Julie Lucas
email: julie.lucas@obs-banyuls.fr

Valentin Logeux

S Rodrigues

Didier Stien

Philippe Lebaron

Alice M S Rodrigues

Exposure to four

Keywords: Danio rerio, artificial sediment, metabolic rate, heart rate, sub-lethal concentration, sunscreen

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The increase of anthropogenic activities has a continuous impact on aquatic ecosystems by the input of many chemicals and the resulting degradation of habitat quality. They also represent a significant threat for aquatic life.

Among them, UV filters are emerging potential pollutants that have received public interest since the last decade (Tovar-Sánchez et al., 2013;Tsui et al., 2014). These chemicals can be found in many personnal care products (sunscreen, cosmetics) and as additive in plastics or clothes to protect from UV radiation damage [START_REF] Fent | Widespread occurrence of estrogenic UV-filters in aquatic ecosystems in Switzerland[END_REF]Waldman and Grant-Kels, 2019). UV filters enter the aquatic ecosystem directly by washing off from the skin especially during summer related activities or indirectly through wastewater (Gioka et al. 2007;[START_REF] Fent | Widespread occurrence of estrogenic UV-filters in aquatic ecosystems in Switzerland[END_REF][START_REF] Ramos | A review of organic UV-filters in wastewater treatment plants[END_REF]. Sixty different UV filters are currently found in the global market [START_REF] Pirotta | The encyclopedia of allowed sunfilters in the world. Skin Care-Househ[END_REF] and it is estimated that tourism in the coastal areas introduces around 14,000 tons of sunscreen in the oceans every year [START_REF] Downs | Toxicological effects of the sunscreen UV filter, benzophenone-2, on planulae and in vitro cells of the coral, Stylophora pistillata[END_REF][START_REF] Downs | Toxicopathological effects of the sunscreen UV filter, oxybenzone (benzophenone-3), on coral planulae and cultured primary cells and its environmental contamination in Hawaii and the U.S. Virgin Islands[END_REF]. Among them, four UV filters commonly found in sunscreens and in environment were selected in this study: benzophenone-3 (BP3), butyl methoxydibenzoylmethane (BM), bis-ethylhexyloxyphenol methoxyphenyl triazine (BEMT) and methylene bis-benzotriazolyl tetramethylbutylphenol (MBBT). Environmental concentrations of these compounds are found in numerous reports, with recent reviews summarizing their concentrations in different matrices/compartments including rivers, lakes, coastal ocean, surface water or sediments (Giokas et al. 2007;[START_REF] Ramos | Advances in analytical methods and occurrence of organic UVfilters in the environmenta review[END_REF][START_REF] Tsui | Occurrence, distribution and ecological risk assessment of multiple classes of UV filters in marine sediments in Hong Kong and Japan[END_REF]Molins-Delgado et al. 2016;[START_REF] Montes-Grajales | Occurrence of personal care products as emerging chemicals of concern in water resources: a review[END_REF]).

This study focused on sediment compartment. Due to their lipopilic properties, organic UV filters were found to accumulate in sediments that is considered to be the major way of contamination [START_REF] Fagervold | Occurrence and environmental distribution of 5 UV filters during the summer season in different water bodies[END_REF]. Of the four UV filters studied, most occurence data are available for BP3 and BM. The average concentration of 49.4 ng/g for BP3 measured in Banyuls bay sediments during summer by [START_REF] Fagervold | Occurrence and environmental distribution of 5 UV filters during the summer season in different water bodies[END_REF] is in the range of concentrations found by Sanchez-Quiles et al. (2015) and Mao et al. (2018). Lower concentrations of 1.3 and 3.6 ng/g were also found in river sediment [START_REF] Kameda | Occurrence and profiles of organic sun-blocking agents in surface waters and sediments in Japanese rivers and lakes[END_REF]) and in a German lake (Kaiser et al. 2012a), respectively. An extreme BP3 concentration of 1600 ng/g was measured in marine sediments in California [START_REF] Schlenk | In vivo bioassay-guided fractionation of marine sediment extracts from the Southern California Bight, USA, for estrogenic activity[END_REF] and a concentration of 1735 ng/g was found in a Japanese river sediment [START_REF] Kameda | Occurrence and profiles of organic sun-blocking agents in surface waters and sediments in Japanese rivers and lakes[END_REF]. BM was detected at concentrations between 2.5 to 64.4 ng/g (Kaiser et al. 2012a, b;Sanchez-Quiles et al. 2015[START_REF] Tsui | Occurrence, distribution and ecological risk assessment of multiple classes of UV filters in marine sediments in Hong Kong and Japan[END_REF], Fagervold et al. 2019). Only few studies measured MBBT and BEMT concentrations. [START_REF] Fagervold | Occurrence and environmental distribution of 5 UV filters during the summer season in different water bodies[END_REF] detected maximum concentrations of 115.0 ng/g BEMT, and 75.2 ng/g MBBT (dry weight sediment). More data are needed, in particular for these last two UV filters. A summary of the UV-filters concentrations measured in environmental sediment is provided in table 1. In the present study, the concentration tested of 10 μg/g is around 100 times higher (but only ~6 to 10 times for BP3) than highest concentrations of UV filters measured in aquatic sediment. This factor corresponds to the secure ratio generally accepted for regulations.

The toxicity of UV filters has been documented in diverse organisms such as algae [START_REF] Gao | Effects of four commonly used UV filters on the growth, cell viability and oxidative stress responses of the Tetrahymena thermophila[END_REF]Esperaza et al. 2019, Mao et al. 2019[START_REF] Thorel | Effect of 10 UV Filters on the Brine Shrimp Artemia salina and the Marine Microalga Tetraselmis sp[END_REF], coral [START_REF] Downs | Toxicological effects of the sunscreen UV filter, benzophenone-2, on planulae and in vitro cells of the coral, Stylophora pistillata[END_REF][START_REF] Downs | Toxicopathological effects of the sunscreen UV filter, oxybenzone (benzophenone-3), on coral planulae and cultured primary cells and its environmental contamination in Hawaii and the U.S. Virgin Islands[END_REF][START_REF] He | Toxicological effects of two organic ultraviolet filters and a related commercial sunscreen product in adult corals[END_REF], Stien et al, 2019), crustacean [START_REF] Sieratowicz | Acute and chronic toxicity of four frequently used UV filter substances for Desmodesmus subspicatus and Daphnia magna[END_REF][START_REF] Chen | Multigenerational effects of 4-methylbenzylidene camphor (4-MBC) on the survival, development and reproduction of the marine copepod Tigriopus japonicus[END_REF][START_REF] Thorel | Effect of 10 UV Filters on the Brine Shrimp Artemia salina and the Marine Microalga Tetraselmis sp[END_REF] or amphibians [START_REF] Martins | Effects of 4-MBC and triclosan in embryos of the frog Pelophylax perezi[END_REF]. On fish, previous studies have demonstrated that some compounds can act as endocrine disruptors leading to adverse reproductive effects on zebrafish, Japanese medaka, rainbow trout or clownfish [START_REF] Schreurs | Estrogenic activity of UV filters determined by an in vitro reporter gene assay and an in vivo transgenic zebrafish assay[END_REF]Corronado et al. 2008;[START_REF] Kinnberg | Endocrine-disrupting effect of the ultraviolet filter benzophenone 3 in zebrafish, Danio rerio[END_REF][START_REF] Lee | Thyroid hormone-disrupting potentials of major benzophenones in two cell lines (GH3 and FRTL-5) and embryo-larval zebrafish[END_REF][START_REF] Barone | Acute toxicity testing of TiO 2 -based vs. oxybenzonebased sunscreens on clownfish (Amphiprion ocellaris)[END_REF]. Moreover, UV filters can impair gene expression [START_REF] Zucchi | Global gene expression profile induced by the UV-filter 2-ethyl-hexyl-4trimethoxycinnamate (EHMC) in zebrafish (Danio rerio)[END_REF][START_REF] Blüthgen | Accumulation and effects of the UV-filter octocrylene in adult and embryonic zebrafish (Danio rerio)[END_REF], development [START_REF] Torres | Screening the toxicity of selected personal care products using embryo bioassays: 4-MBC, propylparaben and triclocarban[END_REF] or fish behavior [START_REF] Araújo | Effects of UV filter 4methylbenzylidene camphor during early development of Solea senegalensis Kaup, 1858[END_REF][START_REF] Barone | Acute toxicity testing of TiO 2 -based vs. oxybenzonebased sunscreens on clownfish (Amphiprion ocellaris)[END_REF]. For a recent review see [START_REF] Lozano | Bioaccumulation and Toxicological Effects of UV-Filters on Marine Species[END_REF].

In the present study, zebrafish Danio rerio was selected as experimental model that have been recommended for chemical risk assessment [START_REF] Lammer | Is the fish embryo toxicity test (FET) with the zebrafish (Danio rerio) a potential alternative for the fish acute toxicity test?[END_REF]OECD 1998;2013a). This species is well described in litterature [START_REF] Kimmel | Stages of Embryonic Development of the Zebrafish[END_REF][START_REF] Lawrence | The husbandry of zebrafish (Danio rerio): A review[END_REF]) and offers multiple advantages for ecotoxicity testing [START_REF] Hill | Zebrafish as a model vertebrate for investigating chemical toxicity[END_REF][START_REF] Spitsbergen | The State of the Art of the Zebrafish Model for Toxicology and Toxicologic Pathology Research : Advantages and Current Limitations[END_REF]. As there is a need for a better understanding of the toxicity of UV filters, the aim of this study was to address the effect of UV filters-contaminated sediment on zebrafish early life stages. Toxic responses of exposed embryos and larvae were examined through physiological endpoints such as survival, hatching success, cardiac frequency and aerobic metabolism.

Material and methods

Chemicals and preparation of spiked artificial sediment

The UV filters used in this study are listed in Table 1. Analytical standards of BEMT and BM were provided by Pierre Fabre Laboratories wheareas BP3, MBBT and dichloromethane were purchased from Sigma Aldrich (Saint-Quentin-Fallavier, France). Artificial sediment was prepared according to Le Bihanic et al. (2014a, b). The reconstituted sediment was composed of sand measuring 0.2-0.5 mm (sable de Loire SCALARE, Aquastore, France), 5 % kaolin clay (Merck, Darmstadt, Germany), and Sphagnum blond peat (Florentaise, Saint-Mars-du-Désert, France). Briefly, peat was dried for 48 h and sieved (0.5 mm). Milli-Q water was added to the peat (12:1, v/v) in glass bottles and shaken for 48 h at 180 rpm at room temperature. Sand and 5 % kaolin clay were then mixed to humid peat and shaken for 24 h. pH was adjusted to 6.5 with 10 % CaCO 3 solution. After 7 days of stabilization at room temperature, 1:4 v/v of milli-Q water was added to the sediment. After a 24 h equilibration period, supernatant water was removed and the sediment was dried at 105 °C for 14 h. After cooling, 4 g artificial sediment added in glass Petri dishes were spiked with one of the UV filters at 10 μg.g -1 dry weight sediment each as follows: a solution of the UV filter in dichloromethane (10 μl, 4 mg.mL -1 ) was mixed to sediment and then solvent was left to evaporated overnight in the dark at room temperature under a fume hood. This ensures complete residual solvent removal. A control with solvent (dichloromethane without UV filters) and sediment was prepared in the same manner. A negative control was tested too (no solvent, only artificial water E3 and sediment).

Zebrafish embryo-larval assay

Maintenance of zebrafish and egg production

Adults Wild-type AB-TU strain (Tübingen, Germany) zebrafish (Danio rerio) came from the Institute of Functional Genomics of Lyon (IGFL). Fish were maintained in groups of 10 individuals in 10 L tanks. Eggs were obtained from this culture by random pairwise mating of zebrafish. One adult male and one female were placed together in spawning boxes (AquaSchwarz, Germany) the evening before eggs were required. Spawning and fertilization took place within 30 min after the onset of light in the morning. Aquaria were filled with a mix of 1/3 reverse osmosistreated water and 2/3 tap water without chlorine, both being filtered beforehand with dust and charcoal filters and using a flow-through system to obtain a pH of 7.8±0.5 and a conductivity of 450 ± 50 μS.cm -1 . Rack aeration allowed an oxygen level ≥80 %. Water and room temperature were maintained at 28±1 °C with a 14-h light/10-h dark photoperiod. Ammonia, nitrite, and nitrate were monitored weekly and remained within recommended ranges [START_REF] Lawrence | The husbandry of zebrafish (Danio rerio): A review[END_REF]). The fish were fed twice a day with commercial granulates (Ocean nutrition Europe, Belgique) and an additional distribution of Artemia sp. nauplii (Artemia EG>225000 nauplii/g) occasionally supplemented with red sludge worms (Boschetto-Frozen fish food).

Zebrafish exposure

Experiments were done for each of the 4 individual UV filters. Fertilized and normally developed eggs were selected at eight-cell stage (1 h 15 post-fertilization) using a binocular loupe. The egg fertilization rate had to be ≥70 %, and a mix of three spawns was used to avoid a potential spawn effect. Artificial water (E3) according to ISO 7346/3 (1996) was used as test medium (0.33 mM CaCl 2 , 0.33 mM MgSO 4 , 5 mM NaCl, and 0.17 mM KCl). Exposure tests were carried out in glass Petri dishes (35-mm diameter) with 4 g dw artificial sediment and 8 mL E3. Thirty embryos were placed randomly on a 1-mm thermoformed Nytex® grid to avoid being buried in the sediment and exposed for 96 h in an incubator at 28±0.5 °C, with the same photoperiod as the rearing room. At the end of exposure, larvae were transferred to new Petri dishes containing only freshly prepared E3 medium until 10 days post fertilization (dpf) for analysis. Larvae were not fed during the experiment.

Exposures were performed using a procedure adapted from Le Bihanic et al. (2014a, b) for the 10-day embryo-larval toxicity assay (including 96 h of exposure, embryos, and pro-larvae) with zebrafish (OECD 1998(OECD , 2013b)).

Survival and hatching success

Embryonic and larvae survival was recorded daily until 10 dpf. For each condition tested, four replicates of 30 individuals were observed. Dead individuals were removed. The survival rate (SR, %) was estimated as: SR = 100×(SN tx / SN t0 ) equation ( 1)

where SN tx and SN t0 are the number of live individuals at time t x (time of the measure, x hours post fertilizationhpf) and t 0 (beginning of the experiment, 0 hpf).

Similarly, hatched individuals were counted between 48 and 72 hpf. At the end of exposure (96 hpf), hatching success (HS, %) was calculated as: HS = 100×(HI t96h /SN t96h ) equation ( 2)

where HI t96h is the number of hatched and living larvae.

Cardiac frequency

In order to assess cardiac frequency f H , a total of 183 larvae were tested individually (nc ontrol =33, n BP3 =39, n BEMT =40, n MBBT =39, n BM =32). The low pigmentation and transparency of larvae at 5 dpf permitted to visually measure f H through the body wall. In order to do so, individuals were placed in lateral position into 3 % methylcellulose on Petri dishes (i.e., 6-7 ml). Two hours-acclimation were necessary to distress from the transfer and reach basal heart rate (Lucas et al. 2014b). Three consecutive videos of 20 s were recorded to observe in vivo beating heart of each larva.

The use of a microscope (Olympus DP71), above which was placed a color camera, permitted to record videos on "Cell" software. Estimation of f H was based on the principle that each passage of blood from the atrium to the ventricle corresponds to a cardiac contraction. In transparent larvae, blood circulation between these two cardiac chambers induces a difference in contrast on which the video analysis is based. f H was manually counted for each video.

Respirometry

Experimental set-up

The set-up consisted of eight independent glass micro respirometer chambers (diameter d = 9 mm, volume V = 0.6 mL; Loligo systems, Denmark). Respirometers were submerged into buffer tanks (depth×length×height 14×35×13 cm) and filled with oxygenated E3 solution at a constant temperature of 28 °C. Peristaltic pumps controlled the water supply from the buffer tank to each respirometer. Each pump was controlled by a timer, allowing intermittent flow respirometry [START_REF] Steffensen | Oxygen consumption in four species of teleosts from Greenland: no evidence of metabolic cold adaptation[END_REF], where phases of oxygen renewal alternated with phases of oxygen consumption measurements with a period of 30:15 min. Each respirometer was equipped with an optic fiber sensor (PreSens) connected to a multichannel oxygen measuring system (Witrox 4, PreSens) to record dissolved oxygen levels. Optic fibers were calibrated at 0 and 100 % of air saturation at a temperature of 28 °C. A conversion factor based on oxygen solubility into water was used to convert oxygen data from percentage saturation to mgO 2 .L -1 (i.e., 100 % was equivalent to 7.94 mgO 2 .L -1 for a 28 °C temperature and a 0 salinity). Oxygen saturation was recorded every 5 s with the program Witroxview (PreSens).

Experimental protocol

Each experimental trial consisted in testing simultaneously four groups of 5 larvae exposed to one of the 4 UV filters tested. Ten groups per filter were tested. Each experimental trial involved two steps. First, to assess active metabolic rate (AMR), metabolism of larvae was increased through chasing with a stick into Petri dishes [START_REF] Schurmann | Effects of temperature, hypoxia and activity on the metabolism of juvenile Atlantic cod[END_REF][START_REF] Lefrançois | Influence of ambient oxygenation and temperature on metabolic scope and scope for heart rate in the common sole Solea solea[END_REF][START_REF] Dupont-Prinet | An investigation of metabolic prioritization in the European Sea Bass, Dicentrarchus labrax[END_REF]Cannas et al. 2013;[START_REF] Clark | Physiological benefits of being small in a changing world: responses of Coho salmon (Oncorhynchus kisutch) to an acute thermal challenge and a simulated capture event[END_REF].

When exhausted (i.e., when fish did not respond to stimulation anymore), each group of five larvae was transferred into one of the four respirometers to assess AMR through a first oxygen consumption over the next 15 min. Second, to estimate standard metabolic rate (SMR), larvae were left undisturbed in the respirometer for 48 h, during which oxygen consumption was regularly and automatically measured. After this period of measurements, larvae were removed from the respirometers and slightly anesthetized with benzocaïne at a concentration of 50 mg.L -1 . The body mass of each individual was determined using a microbalance (Sartorius Secura 26-1S). Before and after each trial, a blank measurement was performed to quantify microbial oxygen consumption in the respirometer. The average of these two values was subtracted from the measured oxygen consumption. Each group of larvae was tested once.

Calculations

Oxygen consumption MO 2 is expressed in mgO 2 .g -1 .h -1 and calculated using the following formula:

MO 2meas = Δ[O 2 ].Δt -1 .V.M meas -1

Equation (3)

where

Δ[O 2 ] (in mgO 2 .L -1
) is the variation in oxygen concentration during the measurement period Δt (in h), V (in L) is the volume of the respirometer minus the volume of the fish and M meas (in g) is the mass of the corresponding group of 5 larvae.

An allometric relationship exists between oxygen consumption and body mass, which encourages correction of MO 2meas using the following formula:

MO 2cor = MO 2meas .(M meas .M cor -1
) 1-b Equation ( 4)

where MO 2cor (in mg O 2 .g -1 .h -1 ) is the oxygen consumption related to a standard fish of 1 g (M cor ), MO 2meas (in mg O 2 .g -1 .h -1 ) is the oxygen consumption estimated with eq. 3 for experimental fish whose mass was M meas (in g) and b

is the allometric scaling exponent describing the relationship between oxygen consumption and mass of fish. In the case of this study, we used b AMR = 0.926 for the correction of active metabolic rate and b SMR = 0.965 for the correction of standard metabolic rate (Lucas et al. 2014a).

AMR was assessed as the maximal MO 2 obtained after chasing the larvae. SMR was estimated according to the method described by [START_REF] Steffensen | Oxygen consumption in four species of teleosts from Greenland: no evidence of metabolic cold adaptation[END_REF] and Lucas et al. (2014a). Aerobic metabolic scope (AMS) was calculated as the difference between AMR and SMR. AMR, SMR, and AMS were assessed for each group.

Statistical analysis

The statistical analysis was carried out using GraphPad Prism software. For all analyses the normality and homoscedasticity were checked through the Shapiro and Barlett tests, respectively. An unpaired t-test was then used to determine which chemical treatment differed significantly from the solvent control. The results were considered to be significantly different when p < 0.05.

Results and discussion

In last decades, several different ecotoxicity tests have been established demonstrating the robustness of fish embryo larval assays to reflect the toxicity of organic pollutants. Our study investigated for the first time the toxicity of BP3, BMDBM, BEMT and MBBT through sediment exposure on embryo larval stages of the zebrafish and through different physiological endpoints.

Sediment contact assays on fish embryos stages have been proposed as an alternative to classic ecotoxicity assays for chemical testing (Barjoux et al. 2012, Le bihannic et al 2014, Viquelin et al. 2011). This king of assay better reproduces the multiple sources of exposure to benthic organisms via sediment, particle and dissolved phase.

For all physiological endpoints, a negative control (sediment) and solvent control (solvent and sediment) were tested in order to estimate the impact of solvent. No difference was observed between these two controls whatever the parameters measured (p>0.05).

Impact on survival and hatching

Survival rate (%) Hatching success (%) .3739). This means that sediment exposure to BP3, BEMT, MBBT or BM did not display any lethal effect in zebrafish embryo larval stage in our conditions of exposure. It also indicates that direct embryonic exposure to these compounds may not impact the fish recruitment in environmental conditions. This is the first report on the effect of BP3, MBBT, BEMT or BM in zebrafish embryo larval stages. Negative control and solvent control presented heartbeat rate of 163.42±18.44 bt.min -1 and 156.44±22.81 bt.min -1 respectively which is in accordance with normal zebrafish heartbeat rate measured in others studies [START_REF] Barrionuevo | O2 consumption and heart rate in developing zebrafish (Danio rerio): influence of temperature and ambient O2[END_REF][START_REF] Burggren | Developing animals flout prominent assumptions of ecological physiology[END_REF][START_REF] Lin | Toxicity and cardiac effects of carbaryl in early developing zebrafish (Danio rerio) embryos[END_REF]Lucas et al., 2014[START_REF] Quintaneiro | Toxicity effects of the organic UV-filter 4-Methylbenzylidene camphor in zebrafish embryos[END_REF]. No significant effect was observed on MBBT (t-test, p=0.5376) and BM (t-test, p=0.2435) treatments compared to the solvent control.

Cardiotoxicity

BP3 exposure induced a significantly 15.8 % decreased heart rate (Fig 1 ; t-test, p < 0.0001). Bradychardia of this kind was also observed on zebrafish for 4-methylbenzylidene camphor (4-MBC) exposures at higher concentration using exposure through water [START_REF] Torres | Screening the toxicity of selected personal care products using embryo bioassays: 4-MBC, propylparaben and triclocarban[END_REF][START_REF] Quintaneiro | Toxicity effects of the organic UV-filter 4-Methylbenzylidene camphor in zebrafish embryos[END_REF]. Similar pattern was observed in studies investigating the cardiotoxicity of polyaromatic hydrocarbons (PAHs) in early life stages of zebrafish directly exposed to contaminated sediment or oil [START_REF] Incardona | Defects in cardiac function precede morphological abnormalities in fish embryos exposed to polycyclic aromatic hydrocarbons[END_REF][START_REF] Incardona | Aryl hydrocarbon receptorindependent toxicity of weathered crude oil during fish de-velopment[END_REF](Incardona , 2011;;[START_REF] Hicken | Sub-lethal exposure to crude oil during embryonic development alters cardiac morphology and reduces aerobic capacity in adult fish[END_REF][START_REF] Perrichon | Influence of sediment composition on PAH toxicity using zebrafish (Danio rerio) and Japanese medaka (Oryzias latipes) embryo-larval assays[END_REF][START_REF] Perrichon | Toxicity assessment of wateraccommodated fractions from two different oils using a zebrafish (Danio rerio) embryo-larval bioassay with a multilevel approach[END_REF]. [START_REF] Li | Joint Effects of Multiple UV Filters on Zebrafish Embryo Development[END_REF] and [START_REF] Quintaneiro | Toxicity effects of the organic UV-filter 4-Methylbenzylidene camphor in zebrafish embryos[END_REF] suggested a direct association between heart rate decrease and heart deformities. In our study, no heart deformities were observed during embryo development (pericardial oedema for example). Additionnal histological analysis of fish heart could show internal lesions characteristics of cardiac insufficiency. Furthermore, cardiac frenquency might be related with the functioning of cardiac muscle, which should be further investigated in future studies as well as cardiac gene expression. In contrast to BP3, we observed that BEMT induced a significant 10.2 % increase of cardiac frequency ( Fig 1,p =0.003). This kind of increase was also observed by Lucas et al. (2014) in zebrafish larvae born from progenitors exposed to polycyclic aromatic hydrocarbons PAHs at environmental concentration (i.e. 12.5 % increased cardiac frequency). This increase was not lethal and maybe only due to a stress reponse but require more investigations to be interpreted. One hypothesis to explain this tachycardia pattern is the production of stress hormones such as cortisol. Pollutant exposure could induce an elevated production of this hormone which could modulate cardiac frequency [START_REF] Mccormick | Influence of cortisol on developmental rhythms during embryogenesis in a tropical damselfish[END_REF][START_REF] Nesan | Embryo exposure to elevated cortisol level leads to cardiac performance dysfunction in zebrafish[END_REF]. Another hypothesis is the stress related to the UV filter. It is known that UV-filters can lead to oxidative stress by increasing production of reactive oxygen species (ROS). This can result in activation of antioxidant defences and/or detoxification system in order to prevent oxidative damage (Liu et al., 2015), therefore increasing heartbeat rate. These processes are energy costly which could have negative impact on aerobic metabolic scope, increasing SMR [START_REF] Lannig | Temperature-dependent stress response in oysters, Crassostrea virginica: Pollution reduces temperature tolerance in oysters[END_REF], Lucas et al. 2014b). [START_REF] Lannig | Temperature-dependent stress response in oysters, Crassostrea virginica: Pollution reduces temperature tolerance in oysters[END_REF], Lucas et al. 2014b) as observed for BP3.

Aerobic metabolic scope

Even if SMR increased AMS was not significantly different from the solvent control. Fishes seem to maintain their aerobic metabolic metabolic scope despite UV filter exposure. Furthermore, although alteration of the cardiovascular system can have harmful consequences on energetic metabolism, the aerobic metabolic rate did not correlate to the effect on cardiac frequency observed with BP3 and BEMT. We hypothesized that larvae may offset by modulating other mechanisms implied in blood circulation and oxygen transport. Moreover, a part of respiration is cutaneous in early life stage, hereby possibly modulating oxygen transport capacity. Overall, the compounds tested in our study did not induce impairment in the mechanisms involved in metabolic regulation at the tested concentration. These results suggest that Danio rerio larvae would not have a reduced capacity to sustain oxygendemanding activities such as locomotion, digestion or growth (Claireaux & Lefrançois, 2007).

Conclusion

In sum, in the tested conditions, BP3 and BEMT significantly altered D. rerio larvae heartbeat rate after exposure of fish eggs to contaminated sediments. BP3 also increased SMR. BP3 seems to be the more toxic UV-filter of the four

Fig 1

 1 Fig 1 Cardiac frequency of 5 dpf larvae zebrafish Danio rerio exposed to one of the following treatment: control, exposure to BP3, MBBT, BEMT or BM. Mean values±standard deviation are plotted and * indicates significant difference with the solvent control (t-test, p < 0.05).

Fig 2

 2 Fig 2 Active metabolic rate (AMR), standard metabolic rate (SMR) and aerobic metabolic scope (AMS) of 5 dpf larvae Danio rerio in control or after exposition to one of the 4 UV filters tested: BP3, BEMT, MBBT, BM. Results

Table 1

 1

							Maximum		Environmental
	Abbr.	Name	Alternative names	CAS #	Formula	concentration in final product a	concentration measured in sediment	References
						USA	EU	Aus.	(in ng/g)
	MBBT	Methylene bis-benzotriazolyl tetramethylbutylphenol	Bisoctrizole, Tinosorb M, Milestab 360	103597-45-1	C41H50N6O2	n.a.	10 %	10 %	48 to 75.2	Fagervold et al 2019 Kaiser et al 2012a
										Fagervold et al 2019
										Kameda et al 2011
	BP3	Benzophenone-3	Oxybenzone	131-57-7	C14H12O3	6 %	6 %	10 %	1.3 to 1735	Kaiser et al 2012a Mao et al 2018
										Sanchez-Quiles et al 2015
										Schlenk et al 2009
										Fagervold et al 2019
	BM	Butyl methoxydibenzoyl methane	Avobenzone	70356-09-1	C20H22O3	3 %	5 %	5 %	2.5 to 64.4	Kaiser et al 2012a Sanchez-Quiles et al 2015
										Tsui et al. 2015
	BEMT	bis-Ethylhexyloxyphenol methoxyphenyl triazine	Bemotrizinol, Tinosorb S, Escalol S	187393-00-6	C38H49N3O5	n.a.	10 %	10 %	48 to 111.5	Fagervold et al 2019 Kaiser et al 2012a, b

UV filters tested. a USA: United States of America, EU: European Union, Aus.: Australia. n.a.: not approved

Table 2

 2 Survival and hatching success on zebrafish assessing UV filters toxicity. Mean values±standard deviation.

		Number of
	Treatment	replicate	96 hpf	10 dpf	96 hpf
	Negative control	3	86.67±6.7	85.58±5.1	95.75±5.68
	Solvent control	4	84.2±7.4	84.2±7.4	92.5±11.2
	BP3	4	82.2±6.9	82.2±6.9	72.6±23.1
	MBBT	4	70.8±11.2	69.2±7.4	85.4±17.7
	BEMT	4	76.7±3.3	74.4±1.9	97.1±2.5
	BM	4	87.8±1.9	86.6±0.0	92.5±6.5
	All results are not significantly different from solvent control experiment (t-test, p < 0.05).
	Survival and hatching success measured following UV filters exposure are shown in Table 2. At 10 µg.g -1 , none of
	the four UV filters tested induced significant mortality, hatching failure, or delayed hatching of embryos and larvae
	compared to the solvent control (t-test, p>0.05 : BP3 p 4j =0.7396, p 10j =1.000; BEMT p 4j =0.0.075, p 10j =0.0721; MBBT
	p 4j =0.0.1683, p 10j =0.1649; BM p 4j =0.4588, p 10j =0

  are expressed as mean values ± standard deviation. Among all the metabolic parameters measured and UV-filters tested, only SMR for BP3 differed significantly from the solvent control (t-test, p<0.05). This is the first work on the impact of UV filters on fish metabolism. Therefore, comparison with other studies are still limited. No effect was observed on SMR, AMR or AMS measured in 5 dpf embryo larvae for exposure to BEMT (Fig 2, t-test, p SMR =0.2578, p AMR =0.9471, p AMS =0.9103), MBBT (Fig 2, t-test, p SMR =0.0.8426, p AMR =0.6822, p AMS =0.9103) and BM (Fig 2, t-test, p SMR =0.2185, p AMR =0.8715, p AMS =0.949) compared to Solvent control. For BP3 exposure, no effect was observed on AMR (Fig 2, t-test, p = 0.0969) and AMS (Fig 2, t-test, p = 0.2113) and thus the decreased cardiac frequency observed for BP3 cannot be associated with lower AMR and AMS as observed for other organic pollutants(Claireaux and Davoodi 2010, Hiken et al. 2011). SMR was significantly higher than the control(Fig 2, p=0.0195). It is known that exposure to pollutant can modulate cardio-respiratory performance and aerobic capacity of fish. Pollutant exposure can induce detoxification processes that are energy costly. This supplementary energetic demand would increase SMR

Acknowledgements

tested. This is consistent with numerous reports on BP3 toxicity on aquatic species (review of Lozano et al. 2020).

This work is the very first report on the effect of BEMT and MBBT on fish development and larval metabolism.

BEMT and BM had no effect on the different parameters tested. In our conditions, these compounds were not toxic for zebrafish embryo larvae. Since environmental toxicity data remain scarce for many UV filters, and considering the intense use of personal care products resulting in notable concentrations in the aquatic environment, more data on ecotoxicological effects of UV filters on fish are urgently needed (review of Lozano et al. 2020), and it is necessary to continue to perform ecotoxicological study and to develop new models.

instrumentation. We also thank LIENSs laboratories and especially Dr. Christel Lefrancois for providing a part of respirometry material. This study was conducted under the agreement of the Animal Care Committee N° A66-01-601. Eau Thermale Avène. It was neither supervised nor audited by Pierre Fabre Laboratories.


Emmanuel Bernuau
email: emmanuel.bernuau@agroparistech.fr

Emmanuel Moulay
email: emmanuel.moulay@univ-poitiers.fr

Patrick Coirault
email: patrick.coirault@univ-poitiers.fr

Fayrouz Isfoula
email: fayrouz.isfoula@univ-poitiers.fr

Practical consensus of homogeneous sampled-data multi-agent systems

Keywords: Multi-agent systems, practical consensus, sampled-data systems, homogeneous systems

The aim of this article is to study the secondorder practical consensus of homogeneous sampled-data multiagent systems. To do this, a new nonlinear emulation strategy based on homogeneity is developed. It is then applied to multiagent systems under synchronously variable sampling. Finally, a comparison with the classical linear strategy is provided in the case of multi-agent systems under synchronously periodic sampling.

I. INTRODUCTION

The second-order consensus for multi-agent systems (MAS) has been solved during the 2000's by using linear controllers [START_REF] Ren | Distributed consensus in multi-vehicle cooperative control[END_REF]. The multi-agent formation control has then attracted a lot of attention [START_REF] Oh | A survey of multi-agent formation control[END_REF]. For real applications involving wireless communications, the problem of the sampled-data inputs must be taken into account [START_REF] Hetel | Recent developments on the stability of systems with aperiodic sampling: an overview[END_REF]. So, the study of sampled-data MAS has been developed in the 2010's [START_REF] Cao | Multi-vehicle coordination for double-integrator under fixed undirected/directed interaction in a sampled-data setting[END_REF], [START_REF] Chen | Quantized consensus of second-order continuous-time multi-agent systems with a directed topology via sampled data[END_REF], [START_REF] Cheng | Sampled-data based average consensus of second-order integral multi-agent systems: Switching topologies and communication noises[END_REF], [START_REF] Gao | Sampled-data based consensus of continuous-time multi-agent systems with time-varying topology[END_REF], [START_REF] Huang | Some necessary and sufficient conditions for consensus of second-order multi-agent systems with sampled position data[END_REF], [START_REF] Qin | A sufficient condition for convergence of sampleddata consensus for double-integrator dynamics with nonuniform and time-varying communication delays[END_REF], [START_REF] Yu | Second-order consensus in multi-agent dynamical systems with sampled position data[END_REF], [START_REF] Wen | Consensus of multi-agent systems with nonlinear dynamics and sampled-data information: a delayed-input approach[END_REF], [START_REF] Qin | Stationary consensus of asynchronous discrete-time second-order multi-agent systems under switching topol-ogy[END_REF]. There are four kinds of samplings that appears for MAS: synchronously periodic sampling (SPS) where all agents have the same periodic sampling on the input; asynchronously periodic sampling (APS) where each agent has its own periodic sampling on the input; synchronously variable sampling (SVS) where all agents have the same aperiodic sampling on the input; and asynchronously variable sampling (AVS) where each agent has its own aperiodic sampling on the input. The reader can find an overview on sampled-data MAS in [START_REF] Ge | Distributed networked control systems: A brief overview[END_REF], [START_REF] Ge | A survey on recent advances in distributed sampled-data cooperative control of multi-agent systems[END_REF].

To tackle the problem of stability of the double integrator with a sampled-data input, a new nonlinear emulation strategy based on homogeneous systems has been proposed in [START_REF] Bernuau | Stability of homogeneous nonlinear systems with sampled-data inputs[END_REF]. With the emulation method, the controller is first designed in continuous time and then implemented as a sampled-data controller [START_REF] Bemporad | Networked control systems[END_REF], [START_REF] Hetel | Recent developments on the stability of systems with aperiodic sampling: an overview[END_REF], [START_REF] Nešić | Explicit computation of the sampling period in emulation of controllers for nonlinear sampled-data systems[END_REF]. The strategy proposed in [START_REF] Bernuau | Stability of homogeneous nonlinear systems with sampled-data inputs[END_REF] for homogeneous nonlinear systems with sampled-data inputs cannot be directly used for MAS. Indeed, this paper only considers control laws leading to asymptotic stability in continuous time, whereas the consensus of MAS do not lead to asymptotic stability in continuous time.

The first task of this article is to adapt the result of [START_REF] Bernuau | Stability of homogeneous nonlinear systems with sampled-data inputs[END_REF] and prove a new theoretical result for the set stability of homogeneous nonlinear systems with sampled-data inputs and relevant for MAS. Then, we use this new result to obtain the practical consensus for homogeneous sampled-data MAS [START_REF] Dong | Practical consensus for highorder linear time-invariant swarm systems with interaction uncertainties, time-varying delays and external disturbances[END_REF] with undirected or directed graphs. We only ensure practical consensus due to the sampling of the inputs which can be arbitrary large. In practice, we need a controller for the continuous plant which renders the closed-loop MAS homogeneous of negative degree. This is precisely what is done by some finite time controllers by using the notion of finite time stability developed in [START_REF] Bhat | Finite-time of continuous autonomous systems[END_REF]. The finite time consensus of MAS has been studied for instance in [START_REF] Li | Finite-time consensus algorithm for multiagent systems with double-integrator dynamics[END_REF], [START_REF] Wang | Finite-time consensus for multi-agent networks with second-order agent dynamics[END_REF] for double-integrator dynamics. In our article, we use the homogeneous nonlinear controllers proposed in [START_REF] Wang | Finite-time consensus for multi-agent networks with second-order agent dynamics[END_REF], [START_REF] Zhao | Distributed finite-time tracking control for multi-agent systems: an observer-based approach[END_REF] in order to obtain the practical consensus of the second-order MAS with SVS, which has been studied with linear controllers in [START_REF] Xiao | Sampled-data consensus for multiple double integrators with arbitrary sampling[END_REF]. The main advantage of the proposed method relies on the fact that the maximum distance between the agents remains bounded when the sampling interval increases, unlike linear strategies. The main drawback is the loss of the exact consensus, replaced by a practical consensus; however, the maximal distance between agents is bounded by an explicit function on the maximum sampling period. The results are illustrated in simulations with the controller proposed in [START_REF] Zhao | Distributed finite-time tracking control for multi-agent systems: an observer-based approach[END_REF] in order to compare with emulated linear strategies, which fail to maintain the consensus when the sampling periods are too high.

The article is organized as follows. After some notations, definitions and preliminary results given in Section II, the main results are addressed in Section III. Then, they are illustrated in simulation in Section IV. Finally, a conclusion is provided in Section V.

II. PRELIMINARY Let us introduce the following notations:

• R + = {x ∈ R : x ≥ 0}, where R is the set of real numbers. • | • | denotes the absolute value in R, for p ≥ 1, • p
denotes the p-norm on R n and • = • 2 denotes the Euclidean norm on R n . • For any closed set Z ⊂ R n and any x ∈ R n , we denote d Z (x) = inf z∈Z xz the distance between x and the closed set Z.

• If x(t) is a curve in R n , we will say that x(t) → Z as t → +∞ when d Z (x(t)) → 0 as t → +∞. • We denote 1 n = (1, . . . , 1) T ∈ R n and ∆ n = Span(1 n ) ⊂ R n .
• r = (r 1 , . . . , r n ) is called a generalized weight if its components are positive numbers. • For any generalized weight r and for any λ > 0, we denote Λ r (λ) = Diag(λ r1 , . . . , λ rn ) where Diag(a 1 , . . . , a n ) denotes the diagonal matrix of dimension n × n with k th diagonal entry a k .

• ⌊x⌉ α = (|x 1 | α sign(x 1 ), . . . , |x n | α sign(x n ))
T where

x = (x 1 , . . . , x n ) T ∈ R n and α > 0.

• If A is an m × n matrix and B a p × q matrix, then the Kronecker product A ⊗ B is the mp × nq block matrix

A ⊗ B =    a 11 B • • • a 1n B . . . . . . . . . a m1 B • • • a mn B   .
• I n denotes the unit matrix of size n.

• If V ⊂ R n , we define the orthogonal complement of V by V ⊥ = x ∈ R n : x T y = 0 for all y ∈ V . Definition 1: Let η and h be positive numbers. An (η, h)sampling sequence is a sequence (t k ) k≥0 such that t 0 = 0 and for all k ≥ 0, η ≤ t k+1t k ≤ h.

A. Graph theory

Let us recall some basic definitions about graph theory given for instance in [START_REF] Ren | Distributed consensus in multi-vehicle cooperative control[END_REF]Appendix B].

A directed graph G N = (V N , E N ) consists of a finite nonempty set of nodes V N = {1, 2, . . . , N } and a set of edges E N ⊂ V N × V N which is a set of ordered pairs of nodes. An edge (i, j) ∈ E N in a directed graph G N denotes that node i communicates with node j, but not conversely. An undirected graph G N = (V N , E N ) also consists of a set of nodes V N = {1, . . . , N } and a set of edges E N ⊂ V N × V N which is an unordered set of pairs of nodes. An edge (i, j) ∈ E N in an undirected graph G N denotes that nodes i and j obtain information from each other. An undirected path is a sequence of edges in an undirected graph of the form (i 1 , i 2 ), (i 2 , i 3 ),• • • . An undirected graph is connected if there is an undirected path between every pair of distinct nodes.

The adjacency matrix of an undirected graph

(V N , E N ) is defined by A N = [a ij ] ∈ R N ×N where a ij = a ji = 1 if (i, j) ∈ E N
and a ij = 0 otherwise. The Laplacian matrix associated with adjacency matrix A N is given as

L N = [ℓ ij ] ∈ R N ×N where ℓ ii = N j=1,j =i a ij and ℓ ij = -a ij if i = j.

B. Lyapunov stability

Consider the following system with f continuous

ẋ = f (x), x ∈ R n . (1)
Let us extend to closed sets the definitions of Lyapunov set stability given for instance in [START_REF] Bhatia | Stability theory of dynamical systems[END_REF] for compact sets.

Definition 2: A closed set C ⊂ R n is:

• stable w.r.t. the system (1) if for any ε > 0 there exists η > 0 such that for any maximal solution x(t) of ( 1), if there exists t 0 such that d C (x(t 0 )) < η, then x(t) is defined for all t ≥ t 0 and d C (x(t)) < ε for all t ≥ t 0 ;

• locally attractive w.r.t. the system (1) if there exists ε > 0 such that for any maximal solution x(t) of ( 1), if there exists t 0 such that d C (x(t 0 )) < ε, then x(t) is defined for all t ≥ t 0 and d C (x(t)) → 0 when t → +∞; • globally attractive w.r.t. the system (1) if it is locally attractive and if the previous point holds for any ε > 0; • locally (resp. globally) asymptotically stable w.r.t. the system (1) if it is stable and locally (resp. globally) attractive w.r.t. the system (1); • unstable if it is not stable.

C. Homogeneity

The most common notion of homogeneity is the weighted homogeneity introduced in [START_REF] Rothschild | Hypoelliptic differential operators and nilpotent groups[END_REF], based on a particular choice of the coordinates, while the most generic one is the geometric homogeneity, which is coordinate free [START_REF] Bhat | Geometric homogeneity with applications to finite-time stability[END_REF], [START_REF] Kawski | Geometric homogeneity and stabilization[END_REF]. We use in the sequel the framework of weighted homogeneity.

Definition 3: Let V be a finite dimensional vector space and A : V → V an endomorphism of V such that the real part of each eigenvalue of A is positive. We will say that a function V :

V → V is A-homogeneous of degree d if V (exp(As)x) = e ds V (x), for all s ∈ R and x ∈ V. A vector field f defined on V is A-homogeneous of degree d if f (exp(As)x) = e ds exp(As)f (x), for all s ∈ R and x ∈ R n . If r = (r 1 , . . . , r n
) is a generalized weight, we will simply say that a function or a vector field defined on R n is r-homogeneous of degree d if it is A-homogeneous of degree d for the endomorphism defined for all i = 1, . . . , n by A(e i ) = r i e i , where e i denotes the i th vector of the canonical basis of R n .

Let us now give the definition of a homogeneous norm detailed for instance in [START_REF] Bernuau | Stability of homogeneous nonlinear systems with sampled-data inputs[END_REF].

Definition 4: A r-homogeneous norm is a positive definite and continuous mapping N : R n → R that is r-homogeneous of degree 1.

D. Technical lemmas

Lemma 5: Let M be a square matrix of size n and K a compact subset of R n such that K ⊂ im M . Then for any curve

x(t) ∈ R n , we have M x(t) → K if and only if x(t) → M -1 K. In particular, we have M x(t) → 0 if and only if x(t) → ker M . Proof. If M x(t) → K, there exists y(t) ∈ K such that d K (M x) = M x-y → 0. Then, there exists z(t) ∈ M -1 K such that M z = y and x -z ∈ (ker M ) ⊥ . Therefore M (x - z) → 0. Given that M is an isomorphism when restricted to (ker M ) ⊥ , we have x -z → 0 and then d M -1 K (x) → 0, that is x(t) → M -1 K. Conversely, if x(t) → M -1 K, there exists z(t) ∈ M -1 K such that x -z → 0. It yields that M x -M z → 0, which, given that M z ∈ K, shows that d K (M x) → 0, that is M x(t) → K. The last statement is immediate taking K = {0}.
Lemma 6: For any curve x(t) ∈ R n of coordinates x i (t), i = 1, . . . , n, we have x(t) → ∆ n if and only if for all i and j we have x i (t)x j (t) → 0.

Proof. Let us denote Π the orthogonal projection matrix onto ∆ ⊥ n . We easily see that Π = V V T with V ∈ R n×(n-1) a matrix of columns denoted v i and given by

v i = 1 √ i + i 2             1 . . . 1 -i 0 . . . 0            
where the -i is located on the (i + 1) th line. Lemma 5 yields x(t) → ∆ n if and only if Πx(t) → 0. But we also have Πx → 0 iff ∀i, v T i x → 0 iff ∀i, j, x ix j → 0, which concludes the proof.

III. MAIN RESULTS

Let us state the main theoretical result of this article. Theorem 7: Let ẋ = f (x, u) be a continuous controlled vector field. Assume that we know a continuous control u(x) such that:

1) there exists a linear subspace E such that for any

x 1 , x 2 , z 1 , z 2 ∈ R n , if x 1 -x 2 ∈ E and z 1 -z 2 ∈ E, then f (x 1 , u(z 1 )) -f (x 2 , u(z 2 )) ∈ E; 2)
the set E is globally asymptotically stable for ẋ = f (x, u(x)); 3) there exists an endomorphism of R n denoted A such that:

• the real part of each eigenvalue of A is positive;

• the linear subspaces E and E ⊥ are invariant subspaces of A; • there exists κ < 0 such that for all x, z ∈ R n and all s ∈ R we have f (exp(As)x, u(exp(As)z)) = e κs exp(As)f (x, u(z)). Then, for any (η, h)-sampling sequence (t k ) k∈N and any Ãhomogeneous norm N , there exists C > 0 such that any solution x(t) of

ẋ(t) = f (x(t), u(x(t k ))), t ∈ [t k , t k+1 ) (2)
converges to {ξ ∈ E ⊥ : N (ξ) ≤ Ch -1/κ } + E, where à denotes the restriction of A to E ⊥ .
Proof. We will denote π the orthogonal projection onto E ⊥ . From assumption 3, we have

A • π = π • A.
We define, for ξ 1 , ξ 2 ∈ E ⊥ the controlled vector field g(ξ 1 , u(ξ 2 )) = π(f (ξ 1 , u(ξ 2 ))). Remark that assumption 1 yields that, for any

ζ 1 , ζ 2 ∈ E, we have g(ξ 1 , u(ξ 2 )) = g(ξ 1 + ζ 1 , u(ξ 2 + ζ 2 )).
Let us first show that we can apply Theorem 12 in [START_REF] Bernuau | Stability of homogeneous nonlinear systems with sampled-data inputs[END_REF] to g.

From assumption 3, for ξ 1 and ξ 2 ∈ E ⊥ we get

g(exp( Ãs)ξ 1 , u(exp( Ãs)ξ 2 )) = π(f (exp( Ãs)ξ 1 , u(exp( Ãs)ξ 2 ))) = π(f (exp(As)ξ 1 , u(exp(As)ξ 2 ))) = π(e κs exp(As)f (ξ 1 , u(ξ 2 ))) = e κs exp(As)π(f (ξ 1 , u(ξ 2 )))
= e κs exp( Ãs)g(ξ 1 , u(ξ 2 )).

Consider ξ(t) ∈ E ⊥ a solution of ξ = g(ξ, u(ξ)) such that ξ(0) = ξ 0 ∈ E ⊥ . The vector field being continuous, ξ is absolutely continuous 1

. Now, denoting p ⊥ E the orthogonal projection onto E, consider a solution of ζ = p ⊥ E f (ξ+ζ, u(ξ+ ζ)) such that ζ(0) = 0. Clearly, ζ(t) ∈ E. The curve ξ(t) + ζ(t) is a solution of ẋ = f (x, u(x)). Indeed, ξ + ζ = g(ξ, u(ξ)) + p ⊥ E f (ξ + ζ, u(ξ + ζ)) = πf (ξ +ζ, u(ξ +ζ))+p ⊥ E f (ξ +ζ, u(ξ +ζ)) = f (ξ +ζ, u(ξ +ζ))
. This shows that, for any solution ξ(t) ∈ E ⊥ of ξ = g(ξ, u(ξ)), there exists a solution x of ẋ = f (x, u(x)) such that πx = ξ. Now, considering a curve x(t) ∈ R n such that πx(t) = ξ(t) and ẋ = f (x, u(x)), by assumption 2 we have x(t) → E and Lemma 5 yields that ξ(t) = πx(t) → 0. We conclude that the vector field defined by ξ = g(ξ, u(ξ)) is Ã-homogeneous and 0 ∈ E ⊥ is a globally attractive equilibrium, which implies it is a globally asymptotically stable equilibrium. Thus, we can apply Theorem 12 in [START_REF] Bernuau | Stability of homogeneous nonlinear systems with sampled-data inputs[END_REF] and there exists a constant C > 0 such that the solution of

ξ(t) = g(ξ(t), u(ξ(t k ))), t ∈ [t k , t k+1 ) (3) converges to the compact set {ξ ∈ E ⊥ : N (ξ) ≤ Ch -1/κ }.
Finally, given a solution x(t) of ( 2), by assumption 1, πx(t) is a solution of (3) and therefore πx(t) → {ξ ∈ E ⊥ : N (ξ) ≤ Ch -1/κ } which in turn shows, by Lemma 5, that

x(t) → π -1 {ξ ∈ E ⊥ : N (ξ) ≤ Ch -1/κ } = {ξ ∈ E ⊥ : N (ξ) ≤ Ch -1/κ } + E which concludes the proof.
Our aim is to apply Theorem 7 to MAS with control laws found in the literature. Let us first set some notations. The system under consideration is given by qi = p i ṗi = u i (4) with q i , p i ∈ R n and i = 1, . . . , N . Usually, a family of control laws u i (q, p) is said to achieve the consensus of the agents if in closed loop, for any i, j, q iq j → 0 and p ip j → 0. The problem with this definition is that it does not allow us to talk about stability. Let us find an equivalent formulation of the consensus. Given that, for any i = 1, . . . , N , q i and p i ∈ R n , let us write the coordinates of the vector q i (resp. p i ) as q k i (resp. p k i ) for k = 1, . . . , n. For a given k, let us denote r k = (q k 1 , . . . , q k N ) ∈ R N . Clearly, q iq j → 0 for all i, j is equivalent to r k ir k j → 0 for all i, j, k. From Lemma 6, this is equivalent to r k → ∆ N for all k. Now, let us remark that, denoting q = (q T 1 , q T 2 , . . . , q T N ) T ∈ R nN , we have q = n k=1 r k ⊗ e k . Hence r k → ∆ N for all k rewrites as q → Span(1 N ⊗e i , i = 1, . . . , n). The same reasoning applied to p = (p T 1 , p T 2 , . . . , p T N ) T ∈ R nN leads to the following reformulation of the consensus in the classical sense.

Definition 8 (Attractive consensus): The system (4) with control laws u i (q, p) is said to achieve the attractive consensus if q p → E where

E = Span 1 N 0 N ⊗ e i , 0 N 1 N ⊗ e j , i, j = 1, . . . , n , (5)
with 1 N = (1, . . . , 1) T ∈ R N and (e i ) 1≤i≤n the canonical base of R n . Definition 9 (Stable consensus): The system (4) with control laws u i (q, p) is said to achieve the stable consensus if the set E defined in ( 5) is globally asymptotically stable.

Definition 10 (Practical consensus): The system (4) with control laws u i (q, p) is said to achieve the practical consensus if there exists a non empty compact set K ⊂ R 2nN such that q p → K + E.

Remark 1: Let us stress that consensus (attractive, stable or practical) does not necessarily imply the convergence of the states of the individual agents. Only the differences between the agents have to decrease and it is very common in practice for the states of the agents to diverge.

Remark 2: For continuous linear controllers which render the closed-loop MAS linear as in [START_REF] Ren | Distributed multi-vehicle coordinated control via local information exchange[END_REF], the attractivity implies the global asymptotic stability and is enough for defining the consensus [START_REF] Ren | Distributed consensus in multi-vehicle cooperative control[END_REF]. With continuous nonlinear systems, though, we know that the mere attractivity does not imply Lyapunov stability [START_REF] Hahn | Stability of motion, ser. Die Grundlehren der mathematischen Wissenschaften[END_REF]Subsection 40]. In the sequel, we will only consider continuous controllers which render the closed-loop system homogeneous and we know that the attractivity implies the global asymptotic stability for homogeneous systems [START_REF] Bhat | Geometric homogeneity with applications to finite-time stability[END_REF]Proposition 6.1]. In this framework of continuous homogeneous systems, Definition 8 is equivalent to Definition 9. Finally, the definition of practical consensus is an extension of the notion of practical consensus used in the literature [START_REF] Dong | Practical consensus for highorder linear time-invariant swarm systems with interaction uncertainties, time-varying delays and external disturbances[END_REF].

We can now state the main result of this article dedicated to MAS.

Corollary 11: Let us consider the MAS (4). Assume known a family of continuous control laws u i (q, p) such that:

1) u i depends only on q iq j and p ip j ; 2) the control laws u i (q, p) applied to (4) achieve stable consensus; 3) there exists 1/2 < r < 1 such that u i (λq, λ r p) = λ 2r-1 u i (q, p). Consider an (η, h)-sampling sequence (t k ) k∈N . We also define A the endomorphism given in the canonical base of R 2nN by the matrix 1 0 0 r ⊗ I nN and à the restriction of A on E ⊥ . Then, for any Ãhomogeneous norm N , there exists C > 0 such that the vector

(q T 1 , . . . , q T N , p T 1 , . . . , p T N ) T ∈ R 2nN converges to {ξ ∈ E ⊥ : N (ξ) ≤ Ch -1/κ } + E, where q i and p i denote the solutions of qi (t) = p i (t) ṗi (t) = u i (q(t k ), p(t k )) , t ∈ [t k , t k+1 ). ( 6
)
Thus the the control laws u i (q, p) applied to (6) achieve practical consensus.

Proof. Let us apply Theorem 7. A direct computation shows that the assumption 1 of Corollary 11 implies the assumption 1 of Theorem 7, and we have obviously assumption 2 of Theorem 7 from assumption 2 of Corollary 11. The endomorphism A is clearly such that -A is Hurwitz. It is also straightforward to show that E is a stable space for A and given that A is symmetric, so is E ⊥ . Only the last point of assumption 3 of Theorem 7 remains to check. But indeed e rs p u(e s q, e rs p) = e rs p e (2r-1)s u(q, p) = e (r-1)s e s p e rs u(q, p) = e (r-1)s exp(As) p u(q, p) which, setting κ = r -1 < 0, concludes the proof.

Remark 3: Let us recall that we assume all agents to update their control simultaneously.

No assumption is made on the graph of the agents, and the result holds for undirected as well as directed graphs, as long as the three assumptions of Corollary 11 are verified.

Assumption 3 of Theorem 7 and Corollary 11 states that the closed loop system is homogeneous of negative degree. Such systems have been notably studied in the literature for their finite-time stability properties. It is easy to see that, should we take h = 0, that is continuous time control, we would recover these finite-time stability properties. Linear control laws would verify Assumption 3 provided that we allow r = 1. In addition, if the control laws are designed such that, in the assumption 3, r > 1, it is possible to show that we get local consensus instead of practical consensus (that is, exact consensus is achieved provided that the initial states of the agents are close enough). However, in practice, when the sampling period becomes large, this requires the initial states of the agents to be very close to consensus, hence drastically limiting the usefulness of this extension.

Remark 4: Let us denote (ε i ) 1≤i≤N the canonical base of R N and

M q ij = ((ε T i , 0 T N ) -(ε T j , 0 T N )) ⊗ I n , M p ij = ((0 T N , ε T i ) -(0 T N , ε T j )) ⊗ I n .
Denoting x = (q T , p T ) T , a direct computation shows that M q ij x = q iq j and M p ij x = p ip j . Moreover, the set E defined in ( 5) verifies E ⊂ i,j ker M p ij ∩ ker M q ij . Therefore, for any compact set K ⊂ E ⊥ , we have x(t) → K + E ⇒ M q ij x(t) = q iq j → M q ij K and then lim sup t→+∞ q iq j ≤ 2 sup z∈K z . Similarly, lim sup t→+∞ p ip j ≤ 2 sup z∈K z . This ensures that, under the conclusions of Corollary 11, the states of the agents remain asymptotically bounded by a class K function of h -1/κ .

Example 12: Suppose that we have an undirected and connected graph and consider the control laws for (4) given in [START_REF] Zhao | Distributed finite-time tracking control for multi-agent systems: an observer-based approach[END_REF] [START_REF] Gao | Sampled-data based consensus of continuous-time multi-agent systems with time-varying topology[END_REF] with 0 < α < 1, ℓ 1 > 0 and ℓ 2 > 0. The conclusions of Corollary 11 apply to these control laws. The second assumption of Corollary 11 is the only non-trivial to show. First notice that j a ij (q iq j ) = ((L N ⊗ I n )q) i and j a ij (p ip j ) = ((L N ⊗ I n )p) i , which allows us to rewrite the system under the form

u i = -ℓ 1     N j=1 a ij (q i -q j )     α -ℓ 2     N j=1 a ij (p i -p j )     2α (1+α)
q = p ṗ = -ℓ 1 ⌊(L N ⊗ I n )q⌉ α -ℓ 2 ⌊(L N ⊗ I n )p⌉ 2α (1+α)
. [START_REF] Huang | Some necessary and sufficient conditions for consensus of second-order multi-agent systems with sampled position data[END_REF] Let us now define the function

V (q, p) = ℓ 1 (L N ⊗ I n )q 1+α 1+α 1 + α + p T (L N ⊗ I n )p 2 .
A direct computation yields

V = ℓ 1 ((L N ⊗ I n )p) T ⌊(L N ⊗ I n )q⌉ α + p T (L N ⊗ I n ) × -ℓ 1 ⌊(L N ⊗ I n )q⌉ α -ℓ 2 ⌊(L N ⊗ I n )p⌉ 2α (1+α) = -ℓ 2 ((L N ⊗ I n )p) T ⌊(L N ⊗ I n )p⌉ 2α (1+α) = -ℓ 2 (L N ⊗ I n )p β β ≤ 0, with β = 3α+1
1+α . We now use the version of the LaSalle principle given in [21, Lemma 1]

: V ≡ 0 ⇒ (L N ⊗ I n )p ≡ 0 ⇒ (L N ⊗ I n ) ṗ ≡ 0 ⇒ (L N ⊗ I n )q ≡ 0.
Therefore E is globally asymptotically stable and Corollary 11 can be applied.

Remark 5: If the graph is supposed to be oriented, the method used in Example 12 does not work anymore. Indeed, the matrix L N is not symmetric and the function V is therefore not positive anymore.

Example 13: Suppose that we have an undirected and connected graph and consider the control laws for (4) given in [START_REF] Wang | Finite-time consensus for multi-agent networks with second-order agent dynamics[END_REF]

u i = -ℓ 1 N j=1 a ij ⌊q i -q j ⌉ α -ℓ 2 N j=1 a ij ⌊p i -p j ⌉ 2α (1+α) . (9)
The conclusions of Corollary 11 apply to these control laws. This is straightforward, given that the article [START_REF] Wang | Finite-time consensus for multi-agent networks with second-order agent dynamics[END_REF] proves the consensus for this control laws family.

IV. SIMULATIONS Consider the second-order MAS (6) with N = 5 and n = 2, i.e. 5 agents in the plane whose communications are described by the undirected graph given on Figure 1. In the following,

t k = kT, T > 0, k ∈ N. (10)
The initial positions q i0 are chosen randomly in the square [0, 100] × [0, 100]. Consider the linear controllers

u i (q, p) = - 5 j=1 a ij (q i -q j ) - 5 j=1 a ij (p i -p j ) (11)
with q i = (q ix , q iy ) and p i = (p ix , p iy ) for i = 1, . . . , 5 which ensure the stable consensus (see for instance [START_REF] Yu | Consensus in multiagent systems with second-order dynamics and sampled data[END_REF]).

The trajectories of the sampled-data closed-loop MAS ( 6)- [START_REF] Wen | Consensus of multi-agent systems with nonlinear dynamics and sampled-data information: a delayed-input approach[END_REF] with SPS (10) are drawn in Figure 2 in the plane (q ix , q iy ) for T = 0.1s. After a while, the trajectories turns into uniform linear motions. The evolutions of positions q ix of the 5 agents are given in Figure 3 and the evolutions of velocities p ix of the 5 agents are given in Figure 4. We know that the linear closed-loop MAS ( 6)- [START_REF] Wen | Consensus of multi-agent systems with nonlinear dynamics and sampled-data information: a delayed-input approach[END_REF] with SPS (10) is asymptotically stable if and only if the matrix Λ(T ) of the linear difference equation associated with ( 6)-( 11)- [START_REF] Yu | Second-order consensus in multi-agent dynamical systems with sampled position data[END_REF] and defined in [START_REF] Hetel | Recent developments on the stability of systems with aperiodic sampling: an overview[END_REF] is Schur. For our 5 agents, Λ(T ) is Schur if and only if T < T Schur = 0.4795s. It means that the fleet of 5 agents with the linear controllers (11)-( 10) is unstable and disperses into the space when T > 0.4795s.

By using the controllers [START_REF] Gao | Sampled-data based consensus of continuous-time multi-agent systems with time-varying topology[END_REF] with ℓ 1 = ℓ 2 = 1, α = 0.5 and SPS [START_REF] Yu | Second-order consensus in multi-agent dynamical systems with sampled position data[END_REF] we obtain the practical consensus of the 5 agents as shown in Figures 56-7 for T = 0.5s. In this case, the behavior of the 5 agents reaching the practical consensus is very close to the behavior of the 5 agents reaching the stable consensus.

We obtain a result of practical consensus for higher sampling periods, as shown in Figures 89-10 for T = 2s, but the Fig. 5. Trajectories of the 5 agents in two-dimensional space (q ix , q iy ) with homogeneous controllers [START_REF] Gao | Sampled-data based consensus of continuous-time multi-agent systems with time-varying topology[END_REF] and T = 0.5s x Fig. 6. Positions q ix of the 5 agents with homogeneous controllers [START_REF] Gao | Sampled-data based consensus of continuous-time multi-agent systems with time-varying topology[END_REF] and T = 0.5s trajectories of the 5 agents are no more uniform linear motions and the velocities fluctuate. However, the fleet of 5 agents with the homogeneous controllers ( 7)-( 10) is not unstable and does not disperse into the space.

V. CONCLUSION

In the article, the second-order practical consensus of MAS under synchronously variable sampling is developed by using a new nonlinear emulation strategy based on homogeneity. The results obtained with this nonlinear strategy are compared in simulation with the linear one.

Fig. 1 .

 1 Fig. 1. Undirected graph of the 5 agents MAS we choose constant sampling periods

Fig. 2 .Fig. 3 .Fig. 4 .

 234 Fig.2. Trajectories of the 5 agents in two-dimensional space (q ix , q ) with linear controllers[START_REF] Wen | Consensus of multi-agent systems with nonlinear dynamics and sampled-data information: a delayed-input approach[END_REF] and T = 0.1s

Fig. 7 .Fig. 8 .Fig. 9 .Fig. 10 .

 78910 Fig. 7. Velocities p ix of the 5 agents with homogeneous controllers (7) and T = 0.5s

See[START_REF] Turner | Mathematical Methods for Robust and Nonlinear Control[END_REF] Page 198]


Alexandre Guthertz

Jonathan Lusseau

Dr Valérie Desvergnes

Dr Stéphane Massip

Prof. Dr Yannick Landais
email: yannick.landais@u-bordeaux.fr

Dr A Guthertz

An Approach towards the Synthesis of the Spiroimine Fragment of 13-Desmethylspirolide C and Gymnodimine A

Keywords: Phycotoxin, Enantioselective Diels-Alder, Wittig-Horner reaction, Spiroimine, nitrile

come

regions of both the toxins and receptors. These studies suggest that the spiroimine moiety is the main pharmacophore and plays an important role in the biological activities of these phycotoxins. 2e The synthesis of spirolide fragments, but also total synthesis of phycotoxins is still at the heart of an intense research activity, both in biology and organic synthesis. While the total synthesis of gymnodimine 2 has been completed by the Romo's group [START_REF] Kong | [END_REF] and that several groups have reported their efforts toward the synthesis of spirolides, [9,10] no total synthesis of spirolide has ever been described. In this context, we recently embarked on the synthesis of the various fragments of 1 and 2, following general strategies depicted in Figure 2. We achieved the synthesis of the bis-spiroketal [11] and tetrahydrofuran [12] cores of phycotoxins 1 and 2 respectively (Figure 2) and report here our efforts to access the spiroimine moiety of 2. Our general approach should in theory allow an access to spiroimines of both phycotoxin's families and relies on the use of the Ward enantioselective Diels-Alder reaction [13] between an acrylate and a suitably functionalized dienol, carried out in the presence of a BINOL-type ligand. Using this strategy, the preparation of a spiroimine skeleton precursor of natural product 2 (closely related to B, Figure 2) was thus achieved through a 16 steps linear sequence and a remarkable 21% overall yield.

Results and Discussion

Enantioselective Diels-Alder based construction of the cyclohexene fragment

The major difficulty in the synthesis of the spiroimine fragment is the introduction of the all-carbon quaternary stereocenter (C29 in 1, C22 in 2). Diels-Alder's reaction has been widely used to meet this synthetic challenge. Kishi's [14] pioneering biomimetic approach to gymnodimine employed a late stage intramolecular Diels-Alder process to assemble 2, albeit with low selectivities. Early studies by White et al. [15] used an intermolecular Diels-Alder reaction between a suitably substituted Meldrum acid as a dienophile partner to construct the cyclohexene fragment of 2, which also led to low diastereocontrol. More recently, however, a copper catalyzed Diels-Alder reaction enabled Romo and co-workers [START_REF] Kong | [END_REF] to set up the quaternary center with the correct configuration of the natural product with very good enantioselectivities. In the spirolide series, Brimble et al. also used a Diels-Alder approach to elaborate a functionalized cyclohexene with the quaternary stereocenter with a satisfying diastereocontrol, using a Danishefsky diene and a chiral dienophile. [10] Other approaches based on nucleophilic processes (Ireland-Claisen rearrangement Birch reductive alkylation, alkylation) respectively by Zakarian, [16] Guillou [17] and Hirama [18] and co-workers were used to elegantly elaborate the 6-membered ring having the all-carbon quaternary stereocenter. Our preliminary experiments were designed to incorporate the quaternary stereocenter during the Diels-Alder process using dienophile 3 [19] and diene 4 [20] as precursors in the presence of MeMgBr and Me2Zn and (R)-BINOL as described in Ward's standard conditions. [13] The expected cyclohexene 5 was obtained in good yield, albeit with low enantiocontrol, after 3 days at room temperature (Scheme 1). Absolute configuration was assumed to be that observed by Ward et al. during their seminal studies using the same (R)-BINOL. [13] Scheme 1. BINOL-mediated Diels-Alder reaction with diene 4.

Anticipating a deleterious effect of the CH2OPMB substituent in 3, which likely hinders the required assembly of the partners around the Mg-Zn-BINOL complex, a second approach was conceived using simple acrylate 6a-b as dienophiles. Formation of 8 would then be followed by alkylation -to the lactone to set-up the quaternary stereocenter. Lewis Acid, T (°C) [a] Yield [%] [b] e.e.

[%] [c] 1 6a 7a Me2Zn, MeMgBr, 20°C 58 70 2 6a 7a Me2Zn, MeMgBr, 20°C [d] [a] Typical conditions: diene 4 (1 equiv.), 6 (15 equiv.), BINOL 7a-d (1 equiv.), MeMgBr (1 equiv.) or AlMe3 (1 equiv.), Me2Zn (1 equiv.) in a 0.02 M CH2Cl2 at the given temperature for 2 days. [b] Yield of isolated product. [c] Enantioselectivity measured through HPLC on a Chiralcel OD-A. [d] 1.3 equiv. of Me2Zn was used.

[e] C = 0.04 M instead of 0.02 M.

These preliminary studies showed that, under standard Ward's conditions, a satisfying yield and a moderate enantioselectivity was observed (Table 1, entry 1). Increasing the amount of zinc reagent led to an improved yield (entry 2), albeit with a drop in enantioselectivity, which may be ascribed to a competitive Zn-catalysis in the absence of the BINOL ligand. Lowering the temperature decreased dramatically the kinetic of the process (entry 3). When the MeMgBr-ZnMe2 couple was replaced with AlMe3, results were slightly improved (compare entry 4 vs 1), but again as in entry 3, lowering the temperature led to rapid decrease of the reactivity (entry 5). Both the yield and enantioselectivity were slightly improved using acrylate 6b (entry 6), emphasizing on the subtle influence of steric effect on the reaction outcome. As above, lowering the temperature had a deleterious effect (entry 7) and the use of AlMe3 did not improve further the process (entry 8). Finally, an increase of the concentration led to quantitative yield, albeit with a modest selectivity (entry 9). Although a reasonable yield and enantioselectivity were attained using 6b as a dienophile in conditions summarized in entry 6, the level of enantioselectivity did not meet the requirement for a total synthesis. We therefore continued further the optimization, looking at the effect of the BINOL structure [21] on the enantioselectivity of the process. Interestingly, the use of BINOL 7b instead of 7a produced the other enantiomer of 8, albeit with a lower selectivity (entry 10). Although we have no clear explanation for this reversal of enantioselectivity, it is anticipated that the increasing steric hindrance in 7b modifies the coordination around zinc, and consequently the topicity of the process. Using AlMe3 as an additive and BINOL-7b restored the enantioselectivity, leading to the same enantiomer as that using 7a (entry 11). Similar trends were followed with BINOL-7c (entries-12-13), and finally sterically hindered BINOL-7d led to nearly racemic compound (entry 14). During the course of this work, Ishihara, Hatakeyama and co-workers [22] reported the same reaction using partially hydrogenated H8-BINOL-7e. [23] In our hand, this ligand, which can now be used in catalytic amount, led to high and reproducible yields and enantioselectivities on scales ranging from 100 mg to 6 g. This improvement in the enantioselectivity may be rationalized invoking an increase of the dihedral angle for H8-BINOL as compared to BINOL. [24] Scheme 2. H8-BINOL 7e-catalyzed Diels-Alder reaction between dienol 4 and acrylate 6b.

We studied in parallel another diene 9 bearing a silyl enol ether, [25] anticipating a higher reactivity of this diene as well as the formation of a more functionalized cyclohexene as final product, which would be beneficial later in the synthesis (for instance for the coupling with the butenolide fragment at C5).

As depicted in Scheme 3, Diels-Alder reaction between diene 9 and acrylate 6a under Ward's conditions effectively led to silyl enol ether 10 in satisfying yield. Unfortunately, we were unable to estimate the enantioselectivity of this reaction, even after hydrolysis of the enol ethers, which led to a mixture of epimeric ketones. Therefore, considering the high yield and good level of enantiocontrol obtained using H8-BINOL 7e in cycloaddition between 4 and 6b, this approach was selected for the remaining part of our study (vide infra).

Scheme 3. Diels-Alder reaction between dienol 9 and acrylate 6a.

Functionalization of the cyclohexene core

The installation of the all-carbon quaternary stereocenter was performed through aldolisation or alkylation of the cyclohexene 8, using LDA as a base and formaldehyde [26] or BOMCl [27] respectively as electrophiles, affording compounds 11 and 12 in excellent yields as single cis-diastereomers (Scheme 4). Protection of the alcohol function of 11 as a PMB ether under acidic conditions led to 5 in high overall yield. Alternatively, aldol reaction from 8, under similar conditions, using acetaldehyde, led to alcohol 13 as a mixture of diastereomers, albeit with low diastereocontrol. Oxidation of 13 under Swern conditions followed by the protection of the keto functional group with ethane-1,2-diol eventually led to cyclic ketal 14 in excellent yield.

Scheme 4. Installation of the all-carbon quaternary stereocenter.

The rapid assembly of fused lactones 5, 12 and 14 opens an access either to the spiroimine moiety of spirolides for the former or that of gymnodimines for the latter. We thus concentrated our efforts on the elaboration of 14 in order to access the spiroimine skeleton of 2. The formation of the spirane system in spiroimine I (Figure 3) would require the incorporation early in the sequence of the C30-C32 aminoalkyl chain. This could be realized either through a nucleophilic addition at C30 of lactone 14 of a fragment containing a nitrile functional group or via the coupling of a vinyl-metal at C30 of the lactone as in III (Figure 3). Addition of a vinyl Grignard reagent was thus attempted after prior reduction of 14 into the corresponding lactol 15 using DIBAL-H (Scheme 5). No conversion of 15 was observed whatever the conditions. The use of vinyl-lithium led to the same observation, emphasizing on the difficult functionalization of this sterically hindered lactone. More satisfying results were obtained using the allyl Grignard analogue, [28] which led to the corresponding diol 16 as a mixture of two diastereomers, albeit with a modest d.r. A high yield and a similar diastereocontrol was observed with lactol 17 issued from 5, which led to the formation of diol 18. Our plan then relying on the oxidative cleavage of the double bond, to access the corresponding aldehyde, deoxygenation at C30 was first required. Selective protection of the primary alcohol was carried out on both compounds 16 and 18 (Scheme 6). Silylation of 16 using standard conditions (TBSOTf, NEt3) affording essentially the bis-silylation product, the selective protection was carried out through the treatment of the with 2 equiv. [29] followed by the addition of 1 equiv of TBSCl at low temperature, leading to 19 in excellent yield. These conditions were inefficient with diol 18, which was finally monosilylated into 20, using the corresponding triflate in the presence of Et3N. Deoxygenation at C30 was then tested. However, all our attempts to remove this hydroxy group, either through radical deoxygenation of a xanthate, or deoxygenation of the corresponding ketone through a Wolff-Kishner process, unfortunately failed. Scheme 6. Selective silylation of diols 16 and 18.

An alternative strategy was thus devised, which would afford the correct chain length, based on the addition of a carbanion -to a nitrile as depicted in Figure 3. It was anticipated that the small size of the CN group would allow the approach of the anion at C30. [START_REF]A-value for the CN group has been estimated to be 0[END_REF] Following this plan, the addition of the C31-C32 aminoalkyl chain was realized by simply adding the anion of acetonitrile onto the model lactone 5 (Scheme 7). This led to the formation of lactol 21, which structure was assigned through 1 H and 13 C NMR as well as X-ray diffraction studies. Our efforts to open the lactol in 21 and protect the alcohol at C8 met with failure. Better results were obtained by directly quenching the reaction mixture after the first step (addition of acetonitrile anion) with a chlorosilane. Following this procedure, nitrile 22 was obtained in modest yield. The same approach was extended to lactone 14, which led first to silylenol ether 23, the treatment of which afforded nitrile 24 using TBAF at low temperature. With the C30-C32 chain installed, the ensuing removal of the free ketone proved to be tedious, as experienced above with the hydroxy group in 16 and 18. Formation of a tosyl-hydrazone or reduction of the ketone with NaBH4 or LiBH4 all failed likely as a result of the steric hindrance. Addition of a 1,2dithiol or the Lawesson reagent also failed, emphasizing on the difficulties to functionalize the carbon center next to the quaternary stereocenter.

The "nitrile" routes. An access to the gymnodimine spiroimine fragment

Considering our difficulty to modify the resident functionalities in the vicinity of the quaternary center at C22, but also the option to add nucleophiles at C30 (Schemes 5 and 7), we envisioned elaborating the C30-C32 aminoalkyl chain through an olefination process at the C30 center. This approach would allow the incorporation of the remaining 2-carbons and avoid a late and tedious manipulation of a functional group at C30. The selective silylation of lactol 15 (Scheme 5) was first attempted to generate the corresponding aldehyde at C30, ready for the olefination process, but this failed whatever the conditions. The lactone precursor 14 was thus reduced instead into the corresponding diol 25 using LiAlH4 (Scheme 8). The selective silylation at C8 was then performed as described above for diol 16, [29] to give the alcohol 26 in high yield. The oxidation of the latter using Dess-Martin periodinane led to the desired aldehyde 27 in 84% yield. The more economical Swern oxidation was however preferred to produce 27 on larger scale, but also in higher yield. 27 was finally submitted to a Wittig-Horner olefination using cyanophosphonate 28, [15] affording the unsaturated nitrile 29 in quantitative yield as a single E-isomer. It is worth noticing that the olefination using the ester analogue of 28 under similar conditions led to full recovery of 27, highlighting the influence of the size of the CN substituent in these nucleophilic additions onto a sterically congested aldehyde. [START_REF]A-value for the CN group has been estimated to be 0[END_REF] Scheme 8. Elaboration of the C30-C32 chain using a Wittig-Horner approach.

The next step involved the chemoselective hydrogenation of the unsaturated nitrile in the presence of the cyclohexene double bond. Various conditions were tested, including single electron transfer process using Mg in MeOH, [START_REF] Ornstein | [END_REF] but the best results were obtained with hydrogenation using H2 under Pd-C catalysis. Treatment of 29 with H2, Pd-C in ethyl acetate led to a mixture of the desired saturated nitrile 30 in equimolar amount with the fully hydrogenated compound. Finally, using Et2NH as a cosolvent led to the selective hydrogenation of the electron-poor olefin affording 30 in a 88% yield (Scheme 9). The nitrile was then reduced into the corresponding amine 31 using LiAlH4 and the free amine protected as a phthalimide to afford 32 in high yield. [32] The deprotection of the primary alcohol in 32 in the presence of an acid-sensitive ketal proved to be more troublesome than expected and after extensive studies, this was finally accomplished using an excess of TBAF (6 equiv.) in acetic acid, with acetic anhydride added to trap residual water present in the ammonium fluoride. [33] 2 g of alcohol 33 could thus be prepared in 90% yield (96% brsm), albeit at a slow rate. 33 was then oxidized using Ley's TPAP reagent [34] to afford aldehyde 34. At this stage, we envisioned the conversion of 34 into the corresponding alkyne (C8-C9), a potent precursor of the vinyl iodide (i.e. spiroimine B, Figure 2) required for the Nozaki-Hiyama-Kishi coupling. All our efforts to transform the C8 aldehyde following Ohira-Bestmann [35] or Corey-Fuchs [36] procedures to install the triple bond however failed, leading to no reaction or complete degradation of 34. The latter was eventually engaged in a Wittig-Horner reaction with phosphonate 35 to give the unsaturated nitrile 36 as a 3:1 mixture of Z/E stereoisomers, [START_REF] Yadav | The vinylic proton of the Z-isomer appears at upper field as compared with the same proton in the E-isomer. For related examples[END_REF] thus installing the C8-C9 fragment. It is worth mentioning at this point that the reaction with the ester analogue of 35 led to degradation of 34. Phosphonium reagents did not react with 34, while Peterson elimination using the enolate of MeCH(SiMe3)CO2Et (LDA, THF, -78°C) led to a 6:4 E/Z mixture of the ester analogue of 36 in only 39% yield. [38] Nitrile 36 was finally converted into spiroimine 38 in high yield, after removal of the phthalimide group using hydrazine, [39] followed by the ketal deprotection and ring closure of the corresponding amine 37 under acidic conditions. The synthesis of 38 has thus been realized through a linear 16 steps sequence in a remarkable 21% overall yield from dienol 4. 38 constitutes a valuable synthon for further incorporation into the preparation of the macrocyclic core of gymnodimine 2.

Scheme 9. Final steps in the synthesis of the spiroimine core of gymnodimine 2.

Conclusions

In summary, we reported a general access to the fully functionalized cyclohexene skeleton of spirolides and gymnodimine, macrocyclic phycotoxins produced by the dinoflagellate Alexandrium ostenfeldii and peruvianium. The key-cyclohexene fragment is elaborated relying on an enantiocontrolled Diels-Alder process between readily available dienol and acrylates in the presence of H8-BINOL as a chiral inducer. This step was performed on multi-gram scale, furnishing the desired cyclohexene in high yield and enantioselectivity. The all-carbon quaternary stereocenter was then installed through an alkylation or an aldolisation with complete diastereocontrol. The C30-C32 aminoalkyl chain was elaborated based on a Wittig-Horner process between a cyanophosphonate and the aldehyde function at C30. A related olefination process was used to install the C8-C9 vinylic moiety. Further manipulation finally allowed the ring closure and access to the spiroimine core of gymnodimine 2. This study thus opens an access to one of the three key-synthons of 2 in high overall yield, also highlighting the difficulties to manipulate functional groups at the neighborhood of an all-carbon quaternary stereocenter. This hurdle was finally circumvented using two different cyanophosphonates to install C8-C9 and C31-C32 fragments relying on the small size of the cyano group. The access to synthon 38 has somewhat modified the original plan based on a Nozaki-Hiyama-Kishi coupling (Figure 2). Although intermediate 38 is not appropriate for a further conversion into the spirolide spiroimine fragment, the present Diels-Alder strategy has the potential for application to the elaboration of other spiroimine systems. Intermediate 8 may for instance be alkylated with the suitable C31-C33 aminoalkyl chain present in 1. Efforts are now underway to isomerize synthon 38 under photochemical conditions [40] and test a direct coupling [41] between the tetrahydrofuran fragment and the C10 nitrile to create the C10-C11 bond and link THF and spiroimine fragments.

Experimental Section Preparation of lactone 5b from acrylate 3b and dienol 4

To a solution of dienol 4 (57.8 mg, 0.589 mmol) in CH2Cl2 (3 mL) was added a 0.85 M solution of Me2Zn in toluene (0.69 mL, 0.589 mmol) at 0 °C. In a separated fask, a 2.5 M solution of MeMgBr in Et2O (0.23 mL, 0.589 mmol) was added at 0°C to a solution of (R)-BINOL (168.9 mg, 0.589 mmol) in CH2Cl2 (3 mL). After 5 min., the solution of the first vessel was added at 0°C to the solution of MeMgBr/BINOL. After 5 min., the resulting reaction mixture was diluted with CH2Cl2 (10 mL) and the acrylate 3b (737 mg, 2.94 mmol) in CH2Cl2 (10 mL) was added. The reaction mixture was allowed to warm to room temperature and stirred for 2 days. The reaction was then quenched by addition of saturated NaHCO3 (ca. 2 mL). The resulting mixture was filtrated over Na2SO4/Celite (1/1), washed with CH2Cl2 and concentrated under vacuum. 4, 159.3, 131.8, 130.0, 129.3, 123.4, 113.1, 73.3, 73.3, 71.1, 55.4, 47.3, 41.9, 24.3, 21.9, 21.6 6, 180.3, 132.0, 131.9, 124.0, 123.5, 123.3, 77.2, 71.8, 71.6, 71.5, 69.7, 51.4, 50.6, 42.7, 40.5, 24.0, 23.8, 21.9, 21.8, 18.8, 18.3; HRMS (ESI) calculated for C11H16O3Na [M+Na] + 219.0991, found : 219.0984.

Unsaturated nitrile 29

To a solution of diethyl cyanomethylphosphonate 28 (0.215 mL, 1.35 mmol,) in THF (6 mL) was added a 2.1 M solution of n BuLi in hexane (0.5 mL, 1 mmol) at 0°C. The resulting mixture was stirred at this temperature for 1 h, then the aldehyde 27 (124 mg, 0.35 mmol) was added at 0°C in THF (3 mL). The reaction mixture was slowly allowed to warm to RT overnight. The reaction mixture was quenched with a sat. aqueous solution of NH4Cl (20 mL), and the aqueous layer extracted with Et2O (3 x 10 mL). The combined organic layers were washed with brine, dried over Na2SO4 and concentrated under vacuum. The crude residue was purified by flash chromatography (SiO2, PE-AcOEt 95:5) to give the 29 (133 mg, 100%) as a colorless oil. 2, 132.5, 123.6, 118.5, 113.2, 98.5, 65.6, 64.2, 62.1, 51.1, 45.1, 26.0, 24.6, 22.9, 22.7, 20.9, -5.4, -5.5; HRMS (ESI) calculated for C21H35NO3SiNa [M+Na] + 400.2278, found : 400.2280.

Spiroimine 38

To solution of protected ketone 37 (36.9 mg, 1.12 mmol) in THF (2.5 mL) was added 1 M HCl in water (0.36 mL, 0.36 mmol) and the mixture was stirred overnight at RT. The reaction mixture was then quenched with a sat. aqueous solution of NaHCO3 (2 mL) to reach pH≈7, and the aqueous layer was extracted with AcOEt (3 x 5 mL) then CH2Cl2 (3 x 5 mL). The combined organic layers were washed with brine, dried over Na2SO4 and concentrated under vacuum. The crude residue was purified by flash chromatography (SiO2, DCM-MeOH-NH4OH 95:4:1) to afford the desired spiromine 38 as a as a yellow oil and a 3:1 mixture of E/Z isomers (28mg, 96%).

[α] 25 D=+55.08 (c=1.3,CHCl3). Rf= 0.5 (DCM-MeOH-NH4OH 95:4:1); IR (ATR) max (cm -1 ) 2930, 2853, 2215, 1643, 1436, 917, 729; 1 H NMR (300 MHz, CDCl3)  (ppm) 6. 25-6.13 (m, 0.22H), 6.04-5.88 (m, 0.78H), 5.54-5.45 (m, 1H), 3.58-3.28 (m, 3H), 2.07-1.96 (m, 5H), 1.91 (s, 2.3H), 1.89 (s, 1.4H), 1.86-1.61 (m, 2.4H), 1.56-1.37 (m, 7.6H); 13 C NMR (75 MHz, CDCl3)  (ppm) 172. 5, 148.0, 147.7, 131.3, 131.1, 123.4, 122.9, 118.2, 111.5, 49.3, 49.2, 48.5, 44.9, 41.5, 31.2, 31.0, 26.7, 26.5, 23.4, 23.4, 22.7, 22.5, 22.0, 22.0, 20.6, 20.0, 19.7, 15.5

Figure 1 .

 1 Figure 1. 13-Desmethylspirolide C 1 and Gymnodimine A 2.

Figure 2 .

 2 Figure 2. General disconnection strategies to access 13-desmethylspirolide C 1 and gymnodimine 2.

Figure 3 .

 3 Figure 3. Introduction of the C30-C32 aminoalkyl chain.

Scheme 5 .

 5 Scheme 5. Allylation of lactones 5 and 14.

Scheme 7 .

 7 Scheme 7. Addition of the acetonitrile carbanion to install the C30-32 chain.

  [α] 25 D=+121.07 (c=1.27, CHCl3). Rf= 0.41 (PE-AcOEt 95:5); IR (ATR) max (cm -1 ) 3956, 2221, 1258, 1092, 1020, 835, 797; 1 H NMR (300 MHz, CDCl3)  (ppm) 7.23 (d, J = 17.1 Hz, 1H), 5.48 (s, 1H), 5.36 (d, J = 17.1 Hz, 1H), 4.00 -3.73 (m, 4H), 3.68-3.57 (m, 2H), 2.28 (s, 1H), 2.13 -1.81 (m, 3H), 1.74 (d, J = 1.8 Hz, 3H), 1.71 -1.65 (m, 1H), 1.17 (s, 3H), 0.88 (s, 9H), 0.02 (d, J = 3.7 Hz, 6H); 13 C NMR (75 MHz, CDCl3)  (ppm) 161.

Table 1 .

 1 Diels-Alder reaction between dienol 4 and acrylates 6a-b.

	Entry	6	BINOL 7

  ; HRMS (ESI) calculated for C10H14O3Na [M+Na] + 325.1410, found : 325.1418.Lactone 13To a solution of diisopropylamine (2.37 mL, 16.76 mmol) in THF (50 mL) was added a 1.8 M solution of n BuLi in hexane (9.3 mL, 16.76 mmol) at -30°C. The reaction mixture was stirred at -30 °C for 15 min. then cooled to -78 °C and a solution of lactone 8 (1.7 g, 11.17 mmol) in THF (35 mL) was added. The light yellow solution was warmed up to -30 °C in 30 min. and cooled to -78 °C again. Acetaldehyde (1.88 mL, 33.51 mmol) was added. The reaction mixture was then warmed up to 0°C overnight. The reaction mixture was quenched with an aqueous solution of NaHCO3 (100 mL), extracted with Et2O (3 x 50 mL). The combined organic layers were washed with brine, dried over Na2SO4 and concentrated under vacuum. The crude residue was purified by flash chromatography, (SiO2, PE-AcOEt 80:20 to 70:30) to give 13 as a light yellow oil and a 1.

	4:1 mixture of 2
	diastereoisomers (2.08 g, 95%). [α] 25 D=-160.75 (c=1.0, CHCl3). Rf= 0.2 (PE-AcOEt 85:15); IR (ATR) max (cm -1 )
	3455, 2918, 1759, 1667, 1375, 1150,1012, 785; 1 H NMR (300 MHz, CDCl3)  (ppm) 5.58 (s, 1H), 4.51 (td, J =
	8.3, 3.8 Hz, 1H), 3.98-3.74 (m, 2H), 3.01-2.76 (m, 2H), 2.68-2.46 (m, 1H), 2.11-1.85 (m, 2H), 1.75-1.54 (m, 5H),
	1.32 (d, J = 6.5 Hz, 1H), 1.22 (d, J = 6.4 Hz, 2H); 13 C NMR (75 MHz, CDCl3)  (ppm) 181.

  ; HRMS (ESI) calculated for C16H22N2Na [M+Na] + 265.1675, found : 265.1678.