Rita Rahmeh

Marjorie Damian

Martin Cottet

Hélène Orcel

Christiane Mendre

Thierry Durroux

Shivaji Sharma

Grégory Durand

Bernard Pucci

Eric Trinquet

K Shivaji

Jurriaan M Zwier

Xavier Deupi

Patrick Bron

Jean-Louis Banères

Bernard Mouillac
email: bernard.mouillac@igf.cnrs.fr

Sébastien Granier
email: sebastien.granier@igf.cnrs.fr

Structural insights into biased G

est destinée au dépôt

Introduction

G protein-coupled receptors (GPCRs) constitute the largest family of membrane proteins. They are responsible for the majority of cellular responses to a broad range of stimuli, including peptide and non-peptide neurotransmitters, hormones, growth factors, odorant molecules and light. GPCRs play critical roles in regulating most physiological functions and, thus, are the targets of 30% of currently marketed drugs [START_REF] Ma | Value of novelty?[END_REF].

To understand the function of GPCRs at the molecular level, it is fundamental to investigate the nature of the structural rearrangements that couple ligand binding to receptor-dependent activation of downstream signaling pathways. It is now clear that a given ligand is able to induce multiple signaling pathways such as activation of G proteins and -arrestin mediated pathways [START_REF] Rajagopal | Teaching old receptors new tricks: biasing seven-transmembrane receptors[END_REF]. Accordingly, the traditional ligand classification into agonists, partial agonists, antagonists and inverse agonists cannot be restricted to activation of a single signaling pathway. For instance, a given ligand can act as an inverse agonist of the Gs pathway and as an agonist of -arrestin signaling cascade [START_REF] Kenakin | Ligand-selective receptor conformations revisited: the promise and the problem[END_REF]. These ligand properties have a potential clinical relevance as suggested by a recent study on the beta-blocker carvedilol [START_REF] Wisler | A unique mechanism of beta-blocker action: carvedilol stimulates beta-arrestin signaling[END_REF]. Initially termed 'agonist-selective trafficking of receptor signaling [START_REF] Kenakin | Agonist-receptor efficacy. II. Agonist trafficking of receptor signals[END_REF], this concept is now also described as the 'functional selectivity or biased agonism' of a GPCR ligand [START_REF] Urban | Functional selectivity and classical concepts of quantitative pharmacology[END_REF].

It is known from several biophysical studies on purified 2-adrenergic receptor ( 2AR), as well as on α 2a -adrenergic receptor in living cells, that binding of different classes of ligands induce distinct conformational changes in these receptors [START_REF] Swaminath | Sequential binding of agonists to the beta2 adrenoceptor. Kinetic evidence for intermediate conformational states[END_REF][START_REF] Zurn | Fluorescence resonance energy transfer analysis of alpha 2a-adrenergic receptor activation reveals distinct agonist-specific conformational changes[END_REF], suggesting a high degree of structural plasticity in GPCRs [START_REF] Kobilka | G protein coupled receptor structure and activation[END_REF]. GPCR conformational changes associated to ligand binding are responsible for G-protein coupling andarrestin recruitment [START_REF] Hoffmann | Conformational changes in G-proteincoupled receptors-the quest for functionally selective conformations is open[END_REF][START_REF] Kobilka | Conformational complexity of G-protein-coupled receptors[END_REF].

Cell-based studies suggest that functional selectivity arises as a result of distinct conformational states of the receptor stabilized by the ligands [START_REF] Azzi | Beta-arrestin-mediated activation of MAPK by inverse agonists reveals distinct active conformations for G protein-coupled receptors[END_REF][START_REF] Kenakin | Functional selectivity through protean and biased agonism: who steers the ship?[END_REF]. In this context, establishing links between functional selectivity and distinct conformational states of GPCRs is of primary importance. However, only limited information is available concerning receptor molecular switches involved in ligand-dependent efficacy and functional selectivity. Moreover, the molecular mechanisms underlying the functional selectivity property of so-called biased agonist ligands remain elusive.

To study the molecular mechanisms responsible for ligand efficacy and functional selectivity, we directly monitored conformational changes induced by either biased or unbiased ligands within a prototypic peptide-activated GPCR, the V2 argininevasopressin (AVP) receptor (V2R) using two fluorescence-based approaches: tryptophan intrinsic fluorescence spectroscopy and lanthanide resonance energy transfer (LRET) [START_REF] Cha | Atomic scale movement of the voltagesensing region in a potassium channel measured via spectroscopy[END_REF]. For LRET measurements, two fluorophores were introduced into domains important for G protein coupling and for interaction with -arrestin [START_REF] Erlenbach | Molecular basis of V2 vasopressin receptor/Gs coupling selectivity[END_REF][START_REF] Benovic | Beta-adrenergic receptor kinase: identification of a novel protein kinase that phosphorylates the agonist-occupied form of the receptor[END_REF]. The donor was fused either to the cytoplasmic end of the transmembrane domain 6 (TM6) or to the cytoplasmic end of the transmembrane domain 7 (TM7) just before the putative helix 8 sequence. The acceptor was attached at the extreme C-terminus domain of the receptor. V2R represents an interesting model to study molecular bases of functional selectivity for several reasons; i) its pharmacology has been well-characterized using a large panel of ligands with different efficacies toward the Gs signaling pathway [START_REF] Manning | Peptide and non-peptide agonists and antagonists for the vasopressin and oxytocin V1a, V1b, V2 and OT receptors: research tools and potential therapeutic agents[END_REF] and ii) several V2R ligands are biased agonists. For instance, while the unbiased natural hormone AVP is a full agonist towards Gs protein and -arrestin (Gs agonist / Arr agonist), two non-peptide biased synthetic ligands, MCF14 (Gs agonist / Arr antagonist) and SR121463 (Gs inverse agonist / Arr partial agonist) have been described [START_REF] Azzi | Beta-arrestin-mediated activation of MAPK by inverse agonists reveals distinct active conformations for G protein-coupled receptors[END_REF][START_REF]Biased agonist pharmacochaperones of the AVP V2 receptor may treat congenital nephrogenic diabetes insipidus[END_REF].

In this study, we first demonstrate that purified V2R functionally reconstituted into a neutral amphipol (NAPol(s)) [START_REF] Bazzacco | Trapping and stabilization of integral membrane proteins by hydrophobically grafted glucose-based telomers. Biomacromolecules 20. Popot JL[END_REF] responds to ligands with the same efficacy profiles towards activation of purified Gs protein and arrestin-2 (β-arrestin-1) as in living cells.

We then analyzed the effects of biased ligands on the receptor structure. We found that changes in the tryptophan intrinsic fluorescence upon ligand binding correlated well with the efficacy profile of ligands towards the G protein signaling pathway.

LRET spectroscopy measurements showed that ligands exhibiting different efficacies (full agonist, antagonist or inverse agonist) towards G protein activation and arrestin recruitment stabilize distinct conformations of the V2R. Taken together, our study demonstrates that ligand-dependent arrestin recruitment by GPCRs is triggered by specific conformational movements that are different from the conformational changes responsible for G protein activation.

RESULTS

Reconstitution of purified V2R in amphipol

The Flag-tagged V2R (Flag-V2R) was expressed in insect cells by using recombinant baculovirus technology. Flag-V2R is properly expressed at the plasma membrane of Sf9 cells, binds [ 3 H]-AVP with an affinity similar to receptor expressed in mammalian cells (Kd = 13.5 ± 0.7 nM) (Supporting Fig. S1a andS1b) and is able to couple to Gs protein (Supporting Fig. S1c). The Flag-V2R was purified by immuno-chromatography using a M1 Flag antibody affinity resin (Fig. 1a) and reconstituted by exchanging detergents for neutral amphipols (NAPol(s), 1:10 protein:NAPol(s) weight ratio) which are known to enhance the stability of membrane proteins (20). In order to obtain a homogeneous fraction of the V2R, the NAPol(s)reconstituted receptors were then loaded onto a size exclusion chromatography column. A receptor population was eluted at 15.5 ml close to the 75 kDa protein standards (Conalbumin) (Fig. 1b). Non-denaturating blue native-PAGE analysis (Fig. 1b) revealed NAPol(s) reconstituted V2R at around 75-80 kDa. These results are consistent with the molecular weight of a monomeric V2R (42 kDa) that would be in complex with at least 3 molecules of NAPol(s) on average [START_REF] Bazzacco | Trapping and stabilization of integral membrane proteins by hydrophobically grafted glucose-based telomers. Biomacromolecules 20. Popot JL[END_REF]. The enriched monomeric fraction was then used throughout the study. The particles containing the V2R were then visually inspected by negative staining electron microscopy. Images analyses from electron micrograph of the monomeric V2R-containing sample are shown in Fig. 1c. It reveals the presence of apparently homogeneous population of particles having a round shape with a diameter ranging from 7 to 10 nm. Finally, we analyzed the pharmacological properties of V2R reconstituted in NAPol(s) using [ 3 H]-AVP saturating experiments (Fig. 1d). We determined an affinity constant of 674 ± 90 nM. Using the calculated Bmax value we evaluated the active fraction of receptor at about 90% of the total protein amount (see supporting information).

Effect of ligands on G protein activation and recruitment of arrestin-2 by

NAPol(s)-reconstituted V2R.

To determine the functionality of NAPol(s)-reconstituted V2R, we first measured ligand-induced incorporation of [ 35 S]GTP S to purified Gαs proteins (1:2.5 receptor:G protein molar ratio). This assay was done using two full agonists for the Gs pathway, the endogenous hormone AVP and the biased ligand MCF14 [START_REF]Biased agonist pharmacochaperones of the AVP V2 receptor may treat congenital nephrogenic diabetes insipidus[END_REF], and also with a Gs inverse agonist (SR121463)(21) (Fig. 2a). The effects of ligands (10 M) were compared to a basal signaling (i.e. basal [35S]-GTP S binding to the subunit Gαs). As shown in Fig. 2b, V2R induced a 1.42 -fold increase in [35S]GTP S incorporation over basal, suggesting a constitutive activity towards Gαs.

The full agonists elicited similar GTP S incorporation (3.22 and 2.95 -fold over basal) whereas the inverse agonist treatment reduced the constitutive activity of the untreated V2R-Gαs complex by 50% (Fig. 2b). These results demonstrate that NAPol-reconstituted V2R interacts with and efficiently activates Gs protein in a ligand-dependent manner. Importantly, the efficacy profiles of the ligands correlated well with those observed in living cells [START_REF]Biased agonist pharmacochaperones of the AVP V2 receptor may treat congenital nephrogenic diabetes insipidus[END_REF][START_REF] Morin | The D136A mutation of the V2 vasopressin receptor induces a constitutive activity which permits discrimination between antagonists with partial agonist and inverse agonist activities[END_REF].

It has been shown that AVP activation of the V2R induces a stable interaction with both arrestin-2 and 3 [START_REF] Oakley | Association of beta-arrestin with G protein-coupled receptors during clathrin-mediated endocytosis dictates the profile of receptor resensitization[END_REF]. In addition, SR121463 which is a V2R inverse agonist for the Gs protein pathway has been described as a partial agonist for the arrestin pathway [START_REF] Azzi | Beta-arrestin-mediated activation of MAPK by inverse agonists reveals distinct active conformations for G protein-coupled receptors[END_REF] defining this compound as an arrestin biased ligand. To determine the ability of ligands to modulate the recruitment of arrestin-2 by the NAPol(s)reconstituted V2R, we measured direct interactions between the receptor and a monobromobimane-labeled purified arrestin-2 (1:1 arrestin:receptor molar ratio, 10 M). Bimane is a small size fluorophore with a high sensitivity to the polarity of its molecular environment that can be used as a sensor to detect interactions between the arrestin and its protein partners, as observed for visual arrestin and rhodopsin [START_REF] Sommer | Dynamics of arrestin-rhodopsin interactions: arrestin and retinal release are directly linked events[END_REF]. As illustrated in Fig. 2c, incubation of arrestin-2 with NAPol(s)reconstituted V2R led to a quenching of 54 ± 5 % of the bimane fluorescence intensity, suggesting a high constitutive interaction of the two proteins. Typical fluorescence spectra of the bimane-labeled arrestin are provided in the supporting figure S2 in the absence or presence of V2R and after AVP treatment. In the presence of the full agonist AVP and the partial agonist (SR121463), we observed a fluorescence quenching of 77 ± 3 % and 65 ± 4 %, respectively. On the contrary, the Gs-biased MCF14, an agonist for the Gs pathway and an antagonist for the V2R-arrestin pathway [START_REF]Biased agonist pharmacochaperones of the AVP V2 receptor may treat congenital nephrogenic diabetes insipidus[END_REF], did not induce a change in fluorescence intensity (Fig. 2c).

Moreover, we found that the AVP-induced fluorescence quenching can be reversed by MCF14 as expected for an antagonist. To rule out the possibility that the NAPol(s) by itself had an effect on the quenching of the fluorophore, the receptor-free amphipol was incubated with bimane-labeled purified arrestin-2 and with or without AVP. No change in the fluorescence of bimane was recorded in these conditions (Fig. 2c).

These results demonstrate that the monomeric NAPol(s)-reconstituted V2R efficiently interacts with arrestin-2 and that the efficacy profiles of the ligands correlate with those observed in living cells. intensity. On the other hand, AVP (Gs agonist / Arr agonist) led to an increase of 8 ± 0.9 % (Fig. 3b). Similarly, MCF14 (Gs agonist /Arr antagonist), induced an increase of 10 ± 2 % (Fig. 3b). As a control, we did not observe any changes in tryptophan intrinsic fluorescence when we added SR121463, AVP and MCF14 to the unfolded receptor (0.61 ± 1 %, 0.6 ± 2 %, 1 ± 0.6 %, respectively compared to the untreated receptor). We also determined the ligand concentration dependence on the tryptophan fluorescence signals and determined EC50s for AVP (1.13 ± 0.5 µM), MCF14 (2.28 ± 0.8 µM) and SR121463 (2.86 ± 0.7 µM) as described in supporting information and in the supporting Figure S3.

Effect of ligands on the intrinsic

Our results thus show that ligands with different efficacies towards Gs and arrestin signaling pathways induce opposite changes in intrinsic tryptophan fluorescence suggesting that these ligands induce distinct conformational changes within the receptor that are necessary for pathway-selective signaling. However, intrinsic tryptophan fluorescence only gives information about global conformational changes in the receptor, but cannot report on molecular movements from specific structural domains.

LRET reveals two basal V2R states

Fluorescence resonance energy transfer (FRET) has been used to study conformational changes in the β 2 AR [START_REF] Granier | Structure and conformational changes in the C-terminal domain of the beta2-adrenoceptor: insights from fluorescence resonance energy transfer studies[END_REF]. This approach requires site-specific labeling of the protein with two different fluorescent probes. We developed a related approach based on resonance energy transfer (RET), known as lanthanide-based or luminescence RET (LRET, see supporting information). We used two mutants for the LRET experiments (depicted schematically in Fig. 4) for which the site-specific labeling and the biochemistry were carefully characterized (supporting information, supporting Fig. S4 andS5). For clarity, the Flag-V2R-A267C-C358A-FlAsH mutant will be noted TM6 sensor and the Flag-V2R-S330C-C358A-FlAsH, TM7-H8 sensor.

Based on ligand binding and Gs-dependent cAMP accumulation in insect cells, the function of both mutants appeared equivalent to that of the wild-type receptor (Supporting Fig. S1c).

LRET measurements were performed as described in the Material and Methods section. Our results show that the fluorescence decay of the donor-only species is adequately fit by a one component exponential function for both sensors (Fig. 4a and4b). In contrast, the fluorescence decay of the donor in the presence of acceptor is best fit with a two component exponential function (Fig. 4a and4b) suggesting the presence of two distinct lifetimes. This measurement is made by monitoring the decay of the sensitized acceptor emission so that only the donor and acceptor engaged in LRET is detected (supporting information), We observed no intermolecular LRET in samples containing receptor labeled only with donor mixed with equivalent amounts of receptor labeled only with acceptor (Supporting Fig. S6).

These LRET data suggests the existence of at least two distinct basal states of the V2R characterized by distinct lifetimes constants of acceptor-sensitized emission (τ AD fast and τ AD slow, fast population and slow population) in both the TM6 and TM7-H8 sensor.

Effect of ligands on the conformational states of V2R revealed by LRET

To study ligand-dependent conformational changes in the V2R, we monitored the effects of AVP (Gs agonist / Arr agonist), SR121463 (Gs inverse agonist / Arr partial agonist) and MCF14 (Gs agonist / Arr antagonist) on the LRET signals for the TM6 sensor and TM7-helix 8 sensor. Fig. 5a and5b show typical acceptor-sensitized fluorescence decays on a linear scale obtained with untreated TM6 sensor and TM7-H8 sensor, respectively (black trace) and after treatment with the full agonist AVP (red trace). The lifetime analysis revealed in this case significant changes in τ AD fast and τ AD slow after AVP treatment (Fig. 5c and5d).

We then analyzed the effect of biased ligands on the LRET signals. For clarity and as discussed in the supporting information, only the effect on the largest population is shown in Fig. 6 (slow population). Importantly, the effects on both receptor populations follow a similar trend for both sensors (Supporting Table S1).

The results show a clear correlation between the ligands efficacy and their effect on lifetimes values (Fig. 6 S1).

To address the contribution of the background labeling on the LRET results, we measured the amount of Lumi4-Tb incorporated into the Flag-V2R-C358A-FlAsH receptor. The ratio of Lumi4-Tb labeling is 0.2 instead of 0.5 to 0.7 for TM6 and TM7-H8 sensors (supporting information). This level labeling level did not allow us to measure any LRET when using a receptor concentration equivalent to that used for TM6 or TM7-H8 sensors (0.1 M). By increasing the amount of protein used in the LRET experiment (0.5 M) we were able to detect significant LRET. In this condition, as for the two other sensors, we measured two lifetime components (Supporting Table 1 and Supporting Fig. S8a). However, we cannot directly compare this background labeling to that of the TM6 and TM7/H8 sensors because in the absence of the most reactive cysteine, the probes will necessarily react with the low reactive cysteines thus increasing the amount of background. In the presence of the reactive cysteine, one would expect a ratio of background labeling much lower than 0.2. It is unlikely that this level of background labeling influences our results since both SR and MCF induce opposite effects on receptor labeled specifically at the TM6 or TM7-H8 sensors.

DISCUSSION

GPCRs are known to activate different signaling pathways that can be differentially regulated by specific ligands. This phenomenon is described as functional selectivity or biased agonism. However, the mechanisms by which biased ligands can control the signaling outcome of a receptor at the molecular level are not yet known. Here, we used the AVP V2R subtype as a GPCR model to address this question. We used tryptophan and LRET spectroscopy, to investigate ligand-specific conformational changes in purified V2R reconstituted into neutral amphipols. Purified V2R reconstituted in neutral amphipols is functional and couples to both Gs protein and arrestin-2 with the same efficacy profiles as observed in living cells.

Using this system, we show that the signaling properties of the receptor are achieved through the stabilization of distinct conformational states by ligands. Our data suggest that movements of TM6-i3 are required for G protein activation/inhibition but may not be involved in the selective recruitment of arrestin. On the other hand, movements between the TM7-helix8 are required for selective recruitment of arrestin. These results provide evidence for a direct conformational link between the ligand binding pocket and the intracellular surface of the receptor, supporting a model whereby binding of structurally different ligands to a common orthosteric site induce or stabilize specific set confomational states that dictate the interactions between the V2R and cytoplasmic signaling molecules.

Consistent with the dynamic nature of GPCR [START_REF] Kobilka | Conformational complexity of G-protein-coupled receptors[END_REF], our LRET data provide evidence for two distinct conformations of V2R in the basal state. Indeed, these two populations are also detected in presence of ligands, and their relative distribution is not significantly affected (Supporting Table S1). It is thus unlikely that they represent equilibrium between an active and inactive form of the receptor.

Using the time constants of acceptor-sensitized emission (τ AD fast and τ AD slow) and donor-only emission (τ D ), we calculated the distances between donor and acceptor both for the TM6 sensor and in the TM7-H8 sensor (Materials and Methods section and Supporting Table S1). According to these calculations, the estimated difference in the two distances (fast and slow) between the TM6 and the C-terminus, or between the TM7-H8 and the C-terminus is around 10 Å. We thus suggest that the position of the C-terminus is the main structural difference between the two conformations present in the basal state, possibly due to some heterogeneity in phosphorylation or palmitoylation; however, we cannot exclude structural heterogeneity in TM6 and in TM7-H8.

The changes in LRET in the presence of different ligands can be attributed to changes in distance between the donor and acceptor probes (Supporting Table S1). While these calculated changes are small (the largest being 2.5Å increase in the distance between cytoplasmic end of TM6 and the extreme C-terminus of the V2R), it is known that molecular movements as little as 1 Å can lead to profound modifications on the activity of enzymes and receptors [START_REF] Koshland | Conformational changes: how small is big enough?[END_REF]. Accordingly, these minor movements may also have important effects in the GPCR-G protein and arrestin coupling/uncoupling mechanisms. In agreement with our data, the recent crystal structures of a nanobody-stabilized β2-AR active state and of the β2-AR-Gs protein complex revealed a large movement of the TM6 in comparison with the inverse agonist-bound structures [START_REF] Rasmussen | Structure of a nanobody-stabilized active state of the beta(2) adrenoceptor[END_REF][START_REF] Rasmussen | Crystal structure of the beta2 adrenergic receptor-Gs protein complex[END_REF]. Although less dramatically the TM7-H8 region is also moving during activation [START_REF] Choe | Crystal structure of metarhodopsin II[END_REF]. These movements were also detected using DEER spectroscopy in rhodopsin [START_REF] Altenbach | High-resolution distance mapping in rhodopsin reveals the pattern of helix movement due to activation[END_REF].

To reconcile our data obtained with the TM6 and the TM7-H8 sensors, we propose that the functional outcome of ligand binding depends on the effect that they trigger in the TM6-icl3 (Gs activity) and in the TM7-H8 (arrestin activity) domains, which can be considered molecular switches for the activation of intracellular partners (Fig. 7).

While the full agonist AVP affects both molecular switches and, thus, is able to activate both signalling pathways, MCF is only able to trigger the TM6 switch and activate Gs. On the other hand, SR is able to reduce the activity of the G protein and promote arrestin recruitment by constraining the TM6 molecular switch and activating the TM7-H8 domain.

Importantly, our data demonstrate that ligand-dependent arrestin recruitment by the receptor is triggered by conformational states that are distinct from those responsible for Gs protein activation, laying the foundation for a structural mechanism of ligandinduced biased signalling.

MATERIALS AND METHODS

Preparation of V2R/NAPol(s) complexes

Construction, expression, solubilization and purification of the receptor mutants from Sf9 insect cells as well as the labeling kinetics methods are described in the supporting information. Purified detergent-soluble receptors were incubated at 4°C in the presence of the amphipols at a 1:10 protein:NAPol(s) weight ratio [START_REF] Bazzacco | Trapping and stabilization of integral membrane proteins by hydrophobically grafted glucose-based telomers. Biomacromolecules 20. Popot JL[END_REF]. After detergent removal with Bio-beads, the sample was subjected to size exclusion chromatography (see supporting information) to isolate the monomeric fraction of V2R. The V2R/NAPol(s) complexes were then characterized by negative-stain transmission EM as described in the supporting information.

[ 35 S]-GTPγS binding and Arrestin-2 recruitement assays

Gαs protein was produced and purified as described in the supporting information.

Binding experiments were performed as described in the supporting information.

The arrestin-2 mutant L68C-R169E was produced in E.coli and purified by IMAC as described in the supporting information. The purified protein was then labelled with monobromobimane as described in Sommer et al. [START_REF] Sommer | Dynamics of arrestin-rhodopsin interactions: arrestin and retinal release are directly linked events[END_REF] and the arrestin recruitement assays were performed as described in the supporting information.

Analysis of LRET Data

Luminescence emission decays were measured at 620 nm and 520 nm and fitted as described in the supporting information. For acceptor-donor we calculated two lifetimes, defined as τ AD fast and τ AD slow. Ligand-induced changes in LRET and distances were measured as described in the supporting information. The proportion of slow and fast populations was calculated as previously described [START_REF] Posson | Small vertical movement of a K+ channel voltage sensor measured with luminescence energy transfer[END_REF] and details on the analysis are available in the supporting information.

  tryptophan fluorescence of the V2R. The fluorescent properties of tryptophans are very sensitive to their local chemical environment. There are 11 tryptophans distributed primarily in the TM segments (TM) including in TMs 2, 4, 5, 6 and 7. Therefore, tryptophan fluorescence can provide information about ligand-specific changes in the transmembrane core of the V2R. As shown in Fig. 3a, addition of SR121463, (Gs inverse agonist / Arr partial agonist), induced a 21 ± 1.7 % decrease of the intrinsic tryptophan fluorescence

  , top panel representing the ligands efficacy, and bottom panel their effects on lifetime measurements). First, for the TM6 sensor, AVP and MCF14 (full agonists towards G protein, Fig. 6a top panel) induced an increase in the lifetime  AD (26 ± 1 % and 23 ± 3 %, respectively, Fig. 6a bottom panel), whereas treatment of the receptor with the Gs inverse agonist SR121463 (Fig. 6a top panel) led to an opposite effect with a decreased lifetime (Fig. 6a, bottom panel). For the TM7-H8 sensor, AVP (Arr full agonist) and SR121463 (Arr partial agonist) (Fig. 6b top panel) induced an increase of 27 ± 7 % and 31 ± 6 % of  AD , respectively (Fig. 6b, bottom panel). Finally, treatment of the receptor with the Arr antagonist, MCF14 (Fig. 6b top panel), did not induce a significant change in lifetime measurements (Fig. 6b, bottom panel). As a control, under the same conditions we did not observe any changes in the lifetimes  AD when we treated the receptor with Leuprolide, an unrelated nonapeptide acting as an agonist for the GnRH receptor(25) (Supporting Table

Figure 1 .

 1 Figure 1. Purification of Flag-V2R, biochemical and pharmacological

Figure 2 .

 2 Figure 2. Chemical structure and effect of ligands on G protein coupling and

Figure 3 .

 3 Figure 3. Effect of ligands on V2R tryptophan intrinsic fluorescence.

Figure 4 .

 4 Figure 4. V2R constructs used in LRET experiments and basal state V2R LRET

Figure 5 .

 5 Figure 5. Changes in lifetimes upon AVP binding

Figure 6 .Figure 7 .

 67 Figure 6. Correlation between ligand efficacy and their effects on lifetime

Figure- 1

 1 Figure-1 (Granier)

ACKNOWLEDGMENTS

This work was supported by research grants from the CNRS, INSERM, and ANR ANR-06-BLAN-0087-03, ANR-10-BLAN-1208-01 and ANR-10-BLAN-1208-02.

FOOTNOTES S.G, B.M., J-L.B. and T.D. designed research. P.B performed the EM experiments. R.R., M.D., M.C., C.M., H.O., S.G. performed research and analyzed data. K.S.S., G.D., and B.P. contribute neutral amphipol reagents. E.T, J.Z contribute lanthanide reagents and help with the lifetimes analyses. X.D. built the receptor models to design the cysteine mutants for receptor labeling and helped with the interpretation of the LRET data. S.G., B.M and R.R. wrote the paper. S.G. originated the project and supervised the research with the help of B.M.


Agathe Chollet
email: achollet01@univ-lr.fr

Guy Wallet
email: guy.wallet@univ-lr.fr

Eric Andres
email: andres@sic.univ-poitiers.fr.

Laurent Fuchs
email: fuchs@sic.univ-poitiers.fr.

Gaëlle Largeteau-Skapin

Aurélie Richard
email: arichard@sic.univ-poitiers.fr.

Ω-Arithmetization of Ellipses

Keywords: discrete geometry, multi-resolution analysis, nonstandard analysis

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

Introduction

Discrete analytical geometry has been somewhat accidently founded in 1988 by Jean-Pierre Reveillès [START_REF] Reveillès | Géométrie Discrète, Calcul en Nombres Entiers et Algorithmique[END_REF] when he proposed the analytical description of a discrete straight line. This was an unexpected result that came out of theoretical research in nonstandard analysis (NSA). Nonstandard analysis [START_REF] Robinson | Non-standard Analysis[END_REF][START_REF] Nelson | Internal set theory: A new approach to nonstandard analysis[END_REF] provides an explicit framework for the manipulation of infinitely large and infinitely small numbers. The authors, in this paper and several previous papers, have decided to go back to the roots of Reveillès' discovery: the arithmetization method. The arithmetization process is a way to discretize a continuous curve that is a solution of a differential equation. The general idea is to transform a classical approximation scheme (such as the Euler scheme for instance) of the continuous solution of a differential equation defining a curve, into an equivalent discrete scheme. This is possible because, given an infinitely large (nonstandard) number ω (the global scale), it is possible to establish an equivalence between the set of limited real numbers and a subset HR ω of Z. The set HR ω , with an additional structure, is called the Harthong-Reeb line. The intuitive idea is that, in some way, the real line R is similar to the discrete line Z seen from far away.

In a previous work, the authors re-examined the circle of Holin [START_REF] Holin | Harthong-Reeb circles[END_REF][START_REF] Holin | Harthong-Reeb analysis and digital circles[END_REF]. The corresponding arithmetization process was based on infinitely large integers that had only an axiomatic status. Hence, the method was not constructive in nature and it was impossible to give an exact numerical representation of the result. In a past paper [START_REF] Chollet | ωarithmetization: a discrete multi-resolution representation of real functions[END_REF], we tried to tackle the issue of the constructivity of our model by using Ω-numbers of Laugwitz and Schmieden [START_REF] Laugwitz | Ω-calculus as a generalization of field extension[END_REF]. Roughly speaking, an Ω-number (natural, integer or rational) is a serie of numbers of same nature, with an adapted equality relation. The sets of Ω-numbers are extending the corresponding sets of usual numbers with the added advantage of providing naturally infinitely large integer numbers: for instance, an Ω-integer α represented by a sequence (α n ) of integers is such that α ≈ +∞ if lim n→+∞ α n = +∞ in the usual meaning. Clearly, these infinite numerical entities are constructive . It was a pleasant surprise to discover that the corresponding Ω-arithmetization method is also a discrete multi-scale representation of the real function on which the method is applied.

The goal of the present paper is to apply the Ω-arithmetization to the case of an ellipse and to study some connectivity property of the corresponding discrete curve. The first result is a constructive algorithm which gives an exact discrete multi-resolution representation of an arc of ellipse. In the Figure 1, we give a graphical illustration of this kind of representation. In fact, this multi-resolution aspect is a normal consequence of the Ω-arithmetization: this is in relation with thenature of the scaling parameter ω of the method. Since ω is now an infinitely large Ω-integer, it encodes an infinity of increasing scales. The arithmetization process gives simultaneously a discretization of the initial real function at each of these scales. Since nowadays many developments in image analysis, geometrical modelling, etc. comprise multi-resolution approaches and must deal with numerical precision problems, the Ω-arithmetization is a new tool which has the interesting property of taking into account these two aspects. The second main result of this paper is about the discrete connectivity: we show that the connectivity of the corresponding discrete ellipse arcs is a global property and that there is a rectangle within these curves are connected at every scale. Such global properties for a step-by-step integration process is unexpected. It extends a similar result we already had for circles but here, the property extends through all the scales. The paper is organized as follows: in Section 2, we introduce the Ω-numbers and the associated Harthong-Reeb line. In Section 3, we propose an Ω-arithmetization of ellipse arcs. Properties and graphical representation are discussed in Section 4. We conclude and provide perspectives for this work in Section 5.

2 Theoretical Basis : the Ω-numbers and the Associated Harthong-Reeb Line

The aim of this section is to present the basis of the nonstandard theory of Laugwitz and Schmieden [START_REF] Laugwitz | Ω-calculus as a generalization of field extension[END_REF][START_REF] Schmieden | Eine erweiterung der Infinitesimalrechnung[END_REF]. Our goal is to implement such a theory using the Ocaml language [START_REF]Le langage Caml[END_REF] and use it to build conic arcs. Globally, this theory permits the definition of a nonstandard model of the integer number set that can be used to build a discrete model of real numbers, this discrete model of the continuum is called the Harthong-Reeb line [START_REF] Harthong | Éléments pour une théorie du continu[END_REF]. The main interest of this nonstandard model (compared to other nonstandard theories such as, for instance, Internal Set Theory [START_REF] Nelson | Internal set theory: A new approach to nonstandard analysis[END_REF]) is that it is constructive [START_REF] Bridges | Constructive mathematics: A foundation for computable analysis[END_REF]. Therefore, the implementation in a programming language is possible. In this section, we will not describe the whole theory but only introduce the basic notions that are essential to understand the Harthong-Reeb line. For more details about our approach please refer to [START_REF] Chollet | ωarithmetization: a discrete multi-resolution representation of real functions[END_REF].

2.1

The Ω-numbers of Laugwitz and Schmieden

To extend a theory of integer numbers, Laugwitz and Schmieden introduce a new symbol, Ω to the classical ones (0, 3, 9, +, /, ...). The only thing that we know about it is that Ω verifies the following propriety named the Basic Definition and called (BD) :

Definition 1. Let S(n) be a statement in N depending of n ∈ N. If S(n) is true for almost n ∈ N, then S(Ω) is true.
We specify that here and in all this article, the expression "almost n ∈ N" means "for all n ∈ N from some level N", i.e. "(∃N ∈ N) such that (∀n ∈ N) with n > N ". Since Ω can be substituted to any natural number, it denotes an Ω-number which is the first example of Ω-integer. Immediately, we can verify that Ω is infinitely large, i.e. greater than every element of N. Indeed, for p ∈ N, we apply (BD) to the statement p < n which is true for almost n ∈ N; thus p < Ω for each p ∈ N. And Ω is the sequence (n) n∈N . Hence, each element a of this theory will be declined as a sequence (a n ) n∈N . To compare such Ω-numbers, we put the following equivalence relation:

Definition 2. Let a = (a n ) n∈N and b = (b n ) n∈N be two Ω-numbers, a and b are equal if it exists N ∈ N such that for all n > N , a n = b n .
The definition of the operations and relations between Z Ω , the set of Ωnumbers are the following:

Definition 3. Let a = (a n ) n∈N and b = (b n ) n∈N two Ω-numbers, -a + b = def (a n + b n ) n∈N and -a = def (-a n ) and a × b = def (a n × b n ) n∈N ; -a > b = def [(∃N ∀n > N ) a n > b n ] and a b = def [(∃N ∀n > N ) a n b n ]; -|a| = def (|a n |).
As in all nonstandard theories, there exist two classes of elements, the standard one and the nonstandard one. We recall that in classical IST nonstandard theory, this second class exists only in an axiomatic way: there are the infinitely small and large numbers. Here two classes of elements which can be distinguished:

-the class of element standard which are the elements α = (α n ) n∈N which verify ∃p ∈ Z such that ∃N ∈ N, ∀n > N, α n = p (example: (2) n∈N ). -the class of element nonstandard which are all the other element of Z Ω (examples : (-1) n n∈N ,(n) n∈N ) Among the nonstandard element, we focus infinitely large numbers which are the sequences α = (α n ) n∈N such that lim n→+∞ α n = +∞ (example: (n) n∈N ).

The Harthong-Reeb Line Based on Ω-numbers

The Harthong-Reeb line (HR ω ) is built upon the usual nonstandard axiomatic theory IST [START_REF] Harthong | Éléments pour une théorie du continu[END_REF]. This kind of formalism was introduced by Diener in [START_REF] Diener | Application du calcul de Harthong-Reeb aux routines graphiques[END_REF]. It is defined as a scaling on the integer numbers. We consider a new unit which is an infinitely large integer named ω = (ω n ) n∈N , which can be Ω himself. This scaling strongly contracts Z so that the result looks like R [START_REF] Fuchs | A first look into a formal and constructive approach for discrete geometry using nonstandard analysis[END_REF]. More formally, we defined the Harthong-Reeb with Ω-numbers as follows.

Definition 4. We consider the set

HR ω = {x ∈ Z Ω , ∃p ∈ N, |x| ≤ p ω}
and the relations, operations and constants on HR ω described by the following definitional equalities: for all (x, y) ∈ HR 2 ω , we set

• (x = ω y) = def (∀p ∈ N) (p|x -y| ≤ ω); • (x > ω y) = def (∃p ∈ N) (p(x -y) ≥ ω) ; • (x = ω y) = def (x > ω y) ∨ (x < ω y); • (x ≤ ω y) = def (∀z ∈ HR ω ) (z < ω x ⇒ z < ω y); • (x + ω y) = def (x + y) and 0 ω = def 0 and -ω x = def -x; • (x × ω y) = def ((x × y) ÷ ω) and 1 ω = def ω and x (-1)ω = def (ω 2 ÷ x) for x = ω 0.
Then, the Harthong-Reeb line is the numerical system

(HR ω , = ω , ≤ ω , + ω , × ω ).
We can say that HR ω is the set of Ω-integers which are limited at the scale ω.

We can describe all these relations and operations with integers, for instance,

x = ω y ⇐⇒ ∀p ∈ N ∃M p ∈ N ∀n ≥ M p p|x n -y n | ≤ ω n
The goal of the construction of this line is to obtain a discrete model of the continuum. To understand that the Harthong-Reeb line is a kind of continuum, we can compare it to (R lim Ω , ≃, , +, ×). This is the set of limited Ω-rational numbers of Laugwitz and Schmieden defined considering sequences of rational numbers. The two following maps:

ϕ ω : HR ω → R lim Ω x → x/ω and ψ ω : R lim Ω → HR ω u → (⌊ωu⌋)
are the isomorphic maps necessary to pass from the classical real world to the discrete one. The following section uses the Harthong-Reeb line to define the arithmetization process based on the well known Euler scheme.

The Discrete Ellipse Arcs

In this section we revisit and extend recent work about the arithmetization method. The arithmetization is basically a way of transforming a continuous Euler scheme into a discrete one. This leads to a step-by-step generation algorithm of a discrete object. Recently the authors have arithmetized differential equations defining a circular arc [START_REF] Richard | Arithmetization of a circular arc[END_REF]. This was done with an axiomatic definition of infinitely large integers. In another recent paper, the authors introduced the Ω-arithmetization method based on Ω-numbers which allows a constructive representation of infinitely large integers. In this section we apply the Ωarithmetization to circles and more generally to ellipses.

We consider an axis-aligned ellipse of equation

x(t) 2 a ′2 + y(t) 2 b ′2 = 1 (1)
with a ′ and b ′ ∈ Q. The parametric form of this set is

x = a ′ cos(t) y = b ′ sin(t) (2)
and is the solution of the differential system

x ′ = -a ′ /b ′ y(t) y ′ = b ′ /a ′ x(t). (3)
As a ′ and b ′ are in Q, so there exist p, q, ∈ Z and r, s ∈ N * such that a ′ = p/r and b ′ = q/s. We can define two integer numbers a an b thus that a/b = a ′ /b ′ and b/a = b ′ /a ′ . So, now only integers are manipulated in the differential system.

To obtain a solution we use the well known Euler method. In this case, we have:

(x 0 , y 0 ) = (0, b ′ ) (x n+1 , y n+1 ) = (x n , y n ) + (- a b h y n , b a h x n ). ( 4
)
We know that the smaller the integration step h, better is the approximation. In some meaning, the better choice is h infinitesimal.

We want to embed this scheme in HR Ω line. Using the idea of the isomorphism ψ ω ψ ω : R lim -→ HR ω x -→ ⌊ωx⌋ and replace the step h by 1/β we can write the following scheme

(x 0 , y 0 ) = (0, ⌊ωb ′ ⌋) (x n+1 , y n+1 ) = (x n , y n ) + ((-ay n ) ÷ (bβ), bx n ÷ (aβ)) (5)
where -∃α such that α, β, ω ∈ HR ω ; α, β, ω ≃ +∞ and ω = αβ (we say that ω is the global scale)

-⌊ωb ′ ⌋ = (⌊ω 0 b ′ ⌋, ⌊ω 1 b ′ ⌋, . . .) -∀u, v ∈ HR ω , u ÷ v = (⌊u 0 ÷ v 0 ⌋, ⌊u 1 ÷ v 1 ⌋, . . .) -(x 0 , y 0 ) ∈ HR 2 ω
As presented in [START_REF] Chollet | Insight in discrete geometry and computational content of a discrete model of the continuum[END_REF][START_REF] Richard | Arithmetization of a circular arc[END_REF], the problem of this scheme is that it generates values that are infinitely far from each other and thus the corresponding discrete curve is strongly (infinitely) non-connected. To avoid this problem, we divide everything by β in order to bring the discrete points close together. It is equivalent to work at a scale α which is named the intermediary scale. 3 Let us introduce the following notations to describe the decomposition x = xβ + x for any integer x ∈ HR ω so x = x ÷ β ∈ HR α and x = x mod β ∈ {0, . . . , β -1}. The integer x ∈ HR α is interpreted as the result of the rescaling on x. This decomposition produces the following scheme:

           ( x 0 , y 0 ) = (0, ⌊ωb ′ ⌋ ÷ β) ( x 0 , y 0 ) = (0, ⌊ωb ′ ⌋ mod β) (f 1 n , f 2 n ) = ((-a( y n β + y n )) ÷ (bβ), b( x n β + x n ) ÷ (aβ)) ( x n+1 , y n+1 ) = ( x n + ( x n + f 1 n ) ÷ β, y n + ( y n + f 2 n ) ÷ β) ( x n+1 , y n+1 ) = (( x n + f 1 n ) mod β, ( y n + f 2 n ) mod β) (6)
3 We could of course as well work at the intermediary scale β. The equations would be slightly different because of the role of β as 1/h but the principle would remain the same.

the relevant variables are x and y ∈ HR α while x and x only conserve the accumulated error. Let us call E α d (0, a, b) the discrete curve defined by the solution ( x n , y n ) of ( 6). E α d (0, a, b) is the arithmetization at the intermediate scale α of the initial ellipse [START_REF] Andres | Discrete circles, rings and spheres[END_REF].

Observe that the algorithm ( 6) is standard. This means that for α, β and ω standard we get a usual scheme in Z 2 . Nevertheless, in this paper, we have to consider that all objects are Ω-numbers, i.e. infinite sequences of integers operating with the relations and operations defined in Section 2, for instance, a = (a) m∈N , b = (b) m∈N 4 and β = (β m ) m∈N and ( x n , y n ) n∈N = (( x m,n ), ( y m,n )) n∈N . Let us note that n represents the iteration in the algorithm and that m the level in the sequence.

Properties and Graphical Results

In this section we present some theoretical results on the connectivity of the elliptical arcs and some graphical illustrations of the algorithm which result of an implementation in Ocaml. Let us first start with an extension of the theorem on the connectivity of circular arcs proposed in [START_REF] Richard | Arithmetization of a circular arc[END_REF] by Richard and al.

Connectivity Properties

Before proving connectivity properties, we need some definitions.

( x m,n ), ( y m,n )) (m∈N, k≤n≤k+p) is 8 Ω -connexe if ∀n, ∀m, |x m,n+1 -x m,n | ≤ 1 and |y m,n+1 -y m,n | ≤ 1.
This is equivalent to the classical 8-connectedness for each level m of the underlying sequences. This is the natural generalization to the Ω-numbers of the discrete connectivity. We also need the definition of a rectangle that is defined in Z 2 Ω . A rectangle in Z 2 Ω , centered in zero and with length 2l = (2l, 2l, ...) and width 2w = (2w, 2w, ...) is defined by : R l,w = {(x, y) ∈ Z 2 Ω ; -l ≤ x < l and -w ≤ y < w}.

If we remember that Ω-numbers are sequences, this definition becomes:

R l,w = {((x m ) (m∈N) , (y m ) (m∈N) ) ∈ Z 2 Ω ; ∀n ∈ N (-l ≤ x m < l and -w ≤ y m < w)}.
The following theorem is an extension of the connectivity theorem given in [START_REF] Richard | Arithmetization of a circular arc[END_REF]. This is a double extention: firstly to the axis aligned elliptical arcs and secondly to the Ω-numbers. The proof is very close to the one proposed in [START_REF] Richard | Arithmetization of a circular arc[END_REF]. It is however remarkable that, since we are working with Ω-integers, the result is a multi-scale result that is valid at all scales at the same time.

Theorem 1. Every arc of E α d (0, a, b) in the square R l,w is 8-Ω-connected for l = aβ ÷ b and bβ ÷ a. Proof. Let Γ = (( x n , y n )) k≤n≤k+p an arc of E α d (0, a, b) such that ( x n , y n ) ∈ R β for each n = k, . . . , k + p.
The proof is in two parts: in part (a) we will give a necessary and sufficient condition for the connectedness of Γ and in part (b) we will show that the condition Γ ⊂ R l,w with (l, w) = (aβ ÷ b, bβ ÷ a) is sufficient.

(a) Equivalent conditions: using the two schemes ( 5) and ( 6) and properties of the Euclidean division, we can see that the following conditions are equivalent:

-1 ≤ xm,n+1 -xm,n ≤ 1 -1 ≤ ( xm,n + f 1 n ) ÷ βm ≤ 1 -βm ≤ xn + f 1 n < 2βm -βm -xm,n ≤ (-a( ym,nβm + ym,n)) ÷ (bβm) < 2βm -xm,n -bβ 2 m -b xm,nβm ≤ -a( ynβm + ym,n) < 2bβ 2 m -b xm,nβm -bβ 2 m -b xm,nβm ≤ -aym,n < 2bβ 2 m -b xm,nβm - b a (β 2 m + xm,nβm) ≤ -ym,n < b a (2β 2 m -xm,nβm)
Hence with a similar proof for ( y m,n+1 -y m,n ), Γ is 8 Ω -connected if and only if, for each n = k, . . . , k + p -1 for all m , we have:

• b a (-2β 2 m + x m,n β m ) < y m,n ≤ b a (β 2 m + x m,n β m ) • a b (β 2 m + y m,n β m ) ≤ x m,n < a b (2β 2 m -y m,n β m ).
(b) Sufficient condition: We prove here that if Γ ⊂ R aβ÷b,bβ÷a , then the previous condition is verified. Since 0 ≤ x n ≤ β -1 and 0 ≤ y n ≤ β -1, we get the two following sequences of inequalities:

b a (-2β 2 m + x m,n β m ) ≤ b a (-β 2 m -β m ) < - b a β 2 m < b a β 2 m ≤ b a (β 2 m + x m,n β m ) - a b (β 2 m + y m,n β m ) ≤ - a b β 2 m < a b β 2 m < a b (β 2 m + β m ) ≤ a b (2β 2 m -y m,n β m ). From - b a β 2 m ≤ y n < b a β 2 m , we derive 5 - b a β 2 m ÷ β ≤ y n ÷ β < b a β 2 m ÷ β which imply - b a β m ≤ y n < b a β m . This implies -(bβ m ÷ a) ≤ y n < bβ m ÷ a.
For x, applying the same property, if -

a b β 2 m ≤ x n < a b β 2 m , then -(aβ m ÷b) ≤ x n < aβ m ÷ b.
This is the condition of the inclusion in the rectangle R aβ÷b,bβ÷a and so for the 8 Ω -connectedness.

The section (4.2) shows some representations and interpretations of this theorem.

Graphical Illustrations

In this section, we present different representations of axis-aligned elliptical arcs which illustrate our connectivity theorem and our arithmetization method in general. The Ω-integers are implemented in Ocaml language. Firstly, some words about this language, Ocaml is a functional programming language i.e. a programming paradigm that treats computation as the evaluation of mathematical functions and avoids state and mutable data. It prefers the application of functions, contrarily to the imperative programming style, which emphasizes changes in state. Hence, objects are functions which permits the manipulation of infinite ones. For instance, Ω-integers are viewed as a function of integers: sequences which associate a value a n to n for all n.

The multiresolution aspect is introduced by the different colors which encode the level m in the Ω-integers. We can observe that there is a correlation between the level of resolution and the size of the error. It is important to understand that the Ω-numbers contain all the scales at the same time and that the model as such is inherently multiscale. In fact, the algorithm handle entities which are infinity, but for the graphical illustration, we just extract some of them.

According to the theorem of Section 4.1, we have here two Ω-arithmetization of circle arcs. In Figure 5(a), the arc is connected and satisfies αR < β, (R, α, β) = ((2) n , (2n) n , (9n) n ), hence we have indeed ∀n ∈ N, Rα n = 2 * 2n = 4n < 9n. The second example, in Figure 5

(b) is not connected. Its parameters are (R, α, β) = ((2) n , (4n) n , (3n) n ).
The Figure 5(b). shows that all the parts (arcs) of the ellipse that are located inside the rectangle R (2β;⌊0.5β⌋) are connected.

In

Conclusion

In the present paper, we have applied the new concept of Ω-arithmetization to the case of an arc of ellipse. This method gives a discrete and multi-scale representation of this kind of Euclidean object. Due to the structure of the Ωintegers, we obtain a completely constructive algorithms which can be exactly translated into functional computer programs. It follows that these programs do not generate any numerical error and they provide an exact discrete multiresolution representation of the given arc of ellipse. Furthermore, we have shown that a discrete ellipse arc is connected inside a global region of the discrete plane. The properties of the obtained one-revolution-connected ellipse are not studied because it is due to the accumulated error of Euler scheme, and does not depend on the applied theory. Moreover, we emphasize that the goal of this work is not to define discrete objects in an intuitive way. That is why we do not com- pare our discrete ellipse definition with others like Bresenham [START_REF] Bresenham | A linear algorithm for incremental digital display of circular arcs[END_REF] or Andres [START_REF] Andres | Discrete circles, rings and spheres[END_REF] definitions.

In future works on this subject, we plan to study systematically this form of multi-resolution analysis for other kinds of continuous curves such as polynomial ones and spline curves. In addition, we intend to change our general theoretical framework; we want to move from the theory of Ω-numbers of Laugwitz-Schmieden to the formalism of constructive type theory of P. Martin-Löf [START_REF] Martin-Löf | Constructive mathematics and computer programming[END_REF][START_REF] Martin-Löf | Intuitionnistic Type Theory[END_REF]. The first reason is that this stark approach of mathematics and computer science is well suited for both developing constructive mathematics and writing programs. Furthermore, Martin-Löf has already developed a nonstandard extension of constructive type theory [START_REF] Martin-Löf | Mathematics of infinity[END_REF] in which we dispose of infinitely large natural numbers. Hence, in this more satisfactory context, it would be possible to define an Harthong-Reeb line as a new model of the continuum and build a general tool for discrete multi-resolution analysis of continuous objects.

Fig. 1 .

 1 Fig. 1. Graphical representations of the multiresolution aspects of the Ωarithmetization of an arc of ellipse. (Full explanation in Section 3).

Fig. 2 .

 2 Fig. 2. Intuitive representation of the Harthong-Reeb line.

Definition 5 .

 5 An arc of E α d (0, a, b) is a sequence of the following form ( x n , y n ) k<n<k+p for k and p fixed in N Then, we define the notion of connectedness with Ω-intergers and call it 8 Ωconnectedness: Definition 6. A curve defined by the Ω-points (

Figure 5 .

 5 we can see in (a) a multiresolution representation of a discrete ellipse E (2n) n∈N d (0, 4, 2) with β = (7n) n∈N . The pictures (b), (c), (d), (e), (f) are the different scales presented separatly in Figure 6.(a). The rectangle R (2β;⌊0.5β⌋) is also displayed for each picture .

Fig. 3 .

 3 Fig. 3. Graphical representations of the Ω-arithmetization of connected and nonconnected circle arcs.

Fig. 4 .

 4 Fig. 4. Illustration of the theorem with an ellipse E (2n) n∈N d (0, 3, 2) with β = (80n) n∈N .

Fig. 5 .

 5 Fig. 5. Graphical representations of ellipse with the associated rectangleconnectedness.

Here a and b are standard Ω-numbers because of ϕΩ, hence the sequences are constant ones.

Usually, the division algorithm operates on integer numbers. Here we need to extend it to rational numbers. Hence we defined u ÷ v where u and v are rational fractions by u ÷ v = def ⌊u/v⌋.


C Carnec
email: c.carnec@brgm.fr

H Modaressi

J Douglas

D Raucoules

E Simonetto

Contribution of space imagery to vulnerability assessment of elements exposed to geological risks

Keywords: Vulnerability, Geohazards, Multi-Risk, Remote Sensing Techniques, Interferometry

Vulnerability assessment of systems exposed to earthquakes has been increasily investigated in the last few years. Most efforts have been devoted particularly to vulnerability reduction of what is commonly called the physical vulnerability, given in terms of expected behaviour.

This includes the damage inflicted on exposed elements. It is however premature to consider that this work has led to a standardised method with respect to seismic risks. In addition, the generalization of this step to other natural phenomena is not often considered. Our work aims to define a common methodology for the assessment of the physical vulnerability of infrastructure elements subject to various natural risks. To do this, remote sensing techniques can provide useful information.

INTRODUCTION

1.1 Interest in homogenised approach for natural risk assessment Generally, only one natural event is considered at a time, leading to a lack of homogeneity in the risk assessment process for all natural hazards in a given area. Adapting methods developed for one specific risk to others is rarely envisaged. Futhermore, data is often difficult to obtain and the same data is sometimes collected many times for different purposes. Our project comes in response to the real need of harmonising the data and methodologies by integrating disparate, multi-and trans-disciplinary applied research into global and operational systems. This should lead to a homogeneous and global methodology for optimal multi-risk assessment. Main objectives are the harmonisation of data and methodologies for 1) hazard assessment and mapping and 2) for vulnerability assessment, by integrating uncertainty quantification and representation, and 3) development of risk scenarios and validation of the homogenised methodology, focusing on a few well-known high-risk areas.

The risk analysis techniques applied to hazards that originate from natural events such as earthquakes, volcanic eruptions, floods, subsidence and landslides (so-called geohazards), share some common features. The general objective of our project is to enhance the generic parts common to the assessment of the risks related to the aforementioned natural events, and to underline the specific characteristics of each one. The overall approach including the generic part as well as all specific characteristics may be interpreted as a multi-risk methodology with a twofold interest: 1) optimising the required data and evaluation tools, and 2) avoiding the overestimation of risks in regions exposed to several natural hazards.

Implementation of a common methodology for vulnerability assessment

The analysis of the physical vulnerability of exposed elements to earthquakes, that is the damage undergone by the element directly exposed to hazard, has undergone considerable development in the last decade. The pioneering work performed in the HAZUS programme (in the USA) and its extension to the specific European context (the RISK-UE project) have permitted the implementation of an easily applicable methodology [START_REF] Mouroux | Projet européen RISK-UE[END_REF][START_REF] References | An advanced approach for earthquake risk scenarios with application to different european towns[END_REF]. When considering the systemic vulnerability evaluation, including social and economical aspects, research efforts have already been carried out in the scientific community, but have not yet produced efficient operational techniques Finally methodologies implemented for vulnerability assessment in the context of seismic hazard, still need to be generalised in order to be applicable in the context of other natural hazards. This is one of the objectives of our project.

VULNERABILITY EVALUATION

Social and environmental vulnerability to natural hazards can be explained by several factors. Recent trends in Europe that increase vulnerability to natural hazards are : population growth and density, rapid urbanisation and unplanned human settlements, occasional poor engineering of construction, lack of adequate infrastructure, inequities in social structure, poverty, and inadequate environmental practices. Reducing social and economic vulnerability to natural hazards requires special attention and actions at these two levels :

• analysis and characterisation of hazards, which entails the assessment of the most vulnerable production areas, settlements and infrastructure, and adoption of risk reduction measures;

• existence of an institutional framework for implementing risk reduction measures using development policy instruments, contingency plans and environmental management tools.

The methods for vulnerability assessment comprise of two specific phases: 1) to evaluate and/or define proper vulnerability relations suitable for different types of buildings and infrastructures, depending on the natural hazard considered, and 2) to improve results obtained in risk scenarios on the selected sites, by providing fast estimation techniques (e.g. space technologies) and by comparing them with conventional techniques.

IMPLEMENTATION OF EFFICIENT OBSERVATION TECHNIQUES

An innovative action which can be outlined in the ambitious task of harmonising methodology for natural risk assessment concerns the use of space technologies to provide fast estimation of the relevant hazard and vulnerability parameters over large areas. Moreover, these technologies simultaneously cover a wide area, which is not practically achievable by other traditional techniques of investigation or monitoring. There are substantial benefits to having pre-emptive knowledge and real-time follow-up on potentially or ongoing damaging hazards. Space technologies can thus drastically improve the results obtained in risk scenarios on high-risk areas and help to implement prevention policies.

Using optical imagery data

The standard processes implemented for exposed elements identification consists in field investigations and available data collection from local actors. The cost of such a task is usually a dissuasive factor in the context of large built-up areas. The availability of remote sensing data over large high-risk areas may constitute an efficient and fast alternative or complement other traditional investigation approaches. They can be helpful as well, in identifying relevant parameters to be considered for the analysis. Recent work led in collaboration with ISTAR (Fig. 1) attempts to provide information on building typology by using Digital Surface Model from very high resolution data (1 m resolution). Considering seismic hazard, among the existing methods used for vulnerability evaluation, optical imagery (airborne or spaceborne) is one of the most advanced techniques able to provide spatialised topographic information. Thanks to the new generation of high/very high resolution sensors (Quickbird, Ikonos, SpotV, and soon PLEIADES), new methodologies and applications can be envisaged, especially in the field of vulnerability evaluation and natural risk management. Among the parameters required to evaluate building vulnerability to earthquakes, the height and type of construction are of significant importance. Studies have shown that use of high resolution digital elevation models can provide useful information for the derivation of the vulnerability parameter associated with the building height. Similarly, due to the historical evolution in building materials, it is anticipated that remote sensing data can be exploited to provide an estimation of the building-type parameter. The most suited data to discriminating between different roof material types and/or ages is hyperspectral imagery which provides information on the spectral response of materials in fine spectral bands across the visible, near-and shortwave infrared parts of the electromagnetic spectrum.

The objectives of our study are the following:

• To evaluate the potential of characteristic material spectral signatures obtained from hyperspectral imagery, in order to improve the determination of building types/ages at a city district and potentially individual building level; this technique is implemented on selected sites where all relevant information are already existing.

•

To evaluate the reliability and improvement of space techniques (high resolution digital elevation models and hyperspectral imagery), by comparing with conventional observation techniques.

•

To improve results obtained in risk scenarios on the selected sites, by providing fast estimation of the building-type vulnerability parameter.

Using radar imagery

The monitoring of ground deformation by means of classical methods such as optical levelling or Global Positioning System (GPS) can reveal deformations with sub-millimetric to centimetric precision on localized areas. However, to provide data on a reasonably dense two-dimensional (2D) benchmark grid, conventional techniques require the test area to be very small, otherwise cost and measurement time are no longer affordable. Moreover, a regular (both expensive and time consuming) intervention on the site is required and a retrospective study of a specific deformation event is usually not possible, since monitoring networks are often implemented after the major events themselves. Synthetic Aperture Radar (SAR) images provide an alternative and complementary method to investigate ground deformation phenomena [START_REF] Carnec | Three years of mining subsidence monitored by SAR interferometry, Gardanne, France[END_REF][START_REF] Le Mouélic | A Leastsquares adjustment of multi-temporal InSAR data -Application to the ground deformation of Paris[END_REF]2005).

Assessment of radar techniques (conventional or advanced-InSAR) for the characterization of the physical behavior of buildings and the supply of vulnerability parameters is ongoing. Methodological examples are generated concerning deformation parameters acting on buildings (linearity of the movements or not, the gradient, and others). First results of these studies are presented here. The demonstration concerns Permanent Scatterer (PS) approach. Space-borne differential radar interferometry provides a unique tool able to give a synoptic view of ground deformation with centimetric to millimetric vertical precision. Approaches for combining a wide range of radar images such as the PS technique allow the estimation of the deformation history of single buildings. A detailed description of this technique can be found in Ferretti et al. (2000Ferretti et al. ( , 2001)). The PS approach has been exploited to investigate a test site particularly exposed to ground deformation hazards, namely the iron mining basin in Lorraine (France) [START_REF] Colesanti | Detection of mining related ground instabilities using the Permanent Scatterers technique -A case study in the East of France[END_REF]. Despite the rural environment, the average PS spatial density resulted in 60 PS.km -2 (Fig. 2). Clear collapse precursor signs have been identified in the 10 months preceding a major collapse event, inducing severe damage to houses (maximal vertical amplitude of 59 cm). The comparison with available levelling measurement after the collapse itself shows a close match (differences in the order of 1-5 mm) between in situ and spaceborne measurements.

Figure 2. Monitoring of ground instabilities due to historical iron mining activity using Permanent Scatterers (PS) technique from ERS SAR images (processing by TRE). Precursor signs of a collapse affecting an area of 300x300m 2 have been identified.

CONCLUSION

In summary, our project addresses multi-risk assessment for natural disasters through homogenised and global methodologies, in order to improve disaster preparedness and mitigation. Focused on seismic, volcanic and ground instability hazard assessment, the project's strategic framework attempts to encompass using a system perspective the whole chain of environmental information flow, from observing, monitoring and other data collection, through assessment, modelling and other value-added data processing. This integrated observing, assessment and reporting framework will be developed at nested geographic scales including global, national, regional and local levels.

The capability of investigating the deformation before a major collapse (when no levelling data were recorded) suggests that a relevant application of the PS technique could be envisaged for the detection of collapse precursors. Anyway, for achieving operational capability, a shorter revisit time would often be required. Moreover, other collapse events occurred (also in similar geological and mining contexts) without clear evidence of known precursor signs. A deeper analysis of several different cases should, therefore, be performed to generalize the approach. Nevertheless, this study clearly shows the important role that the PS technique as well as other advanced multi-interferogram radar interferometry approaches, could play in land-use planning and natural or anthropogenic risk assessment. In particular, these approaches could be exploited systematically on large areas (100x100 km2) to optimize and drive other ground control methods such as optical levelling and microseismicity techniques, which can be used locally to set up a real-time alert system

The demonstration will also include the potential of using subpixel correlation applied to optical imagery to measure lateral displacements. These measurements are complementary to vertical displacements provided by InSAR techniques. The goal is to assess the capacity of new generation instruments (C, L, X radar band and very high resolution optical sensors) to contribute to the estimation of infrastructure vulnerability.

Figure 1 .

 1 Figure 1. Building elevation from a detailed Digital Surface Model (1 meter resolution) for the vulnerability assessment of the buildings on Nice, France. Istar-BRGM, 2003.