García -Fernández

Eugenia De Diego

Juan Collantes

Elena Mendiola

Marta Prim

María Pérez-Fernández

Elia Miguel-Martín

María Nistal

Manuel Hardisson
email: dhardisson.hulp@salud.madrid.org

Eugenia García-Fernández

Juan I De Diego

Elena Collantes-Bellido

Marta Mendiola

María P Prim

Elia Pérez-Fernández

María Miguel-Martín

Manuel Nistal

Dr David Hardisson

Aurora B kinase expression in laryngeal squamous cell carcinoma and its prognostic implications Running title: Aurora B expression in LSCC

Keywords: Aurora B, Survivin, Aurora A, Larynx, Squamous cell carcinoma, Prognostic factor Aurora B, Survivin, Aurora A, Larynx, Squamous cell carcinoma, Prognostic factor

Aims: To investigate the clinical and prognostic significance of Aurora B in laryngeal squamous cell carcinomas (LSCC).

Methods and results: Aurora B protein expression was analyzed in a series of 259 LSCC. In addition, the expression of the proliferation index (Ki67) and other proteins involved in cell cycle control, such as Aurora A, survivin and p53 was also determined. Aurora B was highly expressed in 55.4% of LSCC. High Aurora B expression levels were correlated with tumour recurrence (P = 0.01), dead of disease (P = 0.05), and decreased disease-free survival (P = 0.013) and overall survival (P = 0.04). Survivin expression was neither associated with clinicopathological characteristics nor with survival. However, survivin expression in the nucleus paralleled Aurora B expression (P = 0.014). Aurora A expression was significantly associated with increased tumour grade (P = 0.008). Multivariate analysis indicated that Aurora B was an independent predictor for LSCC-specific disease-free survival (hazard ratio [HR], 2.10; 95% confidence interval [95% CI], 1.25-3.52 [P = 0.005]) and overall survival (HR, 1.91; 95% CI, 1.01-3.34 [P = 0.023]).

Conclusions: Aurora B may be a novel prognostic biomarker for LSCC and a potential therapeutic target in this type of tumour.

INTRODUCTION

Laryngeal squamous cell carcinoma (LSCC) is a common disease of the upper respiratory tract, and approximately 12,000 new cases are estimated to occur yearly in the United States, with nearly 4,000 expected deaths. [START_REF] Jemal | Cancer statistics[END_REF] It is one of the most common malignancies in Europe, with about 45,000 new cases per year and 21,000 deaths. [START_REF] Ferlay | Estimates of the cancer incidence and mortality in Europe in 2006[END_REF] In Spain, about 4,500 cases are diagnosed yearly with 1,500 estimated deaths. [START_REF] Jodra | Cáncer de laringe, tendencia temporal de la mortalidad[END_REF] The prognosis of LSCC is based on commonly recognized prognostic factors, including site of the primary tumour, histological grade, neck metastasis, and stage. However, patients with LSCC of the same clinical stage often show different clinical course, suggesting that the clinicopathological parameters alone may be insufficient to predict the prognosis. Biological markers that can identify the lesions with more aggressive phenotype and worse prognosis still need to be identified.

The genetic alterations associated with head and neck squamous cell carcinoma are numerous and include a variety of different pathways. [START_REF] Hardisson | Molecular pathogenesis of head and neck squamous cell carcinoma[END_REF] The Aurora family of serine/threonine kinases contains three members in mammalian cells, designated Auroras A, B, and C, which share the highest degree of sequence homology in their catalytic domains. [START_REF] Keen | Aurora-kinase inhibitors as anticancer agents[END_REF] However, their expression pattern and cellular localization differ markedly. Auroras A and B are expressed in many different cell types, whereas the expression of Aurora C seems to be restricted to testicular tissue. [START_REF] Kimura | Cell cycle-dependent expression and centrosome localization of a third human aurora/Ipl1-related protein kinase, AIK3[END_REF] Aurora A is required for centrosome maturation and division. [START_REF] Vader | The Aurora kinase family in cell division and cancer[END_REF][START_REF] Liu | Aurora A, mitotic entry, and spindle bipolarity[END_REF][START_REF] Satinover | Aurora-A kinase and inhibitor-2 regulate the cyclin threshold for mitotic entry in Xenopus early embryonic cell cycles[END_REF] Aurora B is localized to the late-interphase chromosomes proximal to the centromere, from where, during mitosis, it regulates the phosphorylation and thus the mitotic activity of many substrates [START_REF] Adams | Essential roles of Drosophila inner centromere protein (INCENP) and aurora B in histone H3 phosphorylation, metaphase chromosome alignment, kinetochore disjunction, and chromosome segregation[END_REF][START_REF] Ota | Increased mitotic phosphorylation of histone 3 attributable to AIM-1/Aurora-B overexpression contributes to chromosome number instability[END_REF] Together with survivin, inner centromere protein, and borealin, Aurora B constitutes a chromosome passenger complex that regulates several aspects of mitosis, such as the spindle Formatted: Highlight Formatted: Highlight Formatted: Highlight Deleted: Carcinogenesis is a multistage process involving the activation of oncogenes and the inactivation of tumour suppressor genes. In this context, most human tumours are characterized by an imbalance of regulatory mechanisms controlling the cell cycle progression, cell death/viability balance, and apoptosis.

Deleted: Subsequently, the divided centrosomes migrate to opposite sites of the dividing cell to define the poles of the bipolar mitotic spindle. [START_REF] Vader | The Aurora kinase family in cell division and cancer[END_REF] This process, as well as the formation of the mitotic spindle, also requires Aurora A function. [START_REF] Liu | Aurora A, mitotic entry, and spindle bipolarity[END_REF] Entry into mitosis is also regulated by Aurora A due to an effect onto the CDK1/cyclinB complexes, making Aurora A also a cell cycle-regulating protein. [START_REF] Satinover | Aurora-A kinase and inhibitor-2 regulate the cyclin threshold for mitotic entry in Xenopus early embryonic cell cycles[END_REF] Deleted: , including histone H3. [START_REF] Adams | Chromosomal passengers and the (aurora) ABCs of mitosis[END_REF] Aurora C has also been reported to localize the chromosome passenger complex, but little is known about its function. [START_REF] Sasai | Aurora-C kinase is a novel chromosomal passenger protein that can complement Aurora-B kinase function in mitotic cells[END_REF] Both Auroras A and B are overexpressed in several tumour types. [START_REF] Gautschi | Aurora kinases as anticancer drug targets[END_REF][START_REF] Boss | Clinical experience with Aurora kinase inhibitors: a review[END_REF] Aurora A is located in chromosome 20q13.2, a region commonly amplified in human tumours, including those in colon, breast, pancreas, ovary, and prostate. [START_REF] Gautschi | Aurora kinases as anticancer drug targets[END_REF] Although aberrant expression of Aurora A has been found in a variety of human tumours, it does not always correlate with gene amplification. In this sense, we have recently demonstrated amplification of AURKA in 27.6% of ovarian carcinomas although 58.8% of tumours overexpressed Aurora A [START_REF] Mendiola | Aurora kinases as prognostic biomarkers in ovarian carcinoma[END_REF] , suggesting that the expression of Aurora A is likely to be regulated not only by gene amplification, but also by other mechanisms that have been previously described such as transcriptional activation and/other suppression of protein degradation. [START_REF] Fu | Targeting Aurora kinases in ovarian cancer[END_REF] The contribution of Aurora B to carcinogenesis has been less studied than that of Aurora A. Aurora B is located in chromosome 17p13.1, a region not typically amplified in human cancer. However, overexpression of Aurora B has been demonstrated in a variety of human cancers, including astrocytomas, [START_REF] Araki | High expression of Aurora-B/Aurora and Ipl1-like midbody-associated protein (AIM-1) in astrocytomas[END_REF] malignant mesotheliomas, [START_REF] Lopez-Rios | Global gene expression profiling of pleural mesotheliomas: overexpression of aurora kinases and P16/CDKN2A deletion as prognostic factors and critical evaluation of microarray-based prognostic prediction[END_REF] seminomas, [START_REF] Chieffi | Aurora B expression in normal testis and seminomas[END_REF] colorectal, [START_REF] Katayama | Mitotic kinase expression and colorectal cancer progression[END_REF] prostate, [START_REF] Chieffi | Aurora B expression directly correlates with prostate cancer malignancy and influences prostate cell proliferation[END_REF] endometrial, [START_REF] Kurai | Expression of Aurora kinases A and B in normal, hyperplastic, and malignant human endometrium: Aurora B as a predictor for poor prognosis in endometrial carcinoma[END_REF] thyroid, [START_REF] Sorrentino | Aurora B overexpression associates with the thyroid carcinoma undifferentiated phenotype and is required for thyroid carcinoma cell proliferation[END_REF] nonsmall cell lung, [START_REF] Vischioni | Frequent overexpression of aurora B kinase, a novel drug target, in non-small cell lung carcinoma patients[END_REF] oral squamous cell, [START_REF] Qi | Aurora-B expression and its correlation with cell proliferation and metastasis in oral cancer[END_REF] and ovarian [START_REF] Mendiola | Aurora kinases as prognostic biomarkers in ovarian carcinoma[END_REF] cancers, at various pathological stages, with a tendency to group in higher grades of malignancy.

Survivin is a multifunctional protein that inhibits apoptosis, regulates cell division and enhances angiogenesis. Although rarely expressed in terminally differentiated adult tissues, upregulation of survivin is found in many cancers, usually associated with increased aggressiveness and decreased survival. [START_REF] Altieri | Validating survivin as a cancer therapeutic target[END_REF] Aurora B protein expression has not been previously characterized in human LSCC by immunohistochemistry, and its role in LSCC prognosis has not been well documented.

In the present study our aim was to investigate Aurora B protein expression in a large To further evaluate the effect on patient prognosis and clinical characteristics of other key players of cell cycle regulation, we also performed immunohistochemistry for Aurora A, survivin, p53 and Ki67, and we examined the relationship between clinicopathological findings and the expression of these biomarkers in tumour cells. Moreover, we compared Aurora B, Aurora A, and survivin with other classical prognostic parameters to evaluate the effectiveness of these molecules as prognostic biomarkers in LSCC.

MATERIAL AND METHODS

Patient selection and tissue samples

Surgical tissue samples of LSCC from 259 patients, obtained in the period between 1990 and 1996, were retrieved from the archive of the Department of Pathology, University Hospital La Paz, Madrid (Spain). None of the patients had been treated previously. Tissue samples were fixed in 10% formalin and embedded in paraffin.

Standard haematoxylin and eosin stains were used to establish the diagnosis and to grade the tumours into well (grade I), moderately (grade II) and poorly (grade III) differentiated, according to the World Health Organization classification. Tumour size and clinicopathological stage of tumours were classified according to the 1988 TNM classification of the American Joint Committee of Cancer (AJCC). [START_REF] Beahrs | Manual for Staging of Cancer[END_REF] Follow-up data from the patients were registered by means of protocols that the otolaryngologists fulfilled throughout the follow-up of the patients. Disease-free survival (DFS) and overall survival (OS) were the main endpoints of this study. DFS was calculated from the day of surgery to the date of recurrence, either locoregional or distant. OS was calculated from the day of surgery to the date of either death or the last follow-up.

Approval for the study was obtained from the Local Ethics Committee (Comité Ético de Formatted: Highlight Deleted: Our results showed a marked up-regulation of Aurora B in tumour cells, and a correlation with disease-free and overall survival, suggesting that Aurora B might be a good drug target in LSCC. 1.

Tissue microarray construction

Representative areas of the tumours were selected on haematoxylin and eosin-stained sections and marked on individual paraffin blocks. Two tissue cores (1 mm in diameter) were obtained from each specimen. The tissue cores were arrayed into a receptor paraffin block using a tissue microarray workstation (Beecher Instruments, Silver Spring, MD), as described previously. [START_REF] Hardisson | Tissue microarray immunohistochemical expression analysis of mismatch repair (hMLH1 and hMSH2 genes) in endometrial carcinoma and atypical endometrial hyperplasia. Relationship with microsatellite instability[END_REF] A haematoxylin and eosin-stained section of the array was reviewed to confirm the presence of morphologically representative areas of the original lesions.

Immunohistochemistry

Immunohistochemistry (IHC) was performed on 4-µm sections of formalin-fixed, paraffin-embedded tissues. Briefly, the tissue sections were deparaffinized and rehydrated in water, after which antigen retrieval was carried out by incubation in EDTA solution, pH 8.2 at 50ºC for 45 minutes in an autoclave. Endogenous peroxidase and non-specific antibody reactivity was blocked with peroxidase blocking reagent (Dako, Glostrup, Denmark) at room temperature for 15 minutes. The sections were then incubated for 60-90 minutes at 4ºC with the following antibodies: Aurora A monoclonal antibody (clone JLM28, Novocastra Laboratories, Newcastle Upon Tyne, UK, dil. 1:50), Aurora B polyclonal antibody (Abcam pcl, Cambridge, UK, dil. 1:50), survivin polyclonal antibody (Abcam pcl, dil. 1:400), p53 monoclonal antibody (clone DO-7, Novocastra Laboratories, dil. 1:100), and Ki67 monoclonal antibody (clone MIB1, Dako, dil. 1:100). Detection was performed with EnvisionPlus Detection System (Dako).

Formatted: Highlight

Deleted: According to our Local Ethics Committee it was not necessary to obtain a verbal or written consent for publication from the patients included in this study considering that this is a retrospective study based on formalin-fixed, paraffinembedded tumour samples obtained between 1990 and 1996.

Analysis of immunohistochemical stains

For Aurora B, evaluation of the IHC slides was performed using a semiquantitative scoring system for both, intensity of stain and percentage of positive cells (labelling frequency percentage). For the intensity, the grading scale ranged from non-detectable signal [0] to strong signal [2] seen at low power. A weak to moderate signal seen at low to intermediate power was designated 1. Labelling frequency was scored on a fourtiered system as 1 (0-25%), 2 (26-50%), 3 (51-75%), or 4 (76-100%). The multiplicative expression index (MEI) was obtained by totalling the scores of intensity and percentage, as previously reported. [START_REF] Lee | Targeting Aurora kinases for the treatment of prostate cancer[END_REF] Expression level of Aurora B was defined as low (MEI, 1-4) or high (MEI, >4). For Aurora A, the MEI could not be established, as the intensity ranged only from non-detectable signal (0) to weak signal (1), and the labelling frequency ranged between 0-60%. Therefore, according to the mean value of the registered scores, we established a cut-off rate of 5% to consider the expression of Aurora A as positive, as previously published. [START_REF] Mendiola | Aurora kinases as prognostic biomarkers in ovarian carcinoma[END_REF] Survivin expression was assessed as the percentage of cells with nuclear and/or cytoplasmic immunostaining. Expression of p53 was evaluated as the percentage of tumour cells with nuclear-positive immunostaining. The Ki67 labelling index was expressed as the percentage of positively stained nuclei in the tumour cells.

Statistical analysis

Statistical analysis was carried out using 9.0 SPSS software for Windows (SPSS Inc, Chicago, IL). All tests were 2-sided and used a significance level of 0.05. Qualitative data were registered as absolute frequencies and percentages; quantitative data were expressed as median, range, and/or mean and standard deviation. Frequency tables were tested by Fisher test for comparison of discrete variables. Correlations between the expression of Aurora A and B, survivin, p53 and Ki67 were examined using Spearman's rank correlation test. Analysis of disease-free and overall survival data were carried out using Kaplan-Meier method plots and log-rank test was used to compare the survival distributions of two samples. Factors with significant effects on disease-free and overall survival were identified with univariate and multivariate Cox proportional hazards regression model. To accommodate non-proportional hazards interactions were built between covariates and time into the Cox regression model as time-dependent covariates. These interactions with a non significant effect were excluded from the final model.

RESULTS

Aurora B, survivin, and Aurora A protein expression and clinicopathological characteristics

Mean expression values for Aurora A, Aurora B, survivin, p53 and Ki67 are shown in Table 2. For Aurora B the staining was predominantly nuclear (Figure 1). Aurora B was highly expressed (MEI >4) in 139 (55.4%) of 251 LSCC in our study. Tumours with high expression of Aurora B (MEI >4) showed a higher rate of recurrence and death of disease than those tumours with a low expression of Aurora B protein (MEI ≤4) (37.2% versus 21.3%, P = 0.01, and 28.7% versus 17.9%, P = 0.05, respectively) (Table 3). Aurora B protein expression was not related to location of tumour (glottic versus nonglottic), T category (T1-2 versus T3-4), tumour grade, regional lymph node or distant metastasis, and tumour stage (I-II versus III-IV) (Table 3).

Survivin expression was assessable in 234 samples. All cases showed cytoplasmic staining for survivin; however, in some tumours, survivin was detected both in the nucleus and the cytoplasm of tumour cells (106 tumours, 45.23%) (Figure 2). Survivin expression was not associated with any of the clinicopathological variables studied (Table 3). However, survivin expression in the nucleus paralleled Aurora B expression The immunohistochemical staining pattern for Aurora A was cytoplasmatic (Figure 3). Aurora A was expressed (>5%) in 57 (22.4%) of 255 tumours. Aurora A protein expression was significantly associated with tumour grade; thus, 33.3% of grade 3 tumours expressed Aurora A compared to 17.5% of grade 1 and 2 tumours (P = 0.008) (Table 3). The expression of Aurora A was not related to location of tumour (glottic versus non-glottic), T category (T1-2 versus T3-4), regional lymph node or distant metastasis, tumour stage (I-II versus III-IV), tumour recurrence or final status of patient (alive versus dead) (Table 3).

Correlation between the expression of Aurora B and Aurora A, survivin, p53 and Ki67 was examined using a Spearman rank correlation test. A positive correlation was obtained between survivin expression both in the nucleus and cytoplasm and the mean expression of Aurora B (P = 0.009, R = 0.18, and P = 0.001, R = 0.23, respectively) and Ki67 (P < 0.001, R = 0.25, and P = 0.015, R = 0.16, respectively). There was also a positive correlation between the expression of Aurora A and p53 (P = 0.003, R = 0.20), and the expression of Aurora A and B and Ki67 (P = 0.011, R = 0.17, and P = 0.003, R = 0.2, respectively).

Association between disease-free survival, overall survival, and Aurora B and A protein expression in LSCC

The mean disease-free survival was 43.54 ± 31.01 and mean overall survival was 46.56 ± 30.7 months. Sixty patients (28.6%) died due to their cancer. In the univariate analysis, Kaplan-Meier method showed that patients with high expression of Aurora B (MEI >4) had a decreased disease-free survival compared with patients whose tumours expressed confidence interval [CI], 62-82, versus 89 months, 95% CI, 81-98; P = 0.013, respectively) (Figure 4). Regarding overall survival, patients with high expression of Aurora B (MEI >4) showed a significant decreased survival time compared to those patients with a low expression of Aurora B protein (MEI ≤4) (mean survival of 84 months, 95% CI, 75-92, versus 122 months, 95% CI, 111-133; P = 0.04, respectively) (Figure 5). There was no correlation between disease-free survival or overall survival and survivin and Aurora A protein expression.

The multivariate analysis using the Cox regression model adjusted to clinicopathological variables (tumour site [glottis versus other], T category [T1-2 versus

T3-4], histological grade (G1-2 versus G3), tumour stage (I-II versus III-IV) and lymph node metastasis) showed Aurora B protein expression as an independent prognostic factor for both disease-free survival (P = 0.005, hazard ratio [HR] = 2.10, 95% CI, 1.25-3.52) and overall survival (P = 0.023, HR = 1.91, 95% CI, 1.01-3.34) (Table 4).

DISCUSSION

This is, to our knowledge, the first study that investigates Aurora B protein expression in LSCC. Using immunohistochemical analysis of a large series of tumours, we showed frequent high expression of Aurora B in LSCC (55.4%), suggesting that Aurora B plays a role as tumour marker in LSCC. Aurora B is reported to form complexes with inner centromere protein and survivin. These complexes are thought to be involved in the regulation of chromosome alignment, segregation, and cytokinesis. [START_REF] Adams | Essential roles of Drosophila inner centromere protein (INCENP) and aurora B in histone H3 phosphorylation, metaphase chromosome alignment, kinetochore disjunction, and chromosome segregation[END_REF][START_REF] Wheatley | Aurora-B phosphorylation in vitro identifies a residue of survivin that is essential for its localization and binding to inner centromere protein (INCENP) in vivo[END_REF] In LSCC, Aurora B was localized in the nucleus, in line with previous reports on other tumour types. [START_REF] Mendiola | Aurora kinases as prognostic biomarkers in ovarian carcinoma[END_REF][START_REF] Chieffi | Aurora B expression in normal testis and seminomas[END_REF][START_REF] Katayama | Mitotic kinase expression and colorectal cancer progression[END_REF][START_REF] Chieffi | Aurora B expression directly correlates with prostate cancer malignancy and influences prostate cell proliferation[END_REF][START_REF] Kurai | Expression of Aurora kinases A and B in normal, hyperplastic, and malignant human endometrium: Aurora B as a predictor for poor prognosis in endometrial carcinoma[END_REF][START_REF] Sorrentino | Aurora B overexpression associates with the thyroid carcinoma undifferentiated phenotype and is required for thyroid carcinoma cell proliferation[END_REF][START_REF] Vischioni | Frequent overexpression of aurora B kinase, a novel drug target, in non-small cell lung carcinoma patients[END_REF][START_REF] Qi | Aurora-B expression and its correlation with cell proliferation and metastasis in oral cancer[END_REF] Since recent studies have identified the histone H3 protein as an important substrate for Aurora B kinase, [START_REF] Ota | Increased mitotic phosphorylation of histone 3 attributable to AIM-1/Aurora-B overexpression contributes to chromosome number instability[END_REF][START_REF] Crosio | Mitotic phosphorylation of histone H3: spatiotemporal regulation by mammalian Aurora kinases[END_REF] nuclear localization of Aurora B seems to be functionally important. Moreover, high expression of Aurora B was correlated with F o r P e e r R e v i e w 11 nuclear expression of survivin. This is consistent with the results of previous studies in non-small cell lung carcinoma [START_REF] Vischioni | Frequent overexpression of aurora B kinase, a novel drug target, in non-small cell lung carcinoma patients[END_REF] and in vitro studies which show that cells lacking survivin expression upon targeting by antisense oligonucleotide treatment have a lower Aurora B kinase activity. [START_REF] Chen | Survivin enhances Aurora-B kinase activity and localizes Aurora-B in human cells[END_REF] Overexpression of Aurora B has been related to poor prognosis in recent reports of endometrial carcinoma, [START_REF] Kurai | Expression of Aurora kinases A and B in normal, hyperplastic, and malignant human endometrium: Aurora B as a predictor for poor prognosis in endometrial carcinoma[END_REF] non-small cell lung carcinoma [START_REF] Vischioni | Frequent overexpression of aurora B kinase, a novel drug target, in non-small cell lung carcinoma patients[END_REF] and mesothelioma. [START_REF] Lopez-Rios | Global gene expression profiling of pleural mesotheliomas: overexpression of aurora kinases and P16/CDKN2A deletion as prognostic factors and critical evaluation of microarray-based prognostic prediction[END_REF] In our study, tumours with high expression of Aurora B showed a decreased disease-free survival and overall survival, thus suggesting that Aurora B plays a role in LSCC tumour progression. Moreover, the multivariate analysis adjusted to tumour site, T category, histological grade, lymph node metastasis, and tumour stage by Cox proportional hazards regression showed Aurora B expression as an independent prognostic factor for both disease-free survival and overall survival. Aurora B has been proposed to promote tumour progression through mechanisms such as the loss of mitotic control and abnormal proliferation. [START_REF] Ota | Increased mitotic phosphorylation of histone 3 attributable to AIM-1/Aurora-B overexpression contributes to chromosome number instability[END_REF] In this regard, we found a positive correlation between Aurora B levels and the tumour proliferation rate assessed by the immunostaining for the proliferation marker Ki67, similar to the data published for astrocytomas, [START_REF] Araki | High expression of Aurora-B/Aurora and Ipl1-like midbody-associated protein (AIM-1) in astrocytomas[END_REF] endometrial, [START_REF] Kurai | Expression of Aurora kinases A and B in normal, hyperplastic, and malignant human endometrium: Aurora B as a predictor for poor prognosis in endometrial carcinoma[END_REF] thyroid, [START_REF] Sorrentino | Aurora B overexpression associates with the thyroid carcinoma undifferentiated phenotype and is required for thyroid carcinoma cell proliferation[END_REF] and non-small cell lung carcinomas. [START_REF] Vischioni | Frequent overexpression of aurora B kinase, a novel drug target, in non-small cell lung carcinoma patients[END_REF] In LSCC, survivin expression has been related to poor patient survival and tumour invasion, suggesting that survivin protein frequently exhibits dysregulated expression and may contribute to the malignant transformation of the laryngeal squamous epithelia, and a possible association with the stronger aggressiveness. [START_REF] Dong | Survivin expression in laryngeal squamous cell carcinoma and its prognostic implications[END_REF][START_REF] Pizem | Survivin expression is a negative prognostic marker in laryngeal squamous cell carcinoma and is associated with p53 accumulation[END_REF][START_REF] Marioni | Expression of the apoptosis inhibitor protein survivin in primary laryngeal carcinoma and cervical lymph node metastasis[END_REF] Thus, it is postulated that the state of survivin expression might be an important discriminator in the progression of LSCC. In our study, survivin expression was not correlated with any patient characteristics of adverse disease. On the other hand, it has been previously reported that the expression of survivin in the nucleus may have prognostic [START_REF] Altura | Nuclear expression of survivin in pediatric ependymomas and choroid plexus tumours correlates with morphologic tumour grade[END_REF][START_REF] Ferrandina | Survivin expression in ovarian cancer and its correlation with clinico-pathological, surgical and apoptosis-related parameters[END_REF] In our cases of LSCC, however, we did not observe any association between localization of survivin (cytoplasmic or nucleocytoplasmic) and survival or other clinicopathological parameters.

We showed that Aurora A expression increases with lower tumour differentiation, consistent with what has been recently reported for the levels of Aurora A mRNA quantified by real-time reverse transcription PCR in head and neck squamous cell carcinomas. [START_REF] Reiter | Aurora kinase A messenger RNA overexpression is correlated with tumour progression and shortened survival in head and neck squamous cell carcinoma[END_REF] These data suggest that expression of Aurora A contributes to poor prognosis in LSCC patients, as described in other tumour types showing a correlation between the up-regulation of Aurora A and tumour progression and clinical aggressiveness. [START_REF] Boss | Clinical experience with Aurora kinase inhibitors: a review[END_REF] Aurora A has been shown to interact with many important cellular proteins, including p53. Aurora A is a key regulatory component of the p53 pathway, and high expression of Aurora A phosphorylates p53, leads to an increased p53 degradation, thus facilitating oncogenic transformation. [START_REF] Katayama | Phosphorylation by aurora kinase A induces Mdm2-mediated destabilization and inhibition of p53[END_REF] In this sense, in this study we found a correlation between expression of Aurora A and p53; nevertheless the interrelationship between these molecules and tumour progression is, however, still unclear.

Targeted therapies alone and in combination with cytotoxic therapy have recently been successfully introduced into the treatment of some solid tumours. In the last years huge progress has been made in both understanding the basic biological role of Aurora A and B kinases and in the development of small molecular inhibitors that seem to have a very interesting preclinical activity. [START_REF] Harrington | VX-680, a potent and selective smallmolecule inhibitor of the Aurora kinases, suppresses tumour growth in vivo[END_REF][START_REF] Keen | Mitotic drivers-inhibitors of the Aurora B kinase[END_REF] These are presently being tested in clinical studies in cancer patients. [START_REF] Garber | Divide and conquer: new generation of drugs targets mitosis[END_REF] In this sense, our results show that Aurora B could be a new therapeutic target in LSCC.

In conclusion, we have reported for the first time the presence of Aurora B expression in a large series of LSCC, and its correlation with adverse tumour features. Our results

F

6 Investigación

 6 Clínica del Hospital Universitario La Paz, ref. PI-349). Detailed patients characteristics and histopathologic data are shown in Table

  Highlight Deleted: , P = 0.54 Deleted: , P = 0.80 Deleted: (P = 0.41), Deleted: (P = 0.67) Deleted: (P = 0.63)Deleted: , P = 0.59 0.014); thus, mean nuclear survivin expression in tumours with high expression of Aurora B was 12.19% ± 17.88 (range 0-100%) versus 6.73% ± 12.64 (range 0-60%) in those tumours with low expression of Aurora B.

  Highlight Deleted: , P = 0.72 Deleted: , P = 0.44 Deleted: (P = 0.13) Deleted: (P = 0.41) Deleted: , P = 0.87 Deleted: (P = 0.86) Deleted: , P = 0.21 at lower levels (MEI ≤4) (mean survival of 72 months, 95%

  of Aurora B in LSCC progression. Studies aimed at further understanding of Aurora B in tumour progression in LSCC are warranted.

Figure 1 .

 1 Figure 1. Expression of Aurora B protein in the nuclei of the tumour cells.

Figure 2 .

 2 Figure 2. Expression of survivin in the tumour cells. Note the strong immunostaining of the nuclei of the tumour cells (same case as depicted in Figure 1).

Figure 3 .

 3 Figure 3. Expression of Aurora A is observed in the cytoplasm of the tumour cells.

Figure 4 .

 4 Figure 4. Kaplan-Meier curve showing the association between the expression of Aurora B and disease-free survival.

Figure 5 .

 5 Figure 5. Kaplan-Meier curve showing the association between the expression of Aurora B and overall survival.

  Expression of Aurora B protein in the nuclei of the tumour cells 76x57mm (300 x 300 DPI) Expression of survivin in the tumour cells. Note the strong immunostaining of the nuclei of the tumour cells (same case as depicted in Figure 1) 76x57mm (300 x 300 DPI) Expression of Aurora A is observed in the cytoplasm of the tumour cells 76x57mm (300 x 300 DPI) curve showing the association between the expression of Aurora B and disease-free survival 76x56mm (600 x 600 DPI)

Table 1 .

 1 Clinicopathological characteristics of the patients with LSCC

	n (%)

Table 2 .

 2 Mean protein expression values for Aurora B, survivin, Aurora A, p53 and Ki67.

		Mean (%)	± SD	Range (%)
	Aurora B	80.29	31.72	0-100
	Survivin (nuclear)	9.81	15.92	0-100
	Survivin (cytoplasm)	93.13	14.91	25-100
	Aurora A	4.86	9.92	0-60
	p53	30.80	37.77	0-100
	Ki67	40.49	25.65	5-100
	SD, Standard deviation

Table 3 .

 3 Relationship between Aurora B, survivin and Aurora A protein expression and clinicopathological variables.

				Aurora B			Survivin			Aurora A
			MEI ≤4 (n, %) MEI >4 (n, %)	P	C (n, %)	N/C (n, %)	P	≤5% (n, %)	>5% (n, %)	P
	Tumour site	Glottic	27 (47.4)	30 (52.6)	0.54	23 (52.3)	21 (47.7)	0.74	47 (79.7)	12 (20.3)	0.72
		Non-glottic	77 (42.5)	104 (57.5)		93 (55.4)	75 (44.6)		140 (76.5)	43 (23.5)
	T category	T1-T2	45 (43.3)	59 (56.7)	0.80	48 (52.7)	43 (47.3)	0.78	82 (75.9)	26 (24.1)	0.44
		T3-T4	63 (45)	77 (55)		70 (55.1)	57 (44.9)		113 (80.1)	28 (19.9)
	Histological	G1-G2	71 (43)	94 (57)	0.41	76 (51.4)	72 (48.6)	0.25	141 (82.5)	30 (17.5)	0.008
	grade	G3	39 (48.8)	41 (51.3)		43 (60.6)	28 (39.4)		52 (66.7)	26 (33.3)
	Lymph node	N 0	74 (43.3)	97 (56.7)	0.67	86 (57)	65 (43)	0.24	142 (81.1)	33 (18.9)	0.13
	metastasis	N +	34 (46.6)	39 (53.4)		32 (47.8)	35 (52.2)		53 (71.6)	21 (28.4)
	Distant	M 0	101 (44.9)	124 (55.1)	0.63	111 (55)	91 (45)	0.44	180 (78.9)	48 (21.1)	0.41
	metastasis	M 1	7 (36.8)	12 (63.2)		7 (43.8)	9 (56.3)		15 (71.4)	6 (28.6)
	Tumour stage	I-II	36 (41.4)	51 (58.6)	0.59	43 (57.3)	32 (42.7)	0.57	72 (79.1)	19 (20.9)	0.87
		III-IV	72 (45.9)	85 (54.1)		75 (52.4)	68 (47.6)		123 (77.8)	35 (22.2)
	Recurrence	No	85 (78.7)	81 (62.8)	0.01	85 (71.4)	68 (71.6)	1.00	134 (71.3)	37 (69.8)	0.86
		Yes	23 (21.3)	48 (37.2)		34 (28.6)	27 (28.4)		54 (28.7)	16 (30.2)
	Final status	Alive	92 (82.1)	97 (71.3)	0.05	91 (75.8)	80 (79.2)	0.63	148 (75.1)	46 (83.6)
		DOD	20 (17.9)	39 (28.7)		29 (24.2)	21 (20.8)		49 (24.9)	9 (16.4)

Table 4 .

 4 Prognostic value of Aurora B expression adjusted for clinicopathologic parameters byCox proportional hazards regression.

	Univariate		DFS			OS
	Prognostic factor	P	HR	95% CI	P	HR	95% CI
	Tumour site	0.016	0.47	0.26-0.87	0.001	0.25	0.11-.58
	T category	0.001	2.52	1.50-4.25	<0.001 4.93	2.49-9.74
	Histologic grade	0.042	1.67	1.02-2.74	0.154	1.48	0.86-2.55
	Lymph node metastasis	<0.001 3.34	2.10-5.44	<0.001 3.43	2.05-5.74
	Tumour stage	<0.001 3.05	1.72-5.42	<0.001 6.09	2.76-13.43
	Aurora B expression	0.019	1.81	1.10-2.90	0.043	1.74	1.02-2.90
	Multivariate		DFS			OS
	Prognostic factor	P	HR	95% CI	P	HR	95% CI
	Tumour site	0.305	0.69	0.34-1.40	0.067	0.4	0.15-1.07
	T category	0.379	1.54	0.59-4.07	0.25	2.03	0.61-6.80
	Histologic grade	0.637	0.88	0.50-1.52	0.26	0.72	0.40-1.28
	Lymph node metastasis	<0.001 2.70	1.48-4.93	0.007	2.28	1.25-4.15
	Tumour stage	0.904	1.07	0.33-3.5	0.535	1.6	0.36-7.10
	Aurora B expression	0.005	2.10	1.25-3.52	0.023	1.91	1.01-3.34
	DFS, Disease-free survival; OS, Overall survival; HR, Harzard ratio; 95% CI indicates
	95% confidence interval.

Published on behalf of the British Division of the International Academy of Pathology Histopathology

ACKNOWLEDGEMENTS

Dr Marta Mendiola is supported by a postdoctoral research contract of Fondo de Investigación Sanitaria (FIS) (Program "Sara Borrell"), Instituto de Salud Carlos III, Ministerio de Ciencia e Innovación, Spain.


Mathieu Tillier

Mathieu Tillier Al-Ahwāz

ouverte pluridisciplinaire HAL, est destinée au dépôt

al-Ahwāz (persan Ahvāz) : ville du Khūzistān (31°19' N, 48°46' E), située de part et d'autre de la rivière Kārūn (aussi appelée Dujayl, « petit Tigre »), à l'endroit où celle-ci entre dans la plaine après avoir traversé une chaîne de collines de grès dont les rapides entravèrent longtemps la navigation. La ville fut probablement implantée sur le site de la Taryana achéménide, où la route royale reliant Suse à Persépolis traversait la rivière grâce à un pont de bateaux. Elle fut reconstruite par le Sassanide Ardashīr I (m. 242) ou par son fils Shāpūr I (r. 240-270) et nommée Hurmuz Shahr ou Hurmuz Ardashīr. Lors de la conquête arabe, la partie administrative fut détruite et le nouveau nom de la ville fit référence aux activités commerciales de sa rive orientale : Sūq al-Ahwāz, ou « Marché des Hūz » (ou des Ḥūz/Khūz, peuple identifié comme les Οὔξιοι/Uxii). « Al-Ahwāz », forme abrégée de ce nom, en vint à désigner également l'ensemble des « Kuwar al-Ahwāz », c'est-à-dire le Khūzistān.

Al-Ahwāz fut conquise par les armées baṣriennes. Selon al-Balādhurī, al-Mughīra b. Shuʿba (m. entre 48 et 51/668-71) la pilla une première fois en 16/637, lorsque ʿUtba b. Ghazwān (m. 17/638) entreprit de conquérir le 376). Al-Hurmuzān, commandant des troupes sassanides, obtint un traité de ṣulḥ lui laissant autorité sur la partie de la ville que les musulmans n'avaient pas prise, ainsi que sur une grande part de la province Au IV e /X e siècle, sa population était considérée comme largement muʿtazilite. Le chiisme y était également bien implanté, notamment sous sa forme ismāʿīlienne, et le dāʿī ʿAbdān (m.

  (al-Ṭabarī, I, 2537). Al-Hurmuzān reprit néanmoins les armes peu après et le calife ʿUmar I (r. 13-23/634-44) envoya contre lui Abū Mūsā al-Ashʿarī (m. v. 42/662), alors gouverneur d'al-Baṣra, qui s'empara d'al-Ahwāz en 17/638 puis acheva la conquête de la province. Comme souvent, les divergences des historiens sur les modalités de la conquête (par la force, ʿanwatan, ou sur la base d'un ṣulḥ) reflètent surtout des discussions postérieures sur le statut de son territoire, qui fut laissé aux populations autochtones en contrepartie du versement du kharāj. Selon al-Ṭabarī, le conquérant de Sūq al-Ahwāz fut Ḥurqūṣ b. Zuhayr al-Saʿdī (m. 38/657), envoyé par ʿUmar pour défaire al-Hurmuzān (al-Ṭabarī, 1:2541). Plaque tournante du commerce terrestre et fluvial entre al-Baṣra, le Fārs et le Jibāl (Muqaddasī, 411), al-Ahwāz s'imposa aux premiers siècles A.H. comme la capitale politique et économique du Khūzistān. Sa position stratégique et sa prospérité excitèrent la convoitise de plusieurs révoltés notoires. Les khārijites l'occupèrent à plusieurs reprises à la fin du VII e siècle et, de là, lancèrent des expéditions contre le Sud de l'Iraq. Nāfiʿ b. al-Azraq (m. 65/685) y implanta son quartier général en 64/684. Sous les derniers Umayyades et les premiers Abbassides, elle tomba successivement aux mains des révoltés chiites ʿAbd Allāh b. Muʿāwiya (m. 129/746-7) et Ibrāhīm b. ʿAbd Allāh (m. 145/763). La ville souffrit surtout de la révolte des Zanj, dans la seconde moitié du III e /IX e siècle. Ceux-ci s'en emparèrent en 256/870 et en 259/872 ; ils la pillèrent à nouveau et massacrèrent ses habitants en 261/874-5 (al-Ṭabarī, III, 1889 ; Popovic, 96, 109, 111). Le rôle politique d'al-Ahwāz s'accrut néanmoins au IV e /X e siècle sous les Būyides. Située à mi-chemin entre les principaux centres du régime (Baghdād et Shīrāz), elle devint une résidence appréciée de l'amīr al-umarā' Muʿizz al-Dawla (m. 356/967) et de ses successeurs. ʿAḍud al-Dawla (m. 372/983) y fit reconstruire son monumental pont de briques (le Qanṭarat Hinduwān), au milieu duquel se dressait une magnifique mosquée. Afin d'accélérer les communications entre al-Baṣra et al-Ahwāz, il fit également relier le Tigre et le Kārūn par un canal.


Lucile Rapin

Paul Metzak

Jennifer C Whitman

Marion Dohen

Hélène Loevenbruck

Todd Woodward

Verbal thought generation and auditory perception in schizophrenia: a follow-up fMRI study

Introduction

Although overactive inner speech has been one of the major theoretical accounts of auditory verbal hallucinations (AVHs) in schizophrenia, inner speech associated with the mental generation of thoughts has only been rarely examined. In this study, we present a follow-up experiment [START_REF] Rapin | Increased Activation of the Auditory Cortex in Schizophrenia in Thought Generation and Perception[END_REF] investigating the neural underpinnings of verbal thought generation and verbal auditory perception using fMRI. Constrained principal component analysis (CPCA;Takane & Hunter, 1991;[START_REF] Woodward | Functional connectivity reveals load dependent neural systems underlying encoding and maintenance in verbal working memory[END_REF]Metzak et al., 2010a,b) was used to identify networks of cerebral activation associated with generating thoughts and auditory perception.

Methods

Two experimental conditions were examined. In the first one, participants were required to mentally generate a definition of a common word presented on the screen. In the second condition, they were required to listen to the definition of a common word presented on the screen. All definitions started with the phrase "Something you" (e.g., pen: something you use for writing). To avoid semantic ambiguity, all words were accompanied by an illustration of the word. Each condition consisted of 30 words. 7 schizophrenia patients with AVHs, 7 schizophrenia patients without AVHs and 14 healthy controls participated. All were righthanded. An event-related fMRI protocol was used in a 3T scanner for a total of 352 images. Preprocessing was performed using SPM5 and statistical analyses were carried out using CPCA with a finite impulse response (FIR) model, and an orthogonal rotation applied on both conditions simultaneously.

Results

Four reliable components were extracted. The first component extracted (Figure 1) included the superior temporal cortex and the visual cortex bilaterally and showed a higher intensity for the auditory perception condition relative to the generation condition, but did not differ between groups. The second component (Figure 2) mainly involved the primary visual cortex (and the superior parietal lobule). It also included deactivations in the ventral anterior cingulate cortex (ACC) bilaterally and showed a greater hemodynamic response (HDR) for the generating condition relative to the auditory perception condition. An interaction between time and group was also present whereby the non hallucinating group had a higher HDR peak value, closely followed by the hallucinating group. The third component (Figure 3) involved activations in the secondary auditory cortex as well as in the insula and deactivations in the dorsal frontal regions and in the premotor and somato-sensory cortices. This component showed higher HDR in the auditory perception condition compared to the generation one. A trend towards higher intensity for the hallucinating group relative to the others was also observed. The fourth component (Figure 4) represented both the task positive and the task negative networks [START_REF] Fox | The human brain is intrinsically organized into dynamic, anticorrelated functional networks[END_REF] and differed between conditions, whereby higher activity was found for the thought generation condition compared to the auditory perception condition.

Conclusions

These results verified our pilot study [START_REF] Rapin | Increased Activation of the Auditory Cortex in Schizophrenia in Thought Generation and Perception[END_REF] by confirming that, regardless of group and relative to the auditory perception condition, the thought generation involves more visual cortex activation as well as an overall intensified task positive network, suggesting increased cognitive effort for this condition. One component, involving the bilateral visual cortices associated with deactivations in the ventral ACC showed a greater HDR for both groups of patients relative to controls regardless of condition, suggesting inefficiency in this network (increased activity required to perform difficult task). Moreover, a trend towards overactivation of the secondary auditory cortex was found in hallucinating patients, supporting the hypothesis of heightened left auditory cortex in hallucinating schizophrenia patients (Hoffman & McGlashan, 1997).

Figure 1 :

 1 Figure 1: Images, plots of BOLD signal, and plots of predictor weights for Component 1.

Figure 2 :

 2 Figure 2: Images, plots of BOLD signal, and plots of predictor weights for Component 2.

Figure 3 :

 3 Figure 3: Images, plots of BOLD signal, and plots of predictor weights for Component 3.

Figure 4 :

 4 Figure 4: Images, plots of BOLD signal, and plots of predictor weights for Component 4.


Isabellea Audo
email: isabelle.audo@inserm.fr

Marie-Elise Lancelot

Saddek Mohand-Saïd

Aline Antonio

Aurore Germain

José-Alain Sahel

Shomi S Bhattacharya

Christina Zeitz
email: christina.zeitz@inserm.fr.

Novel C2orf71 mutations account for ~1% of cases in a large French arRP cohort

Keywords: Rod-cone dystrophies, Retinitis pigmentosa, C2ORF71, genetic prevalence Rod-cone dystrophies, RP, C2orf71

Autosomal-recessive retinitis pigmentosa (arRP) is a genetically heterogeneous group of disorders to which a novel gene, C2orf71, was recently associated. The purpose of our study was to establish the prevalence and nature of C2orf71 mutations in a clinically well-characterized cohort of 345 sporadic and arRP French cases. Direct sequencing of C2orf71 was performed in 209 subjects for whom mutations had previously been excluded by microarray technology and direct sequencing of EYS. Putative pathogenicity of the identified variants was evaluated through cosegregation analysis, screening of more than 188 control chromosomes and prediction programs. We identified two patients compound heterozygous for mutations predicted to lead to a premature stop codon, 3 of which are novel. In addition, 3 patients carried a single variant of likely pathogenicity. Furthermore a large number of novel putative non-disease causing variants were identified, highlighting the extremely polymorphic nature of C2orf71. To our knowledge, our study provides the first large scale screening of C2orf71 in a French arRP cohort through direct sequencing and suggests that it would account for approximately 1% of arRP cases. ©2010 Wiley-Liss, Inc.

INTRODUCTION

Rod-cone dystrophy also known as retinitis pigmentosa (RP) is a heterogeneous group of inherited disorders in most cases primarily affecting rods with secondary cone degeneration. It is the most common inherited form of potentially severe retinal degeneration and blindness, with a frequency of about 1 in 4000 births and more than 1 million individuals affected worldwide [START_REF] Hartong | Retinitis pigmentosa[END_REF]. Patients initially complain of night blindness due to rod dysfunction followed by progressive visual field constriction, abnormal color vision and eventually loss of central vision. Inheritance of RP can be either autosomal recessive (ar), autosomal dominant (ad) or X-linked (xl) with also rare cases of mitochondrial transmission. Recessive and isolated cases account for 50-60% of all forms of RP [START_REF] Hartong | Retinitis pigmentosa[END_REF]). So far, 32 genes have been implicated in arRP and 3 additional loci have been mapped (http://www.sph.uth.tmc.edu/Retnet). Most of the identified genes accounted for only 1-2% of cases each, besides EYS (MIM# 612424) and USH2A (MIM# 608400), which were found to account for up to 15.9% and up to 23%of isolated arRP respectively [START_REF] El-Aziz | EYS, encoding an ortholog of Drosophila spacemaker, is mutated in autosomal recessive retinitis pigmentosa[END_REF][START_REF] Audo | EYS is a major gene for rod-cone dystrophies in France[END_REF]Bandah-Rozenfeld et al.; [START_REF] Barragan | Mutation spectrum of EYS in Spanish patients with autosomal recessive retinitis pigmentosa[END_REF][START_REF] Littink | Mutations in the EYS gene account for approximately 5% of autosomal recessive retinitis pigmentosa and cause a fairly homogeneous phenotype[END_REF]) and [START_REF] Rivolta | Missense mutation in the USH2A gene: association with recessive retinitis pigmentosa without hearing loss[END_REF][START_REF] Bernal | Mutations in USH2A in Spanish patients with autosomal recessive retinitis pigmentosa: high prevalence and phenotypic variation[END_REF][START_REF] Mcgee | Novel mutations in the long isoform of the USH2A gene in patients with Usher syndrome type II or non-syndromic retinitis pigmentosa[END_REF]. Recently, the prevalence of loss-of-function mutations in FAM161A (MIM#613596) was found to be ~12% in arRP patients from Israel and Palestinian territories (Bandah-Rozenfeld et al. 2010b). However, in a similar study the prevalence of FAM161Amutations in arRP patients from Germany was 2%-3% [START_REF] Langmann | Nonsense mutations in FAM161A cause RP28-associated recessive retinitis pigmentosa[END_REF]. Future prevalence studies will show if FAM161A can be also considered as a major gene in other arRP populations.

Two articles in the same issue of the American Journal of Human Genetics report findings on a new causative gene, C2orf71 (MIM#613425), for arRP using a homozygosity mapping approach on consanguineous and nonconsanguineous families [START_REF] Collin | Mutations in C2ORF71 cause autosomal-recessive retinitis pigmentosa[END_REF][START_REF] Nishimura | Discovery and functional analysis of a retinitis pigmentosa gene, C2ORF71[END_REF]. C2orf71 (Chromosome 2 Open reading Frame 71) harbors 2 exons and encodes for a 1288 amino-acid protein. This protein is highly expressed in retina [START_REF] Collin | Mutations in C2ORF71 cause autosomal-recessive retinitis pigmentosa[END_REF][START_REF] Nishimura | Discovery and functional analysis of a retinitis pigmentosa gene, C2ORF71[END_REF]) and seems to play a role in primary cilia of the outer segment [START_REF] Nishimura | Discovery and functional analysis of a retinitis pigmentosa gene, C2ORF71[END_REF], although its precise function remains to be elucidated. Bioinformatic analysis predicts a proline-rich motif at the C-terminus and functional studies suggest post-translational lipid modifications at the second and third amino acid that may be important for the sorting and function of the protein [START_REF] Nishimura | Discovery and functional analysis of a retinitis pigmentosa gene, C2ORF71[END_REF].

To assess the prevalence of C2orf71 pathogenicity, we screened 209 patients of different ethnic origins from a French cohort of 345 arRP who had previously been excluded for known pathogenic variants by applying a commercially available microarray (ASPER Ophthalmics, Tartu, Estonia) and by direct sequencing of EYS [START_REF] Audo | EYS is a major gene for rod-cone dystrophies in France[END_REF].

PATIENTS AND METHODS

Three hundred forty five index patients with a presumed diagnosis of simplex or arRP were ascertained. Informed consent was obtained from each patient and unaffected family member after explanation of the study and its potential outcome. The study protocol adhered to the tenets of the Declaration of Helsinki and was approved by the local ethics committee. Each patient underwent full ophthalmic examination as described earlier [START_REF] Audo | EYS is a major gene for rod-cone dystrophies in France[END_REF]. Total genomic DNA was extracted from peripheral blood leukocytes according to manufacturer recommendation (Puregen Kit, Qiagen, Courtaboeuf, France). Prior to C2orf71 mutation screening, the DNA of all index patients was analyzed for known arRP mutations by microarray analysis on a commercially available microarray (ASPER Ophthalmics, Tartu, Estonia). Samples, which were excluded for known variants were further excluded for mutations in the coding exons and flanking intronic regions of EYS by direct sequencing [START_REF] Audo | EYS is a major gene for rod-cone dystrophies in France[END_REF]. The remaining 209 samples were further investigated for variants in the coding exons and flanking intronic regions of C2orf71 by PCR and direct sequencing. The 2 coding exons, and flanking intronic regions of C2orf71 were PCR amplified in 11 fragments (C2orf71 RefSeq NM_001029883.1) using oligonucleotides reported in Supplementary Table 1, a commercially available polymerase (HotFire, Solis Biodyne, Tartu, Estonia), 2 mM MgCl2 at an annealing temperature of 60°C for 1 minute. PCR products were enzymatically purified (ExoSAP-IT, USB Corporation, Cleveland, Ohio, USA purchased from GE Healthcare, Orsay, France) sequenced and investigated as previously reported [START_REF] Audo | EYS is a major gene for rod-cone dystrophies in France[END_REF]. Nucleotide numbering reflects cDNA numbering with +1 corresponding to the A of the ATG translation initiation codon in the reference sequence, according to journal guidelines (www.hgvs.org/mutnomen). The initiation codon is codon 1. The correct nomenclature for each change was checked applying Mutalyzer (http://www.lovd.nl/mutalyzer/). PolyPhen (Polymorphism Phenotyping, http://genetics.bwh.harvard.edu/pph/; http://genetics.bwh.harvard.edu/pph2/) [START_REF] Ramensky | Human non-synonymous SNPs: server and survey[END_REF][START_REF] Adzhubei | A method and server for predicting damaging missense mutations[END_REF], and SIFT (Sorting Intolerant From Tolerant; http://blocks.fhcrc.org/sift/SIFT.html) (Ng and Henikoff A sequence variant was considered to be pathogenic if it represented a nonsense or frameshift mutation and cosegregated with the phenotype. A sequence variant was considered to be of likely pathogenicity if it affects an evolutionary conserved residue on amino acid and nucleotide level, if it was predicted by one or more programs mentioned above to be pathogenic, appeared only 1-3 times in 209 patients and did not show up in at least 188 control alleles. Variants assumed to be most likely non pathogenic did not affect conserved amino acids or nucleotides, represent silent mutations, did not co-segregate with the phenotype, were predicted by both programs to be benign, appeared in >3 patients or showed up in controls.

RESULTS AND DISCUSSION

Mutation screening of C2orf71 by direct sequencing of all coding exons was performed on 209 index patients with a diagnosis of arRP. Of these, 2 patients were compound heterozygous for mutations, which are predicted to lead to premature stop codons (Table 1, first part, Figure 1). Patient CIC00261 from family 180 (II-1 in Figure 1A) was heterozygous for a deletion (c.946delA) (reviewed but not shown) resulting in a frameshift mutation p.Asn316MetfsX7. This would lead to a truncation of more than 80% of the C2orf71 protein. Co-segregation analysis revealed that this change was inherited from the unaffected mother (I-2, Figure 1A). The same deletion has been reported by Collin et al. as a homozygous mutation in a Dutch family [START_REF] Collin | Mutations in C2ORF71 cause autosomal-recessive retinitis pigmentosa[END_REF]. Our patient's family originated from the Southwest of France but the mother did not precisely know her ancestral history so a Dutch descent with a founder effect could not be excluded. In addition, CIC00261 was heterozygous for a novel nonsense mutation (c.3002G>A transition), which is predicted to lead to a stop codon (p.Trp1001X) (reviewed but not shown). Co-segregation analysis revealed that this change was inherited from the unaffected father (I-1). The unaffected brother (II-3) showed two normal alleles (Figure 1A). Patient CIC02622 from family 927 (II-3 in Figure 1B) was heterozygous for a novel nonsense mutation (c.2950C>T) leading to a stop codon (p.Arg984X) (reviewed but not shown), which was inherited from the unaffected father (I-1, Figure 1B). In addition, the patient was heterozygous for a novel duplication (c.2327dupC), which is predicted to lead to a frameshift and premature truncation of the protein (p.Leu777PhefsX34) (reviewed but not shown). This change was inherited from the unaffected mother (I-2, Figure 1B). Similar to the index patient, the affected brother (II-5, Figure 1B), was heterozygous for both mutations while the unaffected brother (II-1, Figure 1B) carried only the nonsense mutation. 2. Besides a variable age of onset and degree of visual impairment, both patients showed severe retinal dysfunction with marked macular atrophy. Similar clinical findings were reported by Collin and colleagues [START_REF] Collin | Mutations in C2ORF71 cause autosomal-recessive retinitis pigmentosa[END_REF]). C2orf71 mutations are therefore associated with severe RP and early cone involvement, which highlights the importance of this gene for normal photoreceptor function. Received <date>; accepted revised manuscript <date>.

C2orf71 screening of our cohort led to the identification of many other variants apart from these 4 premature stop mutations. These changes were classified as of likely pathogenicity (Table 1, second part), most likely non pathogenic (Table 3) and as known variants (Supplementary Table 2).

The group of variants with likely pathogenicity includes 3 missense mutations (Table 1 second part), which were evolutionary conserved (Supplementary Figure 1) and predicted by at least one program to be pathogenic. Unfortunately, no other affected family member of the respective index patients was available for genetic testing to prove the pathogenic character of those mutations. Patients carrying these mutations were only heterozygous for a single mutation. The p.Val615Asp variant was also detected in a patient in a different study investigating C2orf71 variations in patients with Leber Congenital Amaurosis and arRP [START_REF] Sergouniotis | A survey of DNA variation of C2ORF71 in probands with progressive autosomal recessive retinal degeneration and controls[END_REF]. As described here no second mutation was identified, but the mutation did not show up in control samples. To date, the 5'UTR of C2orf71 is not described. In addition other isoforms of C2orf71 may exist, which may harbor the second missing mutation. Since our screening strategy was based on direct sequencing of currently known coding exons and flanking regions, we may have missed a second change. This may also hold true for a number of isolated variants identified by Sergouniotis and co-workers [START_REF] Sergouniotis | A survey of DNA variation of C2ORF71 in probands with progressive autosomal recessive retinal degeneration and controls[END_REF]). In addition, patients may have large heterozygous deletions, which can not be detected by our direct sequencing approach. Another possibility would be that these 3 variants are not pathogenic, that they represent rare variants or that the patients are only carriers. Our screening together with those by Sergouniotis and co-workers [START_REF] Sergouniotis | A survey of DNA variation of C2ORF71 in probands with progressive autosomal recessive retinal degeneration and controls[END_REF] showed indeed the highly polymorphic nature of C2orf71, which is concluded from the following results. Pursuing our stringent criteria, we identified 24 non-likely pathogenic variants, of which some were also identified by others [START_REF] Collin | Mutations in C2ORF71 cause autosomal-recessive retinitis pigmentosa[END_REF][START_REF] Sergouniotis | A survey of DNA variation of C2ORF71 in probands with progressive autosomal recessive retinal degeneration and controls[END_REF]. They comprise 15 missense changes, 8 silent mutations and 1 duplication and represent either frequent or very rare variants (Table 3). In addition, many patients were found to only carry one or more known SNPs (Supplementary Table 2). We conclude that all of them were not pathogenic since they appeared frequently in patients or controls in our study or in those performed by Sergouniotis and co-workers [START_REF] Sergouniotis | A survey of DNA variation of C2ORF71 in probands with progressive autosomal recessive retinal degeneration and controls[END_REF] or did not affect a conserved amino acid or nucleotide as it is the case for p.His122His Val247Ala and p.Pro1254Leu (data not shown).

All together, our study provides a large scale screening of C2orf71 in a French cohort of arRP patients through direct sequencing. It identifies 2 families with 3 novel and 1 known mutations leading to premature stop codon and the absence of protein production as a putative disease mechanism. Furthermore, our study identifies 3 patients heterozygous for variants with uncertain pathogenic character. C2orf71 would therefore account for at least 0.6-1.7% of arRP. This figure derives from PCR based exonic sequencing of 209 patients from a total of 345 ar and simplex RP patients. All 345 patients had been pre-screened by microarray or direct sequencing approaches for other RP causing mutations/genes. This pre-screening showed at least a single heterozygous putative mutation in other RP causing genes in 136 patients. Some of them carried two mutations and some of the mutations even showed familial segregation. Of note is that some of the heterozygous mutations amongst the 136 patients may have occurred by chance and therefore it is possible that some of these patients may actually carry a putative C2orf71 mutation. If this were to be true then our estimated figure for C2orf71 mutation (0.6-1.7%) is likely to go up since the figure is based on a total of 345 patients and not only on the 209 patients who have been sequenced in C2orf71. Received <date>; accepted revised manuscript <date>.

F o r P e e r R e v i e w

Interestingly, another prevalence study from a cohort with 286 affected individuals with Leber Congenital Amaurosis and arRP identified a large number of variants in C2orf71, however for none of the variants pathogenicity could be proven [START_REF] Sergouniotis | A survey of DNA variation of C2ORF71 in probands with progressive autosomal recessive retinal degeneration and controls[END_REF].

Although our study documents that C2orf71 is not a major gene for arRP, however at least for the French arRP population it is still an important gene to screen and its prevalence is similar to most of the other genes reported in arRP [START_REF] Hartong | Retinitis pigmentosa[END_REF].

  packages as well as evolutionary conservation were used to predict the pathogenic character of the sequence alteration.

Figure 2 :

 2 Figure 2: Colour fundus photographs (A-C) and autofluorescence imaging (B-D) of two index patients with C2orf71 muations leading to a premature stop codon. (A-B) CIC00261 showed little pigmentation in retinal periphery, some optic disc pallor and narrowed retinal vessels. In addition, there were macular atrophic changes which manifests by a patchy loss of autofluorescence; (C-D) CIC02622 showed marked atrophic changes in the posterior pole with large corresponding areas of loss of autofluorescence.

  Figure 1: C2orf71 mutations and cosegregation in 2 families with arRP. Filled symbols represent affected and unfilled unaffected subjects. Squares indicate males and circles females. Arrows reflect the index patients. The equal signs depict the normal allele.

  Received <date>; accepted revised manuscript <date>.

Table 1 :

 1 C2orf71variants which are pathogenic and likely pathogenic Exon

Sequence variation Protein effect Frequency in patient chromosomes Frequency in control chromosomes Polyphen Prediction SIFT Prediction

			F o
				r
				P
				e e r
	1 1 1	c.946delA c.2327dupC p.Leu777Phefs*34 p.Asn316Metfs*7 c.2950C>T p.Arg984*	1/418 1/418 1/418	a 0/242 0/188 0/188 R e v i n.a. n.a. n.a. e w	n.a. n.a. n.a.	Conservation n.a. n.a. n.a.
	1	c.3002G>A	p.Trp1001*	1/418	0/188	n.a.	n.a.	n.a.
	1	c.1114G>A	p.Asp372Asn	1/418	0/188	benign	affects protein	conserved
							function
	1	c.1835T>C	p.Leu612Pro	1/418	0/192	possibly	tolerated	conserved
						damaging
	1	c.1844T>A	p.Val615Asp	2/418	0/192	possibly	tolerated	conserved
				b 1/572	b 0/190	damaging
		a performed by (Collin et al. 2010)
		b performed by (Sergouniotis et al. 2010)

Clinical data of index patients for the 2 families are summarized in table 2 and colour fundus and autofluorescence images are displayed in figure

Table 2 :

 2 clinical data for the 2 patients compound heterozygous for mutations leading to premature stop codons

	Patient	Age at time of diagnosis	Age at time of examination	Sex	Relevant medical and ophthalmic history	Family history	Symptoms	BCVA OD/OS Refraction	Colour vision	Binocular Goldman visual field, III4 isopter	Full field ERG	Multifocal ERG	Fundus and autofluorescence imaging (Figure 2)	Horizontal foveal thickness measured by SD-OCT*
							Night
							blindness						Typical RP
	CIC00261 Family 180 (II-1)	18	35	F	Refractive surgery for myopia at 25	None from Southwest of France	at 18 ; visual field constriction and decreased	20/40 20/50 +0(-1)0° +0(-1.50)170°	tritanopia at the 28 Hue of Roth	40° both horizontally and vertically	UD	UD	fundus changes some macular with little pigment migration and	OD 204µm OS 181µm
							vision at						atrophic changes
							age 25
	CIC02622 Family 927 (II-3)	Early childhood	44	M	none	One brother with RP from the West part of France	Life long night blindness and vision decreased	LP both eyes -3.50(-1)180° -3.25	NP	Inferior to 10°	UD	NP (unstable fixation)	Typical RP changes fundus changes with marked macular atrophic	OD 147µm OS 139µm

BCVA: best corrected visual acuity; OD: Oculis dextra (right eye); OS: Oculis Sinistra (left eye); UD: undetectable; NP: not performed; LP: light perception; RP: retinitis pigmentosa; SD-OCT: spectral-domain Optical Coherence Tomography; *normal value 227.64±11.74µm

[START_REF] Carpineto | Correlation in foveal thickness measurements between spectral-domain and time-domain optical coherence tomography in normal individuals[END_REF]

.

HUMAN MUTATION Mutation in Brief #____ (20XX) Online

© 2010 WILEY-LISS, INC.

Table 3 :

 3 C2orf71 variants that are most likely non pathogenic . § except where it is noted, the variants occurred heterozygously a performed by[START_REF] Collin | Mutations in C2ORF71 cause autosomal-recessive retinitis pigmentosa[END_REF] b performed by[START_REF] Sergouniotis | A survey of DNA variation of C2ORF71 in probands with progressive autosomal recessive retinal degeneration and controls[END_REF]) ° located in a highly polymorphic region with 2 known SNPs in the same region rs72183347 and rs72122505• n.d. not done • n.a. not applicable

					Human Mutation
	Exon	Sequence	Consequence	Frequency in patient	Frequency in	Polyphen	SIFT	Conservation	Co-segregation
		variation		chromosomes §	control	Prediction	Prediction
					chromosomes
	1	c.679G>A	p.Glu227Lys	5 /418	0/190	benign	affects protein	conserved	did not co-segregate in
				b 3/572	b 0/382		function		one family
	1	c.773G>T	p.Arg258Ile	1/418	0/190	possibly	tolerated	same nucleotide	n.d.
						damaging		found in Stickleback
	1	c.935G>A	p.Ser312Asn	1 /418	0/188	benign	tolerated	not conserved on	n.d.
								nucleotide level
	1	c.958C>T	p.Arg320Cys	1 /418	0/188	benign	tolerated	same amino acid	n.d.
								found in rat and
								mouse
	1	c.1132G>A	p.Glu378Lys	1 /418	0/188	benign	tolerated	not conserved on	n.d.
								nucleotide level
	1	c.1545A>G	p.Gln515Gln	1/418	n.d.	n.a.	n.a.	not conserved on	n.d.
								nucleotide level
	1	c.1882G>A	p.Ala628Thr	2/418	0/192	benign	tolerated	not conserved on	n.d.
								nucleotide level
	1	c.2063G>A	p.Cys688Tyr	4/418	5/192	probably	tolerated	not conserved	n.d.
				b 0/572	b 3/190	damaging
	1	c.2502T>C	p.Pro834Pro	2/418	n.d.	n.a.	n.a.	not conserved	n.d.
				b 1/572	b 0/382
	1	c.2600C>T	p.Pro867Leu	1/418 patient	n.d.	benign	tolerated	not conserved	n.d.
				homozygous for this
				variant
				a 1	a 1/180
				b 2/572	b 1/380
	1	c.2860C>T	p.Pro954Ser	1/418	0/190	possibly	tolerated	not conserved on	n.d.
						damaging		nucleotide level
	1	c.2864G>A	p.Arg955Gln	3 /418	2/190	benign	tolerated	not conserved	n.d.
				b 0/572	b 1/190

John Wiley & Sons, Inc. HUMAN MUTATION Mutation in Brief #____ (20XX) Online © 2010 WILEY-LISS, INC.

John Wiley & Sons, Inc.Human Mutation

Contract grant sponsor: Agence Nationale de la Recherche, Foundation Voir et Entendre, Foundation Fighting Blindness (FFB); Grant number: CD-CL-0808-0466-and C-CMM-0907-0428-(FFB), Ville de Paris et Région Ile de France. Short Title: C2orf71 Mutations in a French cohort. Communicated by <Please don't enter>

SUPPORTING FILES:

Supplementary Figure 1

Evolutionary conservation of altered amino acid residues (red) in other orthologs. Multiple amino acid alignments show evolutionary conservation of mutated residues (green).

Supplementary Table 1: primer sequences used for C2ORF71 PCR amplification and sequencing

Primer

Sequence (5'-3') Received <date>; accepted revised manuscript <date>.

Supplementary


A Giovagnoli

H P Wynn

Stochastic orderings for discrete random variables

A number of application areas of statistics make direct use of stochastic orderings.

Here the special case of discrete distributions is covered. For a given partial ordering one can define the class of all -order preserving functions x y ⇒ g(x) ≤ g(y). Stochastic orderings may be defined in terms of :

for all order-preserving g. Alternatively they may be defined directly in terms of a class of functions F: X st Y ⇔ E X g(X) ≤ E Y g(Y ) for all f ∈ F. For discrete distributions Möbius inversions plays a useful part in the theory and there are algebraic representations for the standard ordering ≤ for integer grids. In the general case, based on F, the notion of a dual cone is useful. Several examples are presented.

Introduction

Stochastic orderings and order preserving functions have a long history with perhaps the most famous being those associated with the Gini coefficient. Some general works are [START_REF] Shaked | Stochastic orders and their applications[END_REF] [START_REF] Shaked | Stochastic orders and their applications[END_REF] and [START_REF] Müller | Comparison Methods for Stochastic Models and Risks[END_REF] [START_REF] Müller | Comparison Methods for Stochastic Models and Risks[END_REF] . Sub-areas include group invariant ordering: [START_REF] Marshall | Inequalities: Theory of majorization and its applications[END_REF] [START_REF] Marshall | Inequalities: Theory of majorization and its applications[END_REF], [START_REF] Eaton | Lectures on topics in probability inequalities[END_REF] [START_REF] Eaton | Lectures on topics in probability inequalities[END_REF] and [START_REF] Giovagnoli | G-majorization with applications to matrix orderings[END_REF] [START_REF] Giovagnoli | G-majorization with applications to matrix orderings[END_REF] and orderings related to association and dependence measures: Joe (1987) [START_REF] Joe | Multivariate models and dependence concepts[END_REF] and [START_REF] Haberman | Measures of association[END_REF] [START_REF] Haberman | Measures of association[END_REF]. Applications are to general statistical theory, particularly testing: [START_REF] Bickel | Descriptive statistics for non-parametric models. i introdution[END_REF] [START_REF] Bickel | Descriptive statistics for non-parametric models. i introdution[END_REF] and risk and insurance, eg [START_REF] Artzner | Coherent measures of risk[END_REF] [START_REF] Artzner | Coherent measures of risk[END_REF].

One approach is to begin with a non-stochastic ordering betweeen variables, consider the class of order preserving functions, and then make the variables stochastic and, via expectations, extend the ordering to random variables. These are sometimes called "integral" stochastic orderings and this is the starting point of the present paper.

Most of the existing literature focuses on real (continuous) univariate or multivariate random variables. There is less work on stochastic orderings for discrete random variables. We try to give a general theory that includes a large class of examples. Although one can consider some results for discrete random variables as specialisations of the continuous theory, there are nice features of the discrete case which can be used for some closer analysis, particularly the use of Möbius inversion.

2 Partial orderings and order-preserving functions

Lattices

We shall consider a partially ordered set X , which is finite unless otherwise stated. We shall use X both as the base set of the partial ordering and the support of a probability distribution.

Definition 1 By a partial ordering on the base set X we mean a relationship which satisfies 1. for any x ∈ X , x x, 2. for x, y ∈ X , x y and y x implies x = y, 3. for x, y, z ∈ X , x y and y z implies x z.

In this paper z ≺ y will mean z y, z = y. In some of the examples X will be specialised to the product set

S = N 1 × • • • × N d
where N i = {0, . . . , n i -1}. We shall call the entrywise order x ≤ y ⇔ x i ≤ y i , (i = 1, . . . d) the standard order on the discrete rectangular grid.

Definition 2 A total ordering on X is a partial ordering with the the additional condition that for any x, y ∈ X either x y or y x Definition 3 For two partial orderings σ , τ on X we shall say that σ is a refinement of τ if for all x, y ∈ X x τ y ⇒ x σ y.

In the following schematic array we have d = 2 and n 1 = n 2 = 4. We represent the total order S by the location of the numbers 1 to 16 from the lower left hand corner, representing the point (0, 0), to the top right hand corner, representing [START_REF] Eaton | Lectures on topics in probability inequalities[END_REF][START_REF] Eaton | Lectures on topics in probability inequalities[END_REF]. We see that this total order is a refinement of the standard ordering. Taking any pair x, y with x ≤ y we see that y is allocated a larger number than x.

We can define generalized upper and lower quadrants in X Definition 4 A generalized upper quadrant is a subset of X of the form Q u x 0 = {y ∈ X : x 0 y} for a given x 0 in X . Similarly, a generalized lower quadrant is

Q l x 0 = {y ∈ X : y x 0 }
We shall consider partial orderings such that for any two elements x,y there is always a "minimal element" x ∧ y such that x ∧ y x, x ∧ y y and if z x, z y, then z x ∧ y. Similarly for a "maximal element" x ∨ y: thus the set X equipped with partial order becomes a lattice. The following holds Remark 1. The intersection of generalized upper (lower) quadrants is an upper (lower) quadrant, since

Q u x ∩ Q u y = Q u x∨y Q l x ∩ Q l y = Q l x∧y
The term ideal is used in lattice theory, but we prefer quadrant because of its geometric interpretation in the case of S defined above.

A more general notion is upper and lower sets.

Definition 5 A set U ⊆ X is an upper set if for all x ∈ U and all y ∈ X , if x y, then y ∈ U . Similarly for a lower set L.

It is straightforward to see that all upper (lower) quadrants are upper (lower) sets, but not conversely.

Remark 2 A set U ∈ X is an upper set if and only if it is a union of upper quadrants; that is to say there is a set I * such that

U = ∪ x∈I * Q u x
Similarly a lower set is a union of lower quadrants.

Order preserving functions

Definition 6 Given a partial ordering , a -preserving function g on X is one which satisfies

x y ⇒ g(x) ≤ g(y)

for all x, y ∈ X . A -decreasing function is one which satisfies

x y ⇒ g(x) ≥ g(y)
for all x, y ∈ X .

The characterisation of order preserving functions is a key result. We first need a definition.

Definition 7

For any set U ⊆ X we define the indicator functions

δ U (x) = 1 if x ∈ U 0 otherwise
Theorem 1 Let U be the collection of all upper sets. A real function φ(x) is ≺-order preserving if and only if there are constants α 0 and α U ≥ 0 such that

φ(x) = α 0 + U ∈U α U δ U (x) (1)
Proof. If φ(x) is expressible in the given form then it is clearly order preserving since each δ U (x) is order preserving from Remark 2 above.

In the other direction we proceed by construction in the case when all φ(x) values are distinct. The proof is easily adapted for the case of "ties". Let φ(x) be a -increasing function. Label the x to be increasing by increasing φ(x):

φ(x 0 ) < φ(x 1 ) < . . . < φ(x n-1 ) For any x i ∈ X define U x i = {y ∈ X : φ(x i ) ≤ φ(y)}.
Since the total ordering induced by φ(x) is a refinement of , every U x i is an upper set:

x i y ⇒ φ(x i ) ≤ φ(y) ⇒ y ∈ U x i Set α 0 = φ(x 0 ) and α i = φ(x i ) -φ(x i-1 ), i = 1, . . . , n -1,
and α U = 0 for all other U ∈ U. Then the construction is

α 0 + n-1 i=1 α i U x i (x j ) = φ(x 0 ) for j = 0 φ(x 0 ) + j i=1 {φ(x i ) -φ(x i-1 )} = φ(x i ) for j = 0, 1, ...n -1
This completes the proof.

Stochastic orderings

Order preserving functions with respect to a partial ordering lead naturally to a certain type of stochastic ordering. Let X and Y be two random variables defined with respect to the same base measure on X . We shall use the notation p X (x) = Pr{X = x} and for a subset A ⊂ X we shall sometimes write p X (A) = Pr{X ∈ A}. It is convenient to take the base measure as uniform. We define a partial ordering as above.

Definition 8 For two random variables X, Y on X , and a partial ordering, , on X we define an integral stochastic orderings of Type I as the stochastic ordering st by

X st Y ⇔ E X {g(X)} ≤ E Y {g(Y )}
for all -preserving functions g on X .

The characterisation of order preserving functions in Theorem 2 gives a characterisation of all st .

Theorem 2 For a partial ordering on X and random variables X, Y we have X st Y if and only if for all upper sets U ⊆ X

Pr{X ∈ U } ≤ Pr{Y ∈ U } Proof. Using the characterisation (1) we have E X {φ(X)} = α 0 + U ∈U α U p X (U ) If p X (U ) ≤ p Y (U ) then it follows, since all α U ≥ 0, that E X {φ(X) ≤ E Y {φ(Y )}.
The converse follows by recalling that the indicator functions δ U (x) are order preserving.

The dual ordering ˜ is given by

x ˜ y ⇔ y x
(the authors find it confusing to write ).

Definition 9 For two random variables X, Y on X and on X define the dual Type I integral stochastic -ordering by

X ˜ st Y ⇔ E X {g(X)} ≤ E Y {g(Y )}
for all -decreasing functions g on X .

We obtain the characterisation of ˜ st as in Theorem 1, by replacing upper sets by lower sets.

Type I stochastic ordering is a special case of a more general type of integral stochastic ordering based on some class of functions F.

Definition 10 For two random variables X, Y on X and on X we define integral stochastic orderings of Type II as the stochastic ordering st by

X st Y ⇔ E X {g(X)} ≤ E Y {g(Y )}
for all function g in some class F.

For this more general class the characterisation of the ordering in terms of the probability function p(x) or the cdf F (x) may be harder. There is an important technique for doing this, in some cases, which is that of dual cones. We shall return to this later.

Möbius inversion

Let be a partial ordering on a lattice X . Define the zeta function to be

ζ(x, y) = 1 if x y 0 otherwise (2)
(see for example [START_REF] Graver | Combinatorics with emphasis on the theory of graphs[END_REF]). Define the Möbius function µ by µ(x, x) = 1 and

µ(x, y) = - x z≺y µ(x, z). (3)
Möbius inversion gives the inversion of functions of a particular kind on X .

A C C E P T E D M A N U S C R I P T

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Theorem 3 (Möbius inversion) For functions f, g on S such that

g(x) = z x f (z) = z∈S ζ(z, x)f (z) we have f (x) = z x µ(z, x)g(z). (4)
Thus, µ is the inverse of ζ: let

δ(x, y) = 1, if x = y otherwise , then x z y µ(x, z) = z∈S µ(x, z)ζ(z, y) = δ(x, y)
Definition 11 The cumulative distribution function (cdf ) relative to is

F X (x) = Pr{X x} = Pr{X ∈ Q l x }
and the dual cdf is

FX (x) = Pr{x X} = Pr{X ∈ Q u x } Since F (x) = z x p X (z) = z x ζ(z, x)p X (z),
Equation (??) gives p X (x) = z x F (z)µ(z, x).

The above formulae have duals

p X (x) = z x μ(z, x) FX (x). (5)
It is a little easier to use a matrix notation at this stage. Thus, with a suitable ordering of elements of S, we define the elements of

gives, from Theorem 3

[p] = [µ][F ] (7)
F = [ ζ][p], [p] = [ µ] [F ]
where ζ and μ is for the dual ordering ˜ . Moreover, if we keep the order of the elements in the matrix rows and columns as in the ordering, the duals are obtained by taking the transpose:

[μ] = [ ζ] -1 = [ζ] T -1 = [µ] T .
It is useful to be able to switch between F and F :

[ F ] = [ ζ][µ][F ] = [ζ] T [µ][F ] = ([µ] T ) -1 [µ][F ]

Cone ordering

Using the above representations stochastic ordering, st , can be represented as a matrix cone ordering. We recall that a cone C in R d with apex at the origin is a set with the property u ∈ C ⇒ ρu ∈ C

for any scalar ρ ≥ 0. Define the cone ordering

u C v ⇔ v -u ∈ C
This is a generalisation of the standard ordering for which the cone is the non-negative quadrant in R d : {u : u ≥ 0}. The columns of a d × m matrix generate a cone by taking all non-negative linear combinations.

Starting with the representation in Theorem 2 we collect together all the statements that Pr X (U ) ≤ Pr Y (U ) for all U ∈ U into a single matrix inequality:

A[p X ] ≤ A[p Y ].
This can be written

A([p Y ] -[p X ]) ≥ 0.
This, in turn, says that

[p Y ] -[p X ] ∈ C * (A)
the dual cone to the cone formed by the columns of A. From Möbius inversion we also have a cone for

F [F Y ] -[F X ] ∈ C * (A[µ]).
The dual cone idea is useful for characterising some cases of Type II integral stochastic orderings. This is when the class of function F is itself defined by a collection of linear inequalities. These may be derived from the original lattice representation, as in the case of Type I ordering above or directly in R d .

As a small example consider the restriction of the so-called convex ordering on the nonnegative integers: N = {0, 1, . . . , N -1} . Discrete convexity can be defined in terms of second differences:

F = {f : f (i) -2f (i + 1) + f (i + 2) ≥ 0, i = 0, 1, . . . , N -3}
This can be written in matrix form:

A[f ] ≥ 0.
Consider any ordering which has such a representation. From Definition 10, and using [p X ] and [p Y ] for the vector of probabilities for X and Y , the required condition for

X ≺ st Y is [p X ] T [f ] ≥ A[p Y ] T [f ] ≥ 0 ⇔ ([p Y ] -[p X ]) T [f ] ≥ 0, for all [f ] : A[f ] ≥ 0. These condition says that [f ] ∈ C * (A), the dual cone of C(A) and that [p Y ] -[p X ] ∈ C * * (A) = C(A).
Thus without computing C * (A) we can characterise the ordering in terms of constraints on the differences:

[d] = [p Y ] -[p X ] = A T [α]
for some [α] ≥ 0. Much of the effort of characterising the ordering reduces to a minimal representation, which is equivalent to finding the extreme generators of the cone C(A); some of the rows of A may be redundant. In the above example of convexity the representation, in terms of the components of the differences [d], is

d 0 = α 0 , d 1 = -2α 0 +α 1 , . . . , d i = α i-2 -2α i-1 +α i , . . . , δ N -2 = d N -2 -2α N -1 , d N -1 = α N -1
The condition is that differences of probabilities are the backward second differences of the nonnegative α i sequence.

Where there is a natural partial ordering the cone conditions on the [d] induce cone conditions on D = [F Y ] -[F X ] which we can write

[D] = [ζ][d] = [ζ]A T [α].
In the convex case we have the representation in terms of first differences:

D 0 = α 0 , D 1 = α 1 -α 0 , . . . , D i = α i -α i-1 , . . . , D N -2 = -α N -1 , D N -1 = 0. 5 Algebra for (Z + ) d
Because of its importance in many fields we study in more detail the infinite integer lattice (Z + ) d , namely the set of all non-negative integer d-vectors and the standard ordering ≤.

We show how to "algebraize" the problem. We use a dummy variable x = (x 1 , . . . , x d ) and code α (Z + ) d by the monomial

x α = x α 1 1 • • • x α d d .
A set V ⊂ (Z + ) d , or equivalently its indicator function δ V (x) can be represented by a (multivariate) polynomial generating function

V (x) = α∈V x α
We first note that

Z + d (x) = 1 d i=1 (1 -x i )
For upper and lower quadrants algebraically we have:

Q l α (x) = 0≤β≤α x β = d i=1 1 -x α i +1 i 1 -x i and Q u α (x) = β≥α x β = x α d i=1 (1 -x i )
Motivation is provided by the identity in one-dimension:

1 + x + x 2 + . . . + x n-1 = 1 -x n 1 -x = 1 1 -x - x n+1 1 -x
The left hand side of this identity is V (x) where V = {0, 1, . . . , n -1}. The right hand side has a term x n in the numerator, with an exponent one more than the largest exponent of the left had side. This is expressed as the difference between the generating function of the upper quadrant, in this case a integer "line" starting at n + 1, and that of Z + . Möbius inversion is the relation between a single x α and the generating function for the lower quadrant Q l α given by the expansion of the numerator in

x α = x α d i=1 (1 -x i ) d i=1 (1 -x i )
We are now in a position to describe the upper sets algebraically. This development is partly based on [START_REF] Giglio | Monomial ideals and the Scarf complex for coherent systems in reliability theory[END_REF] [START_REF] Giglio | Monomial ideals and the Scarf complex for coherent systems in reliability theory[END_REF], in the context of system reliability. For two vectors α, α ∈ Z + α ∨ α = max(α 1 , α 1 ), . . . , max(α d , α d ) .

The relation Q u α ∩ Q u α = Q u α∨α carries over to the generating functions:

Q u α ∩ Q u α (x) = Q u α∨α (x)
Now consider an upper quadrant U α = ∪ α∈U * Q u α . We can use an inclusion-exclusion to obtain the generating function U (x). First index the elements of U * in some order: α (1) , . . . , α (|U * |) . Then the inclusion-exclusion lemma is:

U = i U u α 1 - i =j U u α i ∩ U u α j + . . .
We also obtain a formula for the generating function of an order preserving function. Thus, for a ≤-preserving function φ(x) with characterisation φ

(x) = α 0 + U ∈U α U δ U (x) of Theorem 1, φ * (s) = α 0 + U ∈U α U U (s) = α 0 + U ∈U α U W U (s) Z + d (s)
This follows simply by recalling that the generating function of an upper quadrant Q u α is Q u α (x).

Moments

We now take x ∈ (Z + ) d and consider the matrix indexed by x and α ξ(x, α) = x α Consider x as a support point of a random variable X and let the general α-moment be m α = E(X α ), then in the matrix notation:

[m] = [ξ] T f = [ξ] T [µ][F ]
This gives another equivalent condition to stochastic ordering, in terms of moments: Example 2. Lexicographic total ordering ≺ τ . In this case we fix an order of the variables (dimensions). For ease of explanation take the standard order. Then favouring the earlier entries we have in S = N 1 × • • • × N d (0, 0, . . . , 0, ) ≺ τ (1, 0, . . . , 0) ≺ τ . . . ≺ τ (n 1 , 0, . . . , ) ≺ τ (n 1 , 1, . . . , 0) ≺ τ . . . ≺ τ (n 1 , n 2 , 0, . . . , 0) ≺ τ . . . ≺ τ (n 1 , n 2 , . . . , n d -1) ≺ τ (n 1 , . . . , n d )

[m Y ] -[m X ] = [ξ] T [µ][α]

  [p] and [F ] for the vector of values of functions f and F consistent with the ordering. Similarly, we can think of ζ and µ as matrices [ζ] and [µ]. Then [µ] = [ζ] -1 and the equation [F ] = [ζ][p]

6 ExamplesExample 1 .

 61 for [α] ≥ 0. This says that [m Y ] -[m X ] lies in the cone generated by the columns of [ξ] T [µ]. Thus, because of the linear relationship between [M α ] and [p] for both Type I and II we can express stochastic orderings in terms of moments. In the infinite support case it is known that moments do not necessarily determine distributions, so we must restrict to the finite support case or take care with convergence. The standard ordering on S = {0, ..., n -1}. The stochastic ordering is the standard univariate stochastic orderingX ≤ st Y ⇔ F Y (z) ≤ F X (z) ⇔ FX (z) ≤ FY (z),for all z ∈ S. Here the stochastic ordering and its dual are equivalent since F (z) = 1 -F (z -1). The extension to the multivariate case S = N 1 × • • • × N d has been dealt with in Section 5.

A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT

which can be written

U u α i ∨α j (x) + . . .

The key construction in this section is to form a special polynomial W U * (x) which holds the term such as α i ∨ α j in the above expansion, together with the sign given by the inclusionexclusion. Thus

x α i ∨α j + . . .

In the algebraic theory this is called the Taylor polynomial.

Theorem 4 For an upper set U = ∪ α∈U * Q u α and associated Taylor polynomial W U * (x) the following identity holds

,

where

We do not give a formal proof of this but a simple example will make the identity clear. Let d = 2 and U * = {(3, 0), (2, 1), (0, 2)}. Then

Notice the redundancy in the inclusion-exclusion lemma. The formula in Theorem 4 gives

In general and using a compact notation we have

The formula in Theorem 4 becomes

Again this can be verified for the example. In essence the proof of Theorem 4 is an elaboration of this formula. These formulae lead to an algebraic characterisation of the standard ≤ and ≤ st -orderings.

Definition 12

The generating function (gf ) of a function g(x) on Z + is defined to be

The table below shows the order of the entries for when d = 2 and n 1 = n 2 = 4 13 14 15 16 9 10 11 12 5 6 7 8 1 2 3 4

The general statement is that x y if and only if the first nonegative entry in y -x is positive. The stochastic version considers (for X and Y ), the probability that the first nonzero entry of X -z (or Y -z) is positive. We could name this the lexigraphic stochastic ordering.

Example 3. We give an example of Type II integral stochastic ordering which cannot naturally be represented as Type I ordering, namely the supermodular stochastic ordering. Define, for a finite lattice the class of real supermodular functions namely functions satisfying

for all x, y in the lattice X , where ∧ and ∨ are the join and meet on the lattice. For a recent paper on this case see Promislow and Young (2005) [START_REF] Shaked | Stochastic orders and their applications[END_REF]. This leads to a special matrix A as in the discussion in Subsection 4.1. The dual cone analysis there leads to a representation, similar in spirit to the convex function case. Each row of A has two ones and two minus ones corresponding to the condition, and there is one such row for every pair (x, y) in the lattice. Let α(x, y) ≥ 0 be the weighting for such a row. The representation is: for each z in the lattice:

where the sum is over distinct pairs (x, y), ie every pair is visited once. As before we can induce an ordering on the cdf's: which is neatly express by simply replacing the δ functions by ζ functions:

In generalising univariate to multivariate orderings one is naturally led to the "quadrant orderings". We can express these in terms of the cdf's, F , for the lower quadrant and F for the upper quadrant ordering. Thus the lower quadrant ordering is

for all z in the lattice. The task is to express this as a Type II ordering by finding the class F in Definition 10. We use the dual cone argument in reverse. Since the condition on the distributions is

for [α] ≥ 0. This shows that f is the upper accumulation of a non-negative function and we can find the conditions on f by Möbius inversion:

In summary, the class of function F for lower quadrant ordering is the cone of all positive multiples all upper quadrant cdf's. In some sense this is a canonical case of a cone ordering in which Möbius inversion performs operation of computing the relevant dual cone.