Céline Laurencé

Michael Rivard

Thierry Martens
email: martens@icmpe.cnrs.fr

Christophe Morin

Didier Buisson

Sophie Bourcier

Michel Sablier

Mehmet A Oturan
email: mehmet.oturan@univ-paris-est.fr

Anticipating the fate and impact of organic environmental contaminants: A new approach applied to the pharmaceutical furosemide

Keywords: Furosemide, Electro-Fenton, Bioconversion, Toxicity

INTRODUCTION

Emerging pollutants are a class of compounds that result from human activity, are detected in the environment and whose impact on ecosystems remains unknown [START_REF] Levi | Challenges in the assessment and management of health risks associated with emerging water micropollutants[END_REF].

These pollutants consist of compounds that were originally developed as pharmaceuticals and personal care products (PPCPs) [START_REF] Daughton | Environmental stewardship and drugs as pollutants[END_REF][START_REF] Khetan | Human Pharmaceuticals in the Aquatic Environment: A Challenge to Green Chemistry[END_REF]Fatta-Kassinos et al., 2011a) bactericides or industrial additives [START_REF] Jurado | Emerging organic contaminants in groundwater in Spain: A review of sources, recent occurrence and fate in a European context[END_REF][START_REF] Rodil | Emerging pollutants in sewage, surface and drinking water in Galicia (NW Spain)[END_REF].

Regarding the pharmaceuticals, the progress made in trace analysis during the last decade [START_REF] Buchberger | Current approaches to trace analysis of pharmaceuticals and personal care products in the environment[END_REF] has allowed their detection and quantification in the effluents of sewage treatment plants (STPs), rivers, ground water, and sometimes even in drinking water [START_REF] Heberer | Occurrence, fate, and removal of pharmaceutical residues in the aquatic environment: a review of recent research data[END_REF][START_REF] Kuemmerer | Pharmaceuticals in the Environment: Sources, Fate, Effects and Risks[END_REF]. The presence of pharmaceuticals in the environment mainly occurs after their excretion and results from an incomplete degradation in STPs [START_REF] Deblonde | Emerging pollutants in wastewater: A review of the literature[END_REF]. Despite their low environmental concentrations, which are generally measured to be less than human therapeutic levels (from ng/L to g/L), the continuous release of these substances makes them pseudo-persistent, thus presenting a risk of ecotoxicity or chronic toxicity for non-target organisms [START_REF] Daughton | Pharmaceuticals and personal care products in the environment: agents of subtle change?[END_REF].

Assessing the impact of the presence of drugs (i.e., molecules designed to exert a biological effect) on aquatic biota requires an ecotoxicity or a chronic toxicity to be univocally associated with a perfectly identified chemical structure (or alternatively, to a welldefined mixture of molecules). For this purpose, biological models have been applied to molecules whose presence in the environment had been previously highlighted [START_REF] Henschel | Environmental hazard assessment of pharmaceuticals[END_REF]. However, the primary limitation of such studies is that they do not consider the environmental transformation products (ETPs), i.e., compounds stemming from the parent molecule and likely to appear in the environment consecutively to (a)biotic transformations.

Attempts to identify ETPs from the analysis of complex environmental matrices, such as soils, sediments or biota, have been demonstrated to be prone to failure [START_REF] Ibanez | Use of Quadrupole Time-of-Flight Mass Spectrometry in Environmental Analysis: Elucidation of Transformation Products of Triazine Herbicides in Water after UV Exposure[END_REF][START_REF] Richardson | Environmental Mass Spectrometry: Emerging Contaminants and Current Issues[END_REF]. Under these conditions, chromatographic techniques coupled with mass spectrometry in "full-scan" detection mode cannot achieve the detection limits required to characterize these unknown compounds that are present at trace levels. In the absence of characterization, the majority of ETPs remain ill-defined, thus ruling out the evaluation of their (eco)toxicities, which is problematic because the toxicity of ETPs may exceed that of their parent molecules [START_REF] Dirany | Study of the toxicity of sulfamethoxazole and its degradation products in water by a bioluminescence method during application of the electro-Fenton treatment[END_REF]Fatta-Kassinos et al., 2011b;[START_REF] Dirany | Electrochemical wastewater treatment for antibiotic sulfachloropyridazine degradation: kinetics, reaction pathways, and toxicity evolution[END_REF].

In the case of pharmaceuticals, metabolites are a class of ETPs whose structure can be rather well-anticipated [START_REF] Lohmann | Biomimetic modeling of oxidative drug metabolism[END_REF], and these have already been the subject of environmental detections [START_REF] Celiz | Pharmaceutical metabolites in the environment: analytical challenges and ecological risks[END_REF][START_REF] Mompelat | Occurrence and fate of pharmaceutical products and by-products, from resource to drinking water[END_REF]. Further investigations focusing on the identification of new ETPs and the assessment of their impact has led to the development of various strategies [START_REF] Escher | Recent Advances in Environmental Risk Assessment of Transformation Products[END_REF]. With this aim, experiments that mimic the transformations that drugs undergo in the environment have been developed [START_REF] Ziylan | The occurrence and fate of anti-inflammatory and analgesic pharmaceuticals in sewage and fresh water: Treatability by conventional and nonconventional processes[END_REF]. These experiments, which were applied to the parent molecule, resulted in transformation products (TPs) assumed to be real (at least similar to) ETPs. Such experiments typically proceed by electro-Fenton [START_REF] Oturan | Hydroxylation of aromatic drugs by the electro-Fenton method. Formation and identification of the metabolites of Riluzole[END_REF], photodegradation [START_REF] Packer | Photochemical fate of pharmaceuticals in the environment: naproxen, diclofenac, clofibric acid, and ibuprofen[END_REF] or biodegradation [START_REF] Ericson | Evaluation of the OECD 314B Activated Sludge Die-Away Test for Assessing the Biodegradation of Pharmaceuticals[END_REF][START_REF] Gauthier | Biodegradation of pharmaceuticals by Rhodococcus rhodochrous and Aspergillus niger by co-metabolism[END_REF]. Because of such protocols and thanks to the means now available, elucidating the structure of these compounds is possible. Nevertheless, authentication of the structure requires the use of standards either commercially available or synthesized [START_REF] Schulze | Identification of a phytotoxic photo-transformation product of diclofenac using effect-directed analysis[END_REF][START_REF] Svanfelt | Synthesis of substituted diphenylamines and carbazoles: phototransformation products of diclofenac[END_REF]. In the absence of isolated compounds, (eco)toxicological studies are conducted on mixtures of TPs resulting from the degradation of the parent molecule.

Interestingly, examples of (eco)toxicological evaluations of fully characterized TPs obtained and isolated after the degradation of the parent pharmaceutical remain scarce [START_REF] Isidori | A multispecies study to assess the toxic and genotoxic effect of pharmaceuticals: Furosemide and its photoproduct[END_REF].

In this report, we describe a new approach that enables both the identification of plausible ETPs and their toxicological evaluation. Because the main difficulty in studying the fate of drugs in the environment lies in the lack of authentic samples of their ETPs, we propose a new stepped approach. The first step consists of isolating the TPs of a drug that is known to be an environmental contaminant. This step, which requires the tools of organic synthesis, focuses on providing fully characterized standards and the development of methods for their detection in complex mixtures. In the second step, the contaminant is submitted to environmentally relevant transformations, i.e., processes that are representative of the transformations undergone by contaminants in the environment. By highlighting previously characterized TPs, this step allows the identification of plausible ETPs. In the third step, a toxicological study is initiated on each of the TPs that are identified as plausible ETPs.

In a recent work, we described the behavior of furosemide 1 (fig 1) under oxidative conditions [START_REF] Laurence | Preparative access to transformation products (TPs) of furosemide: a versatile application of anodic oxidation[END_REF]. This drug, which has been widely used as a diuretic since the nineteen sixties, is one of the forty compounds having the highest risk with a predicted environmental concentration greater than 100 ng/L [START_REF] Besse | Human pharmaceuticals in surface waters[END_REF]. Poorly metabolized and eliminated either unmodified or as glucuronide conjugate [START_REF] Lee | Pharmacokinetics and pharmacodynamics of furosemide after direct administration into the stomach or duodenum[END_REF] (which releases furosemide after hydrolysis), this drug has been detected at concentration ranges between 61 and 200 ng/L in European rivers [START_REF] Castiglioni | Removal of Pharmaceuticals in Sewage Treatment Plants in Italy[END_REF][START_REF] Khalaf | In vitro analysis of inflammatory responses following environmental exposure to pharmaceuticals and inland waters[END_REF]. During the course of our preliminary study, three TPs of furosemide were identified: furfural 2, saluamine 3 (both resulting from the oxidation of the amino group) and a zwitterionic pyridinium 4 (resulting from the oxidation of the furan ring). These three TPs, which are obtained by anodic oxidation, have been isolated and fully characterized, thereby allowing their detection by LC-MS in complex mixtures.

This study seeks to confirm whether these three TPs, which were obtained without any environmental relevance, are indeed plausible ETPs. For this purpose, environmentally relevant degradation pathways were applied to furosemide. To be representative of the transformations likely to occur in the environment, two protocols were selected. The first protocol is abiotic and uses an electrochemical advanced oxidation process (electro-Fenton); the second protocol is biotic and requires microorganisms. The TPs identified as plausible ETPs were then subjected to a preliminary toxicological study.

MATERIAL AND METHODS

Chemicals

Furosemide (> 99.0%) was purchased from TCI Europe N.V. Furfural (99%) and 1-Methyl-4-phenylpyridinium iodide (MPP + ) were purchased from Sigma-Aldrich. Saluamine 3 and pyridinium 4 employed for the toxicological evaluation and required as standards for the analysis of complex mixtures were prepared as previously described [START_REF] Laurence | Preparative access to transformation products (TPs) of furosemide: a versatile application of anodic oxidation[END_REF]).

Analytical procedure

ESI-MS/MS analyses were conducted using a Q-TOF Premier instrument equipped with a Z-spray electrospray source (Waters, Saint Quentin-en-Yvelines, France) operating in the positive mode. For fragmentation studies, solutions were introduced into the electrospray ionization source using a syringe pump at an infusion rate of 10 µL/min. To investigate the degradation products, the sample was analyzed using a 2690 liquid chromatography module from Waters (Saint Quentin-en-Yvelines, France) coupled with a Q-TOF premier mass spectrometer. The employed analytical column was a C18 Atlantis T3, 3 µm, 150 x 2.1 mm (Waters, Saint Quentin-en-Yvelines, France). The HPLC solvents were acetonitrile with 0.1% formic acid (A) and water with 0.1% formic acid (B). The following linear gradient elution program was applied: 20% of A for 11 min and 100% of A from 11.1 to 20 min. Then, before the next injection cycle, the column was reconditioned with 20% of A for 10 min. The effluent was introduced at a rate of 0.2 mL/min into the Z-spray interface for ionization. The following three acquisition modes were used to characterize each compound: (i) full-scan mode in V-mode; (ii) MS/MS of the precursor ion and (iii) full-scan in the W-mode for high resolution. The full-scan acquisition mode allowed the cone voltage to be optimized to obtain maximum intensity for the precursor ion. Several MS/MS spectra were recorded to obtain the decomposition pathways.

The cone voltage was adjusted for each ion of interest that was generated in the ion source. The ion source parameters were adjusted as follows to optimize the ion signals: the cone voltage was set between 20 to 80 V while the capillary voltage was set to 2.6 kV.

Typical values for the other source parameters were as follows: extraction cone, 1.7 V; ion guide, 3 V; source and desolvation temperatures were set at 80 °C and 250 °C for direct infusion and at 100 °C and 450 °C for LC-MS studies, respectively. Nitrogen was used as both nebulizing and desolvation gases. The gas flow rates were dependent on the introduction mode; they ranged from 20 L/h to 300 L/h for direct introduction and from 70 L/h to 700 L/h for LC-MS. For LC-MS analysis, the injected volume of sample was 10 µL. Argon was used as a collision gas at a flow of 0.28 mL/min, which corresponds to a pressure of ca. 4 10 -3 mbar. To record the MS/MS spectra of ions, the collision energies were varied between 2 to 30 eV for each compound to obtain their primary characteristic ions. For accurate mass measurements, the analyses were performed using W-mode and an independent reference spray via the LockSpray interface. Sulfadimethoxine was used as the lock mass compound at a flow rate of 10 µL/min. The LockSpray frequency was set at 10 s, and data for the reference compounds were averaged over 10 spectra/min. Accurate masses and elemental compositions for all ions were obtained using the MassLynx instrument software. The ion used for the mass correction was m/z 311.0814.

Electro-Fenton

Experiments were conducted in an open, cylindrical and undivided glass cell of 6 cm diameter and 250 mL capacity. This cell was equipped with a cylindrical Pt mesh (4.5 cm height, 3.1 cm internal diameter) as an anode and a large surface area (14 cm x 4. was performed in isocratic mode with a flow rate of 0.7 mL/min. Identification of aromatic intermediates was conducted by comparing the retention times with that of authentic compounds and UV-Vis spectra. The analyses were monitored using the EZChrom Elite software package.

Bioconversion

Microorganisms, including 48 fungi and 24 bacteria, were cultured in liquid medium (25 mL) containing in g/L: yeast extract (Conda) 4, malt extract (Conda) 10, soybean peptones (OrganoTechnie) 5, and glucose 16. The process of selection involved six series of 12 microorganisms. After sufficient growth, the biomasses were harvested by centrifugation (bacteria) or filtration (fungi) and suspended in 12 mL of a 0.1 M sodium citrate buffer (pH 5). For each series, four incubation mixtures (3 fungi and 2 bacteria) were created using the suspensions (4 mL, total volume of 20 mL). Furosemide was dissolved in N,Ndimethylformamide (0.066 mg/L) and 60 µL of this solution were added to the incubation mixtures (final concentration 0.2 mg/ml). Biotransformations were performed under an air atmosphere at 27 °C and 200 rpm and monitored by HPLC and MS. Aliquots (culture medium and biomass; 0.8 mL) were withdrawn every two days, diluted with methanol (200 µL), mixed vigorously and centrifuged at 13,000 g for 5 minutes. The resulting supernatants were micro-filtered (0.45 µm) before analysis.

Toxicity measurements

MTT Cell Viability Assay

Human neuroblastoma cells (SH-SY5Y) (ATCC CRL2266) were cultured in DMEM (in 75 mL flasks), containing 4.5 g/L glucose and supplemented with 1% penicillin/streptomycin and 10% FBS, at 37 °C in a humidified atmosphere of 5% CO2. Cell viability was measured using the MTT assay in 96-well plates at a cell density of 8  10 4 cells/mL. The cells were exposed to different compounds (1, furfural 2, 3 and 4) at various concentrations (from 1 to 1000 µM).

All drugs were dissolved in a mixture of DMSO and DMEM (1/1 (v/v)) in order to obtain a 10 -1 M stock solution. All controls were carried out using the same solvent mixture. The final solution contained no more than 0.5% (v/v) of DMSO. After 24, 48, 72 and 96 h of exposure, MTT (20 µL, 5 mg/mL) was added to each well. Following incubation at 37 °C for 2 h, the supernatants were carefully removed and 200 µL of DMSO were added. The optical density (OD) value of each well was read using a micro-plate reader (TECAN Infinite M1000) at a wavelength of 570 nm. Cell viability was calculated as a percentage of the control.

Caspase-3 activity

The caspase-3 activity was assessed by following the cleavage of the fluorogenic substrates Ac-DEVD-AFC. Treated cells were harvested through trypsinization using 0.05% trypsin and 0.53 mM EDTA at 37 °C in a humidified atmosphere containing 5% CO2 for 4 min. Trypsinization was terminated by addition of 20% FBS, the cell suspension was then centrifuged, the supernatant discarded and the pellet washed with 1 × PBS (CaCl2-, MgCl2free) (Gibco, Invitrogen). The cell pellet was gently suspended in a buffer containing 30 mM HEPES, 0.3 mM EDTA, 100 mM NaCl, 0.15% Triton X-100 and 10 mM DTT and then centrifuged. The supernatant was used for the assay. The fluorogenic substrates were added at a final concentration of 100 mM. The samples were gently mixed and incubated in the dark at 37 °C for 1 h. Fluorescence intensity of the samples was measured at an excitation λ of 400 nm and emission λ of 505 nm using a fluorescence micro-plate reader (TECAN infinite™ M1000).

Data analysis

Data were expressed as the mean ± SEM. Comparisons among groups were conducted using an ANOVA (F-test) test. Bonferroni adjusted t-tests were used for multiple group comparisons and the unpaired t-test was used for single comparisons. A two-tailed p<0.05

was selected to indicate a statistically significant difference. All values were calculated using the GraphPad Prism 5.00 software package (GraphPad Software, San Diego, CA).

RESULTS

Electro-Fenton

In the first step, furosemide was subjected to electro-Fenton oxidative degradation. This method is based on the in situ and catalytic production of hydroxyl radicals (  OH) from the electrochemically assisted Fenton's reaction (Eqs. 1-3).

O2 + 2H + + 2e - H2O2 (1)

Fe 3+ + e - Fe 2+ (2) H2O2 + Fe 2+  Fe 3+ + OH -+ • OH (3)
The hydroxyl radical is a very powerful oxidizing agent, and it non-selectively reacts with organic molecules to lead to the formation of abiotic TPs. In the treatment strategy, the operating parameters were established to further oxidize these intermediates until they reached their ultimate degree of oxidation, i.e., their mineralization (Eq. 4) [START_REF] Oturan | Complete Destruction of p-Nitrophenol in Aqueous Medium by Electro-Fenton Method[END_REF][START_REF] Sires | Catalytic behavior of the Fe3+/Fe2+ system in the electro-Fenton degradation of the antimicrobial chlorophene[END_REF][START_REF] Brillas | Electro-Fenton Process and Related Electrochemical Technologies Based on Fenton's Reaction Chemistry[END_REF]. This process has been successfully applied for the treatment of toxic/persistent pollutants and particularly for the removal of pharmaceutical pollutants from water [START_REF] Ozcan | Degradation of picloram by the electro-Fenton process[END_REF][START_REF] Dirany | Electrochemical abatement of the antibiotic sulfamethoxazole from water[END_REF][START_REF] Sires | Electrochemical degradation of β-blockers. Studies on single and multicomponent synthetic aqueous solutions[END_REF]).

• OH + organic pollutants  abiotic TPs  mineralization (4)

As stated above, the electro-Fenton process was used to mimic the abiotic transformations that are likely to occur in the environment. Contrary to the abatement (treatment) strategy, the inhibition of oxidation kinetics was consequently required to allow the accumulation of the abiotic TPs. Briefly, this inhibition consisted of applying a small current (for weakening the rate of Eqs. 1 and 2) and using a high concentration of catalyst (Fe 2+ ) to favor the waste reaction between  OH and Fe 2+ (Eq. 5), thus lowering the concentration of • OH and consequently the oxidation rate of the intermediates.

 OH + Fe 2+  Fe 3+ + OH - (5)
This strategy was applied under a pH of 3 at room temperature to a 230 mL aqueous solution of furosemide (0.

Bioconversion

Furosemide is poorly metabolized by humans; however, this does not exclude the possibility for non-target organisms disposing of the appropriate enzymatic equipment to achieve the transformation of furosemide. One way to fairly accurately identify the spectrum of biotic TPs arising from furosemide consists of subjecting this molecule to microbial transformation (bioconversion). The enzymes that are present in many microbial strains allow access to a wide range of biotransformation products. This approach, which is especially used in the preparation of mammalian metabolites [START_REF] Smith | Microbial models of mammalian metabolism[END_REF][START_REF] Azerad | Microbial models for drug metabolism[END_REF][START_REF] Vidyavathi | Cunninghamella -A microbial model for drug metabolism studies -A review[END_REF], requires the selection of active microorganisms [START_REF] Li | Metabolite generation via microbial biotransformations with actinomycetes: rapid screening for active strains and biosynthesis of important human metabolites of two development-stage compounds, 5-[(5S,9R)-9-(4-cyanophenyl)-3-(3,5-dichlorophenyl)-1-methyl-2,4-dioxo-1,3,7triazaspiro[.4]non7-yl-methyl]-3-thiophenecarboxylic acid (BMS-587101) and dasatinib[END_REF][START_REF] Marvalin | Microbial production of phase I and phase II metabolites of propranolol[END_REF].

To obtain the biotic TPs of furosemide, a selection of microorganisms among seventytwo strains of bacteria and fungi was performed. To complete the screening using minimal time and assays, we used our recently developed combinatorial approach [START_REF] Fromentin | Yeast-Mediated Xanthone Synthesis through Oxidative Intramolecular Cyclization[END_REF][START_REF] Joyeau | Combinatorial approach to the selection of active microorganisms in biotransformation: Application to sinomenine[END_REF]. This method is based on incubating the substrate in the presence of five microorganisms (three fungi and two bacteria) in a reaction buffer. After furosemide was added to the suspensions, the biotransformations were monitored by HPLC and MS analyses of aliquots collected every forty-eight hours.

This strategy allowed identification of four microorganisms capable of performing the biotransformation of furosemide as well as the observation of two major biotic TPs.

Saluamine 3, which resulted from the hydrolysis of the intermediate hemiaminal, was obtained with a yield of 25-30% after four days of incubation with the bacteria Agrobacterium tumefaciens (Rhizobium radiobacter CIP 67.1) and Arthrobacter ureafaciens CIP 67.3. This result confirmed the ability of the microorganisms to oxidize the -position of the amino group of furosemide, as previously observed [START_REF] Hezari | Microbial models of mammalian metabolism. N-dealkylation of furosemide to yield the mammalian metabolite CSA using Cunninghamella elegans[END_REF]. The absence of furfural may be explained by the ability of some microorganisms to degrade it [START_REF] Koopman | Identification and characterization of the furfural and 5-(hydroxymethyl)furfural degradation pathways of Cupriavidus basilensis HMF14[END_REF]. The fungus Aspergillus candidus ATCC 20023 produced pyridinium 4 in moderate yields after eight days (see SI §4). The fungus Cunninghamella echinulata var. elegans ATCC 9245 fully metabolized furosemide in six days to yield a mixture of both compounds. In some cases, the compound associated with structure A was also observed as a minor biotic TP (see SI §5). Further investigations are in progress to optimize the production of this TP to isolate it and confirm its structure.

Toxicological evaluation of plausible ETPs

The evaluation on the neuroblastoma cell line (SH-SY5Y) of furosemide 1 revealed no activity on the tested cells after 24 and 48 h. However, after 72 h, a slight inhibition of approximately 20% of the cell survival was observed, which is possibly explained by the action of furosemide on the regulation of the cytosolic pH [START_REF] Rakonczay | CFTR gene transfer to human cystic fibrosis pancreatic duct cells using a Sendai virus vector[END_REF]). An extended exposure to furosemide for 96 h did not reveal any increased inhibition.

In the same test, the saluamine 3 was completely inactive up to 96 h at all investigated concentrations. Exhibiting no toxicity after 24 and 48 h, furfural 2 reduced the cell viability by approximately 20% after 72 h at the maximal concentration of 1000 µM. As observed with the furosemide, no increased inhibition was observed after extended exposure. Interestingly, pyridinium 4 also began to have an effect after 72 h of exposure. The effect of pyridinium 4 on the cell mortality was increased from 40 to 60% between 72 to 96 h. The concentration at which 50% of the cells survived (EC50) was estimated to be 973 µM (Fig 3a).

The induction of cell death by exposure to pyridinium 4 was confirmed after evaluation of the caspase-3 activity (Fig 3b). The increased activity (175%) observed after 96 h for the treated cells compared to the untreated ones (100%) allowed the previously observed cell death to be clearly associated with an apoptotic process.

DISCUSSION

Anticipating the fate of emerging pollutants often leads to complexity in investigating and researching ETPs, which for the most part remain unknown. By detecting after degradation of a contaminant by environmentally relevant transformations, compounds that have been fully characterized in a previous step, our approach has demonstrated its ability to univocally identify plausible ETPs of organic pollutants. Interestingly, the alternative approach consisting of applying environmentally relevant transformations to a contaminant to obtain, isolate and characterize its plausible ETPs in a single step might seem intuitively quicker and more appropriate. However, the results of this study suggest that such an approach would have failed to detect plausible ETPs exhibiting significantly different physicochemical properties from those of the parent compound, as in the case of pyridinium 4. Note that saluamine 3 has already been identified as a biotic TP of furosemide after bioconversion with the fungus Cunninghamella elegans [START_REF] Hezari | Microbial models of mammalian metabolism. N-dealkylation of furosemide to yield the mammalian metabolite CSA using Cunninghamella elegans[END_REF]. The fact that pyridinium 4 was not observed in the same study may be explained by the unavailability of any standard. However, Burka observed the formation of pyridinium 4 after the liver microsomal incubation of furosemide [START_REF] Chen | Chemical and Enzymatic Oxidation of Furosemide: Formation of Pyridinium Salts[END_REF], but this was consecutive to its NMR characterization after the oxidation of furosemide with dimethyldioxirane in deuterated acetone.

Because electro-Fenton degradation and bioconversion are representative of the transformations that are likely to occur in the environment, the saluamine 3 and the pyridinium 4, which both result from these two degradation processes, can consequently be considered as plausible ETPs of furosemide. With the help of available standards, the implementation of protocols for the environmental detection of these ETPs can now be initiated. A possible co-product of the oxidation leading to aniline 3, furfural, is therefore also a plausible ETPs of furosemide. The abundant use of this compound stemming from the biomass for industrial purposes makes it nevertheless an unreliable marker of the presence of furosemide in the environment. Additionally, while the growth-inhibiting properties of furfural have already been mentioned [START_REF] Almeida | Metabolic effects of furaldehydes and impacts on biotechnological processes[END_REF], it appears that the concentrations at which furosemide is detected in the environment make the risk associated to its ability to release furfural not significant [START_REF] Wierckx | Microbial degradation of furanic compounds: biochemistry, genetics, and impact[END_REF].

By the observation of compounds A, B or C and the collection of their analytical data, our approach has also allowed the identification of new plausible ETPs, including their detection in any complex mixtures. To complete their characterization, their structure proposed on the basis of CID mass spectrum analysis and structural similarities with provided standards, has to be confirmed or refined (which is why they were indicated with letters, not numbers). This confirmation process requires either their isolation from degradation mixtures or their total synthesis (this work is currently underway). Note that each new compound that is isolated and authenticated as a plausible ETP can be subsequently subjected to environmentally relevant transformations and toxicologically evaluated. Such iterative protocol appears appropriate to anticipate further oxidation states of contaminants and thus, to accurately predict their environmental fate, either on a structural or on a toxicological basis.

Interestingly, anticipating such further oxidation states of contaminants allowed the environmental fate of furosemide to be reconsidered. Indeed, whereas the anodic oxidation, which is focused on the earlier oxidation states, suggested the aforementioned two distinct degradation pathways (proceeding by oxidation of either the amino group or the furan ring), this study revealed the ability of 4 to release saluamine 3 after over-oxidation of its pyridinium ring. Such information may be of interest for investigating the kinetics of the appearance and disappearance of toxicity associated with the transformation of contaminants in the environment.

The final part of this study was devoted to the toxicological evaluation of plausible ETPs, and it highlighted the difference in terms of toxicity between the parent molecules and their

ETPs. The present study has consequently confirmed the necessity of developing new methodologies to anticipate the fate of environmental contaminants. Furthermore, this study has also demonstrated the need to provide suitable samples for biological tests, i.e., obtained on preparative scale and isolated. This latter point was demonstrated to be crucial for univocally associating toxicity to a chemical structure. In the case of furosemide, note that if a genotoxicity has already been mentioned, no molecule has yet been clearly associated with it [START_REF] Rocco | Evaluation of zebrafish DNA integrity after exposure to pharmacological agents present in aquatic environments[END_REF][START_REF] Mondal | Furosemide-induced genotoxicity and cytotoxicity in the hepatocytes, but weak genotoxicity in the bone marrow cells of mice[END_REF]. To date, only a reactive intermediate in the form of an epoxide has been proposed to explain the cytotoxicity of this drug [START_REF] Williams | The metabolism and toxicity of furosemide in the Wistar rat and CD-1 mouse: a chemical and biochemical definition of the toxicophore[END_REF].

The present work may consequently help the mechanism for the toxicity of furosemide to be reconsidered in relation with one of its TPs.

In summary, our multidisciplinary approach for anticipating the fate of environmental contaminants allowed identifying three plausible ETPs of furosemide, and five other ETPs were observed whose chemical structures must be confirmed or refined. The availability of standards now enables the environmental investigations and, possibly, the quantification of aniline 3 and pyridinium 4, which appear as relevant markers of the presence of furosemide in the environment.

The biological results also revealed a significant toxicity for the pyridinium 4. Following this work, assessment of the risk associated with the presence of furosemide in the environment requires a particular focus on this plausible ETP, whose mechanism of action remains unknown and whose ecotoxicity remains to be investigated.

ACKNOWLEDGMENTS

Figure captions

Figure 1: Furosemide 1 and its TPs

1 Cl H 2 NO 2 S CO 2 N OH 1 Cl H 2 NO 2 S CO 2 H H N O Cl H 2 NO 2 S CO 2 H NH 2 O OHC 2 3 4 Figure 2 Cl H 2 NO 2 S O N Cl H 2 NO 2 S CO 2 N OH A Cl H 2 NO 2 S O N OH B1 Cl HO CO 2 N OH C1 B2,3 a) b) OH Cl HO CO 2 N R 2 C2,3 HO OH R 1 HO OH R 2 R 1 2: R 1 = H, R 2 = OH; 3: R 1 = OH, R 2 = H.

  Dulbecco's Minimal Essential Medium (DMEM), fetal bovine serum (FBS), penicillin/streptomycin, trypsin-EDTA solution and phosphate buffered saline (PBS) were purchased from GibcoBRL (France). The caspase fluorogenic substrate N-Acetyl-Asp-Glu-Val-Asp-7-amido-4-trifluoromethylcoumarin (Ac-DEVD-AFC) was purchased

  5 cm each side, 0.5 cm width) carbon felt (Carbon-Lorraine, France) as a cathode. The anode was centered in the electrochemical cell and surrounded by the cathode, which covered the inner wall of the cell. Electrolyses were conducted under a constant current applied by a Hameg HM8040 triple power supply. H2O2 was produced from the reduction of O2 dissolved in the solution. Continuous saturation of this gas at atmospheric pressure was ensured by bubbling compressed air. 220 mL of solutions containing 0.1 to 0.3 mM furosemide and 0.5 mM Fe 2+ at the optimum pH value of 3.0 were electrolyzed at a constant current (30-350 mA) at room temperature. The decrease in the concentration of furosemide and the formation of oxidation intermediates were followed using a Merck Lachrom HPLC chromatograph composed of a degasser (L-7614), an injection pump (L-7100 equipped with a 20 L injection loop), a reverse phase RP-18 Purospher column (5 μm, 4.6 mm× 25 cm) placed into a L-7350 oven set to 40 °C and a photodiode array detector (L-4755) selected at 280 nm. The mobile phase consisted of a mixture of methanol and phosphoric acid (55/45 (v/v)) at a pH of 3. Elution

  3 mM) using a current of 50 mA between a Pt anode and a carbonfelt cathode in the presence of Fe 2+ (0.5 mM) as a catalyst. After 5 min of electrolysis, HPLC analysis of an aliquot revealed the presence of several abiotic TPs. Preparative chromatography on an RP 18 column allowed the isolation of the main compound and its characterization by 1 H-NMR. After comparison with the previously obtained data, the aniline derivative 3 (Fig 1) was unambiguously identified. The co-product of this oxidation process, the furfural, was also identified among the other abiotic TPs by comparison with the HPLC chromatogram and UV spectrum of an authentic sample. Monitoring the electro-Fenton treatment of furosemide by HPLC and analyzing aliquots by LC-MS confirmed the formation of compound 3 with the presence of the m/z 251 protonated molecular ion (see SI §2). Two other compounds with protonated molecular ions at m/z 329 and m/z 345 were also observed. The structure associated with m/z 329 was confirmed by comparison with previously obtained data and unambiguously identified as pyridinium 4 (see SI §4). The CID mass spectrum analysis led to the proposal of structure A (see SI §5, 6) for m/z 345 (Fig 2a).To identify further oxidation products of furosemide, electro-Fenton oxidation was performed on pyridinium 4. Under these conditions, the compound previously associated with structure A was again observed (see SI §5) as well as the aniline derivative 3 (see SI §3). The oxidation of the pyridinium ring of 4, which leads to intermediates such as A, may explain the formation of the latter. Meanwhile, LC-MS analysis of the reaction mixture allowed the observation of two compounds with pseudo-molecular ions at m/z 317 and 282 for which structures B1 and C1 (Fig2b) have been, respectively proposed (see SI §7-10). The same experiment also revealed the presence of two isomers of B1 and C1. The analysis of the CID mass spectra did not allow their univocal characterization, which leads to the proposal of the regioisomers B2 or B3 for m/z 317 and C2 or C3 for m/z 282.

Figure 2 :Figure 3 :

 23 Figure 2: Structures proposed for a) m/z 345, b) m/z 317 and 282

Figure

  Figure 1

  Figure 3

2 . 4 Fig. SM- 4 .

 244 Fig. SM-1. (a) Extracted ion chromatogram (positive ion current) of m/z 251 for electro-Fenton oxidation of furosemide 1. (b) CID mass spectrum of protonated molecular ion of compound 3 (MH + m/z 251), cone =25 V, Ecol = 10 eV. The CID spectrum of the ion at m/z 251 (RT=5.2 min) is the same as that obtained for the MH + of compound 3.

7 . 8 . 4 9 .

 7849 Fig. SM-5. Extracted ion chromatograms (positive ion current) of m/z 317 for electro-Fenton oxidation of (a) furosemide 1 and (b) electro-Fenton oxidation of pyridinium 4. CID mass spectra of ion at m/z 317 at (c) RT=4.0 min and (d) RT=7.0 min. Cone=25 V, Ecol=15 eV. 8. Discussion on the structures of m/z 317, RT=4.0 and 7.0 min

We thank the CNRS for funding (CPDD), MESR for a PhD fellowship and Caroline Bance for technical support in microbiology.


Hsieh Tai-Yun
email: tai-yun.hsieh@umontpellier.fr

Montero Cédric

Alméras Tancrède

Bardet Sandrine

Experimental study and modeling on determining the predictors of creep behavior on small clear wood

B12 Materials and Methods

• All the specimens were scanned at room condition and average grain angle was measured by ImageJ from the photos (Fig. 3 c).

• The specimens were stabilized in climatic chamber at 20 °C and relative humidity (RH) 85% then density (ρ) and specific modulus (E s ) were measured (Tab.1).

• 29 Douglas fir samples, 34 Poplar samples, 33 Beech samples, and 12 Oak samples (108 samples in sum) were chosen in sampling and conducted creep test.

• 7-day 4-point bending test (Fig. 3 a b) with average loading stress 10.52 (±3.05) MPa was conducted in climatic chamber at 20 °c and RH=85%.

• Data was collected through strain gauges (KYOWA KFGS-5-120-C1-11) on both sides of the specimen and connected to the TDS-120 data logger.

Assuming that the creep behavior happens at the cellular level,

The two main hypothesis in this study:

 Delayed deformation is cause by the increased deformation of the cell wall.

• Reorganization of microfibrils.

• Deformation of the amorphous regions in cell wall;

 Creep deformation due to the sliding between the cells

Primary creep Secondary creep

The delayed compliance:

• primary creep: J 1d -J 0 • Secondary creep: J 7d -J 1d
1. Density is negatively proportional to delayed compliance → Creep behavior occurring on the cell wall.

2. Specific compliance = delayed compliance x density → Eliminates the effect of density.

3. The specific compliance has a strong linear relationship with the specific modulus and is more strongly correlated in secondary creep than in primary creep.

(1)

(2)

(3)

The In this study, the use of small clear wood was able to control the macroscopic material properties and reduce its variability. Density corresponds to material properties larger than cellular scale, and specific modulus corresponds to the property of the material that is smaller than the cellular scale, which is the microfibril angle or the grain angle. The assumption of the model in this study is that if delayed compliance is proportional to density, then creep behavior occurs only at the cell wall. It is proved in the linear model. However, we also observed that the magnitude of the effect of these factors on creep behaviors changed over time. Through the discrete compliance model, we found the compliance parameter values at characteristic time and the trend is nonlinear. Therefore, in the multiple regression model, we assume that the relationship between creep parameters and time is a polynomial function, which shows the degree of influence of material properties on the creep behavior change with time. The fitting results show that the effect of density decreases with time, while the effect of specific modulus increases. From this, we can infer that the creep behavior of wood starts with the deformation of the cell walls and then shifts to the intercellular space, which is related to the MFA and the grain angle and is presented by the specific modulus. This work has been supported by French National Research Agency (ANR) during EFEUR5 project (ANR-15-CE08-0027) for the investments into experimental equipment. To understand the relationship between time and predictors, the discrete compliance model assumes that the entire creep model is constructed from 5 kelvin bodies. The whole creep process is divided into 5 parts by chatracteristic time, and J*n is the fitted parameter for each creep data. Through the regression slope of J*n and predictors, we can obtain trends in the degree of impact.

From linear model and discrete compliance model: → The effect of density and specific modulus varies with time.

→ It is present on the subdivision of the factor.

1. J*i is defined as a function of density and specific modulus.

2. Density and specific modulus is defined as polynomial equations of time. 3. 10 parameters (i a , s a , q a , c a , i b , s b , q b , i c , s c , q c ) are calculated by test result. 4. log τ i value: from 2 to 5.5 every 0.1

Number

  Fig 3: (a) 4-point bending test system for small clear wood. (b) Dimension of the loading system. (c) Measuring method of grain angle.

Fig. 4 :

 4 Fig. 4 : Assumptions of creep behavior in the cell scale: deformation of the cell wall (1) and sliding between cells (2), proposition of a model to represent both phenomena.

Fig. 1 :Fig

 1 Fig. 1: Evolution of relative creep after 1 week at various relative humidity levels and for different values of specific modulus (GPa) indicated on each curve (Matar 2003)

  best-fit parameters are shown in Table 2, and some examples of the fit results are shown below. In the figure of b(log τ i ) and c(log τ i ), it shows a similar trend to the previous model, which present that the creep behavior start from the deformation of cell wall, then transfer to intercellular sliding. Best fitting parameters of multiple regression model.

  result: → The predictors have different effects on primary and secondary creep. → The effect of predictors change with time.


Structure and rheology in vertex models under cell shape-dependent active stresses Shao-Zhen Lin, 1 Matthias Merkel, 1 and Jean-François Rupprecht 1, * 1 Aix Marseille Univ, Université de Toulon, CNRS, CPT (UMR 7332), Turing Centre for Living systems, Marseille, France.

(Dated: March 28, 2023) Biological cells can actively tune their intracellular architecture according to their overall shape. Here we explore the rheological implication of such coupling in a minimal model of a dense cellular material where each cell exerts an active mechanical stress along its axis of elongation. Increasing the active stress amplitude leads to several transitions. An initially hexagonal crystal motif is first destabilized into a solid with anisotropic cells whose shear modulus eventually vanishes at a first critical activity. Increasing activity beyond this first critical value, we find a re-entrant transition to a regime with finite hexatic order and finite shear modulus, in which cells arrange according to a rhombile pattern with periodically arranged rosette structures. The shear modulus vanishes again at a third threshold beyond which spontaneous tissue flows and topological defects of the nematic cell shape field arise. Flow and stress fields around the defects agree with active nematic theory, with either contractile or extensile signs, as also observed in several epithelial tissue experiments.

Connecting the single-cell behavior to large-scale mechanical properties of biological tissues is key to understand development, regeneration, and disease [1]. Growing experimental evidence supports the idea that biological cells actively tune their intracellular architecture according to their overall shape. For instance, actomyosin [2][3][4][5] and microtubules (whether in Drosophila [6,7] or in plants [5,8]) tend to align along the direction of cell shape elongation. Such oriented fibers are known to generate anisotropic stress [9]. For instance, during Drosophila germ band extension, cells involved in rosette formation exhibit actin polymerization at opposite corners [10], which effectively creates an extensile active stress, Fig. 1a.

Here, we explore the consequences of such cell-shape feedback for the tissue-scale behavior. We study a minimal model where a bulk cellular active stress σ (act) is created by filaments which in turn align with cell shape, represented by a tensor Q (Fig. 1b-d). To lowest order,

σ (act) = -βQ, (1)
which we incorporate in a computational model for dense epithelial tissue. For β > 0 (resp. β < 0), cells actively push (resp. pull) on their neighbours along their direction of elongation [11,12]. Relations like Eq. ( 1) have been considered in tissue models before. For instance, in active hydrodynamic theories, where both σ (act) and Q are defined by averages over several cells, Eq. ( 1) gives rise to a classical flow instability [13][14][15]. In the absence of confining boundaries, this instability occurs at arbitrarily small activities β > 0. An active stress as in Eq. ( 1) has also been included in cell-based phase field simulations [16]. In these simulations, the transition to spontaneous flows occurs for a critical value of β which (i) is finite and appears to be independent of the system size, contrasting with active hydrodynamic theory results, and (ii) scales linearly with a cell surface tension. So far, while Ref. [16] pro-vided an intuitive conjecture, a precise understanding of both is missing. Is a finite activity threshold a general property of deformable cellular materials?

We address these questions by combining analytical arguments and vertex model simulations. Vertex models describe epithelial tissues as polygonal tilings [17][18][19]. Forces on the polygon vertices are defined by a mechanical energy:

E = 1/2 J [K A (A J -A 0 ) 2 + K P (P J -P 0 ) 2 ]
where the sum is over all cells J of the tissue, and A J and P J are cell area and perimeter, respectively. The parameters A 0 and P 0 are preferred cell area and perimeter with the associated rigidities K A and K P , respectively. A transition occurs at P 0 = P * 0 , with the tissue behaving as a yield stress solid for P 0 < P * 0 and as a fluid for P 0 ≥ P * 0 [20,21]. The numerical value of P * 0 is in the range 3.72 . . . 3.94, with a value that depends on the disorder in the cellular packing [20][21][22][START_REF] Merkel | Manning[END_REF][24] (SM [START_REF]The Supplemental Material at [URL] provides further details of analytical calculations and numerical simulations[END_REF], Sec. I). This solid-to-fluid transition can additionally be driven by cell-based active polar forces [START_REF] Barton | [END_REF][27][28], or by active tension fluctuations at cell-cell junctions [29], or by a chemomechanical feedback loop between tension and myosin ac- tivity along cell-cell junctions [30,31]. The vertex model solid-to-fluid transition is also echoed in experiments on biological tissues [32].

Here we introduce Eq. ( 1) in the vertex model framework, and depending on the value of P 0 , we find different kinds of transitions. For P 0 < P * 0 , we find four different regimes upon increasing β (Fig. 3). For β < β 1 the tissue is solid with isotropic cells. For β 1 < β < β 2 , cell shapes are anisotropic. For β 2 < β < β 3 , we find a regime with rhombile cells, many-fold vertices, long-range crystalline order and finite shear modulus. Finally, at β 3 < β, the tissue turns into an active fluid and displays constant flows that lack large-scale coherence. Conversely, for P 0 ≥ P * 0 , we find that a solid regime with anisotropic cells for β ≤ 0 directly transitions to the active fluid regime for β > 0. We demonstrate why a finite activity threshold β 3 appears in the solid regime, P 0 < P * 0 , in our cell-based model; to overcome the yield stress in the solid phase, a finite β = β 3 is required for active flows to appear, as in Ref. [16]. Meanwhile, active hydrodynamic theories as in [13,14] describe tissues as fluids, which corresponds in our model to P 0 ≥ P * 0 . In this case, active flows can appear without threshold, i.e. β 3 = 0.

Method We implement Eq. ( 1) through the following friction-based dynamics for the vertex positions r i

γ dr i dt = F (svm) i + F (act) i . (2)
Here, γ is a friction coefficient, F

(svm) i = -∂E/∂r i
are the standard vertex model (svm) forces, and F (act) i are the active forces induced by the active anisotropic bulk stresses σ (act) defined for each cell according to Eq. ( 1).

There are different ways to translate the cellular bulk stresses σ (act) into the vertex forces F (act) i [33][34][35][36]. Here we use the approach proposed by Tlili et al. [33,34,36], which relies on Cauchy's stress definition. For the cell shape anisotropy tensor Q J in Eq. ( 1), we use the symmetric, traceless tensor

Q J = 1 P J k l k t k ⊗ t k - I 2 , (3)
where P J is the cell perimeter; the sum is over all sides k of the cell J, and l k and t k denote length and unit tangent vector of side k, respectively. The eigenvalues and principal directions of Q J provide metrics for the cell shape and cell orientation. We measure the cell shape anisotropy by q J = 2tr(Q 2 J ) ∈ [0, 1), with q J = 0 for round cells and q J → 1 for increasingly elongated ones. We initialize the system with cells arranged according to either (i) a regular hexagonal pattern with small random deviations in the vertex positions, or (ii) random Voronoi tessellations (SM [START_REF]The Supplemental Material at [URL] provides further details of analytical calculations and numerical simulations[END_REF], Sec. I). In addition, we perform cell-neighbor exchanges (T1 transitions) for cellcell interfaces shorter than a length threshold ∆ T1 (SM [START_REF]The Supplemental Material at [URL] provides further details of analytical calculations and numerical simulations[END_REF], Sec. I). We use periodic boundary conditions with fixed system size. We set K A = 1, A 0 = 1, K P = 0.02, γ = 1, P 0 = 1, ∆ T1 = 0.01, and N = 10 3 cells if not otherwise stated (SM [START_REF]The Supplemental Material at [URL] provides further details of analytical calculations and numerical simulations[END_REF], Sec. I; Table S1).

Results Increasing β, we observe several rheological and structural transitions (Fig. 2a,b; Movies S1-S3). In Fig. 3 we show the dependence of these transitions on both β and P 0 ; however, in the following, we focus on the case P 0 = 1 (Fig. 2). For small β < β 1 ≈ 0.20 the vertex model tissue is solid (Fig. 2e) with isotropic cell shapes (Fig. 2c), where the average cell elongation is q = 0 (resp. q ≈ 0.16) when using a hexagonal (resp. Voronoi) initial state. When β increases beyond β 1 , cell shapes become anisotropic, as indicated by an increase in q (Fig. 2c). This is accompanied by a decrease in the hexatic order parameter ψ 6 (Fig. 2d), defined as ψ 6 = | Ψ j /N |, where Ψ j = k∈neighbors exp (i6θ jk )/N j and θ jk = arg (r k -r j ) [37][38][39]. While in this regime the shear modulus vanishes for the hexagonal initial state, the tissue remains solid, as verified through the examination of the yield stress (Fig. 2e, SM [START_REF]The Supplemental Material at [URL] provides further details of analytical calculations and numerical simulations[END_REF], Sec. I). The transition point β 1 decreases with increasing P 0 up until the critical point P * 0 (Fig. 3). The cell shape transition at β = β 1 occurs because the β term in Eq. ( 1) effectively corresponds to a negative shear modulus. As a consequence, when β > β 1 , the total cellular shear modulus decreases to zero, destabilizing the isotropic cell shape. To see this, we start from the Batchelor stress of a vertex model cell with perimeter P and area A [36,[40][41][42], whose anisotropic part σ is (SM [START_REF]The Supplemental Material at [URL] provides further details of analytical calculations and numerical simulations[END_REF], Sec. I):

σ = K P P (P -P 0 ) A -β Q. ( 4
)
To obtain the global tissue shear modulus G aff in an analytical mean-field picture, we apply an affine pure shear strain ϵ to an isotropic cell, which creates the cell shape anisotropy q = 3ϵ/2 to lowest order in ϵ (SM [START_REF]The Supplemental Material at [URL] provides further details of analytical calculations and numerical simulations[END_REF], Sec. II). Comparing Eq. ( 4) to σ = 2G aff ϵ, we obtain:

G aff = 3 8 K P P (P -P 0 ) -β . (5)
Here, we have used A ≈ A 0 = 1, which corresponds to the limit of incompressible cells. Testing Eq. ( 5) numerically, where we also include all non-affinities, we find the same result, except for a prefactor: G non-aff ≈ 2G aff /3. To determine the value of the perimeter P appearing in Eq. ( 5), we use that in the isotropic solid regime P = P * 0 ≈ 3.722 for a hexagonal tissue [21,43]. Isotropic cell shape thus becomes unstable for β > β 1 (P 0 ) with (SM [START_REF]The Supplemental Material at [URL] provides further details of analytical calculations and numerical simulations[END_REF], Sec. II):

β 1 (P 0 ) = K P P * 0 (P * 0 -P 0 ). ( 6
)
This equation exactly predicts the stability of the regular hexagonal crystal (white lines in Fig. 3). This meanfield picture also explains how in the regime P 0 < P ⋆ 0 cells elongate for β > β 1 (P 0 ). For an affinely sheared isotropic cell, the perimeter increases quadratically with its shape anisotropy as P = P * 0 (1 + q 2 /3) (SM [START_REF]The Supplemental Material at [URL] provides further details of analytical calculations and numerical simulations[END_REF], Sec. II) [24]. Inserting this in Eq. ( 4) and combining it with σ = (∂E/∂ϵ)/2A, we obtain an effective potential of the cell depending on its shape anisotropy E eff (q), which reads to fourth order in q (SM [START_REF]The Supplemental Material at [URL] provides further details of analytical calculations and numerical simulations[END_REF], Sec. II): The energy minimum for β < β 1 is at q min = 0, while for β > β 1 the minimum is at q min (β) = 3(β -β 1 )/K P P * 0 (2P * 0 -P 0 ), which corresponds to a typical pitchfork bifurcation. Indeed, this predicts well the observed cell elongation in the regime close to β 1 for P < P ⋆ 0 (Fig. 3c, SM [START_REF]The Supplemental Material at [URL] provides further details of analytical calculations and numerical simulations[END_REF], Sec. II). The behavior of cell-shape elongation q is different in the regime P 0 > P ⋆ 0 , where we observe a discontinuous increase in q and coordination number Z as soon as β is increased above zero (Fig. 3a,b, SM [START_REF]The Supplemental Material at [URL] provides further details of analytical calculations and numerical simulations[END_REF], Sec. VI). The discontinuity in q can be understood by first looking at a system with β = 0 and given cell elongation q. In this case, for P 0 > P ⋆ 0 , the system is floppy with vanishing energy E(q) = 0 as long as q < q crit , where the critical q value is q crit ∼ P 0 -P * 0 . Beyond this value, cells and the vertex model tissue start to attain a finite shear modulus [24]. As a consequence, as soon as β is set to a positive value, for q < q crit the energy becomes E(q) ∼ -βq 2 , and so any state q < q crit becomes unstable. Thus, for small positive β, cell elongation will make a jump from zero to a value close to q crit . For P 0 < P * 0 , beyond β > β 2 (with β 2 ≈ 0.24 for P 0 = 1), starts a regime marked by an increasing hexatic order ψ 6 (Fig. 2b-d). For tissues initiated in the near hexagonal pattern, this order appears to be even long-ranged (SM [START_REF]The Supplemental Material at [URL] provides further details of analytical calculations and numerical simulations[END_REF], Sec. I). We call this the rhombile regime, because domains appear where cells attain a rhombic shape, which arrange into a periodic arrangement of six-fold vertices (Fig. 2b). Correspondingly, the rhombile regime is marked by the increase in the average vertex coordination number Z (Fig. 2d).

E eff (q) = 1 3 β 1 (P 0 ) -β q 2 + 1 18
K P P * 0 2P * 0 -P 0 q 4 . (7)
The emergence of the rhombile pattern and manyfold vertices can be understood from single-and four-cell systems. Indeed, at a critical value of β (1) 2 ≈ 0.27 for the single-cell system (β (4) 2 ≈ 0.25 for the four-cell system), the two shortest edges of a hexagonal cell shrink to length zero, resulting in the observed rhombic cell shapes (Fig. 2a; Movie S4-S5; SM [START_REF]The Supplemental Material at [URL] provides further details of analytical calculations and numerical simulations[END_REF] Sec. II).

We find that the shear modulus in the rhombile regime is finite and peaks when the rhombile domain extension is maximal at around β * ≈ 0.3, which coincides with local maxima in the hexatic order ψ 6 and coordination number Z (Fig. 2d). We understand this by considering a single cell: at β * = 0.3, the cell reaches a regular diamond shape with two π/3 and two 2π/3 angles. This shape (also called calisson [START_REF] Alsina | A Mathematical Space Odyssey[END_REF]) is the building block of the crystal rhombile pattern (SM [START_REF]The Supplemental Material at [URL] provides further details of analytical calculations and numerical simulations[END_REF], Fig. S11b). When β < β * = 0.3 (resp. β > β * = 0.3), acute angles above (resp. below) π/3 form in the single cell, leading to frustration in the rhombile domains, which can destabilize them.

However, despite a finite shear modulus, the rhombile regime exhibits a finite steady-state T1 transition rate (Fig. 2d; SM [START_REF]The Supplemental Material at [URL] provides further details of analytical calculations and numerical simulations[END_REF] Sec. III; Movie S2). This may appear paradoxical at first sight, since T1 transitions are expected to relax the applied stress [START_REF] Marmottant | Proceedings of the National Academy of Sciences of the United States of America[END_REF], leading to a longtime fluid material response. Yet such argument may not hold for an active system at steady state, as considered here. Further, we notice that these T1 transitions are generated at the interfaces, rather than in the bulk, of the rhombile crystal domains (Movie S2).

The shear modulus eventually vanishes at β 3 ≈ 0.37 for P 0 = 1. We find that this value matches the one at which the shear modulus of the perfect rhombile crystal vanishes (SM [START_REF]The Supplemental Material at [URL] provides further details of analytical calculations and numerical simulations[END_REF], Sec. III). This suggests that the finite shear modulus of the rhombile regime is caused by rhombile crystal domains.

For β > β 3 , the system flows continuously, where the steady-state T1 transition rate increases with activity (Fig. 2e, 3d and Movie S2). Such flow already appears in a four-cell system for an intermediate activity range (Movie S6). This flowing regime exhibits features of an active nematic material. For instance, coarse-graining of the cell orientation field reveals the presence of ±1/2 topological defects in the fluid regime (SM [START_REF]The Supplemental Material at [URL] provides further details of analytical calculations and numerical simulations[END_REF], Sec. V) [START_REF] Vromans | [END_REF]47]. The stress and velocity patterns around ±1/2 topological defects are qualitatively consistent with those predicted in an incompressible material with extensile nematic activity [15,16,48] (Fig. 3f; SM [START_REF]The Supplemental Material at [URL] provides further details of analytical calculations and numerical simulations[END_REF], Sec. V). We note, however, the presence of a backward flow at the tip of +1/2 defects (Fig. 3f) that is reminiscent of negative wake reported in driven yield-strain materials [49]. A similar negative wake flow pattern is visible in experiments on Madin-Darby Canine Kidney cell monolayers [33,50] (see SM [START_REF]The Supplemental Material at [URL] provides further details of analytical calculations and numerical simulations[END_REF], Sec. VIII).

Taken together, our results show that the existence of a yield stress explains the onset of active flows beyond a finite critical activity β 3 > 0, while active continuum models [13,14], predict active flows at any finite activity, i.e. β 3 = 0. In our vertex model, such a difference arises as a function of P 0 . For P 0 < P ⋆ 0 , where the vertex model behaves as yield stress solid, we showed that β acts as a negative shear modulus, destabilizing the tissue beyond a critical value that increases with the distance to P ⋆ 0 . In contrast, we observe that for the fluid vertex model regime, P 0 > P ⋆ 0 , any positive β destabilizes the tissue, inducing a sharp increase in flow (Fig. 3d, SM [START_REF]The Supplemental Material at [URL] provides further details of analytical calculations and numerical simulations[END_REF], Sec. VI). Comparing both cases, we conclude that while any positive activity leads to flows in a fluid material, a yield stress creates a finite activity threshold to flow. This is likely the case in Ref. [16], because the employed phase field model essentially describes a foam that, expectedly, displays a yield stress.

In contrast to active hydrodynamic theory predictions [13,15,48], the cell-based model by Ref. [16] and the model we discussed so far do not exhibit spontaneous flows in the contractile regime (β < 0). One could take this as an indication that the observed transition to spontaneous flows is qualitatively different from the known instability of active hydrodynamic theory [13,15,48].

However, spontaneous flows can also exist in the contractile regime, β < 0. To show this, we included in our model the tendency of cells to attain a finite elongation q J through an additional energy term, E el = J K Q [q 2 J -q 2 0 ] 2 /8, where K Q ≥ 0 and q 0 refers to a preferred cell elongation parameter.

Using this model, we observe spontaneous tissue flows in both the extensile (β > 0) and the contractile regime (β < 0), with cells displaying rhombic shapes (Fig. 4a, Movie S7) and both the stress field and the flow field qualitatively agree with predictions of active hydrodynamic theories (Figs. 4(b); SM [START_REF]The Supplemental Material at [URL] provides further details of analytical calculations and numerical simulations[END_REF], Sec. VII) as well as experiments [11,[51][52][53][54].

Lastly, we tested how much our results depend on the definition of the Q tensor used in Eq. (1) (SM [START_REF]The Supplemental Material at [URL] provides further details of analytical calculations and numerical simulations[END_REF], Sec. IV). We find that the transitions in cell shape and rigidity are generic, but not the appearance of the rhombile crystal domains. In addition, we also tested the case of no perimeter elasticity (K P = 0) but with a constant cell-cell interfacial tension instead [29]; we find the same results regarding the transition to flows and the onset of a rhombile phase (Fig. S23).

Conclusion We studied how a feedback of cell shape on the cellular active stress generation affects collective tissue dynamics. We show that increasing such a feedback eventually fluidifies the vertex model tissue, yet through a series of intermediate steps displaying hexatic order, enhanced finite shear modulus and spontaneous T1 transitions. We find that spontaneous flows can be thresholdless also for cellular materials and are not limited to the extensile regime.

Our prediction of rhombile domains could potentially also explain why, during the Drosophila germ band extension process [10], cells display rhombic shapes before the onset of six-fold vertices, Fig. 1a. These cell shape changes are shown to be triggered by actin protrusions [10]. At early stages, these protrusions are located at opposite ends of the cell, suggesting that the resulting set of forces could be compatible with the one considered in our extensile active stress model, Fig. 1.

Perspectives While in most instances active cellular materials have been studied in the vertex model through the introduction of traction forces against a resting substrate [28,55], here we introduce activity through momentum-conserving forces F (act) i [36]. Implementing this kind of activity within a Galilean invariant dissipation framework [42] could be useful to describe tissue flows in conditions of low environmental friction e.g. freestanding tissues, pre-implantation embryos or intestinal organoids [56].

  FIG. 1. (a) Experimental images of the basal side of Drosophila epithelial cells during germ band extension with marked GFP-Rac1 marking filament growth activity; six-fold vertex indicated by a square. Scale bar: 5 µm. Experimental images provided by Y. Toyama. (b) A model of increasingly strong filament growth (magenta rods) for increasingly acute cell corners (black wedge). (c) Our model where active stresses are defined by cell shape. (d) Active forces in a cell (index J) of our vertex model, mimicking the effect of growing filaments at wedge-like cell corners.

FIG. 2 .

 2 FIG. 2. (a, b) Morphology of (a) a four-cell system and (b) a cell sheet at different activities β, with an initially hexagon (top) or Voronoi pattern (bottom). In (b), we mark vertices with coordination number Zi > 3. (c) Cell elongation parameter q, and T1 transition rate kT1 versus cell activity β, for an initially hexagonal (H) or Voronoi (V) pattern. The solid black line refers to an analytical approximation of q. (d) Hexatic order parameter ψ6 and average vertex coordination number Z versus cell activity β. (e) Long-time shear modulus Gxy and yield stress σ yield versus cell activity β. Inset: red shaded area indicates the mean ± standard deviation of the shear modulus for the initially hexagonal pattern, estimated from n = 5 simulations. Default parameter values with P0 = 1.

5 -FIG. 3 .

 53 FIG. 3.Diagrams depending on the target shape index P0 (inter-cellular tension) and cell bulk activity β, showing (a) average cell elongation parameter q, (b) average vertex coordination number Z, (c) shear modulus Gxy, and (d) T1 topological transition rate kT1. Here, the solid white curves refer to the theoretical prediction β1(P0), Eq. (6), for the transition between the solid regime with isotropic cells and the solid regime with anisotropic cells. (e) Overall phase diagram. (f) Average isotropic stress (color map) and velocity (black streamlines) fields near +1/2 defects, with P0 = 1 and β = 0.5. Average over n = 44368 defects. Scale bar = 1 cell.

FIG. 4 .

 4 FIG. 4. Tissue flows and topological defects in the contractile regime (β < 0) in the presence of a passive cell elongation energy. (a) Sketch of cell activity leading to cell edge vanishing. Here, we mark the edges to vanish as red ones. Background color: gray for solid regime and light green for fluid regime. (b) Average isotropic stress (colorbar) and the velocity (streamlines) near +1/2 topological defects (n = 25323 defects. Scale bar = 1 cell), where KQ = 10 and β = -1. Other parameters: P0 = 1 and q0 = 0.8.

Acknowledgements We thank the Centre Interdisciplinaire de Nanoscience de Marseille (CINaM) and the Laboratoire Adhésion Inflammation (LAI) for providing office space. J.F. R. and S-Z. L. thank F. Graner, T. Saunders and Y. Toyama for discussions and for the experimental data shown in Fig. 1, as well as B. Ladoux and R.-M. Mège for the experimental data shown in Fig. S22. The project leading to this publication has received funding from France 2030, the French Government program managed by the French National Research Agency (ANR-16-CONV-0001) and from Excellence Initiative of Aix-Marseille University -A*MIDEX. J.F. R. is also funded by ANR-20-CE30-0023 COVFEFE.


Yuting Lou

Sophie Theis
email: sophie.theis@warwick.ac.uk

Jean-Francois Rupprecht
email: jean-francois.rupprecht@univ-amu.fr

Tetsuya Hiraiwa

Timothy E Saunders
email: timothy.saunders@warwick.ac.uk

Curvature-induced cell rearrangements in biological tissues

On a curved surface, epithelial cells can adapt to geometric constraints by tilting and by exchanging their neighbors from apical to basal sides, known as an apicobasal T1 (AB-T1) transition. The relationship between cell tilt, AB-T1 transitions, and tissue curvature still lacks a unified understanding. Here, we propose a general framework for cell packing in curved environments and explain the formation of AB-T1 transitions under different conditions. We find that steep curvature gradients can lead to cell tilting and induce AB-T1 transitions. Conversely, large curvature anisotropy can drive AB-T1 transitions by hydrostatic pressure. The two mechanisms compete to determine the impact of tissue geometry and mechanics on optimized cell rearrangements in 3D.

As the external surfaces and barriers of many organs, epithelial tissues have to mechanically adapt to their environment [START_REF] Guillot | Mechanics of epithelial tissue homeostasis and morphogenesis[END_REF][START_REF] Latorre | Active superelasticity in threedimensional epithelia of controlled shape[END_REF]. Extensive research into cell shape in 2D [START_REF] Aigouy | Cell flow reorients the axis of planar polarity in the wing epithelium of drosophila[END_REF][START_REF] Sagner | Establishment of global patterns of planar polarity during growth of the drosophila wing epithelium[END_REF][START_REF] Khalilgharibi | Stress relaxation in epithelial monolayers is controlled by the actomyosin cortex[END_REF][START_REF] Tetley | Tissue fluidity promotes epithelial wound healing[END_REF][START_REF] Gibson | The emergence of geometric order in proliferating metazoan epithelia[END_REF][START_REF] Farhadifar | The influence of cell mechanics, cell-cell interactions, and proliferation on epithelial packing[END_REF][START_REF] Staple | Mechanics and remodelling of cell packings in epithelia[END_REF][START_REF] Atia | Geometric constraints during epithelial jamming[END_REF] and 3D [START_REF] Du | Computational analysis of three-dimensional epithelial morphogenesis using vertex models[END_REF][START_REF] Krajnc | Theory of epithelial elasticity[END_REF][START_REF] Fletcher | Mechanocel-366 lular models of epithelial morphogenesis[END_REF][START_REF] Merkel | A geometrically con-370 trolled rigidity transition in a model for confluent 3d tis-371 sues[END_REF] has revealed insights into how cells pack and undergo rearrangement during epithelial tissue formation [START_REF] Gibson | The emergence of geometric order in proliferating metazoan epithelia[END_REF][START_REF] Farhadifar | The influence of cell mechanics, cell-cell interactions, and proliferation on epithelial packing[END_REF][START_REF] Staple | Mechanics and remodelling of cell packings in epithelia[END_REF][START_REF] Atia | Geometric constraints during epithelial jamming[END_REF][START_REF] Bertet | Myosin-dependent 373 junction remodelling controls planar cell intercalation 374 and axis elongation[END_REF]. Cellular dynamic processes, like division and apoptosis, can rearrange cell neighbors. T1-transitions -the exchange of neighbors without altering the cell number -is another ubiquitous mechanism of cell rearrangements [START_REF] Honda | Cell move-376 ments in a living mammalian tissue: Long-term observa-377 tion of individual cells in wounded corneal endothelia of 378 cats[END_REF][START_REF] Fletcher | Vertex models of epithelial morphogenesis[END_REF]. T1 transitions are important in mediating planar tissue dynamics. For example, oriented T1 transitions can lead to tissue elongation or flow [START_REF] Bertet | Myosin-dependent 373 junction remodelling controls planar cell intercalation 374 and axis elongation[END_REF][START_REF] Honda | Two different 383 mechanisms of planar cell intercalation leading to tissue 384 elongation[END_REF][START_REF] Sato | Left-right asymmetric cell inter-387 calation drives directional collective cell movement in ep-388 ithelial morphogenesis[END_REF][START_REF] Hiraiwa | Wave propaga-391 tion of junctional remodeling in collective cell movement 392 of epithelial tissue: Numerical simulation study[END_REF], and the energetic barriers for T1 transitions to occur can dictate tissue fluidity/solidity [START_REF] Staple | Mechanics and remodelling of cell packings in epithelia[END_REF][START_REF] Bi | A 395 density-independent rigidity transition in biological tis-396 sues[END_REF][START_REF] Bi | 398 Motility-driven glass and jamming transitions in biolog-399 ical tissues[END_REF][START_REF] Krajnc | Fluidiza-401 tion of epithelial sheets by active cell rearrangements[END_REF].

For a cell monolayer under 3D geometric constraint, cells can undergo apical-basal T1 (AB-T1) transitions (Fig. 1A,top). Different from the planar and dynamic T1-transitions described above, AB-T1 transitions are a static exchange of neighbors from the apical to basal layers of the cell. Such a 3D cellular arrangement, termed as a scutoid in the context of epithelial tissues [START_REF] Gómez-Gálvez | Scutoids are a geometrical solution to 407 three-dimensional packing of epithelia[END_REF][START_REF] Blanchard | A 3d cell shape that enables tube formation[END_REF][START_REF] Gómez-Gálvez | The complex threedimensional organization of epithelial tissues[END_REF] (Fig. 1A), has been observed in foams [START_REF] Matzke | The three-dimensional shape of epidermal cells of the apical meristem of anacharis densa (elodea)[END_REF][START_REF] Mughal | Demonstration and interpretation of 'scutoid' cells formed in a quasi-2d soap froth[END_REF] and biological systems with curved surfaces [START_REF] Rupprecht | Geometric constraints alter cell arrangements within curved epithelial tissues[END_REF][START_REF] Sanchez-Corrales | Radially patterned cell behaviours during tube budding from an epithelium[END_REF][START_REF] Nelson | Epithelial packing: even the best of friends must part[END_REF][START_REF] Perez-Vale | Orchestrating morphogenesis: building the body plan by cell shape changes and movements[END_REF][START_REF] Pérez-González | Mechanical compartmentalization of the intestinal organoid enables crypt folding and collective cell migration[END_REF].

Tissue curvature is proposed to be pivotal in inducing AB-T1 transitions.

In the ellipsoidal early Drosophila embryo, AB-T1 transitions appear most frequently around 20-50µm from the embryo head, a region with low curvature anisotropy but large tilt of cell lateral membranes [START_REF] Rupprecht | Geometric constraints alter cell arrangements within curved epithelial tissues[END_REF] (Fig. 1A). During salivary gland formation in the Drosophila embryo, AB-T1 transitions occur at maximal curvature anisotropy [START_REF] Gómez-Gálvez | Scutoids are a geometrical solution to 407 three-dimensional packing of epithelia[END_REF]. Models have been proposed for cell packing in these specific cases [START_REF] Gómez-Gálvez | Scutoids are a geometrical solution to 407 three-dimensional packing of epithelia[END_REF][START_REF] Rupprecht | Geometric constraints alter cell arrangements within curved epithelial tissues[END_REF],

but there is currently no consensus on how curvature induces AB-T1 transitions.

lateral tilt  () scutoids() 1). (D) Force balance of a curved layer under axisymmetric loads: (left) at the meridional cut (red dashed ring) and (right) along the normal direction of the element surface dA(ϕ, θ). (E) A meridional cross section view of a two-layered prolate ellipse. The black tilted line is the tilted lateral membrane, with the basal end at s and the apical end at s , with the tilt angle φ and apical-basal distance e at s. The orange curves are the accumulated cell number from the head to s at the basal side; the skyblue curve is the accumulated cell number from the head to s at the apical side.

which can be decomposed into a normal part σ N (positive pointing outward) and a tangential part along the meridian σ T (positive pointing to the head) and hence the in-plane shear σ θϕ ≈ 0. The meridional stress σ ϕϕ at any local cut (red ring in Fig. 1D) is balanced in the polar direction by the accumulated force over the revolved surface as:

σ ϕϕ 2πrδsinϕ = ˆs(ϕ) 0 σ N cosϕ + σ T sinϕ 2πrds, ( 1
)
where δ is the thickness of cell membrane, r is the distance to the polar axis from the local surface dA (Fig. 1D) and ds is the meridional arc length; See Supplemental Material (Supp. Mat.) at []) for the derivations of force balance. The circumferential stress σ θθ is derived from force balance along the normal direction of the surface:

C ϕϕ σ ϕϕ + C θθ σ θθ = σ N δ , (2)
where C ϕϕ and C θθ are the principal curvatures along the meridional and circumferential direction, respectively.

AB-T1 transitions:

The stresses in apical or basal layers can induce cell shape changes and cell intercalations. Here, we assume that prior to any applied external load, cells are relaxed to isotropic shapes without any deviatoric strain. AB-T1 transitions will take place most frequently when the apical and basal sides of a cell have oppositely directed deviatoric stresses [START_REF] Duclut | Nonlinear rheology of cellular networks[END_REF] under external loading. In the absence of shear components σ ϕθ , we can define a measure for AB-T1 transitions, γ, as propor-116 tional to the difference of the deviatoric strain between 117 the apical and basal sides: system, the cells are observed to tilt (Fig. 1A). The pro-143 file of external load σ T , σ N is affected by tilt of lateral 144 membranes. We next investigated cell tilting within our 145 model and explain its role in inducing AB-T1 transitions.

118 γ = σ a ϕϕ -σ a θθ µ a - σ b ϕϕ -σ b θθ µ b , (3)
146

The tilted lateral membrane leans to the head by a 147 small angle φ away from the normal direction (illustrated 148 in Fig. 1E) as

149 tanφ(s) ∼ ∆s e(s) ∼ N a (s) -N b (s) 2πr a (s)e(s)ρ a (s) , (4)
where ∆s is the distance between the apical projection

φ(s) = φ * (s)(1 -k), ( 5
)
169
where k 1 is the ratio of tension strength between the 170 lateral and apical/basal layers; φ * is the tilt in the limit 171 of zero lateral tension, depending on the curvature as:

tanφ * (s) ∼ N b (s)(N total -N b (s) H(s 1 ) -H(s 2 ) πr a (s)ρ b (s)N total , (6)
where H(s 1 ) and H(s 2 ) are the mean curvature weighted tilted lateral membranes:

σ a T = T sinφ, σ a N = T cosφ, σ b T = -T sinφ, σ b N = -T cosφ.
The magnitude of σ θθ and σ ϕϕ grows from the head to ences across the cell along the apical-basal axis (Fig. 3D).

233

The tissue height follows Eq. 7. The apical and basal geometries [START_REF] Gómez-Gálvez | Scutoids are a geometrical solution to 407 three-dimensional packing of epithelia[END_REF][START_REF] Rupprecht | Geometric constraints alter cell arrangements within curved epithelial tissues[END_REF].

284

The lateral membranes play an essential role in balanc-

285

ing stress across the cell, thereby regulating cell shape.

286

In particular, lateral membranes with low contractility Though we have focused on a prolate geometry with simple external loads, our formalism can be generalized to a diverse range of tissue geometries observed in vivo.

We expect tilt to occur at the steepest curvature gradient, even for non-axisymmetric and non-closed surface geometries; e.g. the brain and gut. We can also explore the role of in-plane shear and bending within this theoretical framework. Internal cell strain, which is likely significant during cellular process such as cell division [START_REF] Gomez | and 454 D. Iber, 3d cell neighbour dynamics in growing pseudos-455 tratified epithelia[END_REF],

can also be considered as a source of external loading.

Finally, transient and reversible AB-T1 transitions have been observed [START_REF] Martin | Apical constriction: 457 themes and variations on a cellular mechanism driving 458 morphogenesis[END_REF][START_REF] Sun | Basolateral protrusion and apical 461 contraction cooperatively drive drosophila germ-band ex-462 tension[END_REF]; the dynamic aspect of AB-T1 transitions may be relevant to the mechanism of T1 tran-sitions [START_REF] Krajnc | Fluidiza-401 tion of epithelial sheets by active cell rearrangements[END_REF][START_REF] Duclut | Nonlinear rheology of cellular networks[END_REF] and their contributions to processes like

FIG. 1 .

 1 FIG. 1. The effect of curvature on cell packing and cellular forces. (A) Top: Scutoid geometry in epithelial tissues; point Q is the additional point shared by two columnar cells. The AB-T1 transition occurs at the edges highlighted in thick brown, leading to the exchange of neighbor pair from 1-4 (apical) to 2-3 (basal). Bottom: Tilt angle of lateral membrane (yellow) and percentage of scutoids (pink) peak near the head of a wild type Drosophila embryo, adapted from [29] under a Creative Commons License. (B) Two-layered model for curved epithelia on an axisymmetric object and the coordinates for a any local surface dA; (C) Graphical representation of the stress tensor decomposition, Eq. (1). (D) Force balance of a curved layer under axisymmetric loads: (left) at the meridional cut (red dashed ring) and (right) along the normal direction of the element surface dA(ϕ, θ). (E) A meridional cross section view of a two-layered prolate ellipse. The black tilted line is the tilted lateral membrane, with the basal end at s and the apical end at s , with the tilt angle φ and apical-basal distance e at s. The orange curves are the accumulated cell number from the head to s at the basal side; the skyblue curve is the accumulated cell number from the head to s at the apical side.

119 30 bFIG. 2 .

 302 FIG. 2. The lateral tilt angle in the zero-lateral-tension limit, Eq. 6, as a function of the distance to the head along the polar direction d = z/a (Fig. 1E) (A) under varying inverse aspect ratio b/a at ε/a = 0.05, β = 0; (B) under varying thickness modulation β at ε/a = 0.01, b/a = 0.4. Experimental data is shown for the cell tilt angle in the early Drosophila embryo (b/a ∼ 0.4, β ∼ 0.5), with s.d., by grey dots (data from [29]).

  151 of s and the apical end of the tilted lateral membrane s ; 152 e(s) is the distance between the apical and basal layer; 153 ρ a,b (s) is the cell density; r a,b (s) is the distance from 154 s to the polar axis; N a,b (s) = ´s 0 ρ a,b dA a,b are the ac-155 cumulated number of cells from the head apex to the 156 coordinate s on the apical and basal sides, respectively. 157 Although Fig. 1E is illustrated for an prolate ellipsoid, 158 Eq. 4 works for any arbitrary axisymmetric shape. 159 160 The distribution of ρ a,b (s) and e(s) are interdependent, 161 as a consequence of minimizing the system free energy 162 including the contributions from cell lateral membranes 163 (See Supp. Mat. G). If the lateral membrane tensions 164 are weak compared with the apical and basal cell layers, 165 the apico-to-basal density ratio ρ a (s)/ρ b (s) converges to 166 a space-independent constant (Supp. Mat. D). In this 167 limit, the tilt angle 168

174

  by cell numbers in a range of 0 < s 1 < s and s < s 2 < s 0 , 175 respectively (s 0 is the half meridian). For a convex ob-176 ject, a large gradient of H(s) corresponds to a large mag-177 nitude of φ * at s, with the corresponding tilt direction to-178 wards the region of higher positive curvature (See Supp. 179 Mat. D for the derivations). 180 Conversely, if lateral membranes are extremely rigid, 181 the lateral membrane tends to stand perpendicular to the 182 surfaces, and ρ a (s)/ρ b (s) equals inverse apico-to-basal 183 area ratio dA b (s)/dA a (s), hence the tilt vanishes (Supp. 184 Mat. F). To further simply the model, we show that the 185 effect of any cell density inhomogeneity on cell tilt is 186 negligible if cell density changes along the surface more 187 slowly than the curvature does (See Supp. Mat. E for 188 the analysis). We henceforth set a homogeneous density 189 ρ a,b (s) = ρ a,b 0 . 190 Ellipsoid case: We now apply this formalism to a 191 prolate ellipsoidal geometry as shown in Fig. 1E. It has 192 a major half axis a and minor half axis b (see Supp.

193where s 1 / 4

 14 FIG. 3. AB-T1 transition rate γ calculated for a prolate ellipsoid with b/a = 0.4. (A) A schematic illustration of the system under tensile lateral stress T with a constant tissue height ε = 0.05a for panel B and C; (B) In plane, apical and basal stress components normalized by T a/δ as a function of the distance to the head. Left: T perpendicular to the layer (hydrostatic); right: T along tilted lateral membranes. The cyan and magenta curves stand for a sphere (a = b). The tensile stresses have a positive sign (apical,blue) and contractile stress has a negative sign (basal, yellow). (C) The correspondent AB-T1 rate γ for the prolate ellipsoid. (D) A schematic tissue setting under the external loads with pressures and inhomogeneous tissue height (Eq. 7). (E) Profile of the AB-T1 measure, γ, with varying apical pressure difference ∆Π a and with basal pressure difference ∆Π b = 0. Left: the tissue height modulation rate β = 0; right: β = 0.4. Black dots indicate the peak, where the absolute AB-T1 measure |γ| reaches the maximum. (F) The phase diagram for the peakiness of γ, which is calculated as sign(γ peak ) × ||γ peak | -|γ trunk ||. Left: β = 0; right: ∆Π b = 0. The size of the scattered square is ∝ (1 -d peak ) 2 , so positions closer to the head (d = 0) are represented by larger squares. The grey contours separate the trunk (d peak > 0.5) and the head (d peak < 0.5) regions.

223

  the trunk in different manners, depending on whether T 224 is perpendicular to the shells (hydrostatic) or T along 225 the tilted lateral membranes. The resultant AB-T1 transition rate, calculated through Eq. 3, flips its sign with or 227 without the tilt (Fig. 3C). However, this qualitative dif-228 ference will vanish when the surface approaches a sphere 229 (a/b = 1) (Fig. 3B; cyan and magenta lines), leading 230 to no AB-T1 transition at all locations (Fig. 3C; black 231 line). 2 Next, we discuss results with pressure differ-232

234

  membranes are subject to pressure from: the outside 235 P out ; from the internal cavity (e.g. yolk or luminal pres-236 sure) P in ; and inside the tissue P T . The pressure dif-237 ferences at the apical and basal surfaces are given by 238 ∆Π a = P T -P out and ∆Π b = P in -P T respectively, 239 with positive ∆Π pointing towards the outside. Before 240 applying external load, we assume cells have relaxed to 241 their preferred cell shape with no internal strain. The 242 external normal and tangential loads on the apical and 243 basal side are σ (a) N = ∆Π + T cosφ, σ (a) T = ∆Π + T sinφ, 244 σ (b) N = ∆Π + T cosφ and σ (b) T = ∆Π -T sinφ.

245

  The system dominated by pressure (∆Π/T → ∞) cor-246 responds to a hydrostatic limit, Fig.3B(left). In this 247 limit, the profiles of stresses and the consequent spatial 248 distribution of AB-T1 transition frequency do not qual-249 itatively depend on the pressure differences or the cell 250 height profile (See Supp. Mat. C for the derivations). In 251 contrast, strikingly, when the pressure difference is com-252 parable with lateral stress (∆Π ∼ T ), γ is sensitive to 253 the two pressure differences and β, Fig. 3E. ∆Π can be 254 negative (pointing inwards), thus the normal component 255 of lateral tension T can be partly balanced by this pres-256 sure and σ T /σ N becomes much larger as if the cells tilt 257 more significantly. When ∆Π a /T shifts sign from nega-258 tive to positive, at the trunk (d = 1) γ becomes negative, 259 altering the orientation of AB-T1 transitions. 260 To capture the key features of the distribution of γ, 261 we define the peak of γ as where γ peak is the maximal 262 value of |γ| (Fig. 3E) and its value at peak (referred here 263 as the peakiness) as sign(γ peak ) × ||γ peak | -|γ trunk ||. Ac-264 cordingly, we can construct a phase diagram of AB-T1 265 transitions, using the position of the peak and peakiness 266 as the order parameters, Fig. 3F. We show the diagram 267 in the ∆Π a -∆Π b space for β = 0 (left) and in the space 268 of β-∆Π a /T with ∆Π b = 0 (right). The peak in the ten-269 dency of AB-T1 transition switches from trunk to head 270 beyond a critical line β(∆Π) (Fig. 3F). From these phase 271 diagrams, we can estimate mechanical properties (e.g. 272 pressure, lateral tension, or possible external loads) from 273 the geometric cell profiles (e.g. cell tilt, cell height and 274 AB-T1 locations/orientations). 275 Conclusions: We have proposed a model for the on-276 set of cellular tilt within a curved monolayer. We find 277 that the interplay between the lateral cell-cell tension 278 and the cellular tilt leads to a shift in the location at 279 which we expect the number of neighbor rearrangements 280 to be maximal. Our formalism provides predictions for 281 the location of AB-T1 transitions in several geometries 282 that are echoed by experimental observations in various 283

287

  lead to cell tilting, which cooperates with pressure and 288 tissue thickness to result in a rich phase diagram for the 289 tendency of AB-T1 transitions to occur. If the lateral 290 membranes are sufficiently stiff, then the tilt of lateral 291 membranes is suppressed and AB-T1 transitions occur 292 at regions with large curvature anisotropy, following the 293 model prediction in the hydrostatic limit.

  294

has received funding from France 2030, the 312 French Government program managed by the French 313 National Research Agency (ANR-16-CONV-0001) and 314 from the Excellence Initiative of Aix-Marseille Univer-315 sity -A*MIDEX and ANR-20-CE30-0023 grant. T.H. 316 and T.E.S. are funded by Mechanobiology Institute seed 317 grants. T.E.S. is also funded by start-up support from 318 the University of Warwick.

319