Surabhi Sonam

Lakshmi Balasubramaniam

Shao-Zhen Lin

Ying Ming

Yow Ivan

Irina Pi Jaumà

Cecile Jebane

Marc Karnat

Yusuke Toyama

Philippe Marcq

Jacques Prost

René-Marc Mège

Dr Jean-François Rupprecht
email: rupprecht@cpt.univ-mrs.fr.

Benoît Ladoux

Dr Benoit Ladoux
email: benoit.ladoux@ijm.fr

Mechanical stress driven by rigidity sensing governs epithelial stability

Keywords: rigidity, epithelium, MDCK cells, hole formation, polyacrylamide gels, tissue stress and tension

Epithelia act as a barrier against environmental stress and abrasion and in vivo they are continuously exposed to environments of various mechanical properties. The impact of this environment on epithelial integrity remains elusive. By culturing epithelial cells on 2D hydrogels, we observe a loss of epithelial monolayer integrity through spontaneous hole formation when grown on soft substrates. Substrate stiffness triggers an unanticipated mechanical switch of epithelial monolayers from tensile on soft to compressive on stiff substrates. Through active nematic modelling, we find unique patterns of cell shape texture called nematic topological defects that underpin large isotropic stress fluctuations at certain locations thereby triggering mechanical failure of the monolayer and hole opening. Our results show that substrate stiffness provides feedback on monolayer mechanical state and that topological defects can trigger stochastic mechanical failure, with potential application towards a mechanistic understanding of compromised epithelial integrity in bacterial infection, tumor progression and morphogenesis.

The plasticity of epithelial tissue is crucial during animal ontogenesis, and wound healing [START_REF] Ladoux | Mechanobiology of collective cell behaviours[END_REF][START_REF] Campinho | Tension-oriented cell divisions limit anisotropic tissue tension in epithelial spreading during zebrafish epiboly[END_REF][START_REF] Heisenberg | Forces in tissue morphogenesis and patterning[END_REF][START_REF] Yu | Local mechanical forces promote polarized junctional assembly and axis elongation in Drosophila[END_REF] .

Epithelia are cell sheets that act as a covering for most of the internal and external surfaces of the body, playing an important role as protective barriers. As such, epithelial cells are constantly challenged by the environment assaults. In response to these challenging environments, epithelia are subject to constant cell renewal and cell extrusions, whose balance is key for epithelia homeostasis but also modify their shapes [START_REF] Lecuit | Cell surface mechanics and the control of cell shape, tissue patterns and morphogenesis[END_REF] . These morphological changes include the formation of holes between cells which are crucial for tissue development [START_REF] Low | Morphogenesis of neurons and glia within an epithelium[END_REF][START_REF] Dumortier | Hydraulic fracturing and active coarsening position the lumen of the mouse blastocyst[END_REF] but also in macrophage infiltration [START_REF] Akhmanova | Cell division in tissues enables macrophage infiltration[END_REF] or diseases such as diabetes [START_REF] Damms-Machado | Gut permeability is related to body weight, fatty liver disease, and insulin resistance in obese individuals undergoing weight reduction[END_REF] , asthma [START_REF] Xiao | Defective epithelial barrier function in asthma[END_REF] and cancer [START_REF] Soler | Increased tight junctional permeability is associated with the development of colon cancer[END_REF] .

Epithelial integrity is maintained under the guidance of long range and coordinated mechanical forces [START_REF] Ladoux | Mechanobiology of collective cell behaviours[END_REF] . These force patterns give rise to tissue scale stresses that assist in the maintenance of tissue homeostasis by controlling cell division 2 , cell intercalation [START_REF] Heisenberg | Forces in tissue morphogenesis and patterning[END_REF] , cell extrusion [START_REF] Eisenhoffer | Crowding induces live cell extrusion to maintain homeostatic cell numbers in epithelia[END_REF] and cell migration [START_REF] Trepat | Physical forces during collective cell migration[END_REF] . While local compression contributes to cell extrusion [START_REF] Saw | Topological defects in epithelia govern cell death and extrusion[END_REF] , cell division 2 and migration [START_REF] Trepat | Physical forces during collective cell migration[END_REF] occur in a tension dependent manner. In vitro, migrating epithelial cells display a tensile stress gradient that increases as one moves from the front to the middle of the migrating sheet [START_REF] Trepat | Physical forces during collective cell migration[END_REF][START_REF] Serra-Picamal | Mechanical waves during tissue expansion[END_REF] mediated through E-cadherin based adhesions [START_REF] Gayrard | Src-and confinementdependent FAK activation causes E-cadherin relaxation and β-catenin activity[END_REF] . However, excess stress buildup can result in tissue fracture. In vitro studies have shown that tension build up results in tearing of suspended proliferative epithelial monolayers indicating that tensile stresses can contribute to hole formation in growing epithelia [START_REF] Harris | Characterizing the mechanics of cultured cell monolayers[END_REF][START_REF] Fouchard | Curling of epithelial monolayers reveals coupling between active bending and tissue tension[END_REF] . In vivo studies across various organisms such as Drosophila melanogaster, Trichoplax adhaerens and Hydra [START_REF] Prakash | Motility-induced fracture reveals a ductile-tobrittle crossover in a simple animal's epithelia[END_REF][START_REF] Bull | Mobile defects born from an energy cascade shape the locomotive behavior of a headless animal[END_REF][START_REF] Proag | Physical and functional cell-matrix uncoupling in a developing tissue under tension[END_REF][START_REF] Carter | Dynamics of Mouth Opening in Hydra[END_REF] have shown hole formation to be an important developmental feature necessary for growth. This occurs through a build-up of force and stress within the developing epithelia leading to subsequent loss of cell-substrate adhesions [START_REF] Proag | Physical and functional cell-matrix uncoupling in a developing tissue under tension[END_REF] . The rupture is either initiated through tension build-up at cell-cell contacts as in chick lung epithelia [START_REF] Maina | Structural failures of the blood-gas barrier and the epithelialepithelial cell connections in the different vascular regions of the lung of the domestic fowl, Gallus gallus variant domesticus, at rest and during exercise[END_REF] , or tissue level tension as in the peripodial epithelium in Drosophila leg [START_REF] Fouchard | Curling of epithelial monolayers reveals coupling between active bending and tissue tension[END_REF] .

In other studies, the loss of cell-cell junctions in Drosophila [START_REF] Martin | Integration of contractile forces during tissue invagination[END_REF] or reduction of junctional contractility in Tribolium castaneum [START_REF] Benton | Fog signaling has diverse roles in epithelial morphogenesis in insects[END_REF] leads to temporary hole formation and loss of epithelial integrity. During mouse embryogenesis, cracks at cell-cell contacts gives rise to eventual opening up of the lumen and blastocyst formation in a tension dependent manner [START_REF] Dumortier | Hydraulic fracturing and active coarsening position the lumen of the mouse blastocyst[END_REF] . All these results across various model organisms show that hole formation possibly occurs in a tension dependent manner and can be related to the strength of cell-cell and cell-substrate adhesions.

However, the impact of cellular and tissular mechanical forces and stresses on epithelial integrity remains unclear. These mechanical stresses can be modulated by parameters such as substrate geometry and adhesiveness and rigidity [START_REF] Sunyer | Collective cell durotaxis emerges from long-range intercellular force transmission[END_REF][START_REF] Balcioglu | A subtle relationship between substrate stiffness and collective migration of cell clusters[END_REF][START_REF] Chen | Large-scale curvature sensing by directional actin flow drives cellular migration mode switching[END_REF] . Rigidity is known to impact a large variety of biological processes including cell adhesion, migration and differentiation [START_REF] Balcioglu | A subtle relationship between substrate stiffness and collective migration of cell clusters[END_REF][START_REF] Lo | Cell Movement Is Guided by the Rigidity of the Substrate[END_REF][START_REF] Nasrollahi | Past matrix stiffness primes epithelial cells and regulates their future collective migration through a mechanical memory[END_REF][START_REF] Barriga | Tissue stiffening coordinates morphogenesis by triggering collective cell migration in vivo[END_REF][START_REF] Trichet | Evidence of a large-scale mechanosensing mechanism for cellular adaptation to substrate stiffness[END_REF][START_REF] Hadden | Stem cell migration and mechanotransduction on linear stiffness gradient hydrogels[END_REF][START_REF] Shellard | Collective durotaxis along a self-generated stiffness gradient in vivo[END_REF] and stiffness mechanosensing relies on the cross-talk between cell-cell and cell-substrate adhesion [START_REF] Guo | Substrate Rigidity Regulates the Formation and Maintenance of Tissues[END_REF][START_REF] Matte | Matrix stiffness mechanically conditions EMT and migratory behavior of oral squamous cell carcinoma[END_REF][START_REF] Seddiki | Force-dependent binding of vinculin to α-catenin regulates cell-cell contact stability and collective cell behavior[END_REF] . Here, we used substrates of various stiffness on which we cultured epithelial cells as a tool to modulate tissue mechanics. We revealed that substrate stiffness induced a switch from a tensile state on soft to compressive on stiff substrates. Furthermore, this tensile state observed on softer substrates favors the formation of spontaneous holes within epithelial monolayers at nematic topological defects, further confirmed by our modeling approaches combining cell-based simulations and continuum hydrodynamic theory.

Hole formation in epithelial monolayers is rigidity-dependent

We first cultured MDCK epithelial cells on polyacrylamide (PA) gels of distinct stiffnesses, 2.3 and 55 kPa (hereafter referred to as unconfined condition). Surprisingly, we observed the spontaneous formation of holes within intact monolayers on 2.3 kPa gels (Figure 1a, Supplementary movie S1) during their expansion but not on 55 kPa (Figure 1a, Supplementary movie S1). To better control culture conditions, we thus used PA gels with circular micropatterns (1mm diameter) using deep UV patterning (hereafter referred to as confined condition) (Figure 1b), in combination with live cell imaging to study the response of cell monolayers to substrate stiffness (Figure 1c, Supplementary movie S2). We noted a significant increase in the density of holes upon confining the monolayers (Extended Figure 1a) on 2.3kPa gels to ~18±9 holes/mm 2 . On 2.3kPa gels holes formed at the rate of 3.4±1.5 holes/mm 2 /h (n = 15 circles in 2 independent experiments). Similar to the unconfined conditions, MDCK monolayers formed holes in a rigidity-dependent manner and the prevalence of holes monotonically decreased with the increase in gel stiffness (Figure 1d, Supplementary movie S2). Hence the most stable holes were observed on gels of stiffness 2.3 kPa (Figure 1c,d) and thus, we continued to study hole growth and dynamics on 2.3 kPa gels.

Using live imaging of Cy5-tagged fibronectin coated gels, we verified that hole initiation was not associated to any pre-existing inhomogeneity nor progressive loss of fibronectin coating over the time scale of our experiments (Extended Figure 1b, Supplementary movie S3).

A closer examination revealed two types of cellular events prior to hole initiation: ~10% of holes were associated to breakage of junctions between stretched cell (Extended Figure 1c, Supplementary movies S1 and S4) while ~90% (n=30 holes, 2 independent experiments) were formed in the immediate vicinity of a figure of cell division at the time of cytokinesis (or of abscission of the intercellular bridge [START_REF] Lafaurie-Janvore | ESCRT-III Assembly and Cytokinetic Abscission Are Induced by Tension Release in the Intercellular Bridge[END_REF] ) (Extended Figure 1d, Supplementary movie S4), with the hole major axis aligned with the division axis, (Extended Figure 1d,e). Within the preceding 50min to a hole detection event, the nearest cell division formed at a mean distance of 14±4𝜇m, against 36𝜇m if cell division were randomly distributed, see Extended Figure 1f and Supplementary movie S4.

Only in very few instances we observed the formation of holes on stiff substrates and only at higher densities. However, the prevalence of hole formation on 2.3 kPa (soft) gels increased as a function of cell density (Figure 1e) and these holes continue to grow as the density increases (Supplementary Movie S2). As suggested above, hole formation may be induced through two different mechanisms associated to cell stretching and cell division, respectively. Interestingly, when comparing cell proliferation on soft and stiff, we found a significant decrease in cell proliferation on stiff substrates (Extended figures 1g, h).

To delineate the role of cell division we blocked DNA synthesis by mitomycin C treatment.

Upon mitomycin C treatment on 2.3kPa gels we obtained a significant reduction in the number of holes formed per unit surface (Figure 1f), indicating that when cell division is blocked, hole formation is largely blocked as well, but cell stretching events can still lead to hole formation. We generalize our findings by performing complementary experiments on the colorectal adenocarcinoma epithelial cell line Caco2, which displayed a similar behavior as MDCK cells leading to hole formation on soft (2.3 kPa) substrates (Extended Figure 1i, Supplementary movie S5). Altogether, these observations point towards hole formation in epithelial monolayers due to altered cell-cell interactions mediated by substrate stiffness.

Dynamics of hole initiation and growth

Tracking the dynamics of hole opening through a side view angle and imaging MDCK monolayers tagged with actin-GFP showed differences in the curvature formed by the cell edge at the hole periphery during and after hole opening. At early stage of hole initiation, cells spread on the substrate and the cell front formed a contact angle with the substrate lower than 90⁰, suggesting that rupture first occurs on the apical side (Figure 2a(i),b). During hole opening, the local curvature of the cell front changes, evidenced by an increase in contact angle between the cell front and substrate (greater than 90⁰) (Figure 2a(ii),b). The change in curvature of the cell front and the radial speed of hole growth are reminiscent of an active dewetting process, previously described during cell spreading [START_REF] Douezan | Spreading dynamics and wetting transition of cellular aggregates[END_REF][START_REF] Beaune | How cells flow in the spreading of cellular aggregates[END_REF] as a result of low interactions with the underlying surface or increased tension. In addition, at high cell density, a side-view of nuclei-labelled monolayers revealed a preference to form multilayered epithelia at the hole boundary (Figure 2c,d). This shows that cells at high densities prefer to pile up around the hole instead of closing it, pointing towards reduced substrate adhesion.

Along this line, tracking the dynamics of holes revealed two distinct lifetimes: short-lived holes that tend to close in less than 5 hours (Figure 2e (corresponding to approx. 36µm in radius) beyond which the holes tended to be long-lived (Figure 2g). These long-term holes occasionally grew through the fusion of two holes forming a larger one while in other cases, fission of large holes results in the formation of two small holes (Supplementary movie S2).

Reduced substrate adhesion triggers hole formation

Previous studies have shown that substrate stiffness has a strong impact on focal adhesion (FA) mechanosensing at single and collective cell levels [START_REF] Pelham | Cell locomotion and focal adhesions are regulated by substrate flexibility[END_REF][START_REF] Saez | Is the Mechanical Activity of Epithelial Cells Controlled by Deformations or Forces?[END_REF][START_REF] De Rooij | Integrin-dependent actomyosin contraction regulates epithelial cell scattering[END_REF][START_REF] Plotnikov | Force Fluctuations within Focal Adhesions Mediate ECM-Rigidity Sensing to Guide Directed Cell Migration[END_REF][START_REF] Kocgozlu | Epithelial cell packing induces distinct modes of cell extrusions[END_REF] while its role in regulating cell-cell adhesion remains less clear [START_REF] Seddiki | Force-dependent binding of vinculin to α-catenin regulates cell-cell contact stability and collective cell behavior[END_REF][START_REF] Eguiluz | Substrate stiffness and VE-cadherin mechano-transduction coordinate to regulate endothelial monolayer integrity[END_REF] . We first looked at the role of substrate adhesions during hole formation.

To this end, we decreased the fibronectin concentration from 20 µg/ml to 5 µg/ml (see Methods). On 2.3 kPa gels varying fibronectin concentration did not significantly affect hole formation. On stiff (55 kPa) gels the hole formation further diminished to zero on 5 µg/ml compared to close to zero at 20 µg/ml (Extended Figure 2a). However, when soft (2.3 kPa) gels were patterned with different amounts of fibronectin simultaneously (see Methods), holes preferentially formed on regions of low fibronectin coating (Figure 3a, Supplementary movie S7), with 60% probability. No holes were formed in the other cases (n= 10 samples in 2 independent experiments). We then stained for cell-substrate adhesion marker (paxillin), where we noted a marked increase in the length and area of focal adhesions on 55kPa gels in comparison to 2.3kPa gels (Extended Figure 2b-d). In agreement with [START_REF] Kocgozlu | Epithelial cell packing induces distinct modes of cell extrusions[END_REF] , we also showed that both the number and length of substrate adhesions reduced as cell density increases (Extended Figures 2e-g) which may explain the higher propensity of holes to remain open at high density (Supplementary movie S2). Altogether, these results point toward a major contribution of stiffness sensing over purely substrate adhesion mechanisms on hole formation.

Mechanical failure of monolayers grown on soft substrates is unlikely to be explained by a lower strength of cell-cell adhesion as no difference in junctional E-cadherin intensity within MDCK monolayers was observed whether grown on 2.3kPa or 55kPa gels, nor did E-cadherin intensity change prior to hole formation on 2.3kPa (Extended Figure 2h-k). Yet, we observed an increase in the junctional intensity of vinculin on soft substrates, as compared to the stiff ones (Extended Figures 2l,m). This increase in junctional vinculin, a mechanosensitive protein -known to shuttle between cell-cell contacts and cell-substrate adhesions in a force dependent manner [START_REF] Yonemura | α-Catenin as a tension transducer that induces adherens junction development[END_REF][START_REF] Bays | Vinculin in cell-cell and cell-matrix adhesions[END_REF] , accompanied by reduced substrate adhesion point towards a build-up in junctional stresses leading to hole formation on soft (2.3kPa) gels.

We tested this interpretation through an in silico vertex model of epithelial tissues (see Methods) [START_REF] Alt | Vertex models: From cell mechanics to tissue morphogenesis[END_REF] . In such framework, the dynamics the contacts between cell-cell junctions (vertices) results from a balance between a drag friction to the substrate and internal stresses generated within cells; constant tissue flows were achieved through an active stress mechanism (see Methods and next paragraph). Modulating surface friction in our simulations resulted in higher cellular stresses within regions of low substrate friction (Extended Figure 3a,b), in agreement with our experimental observation of hole formation at low fibronectin density (Figure 3a).

Importantly, these results can be recapitulated by considering a minimal Kelvin-Voigt rheological model for cells which accounts for the role of cell-substrate friction. In such model, the stress load imposed by the tissue on an individual cell is shared between the viscous drag on the substrate (damper) and a bulk cell or cell-cell elastic moduli (spring). In regions of reduced adhesion molecule density or low substrate elastic modulus [START_REF] Walcott | A mechanical model of actin stress fiber formation and substrate elasticity sensing in adherent cells[END_REF] , a lower viscous drag on the substrate is expected [START_REF] Tawada | Protein friction exerted by motor enzymes through a weakbinding interaction[END_REF] . In a Kelvin-Voigt model, a weaker damper (low adhesion) results in a higher stress load within cells (spring). Since higher stresses within epithelial monolayers has been attributed to mechanical failure of cell-cell junctions [START_REF] Harris | Characterizing the mechanics of cultured cell monolayers[END_REF] , our Kelvin-Voigt model explains why hole formation is favored by a lower adhesion at the cell-substrate interface in both experiments and simulations.

Monolayer tension on soft gels drives hole formation

Our results thus far, point towards an increase in junctional tension on soft (2.3kPa) substrates.

To assess the contributions of tension during hole formation, we measured traction forces and inferred stress within the monolayer using Traction Force Microscopy (TFM) and Bayesian Inversion Stress Microscopy 52 (BISM), respectively (Figure 3b). In these experiments, we used 30 kPa as stiff substrates instead of 55 kPa due to limitations of TFM use at higher stiffnesses. We first confirmed that cells exerted higher traction forces on stiffer substrates both under confined and unconfined conditions (Figure 3c Our results are in agreement with previous studies that described cellular monolayers to be tensile on soft substrates [START_REF] Trepat | Physical forces during collective cell migration[END_REF][START_REF] Nier | Inference of Internal Stress in a Cell Monolayer[END_REF] , but unveiled an unanticipated switch from a tensile to a compressive state upon increase in substrate stiffness (Figure 3e). We further confirmed our stress inference results using laser ablation on cellular monolayers plated on 2.3 kPa and 30 kPa gels (Supplementary movie S8). The recoil velocity (noise level ~0.6 μm/sec) of monolayers was negligible on 30 kPa gels but significant around 2.5µm/s on 2.3 kPa gels (Figure 3f) confirming the tensile nature of the epithelial monolayers on soft gels.

We then investigated the changes of cell movement and monolayer tension in the vicinity of the hole over time. About 30-50 minutes prior to hole initiation in a region of about 5 cell radii, cells showed outward movement combined by high velocity vectors pointing outwards and positive divergence (Figure 3g top, Extended Figure 3f). Traction force vectors demonstrated a similar pattern, where cells exerted outward forces before hole initiation (Extended Figure 3g). The isotropic stress exhibited a local buildup of tensile stress which relaxed prior to hole initiation (Figure 3g bottom,h, Extended Figure 3h) and about 14% of all tensile regions give rise to hole formation. This local spike in stress prior to hole formation was in striking contrast to the overall monolayer stress that remained unchanged (Figure 3h). We quantified the straightness of E-cadherin junctions, a direct indicator of the local stress in the tissue [START_REF] Hara | Cell Boundary Elongation by Non-autonomous Contractility in Cell Oscillation[END_REF] and found an increased straightness around the hole in comparison to the regions far away from the hole (Extended Figure 3i,j) confirming the implication of local tensile stresses in the process of hole initiation.

To understand the origin of this difference in tissue stresses with substrate stiffness, we turned to a vertex model approach where cell-adhesion energy can be modulated. Beyond the individual cell shape energy function (Method), we expect substrate interactions to modify the cell-substrate adhesion energy [START_REF] Hannezo | Theory of epithelial sheet morphology in three dimensions[END_REF] through a contribution 𝛿𝑈 = -𝛾 𝑏 𝐴, where 𝐴 is the cell area; this results in a modulated spreading area 𝐴 0 * = 𝐴 0 + 𝛾 𝑏 /𝐾 𝐴 , where 𝐾 𝐴 is the standard area elasticity (SI). Thus, having constant cell divisions with a constant spreading area of each daughter cell 𝐴 0 * leads to a simulated tissue under compression (Figure S5). However, a reduced spreading area of daughter cells upon cell division gives rise to a simulated tensile tissue (Figure S5). Such interpretation is consistent with our experimental observations of tensile monolayers on soft substrates with reduced size of focal adhesions.

Nematic ordering of cells initiates hole opening

To understand the origin of this increase in local stress, we looked at local organization of cells since nematic ordering and defects (trefoils or comet shapes, corresponding to -1/2 and +1/2 topological defects) (Figure 4a), with either extensile [START_REF] Saw | Topological defects in epithelia govern cell death and extrusion[END_REF][START_REF] Blanch-Mercader | Turbulent Dynamics of Epithelial Cell Cultures[END_REF][START_REF] Balasubramaniam | Investigating the nature of active forces in tissues reveals how contractile cells can form extensile monolayers[END_REF] or contractile [START_REF] Balasubramaniam | Investigating the nature of active forces in tissues reveals how contractile cells can form extensile monolayers[END_REF][START_REF] Duclos | Topological defects in confined populations of spindle-shaped cells[END_REF] active stresses give rise to characteristic stress profiles around these defects. In particular, isotropic stress exhibits distinct patterns between -1/2 and +1/2 defects as it is tensile around -1/2 defects (Figure 4a left) and compressive at the head portion and tensile at the tail end of a + 1/2 defect (Figure 4a right) [START_REF] Saw | Topological defects in epithelia govern cell death and extrusion[END_REF] . Topological defects are found on both soft and stiff substrates with comparable number of defects formed on soft and stiff substrates as shown in Extended Figure 4a. Tracking the movement of +1/2 (comet) defects revealed monolayers to be extensile on both soft and stiff substrates.(Extended Figure 4b) in agreement with our previous work [START_REF] Saw | Topological defects in epithelia govern cell death and extrusion[END_REF][START_REF] Balasubramaniam | Investigating the nature of active forces in tissues reveals how contractile cells can form extensile monolayers[END_REF] .

Upon calculating the MSD of defects, we found the lifetime of +1/2 defects to be 129±49 mins on soft substrates and 92±90 mins on stiff substrates (Extended Figure 4c), although, -1/2 defects were slower than +1/2 defects on both substrates as shown previously [START_REF] Blanch-Mercader | Turbulent Dynamics of Epithelial Cell Cultures[END_REF][START_REF] Balasubramaniam | Investigating the nature of active forces in tissues reveals how contractile cells can form extensile monolayers[END_REF] . We then compared this to the persistence of tensile hotspots in a region of 51x51µm at random locations and we find that tensile regions are persistent for a longer period on soft substrates in comparison to stiff substrates. The mean lifetime of tensile regions on soft substrates was 97.5±39 min while the mean lifetime of +1/2 defects on soft substrates was 129±49 min thereby showing a strong correlation with the lifetime of +1/2 defects. However, the persistence of tensile regions on stiff substrates was 53±21 min and is lower than the lifetime of +1/2 defects (Extended Figure 4d).

The stress pattern around defects on soft and stiff substrates seem to be different. +1/2 defects have compressive stresses at the head of the defect and tensile stresses around the tail on soft substrates, whereas stresses are overall compressive on stiff substrates at +1/2 defects but less at the tail (Extended Figure 4e,f). Moreover, the cores of -1/2 defects on soft substrates exhibit tensile stresses (Extended Figure 4g) on the contrary the core of -1/2 defects are compressive on stiff substrates.

Upon correlating the location of hole formation with the defect location, we find that the holes form at specific locations around topological defects on soft substrates, with ~70% of holes initiated at locations near -1/2 nematic defects which are under tensile stress prior to hole initiation (Figure 4b left) while the remaining ~30% holes were initiated close to the tensile tail region of a +1/2 defect (Figure 4b right). Altogether, these results show that the formation of holes is favored on soft surfaces (i) due to the overall tensile nature of the monolayer and

(ii) near -1/2 defects or tail region of +1/2 defects due to a local amplification of tensile stresses.

However, dividing these holes into different groups based on their origin (cell division or cell stretching) and correlating them to the formation of topological defects showed no correlation between topological defect shape and either event of hole formation. In our experiments ~50% of hole formation driven by cell division was related to a +1/2 defect and ~50% of holes driven by cell stretching were driven by +1/2 defect formation. (Extended Figure 4h).

We used our cell-based vertex model to explore whether topological defects could underpin the large stresses needed to trigger hole opening. Indeed, as visible in Figure 4a, the mean stress around the topological defect is at a maximum of 60 Pa.µm, significantly lower than the range of stresses observed to trigger hole opening (200-500 Pa.µm, Extended Figure 3d).We utilized a recent method to generate cell-based nematic stresses [START_REF] Tlili | Shaping the zebrafish myotome by intertissue friction and active stress[END_REF][START_REF] Lin | Implementation of cellular bulk stresses in vertex models of biological tissues[END_REF] that were oriented by the cell shape [START_REF] Balasubramaniam | Investigating the nature of active forces in tissues reveals how contractile cells can form extensile monolayers[END_REF][START_REF] Mueller | Emergence of Active Nematic Behavior in Monolayers of Isotropic Cells[END_REF][START_REF] Comelles | Epithelial colonies in vitro elongate through collective effects[END_REF] (SI I). We show that such cell-based active stresses lead to spontaneous tissue flows (Supplementary movie S9) with spontaneous generation of ±1/2 topological defects with average stress patterns that match those observed in experiments (Figure 4c,d). We further found that the 95 th upper percentile (Methods) of the isotropic stress, is in that range of values of stresses needed to create holes. Importantly, it exhibited the same pattern as mean stress both in simulations (Figure 4d,e) and experiments (Figure 4b,f). Together, both experimental and numerical results show that hole formation on soft surfaces emerge in areas of high tension associated to specific location around topological defects.

To further identify the mode of stress-mediated fracture we validated our model prediction that regions of large shear stresses (as quantified through the Von-Mises criterium, Methods) were anti-correlated to the location of hole opening (Extended Figure 5). Hole opening is therefore a mode I (tension mediated) process according to classical terminology in material science [START_REF] Anderson | Fracture Mechanics: Fundamentals and Applications[END_REF] .

Here, the measured stress maps were obtained through coarse-graining over a few cell sizes.

We utilized our cell-based simulation framework to test whether we could test hypothesis at the single cell-cell junction scale. Initiating holes by splitting a vertex into three, we investigated whether cell-cell junction ruptures could either be mediated (i) through normal stresses applied to cell-cell junction or (ii) by the junction elongation rate (Extended Figure 6a,b). We found that only the normal stress hypothesis is consistent with holes forming in regions of extreme isotropic stresses (Extended Figure 6c-f).

Reducing the level of active stress level led to a lower amplitude of stress fluctuations thereby suppressing hole formation in our simulations (SI). This is in qualitative agreement with our experiments where 50µM CK666 (inhibitor of actin polymerization) led to abrogation of hole formation (Extended Figure 7a,b) and treatment with 20 µM blebbistatin (Myosin IIA inhibitor) led to reduced number of holes (Extended Figure 7b,c), in spite of both blebbistatin [START_REF] Saw | Topological defects in epithelia govern cell death and extrusion[END_REF] and CK666 treatment maintaining extensile activity (Extended Figure 7d,e).

Hole dynamics

We then evaluated the dynamics of holes in our simulations using the same parameter set as in Supplementary movie S10). On short times, hole growth was linear with a radial speed of (6.1 ± 0.1) x 10 -3 µm/s (Extended figure 8c), leading to similar values as those reported by Beaune et al [START_REF] Beaune | How cells flow in the spreading of cellular aggregates[END_REF] for spreading sarcoma cell aggregates (9 ± 0.5) x 10 -3 µm/s).

As holes grew and opened, they evolved from a circular to an irregular shape featuring fingerlike structures (Figure 5a 5b,c). This tangential organization continued until the hole reached its maximal area (Figure 5c, Extended Figure 8f). In contrast, closure of these holes was associated to a loss of tangential organization in both experiments and simulations (Extended figures 8a, e).

We propose a simplified coarse-grained model to fit the hole dynamics and the existence of 1) and find an increase of the hole opening rate with the exerted activity stress, see SI. We recapitulated the key features of our model in Figure 5e.

Conclusion

In vivo failure in adherens junctions [START_REF] Martin | Integration of contractile forces during tissue invagination[END_REF][START_REF] Classen | Hexagonal packing of Drosophila wing epithelial cells by the planar cell polarity pathway[END_REF] , increased mechanical load [START_REF] Maina | Structural failures of the blood-gas barrier and the epithelialepithelial cell connections in the different vascular regions of the lung of the domestic fowl, Gallus gallus variant domesticus, at rest and during exercise[END_REF] and post-apoptotic failure in sealing [START_REF] Watson | Mechanisms of Epithelial Cell Shedding in the Mammalian Intestine and Maintenance of Barrier Function[END_REF][START_REF] Bullen | Characterization of epithelial cell shedding from human small intestine[END_REF] leads to hole formation in tissues. During morphogenesis, hole formation is not only essential for Drosophila peripodium elongation [START_REF] Proag | Physical and functional cell-matrix uncoupling in a developing tissue under tension[END_REF] , mouse embryogenesis 7 but is also crucial for macrophage invasion during Drosophila development [START_REF] Akhmanova | Cell division in tissues enables macrophage infiltration[END_REF] . In addition loss of epithelial integrity has been reported under pathophysiological conditions [START_REF] Weiss | Cell detachment and metastasis[END_REF][START_REF] Gelman | Retinal damage induced by internal limiting membrane removal[END_REF] , immune cell [START_REF] Escribano | Balance of mechanical forces drives endothelial gap formation and may facilitate cancer and immune-cell extravasation[END_REF] and tumor extravasation [START_REF] Chen | Mechanisms of tumor cell extravasation in an in vitro microvascular network platform[END_REF] . Since tissue integrity and barrier function [START_REF] Wang | Matrix stiffness regulates vascular integrity through focal adhesion kinase activity[END_REF][START_REF] Mammoto | Control of lung vascular permeability and endotoxin-induced pulmonary oedema by changes in extracellular matrix mechanics[END_REF] are known to be altered by substrate stiffness, our study shows distinct mechanical states of epithelial tissues are triggered by substrate stiffness which in turn modify tissue integrity. Epithelial monolayers grown on soft substrates exhibit the formation of holes in the bulk of the tissues at locations of high tensile stresses.

The emergence of holes within these monolayers relies on an unanticipated behavior of cellular monolayers, being under tension on soft and compression on stiff substrates. Tension has been attributed to tissue tear in suspended monolayers [START_REF] Harris | Characterizing the mechanics of cultured cell monolayers[END_REF] , stretched monolayer [START_REF] Casares | Hydraulic fracture during epithelial stretching[END_REF] , peripodium elongation [START_REF] Proag | Physical and functional cell-matrix uncoupling in a developing tissue under tension[END_REF] , gaps in monolayer of neural progenitor cells [START_REF] Kawaguchi | Topological defects control collective dynamics in neural progenitor cell cultures[END_REF] and in migrating organisms [START_REF] Prakash | Motility-induced fracture reveals a ductile-tobrittle crossover in a simple animal's epithelia[END_REF] . Our results so far show that reduced substrate adhesion in combination with tensile nature of monolayers on soft substrates make them sensitive to local increase in tension, resulting in spontaneous hole formations. These tensile stresses arise from the formation of topological defects and hole formation is initiated within the tensile regions of those defects. Theoretical approaches, corroborated by experiments, predict that hole formation relies on the formation of topological defects where cells experience high tensile normal stresses along cell-cell junctions. We anticipate that our mechanistic in vitro study will bring new perspectives to understand morphological changes of epithelial tissues as well as the maintenance of their integrity during development and morphogenetic events.

Material and Methods

Patterned surfaces on polyacrylamide (PA) gels

Glass coverslips were plasma activated and coated with 0.1mg/ml PLL-g-PEG (SuSoS Technology). Using a quartz mask, circles of 1mm diameter were patterned on the passivated glass coverslips and incubated with 5 or 20µg/ml fibronectin for 30 min. After incubation, glass

Immunostaining and antibodies used

Samples were fixed with 4% paraformaldehyde (Thermo Fischer Scientific, Waltham, MA, USA) for 10 mins and permeabilized with 0.5% Triton-X for 5 mins. Cells were then blocked with 1% BSA/PBS and incubated with primary antibody overnight. Cells were incubated in secondary antibody for 1 hour and then mounted on another glass coverslip using Mowiol 40-88 (Sigma-Aldrich). For immunostaining, the following primary antibodies were used, E-cadherin (24E10-Cell Signaling Technology; DECMA1-Sigma Aldrich, ECCD2, Takara Bio)

(1:100)and paxillin (Abcam, Ab32084) (1:100) vinculin was a gift from Glukhova lab (1:2).

Anti-mouse, anti-rat, and anti-rabbit secondary antibodies conjugated with Alexa (488 or 568) (used at 1:200 dilution), phalloidin (1:20) were purchased from Life Technologies and Hoechst (Thermo Fisher, 1:10000). The fixation was slightly altered to stain vinculin at cell-cell junctions and focal adhesions. Cells were fixed with a mix of 4% PFA and 0.5% Triton-X 100 for 90 seconds, followed by fixation with 4% PFA for 10 min.

Traction force microscopy

PA gels were embedded with 200nm fluorescent beads (Life Technologies, Paisley, UK) during polymerization. The samples were imaged on Biostation IM-Q (Nikon) for 24 hours after which 1ml of 10x SDS was added to the sample in order to obtain the resting position of beads.

Differential fibronectin coating on PA gels

Differential coating of fibronectin was performed using Primo® (Alveole). Glass coverslips were passivated with double coating of 0.1% w/v poly-L-Lysine (PLL) for 30 min and 100mg/ml poly (ethylene glycol) succinimidyl valerate (PEG-SVA) for 1 hour. After washing the glass coverslip carefully with water, they were patterned with evenly spaced grey circular patterns (100µm diameter, 500µm apart) using a UV beam with an energy dosage of 600mJ/mm in presence of a photoinitiator, PLPP (Alveole). After patterning, the sample was washed with 1x PBS and incubated with 20µg/ml fibronectin for 10 min. Excess protein was washed away with 1x PBS and the rest of the unpatterned space (while covering the patterned area with black masks) was exposed to UV beam with an energy of 600mJ/mm using a white pattern. Newly exposed surface was incubated with 50µg/ml fibronectin (Sigma Aldrich) and 1µg/ml cy3-conjugated fibronectin for 10 min. PA gel was casted after washing off excess protein. As mentioned previously, patterned glass coverslip was removed from the sandwich to reveal the patterns on PA gels.

Imaging

Phase contrast long-term imaging of samples was performed with 10x objective on Olympus inverted microscope (IX81). Traction force microscopy was performed on Biostation IM-Q (Nikon) with a 10x objective. Imaging was done for 24 hours with frame rate of 0.2frames/min.

Fluorescent cells were imaged on Zeiss spinning disk confocal (CSU X1) using 40x water objective. To follow the actin dynamics, Actin-GFP MDCK cells were imaged at a frame rate of 0.5frames/min for 12 hours. Fixed samples were imaged with 63x Oil objective on Zeiss

Confocal microscope (LSM 710).

Laser ablation experiments

Wound induction (by ablating a cell) on monolayer tissue of density of about 3000-3500 cells per mm 2 was done on a Nikon A1R MP laser scanning confocal microscope with Nikon CFI Apo LWD Lambda S 40XC WI/1.15 water-immersion objective. After about 10 s of imaging at 1 s interval for a total duration of 5 minutes, an ultraviolet laser (355 nm, 3-5 ns pulse duration, 1-15 Hz repetition rate, Minilite II, High Energy Nd:YAG, Amplitude Laser) was focused on the center of the monolayer within the region of interest for 1 s at a laser power of 450 nW at the back aperture of the objective. The recoil velocity was measured by manually detecting cell edges over time and plotting the change in perimeter of the ablated region.

Data analysis

Images were prepared using ImageJ and analysis was done with the plugins and home-written macros on ImageJ and MATLAB. Cross-sectional views of the tissue were represented by

Imaris. E-cadherin and vinculin intensities were measured by manually drawing a line over the junction and measuring the intensity at cell-cell contacts. Similarly, paxillin size was measured by manual segmenting paxillin molecules obtained from immunostaining and size determination on ImageJ.

Junctional straightness was calculated as a ratio between Euclidean length (measured by drawing a straight line from one end to another) and actual length of cell-cell junctions identified by the E-cadherin staining.

Cell segmentation

Cell segmentation and division tracking shown in Supplementary Movie 4 was manually corrected using Cellpose [START_REF] Stringer | Cellpose: a generalist algorithm for cellular segmentation[END_REF] with an erosion with a 3x3 square pixel kernel applied to each mask;

we used btrack [START_REF] Bove | Local cellular neighborhood controls proliferation in cell competition[END_REF] for the cell tracking. Manual corrections were performed using the Napari software (46 lineage for the soft case, 42 for the stiff case) 77 . Figure 1f is obtained through a standard method in spatial statistics [START_REF] Baddeley | Spatial Point Patterns: Methodology and Applications with R[END_REF] : if the nearest division (with the 50min time window prior to the hole initiation date) is located further away from the hole than the boundary of the image, the hole is not considered in the statistics of

Nematic analysis

In our experiments, nematic detection analysis was done on both untreated and mitomycin-Ctreated samples at low density as cells tend to be more isotropic at higher densities and form 3D structures making it hard to identify the cell shape. Orientation field and defect detection was carried out as previously described [START_REF] Saw | Topological defects in epithelia govern cell death and extrusion[END_REF] . Briefly, the orientation of cells was obtained using the OrientationJ plugin on ImageJ [START_REF] Püspöki | Transforms and Operators for Directional Bioimage Analysis: A Survey[END_REF] where the largest eigenvector of the structure tensor was obtained for each pixel thereby giving us the orientation. The local nematic order parameter tensor Q (averaged over 1-2 cells) (52 µm) was then calculated. The largest eigenvalue of Q was taken to the orientation of 2-3 cells and overlaid over phase contrast images as a red line.

Based on this orientation, winding number parameter [START_REF] Vromans | Orientational properties of nematic disclinations[END_REF][START_REF] Decamp | Orientational Order of Motile Defects in Active Nematics[END_REF] , was used to identify +1/2 and -1/2 defects. In order to reduce noise, we only identify defects that stable over 60min, we obtain the orientation of 1-2 cells. Defect detection was compared against previous studies [START_REF] Duclos | Topological defects in confined populations of spindle-shaped cells[END_REF] . The MSD and lifetime of defects were obtained by using TrackMate plugin in ImageJ [START_REF] Tinevez | TrackMate: An open and extensible platform for single-particle tracking[END_REF] . In the Supplementary Material, we show the robustness of our topological defect analysis. The 95th percentile is the value below which 95% of scores in the isotropic stress (or the von Mises stress) field frequency distribution falls.

Velocity measurements, strain rate, traction force and stress analysis Velocities of the monolayer were obtained through Particle Image Velocimetry (PIV) analysis using PIVlab 83 on MATLAB. Interrogation window of size 41x41µm (strain rate) and 21x21 µm (monolayer) with an overlap of 50% was used for analysis and the outlier vectors were manually removed. Having identified the location of defects, we obtain the average velocity field around defects identified by realigning these defects. The strain rate around these defects was calculated from the gradient of the velocity field as 𝜀̇= ∇𝑣. By plotting the velocity around +1/2 defect, we can characterize the system as an extensile or contractile system.

For force measurements, firstly the reference bead image was concatenated with the bead images containing cells such that the reference frame precedes the bead image with cells. The images were then stabilized using Image Stabilizer plugin in ImageJ (written by Kang Li, CMU) to correct with xy drift following which the intensity correction was applied on these images. The velocities of bead movement were obtained using PIV analysis. Using the FTTC plugin [START_REF] Tseng | Spatial organization of the extracellular matrix regulates cell-cell junction positioning[END_REF] , we obtain traction force fields from bead velocities using a regularization parameter of 9 x 10-9. Using the traction force fields, stress in the monolayer is obtained using Bayesian Inversion Stress Microscopy (BISM) [START_REF] Nier | Inference of Internal Stress in a Cell Monolayer[END_REF] where an unconfined boundary condition was used to obtain stresses. The isotropic stress is defined as half of the stress tensor trace ((𝜎 𝑥𝑥 + 𝜎 𝑦𝑦 ) 2 ⁄ ).

Besides the isotropic stress, we also checked the von Mises stress, which is a scalar invariant value of a stress tensor and defined as 𝜎 𝑉𝑀 = √𝜎 𝑥𝑥 2 -𝜎 𝑥𝑥 𝜎 𝑦𝑦 + 𝜎 𝑦𝑦 2 + 3𝜎 𝑥𝑦 2 . Using the traction force fields, stress in the monolayer is obtained using Bayesian Inversion Stress Microscopy (BISM) [START_REF] Nier | Inference of Internal Stress in a Cell Monolayer[END_REF] where an unconfined boundary condition was used to obtain stresses [START_REF] Saw | Topological defects in epithelia govern cell death and extrusion[END_REF] . As a consequence, we systematically ignored stress values obtained close to the domain boundaries, and considered only stresses in the central part of the domain (previously validated using numerical data [START_REF] Saw | Topological defects in epithelia govern cell death and extrusion[END_REF] ). The heatmaps obtained for strain rate and stress were smoothed through linear interpolation. Tensile hotspots detection was based on the time needed for the stress to change sign over an interrogation window of 51x51 µm.

Vertex-based active nematic model

Please refer to Supplementary Information I

Statistics

Statistical analysis of all the data sets were performed by the student t-tests.

  , Extended Figure 1j, Supplementary movie S6) and long-lived holes that stayed open for a longer duration (at least during the time course of our experiments) (Figure 2f, Extended Figure 1k, Supplementary movie S6). Computing the area of these short-and long-lived holes, revealed a threshold area, around 4000µm 2

,

  Extended figure 3c). Additionally, in confined MDCK monolayers the direction of traction forces along the diameter of the circle changed from soft to stiff substrates, i.e., traction forces pointed inward on soft and outward on stiff substrates (Figure 3d, Extended figure 3d), revealing a potential change in stress patterns within the monolayer. Computing monolayer stresses, we observed a change in the sign of averaged monolayer isotropic stress from positive values (~ 50 Pa.µm) (negative pressure) on soft substrates to a negative values (~ -60 Pa.µm) (positive pressure) on stiff substrates under both confined and unconfined conditions (Figure 3e, Extended figure 3e).

Figure 4 and

 4 Figure 4 and Extended figure 8a. As in experiments, we identified short-lived holes with a lifetime < 5 hours (Figure 2e, Extended figure 8a, Supplementary movie S10) and longlived holes that stayed open for a longer duration (Figure 2f, Extended figure 8b,

,

  Extended figures 8a, e and Supplementary movie S10) both in simulations and experiments. Occasionally, such fingers could reach the other side of the hole and drive fission of a large hole into two smaller ones (Extended figures 8a, e and Supplementary movie S10) both in simulations and experiments. In addition, during the phase of opening, cells at the periphery of hole became increasingly elongated and organized themselves tangentially at the margin of the hole both in simulations and experiments (Figures

  short and long-lived holes (SI). The hole opening rate is determined by a mechanical balance between (i) the local tissue tension around the hole 𝜎, (ii) a resisting line tension, denoted by 𝛾 (amount of mechanical work required to create a new free interface within the tissue), along with stresses associated to cell elongation around the hole, denoted by 𝜁𝛥q where 𝛥q quantifies the local elongation field within a nematic layer of width 𝜆 around the hole and (iii) a bulk viscous dynamic within the monolayer hole. Our analytical prediction for hole opening rate (𝛾 ̃≈ 𝛾 -𝜆𝜁Δ𝑞 a renormalized tension and 𝜉 ̃= 𝜉 ln (2𝑙/𝑟), where 𝑙 = √𝜂 𝜉 ⁄ a hydrodynamic length expressed in terms of the cell-substrate friction 𝜉 and the monolayer viscosity 𝜂. Equation (1) leads to a typical nucleation size 𝑅 𝑐 ≈ 𝛾 ̃𝜎 ⁄ ; for 𝑟 < 𝑅 𝑐 , holes spontaneously close while for a hole of radius r > Rc holes would tend to grow in agreement with the observation of small (resp. large) holes being short-lived (resp. long-lived). Fits of the control experiments based on Eq. (1) are consistent with 𝜎 = 500 Pa.𝜇m, 𝛾 ̃ ≈ 1.5 kPa.𝜇m 2 and 𝜉 = 50 Pa.𝜇m -1 .min (with 𝑙 =1000 𝜇m), Figure 5d. We also compare our vertex model result with Eq. (

  coverslips were rinsed in 1x PBS to remove excess protein. Simultaneously, silanization of another set of glass coverslips were performed by plasma activation of clean coverslips and then incubated with an ethanol solution containing 2% (v/v) 3-(trimethoxysilyl) propyl methacrylate (Sigma-Aldrich, St Louis, Missouri, USA) and 1% (v/v) acetic acid. The silanised coverslips were then heated at 120 o C. Freshly made Polyacrylamide (PA) mix was sandwiched between the patterned glass coverslip and silanized coverslip. Polyacrylamide mix was made as suggested in Plotnikov et al 74 . After polymerization, the patterned coverslips were peeled off to reveal the patterns of protein on PA gels. Samples were kept hydrated with 1x PBS until cell seeding.Cell culture and reagents usedMDCK WT (ATCC CCL-34), stably transfected actin GFP and E-cadherin GFP MDCK cells (kind gift from James Nelson's lab) and Caco2 cells were cultured in 4.5 gl -1 DMEM containing 10% FBS (20% for Caco2) and 1% Penicillin-Streptavidin (Gibco, Thermo Fischer Scientific, Waltham, MA, USA). 100µl cell culture media containing 100,000 cells was seeded on the PA gels for 1 hour. 2 ml fresh media was added into the sample without washing to prevent detachment of loosely attached cells. Samples were imaged the following day once they spread and reached confluency. In some experiments to study the effect of blocking cell division, cells were treated with 10µg/ml of mitomycin-C (Sigma-Aldrich, St Louis, Missouri, USA) for 1 hour and washed prior to imaging. 100µM CK666 (Sigma Aldrich) was added prior to imaging for Arp 2/3 inhibition experiments.

Figure 1f .

 1f Comparison to the random case if achieved by considering a Poisson process of parameters matching the total number of cell division observed during the movie.

Figure 1|

 1| Figure 1| Rigidity based hole formation within MDCK monolayers. (a) Time course of hole formation in an unconfined MDCK monolayer on 2.3kPa and 55kPa PA gels. Red arrows show the holes formed within the monolayer (Scale: 50µm) (b) Schematic of the process of fibronectin patterning on PA gels; top: fibronectin coating on coverslips patterned through deep UV; middle: PA gel is placed between patterned and silanized coverslips; bottom: MDCK cells are seeded on patterned PA gel. (c) Time based montage of MDCK cells forming holes on 2.3kPa PA gels (top panel) and intact monolayers on 55kPa gels (bottom panel). Red arrows show the holes formed in the monolayer (Scale: 100µm). (d) Likeliness of hole formation as a function of PA gel stiffness (n2.3 = 23, n4.1kPa = 23, n8.6 kPa = 24, n16.3 kPa = 8, n30 kPa = 7 and n55 kPa = 15 different circles; ***p < 0.001, ****p < 0.0001) (e) The number of holes formed as a function of cell density on untreated and mitomycin C treated monolayers on soft (2.3kPa) substrates and monolayers on stiff (55kPa) substrates averaged over n= 22 circles and n = 10 circles for untreated and mitomycin treated samples on 2.3kPa and n=14 circles for 55kPa. (f) The number of holes formed within untreated and mitomycin-c treated circles on soft (2.3kPa) gels. (nctrl = 33, nmito = 44, *p < 0.05). All results were obtained from 2 independent experiments and solid bars represent mean and error bars standard deviation.

Figure 2|

 2| Figure 2| Hole characteristics. (a) Cross-sectional view of actin-GFP MDCK cells before (top panel) and after (bottom panel) hole formation on soft (2.3kPa) substrates (Scale: 10 µm) (b) Change of contact angle over time (t0 = initiation of hole) as the hole grows. Blue line represents 90° on the y-axis. (n = 6 different circles) where (i) and (ii) illustrate the regions (i) and (ii) in panel a (c) Illustration of cell density change around the hole periphery on soft (2.3kPa) substrates (d) Post-hole formation, increase in cell density does not contribute to hole closure (n3000 = 16 different circles and n4000 = 15 different circles; ****p <0.0001). (Scale: 20 µm) (e, f) Time dependent change in hole area normalized over maximum area of the hole for experiment and simulation of (e) short-lived holes (<5hr) (n = 36 different circles) and (f) long-lived hole (>5hr) (n = 9 different circles in experiment) on soft (2.3kPa) substrates from 2 independent experiments. (g) Maximal hole opening area for short-lived (n = 42 different circles) and long-lived holes (n = 7 different circles) from 2 independent experiments (****p < 0.0001)). Solid lines represent mean and error bars represent standard deviation.

Figure 3|

 3| Figure 3| Substrate stiffness modulates tissue stress within MDCK monolayers. (a) Actin GFP monolayer on a soft (2.3kPa) substrate with differential coating of fibronectin with 100 µm circular islands of 50µg/ml and surroundings with 200µg/ml allows holes formation in regions of low fibronectin coating. Velocity maps show movement of cells towards the high fibronectin coated spaces. (scale: 20µm) (b) Schematic showing the method of measurement of traction forces and inferred stress (c) Average traction force magnitude increases in MDCK monolayer on soft (2.3kPa) and stiff (30kPa) substrates (n = 114 time averaged for n2.3kPa = 11 different circles, 3 independent experiments & n = 183 time averaged for n30kPa = 12 different circles, 2 independent experiments; ****p <0.0001) (d) Representative graph of a confined monolayer shows that traction forces are oppositely directed on soft (2.3kPa) and stiff (30kPa) gels along the diameter of the circle (e) Average isotropic stress for no-hole regions of confined monolayers on 2.3kPa and 30kPa substrates. Green dashed line represents 0 Pa.µm on the y-axis (n = 50 time averaged for n2.3kPa = 10 different circles, 3 independent experiments & n = 52 time averaged for n30kPa = 8 different circles, 2 independent experiments; ****p < 0.0001, ****p < 0.0001) (f) Recoil velocity after laser ablation of the MDCK monolayers grown on substrates of different stiffnesses. Green dashed line indicated the noise in the recoil velocity measurements (~0.6 μm/sec) (n2.3kPa = 5 & n55kPa = 5 different circles; *p < 0.05) (g) Velocity vectors (top) and isotropic stress (bottom) before and during hole formation in a local region obtained from experiments on 2.3kPa PA gels. Color bar represents the respective scale bar for the values of the particular measurement and white arrow indicates the location of hole formation. (h) Change in isotropic stress (blue) in a local region near a hole and the change in area of a hole (red) as a function of time (additional trajectories are provided in Extended Figure 3). (i) Isotropic stress of the entire monolayer (grey) and isotropic stress in a local region (red) containing the hole as a function of time until the hole is formed on soft (2.3kPa substrates). Solid lines indicate mean values and error bars the standard deviation.

Figure 4| Hole formation is triggered by tensile stresses around

  Figure 4| Hole formation is triggered by tensile stresses around topological defects. (a) Orientation field overlaid over phase contrast images obtained from experiments on soft (2.3kPa) substrates representing -1/2 defect (left column) and +1/2 defect (right column). Scale bar: 50µm. (b) Averaged isotropic stress overlaid with location of hole initiation events with respect to the defect for (left) -1/2 defects (n=2810 defects and n=35 holes) (right) +1/2 defects (n=2924 defects and n=20 holes) obtained from experiments. (c) Typical cell orientation and topological defect detected in our vertex model simulation (d) Averaged isotropic stress overlaid with location of hole initiation events with respect to the defect for (left) -1/2 defects (n=8555 defects and n=1302 holes) (right) +1/2 defects (n=8553 defects and n=1371 holes) obtained from vertex model. (e, f) 95 th percentile of isotropic stress around -1/2 defects (left) and +1/2 defects (right) obtained from vertex simulations (e) and experiments (f). Experimental results were obtained on soft (2.3kPa) substrates. White dots represent center of the defect and scale bars represent limits of stress values in Pa.µm

Acknowledgements

This work was supported by the European Research Council (Grant No. Adv-101019835 to BL), LABEX Who Am I? (ANR-11-LABX-0071 to BL and RMM), the Ligue Contre le Cancer (Equipe labellisée 2019 to BL and RMM), and the Agence Nationale de la Recherche ("MechanoAdipo" ANR-17-CE13-0012 to BL, "Myofuse" ANR-19-CE13-0016 to BL, ANR-16-CONV-0001 to JFR, ANR-20-CE30-0023 to JFR) and the Excellence Initiative of Aix-Marseille University -A*MIDEX to JFR. We acknowledge the ImagoSeine core facility of the

Code Availability Statement

Codes used in this manuscript will be available upon request.


The accelerating open science agenda

From open access to open science • publications ○ issues related to APC, diamond model, text and data mining possibilities, sovereignty at large (who has the corpus?)

• data sets ○ more constrained than publications (sensibility, rights, formats, communities)

• software ○ a beast of its own: specific issues related to authorship, versioning, reproducibility, relation to data

Open science policies: institutions, countries, Europe For more information and evaluations, see [START_REF] Bassinet | Large-scale Machine-Learning analysis of scientific PDF for monitoring the production and the openness of research data and software in France[END_REF]

Mining full-texts for software mentions

  ~10% of dataset mentions have PID [4] and datasets are mostly unnamed, e.g.: "The data has been collected by the UN Comtrade organization, and cleaned by CEPII. "

	A long-standing tradition of openness Yearly submission in arXiv (Ginsparg 2021) Open access to publications in informatics Opening routes (French publications of 2021, BSO) A certain wealth of scholarly tools, services and data Other collections of open bibliographic and citation data Linking publications with datasets and software
	Citations to datasets and software are mostly informal mentions in the text body of articles:
	Central role of software • Corresponding assessment practices: Conferences and workshops: • More publications than in journals • Quality peer-review => significant weight in assessment practices • Strong variations across communities ○ https://github.com/societe-informatique-de-france/referentiel-pratiques-publication-2019 • Difficulty in multidisciplinary assessment contexts, hence: ○ low representation in commercial bibliographic databases (Kuserow & Groppe, 2014) ○ lack of authority data for conferences, which is also a difficult topic (naming, recurrence, evolution/fusion, etc.) The essence of open source software • and correlatively of datasets (test suites, parameters, simulations, data models) And of the necessary infrastructures • from forges to Software Heritage • high usage of publication repositories such as arXiv or ad hoc repositories (cryptology) Inria: 89% of openly available full texts (HAL and arXiv) According to FoS classes (OpenAIRE), two main related fields: Computer & Information Sciences: • 98,078 research products Electrical Engineering, Electronic Engineering & Information Engineering: • 2,026,965 research products • 820,858 of them (40.5%) being open Specific OA profile (with Mathematics), cf. French open science monitor (next slide): • favouring green open access (i.e. author's manuscripts deposited in a publication repository) • low level of presence in gold open access journals (i.e. with author-pays fees, aka APC -Article Processing Charges) ○ central contributor to international efforts within EOSC or RDA • Bibliographic databases and digital libraries ○ DBLP (Schloss Dagstuhl -Leibniz Center for Informatics): since 1993, 6M records of bibliographic references o ensures data is related to research work Semantic Scholar: AI-driven search and discovery tools, open resources ○ Minimal contribution from IEEE… Publications can be used as proxies to the dataset and software usage and creation: ○ IFIP DL: hosted on HAL, 3 year embargo, 18 000 documents research from repositories and journals (300 000 research articles in CS) ○ ACM Digital Library: journals and conferences, coupled with the ACM CCS (Computing Classification System) CORE: collection of open access research papers, collecting and indexing publications ○ CEUR (RWTH Aachen University): open-access publication service, 3423 volumes o Necessity to link dataset to real research work to exploit metadata: this requires ○ CiteSeeX (Pennsylvania State University): links indexed documents with other sources (DBLP, ACM DL) o Deposit of datasets in repositories is often not correlated with actual data production sources) o Granularity issues: 1 dataset with 10,000 images gives 10,000 DOI of type "dataset" (organisations, funders, funding streams, projects, communities, and data o Metadata debt: lack of affiliation and domain information OpenAIRE Graph: multiple research products, with links to various entities and creation: • 65,265 of them (66.5%) being open Generic or thematic infrastructures at the service of the open and free dissemination of their research outputs • Software Heritage (foundation in collaboration with Unesco) ○ 2 billion source files ○ experiment to connect publications and software in HAL o software: only 1-8% of mentions as bibliographic references, 0-0.6% of mentions with PID OpenCitations: harvests and publishes comprehensive metadata describing [2,3] academic publications and associated citations • OpenCitations Index (COCI): citation links • OpenCitations Meta: bibliographic metadata o trained on 4,971 manually annotated o Note: Data repositories turned out to be limited currently for following and analyzing data usage documents (37 annotators)
	o possible to rely on document metadata

• e.g. in France: national open science plan (V2, July 2022) Infrastructures for open science • e.g. in France: HAL, Recherche Data Gouv, Software Heritage The specificities of scholarly practices in informatics ○ Attributing and Referencing (Research) Software: Best Practices and Outlook from Inria. Computing in Science and Engineering, Alliez et al. 2019 ○ Évaluation des Logiciels (Assessing software), Canteaut et al. 2021: ○ Note: Zuo et al. showed that GitHub was the most popular repository for hosting COVID-19 datasets Online proceedings -publication platforms Inherent understanding of open science issues • identification, metadata, licencing, citation, versioning, authorship • e.g. Software Heritage: SoftWare Heritage persistent IDentifiers (SWHIDs), CodeMeta, "Citing software with style" initiative • Softcite: software mention detection o funding Sloan & Moore Foundations, and French Open Science Plan o https://github.com/softcite • Automatic characterization of mentions:

used / created / shared

	Mentions to datasets and software
	# documents	share	successful
				download rate
	Full corpus (2012-2021)	1,426,140	100.00 %
	Full text downloaded → open access → closed access	908,567 → 660,501 → 248,066	63.7 % 46.3% 17.4%	63.7 % 85.4% 38.0%
		# full text documents # mentions
	o trained on 3,643 manually annotated processed with Softcite	742,289	3,567,547
	sentences processed with DataStet		621,306	5,607,080

Publications mentioning sharing their software

Gaps and prospects for further developments

Linking these infrastructures with general research assessment practices

• absence of good linking mechanisms between publications, data and software

○ better cross-referencing mechanisms (software and data citations) ○ automatic detection of data sets and software references in publications

• better coverage of conferences ○ lack of reference authority lists ○ contributions to generic databases (e.g. OpenCitation)

• contribution to the changing assessment landscape (CoARA)

○ Defining CS specific profiles and features ○ Should Informatics Europe take the lead on this?

Dealing with a variety of communities in informatics

From highly theoretical topics

• complexity, graph theory etc.

to practical or experimental areas, often in a multidisciplinary context • computational biology, natural language processing or digital humanities

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domain: phys phys

text:
Faouzi Boussaha

Jie Hu

Paul Nicaise

Jean-Marc Martin

Pham Christine Chaumont

Viet Dung

Josiane Firminy

Florent Reix

Piercarlo Bonifacio

Michel Piat

Photon-counting with single stoichiometric TiN layer-based optical MKIDs

come

Photon-counting with Single Stoichiometric TiN Layer-based Optical MKIDs

Faouzi Boussaha, 1 Jie Hu, 1,2 Paul Nicaise, 1 Jean-Marc Martin, 1 Christine Chaumont, 1 Pham Viet Dung, 2 Josiane Firminy, 1 Florent Reix, 1 Piercarlo Bonifacio, 1 Michel Piat, 2 and Hervé Geoffray 3 1) GEPI Observatoire de Paris -Université PSL, CNRS, 75014 Paris, France 2) Université de Paris Cité, CNRS, Astroparticule et Cosmologie, F-75013 Paris, France 3) French Space Agency CNES, 31400 Toulouse, France (*Electronic mail: faouzi.Boussaha@obspm.fr) (Dated: 9 May 2023)

We demonstrate the single photon counting mode at 405 and 850 nm with stoichiometric TiN-based microwave kinetic inductance detectors realized on sapphire substrate and operated at bath temperatures over 300 mK. The detectors use single 15-25 nm-thick TiN layers featuring a critical temperature in the 2-3K range. We found that the energyresolving power R = E/∆E exhibits an optimum with bath temperature, occurring in the 300-450mK range, which can be almost double compared to those obtained at the lowest temperatures. Furthermore, the single photon regime is observed up to 700 mK. In addition to a high-temperature operation, the single stoichiometric layer would allow achieving a better uniformity in the critical temperature and thus kinetic inductance, compared to the often desired ∼ 1K sub-stoichiometric TiN.

Microwave kinetic inductance detector (MKID) 1 is currently one of the most promising superconducting sensor technologies, capable to address many applications in astronomy [2][3][4][5] and physics. It has already been demonstrated in astrophysics, where several instruments covering a large electromagnetic spectrum ranging from millimetersubmillimeter waves 3,[6][7][8] to optical 2,9 and x-rays 10,11 have been implemented. An MKID is a superconducting LC resonator usually consisting of a meander as the inductor and an interdigital capacitor in the optical and near-infrared range. The detection is achieved when the absorbed photons (hv > 2∆, ∆ = 1.76k B T c is the energy gap of the superconductor of the inductor. k B is the Boltzmann constant and T c is the critical temperature) by the inductor L break the Cooper pairs into quasiparticles, modifying its resonance frequency and impedance. The performance of MKIDs depends to a large extent on the quality of the superconducting layer. The implementation and investigation of suitable superconductors to achieve ultimate performance is currently an important research area. In the optical and near-infrared wavelength range, superconducting materials with high normal resistivity and low critical temperature are favored, such as titanium nitride (TiN), platinum silicide (PtSi) 12 , hafnium (Hf) 13 and beta phase tantalum (β -Ta) 14 to achieve better quantum efficiency as well as higher energy-resolving power as the sensitivity inversely proportional to the critical temperature T c (∝ /T c 2 ). TiN obtained by nitriding titanium can exhibit high resistivity (several hundred of µΩ • cm) and large kinetic inductance L k (a few dozen to a few hundred pH/ ) at a T c that can be tuned between ∼ 0.35 and 4.6 K. T c depends on the amount of nitrogen reacting with Ti to form the TiN during the deposition 15 , leading to two TiN states: sub-stoichiometric and stoichiometric. As reported elsewhere [15][16][17] , one can consider that the stoichiometric state corresponds to the plateau above the steep slope that represents the measured critical temperature as a function of nitrogen flow, while the sub-stoichiometric state mainly concerns values directly in the steep slope, defined as the transition phase. The sub-stoichiometric TiN with T c not lower than 0.8-1 K is preferred to improve the sensitivity as well as to meet cryogenics constraints. The first substoichiometric TiN-based MKIDs demonstrating very promising performances, in terms of intrinsic resonance quality Q i and sensitivity 18,19 have been reported. Nevertheless, it has two main drawbacks that can greatly hamper the realization of large arrays. On the one hand, the sub-stoichiometric TiN layer exhibits inhomogeneities in the critical temperature, and consequently, in the kinetic inductance L k , leading ultimately to the occurrence of overlaps and collisions between resonances. Furthermore, unlike the stoichiometric layer, it usually shows larger losses when deposited on a sapphire substrate than on a silicon substrate due to lattice mismatch, while the latter should not be used especially in the 400-800 nm wavelength range to avoid photon absorption in the substrate, which could produce strong substrate event 18 . On the other hand, the transition from stoichiometric to sub-stoichiometric TiN with respect to nitrogen flow is very sharp, which makes the T c not so controllable. In this case, TiN/Ti/TiN 16,20,21 , PtSi 12 , and Hf are investigated to improve the homogeneity to the superconducting films as well as the energy-resolving power. However, PtSi needs a specific fabrication process whereas Hf, which features a very low critical temperature of ∼ 0.4 K, requires the use of a more powerful cryostat to go down to about 15-20 mK 13 . This can present a significant challenge for building large-scale arrays. So far, most reported MKIDs are deposited with T c ∼ 1 K and are operated at a low bath temperature around T ≈ 100 mK and below, as it is well known that at least T < T c /8 is required to achieve the high detection performance. In this case, a dilution cryostat or an adiabatic demagnetization refrigerator (ADR) is needed.

In this letter, we investigate the possibility of using a singlelayer stoichiometric TiN on sapphire to build sensitive MKID arrays operating in a single photon counting regime at temperatures well above 300 mK. Unlike sub-stoichiometric TiN, the T c of stoichiometric layers varies very little with the fabrication parameters which likely contributes to achieving a better uniformity in the kinetic inductance. We designed an array transition and on the plateau 17 . The first array uses a 15 nmthick TiN layer to achieve a high kinetic inductance within the absorber. However, this thin film yields a lower T c of around 2.1 K, instead of 3.8 K for a 60 nm-thick layer. Its resistivity is measured to be ρ TiN I.

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The arrays are fabricated on c-plane sapphire substrates. They are cooled in an ADR and characterized using the standard homodyne detection 22 in the 50-700 mK temperature range. Using the same fabrication process, the typical internal quality factors Q i are 2.5 × 10 4 and 7.7 × 10 4 for TiN 2.1K and TiN 3.8K MKIDs, respectively. The coupling quality factor Q c are around 2.5 × 10 4 and 4 × 10 4 , respectively. Note that MKIDs fabricated from sub-stoichiometric 60 nm-thick TiN (T c ≈ 1-1.4 K) layer deposited on sapphire substrate gave consistently much shallower resonances of maximum 1-2 dB with lower Q i not exceeding 5×10 3 . The power on the CPW readout line is always judiciously applied to avoid resonance bifurcation due to the non-linear regime.

The illumination is performed with two laser sources at 405 and 850 nm wavelength, respectively, through an optical fiber positioned at 35 mm above the array 22,23 . The MKIDs are subjected to a train of short pulses of 50-150 ns width at a frequency of 250 Hz. FIGs. 1-B-C show the typical histograms obtained for 10 4 optical pulse events at 300 mK with the TiN 2.1K MKID resonating at f r = 3.136 GHz. To assess whether these stoichiometric TiN-based MKIDs are able to achieve photon-counting, we employ the widely used Weiner optimal filter 19,20 readout power on the CPW readout line is P r =-98 dBm. Using both lasers, in FIGs. 1-D-E, we show histograms of the TiN 3.8K MKID, with f r =4.078 GHz, measured at 400 mK and P r =-90 dBm. Due to the difference in T c as well as in volume (both MKIDs obviously have the same geometry), the maximum E/∆E is slightly smaller and is measured to be 3.8 at 405 nm (see FIG. 3). At 850 nm, we measured a maximum E/∆E of 2.2 with the MKID of V=2.2 µm 3 . Depending on the optical power absorbed in the MKIDs, we can notice that up to 4 peaks corresponding to n=0, 1, 2, and 3 photons can clearly be resolved. The full-width-at-half-maximum (FWHM) energy resolution ∆E=1.0 eV and 1.2 eV for n=1 and 2 at 850 nm, respectively. Furthermore, similar histograms are obtained up to 500 mK. The energy-resolving power is admittedly still far below the theoretical Fano limit 24 of more than 100 at 400 nm but can be considered in the same order of magnitude as those usually obtained so far with sub-stoichiometric TiN and TiN/Ti/TiN multilayer 16,20 . Note that this is without the use of quantum-limited parametric amplifier 13 , a phonon trapping layer under the meander 24 or a suspended meander on a thin membrane 25 to trap photons within the meander to further enhance the Cooper pairs breaking, leading to the improvement of the energy-resolving power. Thus, the combination of stoichiometric TiN-based MKIDs with one or several of these approaches will lead to a significant performance improvement. This is particularly important for large meanders where E/∆E can decrease dramatically. For example, the use of a parametric amplifier could greatly enhance E/∆E by almost a factor of two. In FIGs. 2-A-B, we present the single pulse response in the time domain as a function of the temperature of the two MKIDs when illuminated at 405 nm. The corresponding max-imum quasi-particle lifetime τ qp are measured to be around 22 and 13µs for, respectively, TiN 2.1K and TiN 3.8K MKIDs at 50 mK which are similar to those reported elsewhere 15,26 , but much smaller than that of substoichiometric TiN layers which are usually in 50-100 µs 18 range when cooled at <T c /10. TiN 2.1K MKID which presents a higher τ qp decreases with temperature more rapidly than TiN 3.8K MKID to reach their minimum around 1µs from 400 and 700 mK, respectively. These short τ qp can still be easily processed with our straight- forward readout electronics. As we noticed that phase response heights ∆ϕ can increase with temperature as shown in Fig. 2-A-B, we plot as insets in the same figure, their maximum as a function of bath temperature.

The maximum ∆ϕ increases as MKIDs operate in the transient mode. In this case, ∆ϕ is related to the photon energy E and the energy gap ∆(T ) as follows: 19 ∆ϕ ∝ ηE ∆(T )

where η is the photon absorption efficiency. Thus, ∆ϕ increases as ∆(T ) decreases when the bath temperature rises.

It should be noted that for an MKID operating in the steady state in the millimeter wave range 27 , ∆φ is different from (1), which is

∆ϕ ∝ τ qp ηE ∆(T ) (2)
Thus, when the bath temperature rises, τ qp decreases resulting in τ qp /∆(T ) < 1 1 and a reduction of phase response. In FIGs. 2-C-D, we also present the energy-resolving power R = E/∆E along with the signal-to-noise ratio (SNR) which can be estimated as

SNR ∝ P max P n d f (3)
where P max is the pulse maximum as shown in Fig. 2-A-B, and P n is the phase noise shown in Fig. 2-E-F. According to all these plots, we can distinguish 3 stages as a function of bath temperature. The first stage which occurs at T ∼ 0.05T c , up to ∼ 150 mK for the TiN 2.1K MKID and ∼ 200 mK for the TiN 3.8K MKID, E/∆E and ∆ϕ are small and roughly constant. In this temperature range, TLS dominates in two ways. First, it introduces a 1/ f 0.5 frequency-dependent noise for T<T c /10 (see Fig. 2-E-F). Second, it affects the frequency shift through the capacitor, generating an undesired phase response in the opposite direction, which reduces the meander-based phase response, as was experimentally observed by J. Hu et al. 22 . The second stage is triggered when the TLS noise level starts to decrease (i.e., 1/ f 0.5 slope is less pronounced) from ∼ 150 mK and ∼ 200 mK, respectively, for the two MKIDs, while phase responses P max , and therefore SNR, increase with bath temperature. Here, E/∆E exhibits optimum in 200-350 mK and 250-500 mK ranges for, respectively, TiN 2.1K and TiN 3.8K with an improvement factor of 1.5-2 to the E/∆E at 50 mK. This is comparable with the improvement in energy-resolving power obtained by using the superconducting parametric amplifier 13 . The third stage occurs, around T c /8, beyond 350 mK for the TiN 2.1K MKID and 500 mK for the TiN 3.8K MKID where E/∆E and ∆ϕ dramatically decrease due to the degradation of the signal-to-noise ratio caused by the significant increase of the thermal noise. In fact, when the temperature further rises, TLS tends to fully saturate which decreases its effects. However, the total noise keeps increasing due to thermal noise which becomes dominant. In this case, the noise spectrum follows the Lorentzian function of white noise. Thus, the energy-resolving power magnitude is mainly limited by TLS noise when it is present and otherwise, by the signal-to-noise ratio. In FIG. 3, we overlay E/∆E measured at 405 nm of four MKIDs, with two different sizes mentioned above per thickness, on the residual frequency shift ∆ f r / f r as a function of bath temperature. The latter is measured in dark and it indicates the equilibrium state of MKIDs for a given temperature before being hit by photons. As reported elsewhere, a smaller meander size 20 combined with a lower T c improves the energy-resolving power. We can also see that the wider temperature range where E/∆E is optimal for the TiN 3.8K MKID is due to its larger T c . For example, with the TiN 3.8K MKID, we can see that E/∆E ≥ 2 up to 550 mK and ∼ 1.5-1 up to 700 mK are achieved.

In conclusion, we have demonstrated the capability of stoichiometric TiN-based MKIDs to operate in single photon mode. The obtained energy-resolving power E/∆E is comparable to that of sub-stoichiometric TiN 18 or multilayer TiN/Ti/TiN. Furthermore, it can operate at a higher bath temperature well beyond 300 mK, up to 700 mK. On the other hand, we found out that E/∆E shows an optimum with the bath temperature, between two noise minima, that of evanescent TLS and that of increasing thermal noise. In addition to a better understanding of the energy-resolving power-noise relationship, these results allow to consider the use of sensitive stoichiometric TiN-based MKIDs featuring T c > 2K that will pave the way for the use of modest cryogenic systems such as sorption coolers to operate above 300 mK.

  2.1K ≈ 140 µΩ • cm (RRR ≈ 1.1), yielding a L k = ρ h/∆πt ≈ 61 pH/ , where h is the reduced Planck constant and t is the thickness. We carried out this study with two different meander volumes of V=1.3 and 2.8 µm 3 with areas of A=13.8×6.4 µm 2 and 13.8×13.5 µm 2 , respectively. These meanders consist of strips of 2.7 µm-width and total length of 20 and 72 µm, respectively. This leads to resonance frequencies ranging from 1.96 to 3.2 GHz. The second array is made of a thicker TiN layer of 25 nm. In addition to the thickness, we also increased its T c by changing only the nitrogen flow (an increase of ∼ 20%) during the TiN deposition. This allows to achieve T c ≈ 3.8 K (∼ 4.5 K for 60 nm-thick). Its measured ρ TiN 3.8K ≈ 210 µΩ • cm gives L k ≈ 30 pH/ . The meander volumes become 2.16 and 4.6 µm 3 , respectively, and resonance frequencies range from 2.9 to 4 GHz. For both arrays, the low Nb kinetic inductance of ∼ 0.2 pH/ will greatly contribute to preventing the undesired responses from capacitors and the coupling arms. For clarity, the main parameters are summarized in Table

FIG. 3 .

 3 FIG. 3. Evolution of the energy-resolving power R = E/∆E along with residual frequency shift ∆ f r / f r as a function of bath temperature of four MKIDs illumintated at 405 nm. The corresponding frequencies are 2.625, 3.137 GHz made of TiN 2.1K 15 nm-thick and 3.128, 4.078 GHz made of TiN 3.8K 25 nm-thick.

  MKIDs where each MKID is made up of a TiN meander in parallel with an Nb interdigitated capacitor (IDC) as illustrated in FIG.1-(A). The two elements are connected with arms on either side of the meander that are shaped to improve the RF coupling between the resonator and the 50 Ω CPW line made from a 100 nm-thick Nb layer that reads out the resonators. The capacitors of 288 × 282 µm are also made of a 100 nm-thick Nb layer with a finger gap and width of 1.2 µm.

Optical picture showing one of the fabricated 6x6 MKIDs, which uses a meander of 2.8 µm 3 . The volume of the meander is small to improve the sensitivity. (B-C) Histograms of phase response heights measured at 300 mK when the TiN 2.1K MKID with 15 nm-thick, V=1.3 µm 3 and f r =3.137 GHz is illuminated with short pulses at 405 and 850 nm wavelength, respectively. (D-E) Histograms of phase response heights of the TiN 3.8K MKID with 25 nm-thick, V=2.8 µm 3 and f r =4.078 GHz at 400 mK when it also illuminated at 405 and 850 nm, respectively. At these temperatures, their quality factors Q i are ∼ 2.5×10 4 and 7.7×10

4

, respectively. The dashed red-gray lines indicate the Gaussian fits. of 6 × 6 Multiplexing is achieved by removing different portions of the IDC fingers so that each resonator has its unique resonance frequency. We fabricated two arrays that differ only by their thickness and critical temperature of the TiN inductive meander. The chosen T c are, respectively, right after the steep

  Evolution of different measured parameters as a function of the temperature of a pixel of TiN 2.1K -15nm with V=2.8 µm 3 (A, C, E) and a pixel of TiN 3.8K -25nm with V=4.6 µm 3 (B, D, F): (A-B) Single photon pulses versus temperature. The insets show the pulse maximum versus temperature. (C): Energy-resolving power and signal-to-noise SNR versus temperature at 405 nm and 850 nm, respectively with TiN 2.1K -15nm. (D): Energy resolving power and signal-to-noise SNR versus temperature at 405 nm TiN 3.8K -25nm (measurements at 850 nm could not be carried out with the same MKID). (E-F): The phase noise versus temperature.

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to generate photon-counting statistics. The maximum energy-resolving power R = E/∆E are 5 and 2.4 at 405 and 850 nm, respectively (see FIGs. 2-3).

ACKNOWLEDGMENTS

This work is supported by the European Research Council (ERC) and the french space agency (CNES).