Hugo Berthelot
email: hugo.berthelot@univ-brest.fr.

Solange Duhamel
email: duhamel@arizona.edu.

Stéphane L'helguen
email: stephane.lhelguen@univ-brest.fr.

Jean-François Maguer
email: jean-francois.maguer@univ-brest.fr.

Nicolas Cassar
email: nicolas.casssar@duke.edu

Inorganic and organic carbon and nitrogen uptake strategies of picoplankton groups in the northwestern Atlantic Ocean

Keywords: Picoplankton, Nitrogen, Carbon, Leucine, Nitrate, Ammonium, Urea, Prochlorococcus, Synechococcus, nanoSIMS

Picoplankton populations dominate the planktonic community in the surface oligotrophic ocean. Yet, their strategies in the acquisition and the partitioning of organic and inorganic sources of nitrogen (N) and carbon (C) are poorly described. Here, we measured at the single-cell level the uptake of dissolved inorganic C (C-fixation), C-leucine, N-leucine, nitrate (NO3 -), ammonium (NH4 + ) and N-urea in pigmented and non-pigmented picoplankton groups at six low-N stations in the northwestern Atlantic Ocean. Our study highlights important differences in trophic strategies between Prochlorococcus, Synechococcus, photosynthetic pico-eukaryotes and non-pigmented prokaryotes. Non-pigmented prokaryotes were characterized by high leucine uptake rates, non-significant C-fixation and relatively low NH4 + , N-urea and NO3 -uptake rates. Non-pigmented prokaryotes contributed to 7±3%, 2±2% and 9±5% of the NH4 + , NO3 -and N-urea community uptake, respectively. In contrast, pigmented groups displayed relatively high C-fixation rates, NH4 + and N-urea uptake rates, but lower leucine uptake rates than non-pigmented prokaryotes. Synechococcus and photosynthetic pico-eukaryotes NO3 -uptake rates were higher than Prochlorococcus ones. Pico-sized pigmented groups accounted for a significant fraction of the community C-fixation (63±27%), NH4 + uptake (47±27%), NO3 -uptake (62±49%) and Nurea uptake (81±35%). Interestingly, Prochlorococcus and photosynthetic pico-eukaryotes showed a greater reliance on C-and N-leucine than Synechococcus on average, suggesting a greater reliance on organic C and N sources. Taken together, our single-cell results decipher the wide diversity of C and N trophic strategies between and within marine picoplankton groups, but a clear partitioning between pigmented and non-pigmented groups still remains.

Introduction

Primary production is limited by nitrogen (N) availability in large portions of the world ocean [START_REF] Moore | Processes and patterns of nutrient limitation[END_REF]. The scarcity of N resources selects for smaller phytoplankton with larger surface-area-to-volume ratio. This strategy is believed to explain the biomass dominance of picoplankton (< 3 µm) in oligotrophic regions [START_REF] Marañón | Cell Size as a Key Determinant of Phytoplankton Metabolism and Community Structure[END_REF]. Picoplankton encompass a great diversity of populations and ecological functions [START_REF] Massana | Eukaryotic Picoplankton in Surface Oceans[END_REF]) but when analyzed using flow cytometry the populations generally cluster in well-defined groups including the pigmented photosynthetic groups of prokaryotes Prochlorococcus and Synechococcus and pico-eukaryotes, as well as the non-pigmented prokaryotes.

These groups are present in the surface ocean in variable abundances and proportions depending on environmental factors such as temperature, light and nutrient supply [START_REF] Otero-Ferrer | Factors controlling the community structure of picoplankton in contrasting marine environments[END_REF].

Prochlorococcus is present at latitudes lower than 45°N/S and numerically dominates the phytoplankton communities in oligotrophic and warm waters such as the subtropical gyres. Synechococcus is more widespread and is observed in nearly all the surface waters of the world ocean with the exception of the Arctic and Southern Oceans. As opposed to Prochlorococcus, Synechococcus is most abundant in temperate and relatively mesotrophic waters [START_REF] Flombaum | Global picophytoplankton niche partitioning predicts overall positive response to ocean warming[END_REF]. Photosynthetic pico-eukaryotes are ubiquitous in the oceans and their biomass generally dominate over Prochlorococcus and Synechococcus in nutrient rich water and at high latitudes [START_REF] Flombaum | Global picophytoplankton niche partitioning predicts overall positive response to ocean warming[END_REF]. They harbor a great diversity of organisms (including Prasinophyceae, Mamiellophyceae, Haptophyceae, Chrysophyceae, Pelagophyceae) making this group heterogeneous [START_REF] Hernández-Ruiz | Seasonal succession of small planktonic eukaryotes inhabiting surface waters of a coastal upwelling system[END_REF][START_REF] Mucko | Winter picoplankton diversity in an oligotrophic marginal sea[END_REF].

Non-pigmented prokaryotes in surface layers of the oceans are also diverse, mostly composed of bacteria (>90%) [START_REF] Ibarbalz | Global Trends in Marine Plankton Diversity across Kingdoms of Life[END_REF], which are ubiquitous in the ocean at relatively high abundances (10 5 -10 6 cells ml -1 ) [START_REF] Du | Diversity and distribution of pigmented heterotrophic bacteria in marine environments[END_REF]).

Pigmented groups have traditionally been hypothesized to exclusively use dissolved inorganic carbon (C) as their source of C, and sunlight as a source of energy to fix C via photosynthesis (photoautotrophy).

Conversely, non-pigmented prokaryotes are conventionally described as pure heterotrophs, i.e. relying on organic C for their growth, playing a key role in the remineralization of organic matter [START_REF] Azam | Microbial Control of Oceanic Carbon Flux: The Plot Thickens[END_REF]).

However, this idealized conceptual model is questioned by a growing body of evidence showing that mixed trophic regimes are a common feature in the ocean. Some pigmented organisms use organic C as sources of C and energy in addition to dissolved inorganic C, a trophic strategy called mixotrophy. The organic C acquisition can be mediated by a direct uptake of dissolved compounds (osmo-mixotrophy) or by predation on prey (phago-mixotrophy) [START_REF] Sanders | Bacterivory by phototrophic picoplankton and nanoplankton in Arctic waters[END_REF][START_REF] Hartmann | Mixotrophic basis of Atlantic oligotrophic ecosystems[END_REF][START_REF] Muñoz-Marín | Mixotrophy in marine picocyanobacteria: use of organic compounds by Prochlorococcus and Synechococcus[END_REF]. Recent studies have demonstrated the high affinities of several pigmented species for dissolved organic substances, leading to a potential for resource competition with pure heterotrophs [START_REF] Kamjunke | Algae as competitors for glucose with heterotrophic bacteria[END_REF]. As an example, the complete gene pathways for glucose acquisition, and small but significant uptake rates have recently been found in Prochlorococcus and Synechococcus [START_REF] Muñoz-Marín | Mixotrophy in marine picocyanobacteria: use of organic compounds by Prochlorococcus and Synechococcus[END_REF]. Similarly, the use of dissolved inorganic C by non-pigmented organisms through chemoautotrophic processes has been reported in communities present in oceanic surface waters [START_REF] Middelburg | Chemoautotrophy in the ocean[END_REF] This partitioning between organic and inorganic resources is also relevant to N acquisition. While the former traditional view is that pigmented organisms mostly rely on inorganic N, mainly ammonium (NH4 + ) and nitrate (NO3 -), and non-pigmented prokaryotes on organic N substrates, including aminoacids, a number of studies have shown that the N strategies between the two trophic regimes are not completely distinct. In the N-rich sub-Arctic Pacific, non-pigmented prokaryotes have been shown to contribute to ~30% of the NO3 -and NH4 + community uptake rates [START_REF] Kirchman | Uptake of ammonium and nitrate by heterotrophic bacteria and phytoplankton in the sub-Arctic Pacific[END_REF].

Comparable contributions (~40%) were found in a eutrophic coastal Mediterranean lagoon [START_REF] Trottet | Use of inhibitors for coastal bacteria and phytoplankton: Application to nitrogen uptake measurement[END_REF]) and in Sub-Arctic Atlantic [START_REF] Fouilland | Nitrogen uptake by heterotrophic bacteria and phytoplankton in Arctic surface waters[END_REF]) while much smaller (4-14%) were observed in the post-bloom temperate waters of the North Atlantic [START_REF] Kirchman | Biomass and nitrogen uptake by heterotrophic bacteria during the spring phytoplankton bloom in the North Atlantic Ocean[END_REF]. In parallel, an increasing number of studies show that photosynthetic organisms use Dissolved Organic N (DON) compounds for their growth [START_REF] Bronk | DON as a source of bioavailable nitrogen for phytoplankton[END_REF]. For example, urea is thought to fuel the recurrent harmful algal blooms of Aureococcus anophagefferens [START_REF] Berg | Organic nitrogen uptake and growth by the chrysophyte Aureococcus anophagefferens during a brown tide event[END_REF]. The ability of Prochlorococcus to use dissolved free amino acid has been hypothesized to explain its dominance in oligotrophic waters where inorganic N is depleted [START_REF] Zubkov | High rate of uptake of organic nitrogen compounds by Prochlorococcus cyanobacteria as a key to their dominance in oligotrophic oceanic waters[END_REF].

Despite the growing body of evidence for the prevalence of mixed nutritional strategies and the clear implications for N and C resource competition, few studies have simultaneously investigated the uptake of organic and inorganic N and C by pigmented and non-pigmented planktonic communities [START_REF] Bradley | Inorganic and Organic Nitrogen Use by Phytoplankton Along Chesapeake Bay, Measured Using a Flow Cytometric Sorting Approach[END_REF]. The lack of observations can in part be explained by the methodological challenge of measuring these processes in situ at the plankton group scale. The most common approach combines stable or radioactive isotope labelling and post incubation size fractionation where large and small size fractions are attributed to photosynthetic groups and non-pigmented prokaryotes, respectively, with a typical cut-off at 0.7-1 µm [START_REF] Kellogg | Comparison of free-living, suspended particle, and aggregate-associated bacterial and archaeal communities in the Laptev Sea[END_REF][START_REF] Schapira | Metabolism of free-living and particleassociated prokaryotes: Consequences for carbon flux around a Southern Ocean archipelago[END_REF]. However, retention of non-pigmented prokaryotes in the largest size fractions has been shown to be significant, in particular embedded in aggregates and/or attached to the cell surface of large organisms [START_REF] Seymour | Zooming in on the phycosphere: The ecological interface for phytoplankton-bacteria relationships[END_REF].

Similarly, the smallest pigmented groups, such as Prochlorococcus or Ostreococcus, are within the size spectrum of non-pigmented prokaryotes leading to an overall poor specificity of size fractionation, particularly in open ocean waters where small cells dominate photosynthetic communities [START_REF] Casey | Size dependence of metabolism within marine picoplankton populations[END_REF]. Inhibitors specific to plankton groups have also been used to assess the contribution of prokaryotes to the uptake of inorganic nutrients [START_REF] Fouilland | Nitrogen uptake by heterotrophic bacteria and phytoplankton in Arctic surface waters[END_REF]). However, the efficiency and the specificity of the inhibitors in natural planktonic communities is questionable [START_REF] Oremland | Use of "Specific[END_REF]. Flow cytometric cell sorting has been used in combination with radioactive isotope labelling allowing the determination of uptake rates of inorganic or organic substances labelled with 14 C, 33 P, 35 S or 3 H at the group-level (e.g. [START_REF] Jardillier | Significant CO2 fixation by small prymnesiophytes in the subtropical and tropical northeast Atlantic Ocean[END_REF][START_REF] Duhamel | Small pigmented eukaryotes play a major role in carbon cycling in the P-depleted western subtropical North Atlantic, which may be supported by mixotrophy[END_REF]. Approaches using radioactive isotopes present the advantage of higher sensitivity than stable isotopes. They have been used to evidence group specific patterns such as the in-situ Prochlorococcus uptake of amino acids [START_REF] Muñoz-Marín | Mixotrophy in marine picocyanobacteria: use of organic compounds by Prochlorococcus and Synechococcus[END_REF], the ingestion of non-pigmented prokaryotes cells by photosynthetic pico-eukaryotes [START_REF] Hartmann | Mixotrophic basis of Atlantic oligotrophic ecosystems[END_REF][START_REF] Duhamel | Small pigmented eukaryotes play a major role in carbon cycling in the P-depleted western subtropical North Atlantic, which may be supported by mixotrophy[END_REF] or the faster growth of Prochlorococcus and Synechococcus compared to photosynthetic pico-eukaryotes [START_REF] Zubkov | Faster growth of the major prokaryotic versus eukaryotic CO2 fixers in the oligotrophic ocean[END_REF]. Unfortunately, there is no suitable radioactive tracer for N.

In this study, we circumvent these methodological issues by coupling dual stable isotope labelling assays ( 13 C, 15 N) with flow cytometry cell sorting and nanoSIMS to measure at the single cell level the use of inorganic (dissolved inorganic C, NH4 + and NO3 -) and organic (urea, leucine) sources of N and C by Prochlorococcus, Synechococcus, photosynthetic pico-eukaryotes and non-pigmented prokaryotes. We applied our approach in various biomes of the northwestern Atlantic, from the subtropical gyre to the Gulf Stream and Labrador Current. A common characteristic of these regions is a general state of nitrogen or phosphorus limitation [START_REF] Wu | Phosphate depletion in the Western North Atlantic Ocean[END_REF][START_REF] Lipschultz | A time-series assessment of the nitrogen cycle at BATS[END_REF][START_REF] Moore | Relative influence of nitrogen and phosphorus availability on phytoplankton physiology and productivity in the oligotrophic sub-tropical North Atlantic Ocean[END_REF]. While wintertime mixing can bring NO3 -and phosphate (PO4 3-) rich waters to the surface (even in the oligotrophic waters near Bermuda), summer stratification leads to a reduction in nutrient availability in oceanic and coastal waters to levels limiting primary production as demonstrated by nutrient addition bioassays, in particular for N [START_REF] Sedwick | Assessing Phytoplankton Nutritional Status and Potential Impact of Wet Deposition in Seasonally Oligotrophic Waters of the Mid-Atlantic Bight[END_REF]. This study provides a direct in situ comparison of N and C uptake by pigmented and non-pigmented picoplankton at the single cell level, highlighting the contrasting nutritional strategies sustaining the growth of specific picoplankton groups in the ocean.

Materials

Sampling and biogeochemical analyses

The study was conducted in the northwestern Atlantic between Bermuda and the United States New England coast aboard of the R/V Atlantic Explorer in August 2017. Six stations with contrasting biogeochemistry were sampled: two stations in the North Atlantic Gyre (stations A and B) among which one is the Bermuda Atlantic Time-series Study (BATS) station (hereafter Station A), two stations in the Gulf Stream (stations C and D) and two stations on the continental shelf of the coast of New England (stations E and F) (Fig. 1). Surface seawater samples (5 m) were collected using Niskin bottles mounted on a rosette equipped with CTD sensors. At each station, samples for dissolved N nutrients were collected in triplicate and filtered through combusted GF/F filters (4h, 450°C) before being stored at -20°C until further analysis. Care was taken to copiously rinse the filters with ultrapure water and seawater before sampling in order to avoid contaminations. Samples for NO3 -, urea and PO4 3-were collected in acid cleaned (soaked in hydrochloric acid 10%, followed by ultrapure water three times) polypropylene tubes (50 mL or 15 mL) and measured colorimetrically according to [START_REF] Raimbault | Feasibility of using an automated colorimetric procedure for the determination of seawater nitrate in the 0 to 100 n M range: Examples from field and culture[END_REF], [START_REF] Mulvenna | A modified manual method for the determination of urea in seawater using diacetylmonoxime reagent[END_REF] and Strickland and Parsons (1972), respectively. The limits of detection were 10, 60 and 14 nmol N L -1 for NO3 -, urea and PO4 3-, respectively. Samples for NH4 + were collected in 50 mL polypropylene tubes which were conditioned beforehand to reduce risks of contamination: the tubes were first left overnight in hydrochloric acid 10%, rinsed three times with freshly produced ultrapure water and filled in a mixture of ultra-pure water and reagents for NH4 + determination until sampling. Samples were then measured fluorimetrically as described in [START_REF] Holmes | A simple and precise method for measuring ammonium in marine and freshwater ecosystems[END_REF] with standards made from NH4 + -free deep seawater stored in the same conditions as samples.

The limit of detection was 3 nmol N L -1 .

Experimental design

At each station, uptake of dissolved inorganic C (hereafter referred to as C-fixation), NH4 + , NO3 -, urea and leucine were measured using stable isotope tracer incubations both at the plankton-community and at the single-cell levels. For this purpose, seawater was collected at each station from the Niskin bottles in four sets of five acid-cleaned 1.2 L polycarbonate bottles. Isotope-labelled tracers were added in all the bottles directly after collection as follow: 13 C was added under the form of dissolved inorganic C (NaH 13 CO3, 99%, Eurisotop) together with either 15 NO3 -(KNO3, 98%, Eurisotop), 15 NH4 + (NH4Cl, 98%, Eurisotop) or 15 N-urea (98%, Eurisotop) or under the form of dual-labeled 13 C-15 N leucine (99% 13 C and 98% 15 N). C-fixation rates were calculated from the average of incubations performed in the presence of 15 NO3 -, 15 NH4 + and 15 N-urea. Isotopes were added to a final concentration of 30 nmol N L -1 for NH4 + , NO3 -and urea at the gyres and at the Gulf Stream stations and 50 nmol N L -1 at the continental shelf stations. Due to expected relatively low uptake rates, leucine was added at the saturating (or close to saturating) concentrations of 10 nmol L -1 at all stations to ensure significant isotopic enrichments [START_REF] Hill | Low microbial respiration of leucine at ambient oceanic concentration in the mixed layer of the central Atlantic Ocean[END_REF]. In order to determine the initial 13 C and 15 N isotopic abundance in the particulate matter, one bottle from each set was filtered onto combusted GF/F (4 h, 450 °C) directly after the addition of the isotopes. Filters were rinsed with 0.2 µm pore-size filtered seawater and stored at -20 °C. The remaining bottles from the set were placed in an on-deck incubator reproducing the light intensity at the surface and kept at sea surface temperature by a continuous circulation of surface seawater. The incubations lasted 3-8 h (5.5 h on average), except for the leucine treatments for which incubations lasted 22-24 h in order to ensure significant isotopic enrichments. Incubations were stopped by filtering three of the four remaining bottles from each set onto combusted GF/F as described above. The last bottle from each set was used to concentrate the cells for flow cytometry cell sorting. For this purpose, the bottle content was filtered onto 0.2 µm polycarbonate filters. The filtration was stopped just before the filter went dry and ~10 mL of 0.2 µm filtered sea water with PFA (1.6% final concentration) was added on the filter and left for 1 h in the dark. The solution was then filtered, the filters were placed in 5 mL cryotubes filled with 0.2 µm filtered seawater. The cryotubes were vortexed in order to resuspend the cells in the solution, then flash frozen in liquid N2 and stored at -80°C.

Flow cytometry cell sorting and isotopic analyses

Flow cytometry cell sorting and nanoSIMS analyses were conducted as previously described in [START_REF] Berthelot | NanoSIMS single cell analyses reveal the contrasting nitrogen sources for small phytoplankton[END_REF] with a few modifications. Concentrated cells in cryotubes were sorted back onshore using a BD Influx cell sorter equipped with a 70 µm nozzle, with sheath fluid and sample fluid pressure of 30 PSI (207 kPa) and 31 PSI (214 kPa), respectively. The instrument was set at the highest sorting purity (1.0 drop single mode), the drop delay was calibrated using Accudrop Beads (BD Biosciences, USA) and the sorting efficiency was verified manually by sorting a specified number of 1 µm yellow-green microspheres (Polysciences #17154-10) onto a glass slide and counting the beads under an epifluorescence microscope. We systematically recovered 100% of the targeted beads before sorting samples. Using this setup, the sorting purity on our instrument typically exceeds 96% [START_REF] Duhamel | Small pigmented eukaryotes play a major role in carbon cycling in the P-depleted western subtropical North Atlantic, which may be supported by mixotrophy[END_REF]. Prochlorococcus, Synechococcus and photosynthetic pico-eukaryotes were discriminated in unstained samples while non-pigmented prokaryotes were discriminated in a sample aliquot stained with SYBR Green I DNA dye (0.01% final). non-pigmented prokaryotes clustered in two groups: low nucleic acid and high nucleic acid. Using a forward scatter detector with small particle option and focusing a 488 plus a 457 nm (200 and 300 mW solid state respectively) laser into the same pinhole allowed the resolution of dim surface Prochlorococcus population from background noise in unstained samples. However, in stained samples, Prochlorococcus overlapped with high nucleic acid group and therefore, only cells belonging to low nucleic acid group were sorted and further analyzed for isotopic 13 C and 15 N contents and are referred collectively as non-pigmented prokaryotes. Filters containing the sorted cells were analyzed on a CAMECA nanoSIMS 50 using a focused 1.2 pA Cs + ion beam scanning fields of 10 x 10 µm (for non-pigmented prokaryotes and Prochlorococcus), 20 x 20 µm (for Synechococcus) and 30 x 30 µm (for photosynthetic pico-eukaryotes) and recording alternatively the 12 C 14 N -and 12 C 15 N -or 12 C 14 N -and 13 C 14 N -secondary ions using the "peak jumping mode" over at least 20 planes. Mass resolution was >7,000 to resolve the 12 C 15 N -and 13 C 14 N -ions (See [START_REF] Berthelot | NanoSIMS single cell analyses reveal the contrasting nitrogen sources for small phytoplankton[END_REF] for further details). Cells were then identified based on the 12 C 14 N -total ion count images and outlined using the particle detection mode of the LIMAGE software. Each particle detected was individually checked and redrawn if needed or discarded when it was not possible to attribute it to a cell with certainty. Particulate C and N concentrations and isotopic ratios at the community scale were determined on an isotope ratio mass spectrometer coupled to an elemental analyzer (EA-IRMS) from the triplicate GF/F filters. At the average C and N content measured on the samples (~6 µmol C and ~0.9 µmol N), the precision (standard deviation of repeated measurements) of the elemental analyses were 0.08 µmol C and 0.007 µmol N and the precision of the isotopic percent abundances were 0.0004 atom% and 0.0003 atom% for C and N, respectively.

Rate calculations and statistical analyses

For each cell analyzed, the isotopic percent abundances of 13 C (𝐴 𝐶 = 13 𝐶 𝑁 14 -13 𝐶 𝑁 14 -+ 𝐶 𝑁

𝑒𝑙𝑒𝑚𝑒𝑛𝑡 𝑠𝑝𝑒𝑐𝑖𝑓𝑖𝑐 𝑢𝑝𝑡𝑎𝑘𝑒 = 𝐴 𝑐𝑒𝑙𝑙 -𝐴 ̅ 𝑡0 𝐴 𝑠𝑜𝑢𝑟𝑐𝑒 -𝐴 ̅ 𝑡0 * 1 𝑡
where Acell, 𝐴 𝑡0 ̅̅̅̅ , and Asource are the isotopic percent abundances of the cell after incubation (A13C or A15N), of the cells (mean) prior to incubation, and of the source pool, respectively and t is the incubation time. Note that in the case of dissolved inorganic C, the specific uptake rates are termed C-specific Cfixation rate. In addition to C-and N-specific uptake rates, C-fixation based division rates were calculated as follows:

𝐶 𝑓𝑖𝑥𝑎𝑡𝑖𝑜𝑛 𝑏𝑎𝑠𝑒𝑑 𝑑𝑖𝑣𝑖𝑠𝑖𝑜𝑛 = 𝑙𝑜𝑔2 ( 𝐴 𝑠𝑜𝑢𝑟𝑐𝑒 -𝐴 ̅ 𝑡0 𝐴 𝑠𝑜𝑢𝑟𝑐𝑒 -𝐴 𝑐𝑒𝑙𝑙 ) * 1 𝑡
C-fixation based division rates reflect cellular division rates if inorganic C-fixation is the unique source of elemental C to the organism, as would be the case in case of exclusive photoautotrophy. For comparison with the literature, C-fixation division rates are presented in d -1 . As the incubations were not performed from dawn to dusk, hourly rates were converted to daily rates using the model developed by [START_REF] Moutin | Primary production in surface waters of the western Mediterranean sea. Calculation of daily production[END_REF] which account for the variation of daylight intensity at the sampling site and on the sampling day.

Cell-specific uptake rates (amol C cell -1 h -1 and amol N cell -1 h -1 ) were calculated as follows:

𝑐𝑒𝑙𝑙 𝑠𝑝𝑒𝑐𝑖𝑓𝑖𝑐 𝑢𝑝𝑡𝑎𝑘𝑒 = 𝐴 𝑐𝑒𝑙𝑙 -𝐴 ̅ 𝑡0 𝐴 𝑠𝑜𝑢𝑟𝑐𝑒 -𝐴 ̅ 𝑡0 * 1 𝑡 * 𝑄 𝑐𝑒𝑙𝑙
With 𝑄 𝑐𝑒𝑙𝑙 the estimated C and N cell contents. For pigmented organisms, C and N cell contents used were the median values reported by [START_REF] Baer | Stoichiometry of Prochlorococcus, Synechococcus, and small eukaryotic populations in the western North Atlantic Ocean[END_REF] in northwestern Atlantic (5, 23 and 257 fmol C cell -1 and 0.6, 2.4 and 15 fmol N cell -1 for Prochlorococcus, Synechococcus and photosynthetic picoeukaryotes, respectively). For non-pigmented prokaryotes, a C cell content of 1.7 fmol C cell -1 and a C:N ratio of 6.6 were used [START_REF] Fukuda | Direct Determination of Carbon and Nitrogen Contents of Natural Bacterial Assemblages in Marine Environments[END_REF]. The cell-specific uptake rates of dissolved inorganic C are termed cell-specific C-fixation rates. Group uptake rates were obtained by multiplying per-cell rates by cell abundances in the respective group.

Single cell uptake was considered to be above the detection limit when the percent abundance enrichment (𝐴 𝑐𝑒𝑙𝑙 -𝐴 ̅ 𝑡0 ) was higher than two times the standard deviation associated with the Poisson distribution (λ) parameterized as λ = 𝐴 𝑐𝑒𝑙𝑙 * 𝑁 𝐶𝑁 -,𝑐𝑒𝑙𝑙 , where 𝑁 𝐶𝑁 -,𝑐𝑒𝑙𝑙 is the CN -ions counts of the cell.

Similarly, groups were considered as active when the mean cellular percent isotopic abundances enrichment of the groups (𝐴 ̅ 𝑔𝑟𝑜𝑢𝑝 -𝐴 ̅ 𝑡0 ) were two times higher than the standard deviation of the cellular percent abundances. Differences in C or N specific uptake between stations or groups were tested using unpaired Kruskal-Wallis test and considered significant if p < 0.05.

The community C and N uptake rates (nmol C L -1 h -1 or nmol N L -1 h -1 ) were measured from GF/F filters as follows:

𝐶𝑜𝑚𝑚𝑚𝑢𝑛𝑖𝑡𝑦 𝑢𝑝𝑡𝑎𝑘𝑒 = 𝐴 𝑃𝑀 -𝐴 ̅ 𝑡0 𝐴 𝑠𝑜𝑢𝑟𝑐𝑒 -𝐴 ̅ 𝑡0 * 1 𝑡 * 𝑃𝑀
With PM the particulate C or N concentration and 𝐴 𝑃𝑀 the isotopic percent abundance of 13 C or 15 N in the particulate matter.

Due to the low concentrations of NH4 + , NO3 -and urea, isotopes tracers additions exceeded the threshold of 10% of the ambient concentrations for trace level additions. As a result, percent isotopic abundances in the source pool (𝐴 𝑠𝑜𝑢𝑟𝑐𝑒 ) ranged between 41-95%. Two uptake kinetics experiments with increasing nutrient additions at the gyre stations were performed to assess the extent of overestimation of the uptake due to the 15 N added. Rates were then corrected for this overestimation according to [START_REF] Harrison | The kinetics of nitrogen utilization in the oceanic mixed layer: Nitrate and ammonium interactions at nanomolar concentrations[END_REF] to approximate in situ rates as much as possible. This correction resulted in a reduction of NH4 + , NO3 -and urea uptake rates by a factor 1.5, 2.4 and 1.1 on average, respectively. At stations were NO3 - was below the detection limit, we assumed a NO3 concentration of 5 nmol L -1 for the calculations.

Leucine concentrations were not measured but assumed to be lower than 1 nmol L -1 [START_REF] Zubkov | Differential microbial uptake of dissolved amino acids and amino sugars in surface waters of the Atlantic Ocean[END_REF] and uptake rates were calculated assuming percent isotopic abundances of 13 C and 15 N of 95% in the source pool. Leucine rates were not corrected from overestimation due to leucine addition at saturating (or close to saturating) concentrations and should thus be interpreted as "potential" rates. We did not correct for isotope dilution associated with regeneration of NH4 + during the incubations, which would tend to bias our estimates low. In order to limit this bias, the duration of the NH4 + incubations were kept short (≤3.5 h). Minimum quantifiable rates were calculated at each station and for each tracer from the propagation of errors of the different parameters involved in the community uptake rate calculations according to [START_REF] Gradoville | Diversity and activity of nitrogen-fixing communities across ocean basins[END_REF] (Table S1). All community uptake rates were higher than the minimum quantifiable rates. Maximal N fluxes constrained by diffusion-limited N supply to single cells were calculated from the analytical solutions of diffusion to a sphere:

𝜌 𝑚𝑎𝑥 = 4𝜋𝐷𝑟 0 (𝐶 ∞ -𝐶 0 )
where 𝜌 𝑚𝑎𝑥 is the maximal NH4 + , NO3 -or urea uptake rate (nmol s -1 ) of a cell with the equivalent spherical radius 𝑟 0 (cm), 𝐷 is the diffusion coefficient (cm 2 s -1 ) of the considered compound in water, 𝐶 0 is the NH4 + concentration at the cell surface (assumed to be zero) and 𝐶 ∞ is the measured concentration in the ambient water. We assume diffusion coefficients of 1.98 x 10 -5 , 1.90 x 10 -5 and 1.38

x 10 -5 cm 2 s -1 at 25°C for NH4 + , NO3 -and urea, respectively [START_REF] Longsworth | Diffusion in the water-methanol system and the Walden product[END_REF][START_REF] Li | Diffusion of ions in sea water and in deep-sea sediments[END_REF].

Results

Biogeochemistry of the studied area

The biogeochemical characteristics of the sampled stations are presented in Table 1. At the time of sampling, surface NO3 -, NH4 + and urea concentrations were low: NO3 -was below the detection limit (10 nmol L -1 ) and NH4 + and urea concentrations ranged between 11-28 nmol N L -1 and 91-173 nmol N L -1 , respectively, without clear patterns between regions. In contrast, PO4 3-concentrations showed a strong pattern with higher concentrations at the continental shelf stations (110-152 nmol L -1 ) than at the Gulf Stream (40-45 nmol L -1 ) and the gyre (15-18 nmol L -1 ) stations. In the surface waters of the gyre, Particulate C concentrations were ~2 µmol C L -1 . Particulate C concentrations were higher at the Gulf Stream stations (3-4 µmol C L -1 ) and at the continental shelf stations (8-12 µmol C L -1 ).

Table 1. Biogeochemistry of the six stations investigated. All the samples were collected in triplicates (average ± standard deviation), unless otherwise stated. ND: not detected.

The cyanobacteria Prochlorococcus and Synechococcus dominated the surface pigmented pico-plankton community in the gyre with abundances in the same order of magnitude (~10 4 cell mL -1 ), while photosynthetic pico-eukaryotes abundances were consistently lower than 5.10 2 cell mL -1 (Table 1). In the Gulf Stream, abundances were three times higher for photosynthetic pico-eukaryotes, three-to-seven times higher for Synechococcus and four-to-ten times higher for Prochlorococcus (which reached particularly high abundances at station C, >2.10 5 cell mL -1 ) than in the gyre. At the continental shelf stations, Prochlorococcus was not detected, while Synechococcus abundances were in the range of those measured in the Gulf Stream and photosynthetic pico-eukaryotes were more abundant than in the two other regions studied. Non-pigmented prokaryotes abundances ranged between 2.10 5 to 6.10 5 cells mL - 1 with the lowest abundances observed in the gyre. Among non-pigmented prokaryotes, two-subgroups were observed, characterized by their level of green fluorescence, which reflects their nucleic acid content: low nucleic acid content and high nucleic acid content. Low nucleic acid sub-group numerically dominated the non-pigmented prokaryotes group at the gyre stations (59-63%) in contrast to the Gulf Stream and continental shelf stations (22-44%) (Table S2).

Community C-fixation in surface increased from the gyre (<20 nmol C L -1 h -1 ) to the Gulf Stream (~60 nmol C L -1 h -1 ) and to the continental shelf (>120 nmol C L -1 h -1 ) (Table 1). Similarly, community NO3 - uptake rates were 0.4 nmol N L -1 h -1 in the GYR, 0.9-1.4 nmol N L -1 h -1 in the Gulf Stream and reached as much as 3.8 nmol N L -1 h -1 at the continental shelf stations. Community NH4 + and N-urea uptake rates were within the same range at each region (<1.8, 3.0-7.0 and 6.1-23.6 nmol N L -1 h -1 , at the gyre, Gulf

Stream and continental shelf stations, respectively). Community C-and N-leucine potential uptakes rates were low in comparison to the other N compounds investigated (0.2-0.8 nmol C L -1 h -1 and 0.1-0.3 nmol N L -1 h -1 , respectively) but followed the same regional trends with higher rates on the continental shelf than in the Gulf Stream and gyre.

Single cell metabolic rates of the cytometrically sorted groups

Single cell analyses of cytometrically sorted groups allowed the determination of the cellular fixation or uptake rates of the 13 C or 15 N labelled substrates tested (Fig. 2, 3 and Table S3). The results are presented in C-and N-specific uptake rates (h -1 ) to allow for the comparison of metabolic activities between cells with different biomass content. Cell-specific uptake rates (amol C cell -1 h -1 and amol N cell -1 h -1 ) and Cfixation based division rates (d -1 ) are also presented in Table S3, S4 and Fig. S1 for comparison with literature data. Intragroup C-specific C-fixation rates varied greatly, from undetectable to more than 0.1 h -1 (Fig. 2). However, when averaged, clear patterns appeared between the different groups sorted and between regions. C-specific C-fixation was detected for a subset of the non-pigmented prokaryotes cells (15% on average) but at the group scale significant activities were not detected at any stations (see criteria in the experimental procedure section). In contrast, significant C-specific C-fixation was always detected for pigmented groups (averaging 0.016±0.010, 0.022±0.015 and 0.016±0.012 h -1 for Prochlorococcus, Synechococcus and photosynthetic pico-eukaryotes, respectively). C-specific Cfixation rate was on average twice higher in the Gulf Stream than in the gyre for Prochlorococcus.

Similarly, Synechococcus displayed higher activity in the Gulf Stream and the continental shelf as compared to the gyre. In contrast, no clear trends were observed between regions for photosynthetic pico-eukaryotes.

When detected, C-specific leucine potential uptake rates were higher on average in non-pigmented prokaryotes (0.0004±0.0002 h -1 on average) than in pigmented organisms (0.0001±0.0001 h -1 on average). Among the pigmented groups, Prochlorococcus showed the highest C-specific leucine potential uptake rates and the highest proportion of active cells (0.0002±0.0055 h -1 , 53% of active cells), as compared to photosynthetic pico-eukaryotes (0.0001±0.0001 h -1 , 26% of active cells) and Synechococcus (<0.0001 h -1 , 14% of active cells), respectively, and no clear patterns were observed between regions (Fig. 2). Gulf Stream (GULF) and continental shelf (SHELF) in grey, yellow and purple, respectively). Note the order of magnitude difference between the two-logarithm y-scales.

N-specific uptake rates were also highly variable between cells (Fig. 3). In the gyre and Gulf Stream regions where all three groups were detected, N-specific NH4 + uptake rate was on average slightly higher for Synechococcus (0.014±0.008 h -1 ) than for Prochlorococcus (0.013±0.006 h -1 ) and photosynthetic pico-eukaryotes (0.010±0.010 h -1 ) and higher for pigmented groups (0.012±0.004 h -1 ) than for nonpigmented prokaryotes (0.003±0.006 h -1 ) (each group was significantly different from each other, p<0.05). N-specific urea uptake was the highest on average for Synechococcus (0.030±0.018 h -1 ) followed by Prochlorococcus (0.023±0.016 h -1 ), photosynthetic pico-eukaryotes (0.003±0.004 h -1 ) and non-pigmented prokaryotes (0.003±0.002 h -1 ), respectively (each group were significantly different from each other, p<0.05). N-specific NO3 -uptake rates were on average higher for Synechococcus (0.006±0.005 h -1 ) compared to photosynthetic pico-eukaryotes (0.001±0.002 h -1 ), Prochlorococcus (0.001±0.003 h -1 ) and non-pigmented prokaryotes (<0.001 h -1 ), respectively (each group were significantly different from each other, p<0.05). Noticeably, at station C Prochlorococcus N-specific NO3 -uptake peaked at 0.004±0.005 h -1 , a rate much higher than the one observed in photosynthetic picoeukaryotes (0.002±0.003 h -1 ). N-specific leucine potential uptake was an order of magnitude lower compared to the three other N substrates studied and was highest for non-pigmented prokaryotes (0.0013±0.0005 h -1 ) as compared to the phytoplankton groups studied (0.0004±0.0004 h -1 , 0.0001±0.0001 h -1 , 0.0002±0.0003 h -1 for Prochlorococcus, Synechococcus and photosynthetic picoeukaryotes, respectively).

Gulf Stream (GULF) and continental shelf (SHELF) in grey, yellow and purple, respectively).

At the intra group scale, the C-specific C-fixation uptake rates were relatively stable as a function of cell size. However, the higher C-specific C-fixation rate of Synechococcus lead to a peak of rates centered around 1 µm in equivalent spherical diameter when all the groups are considered together (Fig. 4). Cell size rates dependent patterns appeared more clearly at the intragroup levels for the other parameters measured. Intriguingly, the N-specific uptake rates seemed to be more influenced by cell size rather than group identity. For NH4 + , NO3 -and N-urea patterns were similar with rates peaking for cells of size ca 1.5, 1.2 and 1.0 µm equivalent spherical diameter, respectively. For C-and N-specific leucine potential uptake rates, no peaks were observed but a decrease with increasing cell size. Using C and N cell contents estimated from the literature and measured abundances, we computed groups' absolute rates and compared them to the community rates measured on the GF/F filters (Fig. 5).

Pico-plankton (sum of non-pigmented prokaryotes, Prochlorococcus, Synechococcus and photosynthetic pico-eukaryotes) represented a significant fraction of the community C biomass (52±17% on average) and of the community C-fixation (63±27% on average) with large variability between stations (ranging from 35% to more than 100% of the community C-fixation). The contribution of groups to the community N species uptake was also noticeably variable between stations. For example, Prochlorococcus and photosynthetic pico-eukaryotes explained less than 10% of the community NO3 -uptake, except at stations C and E, where these groups contributed to more than 55% of the community NO3 -uptake. Similar patterns were also observed for NH4 + and N-urea uptake at these stations, which were explained by a conjunction of high abundances and high N-specific uptake. On average, the sum of pico-sized pigmented groups accounted for a relatively large fraction of NH4 + uptake (47±27%), NO3 -uptake (62±49%) and N-urea uptake (80±35%). The contribution of non-pigmented prokaryotes to N uptake were much lower (averaging 7±3%, 2±2% and 9±5% for NH4 + , NO3 -and Nurea, respectively). In contrast, this group was the main contributor to the community N-leucine potential uptake (range 42-54%).

Figure 5. Contribution of picoplankton groups to the community C-biomass, C-fixation, nitrate (NO3 -),

ammonium (NH4 + ), N-urea and N-leucine uptake rates at each station investigated. Colors denote the analyzed groups (non-pigmented prokaryotes, Prochlorococcus, Synechococcus and photosynthetic pico-eukaryotes in green, yellow, blue and red, respectively).

Discussion

Methodological considerations

The role of various groups of pico-plankton in ocean C and N cycling remains poorly resolved in part because of a lack of appropriate methodological tools. Since the first applications to environmental microbiology more than a decade ago, nanoSIMS coupled to isotope labelling assays has gained in popularity and has been used to measure the contribution of different microbial groups to the community activity [START_REF] Klawonn | Cell-specific nitrogen-and carbon-fixation of cyanobacteria in a temperate marine system (Baltic Sea)[END_REF][START_REF] Berthelot | NanoSIMS single cell analyses reveal the contrasting nitrogen sources for small phytoplankton[END_REF]. Using this approach, we show in this study that picoplankton account for more than half of the community C-fixation in our study region (63% on average). This result is in line with previous measurements made in this area, and more generally in oligotrophic environments, using 14 C-sodium bicarbonate radioassays coupled with size fractionation or cell sorting [START_REF] Jardillier | Significant CO2 fixation by small prymnesiophytes in the subtropical and tropical northeast Atlantic Ocean[END_REF][START_REF] Duhamel | Small pigmented eukaryotes play a major role in carbon cycling in the P-depleted western subtropical North Atlantic, which may be supported by mixotrophy[END_REF]. It is important to note that the cell-specific and group-specific uptake rates derived from nanoSIMS approaches rely on cell content data independently measured or reported in the literature which can vary by up to an order of magnitude between studies [START_REF] Martiny | Strong latitudinal patterns in the elemental ratios of marine plankton and organic matter[END_REF][START_REF] Baer | Stoichiometry of Prochlorococcus, Synechococcus, and small eukaryotic populations in the western North Atlantic Ocean[END_REF]. In our study, we used biomass cell contents measured from samples obtained in our sampling area, the northwestern Atlantic ocean [START_REF] Baer | Stoichiometry of Prochlorococcus, Synechococcus, and small eukaryotic populations in the western North Atlantic Ocean[END_REF]). The derived cell-specific C-fixation rates (51-102, 317-806 and 2872-5388 amol C h -1 for Prochlorococcus, Synechococcus and photosynthetic pico-eukaryotes, respectively) are in good agreement with values recently reported in the literature [START_REF] Jardillier | Significant CO2 fixation by small prymnesiophytes in the subtropical and tropical northeast Atlantic Ocean[END_REF][START_REF] Zubkov | Faster growth of the major prokaryotic versus eukaryotic CO2 fixers in the oligotrophic ocean[END_REF][START_REF] Duhamel | Small pigmented eukaryotes play a major role in carbon cycling in the P-depleted western subtropical North Atlantic, which may be supported by mixotrophy[END_REF])(Fig. S1).

These cell contents carry uncertainty (coefficient of variation of ±30% to >100%) which can affect the cell and group specific uptake rates to the same extent and may bias the estimated contribution of groups to the community uptake (Fig. 5). The estimated contribution of small cells measured here could also be biased by active cells passing through GF/F filters [START_REF] Bombar | Filtration via Conventional Glass Fiber Filters in 15 N2 Tracer Assays Fails to Capture All Nitrogen-Fixing Prokaryotes[END_REF]. This would lead to an underestimation of the community rates and could explain the picoplankton rates being higher than community rates at some stations (Fig. 5). Using the cells outlined from nanoSIMS images, we measured that 54% of the non-pigmented prokaryotes, 31% of the Prochlorococcus and 7% of the Synechococcus cells had an equivalent spherical diameter lower than the 0.7 µm nominal porosity of GF/F filters (Fig. S2) which is in line with previous reports showing that up to ~50% of the non-pigmented prokaryotes and <10% of the cyanobacteria cells eventually pass through GF/F filters [START_REF] Lee | Imperfect retention of natural bacterioplankton cells by glass fiber filters[END_REF][START_REF] Morán | A comparison between glass fiber and membrane filters for the estimation of phytoplankton POC and DOC production[END_REF][START_REF] Bombar | Filtration via Conventional Glass Fiber Filters in 15 N2 Tracer Assays Fails to Capture All Nitrogen-Fixing Prokaryotes[END_REF]. The combustion of GF/F filters might decrease the nominal pore size [START_REF] Nayar | Relative efficiencies of different filters in retaining phytoplankton for pigment and productivity studies[END_REF]. In the future, the use of silver filters with a pore size of 0.2 µm or the Advantex glass fiber filters with a nominal pore size of 0.3 µm (both compatible with elemental analyzers) could reduce the number of cells passing through the filters [START_REF] Bombar | Filtration via Conventional Glass Fiber Filters in 15 N2 Tracer Assays Fails to Capture All Nitrogen-Fixing Prokaryotes[END_REF].

Based on the relatively low leucine uptake rates in the open ocean [START_REF] Zubkov | Differential microbial uptake of dissolved amino acids and amino sugars in surface waters of the Atlantic Ocean[END_REF], we added leucine at saturating (or close to saturating) concentrations of 10 nmol L -1 in order to ensure significant isotopic signal in our samples. The rates provided thus reflect "potential" rates rather than absolute rates. The leucine community uptake rates measured here (0.1-0.3 nmol leucine L -1 h -1 ) are at the higher end of those reported using trace levels isotopes additions (i.e. additions of leucine < 0.5 nmol L -1 ) in N.

Atlantic with the more sensitive radiotracer assays (~0.01-0.1 nmol leucine L -1 h -1 ) [START_REF] Zubkov | High rate of uptake of organic nitrogen compounds by Prochlorococcus cyanobacteria as a key to their dominance in oligotrophic oceanic waters[END_REF][START_REF] Mary | Light enhanced amino acid uptake by dominant bacterioplankton groups in surface waters of the Atlantic Ocean[END_REF][START_REF] Hill | Low microbial respiration of leucine at ambient oceanic concentration in the mixed layer of the central Atlantic Ocean[END_REF]. Addition of leucine at saturating concentration (20 nmol L -1 ) in the N. Atlantic resulted in a doubling of leucine uptake rates as compared to those obtain from trace level additions (0.4 nmol L -1 ) [START_REF] Hill | Low microbial respiration of leucine at ambient oceanic concentration in the mixed layer of the central Atlantic Ocean[END_REF]. This provides some insight into the extent of the rate overestimation presented in our study.

Could the low N availability explain the dominance of small plankton groups?

The single cell isotopic approach used here provides an estimate of the substrate specific uptake rate. If the only source of C for the pigmented cells is from C-fixation, C-fixation based division rates should reflect division rates at steady-state. For pigmented groups, C-fixation based division rates measured in our study (0.18-0.64 d -1 ) are in line with previous measurements made in the N. Pacific using the same approach (0.32-0.50 d -1 ) [START_REF] Berthelot | NanoSIMS single cell analyses reveal the contrasting nitrogen sources for small phytoplankton[END_REF]. In our study, C-fixation based division rates were higher for Synechococcus (0.45±0.22 d -1 on average) than for the two other pigmented groups (0.28±0.12 d -1 on average) investigated. This is consistent with patterns and values reported in the review of Kirchman (2016).

While C-specific fixation did not appear to scale with cell size at the intra-group level, clear patterns were observed for N-specific uptake rates (Fig. 4). Such a relationship could be explained by nutrient availability. Under nutrient scarcity, small size plankton have a competitive advantage due to their high surface-area-to-volume ratios [START_REF] Naselli-Flores | Shape and size in phytoplankton ecology: do they matter?[END_REF]. At the time of sampling, N species concentrations were low with the sum of NO3 -, NH4 + and N-urea below <200 nmol N L -1 . At such low N concentrations, the uptake is limited by the molecular diffusion of N compounds to their cellular membranes [START_REF] Karp-Boss | Nutrient fluxes to planktonic osmotrophs in the presence of fluid motion[END_REF][START_REF] Olofsson | Nitrate and ammonium fluxes to diatoms and dinoflagellates at a single cell level in mixed field communities in the sea[END_REF]. Using a diffusion model (see details in Materials and Methods section), we calculated that cells with a diameter larger than 5 µm could not maintain N-specific NH4 + uptake as high as those measured here for pico-plankton. Similar thresholds were found for NO3 -uptake (7 µm) and N-urea uptake (12 µm). Below these thresholds, smaller cells still profit from their high surface-area-to-volume ratio which could explain the overall inter-and/or intra-group patterns of increasing N-specific uptake with decreasing cell size up to ~1-2 µm equivalent spherical diameter (Fig. 4). Below this limit, the reduction in size for pigmented organisms is limited by non-scalable cellular components, in particular photosynthetic apparatus [START_REF] Ward | The size dependence of phytoplankton growth rates: A trade-off between nutrient uptake and metabolism[END_REF]. The peak around 1-2 µm equivalent spherical diameter in N-specific rates observed here for pigmented organisms is lower than a previous report in the Mediterranean lagune (2-3 µm) [START_REF] Bec | Growth rate peaks at intermediate cell size in marine photosynthetic picoeukaryotes[END_REF]. This is consistent with an adaptation of the present communities to extremely oligotrophic conditions where further cell size reduction to cope with nutrients scarcity is hindered by minimal maintenance of cellular basal functions. In the case of non-pigmented prokaryote, the more streamlined genome and metabolic functions of this group as compared to cyanobacteria and to a larger extent to photosynthetic picoeukaryotes allow them a smaller cell size [START_REF] Swan | Prevalent genome streamlining and latitudinal divergence of planktonic bacteria in the surface ocean[END_REF]) and could explain their relatively high efficiency at using leucine available at extremely low concentrations in the ocean (<1 nM, [START_REF] Zubkov | Differential microbial uptake of dissolved amino acids and amino sugars in surface waters of the Atlantic Ocean[END_REF]. Taken together these observations largely explain the numerical dominance of pico-plankton in the oceanic regions sampled. It also implies that, to compensate for their lack of N acquisition competitiveness, larger photosynthetic plankton cells have to rely on alternative strategies such as increasing their surface-area-to-volume ratios by developing complex nanostructure shapes [START_REF] Mitchell | The Role of Diatom Nanostructures in Biasing Diffusion to Improve Uptake in a Patchy Nutrient Environment[END_REF], relying on predation [START_REF] Stoecker | Mixotrophy in the Marine Plankton[END_REF] or symbioses with N2 fixing organisms [START_REF] Foster | Nitrogen fixation and transfer in open ocean diatom-cyanobacterial symbioses[END_REF][START_REF] Zehr | Unusual marine unicellular symbiosis with the nitrogen-fixing cyanobacterium UCYN-A[END_REF].

The relative importance of organic and inorganic sources of C

We measured some C-fixation for a subset of the cells of the non-pigmented prokaryotes group at all stations (15% on average, Fig. 2). This C-fixation by the non-pigmented prokaryotes group may stem from the transfer of 13 C fixed by the photosynthetic organisms during the incubation (Arandia-Gorostidi et al. 2017) or an active fixation performed by chemoautotrophs such as nitrifying bacteria [START_REF] Middelburg | Chemoautotrophy in the ocean[END_REF]. Despite conservative sorting procedures, it is also possible that pigmented organisms were missorted in the non-pigmented prokaryotes group. However, at the group scale, C-fixation by nonpigmented prokaryotes were not statistically significant and trivial in comparison to uptake by their photosynthetic counterparts, in line with the expected partitioning between pigmented and nonpigmented organisms with respect to C-fixation.

C-specific leucine potential uptake rates by non-pigmented prokaryotes were on average 4-10 times higher than those of pigmented groups (Fig. 2, Table S3) confirming the competitive advantage of heterotrophs in the acquisition of organic molecules such as leucine. C-specific leucine potential uptake rates in pigmented groups were low but statistically significant at the single cell level for 14-53% of the pigmented cells and at the group level at most sites (see criteria in material and methods section) (Table S3). This use of organic C might explain the survival of photosynthetic pico-eukaryotes in extended darkness such as polar winter [START_REF] Deventer | Effects of prolonged darkness on the relative pigment content of cultured diatoms and green algae[END_REF] or the maintenance of active Prochlorococcus populations at depth when low light levels limit photosynthetic activity [START_REF] Coe | Survival of Prochlorococcus in extended darkness[END_REF]. The leucine uptake measured in pigmented groups can originate either from a direct osmotrophic uptake of leucine, or indirectly from predation on prey which would have assimilated 13 C-leucine during the incubation. Many studies report predation by taxa belonging to the photosynthetic pico-eukaryotes group by phagocytosis [START_REF] Zubkov | Differential microbial uptake of dissolved amino acids and amino sugars in surface waters of the Atlantic Ocean[END_REF][START_REF] Duhamel | Small pigmented eukaryotes play a major role in carbon cycling in the P-depleted western subtropical North Atlantic, which may be supported by mixotrophy[END_REF], which might explain a fraction of the leucine uptake measured here for this group. On the other hand, direct osmotrophic uptake of dissolved organic compounds by pigmented eukaryotes is common and could also explain the leucine uptake observed in photosynthetic pico-eukaryotes observed in our study [START_REF] Ruiz-González | Sunlight Effects on the Osmotrophic Uptake of DMSP-Sulfur and Leucine by Polar Phytoplankton[END_REF]. More studies are needed to assess the relative importance of phagotrophy and osmotrophy in the mixotrophic strategies of photosynthetic pico-eukaryotes. In the cases of Prochlorococcus and Synechococcus, leucine uptake is likely through osmotrophy as direct uptake of organic molecules (e.g. glucose, amino acids) has been reported [START_REF] Muñoz-Marín | Mixotrophy in marine picocyanobacteria: use of organic compounds by Prochlorococcus and Synechococcus[END_REF]) while, to the best of our knowledge, predation has not been observed.

The relative importance of organic and inorganic sources of N

In our study, NH4 + and urea were the dominant sources of N at the community level (Table 1), in agreement with observations made previously in the gyre and in the Gulf Stream [START_REF] Lipschultz | A time-series assessment of the nitrogen cycle at BATS[END_REF][START_REF] Casey | Prochlorococcus contributes to new production in the Sargasso Sea deep chlorophyll maximum[END_REF]. This was also verified at the group level for the pigmented and non-pigmented groups (Fig. 3). The importance of NO3 -uptake was much more reduced but contrasting patterns were observed between groups. Significant NO3 -uptake by Prochlorococcus confirms previous reports [START_REF] Casey | Prochlorococcus contributes to new production in the Sargasso Sea deep chlorophyll maximum[END_REF][START_REF] Berube | Physiology and evolution of nitrate acquisition in Prochlorococcus[END_REF][START_REF] Berthelot | NanoSIMS single cell analyses reveal the contrasting nitrogen sources for small phytoplankton[END_REF]). However, NO3 -only accounted for a small fraction of Prochlorococcus N sources (3.7±8.2%, Table 2), in line with previous results using a similar approach in the North Pacific Gyre (4.5±6.5%) [START_REF] Berthelot | NanoSIMS single cell analyses reveal the contrasting nitrogen sources for small phytoplankton[END_REF]. In contrast, NO3 -represented a larger fraction of N uptake for Synechococcus in the North Atlantic (11.5±12.8% on average, this study) than in the North Pacific (2.9±2.1%, [START_REF] Berthelot | NanoSIMS single cell analyses reveal the contrasting nitrogen sources for small phytoplankton[END_REF]. This difference may be explained by the greater NO3 -concentrations in the Atlantic, since in the North Pacific Gyre, surface NO3 -concentrations remain lower than 10 nmol L -1 [START_REF] Karl | Ecological nitrogen-to-phosphorus stoichiometry at station ALOHA[END_REF]. In our study regions, sampled surface waters were depleted in NO3 -but regular mixing events bring NO3 -concentrations well above 100 nmol L -1 in the mixed layer, even in the GYR [START_REF] Lipschultz | A time-series assessment of the nitrogen cycle at BATS[END_REF][START_REF] Treibergs | Nitrogen isotopic response of prokaryotic and eukaryotic phytoplankton to nitrate availability in Sargasso Sea surface waters[END_REF]. The transiently more available NO3 -might result in the adaptation or selection of Synechococcus populations that are more efficient at using NO3 - when available [START_REF] Casey | Prochlorococcus contributes to new production in the Sargasso Sea deep chlorophyll maximum[END_REF]. This is further confirmed by the generally low δ 15 N signature of Synechococcus and Prochlorococcus, characteristic of a reliance on remineralized N compounds such as NH4 + or N-urea [START_REF] Fawcett | Assimilation of upwelled nitrate by small eukaryotes in the Sargasso Sea[END_REF]). In the presence of NO3 -, the δ 15 N of these groups can increase, confirming the capacity of the organisms to use NO3 -when available in the North Atlantic Gyre [START_REF] Fawcett | Assimilation of upwelled nitrate by small eukaryotes in the Sargasso Sea[END_REF][START_REF] Treibergs | Nitrogen isotopic response of prokaryotic and eukaryotic phytoplankton to nitrate availability in Sargasso Sea surface waters[END_REF].

Table 2. Relative importance of the different N sources investigated (average ± standard deviation between stations, in %) for the N acquisition budget of each group. Pigmented organisms generally outcompeted non-pigmented prokaryotes for the acquisition of these inorganic N species. N-specific NH4 + and NO3 -uptake rates by non-pigmented prokaryotes were indeed generally lower than in pigmented groups (Fig. 3). The uptake of NH4 + and NO3 -by non-pigmented prokaryotes averaged 7±3% and 2±2% of the community uptake, respectively. These uptake estimates fall at the lower end of previous reported contributions ranging from 5 to 60% for NH4 + and 4 to 80% for NO3 - [START_REF] Kirchman | Biomass and nitrogen uptake by heterotrophic bacteria during the spring phytoplankton bloom in the North Atlantic Ocean[END_REF][START_REF] Fouilland | Nitrogen uptake by heterotrophic bacteria and phytoplankton in Arctic surface waters[END_REF][START_REF] Trottet | Use of inhibitors for coastal bacteria and phytoplankton: Application to nitrogen uptake measurement[END_REF]). The differences could be due to regional variability in dissolved inorganic N concentrations, with our study sites displaying lower concentrations than previous studies which were conducted in more N rich waters. We cannot rule out that discrepancies also result from methodological differences, with previous studies relying mostly on size fractionation or inhibitors. Additional single-cell experiments in N rich waters would help to unravel changes in N-species uptake for different groups as a function of N availability.

Conclusions and implications

In this study, we provided a comprehensive analysis of in situ assimilation rates of organic and inorganic C and N sources for different groups of the picoplankton community. A principle component analysis shows the group specific C-and N-trophic strategies (Fig. 6). At the group level pigmented and nonpigmented organisms were clearly partitioned, with the former clustering around C-fixation and NO3 - and N-urea uptake, and the latter around C-and N-leucine uptake. On the other hand, the principal component analysis shows that NH4 + uptake is a poor predictor of the groups' partitions. In contrast to previous findings in the North Pacific [START_REF] Berthelot | NanoSIMS single cell analyses reveal the contrasting nitrogen sources for small phytoplankton[END_REF], Synechococcus appeared to be the group relying the most on NO3 -. This suggests that the same pico-plankton group might adopt different nutrient uptake strategies between oceanic regions. Intriguingly, Prochlorococcus and photosynthetic picoeukaryotes showed higher C-and N-specific leucine potential uptake than Synechococcus. This capacity to diversify their C and N sources, either by osmotrophic uptake or by predation, may allow these microorganisms to maintain their growth under extremely severe nutrient-depleted environments [START_REF] Zubkov | High rate of uptake of organic nitrogen compounds by Prochlorococcus cyanobacteria as a key to their dominance in oligotrophic oceanic waters[END_REF]. This behavior could explain the maintenance of the populations and their dominance in some of the most oligotrophic oceanic regimes [START_REF] Flombaum | Global picophytoplankton niche partitioning predicts overall positive response to ocean warming[END_REF].

Little is known on variations in nutrition acquisition strategies across environmental gradients, and in particular, across light and nutrients gradients. Taken together, our results provide a snapshot of the picoplankton strategies in the uptake of organic and inorganic sources of C and N. While we observed clear distinctions in C and N uptake strategies between pigmented and non-pigmented groups (Fig. 6), our results also highlight contrasting strategies among the pigmented groups and the importance of cell size in osmotrophic nutrient acquisition (Fig. 6). As a result of global warming, picoplankton is likely to become more important in ocean biogeochemistry as oligotrophic waters are predicted to expand due to increased water column stratification [START_REF] Flombaum | Global picophytoplankton niche partitioning predicts overall positive response to ocean warming[END_REF]. A combination of approaches, including those presented here, will be needed to improve our understanding of the key processes (such as nutrient affinity, competition, associations, predation) determining the dynamic of plankton groups and to predict their fate in the context of a changing ocean. Table S2: Abundances of low nucleic acid content and high nucleic acid content sub-groups in the non-pigmented prokaryote groups at each station sampled.

Station

High nucleic acid content sub-group (10 3 cells ml -1 )

Low nucleic acid content sub-group(10 3 cells ml -1 )

North Atlantic Gyre A 100±10 143±3 B 95±13 158±7 Gulf Stream C 404±58 181±29 D 214±9 167±8 Continental Shelf E 425±17 185±5 F 230±6 68±18
Table S3. C-fixation division growth rate (d -1 ) and C-and N-specific C-fixation, ammonium (NH4 + ), nitrate (NO3 -), urea and leucine uptake rates (h -1 ) measured for each group and at each station. Mean and standard deviation are in bold font. Median are in regular font and the number of cells analyzed are in italics. ND: not detected. NA: not analyzed. Table S4: Cell-specific C-fixation rates and ammonium (NH4 + ), nitrate (NO3 -), N-urea, C-leucine, Nleucine uptake rates (average of cells analyzed, in amol C cell -1 h -1 and amol N cell -1 h -1 ) at the sampled stations. ND: not detected. NA: not analyzed.

Figure 1 .

 1 Figure 1. Location of the stations and the oceanic regions sampled in the northwestern Atlantic Ocean (SHELF: continental shelf, GULF: Gulf Stream, GYR: North Atlantic Gyre) superimposed on surface chlorophyll a (Chl a) concentration (in mg.m -3 ) from AQUA/MODIS (composite image of August 2017).

  ) were used to calculate the element (C-or N-) specific uptake rate (h - 1 ):

Figure 2 .

 2 Figure 2. Single cell C-specific C-fixation rates and C-specific leucine potential uptake rates (h -1 ) for each group investigated (NP-P: non-pigmented prokaryotes, Pro: Prochlorococcus, Syn: Synechococcus, PPE: photosynthetic pico-eukaryotes). Each point represents an analyzed cell. Only the cells with detected activity with respect to the process under study are shown. Italic black numbers denote the number of cells analyzed for each group. Grey numbers denote the proportion (in %) of cells for which activity was detected. Colors denote the sampling regions (North Atlantic Gyre (GYR),

Figure 3 .

 3 Figure 3. Single cell N-specific nitrate (NO3 -), ammonium (NH4 + ), urea and leucine specific uptake rates (h -1 ) for each group (NP-P: non-pigmented prokaryotes, Pro: Prochlorococcus, Syn: Synechococcus, PPE: photosynthetic pico-eukaryotes). Each point represents an analyzed cell. Only the cells with detected activity with respect to the process under study are shown. Italic black numbers denote the number of cells analyzed for each group. Grey numbers denote the proportion (in %) of

Figure 4 .

 4 Figure 4. Single cell C-and N-specific C-fixation, nitrate (NO3 -), ammonium (NH4 + ), urea and leucine uptake rates (h -1 ) as a function of cell size. The black lines and the shaded area denote the average and

Figure 6 .

 6 Figure 6. Principal Component Analysis of the C-and N-specific uptake rates highlighting the differences in C and N strategies of the different groups investigated (data were normalized and centered). Each point represents a group at a given sampling station.

Fig S2 .

 S2 Fig S2. Compilation of the cell-specific C-fixation rates (in amol C cell -1 h -1 ) available in the literature for Prochlorococcus, Synechococcus and photosynthetic pico-eukaryotes.

  F4098.8±2520.2 878.5±440.7 59.6±80.9 466.6±289.4 50.0±62.4 11.8±12.6

	D	5387.9±2811.5 449.0±277.8 13.8±18.0	66.7±86.1	3.5±23.8	2.8±5.7
	E	3056.0±1689.5	27.4±42.1	174.1±93.8 437.2±326.6 4.4±12.2	1.6±1.7
						Cell-	Cell-
			Cell-	Cell-	Cell-	specific	specific
	Station	Cell-specific C-fixation	specific NH4 +	specific NO3 -	specific N-Urea	C-leucine	N-leucine
			uptake	uptake	uptake	potential	potential
						uptake	uptake
			Non-pigmented prokaryote
	A	ND	NA	NA	0.7±0.3	0.8±0.2	0.3±0.1
	B	ND	0.8±0.2	0.1±<0.1	1.1±0.2	0.7±0.3	0.4±0.1
	C	ND	0.6±0.5	<0.1	1.2±1.2	0.2±0.2	0.2±0.1
	D	ND	1.0±0.5	<0.1	0.4±0.3	0.7±0.2	0.4±0.1
	E	ND	0.4±0.2	0.1±<0.1	1.1±0.4	0.6±0.2	0.3±0.1
	F	ND	5.3±1.6	0.1±<0.1	2.9±1.4	1.1±0.4	0.5±0.1
			Prochlorococcus
	A	69.5±73.2	7.5±2.9	0.2±0.3	2.9±3.1	0.9±1.1	0.3±0.3
	B	50.6±26.9	6.2±1.7	0.3±0.3	12.3±7.1	0.6±1.3	0.3±0.4
	C	101.8±38.4	7.0±1.3	2.7±3.4	27.4±7.4	1.0±0.5	0.2±0.1
	D	97.9±44.0	13.1±4.3	0.4±0.8	17.7±8.4	0.5±0.5	0.2±0.2
			Synechococcus
	A	419.1±283.7	32.9±11.2	3.4±2.9	44.2±10.8	ND	ND
	B	317.3±196.6	24.1±9.7	20.2±7.1	73.9±25.4	ND	ND
	C	558.4±244.8	44.1±7.0	28.4±9.2	56.1±36.3	0.6±1.8	0.3±0.4
	D	806.3±476.9	41.8±31.6	13.5±6.5	143.4±43.1	0.5±1.1	0.3±0.3
	E	618.4±160.1	24.5±4.3	12.0±5.0	77.5±33.6	0.5±2.2	0.2±0.3
	F	645.4±292.2	83.0±40.4	20.9±9.4	109.8±25.1	3.3±8.7	1.2±1.6
		Photosynthetic pico-eukaryotes
	A	2871.6±2846.0	NA	3.3±6.5	55.7±55.6	24.8±21.1	6.5±5.7
	B	4968.5±3122.3 132.8±79.6	22.1±41.1	47.3±45.0	16.5±40.6 8.7±12.3
	C	2886.8±2411.4	58.0±47.0	38.4±63.2 119.0±116.6 10.1±19.5	3.1±5.0

Acknowledgements

We would like to thank the crew of the R/V Atlantic Explorer for their help during the cruise. We also thank Smail Mostefaoui for his assistance with the nanoSIMS analyses at the French National Ion MicroProbe Facility hosted by the Muséum National d'Histoire Naturelle (Paris). N. C. and H. B. were supported by the "Laboratoire d'Excellence" LabexMER (ANR-10-LABX-19) and co-funded by a grant from the French government under the program "Investissements d'Avenir". SD was funded by the National Science Foundation (OCE-1434916 and OCE-1458070).

Supplementary Information

Table S1: Minimum quantifiable rates (MQR) (as defined by [START_REF] Gradoville | Diversity and activity of nitrogen-fixing communities across ocean basins[END_REF]) of C and N community uptake (in nmol C L -1 h -1 or nmol N L -1 h -1 ) were calculated for each tracer and each station from the propagation of errors of each parameter involved in the uptake rate calculation. For the parameters measured in triplicate, the errors were the standard deviation. For the incubation time, the error was estimated to be ±10 minutes (representing the typical filtration duration). For the isotopic percent abundance of source pool during dissolved inorganic 13 C incubations, the error was estimated to be ±0.5 atom%. measured for each group. The red vertical lines depict the nominal pore size of the GF/F filters (0.7 µm). 54%, 31%, 7% and 0% of the non-pigmented prokaryotes, Prochlorococcus, Synechococcus and photosynthetic pico-eukaryotes cells, respectively, were smaller than 0.7 µm. Note that only low nucleic acid non-pigmented prokaryotes were analyzed. It is likely that the equivalent spherical diameter of the high nucleic acid non-pigmented prokaryotes was higher.


Marcel Ebene Ebene

Yves Maréchal
email: yves.marechal@g2elab.inpg.fr

David Armand

Dimitrios Ladas

M Ebene-Ebene
email: marcel.ebene-ebene@g2elab.inpg.fr

Y Marechal

An Adaptive Remeshing Technique Insuring High Quality Meshes

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I. INTRODUCTION

While finite-elements methods (FEM) are currently employed extensively for electromagnetic analysis and design, the use of adaptive finite-elements methods has increased considerably in recent years (AFEM). The quality of the meshes obtained after AFEM remains improvable, therefore, there is still a need for a procedure which insures good quality's meshes after AFEM iterations.

AFEMs consist in, given an error criterion, finding the set of elements (or nodes) where the error is large, and modifying the mesh at those locations in order to reduce the error. The error is defined as the difference between the estimation of the exact solution and the computed one.

There are two main families of procedure to address AFEMs:

• p-procedures that consist in increasing the degree of the interpolating function. These techniques are difficult to use due to the cost it requires to increase that degree.

• h-procedures that consist in subdividing the mesh where the error criterion is large, by increasing (or eventually decreasing) the discretisation. Conceptually, htype adaptive refinement is straightforward. However an effective implementation can be subject to practical limitations. For instance, the way the new elements are created where the error is large can affect the quality of the resulting mesh.

There are two main ways to add new elements in a mesh: -Using the Delaunay technique: Once an element needs to be modified, a node is inserted at the center of the element or in the middle of its longer edge. All the edges of the elements around this new node are destroyed freeing a certain number of nodes, and a new Delaunay procedure [START_REF] Delaunay | Sur la sphère vide[END_REF] is launched with these Manuscript received June 24, 2007. This work is the product of the collaboration between the G2elab and Cedrat with the goal of a high quality meshes and the implementation of the refinement process.

nodes in order to create new elements. This technique is correct, but does not allow really driving the number of nodes according to the error, since the number of new nodes is the same, whatever the error is.

-Using a subdivision technique where several strategies exist: One method, given in [START_REF] Tani | H-version Adaptative Finite Element Method Using Edge Element for 3D Non-Linear Magnetostatic Problems[END_REF] consists in inserting a certain number of nodes in the edges of an element that need to be modified. The elements with the larger error receive several new nodes, while those with lesser error receive fewer new nodes.

Both these methods share a common drawback: They do not focus on the quality of the resulting mesh at all. The inserted nodes can produce flat, long and thin elements that can lead to poorly conditioned finite element matrices, which in turn, can compromise the accuracy of the solution.

In this paper, we use a Bubble meshing method to remesh once the error criterion is computed. This technique allows obtaining a good initial mesh to start the AFEM, and keep this good quality all along the refinements iterations. It also leads the post AFEM meshes to match the error: the elements are smaller where the error is bigger and larger where the error is smaller.

II. THE BUBBLE MESHING METHOD

In this part, the global principle of the method will be presented. Improvements of this technique will also be presented, improvements that lead to a faster and better quality prone procedure.

A. Description of the technique

The Bubble Meshing Method consists in acting on a given mesh in order to improve its quality, or to create a new mesh insuring high quality elements. After this procedure, the mesh is said as "bubblelized".

The bubble meshing procedure can be summarized as follow [START_REF] Shimada | Bubble Mesh: Automated Triangular Meshing of Non-Manifold Geometry by sphere packing[END_REF], [START_REF] Yokoyama | 3-D automatic mesh generation for FEA using dynamic bubble system[END_REF]: In a given geometry, spheres (or circles in 2D) called bubbles are created in order to fill the whole geometry without gaps. These bubbles are then set into movement with a given interaction force and they reach a stable position, which is when all the bubbles are slightly touching their respective neighbors without significant overlaps and gaps between them. During this process, the control of the population of bubbles is made simultaneously in order to avoid overlaps and holes. Once the bubbles are in a stable position, the nodes of the future mesh are then created at the centre of these bubbles and a mesh with tetrahedra (3D) or triangles (2D) is obtained.

B. Problems faced

This technique has been implemented. While it produces better meshes than initial ones, it also raised some hurdles that need to be crossed in order to have a faster, more efficient and reliable technique. The following part deals with the solutions proposed to improve the technique.

III. IMPROVEMENTS BROUGHT TO THE BUBBLE MESHING METHOD

A. Initial bubbles placement

One of the main problems with this technique is the choice of initial localisation of the bubbles. Some studies packed bubbles directly on the geometry, based on a quadtree approach for instance [START_REF] Shimada | Bubble Mesh: Automated Triangular Meshing of Non-Manifold Geometry by sphere packing[END_REF]. We have chosen in this paper to start with an initial mesh obtained by existing methods such as the well known Delaunay triangulation, each node having a given weight. This discrete set of weights defines a size map that can be interpolated on the whole geometry. Bubbles are created at the nodes of the mesh. During their movement, their size is adapted according to their new location to match the size map.

The main part of this technique, in addition of the definition of the size map, is thus the succession (10 iterations) of movement and control of the population. Those two steps need to be optimized in order to improve the method performances.

B. Adaptive Population Control 1) Ideal number of neighbors for a bubble

In a geometry with a uniform size of bubbles, the ideal number of neighbors for a bubble is six (in 2D) to avoid gaps and overlaps. When the size map is no more constant, it becomes complicated to find out the correct number of neighbors. We use the overlapping ratio i β , introduced in [3] to determine whether the neighborhood of the ith bubble is overloaded (the number of nearby bubbles is superior to i β ) or open (the number of nearby bubbles is inferior to i β ). Thus excess bubbles will be removed and new bubbles will be added. i β is defined as follows:

) . 2 ( . 1 1 ∑ = + + = i N j ij j i i i d r r r β ( 1
)
where i r is the radius of the ith bubble, ij d is the distance between the ith bubble and its jth neighbor, i N being the number of its neighbors

In an ideal situation where all bubbles have the same size and are tightly packed, β = 2.0 for an edge, β = 6.0 on a face and β = 12.0 in a volume.

2) Remaining gaps between bubbles

Toward the end of the movement / control process, an instable situation may appear, bubbles being added in some place whereas few other bubbles being also deleted, leading to some residual gaps between the bubbles. This is solved by deactivating the suppression of bubbles (the control of the population can only add bubbles) in the last four iterations.

C. Optimization of the movement

Whereas the quality of the bubblelized mesh drastically improves, the cost of the whole procedure could be very high in terms of computation time. Two aspects can minimize this drawback: a correct choice of the force and an optimal simulation of the movement.

1) Force model -A repulsive force is probably the best choice if the number of bubbles has to remain constant. The bubbles won't touch each other at the end of the regularization procedure, but will be uniformly spread like gas molecules in a closed box. However, when inserting a new bubble, all the bubbles will move in the next step of movement since they are all disturbed.

-An attractive-repulsive force is ideal when new bubbles need to be inserted as it creates holes and thus facilitates the insertion of new bubbles: only the very close neighbors of the new bubbles will be disturbed, the rest remaining stable. Hence, the next steps of movement will be faster. This is a model we have chosen to use since we also control the population.

2

) Force simulation

The force that governs the movement of a bubble submitted to the influence of its neighbors is the sum of each force generated by each of these neighbors. To gain performance during simulation of movement, we have first modeled the movement of a single bubble attracted by one fixed neighbor. Then this model has been extended to the case with several nearby bubbles.

The force between two bubbles [START_REF] Shimada | Bubble Mesh: Automated Triangular Meshing of Non-Manifold Geometry by sphere packing[END_REF] is such as, at an ideal distance (the sum of the radius of both bubbles), the force is repulsive and becomes attractive over that ideal distance. Therefore, this force is much like the force of a spring. The equation of its movement is then

X c X k X m & & & . . . - - = (2)
where k is the spring constant (directly related to the force), m, the mass of the bubble, and c the coefficient of friction.

These parameters are the ones, in addition to the step time and the integration method, that need to be addressed in order to optimize the movement.

• The spring constant k: We use the value given in [3] that gives for each bubble (3).

• The mass of the bubble m: the mass is chosen as normalized instead of being proportional to the surface (volume) of the bubble. This allows us to keep the step time constant whereas it needs to be adjusted if the mass varies to avoid numerical instability of the integration procedure.

• The coefficient of friction c: The system described is similar to a second order system in Control Theory [5]. Therefore we can make analogy with the transfer function given in [5]. We extract the damping ratio ξ which can be written in our case. To ensure the best compromise between the speed of the system and its stability [5], ξ must be taken so that ξ = 0.7. Given ( 4), all the parameters involved in (5) already described and addressed

k m c . . 2 . ζ = (5)
• The stop criterion: In order to improve the speed of the process, we have developed a criterion that, when a certain limit is reached, stops the movement. It is based on the kinetic energy of the bubbles. At each time step, the total kinetic energy of all bubbles is calculated and compared to an estimated energy threshold, corresponding to a movement of all bubbles of 1/10 th of their radius. The movement is stopped once the energy goes under a fraction (a thousandth) of this value.

• The step time: the settling time of the system, which is the time taken for the mobile to reach 98% of its final position, is known by [5] and is • The integration method: we have done comparisons between the first order Runge-Kutta method (RK1) and the fourth-order Runge-Kutta method (RK4) [6].

The RK4 is four times more costly but is also much more accurate and less unstable. Doing some numerical comparisons, we decided to keep RK4 as an optimal choice.

Results of this Bubble meshing technique are given in part V of this paper.

IV. MESH REFINEMENT

Fig2. The classic refinement method (left) and our proposed one (right).

As we have seen in the previous part, the Bubble Method improves the mesh quality, according to a size map. The fig. 2 shows the difference between the classic h-refinement technique and our technique introducing the Bubble meshing method. In the standard method, the mesh is directly modified as described in part I of this paper. In our method, the distribution of the error on the domain directly gives a new size map that drives the bubble regularization and adaptation process.

This technique, used in the refinement process presents some advantages: First, the size map can either decrease or increase the number of nodes, thus leading to an optimal cost of simulation. Next new refined meshes present a very good overall quality, which also lead to a reduction of the conditioning matrix (compared to a refined with the classic technique mesh).

Next part presents some regularization as well as refinement results.

V. RESULTS

In this part, a quality criterion will be presented, then the way the error affects the size map is shown, and finally results for both the bubble meshing technique and meshes refinement using this technique will be shown.

A. The quality criterion

The optimal shape of an element in 2D is the one where each angle of the triangle is 60°. Therefore, the minimum angle in a triangle appears to be a simple but valuable criterion. The more this minimum angle is, the better the shape of the element is. For each element, we have computed the minimum angle (Am), the quality of the element being determined as follows: Angle min (Am) Am>50° 35°<Am<50° 20°<Am<35° Am<20°

Table 1: definition of the different quality criteria

B. The Bubble meshing method

Fig. 3 represents a quarter of an induction motor. In fig3.a, the initial mesh, together with the level of quality of the elements is showed. The fig 3.b represented the same mesh, after a bubblelisation, the level of excellent elements has risen from 40% to 85%, and the level of good elements has fallen from 55% to 12% as it is shown in table 2 which presents the number of elements (%) in each level for the initial mesh and the bubblelized one. We can safely conclude that both the bubble meshing method and the improvements brought have been well implemented. It appears that the bubblelized versions have more nodes, approximately 10%more, than the initial meshes, but the overall better quality of the elements (4) guarantees a more accurate solution, thus a gain in the future refinement process. To transform the computed error into a size map, we use a function that multiplies the initial weight of a node by a coefficient according to the node error: Fig. 5.a shows the shape of the two transformation functions, linear and parabolic. Fig 5 .b shows the results of their efficiency on the same problem. To reduce the mean error by 80%, the parabolic function is faster and less expensive since it needs only 6 refinements instead of 10 for the linear function. A similar transformation is made in [START_REF] Matsutomo | An error evaluation scheme based on rotation of magnetic field in adaptive finite element analysis[END_REF], but the transformation function is not fully explained there.

D. Mesh refinement

Several error criteria exist in the literature such as in [START_REF] Yamada | Error estimation for Transient Finite Element[END_REF].

Choosing one of them, fig. 4 shows the results of the refinement process on a contactor. Fig. 4a is the initial mesh, fig. 4b, the error map, and fig. 4c the refined mesh after 4 iterations, including the proposed regularization. It can be noticed that the new mesh keeps its quality, while being denser in corners.

Fig4. Mesh refinement of a contactor

VI. CONCLUSION AND PERSPECTIVES

In conclusion, the Bubble Meshing method provides good initial meshes to start with, and the importance of a good initial mesh is well known. After the refinement, the new mesh perfectly matches the distribution of the error in the whole domain. The mesh quality remains good during the refinement process. Even if the paper deals with 2D cases, the procedure can be implemented in 3D since the Bubble meshing method is extensible in 3D.

  proportional to this settling time.

Fig3.

  Fig3. Comparison of an initial mesh and its bubblelized version.C. The effect of the error on the size map

Table 2 :

 2 quality comparison between an initial and a bubblelized mesh