Mette Stoedter

Kostja Renko

Antonia Hoeg

Lutz

Antonia Hög

Dr Lutz Schomburg
email: lutz.schomburg@charite.de.

Selenium controls sex-specific immune response and selenoprotein expression during the acute phase response in mice

Keywords: trace element, LPS, translation, critical illness, sexual dimorphism short Sex-specific selenoprotein regulation in acute phase ERAD, ER-associated protein degradation, GPX, glutathione peroxidase (human), Gpx, glutathione peroxidase

come

Selenium controls sex-specific immune response and selenoprotein expression during the acute phase response in mice

INTRODUCTION

Selenium (Se) is an essential trace element needed for the biosynthesis of a small number of mammalian selenoproteins [START_REF] Hatfield | Selenocysteine incorporation machinery and the role of selenoproteins in development and health[END_REF]. Se intake and personal Se status are implicated in wide-spread human pathologies including cancer, cardiovascular disease or neurodegeneration [START_REF] Papp | From selenium to selenoproteins: synthesis, identity, and their role in human health[END_REF]. Recent research has specifically highlighted the importance of Se status for the activity of the immune system [START_REF] Broome | An increase in selenium intake improves immune function and poliovirus handling in adults with marginal selenium status[END_REF][START_REF] Sheridan | Decreased selenoprotein expression alters the immune response during influenza virus infection in mice[END_REF][START_REF] Hoffmann | The influence of selenium on immune responses[END_REF]. Positive effects of Se supplementation have been observed in a number of clinical trials with patients suffering from sepsis [START_REF] Angstwurm | Selenium in Intensive Care (SIC): results of a prospective randomized, placebo-controlled, multiple-center study in patients with severe systemic inflammatory response syndrome, sepsis, and septic shock[END_REF], human immunodeficiency virus infection [START_REF] Hurwitz | Suppression of human immunodeficiency virus type 1 viral load with selenium supplementation: a randomized controlled trial[END_REF] or autoimmune thyroid disease [8]. The effects of Se are mediated by selenoproteins, a notion which is supported by association studies linking specific selenoprotein genotypes to human diseases [9,10]. To assess an individual's Se status, glutathione peroxidase activity in blood cells (GPX1) or plasma (GPX3) and total Se or circulating selenoprotein P (SEPP) concentrations are measured, respectively [11]. SEPP is mainly derived from liver and seems to represent the best Se marker for it accounts for the majority of circulating Se in blood and responds over a broad range to Se intake [12]. Serum Se is an established negative acute phase reactant and declining serum Se levels correlate to low circulating SEPP in sepsis [13,14]. Most importantly, the lower the Se status during critical illness the more likely the patients will not survive [START_REF] Angstwurm | Selenium in Intensive Care (SIC): results of a prospective randomized, placebo-controlled, multiple-center study in patients with severe systemic inflammatory response syndrome, sepsis, and septic shock[END_REF][15][16][17]. Some but not all clinical trials have proven effective to improve outcome of critically ill patients by Se supplementation but the best application regimen, most suitable Se compound and the mechanisms of action are a matter of discussion [18][19][20]. One possible molecular component linking Se status with the immune system was recently identified when single nucleotide polymorphisms in the human gene encoding selenoprotein S (SEPS) were correlated to serum concentrations of proinflammatory cytokines, e.g., IL-6, IL-1 and TNF- [21]. SEPS encodes an essential factor for retrotranslocation of misfolded proteins from the endoplasmic reticulum (ER) into the cytosol for proteasomal degradation (ERAD, ER-associated protein degradation) [22,23]. Hereby, SEPS is connected to a number of physiological pathways that modulate inflammation, e.g., NF B and JNK activation and induction of acute-phase proteins [24]. Insufficient Se supply might cause impaired SEPS expression leading to reduced ERAD activity and cell dysfunction upon ER stress. Accordingly, proinflammatory cytokines, glucose deprivation or chemically induced ER stress induce SEPS gene expression in vitro [25][26][27]. A molecular circuit can be envisaged in which inflammatory stimuli induce SEPS biosynthesis as a positive acute phase reactant to ensure efficient ERAD activity, stress relaxation and prevention of apoptosis [26]. We wondered about hepatic SEPS regulation in vivo during the acute phase response since the hepatic Se status and hepatic enzymes involved in selenoprotein biosynthesis are reduced under these conditions [28]. Moreover, we wanted to test the suitability of a short-term Se supplementation strategy as a preventive measure to positively affect the cytokine response pattern and increase selenoprotein expression in order to prepare an individual for an impending sepsis thread. This scenario can be envisaged for a regular patient in the clinics in face of e.g. a surgical intervention [29]. To this end, we have analysed the endotoxin (LPS)-based murine model of septic shock and studied both male and female mice for their regulation of selenoprotein expression differs between the sexes [30]. Our findings indicate mitigating effects of Se supplementation on the LPS-dependent rise of certain cytokines in blood especially in male mice. This response is paralleled by specific changes in selenoprotein expression in hepatocytes in favour of improved ER function during the acute phase and at the expense of SepP production. We conclude that availability of Se is of prime importance for hepatic SepS expression in vivo. This connection might help to explain some of the discrepant findings on the given or missing associations of SEPS genotype with inflammatory diseases in recent clinical studies for these analyses failed to correct for the actual Se status of the probands.

EXPERIMENTAL

Materials

All chemicals were of analytical grade and obtained from Sigma (München, Germany). Oligonucleotides were from Invitrogen, restriction endonucleases were from New England Biolabs (Frankfurt, Germany) and Taq polymerase was from Qiagen (Hilden, Germany).

Animal experimentation

Wild-type C57BL/6J mice were obtained from Charles River (Sulzfeld, Germany) and were bred on regular lab chow (Altromin, Lage, Germany). A time-course analysis of selenoprotein expression after a single injection of LPS was conducted as described before [28]. In addition, transgenic mice on C57BL/6 background carrying a removable gene for the selenocysteine-specific tRNA (floxed Trsp) have been generously supplied by D.L. Hatfield et al. [31] and were bred, genotyped and studied as described [32]. In these mice, selenoprotein expression can become specifically impaired in hepatocytes by crossing these mice with a strain expressing Cre recombinase under an albumin-promoter (Alb-Cre) [33]. In a third experiment, groups of male and female mice (n=6 per group) were transferred at the age of 45 days (P45) to a Se-poor diet (C1045, Altromin) containing 0.1 ppm Se on average. Then, they received either regular tab water or 100 µM Na 2 SeO 3 in tab water for three days. 24 h before analysis, all of these mice were injected intraperitoneally once with sterile saline or 100 µg LPS per animal (E. coli 55:B5, Sigma) as shown in Fig. 2. Necroscopy, tissue homogenization, RNAisolation, Northern and Western blot analyses were performed as described [28]. The experimental animals received humane care in compliance with the "Guide for the Care and Use of Laboratory Animals" prepared by the National Academy of Sciences, and experiments had been approved by the local governmental authorities in Berlin, Germany.

Enyzme assays

Hepatic Gpx and thioredoxin reductase (TxnRd) activities were determined as described [30,34]. Briefly, livers were homogenized on ice in 250 mM sucrose, 20 mM HEPES, and 1 mM EDTA, pH 7.4 and cleared by centrifugation. Protein concentrations were determined in the supernatants by a modified Bradford assay using IgG as standard (Bio-Rad, Muenchen, Germany). NADPH consumption by GSH reductase was recorded at 340 nm as a measure of oxidized GSH formation in the Gpx reaction with t-butylhydroperoxide as substrate [30]. TxnRd activity was determined by monitoring the NADPH-dependent reduction of 5,5'-dithiobis-(2-nitrobenzoic acid, DTNB) determined as the increase in absorbance at 412 nm as described [34].

Determination of circulating cytokines

Aliquots of serum (10 l/sample) were analyzed by a multiplex analyzer system (Luminex 200, Luminex Corp., Austin, TX) in combination with LINCOplex cytokine immunoassays (Linco Research, St. Charles, MO) as described [30]. The tests for murine Il-6, Mcp1 and TNF-were conducted according to the manufacturer's instructions and calibration curves were generated with seven commercial standard samples for each cytokine covering the concentration range from 0.0 to 10,000 pg/ml. The intra-and inter-assay precision of these cytometric bead assays typically range below 10% [35].

Expression of recombinant SepS

SepS coding sequence was amplified from a murine liver cDNA library (f-primer: AAGCTTAGAGCAGAGAGGAGCAGC, r-primer: CTCGAGCCGCCAGATGATGGGC) with proper restriction sites for in-frame cloning into pcDNA3.1-V5-His-TOPO (Invitrogen). Correct sequence and tag position were verified by DNA sequencing. The resulting construct was transfected or not into human hepatoma Huh7 cells by Ca-phosphate. Cells were harvested 48 h after transfection. The resulting cell lysates containing or not recombinant SepS were used as positive and negative controls in the Western Blot analyses, respectively.

Western Blot Analyses

Serum samples were collected and protein homogenates were prepared from different tissues and separated by 10% (SepP) or 15% (SepS) SDS-PAGE before being transferred to a nitrocellulose membrane (Protran, Schleicher & Schuell, Dassel, Germany) as described [28]. Membranes were stained by Ponceau S (PonS) in order to control complete transfer and equal loading of the lanes. Relative serum albumin concentrations were determined by densitometric quantification of the major band at 67 kD. A polyclonal antiserum ( -SepS) was generated in rabbits by immunization with a synthetic SepS-specific peptide (Ac-GRSYKRNSGRPQEEDGPGC-amid, corresponding to residues 125-142 of murine SepS, Mus musculus histocompatibility H47 mRNA, NM_024439.3). Antibodies specific for murine SepP have been characterized before and were used as described [28]. V5-specific antibodies were purchased and used according to the manufacturer's instruction (Invitrogen).

Se determination by total reflection X-ray fluorescence analysis (TXRF)

Aliquots of tissue homogenates (100 µl) were digested with nitric acid (4 volumes, 65%) and spiked with an internal gallium standard (Sigma). Mouse serum samples (10 µl) were diluted with distilled water (40 µl) and spiked as above. Samples were applied to quartz glass carriers and left to dry. TXRF analyses were performed as recommended using a Picofox A c c e p t e d M a n u s c r i p t Licenced copy. Copying is not permitted, except with prior permission and as allowed by law.

Wisconsin, USA). Accuracy of Se determination was verified with a mouse serum pool and a specified human serum sample (Seronorm, Billigstad, Norway). The inter-assay coefficient of variation was below 10% for samples containing Se concentrations > 100 µg/l.

Statistical analysis

Data are expressed as mean ± SEM. Comparisons of two groups were done with Student's t-test. Differences between treatment groups were analyzed by one-way analysis of variance (ANOVA) and Dunnett's or Bonferroni post hoc tests for significance as indicated in the figure legends using Graphpad Prism v4.0 software (Graphpad Software, San Diego, CA, USA). Statistical significance was defined as p < 0.05 (*), as p < 0.01 (**) or as p < 0.001 (***).

RESULTS

Liver-specific induction of SepS during the acute phase response in vivo

SepS has been shown to be dynamically regulated by divergent stressors in vitro [25][26][27]. Therefore, we aimed to test whether SepS expression also changes during the acute phase response in vivo. To this end, a SepS-specific antiserum ( -SepS) was raised in rabbits and tested in Western Blot analyses. A single band of approx. 27 kD was detected from cells transfected with a recombinant tagged SepS protein but not in control cells (Fig. 1A). Identity of the band was verified by antibodies directed against the V5 tag of the recombinant SepS protein. In vivo, a single injection of LPS induced SepS expression in liver mainly on the protein level (Fig. 1B). Relative transcript concentrations of SepS were determined by densitometric analysis of the Northern blots and remained constant in relation to -actin signals ( -act.) during the study period of 24 h after LPS injection (Fig. 1B). In comparison to liver, no effects on SepS protein concentrations were observed in brain, spleen or testes of the same mice over the time period analysed (Fig. 1C). In order to identify the cell type responsible for the upregulation of SepS protein levels in liver, a transgenic mouse line with hepatocyte-specific genetic inactivation of selenoprotein expression was studied [36]. LPS had no effects on SepS protein concentrations in the livers of these mice (Fig. 1D). These results indicate that hepatocytes are responsible for the LPS-dependent induction of SepS in liver during the acute phase response.

Se-dependent sex-specific modulation of the immune response

Se has been shown to modulate the LPS-induced oxidative stress response in murine macrophages in vitro [37]. Moreover, mice with altered selenoprotein expression display a different cytokine response than controls upon influenza virus infection [START_REF] Sheridan | Decreased selenoprotein expression alters the immune response during influenza virus infection in mice[END_REF]. Yet the selenoproteins are sex-specifically expressed [30,38]. We thus hypothesized that both sex and Se are important parameters for selenoprotein regulation and immune response. Therefore, we conducted a short-term Se supplementation experiment to specifically elucidate the importance of Se availability for the cytokine response and hepatic SepS expression. To this end, male and female mice were raised on a Se-poor diet during the 3 weeks before challenge to develop a marginal Se status. Then, mice were supplemented or not during their last 3 days with sodium selenite (100 µM) via their drinking water and received a single injection of NaCl or LPS (100 µg/animal) 24 h before being sacrificed and analysed (Fig. 2). Circulating levels of Il-6, Mcp1 and TNF-were analyzed in the individual serum samples. The LPS-injected mice displayed strongly-increased cytokine concentrations compared to the control animals in both the male and female groups (Fig. 3). Se supplementation decreased the fulminant immune response in male mice; here, the LPS-induced TNF-response was decreased with borderline significance whereas the LPS-dependent inductions of Il-6 and Mcp1 were significantly decreased on average to 50% compared to the Se-poor male individuals. In the female groups, Se had no statistical significant effects on the LPS-induced cytokines under these conditions.

Efficiency of a short term Se supplementation regimen to increase the Se status

In order to test whether this short-term Se supplementation was effective to raise biomarkers of the Se status in the experimental mice, hepatic TxnRd, hepatic Gpx and serum Gpx3 enzyme activities and concentrations of hepatic SelS and serum SepP along with serum and hepatic Se concentrations were analysed. As expected, hepatic Gpx activities, serum and hepatic Se along with circulating SepP concentrations increased strongly in response to Se supplementation in both female and male mice (Fig. 4A-D). In contrast, hepatic TxnRd remained unaltered. The Se-dependent increase in serum Gpx3 activity was sex-specific under these conditions and statistically significant in males only (Fig. 4E-F). The strong effects on hepatic and circulating Se, hepatic Gpx activity and serum SepP concentrations indicate that application modus, dosage and time were sufficient to allow efficient Se uptake and improved hepatic selenoprotein biosynthesis and secretion under these conditions. Circulating SepP and serum Se turned out to represent robust markers of a short-term Se supplementation effect and faithfully mirrored changes in hepatic Se availability and hepatic Gpx enzyme activities while Gpx3 proved less suitable to monitor supplementation success under these conditions.

Comparison of different Se-dependent biomarkers during the acute phase response

A single sub-lethal LPS-injection represents a reliable model to analyse the acute phase response of Se and selenoproteins in experimental rodents [28,39]. As observed before, the inflammatory stimulus caused strongly declining serum Se and SepP concentrations but had only marginal effects on serum albumin concentrations (Fig. 4A-B). Hepatic SepP mRNA concentrations remained unaffected under these conditions (data not shown) as studied before in detail [28]. LPS-dependent down regulation of serum Se and SepP concentrations were especially pronounced in the supplemented mice with higher Se status. These results verify that serum Se and SepP represent robust negative acute phase reactants. Hepatic Se concentrations were similarly strongly affected by the LPS stimulus in both sexes (Fig. 4C). In comparison, LPS affected hepatic Gpx activity in Se-supplemented male mice only (Fig. 4D). TxnRd activities in liver were not significantly affected by LPS under these conditions (Fig. 4E). The activity of kidney-derived Gpx3 remained largely unaltered by LPS in the Se-supplemented or Se-poor male or female mice (Fig. 4F). These results are consistent with the different dynamics and half-lives of circulating SepP and Gpx3 [40]. Moreover, this finding is in line with the results from a recent clinical study highlighting circulating SEPP as the presumably best biomarker for Se metabolism in sepsis [14]. In summary, SepP as a mainly hepatic product appears to be directly and strongly affected while Gpx3 as a kidney-derived and SepP-dependent secondary parameter appears to respond only poorly to LPS and the decreasing serum Se and SepP concentrations.

Importance of sex and Se for SelS expression and LPS-dependent maximal induction

In general, ample Se as substrate is needed in the form of selenocysteine-loaded tRNA in order to support biosynthesis of selenoproteins. Accordingly, SepS protein levels increased significantly in Se supplemented mice as compared to Se-poor animals (Fig. 5). In contrast to the declining hepatic Se and circulating SepP concentrations [28], LPS injection increased SepS expression in liver in both male and female animals (Fig. 1B, D and Fig. 5). Notably, a robust induction was observed in the Sepoor animals indicating that certain mechanisms within the hepatocytes can support SepS expression even against overall low Se status in a prioritized manner. In male mice, SepS expression was strongly induced by LPS under Se-poor conditions but not under Se supplemented conditions under which SepS apparently already had reached peak concentrations (Fig. 5). In contrast, maximal SepS expression in females was only observed upon LPS stimulation and Se supplementation, indicating that female mice needed higher Se concentrations or longer supplementation times in order to obtain maximal SepS expression.

DISCUSSION

The acute phase response in sepsis is a life-threatening process aiming to reprogram the organism's metabolism for defence against invading pathogens. Under such conditions, e.g. serum iron becomes strongly reduced as a meaningful reaction to deprive the proliferating microbes from one of their essential trace elements [41]. Likewise, Se in serum is known to decline as a negative acute phase reactant [39,42]. The physiological purpose for this phenomenon is currently unknown but the absence from the circulation might again deprive the pathogens from an essential component [43]. Alternatively, it might improve the ability of immune cells to fight invaders by e.g. oxidative burst processes whose effects would otherwise be dampened by selenoproteins [44]. These theories still need experimental proof but are in line with the reduced serum Se and SEPP concentrations under inflammatory conditions in humans [START_REF] Angstwurm | Selenium in Intensive Care (SIC): results of a prospective randomized, placebo-controlled, multiple-center study in patients with severe systemic inflammatory response syndrome, sepsis, and septic shock[END_REF][13][14][15][16]] and declining serum Se and SepP levels in experimental rodents [28,39]. But what processes control the negative acute phase responses of serum Se and SEPP? Comparing LPS-dependent SepP decline in serum of Se-supplemented and Se-poor mice (Fig. 3B), it becomes obvious that a loss of SepP into other tissues or an extensive binding onto the surface of the blood vessels is unlikely. Otherwise a stronger and more uniform decline would have been observed, especially in the male and female animals with low Se status. Since SepP expression depends on the Se status, the declining Se concentrations in liver (Fig. 4C) argue more in favour of a reduced hepatic SepP biosynthesis rate during the acute phase response. This notion is in line with the down-regulation of important factors involved in hepatic SepP biosynthesis under inflammatory conditions [28] and the rapid and specific degradation of another hepatic selenoprotein, i.e., Sep15, during acute ER stress [45]. These findings imply that the reducing Se concentrations in serum rather represent a side-effect secondary to an altered metabolic program in liver. This interpretation is corroborated by the better survival odds of sepsis patients with high Se status and with the positive effects of Se supplementation in critically ill patients [START_REF] Angstwurm | Selenium in Intensive Care (SIC): results of a prospective randomized, placebo-controlled, multiple-center study in patients with severe systemic inflammatory response syndrome, sepsis, and septic shock[END_REF]15]. In this light, the concurrently rising SepS concentration in liver as a positive acute phase reactant might offer a causal explanation on the fate of the missing serum Se especially in Se-poor organisms. Of note, the effects of LPS on the biosynthesis of SepP and SelS are elicited largely independent of their transcript concentrations. We have shown before that LPS impairs expression of central components of the selenoprotein biosynthesis machinery and reduces hepatic SepP production without affecting SepP mRNA concentrations [28]. Here, we report a similarly strong effect on SepS expression in liver, albeit in the opposite direction, i.e., increased protein levels again without respective LPS-effects on SepS transcript concentrations (Fig. 1B). Moreover, using the transgenic mice with cell-type specific impaired selenoprotein biosynthesis, both effects proceed simultaneously in the same cells, i.e., in hepatocytes. Obviously, hepatocytes shuttle their available Se into SepS and away from the biosynthesis of other less vital selenoproteins during the acute phase response. Such a posttranslational regulatory circuit nicely complements the diverse molecular pathways known which finally translate ER stress into the inflammatory response, i.e., rearranging the protein biosynthesis pattern via eIF2 phosphorylation, translational frame-shifting and activation of XBP1 and proteolytic generation of nuclear ATF6-p50 [24]. In contrast to these effects which relay a stress signal from ER into cytoplasm and nucleus, increased biosynthesis of SepS directly improves functioning of the critical cellular compartment, i.e. ER, via its important role during the removal of misfolded proteins during ERAD [25,27,46]. It is the nature of SepS as a selenoprotein to depend on a sufficiently high Se status needed for increased biosynthesis. Apparently, the redirection of Se away form other selenoproteins towards SepS expression fulfils this demand both under adverse circumstances as is the case in a poorly supplied individual and during the acute phase response. The concurrent decline of serum Se might represent the price for the prioritized hepatic SepS biosynthesis under inflammatory conditions, i.e., a side-effect. But an efficient selenoprotein biosynthesis usually profits from increased Se supply. Such support can be provided by a respectively tailored immunonutrition [47]. The positive effects of Se supplementation in the clinics imply that a readily available Se source, e.g. sodium selenite, improves acute phase response and outcome [START_REF] Angstwurm | Selenium in Intensive Care (SIC): results of a prospective randomized, placebo-controlled, multiple-center study in patients with severe systemic inflammatory response syndrome, sepsis, and septic shock[END_REF]. While the best regimen and selenocompound in terms of application route and dosage is a matter of discussion, a low Se status during septicaemia has uniformly been proven as a negative prognostic marker for survival odds [48,[START_REF] Berger | Antioxidant supplementation in sepsis and systemic inflammatory response syndrome[END_REF]. Therefore, we have attempted a short term preventive Se supplementation effort aiming to improve Se status and selenoprotein biosynthesis fast via oral application in our experiments. The data clearly indicate that this effort was successful and especially the cell-type specific up-regulation of SepS in hepatocytes relied stringently on the Se status. Both male and female mice took advantage of an increased Se supply in order to maximize SepS expression albeit with sexual dimorphic dynamics. In response to supplementation, maximal SepS levels were observed already under basal conditions in male but not in female mice. Likewise, concentrations of circulating cytokines were mitigated by Se especially in male mice. Even though these mechanistic insights have been deduced from rodent models and not human patients, the interrelation of Se status and SepS expression might help to explain divergent findings in genetic association studies of SEPS genotype and inflammatory diseases. In this respect, strong A c c e p t e d M a n u s c r i p t Licenced copy. Copying is not permitted, except with prior permission and as allowed by law.

associations were reported between the well characterized -105 SEPS promoter polymorphism [21] and gastric cancer susceptibility [START_REF] Shibata | Selenoprotein S (SEPS1) gene -105G>A promoter polymorphism influences the susceptibility to gastric cancer in the Japanese population[END_REF], coronary heart disease and stroke [51] and preeclampsia [START_REF] Moses | Genetic association of preeclampsia to the inflammatory response gene SEPS1[END_REF]. Together with the initial characterization of the interrelation of SEPS and inflammatory cytokines [21], a general effect of SEPS genotype on inflammatory diseases had been assumed. Therefore, a lack of SEPS genotype-disease association as recently reported for type 1 diabetes, rheumatoid arthritis, inflammatory bowel disease or ulcerative colitis [START_REF] Martinez | Polymorphisms in the selenoprotein S gene: lack of association with autoimmune inflammatory diseases[END_REF] was surprising. Here, the generalized conclusion might be misleading since the data have not been corrected for by sex and the actual Se status of the probands. Our results highlight the importance of these parameters for immune response, cytokine patterns and SEPS induction. Congruent data on the interaction of genotype, sex and Se status were recently reported for SEPP isoform expression in human cancer [START_REF] Meplan | Relative abundance of selenoprotein P isoforms in human plasma depends on genotype, se intake, and cancer status[END_REF]. Since the observed inflammatory effects on SEPP and SELS biosynthesis are mainly exerted on the post-transcriptional level, the Se status and the concentration of selenocysteine-loaded tRNA are decisive parameter for the diseasespecific response. According to our results, it seems advisable to determine the Se status along with the SEPS genotype in future studies to be better able to compare the genotype-disease association in men and women and gain knowledge on the potential benefit of a personally tailored Se supplementation in a specific pathology. The herein proposed preventive measure to increase the Se status fast via an orally available supplement might prove useful in the clinics to ensuring optimal physiological ER stress response by high SEPS expression prior to a potential challenge. At the same time the unfavourable low circulating Se concentrations can be avoided. Respective clinical studies taking the Se status and sex into account appear necessary and are highly recommended. Experimentation scheme to study Se-dependent acute phase response Male and female mice were bred on regular diet and transferred to a Se-poor diet (0.1 ppm Se) at the age of 45 days (P45) on which they remained for the next 3 weeks. Then, groups of animals (n=6 per sex) were supplemented or not with selenite in drinking water (100 µM) for 2 days prior to NaCl (control) or LPS injection (100 µg/animal, i.p.) yielding 4 experimental groups (far right). Acute phase responses of cytokines, Se and selenoproteins were studied 24 h after injection of NaCl or LPS.

Figure 3

Inflammatory cytokines in control and Se supplemented mice after LPS injection Serum samples from control and experimental female (left row) and male (right row) mice (n=6 per group) were analysed for circulating concentrations of Il-6, Mcp1 and TNF-. The LPS injection increased serum cytokine concentrations in mice independent of sex. Supplementation of Se mitigated the profound immune response in male but not in female mice, especially pronounced in case of Il-6 and Mcp1. *P <0.05, **P <0.01 (Student's t-test).

Figure 4

Biomarkers of Se status after Se supplementation and LPS injection (A) Selenite supplementation increased serum Se concentrations in male and female mice (n=6 per group). LPS injection reduced serum Se strongly in the Se-supplemented mice of both sexes while the effect was not significant in Se-poor animals. (B) Serum levels of SepP increased after Se supplementation as determined by Western blot analysis with antibodies against murine SepP ( -SepP). Ponceau S (PonS) staining (top) verifies constant loading of the serum samples and comparable serum protein amounts in the lanes. Western result of pooled serum samples is shown (top). Quantification of the effects (bottom) was based on additional Western blot analyses with individual samples (n=6 per group). A marginal but significant effect of LPS was observed on albumin concentrations in serum of male mice (*P <0.05, Student's t test). Albumin concentrations in female mice were not affected. LPS caused a significant decline in circulating SepP especially in the Sesupplemented male and female mice. **P <0. Licenced copy. Copying is not permitted, except with prior permission and as allowed by law.

  S2 instrument (Bruker AXS Inc. Madison, Biochemical Journal Immediate Publication. Published on 06 Apr 2010 as manuscript BJ20091868 THIS IS NOT THE VERSION OF RECORD -see doi:10.1042/BJ20091868

Biochemical

  Journal Immediate Publication. Published on 06 Apr 2010 as manuscript BJ20091868 THIS IS NOT THE VERSION OF RECORD -see doi:10.1042/BJ20091868

Figure 1

 1 Figure 1 Tissue-specific acute phase regulation of SepS expression in vivo (A) The new SepS-antiserum yields strong signals in cells positively transfected with recombinant V5tagged SepS (lanes 2-5) but not in control cells (lanes 1, 6). Position of immunoreactive SepS colocalized with V5-signals and the theoretical size of the recombinant protein verifying suitability of the antibodies for SepS analysis. (B) Time-course analysis of hepatic SepS protein (top) and mRNA (bottom) concentrations after LPS-injection. Male mice (n = 4 per group) received a single injection of saline (C) or LPS (100 µg/mouse) and were sacrificed after 6 h (all controls) or after the indicated periods of time. Samples were analyzed by two Western and two Northern blots of which a representative result is shown. Liver protein extracts from the individual mice were analyzed by Western blotting (top) and SepS amounts were quantified by densitometry. Ponceau S (PonS) staining served to control for efficient protein transfer onto the membrane and equal loading of the lanes. RNA (bottom) was isolated from the same livers and used for Northern analysis of SepS and -actin ( -act). *P <0.05, **P <0.01 (one-way ANOVA with Dunnett's post hoc test). (C) Tissue-specificity of LPS effects on SepS up-regulation. Protein extracts (n=3) from brain, spleen and testes of the experimental mice in (B) were analyzed by Western blot at different time intervals after injection of LPS. Recombinant SepS served as positive control (+). No significant effects of LPS on SepS protein concentrations were observed in these tissues. (D) Effects of LPS on SepS expression in wild-type (wt) and transgenic (KO) mice. Protein extracts from livers of treated (LPS) and control (C) mice were analyzed by Western blot. Recombinant SepS protein served as positive control (+). Transgenic mice (n=5) with albumin-promoter driven Cre expression causing genetic deletion of the Sec-specific tRNA in hepatocytes express little SepS compared to wt (n=5) animals (wt C vs. KO C). SepS expression is not stimulated by LPS in livers of KO mice (n=3) compared to saline-injected KO controls (n=2) indicating that hepatocytes account for increased SepS expression in liver of wt mice during an acute phase response in vivo. **P <0.01 (one-way ANOVA with Bonferroni's post hoc test).

Figure 2

 2 Figure 2Experimentation scheme to study Se-dependent acute phase response Male and female mice were bred on regular diet and transferred to a Se-poor diet (0.1 ppm Se) at the age of 45 days (P45) on which they remained for the next 3 weeks. Then, groups of animals (n=6 per sex) were supplemented or not with selenite in drinking water (100 µM) for 2 days prior to NaCl (control) or LPS injection (100 µg/animal, i.p.) yielding 4 experimental groups (far right). Acute phase responses of cytokines, Se and selenoproteins were studied 24 h after injection of NaCl or LPS.

  Figure 4Biomarkers of Se status after Se supplementation and LPS injection (A) Selenite supplementation increased serum Se concentrations in male and female mice (n=6 per group). LPS injection reduced serum Se strongly in the Se-supplemented mice of both sexes while the effect was not significant in Se-poor animals. (B) Serum levels of SepP increased after Se supplementation as determined by Western blot analysis with antibodies against murine SepP ( -SepP). Ponceau S (PonS) staining (top) verifies constant loading of the serum samples and comparable serum protein amounts in the lanes. Western result of pooled serum samples is shown (top). Quantification of the effects (bottom) was based on additional Western blot analyses with individual samples (n=6 per group). A marginal but significant effect of LPS was observed on albumin concentrations in serum of male mice (*P <0.05, Student's t test). Albumin concentrations in female mice were not affected. LPS caused a significant decline in circulating SepP especially in the Sesupplemented male and female mice. **P <0.01 (Student's t-test). (C) Se concentrations in livers mirror changes in circulating SepP upon selenite and LPS treatment. Se supplementation increased hepatic Se concentrations in all groups of mice. LPS injection caused a strong decline in hepatic Se

Figure 5

 5 Figure 5Sex-specific regulation of hepatic SepS in response to selenite and LPS treatment LPS injection induced maximal SepS concentrations in Se-supplemented mice indicating the importance of the Se status for SepS expression. A Western blot of pooled homogenates from the livers of the experimental mice is depicted (top) in order to provide an overview. SepS protein concentrations were determined from two additional Western analyses from samples of individual mice which are shown (bottom) together with the quantification (graph). Characteristic sex-specific differences were observed with respect to both basal SepS expression levels and responsiveness to Se and LPS. *P <0.05, **P <0.01, ***P <0.001 (Student's t-test).

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Acknowledgements

The excellent technical assistance of Anita Kinne, Birgit Hollenbach and Jazmin Chiu is gratefully acknowledged. We thank Drs. D.L. Hatfield and B.A. Carlson for generously sharing the transgenic mice, Drs. J. Köhrle, U. Schweizer, P.J. Hofmann and E.K. Wirth for stimulating discussions and help with the experiments, and Deutsche Forschungsgemeinschaft DFG [Scho849/2-2, GraKo1208], Charité Berlin [personal grant to LS] and Deutsche Krebshilfe [10-1792 SchoII] for financial support.


Audrey Finkler

Goodness of fit statistics for sparse contingency tables

Statistical data is often analyzed as a contingency table, sometimes with empty cells called zeros. Such sparse tables can be due to scarse observations classified in numerous categories, as for example in genetic association studies. Thus, classical independence tests involving Pearson's chi-square statistic Q or Kullback's minimum discrimination information statistic G cannot be applied because some of the expected frequencies are too small. More generally, we consider goodness of fit tests with composite hypotheses for sparse multinomial vectors and suggest simple corrections for Q and G that improve and generalize known procedures such as Ku's. We show that the corrected statistics share the same asymptotic distribution as the initial statistics. We produce Monte Carlo estimations for the type I and type II errors on a toy example. Finally, we apply the corrected statistics to independence tests on epidemiological and ecological data.

Introduction and notations

Physical, sociological or biological surveys often lead to data presented as contingency tables. These surveys aim at studying relationships such as total, mutual, partial or conditional independence between several characters in a population. Table notations are very useful to formulate the test and make the corresponding hypotheses explicit, but can be cumbersome when the number of characters exceeds three. For theoretical results, we therefore use vector notations instead, and we reformulate the independence test as a multinomial goodness of fit test.

Goodness of fit tests

Let p = (p 1 , . . . , p R ) be a probability vector of dimension R where R 2 is the total number of cross-classifying categories. Let n be the sample size and x = (n 1 , . . . , n R ) the vector of observed frequencies, realization of X = (N 1 , . . . , N R ) with distribution M(n; p), multinomial distribution with parameters n and p. We denote by p 0 the probability vector under the null 1 hypothesis and consider the following test : H 0 : p = p 0 against H 1 : p = p 0 .

(1.1)

Popular goodness of fit statistics include Pearson's chi-square statistic Q and Kullback's minimum discrimination information statistic G, defined in [START_REF] Good | Probability and the weighing of evidence[END_REF][START_REF] Pearson | On the criterion that a given system of deviations from the probable in the case of a correlated system of variables is such that it can be reasonably supposed to have arisen from random sampling[END_REF]. For two probability distributions p and p ′ , these are written :

Q p (p ′ ) = n R r=1 (p ′ r -p r ) 2 p r , (1.2)
and

G p (p ′ ) = 2n R r=1 p ′ r ln p ′ r p r . (1.3)
They belong to the power divergence statistics family {RC λ , λ ∈ R} defined by Read and Cressie in [START_REF] Cressie | Multinomial goodness-of-fit tests[END_REF], respectively for λ = 1 and λ → 0, where :

RC λ p (p ′ ) = 2n λ(λ + 1) R r=1 p ′ r p ′ r p r λ -1 . (1.4)
For a review on goodness of fit testing methods and statistics, see [START_REF] Bishop | Discrete multivariate analysis: theory and practice[END_REF]. The vector p 0 is not always completely specified. We assume that p 0 is a known function of an unknown parameter θ of Θ ⊆ R s with s < R -1, and denote p 0 = p 0 (θ) with θ = (θ 1 , . . . , θ s ). We suppose that the functions θ → p 0 (θ) we consider here are bijective and estimate p 0 (θ) by p * 0 = p 0 (θ * ) where θ * is the maximum likelihood estimator of θ. As the probability p is generally unknown, we also estimate the p r for r in {1, . . . , R} by their maximum likelihood estimators p * r = n r /n. Throughout this paper, underlying indexes n are omitted for simplicity of notation, and we consider the statistics Q p * 0 (p * ) and G p * 0 (p * ).

Read and Cressie show that under Birch's regularity conditions, see [START_REF] Birch | A new proof of the Pearson-Fisher theorem[END_REF], the statistics RC λ are asymptotically equivalent and share a common chisquare limit distribution:

Theorem 1. ∀λ ∈ R, lim n→+∞ L H 0 (RC λ p * 0 (p * )) = χ 2 R-s-1 . (1.5)
This implies the following result :

Corollary 1. lim n→+∞ L H 0 (Q p * 0 (p * )) = lim n→+∞ L H 0 (G p * 0 (p * )) = χ 2 R-s-1 . (1.6)
This asymptotic result is a consequence of the Central Limit Theorem. Classical empirical limitations include that the sample size n must be over 30 and that all expected frequencies must be over 5. Usually G p * 0 (p * ) is preferred to Q p * 0 (p * ) because it is less sensitive to small cell frequencies. Read and Cressie recommend the use of RC 2/3 p * 0 (p * ) instead for n 10 and a minimum expected frequency over 1. Sometimes, even the lower bound 0.5 is accepted for the expected frequencies. A review on these conditions can be found in [START_REF] Conover | Practical nonparametric statistics[END_REF].

Sparse tables

When there are too few subjects in the study or when the classifying categories are too numerous, the table comprises one or several empty cells called random zeros. The table is then called sparse and it is likely that at least one cell has an expected frequency below 0.5. Random zeros would not appear if the sample was of sufficient size. Structural zeros, corresponding to cells with an expected probability of zero, are not considered here and should be suppressed. We therefore assume that the following condition is satisfied :

p * 0 r = 0, r ∈ {1, . . . , R}. (1.7)
Sparse tables can nonetheless be tested for independence, for example by regrouping cells so that the condition on the expected frequencies is satisfied. However this procedure is not always data relevant. Fisher's exact test given in [START_REF] Fisher | On the interpretation of χ 2 from contingency tables, and the calculation of P[END_REF] applies without restrictions, except that it becomes numerically unmanageable when the table dimension grows. This leads to the use of Monte Carlo simulation methods as explained in [START_REF] Agresti | Categorical data analysis[END_REF].

The approach we propose here consists in correcting the historical statistics Q p * 0 (p * ) and G p * 0 (p * ) according to the number of zero cells, by generalizing and improving a method designed by Ku in [START_REF] Ku | A note on contingency tables involving zero frequencies and the 2 Î test[END_REF].

Corrections for Pearson's and Kullback's statistics

Let C be the random variable giving the number of zeros in the vector or the contingency table, and c a realization of C with R -c 1. For simplicity, we assume that n 1 = n 2 = • • • = n c = 0 and that n j 1 for all j in {c + 1, . . . , R}. The maximum likelihood estimator p * = (0, . . . , 0, n c+1 /n, . . . , n R /n) underestimates the p i for i in {1, . . . , c} and overestimates the p j for j in {c + 1, . . . , R}. Its use when c = 0 thus has consequences on the statistics Q p * 0 (p * ) and G p * 0 (p * ).

Ku's correction for one zero

Ku argues in [START_REF] Ku | A note on contingency tables involving zero frequencies and the 2 Î test[END_REF] that G p * 0 (p * ) tends to inflate with respect to Q p * 0 (p * ) when C grows. He then proposes to subtract 1 from G p * 0 (p * ) for each zero, that is c in total. He proves the asymptotic equivalence between G p * 0 (p * ) and its corrected version only for c = 1. I will explain why his reasoning is inconsistent. First, he considers a new statistic which is not a member of the power divergence family. Moreover, he uses the straightforward Lemma 1 to deduce the approximation:

2n r ln n r np * 0 r ≃ n 2 r -(np * 0 r ) 2 np * 0 r , (2.1)
for a = n r /n, b = p * 0 r and n r = 0, despite the fact that a = 0 and 1/a is not bounded.

Lemma 1. For each a, b > 0 such that a < 2b, b < 2a and the quantities 1/a and 1/b are bounded, we have :

ln a b = a 2 -b 2 2ab + o(a -b).
The expression on the left in (2.1) is null whereas the one on the right is negative. The sum over r of the left-hand side gives G p * 0 (p * ) and we recognize Q p * 0 (p * ) in the sum of the right-hand side. However, unlike Ku seems to think, zero and non-zero cells tend to compensate for each other and the approximation (2.1) can not be extended to the corresponding sum. There is therefore no behavior of G and Q that we can deduce from this. Finally, he illustrates this asymptotic result on a small sample of size n = 10.

We propose new corrections for both statistics, based on Ku's correction and a likelihood inequality that we present in the next subsection.

Likelihood inequality

Let us consider the following inequality coming from a likelihood reasoning : the sample vector we observe can be thought of more likely to happen than any other possible vector, since it is the one we actually observed. With exactly c zeros and R -c non-zero cells observed, so for all m c and n

′ j such that n ′ j n j , ∀j ∈ {c + 1, . . . , R} and n = m i=1 n ′ i + R j=c+1 n ′ j : P(N 1 = 0, . . . , N c = 0, N c+1 = n c+1 , . . . , N R = n R ) P(N 1 = n ′ 1 , . . . , N m = n ′ m , N m+1 = . . . = N c = 0, N c+1 = n ′ c+1 , . . . , N R = n ′ R ). (2.2)
We give in Proposition 1 a sufficient condition on the p r for (2.2) to be satisfied, and prove this statement in Appendix A.

Corrections

We propose an estimator p for p, different from the maximum likelihood estimator, with the following form :

pi = a, ∀i ∈ {1, . . . , c}, pj = n j n b -d, ∀j ∈ {c + 1, . . . , R}, (2.4
pab j = n j n b - ac + n 1-b -1 R -c , ∀j ∈ {c + 1, . . . , R}. (2.5)
Note that for c = 0, we fix a = 0 and b = 1.

Before considering the other conditions on pab , let us first give some notations. Let n be min j∈{c+1,...,R} {n j } and n be max j∈{c+1,...,R} {n j }. Let n stand for n -n(R -c) and n for n(R -c) -n. We dismiss the uniformly distributed case where :

n = n = n j = n R -c , ∀j ∈ {c + 1, . . . , R}, (2.6)
thus guaranteeing that n and n are positive. Let

b min = max 0, ln n/(R -1) ln(n) , ln(n) ln(n) , ln (n -n) ln(n) , (2.7
)

a min (b) = max 0, (n -n b )(R -c) + n b cn b , (2.8)
and

a max (b) = min 1, n b -n cn b , n b -n n b (n(R -c) + c) , (2.9)
Proposition 1 is applied to pab . Together with inequalities 0 < pab r < 1, for all r in {1, . . . , R} it is then equivalent to these conditions on a and b:

b min < b < b max = 1, (2.10)
and a min (b) < a < a max (b).

(2.11)

We want to make a practical choice among possible values of pab ensuring us that it is as far from p * as possible. We therefore fix b in (2.10) quite far from 1 as a convex combination of b min and b max with an empirical parameter h equal to 0.1. For this value of b we choose a near the upper limit of the interval in (2.11), that is :

b = hb max + (1 -h)b min and a = a max (b) -ǫ, (2.12)
where ǫ is a small constant designed to eliminate boundary effects.

The final expressions we get for the corrected statistics

Q ab = Q p * 0 (p ab ) and G ab = G p * 0 (p ab ) of Q = Q p * 0 (p * ) and G = G p * 0 (p * ) are: Q ab = n 2(1-b) Q -f (a, b), (2.13) with f (a, b) = n   1 -n 2(1-b) + 2n 1-b (ac + n 1-b -1) R -c R j=c+1 n j np * 0 j -a 2 c i=1 1 p * 0 i - ac + n 1-b -1 R -c 2 R j=c+1 1 p * 0 j   , (2.14)
and

G ab = n 1-b G -g(a, b), (2.15)
where

g(a, b) = 2n   ac + n 1-b -1 R -c R j=c+1 ln n j (R -c) -n b (ac + n 1-b -1) p * 0 j n b (R -c) -a c i=1 ln a p * 0 i -n 1-b R j=c+1 n j n ln n j (R -c) -n b (ac + n 1-b -1) n j n b-1 (R -c)   .
(2.16)

Convergence

In this paragraph, we show the convergence in distribution of Q ab and G ab to a chi-square distribution. Let us first study the parameter b. Proof. Inequalities n(R-c) < nR, n-n(R-c) < n and n-n < n show that all three components of the maximum defining b min are strictly less than 1.

Their order 1 developments as n tends to +∞ give their convergence to 1, with the quantities R -1, R -c and R -c -1 bounded.

We also study the asymptotic behavior of C, for once denoted C n , and state the following lemma which proof appears in Appendix A.

Lemma 2. The number of zeros C n converges almost surely to 0 as n tends to +∞.

We recalled in section 1 the convergence of Pearson's and Kullback's statistics to a chi-square distribution. In Theorem 2, we state a similar property for the corrected statistics Q ab and G ab . Theorem 2. Under Birch's regularity conditions in [START_REF] Birch | A new proof of the Pearson-Fisher theorem[END_REF], the estimator pab defined by (2.5), (2.10) and (2.11) is such that :

lim n→+∞ L H 0 (Q ab ) = lim n→+∞ L H 0 (G ab ) = χ 2 R-s-1 .
(2.17)

Proof. We deduce from Lemma 2 the existence of a set Ω ′ of probability 1 on which we can find a rank n 0 such that C n = 0 for n n 0 . The variable a is then set equal to 0 and the variable b equal to 1. Hence, the estimates p * and pab match, so that Q ab = Q and G ab = G. Theorem 1 then completes the proof.

The final two sections are dedicated to proving the relevancy of our corrections through simulations and real data analyses.

Simulations

In this section, simulations confirm the necessity to correct not only G but also Q. We compute the statistics Q, G, RC 2/3 = RC 2/3 p * 0 (p * ), Q ab and G ab on 1 000 vectors of length R = 100 of total frequency n = 400 for each of the four multinomial distributions defined in Table 1 by f 1 to f 4 . Let E r denote the expected frequencies for r in {1, . . . , 100}.

For each set of vectors sharing the same c, we give the quantile of order 1 -α = 95% for the five statistics considered. Results are displayed in ) figure 1, as well as a line indicating the chi-square quantile of order 1 -α = 95% with R-1 = 99 degrees of freedom χ 2 0.95,99 = 123.22. Only the center of each graph should be considered because the quantiles for extreme numbers of zeros are computed on too few observations, sometimes only on one or two among the 1 000 simulations in total.

Quantile values of Q tend to explode as c grows whereas G stays quite stable around χ 2 0.95,99 . This behavior is the opposite of the one predicted by Ku. For f 1 , statistics Q ab and G ab lead to the rejection of the null hypothesis. For f 2 to f 4 however, their quantiles lie below the critical line and H 0 is accepted. We thus have compensated for the rise of Q, and both our corrected statistics are stable.

This analysis is confirmed by the computation of empirical risks of type I for α = 0.01, 0.05 and 0.1 as showed in Table 2, and by the power study below. Probabilities f 1 to f 4 are perturbed into f ′ 1 to f ′ 4 such that for all j in {1, 2, 3, 4}:

∀i ∈ {1, . . . , 10}, f ′ j (i) = f j (i) + 1/300, (3.1) ∀i ∈ {11, . . . , 90}, f ′ j (i) = f j (i), (3.2
) ∀i ∈ {91, . . . , 100}, f ′ j (i) = f j (i) -1/300. (3.3)
Vectors are simulated with probabilities f 1 to f 4 , and goodness of fit for f ′ 1 to f ′ 4 is tested. The Tables 2 and3 show that the empirical type I risks are lower for our corrections compared to the classical statistics when c is quite important, whereas the empirical power is much higher for our corrections when c is small.

Applications

We apply the total independence test using the corrected statistics Q ab and G ab to two datasets involving two-dimensional tables. For such tables, the

Q Q ab G G ab RC 2 3
Figure 1: Quantiles of order 0.95 for Q, Q ab , G, G ab and RC 2/3 as functions of c, under null multinomial probabilities f 1 to f 4 , for 1 000 samples of size n = 400 and R = 100 categories. The line represents the threshold χ 2 0.95,99 .

Table 2: Empirical type I risks for Q, Q ab , G, G ab and RC 2/3 , for 1 000 samples of size n = 400 and R = 100 categories, at levels α = 0.01, 0.05, 0.1, for vector probabilities f 1 to f 4 . 3: Empirical powers for Q, Q ab , G, G ab and RC 2/3 , for 1 000 samples of size n = 400 and R = 100 categories, at levels α = 0.05 for simulated vector probabilities f 1 to f 4 , and null vector probabilities

α H 0 mode(c) Q Q ab G G ab RC 2/3 0.
f ′ 1 to f ′ 4 . H 0 H 1 mode(c) Q Q ab G G ab RC 2/3 f 1 f ′ 1 19 0.
H 0 : p ij = p i+ p +j against H 1 : ∃ (i 1 , j 1 ) ∈ I × J, p i 1 j 1 = p i 1 + p +j 1 , (4.1
) where p i+ and p +j are the marginal distributions for the two characters featured in the table. To ensure the condition (1.7) we remove the empty lines i of {1, . . . , I} such that n i+ = 0, and the empty columns j of {1, . . . , J} such that n +j = 0.

Multi-marker approach for Systemic Sclerosis

Table 4 is the diplotype table obtained from an association study in Humans looking for an association between three genetic markers on the gene TNFAIP3 and Systemic Sclerosis presented in [START_REF] Guedj | Association of TNFAIP3 rs5029939 variant with systemic sclerosis in European Caucasian population[END_REF]. Empty columns have been removed. A haplotype is the allelic distribution of markers on a chromosome, and a diplotype is the combination of both parental haplotypes. Diplotype tables, though more interesting than haplotypic tables because they take into account more information, are usually trickier to handle because they are sparse. Our corrected statistics can therefore be helpful in such situations.

Though nine haplotypes theoretically exist, only eight are observed, denoted H1 to H8, leading to 8 2 = 64 diplotypes Hi/Hj. Two samples are compared, affected versus sound subjects, on which we test the independence between the diplotype configuration and the health status of n = 794 individuals.

The table is of dimension 2×16, that is R = 32 categories with c = 1 zero and s = 16 parameters. There are exactly 16 expected frequencies below 5. The chi-square quantile χ 2 0.95,15 = 24.99 is compared to the statistics : Q = 14.62, Q ab = 20.76, G = 15.82, G ab = 28.43, RC 2/3 = 14.85. (r, p, e) Trophic level (0, 0, 0) (1, 0, 0) (0, 1, 0) (0, 0, 1) (1, 1, 0) (0, 1, 1)

Oligotrophic 0 0 3 0 3 2 Mesotrophic 2 1 0 2 1 0 Eutrophic 2 0 3 1 1 0
Only G ab leads to reject the null hypothesis of independence. This seems to be the right decision since it is confirmed by the single-markers approaches and haplotype tests in [START_REF] Guedj | Association of TNFAIP3 rs5029939 variant with systemic sclerosis in European Caucasian population[END_REF], all showing a significative association between the markers and the disease.

Trophic level and vegetables in the rivers of the Petite Camargue Alsacienne

The search for a link between the trophic level and the vegetable composition of some rivers of the Petite Camargue Alsacienne in North-East France leads to Table 5. A river can be either oligotrophic, mesotrophic or eutrophic, if its nutritive content is respectively poor, intermediate or high. Uncommon vegetables are considered rare, exotic or polluo-tolerant. To each river, a triplet of binary characteristics (r, p, e) is assigned, indicating the presence (1) or the absence (0) of rare (r), exotic (e) and polluo-tolerant (p) species.

The original ecological study can be found in [START_REF] Trémolières | Conservation status assessment of aquatic habitats within the Rhine floodplain using an index based on macrophytes[END_REF]. We consider n = 21 different rivers. Two empty columns were removed from the original table, leading to Table 5 of dimension 3 × 6, that is R = 18 categories and c = 7 zeros, with s = 17 parameters. Here are 3 expected frequencies below 0.5. Test statistics are compared to the chi-square quantile χ 2 0.95,10 = 18.31: Q = 14.38, Q ab = 20.68, G = 18.67, G ab = 26.05, RC 2/3 = 14.84.

Both corrected statistics Q ab and G ab as well as G lead to reject the null hypothesis, indicating an association between trophic level and vegetable composition. A thorough study of the table shows that rare species tend to settle preferentially in oligotrophic rivers. They are indeed better adapted to this kind of environment which tends to disappear from the rivers in the study. Moreover, polluo-tolerant species constitute the majority of the vegetables in eutrophic rivers. A eutrophic environment is competitive and these resistant species tend to get the best of it.

Discussion

We suggest to compute both Q ab and G ab , and to reject the null hypothesis if at least one of them is larger than the chi-square threshold.

  )where a = a n , b = b n and d = d n are random variables depending on n designed to compensate for the under-and overestimations due to p * . We thus take them positive with 0 < b < 1, such that b inflates the modified maximum likelihood estimator n j /n b and such that d controls the related rise.This new probability vector allows us to define corrected statistics Q p * 0 (p) and G p * 0 (p), provided that the parameters a, b and d satisfy several conditions. Forcing the summation of the pr to 1 implies that (R -c)d = ac + n 1-b -1, and thus allows us to define Q p * 0 (p ab ) and G p * 0 (p ab ) with pab such that : ∀i ∈ {1, . . . , c},

Property 1 .

 1 Bound b min is strictly less than 1. Moreover, if n = o(n) and n ∼ n when n tends to +∞, then b min and b tend to 1.

  Proposition 1. The inequality (2.2) is satisfied under the following assumption :

	p i	p j n	, ∀i ∈ {1, . . . , c}, ∀j ∈ {c + 1, . . . , R}.	(2.3)

Table 1 :

 1 Multinomial probabilities f 1 to f 4 .

	H 0 |{r; E r < 0.5}|	Probability
	f 1	20	(0.0002, . . . , 0.0002	, 0.01245, . . . , 0.01245	)
			20 times			80 times
	f 2	50	(0.0002, . . . , 0.0002	, 0.0198, . . . , 0.0198	)
			50 times			50 times
	f 3	70	(0.0002, . . . , 0.0002	, 0.03286667, . . . , 0.03286667	)
			70 times			30 times
	f 4	90	(0.0002, . . . , 0.0002	, 0.0982, . . . , 0.0982
			90 times			10 times

Table 4 :

 4 Diplotype table for the association between TNFAIP3 and Systemic Sclerosis..

			233 0.853 0.322 0.983 0.157
	f 2 f ′ 2 f 3 f ′ 3 f 4 f ′ 4	47 64 84	0.087 0.026 0.009 0.061 0.009 0.229 0.005 0 0 0.027 0.089 0 0 0 0

Table 5 :

 5 Contingency table for the joint study of trophic level and vegetable composition in rivers.

Acknowledgements

I want to thank I. Combroux and M. Guedj for letting me use their data sets as illustrations for this work. I am also truly grateful to P. Nobelis for his help and his advice.

Our results tend to prove that this approach is relevant and our corrections efficient in sparse tables. They are all the more interesting for the fact that sparse tables are usually left aside because the hypotheses needed to apply classical chi-square tests are not satisfied.

A Appendix section

Proof of Proposition 1. Assume that the first m of the c frequencies n i , 1 i c are modified. Let n ′ j , j ∈ {c + 1, . . . , R} compensate for these changes. The likelihood inequality (2.2) is then equivalent to :

Let us show that (2.3) implies (A.1). Applying (2.3) to each element of the left hand side of (A.1) with multiplicities such that : R j=c+1

we get :

As (n j -n ′ j + 1)! n (n j -n ′ j ) for all j in {c + 1, . . . , R} and n ′ i ! 1 for all i in {1, . . . , m}, we deduce (A.1) and equivalently (2.2).

Proof of Lemma 2. Let us first show that:

For each n we compute P(C n = c) for c in {0, . . . , R -1}. Let p 0 be the probability under the null hypothesis p 0 = (p 0 1 , . . . , p 0 c , p 0 c+1 , . . . , p 0 R ). A subject belongs to one of the first c cells with probability q 0 = p 0

and to one of the R -c last cells with probability 1 -

Let us consider now the binomial distribution B(n; q 0 ) and write the probability P(C n = c) of obtaining a table containing exactly c zeros placed anywhere :

Then :

As R c is bounded and (1 -q 0 ) n tends to 0, the probability P(C n = c) also converges to 0 for all 1 c R -1 when n tends to +∞, and so does the corresponding sum over c.

We now use Borel-Cantelli's Lemma to conclude that C n converges to 0 almost surely. Indeed, for ǫ > 0:

(A.7)


Anh Tai Ho

Jean-François Helard
email: jean-francois.helard@insa-rennes.fr

Youssef Nasser
email: youssef.nasser@insa-rennes.fr

Yves Louët
email: yves.louet@rennes.supelec.fr

A novel combined PAPR reduction and channel estimation approach for OFDM systems

Keywords: DVB-T2, OFDM, PAPR, tone reservation, channel estimation. I

In this paper, a novel combination of peak-to-average power ratio (PAPR) reduction and channel estimation techniques for orthogonal frequency division multiplexing (OFDM) systems is addressed. In order to reduce the spectrum efficiency loss due the insertion of dedicated pilots for PAPR reduction issues, we propose to use some pilots dedicated for channel estimation to reduce the PAPR value. These pilots follow particular laws which allow their blind detection at the receiving side and avoid sending side information. At the receiver, based on these properties, the pilots are detected and channel estimation is then performed. The proposed laws operate in discrete domain to mitigate the performance degradation due to residual estimation error in continuous domain. Simulation results performed using the new DVB-T2 standard parameters show that the proposed approach gives better performance and higher spectral efficiency when compared to conventional Tone Reservation (TR) method.

INTRODUCTION

Orthogonal Frequency Division Multiplexing (OFDM) has been the subject of numerous dissertations in recent years, mainly due to its several advantages for mobile wireless communications such as high data rate transmission with high bandwidth efficiency, exceptional robustness to multi-path fading and low implementation complexity. It has been adopted in several standards such as IEEE 802.11a, IEEE 802.16a, ETSI HIPERLAN/2 and DVB [START_REF] Reimers | Digital Video Broadcasting[END_REF][START_REF] Van Nee | OFDM for Wireless Multimedia Communications[END_REF]. However, OFDM system is not without its limitations [START_REF] Van Nee | OFDM for Wireless Multimedia Communications[END_REF]. The composite time signal in OFDM system is formed from the linear addition of independently modulated subcarriers. As a result, when the number of subcarriers is large, high amplitude peaks appear in the OFDM signal which induces a large Peakto-Average Power Ratio (PAPR). Usually, the OFDM signal passes through a non-linear power amplifier before its transmission over the channel. High PAPR value implies sophisticated high power amplifier (HPA) operating on a very large linear range at the transmitter. Otherwise, the nonlinearity of the HPA leads to in-band distortion, which increases the bit error rate (BER) of the system, and the out-of-band (OOB) distortion, which introduces high adjacent channel interference.

Various approaches have been proposed as resumed in [START_REF] Han | An overview of Peak-to-Average Power Ratio reduction techniques of multicarrier transmission[END_REF] to mitigate the PAPR of an OFDM signal. Among them, clipping and filtering technique is easy to implement. However, this scheme decreases the system performance since clipping is a nonlinear process [START_REF] Li | Effect of Clipping and Filtering on the performance of OFDM[END_REF]. An alternative method is based on coding, in which a data sequence is embedded in a longer sequence and only a subset of all these possible sequences is used in order to exclude patterns generating high PAPR [START_REF] Jones | Block coding scheme for reduction of Peak to Mean envelope Power Ratio of Multicarrier Transmission scheme[END_REF]. Moreover, other methods such as partial transmit sequence [START_REF] Muller | OFDM with reduced Peak-to-Average Power Ratio by Optimum Combination of Partial Transmit Sequences[END_REF], selected mapping [START_REF] Bauml | Reducing the Peak-to-Average Power Ratio of multicarrier Modulation by Selected Mapping[END_REF] and interleaving [START_REF] Jayalath | Reducing the Peak-to-Average Power Ratio of Orthogonal Frequency Division Multiplexing signal through bit of symbol interleaving[END_REF] are also proposed. The main drawback of these methods is the necessity to transmit side information (SI) to the receiver, resulting in some loss of throughput efficiency. Some methods recently proposed do not need this SI transmission [START_REF] Krongold | PAR reduction in OFDM via Active Constellation Extension[END_REF][START_REF] Tellado-Mourelo | Peak-to-Average Power Ratio reduction for multicarrier Modulation[END_REF]. Indeed, the active constellation extension (ACE) method is proposed in [START_REF] Krongold | PAR reduction in OFDM via Active Constellation Extension[END_REF]. It reduces PAPR by changing the constellation of the signal without modifying the minimum distance. The tone reservation (TR) method is proposed by Tellado [START_REF] Tellado-Mourelo | Peak-to-Average Power Ratio reduction for multicarrier Modulation[END_REF] which uses allocated subcarriers to generate additional signal that minimizes PAPR. These two methods have been already adopted for DVB-T2, the evolution of DVB-T standard. However, ACE technique increases the transmit signal power and its performance is rather restricted for a modulation with a large constellation size, while TR technique reduces the spectrum efficiency by the dedication process.

The minimization of PAPR in the TR method can be formulated as a convex optimization problem, which is modelized as a second order cone programming (SOCP) [START_REF] Mahafeno | Peak-to-average power ratio reduction using second order cone programming based tone reservation for terrestrial digital video broadcasting systems for Terestrial DVB Systems[END_REF]. In this paper, we continue the work of [START_REF] Mahafeno | Peak-to-average power ratio reduction using second order cone programming based tone reservation for terrestrial digital video broadcasting systems for Terestrial DVB Systems[END_REF] by investigating the TR method. The contribution of this paper is multifold. First, instead of using dedicated pilots for PAPR reduction, we propose to use existing pilots dedicated for channel estimation. In this way, we avoid the use of TR pilots and improve spectral efficiency of the system. Second, in order to allow their recovery serving for channel estimation process at the receiver, these pilots have to follow particular laws. Additive law and multiplicative law are then proposed and investigated in this work. At the receiver, laws relating pilots at the transmitter are applied to detect and estimate transmitted pilots. Since the detection and the estimation of these pilots in continuous domain cause degradation, we propose to apply these laws in discrete domain. Third, we show by simulations the validity of our technique using DVB-T2 standard parameters. Its performance is compared to the SOCP technique proposed in [START_REF] Mahafeno | Peak-to-average power ratio reduction using second order cone programming based tone reservation for terrestrial digital video broadcasting systems for Terestrial DVB Systems[END_REF] and the Walsh-Hadamard sequences technique proposed in [START_REF] Garcia | Peak Power Reduction for OFDM Systems with Orthogonal Pilot Sequences[END_REF]. This paper is organized as follows. Section II briefly presents the OFDM system model and the PAPR statistics. In section III, a novel combined PAPR reduction and channel estimation approach is presented. Simulation results and discussions are addressed in section IV. Finally, conclusions are drawn in section V. , N denotes the number of subcarriers and T is the original complex signal duration. In practice, only NL equidistant samples of ) (t x are considered, where L represents the over-sampling factor. The over-sampled signal is then given by ] 1 ,..., 0

II. OFDM SIGNAL AND PAPR

Let ] ... [ 1 0 - = N X X X
[ 1 ] / [ 1 0 2 - ∈ ∀ = ∑ - = NL n e X N L n x N k LN nk j k π (2)
When the over-sampling factor L is large enough, the PAPR of the OFDM signal is defined as the ratio of its maximum power divided by its average power. It is expressed as [START_REF] Sharif | On the peak-toaverage power of OFDM signals based on oversampling[END_REF] {

} { } { } 2 2 max 4 , ) ( L n n L x E x L x PAPR t x PAPR = ≥ ≈ (3)
where

L L L X Q x = , L X is the zero-padded vector of X ,
{} .

E denotes expectation operation and L

Q is the inverse discrete Fourier transform matrix of size NL . L Q is given by

              = - - - - ) 1 ).( 1 ( 2 1 ). 1 ( 2 ) 1 .( 1 2 1 . 1 2 ... 1 ... ... 1 1 ... 1 1 1 NL NL NL j NL NL j NL NL j NL j L e e e e N Q π π π π M M M (4)
In this study, the PAPR performance is evaluated using the complementary cumulative distribution function (CCDF). It is defined by the probability that the PAPR value exceeds a given threshold γ . For 1 = L , it can be expressed as [START_REF] Mahafeno | Peak-to-average power ratio reduction using second order cone programming based tone reservation for terrestrial digital video broadcasting systems for Terestrial DVB Systems[END_REF]

{ } { } [ ] N t x PAPR e L t x PAPR CCDF ) 1 ( 1 1 , ) ( Pr ) ( γ γ - - - ≈ = > = (5)
It is demonstrated in [START_REF] Van Nee | Reducing the peak-to-average power ratio of OFDM[END_REF] that real PAPR can be approximated when

4 = L by { } { } [ ] N t x PAPR e L t x PAPR CCDF 8 . 2 ) ( ) 1 ( 1 4 , ) ( Pr γ γ - - - ≈ = > = (6)
For simulations, we consider

   ∈ = not if X P k if C S k k k (7)
The OFDM baseband signal becomes

C Z x X C Q S Q s L L k L L L L + = + = = ) ( (8)
where L Z is given by , this is formulated as a SOCP problem [START_REF] Mahafeno | Peak-to-average power ratio reduction using second order cone programming based tone reservation for terrestrial digital video broadcasting systems for Terestrial DVB Systems[END_REF]. At the receiver, C has to be recovered in order to allow channel estimation operation. So, these pilots must follow particulars law. We consider two cases [START_REF] Tellado-Mourelo | Peak-to-Average Power Ratio reduction for multicarrier Modulation[END_REF] then

              = - - ) 1 .( 2 1 . 2 ) 1 .( 2
Additive law: ] 2 ,..., 0 [ 1 - ∈ ∀ Ω + = + M k C C k k
Ω + = k C C k 0 Multiplicative law: ] 2 ,..., 0 [ 1 - ∈ ∀ Ω × = + M k C C k k (11) then 0 C C k k Ω =
where Ω denotes step between two consecutive pilots.

In comparison with [START_REF] Mahafeno | Peak-to-average power ratio reduction using second order cone programming based tone reservation for terrestrial digital video broadcasting systems for Terestrial DVB Systems[END_REF], this operation corresponds to an additional constraint on SOCP problem. At the receiver, estimating C and Ω in continuous domain can cause degradation of the system performance due to residual estimation error. We propose to perform this in discrete domain. The laws become: At the receiver, C and Ω are detected and estimated. By quantifying them in discrete domain, λ , η , φ and ∆ can be simply recovered.

Additive law: ] 2 ,..., 0 [ 0 - ∈ ∀ Ω + = M k k C C k (12)

B. Pilot recovery and channel estimation

Assuming a complex multiplicative channel, the received symbols in frequency domain are expressed as

l l l l W S H R + × = (14)
where l refers to OFDM symbol index, For the second step, the channel response in the frequency domain is estimated at pilot positions as follows:

P k C R H l k l k l k ∈ ∀ = ˆ (16)
Then, channel frequency response for all subcarriers can be classically deduced by interpolation based on Wiener filtering.

IV. SIMULATIONS AND DISCUSSION

Simulation results are performed using DVB-T2 parameters. Some of the main parameters are resumed in table 1. The PAPR parameters of the proposed method, λ , η , φ and , are selected according to the simulation results. They will be specified in each scenario.

A. Comparison between additive and multiplicative laws

Firstly, we compare the performance of additive and multiplicative laws. Let's consider SOCP solution in the continuous domain with M=8. φ varies from 0 to 2π with a step of π/8, while ∆ varies from 0 to 2π with variable steps, of π/2, π/4, π/8 and π/16. The results are presented in figure 1. Multiplicative law gives better performance than the additive law. This is because in multiplicative law, all pilot subcarriers are modulated with the same power. Moreover, multiplicative law is performed with φ and ∆ in the discrete domain while additive law is performed in continuous domain. So in the discrete domain, this difference is still more important. Then, the multiplicative law presents better performance than the additive law in discrete domain. For the rest of this paper, we investigate the evaluation of the performance for multiplicative law in discrete domain.

B. Multiplicative law performance in discrete domain

We consider now multiplicative law in the discrete domain. Firstly, we present CCDF performance for fixed value of λ and for φ and ∆ varying from 0 to 2π with a step of π/8. The number of subcarriers used for PAPR reduction is always M=8. In this case, reducing PAPR is not a SOCP problem anymore but becomes a research of the optimum combination minimizing PAPR. At the receiver, pilots are not recovered by a systematic research (as the one presented in [START_REF] Garcia | Peak Power Reduction for OFDM Systems with Orthogonal Pilot Sequences[END_REF]) but by a detection and estimation algorithm. These results are shown in figure 2. We note that the performance in discrete domain is slightly worse than the one of SOCP solution and it depends on the value of λ . Then, we present in figure 3 the PAPR gain in terms of CCDF, in comparison with theoretical PAPR, as a function of λ . φ and ∆ varying from 0 to 2π with step of π/8, π/16 and π/32, are considered. This gain is evaluated at CCDF=10 -3 . It is also compared with the SOCP solution for multiplicative law ( λ continuous) and the Walsh-Hadamard sequences method presented in [START_REF] Garcia | Peak Power Reduction for OFDM Systems with Orthogonal Pilot Sequences[END_REF], named OPS method.

The system performance improves when the step of φ and ∆ becomes smaller. This is because the number of pilot combinations increases when this step decreases. For each value of this step, there exists one optimum value of λ for which the PAPR gain in terms of CCDF reaches the maximum. Besides, we note that the performance of the proposed method is much better than the one of the OPS method proposed in [START_REF] Garcia | Peak Power Reduction for OFDM Systems with Orthogonal Pilot Sequences[END_REF]. Finally, the performance in terms of PAPR gain is presented in figure 5, for M=8, 16 and 32, for φ and ∆ varying from 0 to 2π with step of π/8, π/16 and π/32, as a function of λ . increases, but we note that the optimum value of λ also decreases when M increases. This is important because it means that we can reduce pilot's power when increasing the number of subcarriers used for PAPR reduction.

Simulations results also show that the channel response is well estimated using this approach. As a consequence, this method allows avoiding the use of pilots dedicated for PAPR reduction, as TR pilots in DVB-T2 standard, improving also spectral efficiency.

V. CONCLUSIONS

In this paper, we have proposed a novel combined PAPR reduction and channel estimation method. Pilots dedicated for channel estimation have been used for both PAPR reduction and channel estimation purposes. These pilots followed particular laws in order to allow their detection at the receiver. Additive and multiplicative laws have been then investigated. By operating these laws in discrete domain, the system performance is improved because residual estimation error in continuous domain is avoided. Simulations showed that in discrete domain, multiplicative law is much better than additive law. Performance of the proposed method with multiplicative law in discrete domain was than evaluated using DVB-T2 parameters. Simulations shoed that it can achieve until 2.7dB in terms of PAPR gain. In comparison with DVB-T2 standard, proposed method presented better PAPR gain than the gradient method initially proposed in DVB-T2. In terms of spectral efficiency, proposed method avoided the use of dedicated pilots, which are equal to 1% of subcarriers in DVB-T2 standard specifications, improving also spectral efficiency of the system.

  approximation as a reference PAPR value of the original OFDM signal.III. NOVEL COMBINED PAPR REDUCTION WITH CHANNEL ESTIMATION APPROACHA. PAPR reductionIn this section, we propose to use the pilots dedicated to channel of M pilot positions dedicated to channel estimation to be used for PAPR reduction, of M pilots transmitted on these positions. Then, N modulated symbols { }

  the transmitter, reducing PAPR turn out to the research of C minimizing

  λ , η , φ and ∆ are in discrete domain. "Discrete domain" means that instead of searching SOCP solution in C space, we search it in a predefined discrete set.

  matrix containing the complex channel attenuation per subcarrier and l W is the additive white Gaussian noise vector.Pilot recovery and channel estimation procedures are performed in two steps: order to accomplish the first step, we assume that the channel slightly change between two successive OFDM symbols, i.e.

C

  by a simple quantification operation.

Figure 1 .

 1 Figure 1. CCDF performance of additive and multiplicative laws, SOCP solution.

Figure 2 .

 2 Figure 2. CCDF performance of multiplicative law in discrete domain.

Figure 3 .

 3 Figure 3. CCDF performance as a function of λ , M=8.

TABLE I

 I

	.	SIMULATION PARAMETERS, EXTRACTED FROM DVB-T2
		STANDARD.
	Mode		2K
	Number of subcarriers	NFFT=2048
	Guard Interval length	GI=1/8
	Modulation	16-QAM
	Coding rate	CR=1/2
	Over-sampling factor	L=4
	Number of subcarriers used	M=8, 16, 32
	for PAPR reduction

TABLE II .

 II PERFORMANCE SUMMARY OF DIFFERENT METHODS, DVB-T2 PARAMETERS, 2K MODE, 16-QAM MODULATION.

		Methods	∆G	λ opt
			(dB)	(dB)
		M=8, µ =π/16	2.58	20
	Proposed method	M=8, µ =π/32 M=16, µ =π/16 M=16, µ =π/32	2.66 2.07 2.17	20 14 14
		M=32, µ =π/16	1.61	6
		M=32, µ =π/32	1.88	8
	Gradient	CLIP=3 dB	1.62	-
	method	CLIP=5 dB	2.01	-
	The PAPR gain performance slightly decreases when M