Iryna Moskalevska

Virginie Faure

Lucie Haye

Marion Mercey-Ressejac

Arindam K Dey

Benoit Chovelon

Kevin Soro

Loïc J Charbonnière

Andreas Reisch

Andrey S Klymchenko

Patrice N Marche

Jean-Luc Coll

Zuzana Macek Jilkova

Xavier Le Guével

Intracellular accumulation and immunological response of NIR-II polymeric nanoparticles

Keywords: NIR-II, immune response, gold nanoclusters, nitric oxide

The innate immune system presents the first line of host defence. When designing new nanomedicine probes, it is therefore crucial to understand how these probes interact with it. We prepared highly monodisperse, well defined, and stable polymeric nanoparticles (NPs) loaded with gold nanoclusters (Au NCs) which provide fluorescence emission in the NIR-II (900-1700 nm) optical window suitable for in vivo imaging. Au NCs loaded in polymeric NPs are potentially of great interest as they could be used as biosensors, delivery and therapeutic agents. After their molecular characterization, their interaction with macrophages and their immunological properties were investigated in detail. The NIR-II emitting polymeric NPs showed an efficient cellular uptake to more than 70% in time and in dosedependent manner. While the NPs did not produce any immunotoxicity, it was reported an anti-inflammatory response and a strong down-regulation of nitric oxide level after LPS stimulation.

INTRODUCTION

The design of biodegradable and biocompatible polymer nanoparticles has greatly expanded the range of applications of nanomedicines in the fields of bioimaging and therapeutical applications against cancers [START_REF] Aghebati-Maleki | Nanoparticles and cancer therapy: Perspectives for application of nanoparticles in the treatment of cancers[END_REF]. With the capacity to tailor polymeric NPs as swiss army knives with imaging modalities, drug delivery and targeting features, perspectives are opened to offer new theranostic agents for personalized medicine [START_REF] Johnson | Preclinical Cancer Theranostics-From Nanomaterials to Clinic: The Missing Link[END_REF]. Optical modalities to monitor non-invasively theranostic agents are highly desirable and the near-infrared window between 900 and 1700 nm, called NIR-II, offers a greater detection indepth compared to the visible and to the first infrared window (700-900 nm) [START_REF] Liu | The Near-Infrared-II Fluorophores and Advanced Microscopy Technologies Development and Application in Bioimaging[END_REF][START_REF] Zhu | Near-Infrared-II Molecular Dyes for Cancer Imaging and Surgery[END_REF][START_REF] Musnier | Optimization of spatial resolution and scattering effects for biomedical fluorescence imaging by using sub-regions of the shortwave infrared spectrum[END_REF][START_REF] Kong | A NIR-II-emitting gold nanocluster-based drug delivery system for smartphone-triggered photodynamic theranostics with rapid body clearance[END_REF]. Various NIR-II contrast agents have been developed over the past few years made of organic dyes (cyanines, BODIPY) [START_REF] Godard | Water-Soluble Aza-BODIPYs: Biocompatible Organic Dyes for High Contrast In Vivo NIR-II Imaging[END_REF][START_REF] Carr | Shortwave infrared fluorescence imaging with the clinically approved near-infrared dye indocyanine green[END_REF], inorganic fluorophores (SWCNs, Lanthanides) [START_REF] Yang | Recent advances in design of lanthanide-containing NIR-II luminescent nanoprobes[END_REF][START_REF] Hong | Through-skull fluorescence imaging of the brain in a new nearinfrared window[END_REF] or nanocrystals (Quantum Dots) [START_REF] Bruns | Next-generation in vivo optical imaging with short-wave infrared quantum dots[END_REF][START_REF] Zebibula | Ultrastable and Biocompatible NIR-II Quantum Dots for Functional Bioimaging[END_REF], each of them presenting their advantages and their drawbacks in terms of brightness, biocompatibility and photostability [START_REF] Li | Recent advances in the development of NIR-II organic emitters for biomedicine[END_REF]. Gold nanoclusters (Au NCs) are an interesting class of NIR-II emitters with an ultra-small size below 3 nm. Their NIR-II photoluminescence (PL) is tunable by controlling their number of atoms and their surface chemistry [START_REF] Chen | Shortwave Infrared in Vivo Imaging with Gold Nanoclusters[END_REF][START_REF] Musnier | High photoluminescence of shortwave infrared-emitting anisotropic surface charged gold nanoclusters[END_REF][START_REF] Yu | High-Resolution Shortwave Infrared Imaging of Vascular Disorders Using Gold Nanoclusters[END_REF][START_REF] Liu | Atomic-Precision Gold Clusters for NIR-II Imaging[END_REF][START_REF] Guével | Tailoring the SWIR emission of gold nanoclusters by surface ligand rigidification and their application in 3D bioimaging[END_REF]. Furthermore, the ability to functionalise their surface is favourable for high loading in numerous nanocarriers [START_REF] Guével | NIR-emitting fluorescent gold nanoclusters doped in silica nanoparticles[END_REF][START_REF] Tseng | Synthesis of gold nanoclusters-loaded lysozyme nanoparticles for ratiometric fluorescent detection of cyanide in tap water, cyanogenic glycosidecontaining plants, and soils[END_REF] as well as encapsulation into biomolecules such as proteins [START_REF] Wang | Engineering the Protein Corona Structure on Gold Nanoclusters Enables Red-Shifted Emissions in the Second Near-infrared Window for Gastrointestinal Imaging[END_REF][START_REF] Franck | Research Square[END_REF][START_REF] Xie | Protein-Directed Synthesis of Highly Fluorescent Gold Nanoclusters[END_REF]. Polymeric NPs are highly suitable nanosystems to act both as delivery system and carry high numbers of fluorescent emitters [START_REF] Reisch | Fluorescent Polymer Nanoparticles Based on Dyes: Seeking Brighter Tools for Bioimaging[END_REF][START_REF] Shi | Cancer nanomedicine: progress, challenges and opportunities[END_REF]. Based on the high versatility to tailor size, surface charge and loading of polymeric NPs, we designed monodisperse poly(ethyl methacrylate (PEMA) NPs loaded with hydrophobic photoluminescent Au NCs. [START_REF] Reisch | Tailoring Fluorescence Brightness and Switching of Nanoparticles through Dye Organization in the Polymer Matrix[END_REF][START_REF] Rosiuk | Controlling Size and Fluorescence of Dye-Loaded Polymer Nanoparticles through Polymer Design[END_REF]. Because the first line of cells in contact in the body are innate immune cells such as macrophages, we investigated the interaction and the intracellular accumulation of these NIR-II emitting PEMA NPs with macrophages. We also evaluated the capacity of PEMA NPs to modulate immunological responses of macrophages with or without stimulation with lipopolysaccharide (LPS).

EXPERIMENTAL SECTION

All chemical products were purchased from Sigma-Aldrich (France). PEMA bearing 5 mol% of methacrylic acid was synthetized through free radical polymerization as described previously [START_REF] Rosiuk | Controlling Size and Fluorescence of Dye-Loaded Polymer Nanoparticles through Polymer Design[END_REF][START_REF] Combes | Protein-like particles through nanoprecipitation of mixtures of polymers of opposite charge[END_REF].

Synthesis and characterisations of particles.

Gold nanoclusters stabilized by dodecanethiol (DDT) called here AuDDT were prepared as described elsewhere [START_REF] Li | Controllable synthesis of gold nanoparticles with ultrasmall size and high monodispersity via continuous supplement of precursor[END_REF]. Briefly, 46.4 mg (93.75 mol) AuClPPh 3 was dispersed in ethanol with the addition of 81.5 mg (0.94 mol) tert-butylamine-borane for 45 min with fast stirring. The colour of the solution changed from orange-red to black after 5 min. Then 12 L of DDT were added and the mixture was left to stir for another 3 hours. The dark solution was then filtered 3 times with filters at 3kDa cutoff to remove unreactive species and kept refrigerated. Preparation of NPs: Stock solutions of PEMA were prepared at a concentration of 10 g.L -1 in acetonitrile. These solutions were diluted to 2 g.L -1 in acetonitrile, with or without 10 wt% of AuDDT (relative to the polymer). These solutions were quickly added to a 9-fold volume excess of phosphate buffer (20 mM, pH 7.4) under shaking (Thermomixer comfort, Eppendorf, 1050 rpm at 21°C), followed by further dilution to the desired concentration. A similar procedure was used to load the dye salt R18/F5-TPB (rhodamine B octadecyl ester with tetrakis(pentafluorophenyl)borate as counterion) in PEMA NPs of the same particle size, as previously described [START_REF] Reisch | Collective fluorescence switching of counterion-assembled dyes in polymer nanoparticles[END_REF].

Dynamic light scattering (DLS) and ζ-potential:

The size and ζ-potential of the mixed polymer NPs were measured on a Zetasizer Nano series ZSP (Malvern Instruments S.A.). For size determination, each sample was measured 10 times with a run length of 10 s each. The volume average values, determined by the Zetasizer software (Malvern) based on Mie theory, were used. Mean values give the average over at least three independent preparations, error bars correspond to standard error of the mean. For ζ-potential determination three successive measurements combining electrophoretic mobility and laser Doppler velocimetry with > 10 runs each were carried out with an applied potential of ±150 V. Transmission electron microscopy (TEM): Solutions of NPs (5 μL) were deposited onto carbon-coated copper-rhodium electron microscopy grids following air or amylamine glowdischarge. They were then treated for 20 s with a 2% uranyl acetate solution for staining. The obtained grids were observed using a Tecnai F20 Twin transmission electron microscope (FEI Eindhoven Holland) operating at a voltage of 200 kV. Images (2,048 pixels × 2,048 pixels) were recorded using a US1000 camera (Gatan) and analyzed using the Fiji software. At least 200 particles per condition were analyzed. Spectroscopic properties: Absorption spectra were recorded on a Cary 4000 Scan ultraviolet-visible spectrophotometer (Varian). Emission spectra were performed with 10 × 10 mm² quartz suprasil certified cells (Helma Analytics). The steady state emission spectra were recorded on an Edinburgh Instrument FLP920 spectrometer working with a continuous 450-W Xe Lamp and a Hamamatsu R5 509-72 photomultiplier for NIR detection. Then, 850nm high-pass filter was used to eliminate the second-order artefacts when recording the NIR spectra. Excitation was performed at 415nm. Gold loading in polymeric NPs was determined using a quadrupole Inductively coupled plasma mass spectrometry (ICP-MS) (Perkin Elmer NexION 2000, Waltham, MA, USA). Standard solutions were prepared in aqua regia 3% (v/v). 197 Au was measured using 103 Rh at 141 nmol/L was used as internal standard.

Cell culture. Both the J774.1A murine macrophage cell line (ATCC) and the NIH-3T3 murine embryonic fibroblast cell line (ATCC) were cultured in DMEM supplemented with 10% foetal calf serum at 37°C, 5% CO 2 .

Cytotoxicity. Cytotoxicity was assessed by two different methods: PrestoBlueTM and lactic dehydrogenase (LDH) release assays. Cells were seeded in 96 well-plates at 3000 cells/well for J774.1A and 5000 cells/well for 3T3. After 24 h, cells were treated with different concentrations of nanoparticles (5, 10, 25, 50 and 100 µg/mL for PEMA NPs) for 24 or 48 h. PrestoBlueTM (Thermofisher cat n° A13261) was added at 1:10 and incubated for 3h. The absorbance was read at 570 nm using a 595 nm reference with the SunriseTM reader (Tecan). The released LDH from cells with damaged membranes was quantified using the CytoTox-ONE assay kit according to the manufacturer's protocol (Promega cat n° G7891). The fluorescence signal was read at 560 nm for an excitation at 590 nm with CLARIOstar® Microplate Reader (BMG Labtech).

Cellular uptake. The cellular interaction kinetics of loaded PEMA NPs were assessed by flow cytometry using R18/F15-TPB loaded PEMA NPs. J774.1A were seeded at 10 5 cells/mL and, after 24h, incubated for different periods from 0 to 24h and 0 to 100 µg/mL R18/F15TPB-PEMA NPs concentrations. The cells were fixed with paraformaldehyde 4% and the cellular interaction was assessed using BDTM LSR II (BD Biosciences). The fluorescence of R18/F15TPB -PEMA NPs was detected using the band pass 525/50 upon a 488 nm excitation. The cellular uptake of Au-PEMA NPs was validated by confocal microscopy. J774.1A were first seeded at 5x10 4 cells/mL on coverslips and then incubated with Au-PEMA NPs (50µg/mL) for 24 h at 37°C. Cells were fixed in paraformaldehyde 4% and permeabilized by Triton 0.2% BSA 4%. They were then labelled with Alexa FluorTM 488 Phalloidin (Thermofisher cat n°: A12379).

The intracellular fate of the nanocarriers was studied by live-cell confocal microscopy. J774.1A cells were seeded at 5x10 4 cells/mL in 4-well LabTeck and after 24h incubated with Au-PEMA NPs (50 µg/mL) for 24 h. The cells were labelled with Cell mask green plasma membrane (Thermofisher cat n°: C37608). All confocal measurents were done using an LSM710 NLO confocal microscope (Carl Zeiss) with an objective Plan Apochromat 20×/0.8 NA in air and an objective Plan Apochromat 63×/1.4 NA in oil. Near-infrared signal from Au NCs was captured using APD detector with a 405 nm excitation and a LP750 filter. Image treatment was done using the ImageJ software. The gold uptake quantification was determined by ICP-MS. Cells were mineralized at atmospheric pressure in aqua regia (nitric acid and hydrochloric acid in a molar ratio of 1:3), for 48 h at room temperature. The mineralization is diluted to reach a concentration of 3% of aqua regia before analysis. Standard solutions were prepared in aqua regia 3% (v/v).

Immunotoxicity. J774.1A were seeded at 2.5x10 5 cells/mL. After 2 h, the cells were treated with different concentrations of nanocarriers (10 and 50 µg/mL) for 24h. The nanocarrier exposed cells were stimulated with E. coli lipopolysaccharide (LPS) at 2 µg/mL for 24 h at 37°C, 5% CO 2 . Macrophages were identified by the expression of markers CD11b (BioLegend cat n°: 101226) and F4/80 (BioLegend cat n°: 123152), and their activation was evaluated by the expression of I-A b (BioLegend cat n°: 116410) and CD86 (BioLegend cat n°: 105008). Live cells were selected by Zombie VioletTM staining (BioLegend cat n°: 423113). Fluorescence labelled cells were analysed by flow cytometry using LSR II (BD Biosciences) using FCS Express V6 software (De Novo Software). Cytokines production was measured in the supernatant of cell cultures using IL-6, TNF-α, and IL-10 mouse ELISA kits following the manufacturer's protocols (Thermofisher cat n°: 887064, 88732488, 88710588). The absorbance was read with CLARIOstar® Microplate Reader (BMG Labtech) at 450 nm using 570 nm as a reference. The NO production was evaluated by measuring the nitrite concentration in the cell culture supernatant using Griess assay. 50 µL of supernatant was incubated with the same volume of Sulphanilamide 1% (w/v) (Sigma cat n°: S9251) for 10 min under shaking at 37°C. 50 µL of N-alpha-naphthyl-ethylenediamine (NED) 0.1% (w/v) (Sigma cat n°: 222488) was added and incubated for 10 min at 37°C in the dark. The absorbance was read with CLARIOstar® Microplate Reader (BMG Labtech) at 540 nm. Production of ROS in the cell culture supernatant was estimated following the manufacturer's protocol (ENZO ROS/superoxide detection kit ref: ENZ-51010).

Statistical analysis.

The values are presented by mean ± standard error of mean (SEM). At least three independent experiments were performed. Data were tested for normality and the appropriate statistic test was chosen. The statistical analysis was performed by Kruskal-Wallis test for multiple means comparisons of non-parametric data using GraphPad Prism software version 8.0. The probability was set at *p-value < 0.05, **p-value < 0.01, ***pvalue < 0.001.

RESULTS & DISCUSSION

1-Characterisation of NIR-II polymeric NPs

AuDDT prepared by wet chemistry (see experimental detail, Fig. S1A) possessed a core size smaller than 2.5 nm as confirmed by HRTEM (Fig. S1B). Unlike others Au NCs, AuDDT showed in solution multiple absorption bands at 390 nm, 420 nm, 450 nm, 545nm, 700 nm, 750 nm (Fig. S1C). To note that the absorption bands at 390 nm, 450 nm and 670 nm and the shoulder at 800 nm are usually attributed to the molecular species Au 25 SR 18 [START_REF] Jin | Atomically Precise Colloidal Metal Nanoclusters and Nanoparticles: Fundamentals and Opportunities[END_REF] (SR= DDT in our case) which suggested a high population of Au 25 DDT 18 within the AuDDT sample. AuDDT showed a good stability in ethanol, chloroform, and acetonitrile during a few weeks at room temperature enabling their loading in the polymer PEMA. The AuDDT NCs were then encapsulated into PEMA polymer NPs through nanoprecipitation (Fig. 1A) at a loading of 10 wt% relative to the polymer (Au-PEMA). The loading was confirmed by ICP-MS measurement. DLS measurements in water indicated an hydrodynamic diameter of Au-PEMA NPs around 333 nm (Fig. 1A) in agreement with the size measured by TEM at 265 nm (Fig. 1B,C). Both, the polydispersity index from DLS (0.20) and size distribution observed in TEM, indicated a monomodal population with low polydispersity. The surface charge determined by zeta potential confirmed also their negative charge (-31 mV) in water at pH 7, in line with the observed good colloidal stability over several months (at 4°C). Loading AuDDT in a polymeric matrix did not affect their absorbance properties as shown by a similar absorption profile for AuDDT in acetonitrile and loaded in PEMA NPs as illustrated in Fig. 1D. Direct dispersion of AuDDT in water (without polymer) led to cloudy samples with a broad absorbance spectrum, indicating strong scattering and aggregation of the NCs (Fig. 1D). The fact that this behaviour was not observed when using AuDDT with the polymer supported the efficient entrapment of AuDDT in the polymer NPs. Au NCs have shown recently high interest as NIR-II emitters due to their broad absorption in the visible and the red/near-infrared region (600-900 nm) and their relatively intense NIR-II emission that could be tuned by controlling their core size and surface chemistry [START_REF] Musnier | Optimization of spatial resolution and scattering effects for biomedical fluorescence imaging by using sub-regions of the shortwave infrared spectrum[END_REF][START_REF] Kong | A NIR-II-emitting gold nanocluster-based drug delivery system for smartphone-triggered photodynamic theranostics with rapid body clearance[END_REF][START_REF] Chen | Shortwave Infrared in Vivo Imaging with Gold Nanoclusters[END_REF][START_REF] Yu | High-Resolution Shortwave Infrared Imaging of Vascular Disorders Using Gold Nanoclusters[END_REF][START_REF] Liu | Atomic-Precision Gold Clusters for NIR-II Imaging[END_REF][START_REF] Li | Gold Nanoclusters for NIR-II Fluorescence Imaging of Bones[END_REF]. AuDDT showed broad NIR-II photoluminescence (PL) emission typical of Au NCs with a shoulder at 950 nm, a maximum peak at 1050 nm and a second shoulder at 1200 nm which have been assigned to interdependent energy transfer taking place with the gold core and at the surface of the core (Fig. 1E) [START_REF] Liu | Atomic-Precision Gold Clusters for NIR-II Imaging[END_REF][START_REF] Guével | Tailoring the SWIR emission of gold nanoclusters by surface ligand rigidification and their application in 3D bioimaging[END_REF][START_REF] Zhou | Origins of Visible and Near-Infrared Emissions in [Au25(SR)18]-Nanoclusters[END_REF]. The fluorescence emission of Au-PEMA NPs seems not sensitive to the excitation wavelength with the same profile at 405 nm and 730 nm excitation (Fig. S1D). Compared to free AuDDT in acetonitrile, PL of Au-PEMA NPs has higher intensity (2.3-fold) with broader emission among the longer wavelength. We confirmed again the poor stability of free AuDDT in water which led to the fall of the PL as depicted in Fig. 1E. We did not notice any PL decrease over a period of a few weeks nor photobleaching during the cellular studies that demonstrated the robustness of this protocol to load hydrophobic Au NCs such as AuDDT inside these 30 nm polymeric PEMA NPs.

2-Cytotoxicity and cellular uptake

Cell viability of J774.1A macrophages in presence of Au-PEMA NPs was evaluated by presto-blue assays after 24h and 48h incubation using particle concentrations between 1 to 100 g/mL. No cytotoxicity was observed in such conditions (Fig. 2A) using also LDH assays and with another cell lines such as mouse embryonic fibroblast (Fig. S2). This enabled us to move forward to study the cellular uptake of Au-PEMA NPs in macrophages as a function of the particle concentration and the incubation time. Cellular interaction of 30 nm fluorescent PEMA NPs was assessed by flow cytometry as a function of time up to 24h at 50 g/mL. Because the Au NC fluorescence emission is out of the range of the flow cytometry detection window, we used rhodamine dye (R18 F5-TPB) loaded PEMA NPs of similar size (223 nm; PDI = 0.17), which were prepared using the same protocol. However, taking into account previous works on fluorophore loaded PEMA and other polymer NPs and the particle characterization performed in Au-PEMA NPs, we assumed a similar cellular behaviour of the 30 nm PEMA NPs independently of the loading. [START_REF] Reisch | Fluorescent Polymer Nanoparticles Based on Dyes: Seeking Brighter Tools for Bioimaging[END_REF][START_REF] Combes | Protein-like particles through nanoprecipitation of mixtures of polymers of opposite charge[END_REF][START_REF] Andreiuk | Fluorescent Polymer Nanoparticles for Cell Barcoding In Vitro and In Vivo[END_REF] There was a clear increase, over the time of exposure, of the quantity of fluorescence associated with macrophages and of the number of macrophages associated with PEMA NPs, reaching 69.8% after 24h as seen in Fig. 2B andS3. In order to confirm this result, we quantified the amount of gold from Au-PEMA in macrophages using the same experimental conditions by ICP-MS. Results indicated an increase in the gold content in cells with time and with the particle concentration (Fig. 2C). In another series of experiments, we looked at the influence of the particle concentration on the cellular interaction. Flow cytometry experiment (Fig. 2D,S3) performed at 24h using a wide range of fluorescent PEMA particle concentrations from 0 to 100 g/mL showed a constant increase of the cellular interaction up to 76% which also agreed with the trend observed by ICP-MS at two different particle concentrations (10 and 50 g/mL). In order to determine whether Au-PEMA NPs accumulated on the membrane or inside the cells, we conducted confocal fluorescence microscopy. We could track the Au-PEMA NPs thanks to a highly sensitive APD camera enabling to detect fluorescence signal up to 1000 nm suitable for the NIR-II emission of the Au NC emitters. In Fig. 3A, the accumulation of AuPEMA NPs (in red) in the cytoplasm was clearly observed after 24 h incubation at 50 g/mL. Kinetic experiments conducted at short incubation times (5 min, 30 min) indicated a fast interaction of the NPs with the cellular membrane after 5 min, followed by their internalization as seen at 30 min (Fig. 3B). This particle uptake is commonly associated with an endocytic process where particles are first interacting with the cell membrane and then are engulfed into endosomes vesicles until their late storage in lysosomes [START_REF] Kettler | Cellular uptake of nanoparticles as determined by particle properties, experimental conditions, and cell type[END_REF]. 1A macrophages were incubated with Au-PEMA (red) at 50µg/mL for 24h and labelled for actin (in green). B) Measurements performed at t = 0, 5min, 30 min after incubating Au-PEMA (50µ/mL in DMEM with 10% FBS; in red) with J774.1A cells (membrane in green).

3-Immunological response

We then investigated the immunological response of J774.1A macrophages in presence of Au-PEMA NPs (303 nm) or with the unloaded PEMA NPs (271 nm) by first looking at their activation and their cytokine secretion. We observed a slight activation of the macrophages in presence of the unloaded PEMA NPs at high concentration (50 g/mL) from 60% to 90% while no significant effect is detected using the loaded Au-PEMA NPs as seen in Fig. 4A. In the second set of experiments, we intended to determine if the presence of NPs modifies the macrophage activation after a stimulation with LPS. Results presented in Fig. 4A show no significant impact of PEMA NPs or Au-PEMA NPs at 10 and 50 g/mL of NPs. This suggests that the polymer with or without Au NCs did not alter the activation of these macrophages at this concentration range. To determine whether these NPs elicited proinflammatory or an anti-inflammatory response, we then checked the levels of indicative cytokins (IL-6, TNF-, IL-10) secreted by macrophages. Results depicted in Fig. 4B,C,D show almost undetectable levels of cytokines secreted by macrophages exposed to PEMA NPs or Au-PEMA NPs. After LPS stimulation, IL-6 and TNF- levels slightly decreased for the loaded and the unloaded NPs at the highest particle concentration when compared to LPS-stimulated cells without any NP, but these variations were not statically significant. These data suggest a moderate, if any, anti-inflammatory effect after stimulation due to the presence of the polymer PEMA and independently of the loading in our case. This antiinflammatory effect has been reported with others polymeric NPs such as PLGA and PLA NPs [START_REF] Hao | Neutrophils, as "Trojan horses", participate in the delivery of therapeutical PLGA nanoparticles into a tumor based on the chemotactic effect[END_REF][START_REF] Saito | Designing drug-free biodegradable nanoparticles to modulate inflammatory monocytes and neutrophils for ameliorating inflammation[END_REF]. This trend was also observed for IL-10 which is associated with regulatory response [START_REF] Iyer | Role of interleukin 10 transcriptional regulation in inflammation and autoimmune disease[END_REF]. To further seek for potential anti-inflammatory effect of the Au-PEMA, we analysed the production of reactive oxygen species (ROS) and of nitric oxide (NO), which both play an essential role in cell regulation and are associated to inflammation [START_REF] Zhao | Interplay among nitric oxide and reactive oxygen species: a complex network determining cell survival or death[END_REF][START_REF] Sadhu | Nitric oxide and ROS mediate autophagy and regulate Alternaria alternata toxin-induced cell death in tobacco BY-2 cells[END_REF]. We did not observe any detectable NO production by unstimulated macrophages, even in the presence of Au-PEMA NPs demonstrating the lack of proinflammatory potential. During the LPS-induced inflammatory response, macrophages release NO which is a key proinflammatory mediator [START_REF] Chi | Regulation of nitric oxide production from macrophages by lipopolysaccharide and catecholamines[END_REF]. As expected, cells stimulated with LPS produced a high NO level (30M), Fig. 5A. However, we noticed a striking decrease of the NO production in presence of Au-PEMA. This trend was not affected by the Au loading of the PEMA, since we observed a similar decrease for PEMA NPs and for Au-PEMA NPs in a dose-dependent manner (Refere to new FigS). For instance, NO production decreased by 30% at 10 g/mL of Au-PEMA and by more than 85% at 50 g/mL of Au-PEMA. These findings corroborated the previous results suggesting the possible anti-inflammatory effect of NPs.

Next, we focused on ROS. We observed no difference in the ROS production by the unstimulated macrophages exposed to Au-PEMA (Fig. 5B) nor to PEMA (Refer to new FigS) compared to control cells, Contrarily to NO production after LPS stimulus, there were no significant changes of ROS production detected by our experimental aproach.

At first, we had expected a similar behaviour between the NO and the ROS level with several examples reporting the scavenging of ROS in cells by some NPs leading to a fall of NO level, as reviewed by Mauricio et al. [START_REF] Mauricio | Nanoparticles in Medicine: A Focus on Vascular Oxidative Stress[END_REF]. PEMA NPs seemed then to regulate NO level by others pathways. For example, Qureshi et al. reported a concomitant inhibition of NO and inflammatory cytokines in LPS-stimulated murine macrophages. They hypothesized that various tested drugs, such as Resveratrol, inhibit proteasomal enzymatic activity, the production of inflammatory cytokines NO in response to LPS activation [START_REF] Qureshi | Inhibition of nitric oxide and inflammatory cytokines in LPSstimulated murine macrophages by resveratrol, a potent proteasome inhibitor[END_REF]. We reckon at this stage that more works are requested to fully elucidate the immuno-metabolism mechanism occurring in macrophages in the presence of PEMA NPs. The future effort may focus on in vivo studies to evaluate the whole-body immunological response to PEMA NPs. Altogether these results suggest that the exposure of macrophages to PEMA NPs may decrease their inflammatory response in dose-dependent manner without compromising their activation status.

CONCLUSION

We reported here the simple design of NIR-II emitting polymeric NPs that were efficiently taken up by murine macrophages using endocytic processes in a time and dose dependant manner. We found that the nature of the polymer matrices made of polyethylmethacrylate can lead to an anti-inflammatory effect accompanied by a down regulation of NO level after LPS stimulation. These interesting results suggest a potential use of such nanocarriers for immunoregulatory applications with the possibility to track these nanosystems non-invasively in small animals by NIR-II imaging.

Figure 1 .

 1 Figure 1. NP characterization. A) Scheme of AuDDT loading in PEMA NPs. B) TEM image of Au-PEMA NPs in water and C) their corresponding particle size distribution (n NPs >100). D) Absorbance and E) Fluorescence spectra of AuDDT dispersed in acetonitrile, in water, and Au-PEMA in water ( exc. 415 nm).

Figure 2 .

 2 Figure 2. Flow cytometry. A) Evaluation of Au-PEMA toxicity on J774.1A macrophages. Cell viability was analyzed by PrestoBlue after 24h and 48h incubation with Au-PEMA NPs. Mean ± SE, n = 3. B) Uptake of NP loaded with R18 F5-TPB as a function of time at 50 µg/mL, using the YFP filter. C) ICP-MS measurements to determine the gold content in J774.1A macrophages incubated with Au-PEMA at different concentrations and incubation times. D) Uptake of NP loaded with R18 F5-TPB as a function of concentration after 24h of incubation, using the YFP filter.

Figure 3 .

 3 Figure 3. Confocal microscopy. A) CLSM of J774.1A macrophages were incubated with Au-PEMA (red) at 50µg/mL for 24h and labelled for actin (in green). B) Measurements performed at t = 0, 5min, 30 min after incubating Au-PEMA (50µ/mL in DMEM with 10% FBS; in red) with J774.1A cells (membrane in green).

Figure 4 .

 4 Figure 4. Effect of the Au-PEMA on macrophage activation . A) Percentage of activated J774.1A macrophages determined as the percentage of cells expressing the CD86 and MHCII markers after a 24h exposure to NPs, followed by a 24h of LPS stimulation. Secretion of cytokines B) IL-6, C) TNF-α and D) IL-10 by J774.1A after a 24h exposure to NPs, followed by a 24h of LPS stimulation. Mean ± SE. PEMA control groups n = 3; other groups n= 7-8.

Figure 5 .

 5 Figure 5. Effect of NP on NO and ROS production. A) NO production by J774.1A macrophages after 24h exposure to NPs, followed by a 24h of LPS stimulation. Mean ± SE. PEMA control groups n = 3; other groups n= 7-8. B) Fold-change in ROS production by J774.1A macrophages after 24h exposure to NPs, followed by a 24h of LPS stimulation. Data are normalized to unstimulated cells. Mean ± SE, n = 5.

Acknowledgements

XLG would like to thank ligue contre le cancer (R21004CC) for their financial support. XLG, AR, LH, KS, LJC thanks ANR LAPIN (ANR-20-CE09-0021-04)) for their financial support. The authors also acknowledge the support and the use of resources of the French Infrastructure for Integrated Structural Biology FRISBI ANR-10-INBS-05 and of Instruct-ERIC and especially C. Crucifix for help with electron microscopy.


Dr Chen Xie

Dr Yue Wu

Ho-Fai Chau

Dr Tao Zhang

Prof Carlos Tinlong Wong

Yik-Hoi Yeung

Pak-Lun Lam

Jean-Claude Bünzli

Dr Loïc J Charbonnière

Ho-Fai Chau

C Carlos

Tinlong Wong

Yik-Hoi Yeung

Pak-Lun Lam

Jean-Claude G Bünzli

Ga-Lai Law
email: ga-lai.law@polyu.edu.hk

Ka-Leung Wong
email: klwong@hkbu.edu.hk

Dr H.-F Chau

Y.-H Yeung

Prof P.-L K Lam

-L Wong

G.-L Law

Prof J.-C Bünzli

Multimodal Visualization of Bioorthogonal Systems by Off-On Luminescence and Enhanced Magnetic Resonance Imaging

Keywords: lanthanide, luminescence, bioorthogonal chemistry, magnetic resonance imaging

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Introduction

It is monumental when Bertozzi et al. introduced the term -Bioorthogonal Chemistry in 2003. [START_REF] Hang | Proc. Natl. Acad. Sci. U S A[END_REF] Those scientists proposed that researchers would be able to perform reactions in complex biological environments rather than under harsh conditions in organic solvents.

There have been many impressive applications of bioorthogonal chemistry in recent years, including molecular imaging, protein lipidation, and in vivo diagnosis. [START_REF] Hang | [END_REF][START_REF] Devaraj | Proc. Natl. Acad. Sci. U S A[END_REF][START_REF] Dong | [END_REF] Although it is common to probe and monitor cellular processes by bioorthogonal reactions, the real-time monitoring of the bioorthogonal reaction progress is also important and usually achieved by organic fluorophores. [5] Lanthanide luminescence shows its great potential for biomedical imaging and analysis. [6] Emission from lanthanides mostly features much longer lifetimes (microsecond-scale) than those from common organic luminophores (nanosecond-scale), so the time-resolved detection is available which largely improves the signal-to-noise ratio. Although lanthanide trivalent ions show impressive luminescence properties, their absorption coefficient is quite low. Therefore, a suitable antenna, which can harvest excitation energy and transfer it efficiently to the lanthanide excited states, is required. [7] There are only few reports concerning the application of luminescent lanthanide complexes in bioorthogonal chemistry, and few of them feature off-on properties. In 2016, O'Malley et al.

prepared Tb(III) complexes that can conjugate to biomolecules through a click reaction between an alkyne group and an azide group. [8] The successful conjugation would result in a significant Tb(III) emission enhancement. However, the usage of Cu(I) catalyst inevitably causes cytotoxicity from the resulting Cu(I) salts. Another example was reported by Zheng and his co-workers in 2020. [9] A biocompatible Tb(III) complex underwent ligation with cell surface glycans which resulted in a large increase in luminescence. It is because the cycloaddition reaction changed the electronic structures of the linked quencher, and the energy transfer from the antenna to Tb(III) was promoted. Jin et al. recently synthesized and characterized a series of lanthanide-based bioorthogonal probes for NIR cell imaging. [10] Noteworthily, most reported examples always feature emitters working independently which generate light before the bioorthogonal reactions. It possibly brings wrong interpretations on the targeting sites and reaction progress while an off-on probe can avoid these problems.

Click reaction is ideal to be applied in bioorthogonal chemistry because it is usually considered to be highly specific and requires no heating. [11] For example, Cu(I)-catalyzed azide-alkyne cycloaddition (CuAAC) is commonly used for nucleic acid labelling. [12] However, "copper-free" click chemistry is much more attractive to bioorthogonal chemists from the view of biocompatibility. There are several strategies to process click reactions without copper catalysts, including the strain-promoted azide-alkyne cycloadditions (SPAAC). [13] The cyclooctynes with high strain, which are eight-membered ring cyclooctynes in most cases, tend to react with azide spontaneously without the aid of metal catalysts.

Although the second-order rate constant of the early reported cyclooctynes was only 0.0012 M -1 s -1 , much lower than those of CuAAC (k 2 = 10-200 M -1 s -1 ), [14] the cyclooctynes with higher reactivity, like dibenzocyclooctyne (DBCO; also called aza-dibenzocyclooctyne, DIBAC) were developed as representatives. At a physiological temperature and pH range, DBCO does not easily react with endogenous amines/hydroxyls which means they are relatively inert to most biomolecules, while they are reacting with azide groups at much higher rates than with thiol groups. [15] This reagent allows copper-free click reactions in living organisms and prefers cycloadditions with azide groups. It features the fusion of dibenzo systems which improves the reactivity of its cycloalkynes. [16] Besides, a nitrogen atom on its strained ring also decreases the molecular hydrophobicity and eases the probe conjugation which facilitates specific reactions with azides. [17] The second-order reaction rate constant for DBCO groups with azide groups is around 1 M -1 s -1 , nearly one thousand times higher than some early reported cyclooctynes. [18] We herein reported lanthanide-based complexes, LnBT, featuring a Ln(III)-coordinated DO3A-azidopyridine conjugate. It can specifically perform metal-free SPAAC reactions with DBCO derivatives, where DBCOs are readily available and capable of subsequent conjugation with other biomolecules through their carboxylic groups. The second-order reaction rate constant of the cycloaddition between LnBT and DBCO derivatives was determined as 0.39 ± 0.007 M -1 min -1 by nuclear magnetic resonance (NMR) techniques with lanthanum(III) counterpart LaBT. Prior to SPAAC reactions, the Eu complex, EuBT, emits negligible light with an excitation wavelength from 200 nm to 400 nm. After the cycloaddition with DBCOs, the formed triazolyl links the pyridine ring of the lanthanide complex and two benzene rings of DBCO together to give an expanded conjugated system. The expanded conjugated system featured a lower excited energy level and therefore enabled

Results and Discussion

The LnBT was obtained by a concise synthetic route (Figure . Although the kinetics of most DBCO derivatives have been fully investigated in previous studies, here the azido group in LnBT is linked to an aromatic system rather than to an alkyl chain in most other cases. Therefore, the reaction rate constants between DBCOs and LnBT should be revisited. We performed a series of automated [START_REF] Hang | Proc. Natl. Acad. Sci. U S A[END_REF] The reaction rate constant k was determined from the slope as 3.92×10 -4 mM -1 min -1 (0.39 M -1 min - 1 ). This value is smaller than those in the literature for SPAAC between azide and DBCO, because most previous works studied alkyl azides, whereas we focused on azidopyridine where the electrons were stabilized on the conjugated π-bond system and less reactive. surface. [19] The retention time of GdBT and DBCO-Man is 14.7 min and 39.3 min, respectively, under given HPLC conditions (Table S1). The GdBT was mostly consumed after the reaction mixture was placed at room temperature for 24 h, while excess DBCO-Man remained. Two newly formed peaks, in very close retention time, were found at 26.0 min and 27.1 min, respectively. The structures of two peaks were characterized by mass spectrometry and confirmed as two configurational isomers of expected post-click products.

Lanthanide luminescence is the key indicator for the off-on property and the evaluation of the bioorthogonal progress in our case. One of the important advantages of our EuBT complex is its off-on luminescent properties. Emission spectra were obtained from a reacting solution of EuBT and DBCO-Man at different time points (Figure . 4a). With 320 nm light irradiation, the EuBT alone has no light emission before the conjugation, but after the addition of DBCO derivatives and then the successful SPAAC reaction, the Eu(III) center will be effectively sensitized and produce long-lived, characteristic visible emission. The emission intensity is enhanced with the progression of the reaction and the formation of conjugated products.

Strong Eu(III) emission was observed after 5 minutes of the reaction. After 240 min, EuBT had mostly been reacted and the emission intensity was 9.2 times stronger compared with the signal at t = 2 min by monitoring 699 nm emission. Emission intensities at 615 and 699 nm were plotted and well fitted with single-exponential functions (Figure . 4b). The excitation spectra were obtained by monitoring the 615 nm Eu(III) hypersensitive emission peak. We had no observable excitation peak for EuBT. However, for the bioorthogonal product from EuBT and DBCO-Man, it was found to be sensitized by light ranging from 230 to 350 nm (Figure. S4a). The EuBT had no observable emission peaks when excited either by 260 nm or 320 nm (Figure. S4b andS4c). On contrary, the bioorthogonal product showed emission peaks at 578, 588, 615, 652, and 699 nm assigned to Eu(III) transitions 5 D 0 → 7 F J (J = 0, 1, 2, 3, 4) respectively (Figure . 4a). In a majority of luminescence spectra for Eu(III) materials, the 5 D 0 → 7 F 2 transitions feature the most intense emission peaks. However, in our case, the 5 D 0 → 7 F 4 transition dominated while 5 D 0 → 7 F 2 was much weaker. This phenomenon was also found in other Eu(III)-DO3A complexes. [20,21] Binnemans et al. [22] claimed that Eu(III)complexes with D 4d symmetry have intense 5 D 0 → 7 F 4 transitions because of the absence of a center of symmetry but the 5 D 0 → 7 F 2 transition is forbidden. As for Ln(III)-tetrasubstituted cyclen chelates, they usually adopt either a monocapped twisted square antiprismatic (TSAP) geometry or a monocapped square antiprismatic (SAP) geometry where SAP geometry has a perfect D 4d symmetry. Even though our complex does not feature a pure SAP geometry, the partial similarity has already resulted in an intense 5 D 0 → 7 F 4 emission peak. We further investigated the change of magnetic relaxivity of the analog, GdBT, that coordinates with gadolinium. Gd 3+ is commonly used for magnetic resonance imaging because this paramagnetic element can enhance nuclear relaxation rates. [23] We performed in vitro MR imaging for different samples before the bioorthogonal reaction or at the different extents of reactions (Figure . 5a), and the obvious brightening effects could be found after 4 hours. The r 1 relaxivity was calculated (Table S2), and that of GdBT was determined as 3.2 mM -1 s -1 prior to the bioorthogonal reaction. By mixing with DBCO-Man in an aqueous solution, the r 1 relaxivity was increasing as the progress of a bioorthogonal reaction. After 4 h reaction, the r 1 relaxivity of the reaction mixture approached 4.4 mM -1 s -1 , which is similar to Gd-DOTA, a commonly used MRI agent (r 1 = 4.5 mM -1 s -1 in our measurement). Noteworthily, ManNAc (N-acetyl-D-mannosamine monohydrate), featuring the same sugar unit as Man, has been reported with a half-life of ~2.4 hours by oral administration, and it reached the maximal plasma concentration after 4 hours [24] , which is adequate for the bioorthogonal reaction revealing enhanced magnetic resonance contrast. After 24 h reaction, the r 1 relaxivity of the reaction mixture is up to 5.5 mM -1 s -1 , which showed around 72 % enhancement over bioorthogonal reaction (Figure . 5b and5c). The signal enhancement resulted from the increase of rotational correlation time due to the enlargement of molecular weight over reaction progress. Through bioorthogonal reactions, our Gd(III)-analog revealed alternative imaging capacity in addition to light signals, and specific MRI/dual-modal imaging could be developed.

Conclusion

In this work, to address the limitations in reaction-progress monitoring for bioorthogonal chemistry, we successfully applied expanding-antenna strategies on a series of lanthanidebased complexes by specific bioorthogonal reactions. A non-emissive complex, EuBT, can generate highly differentiated light signals while being involved in bioorthogonal reactions.

The relatively long-lived signal features both off-on properties and quantification of the products. Its gadolinium counterpart, GdBT, also showed MRI signal enhancement over bioorthogonal-reaction progress. Hence, we report a novel bioorthogonal model that could facilitate solving emerging problems in biomedical imaging.

Supporting Information

Supporting Information is available from the Wiley Online Library or from the author.

  the efficient energy transfer to the Eu(III) center to generate strong visible light (Figure. S1), where the emission intensity was proportional to the extent of this bioorthogonal reaction (Figure. 1a). This reaction and the antenna-expanding strategy show broad application and fill the gap of monitoring bioorthogonal chemistry by off-on luminescence. Furthermore, the Gd(III) counterpart GdBT also showed enhancement in magnetic relaxivity over cycloaddition. We presented a series of kinetic studies by applying our complexes to DBCO-COOH or DBCO-Man (DBCO-mannosamine conjugate). The progressive signaling could realize the visualization of the bioorthogonal reaction process by instant luminescence changes, and also exhibits the potential of our modal serving as a promising agent in targeted diagnosis.

Figure 1 .

 1 Figure 1. a) The illustration of the off-on luminescence generation process. The SPAAC reaction expands the antenna and makes it suitable to transfer excitation energy to the Eu(III) center. Also, for Gd(III) counterpart, the SPAAC reaction enlarges the molecular weight that

  1b). The commercially available starting material (4-bromopyridin-2-yl)methanol (1) was converted into (4azidopyridin-2-yl)methanol (2) by heating with sodium azide in the water-DMF mixed solvent. 2 was then methanesulfonated and installed onto DO3A core by S N 2 reaction to give 3. The desired LnBT complexes were obtained after deprotection of tert-butyl of 3, then coordinated with corresponding lanthanide salt.

Figure 2 .

 2 Figure 2. a) Overlapping 1 H NMR spectra of LaBT and DBCO-COOH mixing solutions in d 4 -methanol at different time points. b) The plot of the calculated concentrations of LaBT (by the NMR peak integrals at 8.5 ppm) vs. reaction time. c) Logarithmic plot of the reaction of LaBT with DBCO-COOH in d 4 -methanol.

  H-NMR spectroscopic measurements by mixing the DBCO-COOH and LaBT in d 4 -methanol. The diamagnetic lanthanum counterpart was used for avoiding signal interference found in most other paramagnetic lanthanides. Although DBCO-Man and aqueous solutions were studied for other studies in our work, they were not ideal in NMR studies requiring much higher concentrations and bringing solubility problems. The samples were mixed in an NMR tube and monitored by the NMR signals for up to 780 minutes (Figure. 2a). Before the reaction started (0 min), there was a doublet peak located at around 8.5 ppm attributed to the proton on the ortho-position of the nitrogen atom of pyridinyl group (Figure. 2a, red spot). With the reaction processing, the doublet peak shifted downfield and turned to be a pair of broad peaks, which represent the two configurational isomers of post-click products. After 600 minutes, the original peak at around 8.5 ppm was completely shifted to around 8.7 ppm, which means the LaBT was fully consumed. Considering the peak integral at t = 0 corresponding to the initial concentration of LaBT, the instantaneous concentrations can be calculated at different time points and plotted (Figure. 2b). The curve exhibits a nonlinear decay which suggests the reaction was not of zero order. Considering a standard second-order reaction, A + B → Product, we have Integrating the equation from 0 to t, we have where x is the concentration of each species reacted at time t, so (dx/dt) represents the instantaneous reaction rate ; [A] and [B] are the concentrations of LaBT complex and DBCO-COOH respectively. We can obtain the instant [A] by comparing the NMR peak integrals and calculate the concentration of B by By plotting the logarithmic expression of the concentrations versus time (t), we obtained the reaction rate constant k from its slope (Figure. 2c) in the dynamic range (0-270 minutes).

Figure 3 .

 3 Figure 3. HPLC chromatograph of GdBT (top), DBCO-Man (middle), and post-click reaction mixture (bottom) with online UV-vis spectra of the corresponding peaks and ESI-MS spectra of two starting materials and reacting mixture.

Figure 4 .

 4 Figure 4. a) Emission spectra of the solution for an ongoing bioorthogonal reaction between EuBT (2 mM) and DBCO-Man (oversaturated) at different time points. The reaction and the measurement were both performed at room temperature in HPEPS, λ ex = 320 nm. b) The changes in the intensity vs. time of emission peaks at 615 nm and 699 nm. Both curves are fitted with single exponential functions.

Figure 5 .

 5 Figure 5. The change of relaxivity of GdBT reacting with DBCO-Man, measured in water, at 25 °C, 1.0 T. a) The reaction was visualized by in vitro MR imaging. b) The plot for linear fitting of the 1/T 1 and the concentration of Gd containing substances. c) The change of r 1 relaxivity over time.

  and Gd) and biomolecules can serve as a rare example of achieving real-time monitoring of bioorthogonal reactions by off-on luminescent signals from Eu(III). The Gd(III)-analog is equally promising where much enhanced molecular relaxivity suggests the potential of our designs in multi-modal diagnosis. Chen Xie, a,b Yue Wu, a Ho-Fai Chau, a Tao Zhang,* c Carlos Tinlong Wong, d Yik-Hoi Yeung, a Pak-Lun Lam, a Jean-Claude G. Bünzli, e Loïc J. Charbonnière,* b Ga-Lai Law,* d Ka-Leung Wong* a Multi-modal Visualization of Bioorthogonal Systems by Off-On Luminescence and Enhanced Magnetic Resonance Imaging

Acknowledgements G.-L. L. gratefully acknowledges The Hong Kong Polytechnic University ((a) University Research Facility in Chemical and Environmental Analysis (UCEA); (b) University Research Facility in Life Sciences (ULS)) and the Hong Kong Research Grants (15300919, 15301120, 153021/18P). K.-L. W. gratefully acknowledges the financial assistance from the Hong Kong Research Grants Council Grant No. 12300021, Dr. Mok Man Hung Endowed Professorship in Chemistry, HKBU -Durham Lanthanide Joint Research Center -Lanthanide Tools for Systems Medicine (SDF19-1011-P02) and the Centre for Medical Engineering of Molecular and Biological Probes (AoE/M-401/20). Chen Xie and Yue Wu contributed equally to this work.

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Gilles Chabert
email: gilles.chabert@emn.frxavier.lorca@emn.fr

Jaulin Luc
email: luc.jaulin@ensieta.fr

Xavier Lorca

A Constraint on the Number of Distinct Vectors with Application to Localization

This paper introduces a generalization of the nvalue constraint that bounds the number of distinct values taken by a set of variables.The generalized constraint (called nvector) bounds the number of distinct (multi-dimensional) vectors. The first contribution of this paper is to show that this global constraint has a significant role to play with continuous domains, by taking the example of simultaneous localization and map building (SLAM). This type of problem arises in the context of mobile robotics. The second contribution is to prove that enforcing bound consistency on this constraint is NP-complete. A simple contractor (or propagator) is proposed and applied on a real application.

Introduction

This paper can be viewed as a follow-up of [START_REF] Jaulin | Localization of an Underwater Robot using Interval Constraint Propagation[END_REF] on the application side and [START_REF] Beldiceanu | Pruning for the minimum Constraint Family and for the number of distinct values Constraint Family[END_REF] on the theoretical side. It proposes a generalization of the nvalue global constraint in the context of a relevant application. The nvalue constraint is satisfied for a set of variables x (1) ,. . ., x (k) and an extra variable n if the cardinality of {x (1) , . . . , x (k) } (i.e., the number of distinct values) equals to n. This constraint appears in problems where the number of resources have to be restricted. The generalization is called nvector and matches exactly the same definition, except that the x (i) are vectors of variables instead of single variables. Of course, all the x (i) must have the same dimension (i.e., the same number of components) and the constraint is that the cardinality of {x (1) , . . . , x (k) } must be equal to n. We first show that this new global constraint allows a much better modeling of the SLAM (simultaneous localization and map building) problem in mobile robotics. In fact, it allows to make automatic and thus robust a process part of which was performed by hand. Second, we classify the underlying theoretical complexity of the constraint. Finally, a simple algorithm is given and illustrated on a real example. Since the application context involves continuous domains, we shall soon focus on the continuous case although the definition of nvector does not depend on the underlying type of domains.

The application is described in Section 2. We first give an informal description of what SLAM is all about. The model is then built step-by-step and its main limitation is discussed. Next, the constraint itself is studied from Section 3 to Section 5. After providing its semantic (Section 3), the complexity issue is analyzed (Section 4), and a simple contractor is introduced (Section 5). Finally, Section 6 shows how the nvector constraint is used in the SLAM problem modeling, and the improvements obtained are illustrated graphically. In the rest of the paper, the reader is assumed to have basic knowledge on constraint programming over real domains and interval arithmetics [START_REF] Moore | Interval Analysis[END_REF]. Domains of real variables are represented by intervals. A Cartesian product of intervals is called a box. Intervals and boxes are surrounded by brackets, e.g., [x]. Vectors are in boldface letters. If x is a set of vectors, x (i) stands for the i th vector and x (i) j for the j th component of the i th vector. The same convention for indices carries over vector of boxes: [x (i) ] and [x (i) j ] are respectively the domains of x (i) and x

Application Context

The nvector constraint can bring substantial improvements to constraint-based algorithms for solving the simultaneous localization and map building (SLAM) problem. This section describes the principles of the SLAM and provides a constraint model of this application.

Outline of the SLAM problem

The SLAM problem can be described by an autonomous robot moving for a period of time in an unknown environment where the self-location cannot be performed accurately with the help of external equipments (typically, the GPS), and the observation of the environment can only be performed by the robot itself. Examples of such environments include under the sea and the surface of other planets (where the GPS is unavailable). Both limitations are also present indoor where the GPS is often considered unreliable. In the SLAM problem, we have to compute as precisely as possible a map of the environment (i.e., the position of the detected objects), as well as the trajectory of the robot. The input data is the set of all measures recorded by the robot during the mission from its embedded sensors. These sensors can be divided into two categories: proprioceptive, those that allow to estimate the position of the robot itself (e.g.: a gyroscope) and exteroceptive, those that allows to detect objects of the environment (e.g.: a camera). Note that the positions of the robot at the beginning and the end of the mission are usually known with a good precision. Objects are called landmarks (or seamarks under the sea).

Since nothing can be observed from outside (should it be a landmark or the robot itself), uncertainties of measures get accumulated every time step. The more the mission lasts, the more the robot gets lost. Likewise, detection of landmarks is achieved with less and less accuracy. However, when the robot detects again a landmark that was already placed with a good accuracy on the map, its position can be adjusted from that of the landmark and the whole process of estimation (trajectory and map building) can be refined (see Figure 1). Hence, localization and map building are two connected goals. This interdependence is one of the reason that makes traditional probabilistic approaches inadequate. In contrast, it makes no difficulty in the constraint programming framework, as shown below.

(b) (d) (c) (a)

Basic Constraint Model

Let us now focus on a basic modeling of the SLAM. Many details on a real experiment (description of the robot, experimental setup, full constraint model, etc.) can be found in [START_REF] Jaulin | Localization of an Underwater Robot using Interval Constraint Propagation[END_REF] and [START_REF] Jaulin | A Nonlinear Set-membership Approach for the Localization and Map Building of an Underwater Robot using Interval Constraint Propagation[END_REF] that deal with SLAM in a submarine context.

The SLAM problem is cast into a CSP as follows. First, the motion of the autonomous robot obeys a differential equation: p ′ (t) = f (u(t)), where p(t) is the position of the robot in space, u(t) a vector of m inputs (speed, rotation angles, etc.) and f a mapping from R m to R 3 . This equation can be cast into a CSP using a classical interval variant of the Euler method. Details can be found in [START_REF] Jaulin | Localization of an Underwater Robot using Interval Constraint Propagation[END_REF] and [START_REF] Jaulin | A Nonlinear Set-membership Approach for the Localization and Map Building of an Underwater Robot using Interval Constraint Propagation[END_REF]. The discretization introduces a set of (N + 1) variables p (0) ,. . ., p (N ) where δ t is the time lapse between to measures and N δ t the total duration of the mission. These variables represent a discretization of the trajectory p, i.e., ∀i,

0 ≤ i ≤ N, p (i) = p(t 0 + iδ t ) (1)
has to be fulfilled. The discretization also introduces N constraints. Thus, the CSP provides a rigorous enclosure of the trajectory, i.e., for every possible input u(t) there exists a feasible tuple (p (0) , . . . , p (N ) ) such that the trajectory p corresponding to the inputs satisfies (1).

Introducing Detections

Now that the motion of the vehicle has been cast into a CSP, let us take into account detections. In the mission, n landmarks have to be localized. Their coordinates will be denoted by o (1) , . . . , o (n) . Once the mission is over, a human operator scans the waterfall of images provided by exteroceptive sensors. When a group of pixels are suspected to correspond to a landmark, a box encompassing the corresponding area is entered as a potential detection. The position of a pixel on the image can be directly translated into a distance between the landmark and the robot. First, the detection time τ (i) (i.e., the number of time steps since t 0 ) and the distance r i are determined. Then, the landmark number σ(i) is identified which amount to match detections with each others. Finally, a distance constraint dist(p (τ (i)) , o (σ(i)) ) = r i between the landmark and the robot is added into the model. Therefore, in [START_REF] Jaulin | Localization of an Underwater Robot using Interval Constraint Propagation[END_REF], the model was augmented as follows:

(P ′ )                additional variables: o (1) ∈ [o (1) ], . . . , o (n) ∈ [o (n) ] domains: [o (1) ] := (-∞, +∞), . . . , [o (n) ] := (-∞, +∞) additional constraints: dist(p (τ (i)) , o (σ(i)) ) = r i (i = 1..k)
Humans are subject to three types of mistakes: (1) Omission, a landmark on the waterfall is missed by the operator; (2) Illusion, the operator adds a detection where there is no landmark; (3) Mismatching, the operator makes a wrong identification (do not match detections properly). On the one hand, the two first types of mistakes have been experimentally proven as irrelevant 1 . On the other hand, if identifying the type of a landmark is fairly easy, recognizing one particular landmark is very hard. In other words, the main difficulty in the operator's task is matching landmarks with each others. The bad news is that mismatching makes the model inconsistent. Hence, the third type of mistakes is critical and requires a lot of energy to be avoided. Up to now, in the experiments made in [START_REF] Jaulin | Localization of an Underwater Robot using Interval Constraint Propagation[END_REF][START_REF] Jaulin | A Nonlinear Set-membership Approach for the Localization and Map Building of an Underwater Robot using Interval Constraint Propagation[END_REF], matching was simply performed using a priori knowledge of seamark positions.

Our contribution

The ambition of our work is simply to skip the operator's matching phase. The idea is to use the knowledge on the number of landmarks to make this matching automatically by propagation. This is a realistic approach. In all the missions performed with submarine robots, a set of beacons is dropped into the sea to make the SLAM possible. The positions of the beacons at the bottom is not known because of currents inside water (consider also that in hostile areas they may be dropped by plane) but their cardinality is.

The Number of Distinct Vectors Constraint

Consider k vectors of variables x (1) , . . . , x (k) of equal dimension (which is 2 or 3 in practice) and an integer-valued variable n. The constraint nvector(n, {x (1) , . . . , x (k) }) holds if there is n distinct vectors between x (1) and x (k) which can be written:

nvector(n, {x (1) , . . . , x (k) }) ⇐⇒ |{x (1) , . . . , x (k) }| = n, where |{x (1) , . . . , x (k) }| stands for the cardinality of the set of "values" taken by x (1) , . . . , x (k) . An equivalent definition that better conform to the intuition behind is that the number of distinct "values" in {x (1) , . . . , x (k) } equals to n.

A family of constraints can be derived from nvector in the same way as for nvalue [START_REF] Bessière | Filtering Algorithms for the NValue Constraint[END_REF]. The constraint more specifically considered in this paper is atmost nvector that bounds the number of distinct vectors:

atmost nvector(n, {x (1) , . . . , x (k) }) ⇐⇒ |{x (1) , . . . , x (k) }| ≤ n.

It must be pointed out that, with continuous domains, the constraints nvector and atmost nvector are operationally equivalent. Indeed, excepted in very particular situations where some domains are degenerated intervals (reduced to single points), variables can always take an infinity of possible values. This means that any set of k non-degenerated boxes always share at least n values. In the nvector constraint, only the upper bound ("at most n") can actually filters. This remark generalizes the fact that all diff is always satisfied with nondegenerated intervals. Figure 2a shows an example with n = 2 (the variable is ground) and k = 5 in two dimensions. Figure 2b shows the corresponding result of a bound consistency filtering.

(i) is the cross product of two intervals [x (i) ] = [x (i) 1 ] × [x (i) 2 ]
. The constraint is satisfiable because there exists two vectors v (1) and v (2) such that ∀i, 1 ≤ i ≤ n, [x (i) ]∩{v (1) , v (2) } = ∅. The set of all such pairs (v (1) , v (2) ) is represented by the two little dashed boxes. (b) Result of the bound consistency with respect to the atmost nvector constraint. The domain of [x (4) ] encloses the two little rectangles.

Operational Complexity

Enforcing generalized arc consistency (GAC) for the atmost-nvalue constraint is NP-hard [START_REF] Bessière | The Complexity of Global Constraints[END_REF]. Worse, simply computing the minimum number of distinct values in a set of domains is NP-hard [START_REF] Bessière | Filtering Algorithms for the NValue Constraint[END_REF]. However, when domains are intervals, these problems are polynomial [START_REF] Beldiceanu | Pruning for the minimum Constraint Family and for the number of distinct values Constraint Family[END_REF][START_REF] Gupta | Efficient Algorithms for Interval Graphs and Circular-Arc Graphs[END_REF]. The question under interest is to know if atmost-nvector is still a tractable problem when domains are boxes (vector of intervals), i.e., when the dimension is greater than 1. We will show that it is not. Focusing on this type of domains is justified because continuous constraints are always handled with intervals. This also implies that bound consistency is the only acceptable form of filtering that can be applied with continuous domains. 2 We shall restrict ourselves to the calculation of the minimum number of distinct values (also called the minimum cardinality) and to boxes of dimension 2, i.e., rectangles. As noticed in [START_REF] Bessière | Filtering Algorithms for the NValue Constraint[END_REF], the minimum number of distinct values shared by a set of rectangles is also the cardinality of the minimum hitting set. A hitting set is a collection of points that intersects each rectangle. In the two following subsections, we consider the equivalent problem of finding the cardinality of the minimum clique partition of a rectangle graph.

Rectangle Graphs and Clique Partitions

A rectangle graph is a n-vertices graph G r = (V r , E r ) that can be extracted from n axis-aligned rectangles in the plane by (1) creating a vertex for each a rectangle, and (2) adding an edge when there is a non-empty intersection between two rectangles. Figure 3 depicts the two views of a rectangle graph. A clique partition of a graph is a collection of complete subgraphs that partition all the vertices. A minimal clique partition is a clique partition with the smallest cardinality (i.e., with the smallest number of subgraphs). Since any set of pairwise intersecting rectangles intersect all mutually (by Helly's theorem), a k-clique in a rectangle graph represents the intersection of k rectangles in the geometrical view. For instance, the 3-clique S1 of G r in Figure 3a

V r be partition into k disjoint sets V 1 , . . . , V k such that ∀i, 1 ≤ i ≤ k the subgraph induced by V i is a complete graph? Proposition 1. RCP is NP-complete.
The fact that RCP belongs to NP is easy to prove: the given of a k-partition is a certificate that can be checked in polynomial time. The rest of the proof, i.e., the reduction of a NP-complete problem to RCP is given below. Note that the transformation below is inspired from that in [START_REF] Rim | On Rectangle Intersection and Overlap Graphs[END_REF] but contains fundamental differences.

Building a Rectangle Graph from a Cubic Planar Graph

The problem we will reduce to RCP involves planar graphs which require the introduction of extra vocabulary. An embedding of a graph G on the plane is a representation of G (see Figure 4) in which points are associated to vertices and arcs are associated to edges in such a way:

the endpoints of the arc associated to an edge e are the points associated to the end vertices of e, -no arcs include points associated to other vertices, -two arcs never intersect at a point which is interior to either of the arcs.

A planar graph (Figure 4a) is a graph which admits an embedding on the plane (Figure 4b), and a cubic graph is a 3-regular graph, i.e., a graph in which every vertex has 3 incident edges. A rectilinear embedding is an embedding where every arc is a broken line formed by horizontal and vertical segments (Figure 4c).

v 1 v 3 v 2 v 4
(a) Cubic Planar graph. We are now in position to discuss about the transformation itself. Consider a planar cubic graph G P . First, Tammassia and al. gave a polytime algorithm in [START_REF] Tamassia | Planar Grid Embedding in Linear Time[END_REF] for computing a 5-rectilinear embedding of a planar graph, i.e., a rectilinear embedding where each broken line is formed by 5 segments. This is illustrated by Figure 5. Second, this embedding can be transformed into a rectangle graph using the patterns depicted on Figures 6a and6b: -Every segment of the 5-rectilinear embedding of G p is replaced by a rectangle such that if two segments do not intersect in the 5-rectilinear embedding, the corresponding rectangles do not intersect (should they be flat). -For segments having a vertex at a common endpoint, the corresponding rectangles intersect all mutually in the neighborhood of this vertex (Figure 6a). Remember that the case of a leaf node is irrelevant because G p is cubic. -For segments having a bend in common, the rectangles are disjoint.

-An extra rectangle is added in the neighborhood of each bend. This rectangle intersects the two rectangles associated to the segments (Figure 6b). The three rectangles cannot intersect all mutually due to the previous point.

Reduction From Cubic Planar Vertex Cover

Consider the well-known vertex cover problem. This problem remains NP-Complete even for cubic planar graphs [START_REF] Uehara | NP-Complete Problems on a 3-connected Cubic Planar Graph and their Applications[END_REF]. It is formally stated as follows:

Cubic Planar Vertex Cover (CPVC)

-Instance: A cubic planar graph G p = (V p , E p ) and k ≤ |V p |.
-Question: Is there a subset V ′ ⊆ V p with |V ′ | ≤ k such each edge of E p has at least one of its extremity in V ′ ?

Lemma 1. Let G p be a n-vertex m-edges cubic planar graph and an integer value k ≤ n. Let G r be the rectangle graph obtained by the transformation of §4.2.

The answer to CPVC with (G p , k) is yes iff the answer to RCP with

(G r , k + 4 × m) is yes.
Proof. This proof is illustrated in Figure 7.

Forward implication: Assume a n-vertex m-edges cubic planar graph G p has a vertex cover V ′ of cardinality k. We shall build a partition P of G r , initially empty. To each edge e = (v i , v j ) of E p corresponds a 2-degree chain c e in G r , with exactly nine vertices, from a vertex of the 3-clique p i (associated to v i ) to a vertex of the 3-clique p j (associated to v j ). First, for every v i ∈ V ′ , add the 3-clique p i in P. Since V ′ is a covering set, every chain has now one of its extreme vertex inside a clique of P. The 8 other remaining vertices of the chain can then easily be partitioned into 4 additional cliques. Add these cliques to P. Once all these additional cliques are added to P, the latter is a partition of G r whose size is k + 4 × m.

v i v j e (a) Edge of Gp. Backward implication: Assume G r can be partition into k + 4 × m cliques and let P be such a partition. We shall call a free clique a clique that only contain vertices of the same given chain. Similarly, a shared clique is a clique involving (extreme) vertices of at least two different chains. Every chain contains 9 vertices. One can easily see that it requires at least 5 cliques to be partitioned, 4 of which are free. For every chain c, remove 4 free cliques of c from P. Then there is still (at least) one clique left in P that involves a vertex of c. When this process is done, the number of remaining cliques in P is k + 4 × m -4 × m = k. Now, for every clique C of P: if C is shared, put the corresponding vertex of G P in V ′ . If C is free, consider the edge e of G P associated to the chain and put anyone of the two endpoint vertices of e into V ′ . We have |V ′ | = k and since every chain has a vertex in the remaining cliques of P, every edge of G P is covered by V ′ . ⊓ ⊔

A Very First Polytime Contractor

We shall now propose a simple algorithm for computing bound consistency with respect to atmost nvector. Let us denote by dim the dimension of the vectors.

common intersection denoted by [G]. Furthermore, this process results in a set of n disjoint intervals [G 1 ], . . . , [G n ] that we will call kernels.

In the favorable case of n groups, the last step can be improved. If two variables x

i belong to the same group G, the whole vector variables x (j) and x (l) are actually constrained to be equal. Indeed, assume that x (j) and x (l) could take two different vectors. The k -2 other variables necessarily share n -1 different vectors because their i th components must "hit" the n -1 (disjoint) kernels [G ′ ] with G ′ = G. This means that the overall number of distinct vectors is at least (n -1) + 2 > n. Hence, the algorithm of atmost nvalue can be modified to intersect boxes (instead of just one component). This multiplies the complexity by dim (which is small in practice).

Experimental Evaluation: the SLAM Problem

This section shows how atmost nvector constraint allows to improve the modeling and resolution of the SLAM problem. We propose an extension of the original model given in Section 2, and provide a graphical validation. We introduce k 3-dimensional variables d (1) , . . . , d (k) related to all the detections. A distance constraint involving each of the latters is added into the model, as well as a nvector constraint capturing the fact that only n landmarks exist. ) ] domains:

(P ′′ )                    additional variables: d (1) ∈ [d (1) ], . . . , d (k) ∈ [d (k
[d (1) ] := (-∞, +∞), . . . , [d (k) ] := (-∞, +∞) additional constraints: dist(p τ (i) , d (i) ) = r i (i = 1..k) atmostnvector(n, {d (1) , . . . , d (k) })

The introduction of the atmost-nvector constraint has provided the expected results. Let us first explain how the benefits of this constraint can be quantified.

In the context of differential equations, we are dealing with a very large number of variables and a sparse system with an identified structure. Moreover, the system is subject to many uncertainties. Under these conditions, the quality of the result is more relevant than the computation time, as we justify now.

No choice point. In applications, the number of variables can be huge (more than 1140000 in [START_REF] Jaulin | Localization of an Underwater Robot using Interval Constraint Propagation[END_REF][START_REF] Jaulin | A Nonlinear Set-membership Approach for the Localization and Map Building of an Underwater Robot using Interval Constraint Propagation[END_REF] in order to represent the three coordinates of the robot, the Euler angles, the speed, the altitude and the components of the rotation matrix at each time step). This prevents us for making choice points. Note that the situation is even worse since the solution set is not a thin trajectory (formed by isolated points). The numerous uncertainties make the best solution ever a thick beam of trajectories (as depicted in Figure 9). Continuums of solutions usually dissuade one from making choice points, even for much smaller problems.

Handcrafted propagation. The size of our problem also prevents us for using a general-purpose AC3-like propagation algorithm, since building an adjacency matrix for instance would simply require too much memory. Furthermore, we have a precise knowledge of how the network of constraints is structured. Indeed, the discretization of the motion yields a single (long) chain of constraints and cycles in the network only appear with detection constraints. An appropriate strategy is then to base propagation on the detection constraints, which are much fewer. Every time a detection reduces significantly the domain of a position p (i) , the motion constraints are propagated forward (from p (i) to p (N ) ) and backward (from p (i) downto p (0) ). In a nutshell, propagation in this kind of problems is guided by the physics.

Irrelevance of computation time. When the application is run, it takes a couple of seconds to load data of the sensors (which amount to initialize the domains of variables representing input vectors) and to precalculate expressions (e.g., rotation matrices). With or without the nvector constraint, propagation takes a negligible time (less than 1%) in comparison to this initialization. Therefore, focusing on computation time is not very informative.

Quality of the result. Our contribution is to make propagation for the SLAM problem automatic whereas part of it was performed by a human operator so far. This is a result in itself. However, one may wonder which between the automatic matching and the operator's is most competitive. This, of course, is hard to evaluate a priori. In the experiment of [START_REF] Jaulin | Localization of an Underwater Robot using Interval Constraint Propagation[END_REF][START_REF] Jaulin | A Nonlinear Set-membership Approach for the Localization and Map Building of an Underwater Robot using Interval Constraint Propagation[END_REF], both have provided the optimal matching. The question that still remains is to know the extent to which our matching (the one provided by the algorithm in Section 5) improves the "quality" of the result, i.e., the accuracy of the trajectory.

The idea was to make the robot looping around the same initial point so that many detections would intersect. 3 For this purpose, we have controlled the robot with a classical feedback loop which gives the expected cycloidal trajectory. Four landmarks have been placed in the environment and we have basically considered that a landmark is detected everytime the distance between the robot and itself reaches a local minimum (if less than a reasonable threshold). The estimation of the landmark position is then calculated from this distance (with an additional noise) and a very rough initial approximation. Figure 9 illustrates the effect of automatic matching on the estimation of the trajectory and the positions of the landmarks. All the results have been obtained in a couple of seconds by the Quimper system [START_REF] Chabert | Contractor Programming[END_REF]. the fixpoint of all the contractors: the four landmarks are very well localized and the trajectory is much thiner (after 4000 iterations, the largest diameter is 10 times less than in Figure 9(c)). Our algorithm has contracted many detections to optimal boxes around landmarks. We can also observe the weakness of our algorithm which has poorly reduced boxes whose projections on both axis encompass two projections of landmarks. This however exclude a detection which really encloses two landmarks: in this case, either there is a real ambiguity or it is the propagation to blame.

Conclusion

A somewhat natural generalization of the nvalue constraint, called nvector has been proposed. The nvector constraint can help modeling and solving many localization problems where a bound on the number of landmarks to localized is known. This has been illustrated on the SLAM problem and applied on a real experiment. We have also analyzed the complexity of this global constraint and given a simple contractor. The benefit of this constraint in terms of modeling has a direct impact on the way data of the experiments have to be processed. Indeed, the constraint allows to avoid requiring someone that matches landmarks by hand. Hence, it reduces considerably the amount of work and the probability of mistake this operation entails.

The field of application is not restricted to the SLAM problem. Ongoing works show that the nvector is as crucial for the passive location of vehicles using TDOA (time difference of arrival) in signal processing. All these problems involve real variables. Hence, as a side contribution, this paper also offsets the lack of activity about global continuous constraints. Future works include the design of more sophisticated contractors with benchmarking. The nvector constraint also leads up to the study of other global constraints. As soon as several estimations of the same landmark position are matched by nvector, this position satisfies indeed a global constraint (namely, the intersection of several spheres if estimations result from distance equations).

  (i) j . Given a mapping f , range(f, [x]) denotes the set-theoretical image of [x] by f and f ([x]) denotes the image of [x] by an interval extension of f .

Fig. 1 :

 1 Fig. 1: A simple SLAM example. The robot is a mouse and landmarks are trees. Uncertainties on the robot position (resp. trees) are represented by blank (resp. hatched) ellipsis. (a) At initial time, the robot knows where it is. (b) The uncertainty on the robot position increases while it moves. When a second tree is detected, there is a significant uncertainty on its real position. (c) The first tree is met again and used to adjust the position of the robot. (d) With a backward computation of the trajectory, uncertainties on the previous positions and observations are reduced.

Fig. 2 :

 2 Fig. 2: (a) The atmost nvector constraint with n = 2 and k = 5. The domains of x (i) is the cross product of two intervals [x (i) ] = [x

Fig. 3 :

 3 Fig. 3: A rectangle graph and its geometrical representation (axis-aligned rectangles).

Fig. 4 :

 4 Fig.4: A (cubic) planar graph, one of its embedding on the plane, and one of its rectilinear embedding.

p 3 p 1 p 4 p 2 Fig. 5 :

 425 Fig. 5: A 5-rectilinear embedding of the cubic planar graph given in Figure 4 (segments are delineated by arrows).

  Transformation of segments having a bend in common.

Fig. 6 :

 6 Fig. 6: Atomic operations to transform the 5-rectilinear embedding of a cubic planar graph (left) into the geometrical view (middle) of a rectangle graph (right).

  chain (subgraph of Gr).

Fig. 7 :

 7 Fig.7: Transforming an edge of the cubic planar graph GP into, first, its rectilinear view, second, a rectangle graph Gr.

  (a) Basic SLAM: trajectory and detections (1200 iterations). (b) Basic SLAM: Detections (4000 iterations). (c) Basic SLAM: trajectory (4000 iterations). (d) Improved SLAM: trajectory and detections (4000 iterations).

Fig. 9 :

 9 Fig.9: Comparing SLAM with the atmost-nvector contractor (Improved SLAM) and without(Basic SLAM). The trajectory is represented by gray filled boxes, detections by thick-border rectangles and landmarks by the little black squares. Figure9(d) depicts the fixpoint of all the contractors: the four landmarks are very well localized and the trajectory is much thiner (after 4000 iterations, the largest diameter is 10 times less than in Figure9(c)). Our algorithm has contracted many detections to optimal boxes around landmarks. We can also observe the weakness of our algorithm which has poorly reduced boxes whose projections on both axis encompass two projections of landmarks. This however exclude a detection which really encloses two landmarks: in this case, either there is a real ambiguity or it is the propagation to blame.

The overall accuracy may suffer from a lack of detections but the consistency of the model is always maintained (at any time, many landmarks are anyway out of the scope of the sensors and somehow "missed"). Besides, perception is based on very specific visual patterns that rule out any confusion with elements of the environment.

Allowing gaps inside intervals and applying GAC filtering leads to unacceptable space complexity[START_REF] Chabert | Techniques d'Intervalles pour la Résolution de Systèmes d' Équations[END_REF].

The experiment of[START_REF] Jaulin | Localization of an Underwater Robot using Interval Constraint Propagation[END_REF][START_REF] Jaulin | A Nonlinear Set-membership Approach for the Localization and Map Building of an Underwater Robot using Interval Constraint Propagation[END_REF] was not appropriate for this illustration because matching seamarks was actually too easy (6 seamarks and a rectangle graph of detections with 6 strongly connected components).

The contractor is derived from the following implication: atmost nvector(n, {x (1) , . . . , x (k) }) =⇒ atmost nvalue(n, {x Therefore, applying a contractor for atmost nvalue with the projections of the boxes onto each dimension in turn gives a contractor for atmost nvector. Since a contractor for atmost nvalue enforcing GAC (hence, bound consistency) when domains are intervals already exists [START_REF] Beldiceanu | Pruning for the minimum Constraint Family and for the number of distinct values Constraint Family[END_REF], we are done. This is achieved in O(dim× k log(k)). Although the purpose of this paper is not to describe an efficient contractor, we shall introduce right away a first (but significant) improvement of our naive algorithm. (5) ] [x (6) ] [x (7) ] [x (2) ] [x (1) ] [x (3) ]

[x

[

Fig. 8: Illustration of what the at most nvalue algorithm yields. Here, k = 7, dim = 2, n = 3 and the algorithm is run for the first projection (i.e., onto the horizontal axis). Three groups, represented by gray ellipsis, are identified, namely G1 = {x

1 } and G3 = {x [START_REF] Jaulin | Localization of an Underwater Robot using Interval Constraint Propagation[END_REF] 1 }. The variables of the same group are proven to be all equal (otherwise, the constraint is violated). E.g., x

(1) 1 and x

(2)

1 . Domains for the variables of a given group can be replaced by the corresponding kernel (e.g., [x

1 ] can be set to [G1]). Notice that the kernels are all disjoint. In our suggested improvement, the whole boxes are intersected instead of the first components only. Hence the domains of [x (1) ] and [x (2) ] are intersected, which gives one of the hatched rectangles. The other hatched rectangle is [x (4) ] ∩ [x (5) ].

The reader must simply admit that the contractor of atmost nvalue(n, {x

i }) works in two steps (see Figure 8). First, it builds groups of variables. If the number of groups turns to be n, it is then proven that for all x (j) i and x (l) i that belong to the same group, x i . Hence, the domains of all the variables of the same group G can be shrunk to their