Janne Prawitt

Mouaadh Abdelkarim

Johanna H M Stroeve

Iuliana Popescu

Helene Duez

Vidya R Velagapudi

Julie Dumont

Emmanuel Bouchaert

Theo H Van Dijk

Anthony Lucas

Emilie Dorchies

Mehdi Daoudi

Frank J Gonzalez

Matej Oresic

Bertrand Cariou

Folkert Kuipers

Sandrine Caron

Bart Staels
email: bart.staels@pasteur-lille.fr

Farnesoid X receptor deficiency improves glucose homeostasis in mouse models of obesity

Keywords: FXR, bile acids, obesity, glucose homeostasis, insulin resistance, energy metabolism, triglyceride metabolism, bile acid sequestrants

Objective: Bile acids (BA) participate in the maintenance of metabolic homeostasis acting through different signaling pathways. The nuclear BA receptor farnesoid X receptor (FXR) regulates pathways in BA, lipid, glucose and energy metabolism which become dysregulated in obesity. However, the role of FXR in obesity and associated complications, such as dyslipidemia and insulin resistance, has not been directly assessed.

Research Design and Methods: Here, we evaluate the consequences of FXR-deficiency on body weight development, lipid metabolism and insulin resistance in murine models of genetic and diet-induced obesity.

Results: FXR-deficiency attenuated body weight gain and reduced adipose tissue mass in both models. Surprisingly, glucose homeostasis improved due to an enhanced glucose clearance and adipose tissue insulin sensitivity. In contrast, hepatic insulin sensitivity did not change, and liver steatosis aggravated due to the repression of β-oxidation genes. In agreement, liverspecific FXR-deficiency did not protect from diet-induced obesity and insulin resistance indicating a role for non-hepatic FXR in the control of glucose homeostasis in obesity.

Decreasing elevated plasma BA concentrations in obese FXR-deficient mice by administration of the BA sequestrant colesevelam improved glucose homeostasis in a FXRdependent manner indicating that the observed improvements by FXR-deficiency are not due to indirect effects of altered BA metabolism.

Conclusions: Overall, FXR-deficiency in obesity beneficially affects body weight development and glucose homeostasis.

Obesity is characterized by an excess of adipose tissue mass which predisposes to metabolic perturbations, including insulin resistance, hyperlipidemia and fatty liver, promoting the development of type 2 diabetes and cardiovascular disease. Dysregulation of various metabolic pathways in tissues such as adipose tissue, skeletal muscle, the pancreas and the liver lay at the basis of these metabolic complications, but the exact mechanisms are still not well understood [START_REF] Cornier | The metabolic syndrome[END_REF].

Nuclear receptors are ligand-activated transcription factors, which regulate gene expression by binding to specific response elements in the promoters of target genes. The farnesoid X receptor (FXR) is a key regulator of bile acid (BA) metabolism. By promoting BA efflux from the liver, inhibiting hepatic BA synthesis and intestinal absorption, FXR controls the enterohepatic cycling of BA [START_REF] Lefebvre | Role of bile acids and bile acid receptors in metabolic regulation[END_REF]. Additionally, FXR regulates lipid metabolism, insulin sensitivity and energy homeostasis [START_REF] Prawitt | How to modulate FXR activity to treat the Metabolic Syndrome[END_REF]. FXR activation lowers plasma and liver triglycerides by repressing sterol regulatory element binding protein (SREBP)1c-mediated hepatic lipogenesis [START_REF] Watanabe | Bile acids lower triglyceride levels via a pathway involving FXR, SHP, and SREBP-1c[END_REF] and by enhancing plasma triglyceride clearance via apolipoprotein (apo)CII induction [START_REF] Kast | Farnesoid X-activated receptor induces apolipoprotein C-II transcription: a molecular mechanism linking plasma triglyceride levels to bile acids[END_REF] and apoCIII inhibition [START_REF] Claudel | Farnesoid X receptor agonists suppress hepatic apolipoprotein CIII expression[END_REF] in the liver, resulting in an increased lipoprotein lipase (LPL) activity. Consequently, FXR-deficient (FXR -/-) mice are dyslipidemic, displaying elevated plasma triglycerides, total and HDL-cholesterol [START_REF] Sinal | Targeted disruption of the nuclear receptor FXR/BAR impairs bile acid and lipid homeostasis[END_REF][START_REF] Cariou | The farnesoid X receptor modulates adiposity and peripheral insulin sensitivity in mice[END_REF][START_REF] Lambert | The farnesoid X-receptor is an essential regulator of cholesterol homeostasis[END_REF].

The characterization of FXR -/-mice further revealed an important function of FXR in the control of glucose homeostasis. FXR -/-mice are transiently hypoglycemic when fasted [START_REF] Cariou | The farnesoid X receptor modulates adiposity and peripheral insulin sensitivity in mice[END_REF][START_REF] Van Dijk | An increased flux through the glucose 6phosphate pool in enterocytes delays glucose absorption in Fxr-/-mice[END_REF], exhibit a delayed intestinal glucose absorption [START_REF] Van Dijk | An increased flux through the glucose 6phosphate pool in enterocytes delays glucose absorption in Fxr-/-mice[END_REF] and a reduced hepatic glycogen content [START_REF] Van Dijk | An increased flux through the glucose 6phosphate pool in enterocytes delays glucose absorption in Fxr-/-mice[END_REF][START_REF] Duran-Sandoval | The farnesoid X receptor modulates hepatic carbohydrate metabolism during the fasting-refeeding transition[END_REF][START_REF] Cariou | Transient impairment of the adaptive response to fasting in FXR-deficient mice[END_REF]. Hepatic gluconeogenic gene expression is decreased in the absence of FXR [START_REF] Duran-Sandoval | The farnesoid X receptor modulates hepatic carbohydrate metabolism during the fasting-refeeding transition[END_REF][START_REF] Cariou | Transient impairment of the adaptive response to fasting in FXR-deficient mice[END_REF][START_REF] Ma | Farnesoid X receptor is essential for normal glucose homeostasis[END_REF] and induced upon FXR activation in vivo [START_REF] Stayrook | Regulation of carbohydrate metabolism by the farnesoid X receptor[END_REF]. Moreover, FXR -/-mice display peripheral insulin resistance with an impaired insulin signaling response in adipose tissue and skeletal muscle [START_REF] Cariou | The farnesoid X receptor modulates adiposity and peripheral insulin sensitivity in mice[END_REF][START_REF] Ma | Farnesoid X receptor is essential for normal glucose homeostasis[END_REF]. Additionally, FXR may control adipose tissue biology. FXR -/-mice exhibit a decrease in adipose tissue mass with smaller adipocytes [START_REF] Cariou | The farnesoid X receptor modulates adiposity and peripheral insulin sensitivity in mice[END_REF]. Conversely, FXR activation in vitro stimulates adipocyte differentiation [START_REF] Rizzo | The farnesoid X receptor promotes adipocyte differentiation and regulates adipose cell function in vivo[END_REF] by promoting PPARγ activity and inhibiting the Wnt/β-catenin pathway [START_REF] Abdelkarim | The Farnesoid X Receptor Regulates Adipocyte Differentiation and Function by Promoting Peroxisome Proliferator-activated Receptor-{gamma} and Interfering with the Wnt/{beta}-Catenin Pathways[END_REF].

BA act as regulators of metabolic homeostasis also through FXR-independent pathways [START_REF] Thomas | Targeting bile-acid signalling for metabolic diseases[END_REF].

Apart from FXR, BA have been reported to activate the G-protein coupled BA receptor (TGR5), the pregnane X receptor (PXR), the vitamin D receptor (VDR) or the formyl peptide receptor (FPR) [START_REF] Thomas | Targeting bile-acid signalling for metabolic diseases[END_REF]. TGR5 activation by BA increases energy expenditure in thermogenic tissues by inducing deiodinase 2 (D2) [START_REF] Watanabe | Bile acids induce energy expenditure by promoting intracellular thyroid hormone activation[END_REF] and improves glucose homeostasis by stimulating glucagon-like peptide 1 (GLP-1) secretion from intestinal L-cells [START_REF] Katsuma | Bile acids promote glucagon-like peptide-1 secretion through tgr5 in a murine enteroendocrine cell line stc-1[END_REF][START_REF] Thomas | TGR5-mediated bile acid sensing controls glucose homeostasis[END_REF]. Hence, since FXR controls plasma BA concentrations, the metabolic actions of FXR can be due to direct FXRmediated and/or indirect BA-mediated effects [START_REF] Lefebvre | Role of bile acids and bile acid receptors in metabolic regulation[END_REF]. BA metabolism can be pharmacologically modulated by BA sequestrants (BAS), such as colesevelam, which bind BA in the intestinal lumen and interrupt their enterohepatic cycling. BAS lower plasma cholesterol levels by stimulating hepatic BA synthesis from cholesterol and increasing LDL receptor activity [START_REF] Prawitt | How to modulate FXR activity to treat the Metabolic Syndrome[END_REF]. In diabetic patients, BAS improve glycemic control [START_REF] Staels | Bile acid sequestrants for lipid and glucose control[END_REF], associated with an activation of the incretin system due to an enhanced GLP-1 secretion as shown in rodent models of diabetes [START_REF] Chen | Cholestyramine reverses hyperglycemia and enhances GLP-1 release in Zucker Diabetic Fatty rats[END_REF][START_REF] Shang | Colesevelam improves insulin resistance in a dietinduced obesity (F-DIO) rat model by increasing the release of GLP-1[END_REF].

Studies on the metabolic function of FXR have so far been limited to lean FXR -/-mice.

However, the role of FXR in the adaptation to obesity and its metabolic complications has not yet been assessed. To address this issue, we investigated the impact of FXR-deficiency in murine models of genetic (ob/ob) and diet-induced obesity. Our results show that FXRdeficiency protects from excessive body weight gain in both models by reducing adipose tissue mass. Interestingly, hyperglycemia and glucose tolerance improved associated with improved peripheral glucose clearance and adipose tissue insulin sensitivity. Liver insulin sensitivity was not altered by FXR-deficiency and hepatic steatosis became even more pronounced associated with elevated plasma triglyceride levels and reduced expression of genes implicated in β-oxidation. Furthermore, liver-specific FXR-deficient mice were not protected from diet-induced obesity and insulin resistance, indicating a role for non-hepatic FXR in the improvement of glucose metabolism. Finally, to assess the contribution of elevated plasma BA concentrations observed in FXR-deficient models of obesity, plasma BA levels were reduced by administration of the BAS colesevelam. Colesevelam treatment improved glucose homeostasis in genetic obesity only in the presence of FXR, indicating that the changes in BA concentrations are not responsible for the improved glucose homeostasis in obese FXR-deficient mice. Thus, selective FXR antagonism or elimination of ligands, e.g. by BAS, may be an interesting pharmacological option for the therapy of metabolic disorders associated with obesity.

Research Design and Methods

Animals.

All experiments were performed with the approval of the Institut Pasteur de Lille review board. Animals were housed in a 12h-light/12h-dark cycle and maintained on a standard chow (A03, SAFE). Unless stated otherwise, male mice were used for experimentation. FXR -/- [START_REF] Sinal | Targeted disruption of the nuclear receptor FXR/BAR impairs bile acid and lipid homeostasis[END_REF] and FXR-floxed mice [START_REF] Sinal | Targeted disruption of the nuclear receptor FXR/BAR impairs bile acid and lipid homeostasis[END_REF] were backcrossed for seven or ten generations, respectively, onto a C57Bl/6J background. FXR +/+ ob/ob and FXR -/-ob/ob mice and lean littermates were generated by crossing FXR -/-and leptin-deficient mice (B6.V-Lep ob /J, Charles River). Liver-specific FXR -/-mice (FXR flox/flox Alb-Cre +/-=LFXR -/-) and wildtype littermates (FXR flox/flox Alb-Cre -/-=LFXR +/+ ) were generated by crossing FXR-floxed and Albumin-Cre recombinase expressing mice (C57BL/6-Tg(Alb-cre)21Mgn/J, Charles River). To study diet-induced obesity, mice received a high-fat diet (HFD) (D12492, Research Diets; 60% kcal fat) and controls a low-fat diet (D12450B, Research Diets; 10% kcal fat), 20w-old FXR +/+ and FXR -/-mice for 20w and 10w-old LFXR -/-and LFXR +/+ mice for 9w.

The BAS colesevelam (Daiichi Sankyo) was administered with the diet (2% in standard chow) ad libitum to 10w-old FXR +/+ ob/ob and FXR -/-ob/ob mice for 3w. Body weight was monitored weekly. Fat and lean mass were determined by dual x-ray absorptiometry (Piximus, GE Medical Systems LUNAR). Food intake was estimated during 6 periods of 2d.

Animals were sacrificed after a 6h-fast by cervical dislocation. Tissues were removed and snap-frozen in liquid nitrogen.

Plasma parameters.

Blood was sampled after a 6h-fast and parameters determined as followed: blood glucose (Accu-Check, Roche Diagnostics), insulin (Mercodia), non-esterified fatty acids (FFA) (Wako), triglycerides (Biomerieux), leptin (R&D Systems) and total BA as described [START_REF] Murphy | A fluorimetric and enzymatic method for the estimation of serum total bile acids[END_REF]. Lipoprotein profiles in plasma pools were generated by fast protein liquid chromatography (FPLC; Superose 6 HR 10/30, Amersham GE Healthcare) with subsequent determination of triglycerides. Indirect calorimetry. Mice were housed individually in metabolic cages (Oxylet, PanLab) for 48h to measure O 2 -consumption (V O2 ) and CO 2 -production (V CO2 ). Energy expenditure was calculated as described [START_REF] Elia | Energy expenditure and fuel selection in biological systems: the theory and practice of calculations based on indirect calorimetry and tracer methods[END_REF].

VLDL production. Mice were intravenously injected with 500mg/kg Tyloxapol (Sigma) after a 6h-fast and triglycerides were measured in plasma sampled as indicated.

Assessment of glucose metabolism. For tolerance tests 1g/kg glucose or 2IU/kg insulin (Actrapid Penfill, Novo Nordisk), respectively, were injected intraperitoneally and blood glucose measured as indicated. The metabolic clearance rate (MCR) was determined by stable isotope dilution as described [START_REF] Laskewitz | Chronic prednisolone treatment reduces hepatic insulin sensitivity while perturbing the fed-to-fasting transition in mice[END_REF]. To measure the insulin signaling response, tissues were harvested 10min after intraperitoneal injection of 1IU/kg insulin (Actrapid Penfill, Novo Nordisk) or 0.9% NaCl into overnight-fasted female mice. Serine473-phosphorylated Akt and total Akt protein expression (Cell Signaling Technology) were detected by Western Blot and signals quantified by GeneTool software (Syngene). Pancreas morphometry was analyzed as described [START_REF] Popescu | The nuclear receptor FXR is expressed in pancreatic beta-cells and protects human islets from lipotoxicity[END_REF].

Quantitative RT-PCR (qPCR). Total RNA was isolated from tissue samples by GSCN/phenol/chloroform extraction [START_REF] Chomczynski | Single-step method of RNA isolation by acid guanidinium thiocyanate-phenol-chloroform extraction[END_REF] and reverse transcribed into cDNA (Superscript II kit, Applied Bioscience). QPCR was performed with the BrilliantII SYBR Green QPCR Master Mix (Agilent) and specific primers (Supplementary Table1) on a mx3005 (Agilent).

Gene expression was normalized to 28S and the value in FXR +/+ ob/ob samples set to 1.

Lipidomics. Lipidomics analysis of adipose tissue was performed as described [START_REF] Laaksonen | A systems biology strategy reveals biological pathways and plasma biomarker candidates for potentially toxic statin-induced changes in muscle[END_REF].

Statistical analysis.

All values are reported as means±SEM. Statistical significance was analyzed as indicated by 2way-ANOVA and Bonferroni posthoc test or Mann-Whitney test using Prism (GraphPad). Differences were considered significant when p<0.05. Partial least squares discriminant analysis (PLS/DA) was performed using Matlab, version 7.5 (Mathworks) and PLS Toolbox, version 4.2 (Eigenvector Research).

Results

FXR-deficiency attenuates weight gain in genetic obesity.

As a first approach to assess the role of FXR in obesity, FXR -/-mice were crossed into the ob/ob background to produce FXR -/-ob/ob mice which were subsequently compared to FXR +/+ ob/ob controls.

The weight gain observed in FXR +/+ ob/ob mice was significantly attenuated by FXRdeficiency, already from 7w of age onwards (Fig. 1A) (p<0.0001; 2way-ANOVA). Dual x-ray absorptiometry revealed a decrease in fat (22%) and less pronounced in lean (10%) mass of FXR -/-ob/ob mice (Fig. 1B). Concordantly, the estimated food intake tended to decrease in FXR -/-ob/ob mice (15.5±1.1 vs. 18.9±0.8 kcal/d/mouse). Indirect calorimetry demonstrated that O 2 -consumption (V O2 ) and CO 2 -production (V CO2 ) were equally reduced in FXR -/-ob/ob mice compared to FXR +/+ ob/ob controls (Fig. 1C and1D), resulting in nearly identical values for the respiratory quotient (data not shown). Consistent with the decrease in V O2 and body weight, calculated energy expenditure was lower in FXR -/-ob/ob than in FXR +/+ ob/ob mice (7.1±0.3 vs. 8.7±0.7 kcal/d/mouse) even though the difference did not reach statistical significance.

Overall, FXR-deficiency leads to a decreased storage of energy in form of adipose tissue resulting in a lower total body weight.

FXR-deficiency improves glucose homeostasis in genetic obesity by increasing peripheral glucose disposal.

Since lean FXR -/-mice develop peripheral insulin resistance [START_REF] Cariou | The farnesoid X receptor modulates adiposity and peripheral insulin sensitivity in mice[END_REF][START_REF] Ma | Farnesoid X receptor is essential for normal glucose homeostasis[END_REF], it was next assessed whether FXR-deficiency in obesity affects glucose homeostasis. In contrast to hyperglycemic FXR +/+ ob/ob mice, FXR -/-ob/ob mice displayed normal blood glucose levels (Fig. 2A) (p<0.0001; 2way-ANOVA). In addition, the hyperinsulinemia typically observed in FXR +/+ ob/ob mice was significantly reduced in the absence of FXR (Fig. 2B). Concordant with the decreased hyperglycemia and hyperinsulinemia, glucose tolerance was significantly improved in FXR -/-ob/ob mice (Fig. 2C), due to a combination of lower fasting glycemia and improved glucose handling as evident from the significantly smaller integrated area under the curve (iAUC) (Fig. 2D). The MCR of glucose was further significantly improved in FXR -/- ob/ob mice (Fig. 2E) indicative of a higher glucose disposal in the periphery. In line, an insulin tolerance test showed that FXR -/-ob/ob mice are more insulin sensitive than FXR +/+ ob/ob mice (Fig. 2F and2G) with a more drastic decline of blood glucose levels in response to the administered insulin bolus (p<0.0001; 2way-ANOVA). Insulin resistant FXR +/+ ob/ob mice exhibited enlarged islets of Langerhans (Fig. 2H, compare arrows) due to compensatory hypertrophy. In contrast, islets of FXR -/-ob/ob mice were significantly smaller likely (Fig. 2I) reflecting the relieved metabolic pressure due to the improved insulin sensitivity. These findings indicate that, in contrast to lean mice, FXR-deficiency might improve glucose homeostasis in genetic obesity.

Adipose tissue, but not hepatic insulin sensitivity is improved in FXR -/-ob/ob mice.

To analyze the contribution of peripheral tissues and the liver to the improvement of insulin sensitivity, the phosphorylation of Akt, a key mediator of insulin's metabolic action, was measured in response to an in vivo administered insulin bolus (Fig. 3A). Akt phosphorylation was increased in adipose tissue of FXR -/-ob/ob compared to FXR +/+ ob/ob mice (2.8x vs. 1.5x).

In contrast, the response in skeletal muscle (4.5x vs. 4.8x) and liver (2.0x vs. 1.7x) was not different between both genotypes. Hepatic insulin resistance is often associated with a fatty liver [START_REF] Kotronen | Fatty liver: a novel component of the metabolic syndrome[END_REF]. Since FXR regulates lipogenesis in lean mice (2), hepatic gene expression and lipid content were determined. No changes were seen in the mRNA expression of the key regulator of lipogenesis, SREBP1c (Fig. 3B). Expression of genes involved in FA synthesis were only modestly induced (ACC1, FAS, SCD1) in livers of FXR -/-ob/ob mice (Fig. 3B). Nevertheless, the hepatic triglyceride content was doubled in the absence of FXR (Fig. 3C). The expression of PPARα, a key regulator of FA oxidation, and CPT1A, which ensures the FA entry into the mitochondria, were significantly decreased in FXR -/-ob/ob compared to FXR +/+ ob/ob mice (Fig. 3B).

Expression of ACC2, which indirectly inhibits CPT1A, was significantly higher, indicating that a reduced FA oxidation likely contributes to the increased steatosis in FXR -/-ob/ob mice.

Thus, an improvement of adipose tissue insulin sensitivity seems to account for the observed changes in glucose homeostasis. Hepatic insulin sensitivity was not altered by FXRdeficiency, even though liver steatosis increased.

FXR -/-ob/ob mice exhibit hypertriglyceridemia due to an impaired intravascular triglyceride clearance.

Since FXR -/-ob/ob mice display increased liver steatosis and reduced adipose tissue mass, it was tested whether this phenotype was associated with alterations in triglyceride or FA metabolism. Plasma FFA concentrations were not different between genotypes (Fig. 4A). In contrast, plasma triglyceride levels were strongly elevated in FXR -/-ob/ob compared to FXR +/+ ob/ob mice (Fig. 4B). Analysis of the lipoprotein profile revealed that triglycerides were mainly increased in VLDL, but also IDL/LDL-sized particles (Fig. 4C). To determine whether this was due to altered hepatic triglyceride secretion, plasma triglyceride accumulation was assessed after inhibition of intravascular lipolysis (Fig. 4D). Hepatic VLDL production was similar between FXR -/-ob/ob and FXR +/+ ob/ob mice, suggesting that an impairment of intravascular triglyceride clearance by LPL underlies the elevated plasma triglycerides. Consistently, hepatic mRNA levels of apoCII and apoAV, activators of LPL, were markedly repressed (by 65% and 74%, respectively) in FXR -/-ob/ob mice (Fig. 4E).

Hence, the repression of plasma triglyceride clearance regulators combined with the reduction of adipose tissue mass might result in the development of hypertriglyceridemia.

Total but not liver-specific FXR-deficiency protects from diet-induced obesity and insulin resistance.

To assess the impact of FXR-deficiency in obesity in a model expressing functional leptin, lean FXR -/-and FXR +/+ mice were fed a HFD for 20w. Whereas FXR +/+ mice rapidly gained weight upon HFD-feeding, body weight remained significantly lower in FXR -/-mice (Fig. 5A).

Consistently, plasma leptin levels were lower in FXR -/-mice (Fig. 5B). Plasma triglycerides were significantly higher in FXR -/-mice upon HFD-feeding (Fig. 5C). FXR -/-mice further displayed lower blood glucose (Fig. 5D) and plasma insulin (Fig. 5E) concentrations and showed an improved glucose tolerance (p<0.0001; 2way-ANOVA) (Fig. 5F andG). Hence, similar as in ob/ob mice, FXR-deficiency protects from diet-induced obesity and insulin resistance.

To determine the contribution of hepatic FXR to the protection from diet-induced obesity and insulin resistance, liver-specific FXR -/-mice (LFXR -/-) and wildtype littermates (LFXR +/+ ) were fed a HFD for 9w. Both LFXR -/-and LFXR +/+ mice developed obesity (Fig. 6A) and displayed significantly elevated plasma leptin levels upon HFD-feeding (Fig. 6B). LFXR -/- mice had significantly higher basal plasma triglycerides than LFXR +/+ littermates but were unaffected by HFD-feeding (Fig. 6C). HFD-fed LFXR -/-and LFXR +/+ mice became insulin resistant displaying hyperglycemia (p<0.0001; 2way-ANOVA) (Fig. 6D) and elevated plasma insulin levels (Fig. 6E). In addition, the diet-induced glucose intolerance was similar in both genotypes (Fig. 6F and6G). These findings indicate that liver-specific FXR-deficiency does not protect from diet-induced obesity and insulin resistance.

Modulation of BA metabolism by sequestration does not affect glucose homeostasis in FXR -/-

ob/ob mice.

FXR-deficiency led to elevated plasma BA concentrations in genetic (394±86µM vs. 44±13µM, p<0.001) and diet-induced obesity (119±50µM vs. 22±6µM). Since BA exert regulatory functions independent of FXR [START_REF] Thomas | Targeting bile-acid signalling for metabolic diseases[END_REF], we evaluated whether this increase may account for the observed alteration of body weight and glucose homeostasis.

The increased availability of plasma BA may lead to the activation of the TGR5 signaling pathway subsequently increasing energy expenditure in brown adipose tissue (BAT) [START_REF] Watanabe | Bile acids induce energy expenditure by promoting intracellular thyroid hormone activation[END_REF].

However, the expression of the TGR5 target genes UCP1 and D2 was decreased or unchanged, respectively, in BAT of FXR -/-ob/ob compared to FXR +/+ ob/ob mice (Fig. 7A), indicating that TGR5 activity was not induced.

To assess whether reducing the elevated plasma BA levels in FXR -/-ob/ob mice would affect glucose homeostasis, the BAS colesevelam was administered to FXR -/-ob/ob and FXR +/+ ob/ob mice for 3w. Colesevelam successfully reduced plasma BA concentrations in both genotypes by around 80% already within 9d of treatment (Fig. 7B). Body weight slightly but significantly increased with colesevelam treatment only in FXR -/-ob/ob mice (Fig. 7C). Colesevelam decreased blood glucose levels in FXR +/+ ob/ob, but not FXR -/-ob/ob mice (Fig. 7D). The glucose intolerance observed in untreated FXR +/+ ob/ob mice improved upon colesevelam treatment, whereas such effect was not observed in FXR -/-ob/ob mice (Fig. 7E and7F).

However, both genotypes displayed increased plasma insulin concentrations upon treatment (Fig. 7G). Since BAS may lower glycemia by activating the incretin system [START_REF] Shang | Colesevelam improves insulin resistance in a dietinduced obesity (F-DIO) rat model by increasing the release of GLP-1[END_REF], the expression of implicated intestinal genes was measured (Supplementary Table2). The expression of proglucagon, coding for GLP-1, and ProGIP, coding for GIP, were induced and repressed, respectively, by FXR-deficiency. However, only PC1/3, the enzyme activating both incretins, was moderately induced by colesevelam in FXR +/+ ob/ob mice, unlikely explaining the observed improvement of glucose homeostasis. Thus, reducing plasma BA concentrations had no effect on glucose homeostasis in the absence of FXR, suggesting that the alterations of glucose metabolism observed in FXR-deficiency are not mediated by circulating BA.

FXR-deficiency differentially affects the abundance of long-and medium-chain triglyceride species in adipose tissue of lean and ob/ob mice.

FXR-deficiency improves glucose homeostasis in obesity, due at least in part to an increase in adipose tissue insulin sensitivity. To investigate changes in adipose tissue lipid composition linked to FXR-deficiency before the metabolic phenotype fully develops, high-resolution lipidomics was performed in subcutaneous adipose tissue from 6w-old lean and ob/ob (obese) FXR -/-and FXR +/+ mice.

Overall, 133 lipid species were quantified. Total phospholipid and sphingomyelin content was significantly lower in obese than in lean mice, whereas total triglycerides were similar (Supplementary Fig. 1). These lipid classes were unaffected by FXR-deficiency. Of interest, a number of triglyceride species was differentially affected by FXR-deficiency in lean and obese mice. The concentration of medium-chain triglycerides with a high saturation level was significantly higher in obese than lean FXR +/+ mice (Fig. 8A-C) and tended to increase by FXR-deficiency in both. In contrast, long-chain, unsaturated triglycerides were also significantly elevated in obese compared to lean FXR +/+ mice, however increased by FXRdeficiency in lean but decreased in obese mice (Fig. 8D-F). Further, long-chain triglycerides containing few double-bounds, which were similar between lean and obese FXR +/+ mice, only decreased by FXR-deficiency in lean mice (Fig. 8G-I).

PLS/D analysis revealed that the global subcutaneous adipose tissue lipid profiles clustered according to their genotype (Fig. 8J). Lipid profiles are significantly different between lean and obese mice whether deficient or not for FXR, visualized as a clear separation by the latent variable (LV)1 plotted on the x-axis. Interestingly, the LV2 demonstrated an additional separation of lean FXR +/+ and obese FXR -/-from lean FXR -/-and obese FXR +/+ mice, indicating that FXR-deficiency modulates lipid profiles in an opposite manner in the lean versus obese background.

Discussion

The nuclear receptor FXR is a metabolic regulator which controls BA and lipid as well as glucose and energy metabolism [START_REF] Lefebvre | Role of bile acids and bile acid receptors in metabolic regulation[END_REF]. FXR can exert direct transcriptional control or indirectly influence metabolic pathways by modulating the homeostasis of BA, which control metabolism also independent of FXR [START_REF] Thomas | Targeting bile-acid signalling for metabolic diseases[END_REF]. The characteristics of FXR -/-mice -dyslipidemia (7-9), transient hypoglycemia upon fasting [START_REF] Cariou | The farnesoid X receptor modulates adiposity and peripheral insulin sensitivity in mice[END_REF][START_REF] Van Dijk | An increased flux through the glucose 6phosphate pool in enterocytes delays glucose absorption in Fxr-/-mice[END_REF], peripheral insulin resistance [START_REF] Cariou | The farnesoid X receptor modulates adiposity and peripheral insulin sensitivity in mice[END_REF][START_REF] Ma | Farnesoid X receptor is essential for normal glucose homeostasis[END_REF] and reduced adipose tissue mass [START_REF] Cariou | The farnesoid X receptor modulates adiposity and peripheral insulin sensitivity in mice[END_REF][START_REF] Rizzo | The farnesoid X receptor promotes adipocyte differentiation and regulates adipose cell function in vivo[END_REF] -reflect the importance of FXR for the maintenance of metabolic homeostasis. Yet, the impact of FXR-deficiency on obesity and associated metabolic disorders has not been assessed. In the current study we demonstrate, that FXRdeficiency in murine models of genetic and diet-induced obesity attenuated weight gain by reducing adipose tissue mass. FXR-deficiency further improved glucose homeostasis by enhancing peripheral glucose disposal and increasing adipose tissue insulin sensitivity. This improvement is not mediated by hepatic FXR, since liver insulin sensitivity was unchanged and liver-specific FXR -/-mice were not protected from diet-induced obesity and insulin resistance. The administration of the BAS colesevelam, which reduced the elevated plasma BA levels resulting from FXR-deficiency in obesity, further revealed that the improved glucose homeostasis was independent of BA-mediated pathways. FXR-deficiency thus exerts a beneficial effect on body weight development and glucose homeostasis in obesity.

FXR -/-mice display peripheral insulin resistance, but reported results on the liver are contradicting [START_REF] Cariou | The farnesoid X receptor modulates adiposity and peripheral insulin sensitivity in mice[END_REF][START_REF] Ma | Farnesoid X receptor is essential for normal glucose homeostasis[END_REF][START_REF] Zhang | Activation of the nuclear receptor FXR improves hyperglycemia and hyperlipidemia in diabetic mice[END_REF]. Surprisingly, we observed an improvement of glucose homeostasis by FXR-deficiency in genetic and diet-induced obesity, revealing a differential role for FXR in glucose homeostasis in basal (lean) and challenged (obese) metabolic conditions. The reduction of hyperglycemia in FXR -/-ob/ob mice resulted from an increased peripheral glucose disposal associated with an improved adipose tissue insulin sensitivity. This suggests that FXR in adipose tissue contributes to the dysregulation of glucose metabolism in obesity.

Hypertrophic adipocytes develop insulin resistance [START_REF] Biddinger | From mice to men: insights into the insulin resistance syndromes[END_REF]. Hence the reduced adipocyte size in FXR -/-ob/ob mice ( 16) might explain the increased insulin sensitivity. In addition to adipose tissue, skeletal muscle is the main peripheral tissue accounting for glucose uptake. However, the insulin signaling response in skeletal muscle was unaffected by FXR-deficiency. Moreover, FXR is not expressed in skeletal muscle [START_REF] Cariou | The farnesoid X receptor modulates adiposity and peripheral insulin sensitivity in mice[END_REF] and muscle mass strongly reduced in genetic obesity. We can therefore exclude the involvement of skeletal muscle in the FXRdeficiency-mediated changes of glucose homeostasis.

FXR exerts a key function in the control of hepatic glucose and lipid metabolism [START_REF] Lefebvre | Role of bile acids and bile acid receptors in metabolic regulation[END_REF]. In hepatic insulin resistance, increased endogenous glucose production contributes to hyperglycemia and the continuous insulin-mediated activation of lipogenesis leads to hyperlipidemia and may cause a fatty liver [START_REF] Biddinger | From mice to men: insights into the insulin resistance syndromes[END_REF]. In contrast to adipose tissue, FXR-deficiency did not improve hepatic insulin sensitivity in genetic obesity and liver-specific FXRdeficiency did not protect from diet-induced insulin resistance, a rather surprising result regarding the importance of hepatic FXR for the metabolic control of lean mice. Further, hepatic steatosis often associated with hepatic insulin resistance increased in FXR -/-ob/ob mice due to a repression of β-oxidation. Moreover, the repression of hepatic LPL cofactors led to an impaired intravascular triglyceride clearance resulting in elevated plasma triglycerides. Overall, the changes in lipid metabolism appear to be primarily due to altered liver function, while the results on insulin sensitivity emphasize a role for non-hepatic FXR in the dysregulation of glucose homeostasis in obesity -most probably in adipose tissue.

Several studies have shown that FXR controls adipocyte differentiation and function [START_REF] Cariou | The farnesoid X receptor modulates adiposity and peripheral insulin sensitivity in mice[END_REF][START_REF] Rizzo | The farnesoid X receptor promotes adipocyte differentiation and regulates adipose cell function in vivo[END_REF][START_REF] Abdelkarim | The Farnesoid X Receptor Regulates Adipocyte Differentiation and Function by Promoting Peroxisome Proliferator-activated Receptor-{gamma} and Interfering with the Wnt/{beta}-Catenin Pathways[END_REF]. FXR expression occurs early and transiently during adipocyte development, inhibiting the Wnt/β-catenin pathway, thus allowing optimal PPARγ activation [START_REF] Abdelkarim | The Farnesoid X Receptor Regulates Adipocyte Differentiation and Function by Promoting Peroxisome Proliferator-activated Receptor-{gamma} and Interfering with the Wnt/{beta}-Catenin Pathways[END_REF]. Since the lack of hepatic FXR does not seem to mediate the improvement of glucose homeostasis in obesity, it would be of great interest to study adipose tissue-specific FXR-deficiency in obesity. Unfortunately, in the currently available mouse models Cre recombinase expression is driven by promoters from late and/or non-specific adipocyte genes (i.e. adiponectin or aP2), rendering them inappropriate to study the influence of genes acting earlier in the differentiation process, such as FXR. FXR-deficiency in obesity was associated with elevated plasma BA levels. BA have been reported to improve glucose homeostasis independent of FXR by TGR5-mediated stimulation of GLP-1 secretion [START_REF] Katsuma | Bile acids promote glucagon-like peptide-1 secretion through tgr5 in a murine enteroendocrine cell line stc-1[END_REF][START_REF] Thomas | TGR5-mediated bile acid sensing controls glucose homeostasis[END_REF]. The administration of the BAS colesevelam to ob/ob mice successfully reduced plasma BA concentrations, but had no effect on glucose homeostasis in the absence of FXR. Thus, elevated circulating BA in FXR-deficiency do not mediate the observed improvement in glucose homeostasis. BAS improve glycemic control in type 2 diabetes, partly via a TGR5-mediated increase of GLP-1 secretion [START_REF] Chen | Cholestyramine reverses hyperglycemia and enhances GLP-1 release in Zucker Diabetic Fatty rats[END_REF][START_REF] Shang | Colesevelam improves insulin resistance in a dietinduced obesity (F-DIO) rat model by increasing the release of GLP-1[END_REF]. The observed FXR-dependent improvement of glycemic control by colesevelam in ob/ob mice suggests that FXR-mediated pathways contribute to the beneficial effect of BAS. Contradictory to our observation that FXR-deficiency is beneficial for glucose homeostasis in obesity, FXR activation has been shown to improve insulin resistance in rodent models of diabetes [START_REF] Cariou | The farnesoid X receptor modulates adiposity and peripheral insulin sensitivity in mice[END_REF][START_REF] Zhang | Activation of the nuclear receptor FXR improves hyperglycemia and hyperlipidemia in diabetic mice[END_REF][START_REF] Cipriani | FXR activation reverses insulin resistance and lipid abnormalities and protects against liver steatosis in Zucker (fa/fa) obese rats[END_REF]. The reason for this apparent inconsistency is currently unclear. However, the pharmacological activation of a nuclear receptor does not always and necessarily have the opposite effects than the knockout of its expression. The specific pharmacodynamics of the used compounds may also affect FXR in tissue-specific manner (sBARM effect), whereas our study focused on whole-body FXR-deficiency.

Both genetic and diet-induced obesity are attenuated in the absence of FXR. FXR-deficiency leads to reduced adipose tissue mass [START_REF] Cariou | The farnesoid X receptor modulates adiposity and peripheral insulin sensitivity in mice[END_REF][START_REF] Rizzo | The farnesoid X receptor promotes adipocyte differentiation and regulates adipose cell function in vivo[END_REF][START_REF] Abdelkarim | The Farnesoid X Receptor Regulates Adipocyte Differentiation and Function by Promoting Peroxisome Proliferator-activated Receptor-{gamma} and Interfering with the Wnt/{beta}-Catenin Pathways[END_REF] and adipocyte size [START_REF] Abdelkarim | The Farnesoid X Receptor Regulates Adipocyte Differentiation and Function by Promoting Peroxisome Proliferator-activated Receptor-{gamma} and Interfering with the Wnt/{beta}-Catenin Pathways[END_REF] in lean and ob/ob mice limiting energy storage capacity. Since food intake and energy expenditure only tended to decrease in FXR -/-ob/ob mice, the decreased adipose tissue mass most likely accounts for the reduced weight gain. With regard to the moderate decrease in energy expenditure, a central effect of FXR seems unlikely, additionally since FXR expression in hypothalamus is only very weak [START_REF] Gofflot | Systematic gene expression mapping clusters nuclear receptors according to their function in the brain[END_REF]. Further, the lack of TGR5 target gene induction in BAT of FXR -/-ob/ob mice argues against the implication of a TGR5-cAMP-D2-mediated mechanism [START_REF] Watanabe | Bile acids induce energy expenditure by promoting intracellular thyroid hormone activation[END_REF]. If the reduced body weight was a direct effect of a decreased food intake in the hyperphagic model of FXR -/- ob/ob mice, the hepatic triglyceride content would have also been reduced and not elevated as observed.

Interestingly, the analysis of adipose tissue lipid profiles by lipidomics revealed a differential alteration of certain triglyceride classes in lean and obese mice by FXR-deficiency, e.g. longchain, unsaturated triglycerides are lowered by FXR-deficiency in obese but increased in lean mice. This observation demonstrates that FXR-deficiency specifically and opposingly affects adipose tissue metabolism as it does glucose homeostasis in lean and obese conditions.

Overall, in contrast to lean mice, FXR-deficiency in obesity improves energy and glucose homeostasis. The selective inhibition of FXR activity may thus be an appealing option for the therapy of obesity and the related dysregulation of glucose homeostasis. plasma TG, D. blood glucose and E. plasma insulin were determined at the end of the feeding period (n=4-9/group). F. Blood glucose excursion and G. integrated AUC after administration of an i.p. glucose bolus (1g/kg glucose) were measured at the end of the feeding period (n=6-9/group). Significance of the overall effect of genotype (p<0.0001) and time (p<0. LFXR -/-mice and LFXR +/+ littermates (n=7-12/group) were fed a high-fat diet (HFD) or a control diet (CD) for ten weeks (LFXR +/+ CD = vertically-hatched bars or circles; LFXR -/-CD = diagonally-hatched bars or squares; LFXR +/+ HFD = white bars or diamonds; LFXR -/-HFD = black bars or triangles). A. Body weight was monitored weekly. Significance of the effect of diet (p<0.0001) and age (p<0.0001) as well as their interaction (p<0.0001) was calculated by 2way-ANOVA. B. Plasma leptin and C. plasma TG were determined at the end of the feeding period. D. Fasting blood glucose was measured weekly. Significance of the overall effect of diet (p<0.0001) and age (p<0.0001) as well as their interaction (p=0.0217) was calculated by 2way-ANOVA. E. Plasma insulin was measured after six weeks of feeding. F. Blood glucose excursion and G. integrated AUC after administration of an i.p. glucose bolus (1g/kg glucose) were measured after five weeks of feeding (n=6/group). Values are means±SEM. Differences between diet groups over time were analyzed by 2way-ANOVA and Bonferroni post-hoc test, differences between genotypes or diet groups were calculated by Mann-Whitney test (*compares genotypes of the same diet group; $ compares diet groups for LFXR +/+ ; § compares diet groups for LFXR -/-; $ or §<0.05, ** or $$ or § § <0.01, $$$ or § § §<0.001). TG, triglyceride.

Figure 1 .

 1 Figure 1. FXR-deficiency attenuates weight gain in ob/ob mice.A. FXR +/+ ob/ob (white bars or diamonds) and FXR -/-ob/ob mice (black bars or triangles)

Figure 2 .

 2 Figure 2. FXR-deficiency improves glucose homeostasis in ob/ob mice. A. Fasting blood glucose was measured weekly in FXR +/+ ob/ob (white bars or diamonds) and FXR -/-ob/ob mice (black bars or triangles) (n=9-10/group). Significance of the overall effect

Figure 3 .

 3 Figure 3. FXR-deficiency improves adipose tissue, but not hepatic insulin sensitivity in

Figure 4 .

 4 Figure 4. FXR-deficiency in ob/ob mice results in the accumulation of plasma

Figure 5 .

 5 Figure 5. FXR-deficiency protects from diet-induced obesity and insulin resistance.

  FXR +/+ (white bars or diamonds) and FXR -/-mice (black bars or triangles) (n=7-11/group) were fed a high-fat diet (HFD) for twenty weeks. A. Body weight was monitored weekly (n=7-11/group). Significance of the overall effect of genotype (p<0.0001) and age (p<0.0001) as well as their interaction (p<0.0001) was calculated by 2way-ANOVA. B. Plasma leptin, C.

  0001) was calculated by 2way-ANOVA. Values are means±SEM. Differences between genotypes over time were analyzed by 2way-ANOVA and Bonferroni post-hoc test, differences between genotypes were calculated by Mann-Whitney test (*<0.05, **<0.01, ***<0.001). TG, triglyceride.

Figure 6 .

 6 Figure 6. Liver-specific FXR-deficiency does not protect from diet-induced obesity and

Figure 7 .

 7 Figure 7. Modulation of bile acid metabolism does not affect glucose homeostasis in FXR -/-ob/ob mice. Ten week old FXR +/+ ob/ob and FXR -/-ob/ob mice were administered 2% colesevelam mixed in the diet or plain diet for three weeks (n=13/group) (FXR +/+ ob/ob control = white bars or diamonds; FXR +/+ ob/ob treated = vertically-hatched bars or circles; FXR -/-ob/ob control = black bars or triangles; FXR -/-ob/ob treated = diagonally-hatched bars or squares). A. TGR5

Figure 8 .

 8 Figure 8. FXR-deficiency differentially affects the abundance of long-chain, unsaturated

Figure 7 AFXR+

 7

Table 1 . Primer pairs used for quantitative RT-PCR.

 1

	Prawitt et al.

Diabetes

Acknowledgements

This study was supported by the EU Grant HEPADIP (N° 018734), the Agence Nationale de la Recherche (No.A05056GS) and COST (Action BM0602). We are grateful to Daiichi Sankyo for providing colesevelam and an unrestricted scientific grant.

Author Contributions

J.P. performed experiments and wrote the manuscript. M.A., J.H.M.S., I.R.P., H.D., V.R.V., Lipidomics analysis was performed in white adipose tissue of six week old FXR +/+ ob/ob (white bars), FXR


Pierre Alquier

Mohamed Hebiri

Generalization of ℓ 1 constraints for high dimensional regression problems

Keywords: High-dimensional data, LASSO, Restricted eigenvalue assumption, Sparsity, Variable selection AMS 2000 subject classifications: Primary 62J05, 62J07; Secondary 62F25

We focus on the high dimensional linear regression Y ∼ N (Xβ * , σ 2 I n ), where β * ∈ R p is the parameter of interest. In this setting, several estimators such as the LASSO [Tib96] and the Dantzig Selector [CT07] are known to satisfy interesting properties whenever the vector β * is sparse. Interestingly both of the LASSO and the Dantzig Selector can be seen as orthogonal projections of 0 into DC(s) = {β ∈ R p , X ′ (Y -Xβ) ∞ ≤ s} -using an ℓ 1 distance for the Dantzig Selector and ℓ 2 for the LASSO. For a well chosen s > 0, this set is actually a confidence region for β * . In this paper, we investigate the properties of estimators defined as projections on DC(s) using general distances. We prove that the obtained estimators satisfy oracle properties close to the one of the LASSO and Dantzig Selector. On top of that, it turns out that these estimators can be tuned to exploit a different sparsity or/and slightly different estimation objectives.

Introduction

In many modern applications, one has to deal with very large datasets. Regression problems may involve a large number of covariates, possibly larger than the sample size. In this situation, a major issue lies in dimension reduction which can be performed through the selection of a small amount of relevant covariates. For this purpose, numerous regression methods have been proposed in the literature, ranging from the classical information criteria such as C p , AIC and BIC to the more recent regularization-based techniques such as the ℓ 1 penalized least square estimator, known as the LASSO [START_REF] Tibshirani | Regression shrinkage and selection via the lasso[END_REF], and the Dantzig selector [START_REF] Candes | The dantzig selector: statistical estimation when p is much larger than n[END_REF]. These ℓ 1 -regularized regression methods have recently witnessed several developments due to the attractive feature of computational feasibility, even for high dimensional data when the number of covariates p is large.

Consider the linear regression model

Y = Xβ * + ε, (1)
where Y is a vector in R n , β * ∈ R p is the parameter vector, X is an n × p realvalued matrix with possibly much fewer rows than columns, n ≪ p, and ε is a random noise vector in R n . Here, for the sake of simplicity, we will assume that ε ∼ N (0, σ 2 I n ). Let P denote the probability distribution of Y in this setting. Moreover, we assume that the matrix X is normalized in such a way that X ′ X has only 1 on its diagonal. The analysis of regularized regression methods for high dimensional data usually involves a sparsity assumption on β * through the sparsity index β * 0 = j=1,...,p I(β * j = 0) where I(•) is the indicator function. For any q ≥ 1, d ≥ 0 and a ∈ R d , denote by a q q = d i=1 |a i | q and a ∞ = max 1≤i≤d |a i |, the ℓ q and the ℓ ∞ norms respectively. When the design matrix X is normalized, the LASSO and the Dantzig selector minimize respectively Xβ 2 2 and β 1 under the constraint X ′ (Y -Xβ) ∞ ≤ s where s is a positive tuning parameter (e.g. [START_REF] Osborne | On the LASSO and its dual[END_REF]Alq08] for the dual form of the LASSO). This geometric constraint is central in the approach developed in the present paper and we shall use it in a general perspective. Let us mention that several objectives may be considered by the statistician when we deal with the model given by Equation (1). Usually, we consider three specific objectives in the high-dimensional setting (i.e., p ≥ n):

Goal 1 -Prediction: The reconstruction of the signal Xβ * with the best possible accuracy is first considered. The quality of the reconstruction with an estimator β is often measured with the squared error X β -Xβ * 2 2 . In the standard form, results are stated as follows: under assumptions on the matrix X and with high probability, the prediction error is bounded by C log (p) β * 0 where C is a positive constant. Such results for the prediction issue have been obtained in [BRT09, [START_REF] Bunea | Consistent selection via the Lasso for high dimensional approximating regression models[END_REF][START_REF] Bunea | Sparsity oracle inequalities for the lasso[END_REF] for the LASSO and in [BRT09] for the Dantzig selector. We also refer to [START_REF] Koltchinskii | The Dantzig selector and sparsity oracle inequalities[END_REF][START_REF] Koltchinskii | Sparse recovery in convex hulls via entropy penalization[END_REF][START_REF] Meier | High-dimensional additive modeling[END_REF][START_REF] Van De Geer | High-dimensional generalized linear models and the lasso[END_REF][START_REF] Dalalyan | Aggregation by exponential weighting and sharp oracle inequalities[END_REF][START_REF] Chesneau | Some theoretical results on the grouped variables lasso[END_REF] for related works with different estimators (non-quadratic loss, penalties slightly different from ℓ 1 and/or random design). The results obtained in the works above-mentioned are optimal up to a logarithmic factor as it has been proved in [BTW07a]. See also [START_REF] Van De Geer | On the conditions used to prove oracle results for the lasso[END_REF]BC11] for very nice survey papers, or the introduction of [START_REF] Hebiri | Quelques questions de sélection de variables autour de l'estimateur LASSO[END_REF].

Goal 2 -Estimation: Another wishful thinking is that the estimator β is close to β * in terms of the ℓ q distance for q ≥ 1. The estimation bound is of the form C β * 0 (log (p)/n) q/2 where C is a positive constant. Such results are stated for the LASSO in [BTW07a, BTW07b] when q = 1, for the Dantzig selector in [START_REF] Candes | The dantzig selector: statistical estimation when p is much larger than n[END_REF] when q = 2 and have been generalized in [BRT09] with 1 ≤ q ≤ 2 for both the LASSO end the Dantzig selector.

Goal 3 -Selection: Since we consider variable selection methods, the identification of the true support {j : β * j = 0} of the vector β * is to be considered. One expects that the estimator β and the true vector β * share the same support at least when n grows to infinity. This is known as the variable selection consistency problem and it has been considered for the LASSO and the Dantzig Selector in several works [Bun08, Lou08, MB06, MY09, Wai06, ZY06].

In this paper, we focus on variants of Goal 1 and Goal 2, using estimators β that also satisfy the constraint X ′ (Y -X β) ∞ ≤ s. It is organized as follows. In Section 2 we give some general geometrical considerations on the LASSO and the Dantzig Selector that motivates the introduction of the general form of estimator:

Argmin β∈ X ′ (Y -Xβ) ∞ ≤s β
for any semi-norm • . In Section 3, we focus on two particular cases of interest in this family, and give some sparsity inequalities in the spirit of the ones in [BRT09]. We show that under the hypothesis that F β * is sparse for a known matrix F , we are able to estimate properly β * . Some application to a generic inverse problem are provided with numerical experiments. Finally, Section 4 is dedicated to proofs.

2 Some geometrical considerations Definition 2.1. Let us put, for any s > 0,

DC(s) = {β ∈ R p : X ′ (Y -Xβ) ∞ ≤ s}. Lemma 1. For any s > 0, P(β * ∈ DC(s)) > 1 -p exp(-s 2 /(2σ 2 )).
This means that DC(s) is a confidence region for β * . Moreover, note that DC(s) is convex and closed. Let • be any semi-norm in R p . Let Π s

• denote an orthogonal projection on DC(s) with respect to • :

Π s • (b) ∈ Argmin β∈DC(s) β -b .
From properties of projections, we know that

β * ∈ DC(s) ⇒ ∀b ∈ R p , Π s • (b) -β * ≤ b -β * .
There is a very simple interpretation to this inequality: if b is any estimator of β * , then, with probability at least 1p exp(-s 2 /(2σ 2 )), Π s

• (b) is a better estimator. In order to perform shrinkage it seems natural to take b = 0. Definition 2.2. We define our general estimator by

β • s = Π s • (0) ∈ Argmin β∈DC(s)
β .

We have the following examples:

1. for • = • 1 , we obtain the definition of the Dantzig Selector given in [START_REF] Candes | The dantzig selector: statistical estimation when p is much larger than n[END_REF].

2. for β = Xβ 2 , we obtain the program Argmin β∈DC(s) Xβ 2 . It was proved in [START_REF] Osborne | On the LASSO and its dual[END_REF] for example that a particular solution of this program is Tibshirani's LASSO estimator [START_REF] Tibshirani | Regression shrinkage and selection via the lasso[END_REF] known as

βL s = Argmin β∈R p Y -Xβ 2 2 + 2s β 1 .
3. for β = X ′ Xβ q with q > 0, it is proved in [Alq08] that the solution coincides with the "Correlation Selector" and it does not depend on q.

In the next Section, we exhibit other cases of interest and provide some theoretical results on the performances of the estimators.

3 Generalized LASSO and Dantzig Selector

Definitions

Let F be an application R + → R + with the restriction that F (x) = 0 may be equal to 0 only for x = 0. Note that X ′ X may be written, for some orthogonal matrix Q,

Q ′    λ 1 . . . 0 . . . . . . . . . 0 . . . λ p    Q, then we put F (X ′ X) = Q ′    F (λ 1 ) . . . 0 . . . . . . . . . 0 . . . F (λ p )    Q.
The idea is that, for a well chosen norm • , we will build estimators that will be useful to estimate β * when F (X ′ X)β * is sparse, in the sense that they will be close to β * with respect to the semi-norm induced by G(X ′ X) for G(x) = xF (x).

Definition 3.1. We define the "Generalized Dantzig Selector", βGDS

s , as β • s for b = F (X ′ X)b 1 , and the "Generalized LASSO", βGL s , for b = (b ′ G(X ′ X)b) 1/2 . Remark 1.
In the case where the program min β∈DC(s) β ′ G(X ′ X)β has multiple solutions we define βGL s as one of the solutions that minimizes F (X ′ X)β 1 among all the solutions β. The case where the program min β∈DC(s) F (X ′ X)β 1 has multiple solution does not cause any trouble: we can take βGDS s as any of these solution without any effect on its statistical properties.

Sparsity Inequalities

We now present the assumptions we need to state the Sparsity Inequalities. Assumption A(c) for c > 0: for any α ∈ R p such that

j:(F (X ′ X)β * ) j =0 |α j | ≤ 3 j:(F (X ′ X)β * ) j =0 |α j | ,
we have, for H(x) = x/F (x) (with the convention 0/0 = 0),

j:(F (X ′ X)β * ) j =0 α 2 j ≤ cα ′ H(X ′ X)α.
This assumption can be seen as a modification of assumptions in [BRT09]: if we put F (x) = 1, F (X ′ X) = I p and H(X ′ X) = X ′ X and we obtain exactly the same assumption that in [BRT09]. For the sake of shorteness, we put

F = F (X ′ X), G = G(X ′ X) and H = H(X ′ X).
Theorem 1. Let us take ε ∈]0, 1[ and s = 2σ(2 log(p/ε)) 1/2 . Assume that Assumption A(c) is satisfied for some c > 0. With probability at least 1ε we have simultaneously:

         ( βGDS s -β * ) ′ G( βGDS s -β * ) ≤ 72σ 2 c F β * 0 log(p/ε), F ( βGDS s -β * ) 1 ≤ 18 √ 2σ F β * 0 c log(p/ε), ( βGL s -β * ) ′ G( βGL s -β * ) ≤ 128σ 2 c F β * 0 log(p/ε), F ( βGL s -β * ) 1 ≤ 32 √ 2σ F β * 0 c log(p/ε).
In the case F (x) = 1, we obtain the same result as in [BRT09]. However, it is worth noting that the use of βGL s is particularly useful when F β * is sparse for a non-constant F (x), and β * is not. In this case the errors of the LASSO and the Dantzig Selector are not controlled anymore. This generalization is also of some interests especially when Assumption A(c) is satisfied for H, but not satisfied if we replace H by X ′ X. We now give an exemple.

Application to a generic inverse problem

In statistical inverse problems, one usually has to deal with the following regression problem: Y ∼ N (Xβ * , σ 2 I n ) with a known σ 2 , X a symmetric operator (for example a convolution operator) and a regularity assumption on β * . This assumption is often that β * belongs to the range of X or of a power of X: β * = X α g. See for example [START_REF] Cavalier | Inverse problems in statistics[END_REF] and the references therein.

We will now assume that g is sparse. In this case, note that X ′ X = X 2 . As X -α β * is sparse, we put F (x) = x -α/2 . So F = X -α and G = X 2-α . In this case, Theorem 1 gives for example

( βGL s -β * ) ′ G( βGL s -β * ) ≤ 128σ 2 c g 0 log(p/ε),
under an assumption on H = X 2+α (it is worth mentionning that in the case where α = -2, H = I n and so Assumption A(c) is always satisfied with c = 1, even if the case α > 0 is more meaningful).

We now provide a very short empirical comparison of the LASSO and Generalized LASSO approach in a toy example of such a model. Note that for α ≥ 0, β * = X α g being a smoothed version of g, is "almost sparse", so a comparison with the LASSO makes sense. We propose the following setting: let M(ρ) = (ρ |i-j| ) 1≤i,j≤n , and X = M 1/2 . We take g = (7, 0, 0, 0, 0, 5, 0, 0, 0, 0, 7, 0, 0, 0, 0, 5, 0, 0, 0, 0), n = 20, ρ = 0.5, Figure 1 gives the different values of β * = X α g for various values of α.

We compute the LASSO and Generalized LASSO in each case, and report the performance of the oracle with respect to the regularization parameter s:

Perf.GL = inf s>0 X βGL s -Xβ * and Perf.L = inf s>0 X βL s -Xβ * .
Of course, in practice, the optimal s in unknown and may be estimated by crossvalidation for example. We test both estimators with several values for the parameters α and σ 2 . For each value of these parameters, we run 20 experiments and report the mean performances for both estimators. The results are given in Table 1. We can see that the results seem coherent with Theorem 1: there seems to be an advantage in practice to consider the Generalized LASSO in the cases where alpha = 0.

Proofs

Proof of Lemma 1

We have

Y ∼ N (Xβ * , σ 2 I n ) and so Y -Xβ * ∼ N (0, σ 2 I n ) and finally X ′ (Y - Xβ * ) ∼ N (0, σ 2 X ′ X). Let us put V = X ′ (Y -Xβ *
) and let V j denote the j-th coordinate of V . Note that X ′ X is normalized such that for any j, V j ∼ N (0, σ 2 ), so:

P (|V j | > s) ≤ exp(-s 2 /(2σ 2 )). Then P ( V ∞ > s) ≤ p exp(-s 2 /(2σ 2 )).

Proof of Theorem 1

We use arguments from [BRT09]. From now, we assume that the event

{β * ∈ DC(s/2)} = { X ′ (Y -Xβ * ∞ < s/2} is satisfied. According to Lemma 1, the probability of this event is at least 1 -p exp(-s 2 /(8σ 2 )) = 1 -ε as s = 2(2 log(p/ε)) 1/2 .
Proof of the results on the Generalized Dantzig Selector.

We have

( βGDS s -β * ) ′ G( βGDS s -β * ) = ( βGDS s -β * ) ′ X ′ XF ( βGDS s -β * ) ≤ X ′ X( βGDS s -β * ) ∞ F ( βGDS s -β * ) 1 ≤ X ′ (Y -Xβ * ) ∞ + X ′ (Y -X βGDS s ) ∞ F ( βGDS s -β * ) 1 ≤ (s/2 + s) F ( βGDS s -β * ) 1 since βGDS s ∈ DC(s), and {β * ∈ DC(s/2)} is satisfied. By definition of βGDS s , 0 ≤ F β * 1 -F βGDS s 1 = (F β * ) j =0 |(F β * ) j | - (F β * ) j =0 |(F βGDS s ) j | - (F β * ) j =0 |(F βGDS s ) j | ≤ (F β * ) j =0 |(F β * ) j -(F βGDS s ) j | - (F β * ) j =0 |(F β * ) j -(F βGDS s ) j |.
This means that

F ( βGDS s -β * ) 1 ≤ 2 (F β * ) j =0 |(F β * ) j -(F βGDS s ) j |.
We can summarize all that we have now:

( βGDS s -β * ) ′ G( βGDS s -β * ) ≤ 3s 2 F ( βGDS s -β * ) 1 ≤ 3s (F β * ) j =0 |(F β * ) j -(F βGDS s ) j |. (2)
Let us remark that Inequality (2) implies that the vector α = F ( βGDS s β * ) may be used in Assumption A(c). This leads to

( βGDS s -β * ) ′ G( βGDS s -β * ) ≤ 3s (F β * ) j =0 |(F β * ) j -(F βGDS s ) j | ≤ 3s F β * 0 (F β * ) j =0 [(F β * ) j -(F βGDS s ) j ] 2 ≤ 3s F β * 0 c(F βGDS s -F β * ) ′ H(F βGDS s -F β * ) = 3s F β * 0 c( βGDS s -β * ) ′ G( βGDS s -β * ). (3)
As a consequence,

( βGDS s -β * ) ′ A( βGDS s -β * ) ≤ 9s 2 P β * 0 c = 72σ 2 c P β * 0 log(p/ε).
Plugging this result into Inequality (3) and using Inequality (2) again, we obtain:

P ( βGDS s -β * ) 1 ≤ 18 √ 2σ P β * 0 c log(p/ε).
Proof of the results on the Generalized LASSO.

Step 1. As a fist step, we establish an important property of the Generalized LASSO estimator. We prove that

∀β ∈ R p , |Y -XF βGL s 2 2 + 2s F βGL s 1 + ( βGL s ) ′ F (H -X ′ X)F βGL s ≤ Y -XF β 2 2 + 2s F β 1 + β ′ F (H -X ′ X)F β. (4)
To prove Inequality (4), we write the Lagrangian of the program that defines βGL s :

L(β, λ, µ) = β ′ Gβ + λ ′ [X ′ (Xβ -Y ) -sE] + µ ′ [X ′ (Y -Xβ) -sE] ,
where E = (1, . . . , 1) ′ , λ and µ are vectors in R p . Any solution β = β(λ, µ) must satisfy, for some λ j ≥ 0, µ j ≥ 0 and λ j µ j = 0,

0 = ∂L ∂β (β, λ, µ) = 2Gβ + X ′ X(λ -µ),
and then Gβ = (X ′ X)(µ-λ)/2 . Note that λ j ≥ 0, µ j ≥ 0 and λ j µ j = 0 imply that there is a γ j ∈ R such that γ j = (µ jλ j )/2, |γ j | = (λ j + µ j )/2. Hence λ j = 2(γ j ) - and µ j = 2(γ j ) + , where for any a, (a) + = max(a; 0) and (a) -= max(-a; 0). Let also γ denote the vector which j-th component is exactly γ j , we obtain:

Gβ = (X ′ X)γ. (5)
Then we have easily β ′ F β = β ′ (X ′ X)γ = γ ′ Hγ. Using these relations, the Lagrangian may be written:

L(β, λ, µ) = γ ′ Hγ + 2γ ′ X ′ Y -2γ ′ (X ′ X)β -2s p j=1 |γ j | = 2γ ′ X ′ Y -γ ′ Hγ -2s γ 1
Note that λ and β, and so γ, should maximize this value. Hence, γ is to minimize

-2γ ′ X ′ Y + γ ′ Hγ + 2s γ 1 + Y ′ Y Now, note that Y ′ Y -2γ ′ X ′ Y = Y -Xγ 2 2 -γ ′ (X ′ X)γ and then γ also minimizes Y -Xγ 2 2 + 2s γ 1 + γ ′ [H -(X ′ X)] γ.
We end the proof of (4) by noting that for every b such that

F b = γ, then b is to minimize Y -XF b 2 2 + 2s F b 1 + (F b) ′ [H -(X ′ X)] (F b). (6)
and that βGL s is such a b.

Step 2. The next step is to apply Equation (4) with β = β * to obtain

Y -XF βGL s 2 2 + 2s F βGL s 1 + ( βGL s ) ′ F (H -X ′ X)F βGL s ≤ Y -XF β * 2 2 + 2s F β * 1 + (F β * ) ′ (H -X ′ X)F β * .
For the sake of simplicity, we can define γ = F βGL s (following the notations of Step 1) and γ * = F β * and we obtain

Y -X γ 2 2 + 2s γ 1 + γ′ (H -X ′ X)γ ≤ Y -Xγ * 2 2 + 2s γ * 1 + (γ * ) ′ (H -X ′ X)γ * .
Computations lead to

X(γ -γ * ) 2 2 + 2s γ 1 + γ′ (H -X ′ X)γ -2(Y -Xγ * ) ′ X γ + 2(γ * ) ′ (H -X ′ X)(γ * -γ) ≤ 2s γ * 1 + (γ * ) ′ (H -X ′ X)γ -2(Y -Xγ * ) ′ Xγ * ,
and then

X(γ -γ * ) 2 2 ≤ 2s( γ * 1 -γ 1 ) + 2(Y -Xγ * ) ′ X(γ -γ * ) -(γ * -γ) ′ (H -X ′ X)(γ * -γ).
As a consequence [Alq08] P. Alquier. Lasso, iterative feature selection and the correlation selector: Oracle inequalities and numerical performances.

(γ * -γ) ′ H(γ * -γ) ≤ 2s( γ * 1 -γ 1 ) + 2(Y -Xγ * ) ′ X(γ -γ * ) ≤ 2s p j=1 (|γ * j | -|γ j |) + 2 X ′ (Y -Xβ * ) ∞ p j=1 |γ j -γ * j | ≤ 2s

Figure 1 :

 1 Figure1: The parameter β * = X α g for different values of α. In black, α = 0, so β * = g is sparse. In red, α = 1, β * is a bit smoothed, but still can be approximated by a sparse signal. In green, α = 2, β * is smoother, and approximation by a sparse signal do not hold any longer.

( 2 ≤

 2 j | -|γ j |) + s p j=1 |γ jγ * j |.So we obtain(γ *γ) ′ H(γ *γ) + s |γ j | -|γ * j |) + 2s j:γ * j =0 |γ jγ * j | = 4s j:γ * j =0 |γ jγ * j |. (7)In particular, Equation (7) implies thatj:γ * j =0 |γ jγ * j | ≤ 3 j:γ * j =0 |γ jγ * j |,and so α = γjγ * j may be used in Assumption A(c). Then Inequality (7) becomes(γ *γ) ′ H(γ *γ) ≤ 4s j:γ * j =0 |γ jγ * j | ≤ 4s γ * 0 j:γ * j =0 (γ jγ * j ) 4s γ * 0 c(γ *γ) ′ H(γ *γ).That leads to( βGL s β * ) ′ G( βGL s β * ) = (γ *γ) ′ H(γ *γ) ≤ 128σ 2 c F β * 0 log(p/ε).(8)We plug (8) into (7) again to obtain γγ * 1 ≤ 32 √ 2σ P β * 0 c log(p/ε).

Table 1 :

 1 The mean results for 20 experiments for each value of (α, σ 2 ).

	α	σ 2	mean of Perf.L	mean of Perf.GL
		0.01	0.167	0.118
	-2	0.30	4.792	3.076
		1.00	16.636	10.328
		0.01	0.194	0.097
	-1	0.30	5.624	2.911
		1.00	14.386	8.56
		0.01	0.098	0.098
	0	0.30	2.835	2.835
		1.00	9.012	9.012
		0.01	0.196	0.094
	+1	0.30	5.144	2.517
		1.00	13.232	8.597
		0.01	0.199	0.101
	+2	0.30	5.589	3.018
		1.00	17.957	10.228
		0.01	0.183	0.102
	+3	0.30	5.538	3.175
		1.00	19.133	10.371

  Electron. J. Stat., pages 1129-1152, 2008. [BC11] A. Belloni and V. Chernozhukov. High dimensional sparse econometric models: An introduction. In P. Alquier, E. Gautier, and G. Stoltz, editors, Inverse Problems and High-Dimensional Estimation. Springer Lecture Notes in Statistics, 2011. [BRT09] P. Bickel, Y. Ritov, and A. Tsybakov. Simultaneous analysis of lasso and Dantzig selector. Ann. Statist., 37(4):1705-1732, 2009. [BTW07a] F. Bunea, A. Tsybakov, and M. Wegkamp. Aggregation for Gaussian regression. Ann. Statist., 35(4):1674-1697, 2007.


Dr Philippe Borsa
email: philippe.borsa@noumea.ird.nc

Daniel Robineau

Blainville's beaked whale, Mesoplodon densirostris, in New Caledonia 1

. As for all beaked whales, most of the current knowledge on the distribution, biology, and ecology of this species comes from strandings. Strandings or sightings of Blainville's beaked whales have been reported from oceanic islands in the tropical

Pacific, including the Solomon Islands, New Britain, and French Polynesia [START_REF] Reeves | Marine mammals in the area served by the South pacific Regional Environment Programme (SPREP)[END_REF]) and the Cook Islands (N. Hauser, unpubl. data available at http://www.whaleresearch. org/). Strandings of Blainville's beaked whales have also been reported from Queensland [START_REF] Paterson | A humpback whale calf and two subadult dense-beaked whales recently stranded in southern Queensland[END_REF]). To our knowledge, there is only one previous mention of Blainville's beaked whale in New Caledonia, in the form of ''teeth . . . recovered from a reef flat'' [START_REF] Garrigue | Cetacean records for the New Caledonian area (Southwest Pacific Ocean)[END_REF]. No other details were given by those authors on the material or on the circumstances of the finding. Here we present all the information we have collected so far on Ziphiidae in New Caledonia. This comes from two strandings, one live sighting, and the remains (rostra) of two sunk individuals. All the data and material refer to

M. densirostris.

The first known stranding of a Ziphiidae in New Caledonia occurred in November 1997 at Ile Ouen (22°28'S, 168°48'E), a coastal island located within the southern coral-reef lagoon of New Caledonia's Grande Terre. Locals noticed the two protruding teeth of the animal, which they called ''cornes de poissons'' (''fish horns'') because they were positioned above the head. The two teeth were sold to a curio shop and are currently owned by a private collector in Nouméa. The shape of the teeth (Figure 1) unambiguously pointed to a male Mesoplodon densirostris [START_REF] Ross | The smaller cetaceans of the south east coast of southern Africa[END_REF][START_REF] Mead | Beaked whales of the genus Mesoplodon[END_REF][START_REF] Miyazaki | Teeth[END_REF]. These are the only remains known from this individual because the carcass was, reportedly, intentionally destroyed by fire.

The second stranding occurred on 7 June 2001 at Saint-Louis on the southwestern coast of New Caledonia's Grande Terre (22°14'S, 166°33'E) and was reported by the local daily newspaper, although no identification to species was given (Anonymous 2001). Only a few pictures and a sample of skin were saved from this stranding. The carcass was buried on the beach but the head was cut off and separately buried at Ateliers Municipaux in Mont-Dore. Unfortunately, the precise location of burying has since been lost (Anonymous 2002). On the day it was discovered, the animal was already dead and swollen and bore numerous round scars, presumably caused by cookie-cutter sharks, Isistius sp. Most of this scarring was fresh and was located on the back of the animal, suggesting that it was adrift offshore, floating upside down, for some time before stranding. Close-up pictures of the anterior part of the body (Figure2) show the exaggerately arched mandible characteristic of an adult Mesoplodon densirostris [START_REF] Leatherwood | The Sierra Club handbook of whales and dolphins[END_REF]. No erupted tooth was visible, indicating that the individual was a female or an immature male. The identification of the individual as M. densirostris was subsequently confirmed by nucleotide-sequence analysis of a portion of the control region of the mitochondrial DNA (M. L. Dalebout, pers. comm.).

The only live sighting of a beaked whale so far reported for New Caledonia occurred on 9 August 2003 off the eastern coast of New Caledonia's Grande Terre (21°40'S, 166°50'E) and consisted of a loose pod of five animals swimming near or at the surface of the sea. The sea was calm and the weather was sunny. The pod was first spotted from a helicopter cruising at an altitude of ca. 330 m and subsequently observed from an altitude of ca. 50 m for one more surfacing, when all five individuals sounded simultaneously (L. Farines, pers. comm.). A photograph among the series taken then by the helicopter's crew shows the back and the right profile of an individual of the pod (Figure 3). This whale was identified as a Blainville's beaked whale (and excluded from all other known beaked whales) by its flat forehead and typically arched maxilla topped at its middle by a white bulge [START_REF] Leatherwood | The Sierra Club handbook of whales and dolphins[END_REF][START_REF] Cawardine | Eyewitness handbook:Whales, dolphins and porpoises[END_REF], features that were both described by the helicopter's crew (R. Barthe and L. Farines, pers. comm.) and visible on Figure 3 despite the poor quality of the details on this picture. This individual and at least two other individuals in the pod bore numerous pale scars on the back and flank, attributable to cookie-cutter sharks. Pale blotches were visible at the base of the dorsal fin and at the posterior part of the jaw (Figure 3). The back and flanks were beige or brown and the dorsal side of the caudal fin was dark brown. The whitish belly was clearly delineated from the flank along a concave demarcation line running approximately from the anal region to the flipper. The dorsal fin was moderately tall and falcate. Although this individual was not ''black or charcoal gray on the back, slightly lighter on the abdomen'' as described in [START_REF] Leatherwood | The Sierra Club handbook of whales and dolphins[END_REF] and other identification guides (e.g., [START_REF] Watson | Whales of the world[END_REF][START_REF] Jefferson | FAO species identification guide: Marine mammals of the world[END_REF][START_REF] Cawardine | Eyewitness handbook:Whales, dolphins and porpoises[END_REF], the coloration and pigmentation patterns shown in Figure 3 match those of an individual in a pod of Blainville's beaked whales that was filmed in the Bahamas in 1998 (N. Hauser and H. Peckham, unpubl. data).

Two rostra of mesoplodont whales were collected during dredging of the summit of Seamount 1 (25°13'S, 170°22'E; depth, 900-925 m) on 23 November 1993 during the BATHUS III cruise of RV Alis (Richer de Forges and Chevillon 1996). These two rostra, of bone of an extreme density, were stained dark, a coloration presumably caused by a film of ferromanganese deposited on their surface.

The two rostra have been deposited at Museum National d'Histoire Naturelle, Paris, under registration numbers CG 2003-439 and CG 2003-440. Specimen CG 2003-439 was 540 mm long and weighed 3.7 kg;specimen CG 2003-440 was 404 mm long and weighed 2.2 kg. The eroded state of rostrum CG 2003-440, apparently due to drilling or boring invertebrates, suggests that it had lain on the sea floor for years, although from the information currently available it is not possible to estimate its age.

Rostrum CG 2003-439 was almost intact and bore only a few fixed invertebrates, including small bryozoans colonies and small serpulid polychaetes. This suggests that this specimen was very recent, perhaps only a few months or a few years old, when it was collected. Taken together, the following features are diagnostic of the rostrum of Mesoplodon densirostris: it is deeper than wide; in lateral view, the dorsal profile is approximately rectilinear, the ventral profile is slightly convex, and a lateral maxillary ridge is present, forming a smooth sigmoid curve; in dorsal and ventral views, a swelling of the median portion is noticeable (Van Beneden and Gervais 1868-1880[START_REF] Harmer | On Mesoplodon and other beaked whales[END_REF][START_REF] Raven | On the structure of Mesoplodon densirostris, a rare beaked whale[END_REF]

Figure 1 .

 1 Figure 1. Mesoplodon densirostris. Teeth of specimen stranded at Ile Ouen in the southern lagoon of New Caledonia (November 1997; 22°28'S, 168°48'E).

Figure 2 .

 2 Figure 2. Blainville's beaked whale stranded at Saint-Louis (22° 14'S, 166°33'E), southern lagoon of New Caledonia ( June 2001). (Photograph courtesy of C. Ledru.)

Figure 4 .

 4 Figure 4. Mesoplodon densirostris. Rostrum dredged on summit of seamount 1 (23°53'S, 169°46'E, depth = 625-640m) during cruise BATHUS III of R.V. Alis. Muséum National d'Histoire Naturelle, Paris, specimen no. CG 2003-339.

 Moore 1966[START_REF] Mead | Beaked whales of the genus Mesoplodon[END_REF]). The rostrum of M. densirostris is also remarkable by its extreme density (de Buffrénil and Casinos 1995). All the foregoing characteristics were present in both specimens CG 2003-439 (Figure 4) and CG 2003-440, despite the eroded state of the latter, allowing their identification as M. densirostris. In addition to these characteristics, specimen CG 2003-439 was also remarkable by the prominence of its lateral maxillary ridge, the widening of its median and posterior parts, and a relatively strong development of the posterior portion of the mesorostral bone (Figure 4). The latter peculiarities resemble the changes that occur in old male Ziphius cavirostris (Fraser 1942, Robineau andDi Natale 1995), although the rostrum of Z. cavirostris cannot be confused with that of M. densirostris, because the former is wider than deep and narrows steadily from base to tip (e.g., Jefferson et al. 1993). We therefore propose that rostrum CG 2003-439 may be that of an old male M. densirostris.

The marine mammal fauna of the New Caledonia region in the southwestern Pacific is still poorly known (Bannister et al. 1996, Reeves et al. 1999, Garrigue and Greaves 2001). Based on the proximity of sightings, strandings, and catches, New Caledonian waters potentially lie within the distributional range of several Ziphiidae species, including Mesoplodon densirostris; Layard's beaked whale, M. layardii; Cuvier's beaked whale, Ziphius cavirostris; Longman's beaked whale, Indopacetus pacificus; and perhaps Japanese beaked whale, M. ginkgodens (Paterson and Van Dyke 1990, Reeves et al. 1999, Pitman 2002, Rudolph and Smeenk 2002, and references therein). Materials and data on Ziphiidae in the tropical southwestern Pacific include one skull of I. pacificus from Queensland (the holotype of the species [Longman 1926]); strandings of M. densirostris, M. layardii, and Z. cavirostris in Queensland (Paterson and Van Dyke 1990, Paterson et al. 1993); one stranding and several sightings of M. densirostris and at least one stranding of Z. cavirostris in the Cook Islands (N. Hauser, unpubl. data); a specimen of M. densirostris stranded in New Britain (Reeves et al. 1999); and strandings and a few sightings and skulls of Z. cavirostris from New Ireland (Hale 1931), New Britain, and the Solomon Islands (Reeves et al. 1999). The data on Ziphiidae collected so far in New Caledonia refer to M. densirostris only (Garrigue and Greaves 2001; this report). Considering that the reports of M. densirostris given here concern both stranded or sunk animals and a live sighting, this may reflect higher abundance of this species in the New Caledonian archipelago and adjacent waters than other