFP (magenta). Yellow lines indicate the time of photobleaching. (G-I) Plots of the quantitation of fluorescence recovery of ATPα::YFP (G-H) or Bark::GFP (I) after photobleaching over time. Black, blue and red curves correspond to 1 µm, 2 µm below AJ level and midbody level, respectively (G, n=10; H, n=16; I, n=12). Time is in hour:minute (A-C) or in minutes (D-I), with t=0 min corresponding to the time of photobleaching. Distances correspond to the position relative to the plane of AJ labeled with MyoII::RFP. Scale bars represent 3 μm.

Figure 6 :

 6 Figure 6: SJ components regulate midbody basal displacement. (A-B, F-G) Time-lapse of wild-type (A, F), nrv2 k13315 mutant (B) or bark L200 mutant (G) dividing cells expressing MyoII::RFP (magenta, with nls-RFP) with ATPα::YFP (green). (C, H) Plots of the quantitation of fluorescence recovery of ATPα::YFP after photobleaching over time in wild-type (black, C, n=10; H, n=16), nrv2 k13315 mutant (red, C, n=10) or bark L200 mutant (blue, H, n=7) cells 1 µm below AJ. (D, I) Time-lapse of nrv2 k13315 mutant (D) or bark L200 mutant (I) dividing cells expressing MyoII::RFP (magenta, with nls-RFP) with ATPα::YFP (D, green) or E-Cad::GFP (I, cyan); see also Movie S9. (E, J) Plots of the quantitation of the apical to basal movement of the midbody over time relative to AJ (cyan line: 0 to -0.5 µm) for wild-type (black, n=20), nrv2 k13315 mutant (C, red, n=20) or bark L200 mutant (F, blue, n=20) cells. Time is in hour:minute (A-B, D, F-G, I) or in minutes (C, E, H, J), with t=0 min corresponding to anaphase onset (A-B, D-G, I-J) or the time of photobleaching (C, H). Yellow lines delineate the clone border and mutant cells are identified by the loss of nls-RFP (yellow stars), cyan lines highlight the level of AJ and white arrows point to the midbody. Scale bars represent 3 μm. See also Figure S6.

Figure 7 :

 7 Figure 7: Model of SJ remodeling during cytokinesis.

Figure 1  2 Figure 3 Figure 4  5 Figure 6 

 123456 Figure 1  (A) y, w, sqh AX3 ; DE-Cad::GFP GE23 /sqh-Sqh::RFP  (B) y, w, sqh AX3 ; sqh-Sqh::RFP/+; ATPα::YFP/+  (C) sqh::RFP crispr  (D) y, w, sqh AX3 ; sqh-Sqh::RFPap-GAL4, Sep2::GFP/+; UAS-mCherry::Pnut/+ap-GAL4, Sep2::GFP/UAS-LifeAct::RFPy, w, sqh AX3 ; sqh-Sqh::RFP/+; Jupiter::YFP/+  (E-F) y, w, sqh AX3 ; ap-GAL4, Sep2::GFP/sqh-Sqh::RFP

  photobleached midbody/junction values are subtracted from the averaged backgrounds, normalized with 3 pre-FRAP values to 1 and divided by the trend over time to give the fluorescence ratio. For FRAP experiments on ATPα::YFP in wild-type and bark L200 mutant cells and on Bark::GFP, fluorescence ratios were approximated with a linear model of type: 𝑦 = 𝑎𝑥 + 𝑏, where a and b are 2 constants. For FRAP experiments on ATPα::YFP in nrv2 k13315 mosaic tissue, fluorescence ratios were approximated with an exponential model of type: 𝑦 = 𝑎 𝑒 -𝑥 𝜏 ⁄ + 𝑌 0 , where a, τ and Y0 are 3 constants. The Y0 parameter has been added to the usual exponential model, because data do not start to 0. Finally, halftimes were calculated with the equation: 𝑡 1/2 = -ln(0.5) / 𝜏. Statistical tests All information concerning the statistical details are provided in the main text and in figure legends, including the number of samples analyzed for each experiment. Histograms use the following standards: boxes limits indicate the means and errors bars represent the standard deviations. Boxplots use the following standards: centerlines of boxes show the medians; boxes limits indicate the 25 th and 75 th percentiles; whiskers extend 1.5 times the interquartile range

TABLE REAGENT

 REAGENT

	or RESOURCE SOURCE	IDENTIFIER
	Antibodies
	Mouse anti-Coracle (1:200) DSHB	C615.16
	Mouse anti-Fasciclin 3 (1:200)	DSHB	7G10
	Mouse anti-Gliotactin (1:100)	[39]	1F6.3
	Cy2-, Cy3-and Cy5-coupled secondary antibodies (1:250)	The Jackson Laboratory	N./A.
	Chemicals, Peptides, and Recombinant Proteins
	Cacodylate	Sigma	C4945
	Cyanoacrylate	Manutan	A 008749
	Epon Araldyte	Sigma	45345
	Glutaraldehyde	EMS	19344-10
	Osmium tetroxide	EMS	19152
	Paraformaldehyde	EMS	19340-72
	Phosphate Buffered Saline Lonza	BE17-515F
	Potassium ferricyanide	EMS	26604-1A
	Tannic Acid	Sigma-Aldrich	403040
	Triton X-100	Euromedex	2000 B
	Uranyl acetate	Laurylab	48/851/13
	Voltalef	VWR	24627.188
	Experimental Models: Organisms/Strains
	D. melanogaster: w 1118	BDSC	BL3605
	D. melanogaster: DE-Cad::GFP GE23	[50]	N./A.
	D. melanogaster: tub-RFP::PHGRP1	[51]	N./A.

ACKNOWLEDGMENTS

We thank S. Luschnig, R. Karess, F. Payre, the Bloomington Drosophila Stock Center, the Drosophila Genetic Resource Center, the National Institute of Genetics Fly Stock Center, and the Developmental Studies Hybridoma Bank, for providing fly stocks, antibodies. We also thank the Microscopy Rennes Imaging Center-BIOSIT facility; Y. Le Cunff for statistical analyses and T. Garber for 3D modeling; and T. Starborg from the EM facility of the University of Manchester (U.K.). We thank A. Pacquelet and members of R.L.B.'s laboratory for critical reading of the manuscript. This work was supported by the Ministère de la Recherche (E.D. and M.D.), the Canadian Institutes of Health Research (V.A.), the Ligue Nationale contre le Cancer-Equipe Labellisée (EL2014-LNCC to R.L.B.), the Fondation ARC contre le Cancer (PJA20151203510 to R.L.B.) and the ANR (ANR-16-CE13-004-01 to R.L.B.).

AUTHOR CONTRIBUTIONS

Declaration of Interests

The authors declare no competing interests.

D. melanogaster: nrv2 k13315

. Abscission is temporally uncoupled to Shrb recruitment, related to Figure 2 (A) TEM micrographs of ultrathin serial sections of WID epithelial cells sectioned parallel to the intercellular bridge (n>20). Blue line labels the plasma membrane and numbers refer to the distance (nm) that separates the sections.

(B-E) TEM micrographs of epithelial cells sectioned parallel to degrading intercellular bridges in notum (B-D, n>20) or WID (E, n>20). The cytoplasm surrounding the degraded bridges is colored in cyan and red arrows show degrading bridges in E. C is a high magnification of the intercellular bridge depicted in the red inset in B.

(F) Maximal projection of orthogonal sections showing the distribution of intercellular bridges (yellow) and degrading bridges (blue) identified using the SFB-SEM volume reconstitution in WID.

(G-H) Time-lapse of dividing cells expressing ATPα::YFP (green) together with the photoconvertible probe KAEDE. Green to red (magenta) photoconversion of KAEDE was done (yellow lightning) in randomly chosen cells from pupae at 15h APF prior to it enters in mitosis. Yellow dashed line delineates the photoconverted dividing cell and its daughters. KAEDE did not diffuse to several cell diameters. Instead, photoconverted KAEDE was either diffusing and equilibrating with only one next neighbor (G, n=4 from 175 to 75 min) or was remaining restricted to the photoconverted cell (H). In the latter case, cells were entering mitosis shortly after photoconversion (H, n=5 from 70 to 5 min). Thus, this assay is faithfully reporting cytoplasmic isolation between the daughter cells.


Warith Harchaoui
email: warith.harchaoui@oscaro.com

Pierre-Alexandre Mattei

Andrés Alamansa
email: andres.almansa@parisdescartes.fr

Charles Bouveyron
email: charles.bouveyron@math.cnrs.fr

Wasserstein Adversarial Mixture Clustering

Clustering complex data is a key element of unsupervised learning which is still a challenging problem. In this work, we introduce a deep approach for unsupervised clustering based on a latent mixture living in a low-dimensional space. We achieve this clustering task through adversarial optimization of the Wasserstein distance between the real and generated data distributions. The proposed approach also allows both dimensionality reduction and model selection. We achieve competitive results on difficult datasets made of images, sparse and dense data.

Introduction

Clustering [START_REF] Duda | Pattern classification[END_REF] is the task of making groups without the need of any manual annotations. Dimensionality reduction give clues to the underlying structure of the data. Along with dimensionality reduction, clustering is a desirable goal in data analysis, visualization and is often a preliminary step in many algorithms for example in computer vision [START_REF] Ponce | Computer vision: a modern approach[END_REF] and natural language processing [START_REF] Goldberg | Neural network methods for natural language processing[END_REF].

In this paper, the clustering is achieved from a new space through a combination of a Wasserstein Generative Adversarial Network [START_REF] Arjovsky | Wasserstein generative adversarial networks[END_REF][START_REF] Goodfellow | Generative adversarial nets[END_REF][START_REF] Gulrajani | Improved training of wasserstein gans[END_REF] and a Gaussian mixture model [START_REF] Fraley | Model-based clustering, discriminant analysis, and density estimation[END_REF] optimized in a stochastic gradient fashion. We propose an algorithm to perform unsupervised classification (a.k.a. clustering) within this framework that we call "WAMiC" for Wasserstein Adversarial Mixture Clustering. We postulate that a generative approach, namely a tuned Gaussian mixture model, can capture an explicit latent model that is the cause of the observed data, in the original space through a latent embedding.

Related Work

The purpose of this research is to build a linear-complexity algorithm that uses a non-linear embedding into a code space. Indeed, in the clustering literature, one can distinguish two kinds of clustering algorithms with respect to their speed and memory complexity in the number of examples. On one side, we have linear algorithms such as k-means (KM) and Preprint. Work in progress.

Gaussian Mixture Models (GMM), which usually work directly on the data (i.e. without any medium such as embeddings). On the other side, we also have quadratic and cubic algorithms such as Hierarchical Clustering [START_REF] Duda | Pattern classification[END_REF] and Spectral Clustering [START_REF] Andrew | On spectral clustering: Analysis and an algorithm[END_REF][START_REF] Zelnik | Self-tuning spectral clustering[END_REF][START_REF] Von | A tutorial on spectral clustering[END_REF] that use pairwise similarities to emphasize the latent clustering structure lying on the data. The proposed approach belongs to the first category although similar approaches to ours such as [START_REF] Yang | Joint Unsupervised Learning of Deep Representations and Image Clusters[END_REF] fall in the second category. Our proposed WAMiC algorithm is meant to work in a scalable fashion with any cluster shapes thanks to a latent space equipped with a tunable mixture distribution.

Clustering-aware representation learning

The importance of finding a suitable representation for clustering was first highlighted by Chang [START_REF] Chang | On using principal components before separating a mixture of two multivariate normal distributions[END_REF], who showed that embeddings based on principal component analysis were often unfit for clustering purposes.

The problem of learning representations from data in an unsupervised manner is a longstanding problem in machine learning [START_REF] Bengio | Representation learning: A review and new perspectives[END_REF][START_REF] Lecun | Deep learning[END_REF]. Principal Components Analysis (PCA) and auto-encoders (AE) which can be seen as non-linear extension of PCA [START_REF] Baldi | Neural networks and principal component analysis: Learning from examples without local minima[END_REF] have been used for representing faces [START_REF] Matthew | Face recognition using eigenfaces[END_REF] or to produce a hierarchy of features [START_REF] Chan | Pcanet: A simple deep learning baseline for image classification[END_REF]. Other techniques have been used such as sparse coding [START_REF] Mairal | Sparse representation for color image restoration[END_REF] where the representation of one image is a linear combination of a few elements in a dictionary of features. More recently Bojanowski and Joulin [START_REF] Bojanowski | Unsupervised learning by predicting noise[END_REF] learned features unsupervisedly by a procedure that consists in mapping a large collection of images to noise vectors through a deep convolutional neural networks. Their work has a clustering objective but they do not report clustering results as their real goal is unsupervised feature learning.

When it comes to compressing data while limiting loss of reconstruction information, autoencoders have proved efficient [START_REF] Vincent | Stacked denoising autoencoders: Learning useful representations in a deep network with a local denoising criterion[END_REF]. Briefly, an auto-encoder is a neural network made of two parts: (i) the encoder maps the data in a low-dimension space, (ii) the decoder maps them back to the original space. An auto-encoder is usually trained to reconstruct the data in the original space in a least squares fashion. At the end, if the reconstruction error is low, one can consider that the code resulting from the encoder has compressed the data without loosing too much information. The assumption is that the input data space of high dimensionality contains structure that could be successfully embedded in a lower-dimensionality manifold [START_REF] Alain | What regularized auto-encoders learn from the data-generating distribution[END_REF][START_REF] Sonoda | Decoding stacked denoising autoencoders[END_REF].

In our work, we try to accomplish representation learning for clustering. The first work we saw doing clustering in the code space of an auto-encoder is the one of Song et al. [START_REF] Song | Deep auto-encoder based clustering[END_REF] and more recently, but independently, Yang et al. [START_REF] Yang | Towards k-means-friendly spaces: Simultaneous deep learning and clustering[END_REF]. More precisely, they considered a KM-regularized auto-encoder loss to get a code space that is more easily clustered with KM namely their loss is the sum of the reconstruction and the KM residual. This philosophy is the one adopted for our own approach but with GMM. In our experiments, we found that optimizing the KM objective (online) when doing joint clustering and feature learning did not work well. We believe this is because it creates high magnitude gradients for points that are far away from cluster centers and there are sharp discontinuities at cluster boundaries whereas GMM diminishes that effect thanks to low density/probability values for far points. In a similar spirit, Xie et al. [START_REF] Xie | Unsupervised deep embedding for clustering analysis[END_REF] embrace the t-SNE framework [START_REF] Van Der Maaten | Visualizing data using t-sne[END_REF] in a clustering context through an auto-encoder in a non-model-based fashion. All these works tend to show that simultaneous representation learning and clustering do help each other. The reader will find an excellent review in the work of Aljalbout et al. [START_REF] Aljalbout | Clustering with deep learning: Taxonomy and new methods[END_REF].

In our preliminary experiments, a Gaussian Mixture Model trained with Expectation-Maximization [START_REF] Arthur P Dempster | Maximum likelihood from incomplete data via the EM algorithm[END_REF] on codes coming from a vanilla MLP1 auto-encoder without convolutions is able to reach approximately 80% of unsupervised clustering accuracy on the famous digits MNIST images dataset2 directly on raw pixels. The idea of our work is based on the fact that clustering and compression (dimensionality reduction) seem to be closely related tasks that auto-encoders accomplish well. Our approach is an attempt to take advantage of that empirical fact.

Deep Generative Models

In the recent literature, we observe three kinds of inference techniques for Deep Generative Models (DGM) based on likelihood, variational auto-encoders and GANs. The goal of the likelihood-based approach is to minimize the Kullback-Leibler divergence between the original data distribution and the generated data distribution. A recent example of that kind is the work of Dinh et al. [START_REF] Dinh | Density estimation using real nvp[END_REF]. On the other hand, Kingma and Welling [START_REF] Diederik | Auto-encoding variational bayes[END_REF] allow to directly specify a prior distribution over the code space of a variational auto-encoder (VAE). Inference is done using stochastic gradient variational Bayes (SGVB), a method based on a reparametrization of the variational lower bound. Deep generative models for clustering may be built using a mixture model as prior distribution. This approach was recently explored by [START_REF] Dilokthanakul | Deep unsupervised clustering with gaussian mixture variational autoencoders[END_REF] and [START_REF] Jiang | Variational deep embedding: A generative approach to clustering[END_REF] who used a GMM prior. Finally, GAN approaches minimize the Jensen-Shannon divergence in the original paper of GAN [START_REF] Goodfellow | Generative adversarial nets[END_REF] and more recently the Wasserstein distance [START_REF] Gulrajani | Improved training of wasserstein gans[END_REF] through the Wasserstein GAN (WGAN). Inspired by recent advances in Optimal Transport theory [START_REF] Villani | Optimal transport: old and new[END_REF], Cuturi [START_REF] Cuturi | Sinkhorn distances: Lightspeed computation of optimal transport[END_REF] has developped an efficient and differentiable way to compute a regularized version of the Wasserstein distance between two empirical distributions [START_REF] Genevay | Learning Generative Models with Sinkhorn Divergences[END_REF] called the Sinkhorn algorithm. For this current work, we have chosen the WGAN approach because of the remarkable quality of the data generation.

Wasserstein Adversarial Mixture Clustering

Clustering is the machine learning task of inferring a function to describe hidden structure from unlabeled data. We aim at clustering a dataset of N points x 1 , ..., x i , ..., x N samples of the random variable x living in a space X (say a D-dimensional space or even more complex). At the heart of our model, there is an auto-encoder made of: (i) an encoder network E (parametrized by θ E ) and (ii) a decoder network D (parametrized by θ D ). That auto-encoder plays the role of a bridge between the data space and a latent space more akin to clustering.

Concretely, clustering is the task of gathering the data in K homogeneous groups. Most of the time even the number K of groups can be found through model selection. This model selection step is discussed at the end of this section and briefly illustrated on synthetic data in section 3.4.

Model

We build a generative model based on two assumptions. First, we believe the data are living on a low-dimensional manifold which is a common assumption in the Machine Learning literature. Thus, there exists a latent code space Z of a low dimension d (say d = 10) such that there is a mapping D from Z to X connecting the random variable x in X to its latent counterpart z in Z. Second, we also assume that z follows a mixture distribution.

In details, our model consists in saying that the data have been generated as follows. First the clustering variable c c ∼ Cat(π)

corresponds to a categorical (multinomial) random variable for clustering with proportions defined in vector π. In this generative process, once the cluster k is chosen, one can generate a code in Z which implies a mixture marginal for z:

z|c = k ∼ g k z ∼ K k=1 π k g k
with proportions π and components (g k ) K k=1 . Ultimately a point in X is generated:

x|z ∼ N (D(z), σ 2 I p ) (2)
To translate the assumption that the code data lie extremely close to a low-dimensional manifold, we further assume that σ → 0. In this context, the posterior probability needed to cluster x is given by

p(c = k|x) = E z∼p(z|x) [p(c = k|z)] (3)

Inference

In order to fit our generative model, we build a random variable y to match x in terms of Wasserstein distance in an adversarial fashion. We only have access to samples of x (through the dataset) and y (through a random generator and D), thus we use a WGAN to fit our model. Fig. 1 summarizes the modeling proposed here.

The posterior probability p(z|x) is intractable in Eq. (3) because of its decoder part. But, as in the work of Kingma and Welling [START_REF] Diederik | Auto-encoding variational bayes[END_REF], we can approximate it using a variational inference network q(z|x) built according to the encoder E. As emphasized by Kingma and Welling [START_REF] Diederik | Auto-encoding variational bayes[END_REF], minimizing the Kullback-Leibler divergence between the true posterior and q(z|x) leads to minimizing a penalized quadratic auto-encoder loss. Since σ → 0, the dominating term in this loss will precisely be the loss of a vanilla auto-encoder which is what we do in practice for the sake of simplicity. Eventually, we can compute an approximation of p(c = k|x) by simply replacing the true posterior by the approximation, which leads to using the maximum-a-posteriori (MAP) rule in code space:

p(c = k|x) ≈ π k N (m k , Σ k )(E(x)) K k =1 π k N (m k , Σ k )(E(x)) . ( 4
)
Given a real data distribution p x (generating variable x) and a generated data distibution p y (generating variable y), a WGAN [START_REF] Arjovsky | Wasserstein generative adversarial networks[END_REF][START_REF] Gulrajani | Improved training of wasserstein gans[END_REF] minimizes the Kantorovich-Rubinstein duality of the Wasserstein distance between these two distributions:

max ∇f ≤1 E x [f (x)] -E y [f (y)] (5)
with f called the critic and implemented in practice by a neural network of parameters θ f and constrained by an augmented Lagrangian (see [START_REF] Gulrajani | Improved training of wasserstein gans[END_REF] for details):

max θ f E x [f (x)] -E y [f (y)] -λE x ( ∇f (x) -1) 2 (6)
where x is sampled from segments between x and y.

In the work of Gulrajani et al. [START_REF] Gulrajani | Improved training of wasserstein gans[END_REF], a WGAN usually uses a simple fixed distribution (Gaussian or uniform) for the random noise generator that is transformed by a neural network called the generator to fit the data distribution in the Wasserstein sense. We use a tunable mixture distribution instead for clustering purposes.

Inspired by the work of Kingma and Welling [START_REF] Diederik | Auto-encoding variational bayes[END_REF], we need to specify for each cluster k a differentiable reparametrization. The codes z k coming from each Gaussian with full covariance matrices Σ k follow: z k = S k e + m k (7) and y k = D(z k ) (8) are our generated data for each cluster k where: e ∼ N (0, I) and S k is a Cholesky decomposition (with only non-zeros in the lower-triangular part) of the full covariance Σ k (with positive diagonal entries in S k parametrized with exponentials). The Kantorovich-Rubinstein duality Eq. ( 5) becomes:

max ∇f ≤1 E x [f (x)] - K k=1 π k E y k [f (y k )] (9)
and:

max θ f E x [f (x)] - K k=1 π k E y k [f (y k )] -λE x ( ∇f (x) -1) 2 (10)
is the regularized version of Eq. ( 9) for unconstrained optimization purposes.

Following the work of Gulrajani et al. [START_REF] Gulrajani | Improved training of wasserstein gans[END_REF] for WGAN, we are trying to make our generated data "indistinguishable" from the real data in the Wasserstein distance sense. In our case, the source of generated noise is coming from our mixture M through the generator/decoder D. Thus, we optimize: min

θ D ,θ M L(θ D , θ M ) (11)
where:

L(θ D , θ M ) = max θ f E x [f (x)] - K k=1 π k E y k [f (y k )] -λE x ( ∇f (x) -1) 2 (12)

Algorithm

Our WAMiC algorithm can be decomposed in four different steps:

1. Auto-Encoder Initialization: We train a classic auto-encoder (E, D) (for an encoder E and decoder D):

(θ 0 E , θ 0 D ) = arg min θ E ,θ D E x∼Data D • E(x) -x 2 2 (13)

EM-based Gaussian Mixture Initialization:

We fit a Gaussian Mixture Model M with the Expectation-Maximization algorithm [START_REF] Arthur P Dempster | Maximum likelihood from incomplete data via the EM algorithm[END_REF] on the encoded data:

θ 0 M = arg max θ M E x∼Data log K k=1 π k N (m k , Σ k ) E 0 (x) (14)
3. Clustered Data Generation: We optimize the mixture M and the decoder D (that now plays the role of a generator) to minimize the regularized Wasserstein distance presented above with the initialization coming from the two previous steps;

( θD , θM ) = arg min

θ D ,θ M L ( 15
)
from Eq. ( 12) 4. Clustering Decoding: We train a new encoder E with an auto-encoder quadratic loss with the decoder D just found maintained fixed.

θE = arg min

θ E E x∼Data D • E(x) -x 2 2 ( 16
)
Steps 1 and 2 are just intuitive initialization of step 3.

Step 4 can be seen as an ad hoc way of inverting the decoder D. Several other strategies have been proposed for that purpose (e.g. [START_REF] Vincent Dumoulin | Adversarially learned inference[END_REF][START_REF] Tolstikhin | Wasserstein auto-encoders[END_REF]). We use the MAP rule in the code space in order to finally cluster the data points which makes our simple approach particularly fit for clustering.

Model Selection

Step 3 of section 3.3 minimizes an estimated Wasserstein distance between the real and generated data distribution. That step's role is to generate clustered data close to the real data in terms of distributions for the Wasserstein criteria. Once trained, we can measure the Wasserstein distance between generated data and some held-out validation data to check under/over-fitting and ultimately choose the number of classes.

Traditionally, model selection is accomplished according to the likelihood (or completed likelihood) itself related to the Kullback-Leibler divergence (see [START_REF] Schwarz | Estimating the dimension of a model[END_REF] for example). One of the originality of our approach is that our model selection technique is done according to an other divergence which is the Wasserstein distance. This parallel allows us to use the historical likelihood-based literature by replacing the well-known Kullback-Leibler divergence by the Wasserstein distance.

More concretely, to choose the number of clusters, we first train step 1 of section 3.3 once for all, then for each number of clusters, we do steps 2 and 3 in parallel. At the end, one can take the best Wasserstein distance and do step 4 to get the best clustering according to these models. For our numerical optimizations, we estimate the Wasserstein distance on batches only. So, the Wasserstein score that we use for model selection is simply the mean of our batch Wasserstein estimations once our f function is finally trained.

Numerical Experiments

Implementation details

We did our experiments in Python by using the PyTorch [START_REF] Paszke | Automatic differentiation in pytorch[END_REF] and Scikit-learn [START_REF] Pedregosa | Scikit-learn: Machine learning in Python[END_REF] libraries. with the same learning rate of 10 -5 with the Adam optimization strategy [START_REF] Diederik | Adam: A method for stochastic optimization[END_REF] everywhere.

Given a clustering result and the ground truth, one can measure the quality of the clustering result according to what we call the "accuracy" computed with the Hungarian Method [START_REF] Kuhn | The hungarian method for the assignment problem[END_REF][START_REF] Stephens | Dealing with label switching in mixture models[END_REF] on the confusion matrix.

Synthetic Experiment

To illustrate our approach, we first work on a toy dataset that we call "Three Moons" with 1000 points for each of the 3 classes in 2 dimensions as presented in Fig. 2(a). With a code space of dimension 1, even though the clusters are not linearly separable in the original data space, our system is able to cluster them with 100% of accuracy.

For that simple toy dataset, most of the clustering work is done by the vanilla auto-encoder. Indeed, once the auto-encoder is trained, we observed that the 1D codes are already separated according to the cluster labels. Here, we just wanted to see if model selection was plausible in a simple scenario.

For the number of clusters, without labels in our unsupervised context, while a classificationscore-based cross-validation is not an option, one can still measure an estimate of the Wasserstein distance (our loss) but on a validation set. The intuition behind is that if we did not overfit, our loss function will still be satisfactory on that validation set (that was not seen during training). After running our four steps algorithm, we can train a new neural network f to measure the Wasserstein distance between a held-out validation dataset and some generated data empirical distributions. More precisely, we find the results presented in Fig. 2(b) actually selecting 3 clusters which is satisfactory.

Real-world Experiments

For the real-world experiments, we were interested in 4 datasets that are different in nature (images, sparse and dense data):

• MNIST: 70 000 handwritten digits images dataset living in dimension 784 (for 28×28 pixels); • Reuters: English news stories labeled with a category tree [START_REF] David D Lewis | Rcv1: A new benchmark collection for text categorization research[END_REF]. Following DEC [START_REF] Xie | Unsupervised deep embedding for clustering analysis[END_REF], we used 4 root categories: corporate/industrial, government/social, markets, and economics as labels and discarded all documents with multiple labels. We computed tf-idf features on the 2000 most frequent words to represent all articles;

• Reuters-10k: a random subset of Reuters with only 10 000 examples (selected with the same random seed as DEC);

• HHAR: The Heterogeneity Human Activity Recognition (HHAR) dataset [START_REF] Stisen | Smart devices are different: Assessing and mitigatingmobile sensing heterogeneities for activity recognition[END_REF] contains 10299 sensor records from smart phones and smart watches. All samples are partitioned into 6 categories of human activities and each sample is of 561 dimensions.

Throughout the experiments, we used the same architecture from Xie et al. [START_REF] Xie | Unsupervised deep embedding for clustering analysis[END_REF] D-500-500-2000-d (D is the dimensionality of the input space e.g. 784 for MNIST and d = 10 is the dimensionality of the code space) for the encoder for fair comparisons with VaDE [START_REF] Jiang | Variational deep embedding: A generative approach to clustering[END_REF].

The most difficult and longest part of the optimization is step 3 of section 3.3. Indeed, this is a min-max optimization that is inherent to WGANs. As we can see in Fig. (3), at the beginning, the Wasserstein distance is not correctly estimated and is even negative: one has to wait for a good critic in the first 50k iterations before seeing the loss actually decreasing meaningfully. 4), we generated data further and further in random directions from the centroids: we see that the digits get fancier away from the centroids. The good quality of the generation is comparable to those of regular GANs but in a cluster-wise fashion.

Results

The results of our approach compare favorably to the deep clustering state-of-the-art in Table . (1). First, we observe that there is an improvement over standard baseline algorithms (GMM and KM) when fed with the output of an AE which is coherent with the results of Xie et al. [START_REF] Xie | Unsupervised deep embedding for clustering analysis[END_REF]. Furthermore, there is a supplementary significant adversarial improvement for WAMiC . 1: Experimental accuracy results (%, the higher, the better) based on the Hungarian method. The top four lines correspond to the same architecture. networks with just a few layers of the 1990s3 . On MNIST, our unsupervised neural network is almost competitive with its supervised counterparts which is surprising and tells us this dataset is not anymore a difficult one even in the non-convolutional unsupervised context.

WAMiC compares favorably to the other frameworks that share the same network architecture (VaDE and DEC). On MNIST, the important performance gap between VaDE and GMAC can be explained by the fact that variational auto-encoders have difficulties with data that lie extremely close to low-dimensional manifolds, like images (see e.g. [START_REF] Arjovsky | Wasserstein generative adversarial networks[END_REF]). In that regard, our algorithm inherits the strenghts of WGAN and models image data much more faithfully.

Conclusion

We present WAMiC a mixture-based Deep Generative Model for Clustering by combining WGAN and variational inference. Empirically, we observe some symbiosis operating between clustering and non-linear embedding. For future work, we want to explore a great advantage of our generative approach: Wasserstein model selection on real data for selecting the number of clusters.

Figure 1 :

 1 Figure 1: WAMiC Optimization Scheme

Figure 2 :

 2 Figure 2: Wasserstein model selection on the three moons dataset for the number of clusters on a validation dataset, (on 30 runs for each number of clusters)

Figure 3 :

 3 Figure 3: Mini-Batch Wasserstein Distance Estimated on Reuters-10k through Iterations (with rolling mean in red)

Figure 4 :

 4 Figure 4: Generated digits images. From left to right, we have the ten classes found by WAMiC and ordered thanks to the Hungarian algorithm. From top to bottom, we go further and further in random directions from the centroids (the first row being the decoded centroids). More specifically, u is sampled from the uniform random density on the unit hypersphere in the code space.

  These good results are comparable to those of supervised non-convolutional

	Datasets	MNIST Reuters Reuters-10k HHAR
	WAMiC (ours)	98.42	79.24	79.87	81.42
	VaDE [22]	94.06	79.38	79.83	84.46
	DEC [45]	84.30	75.63	72.17	79.82
	AE + GMM (full covariance)	82.56	70.98	70.12	78.48
	IMSAT ([21], different architecture)	98.40	-	71.00	-
	GAR ([23], convolutional net)	98.32	-	-	-
	DEPICT ([13], convolutional net)	96.50	-	-	-
	GMM (diagonal covariance)	53.73	55.81	54.72	60.34
	KM	53.47	53.29	54.04	59.98
	Table

Multi-Layered Perceptron

http://yann.lecun.com/exdb/mnist/

see http://yann.lecun.com/exdb/mnist for a complete MNIST benchmark

Acknowledgments

The authors would like to thank Vincent Delaitre from Deepomatic.com, Olivier Grisel from INRIA, Georges Oppenheim from Université de Paris Sud and Université Paris Est and Pierre Roussillon from Université Paris Saclay for fruitful discussions and valuable inputs. We gratefully acknowledge the support of NVIDIA Corporation with the donation of the Titan Xp GPU used for this research. Finally, we thank Oscaro.com for the funding and support.