Emilien Segret
email: emilien.segret@inrae.fr@33633782979

Emilie Cardona

Sandrine Skiba-Cassy

Frédéric Cachelou

Julien Bobe
email: julien.bobe@inrae.fr

Keywords: ultrasound staging, non-invasive phenotyping, egg viability, VisEgg, ion and water transport

Highlights:

-Low water salinity impairs final oocyte maturation and egg quality in rainbow trout -Low water salinity induces delayed ovulation and impaired oocyte hydration -Low water salinity induces a dysregulation of several key ovulatory genes -Monitoring of final oocyte maturation can be performed using ultrasound staging

Water salinity is an important environmental factor known to have detrimental effects on salmonid reproduction, mostly when migrating female broodfish are held in sea water. In contrast, data obtained in freshwater are scarce and the impact of low water salinity, and more specifically of low water concentrations in sodium, chloride and potassium, during reproduction in freshwater is currently unknown. For this reason, and because ion and water fluxes are critical for the final steps of the female gamete formation, including oocyte hydration and ovulation, the aim of the present study was to investigate the impact of low salinity water on final oocyte maturation, ovulation and, ultimately, on egg quality, using rainbow trout as a physiological model and relevant aquaculture species.

Fish from the same commercial strain were raised either in a site characterized by low concentrations of Na+, K+, and Cl-ions in the water or in a closely located control site exhibiting higher concentration in these elements. Egg quality and duration of final oocyte maturation were investigated using innovative phenotyping tools such as automatic assessment of egg viability using the VisEgg system and non-invasive echograph-based monitoring of final oocyte maturation duration, respectively.

Oocyte hydration during final oocyte maturation and after ovulation was also investigated. Finally, molecular phenotyping was performed using real-time PCR-based monitoring of several key players of final oocyte maturation and ovulation associated with ion and water transport, inflammation, proteolytic activity, and coagulation. Oocyte hydration and gene expression data were analyzed in the light of the duration of final oocyte maturation.

Here we show that low water salinity (i.e., low water concentration in chloride, sodium and potassium) negatively influences final oocyte maturation, ovulation and, ultimately, egg quality. Low water salinity triggered delayed ovulation and lower oocyte viability. When investigating the impact of low water salinity on final oocyte maturation duration, individuals presenting the most severe phenotypes exhibited impaired oocyte hydration and abnormally reduced gene expression levels of several key players of the ovulatory process. While the under expression of water (i.e., aquaporins) and ion (i.e., solute carriers) transporters is consistent with impaired oocyte hydration, our observations also indicate that the overall ovulatory gene expression program is disrupted. Our results raise the question of the mechanisms underlying the negative influence of low salinity water on the dynamics of the preovulatory phase, on the control of the oocyte homeostasis, including hydration, and on the overall success of the maturation-ovulation process.

1 Fish egg quality is defined as the egg ability to be fertilized and to subsequently develop into a normal embryo [START_REF] Bobe | Egg and sperm quality in fish[END_REF]. Egg quality can be highly variable in fish under both natural and aquaculture conditions. In most species, a lack of knowledge of mechanisms influencing egg quality is a limiting factor for the mass production of fry (Migaud et al. 2013). Many external factors such as broodstock nutrition, temperature and photoperiod, to name just a few, are known to have a major impact on egg quality in various fish species [START_REF] Aegerter | Effects of post-ovulatory oocyte ageing and temperature on egg quality and on the occurrence of triploid fry in rainbow trout, Oncorhynchus mykiss[END_REF][START_REF] Bonnet | Effect of photoperiod manipulation on rainbow trout (Oncorhynchus mykiss) egg quality: A genomic study[END_REF][START_REF] Carrillo | The effect of modifications in photoperiod on spawning time, ovarian development and egg quality in the sea bass (Dicentrarchus labrax L.)[END_REF][START_REF] Izquierdo | Effect of broodstock nutrition on reproductive performance of fish[END_REF][START_REF] Pankhurst | Effect of holding temperature on ovulation, egg fertility, plasma levels of reproductive hormones and in vitro ovarian steroidogenesis in the rainbow trout Oncorhynchus mykiss[END_REF]. In contrast, other factors such as composition and physico-chemical parameters of the water have received far less attention [START_REF] Brooks | Egg quality in fish: What makes a good egg?[END_REF]Bobe and Labbe, 2010). Ionic composition of the water is however likely to influence the final steps of oogenesis (i.e., final oocyte maturation -FOM) given the importance of water exchange during this critical period [START_REF] Dolomatov | The regulation of osmotic and ionic balance in fish reproduction and in the early stages of ontogeny[END_REF]. Among mineral elements, chloride, sodium and potassium are of specific interest because (i) they are the most abundant electrolytes in the body of living organisms and (ii) they serve a vital function in controlling osmotic pressure and acid-base equilibrium [START_REF] Lall | The Minerals[END_REF].

It is commonly accepted that deficiencies in chloride, sodium and potassium are difficult to trigger because fish derive these mineral elements from surrounding water [START_REF] Lall | The Minerals[END_REF]. To our knowledge, the impact of a low ionic composition in these three key mineral elements on female reproduction was never investigated.

Several concomitant, yet distinct, events take place in the full-grown ovarian follicle during FOM. The follicle-enclosed oocyte progressively acquires the ability to resume meiosis whereas the follicle is preparing the release of oocyte from surrounding somatic layers at ovulation [START_REF] Grier | Germinal epithelium, folliculogenesis, and postovulatory follicles in ovaries of rainbow trout, Oncorhynchus mykiss (Walbaum, 1792)[END_REF][START_REF] Lubzens | Oogenesis in teleosts: How fish eggs are formed[END_REF]. In fish, as in other vertebrates, the mechanisms involved in the ovulatory process are described as inflammatory-like [START_REF] Espey | Current status of the hypothesis that mammalian ovulation is comparable to an inflammatory reaction[END_REF][START_REF] Thibault | Ovulation[END_REF]. In contrast to meiosis resumption and ovulation that are common to all vertebrates, FOM in fish is associated with a significant hydration of the oocyte. This phenomenon results in a dramatic increase of oocyte volume in marine species and is triggered by water entry due to the increase in free amino acid content resulting from yolk protein processing [START_REF] Finn | In vivo oocyte hydration in Atlantic halibut (Hippoglossus hippoglossus); proteolytic liberation of free amino acids, and ion transport, are driving forces for osmotic water influx[END_REF]. In the saltmarsh species, Fundulus heteroclitus, the influx in potassium and sodium is a major cause of the uptake of osmotically obligated water and subsequent volume increase experienced by maturing oocytes [START_REF] Wallace | Analytical and experimental studies on the relationship between Na+, K+, and water uptake during volume increases associated with Fundulus oocyte maturation in vitro[END_REF]. In fresh water species, a more limited, yet significant, increase in water content is observed during FOM that could contribute mechanically to the ovulatory process, ultimately leading to oocyte release from surrounding somatic follicular layers [START_REF] Craik | Biochemical changes occurring during final maturation of eggs of some marine and freshwater teleosts[END_REF][START_REF] Milla | Hydration of rainbow trout oocyte during -dihydroxy-4-pregnen-3-one and cortisol[END_REF].

Rainbow trout (Oncorhynchus mykiss) is a major aquaculture species that produces eggs every year starting at age 2 (Bromage et al., 1992). In this species oocyte maturation, ovulation [START_REF] Cerdà | Physiological and molecular basis of fish oocyte hydration. Fish Oocyte From Basic Stud[END_REF][START_REF] Grier | Germinal epithelium, folliculogenesis, and postovulatory follicles in ovaries of rainbow trout, Oncorhynchus mykiss (Walbaum, 1792)[END_REF][START_REF] Jalabert | The follicular sensitivity in vitro to maturation-inducing hormones in rainbow trout, Salmo gairdnerii: role of oestradiol[END_REF][START_REF] Lubzens | Oogenesis in teleosts: How fish eggs are formed[END_REF], and egg quality [START_REF] Bobe | Egg quality in fish: Present and future challenges[END_REF][START_REF] Bobe | Egg and sperm quality in fish[END_REF]Bromage and Jones, 1992;[START_REF] Brooks | Egg quality in fish: What makes a good egg?[END_REF] have been extensively studied, including at the molecular level [START_REF] Bobe | Targeted gene expression profiling in the rainbow trout (Oncorhynchus mykiss) ovary during maturational competence acquisition and oocyte maturation[END_REF][START_REF] Bobe | Identification of new participants in the rainbow trout (Oncorhynchus mykiss) oocyte maturation and ovulation processes using cDNA microarrays[END_REF][START_REF] Bobe | Ovarian function of the trout preovulatory ovary: New insights from recent gene expression studies[END_REF][START_REF] Yamashita | Molecular mechanisms of the initiation of oocyte maturation: general and species-specific aspects[END_REF]. As previously documented, exposition to stress factors (e.g., temperature) during FOM can alter the quality of unfertilized eggs released from the ovary at ovulation [START_REF] Aegerter | Effects of post-ovulatory oocyte ageing and temperature on egg quality and on the occurrence of triploid fry in rainbow trout, Oncorhynchus mykiss[END_REF][START_REF] Bobe | Egg and sperm quality in fish[END_REF][START_REF] Colson | Maternal temperature exposure impairs emotional and cognitive responses and triggers dysregulation of neurodevelopment genes in fish[END_REF].

For these reasons, and because water and ion fluxes are important for several key steps of oocyte development during FOM, the aim of the present study was to investigate the impact of water ionic composition, and more specifically low concentrations of sodium, chloride and potassium on FOM, ovulation and, ultimately, on egg quality, using rainbow trout as a physiological model and relevant aquaculture species. To achieve this objective, we used two closely located production sites in the French Pyrenees where the stream water exhibits dramatically different concentrations of chloride, sodium and potassium all year round, including during the reproductive period. The experimental protocol registered under the reference #201904021747602 was specifically approved by INRAE-LPGP Animal Care and Use Committee. For oocyte sampling, echograph-based monitoring and ovulation detection, fish were anaesthetized in tricaine metanesulfonate (MS222) at a concentration of 130 mg/liter. For gonad sampling, fish were euthanized in MS222 at a concentration of 300mg/liter, until 10 minutes after cessation of opercular movement.

Experimental sites and water characterization

Two experimental sites located in Rébénacq [43.155948,-0.405067] and Sarrance [43.049999,-0.6014075] (French Pyrenees), and presenting similar water temperature [9°-10°C] and oxygen conditions were used in the present study. The two sites are close (<20 km) but located in different valleys (i.e., hydrogeologic systems) differing mostly by ionic abundance in Sodium (Na+), Chloride (Cl-) and Potassium (K+) in the water (Fig 1 .A , Supplemental datafiles 1 and 2). The site presenting n was significantly higher, will be For both sites, water composition analyses were performed monthly throughout the year, from the water flowing into the experimental raceways using a dedicated water sampling flask to avoid any contamination. Samples were analyzed by a specialized analytical laboratory (Ultra Traces Analyses Aquitaine, Pau, France) that provided the sampling flasks. Ionic abundance analysis was performed using inductively coupled plasma atomic emission spectroscopy (ICP-AES) technique for Na+ and K+ and by ionic chromatography for Cl-. Data obtained were not normally-distributed (Shapiro-Wilk test, p > 0.005). Ionic concentration in the 2 sites was compared using non parametric U-tests. A comprehensive water analysis of both sites is available in supplemental data files 1 and 2.

Fish rearing

Three years old female rainbow trout (Oncorhynchus mykiss) from the same commercial strain (Viviers SA Company strain, France) were used in this study. After rearing in a nursery site, fish were transferred to either Rébénacq (control site) or Sarrance (LSW site) locations at 6 months post-fertilization. On both sites, fish were reared under natural photoperiod in 125m 3 raceways at a 60kg.m -3 density, with a water renewal of 75% per hour until the beginning of the experiment. Stream water was used in an open circulation system without any recirculation. Similar feeding protocols, including feeding rate and diet composition, were applied for both sites throughout their lifecycle. Fish were fed at an identical daily rate (0.6% of the live weight) with the same commercial diet (46% crude protein, 19% crude fat, 0.9% phosphorus, 11.5% ash, 19 MJ/kg gross energy, Omega-3 4.8%, Omega-6 1.25%) (Le Gouessant, Lamballe, France). Following a first reproduction at age 2, fish were held under natural photoperiod during their second reproductive cycle. Spawning naturally occurred during November-December.

Experimental design

Approximately two weeks before expected ovulation (estimated based on previous ovulation dates), fish were transferred into a 8 m 3 raceway to facilitate manipulations. Experimental raceways on both sites were of similar dimensions (8 m in length, 1 m in width and 1 m in depth) with a water renewal of 200% per hour originating directly from the stream water. Ovulation was monitored three time a week. Feeding was stopped when first ovulations were detected. On the control site, a total of 32 fish were used including 18 for egg quality assessment using the VisEgg system [START_REF] Cardona | VisEgg: a robust phenotyping tool to assess rainbow trout egg features and viability[END_REF]) (4 of them being also used for the oocyte hydration study and 11 of them being also used for blood sampling) and 14 for the molecular analysis. On the LSW site, a total of 46 fish were used including 30 for egg quality assessment using the VisEgg system (4 of them being also used for the oocyte hydration study and 9 of them being also used for blood sampling) and 16 for the molecular analysis.

Egg quality assessment using VisEgg

In both groups, specific fish (control, n=18; LSW, n=30) were individually monitored until ovulation.

Ovulations were checked every 2-3 days, under anesthesia, by gentle pressure on the abdomen.

Oocyte viability evaluation was systematically performed using the VisEgg protocol when ovulation occurred. This recently developed tool allows a robust assessment of different egg features including egg viability [START_REF] Cardona | VisEgg: a robust phenotyping tool to assess rainbow trout egg features and viability[END_REF]. The process uses a standardized image-based batch analysis method of 24h-hydrated unfertilized eggs and an image processing algorithm. Outputs are the presence and percentage of whitening (i.e., non-viable) eggs within the batch. These features can be used to reliably assess egg integrity and viability. Analyze of presence of white eggs depending on rearing sites or the maturation duration yielded contingency tables analyzed using Khi-2 tests. Data obtained from VisEgg were not normally distributed (Shapiro-Wilk test, p < 0.001). Comparison of two groups were performed using non parametric U-tests.

Oocyte development staging during FOM

A non-invasive to the female ultrasound-based staging was used to estimate progress into FOM. This approach, previously described for assessing reproductive stages in wild salmons [START_REF] Nevoux | Field assessment of precocious maturation in salmon parr using ultrasound imaging[END_REF], was used here for the first time to assess progression into FOM in rainbow trout. An ultrasound scanner M-Turbo (Sonosite) with a 5-10MHz linear transducer (multiple frequency) was used. Anesthetized fish were placed in a plastic tray for a short duration (<30 sec) and sagittal images of the abdominal cavity were obtained by aligning the transducer with the lateral line of the fish. Late vitellogenic oocytes appeared white on the screen, while a dark area progressively appeared at the center of the oocyte as the oocytes progressed into FOM (Fig 3 .A).

In order to validate echograph-based staging, direct brightfield observation was performed. In rainbow trout, late-vitellogenic (LV) oocytes observed with the naked eye appear opaque and no noticeable changes in yolk structure or any sign of meiosis resumption can be observed. Post-vitellogenesis (PV) oocytes exhibit a central yolk clarification as meiosis progressively resumes. Germinal vesicle is still visible and lipid droplets remain relatively small. During oocyte maturation (Mat), germinal vesicle breakdown (GVBD) progressively occurs and oocytes become totally clear with well-defined lipid droplets [START_REF] Grier | Germinal epithelium, folliculogenesis, and postovulatory follicles in ovaries of rainbow trout, Oncorhynchus mykiss (Walbaum, 1792)[END_REF][START_REF] Jalabert | The follicular sensitivity in vitro to maturation-inducing hormones in rainbow trout, Salmo gairdnerii: role of oestradiol[END_REF][START_REF] Milla | Hydration of rainbow trout oocyte during -dihydroxy-4-pregnen-3-one and cortisol[END_REF][START_REF] Patiño | Ovarian follicle growth, maturation, and ovulation in teleost fish[END_REF][START_REF] Hurk | Cyclic changes in the ovary of the rainbow trout, Salmo gairdneri, with special reference to sites of steroidogenesis[END_REF]Wallace and Selman, 1990). The above description of phenotypic changes occurring in the rainbow trout oocyte during FOM was used to validate the echograph-based staging.

In 100% of the cases (n= 57 fish) the direct observation confirmed the echograph-based staging of the oocyte during FOM. Echograph-based oocyte staging was preferred due to its non-invasiveness nature and was thus subsequently used to measure the duration of the oocyte maturation phase (i.e., between fully transparent oocyte (Mat) stage and detected ovulation) in the present study. The duration of the maturation phase was temperature-normalized using the actual temperature recorded on both sites during experiments and expressed in hours at 10°C in both experimental groups.

Comparisons of the two groups were performed using non-parametric U-tests.

Plasma sampling and analysis

Blood was sampled during oocyte maturation on both sites (control, n=9; LSW, n=11). Blood samples were centrifuged (1000 g, 10 min, 4 analysis. Plasma samples were analyzed by a specialized analytical laboratory (Ultra Traces Analyses Aquitaine, Pau, France). Ionic abundance analysis was performed using inductively coupled plasma atomic emission spectroscopy (ICP-AES) technique for Na+ and K+ and by ionic chromatography for Cl-. Comparison of two groups were performed using non parametric U-tests.

Hydration measurement

For analytical purposes, individual fish originating from the LSW site were separated in two groups, based on the duration of the maturation phase [Mat<64H at 10°C (n=10), Mat 64H at 10°C (n=21)].

Oocyte hydration was measured (n=4 individuals of each group) and monitored from late vitellogenesis stage until ovulation. The method used has previously been described [START_REF] Milla | Hydration of rainbow trout oocyte during -dihydroxy-4-pregnen-3-one and cortisol[END_REF]. Briefly, gentle manual striping was performed under anesthesia to collect oocytes, from late vitellogenesis stage until ovulation. As previously described [START_REF] Finet | Salmo gairdneri[END_REF], non-ovulated oocytes were expelled from the follicle layers using two fine tip tweezers. Oocyte development staging was performed under a binocular microscope. A total of 60 to 80 oocytes per female were obtained and rapidly weighted to obtain wet mass (WM). After heating these oocytes 24 h at 105°C in a drying oven (Memmert, Buechenbach, Germany), dry mass (DM) was measured. The water content (WC) was measured from subtracting DM from WM. For each individual, WC was measured (n=4 individuals of each group) and monitored from late vitellogenesis stage until ovulation. Analysis of the increase in water content (WC) in the two experimental groups (Mat <64H ; Mat 64H) was performed using non parametric U-test.

2.9 Molecular analysis 2.9.1 Tissue collection Ovaries from control (n=4 late-vitellogenesis stage and n=10 oocyte maturation stage) and LSW (n=5 late-vitellogenesis stage and n=11 oocyte maturation stage) groups were sampled for RNA extraction.

After euthanasia, ovaries were dissected out of the abdominal cavity. Oocyte staging was performed using ultrasound images prior to euthanasia followed by macroscopic validation of oocyte stage under microscope after dissection. Ovarian follicles were deyolked as previously described [START_REF] Bobe | A tumor necrosis factor decoy receptor homologue is up-regulated in the brook trout (Salvelinus fontinalis) ovary at the completion of ovulation[END_REF] by pressing the entire tissue between two stainless steel screens while continuously applying ice-cold Cortland medium with a squirt bottle, then immediately frozen in liquid Nitrogen and stored at -80°C until RNA extraction.

Total RNA extraction and reverse transcription

Ovarian tissue was homogenized in Trizol reagent (Molecular Research Center, Cincinnati, USA) at a ratio of 1ml per 100mg of tissue. Tissue was grinded using bead beating technology (Precellys Evolution, Bertin Technologies, Saint-Quentin en Yvelines, France). Total RNA extraction was performed following the Briefly, 2 µg of total RNA were reverse transcribed using Maxima First Strand cDNA Synthesis Kit (Maxima Reverse Transcriptase enzyme, derived from Moloney murine Leukemia virus enzyme). Reverse transcription was performed by 10 minutes of incubation at 25°C followed by a 15 minutes step at 60°C and a 5-minute step at 75°C.

Control reactions were run without Maxima reverse transcriptase and used as negative controls in the real-time polymerase chain reaction study.

Real-time PCR analysis

Real-time PCR was performed using a LightCycler480 (Roche, Switzerland). Reverse transcription products were diluted to 1/30. Technical replicates of each biological sample were run. Runs were performed in 5µL reactions containing 2µL of RT product, 250 nM of each primer and 1X PowerUp SYBR Green Master Mix (Applied Biosystems). The program used was: 50°C for 2 minutes, then 95°C

for 2 minutes for initial denaturation, 40 cycles of 95°C for 3 seconds and 60°C for 30 seconds. After amplification, a fusion curve was obtained by 15-second step followed by a 0.5°C increase, repeated 70 times, and starting at 60°C

Gene expression analysis

Several concomitant events occur in the ovary during FOM that involve distinct biological processes including ovulation, an inflammatory-like process characterized by major tissue rearrangements, blood coagulation, detachment of somatic follicular cells from the oocyte, and oocyte hydration which involves specific water and ion transports. Genes used for the molecular analysis were thus selected for their putative role in inflammation, proteolytic activity, or water and ion transport and were previously reported to be markedly differentially up-regulated during FOM in rainbow trout (Bobe et al., 2006). Primers used in this prior study [START_REF] Bobe | Identification of new participants in the rainbow trout (Oncorhynchus mykiss) oocyte maturation and ovulation processes using cDNA microarrays[END_REF] were used here to specifically monitor the expression of the following genes. including poor semen production for males and presence of atretic follicles in females [START_REF] Valdebenito | Gamete quality in fish: Evaluation parameters and determining factors[END_REF]. Similarly, other studies reported that rearing of Atlantic salmon in sea water induces delays and variability in the timing of ovulation [START_REF] Haffray | Influence du maintien en mer ou de la période de transfert en eau douce des reproducteurs de saumon atlantique Salmo salar sur la maturation sexuelle et la qualité des gamètes[END_REF] and poor gamete quality [START_REF] Maisse | Influence du maintien en mer des mâles de saumon atlantique (Salmo salar), pendant la période de reproduction, sur la qualité du sperme[END_REF].

In Arctic charr (Salvelinus alpinus), gamete quality is improved for broodstock transferred in saltwater during their gametogenesis [START_REF] Atse | Effects of temperature and salinity on the reproductive success of Arctic charr, Salvelinus alpinus (L.): Egg composition, milt characteristics and fry survival[END_REF]. Together, these studies indicate that water salinity is a key factor for the reproduction success of salmonids that all reproduce in freshwater even though a part of their life cycle occurs in sea water in some species. Most studies have however focused on the impact of high water salinity during final oocyte maturation. In contrast, data on low salinity water remain scarce. In addition, most studies have focused on migrating species that return to freshwater to reproduce. In the present study, we confirm that water salinity is a key factor of reproductive success and show that, similarly to high salinity, low salinity can also have a detrimental effect on reproductive success.

Low water salinity triggers delayed ovulation

Together, our results indicate that low water concentration in chloride, sodium and potassium triggers a significant decrease in egg viability at ovulation. In order to shed light on underlying mechanisms, we aimed at investigating the preovulatory period that is known to be critical for egg quality. Given the favorable temperature range (i.e., 9-10°C) and similar feeding regime throughout the reproductive cycle, we reasoned that LSW could perturbate FOM and investigated the duration of the maturation Several studies described the temporal aspects of final oocyte maturation in rainbow trout. The in vivo duration of FOM, including progressive central clearing of the oocyte, ranged from 48 to 72 hours at 13°C [START_REF] Bry | Temporal aspects of macroscopic changes in rainbow trout (Salmo gairdneri) oocytes before ovulation and of ova fertility during the post-ovulation period; effect of treatment with -hydroxy--dihydroprogesterone[END_REF]. The detachment of the mature oocyte from surrounding somatic follicular layers was reported to occur between 72 and 96 hours after the first observable signs of oocyte maturation (i.e., central clarification) at 10°C [START_REF] Jalabert | Production of fertilizable oocytes from follicles of rainbow trout (Salmo gairdnerii) following in vitro maturation and ovulation[END_REF]Jalabert and Szöllösi, 1975). Other authors showed that ovulation occurs generally 48-96 hours at 14°C after the peak concentration of maturation-inducing steroid [START_REF] Fostier | Evolution des niveaux plasmatiques de la gonadotropine glycoprotéique et de la 17 alpha hydroxy-20 beta dihydroprogestérone au cours de la maturation et de l'ovulation chez la Truite arc-en-ciel. Salmo gairdnerii[END_REF]. Another study shown a duration of 48-72 hours at 12°C between maturation and detected ovulation for rainbow trout [START_REF] Breton | GTH I and GTH II secretion profiles during the reproductive cycle in female rainbow trout: Relationship with pituitary responsiveness to GnRHa stimulation[END_REF].

The duration of oocyte maturation, from full oocyte cytoplasm clearing to ovulation, reported here for the control group, is therefore fully consistent with the duration of FOM reported in these prior studies that usually included the post-vitellogenic phase (i.e., germinal vesicle migration towards the periphery of the oocyte). Our observations however suggest that the duration of FOM in the LSW group is higher or in the upper range of what is classically observed in rainbow trout at 10°C. In addition, the LSW group exhibits a strong interindividual variability with individuals that ovulated 97 hours (2 individuals) and 130 hours (1 individual) after detected oocyte maturation (Fig 3 .C). These individuals presented the usual phenotypic traits of maturation (i.e., color pattern and body shape), but ovulation occurred drastically later. Heterogeneity in FOM duration and, in some cases, absence of ovulation has often been reported in trout and other fish. This heterogeneity has however systematically always been associated with premature induction of FOM and ovulation, typically when FOM was triggered in maturationally incompetent (i.e., vitellogenic) follicles in vivo or in vitro. This is for instance the case for hormonally injected fish [START_REF] Bry | Temporal aspects of macroscopic changes in rainbow trout (Salmo gairdneri) oocytes before ovulation and of ova fertility during the post-ovulation period; effect of treatment with -hydroxy--dihydroprogesterone[END_REF][START_REF] Goetz | The effects of steroids on final maturation and ovulation of oocytes from brook trout (Salvelinus fontinalis) and yellow perch (Perca flavescens)[END_REF][START_REF] Jalabert | Production of fertilizable oocytes from follicles of rainbow trout (Salmo gairdnerii) following in vitro maturation and ovulation[END_REF]. Using non-invasive ultrasound-based FOM monitoring, we were able to demonstrate here that oocyte maturation (i.e., oocyte clearing) occurred but that ovulation was delayed. Previous studies have demonstrated that oocyte maturation and ovulation, while being interconnected, were two independent processes. In rainbow trout it is possible to trigger ovulation without oocyte maturation [START_REF] Jalabert | Maturation et ovulation in vitro des ovocytes de la truite arc-en-ciel Salmo gairdneri[END_REF]. Together, our results show that low salinity (i.e., low water concentration in chloride, sodium and potassium)

during FOM triggers delayed ovulation, sometimes by several days. Our observations also indicate that, while oocyte maturation occurs normally, the ovulatory process appears impaired or inhibited, ultimately leading to abnormally long FOM.

Reduced egg viability in low salinity water is linked with delayed ovulation

Together our results show that low water concentration in chloride, sodium and potassium triggered reduced egg viability and delayed ovulation. We also noticed a high inter-individual variability in both egg viability and maturation phase duration in the LSW group (Figs 2.B & 3.C). We reasoned that delayed ovulation could be at the origin of lower egg viability in the LSW group and investigated further any link between egg viability and maturation phase duration. For data analysis, LSW fish were separated in two groups, based on the duration of the maturation phase. We observed a dramatic and These results strongly support our hypothesis of a significant loss of egg quality after delayed ovulation.

A possible explanation for the higher rate of non-viable eggs would be the ageing of the oocytes. This well-known phenomenon has been extensively described [START_REF] Aegerter | Effects of post-ovulatory oocyte ageing and temperature on egg quality and on the occurrence of triploid fry in rainbow trout, Oncorhynchus mykiss[END_REF][START_REF] Billard | Stress, environment and reproduction in teleost fish. Stress and fish[END_REF][START_REF] Craik | Biochemical changes occurring during final maturation of eggs of some marine and freshwater teleosts[END_REF]Escaffre, 1976;[START_REF] Lahnsteiner | Morphological, physiological and biochemical parameters characterizing the over-ripening of rainbow trout eggs[END_REF][START_REF] Samarin | Fish oocyte ageing and its effect on egg quality[END_REF][START_REF] Samarin | Influence of the time to egg stripping on eyeing and hatching rates in rainbow trout Oncorhynchus mykiss under cold temperatures[END_REF][START_REF] Springate | The timing of ovulation and stripping and their effects on the rates of fertilization and survival to eying, hatch and swim-up in the rainbow trout (Salmo gairdneri R.)[END_REF] and is known to trigger major egg quality defects. All existing studies on oocyte ageing prior to fertilization were however conducted after post-ovulatory ageing. To our knowledge, oocyte ageing prior to ovulation (i.e., within the ovarian follicle) is reported here for the first time in a fish species. In salmonids, in contrast to other fish species, post-ovulatory ageing does not induce a rapid decrease of egg quality. In rainbow trout, ovulated eggs can be held in the body cavity for 5 days at 12°C without any significant decrease of their developmental success (Aegerter et al., 2005). It is therefore unlikely that the modest intra follicular post-meiotic ageing observed here in the LSW group, in which ovulation is delayed, on average, by 17 hours, is sufficient to trigger alone the drop of egg quality observed in LSW. The osmotic properties of the rainbow trout egg have been thoroughly investigated in rainbow trout [START_REF] Gray | The osmotic properties of the eggs of the trout (Salmo Fario)[END_REF]) and another explanation for the increase in whitening egg occurrence could be a loss of the oocyte homeostasis in LSW ultimately leading to non-viable eggs. In brook trout, a diminution of ionic concentration (Cl-) in surrounding medium has as direct negative effect of membrane resistance [START_REF] Marshall | Electrophysiology of oocytes during meiotic maturation and after ovulation in brook trout (Salvelinus fontinalis)[END_REF]. Disruption of the oocyte homeostasis could therefore originate from a fragile oocyte vitelline membrane. In addition, the loss in oocyte homeostasis could originate from difficulties (i.e., high energetic cost) in maintaining the osmotic difference across oocyte membrane in a hypotonic external medium.

Together, our results show that the reduced egg viability observed in LSW is linked to delayed ovulation. This reduced egg viability can be due, at least in part, to intra-follicular oocyte ageing. We cannot, however, rule out the possibility that LSW also affects egg viability through other mechanisms that could disrupt the maintenance of oocyte homeostasis and ultimately decrease egg viability.

Further investigations are needed to unravel the mechanisms triggering decreased egg viability when FOM is occurring in LSW.

Low water concentration in chloride, sodium and potassium results in reduced oocyte hydration during FOM

The release of the oocyte from surrounding follicular layers at the time of ovulation results from the ovulatory process that has originally been described in mammals as an inflammatory-like reaction. In fish, in which the oocyte is directly in contact with surrounding follicular layers, a prerequisite is the separation of the oocyte from surrounding granulosa layers [START_REF] Jalabert | In vitro oocyte maturation and ovulation in rainbow trout (Salmo gairdnerii), Northern Pike (Esox lucius), and Goldfish (Carassius auratus)[END_REF][START_REF] Lubzens | Oogenesis in teleosts: How fish eggs are formed[END_REF]. This phenomenon has been thoroughly described in rainbow trout in which suppression of tight junctions between granulosa cells and the oocyte results in the apparition of a thin periplasmic space separating the oocyte from surrounding somatic layers [START_REF] Lubzens | Oogenesis in teleosts: How fish eggs are formed[END_REF]. Many observations, including direct brightfield in vitro follicle observation, have shown that preovulatory rainbow trout oocyte exhibits a significant mechanical pression onto surrounding somatic layers [START_REF] Lubzens | Oogenesis in teleosts: How fish eggs are formed[END_REF]. Somatic follicular layers appear to be stretched around the oocyte at that stage. Because of the angiotensinlike nature of the ovulatory process and the contractile actions of theca layers during ovulation [START_REF] Hajnik | Characterization of a ribonucleic acid transcript from the brook trout (Salvelinus fontinalis) ovary with structural similarities to mammalian adipsin/complement factor D and tissue kallikrein, and the effects of kallikrein-like serine proteases on follicle contraction[END_REF][START_REF] Hsu | Angiotensins stimulate in vitro ovulation and contraction of brook trout (Salvelinus fontinalis) follicles[END_REF], postovulatory follicles appear shrinked after ovulation. It was hypothesized that the modest (i.e., 20%), yet significant, increase in water content observed during rainbow trout FOM would contribute to the increasing pressure applied by the oocyte onto surrounding somatic layers and would ultimately mechanically facilitate the exit of the oocyte through a hole in somatic layers [START_REF] Milla | Hydration of rainbow trout oocyte during -dihydroxy-4-pregnen-3-one and cortisol[END_REF]. Mechanic pressure applied by the oocyte swelling is one cause of follicle rupture and the completeness of hydration facilitates the complete liberation of unfertilized eggs into the abdominal cavity. We reasoned that a lower hydration would decrease the efficiency of ovulation and investigated hydration in the two above defined sub-groups exhibiting respectively <64h and 64h maturation phase duration. Oocyte water content (WC) was monitored throughout FOM in both groups. In the <64h group, WC progressively increased throughout FOM in comparison to the late vitellogenic (LV) stage, with a 15% increase during oocyte maturation and a 27% increase at ovulation to reach 36% during the post-ovulatory period (Fig 5,light blue). In contrast, a much more limited increase in water content was observed in the 64h group. In comparison to the LV stage, a 2% increase in water content was observed during oocyte maturation while a 13% increase was monitored at ovulation that did not further increase during the post-ovulatory period.

Observations in the <64h group are fully consistent with existing data in rainbow trout that were obtained using the same methodology [START_REF] Milla | Hydration of rainbow trout oocyte during -dihydroxy-4-pregnen-3-one and cortisol[END_REF]. In contrast the hydration in the 64h appears to be drastically reduced. At all stages, the water increase was significantly lower in the 64h group in comparison to the <64h group. Together these results indicate that oocyte hydration is impaired during FOM occurring in low salinity water.

Oocyte hydration has been described for several species [START_REF] Finn | In vivo oocyte hydration in Atlantic halibut (Hippoglossus hippoglossus); proteolytic liberation of free amino acids, and ion transport, are driving forces for osmotic water influx[END_REF]LaFleur and Thomas, 1991;[START_REF] Milla | Hydration of rainbow trout oocyte during -dihydroxy-4-pregnen-3-one and cortisol[END_REF][START_REF] Watanabe | Water and ion balance in hydrating oocytes of the grey mullet[END_REF]. Liberation of free amino acids and an uptake of inorganic ions occur during maturation. These events create a change in osmotic equilibrium, resulting in the passive transport of water molecule into the oocyte. This transport mechanism uses several proteins activity such as solute carriers, Na+/K+ ATPases, Aquaporins, and demonstration was made that protein activity and hydration were closely dependent on K+ concentration in the medium [START_REF] Ecker | Influence of exogenous ions on the events of maturation in Rana pipiens oocytes[END_REF]LaFleur and Thomas, 1991). K+ and Na+ accumulation in the oocyte during maturation plays an important role in the oocyte hydration process. For Fundulus heteroclitus, oocytes Na+ and K+ uptakes by gap junctions [START_REF] Lubzens | Oogenesis in teleosts: How fish eggs are formed[END_REF] are main responsible of osmolality increase [START_REF] Greeley | Changes in size, hydration and low molecular weight osmotic effectors during meiotic maturation of Fundulus oocytes in vivo[END_REF][START_REF] Wallace | Analytical and experimental studies on the relationship between Na+, K+, and water uptake during volume increases associated with Fundulus oocyte maturation in vitro[END_REF]. For Atlantic halibut Hippoglossus hippoglossus, Cl-uptakes also contributes to the increase of oocyte osmolality [START_REF] Finn | In vivo oocyte hydration in Atlantic halibut (Hippoglossus hippoglossus); proteolytic liberation of free amino acids, and ion transport, are driving forces for osmotic water influx[END_REF]. Together these studies demonstrate the importance of Na+, K+ and Cl-ions for the oocyte hydration process during FOM. In contrast, the impact of low water salinity was never investigated, mostly because these experiments were often conducted in marine species. We hypothesize that FOM in low salinity water is associated in reduced ions uptake during maturation ultimately leading to incomplete oocyte hydration and thereby causing a large variability in the timing of ovulation. Further investigations are needed to elucidate underlying mechanisms.

Low water concentration in chloride, sodium and potassium induces dysregulated expression of ovulation genes

The ovulatory process involves the activation of a specific gene expression program that has been extensively documented in mammals [START_REF] Richards | Ovulation: New dimensions and new regulators of the inflammatory-like response[END_REF] and in fish [START_REF] Bobe | Ovarian function of the trout preovulatory ovary: New insights from recent gene expression studies[END_REF][START_REF] Bobe | Identification of new participants in the rainbow trout (Oncorhynchus mykiss) oocyte maturation and ovulation processes using cDNA microarrays[END_REF][START_REF] Cerdà | Physiological and molecular basis of fish oocyte hydration. Fish Oocyte From Basic Stud[END_REF][START_REF] Fabra | Marine fish egg hydration is aquaporin-mediated[END_REF][START_REF] Hajnik | Characterization of a ribonucleic acid transcript from the brook trout (Salvelinus fontinalis) ovary with structural similarities to mammalian adipsin/complement factor D and tissue kallikrein, and the effects of kallikrein-like serine proteases on follicle contraction[END_REF][START_REF] Knoll-Gellida | Molecular phenotype of zebrafish ovarian follicle by serial analysis of gene expression and proteomic profiling, and comparison with the transcriptomes of other animals[END_REF]. The inflammatory-like nature of the ovulatory process has been well characterized and the expression of many inflammationrelated genes have been reported in various fish species [START_REF] Hagiwara | Inhibition of medaka ovulation by gap junction blockers due to its disrupting effect on the transcriptional process of LH-induced Mmp15 expression[END_REF][START_REF] Ohnishi | Functions for proteinases in the ovulatory process[END_REF][START_REF] Takahashi | Follicle rupture during ovulation with an emphasis on recent progress in fish models[END_REF] including in rainbow trout [START_REF] Bobe | Identification of new participants in the rainbow trout (Oncorhynchus mykiss) oocyte maturation and ovulation processes using cDNA microarrays[END_REF]. A subset of these previously identified genes related to inflammation, proteolytic activity, or ion and water transport was monitored to characterize the ovulatory process in low salinity water.

Solute transporters and hydration-related genes are under-expressed in low salinity water

Aquaporin are molecular channels responsible of passive water transport across membranes. The role of aquaporins during fish oocyte hydration has been described by several studies. The up-regulation of aquaporin 4 (aqp4) was reported during rainbow trout follicular maturation [START_REF] Bobe | Identification of new participants in the rainbow trout (Oncorhynchus mykiss) oocyte maturation and ovulation processes using cDNA microarrays[END_REF], suggesting that oocyte hydration of fresh water fish is aquaporin mediated, in consistency with the demonstrated role of aquaporin1-like in oocyte hydration of saltwater fish [START_REF] Cerdà | Molecular pathways during marine fish egg hydration: The role of aquaporins[END_REF][START_REF] Fabra | Marine fish egg hydration is aquaporin-mediated[END_REF].

The solute carrier family 26 member 4 gene (slc26a4), previously known as pendrin, codes for a transmembrane anion exchanger and is involved in osmoregulation mechanisms [START_REF] Nakada | Roles of Slc13a1 and Slc26a1 sulfate transporters of eel kidney in sulfate homeostasis and osmoregulation in freshwater[END_REF].

Identified as dramatically up-regulated in the rainbow trout ovary during maturation, slc26a4 is believed to play an important role in the osmotic regulation, causing water influx into oocyte during meiotic maturation. In the present study, a marked under expression of slc26a4 and aqp4 is observed in the ovary when FOM occurs in LSW (Fig 6). Given the role of both genes in ion and water transport, this under expression is fully consistent with the impaired oocyte hydration observed in LSW, in comparison to the control group.

Vasotocin-neurophysin VT 1 (vt1 gene) has two key roles in FOM-ovulation in catfish [START_REF] Joy | General and Comparative Endocrinology Vasotocin A new player in the control of oocyte maturation and ovulation in fish[END_REF]. It is an important link in the gonadotropin cascade controlling FOM and ovulation. By a common action with maturation inductions hormones, VT 1 is a regulator of germinal vesicle breakdown and meiosis resumption. VT 1 also plays a key role in follicular hydration by regulating the aquaporin action [START_REF] Balment | Arginine vasotocin a key hormone in fish physiology and behavio[END_REF]. Here we show a marked under expression of vt1 observed in the LSW group in comparison to the control group (Fig 6). This profile is fully consistent with both the observed impaired ovulatory process and the under expression of aquaporin 4. In addition, the position of vt1 in the gonadotropin-dependent regulatory cascade controlling oocyte maturation and ovulation suggests that it could be a relay in mediating the inhibitory effect of LSW.

Inflammation-related genes are under-expressed in low salinity water

Ovulation is accompanied by a broad-spectrum proteolysis within follicular layers that involves a large number of proteases. Members of A Disintegrin and Metalloproteinase gene family are dramatically up-regulated during trout FOM-ovulation phases [START_REF] Berndtson | Protease activity in brook trout (Salvelinus fontalis) follicle walls demonstrated by substrate-polyacrylamide gel electrophoresis[END_REF][START_REF] Berndtson | Metallo-protease activity increases prior to ovulation in brook trout (Salvelinus fontinalis) and yellow perch (Perca flavescens) follicle walls[END_REF][START_REF] Bobe | Identification of new participants in the rainbow trout (Oncorhynchus mykiss) oocyte maturation and ovulation processes using cDNA microarrays[END_REF].

Adam22 plays an important role in cell-to-cell and cell-to-matrix interaction regulation (White, 2003).

By regulating or processing the deposit of the collagen fibrils, ADAMs proteases cause the extra-cellular matrix proteolysis [START_REF] Ohnishi | Functions for proteinases in the ovulatory process[END_REF].

CXC chemokines stimulate recruitment of leukocytes and play a role of pro-inflammatory mediators.

Cxcl14 implication in immune related functions has been described in fish [START_REF] Baoprasertkul | Constitutive expression of three novel catfish CXC chemokines: Homeostatic chemokines in teleost fish[END_REF].

The over-expression of cxcl14 during FOM-ovulation phases is fully in line with the inflammatory-like nature of the ovulatory process.

Coagulation factor V (f5 gene) is involved in synthesis of blood coagulant factors and has been characterized in mammals and fish [START_REF] Hanumanthaiah | Comprehensive analysis of blood coagulation pathways in teleostei: Evolution of coagulation factor genes and identification of Zebrafish factor VIIi[END_REF][START_REF] Rothberger | Generation of coagulation factor V activity by cultured rabbit alveolar macrophages[END_REF]. It was speculated that synthesis of coagulation factor V prevents bleeding caused by rupture of follicle layers during ovulation.

Here we show that cxcl14, adam22, and f5 are dramatically under expressed during oocyte maturation in the LSW group in comparison to the control group (Fig 6). These expression profiles are consistent with the delayed ovulation observed in LSW and suggest that the inflammatory nature of ovulation is affected. This is especially true for cxcl14 for which a 5-fold decrease in gene expression is observed in the LSW group in comparison to the control group. Together, these results show that several important components of the ovulatory process are toned down when FOM occurs in low salinity water. It is noteworthy that most, if not all, aspects of the ovulatory process including inflammation, coagulation, swelling, water and ion transport as well and proteolysis are under activated. Even if previous studies have shown gene expression changes when fish adapt to water salinity change (e.g. migration, precipitation) for salmonids [START_REF] Morro | Plasma proteome profiling of freshwater and seawater life stages of rainbow trout (Oncorhynchus mykiss)[END_REF][START_REF] Norman | Transcriptomics of salinity tolerance capacity in Arctic charr (Salvelinus alpinus): a comparison of gene expression profiles between divergent QTL genotypes[END_REF] and other species [START_REF] Fiol | Regulation of osmotic stress transcription factor 1 (Ostf1) in tilapia (Oreochromis mossambicus) gill epithelium during salinity stress[END_REF][START_REF] Pujante | Effect of different salinities on gene expression and activity of digestive enzymes in the thick-lipped grey mullet (Chelon labrosus)[END_REF], evidence of such an impact of low water salinity on the overall ovulatory process were, to our knowledge, previously unreported. This raises the question of how low water salinity can inhibit the overall ovulatory process and not only osmotic exchanges.

3.6 Lower blood plasma levels of sodium, chloride and potassium are observed in the

LSW site

In order to assess the impact of water concentration in chloride, sodium and potassium on fish homeostasis, we measured blood plasma concentration in these three elements in both control and LSW sites. We observed a significantly lower plasmatic concentration of these three mineral elements in fish of the LSW site in comparison to the control site (Fig 1 .B). This indicates that the water composition can lead to significant differences in plasma concentration in these elements. This also further supports our hypothesis that reproduction in fresh water exhibiting low sodium, chloride and potassium concentration perturbates the fish oocyte osmolarity and the hydration process during final oocyte maturation and ovulation.

Role of other water components

While marked and highly significant differences in water concentration of chloride, sodium and potassium are found between the two experimental sites, other differences exist in water composition that could also contribute, at least in part, to the impaired FOM and ovulation and reduced egg quality phenotype observed in the LSW site. A lower concentration in Iron (Fe) and Manganese (Mn) is observed in the LSW site in comparison to the control site. For Iron, the difference is however not significant when only the reproductive period is considered (Supplemental datafile 2). Manganese (Mn) is known to inhibit ovulation in rainbow trout (Jalabert and Szollosi, 1975) and inhibit meiotic maturation in terrestrial animals [START_REF] Bilodeau-Goeseels | Manganese inhibits spontaneous nuclear maturation in bovine cumulusenclosed oocytes[END_REF]. It is therefore difficult to link low Mn concentration with impaired FOM and ovulation. In addition, it is commonly accepted that the diet is considered to be the major source of Manganese and Iron (Lall, 2002). It is therefore extremely unlikely that the lower water concentration in Mn and Fe found in the LSW site is responsible for the impaired reproductive phenotype observed in the LSW site. Similarly, a lower nitrate concentration was observed in the LSW site. While we cannot rule out that this could contribute to the abnormal reproductive phenotype observed in the LSW site, this seems however, to our knowledge and in the light of previous studies led on salmonids and others fish species [START_REF] Dolomatov | Features of nitrogen metabolism in fishes[END_REF][START_REF] Tripathi | Changes in nitric oxide (NO) synthase isoforms and NO in the ovary of Heteropneustes fossilis (Bloch.) during the reproductive cycle[END_REF], extremely unlikely.

4

Here we show that low water concentration in chloride, sodium and potassium negatively influences final oocyte maturation, ovulation and, ultimately, egg quality. Low salinity water triggers delayed ovulation and lower oocyte viability. When investigating the impact of LSW on FOM duration, individuals presenting the most severe phenotypes exhibited impaired oocyte hydration and abnormally reduced gene expression levels of several key players of the ovulatory process. While the under expression of water (i.e., aquaporins) and ion (i.e., solute carriers) transporters is consistent with impaired oocyte hydration, our observations also indicate that the entire ovulatory gene expression sequence is disrupted. Our results raise the question of the mechanisms underlying the negative influence of low water concentration in chloride, sodium and potassium on the dynamics of the preovulatory phase, on the control of the oocyte homeostasis, including hydration, and on the overall success of the maturation-ovulatory process. A global analysis of differentially regulated genes in LSW during final oocyte maturation is needed to unravel these mechanisms.

5

Competing interests

The authors declare no competing interest.

Authors' contributions

ES conceived the study, performed experiments and data analysis and drafted the manuscript. EC participated in experiments. SS-K and FC participated in the design of the study. JB conceived the study and participated in manuscript writing and data analysis. Funding sources have no involvement in the conduct of the research.

Funding

6

Color should be used for all figures in print. Supplemental data file 2: Water composition in control (Rébénacq) and LSW (Sarrance) site during the reproductive period (October-December).

Mean (N= 3) values (from 2020-10-15 to 2020-12-15) and standard deviation are shown. Difference between LSW and control water were analyzed using non-parametric U-tests.

* denotes significant differences.

  statement Investigations were conducted in compliance with EU Directive 2010/63/EU for animal experiments.

  aquaporin 4 (aqp4, GenBank accession # BX885214), vasotocinneurophysin VT 1 (vt1, GenBank accession # CA375992) and solute carrier family 26 (slc26a4, GenBank accession # BX873066) are involved in ion and water transport processes whereas A Disintegrin And Metalloproteinases 22 (adam22, GenBank accession # CA363158), CXC chemokine L14 (cxcl14, GenBank accession # BX868653) and coagulation factor V (f5, GenBank accession # BX879767) are inflammation-related genes. 18S RNA level was measured in each sample and used for messenger RNA abundance normalization. Expression levels at the late-vitellogenic (LV) stage are not presented but were not significantly different between groups and were arbitrarily set to 1 for each group. Expression data at the oocyte maturation stage were expressed as a percentage of the transcript abundance at the LV stage. Difference between LSW and control groups at oocyte maturation stage were analyzed using non-parametric U-tests.

33. 1

 1 Low water salinity results in lower egg viabilityThe automatic phenotyping of egg viability using the VisEgg system (Fig 2.A) revealed that 60% of the females exhibited non-viable eggs at the time of ovulation in the LSW group in sharp contrast with the control group in which only 18% of the females exhibited at least one non-viable egg at ovulation. More specifically, the percentage of non-viable eggs was dramatically increased (+378%) in the LSW group in comparison to the control group (Fig 2.B). Salinity is known to have an impact on reproduction in fish, including in salmonids. In Atlantic salmon (Salmo salar) and Coho salmon (Oncorhynchus kisutch), whole-life breeding in freshwater can result in the production of poor quality gametes

  phase from meiosis resumption to ovulation. Using non-invasive echograph-based analysis we could measure the duration of the oocyte maturation phase. After monitoring the first signs of intra-oocyte clearing (i.e., post-vitellogenic stage) (Fig 3.A and B), oocytes became fully transparent (i.e., maturation stage) until they were released from the ovary at ovulation. The duration of the maturation phase was calculated in each group after normalization by the daily temperature on site. We observed (Fig 3.C) that the maturation phase was significantly longer in the LSW group in comparison to the control group. The duration of oocyte maturation was on average 53 hours (at 10°C) in the LSW group while it was 70 hours (at 10°C) in the control group, which corresponds to a 32% increase in FOM duration in the LSW group.

  highly significant difference in egg viability in the LSW group depending on the duration of the maturation phase(Fig 4). A very high proportion (85%) of the fish exhibiting a maturation phase duration of 64 hours or more exhibited non-viable eggs at ovulation, while only 10% of the fish that underwent oocyte maturation in less than 64 hours exhibited this feature (Fig 4.A). In addition, the percentage of non-viable eggs was dramatically increased in the 64h group in comparison to the <64h group(Fig 4.B).

  This work was supported by the National Association for Technologic Research (ANRT) [grant number 2018/0457]; the Nouvelle-Aquitaine regional council [grant number 3379120].

Figure 1 :

 1 Figure 1: Water and plasma composition on both experimental sites

Figure 2 :

 2 Figure 2: Egg quality in the two experimental sites

Figure 3 :

 3 Figure 3: Ultrasound staging for measure of maturation duration

Figure 4 :

 4 Figure 4: Egg quality depending on the duration of oocyte maturation

Figure 5 :

 5 Figure 5: Hydration of oocytes during final oocyte maturation

Figure 6 :

 6 Figure 6: Ovarian expression profiles of genes involved in FOM and ovulation

  Mean(N= 12) yearly (2020-2021) values and standard deviation are shown. Difference between LSW and control water were analyzed using non-parametric U-tests.

	7	Potassium	0.60	0.19	0.18		0.04	*
		Sodium	2.02	0.17	0.99		0.18	*
		CONTROL 3.60 0.09 Mean SD 9.70 0.65 264,00 41,64 Nutrients, in milligrams per liter Sulfur Sulfate Conductivity (µS/cm) Nitrate 4.09 1.79 Major ions, in milligrams per liter Bicarbonate 145,50 17,77 Trace elements, in micrograms per liter Calcium 44,79 8,57 Copper 0.26 0.15 Chloride 2,29 0,84 Iron 8.07 6.72 Magnesium 4,29 0,51 Manganese 0.70 0.33 Potassium 0,47 0,13 Selenium 0.13 0.01 Sodium 1,83 0,28 Iode 1.45 0.50 Sulfur 3,03 0,79 Zinc 1.26 0.56	3.10 Mean LSW 9.5 248,92 1.11 140,00 43,60 0.16 1,24 2.00 5,16 0.12 0,21 0.17 1,09 1.15 5,08 1.44	2.82 8.37 SD 34,07 0.09 17,04 0.13 5,78 0,39 0.44 0.07 2,41 0,09 -0,27 0.08 3,32 0.54	* * * * * *
		Sulfate	8,33	2,06	13,05		9,54
		Nutrients, in milligrams per liter
		Nitrate	2,92	1,28	1,03		0,27	*
		Trace elements, in micrograms per liter
		Copper	0,22	0,09	0,14		0,07
		Iron	14,99	14,12	3,00		4,58	*
		Manganese	1,47	1,28	0,15		0,18	*
		Selenium	0,13	0,02	0,15		0,05
		Iode	1,35	0,53	1,04		0,18
		Zinc	1,37	0,70	1,35		0,83
	Supplemental data file 1: Water composition in control (Rébénacq) and LSW
	(Sarrance) sites.
			CONTROL	LSW
			Mean	SD	Mean		SD
		Conductivity (µS/cm)	282.00	20.785	242.67	29.48
		Major ions, in milligrams per liter
		Bicarbonate	154.50	25.05	143.67	10.97
		Calcium	48.97	5.11	42.43		4.66
		Chloride	2.98	0.45	1.47		0.06	*
		Magnesium	4.59	0.32	3.51		2.66

* denotes significant differences.

Acknowledgements

The authors thank Floriane Colon and Viviers Company staff for their help in fish rearing. The authors thank the Nouvelle-Aquitaine regional council and ANRT for the financial support which allowed the study.

Declaration of interests

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:

Authors statement

Emilien Segret: Conceptualization, formal analysis, investigation, data curation, writing original draft.

Emilie Cardona: Investigation, formal analysis.

Sandrine Skiba-Cassy: Conceptualization, supervision, project administration Frederic Cachelou: Conceptualization, supervision, project administration, funding acquisition.

Julien Bobe: Conceptualization, methodology, validation, visualization, writing review and editing, supervision, funding acquisition.


Quentin Chauleur

THE ISOTHERMAL LIMIT FOR THE COMPRESSIBLE EULER EQUATIONS WITH DAMPING

We consider the isothermal Euler system with damping. We rigorously show the convergence of Barenblatt solutions towards a limit Gaussian profile in the isothermal limit γ → 1, and we explicitly compute the propagation and the behavior of Gaussian initial data. We then show the weak L 1 convergence of the density as well as the asymptotic behavior of its first and second moments.

Introduction

We consider the isentropic compressible Euler equations with frictional damping:

(1.1a) (1.1b)      ∂ t ρ + ∂ x m = 0, ∂ t m + ∂ x m 2 ρ + ∂ x ρ γ + m = 0,
with ρ(0, x) = ρ 0 (x) ≥ 0 and m(0, x) = m 0 (x) for x ∈ R, t ≥ 0, and the adiabatic gas exponent γ > 1. Such a system appears in the mathematical modeling of compressible flow through a porous medium, and its study has drawn a lot of attention over the last decades [START_REF] Marcati | Cauchy problem for compressible Euler equations with damping[END_REF]. The global existence of L ∞ weak entropy solutions to the Cauchy problem of (1.1) is now well established (see for instance [START_REF] Nishida | Nonlinear hyperbolic equations and related topics in fluid dynamics[END_REF] and [START_REF] Huang | Asymptotic behavior of the solutions to the damped compressible Euler equations with vacuum[END_REF]), and it is known that their long-time asymptotic behavior [START_REF] Huang | Large time behavior for compressible Euler equations with damping and vacuum[END_REF] is governed by the limit diffusive profile:

(1.2a)

(1.2b)

∂ t ρ = ∂ 2 x ρ γ , m = -∂ x ρ γ ,
where equation (1.2a) is the porous media equation, whose fundamental solutions are called Barenblatt solutions [START_REF] Barenblatt | On a class of exact solutions of the plane one-dimensional problem of unsteady filtration of a gas in a porous medium[END_REF], and equation (1.2b) is the famous Darcy law. In [START_REF] Liu | Compressible flow with damping and vacuum[END_REF], the author constructs a 1 class of particular solutions for (1.1) which tend to the Barenblatt solutions ρ asymptotically in time, with an explicit rate log(t)/t. In [START_REF] Huang | Convergence to the Barenblatt solution for the compressible Euler equations with damping and vacuum[END_REF], the authors find the explicit rates

ρ -ρ 2 L 2 (R) ≤ C(1 + t) -k1+ε if 1 < γ ≤ 2, and ρ -ρ γ L γ (R) ≤ C(1 + t) -k2+ε if γ > 2, for any ε > 0, with k 1 = min γ 2 (γ+1) 2 , γ-1 γ and k 2 = min γ 2 (γ+1) 2 , 1
γ , for every L ∞ weak entropy solution (ρ, m) of (1.1). Unfortunately, the decay rates are not in L 1 -norm, which is the natural norm as (1.1a) and (1.2a) both satisfies the conservation of norm ρ(t, .) L 1 (R) = ρ 0 L 1 (R) and ρ(t, .) L 1 (R) = ρ 0 L 1 (R) for all t ≥ 0.

Decay rates in L 1 -norm were achieved for a particular range of γ in [START_REF] Huang | L 1 convergence to the Barenblatt solution for compressible Euler equations with damping[END_REF], where the authors show that ρ -ρ L 1 (R) ≤ C(1 + t) -1 4(γ+1) +ε if 1 < γ < 3, which was recently improved and extended in [START_REF] Geng | L 1 -convergence rates to the Barenblatt solution for the damped compressible Euler equations[END_REF] with the estimate

ρ -ρ L 1 (R) ≤ C(1 + t) -1 (γ+1) 2 +ε for all γ > 1.
Throughout the years, a lot of effort has been put to extend the range of γ to finally achieve the full range γ > 1 of barotropic pressure laws P (ρ) = ρ γ . The next natural step is the study of the case γ = 1, which leads to the isothermal pressure law P (ρ) = ρ. However, in the compressible fluid literature, much fewer results are known for the isothermal Euler system with damping

(1.3a) (1.3b)      ∂ t ρ + ∂ x m = 0, ∂ t m + ∂ x m 2 ρ + ∂ x ρ + m = 0,
which stands as the limit γ → 1 of the isentropic system (1.1). In [START_REF] Huang | Asymptotic behavior of the solutions to the damped compressible Euler equations with vacuum[END_REF], the authors show the existence of L ∞ entropy weak solutions to the Cauchy problem of (1.3). They also prove, up to a scaling in space z = x/ √ 1 + t, the L p loc convergence of the density ρ towards a diffusive profile ρ which satisfies the heat equation

(1.4) ∂ t ρ = ∂ 2
x ρ, in the case max(ρ + , ρ -) > 0, where ρ(±∞) = ρ ± . However, time dependent Gaussian functions, which stand as particular solutions of the heat equation on the whole space, do not satisfy this condition. In a bounded domain Ω ⊂ R d , the author shows in [START_REF] Zhao | On the isothermal compressible Euler equations with frictional damping[END_REF] the exponential convergence of any global solution (ρ, m) with small initial data (ρ 0 , m 0 ) towards the limit profile ( ρ 0 L 1 /|Ω|, 0). In fact, at least formally, equation (1.3) satisfies the energy inequality

(1.5) 1 2 R m 2 ρ + R ρ log ρ + t 0 R m 2 ρ ≤ E 0 for all t ≥ 0,
but the left-hand side has no definite sign due to the logarithmic contribution in the potential energy, a property that causes many technical difficulties such as a lack of compactness on the whole space R when the density t → ρ(t, .) ∈ L 1 (R) vanishes at infinity. In [START_REF] Carles | Rigidity results in generalized isothermal fluids[END_REF], in the case of the compressible Euler equations without frictional damping (µ = 0), the authors show that Gaussian functions stand as particular solutions of their system. They also prove that every global weak solution disperses and converges to a universal asymptotic Gaussian profile, up to a rescaling by their dispersion rate, in the weak L 1 topology, under some integrability assumptions on the moment of order 2 of the initial data ρ 0 . We will show in this paper that this kind of property still holds in the case with damping (1.3), and we will study the asymptotic behavior of Gaussian solutions and moments of order 1 and 2 of the density ρ as they stand as a key ingredient for the proof of this kind of feature.

The formal limit γ → 1 is singular regarding many features of the isentropic case, especially for the energy inequality of system (1.1):

(1.6) 1 2 R m 2 ρ + 1 γ -1 R ρ γ + t 0 R m 2 ρ ≤ E 0 .
In this paper we propose to give sense to the formal isothermal limit γ → 1, in particular we will rigorously show and illustrate the convergence of the Barenblatt solutions of (1.2a) to the Gaussian solutions of (1.4), a feature which does not seem to be so much known in the community to the best of the author knowledge.

From now on, (ρ γ , m γ ) with γ > 1 will denote a solution of the isentropic Euler system (1.1), whereas (ρ, m) will denote a solution of the isothermal Euler system. The overline bar will be use for solutions of the different limit diffusive profiles. The notation C will denote a generic constant C > 0.

This paper is organized as follows. In Section 2, we provide energy estimates, assumptions about existence and regularity of solutions of equations (1.3), and we state the main results of this paper. In Section 3, we show that Barenblatt solutions converge to a Gaussian profile as γ → 1. In Section 4, we explicitly compute the behavior of Gaussian solutions of (1.3). In Section 5, we study the evolution of the first and second moments of every weak solution of (1.3). In Section 6, we prove the weak L 1 convergence of every solution towards a universal Gaussian profile. Finally, in Section 7, we give some perspectives about the open question of explicit convergence rate in the isothermal case.

Assumptions and main results

We first give a notion of global weak solution for the 1-dimensional isothermal Euler equations with damping, and we will assume the global existence of this kind of solutions through the rest of the paper: Definition 2.1. We say that (ρ, m) is a weak solution of system (1.3) in [0, T [ with initial data (ρ 0 , m 0 ) ∈ L 1 (R) × L 1 (R), if there exists locally integrable functions √ ρ, Λ such that, by defining ρ := √ ρ 2 and m = √ ρΛ, the following holds:

(i) The global regularity:

√ ρ ∈ L ∞ ([0, T [ ; L 2 (R)), Λ ∈ L 2 ([0, T [ ; L 2 (R)),
with the compatibility condition √ ρ ≥ 0 a.e. on (0, ∞) × R, Λ = 0 a.e. on {ρ = 0} .

(ii) For any test function η ∈ C ∞ 0 ([0, T [ × R), T 0 R (ρ∂ t η + m∂ x η)dxdt + R ρ 0 η(0)dx = 0,
and for any test function

ζ ∈ C ∞ 0 ([0, T [ × R; R), T 0 R m∂ t ζ + Λ 2 ∂ x ζ + ρ∂ x ζ -mζ dxdt + R m 0 ζ(0)dx = 0. Assumption 2.2. Let (ρ 0 , m 0 ) ∈ L 1 (R) × L 1 (R).
We assume that there exists a weak solution (ρ, m) of system (1.3) with initial data (ρ 0 , m 0 ) in the sense of Definition 2.1 which satisfies, for all t ≥ 0, the energy estimate

1 2 R m 2 (t, x) ρ(t, x) dx + R ρ(t, x) log(ρ(t, x))dx + t 0 R m 2 (s, x) ρ(s, x) dxds ≤ C,
with the additional regularity

(t, x) → x 2 ρ(t, x) ∈ L ∞ ([0, T [ ; L 1 (R)).
Remark 2.3. The L ∞ entropy weak solutions of system (1.3) described in [START_REF] Huang | Asymptotic behavior of the solutions to the damped compressible Euler equations with vacuum[END_REF] are in fact weak solutions of (1.3) when the initial data

(ρ 0 , m 0 ) ∈ L ∞ (R) × L ∞ (R) satisfies the estimates 0 ≤ ρ 0 ≤ C and |m 0 | ≤ Cρ 0 | log ρ 0 |
(see [START_REF] Huang | Asymptotic behavior of the solutions to the damped compressible Euler equations with vacuum[END_REF]Definition 1] for a precise definition of L ∞ entropy weak solutions of system (1.3)). Note that the existence of L ∞ entropy weak solutions of the isothermal Euler system without damping is proved in [START_REF] Lefloch | Symmetries and global solvability of the isothermal gas dynamics equations[END_REF] under the same initial data conditions.

We now have to introduce several quantities in order to state our results. Let τ be the unique

C ∞ ([0, ∞)) solution of the differential equation τ = 2 τ -τ , τ (0) = 1, τ (0) = 0.
From [START_REF] Chauleur | Dynamics of the Schrödinger-Langevin equation[END_REF], we know that this function satisfies, as t → +∞,

τ (t) ∼ 2 √ t and τ (t) ∼ 1 √ t .
We also introduce the Gaussian function

Γ := e -y 2 .
We make the change of variable y = x/τ (t),

ρ(t, x) = 1 τ (t) R t, x τ (t) and m(t, x) = 1 τ (t) 2 M t, x τ (t) + τ (t) τ (t) yR t, x τ (t)
.

We first remark that this change of variable preserves the L 1 -norm, so for all t ≥ 0, at least formally

R R(t, y)dy = R ρ(t, x)dx = R ρ 0 (x)dx.
System (1.3) becomes, in the terms of the new unknown (R, M ),

(2.1a) (2.1b)        ∂ t R + 1 τ 2 ∂ y M = 0, ∂ t M + 1 τ 2 ∂ y M 2 R + ∂ y R + 2yR + M = 0,
which have the following energy inequality:

(2.2)

E(t) + t 0 τ (t) τ 3 (t) R M 2 R dyds ≤ C, where (2.
3)

E(t) = 1 2τ (t) 2 R M 2 R dy + R R log R Γ dy.
In fact, differentiating with respect to time the left part of (2.2), using equations (2.1a) and (2.1b) and by integration by parts, we get that

d dt 1 2τ (t) 2 R M 2 R + R R log R + R y 2 R + t 0 τ (s) τ (t) 3 R M 2 R ds = - 1 τ 2 R M 2 R ≤ 0.
Note that from the Csiszár-Kullback inequality (see e.g. [START_REF] Ané | Sur les inégalités de Sobolev logarithmiques, volume 10 of Panoramas et Synthèses [Panoramas and Syntheses[END_REF])

R R log R Γ ≥ 1 2 R 0 L 1 R -Γ 2 L 1 ≥ 0,
we direclty get that for all t ≥ 0, E(t) ≥ 0.

Denoting by E kin the kinetic energy

E kin (t) := 1 τ 2 (t) R M 2 (t, y) R(t, y)
dy for all t ≥ 0, the asymptotics of τ and τ and the following Lemma 5.2 give that

E kin ∈ L ∞ (R + ) and ∞ 0 E kin (t) 1 + t dt < ∞,
so we know there exists a sequence (t n ) n , such that t n → ∞ and

E kin (t n ) → 0 as n → ∞.
Unfortunately, we have no proof that this property is actually satisfied uniformly in time, even if it seems to be a reasonable assumption that we state in the following:

Assumption 2.4. We assume that

E kin (t) = 1 τ 2 (t) R M 2 (t, y) R(t, y) dy → 0 as t → ∞.
This assumption is for instance satisfied for Gaussians solutions of system (1.3), with an explicit convergence rate in t -1/2 (see Remark 5.4 below). We now state the two main results of this paper: Proposition 2.5. Under Assumption 2.2 and Assumption 2.4, we have

R   1 y y 2   R(t, y)dy -→ R   1 y y 2   Γ(y)dy as t → ∞.
Proposition 2.6. Under Assumption 2.2, we have

R(t) t→∞ Γ weakly in L 1 (R).
Remark 2.7. In Proposition 2.5, Assumption 2.4 is only required to show the convergence of the moment of order 2. Without this assumption, we are only able to show that the moment of order 2 is uniformly bounded (see Lemma 5.2).

Remark 2.8. Note that Propositions 2.5 and 2.6 are also satisfied in higher space dimensions R d with d ≥ 1, in the sense that

R d   1 y |y| 2   R(t, y)dy -→ R d   1 y |y| 2   Γ(y)dy as t → ∞ with Γ = e -|y| 2 , and R(t) t→∞ Γ weakly in L 1 (R d ).
Of course, our notion of global weak solution has to be adapted to the d-dimensional case, as well as system (1.3), and we refer to [START_REF] Carles | Rigidity results in generalized isothermal fluids[END_REF] and [START_REF] Chauleur | Dynamics of the Schrödinger-Langevin equation[END_REF] for some d-dimensional analogue of Definition 2.1.

In the same vein, the following discussion about the particular Gaussian solutions of (1.3) can easily be generalized to any dimension d by a tensorization property. We also refer the reader to [START_REF] Carles | Rigidity results in generalized isothermal fluids[END_REF] and [START_REF] Chauleur | Dynamics of the Schrödinger-Langevin equation[END_REF] for some d-dimensional analogue of system (4.1)-(4.2)-(4.3).

The limit γ → 1 of Barenblatt's solutions

We consider the unique fundamental solution of the porous media equation (with the Dirac delta function as initial data)

(3.1) ∂ t ρ γ = ∂ 2 x ρ γ γ , ρ γ (-1, x) = λδ(x), λ > 0,
which is called the Barenblatt solution [START_REF] Aronson | The porous medium equation[END_REF]. Note that we take the initial data at t = -1 to avoid the singularity at t = 0. We recall that this solution can be written

(3.2) ρ γ (t, x) = (1 + t) -1 γ+1 A -Bξ 2 1 γ-1 + with ξ = x(1 + t) -1 γ+1 , [f ] + = max {f, 0}, B = γ-1 2γ(γ+1)
and A determined by

(3.3) 2A γ+1 2(γ-1) B -1 2 π 2 0 (cos θ) γ+1 γ-1 dθ = λ.
Moreover, ρ γ is continuous on R, it satisfies the conservation of mass R ρ γ (t, x)dx = λ for all t ≥ -1, and has compact support for any finite time T > 0, namely

ρ γ = 0 if |ξ| ≥ A/B.
The power 1 γ-1 in (3.2) could make the limit γ → 1 unclear, however we can show:

Proposition 3.1. We make the change of variable ξ = x(1 + t) -1 γ+1 and

ρ γ (t, x) = (1 + t) -1 γ+1 B γ x(1 + t) -1
which preserves the L 1 -norm. Then,

B γ → λ 2 √ π e -ξ 2 4 in L ∞ (R) as γ → 1.
In particular, the L ∞ estimate

|ρ γ (t, x)| ≤ A 1 γ-1 (1 + t) -1 γ+1
has the following continuous limit when γ → 1:

|ρ(t, x)| ≤ λ 2 √ π (1 + t) -1 2 ,
where ρ denotes the Gaussian limit of the Barenblatt profile ρ γ as γ → 1 (namely ρ := ρ 1 , we omit the index γ when γ = 1).

Proof. We have, for |ξ| < A/B,

B γ (ξ) = A -Bξ 2 1 γ-1 + = exp 1 γ -1 log A + 1 γ -1 log 1 - B A ξ 2 ,
as A > 0 from expression (3.3) and B A ξ 2 < 1. We have now two terms to handle when γ → 1. Taking the logarithm in equation (3.3), and recalling that B = γ-1 2γ(γ+1) , we have

1 γ -1 log A = 2 γ + 1 log λ 2 2γ(γ + 1) -log 1 √ γ -1 π 2 0 (cos θ) γ+1 γ-1 dθ .
We denote ε = γ -1, so we look at the limit ε → 0 of the following Wallis integral

W 1+ 2 ε = π 2 0 (cos θ) 1+ 2 ε dθ = π 2 0 (cos θ) γ+1 γ-1 dθ.
We recall the well-known equivalent of Wallis integral when ε → 0,

W 1+ 2 ε = π 2 1 + 2 ε + O 1 1 + 2 ε = πε 4 + 2ε + O(ε),
so it is easy to check that

1 √ γ -1 π 2 0 (cos θ) γ+1 γ-1 dθ = π 4 (1 + O(ε))
as γ → 1, and then

exp 1 γ -1 log A = λ 2 √ π 1 1 + O(ε)
.

For the second term, as we have

B A = B 2W 1+ 2 ε λB 1 2 2ε 2+ε = ε 4 + O(ε 2 )
from the previous equivalent, we write

1 ε log 1 - B A ξ 2 = - 1 ε n≥1 1 n B A ξ 2 n ∼ - 1 4 ξ 2 + ε n≥2 1 n ξ 2 4 n ε n-2 , so finally exp 1 γ -1 log 1 - B A ξ 2 -→ e -ξ 2 4 ,
and

B γ (ξ) - λ 2 √ π e -ξ 2 4 = λ 2 √ π e -ξ 2 4 1 1 + O(ε) e O(ε) -1 = O(ε).
On |ξ| ≥ A/B, we know that B γ (ξ) = 0, and as ξ → e -ξ 2 /4 is a Gaussian,

sup |ξ|≥ √ A/B B γ (ξ) - λ 2 √ π e -ξ 2 4 = λ 2 √ π e -A B ≤ Ce -1 ε , so finally sup ξ∈R B γ (ξ) - λ 2 √ π e -ξ 2 4 ≤ Ce -1 γ-1 + O(γ -1) -→ 0 as γ → 1,
which ends the proof.

In order to illustrate this result, we make the following numerical simulation of the function B γ for several values of γ (namely γ = 2, 1.5 and 1.1), and the initial condition constant λ = 1. We also plot the limit Gaussian profile e -ξ 2 /4 /2 √ π, which corresponds to case γ = 1.

Figure 1. Convergence of B γ towards its limit Gaussian profile.

Gaussian solutions

In this section, following [START_REF] Carles | Rigidity results in generalized isothermal fluids[END_REF], we seek for particular Gaussian solutions of (1.3) of the form ρ(t, x) = b(t)e -α(t)(x-x(t)) 2 and m(t, x) = (β(t)x + c(t))ρ(t, x),

with the initial conditions b(0) = b 0 > 0, α(0) = α 0 > 0, β(0) = β 0 ∈ R, c(0) = c 0 ∈ R. As system (1.
3) is invariant by translation, we assume that x(0) = 0. We also denote ρ 0 = ρ(0, .) and m 0 = m(0, .). Plugging these expressions into (1.3), we obtain the following set of differential equations:

(4.1) α + 2αb = 0, β + β 2 + β = 2α, (4.2) ẋ = βx + c, ḃ = b( αx 2 + 2αcx( ẋ -c) -β), (4.3
) ċ + βc + c = -2αx.
In order to solve this system, mimicking [START_REF] Li | Blowup phenomena of solutions to the Euler equations for compressible fluid flow[END_REF], we can check that the two equations of (4.1) are satisfied if and only if α and β are of the form

α(t) = α 0 τ (t) 2 , β(t) = τ (t) τ (t)
,

where τ is the global solution of the differential equation

(4.4) τ = 2α 0 τ -τ , τ (0) = 1, τ (0) = β 0 .
We recall (see [START_REF] Chauleur | Dynamics of the Schrödinger-Langevin equation[END_REF]) that there exists a unique global solution τ ∈ C ∞ ([0, ∞)) to this nonlinear ODE, and that this solution remains uniformly bounded from below by a strictly positive constant. Plugging these expressions into the second equation of (4.2), we also get that

b(t) = b 0 τ (t) .
We also get an expression of c in terms of the center x of our Gaussian:

c(t) = c 0 -1 + τ τ x.
Let first show that the center of the Gaussian has an explicit expression, which does not depend on the function τ : Proposition 4.1. We have

x(t) = 1 ρ 0 L 1 (R) R xρ 0 -(1 -e -t ) R m 0 .
In particular, there exists x ∞ ∈ R such that

x(t) → x ∞ as t → ∞.
Proof. We know that ρ is a Gaussian function centered in x, so we get that for all t ≥ 0, R (x -x(t))ρ(t, x)dx = 0, hence we only have to study the first moment of ρ keeping in mind that

x(t) = 1 ρ 0 L 1 (R) R xρ(t, x)dx.
Integrating equation (1.3b) over R we get that

d dt R m(t, x)dx = R ∂ t m(t, x)dx = - R m(t, x)dx,
so we get by integration by parts that

d dt R xρ(t, x)dx = R x∂ t ρ(t, x)dx = - R m(t, x)dx = -e -t R m 0 (x)dx,
so finally integrating this expression over [0, t] we get the result.

In order to know the behavior of our Gaussian functions when t → ∞, we only need to get some equivalents of the real function τ , which is the goal of the following proposition. We refer to [START_REF] Chauleur | Dynamics of the Schrödinger-Langevin equation[END_REF] for a complete study of the differential equation (4.4). Proposition 4.2. Let τ be the unique C ∞ ([0, ∞)) solution of the differential equation (4.4), then we have

τ (t) ∼ 2 √ α 0 t and τ (t) ∼ α 0 t .
In particular,

α(t) ∼ 1 4t , β(t) ∼ 1 2t , b(t) ∼ b 0 2 √ α 0 t and c(t) → c 0 -x ∞ .

Evolution of certain quantities

In this section, we are going to give two propositions that describe the behavior of respectively the first and second moment of the renormalized density R. Proposition 5.1. We denote

I 1 = R M dy and I 2 = R yRdy.
Then, there exists C ≥ 0 such that

I 1 (t) = e -t I 1 (0) and I 2 (t) ∼ C √ t as t → ∞.
Proof. First off, mimicking the calculus of the proof of Proposition 4.2, we have by integrating equation (1.3b) over R and integration by parts that

d dt R M dy = - R M dy,
so we easily get that I 1 (t) = e -t I 1 (0). Now, by integration by parts, and using (2.1), we get the system of coupled differential equations:

İ1 = R ∂ t M = - R 1 τ 2 ∂ x M 2 R + ∂ y R + 2yR + M = -I 1 -2I 2 , and İ2 = R y∂ t R = - R y ∂ x M τ 2 = 1 τ 2 I 1 .
We denote Ĩ2 = τ I 2 , so that

İ2 = τ I 2 + τ İ2 = τ I 2 + 1 τ I 1 , and
Ï2 = τ I 2 + τ İ2 - τ τ I 1 + 1 τ İ1 = -( τ I 2 + 1 τ I 1 ) = -İ2 , hence İ2 (t) = İ2 (0)e -t
and Ĩ2 (t) = Ĩ2 (0) + İ2 (0)(e -t -1).

We easily compute the initial conditions İ2 (0) = I 1 (0) and Ĩ2 (0) = I 2 (0), so finally

I 2 (t) = 1 τ (t) I 2 (0) -I 1 (0)(1 -e -t ) ∼ 1 τ (t) [I 2 (0) -I 1 (0)]
when the initial condition are not well prepared in the sense that I 2 (0) = I 1 (0). Note that if I 2 (0) = I 1 (0), we have

I 2 (t) = e -t τ (t) I 1 (0).
We now give an useful lemma, which induces the boundedness of the second moment of the density R uniformly with respect to time: Lemma 5.2. There holds

(5.1) sup t≥0 R R(t, y)(1 + y 2 + | log R(t, y)|)dy < ∞ and (5.2) ∞ 0 τ (t) τ 3 (t) R M 2 R dydt < ∞.
Proof. Since E(t) ≥ 0, (5.2) follows from (2.2). We define

E + := 1 2τ 2 R M 2 R dy + R>1 R log R + R y 2 R, such that Equation (2.2) gives E + (t) + t 0 τ (t) τ 3 (t) R M 2 R dyds ≤ C + R<1 R log 1 R .
The last term is controlled by

R<1 R log 1 R ≤ C ε R R 1-ε
for all ε > 0 arbitrary small. By an interpolation formula, we get that

R R 1-ε R 1-3ε 2 L 1 y 2 R ε/2 L 1
for all 0 < ε < 2 3 . This implies that for all t ≥ 0,

E + (t) + t 0 τ (t) τ 3 (t) R M 2 R dyds 1 + E ε/2 + (t), thus E + (t) ∈ L ∞ (R + )
, and equation (5.1) follows.

With Assumption 2.4, we can then show a much better result on the second moment of the density R than its boundedness, which is the convergence of this second moment towards the second moment of the Gaussian Γ: Proposition 5.3. We denote

J 1 = R y 2 (R -Γ)dy. Then, under Assumption 2.4, |J 1 (t)| → 0 as t → +∞.
Proof. Denoting

J 2 = R yM dy,
we first write the system of differential equations satisfied by J 1 and J 2 . By integration by parts and using equation (2.1a), we compute

J1 = d dt R y 2 ∂ t (R -Γ) = - R y 2 ∂ y M τ 2 = 2 τ 2 R yM = 2 τ 2 J 2 .
Still by integration by parts, and using equation (2.1b), we get that

J2 = R y∂ t M = - R y 1 τ 2 ∂ y M 2 R + ∂ y R + 2yR + M = 1 τ 2 R M 2 R + R R - R 2y 2 R - R yM.
Here we introduce the function

Γ, writing R R -2 R y 2 R = R Γ -2 R y 2 Γ - R 2y 2 (R -Γ),
and we can remark by an easy calculation that

R y 2 Γ = 1 2 R Γ, so that finally J2 + J 2 + 2J 1 = 1 τ 2 R M 2 R .
We denote J1 = τ J 1 , so that

J1 = τ J 1 + τ J1 = τ J 1 + 2 τ J 2 ,
and

J1 = τ J 1 + τ J1 - 2 τ τ J 2 + 2 τ J2 = -J1 - 2 τ J 1 + 2 τ 3 R M 2 R , so we write J1 + J1 + 1 τ 2 J1 = 2 τ 3 R M 2 R .
Rather than trying to solve directly this non-autonomous differential equation of order 2, we will work on the following approximate equation, for t ≥ 1,

(5.3) f + ḟ + 1 4t f = 2 τ E kin .
In fact, denoting w = J1 -f , we see that w satisfies the differential equation

ẅ + ẇ + 1 4t w + 1 τ 2 - 1 4t J1 = 0, with 1 τ 2 - 1 4t = (τ -2 √ t)(τ + 2 √ t) 4tτ 2 = O 1 t 3 2
as we actually know from [START_REF] Chauleur | Dynamics of the Schrödinger-Langevin equation[END_REF] that τ (t) -2 √ t is uniformly bounded. The homogeneous part of equation ( 5.3) can be written under a Kummer's type equation

t f + t ḟ + 1 4 f = 0
which has two fundamental independent solutions (see [START_REF] Abramowitz | Handbook of mathematical functions with formulas, graphs, and mathematical tables[END_REF])

f 1 (t) = tM 5 4 , 2, -t = te -t M 3 4 , 2, t and
f 2 (t) = e -t U - 1 4 , 0, t = te -t U 3 4 , 2, t ,
where M(a, b, z) and U(a, b, z) respectively denotes the Krummer's and the Tricomi's function, which both stand as confluent hypergeometric functions and are independent solutions of the Kummer's equation

z d 2 w dz 2 + (b -z) dw dz -aw = 0.
In particular, from the asymptotic properties [START_REF] Abramowitz | Handbook of mathematical functions with formulas, graphs, and mathematical tables[END_REF] M(a, b, z

) = Γ(b) Γ(a) e z z a-b 1 + O |z| -1
and U(a, b, z) = z -a 1 + O |z| -1 , we get the following asymptotic for our fundamental solutions (5.4)

f 1 (t) ∼ C t 1 4 1 + O 1 t and (5.5) f 2 (t) ∼ e -t t 1 4 1 + O 1 t .
From the classical theory of linear differential equations, we know that every solution of equation ( 5.3) can be written under the form (5.6)

f (t) = c 1 f 1 (t) + c 2 f 2 (t) + f 0 (t),
where c 1 and c 2 denote two real numbers, and f 0 is a particular solution of (5.3). We easily compute the Wronskian function W (t) = f 1 (t) ḟ2 (t) -ḟ1 (t)f 2 (t) by solving the differential equation

Ẇ = ḟ1 ḟ2 + f 1 f2 -f1 f 2 -ḟ1 ḟ2 = f 1 (-ḟ2 - 1 τ 2 f 2 ) + ( ḟ1 + 1 τ 2 f 1 )f 2 = -W, which leads to W (t) = W 0 e -t .
Then, by a variation of constant formula, we can find a particular solution of (5.3) under the form

f 0 (t) = f 2 (t) t 0 f 1 (s) W (s) E kin (s) τ (s) ds -f 1 (t) t 0 f 2 (s) W (s) E kin (s) τ (s) ds,
so that using Assumption 2.4 and the equivalents (5.4) and (5.5), we get that every solution of (5.3) has the asymptotic:

|f (t)| = o( √ t).
In fact, in the expression of (5.6),

f 1 (t) t 0 e s f 2 (s) E kin (s) τ (s) ds 1 t 1 4 t 1 s -1 4 E kin (s)ds = o( √ t),
and every other terms is bounded as t → ∞. The same result applies for the function w = J1 -f , so we can write that for all t ≥ t 0 ≥ 0,

(5.7) J1 (t) = f (t) + c 1 f 1 (t) + c 2 f 2 (t) + f 2 (t) t t0 f 1 (s) W (s) 1 4s - 1 τ (s) 2 J1 (s)ds -f 1 (t) t t0 f 2 (s) W (s) 1 4s - 1 τ (s) 2 J1 (s)ds.
We already know from Lemma 5.2 that J1 is O( √ t), and as we have just shown that f is o( √ t), for ε > 0 fixed, there exists t 0 ≥ 0 such that, for all t ≥ t 0 ,

| J1 (t)| ≤ C √ t, 1 4s - 1 τ (s) 2 ≤ Ct -3 2 and 1 √ t |f (t) + c 1 f 1 (t) + c 2 f 2 (t)| ≤ ε 2 .
Injecting these inequalities in the right-hand side of (5.7), we have

1 √ t | J1 (t)| ≤ ε 2 + C t 1 4 + C t 3 4
≤ ε for t large enough, so we finally get by a bootstrap argument that

| J1 (t)| = o( √ t).
Finally, recalling that J1 = τ J 1 and that τ ∼ 2 √ t, we can conclude that

|J 1 | = R y 2 (R -Γ)dy = o(1),
which ends the proof.

Remark 5.4. Note that in the Gaussian case, we can explicitly compute the kinetic energy

E kin (t) = 1 τ 2 (t) R M 2 (t, y) R(t, y) dy = c(t) τ (t) ρ 0 L 1 (R) = O 1 √ t ,
unless the initial data are well prepared in the sense that c 0 = x ∞ (which would actually improve this rate). This feature gives an explicit rate of convergence of E kin towards 0 that can be propagated to the convergence of the second momentum of R by adapting the proof of Proposition 5.3, namely

|J 1 | = R y 2 (R -Γ)dy = O 1 √ t .

Convergence

Proof of Proposition 2.6. We are first going to try to eliminate the momentum M of our target equation. Differentiating equation (2.1b) with respect to y, and using equation (2.1a) in order to express

∂ t (∂ y (RM )) = -∂(τ 2 ∂ t R), we get that -∂ t (τ 2 ∂ t R) -τ 2 ∂ t R + LR = - 1 τ 2 ∂ y M 2 R .
where we have defined the Fokker-Planck operator L := ∂ 2 y + 2∂ y (y•). Following [START_REF] Chauleur | Dynamics of the Schrödinger-Langevin equation[END_REF], we introduce another scaling in time s defined by (6.1)

∂ s = τ 2 ∂ t ,
and the notation R(s(t), y) := R(t, y).

We calculate the quantities

∂ t (τ 2 ∂ t R) = 1 τ 2 ∂ 2 s R and τ 2 ∂ t R = ∂ s R,
hence we obtain the following equation:

(6.2) - 1 τ 2 ∂ 2 s R -∂ s R + L R = - 1 τ 2 ∂ y M 2 R .
Now we remark that equation (5.1) induces

(6.3) sup s≥0 R R(s, y)(1 + |y| 2 + | log R(s, y)|)dy < ∞, that equation (5.2) gives (6.4) ∞ 0 τ (t) R M 2 R dydt < ∞,
and that τ (s) ∼ 2e 2s and τ (s) ∼ e -2s , so we can conclude like in [START_REF] Carles | Universal dynamics for the defocusing logarithmic Schrödinger equation[END_REF]. Let a sequence s n → ∞, take s ∈ [-1, 2], and denote Rn (s, y) := R(s + s n , y).

From (6.3) along with the de la Vallée-Poussin [START_REF] Dellacherie | Probabilités et potentiel[END_REF] and Dunford-Pettis theorems [START_REF] Dunford | Linear Operators. I. General Theory[END_REF], we get the following weak convergence (up to a subsequence, not relabeled for reader's convenience), for all p ∈ [1, ∞), Rn t→∞ R∞ in L p (-1, 2; L 1 (R)).

We also get the weak convergence of the initial datum, up to another subsequence:

Rn (0) t→∞ R0,∞ in L 1 (R).
Thanks to (6.3), we also get that the family ( R(s n , .)) n is tight, so Then, denoting τ n (s) = τ (s + s n ), equation (6.4) implies that

- 1 τ 2 n ∂ y M 2 n
Rn t→∞ 0 in L 1 (-1, 2; W -2,1 (R)).

In addition, in (6.2), all the other terms but two obviously go weakly to zero, which yields (6.5) ∂ s R∞ = L R∞ in D ((-1, 2) × R), with R∞ (0, ; ) = R0,∞ ∈ L 1 (R). Thanks to the above bounds on R0,∞ , it is known (see [START_REF] Arnold | On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations[END_REF]) that the solution R∞ to (6.5) is actually defined for all s ≥ 0 and satisfies (6.6) R∞ -Γ L 1 (R) -→ t→∞ 0.

Going back to system (2.1), we need to show that R∞ is independent of s. In the s variable, equation (2.1a) becomes (6.7) ∂ s R + ∂ y M = 0, and (6.4) implies that M ∈ L 2 (-1, 2; L 1 (R)). With Mn (s) := M (s + s n ), we have

∂ y Mn -→ n→∞ 0 in L 2 (-1, 2; W -1,1 (R)), so ∂ s R∞ = 0.
Combining this last equality with equation (6.6), we infer that R∞ = Γ. The limit being unique, no extraction of a subsequence is needed, and we conclude that R(s) s→∞ Γ weakly in L 1 (R).

Conclusion

We have shown that Gaussian functions play an important role in the study of the isothermal compressible Euler equation, standing both as particular solutions of the system (1.3) and timeasymptotic limit of solutions of this system. They also make the link between the isentropic and the isothermal system, as limit of the Barenblatt solutions of (1.2a).

The next step of the analysis of the long-time behavior of this system would be to find an explicit convergence rate of any L ∞ weak solution of (1.3) towards the limit Gaussian profile, adapting the work from [START_REF] Huang | Convergence to the Barenblatt solution for the compressible Euler equations with damping and vacuum[END_REF], [START_REF] Huang | L 1 convergence to the Barenblatt solution for compressible Euler equations with damping[END_REF] and [START_REF] Geng | L 1 -convergence rates to the Barenblatt solution for the damped compressible Euler equations[END_REF]. A good approach seems to be the use of the Csiszár-Kullback inequality that gives a lower bound of the entropy by the L 1 -norm of ρ -ρ:

ρ -ρ 2 L 1 ≤ 2 ρ 0 L 1 R ρ log ρ ρ = 2 ρ 0 L 1 R R log R Γ .
Unfortunately, no decreasing rate of the entropy function is currently known. This is still an open question that appears in other fields of the analysis of PDEs, for instance the study of the logarithmic Schrödinger equation [START_REF] Carles | Universal dynamics for the defocusing logarithmic Schrödinger equation[END_REF]. Note that in [START_REF] Zhao | On the isothermal compressible Euler equations with frictional damping[END_REF], the author indeed has an upper bound for the entropy, assuming ρ ≥ c > 0, which is true on the compact set Ω ⊂ R but obviously false on the whole space R.

  (y)(1 + |y| 2 + | log R0,∞ (y)|)dy < ∞.