He ´lène Arambourou
email: helene.arambourou@irstea.fr

Inmaculada Fuertes

Emmanuelle Vulliet

Gae ¨lle Daniele

Patrice Noury

Nicolas Delorme

Khedidja Abbaci

Carlos Barata

Fenoxycarb exposure disrupted the reproductive success of the amphipod Gammarus fossarum with limited effects on the lipid profile

Insect growth regulator insecticides mimic the action of hormones on the growth and development of insect pests. However, they can affect the development of non-target arthropods.

In the present study, we tested the effects of the growth regulator insecticide fenoxycarb on several endpoints in the freshwater crustacean Gammarus fossarum (Amphipoda). Females carrying embryos in their open brood pouch were exposed to 50 μg L -1 fenoxycarb throughout the entire oogenesis (i.e. 21 days). After exposure, newborn individuals from exposed embryos were removed from the maternal open brood pouch for lipidomic analysis, while males were added to assess the reproductive success. After fertilization, the lipid profile, energy reserve content (lipids, proteins and glycogen), and activity of phenoloxidase an enzyme involved in the immune responsewere measured in females. No significant effect of fenoxycarb exposure was observed on the lipid profile of both newborn individuals and females, while reproductive success was severely impaired in exposed females. Particularly, precopulatory behavior was significantly reduced and fertilized eggs were unviable. This study highlighted the deleterious effects of the insect growth regulator fenoxycarb on gammarid reproduction, which could have severe repercussions on population dynamics.

Introduction

The use of growth regulator insecticides having juvenoid activity is increasing because they have a selective mode of action and some pests have developed resistance against classical neurotoxic insecticides [START_REF] Nauen | New mode of action of insecticides[END_REF]. They are used in agriculture for pest management [START_REF] Carlson | The chemical and biological properties of methoxyfenozide, a new insecticidal ecdysteroid agonist[END_REF], in public health for vector control [START_REF] Kroeger | Effective control of dengue vectors with curtains and water container covers treated with insecticide in Mexico and Venezuela: cluster randomised trials[END_REF] and in veterinary treatment to prevent pet infestation [START_REF] Beugnet | Insecticide and acaricide molecules and/or combinations to prevent pet infestation by ectoparasites[END_REF]. Juvenoid insecticides mimic the action of juvenoid hormones on the growth and development of insect pests. Two hormonally regulated pathways are generally targeted: the juvenile hormone pathway and the ecdysteroid hormone pathway [START_REF] Dhadialla | New insecticides with ecdysteroidal and juvenile hormone activity[END_REF]. In insects, the juvenile hormone regulates both metamorphosis and reproduction. Therefore, by acting on these hormonally regulated functions, juvenoid insecticides jeopardize successful molting, metamorphosing and reproduction. Nevertheless, juvenoid insecticides can affect the development of non-target arthropods, such as crustaceans. Indeed, crustaceans have a juvenile hormone methyl farnesoate that has a similar chemical structure and may play a role in development similar to the juvenile hormone in insects [START_REF] Miyakawa | A mutation in the receptor Methoprene-tolerant alters juvenile hormone response in insects and crustaceans[END_REF]. In line with this, lipids profile perturbation [START_REF] Jordão | Compounds altering fat storage in Daphnia magna[END_REF], morphological abnormalities [START_REF] Navis | Timing matters: Sensitivity of Daphnia magna dormant eggs to fenoxycarb exposure depends on embryonic developmental stage[END_REF], reproduction impairment [START_REF] Ginjupalli | The time-and age-dependent effects of the juvenile hormone analog pesticide, pyriproxyfen on Daphnia magna reproduction[END_REF] and reduced vitellogenin genes expression [START_REF] Tokishita | Organization and repression by juvenile hormone of a vitellogenin gene cluster in the crustacean, Daphnia magna[END_REF][START_REF] Toyota | Comparative ovarian microarray analysis of juvenile hormone-responsive genes in water flea Daphnia magna: potential targets for toxicity[END_REF] were reported in the cladoceran Daphnia magna exposed to juvenoid insecticides. In addition, the juvenoid insecticide fenoxycarb was responsible for both growth and lipid content impairment in the mud crab Rhithropanopeus harrisii [START_REF] Nates | Growth, lipid class and fatty acid composition in juvenile mud crabs (Rhithropanopeus harrisii) following larval exposure to Fenoxycarb®, insect juvenile hormone analog[END_REF] and for reduced fertility in the hymenoptera Aphytis melinus [START_REF] Biondi | Life stage-dependent susceptibility of Aphytis melinus DeBach (Hymenoptera: Aphelinidae) to two pesticides commonly used in citrus orchards[END_REF]. Moreover, tissue alterations and delayed hatching were observed in newborn individuals of the amphipod Gammarus fossarum from embryos directly exposed to fenoxycarb [START_REF] Arambourou | Phenotypic defects in newborn Gammarus fossarum (Amphipoda) following embryonic exposure to fenoxycarb[END_REF]. The juvenile hormone is also suspected of playing a key role on the arthropod immune system. In accordance with this, an alteration of the activity of phenoloxidase-an enzyme involved in melanization and arthropod immunity-was reported in the insects Apis mellifera [START_REF] Bitondi | The Apis mellifera pupal melanization program is affected by treatment with a juvenile hormone analogue[END_REF] and Rhodnius prolixus [START_REF] Nakamura | Effects of retinoids and juvenoids on moult and on phenoloxidase activity in the blood-sucking insect Rhodnius prolixus[END_REF] exposed to juvenoid insecticides.

In the present study, we tested the effects of the insecticide fenoxycarb, an analog of the insect juvenile hormone, on two life stages (embryo and adult) of the amphipod G. fossarum. Gammarus sp. are widely distributed in freshwater systems and play a major role in food webs: they not only contribute to the degradation of organic matter, but they also serve as food for many macroinvertebrates, fishes and amphibians [START_REF] Gerhardt | Gammarus: Important Taxon in Freshwater and Marine Changing Environments[END_REF].

In gammarids, both oocyte development and embryo development are synchronized [START_REF] Geffard | Ovarian cycle and embryonic development in Gammarus fossarum: Application for reproductive toxicity assessment[END_REF]. Females of G. fossarum carrying embryos in their ventral brood pouch were exposed to 50 μg L -1 fenoxycarb throughout the entire oogenesis/embryogenesis cycles (Fig 1). In our previous study, a fenoxycarb concentration of 50 μg L -1 was shown to disturb the embryogenesis of G. fossarum [START_REF] Arambourou | Phenotypic defects in newborn Gammarus fossarum (Amphipoda) following embryonic exposure to fenoxycarb[END_REF]. The study concentration is high but could be detected in water bodies surrounding agricultural areas [START_REF] Food | Conclusion on the peer review of the pesticide risk assessment of the active substance fenoxycarb[END_REF] . In the present study, given the potential effect of fenoxycarb on hormonally regulated functions, three endpoints related to reproductive success were evaluated in females previously exposed to fenoxycarb: i) precopulatory behavior, ii) success of fertilization and iii) post-fertilization embryo viability. Furthermore, the effects of fenoxycarb exposure on the lipid profiles of both newborn individuals from exposed embryos and freshly fertilized females were measured. Indeed, given that lipid metabolism is hormonally regulated and that lipids can be mobilized for detoxification processes under toxic exposure, we hypothesize that the lipid profile will be disturbed by the fenoxycarb exposure. Finally, as juvenoid insecticides can interfere with arthropod immunity, phenoloxidase activity was also measured in exposed females.

Materials and methods

Collection and maintenance of G. fossarum

Specimens of G. fossarum were collected (using a hand net) from a source population used in previous studies in our laboratory [START_REF] Nakamura | Effects of retinoids and juvenoids on moult and on phenoloxidase activity in the blood-sucking insect Rhodnius prolixus[END_REF], which inhabits La Bourbre River at Saint-Clair-de-la-Tour (45˚30'N, 5˚30'E, France). As this species is not considered as protected in France, specific collection permission for specimens was not required. Organisms with a size ranging from 2.0 mm and 2.5 mm were sorted by sieving and quickly transported to the laboratory in plastic vessels, where they were kept at 14˚C with a 16:8-h light:dark photoperiod in 30-L tanks, continuously filled with groundwater and aerated. After 10 days of rearing, precopula pairs were placed in an aquarium containing synthetic water at 14˚C. Individuals were monitored daily until the males had fertilized and released the females. Synthetic water used in the present study was the ADaM water [START_REF] Klu ¨ttgen | ADaM, an artificial freshwater for the culture of zooplankton[END_REF], composed of: 1.84 mM (CaCl 2 .H 2 O), 0.66 mM NaHCO 3 , 0.001 mM SeO 2 and 0.333 g.L -1 sea salts (Sigma Aldrich) (electrical conductivity = 1000 μS.cm -1 ). Animals were fed ad libitum on alder leaves (Alnus glutinosa). Alder leaves were collected in nearby Lyon (France), dried at room temperature, and soaked in groundwater for two weeks before use.

Experimental set up

A stock solution of fenoxycarb was prepared by adding fenoxycarb (pestanal, Sigma-Aldrich) to acetone (for HPLC, Carbo-Elba). Then, the fenoxycarb concentration of 50 μg L -1 was made by diluting the stock solution in synthetic water. Acetone concentration (0.005%, v/v) was kept constant in the fenoxycarb and control conditions. Previous experiments using the same gammarid population showed no significant effect of acetone at this concentration of exposure [START_REF] Geffard | Ovarian cycle and embryonic development in Gammarus fossarum: Application for reproductive toxicity assessment[END_REF][START_REF] Xuereb | Acetylcholinesterase activity in Gammarus fossarum (Crustacea Amphipoda): Linking AChE inhibition and behavioural alteration[END_REF]. Two days after fertilization, females carrying embryos in their open brood pouch were placed individually in 125 mL glass jars filled with 100 mL of the test medium (either the solvent control or a 50 μg L -1 fenoxycarb solution, n control = 45, n fenoxycarb = 58). Females carrying embryos were exposed for 21 days at 14±1˚C with a 16:8 h light:dark photoperiod. Half of the medium was renewed every 3 days. Animals were fed ad libitum on alder leaves. At 14˚C, embryos hatched in the brood pouch 23 days after fertilization [START_REF] Arambourou | Phenotypic defects in newborn Gammarus fossarum (Amphipoda) following embryonic exposure to fenoxycarb[END_REF]. Hatched embryos rest inside the maternal brood pouch before being released. To ensure that they had not eaten before the lipid analysis, newborns were gently removed from the brood pouch 23 days after female fertilization (Fig 1) by sweeping the blunted end of a Pasteur pipette through the pouch in the posterior-to-anterior direction.

Once newborns have been removed, each female was placed individually in a 125 mL glass jar filled with 100 mL of synthetic water, and one mature male (i.e. males which were in precopula in the gammarid culture) was added to promote fertilization. Animals were fed ad libitum on alder leaves and 50% of the medium was renewed every 3 days. Three endpoints related to reproductive success were assessed: i) precopulatory behavior, ii) success of fertilization and iii) post-fertilization embryo viability. Precopulatory behavior refers to male clasp and swimming with the female until female molting. This behavior ensures successful sexual contact during the post ecdysis period when mating can occur [START_REF] Hartnoll | An experimental study of sex discrimination and pair formation in Gammarus duebenii (Amphipoda)[END_REF]. Each couple was monitored daily during the reproductive period (i.e. for 7 days) to ensure that the males had fertilized and released the females. Fertilization was considered successful when fertilized eggs were released into the female brood pouch. Each day, newly fertilized females were placed in tanks containing synthetic water and embryos were gently removed from the maternal brood pouch following two days of fertilization by sweeping the blunted end of a Pasteur pipette through the pouch [START_REF] Rehm | Fixation and dissection of Parhyale hawaiensis embryo[END_REF]. Fifty newborn individuals were then placed in 25 ceramic well-plates in saline synthetic water at 14˚C for culturing (Fig 1 ), as described in Arambourou et al. [START_REF] Arambourou | Phenotypic defects in newborn Gammarus fossarum (Amphipoda) following embryonic exposure to fenoxycarb[END_REF]. Females were frozen at -80˚C until further analysis. Half of the embryo medium was renewed every 3 days and embryos were considered viable when they successfully broke their chorion.

Fenoxycarb analyses

Fenoxycarb concentrations in the water phase were checked at the beginning of the exposure and after 3 days of exposure, just before the media was renewed. Fenoxycarb analysis was performed by liquid chromatograph-mass spectrometer (LC-MS/MS) on an H-Class UPLC system (Waters) coupled to a Xevo TQ-S triple quadrupole mass spectrometer (Waters). The chromatographic column was a Kinetex 1.7 μm EVO C18 100A (Phenomenex). The mobile phase (A) was composed of 0.4 mM ammonium acetate + 0.01% acetic acid in water and the mobile phase (B) was methanol. The injection volume was set at 2 μL. MS/MS detection was performed in multiple reaction monitoring (MRM) mode with the electrospray ionization source operating in the positive mode (electrospray ionization, ESI+). The MRM transitions were 302 ! 116 (quantification transition) and 302 ! 88 (confirmation transition). The quantification limit for fenoxycarb was 0.04 μg L -1 . At the beginning of exposure, measured concentrations were <0.04 μg L -1 and 61 μg L -1 for the control and the fenoxycarb exposed groups, respectively. After 3 days of exposure, measured concentrations were <0.04 μg L -1 and 41 μg L -1 for the control and the fenoxycarb exposed groups, respectively.

After reproductive success observations, fenoxycarb concentration was also measured in females. For that purpose, three pooled females were ground and homogenized. Then, a QuE-ChERS extraction, combining an initial liquid-liquid extraction in acetonitrile, water, heptane and a citrate buffering salt, was carried out. This was followed by a purification step of the acetonitrile phase on a PSA/C18 dispersive solid extraction phase. Fenoxycarb was quantified with LC-MS/MS using a matrix-matched calibration, under the same conditions as those described for fenoxycarb measurement in the water phase. Three replicates per condition were analyzed.

Energy reserve content and phenoloxidase activity in females

Frozen females were analyzed for lipid content, protein content and phenoloxidase activity by spectrometry as described in Arambourou and Stoks [START_REF] Arambourou | Combined effects of larval exposure to a heat wave and chlorpyrifos in northern and southern populations of the damselfly Ischnura elegans[END_REF]. Glycogen content was measured by spectrometry according to the method reported by Go ´mez-Lechon et al. [START_REF] Go ´mez-Lechon | A Microassay for Measuring Glycogen in 96-Well-Cultured Cells-ScienceDirect[END_REF].

Lipidomic analysis in both newborns and females

Lipidomic analysis was performed as described by Jordão et al. [START_REF] Jordão | Obesogens beyond vertebrates: Lipid perturbation by tributyltin in the crustacean Daphnia magna[END_REF] with minor modifications. Three replicates, consisting of a pool of 40 newborn individuals, per condition were analyzed. For females, three replicates of two pooled individuals per condition were considered. Briefly, each replicate was homogenized in 1 mL of chloroform:methanol (2:1) with 2,6-di-tert-butyl-4-methylphenol (BHT; 0.01%) as an antioxidant. Lipid extraction was performed using a modification of Folch's method [START_REF] Folch | A simple method for the isolation and purification of total lipides from animal tissues[END_REF]. Briefly, 100 μL of the homogenized sample was mixed with 750 μL of chloroform and 250 μL of methanol. To semi-quantify the lipids, internal standards (S1 Table ) were also added. Samples were then dried under N 2 . Lipid extracts were solubilized in 300 μL methanol. The LC-MS/MS consisted of a Waters Aquity UPLC system connected to a LCT premier orthogonal accelerated time-of-flight mass spectrometer (Waters) operated in positive and negative ESI mode. Full-scan spectra from 50 to 1,800 Da were obtained. Mass accuracy and reproducibility were maintained using an independent reference spray (Lock-Spray; Waters). A 100-mm × 2.1-mm i.d., 1.7-μm C8 Acquity UPLC BEH (Waters) analytical column was used. In newborn individuals, a total of 129 lipids, distributed as follows, were identified and semi-quantified: 70 triacylglycerols (TAG), 10 diacylglycerols (DAG), 34 phosphatidylcholines (PC), 3 lysophosphatidylcholines (LPC), 5 phosphatidylethanolamines (PEA), 3 sphingomyelins(SM) and 3 phosphatidylserine (PS). In females, a total of 122 lipids distributed as follows, were identified and semi-quantified: 62 TAGs, 10 DAGs, 34 PCs, 4 LPCs, 5 PEAs, 3 SMs and 4 PSs.

Mass and lipid droplets in newborns

Newborn gammarids freshly extracted from the female brood pouch were gently dried on a paper, pooled in groups of five individuals, and weighed using an ultra-microbalance (Sartorius Ultra Micro Balance MSU2.7S000DM, readability 0.1 μg, repeatability ± 0.25 μg).

Lipid droplets in newborn individuals were visualized using a stock solution of Nile red staining (technical grade, Sigma-Aldrich) prepared with 10 mg of Nile red to 100 mL of acetone (for HPLC, Carbo-Elba). Just before use, the working solution was prepared by diluting the stock solution to 2 mg L -1 in synthetic water. Live individuals were then exposed to the Nile red working solution in the dark for 3 h at 14˚C. Stained individuals were immobilized between the microscope slide and the cover slide, and images of the entire animal were taken at 5 × magnification for visualization of lipid droplets. Fluorescence images were obtained using a Leica DM 2500 with a L5 filter cuber (EX 480/40, EM 527/30) [START_REF] Wayama | Three-dimensional ultrastructural study of oil and astaxanthin accumulation during encystment in the green alga Haematococcus pluvialis[END_REF]. Twenty individuals per condition were analyzed.

Statistical treatment

All statistical analyses were performed using R software [START_REF]R: a language and environment for statistical computing[END_REF]. Differences in juvenile production, mass, lipid droplets, energy reserve content and phenoloxidase activity between the control group and the fenoxycarb-exposed group were assessed using a non-parametric Mann-Whitney-Wilcoxon test, which is a robust test requiring minimal assumptions [START_REF] Whitley | Statistics review 6: Non parametric methods[END_REF]. For testing differences of mortality and viability a Chi-squared proportion test was used. Lipidome composition (lipid concentration expressed as pmol/mg of fresh tissue) was visualized with a principal component analysis (PCA) using the FactoMineR package and the difference between groups was assessed with an ANOVA on the principal component scores, followed by a posthoc Tuckey test.

Results

Life-history traits

After 21 days of exposure, the adult female survival rate in the fenoxycarb-exposed group (90%) was not significantly different from the control (89%) (Proportion test, χ 2 = 8.10 -31 , p = 1; Table 1). A fenoxycarb concentration of 1.9 ng g -1 of fresh tissue was detected in the exposed females (Table 1). The bioconcentration factor-which is the concentration in the tissues relative to the concentration in the surrounding medium (water)-was 0.03 L kg -1 . No significant effects of fenoxycarb exposure on either female mass (Mann-Whitney-Wilcoxon test, W = 69, p = 0.95; Table 1) or newborns production (Mann-Whitney-Wilcoxon test, W = 130, p = 0.36; Table 1) were found.

Reproductive success and embryos viability

Females previously exposed to fenoxycarb and maintained in control water with males for reproduction, exhibited an altered precopulatory behavior. Indeed, within 7 days after introducing the males, 88% of the gammarids were in precopula in the control group while only 46% of the gammarids were in precopula in the fenoxycarb-exposed group (Proportion test, χ 2 = 15, p<0.001; Fig 2A). Moreover, average number of embryos produced was reduced from 11 in the control group to six in the fenoxycarb-exposed group (Mann-Whitney-Wilcoxon test, W = 206, p = 0.001; Fig 2C). Survival of embryos cultured in well-plates was 60% in the control group, which is of the same order of magnitude as our previous observations [START_REF] Arambourou | Phenotypic defects in newborn Gammarus fossarum (Amphipoda) following embryonic exposure to fenoxycarb[END_REF]. Embryos produced by females exposed during oogenesis were not viable: all were dead after 21 days of rearing (proportion test, χ 2 = 40, p<0.001, Fig 2D).

Lipidomic analysis in both newborns and females

Abundant lipids in both females and newborns included TAGs (energy reserve) and PCs (component of biological membrane) (S2 Table ). The two first axes of the PCA explained 88% of the total variance (Fig 4). The first axis is positively correlated with TAG compounds and negatively correlated with DAG compounds, while the second axis is positively correlated with PC compounds. They were differences in lipidome composition between newborn individuals and the females (ANOVA on the PC1 coordinates, F 3,8 = 39, p<0.01; Fig 4). Newborns exhibited a higher content of TAGs and a lower content of DAGs than females. No significant differences in lipid profile were observed between control and fenoxycarb-exposed groups for either newborn individuals or females (post hoc Tukey tests, both p>0.05).

Mass and lipid droplets in newborns

No significant difference in newborns mass was observed between the control and the fenoxycarb-exposed group (Mann-Whitney-Wilcoxon test, W = 446, p = 0.38; Table 1). Nile red staining revealed an accumulation of lipid droplets in the hepatopancreas (Fig 5B ). No

Discussion

The reproductive success of G. fossarum was strongly affected by exposure to 50 μg L -1 of fenoxycarb, while both exposed newborn individuals and females did not exhibit any significant alteration of their lipid profiles. There was little accumulation of fenoxycarb in female tissue. Yet, given the low solubility of fenoxycarb in water (log Kow = 4.07), this compound is expected to be accumulated in tissues rich in lipids. This result suggested metabolization/excretion of fenoxycarb, which is not surprising because fenoxycarb is suspected of mimicking the action of the crustacean hormone methyl farnesoate, and thus might be metabolized by methyl farnesoate esterase.

Interference of fenoxycarb with hormonally regulated pathways might be responsible for the impairment of precopulatory behavior in exposed females. In amphipods, precopulatory behavior has been shown to be severely altered by insecticide exposure [START_REF] Blockwell | The acute toxicity of lindane to Hyalella azteca and the development of a sublethal bioassay based on precopulatory guarding behavior[END_REF][START_REF] Heckmann | Relationship between biochemical biomarkers and pre-copulatory behaviour and mortality in Gammarus pulex following pulse-exposure to lambda-cyhalothrin[END_REF]. Borowsky and Borowsky [START_REF] Borowsky | The reproductive behaviors of the amphipod crustacean Gammarus palustris (Bousfield) and some insights into the nature of their stimuli[END_REF] showed that the most important stimulus for male reproductive behavior in Gammarus palustris is the female exoskeleton, suggesting contact pheromones in this species may exist. Evidence for pheromonal control of mating behavior is, however, conflicting in gammarids. For example, Lyes [START_REF] Lyes | The reproductive behaviour of Gammarus duebeni (Lilljeborg), and the inhibitory effect of a surface active agent[END_REF] noted that pair formation in Gammarus duebeni was regulated by a pheromone produced by the female, while Hartnoll and Smith [START_REF] Hartnoll | An experimental study of sex discrimination and pair formation in Gammarus duebenii (Amphipoda)[END_REF] were unable to demonstrate sex pheromone activity in G. duebeni. More recently, Cornet et al. [START_REF] Cornet | Influence of female moulting status on pairing decisions and sizeassortative mating in amphipods[END_REF] stated that the female molt stage, which varies in levels of ecdysone and vitellogenin, influences pairing decisions. Therefore, reduced ability of gammarids to enter a precopulatory state following female exposure to fenoxycarb might result from either an alteration of pheromone production or a perturbation of the exoskeleton status related to impairment of ecdysone and vitellogenin metabolic pathways. A previous study using specimens of the same gammarid population showed a significant reduction of vitellogenic-like proteins in females exposed to juvenile hormone methyl farnesoate [START_REF] Jubeaux | Vitellogenin-like proteins in the freshwater amphipod Gammarus fossarum (Koch, 1835): Functional characterization throughout reproductive process, potential for use as an indicator of oocyte quality and endocrine disruption biomarker in males[END_REF]. Similarly, an impairment of vitellogenin genes expression was reported in the cladorecan D. magna exposed to fenoxycarb [START_REF] Tokishita | Organization and repression by juvenile hormone of a vitellogenin gene cluster in the crustacean, Daphnia magna[END_REF][START_REF] Toyota | Comparative ovarian microarray analysis of juvenile hormone-responsive genes in water flea Daphnia magna: potential targets for toxicity[END_REF].

Juvenoids have also been shown to disrupt lipid storage-which is a hormonally regulated function-in the crustaceans D. magna [START_REF] Jordão | Mechanisms of action of compounds that enhance storage lipid accumulation in Daphnia magna[END_REF] and R. harrisii [START_REF] Nates | Growth, lipid class and fatty acid composition in juvenile mud crabs (Rhithropanopeus harrisii) following larval exposure to Fenoxycarb®, insect juvenile hormone analog[END_REF]. In the present study, no significant difference in the lipid profile was observed between females from the control group and females from the fenoxycarb-exposed group. Females were analyzed just after fertilization, which means after having transferred the vitellus to the freshly fertilized eggs. If fenoxycarb mainly affects vitellogenesis processes as stated above, it is likely that the lipid profile was merely altered in the fertilized eggs. Therefore, further studies could perform a lipidomic analysis in freshly fertilized eggs. Given that in gammarids the vitellus is the main energetic pathway that fuels embryo development [START_REF] Clarke | Lipid biochemistry and reproductive biology in two species of Gammaridae (Crustacea: Amphipoda)[END_REF], the disruption of vitellogenesis in females exposed to fenoxycarb might also explain the lack of viable embryos produced. We did not detect any significant alteration of the phenoloxidase activity in females previously exposed to fenoxycarb. In the literature, phenoloxidase activity impairment was reported in insects exposed to juvenoid insecticides [START_REF] Bitondi | The Apis mellifera pupal melanization program is affected by treatment with a juvenile hormone analogue[END_REF][START_REF] Nakamura | Effects of retinoids and juvenoids on moult and on phenoloxidase activity in the blood-sucking insect Rhodnius prolixus[END_REF] but for exposure concentrations much higher than used in the present study. In addition, given that females were placed with males in water without fenoxycarb to assess the reproductive success for a 7-day period, phenoloxidase might have recovered from the induced toxic effect. Further experiments are needed to test the direct effect of fenoxycarb on phenoloxidase activity.

As reported for females, the lipid profile of newborn individuals from exposed embryos was not significantly altered by the fenoxycarb exposure. Hamdoun and Epel [START_REF] Hamdoun | Embryo stability and vulnerability in an always changing world[END_REF] suggested the embryo stage might not be the most sensitive stage within the life cycle of an organism, as is often claimed. The embryo stage has been shown to be less sensitive to toxic exposure than the adult stage in the cladoceran D. magna exposed to heavy metals [START_REF] Bodar | Toxicity of heavy metals to early life stages of Daphnia magna[END_REF] and in the isopod Asellus aquaticus exposed to cadmium [START_REF] Green | The acute and chronic toxicity of Cadmium to different life history stages of the freshwater crustacean Asellus aquaticus (L)[END_REF]. Embryos that are directly in contact with the surrounding medium, such as in the amphipod G. fossarum, appear resistant to environmental stress by pathogens, UV radiation and xenobiotic chemicals [START_REF] Hamdoun | Embryo stability and vulnerability in an always changing world[END_REF]. G. fossarum embryos are protected by an outer layer, the chorion, which might limit fenoxycarb uptake, as has been observed in embryos of the brine shrimp Artemia exposed to chlorpyrifos [START_REF] Varo | Assessment of the efficacy of Artemia sp (Crustacea) cysts chorion as barrier to chlorpyrifos (organophosphorus pesticide) exposure. Effect on hatching and survival[END_REF]. In the fenoxycarb-exposed dormant eggs of D. magna, despite the outer layer, fenoxycarb was taken up to a concentrations of 13 ng/egg [START_REF] Navis | Timing matters: Sensitivity of Daphnia magna dormant eggs to fenoxycarb exposure depends on embryonic developmental stage[END_REF]. Nonetheless, as suggested by the authors, measured toxicant levels may be influenced by chemicals adsorbed on the outer layer.

Conclusions

Reproductive success, especially reproductive behavior, was severely impaired in females of G. fossarum exposed to fenoxycarb. Reproductive behavior disruption might be caused by an interference of fenoxycarb with hormonally regulated pathways, which could also reduce embryo viability although further experiments are needed to test this hypothesis. Overall, these results underlined the substantial sensitivity of the oogenesis period in G. fossarum. By dramatically reducing reproductive success, juvenoid exposure could have severe repercussions on the population dynamics of gammarids, and consequently on ecosystem functioning.

Fig 1 .

 1 Fig 1. Experimental design and measured endpoints in G. fossarum exposed to 50 μg L -1 fenoxycarb. https://doi.org/10.1371/journal.pone.0196461.g001

Table 1 .Fig 2 .

 12 Fig 2. Reproductive parameters in the control group and in the 50 μg L -1 fenoxycarb-exposed group. Pairing success in percent (A), success of fertilization in paired gammarid in percent (B), number of fertilized eggs per female (mean±standard deviation; numbers above the bars indicate the number of females that were observed) (C) and embryo viability in percent (50 embryos/condition) (D). Asterisks indicate significant differences compared to the control group. https://doi.org/10.1371/journal.pone.0196461.g002

Fig 3 .

 3 Fig 3. Biochemical markers in control group and in the 50 μg L -1 fenoxycarb-exposed group. Phenoloxidase activity (A), lipid content (B), protein content (C) and glycogen content (D). Ten females and 15 females were analyzed in the control and fenoxycarb-exposed groups, respectively. https://doi.org/10.1371/journal.pone.0196461.g003

Fig 4 .Fig 5 .

 45 Fig 4. PCA scores of the lipid profiles plotted on the two first components of the PCA. Females in the control (green) and fenoxycarb-exposed group (pink) and newborns in the control (blue) and fenoxycarb-exposed group (orange). https://doi.org/10.1371/journal.pone.0196461.g004

PLOS ONE | https://doi.org/10.1371/journal.pone.0196461April 27, 2018

This study was funded by the French EC2CO program (INSU-CNRS) to He ´lène Arambourou and the French Institute of Science and Technology for the Environment and Agriculture (Irstea). Lipidomic analyses were supported by the Spanish grant CTM2014-51985-R to Carlos Barata. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

All data are contained within the manuscript and the supporting information.

Resources: He ´lène Arambourou.

Supervision: He ´lène Arambourou.

Validation: He ´lène Arambourou, Inmaculada Fuertes, Emmanuelle Vulliet, Gae ¨lle Daniele, Patrice Noury, Khedidja Abbaci, Carlos Barata.

Writing -original draft: He ´lène Arambourou, Carlos Barata.


David Ginsbourger

Céline Helbert

Laurent Carraro

Discrete Mixtures of Kernels for Kriging-based Optimization

Keywords: Gaussian Processes, Global Optimization, Kernel Selection, Mixture of Experts

Kriging-based exploration strategies often rely on a single Ordinary Kriging model which parametric covariance kernel is selected a priori or on the basis of an initial data set. Since choosing an unadapted kernel can radically harm the results, we wish to reduce the risk of model misspecification. Here we consider the simultaneous use of multiple kernels within Kriging. We give the equations of discrete mixtures of Ordinary Krigings, and derive a multikernel version of the expected improvement optimization criterion. We finally provide an illustration of the Efficient Global Optimization algorithm with mixed exponential and Gaussian kernels, where the parameters are estimated by Maximum Likelihood and the mixing weights are likelihood ratios.

The global optimization of numerical simulators is a challenging problem since the number of runs is severely limited by computation time. Furthermore, the derivatives are generally not available. For the past decade, Kriging-based derivative-free algorithms like EGO ( [START_REF] Jones | Efficient global optimization of expensive blackbox functions[END_REF]) have been developed to address this issue. Kriging metamodels are indeed convenient for building exploration strategies since they provide for every potential input vector both a mean predicted response value (Kriging mean) and an associated measure of accuracy (Kriging variance). Along this paper, the simulator is seen as a determinist numerical black-box function y with d-dimensional input:

y : x ∈ D ⊂ R d → y(x) ∈ R (1)
y is known at first on the initial Design of Experiments X = {x 1 , ..., x n0 }, where n 0 ∈ N is the number of initial runs. We denote by Y = {y(x 1 ), ..., y(x n0 )} the set of observations made by evaluating y at the points of X. In almost all Kriging models, the starting point is to make the assumption that y is one realization of a random process of the form

Y (x) = µ(x) + ε(x) (2)
where µ(x) is a deterministic trend function, ε(x) is a centered stationary random field with covariance function k, here assumed to belong to a set of positive definite stationary kernels:

K = {k (r,σ 2 ,ψ) : h ∈ D -D -→ σ 2 r(h; ψ), r ∈ R, σ 2 ∈ R + , ψ ∈ Ψ r } (3)
K is indexed by a finite set R of correlation kernel parametric families, by their respective continous hyperparameters ψ ∈ Ψ r (e.g. correlation lengths), and by a positive parameter σ2 (the process variance). In the following, χ = (r, σ 2 , ψ) denotes the tree-structured covariance 1 parameters. In many industrial applications, r is arbitrarily chosen to belong to a parametric family (exponential, Gaussian, Matèrn, etc...), (σ 2 , ψ) are then fitted to the data using automatic estimation procedures, and χ is finally plugged in as if it were known. Our particular concern here is to review and extend the EGO Algorithm ( [START_REF] Jones | Efficient global optimization of expensive blackbox functions[END_REF]) in taking the risk of model into account.

After recalling some basics about Gaussian processes and metamodel-based optimization, we propose an adaptation of Ordinary Kriging with mixed kernels. We then derive an optimization criterion based on a discrete mixture of Kriging, and finally illustrate its efficiency by applying EGO with two simultaneous kernels to a classical test case function. The weights depend on the distance between the prediction point x and the design of experiments X through the chosen covariance kernel. Here we give the equations of Kriging for a fixed kernel k χ (.) = σ 2 r(.; ψ). The Kriging mean m χ and mean squared error (or variance) s 2 χ at x are the following functions (see [START_REF] Cressie | Statistics for spatial data[END_REF] for pointwise derivation):

m χ (x) = μχ + k χ (x) T K -1 χ (Y -μχ 1 n ) s 2 χ (x) = σ 2 -k χ (x) T K -1 χ k χ (x) + 1 T n K -1 χ 1 n -1 1 -1 T n K -1 χ k χ (x) 2 (4)
where we recall that χ = (k, σ 2 , ψ). K χ and k χ (x) are the matrices1 :

K χ =     k χ (0) k χ (x 1 -x 2 ) ... k χ (x 1 -x n ) k χ (x 2 -x 1 ) k χ (0) ... k χ (x 2 -x n ) ... ... .... .... k χ (x n -x 1 ) ... .... k χ (0)     and k χ (x) =     k χ (x -x 1 ) k χ (x -x 2 ) ... k χ (x -x n )    
and μχ is given by: μχ = (1

T K -1 χ 1) -1 1 T K -1 χ Y.
The classical geostatistical interpretation of eq. ( 4) is to see m χ (x) as the Best Linear Unbiased Predictor (BLUP) of Y (x), under the hypothesis that eq. ( 2) holds with µ(x) being an unknown constant µ estimated by maximum likelihood. Here we find more convenient to consider an interpretation of OK in terms of Gaussian processes, in the flavour of ( [START_REF] Rasmussen | Gaussian Processes for Machine Learning[END_REF]). Assuming that ε(x) is a centered stationary Gaussian process with known covariance function k χ (.) and that µ is an unknown constant with improper uniform prior distribution µ ∼ U(R), we obtain the following conditional distribution for Y (x)

Y OK χ (x) := [Y (x)/Y (X) = Y] ∼ N m χ (x), s 2 χ (x) (5)
This approach allows the analytical calculation of various quantities involving Y (x) knowing the observations, as well as conditional simulations 2 of Y , which ensures for instance that the expectation of any fonction involving Y (x)/Y (X) = Y can be estimated by Monte-Carlo. In the practice, one chooses a parametric correlation kernel r, then estimates the parameters (σ 2 , ψ) , and finally plugs in the estimated values in the formulas. It seems however that such a sketch forgets the uncertainty associated with the choice of r and the estimation of (σ 2 , ψ) and hence underestimates the modeling uncertainty. Assessing uncertainty with a variance s 2 χ obtained by plugging in χ in the Kriging equations entails a hidden risk of trusting too much a "bad" model.

Likelihood maximization (ML) is one of the standard ways, along with cross-validation, to estimate µ and (σ 2 , ψ) on the basis of observations (see [START_REF] Rasmussen | Gaussian Processes for Machine Learning[END_REF] for instance). It relies on the hypothesis that Y is a gaussian vector with mean µ and covariance matrix K χ (this can be seen as a direct consequence of the Gaussian process interpretation of Kriging). Considering that r is known, one searches for the parameters (μ, ψ, σ2 ) that give the largest density value to Y. Noting3 R χ = 1 σ 2 K χ , MLE then relies on the maximization of the Gaussian likelihood function:

L(σ 2 , ψ, µ; Y) = f (Y/σ 2 , ψ, µ) = 1 (2π) d 2 (σ 2 ) d 2 det(R χ ) 1 2 e - » (Y-µ1) R -1 χ (Y-µ1) 2σ 2 - (6)
or equivalently on the minimization of -2 × log(L(σ 2 , ψ, µ; Y)). It can be shown that for every fixed ψ, the optimal µ and σ 2 are given by µ = μχ 4 and:

σ2 (ψ) = (Y -μψ 1) T R -1 χ (Y -μψ 1) d (7)
After some direct calculations, ML can be restricted to the p r -dimensional minimization problem:

min ψ∈Ψr {log(|K (r, σ2 (ψ),ψ) |)} (8)
This optimization problem is generally solved numerically, which adds both computational complexity and randomness in the result. The last point has been studied in detail in the theory of likelihood, and more recently discussed in this particular framework in ( [START_REF] Ginsbourger | On the selection and the estimation of kriging models for deterministic computer experiments[END_REF]).

Once a kriging interpolator and its associated uncertainty are computed, one dispose of a metamodel that may be used to explore the simulator with a design dedicated to optimize y. One way to derive such a design is by iteratively maximizing a figure of merit based on the Kriging metamodel. ( [START_REF] Jones | A taxonomy of global optimization methods based on response surfaces[END_REF]) and ( [START_REF] Michael | Exploration of metamodeling sampling criteria for constrained global optimization[END_REF]) provide a review of most used Kriging-based optimization criteria.

The expected improvement (EI) is a broadly used optimization criterion that makes a tradeoff between promizing (with low predictions, for minimization) and uncertain zones. Let y min = min{y(x 1 ), ..., y(x n )} be the minimum of the currently known observation values. Let x ∈ D be a candidate point for a next evaluation of y. In the end, evaluating y at x would bring an improvement of y min -y(x) if y(x) is below y min and no improvement if y(x) is above y min . Of course, this improvement (y min -y(x))

+ cannot be known without evaluating y (else, we could directly find the minimum). But eq. ( 5) makes it possible to know the statistical distribution of the random variable improvement (y min -Y (x))

+ conditionally on the observations Y (X) = Y. In particular, the expected improvement is defined as the following function of x:

C EI χ (x) = E y min -Y OK χ (x) + = E (y min -Y (x)) + /Y (X) = Y (9)
Note that the expression C EI χ (x) can be calculated analytically: Efficient Global Optimization (EGO) is an algorithm proposed by ( [START_REF] Jones | Efficient global optimization of expensive blackbox functions[END_REF]). It relies on a sequential exploration based on OK and on the maximization of the expected improvement criterion.

C EI χ (x) = (y min -m χ (x)) Φ y min -m χ (x) s χ (x) + s χ (x)φ y min -m χ (x) s χ (x) (10
Algorithm 1 The E.G.O. Algorithm

1: function EGO(X, Y, q) 2:
for i ← 1, q do 3:

(σ 2 * , ψ * ) = argmax (σ 2 ,ψ)∈R + ×Ψr L(σ 2 , ψ; Y (X) = Y) Estimating (σ 2 , ψ) by ML 4:
x * = argmax x∈D {C EI (r,σ 2 * ,ψ * ) (x)} Maximizing the Expected Improvement Here we consider the case in which the number of runs q ∈ N is fixed in advance. EGO has been found to be a competitive global optimization algorithm, in particular in the fields of automotive and aerospace engineering, with objective functions having one to six inputs (see ( [START_REF] Jones | A taxonomy of global optimization methods based on response surfaces[END_REF]), ( [START_REF] Michael | Exploration of metamodeling sampling criteria for constrained global optimization[END_REF])). In other respects, ( [START_REF] Henkenjohann | An adaptive sequential procedure for efficient optimization of the sheet metal spinning process[END_REF]) proposes a adaptation of EGO for physical experiments.

Mixtures of Kriging for prediction and optimization

Let us focus on the situation in which several kernels are in competition to model a sample of observed data. This is typically the case when different methods are available for the estimation of (σ 2 , ψ) (e.g. ML with different initial values, ML and cross-validation, etc...), or when there is a choice to make between a set of functional forms for the correlation kernel r. Let us assume that the function y has already been evaluated at a finite set of points X obs , and denote by Y obs the associated responses. We now consider that M ∈ N experts {E i , i ∈ [1, M ]} are at disposal to estimate χ on the basis of the observations. The E i 's are functional estimators, providing at the same time a correlation structure and its associated parameters. They are defined as follows:

∀i ∈ [1, M ], E i : (X obs , Y obs ) ∈ +∞ k=n0 D k × R k -→ χ i = E i (X obs , Y obs ) ∈ K (11)
For the sake of convenience, we identify here K with the set of possible χ's (there is an obvious one-to-one mapping between both sets). Given the set of kernels {χ 1 , ..., χ M } delivered by the experts {E i , i ∈ [1, M ]}, and W = {w 1 , ..., w M } a set of weights meant to quantify the respective relevance levels of the M experts, we study the idea of replacing the classical approach of kernel selection by a mixture of kernels: instead of keeping the best kernel and dropping off the others, we propose to keep them all and integrate them within Ordinary Kriging in probabilizing χ.

Discrete mixture of Gaussian processes: the unknown function y is now seen as one path of a random field associated with an OK model, which underlying kernel is independently chosen at random following a discrete law supported by the set of kernels delivered by the M experts:

[Y OK mix /χ] = Y OK χ P (χ = χ i ) = w i (12)
Note that the proposed approach is not strictly bayesian: the prior distribution on χ (in the sense of a bayesian framework) would be here depending on the data (we focus only on M experts, who are defined on the basis of observations). Here we first select a set of kernels and then mix them. Following eq. ( 12), the conditional distribution of Y (x) (x ∈ D) is a mixture of Gaussians:

Y OK mix (x) := [Y (x)/Y (X) = Y] with density function M j=1 w j p N " mχ j (x),s 2 χ j (x) " (.) (13)
Ordinary Kriging with a mixed kernel: Following eq. ( 13), Y OK mix is a field of Gaussian mixtures. This entails the equations of the mixed mean5 and variance6 :

m mix (x) = E[E[Y OK mix (x)/χ]] = M i=1 w i m χi (x) (14)
Hence, the mean of the resulting metamodel is the weighted average of the means associated with the different krigings (which coincides with the concept of weighted average surrogate model developped in [START_REF] Goel | Simultaneous use of multiple surrogates[END_REF]). Furthermore, the corresponding variance is given by

s 2 mix (x) = V ar[Y OK mix (x)] = E[V ar[Y OK mix (x)/χ]] + V ar[E[Y OK mix (x)/χ]] = M i=1 w i s 2 χi (x) + M i=1 w i [(m χi (x) -m mix (x)) 2 ] ( 15
)
The first term is a linear combination of the model variances weigthed by the w i s, whereas the second term reflects the dispersion between the different kriging means. The latter plays a capital role since it introduces heteroscedasticity in the Kriging variance: contrarily to the case of regular OK, the variance now depends on the observations Y through the second term.

Optimization under a mixture of kernels: when several values of χ are possible, finding the next most promizing point with a kriging-based optimization criterion (say C EI χ ) becomes a multicriteria decisional problem. Our approach here is to combine all C EI χi 's to provide a unified criterion that takes into account both sources of randomness. Using again the so-called law of total expectation, we derive the mixed expected improvement:

C EI mix (x) : = E y min -Y OK mix (x) + (16) = E E y min -Y OK mix (x) + /χ (17) = E E y min -Y OK χ (x) + = M i=1 w i C EI χi (x) ( 18
)
and hence, the expected improvement function under a mixture of kernels is simply the convex combination of the M expected improvement functions weighted by the {w i , ∈ [1, M ]}. Note that any integral criterion under a mixture of Krigings can be calculated in the same way.

Selecting a benchmark of experts: replacing the step of model selection by a step dedicated at choosing a set of models may seem at first to create more problems than it solves indeed. For instance, if we consider several families of correlation kernels (e.g. a gaussian, an exponential, and even nonstationary correlation kernels -as in [2]-) and estimate each set of kernel parameters by ML, it naturally increases the computational amount. The price for mixing is in that case to multiply the time needed for model inference by the number of experts. In this flavour, possible approaches would be to consider simultaneously experts relying on the same correlation kernel but with hyperparameters inferred using different methods (e.g. mixing the ML and the LOO "best" models), or even getting several candidate hyperparameter sets by parametric bootstrap. Ideally, we would like to have all relevant classes of experts represented in a small set. One of the future issues to be addressed seem to be the selection of sets with dissimilar good experts.

Setting a probability measure over the set of experts: Once a set of experts is chosen, probability weights have to be defined. The most naïve way of probabilizing the models is to put a uniform distribution on them. This approach may be relevant when mixing models obtained by maximizing different criteria (e.g.: a 50% -50% mix of the "best ML model" and the "best LOO model"). In an opposite fashion, it is also possible to consider the density of the mixture of models as a function of both the covariance parameters and parametric weights and then to perform likelihood maximization over all parameters including the w i s. Such problems are typically numerically solved using an Expectation-Maximization (EM) algorithm. We propose a way inbetween, more informative than a raw uniform distribution and yet computationally cheaper than EM. Since we have a criterion of fit (the Gaussian likelihood), why not use it to weight models? At first, we propose a Kriging mixture with weights based upon the likelihood criterion. In what follows, we consider Akaike weights (see [START_REF] Burnham | Model Selection and Multimodel Inference[END_REF]):

∀i ∈ [1, M ], w i = L(ψ i , σψi , μψi /Y, k i ) N i=1 L(ψ i , σψi , μψi /Y, k i ) (19)
Note that the w i 's are simply likelihood profile values divided by a normalization coefficient.

EGO with mixed kernels, applied to Branin's function

The Branin-Hoo function has been intensively studied in the litterature of global optimization of black-box functions ( [START_REF] Jones | Efficient global optimization of expensive blackbox functions[END_REF]). It is a smooth two-variable function defined by: The experts considered here are the two parametric correlation kernels r 1 and r 2 with their respective parameters estimated by ML:

y(x 1 , x 2 ) = x 2 - 5
E 1 : (X obs , Y obs ) → χ 1 = (r 1 , σ * 1 , ψ * 1 ), where (σ * 1 , ψ * 1 ) = argmax σ,ψ [L(σ, ψ; Y obs , r 1 )] E 2 : (X obs , Y obs ) → χ 2 = (r 2 , σ * 2 , ψ * 2 ), where (σ * 2 , ψ * 2 ) = argmax σ,ψ [L(σ, ψ; Y obs , r 2 )] (20)
All the algorithms and computations have been implemented in the frame of the MatLab free toolbox "Gaussian Processes for Machine Learning" (illustrating the book [START_REF] Rasmussen | Gaussian Processes for Machine Learning[END_REF]). In both cases, the kernel hyperparameters intitial values are fixed to (1, 10, 0.5, 1).

The results are summarized on fig. for i ← 1, q do 3:

for j ← 1, 2 do 4:

(σ 2 * j , ψ * j ) = argmax (σ 2 j ,ψj )∈R + ×Ψr j L(σ 2 j , ψ j ; Y (X) = Y, r j ) MLE 5:
end for 6:

for j ← 1, 2 do 7:

w j = L(σ 2 * j ,ψ * j ;Y (X)=Y,rj ) P 2 j=1 L(σ 2 * j ,ψ * j ;Y (X)=Y,rj )
Computing the mixing weights 8:

end for 9:

x

* = argmax x∈D 2 j=1 w j C EI (rj ,σ 2 * j ,ψ * j ) (x)
Maximizing the mixed EI end for 12: end function expert. In particular, the three zones of local optima are visited during the 25 first iterations. This similarities can be understood by looking at the sequences of weights plotted on the right figure. The two curves on the graphic below represent the log-likelihood associated with both experts as functions of the number of EGO iterations. Note that the likelihood of expert 2 is greater than the likelihood of expert 1 until the number of iterations reaches 6, and then becomes significantly lower than the other one. The likelihood ratios plotted on the graphic above show more precisely how the exponential kernel prevails at the beginning of the EGO algorithm, and is later dropped in favour of the Gaussian kernel. This kind of automatic selection seems due to an asymptotical steep decrease of the likelihood ratio. Using Akaike weights to mix kernels within a sequential exploration seems here to be a useful means to automatically select an expert without making a decision based on the initial design of experiments only. Hence, the proposed approach appears to be a sound option to increase EGO's robustness to modeling uncertainty.

Conclusions

We have derived and discussed an optimization criterion, the mixed expected improvement, relying on discrete mixtures of Ordinary Kriging metamodels with different covariance kernels. The presented framework of multiple experts allows one to handle several parametric correlation structures and/or different parameter estimation techniques within the same Kriging-based procedure. The application of the latter to the optimization of the Branin-Hoo function provided a first example, where mixing appears to be a succesful alternative to model selection based on initial data. It is illustrated that, possibly after an oscillating behaviour for a few iterations, mixing two experts within the EGO algorithm (Ordinary Krigings with Gaussian and Exponential correlation structures, with Akaike weights) may ultimately lead to an automatic selection of a unique metamodel. The issues of selecting parsimonious benchmarks of experts, and of using weighting methods dedicated to different purposes are to be addressed in forthcoming works.

Aknowledgements: This work was conducted within the frame of the DICE (Deep Inside Computer Experiments) Consortium between ARMINES, Renault, EDF, IRSN, ONERA, and Total S.A. We wish to thank Raphael T. Haftka and Victor Picheny for their help and useful comments. Special thanks to the R project people for developing such a useful freeware.

  )where φ and Φ are the probability density and the cumulative distribution function of the standard Gaussian law. This expression sheds light on the trade-off between promising and uncertain zones: the first term of the sum enhances local search via the mean prediction m χ (x), whereas the second term puts more emphasis on global search via the prediction variance. ([START_REF] Jones | A taxonomy of global optimization methods based on response surfaces[END_REF]), and ([START_REF] Michael | Exploration of metamodeling sampling criteria for constrained global optimization[END_REF]) intensively commented the criterion of expected improvement.

Figure 1 :

 1 Figure 1: Left: OK (in red, with green 95% confidence intervals) of y (in black), a realization of a Gaussian process with known covariance function (cubic correlation, variance 4, scale 0.6). Right: 4 iterations of EGO applied to y. The blue points represent the visited sites, the red points (and their associated vertical lines) are the current maximizers of the expected improvement. The horizontal blue line represents the current y min values. This example illustrates how an EGO sequences explores the objective function without getting trapped in the zones of local optimum.

5 :X

 5 = X {x * } and Y = Y {Y (x * )} Updating the Design of Experiments 6: end for 7: end function In the exact version ([5]), the algorithms loops until (max x∈D {C EI ψ * (x)} > δ) for a δ fixed by the user (in the original E.G.O., δ is 0.01 times the value of max x∈D {C EI ψ * (x)} at the previous iteration).

  1 ) + 10, (x 1 , x 2 ) ∈ [-5, 10] × [0, 15]y has three global minimizers (-3.14, 12.27), (3.14, 2.27), (9.42, 2.47), and the global minimum is approximately equal to 0.4. We normalized the variables between 0 and 1. Now we wish to illustrate, and compare EGO with different kernels and kernel mixtures.

Figure 2 :

 2 Figure 2: 25 iterations of the EGO algorithm applied to the Branin-Hoo function, with both experts E 1 and E 2 (see eq.(20)) and initial design X (in dark blue dots). Left: the path of EGO with expert E 1 is represented by black thin stars. Right: the path of EGO with expert E 2 is represented by light green squares. One of the three zones of minimum (upper left) is not visited.

( 3 )

 3 . The left figure illustrates 25 iterations of EGO with mixed experts. The pattern of the visited points is close to the trajectory of EGO with Gaussian Algorithm 2 The E.G.O. Algorithm with 2 mixed kernels weighted by their likelihood ratios 1: function EGO(X, Y, q) 2:

10 :X

 10 = X {x * } and Y = Y {y(x * )} Updating the Design of Experiments 11:

Figure 3 :

 3 Figure 3: 25 iterations of EGO applied to the Branin-Hoo function, with a mixture of experts E 1 and E 2 weighted by Akaike weights. Left: the path of EGO with the mixture of experts is represented by black filled stars. Right: evolution of both sequences of log-weights (lower graphic), and the associated series of likelihood ratios (upper graphic).

1

  Gaussian Processes and Metamodel-based Optimization Ordinary Kriging (OK) is a spatial interpolator developed by G. Matheron and named after the mining engineer D.G. Krige. It provides at each point x ∈ D a prediction of Y as a linear combination of the observed values Y.

In the deterministic case, these equations are often written in an equivalent way using correlation matrices.

A conditional simulation on a fine grid covering D can only be performed when D is low-dimensional (typically up to d =

3). However, conditional simulations at a small set of points are affordable whatever the dimension d.

There is implicitely no "nugget" effect since we work here with deterministic experiments

Directly maximizing the likelihood with respect to µ and (σ 2 , ψ) delivers the same value of µ as in the frame of OK, μ = μχ. Note however that OK includes the variability due to µ's estimation in its variance term.

Using the law of total expectation:E[X 1 ] = E[E[X 1 /X 2 ]].

Using the law of total variance: V ar[X 1 ] = E[V ar[X 1 /X 2 ]] + V ar[E[X 1 /X 2 ]]


Géraldine Nogaro
email: geraldine.nogaro@edf.fr

Annabelle Tollié

Christian Chauvin

TEMPORAL EVOLUTION OF MACROPHYTE COMMUNITIES RELATED TO ENVIRONMENTAL CHANGES IN TWO FRENCH LARGE RIVERS

Results

 Site 1 (Garonne River)  High macrophyte abundance in Sept. 2012, sharp decrease in Sept. 2013and 2014 (e.g., for Monestié only), and high abundance in Aug. 2015 and2016 (Figs. 3A &3B).  Higher floristic abundance and diversity in the Malause Station (i.e., low flow conditions and fine substrate) compared to the Monestié Station (i.e., high flow conditions and coarse substrate).

Material and Methods

 Studied sites on the Garonne River and the Rhône River (Fig. 1)

 Data collection  Macrophyte coverage (%) using the transect method and biomass (Kg of wet mass) on 1 m² sampling points (Fig. 2)



Conclusions

 Our results showed that higher water temperature during the early growing season (e.g., spring) were positively correlated with the macrophyte abundance and especially the increasing abundance of exotic species.  Variations of river discharge also affected macrophyte abundance but was modulated by the intensity, duration, and seasonality of flood events.  Our study highlights the need for long-term monitoring data to determine the spatial and temporal evolution of macrophyte communities related to environmental changes such as temperature rise, flood events or eutrophication.

Acknowledgments. We thank Christine Bertier and Lancelot Guillou for their help with the mapping of the sampling locations.



  Positive linear trend between the macrophyte coverage (%) and the accumulated water T°C > 15°C during the growing season (spring) and negative linear trend between % coverage and deviation from average water discharge (Figs. 4A & 4B).  Warm T°C and low flow conditions in the spring => key factors affecting annually the maximum abundance of macrophytes during the summer.  Site 2 (Rhône River)  Peak of floristic abundance in Sept. 2011, sharp decrease in 2013 and the following years (Fig. 5A)  Positive linear correlation between floristic coverage in summer and accumulated water T°C during the growing season (Fig. 5B).  As for the Garonne River, warm water T°C and low discharge during the growing period (spring) can affect the intensity of annual macrophyte abundance on the Rhône River.

  Environmental parameters (water temperature, discharge, and water physico-chemistry)  Data analyses  Temporal evolution of the macrophyte community abundance and diversity at each site  Search of metrics for water temperature and discharge affecting annually the magnitude of macrophyte abundance. climate change), which can influence the structure and evolution of macrophyte communities.  Spatial and temporal dynamics of the macrophyte communities in interaction with environmental parameters (temperature, discharge, nutrients, etc.) were analyzed on several stations located on two large rivers in France (Rhone River and Garonne River).  The goal of this study is to determine the key environmental factors impacting the annual growth, abundance and diversity of the macrophyte community.

Figure 3 .Figure 4 .Figure 5 .

 345 Figure 3. Floristic abundance and diversity in (A) Monestié and (B) Malause stations on the Garonne River.

Figure 1 .

 1 Figure 1. Location of the studied sites in (A) Europe, (B) France, (C) site 1 on the Garonne River, and (D) site 2 on the Rhône River.

Figure 2 .

 2 Figure 2. Macrophyte sampling for coverage and biomass.