Valeria Davì

Hirokazu Tanimoto

Dmitry Ershov

Armin Haupt

Henry De Belly

Rémi Le Borgne

Etienne Couturier

Arezki Boudaoud
email: arezki.boudaoud@ens-lyon.fr

Nicolas Minc
email: nicolas.minc@ijm.fr

Remi Le Borgne

Mechanosensation Dynamically Coordinates Polar Growth and Cell Wall Assembly to Promote Cell Survival

Keywords: Cell Growth, Mechanosensing, Cell Wall, Polarity, Fungi, Fission Yeast

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

Growth control is of fundamental importance in biology, from the regulation of macromolecular assembly, cell physiology, up to organ development [START_REF] Holley | Control of growth of mammalian cells in cell culture[END_REF][START_REF] Hong | Variable Cell Growth Yields Reproducible OrganDevelopment through Spatiotemporal Averaging[END_REF][START_REF] Lander | Pattern, growth, and control[END_REF][START_REF] Mahadevan | Cell biology: powerful curves[END_REF]. The growth of individual cells, for instance, has crucial implications for cell size determination, tissue homeostasis or cancer progression (DeBerardinis et al., 2008a;Deberardinis et al., 2008b;[START_REF] Fantes | Control of cell size at division in fission yeast by a growthmodulated size control over nuclear division[END_REF][START_REF] Marshall | What determines cell size?[END_REF][START_REF] Thompson | Developmental control of cell growth and division in Drosophila[END_REF]. To date, however, the mechanisms which control cellular growth remain poorly understood. This is because growth is an integrated output of multiple intertwined biochemical and biomechanical elements which dynamically probe and alter the cell surface to accommodate surface expansion. Those may include surface material synthesis mediated by processes such as exocytosis and endocytosis [START_REF] Hepler | Polarized cell growth in higher plants[END_REF][START_REF] Novick | Secretion and cell-surface growth are blocked in a temperature-sensitive mutant of Saccharomyces cerevisiae[END_REF], as well as osmotic forces and mechanical elements which set the elasticity of the cell surface, like the actin cortex, the glycocalyx or the CW [START_REF] Davi | Mechanics and morphogenesis of fission yeast cells[END_REF][START_REF] Huang | The structural and mechanical complexity of cell-growth control[END_REF][START_REF] Salbreux | Actin cortex mechanics and cellular morphogenesis[END_REF]. How those modules which act at various time and length scales may dynamically feedback onto each other to control the rate of cell surface expansion remain an outstanding open question.

Cell growth has been admittedly best documented in single walled cells, such as bacteria, fungi, or pollen tubes [START_REF] Harris | Relative Rates of Surface and Volume Synthesis Set Bacterial Cell Size[END_REF][START_REF] Mitchison | Growth in cell length in the fission yeast Schizosaccharomyces pombe[END_REF][START_REF] Qin | Rapid tip growth: insights from pollen tubes[END_REF][START_REF] Rojas | Response of Escherichia coli growth rate to osmotic shock[END_REF] . Those are some of the fastest growing cells, with some fungal cells elongating at rates up to several tens of microns per minute [START_REF] Lew | How does a hypha grow? The biophysics of pressurized growth in fungi[END_REF][START_REF] Qin | Rapid tip growth: insights from pollen tubes[END_REF].

Size expansion in those cells is irreversible and limited by the synthesis of a thin and rigid sugar-made CW around the plasma membrane. The growth of the CW is thought to involve a complex balance between sugar synthesis which build the wall, and mechanical expansion driven by large internal turgor pressure which put the wall under tension to deform it [START_REF] Cosgrove | Growth of the plant cell wall[END_REF][START_REF] Davi | Mechanics and morphogenesis of fission yeast cells[END_REF][START_REF] Harold | Force and compliance: rethinking morphogenesis in walled cells[END_REF][START_REF] Mckenna | Exocytosis precedes and predicts the increase in growth in oscillating pollen tubes[END_REF] . Because the CW bears large turgor-derived stress, it also provides mechanical integrity to those cells: removal or weakening of the CW yields near-immediate cell lysis and death. Given those considerations, cell growth may be seen as a dangerous process, as uncontrolled expansion and consequent thinning of the CW could compromise cell survival. To date, however, our understanding of how CW composition and thickness may be dynamically controlled during cell growth has been limited, as historically it has been mostly studied with Electron Microscopy, which has biased our appreciation of the CW towards a static structure.

The biochemistry and genetics which support CW growth and composition, has been extensively studied in model yeast cells [START_REF] Lipke | Cell wall architecture in yeast: new structure and new challenges[END_REF][START_REF] Perez | Cell wall analysis[END_REF]. The rod-shaped fission yeast Schizosccaromyces pombe, for instance, serve as a prime model for walled cell tip growth, stereotypical of many bacteria, fungi and plant cells [START_REF] Chang | How and why cells grow as rods[END_REF][START_REF] Chang | Shaping fission yeast with microtubules[END_REF][START_REF] Davi | Mechanics and morphogenesis of fission yeast cells[END_REF]. The fission yeast CW behaves as a thin elastic shell with an elastic modulus estimated to be around 30-50MPa which resists an internal turgor pressure of 1-1.5 MPa [START_REF] Abenza | Wall mechanics and exocytosis define the shape of growth domains in fission yeast[END_REF][START_REF] Atilgan | Morphogenesis of the Fission Yeast Cell through Cell Wall Expansion[END_REF][START_REF] Minc | Mechanical forces of fission yeast growth[END_REF]. In those cells, CW synthesis is restricted to cell tips during phases of cell elongation and to the cell middle for septation. At cell tips, wall synthesis is catalyzed by one α-glucan (Ags1) and three β-Glucan (Bgs1, Bgs3 and Bgs4) synthases, which elongate glucan fibers, as well as glucanosyl-transferases enzymes (Gas1 and Gas2) and one putative exo-glucanase, Exg2, which may remodel and/or crosslink glucan fibers at cell tips [START_REF] Davi | Mechanics and morphogenesis of fission yeast cells[END_REF][START_REF] Perez | Cell wall analysis[END_REF]. Those enzymes are under the control of the activity of the highly conserved Rho-GTPases, Rho1 and Rho2, which are activated by the GEFs Rgf1 and Rgf2 [START_REF] Arellano | Schizosaccharomyces pombe protein kinase C homologues, pck1p and pck2p, are targets of rho1p and rho2p and differentially regulate cell integrity[END_REF][START_REF] Garcia | Rgf1p is a specific Rho1-GEF that coordinates cell polarization with cell wall biogenesis in fission yeast[END_REF]. Damages on the CW , such as those caused by antifungal agents are monitored by the Cell Wall Integrity pathway which can activate Rho1 and Rho2, and the expression of a set of CW repair genes through the Pmk1 MAPK [START_REF] Perez | Cell integrity signaling and response to stress in fission yeast[END_REF].

To understand how those different biochemical layers dynamically influence and probe the mechanics of the CW, we here introduce a novel microscopy method to directly measure CW thickness all around live growing fission yeast cells. We find that CW thickness at growing cell tips is highly dynamic in time, with phases of thickening followed by thinning phases, and vice e versa. We demonstrate a homeostatic system which accounts for thickness oscillations and stable values at the cell population. Homeostasis is supported by mechanosensing activities of the CWI which probe cellular growth as a mechanical stress and dynamically adjust synthesis to growth rates. Mutants defective in homeostatic thickness lyse by over-thinning the CW. This work evidence mechanochemical feedbacks promoting growth control and cell survival.

RESULTS

A sub-resolution method to monitor cell wall thickness dynamics in living cells

The fission yeast CW is a thin layer of typically 100-200nm, below the resolution of lightmicrosopy. Given large internal turgor pressure, the plasma membrane is plastered against the internal face of the CW [START_REF] Osumi | Visualization of yeast cells by electron microscopy[END_REF]. In addition, glycosylated galactomannan proteins are secreted and retained at the most outer surface of the wall, and can be detected with specific lectins [START_REF] Horiseberger | Localization of alpha-Galactomannan on the surface of Schizosaccharomyces pombe cells by scanning electron microscopy[END_REF]. We exploited those properties to develop a subresolution method to compute CW thickness all around single live growing fission yeast cells.

We labeled the inner and outer faces of the CW respectively using an integral membrane SNARE protein tagged with a GFP at its intracellular tail (GFP-psy1) [START_REF] Nakamura | The Schizosaccharomyces pombe spo3+ gene is required for assembly of the forespore membrane and genetically interacts with psy1(+)-encoding syntaxin-like protein[END_REF] and by adding Lectins from Griffonia simplicifolia that specifically bind the most outer face of the fission yeast CW, labeled with a different emitting fluorophore. Following image registration and cell contour segmentation, we computed the distance between the centers of the two fluorescent signals along lines bisecting the cell surface, which provided a local measurement of CW thickness [START_REF] Chugh | Actin cortex architecture regulates cell surface tension[END_REF][START_REF] Clark | Monitoring actin cortex thickness in live cells[END_REF]. Thickness measured by those means could be calculated with a precision around ~30 nm and a lateral resolution of ~500 nm, which we represented as colored cellular maps (Figure 1A To validate our approach, we first compared it with Transmission Electron Microscopy (TEM) measurements, which serve as the current standard to image the CW [START_REF] Osumi | Visualization of yeast cells by electron microscopy[END_REF].

The mean CW thickness in a wild-type population measured by our method was 170 ± 23 nm.

This was comparable to values obtained from High Pressure Freezing TEM (172 ± 14 nm) and ~16% higher than values obtained in chemical fixation TEM (143 ± 19 nm), plausibly because chemical fixation may not fully protect the CW against subsequent resin embedding (Osumi, 2012) (Figure 1B). Next, we used Correlative Light Electron Microscopy to image and compute CW thickness in the same cell with TEM and our method. This yielded nearly similar patterns of thickness along the cell boundary, supporting the reliability of our method as a local measurement of CW thickness (Figure 1C and Figure S2D-S2G).

To assess the accuracy, precision and range of this live imaging method, we next tested it against conditions thought to affect CW thickness. As previously suggested, we detected a dose-dependent increase in CW thickness with temperature or as a result of prolonged starvation (Figure 1D-E) [START_REF] Cassone | Ultrastructural changes in the cell wall of Candida albicans following cessation of growth and their possible relationship to the development of polyene resistance[END_REF]. Cells treated with zymolyase, a CW digesting enzyme, exhibited significant CW thinning (Figure 1F). Finally, thickness could be imaged in the same cell over several hours with no major cellular damage and no impact of bleaching on the measurement (Figure 1G, Figure S3A-S3B and movie S1 and S2). Thus, it was possible to quantify the dynamics of CW thickness and assembly with unprecedented accuracy in populations of live growing cells.

Polar growth sites are associated with thinner and softer cell walls.

Given that wall assembly in fission yeast is restricted to growing cell tips during interphase and to the cell middle during septation, we computed CW thickness patterns in cell populations [START_REF] Chang | Shaping fission yeast with microtubules[END_REF][START_REF] Davi | Mechanics and morphogenesis of fission yeast cells[END_REF]. Birth scars which are inherited from previous cell division were associated with a significantly thicker CW, with a thickness of hscar = 224 ± 20nm. Interestingly, we observed that the CW was consistently thinner at the old end, the end which grows most during interphase, hoe = 130 ± 26 nm, as compared to cell sides, hside = 170 ± 20 nm. Thickness at the new end, which only grows moderately after New End take Off (NETO) was in contrast only slightly thinner than cell sides, hne = 167 ± 14 nm (Figure 2A). We also found that monopolar tips of outgrowing spores, spheroplasts, mating projections or of tea1Δ mutants branching from cell sides were also markedly thinner at the front growing site than on the non-growing back [START_REF] Bonazzi | Symmetry Breaking in Spore Germination Relies on an Interplay between Polar Cap Stability and Spore Wall Mechanics[END_REF][START_REF] Kelly | De novo growth zone formation from fission yeast spheroplasts[END_REF][START_REF] Mata | tea1 and the microtubular cytoskeleton are important for generating global spatial order within the fission yeast cell[END_REF][START_REF] Petersen | Conjugation in S. pombe: identification of a microtubule-organising centre, a requirement for microtubules and a role for Mad2[END_REF] (Figure 2B). Not surprisingly, these thin wall regions at sites of polar growth co-localized with the downstream polarity regulator GTP-Cdc42 (visualized with a CRIB domain fused to td-Tomato) and with the β-Glucan synthase Bgs4 [START_REF] Cortes | The novel fission yeast (1,3)beta-D-glucan synthase catalytic subunit Bgs4p is essential during both cytokinesis and polarized growth[END_REF][START_REF] Tatebe | Pom1 DYRK regulates localization of the Rga4 GAP to ensure bipolar activation of Cdc42 in fission yeast[END_REF] (Figure 2C). Furthermore, time-lapse of polarized regrowth in tea1Δ cells recovering from starvation evidenced the stabilization of a Bgs4 polar domain, followed by concomitant CW thinning and emergence of a new growing tip (Figure 2D and Movie S3). Importantly, full depolymerization of actin with 100µM Latrunculin A or the use of an orb6-25 thermosensitive allele affecting Cdc42-based polarity, both yielded the detachment of GTP-Cdc42 polar domains from cell tips, and consequent alteration of CW thickness patterns within tens of minutes [START_REF] Bendezu | Actin cables and the exocyst form two independent morphogenesis pathways in the fission yeast[END_REF][START_REF] Das | The conserved NDR kinase Orb6 controls polarized cell growth by spatial regulation of the small GTPase Cdc42[END_REF] (Figure 2E).

We also found that the CW was softer at growing cell tips. This was evidenced by computing CW elastic moduli, Y, which characterizes the bulk mechanical properties of the CW. To this aim, we plasmolyzed cells by rapidly adding 1.5M sorbitol to the medium in microfluidic flow channels which causes a rapid drop in turgor pressure, and consequent cell shrinkage [START_REF] Abenza | Wall mechanics and exocytosis define the shape of growth domains in fission yeast[END_REF][START_REF] Atilgan | Morphogenesis of the Fission Yeast Cell through Cell Wall Expansion[END_REF][START_REF] Bonazzi | Symmetry Breaking in Spore Germination Relies on an Interplay between Polar Cap Stability and Spore Wall Mechanics[END_REF]. Using the changes in cell shapes between inflated and plasmolyzed states and local thickness values, we could extract values of CW elastic moduli at cell tips and at cell sides, to be Ytip=45.3 +/-0.7 MPa; and Yside=72.5 +/-1.15 MPa, yielding a ratio in surface moduli (σ=hY) of σ tip/ σ side= 1,7+/-0.2 (Figure 2F and Figure S3C-S3D). Together, those results suggest that internal polarity which directs growth and CW synthesis causes sites of polar growth to have thinner and softer CW.

Contribution of synthesis and growth to CW thickness at cell tips

Those findings prompted us to assess how CW thickness and assembly may be determined and regulated at sites of polar growth. We first considered a minimal theoretical model for thickness regulation at cell tips (Figure 3A). We posited that the CW thickens through synthesis and thins through cell elongation, which, given mass conservation yields a dynamic evolution of htip:

dℎ 𝑡𝑖𝑝 dt = 𝑆 -𝛾 ℎ 𝑡𝑖𝑝 (Eq. 1)
where S is a rate of wall synthesis per unit surface and γ the strain rate of the CW, which is related to the cell elongation rate, Gr, by the curvature radius at the tip, Rc:

Gr = dL/dt = q𝑅 𝑐 𝛾 , (Eq. 2)
with L the length of the cell and q a numerical geometrical pre-factor.

To assay the contribution of synthesis, we performed experiments to manipulate synthase activities. In cells overexpressing Pck2, a positive regulator of glucan synthesis, one major component of the fission yeast CW, we could detect significant thickening, up to ~900 nm in some cells [START_REF] Arellano | Schizosaccharomyces pombe protein kinase C homologues, pck1p and pck2p, are targets of rho1p and rho2p and differentially regulate cell integrity[END_REF]. Similar thickening was also observed by over-expressing a constitutively active Rho1 allele, rho1-G15V [START_REF] Arellano | Rho 1 GTPase activates the (1-3)beta-D-glucan synthase and is involved in Schizosaccharomyces pombe morphogenesis[END_REF] (Figure 3B). Conversely, a cwg1-1 thermo-sensitive allele of the glucan synthase Bgs4, exhibited significant thinning at growing cell tips, consistent with previous TEM observations [START_REF] Munoz | Extracellular cell wall beta(1,3)glucan is required to couple septation to actomyosin ring contraction[END_REF] (Figure 3C). As a more direct assay, we also reduced CW synthesis with low doses of Caspofungin, a drug that specifically impairs β-glucan synthase activity [START_REF] Martins | Differential activities of three families of specific beta(1,3)glucan synthase inhibitors in wild-type and resistant strains of fission yeast[END_REF]. This treatment led to a net thickness decrease at growing ends over tens of minutes, eventually causing aberrant cell bulging at longer times (Figure 3D and movie S4).

Using those Caspofungin treated cells, in which synthesis is partially impaired, we also computed the contribution of growth to thickness variation. We measured the relative changes in thickness, Δhoe/hoe, which, given Eqs. 1 and 2, is predicted to linearly scale with -ΔLoe/Rc.

This analysis yielded a negative linear correlation with a slope of -0.81 (Correlation coefficient, r=-0.57), and a positive Y-intersect, which could correspond to remaining CW αglucan synthesis in the presence of this drug (Figure 3E). Together, those data support a simple model for CW thickness evolution positively regulated by synthesis and negatively by growth.

Influence of cell wall thickness on cell growth rates

As the balance between turgor and CW mechanics has been proposed to set growth rates in walled cells [START_REF] Bonazzi | Symmetry Breaking in Spore Germination Relies on an Interplay between Polar Cap Stability and Spore Wall Mechanics[END_REF][START_REF] Minc | Mechanical forces of fission yeast growth[END_REF], we sought to investigate the influence of thickness on strain and growth rates. Using a simple model for wall rheology [START_REF] Minc | Mechanical forces of fission yeast growth[END_REF][START_REF] Rojas | Chemically mediated mechanical expansion of the pollen tube cell wall[END_REF], we posited that the strain rate could be written as:

𝛾 = 𝑟 𝑃𝑅 𝐶 𝑌ℎ 𝑡𝑖𝑝 (Eq. 3)
With r a wall remodeling rate, Y the elastic modulus of the wall, and P the turgor pressure (We here neglect a possible threshold in turgor for growth to occur). Given Eq. 2, this assumption implies that the elongation rate Gr scales with 𝑅𝑐 2 /ℎ 𝑡𝑖𝑝 . This scaling was confirmed using 3D numerical simulations of a mechanical model of fission yeast growth [START_REF] Abenza | Wall mechanics and exocytosis define the shape of growth domains in fission yeast[END_REF] (Figure 3F).

Experimentally, we used time-lapse movies to compute and plot elongation rates, as a function of initial tip curvature radii and CW thickness. To vary Rc, we used wild-type cells and mutants with larger (rga4Δ) and smaller diameters (rga2Δ), as well as microfabricated channels to physically diminish the diameter of rga4Δ cells [START_REF] Das | Regulation of cell diameter, For3p localization, and cell symmetry by fission yeast Rho-GAP Rga4p[END_REF][START_REF] Villar-Tajadura | Rga2 is a Rho2 GAP that regulates morphogenesis and cell integrity in S. pombe[END_REF][START_REF] Zegman | Measurement and manipulation of cell size parameters in fission yeast[END_REF]. This analysis yielded, as predicted, a robust linear dependency between Gr and 𝑅𝑐 2 /ℎ 𝑡𝑖𝑝 indicating that CW thickness could serve as a major determinant influencing elongation rates (Figure 3G-3I).

Homeostasis in Cell wall thickness through strain mechanosensing

The above findings suggest that without regulatory layers, growth and thickness may negatively regulate each other, which could potentially lead to catastrophic situations of cell lysis through CW thinning or growth arrest through thickening. We thus computed tip thickness evolution in single cells over 1h, with a high temporal resolution of 6min, focusing on the old end. Strikingly, this revealed a thickness oscillatory behavior with phases of CW thickening, followed by thinning phases, and so on. Thickness on cell sides, was in contrast almost completely stable (Figure 4A-4B and Figure S4A-S4D). Those oscillating variations at cell tips, suggested the presence of a delayed feedback dynamically regulating CW thickness and assembly at growing tips.

Based on reported observations in fission yeast and other cell types [START_REF] Bonazzi | Symmetry Breaking in Spore Germination Relies on an Interplay between Polar Cap Stability and Spore Wall Mechanics[END_REF][START_REF] Nakayama | Mechanical regulation of auxin-mediated growth[END_REF], we assayed a potential positive feedback between growth, or equivalently strain rate, on polarized synthesis. To reduce strain rate, we rapidly decreased turgor, by rinsing cells with medium containing 0.5M sorbitol using a gpd1 background to prevent turgor adaptation [START_REF] Minc | Mechanical forces of fission yeast growth[END_REF]. This caused an immediate growth arrest, and the rapid detachment of active-Rho1 polar domains [START_REF] Davidson | Regulation of Rho-GEF Rgf3 by the arrestin Art1 in fission yeast cytokinesis[END_REF] followed by the detachment of Bgs4 domain over the subsequent minutes (Figure 4C-4D and Figure S4E).

Those data suggest that CW synthase activity and localization may be actively sensitive to growth or strain rates.

To incorporate mechanical sensing in our model, we used a general delayed dependence, between strain rate and synthesis, of the form: S(t)=S0+λ*γ(t-T), with S0 a basal synthesis rate, λ a positive parameter characterizing the strength of the sensing and T a delay in the response between strain and synthesis, so that:

dℎ 𝑡𝑖𝑝 dt (𝑡) = ( 𝑆 0 - 𝑟𝑃𝑅 𝐶 𝑌 ) + λ 𝑟𝑃𝑅 𝐶 𝑌 1 ℎ 𝑡𝑖𝑝 (𝑡-𝑇) (Eq. 4)
Using the evolution of old end thickness in single cells, we thus evaluated experimentally the dependence of the thickness rate, dhoe/dt (t0+T) computed at different delays, on the inverse of initial thickness 1/hoe (t0), measured at various t0 in time-lapses. We found a positive correlation at delays starting around 12 min (r=0.4) which improved for longer delays, yielding highest correlations at delays around 30-40 min (r=0.8) (Figure 5A and

The Cell Wall Integrity (CWI) pathway mediates strain mechanosensing for thickness homoestasis

A prime candidate system which may sense CW properties and influence synthesis is the cell wall integrity pathway (CWI), a signaling cascade involved in general CW stress response [START_REF] Perez | Cell integrity signaling and response to stress in fission yeast[END_REF]. In fission yeast, this pathway is thought to be activated at the cell surface, by the putative trans-membrane stress sensors Wsc1 and Mtl2 which interact with the CW plausibly through long serine/threonine rich domains extending in the CW matrix [START_REF] Cruz | The fission yeast cell wall stress sensor-like proteins Mtl2 and Wsc1 act by turning on the GTPase Rho1p but act independently of the cell wall integrity pathway[END_REF]. Those factors signal downstream to Rho-GEFs Rgf1 and Rgf2 which may directly regulate the Rho-GTPases Rho1 and Rho2 for glucan synthase activation (Garcia et al., 2009a;[START_REF] Garcia | Rgf1p is a specific Rho1-GEF that coordinates cell polarization with cell wall biogenesis in fission yeast[END_REF][START_REF] Mutoh | Rho1-GEFs Rgf1 and Rgf2 are involved in formation of cell wall and septum, while Rgf3 is involved in cytokinesis in fission yeast[END_REF] . The CWI also triggers the activation of gene transcription for CW repair, through the Pmk1 MAPK cascade [START_REF] Garcia | The Rho1p exchange factor Rgf1p signals upstream from the Pmk1 mitogen-activated protein kinase pathway in fission yeast[END_REF][START_REF] Perez | Cell integrity signaling and response to stress in fission yeast[END_REF]. By performing a candidate screen on single mutants of the CWI, we found that wsc1Δ, rgf1Δ and pmk1Δ, were severely defective in thickness homeostasis, with changes in thickness being mostly independent on previous thickness values (Figure 5C-5D and Figure S6A-B). In addition, the detachment of Bgs4 and active-Rho1 polar domains as a response to pressure and strain reduction in a gpd1Δ wsc1Δ mutant was much less pronounced than in a gpd1Δ alone, suggesting that Wsc1 may directly probe CW strain rate to tune synthesis through the CWI (Figure 5E-5G and Figure S6C).

We then assayed if the implication of the CWI pathway in thickness homeostasis could account for the lysis defect of some CWI mutants during normal growth [START_REF] Cruz | The fission yeast cell wall stress sensor-like proteins Mtl2 and Wsc1 act by turning on the GTPase Rho1p but act independently of the cell wall integrity pathway[END_REF][START_REF] Garcia | Rgf1p is a specific Rho1-GEF that coordinates cell polarization with cell wall biogenesis in fission yeast[END_REF]. By imaging thickness evolution in rgf1Δ, we observed in ~30% of cases, cells that started with a relative thick wall and little growth, which switched incidentally to a phase of unusually rapid growth concomitant with rapid CW thinning eventually yielding to tip lysis and cell death (Figure 6A-6B). We noted, however, that the thickness at which lysis occurred in some of those cells could have values similar to wild-type tips. As the failure strain of the CW yielding to lysis may depend on the surface modulus, σ=Yh, which has contributions from both thickness and elastic modulus, we suspect that those cells have lower bulk modulus, arising from mis-regulated CW composition. This was supported by the similar growth rate values reached before lysis, which is predicted to inversely depend on the surface modulus (Figure 6B). Similar lysis phenotype was also observed in wsc1Δ, but to a much lesser extent which precluded a careful analysis of thickness dynamics. Together those findings demonstrate that the CWI may directly probe cell wall strain rates to adjust synthesis, thereby coordinating growth and CW assembly needed for cell survival (Figure 6C).

DISCUSSION

The mechanical properties of the CW and turgor values underlie walled cells growth and shapes, and have been redundantly used as tunable cellular properties to survive, colonize, infect or reproduce [START_REF] Bastmeyer | Force exertion in fungal infection[END_REF][START_REF] Dagdas | Septin-mediated plant cell invasion by the rice blast fungus, Magnaporthe oryzae[END_REF][START_REF] Dudin | A formin-nucleated actin aster concentrates cell wall hydrolases for cell fusion in fission yeast[END_REF][START_REF] Harold | Force and compliance: rethinking morphogenesis in walled cells[END_REF][START_REF] Keegstra | Plant cell walls[END_REF][START_REF] Lew | How does a hypha grow? The biophysics of pressurized growth in fungi[END_REF][START_REF] Silhavy | The bacterial cell envelope[END_REF]. This importance contrasts with the limited knowledge we have on the dynamic regulation of those mechanical parameters in growing and dividing cells. By developing and validating here the very first approach to monitor CW thickness in live cells, we evidence key contributions of the polarized dynamics of the CW for growth and viability. Our approach overcomes numerous limitations of EM studies, such as alterations in cell shape and CW caused by fixation [START_REF] Osumi | Visualization of yeast cells by electron microscopy[END_REF] , and brings the possibility to visualize the CW in a planar section, in large populations of living cells. We foresee that this approach could serve as a novel standard for studying CW function in processes such as growth, reproduction or infection in bacteria, fungi or plants, with potential therapeutic values for anti-fungal and antibiotic chemical screens.

Spatio-temporal Cell Wall dynamics

One important output of our study, is to find that CW thickness and mechanics are polarized, with growing poles being typically twice softer and thinner than the rest of the cell. Those mechanical anisotropies, which here derive from internal polarity; have long been speculated to be required for tip elongation and rod-shape morphogenesis [START_REF] Boudaoud | Growth of walled cells: from shells to vesicles[END_REF][START_REF] Drake | Model of fission yeast cell shape driven by membranebound growth factors and the cytoskeleton[END_REF][START_REF] Rojas | Chemically mediated mechanical expansion of the pollen tube cell wall[END_REF]. Patterns of wall elasticity may also influence the morphogenesis of other tip growing cells [START_REF] Ma | Surface ultrastructure and elasticity in growing tips and mature regions of Aspergillus hyphae describe wall maturation[END_REF][START_REF] Yanagisawa | Patterning mechanisms of cytoskeletal and cell wall systems during leaf trichome morphogenesis[END_REF], as well as that of multicellular plant tissues [START_REF] Kierzkowski | Elastic domains regulate growth and organogenesis in the plant shoot apical meristem[END_REF][START_REF] Milani | Matching Patterns of Gene Expression to Mechanical Stiffness at Cell Resolution through Quantitative Tandem Epifluorescence and Nanoindentation[END_REF][START_REF] Peaucelle | Pectin-induced changes in cell wall mechanics underlie organ initiation in Arabidopsis[END_REF]. Given the large variety of shapes found in fungal species and mutants, it will be interesting to systematically compare spatial patterns of CW mechanics and thickness with cell shape parameters, to identify generic biomechanical principles guiding the morphogenesis of walled cells. Animal cells also exhibit anisotropic mechanical properties needed for shape changes, cell migration or tissue morphogenesis, which rest on the organization of internal polarity and cytoskeleton [START_REF] Levayer | Biomechanical regulation of contractility: spatial control and dynamics[END_REF][START_REF] Mogilner | The shape of motile cells[END_REF]. Thus, our data reinforces a concept that morphogenesis may primarily emerge from spatial surface mechanical properties patterned from intracellular biochemical organization.

We also discovered that the CW is highly dynamic at sites of polar growth, with variations in thickness amplitude reaching up to 30% of the mean, on a time-scale of few minutes.

Thickness on cell sides appeared comparatively more stable, although we did note some patterns of thickness translating away from cell tip (movie S1 and S2), plausibly reflecting CW material flowing during cell growth [START_REF] Abenza | Wall mechanics and exocytosis define the shape of growth domains in fission yeast[END_REF]. We suggest that thickness oscillations are indicative of a homeostatic system correcting thickness changes overtime, needed to maintain CW mechanical parameters in an optimal range for growth and viability.

Although the current state of the art precludes from dynamically computing changes in elastic moduli in growing cells, we suspect that similar homeostatic systems could also influence composition and bulk elasticity in addition to thickness. CW mechanical oscillations have been speculated to exist in rapidly growing cells such as pollen tubes and fungi; and to be related to cycles of Rho-GTPase activation or Calcium signaling and variations in growth rates [START_REF] Qin | Rapid tip growth: insights from pollen tubes[END_REF][START_REF] Rojas | Chemically mediated mechanical expansion of the pollen tube cell wall[END_REF]. Mechanical oscillations may thus represent a general read out for negative-feedbacks acting at the biochemical or biomechanical level, instrumental to confer robustness to growth and viability [START_REF] Howell | Negative feedback enhances robustness in the yeast polarity establishment circuit[END_REF].

Mechanical homeostasis of the CW by the CWI

By screening through mutants, we uncovered a primary role for components of the CWI in probing CW strain and cell growth. Although the role of the CWI in general CW stress response is well established in yeast, fungi and plants, to our knowledge, those are the very first direct evidence of a direct role in dynamically controlling CW properties during normal cell growth. The implication of Wsc1 in thickness homeostasis and in probing CW mechanical strain to regulate synthase polar domains localization and activity, provides strong support that this protein may act as a direct CW mechanical sensor. Wsc1 localizes in patches at cell tips, and has a predicted serine/threonine rich domain spanning the CW [START_REF] Cruz | The fission yeast cell wall stress sensor-like proteins Mtl2 and Wsc1 act by turning on the GTPase Rho1p but act independently of the cell wall integrity pathway[END_REF]. It remains a puzzle to understand how such configuration could directly probe lateral strain in the CW. Studies of S. Cerevisiae orthologues have proposed that WSC clustering may underlie mechanosensory activities [START_REF] Heinisch | Single-molecule atomic force microscopy reveals clustering of the yeast plasma-membrane sensor Wsc1[END_REF]; further work will be needed to understand how those domains may probe the dynamic mechanical properties of the CW.

Mutants in the second fission yeast stress sensor, Mtl2, did not exhibit strong defects in homeostasis, suggesting a dominant function of Wsc1, here. This may be in agreement with the more diffuse localization of Mtl2 all around the cell surface [START_REF] Cruz | The fission yeast cell wall stress sensor-like proteins Mtl2 and Wsc1 act by turning on the GTPase Rho1p but act independently of the cell wall integrity pathway[END_REF].

Surprisingly, we found that loss of Rgf2, which has been suggested to physically interact with Wsc1 [START_REF] Cruz | The fission yeast cell wall stress sensor-like proteins Mtl2 and Wsc1 act by turning on the GTPase Rho1p but act independently of the cell wall integrity pathway[END_REF], only had a mild effect on wall thickness homeostasis. Rgf1Δ mutants exhibited the most pronounced effect, which may account for its severe lysis phenotype. This suggests a potential link between Wsc1 sensing and Rgf1 activation in wall homeostasis, and a dominant role for Rgf1p in activating Rho1 in this response (Garcia et al., 2009a). However, we note that 70% of rgf1Δ mutant cells still survived, suggesting that a secondary system, likely based on Rgf2 may maintain some form of mechanical homeostasis, plausibly at longer time-scales. Finally, we also found that Pmk1 was required for homeostasis. This suggest that proper homeostasis involves at least partially the expression of a set of regulatory genes for CW construction. This would imply that the Pmk1 cascade acts at short time-scales of several minutes, similar to turgor pressure-regulating Sty1/Hog1 MAPK [START_REF] Miermont | The Dynamical Systems Properties of the HOG Signaling Cascade[END_REF]. Thus, the dynamic regulation of walled cells mechanical properties, through rapid adaptation systems, appear as a central element to support viability not only in response to stress, but also during normal growth and division.

Growth control through mechanochemical systems

The notion that surface mechanics can alter the core spatio-temporal organization of cells, through mechanosensing systems is becoming increasingly important in biology [START_REF] Dufort | Balancing forces: architectural control of mechanotransduction[END_REF]. Examples range from the regulation of cell polarity, migration, cell division up to tissue homeostasis [START_REF] Bonazzi | Symmetry Breaking in Spore Germination Relies on an Interplay between Polar Cap Stability and Spore Wall Mechanics[END_REF][START_REF] Fink | External forces control mitotic spindle positioning[END_REF][START_REF] Houk | Membrane tension maintains cell polarity by confining signals to the leading edge during neutrophil migration[END_REF]. In here, we uncovered a core function of mechanosensing in the control of growth rates. In that view, growth is directly perceived as a mechanical signal straining the CW, without any requirement for interactions with neighboring cells or cell environment. When a cell begins to elongate rapidly, the strain rate increases, which turns on synthase activation by the CWI to assemble more wall, and vice e versa. This homeostatic system thus compensates surface material dilution associated with growth, to maintain its thickness around values compatible with survival. As thickness influences elongation rates, this system may also work towards controlling growth rate values around a defined range. Those findings thus have important implications for cell cycle control, as well as size homeostasis, which are both influenced by growth rate values. We speculate that cells may have evolved built-in mechanical properties and sensing elements which allow them to control expansion speeds adapted to their lifestyles. Future work studying similar problems of surface expansion control will inform on how those elements have been modified to guide the growth of rapidly expanding cells like fungi and plant, or slowly growing cells like those of animals. for 6h and 2h, respectively, before observation. Spores were obtained from homothallic h90 strains, or from heterothallic crosses. Freshly growing cells were placed on malt extract (ME) solid media for 3 days. Mating mixtures were then digested for 1h at room temperature in a 1/200 glusulase solution in water to kill vegetative cells, washed three times in water, and incubated for >7h in YE5S before observation [START_REF] Bonazzi | Symmetry Breaking in Spore Germination Relies on an Interplay between Polar Cap Stability and Spore Wall Mechanics[END_REF]. Spheroplasts were generated from exponentially growing cells in YE5S, which were washed twice in digestion buffer (0.1M Citric Acid 0.1 M NaCitrate pH = 5.8; 1.2 M Sorbitol) and subsequently treated with Lallzyme (0.1 g/ml) during 1h at room temperature. Newly formed spheroplasts, were washed twice in YE5S+1.2 M Sorbitol, and let to regenerate at 25°C for >16h, until outgrowing protrusions appeared [START_REF] Flor-Parra | A Lallzyme MMX-based rapid method for fission yeast protoplast preparation[END_REF]. To image mating protrusions, homothallic cells growing exponentially in liquid YE5S were placed on solid ME until mating protrusion appeared (>5h). Cells were then harvested, labeled and imaged directly.

Pharmacological inhibitors:

Cell wall digestion was achieved by incubating cells with 5000U/ml Zymolyase (ZymoResearch) at room temperature, and imaged immediately after treatment and 1h after.

LatrunculinA (Sigma) was used at a final concentration of 100 M from a 100X stock in DMSO. Cells were incubated for 30' at room temperature before imaging. For Caspofungin experiments, cells were plated on 2% agar pads made with YE5S+10 mg/ml Gs-IB4 -Alexafluor647 and a final concentration of 5M Caspofungin.

METHOD DETAILS

Microscopy:

Live For cell wall thickness measurements, cells were pre-labeled in growth media containing 5 mg/ml of labelled lectin from Griffonia simplicifolia Gs-IB4-Alexafluor647 or Bs-IB4-TRIC [START_REF] Horiseberger | Localization of alpha-Galactomannan on the surface of Schizosaccharomyces pombe cells by scanning electron microscopy[END_REF]. For single-time imaging, cells were placed between a glass slide and a coverslip and imaged within 20 min. For time-lapse imaging and for the experiments, cells were placed on 2% agar-pads made of YE5S containing 10 mg/ml Gs-IB4 -Alexafluor647 and covered with a coverslip. We detected small differences in thickness between experiments performed in glass slides (202±19 nm, n=169) and in agar pads (177±28 nm, n=66), with glass slides generally yielding thicker values, which could be attributed to small cell deformation or starvation. For this reason, all experiments done in mutants, drugs or by modifying protocols are associated with a control measured in the same manner.

Comparison with electron microscopy was performed on agar pads. Imaging was generally performed at room temperature (22-26 °C), with controlled humidity (>30%). For temperature-sensitive mutants, the temperature of the sample was kept at 36 °C by using either a temperature controlled cage or an objective heater (Bioptechs).

Transmission Electron Microscopy:

For high pressure freezing (Figure S3A), cells were grown in liquid and harvested on a vacuum filter. The pellets were transferred to 100µm deep flat carriers (Leica Microsystems).

Samples were then cryo-immobilized by HPF (EM-Pact2, Leica microsystems). Freeze substitution was performed with an AFS2 (Leica Microsystems) in 2% Osmium, 0,1% Uranyl acetate in pure acetone following protocols described in [START_REF] Murray | High pressure freezing and freeze substitution of Schizosaccharomyces pombe and Saccharomyces cerevisiae for TEM[END_REF]. Samples were rinsed three times with acetone and infiltrated with gradually increasing concentrations of an Epon resin mix (Agar scientific) and polymerized for 24h at 60°C. Ultrathin (70 nm) sections were generated with an ultramicrotome (UC6, Leica Microsystems) and collected on formvar/carbon-coated grids. Sections were post-stained by aqueous 2% uranyl acetate and lead citrate before imaging.

For chemical fixation (Figure S3A), samples were fixed with 2% glutaraldehyde in Phosphate Buffer 0.2M buffer, for 1h at room temperature followed by overnight fixation at 4°C and post-fixed with 2% osmium tetroxide in water. Samples were dehydrated through a graded series of ethanol and propylene oxide and embedded in epoxy resin. Ultrathin (70 nm) sections were generated with an ultramicrotome (UC6, Leica Microsystems) and collected on formvar/carbon-coated copper grids. Sections were then post-stained by aqueous 4% uranyl acetate and lead citrate.

For Correlative Light Electron Microscopy (CLEM) cells were immobilized on specialized bottom glass petri-dish with micro-grids for image position registrations (MatTek) (Figure S3D). Dishes were pre-treated with 1mg/ml poly-lysine and 0.1 mg/ml Bs-IB4-lectine and rinsed with water and YE5S. Cells, pre-labeled with Gs-IB4-Alexafluor647, were placed on the dishes, let to sediment and stick for ~15 min, and were subsequently rinsed vigorously with YE5S to detach non-sticking cells. Cells were then fixed with 2% glutaraldehyde in YE5S media for 2h at 4°C. Cells were then imaged in light microscopy and their positions were recorded using marks on the micro-grid. The samples were post-fixed with 2% osmium tetroxide in water and dehydrated through a graded series of ethanol and embedded in epoxy resin, which tends to stick to the micro-grid. Ultrathin (70 nm) sections of selected regions of the micro-grid, were generated with an ultramicrotome (Ultracut UC6, Leica) and collected on formvar/carbon-coated copper grids. Sections were then post-stained by aqueous 4% uranyl acetate and lead citrate. We observed often a partial cell deformation recorded in TEM, plausibly due to resin embedding. In spite of this the cell wall remained intact in most cases.

All samples were observed in a Tecnai12 (FEI, The Netherlands) transmission electron microscope at 80 kV equipped with a 1K×1K Keen View camera.

Microfabricated channels:

Microchannels used to manipulate cell diameter, were fabricated in PDMS from a silicon wafer printed with classical soft-lithography procedures, as described in (24) (Figure 3H).

Homothallic rga4 spores were inserted into the PDMS microchannels, let to germinate for >20h at 25°C, and subsequently rinsed with YE5S + Gs-IB4 -Alexafluor647 for 2h before imaging.

Turgor pressure manipulation by sorbitol addition:

For sorbitol treatments cells were pre-labeled in growth media containing 5 mg/ml of labelled lectin Gs-IB4 -Alexafluor647, then placed in PDMS microfluidic chambers between a dialysis membrane and a coverslip, which allowed live fluid exchange [START_REF] Charvin | A microfluidic device for temporally controlled gene expression and long-term fluorescent imaging in unperturbed dividing yeast cells[END_REF]. Cells were left in the chamber 40 minutes for adaptation before starting the assay. For Young modulus estimations cells were imaged, then immediately rinsed in the same media containing 1.5 M sorbitol and imaged again within the next 4 minutes. For growth rate manipulation, cells were imaged every 2 minutes for 10 minutes in YE5S + Gs-IB4 -Alexafluor647, then washed in the same media containing 0.5M Sorbitol and imaged for 30-60 minutes.

QUANTIFICATION AND STATISTICAL ANALYSIS

Image analysis:

All image analysis scripts developed in the context of this study, can be made available upon request.

Cell Wall thickness measurement: The analysis pipeline to compute CW thickness used was fully automated with a manual check option for each step. All the analyses were performed using custom scripts written in MATLAB (Mathworks, R2013a equipped with Image Processing and Statistics toolboxes). Initial cell segmentation was performed using either fluorescent membrane or lectin signals. The signal was first binarized using a threshold determined with the graythresh function. The cell contour was defined as the most outer pixels of the binarized signal. The coordinates corresponding to the points along the cell contour were then equally spaced with an interval of 1 pixel using a spline function.

Typically, a single cell contour was represented by ~ 600-800 points for a pixel size of 43 nm.

At each point of the contour, a 40 pixels long line perpendicular to the cell contour was defined as the normal to the tangent of the curve determined using 11 neighboring points Chromatic shift correction: To correct for the chromatic positional XY shifts between the channels used for the lectin and membrane signals across the field of view we used a "pointby-point correction" approach. In contrast to standard "whole image" alignment approaches [START_REF] Clark | Monitoring actin cortex thickness in live cells[END_REF], this produces a continuous vector field containing shifts for all pixels in This sparse array is then used to generate a continuous point-by-point shift map. For this, each of the two 3-dimentional sets of data: (X1, Y1, Vx) and (X1, Y1, Vy) are fitted with a plane, which allows for computing a point-to-point shift vector Vi at any position (Xi, Yi) with subpixel resolution close to the particle center detection limit (~20-25nm) (Figure S1A). This map is finally used as an input in the cell wall thickness measurement scripts described above (Figure S1C).

To assess the chromatic shifts in Z, we imaged a 3 µm deep Z stack with 10 nm spacing between slices of the same 200nm beads in 5 different positions in the field of view in two wavelengths channels. To reproduce the distance of the cell center from the cover glass (about 2 m), beads were embedded in 2% agar pad, and only beads at about 2 m from the glass surface were imaged. For each stack the fluorescent peaks were detected using a Gaussian fit, and the shift computed from the distance between the peaks. We found that this shift ranged typically from 0.1 to 0.4 m, which corresponds to a projected shift in thickness smaller than 30nm that we neglected for the measurement (Figure S1B).

Generation and analysis of simulated images: Simulated images were generated as described in [START_REF] Clark | Monitoring actin cortex thickness in live cells[END_REF], using custom scripts written in MATLAB. The simulation starts with two dimensional data, corresponding to a simple spherical cell shape of 3 m diameter, drawn with a 5 nm resolution. The simulated cell has a cell membrane thickness of 15 nm and a cell wall thickness of 200 nm. To best represent the experimental imaging conditions, the cell membrane was assumed to be uniformly stained, and only 5 nm of the most outer part of cell wall was uniformly stained in another color. To account for the experimental noise and finite background signals inside and outside of the cell, two different Gaussian noises, were added.

The mean and standard deviation of the noise was determined so that the noise in the final image matches typical experimental signals. The original image was then convoluted with a Gaussian function with a standard deviation of 200 nm corresponding to the point spread function. The smoothed data were sampled at 40 nm to generate the final pixelized image. To systematically test for the effects of finite signals in the cytoplasm, we kept the standard deviation of the noise and changed only its mean values. The generated images were analyzed using the same scripts as for the experimental data described above.

The simulation results suggest that the inside-outside difference in the background signal could potentially influence cell wall thickness measurement (Figure S1D). By running a systematic modification of background difference between inside and outside, and signal/noise ratio in simulations, we thus computed the influence of those parameters on the error in thickness measurement (Figure S1E). Those results defined a region in which the predicted error is below 30 nm, which is less than 20% of the typical experimental thickness.

We thus used this simulation result as a criterion for image quality, and discarded images or regions of images where the predicted error is out of this accepted region (Figure S1E).

TEM and CLEM images: The full cell wall in electron microscopy images was detected using custom scripts written in MATLAB, by using a threshold value for the signal intensity. In some images with inhomogeneous contrast, the results were manually updated. The edge of the binarized signal were traced to define the inside and outside boundaries of the cell wall.

The normal direction was defined based on the inside cell wall boundary, and the cell wall thickness was defined as the distance between inside and outside boundaries along the normal direction. To compare EM and light microscopy images in CLEM analyses (Figure 1B Cell shape, growth and polarity factor concentration measurements: Cell length was generally measured using lectin signals in ImageJ. To compute old end elongation rates visible birth scars were used as fiducial reference landmarks. Tip curvature was computed from the lectin contour defined for cell wall measurements, by fitting a local circle in Matlab [START_REF] Bonazzi | Actin-Based Transport Adapts Polarity Domain Size to Local Cellular Curvature[END_REF].

To analyze changes in cell shape, growth rates and localization of polar factors following turgor reduction by sorbitol addition, we developed dedicated Matlab scripts. We first segmented cells using the signal from the lectin-labelled cell wall. To this aim, we first smoothed the image with median and Gaussian filters and detected cell edges using the Laplacian of the Gaussian filter. The resultant binary image was then filtered to remove small segments. Given that the signal of the labelled cell wall has a finite thickness, we detected the inner and the outer border of this signal. All spaces in this image are then filled in white except for the spaces between the inner and outer border of the wall, yielding a black band representing the cell wall. Using the watershed algorithm, we finally extracted the whole-cell contour defined as the middle of this band. To compute cell length, we fitted the long axis of the segmented cell with a polynomial of degree 3. This fit was then used to define a "cell spine" and its length was calculated, and used as a measure for cell length.

The whole-cell contour could then be manipulated using morphological and logical operations to obtain a set of arbitrary regions (tips, cytoplasm, etc.). The tip regions are for instance shaped as a cut off from the whole-cell mask perpendicular to the cell spine at specific distances along the spine. A segment outside of the cell can be shaped to compute the background.

Fluorescent signals of interest are then extracted from fluorescent images by using a mask based on corresponding sub-regions. The signal from the tips is corrected for the background signal and for bleaching. This is done by normalizing the background corrected tip signal with the background corrected cytoplasm signal:

𝐼 𝑡𝑖𝑝 = 𝐼 𝑡𝑖𝑝 𝑟𝑎𝑤 -𝐼 𝑏𝑔 𝐼 𝑐𝑦𝑡𝑜𝑝𝑙𝑎𝑠𝑚 -𝐼 𝑏𝑔

Strain-stress assay simulation

The estimation procedure of the mechanical parameters is similar to the one described in the paragraph "Measurement of material properties" in Abenza-Martinez et al. [START_REF] Abenza | Wall mechanics and exocytosis define the shape of growth domains in fission yeast[END_REF] except herein the cell wall thickness has been measured all around the contour. For each of the 21 cells, the steps are:

1. Symmetrization of both the plasmolized contour and the turgid contour of each cell; the same symmetrization is kept for the thickness profile.

2. The plasmolyzed contour is swelled at different pressure steps (pressure is normalized by the Young modulus at the equator) supposing a distribution of thickness, and a distribution of mechanical parameters away from the equator; two different distributions of thickness were tested: averaged plasmolyzed thickness and measured plasmolyzed thickness; three distributions of Young modulus were tested: a zeroparameter (i.e constant Young modulus) and two two-parameters distributions ("cos" and "sigmoid"), one parameter describing the tip width, the other parameter being the ratio between the Young modulus at the tip and the Young modulus at the equator. As two recent articles [START_REF] Abenza | Wall mechanics and exocytosis define the shape of growth domains in fission yeast[END_REF][START_REF] Atilgan | Morphogenesis of the Fission Yeast Cell through Cell Wall Expansion[END_REF] consistently reported a zero-value of the in-plane Poisson ratio, the in-plane Poisson ratio was settled to 0.01 for all the simulations.

3. Among the pressure steps and the tested parameters, the ones for which the swelled plasmolyzed contour best fit the turgid contour are retained to estimate the mechanical parameters. Whatever is the chosen Young modulus distribution, the selected normalized pressure step provides the ratio between the Young modulus at the equator and the internal turgor. The turgor pressure inside the cell has been estimated to be 1.5 MPa in Atilgan [START_REF] Atilgan | Morphogenesis of the Fission Yeast Cell through Cell Wall Expansion[END_REF] providing an absolute estimate of the Young moduli.

Variation in local Young moduli where simulated using: The correlation between Etip and Ecyl is not significant for d below 3 (p-value>0.5 for both the Pearson test and the Spearman test) whereas it becomes significant for d=3.5 (p-value 0.02 Pearson, 0.11 Spearman) which means the tip width has been chosen too large and that the estimations of both the tip and the equatorial Young modulus are combined in the measurements. b) A "cos" distribution: Y(s)=Ytip +(Ycyl-Ytip)(0.5(1-cos(2πs/arc))) d . The "cos" distribution is narrower around the equator than the sigmoid one especialy for d>1. Similarly to the sigmoid distribution, the best fit are obtained when the Young modulus decreases in the more slowest manner from the equator (d=0.5, d=1). The estimations of Ycyl/P are very sensitive to d as the decrease from the equator becomes very sharp when d=2, they also are higher than with the "sigmoid" fit in order to compensate the quicker decrease at the equator. For the best fit (d=0.5, d=1), the estimated ratio of Ytip/Yside are slightly lower than the "sigmoid" ones.

Obtained values are for exponent d of 0.5; 1 and 2 respectively: Yside/P : 48.6±0.5; 54.9±0.5 and 73.6±0; Ytip/Ycyl: 0.46±0.02; 0.49±0.07 and 0.39 with distances from best-fit of 0.045; 0.048 and 0.065 Analytical Models for tip thickness oscillations: Eq (4) of the main text can be written in the form: Therefore, a decrease in λ yields an increase in 𝐴, and hence to oscillations with higher amplitude.

3D cell growth model:

The growth model developed in Abenza-Martinez et al [START_REF] Abenza | Wall mechanics and exocytosis define the shape of growth domains in fission yeast[END_REF]was modified to include varying thickness and varying young modulus. The parameters h (thickness) and Y (Young modulus) (Formula 36-38 in the supplemental material of [START_REF] Abenza | Wall mechanics and exocytosis define the shape of growth domains in fission yeast[END_REF]) are no more constant.

The thickness distribution shape has been selected by the following procedure: all the symmetrized thickness profiles were linearly reparametrized from new end to old end so that the new end lays at curvilinear abscissa 0 and the old end at a curvilinear abscissa of 13.9.

After this operation all the thickness profiles were averaged, the average was very well fitted by a three-parameters linear combination of sigmoids. The three parameters h NE , h OE and hside are the cell wall thickness at the new end, at the old end and at the equator.

Only the modification to the original algorithm are detailed here (Supplemental material of Abenza-Martinez et al [START_REF] Abenza | Wall mechanics and exocytosis define the shape of growth domains in fission yeast[END_REF]):

-At step n (iii), once the new curvilinear abscissa sn has been calculated using formula 46, arcn, the total arclength is updated and the thickness profile can also be updated using the shape family ℎ 𝑛 (𝑠 𝑛 ) = h NE + h side -h NE 1 + e -2.20(𝑠 𝑛 -2.36) -h side -h OE 1 + e -73.3(𝑠 𝑛 -𝑎𝑟𝑐 𝑛 +2.36) 𝑎𝑟𝑐 𝑛 ⁄ as well as the Young modulus distribution: 𝑌 𝑛 (𝑠 𝑛 ) = 62 + 38 1 + e -2.20(𝑠 𝑛 -2.36) -30 1 + e -73.3(𝑠 𝑛 -𝑎𝑟𝑐 𝑛 +2. ) where D=1.17, 1.06 and 0.93 depending on which strain is modeled rga4Δ, wt and rga2Δ.

Each of the D constants have been chosen such that the asymptotic radius of convergence of the cell corresponds to a value close to the experimental tip radii 1.76 μm (rga4Δ), 1.62 μm (wt) and 1.46 μm (rga2Δ). The cell wall tip thicknesses of rga4Δ, wt and rga2Δ are 0.149 μm, 0.166 μm and 0.179 μm.

KEY RESOURCES TABLE

REAGENT or RESOURCE SOURCE IDENTIFIER Experimental Models: S. pombe Strains h-cut11-RFP:HygR This study VD11 h-CRIB-tdTomato:ura This study VD20 h-tea1::NatMX CRIB-tdTomato:ura This study VD31 h+CRIB-tdTomato:ura GFP-psy1:leu (leu1-32 ura4-D18) This study VD39 h-CRIB-tdTomato:ura This study VD50

h+ GFP-psy1:ade bgs4::ura4 RFP-bgs4-Leu (leu1-32 ura4-D18 ade6)

This study VD57

h-GFP-psy1:ade bgs4::ura4 RFP-bgs4-Leu Leu (leu1-32 ura4-D18 ade6)

This study VD58 This study VD72 pmk1:ura This study VD81 rgf1::his3 leu1:rgf1-GFP (his3D1 leu1-32 ade6M210 ura4D-18) Y. Sanchez* VD93 h+ pRep3X- This study VD105 h+ Leu: This study NM468 h-Leu: This study NM469 h+ shk1: This study AH159 gpd1::NatMX GFP-bgs4:leu bgs4::ura (leu1-32 ura4-D18 ade6)

This study AH328

  and Figure S1A-S1E) (see STAR methods). Measurements were slightly sensitive to growth environment and media but robust to variations in lectin-bound fluorophores or membrane associated GFP signals (Figure S2A-S2C).

  . In this plot, the X-intercept corresponded to a homeostatic thickness of hoe* = 119 nm, close to the mean old end thickness at the population level. Importantly, similar homeostasic behavior was observed in Cdc10-M17 and Cdc25-22 cell cycle mutants respectively blocked in G1 or in G2 phase, ruling out putative contribution of cell-cycle transitions to this behavior (Figure5Band FigureS5F-S5H). Solving the delayed differential Eq. 4 for h(t) yielded, in a range of parameter values, oscillatory behavior, reminiscent of experimental observations (FigureS5I-S5K and supplementary methods). Together those findings support the existence of a homeostatic mechanism, in which an overshoot (or an undershoot) in thickness is corrected by thinning (respectively thickening) the wall over the subsequent tens of minutes to maintain it around a narrow range of values.

Figure 1 .

 1 FIGURE LEGENDS:

Figure 2 .

 2 Figure 2. Cell polarity and cell wall thickness patterns. (A) Thickness color map of a typical interphase cell, and thickness values at the old end (n=50), new end (n=41), cell sides (n=51) and birth scars (n=47). (B) Representative thickness color maps and thickness polarity in wt interphase cells (n=41), outgrowing spores (n=24), mating tips (n=18), recovering spheroplasts (n=22), and tea1Δ cells recovering from starvation (n=29). Arrows indicate growth directions. h polarity is defined as a tip/side ratio in cells using an average of both tips values, and tip/back ratio in the other cases. (C) Superimposition of h map and polar growth markers (arrowheads). (D) Mid-slice confocal time-lapse and h maps of a tea1Δ cell recovering from starvation. Arrows indicate growth direction and arrowheads point at RFP-Bgs4 polar cap. (E) (Left) Midslice confocal image and h maps in the indicated conditions.

Figure 3 .

 3 Figure 3. Dynamics of cell wall assembly at polar growth sites. (A) Simple model for tip thickness, htip, dynamics. S is a rate of wall synthesis per unit surface, γ the strain rate and P the turgor pressure. (B) Representative color maps and mean h measurement of controls (n=59 cells), the constitutively active Rho1-G15V (n=27), and cells over-expressing Pck2 (n=59). (C) Local tip thickness in wt (n=27) and cwg1-1 (n=29) grown at restrictive temperature. (D) (Top) Representative hoe maps following Caspofungin treatment, and evolution of hoe in time (Bottom, n=11 cells). (E) Δhoe/hoe, measured in Caspofungin treated wt cells and plotted as a function of ΔL/Rc (n=44 in 12 cells). The black line is a linear fit. (F) Growth simulations: plain and dashed lines mark cell shapes at the beginning and the end of the simulation. (G) Midslice confocal images of typical rga4Δ, wt and rga2Δ cells expressing the membrane marker GFP-Psy1 and close-up local h color maps at the old end. (H) Schematic for diameter manipulation in microchannels, and midslice confocal images of rga4Δ cells grown in microchannels with different diameters and close-up local h color maps at the old end. The yellow dashed lines mark the border of the microchannels. (I) Normalized growth rate at the old end plotted as a function of Rc 2 /hoe, for wt (n=36), rga2Δ (n=26), rga4Δ (n=37) and, rga4Δ cells grown in microchannels (n=33, binned on 11, 10, 13 cells). The dotted line is a fit of the growth simulation results. Whiskers plots show median and full data set range. Error bars correspond to SD. ** P < 0.0001. Scale bars 2 µm in confocal images and full cell color maps, and 1 µm in insets.

Figure 4 .

 4 Figure 4. Cell wall thickness oscillations and strain rate-dependent CW synthase activation and localization. (A) Close up views on thickness maps at the old end of a growing wt cell, and corresponding plots of thickness at the old end and on the cell sides. Blue dots correspond to time points shown in color plots. (B) Evolution of thickness at the old end hoe and on cell sides hside imaged at 6 min interval in 6 wt cells (colors of the lines correspond to the same cell in both plots). (C) GFP-Bgs4 and Active-Rho1-Citrin polar

Figure 5 .

 5 Figure 5. The Cell Wall Integrity pathway mediates cell wall thickness homeostasis. (A) Thickness homeostasis plot. Δhoe/Δt computed as (hoe(t0+42')-hoe(t0))/42 , for various t0 and plotted as a function of 1/ hoe(t0), for wt (n=131 cells in 25 cells). The plain line is a linear fit. (B) Correlation coefficient for thickness homeostasis for cells of wt (n=82 in 21 cells), cdc10-M17 (n=48 in 13 cells) and cdc25-22 (n=80 in 20 cells), grown at restrictive temperature (36°C). (C) Thickness homeostasis plot for wsc1Δ (n=56 in 14 cells) and rgf1Δ (n=32 in 8 cells) mutants. Palin lines are linear fits. (D) Correlation coefficient for thickness homeostasis for wt (n=59 cells), mtl2Δ (n=56 in 14 cells), pck2Δ (n=40 in 10 cells), rgf2Δ (n=52 in 13 cells) and pmk1Δ (n=39 in 10 cells). (E) Defect in GFP-Bgs4 detachment in response to turgor reduction in a gpd1Δwsc1Δ double mutant. Note the remaining signal as compared to a gpd1Δ single mutant shown in Figure 4C. (F) Evolution of normalized tip signal of GFP-Bgs4 (n=19) in gpd1Δ and gpd1Δwsc1Δ cells rinsed with sorbitol at t=0. (G) Ratio between tip signals after and before sorbitol treatment in gpd1Δ and gpd1Δwsc1Δ cells (n=14, 19, 18, 18). Whiskers plots show median and full data set range. Scale bars, 2 µm; Error bars represent SD.

Figure 6 .

 6 Figure 6. Implication of thickness homeostasis for cell survival during normal growth. (A) Time-lapse of GFP-Psy1 and thickness maps for a representative lysing rgf1Δ cell. The arrowhead points at membrane leaking at cell tips after lysis. Evolution of old end thickness hoe and growth rates Groe in lysing rgf1Δ cells. The last point is recorded before lysis. (C)Mechanosensing-based feedback for CW thickness homeostasis during cell growth. In absence of feedback, growth causes cell wall thinning which promote faster growth and further CW thinning and so on, until the CW ruptures and cells lyse. The cell wall integrity pathway probes strain rate (or equivalently growth rates) through the surface protein Wsc1 that promotes the activation (directly or indirectly) of the Rho1-GEF Rgf1 to activate of Rho1 and cell wall synthase (CWs). Rho1 can directly act on synthesis or through the Pmk1 MAPK

  -cell imaging was performed on two different inverted spinning-disk confocal microscopes equipped with motorized stages, automatic focus and controlled with MetaMorph® (Microscopy Automation & Image Analysis Software). The first one (Nikon Ti-Eclipse), is equipped with a Yokogawa CSU-X1FW spinning head, and an EM-CCD camera (Hamamatsu), a 100× oil-immersion objective (Nikon CFI Plan Apo DM 100×/1.4 NA) and a 2.5× magnifying lens, yielding a pixel size of 43 nm. The second one (Leica DMI8), is equipped with a Yokogawa CSU-W1 spinning head, and a sCMOS Orca-Flash 4 V2+ (Hamamatsu) a 100× oil-immersion objective (Leica Plan Apo DM 100×oil/1.4 NA), yielding a pixel size of 70nm.Image registration was performed by imaging a slide containing a solution of 0.2 µm TetraSpeck TM microspheres (Thermofisher). A field containing one bead, was moved sequentially using the Metamorph function "ScanStage" coupled with a home built plugin to allow multidimensional acquisition. This generated an array of fluorescent spots imaged at multiple wavelength, for image registration (FigureS1A, see below). Alternatively, single images of a dense field of non-aggregated beads, could be used to generate the vectorial map.

(

  Figure 1A second panel, only 1 out 10 of those lines are drawn, for clarity). The position of the cell membrane along the normal lines was defined as the intersection point between the normal line and the cell membrane. Finally, the local cell wall thickness h was computed as the distance between lectin and membrane coordinates along the normal line (Figure 1A third panel insert). Lateral resolution was estimated based on the lateral precision with which we could detect birth scars (~500nm) (Figure 1A fourth panel, Figure 1C and Figure S2E-S2G).

  the field of view, and can thus account for non-uniformity in the positional drift. Starting from arrays of 200nm beads imaged at different wavelengths (see above), we first detect the particles positions with a sub-pixel precision (~20-25 nm) using a modified Matlab script available from M. Kilfoil lab (http://people.umass.edu/kilfoil/). The positions of the particles in the two channels (X1, Y1) and (X2, Y2) are then used to compute a sparse map of pointby-point shift vectors, defined as: V = (Vx, Vy) = (X2 -X1, Y2 -Y1) (with one channel used as the reference and features in the second one are being aligned) (FigureS1Aright panel).

  first panel and S2E-G), we defined four morphologically distinguishable points (two tips and two scars), and used these points to overlay the two images by matching the distance between them. Comparison on larger region, was done by measuring thickness in EM pictures manually within 2m regions at 8 individual positions, and compared with an averaged computed from light-microscopy measurement in the Matlab script (Figure 1B third panel).

  a) A "sigmoid" distribution on the form: Y(s)=Ytip+(Yside-Ytip)(sigmf(s,[5 d])-sigmf(s,[5 arcd]))). With s, the curvilinear coordinate and d the width along cell tip. The "sigmoid" distribution is flat at the equator, decreases sharply at a distance d from the tips, then is flat again at the tips.Deviation from the compuatiaonnaly swollen cell and the orginal cell before turgor reduction was computed as a distance from the best-fit. For tip size of 2.5μm; 3 μm and 3.5μm we obtained respective values for Yside/P of 48.1±0.7; 49.0±0.8 and 48.9±0.8 ; for Ytip/Yside = 0.60±0.02; 0.63±0.02 and 0.68±0.02 with distances to best fits of 0.049±0.001; 0.050±0.001; 0.053±0.001.Those values suggest that the lowest distance from the best fit are obtained for tip-width d small enough. The value of Ecyl/P are not very sensitive to d whereas Etip/Ecyl increases with d.

  a unit of time and the homeostatic thickness value b/a as a unit of thickness, this equation can be rewritten as: 𝑡 ̅ = 𝑡/𝑇, and 𝐴 = 𝑎 2 𝑇/𝑏. Numerical solutions of this ordinary differential equation were obtained using the NDSolve function of Mathematica (Wolfram Research Inc.).Eq. S2 has damped solutions with no oscillations for 𝐴 < 𝐴 1 , oscillating damped solutions for 𝐴 1 < 𝐴 < 𝐴 2 , and sustained oscillations for 𝐴 > 𝐴 2 . The values of 𝐴 1 and 𝐴 2 increase when ℎ ̅ 𝑡𝑖𝑝 (0) departs from 1. Analytical calculations show that when ℎ ̅ 𝑡𝑖𝑝 (0) is close to 1, 𝐴 1 = exp(1) and 𝐴 2 = 𝜋/2. Optimal homeostasis corresponds to maximal damping, which occurs for, 𝐴 = 𝐴 1 . Coming back to initial model parameters, 𝐴 = ( 𝑆 0 -

  36) 𝑎𝑟𝑐 𝑛 ⁄ -The amplitude of each step of the growth in formula (47-48) is: δ=1.2 -The model for the fluorescence profile K of the polarity marker:

  h-rga2::KanMX bgs4::ura4 RFP-bgs4-Leu GFP-psy1:ade (leu1-32 ura4-D18) This study VD72 h-rga4::KanMX bgs4::ura4 RFP-bgs4-Leu GFP-psy1:ade (leu1-32 ura4-D18) This study VD81 h-GFP-psy1:ade pREP3X-pck2 (leu1-32 ade6) This study VD93 h-GFP-psy1:ade pRep3X empty vector (leu1-32 ade6) This study VD105 h-tea1::NatMX GFP-psy1:ade bgs4::ura4 RFP-bgs4:leu (leu1-32 ura4-D18 ade6) This study VD110 cwg1-1 GFP-psy1:leu CRIB-tdTomato:ura (leu1-32 ura4-D18) This study VD115 wsc1::KanMX GFP-psy1:ade bgs4::ura4 RFP-bgs4:leu (leu1-32 ura4-D18 ade6) This study VD158 mtl2::KanMX GFP-psy1:ade bgs4::ura4 RFP-bgs4:leu (leu1-32 ura4-D18 ade6) This study VD160 h 90 rga4::KanMX GFP-psy1:ade bgs4::ura4 RFP-bgs4:leu (leu1-32 ura4-D18 ade6) This study VD162 rga4::KanMX GFP-psy1:ade bgs4::ura4 RFP-bgs4:leu (leu1-32 ura4-D18 ade6) This study VD163 rgf2::ura GFP-psy1:ade (leu1-32 ura4-D18 ade6) This study VD164 h+ rgf1::KanMX GFP-psy1:ade (leu1-32 ura4-D18 ade6)

  gpd1::NatMX wsc1::KanMX GFP-bgs4:leu bgs4::ura (ade6) This study AH330 h-gpd1::NatMX P3nmt1-pkc1(HR1-C2)-ECitrine (ura4-D18) This study AH385 gpd1::NatMX wsc1::KanMX P3nmt1-pkc1(HR1-C2)-ECitrine (leu1-32 ura4-D18 ade6) This study AH401 cdc10-M17 GFP-psy1:ade bgs4::ura4 RFP-bgs4-Leu (ade6) This study VD122 cdc25-22 GFP-psy1:ade bgs4::ura4 RFP-bgs4-Leu (leu1

ACKNOWLEDGEMENTS

The authors acknowledge P. Perez, Y. Sanchez, H. Valdivieso, T. Toda, T. Nakamura (YGRC/NBRP Japan) and J.Q. Wu, for sharing material. We acknowledge the ImagoSeine facility, a member of France BioImaging (ANR-10-INSB-04). Our laboratory is supported by the CNRS and grants from the FP7 CIG program, ITN ''FungiBrain'', and the European Research Council (CoG Forcaster no. 647073).


Adrien Meynard

Joint nonstationary blind source separation and spectral analysis

We address a nonstationary blind source separation (BSS) problem. The model includes both nonstationary sources and mixing. Therefore, we introduce an algorithm for joint BSS and estimation of stationarity-breaking deformations and spectra. Finally, its performances are evaluated on a synthetic example.

Introduction: model and background

The BSS problem, originally introduced in a stationary context, has also been discussed in nonstationary situations. Extensions to nonstationary signals have been proposed, based on time-frequency analysis (see [START_REF] Belouchrani | Source separation and localization using time-frequency distributions: An overview[END_REF], chap. 9 in [START_REF] Jutten | Séparation de sources 2. Au-delà de l'aveugle et applications[END_REF] and references therein), or based on mutual information [START_REF] Pham | Blind separation of instantaneous mixtures of nonstationary sources[END_REF]. The BSS of a nonstationary mixtures of stationary signals have also been studied. For instance, in [START_REF] Parra | Convolutive blind separation of non-stationary sources[END_REF], the authors explore the convolutive BSS problem. In the following, we tackle a doubly nonstationary BSS problem, and propose a demultiplexing algorithm adapted to a specific class of nonstationary signals mixed by a instantaneous nonstationary mixing matrix.

Nonstationarity

The nonstationary signals of interest here are deformed versions of stationary signals.

Let x denote a stationary signal, modeled as a realization of a stationary random process with power spectrum denoted by S X . Acting on x with a stationarity-breaking operator yields a nonstationary signal denoted by y. Various classes of stationarity-breaking operators are relevant to model physical phenomena (e.g. frequency modulation [START_REF] Meynard | Spectral estimation for non-stationary signal classes[END_REF], amplitude modulation [START_REF] Meynard | Spectral Analysis for Nonstationary Audio[END_REF]). We focus here on the time warping operator denoted by D γ and defined by:

y(t) = D γ x(t) = γ ′ (t) x(γ(t)) , (1)
where γ ∈ C 2 is a strictly increasing smooth function. Such deformations can model nonstationary physical phenomena as diverse as Doppler effect, speed variations of an engine, animal vocalization or speech [START_REF] Stowell | Computational Analysis of Sound Scenes and Events, ch. Computational Bioacoustic Scene Analysis[END_REF][START_REF] Meynard | Spectral Analysis for Nonstationary Audio[END_REF]. The wavelet transform is a natural tool to analyze such signals. Hence, the wavelet transform W x of the signal x is defined by:

W x (s, τ ) = R x(t)q -s/2 ψ t -τ q s dt with q > 1 . (2)
In that framework, it can be shown that the respective wavelet transforms W y and W x of y and x are approximately related by W y (s, τ ) ≈ W x (s + log q (γ ′ (τ )), γ(τ )) .

In the following, we make the assumption that x is a realization of a stationary random process X. In such a setting, the approximation error can be controlled thanks to the decay properties of the wavelet ψ, and the variations of γ ′ . In [START_REF] Meynard | Spectral estimation for non-stationary signal classes[END_REF][START_REF] Meynard | Spectral Analysis for Nonstationary Audio[END_REF], corresponding quantitative error bounds are given.

Blind source separation

The problem we consider is the BSS of nonstationary signals modeled by equation [START_REF] Belouchrani | Source separation and localization using time-frequency distributions: An overview[END_REF]. We investigate the case where the number of sources and the number of observations are equal and denoted by N . The sources are additionally assumed to be independent. Let y(t), z(t) ∈ R N denote the column vectors containing respectively all the sources and observations at time t. Then, the mixture is written as

z(t) = A(t)y(t) , (4)
where A(t) ∈ R N ×N denotes the time varying mixing matrix, assumed to be invertible. This model generalizes the amplitude modulation model in the case N = 1 detailed in [START_REF] Meynard | Spectral Analysis for Nonstationary Audio[END_REF]. For example, this model can be appropriate in bioacoustics to describe the BSS of a howling wolf pack [START_REF] Passilongo | Visualizing sound: counting wolves by using a spectral view of the chorus howling[END_REF][START_REF] Papin | Acoustic localization at large scales: a promising method for grey wolf monitoring[END_REF]. Our goal is to determine jointly the mixing matrix A(t), the time warping functions γ i (t), and the spectra of the stationary sources S Xi for i = 1, . . . , N from the observations z(t).

Let us consider a fixed time τ , then for each observation z i , we denote by w zi,τ = W zi (s, τ ) the row vector containing the values of the wavelet transform for a vector of scales s (of size denoted by M s ). Then, all these vectors are gathered into a

N × M s matrix w z,τ such that w z,τ = w T z1,τ • • • w T zN ,τ T .
The same operation is applied to the wavelet transform of the sources. The matrix A(t) is assumed to vary slowly with respect to the oscillations of the signals. It can be shown that the linear relation (4) becomes in this new setting a relationship between the wavelet transforms of y and z of the form

w z,τ ≈ A(τ )w y,τ . (5)
Aside from the terms controlling the error bound in (3), the error bound in ( 5) is also controlled by the variations of the mixing matrix coefficients.

Estimation procedure

Approximation equations ( 3) and ( 5) allow us to write an approximate likelihood in the Gaussian case (see [START_REF] Cardoso | Blind signal separation: statistical principles[END_REF] for more details on this approach). The estimation procedure is based upon discrete wavelet transforms, time-varying parameters are therefore estimated on a discrete time grid D. In the following, the estimation procedure is described for a given τ ∈ D. For the sake of simplicity, we introduce the following notations:

B τ = A(τ ) -1 , θ i,τ = log q (γ ′ i (τ )) and θ τ = (θ 1,τ • • • θ N,τ ) T .

Probabilistic setting

It follows from the Gaussianity assumption on X that w yi,τ ∼ N (0, Σ i (θ i,τ )), where

[Σ i (θ i,τ )] kk ′ = q s k +s k ′ 2 R S Xi (q -θi,τ ξ) ψ (q s k ξ) ψ (q s k ′ ξ) dξ.
Let p V denote generically the probability density function of a random vector V . Then, the source independence hypothesis

gives the following opposite of the log-likelihood:

ℓ τ (B τ , θ τ ) ∆ = -log(p wz,τ |(Bτ ,θτ ) (w z,τ ; B τ , θ τ )) + c = -M s log | det(B τ )|+ 1 2 N i=1 log |det Σ i (θ i,τ ))| + 1 2 N i=1 [B τ w z,τ ] i• Σ i (θ i,τ ) -1 [B τ w z,τ ] H i• ,
where [M] i• denotes the i-th line of the matrix M, and M H is its conjugate transpose. Maximum likelihood (ML) estimates, i.e. minimizers of ℓ τ (B τ , θ τ ), can be evaluated numerically. However, in order to take into account the smoothness assumption on the mixing matrix with respect to time, we switch to the Bayesian framework and introduce a prior p Bτ on the unmixing matrix B τ (assuming i.i.d. matrix coefficients). We choose for p Bτ a uniform distribution centered on B τ -∆τ , and with support 2ǫ B ∆ τ . Then, the maximum a posteriori (MAP) estimate Bτ can be written as the solution of the problem Bτ = arg min

Bτ ℓ τ (B τ , θ τ ) s.t. B τ -B τ -∆τ ∞ ≤ ǫ B ∆ τ .
(6) This problem is consistent with the smoothness hypothesis on B τ . Indeed, assuming ∆ τ is small, the constraint in equation ( 6) is almost equivalent to B ′ τ ∞ ≤ ǫ B . Concerning the time warping estimation, we choose not to give a prior on θ τ . Thus, θτ is the ML estimation of θ τ .

Estimation algorithm

The estimation strategy is to alternate the estimations of B τ , θ τ and the spectra. The algorithm 1 (named JEFAS-BSS) synthesizes all the estimation steps which are described below.

• Mixing matrix estimation. In practice, we numerically solve the problem [START_REF] Meynard | Spectral Analysis for Nonstationary Audio[END_REF]. Besides, because of the assumption of slow variations of the matrix coefficients, we make the approximation that B τ is constant on the interval I τ = [τ -∆ τ /2, τ + ∆ τ /2[. Finally, the estimated sources ỹτ are obtained via ỹτ (t) = Bτ z(t) where t ∈ I τ . Notice that for each interval I τ , a new matrix Bτ is applied to the observations. Due to the source ordering indeterminacy, a reordering method has to be introduced to connect consecutive segments of each source signal. We use for that the Gale-Shapley stable marriage algorithm [START_REF] Gale | College admissions and the stability of marriage[END_REF] which constructs stable matchings between consecutive time slices source estimations. The ranking criterion is based on the comparison of the dot products between normalized Fourier spectra of these slices. • Deformations and spectra estimations. For each source, the joint estimation of {θ i,τ } τ ∈D and S Xi is obtained via the JEFAS algorithm (which is detailed in [START_REF] Meynard | Spectral Analysis for Nonstationary Audio[END_REF]). For this purpose, the input wavelet transform w y of the source y i is replaced with its estimate {B τ w z,τ } τ ∈D . Regarding initialization, a basic method is to use a stationary BSS method on observations to obtain a first unmixing matrix estimate. For instance, SOBI [START_REF] Belouchrani | A blind source separation technique using second-order statistics[END_REF] is a stationary BSS algorithm which can give an initial unmixing matrix. A better initial matrix can be obtained by piecewise SOBI estimates on non overlapping segments (called p-SOBI), where the stationarity assumption makes more sense.

The convergence is monitored using the Source to Interference Ratio (SIR) introduced in [START_REF] Vincent | Performance measurement in blind audio source separation[END_REF]. For a given estimated source, SIR quantifies the presence of interferences from the other true sources. As we do not have access to the ground truth sources, we use as stopping criterion the SIR between ỹ(k-1) and ỹ(k) (instead of y) which gives an evaluation of the BSS update, and is therefore a relevant convergence assessment. We construct a synthetic example to evaluate the performances of the algorithm. The two sources are band-pass filtered white noise, with time-varying bandwidth. The mixing matrix coefficients are sinusoidally varying over time. On the left of figure 1, the wavelet transforms of both observations are displayed.

The evolution of the convergence criterion through iterations of JEFAS-BSS is displayed in figure 1 (top-right). We can empirically note that our algorithm converges in a small number of iterations. Indeed, after 15 iterations the convergence criterion is around 100 dB meaning the BSS update is negligible.

Finally, we evaluate the performances of the BSS algorithms (we refer to [START_REF] Meynard | Spectral Analysis for Nonstationary Audio[END_REF] for the evaluation of the performances of the deformations and spectra estimations). The Amari index [START_REF] S.-I. Amari | A new learning algorithm for blind signal separation[END_REF] is a measure of divergence between the matrix Bτ A τ and the identity matrix. The closer to zero the Amari index the better. On the bottom-right of figure 1, we display the evolution of the Amari index through time for each BSS algorithm. In table 1, we also compare the SIR, the SDR (Source to Distortion Ratio [START_REF] Vincent | Performance measurement in blind audio source separation[END_REF]), and the time-averaged Amari index of the BSS algorithms. Those different criteria show that BSS-JEFAS performances are higher than those of SOBI and p-SOBI. Besides, in average, p-SOBI gives a better Amari index than SOBI, which is understandable because it takes into account the nonstationarity of the mixing matrix. Nonetheless, the SIR and SDR of p-SOBI are worse than those of SOBI. Indeed, because this method does not take into account the regularity of B τ , the connections between slices are sensitive to discontinuities and create distortion in the estimated sources.

Figure 1 :

 1 Figure 1: Synthetic example. Left: scalograms of the observations. Top-right: Convergence criterion evolution. Bottom-right: Amari index evolution. Criterion SOBI p-SOBI JEFAS-BSS SIR (dB) 28.55 15.04 46.55 SDR (dB) 16.60 -4.53 37.69 Amari index 4.63 × 10 -2 1.74 ×10 -2 1.40 × 10 -4

Table 1 :

 1 Comparison of the performances between BSS algorithms: standard SOBI, piecewise SOBI and the proposed algorithm.


Aline Brunon
email: aline.brunon@ifsttar.fr

Karine Bruyère-Garnier

Michel Coret

INFLUENCE OF THE BIOLOGICAL ORIGIN OF THE LIVER ON THE CONSTITUTIVE AND FAILURE PROPERTIES OF THE HEPATIC PARENCHYMA

The aim of this study is to investigate the influence of the biological origin and the preservation method (freezing or not) on the hepatic parenchyma mechanical properties. Uniaxial tensile tests are conducted on human and porcine liver samples to characterize the parenchyma constitutive and failure properties. A statistical analysis revealed that the ultimate true stress is significantly higher on porcine parenchyma; the modulus also seems to be affected. For both origins, the ultimate stress is clearly modified by freezing. Microstructure observations revealed a large difference between porcine and human parenchymas' failure mechanisms, explaining the gap between the ultimate stresses. Using porcine parenchyma to mimic human parenchyma is therefore not satisfactory, especially in terms of failure properties.

INTRODUCTION

Predicting the occurrence of liver injuries is an important matter in the field on transport safety and requires knowledge about the failure properties of the hepatic tissues. In particular, literature provides data about the parenchyma behavior [1,[START_REF] Tamura | Stapp car crash journal[END_REF] but very few about its failure [3]. Potentialities of animal testing lead often to study animal tissues before extrapolation to human tissues, thus most of the available data have been determined on animal livers, especially porcine livers [4]. Moreover, testing soft tissues being complicated leads to the use of freezing in many cases, but the influence of this preservation method is not clear yet [START_REF] Tamura | Stapp car crash journal[END_REF]3]. This work aims at comparing human and porcine parenchyma in terms of mechanical properties, particularly at failure, and in terms of microstructure. The influence of freezing on the tissue is also investigated.

METHODS

Seven adult pig livers were obtained from the local butchery, within 4 or 5 days after euthanasia. Four livers were removed from post-mortem human subjects (PMHS) from the Department of Anatomy of University of Lyon through the French voluntary corpse donation to science program. In both cases, the whole livers or parts of them were either stored in the freezer until the cutting phase (from 24h to a few days, -18°C), or immediately cut into samples and tested within 5 days after death. Two types of I-shape samples were cut from the livers: parenchyma samples and samples made of parenchyma and capsule ("bimaterial" sample). As described on Figure 2, the failure mechanism of bimaterial samples allowed characterizing both capsule [START_REF] Brunon | Mechanical parameters determined on human and porcine parenchyma; values with the same superscript are significantly different with respect to the changing parameter (e.g. (a) : porcine fresh parenchyma modulus is significantly higher than human fresh parenchyma modulus). Biological origin Fresh Number of samples Frozen Number of samples Modulus (kPa) Human 420 ± 188 (a)[END_REF] and parenchyma.

The two ends of the sample were clamped; sliding was prevented by using sand-paper stuck on the clamps. Clamps were set up vertically on a Deltalab© EM550 electromechanical testing machine. Samples were loaded until complete failure with a displacement speed of 0.5mm/s, which implied a range of longitudinal strain rate of 10 -3 to 10 -2 s -1 depending on the sample initial length, remaining in quasistatic domain. Elongation of the sample on the front and the side faces was recorded with two Visario© 1500 Speed Cam digital video cameras (Figure 1). Clamp-to-clamp displacement and tensile load on the upper clamp were measured by using a cable actuated position sensor and a 3daN-single point load cell. Constitutive and failure properties of human and porcine parenchymas were compared using a Mann-Whitney test. Significant differences were considered for p<0.05.

For microstructure observation, histological slices of parenchyma were made after testing to observe specifically the failure area (Figure 4).

RESULTS AND DISCUSSION

Figure 2 illustrates the response of bimaterial samples: once the capsule is completely broken, the remaining tissue on the capsule failure area is only parenchyma. The ratio of load sustained by the parenchyma at the capsule failure is supposed to be the same during all the bimaterial elongation, and the parenchyma geometry during the test is caught by the video cameras. True stress and global strain can be determined for both types of samples (Figure 3). The stress vs. strain curves exhibit a typical toe-region that can be found in many biological soft tissues. The linear part occurring after this toe-region allows the determination of the modulus, which values are displayed in Table 1. Ultimate Green-Lagrange longitudinal strain and ultimate true stress were also measured.

CONCLUSIONS

This work shows that using porcine liver to mimic the mechanical properties of human liver parenchyma is not satisfactory in terms of modulus and ultimate stress, which can be explained by microstructure differences. Freezing the tissue affects significantly the ultimate stress in both human and porcine tissues.

Figure 1 :

 1 Figure 1: Upper view of the experimental setup.

Figure 2 :

 2 Figure 2: Failure mechanism of the bimaterial samples.

Figure 3 :Figure 4 :

 34 Figure 3: True stress versus global strain for a human liver Statistical test shows that porcine and human parenchymas are significantly different in terms of ultimate stress. It is also true for the modulus but the smaller amount of samples does not allow drawing a strong conclusion. In addition, for both porcine and human parenchyma, freezing the tissue


A Brunon
email: aline.brunon@inrets.fr

M Coret

K Bruyere
email: karine.bruyere@inrets.fr

A Combescure
email: alain.combescure@insa-lyon.fr

Analytical description of biological fibrous tissue failure

In the field of traumatology, human body Finite Element Models are used to predict injury occurence. This requires the development of constitutive laws to be integrated to these models.

Many fibrous membranes can be found in the human body, like the skin or the capsule surrounding most of the abdominal organs. Liver in particular is surrounded by Glisson's capsule, made mostly of collagen fibers with more or less preferred directions (see Fig. 1). Under impact, one of the failure types of the liver is surface laceration, which involves the capsule and the underlying tissue called parenchyma. Thus, predicting liver surface laceration first requires a model of fibrous tissue failure. Among the previous studies focusing on fibrous tissue modeling, two main directions can be found: continuum models [START_REF] Rigbi | Some Thoughts Concerning the Existence or Otherwise of an Isotropic Bimodulus Material[END_REF][START_REF] Curnier | Conewise linear elastic materials[END_REF] and models based on a microstructural representation of the fiber network [START_REF] Ateshian | Anisotropy of fibrous tissues in relation to the distribution of tensed and buckled fibers[END_REF][START_REF] Lanir | Constitutive equations for fibrous connective tissues[END_REF][START_REF] Soulhat | A fibril-network-reinforced biphasic model of cartilage in unconfined compression[END_REF][START_REF] Cacho | A constitutive model for fibrous tissues considering collagen fiber crimp[END_REF]. The second orientation makes it possible to follow the evolution of the structure (failure of fibers, failure of bonds between fibers, realignement, etc) and is a more precise and more general description of the tissue, but its compilation is time consuming. Most of the work done so far has focused on fibrous tissue elastic behavior.

The present work aims at describing fibrous tissue failure with a homogeneous damage model. It is based on a microstructural representation of a 2 -D tissue made of elastic fragile fibers with a maximum strain before failure of ε r , whom angular distribution is constant or variable. This model links global strain load to the occurrence of fiber failure (see Fig. 2) : the strain is supposed to be homogeneous in the tissue and each fiber bears a longitudinal strain of n.E.n with E the global strain tensor and n the unit vector in the fiber direction. Describing the equivalent homogeneous damage model requires the determination of the elastic domain and its evolution during damage. 2 -D yield surfaces, displaying the maximum shear strain for one fiber to break, are determined in the strain space, using this fiber-by-fiber calculation (see Fig. 3). They show that while the tissue behavior is isotropic if the fiber angular distribution is constant, it becomes

Figure 1 :

 1 Figure 1: a) Surface laceration on human liver -b) Histological section of human liver capsule (x40)

Figure 2 :

 2 Figure 2: Fibrous tissue, microstructural description of the fibrous tissue and macroscopic strain load

anisotropic as fibers break. Every damaging loading case implies therefore a different yield surface. Then, an analytical homogeneous model is proposed, describing the distorsion of the yield surface of the tissue, until its complete failure . Homogenization of the stress finally allows the determination of the constitutive law during damaging. It aims at skiping the microstructural part of the calculation and being integrated to an existing FE-model. To improve this model, further work will consist in enriching the fiber behavior description with a hyperelastic constitutive law, and adding phenomenons already included in behavior models from other authors : taking into account fiber bonds and their failure [START_REF] Isaksson | Evolution of bond fractures in a randomly distributed fiber network[END_REF], including a matrix in the model and its relation to the fibers during damage (sliding, wrenching), considering fibers alignement [START_REF] Gilbert | Collagen fiber alignment and biaxial mechanical behavior of porcine urinary bladder derived extracellular matrix[END_REF][START_REF] Kl Billiar | A method to quantify the fiber kinematics of planar tissues under biaxial stretch[END_REF] and crimp [START_REF] Cacho | A constitutive model for fibrous tissues considering collagen fiber crimp[END_REF] under loading.