Dr Joanne Ryan
email: joanne.ryan@inserm.fr.

Marie-Laure Ancelin

Polymorphisms of Estrogen Receptors and Risk of Depression: Therapeutic Implications Running Title: Estrogen receptor polymorphisms & depression

Keywords: Estrogen receptor polymorphisms, ESR1, ESR2, depression, depressive symptoms, major depressive disorder, anxiety, post-menopausal, hormone treatment

Accumulating evidence suggests the involvement of estrogen in depression. Estrogen can modulate neurotransmitter turnover, enhancing the levels of serotonin and norepinephrine and it is involved in the regulation of serotonin receptor number and function. Across the female reproductive life, fluctuating estrogen levels and low levels have been associated with depressed mood and there is strong support for a beneficial effect of estrogen-containing hormone treatment in depressed peri-menopausal women. Estrogen exerts its biological effects in large part through intracellular activation of its principal receptors, estrogen receptor α (ESR1) and estrogen receptor β (ESR2). Genetic variation in the estrogen receptors may therefore modify estrogen signalling, thus influence a woman's susceptibility to developing depression. This systematic review provides a synthesis of studies which have examined the association between estrogen receptor polymorphisms and depression-related mood disorders across the lifetime. Studies were identified through a search of the literature from January

The studies conducted to date have produced inconsistent findings, which likely relates to the large heterogeneity in terms of the populations, study design and depression measures used. It appears unlikely that the common ESR1 variants rs2234693 and rs9340799 are associated with moderate depressive symptoms in women; however there is some evidence which indicates a significant association with more severe depressive symptoms, major depressive disorder and anxiety. There are too few studies of ESR2 polymorphisms to draw any definite conclusions; however preliminary evidence suggests that specific variants may modify the risk of depression associated with the use of hormone treatment in women.

Few studies have investigated associations in men, and they have focused almost exclusively on ESR1, but all report non-significant findings. Much work is therefore still needed in this field.

If it is confirmed that specific estrogen receptor polymorphisms are associated with the risk of depression, this could have important preventative and therapeutic implications with the potential to develop targeted estrogen receptor agonists and antagonists. Furthermore, it is possible that such therapies may be more effective in treating particular people with depression based on their genetic profile, which is an exciting prospect given that many people do not respond to current antidepressant treatments.

INTRODUCTION 1.1 Depression

Depression is a prevalent psychiatric condition affecting up to 20% of the general population [START_REF] Boyd | Epidemiology of affective disorders. A reexamination and future directions[END_REF][START_REF] Bromberger | History of depression and women's current health and functioning during midlife[END_REF] and 10-20% are diagnosed with a major depressive disorder (MDD) across their lifetime. [START_REF] Alonso | Prevalence of mental disorders in Europe: results from the European Study of the Epidemiology of Mental Disorders (ESEMeD) project[END_REF][START_REF] Patten | Accumulation of major depressive episodes over time in a prospective study indicates that retrospectively assessed lifetime prevalence estimates are too low[END_REF] It is considered a major public health problem, not only due to its high prevalence, but also because the frequency of comorbidity is high, [START_REF] Zender | Women's mental health: depression and anxiety[END_REF][START_REF] Wilhelm | Prevalence and correlates of DSM-IV major depression in an Australian national survey[END_REF][START_REF] Gili | Medical comorbidity in recurrent versus first-episode depressive patients[END_REF] and this further worsens the outcome of these health problems and increases the risk of mortality. [START_REF] Ryan | Late-life depression and mortality: influence of gender and antidepressant use[END_REF] Depression has been associated with poorer social functioning and a greater risk for physical symptoms and body pain. [START_REF] Bromberger | History of depression and women's current health and functioning during midlife[END_REF] The quality of life for the people concerned is severely diminished, [START_REF] Alexander | Women, anxiety and mood: a review of nomenclature, comorbidity and epidemiology[END_REF] creating a substantial burden and cost to the community. [START_REF] Sobocki | Cost of depression in Europe[END_REF] Women have a higher prevalence of major depression compared to their male counterparts, [START_REF] Alonso | Prevalence of mental disorders in Europe: results from the European Study of the Epidemiology of Mental Disorders (ESEMeD) project[END_REF][START_REF] Wilhelm | Prevalence and correlates of DSM-IV major depression in an Australian national survey[END_REF] indicating that female gender is a risk factor. Evidence from family studies suggests there is a genetic component to depressive disorders, however despite the large number of genetic association studies, very few candidate genes have so far been identified. [START_REF] Lopez-Leon | Meta-analyses of genetic studies on major depressive disorder[END_REF] 1.2

Involvement of estrogen

Accumulating evidence suggests the involvement of estrogen in depression. [START_REF] Ancelin | Hormonal therapy and depression: Are we overlooking an important therapeutic alternative?[END_REF][START_REF] Osterlund | Underlying mechanisms mediating the antidepressant effects of estrogens[END_REF] Depression results, at least in part, from a disruption of the normal brain neurochemistry, which includes a change in the level of neurotransmitters and abnormal functioning of the hypothalamicpituitary-adrenal axis. [START_REF] Mokrani | HPA axis dysfunction in depression: correlation with monoamine system abnormalities[END_REF] Serotonin (5-hydroxytryptamine, 5-HT) and norepinephrine are thought to play key roles [START_REF] Blier | Crosstalk between the norepinephrine and serotonin systems and its role in the antidepressant response[END_REF][START_REF] Meltzer | Role of serotonin in depression[END_REF] and treatments which increase the brain levels of these neurotransmitters can improve depressive symptoms. [START_REF] Hirschfeld | History and evolution of the monoamine hypothesis of depression[END_REF] Estrogen can modulate neurotransmitter turnover, which includes enhancing the levels of serotonin and norepinephrine and it is involved in the regulation of serotonin receptor number and function, [START_REF] Andrade | Anxiolytic effect of estradiol in the median raphe nucleus mediated by 5-HT1A receptors[END_REF][START_REF] Matsuda | Gonadal hormones affect the hypothermia induced by serotonin1A (5-HT1A) receptor activation[END_REF][START_REF] Summer | Estrogen increases the density of 5-hydroxytryptamine(2A) receptors in cerebral cortex and nucleus accumbens in the female rat[END_REF] thus controlling the activity of serotonergic neurons. Across the female reproductive life, fluctuating estrogen levels and low levels have been associated with depressed mood [START_REF] Halbreich | The prevalence, impairment, impact, and burden of premenstrual dysphoric disorder[END_REF][START_REF] Ryan | A prospective study of the association between endogenous hormones and depressive symptoms in postmenopausal women[END_REF][START_REF] Soares | Reproductive hormone sensitivity and risk for depression across the female life cycle: a continuum of vulnerability?[END_REF][START_REF] Ryan | Lifetime hormonal factors may predict late-life depression in women[END_REF] and changes in serotonergic activity during the female hormone cycle appear correlated with variations in estrogen levels. [START_REF] Rapkin | The role of serotonin in premenstrual syndrome[END_REF] The menopause transition, which is characterised by fluctuating and declining estrogen levels, is associated with an increased risk of new onset and recurrent depressive episodes. [START_REF] Cohen | Risk for new onset of depression during the menopausal transition: the Harvard study of moods and cycles[END_REF][START_REF] Freeman | Associations of hormones and menopausal status with depressed mood in women with no history of depression[END_REF][START_REF] Bromberger | Depressive symptoms during the menopausal transition: the Study of Women's Health Across the Nation (SWAN)[END_REF][START_REF] Freeman | Associations of depression with the transition to menopause[END_REF] Finally, there is strong support for a beneficial effect of estrogen-containing hormone treatment (HT) in depressed perimenopausal women, [START_REF] Zweifel | A meta-analysis of the effect of hormone replacement therapy upon depressed mood[END_REF] although its use in postmenopausal women remains less certain. [START_REF] Cohen | Short-term use of estradiol for depression in perimenopausal and postmenopausal women: a preliminary report[END_REF][START_REF] Scali | A prospective study on hormonal treatment and depression in community-dwelling elderly women (The Three City Study)[END_REF] Estrogen exerts its biological effects in large part through intracellular activation of its principal receptors, estrogen receptor α (ESR1) and estrogen receptor β (ESR2). [START_REF] Heldring | Estrogen receptors: how do they signal and what are their targets[END_REF] Genetic variation in the estrogen receptors (ESR) may therefore modify estrogen signalling, thus influence a woman's susceptibility to developing depression. The aim of this review is firstly to provide a synthesis of studies published in the literature which have examined the association between ESR polymorphisms and depression; and secondly to evaluate the overall strength of evidence for such an association. The potential therapeutic implications stemming from this research will then be discussed.

METHODOLOGY

A systematic literature search was conducted using the MEDLINE, Web of Knowledge, Cochrane Library and PsychINFO databases, covering the period from January 1980 until March 2012 and combining the terms 'depression' or 'depressive disorder' and 'estrogen receptor polymorphisms' or 'estrogen receptor variants'. Citations were limited to articles and reviews written in English and those involving humans. Studies were also identified by manually searching the reference list of relevant articles, as well as previous reviews on this topic. The studies that were included in this review were those that involved the assessment of a mood disorder (depressive symptoms, MDD, premenstrual dysphoric disorder, puerperal psychosis, and bipolar disorder) or other comorbid affective disorders such as anxiety.

ESTROGEN RECEPTOR

ESRs are expressed in a variety of tissues including the brain, where they predominate in limbic-related areas known to be important for emotion, cognition and behaviour. [START_REF] Osterlund | Estrogen receptors in the human forebrain and the relation to neuropsychiatric disorders[END_REF][START_REF] Ostlund | Estrogen receptor gene expression in relation to neuropsychiatric disorders[END_REF] ESR1 is translated from an eight exon gene located on chromosome 6q25.1. It is found in high concentrations in the hypothalamus, hypothalamic preoptic area and the amygdala, and in lower concentrations in the hippocampus and cerebral cortex. [START_REF] Osterlund | Estrogen receptors in the human forebrain and the relation to neuropsychiatric disorders[END_REF] The more recently identified ESR2 gene [START_REF] Kuiper | Cloning of a novel receptor expressed in rat prostate and ovary[END_REF] is located on chromosome 14q23.2. It appears to be expressed throughout the brain, although dominant areas are the hippocampus, entorhinal cortex and thalamus. [START_REF] Osterlund | Estrogen receptors in the human forebrain and the relation to neuropsychiatric disorders[END_REF] This localisation could indicate that ESR1 plays a more important role than ESR2 in emotive functions and mood. [START_REF] Osterlund | Estrogen receptors in the human forebrain and the relation to neuropsychiatric disorders[END_REF] Indeed, patients with MDD have reduced ESR1 mRNA levels in the amygdala than controls [START_REF] Perlman | Reduced glucocorticoid and estrogen receptor alpha messenger ribonucleic acid levels in the amygdala of patients with major mental illness[END_REF] and the effects of estradiol on the serotonin 1A receptor have been shown to be mediated via ESR1. [START_REF] Wissink | Synergistic activation of the serotonin-1A receptor by nuclear factor-kappa B and estrogen[END_REF] On the other hand, more recent work in ovariectomized rats suggests that estrogen's beneficial effect on depression involves the action of ESR2 in the hippocampus [START_REF] Walf | Administration of estrogen receptor beta-specific selective estrogen receptor modulators to the hippocampus decrease anxiety and depressive behavior of ovariectomized rats[END_REF] and in mice, estradiol's antidepressant-like effect was also shown to be mediated through ESR2 activation. [START_REF] Rocha | 17 Beta-estradiolinduced antidepressant-like effect in the forced swim test is absent in estrogen receptor-beta knockout (BERKO) mice[END_REF] Overall, the ESR subtypes share a high degree of sequence homology. With an almost homologous DNA-binding domain (95% identity), they bind to the same DNA response elements. The subtypes also share many functional similarities in terms of substrate binding affinities, including that for estradiol, [START_REF] Kuiper | Cloning of a novel receptor expressed in rat prostate and ovary[END_REF] with 60% identity in the ligand-binding domain.

However subtype-selective differences in ligand binding and transcriptional potency have also been reported for novel non-steroidal ligands. [START_REF] Heldring | Estrogen receptors: how do they signal and what are their targets[END_REF]

Signalling

Both ESR subtypes are members of the superfamily of nuclear receptors that regulate transcription in target genes containing estrogen response elements (ERE). In liganddependent cell signalling, estrogen, which readily crosses the blood brain barrier, binds to both ESRs inside the cell and the complex then diffuses into the nucleus and binds to a number of specific ERE on various genes. This complex is capable of regulating the expression of hundreds of genes, including genes involved in regulating the synthesis and metabolism of several neurotransmitter systems. [START_REF] Nilsson | Mechanisms of estrogen action[END_REF][START_REF] Mcewen | Ovarian steroids and the brain: implications for cognition and aging[END_REF] The intracellular concentration of the ESRs appears to be correlated to the cellular response to estrogens, [START_REF] Katzenellenbogen | Dynamics of steroid hormone receptor action[END_REF] with both receptors being upregulated in response to bioavailable estradiol levels. [START_REF] Brann | Neurotrophic and neuroprotective actions of estrogen: basic mechanisms and clinical implications[END_REF] The relative levels of ESR1 and ESR2 in a given cell also influences the estrogen-response. Increased expression of ESR2 can antagonise ESR1 mediated transcription [START_REF] Matthews | Estrogen receptor (ER) beta modulates ERalpha-mediated transcriptional activation by altering the recruitment of c-Fos and c-Jun to estrogen-responsive promoters[END_REF][START_REF] Pettersson | Estrogen receptor beta acts as a dominant regulator of estrogen signaling[END_REF] and the two receptors can have opposing effects on gene expression. [START_REF] Liu | Opposing action of estrogen receptors alpha and beta on cyclin D1 gene expression[END_REF] They can however, also dimerise and work co-operatively. [START_REF] Pace | Human estrogen receptor beta binds DNA in a manner similar to and dimerizes with estrogen receptor alpha[END_REF]

Estrogen receptor polymorphisms

Given the putative role for estrogen in depression, variants in the genes coding for the ESR are good candidates to help explain an individual's susceptibility to developing depression.

Furthermore, allelic variants in these genes could explain the differential response of women to estrogen-containing HT. [START_REF] Almeida | Genetic variation of estrogen metabolism and the risks of cardiovascular disease[END_REF] Polymorphisms are the presence of more than one allele at a gene locus with a frequency of greater than one percent, and a single nucleotide polymorphism (SNP) more specifically, results from a single nucleotide change at the level of the DNA sequence. The two most frequently studied polymorphisms are ESR1 rs2234693 and rs9340799, otherwise known as PvuII and XbaI due to the creation of the aforementioned restriction site. They are located at position 397 and 351 of intron 1 respectively and are in strong linkage disequilibrium, [START_REF] Ryan | Hormone treatment, estrogen receptor polymorphisms and mortality: a prospective cohort study[END_REF][START_REF] Silvestri | Estrogen receptor alpha and beta polymorphisms: is there an association with bone mineral density, plasma lipids, and response to postmenopausal hormone therapy?[END_REF] meaning that certain combinations of alleles occur more frequently in the population than would be expected by random assortment. There is evidence that these polymorphisms are functionally significant, as they may alter transcription factor binding [START_REF] Herrington | Common estrogen receptor polymorphism augments effects of hormone replacement therapy on E-selectin but not C-reactive protein[END_REF] and gene expression. [START_REF] Maruyama | Lack of an association of estrogen receptor alpha gene polymorphisms and transcriptional activity with Alzheimer disease[END_REF] These polymorphisms have also been associated

frequently, but not consistently, with a variety of hormone-related health outcomes, such as bone mineral density, [START_REF] Greendale | The association of bone mineral density with estrogen receptor gene polymorphisms[END_REF][START_REF] Kobayashi | Estrogen receptor alpha polymorphism as a genetic marker for bone loss, vertebral fractures and susceptibility to estrogen[END_REF] vasomotor symptoms, [START_REF] Malacara | The relationship of estrogen receptor-alpha polymorphism with symptoms and other characteristics in postmenopausal women[END_REF] age at menopause, [START_REF] Weel | Estrogen receptor polymorphism predicts the onset of natural and surgical menopause[END_REF] breast cancer [START_REF] Kjaergaard | Estrogen receptor alpha polymorphism and risk of cardiovascular disease, cancer, and hip fracture: cross-sectional, cohort, and case-control studies and a meta-analysis[END_REF] and cardiovascular disease, [START_REF] Alevizaki | Severity of cardiovascular disease in postmenopausal women: associations with common estrogen receptor alpha polymorphic variants[END_REF][START_REF] Shearman | Estrogen receptor alpha gene variation and the risk of stroke[END_REF] as well as other brain disorders like Alzheimer's disease. [START_REF] Boada | Estrogen receptor alpha gene variants are associated with Alzheimer's disease[END_REF] These polymorphisms are in strong linkage disequilibrium with the tandem variable repeat TA polymorphism, [START_REF] Becherini | Evidence of a linkage disequilibrium between polymorphisms in the human estrogen receptor alpha gene and their relationship to bone mass variation in postmenopausal Italian women[END_REF] located in the promoter region of ESR1. The TA-repeat can influence tissue-specific gene expression [START_REF] Del Senno | Dinucleotide repeat polymorphism in the human estrogen receptor (ESR) gene[END_REF] and has also been associated with bone mineral density [START_REF] Yim | Association of estrogen receptor alpha gene microsatellite polymorphism with annual changes in bone mineral density in Korean women with hormone replacement therapy[END_REF] and cardiovascular disease. [START_REF] Kunnas | Association of polymorphism of human alpha oestrogen receptor gene with coronary artery disease in men: a necropsy study[END_REF] In a similar manner, associations have been reported between various ESR2 polymorphisms and these same health outcomes. [START_REF] Greendale | The association of bone mineral density with estrogen receptor gene polymorphisms[END_REF][START_REF] Domingues-Montanari | Association between ESR2 Genetic Variants and Risk of Myocardial Infarction[END_REF][START_REF] Giedraitis | Genetic analysis of Alzheimer's disease in the Uppsala Longitudinal Study of Adult Men[END_REF][START_REF] Gold | Estrogen receptor genotypes and haplotypes associated with breast cancer risk[END_REF][START_REF] He | Association study of the oestrogen signalling pathway genes in relation to age at natural menopause[END_REF][START_REF] Rexrode | Polymorphisms and haplotypes of the estrogen receptor-beta gene (ESR2) and cardiovascular disease in men and women[END_REF][START_REF] Sowers | Endogenous estradiol and its association with estrogen receptor gene polymorphisms[END_REF][START_REF] Takeo | Association of cytosine-adenine repeat polymorphism of the estrogen receptor-beta gene with menopausal symptoms[END_REF][START_REF] Yaffe | Estrogen receptor genotype and risk of cognitive impairment in elders: findings from the Health ABC study[END_REF] It therefore seems likely that both ESR1 and ESR2 polymorphisms could also influence an individual's risk of depression. including a study of depressive symptoms in participants aged at least 55 years and our two studies of severe depressive symptoms and MDD in the elderly. [START_REF] Ryan | Oestrogen receptor polymorphisms and late-life depression[END_REF][START_REF] Ryan | Estrogen receptor alpha gene variants and major depressive episodes[END_REF]

ESTROGEN RECEPTOR POLYMORPHISMS AND DEPRESSION

ESR1 and depression in women

There is no support for an association between mild or moderate depressive symptoms and rs2234693 or rs9340799 in women across the lifetime, irrespective of ethnicity, age, depression evaluation, sample and study design. [START_REF] Malacara | The relationship of estrogen receptor-alpha polymorphism with symptoms and other characteristics in postmenopausal women[END_REF][START_REF] Prichard | Association of polymorphisms of the estrogen receptor gene with anxiety-related traits in children and adolescents: a longitudinal study[END_REF][START_REF] Kravitz | Sex steroid hormone gene polymorphisms and depressive symptoms in women at midlife[END_REF][START_REF] Zhou | The symptomatology of climacteric syndrome: whether associated with the physical factors or psychological disorder in perimenopausal/postmenopausal patients with anxiety-depression disorder[END_REF][START_REF] Tiemeier | Estrogen receptor alpha gene polymorphisms and anxiety disorder in an elderly population[END_REF][START_REF] Yalamanchili | Treatment with hormone therapy and calcitriol did not affect depression in older postmenopausal women: no interaction with estrogen and vitamin D receptor genotype polymorphisms[END_REF] The only two studies to examine other SNPs (rs728524 and rs3798577) [START_REF] Kravitz | Sex steroid hormone gene polymorphisms and depressive symptoms in women at midlife[END_REF] and the dinucleotide TA repeat [START_REF] Prichard | Association of polymorphisms of the estrogen receptor gene with anxiety-related traits in children and adolescents: a longitudinal study[END_REF] also reported no significant association with depressive symptoms. In accordance with these findings, we reported no significant association between either rs2234693 or rs9340799 and mild depressive symptoms among 3525 elderly women. [START_REF] Ryan | Oestrogen receptor polymorphisms and late-life depression[END_REF] On the other hand, when we examined severe depressive symptoms (CES-D ≥23), we found an increased risk for women TT for rs2234693 or AA for rs9340799 (p=0.003 and p=0.009 respectively), compared to women homozygous for the alternative allele. [START_REF] Ryan | Oestrogen receptor polymorphisms and late-life depression[END_REF] Importantly this association held even after adjustment for numerous potential confounders including past depression.

Of the studies which have examined a diagnosis of MDD in women, equal numbers involved Caucasian and Asian populations. Significant associations were reported with rs2234693 or rs9340799 in five of six studies. [START_REF] Ryan | Oestrogen receptor polymorphisms and late-life depression[END_REF][START_REF] Ryan | Estrogen receptor alpha gene variants and major depressive episodes[END_REF][81][START_REF] Mill | Association study of the estrogen receptor alpha gene (ESR1) and childhood-onset mood disorders[END_REF][START_REF] Tsai | Association study of oestrogen receptor alpha gene polymorphism and suicidal behaviours in major depressive disorder[END_REF] In a Korean case-control study of postmenopausal women, the 43 depressed women were significantly more likely to be heterozygote for rs2234693 and rs9340799 compared to the 63 non-depressed controls (p≤0.001). [81] By contrast, among 215 middle-aged Chinese women, the C and G alleles of rs2234693 and rs9340799 respectively were more frequent in women with MDD (p=0.004 and p=0.02 respectively). [START_REF] Tsai | Association study of oestrogen receptor alpha gene polymorphism and suicidal behaviours in major depressive disorder[END_REF] In both studies only non-adjusted results were reported. Our study of the association between rs2234693 and rs9340799 with lifetime MDD in older women, supports the findings of Tsai et al. [START_REF] Tsai | Association study of oestrogen receptor alpha gene polymorphism and suicidal behaviours in major depressive disorder[END_REF] , even after adjustment for age, education level, the use of antidepressants or HT and bilateral oophorectomy. [START_REF] Ryan | Estrogen receptor alpha gene variants and major depressive episodes[END_REF] Furthermore, our study showed that the rs2234693 GG genotype was particularly associated with an increased risk of recurrent depressive episodes, regardless of the timing of episodes in relation to the menopause (p=0.02 and p=0.006 for episodes occurring before or after the menopause, respectively). No significant association was found with a single MDD episode. Other ESR1 polymorphisms have also been associated with childhood-onset mood disorders in girls [START_REF] Mill | Association study of the estrogen receptor alpha gene (ESR1) and childhood-onset mood disorders[END_REF] and premenstrual dysphoric disorder. [START_REF] Huo | Risk for premenstrual dysphoric disorder is associated with genetic variation in ESR1, the estrogen receptor alpha gene[END_REF] However, no significant association has been found between either the common or rare ESR1 variants and Puerperal Psychosis, or bipolar 1 disorder. [START_REF] Jones | Molecular genetic studies of bipolar disorder and puerperal psychosis at two polymorphisms in the estrogen receptor alpha gene (ESR 1)[END_REF][START_REF] Feng | Scanning of estrogen receptor alpha (ERalpha) and thyroid hormone receptor alpha (TRalpha) genes in patients with psychiatric diseases: four missense mutations identified in ERalpha gene[END_REF][START_REF] Kishi | Association analysis of functional polymorphism in estrogen receptor alpha gene with schizophrenia and mood disorders in the Japanese population[END_REF]

ESR2 and depression in women

Fewer studies have investigated whether ESR2 polymorphisms are associated with depression. Two studies of Asian populations found that the dinucleotide CA length was significantly associated with depression. In a small study of postmenopausal women, those reporting depressive symptoms were more likely to have one extremely short and one long allele (p<0.01), [START_REF] Takeo | Association of cytosine-adenine repeat polymorphism of the estrogen receptor-beta gene with menopausal symptoms[END_REF] while a slightly larger study of female adolescents reported a higher frequency of short alleles in those diagnosed with first onset MDD (p<0.001). [START_REF] Geng | Comparison of the polymorphisms of androgen receptor gene and estrogen alpha and beta gene between adolescent females with first-onset major depressive disorder and controls[END_REF] Of the other SNPs which have been investigated in more than one study, no significant association has been found between either rs4986938 or rs1256049 and severe depressive symptoms in women. [START_REF] Ryan | Oestrogen receptor polymorphisms and late-life depression[END_REF]81] However, when we considered the use of HT among elderly women, [START_REF] Ryan | Oestrogen receptor polymorphisms and late-life depression[END_REF] the A allele of rs1256049 was associated with an increased risk of severe depressive symptoms among women who were not using HT (p=0.005). For women currently taking HT, the A allele was not associated with an increased depression risk. This suggests a gene-environment interaction whereby some genetically predisposed women may benefit most from HT. To date, no other study has explored the potential interaction between ESR genetic variability and HT use in terms of depression risk.

Estrogen receptor polymorphisms and depression in men

Given that estrogen is the principal female reproductive hormone, and that estrogencontaining hormone treatment is used by menopausal women to relieve symptoms associated with low estrogen levels, it is probably not surprising to find that the majority of studies in this area have focused only on females. The six studies involving men which have examined whether the ESR1 rs2234693 and/or rs9340799 polymorphisms were associated with depressive symptoms or MDD, all failed to find a significant association irrespective of age and ethnicity. [START_REF] Ryan | Oestrogen receptor polymorphisms and late-life depression[END_REF][START_REF] Prichard | Association of polymorphisms of the estrogen receptor gene with anxiety-related traits in children and adolescents: a longitudinal study[END_REF][START_REF] Tiemeier | Estrogen receptor alpha gene polymorphisms and anxiety disorder in an elderly population[END_REF][START_REF] Mill | Association study of the estrogen receptor alpha gene (ESR1) and childhood-onset mood disorders[END_REF][START_REF] Tsai | Association study of oestrogen receptor alpha gene polymorphism and suicidal behaviours in major depressive disorder[END_REF][START_REF] Kishi | Association analysis of functional polymorphism in estrogen receptor alpha gene with schizophrenia and mood disorders in the Japanese population[END_REF] This was also the case for the ESR1 dinucleotide TA repeat. [START_REF] Prichard | Association of polymorphisms of the estrogen receptor gene with anxiety-related traits in children and adolescents: a longitudinal study[END_REF][START_REF] Comings | Potential role of the estrogen receptor gene (ESR1) in anxiety[END_REF] Our study was the only one to investigate ESR2 polymorphisms in men [START_REF] Ryan | Oestrogen receptor polymorphisms and late-life depression[END_REF] and we reported no strong evidence of an association with severe depressive symptoms. The AA genotype of rs4986938 however, was associated with an increased risk that was of borderline significance (p=0.05).

Estrogen receptor polymorphisms and other affective disorders

Five of the studies described above also examined associations with anxiety disorder. Two studies found no significant associations with either depression or anxiety in women, [START_REF] Malacara | The relationship of estrogen receptor-alpha polymorphism with symptoms and other characteristics in postmenopausal women[END_REF][START_REF] Zhou | The symptomatology of climacteric syndrome: whether associated with the physical factors or psychological disorder in perimenopausal/postmenopausal patients with anxiety-depression disorder[END_REF] while the others reported significant findings with anxiety only. [START_REF] Prichard | Association of polymorphisms of the estrogen receptor gene with anxiety-related traits in children and adolescents: a longitudinal study[END_REF][START_REF] Tiemeier | Estrogen receptor alpha gene polymorphisms and anxiety disorder in an elderly population[END_REF][START_REF] Comings | Potential role of the estrogen receptor gene (ESR1) in anxiety[END_REF] The Rotterdam study of men and women aged 55 years and older found that a haplotype consisting of the T and A alleles of rs2234693 and rs9340799 respectively was associated with an increased risk of anxiety in women (p=0.02), and in particular anxiety without comorbid depression symptoms. [START_REF] Tiemeier | Estrogen receptor alpha gene polymorphisms and anxiety disorder in an elderly population[END_REF] This is supported by our study of elderly women where we found that the C and G alleles of rs2234693 and rs9340799 were associated with a decreased risk of phobia independently of comorbid depression (p=0.003 and p=0.03 respectively). [START_REF] Ryan | Estrogen receptor gene variants are associated with anxiety disorders in older women[END_REF] Conversely, a longitudinal study of children reported higher anxiety scores in those homozygous for the C and G alleles of rs2234693 and rs9340799, if they also had the short TA-repeat polymorphisms (p<0.05). [START_REF] Prichard | Association of polymorphisms of the estrogen receptor gene with anxiety-related traits in children and adolescents: a longitudinal study[END_REF] But this is again contradicted by a study in adult male patients where the short TA allele was associated with reduced anxiety scores (p<0.002). [START_REF] Comings | Potential role of the estrogen receptor gene (ESR1) in anxiety[END_REF] In regards to other anxiety disorders, a significant association was found between ESR1 rs34535804 and an obsessive-compulsive disorder subphenotype in adults (p=0.007), [START_REF] Alonso | Variants in estrogen receptor alpha gene are associated with phenotypical expression of obsessivecompulsive disorder[END_REF] and the A allele of ESR2 rs1256049 was associated with generalized anxiety disorder in elderly women (p=0.02). [START_REF] Ryan | Estrogen receptor gene variants are associated with anxiety disorders in older women[END_REF]

Summary

Studies which have investigated the association between ESR1 and depression have produced inconsistent findings, which likely relates to the differences in populations and depression measures. It appears unlikely that the ESR1 variants rs2234693 and rs9340799 are associated with moderate depressive symptoms in women; however an association with more severe depressive symptoms, MDD and anxiety is possible. There are a lack of studies on later-life depression, which is distinct from depression at younger ages, and likewise, few studies of older postmenopausal women who have low endogenous estrogen levels. There are too few studies of ESR2 polymorphisms to draw any conclusions, although animal data clearly suggests an important role for this receptor in psychiatric disorders (see for review [START_REF] Osterlund | Underlying mechanisms mediating the antidepressant effects of estrogens[END_REF] ).

Results originating from our own study suggest that HT use may modulate the increased risk of late-life depression associated with a specific ESR2 variant, however no subsequent study has yet attempted to replicate these findings. Studies have also failed to test whether the ESR1 and ESR2 receptors could interact to influence the risk of depression, although these receptors can form heterodimers [START_REF] Pace | Human estrogen receptor beta binds DNA in a manner similar to and dimerizes with estrogen receptor alpha[END_REF] . Such a gene-gene interaction has been shown to modify the risk of Alzheimer's disease [START_REF] Lambert | Are the estrogen receptors involved in Alzheimer's disease?[END_REF] and we have recent unpublished results suggesting a significant ESR1-ESR2 interaction on the risk of late-life depression.

Only a small number of studies have investigated the association between ESR polymorphisms (almost exclusively ESR1) and depression in men. All report non-significant findings. A gender-specific association between ESRs and depression could be explained by differences in receptor subtype expression levels in various brain regions, as well as sex dimorphism in brain morphology, neurochemistry and neuronal wiring. [START_REF] Gillies | Estrogen actions in the brain and the basis for differential action in men and women: a case for sex-specific medicines[END_REF] However, more research using independent samples is required in this field before definite conclusions can be drawn.

THERAPEUTIC IMPLICATIONS

Therapeutic Potential

If it is confirmed that specific ESR polymorphisms are associated with the risk of depression, this could have important therapeutic implications. This includes the possibility of screening for people who are most at risk of depression and the potential to develop targeted ESR agonists and antagonists. Another exciting prospect is pharmacogenetics, with the possibility of predicting genetically which women would benefit most from HT in terms of depressed mood.

The regulation of ESR1 and ESR2 is complex and the exact mechanisms by which ESR polymorphisms can modify estrogen signalling remain unknown. The most commonly studied variants, ESR1 rs2234693 and rs9340799 for example, are silent mutations but they could still have important functional consequences by affecting mRNA structure, [START_REF] Shen | Single-nucleotide polymorphisms can cause different structural folds of mRNA[END_REF] stability and receptor synthesis, [START_REF] Duan | Synonymous mutations in the human dopamine receptor D2 (DRD2) affect mRNA stability and synthesis of the receptor[END_REF] and they may involve regulatory sequences. [START_REF] Alonso | Variants in estrogen receptor alpha gene are associated with phenotypical expression of obsessivecompulsive disorder[END_REF][START_REF] Aronow | Evidence for a complex regulatory array in the first intron of the human adenosine deaminase gene[END_REF] Indeed, these polymorphisms have been shown to regulate the expression of the ESR1 [START_REF] Maruyama | Lack of an association of estrogen receptor alpha gene polymorphisms and transcriptional activity with Alzheimer disease[END_REF] and alter transcription factor binding [START_REF] Herrington | Common estrogen receptor polymorphism augments effects of hormone replacement therapy on E-selectin but not C-reactive protein[END_REF] and thus could modify estrogen-mediated signaling. Specific alleles of these polymorphisms (C of rs2234693 and G of rs9340799) may be associated with increased serum estradiol levels, [START_REF] Sowers | Endogenous estradiol and its association with estrogen receptor gene polymorphisms[END_REF][START_REF] Schuit | Estrogen receptor alpha gene polymorphisms are associated with estradiol levels in postmenopausal women[END_REF] but this has not been found consistently. [START_REF] Malacara | The relationship of estrogen receptor-alpha polymorphism with symptoms and other characteristics in postmenopausal women[END_REF][START_REF] Yaffe | Estrogen receptor 1 polymorphisms and risk of cognitive impairment in older women[END_REF] Futhermore, studies which have examined whether these same alleles are associated with more favourable estrogen-dependent outcomes, [START_REF] Alonso | Variants in estrogen receptor alpha gene are associated with phenotypical expression of obsessivecompulsive disorder[END_REF][START_REF] Gennari | Steroid hormone receptor gene polymorphisms and osteoporosis: a pharmacogenomic review[END_REF]100] including the studies of depression and anxiety presented here, have reported conflicting results. The exact reasons for these discrepancies remain to be elucidated, but alterations in ESR1 mRNA expression in the brain may be region, diagnosis and gender-specific. [START_REF] Perlman | Reduced glucocorticoid and estrogen receptor alpha messenger ribonucleic acid levels in the amygdala of patients with major mental illness[END_REF]101] These ESR1 polymorphisms may have reverse effects in different ethnic populations, [102] and there may be differential interactions with other genes and environmental factors. In terms of other SNPs, there is variable evidence for a direct biological significance or an association with human disease.

Further knowledge of the functional consequences of ESR polymorphisms will allow a better understanding of the mechanisms by which they can influence estrogen signalling and thus depression. This will be invaluable for the development of new and effective ESRsubtype selective agonists and antagonists, [103] in particular those that target the brain, while avoiding activation of ESR in other tissues. [104] This will enable future improvements in clinical applications by allowing tailored treatment based on an individual's genetic make-up.

Considering that ESR1 and ESR2 appear to have overlapping but also unique biological functions, it is most likely that receptor subtype-specific coregulators exist. Identification of new coregulators may also provide novel targets for the development of new classes of therapeutic drugs with potential use in the treatment of diseases involving ESR signalling, which could include depression and anxiety.

Pharmacogenomics

There is a growing body of literature which suggests that individuals have a variable sensitivity to HT, which alters the efficacy or toxicity of such treatments. ESR polymorphisms have been shown to modulate the effect of HT on a variety of health outcomes including bone mineral density, [START_REF] Kobayashi | Estrogen receptor alpha polymorphism as a genetic marker for bone loss, vertebral fractures and susceptibility to estrogen[END_REF]105] breast cancer, [106] cholesterol levels [START_REF] Silvestri | Estrogen receptor alpha and beta polymorphisms: is there an association with bone mineral density, plasma lipids, and response to postmenopausal hormone therapy?[END_REF]107] and cardiovascular disease. [49] Pharmacogenetic studies in osteoporosis for example, suggest that women with the C and G alleles of rs2234693 and rs9340799 respectively may have a greater sensitivity to HT, [START_REF] Gennari | Steroid hormone receptor gene polymorphisms and osteoporosis: a pharmacogenomic review[END_REF] benefiting more from its protective effects. Women homozygous for these alleles have also been shown to be less susceptible to the negative effects of HT on the breast [108] and may have a reduced risk of all-cause deaths with the use of HT. [START_REF] Ryan | Hormone treatment, estrogen receptor polymorphisms and mortality: a prospective cohort study[END_REF] In terms of depression and anxiety, there are only some very preliminary results of an interaction between HT and ESR polymorphisms, [START_REF] Ryan | Oestrogen receptor polymorphisms and late-life depression[END_REF][START_REF] Ryan | Estrogen receptor gene variants are associated with anxiety disorders in older women[END_REF] but these suggest the possibility that some women may be genetically more susceptible than others to the psycho-protective effects of HT. This is an exciting prospect and could help explain the variable effects of HT when given to peri-and postmenopausal women.

CONCLUSION

Much work is still needed to clearly define the role of ESR polymorphisms in depression and the potential for preventative or therapeutic interventions. There is some support for an association between severe depression or MDD and common ESR1 polymorphisms in midlife and older women. There are too few studies to draw conclusions regarding other polymorphisms or associations in men. There is also preliminary evidence that ESR polymorphisms can modify the effect of HT on numerous health outcomes, including depression and anxiety, however this requires replication. If ESR polymorphisms were established to be associated with depression risk, the clinical use of these polymorphisms to help identify people most at risk of depression would be limited, if the effects of these polymorphisms on disease risk were relatively modest. However, an association between ESR polymorphisms and depression could still have potentially important therapeutic implications, such as the development of ESR-specific targeted hormone therapies. Furthermore, it is possible that such therapies may be more effective in treating particular people with depression based on their genetic profile, but this would first require confirmation through clinical trials. This is an exciting prospect given that over 35% of people do not respond to current antidepressant treatments [109] and antidepressants may be less effective in postmenopausal women in particular, [110,111] due to the moderating effects of low estrogen levels on serotonergic neurotransmission. [112] Estrogen-based HT and novel non-steroidal ligands would thus be of particular benefit to depressed persons who are unresponsive to other forms of treatment, or it could be given concurrently with antidepressants, which has already proven to be effective in peri-and post-menopausal women. [113][114][115][116] [START_REF] Zhou | The symptomatology of climacteric syndrome: whether associated with the physical factors or psychological disorder in perimenopausal/postmenopausal patients with anxiety-depression disorder[END_REF] Case-control population-based study of peri-and post-menopausal women, [START_REF] Malacara | The relationship of estrogen receptor-alpha polymorphism with symptoms and other characteristics in postmenopausal women[END_REF] Observational population-based study of 177 postmenopausal women ≥45 yrs (53 +/ 5 yrs) Mexico (not stated)

ESR1: rs2234693, rs9340799

Depressive symptoms using the modified 9-item HDRS -No significant association Takeo et al., 2005 [START_REF] Takeo | Association of cytosine-adenine repeat polymorphism of the estrogen receptor-beta gene with menopausal symptoms[END_REF] Observational study of outpatients. 51 postmenopausal women ≥45 yrs.

Japan (Japanese)

ESR2 : CA-repeat Menopausal depressive symptoms assessed using a questionnaire.

-Increased frequency of genotype EL -one extremely short and one long allele (p<0.01) Tiemeier et al., 2005 [START_REF] Tiemeier | Estrogen receptor alpha gene polymorphisms and anxiety disorder in an elderly population[END_REF] Observational population-based study of 2268 men & postmenopausal women ≥55 yrs Netherlands (Caucasian)

ESR1: rs2234693, rs9340799

Depressive symptoms assessed with CES-D ≥ 16 (7.8%) and depressive disorder based on DSM-IV (3.2%)

-No significant association in men or women Yalamanchili, Gallagher, 2012 [START_REF] Yalamanchili | Treatment with hormone therapy and calcitriol did not affect depression in older postmenopausal women: no interaction with estrogen and vitamin D receptor genotype polymorphisms[END_REF] Secondary analysis of data from a double-blinded placebo-controlled trial of 412 postmenopausal women. 65-77 yrs (71.4 +/-3.6 yrs).

America (not stated)

ESR1: rs2234693, rs9340799

Depressive symptoms using the Geriatric Depression Scale (GDS), Long Form 30, ≥11

-No significant associations

SEVERE DEPRESSIVE SYMPTOMS AND MDD

Mill et al., 2008 [START_REF] Mill | Association study of the estrogen receptor alpha gene (ESR1) and childhood-onset mood disorders[END_REF] Family [START_REF] Huo | Risk for premenstrual dysphoric disorder is associated with genetic variation in ESR1, the estrogen receptor alpha gene[END_REF] Case-control population-based study of menstruating women. 91 [START_REF] Kishi | Association analysis of functional polymorphism in estrogen receptor alpha gene with schizophrenia and mood disorders in the Japanese population[END_REF] Case-control study of adults, 325 patients, 802 controls, 51% women. (40.4 +-/ 14.8 yrs). Menopause status not detailed. Japan (Japanese)

ESR1: rs2234693

MDD based on DSM-IV -No significant association in men and women.

Tsai et al., 2003 [START_REF] Tsai | Association study of oestrogen receptor alpha gene polymorphism and suicidal behaviours in major depressive disorder[END_REF] Case-control study of adults, 154 patients, 226 controls, 57% women. (47 +/-17 yrs). Menopause status not detailed. .

-

  In women only, the rs2234693 C allele (p=0.004), the CC genotype (p=0.01) & rs9340799 G allele was more frequent (p=0.02)

Table 1

 1

presents a summary of the 19 studies which have examined the association between ESR polymorphisms and mood disorders across the lifetime. The studies vary considerably regarding the populations (ethnicity, age, gender), and the study size and design (case-control, cohort, clinical sample, general population), which may help explain the mixed findings. The assessment of depression also varies, with eight studies focusing on MDD, diagnosed according to the Diagnostic and Statistical Manual of Mental Disorders (DSM-IV), and a similar number on depressive symptomatology. This large heterogeneity between studies makes direct comparisons difficult. Only three studies examined depression in later-life,

Table I : Studies across all ages investigating the association between estrogen receptor gene variants and depression measures

 I

	Publication	Study characteristics	Country	Gene variant	Depression measure	Main finding : genetic profile of « depressed »
		(mean age +/ SD yrs)	(ethnicity)		(reported prevalence)	subjects
			MILD DEPRESSIVE SYMPTOMS
	Prichard et al.,	Longitudinal cohort study of 680	Australia	ESR1:	Depressive symptoms using	-No significant associations in boys or girls
	2002 [76]	boys and girls followed from 4-6	(mostly	rs2234693,	the DSM-III-R derived scale
		months until 15-16 yrs	Caucasian)	rs9340799,	of depression and the Short
				TA-repeat	Mood Feelings Questionnaire
	Comings et	Observational patient-based study,	America	ESR1:	Depressive symptoms using	-No significant association
	al., 1999 [89]	179 adult male from an Addiction	(19% African,	TA-repeat	The Symptom Checklist 90
		Treatment Unit (40.9 +/-6.9 yrs)	15% Hispanic		(SCL-90)
			66% other
			Caucasian)
	Kravitz et al.,	Observational population-based study	America	ESR1:	Depressive symptoms	-No significant associations in any ethnic group
	2006 [77]	of 1435 pre-and peri-menopausal	(African	rs2234693,	assessed with CES-D ≥ 16
		women 42-52 yrs (46 yrs)	American,	rs728524,	(13-26% depending on ethnic
			Caucasian,	rs3798577	group)
			Chinese,	ESR2:
			Japanese)	rs1256030
	Zhou et al.,
	2012

  -based study of 460 affected boy and girl patients aged 7-16 yrs and their families

			Hungary (94%	ESR1:	Childhood onset mood	-No significant association with any individual
			Caucasian)	rs2234693,	disorder (COMD) meeting	SNPs but haplotype of 3 SNPs (rs2077647,
				10 other SNPs	DSM-IV criteria, using The	rs746432, rs532010) associated with the risk of
					Interview Schedule for	Childhood onset mood disorder in females only
					Children and Adolescents	(p=0.05)
					Diagnostic Version
	Geng et al.,	Case-control study of female	China (Han	ESR1:	First-onset MDD based on	-No significant association with ESR1
	2007 [88]	adolescents, 102 patients, 150 control	Chinese)	TA-repeat	DSM-IV	-Higher frequency of short ESR2 alleles (p<0.001)
		volunteers, 15-18 yrs (15 yrs)		ESR2:
				CA-repeat
	Huo et al.,
	2007

Acknowledgments

No sources of funding were used to assist in the preparation of this review. Joanne Ryan is the holder of an NHMRC Training (Postdoctoral) Fellowship (Overseas Public Health,

ESR1 rs2234693, rs9340799 ESR2: rs1256049, rs4986938

Depression assessed using the Beck Depression Inventory (BDI) >21 and DSM-IV diagnosis of MDD or depressive disorder -Higher frequency of rs2234693 & rs9340799 heterozygotes (p≤0.001) Ryan et al., 2011 [START_REF] Ryan | Oestrogen receptor polymorphisms and late-life depression[END_REF] Observational population-based study of 6017 elderly men and women ≥65 yrs (73.5 +/-4.9 yrs). -Depressive symptoms, CES-D ≥ 16 (27.3%) -Among women, higher frequency of rs2234693 TT (p=0.003) or rs9340799 AA (p=0.009) -Higher frequency of A allele of rs1256049 in women not using hormone treatment (p=0.005) -Higher frequency of rs4986938 AA genotype in men (p=0.05) -No significant association with mild depressive symptoms in either gender. Ryan et al., 2012 [START_REF] Ryan | Estrogen receptor alpha gene variants and major depressive episodes[END_REF] Observational population-based study of 3987 elderly women ≥65 yrs (73 +/-5.5 yrs).

France (likely Caucasian population 2 )

ESR1: rs2234693, rs9340799

Lifetime MDD assessed using the MINI according to DSM-IV criteria (11.7%) -Higher frequency of homozygotes for G allele of rs9340799 (p=0.009) for recurrent episodes in particular, regardless of the age at first onset relative to the menopause.

PUERPERAL PSYCHOSIS AND BIPOLAR I DISORDER

Jones et al., 2000 [START_REF] Jones | Molecular genetic studies of bipolar disorder and puerperal psychosis at two polymorphisms in the estrogen receptor alpha gene (ESR 1)[END_REF] Case-control study of adult men and women. 210 patients (26 women with Puerperal Psychosis), 208 controls (45 +/-11 yrs) United Kingdom (Caucasian)

ESR1: rs2234693, rs9340799

Bipolar 1 disorder based on DSM-IV and self-reported Puerperal Psychosis based on DSM-IV for parous women -No significant associations in men or women for Bipolar disorder or Puerperal Psychosis.

Feng et al., 2001 [START_REF] Feng | Scanning of estrogen receptor alpha (ERalpha) and thyroid hormone receptor alpha (TRalpha) genes in patients with psychiatric diseases: four missense mutations identified in ERalpha gene[END_REF] Mixed patient populations. [START_REF] Kishi | Association analysis of functional polymorphism in estrogen receptor alpha gene with schizophrenia and mood disorders in the Japanese population[END_REF] Case-control study of adults, 155 patients, 802 controls, 52% women. (38.8 +-/ 14.2 yrs). Menopause status not detailed. Japan (Japanese)

ESR1: rs2234693

Bipolar based on DSM-IV -No significant association in men and women [START_REF] Boyd | Epidemiology of affective disorders. A reexamination and future directions[END_REF] Did not include either rs2234693 or rs9340799; 2 French law prohibits the collection of ethnic-related data however genotype frequencies were similar to those previously


C Desceliers

G Bonnet
email: guy.bonnet@univ-paris-est.fr

S Hamza

P Delmotte

Mixed nonparametric-parametric probabilistic model for earthquake reliability of an inelastic reinforced concrete frame structure

Keywords: Probabilistic model, earthquake reliability, nonlinear dynamics, inelastic material

The paper is devoted to the study of uncertainties when studying buildings under seismic loading. These uncertainties are related to the simplifications used when constructing the model (model uncertainties) and to the numerical data needed at the computation stage (data uncertainties). It has been shown in previous papers that nonparametric models are able, in the case of linear dynamics, to deal simultaneously with these two kinds of uncertainties. The paper presents an extension of this kind of model by taking into account a "mixed" approach for concrete frame structures, which uses a nonparametric model for the part of the structure which behaves linearly and a parametric approach for the parts of the structure (plastic hinges) which behave non-linearly. A numerical application is presented in the case of a residential building.

Introduction

The estimation of the response of existing buildings when submitted to seismic loads led to the use of pseudo-static methods and of fully non-linear dynamic formulations. In this last case, a mechanical model of the building has to be constructed to predict its transient nonlinear dynamical response. However, the position of the parts of the structure needing a nonlinear modelling is usually known, because it involves plastic hinges which are known from the engineer. For instance, in the case of a building with a reinforced concrete frame structure, the nonlinearities are related to the inelastic behaviour of the material at the ends of the beams and columns [START_REF]European Committee for Standardization Design of structures for earthquake resistance. Part 3: Strengthening and repair of buildings[END_REF][START_REF] Pauley | Seismic design of concrete and masonry buildings[END_REF]. The complexity level of a real building is so high that modelling approximations are necessarily introduced, which yield a model built by the engineer called thereafter "simplified nonlinear mechanical model". Such a model can be improved by taking into account all the uncertainties through reliability analyses (see for instance [START_REF] Ibrahim | Structural dynamics with parameters uncertainties Applied Mechanics Reviews[END_REF][START_REF] Pinto | Reliability methods in earthquake engineering Progress in Structural Engineering and Materials[END_REF] ) and by considering that the simplified model is the "mean model" in the probabilistic sense. The sources of uncertainties are related to the data of the model (for instance, the mechanical parameters of the inelastic constitutive equations) and to the model approximations introduced when constructing the simplified nonlinear mechanichal model (for instance, suppression of "non-significant" structural elements). These uncertainties are very significant in the case of existing buildings due to the lack of data concerning the original design and the historical evolution of the mechanical properties of the materials (concrete damage, corrosion...).

In this paper, a residential building is considered for applying the proposed methodology. A simplified nonlinear mechanical model is built by using the finite element method. In this model, the reinforced concrete frame structure of the building is meshed and secondary parts are represented by their masses. The Takeda nonlinear model [START_REF] Takeda | Reinforced concrete response to simulated earthquakes[END_REF] is used to describe the inelastic constitutive equations of the material at the ends of the beams and columns of the frame structure. These inelastic constitutive equations are mainly relevant for lateral loads as induced by horizontal ground accelerations. A local-to-global approach [START_REF] Combescure | Application of the local-to-global approach to the study of infilled frame structures under seismic loading[END_REF] is used to compute the parameters of the inelastic constitutive equations at the macroscopic scale. Other parts of the structure is assumed to be made up of a linear viscoelastic medium. Then, a reduced nonlinear model is built by using the eigenvectors of the associated undamped linearized model and a stochastic model is developed to take into account the uncertainties by modelling [START_REF] Combescure | Application of the local-to-global approach to the study of infilled frame structures under seismic loading[END_REF] the parameters of the inelastic constitutive equations as random variables (parametric approach) and (2) the generalized mass, stiffness and damping matrices as random matrices (nonparametric approach). The probabilistic model of these random variables and random matrices are built by using the information theory (see for instance [7-9, 13, 14, 16]). According to this theory, the entropy of a random system is a functional of the probability density function related to the uncertain physical parameters. The information theory states that each probability density function must be chosen as the one which maximises the entropy of the system with the constrains to comply with all known informations (mean values, standard deviations, etc). Thus, we present a mixed nonparametric-parametric probabilistic approach for modelling the uncertainties of a nonlinear dynamical system. Such a formulation has already been proposed for instance in [START_REF] Desceliers | Nonparametric-parametric model for random unicertainties in non-linear structural dynamics: Application to earthquake engineering[END_REF][START_REF] Soize | Nonlinear dynamical systems with nonparametric model of random uncertainties Uncertainties in[END_REF]) in the case of geometric nonlinearities. In this paper, we propose to develop such a formulation for structures where nonlinearities are related to the elastoplastic behaviour of constitutive materials.

Description of the structure and of the seismic motion

The methodology developed in this paper is applied to a relatively simple existing residential building (see Fig. 1). The building under consideration is a reinforced concrete frame structure. The strength of the structure is mainly due to the columns whose sections are between 15cmx15cm and 50cmx15cm. It is assumed that the floor and the roof are made up of a linear homogeneous isotropic material and that the frames (beams and columns) are made up of an inelastic heterogenous anisotropic material.

Let (O, X, Y, Z) be an orthogonal reference frame such that (O, X, Y) is the horizontal plane of the structure. The structure is submitted to a seismic loading that induces a ground acceleration along direction X and for which the time history is shown in Fig. 2-a As explained in the introduction, the inelastic behaviour of the material at the ends of the beams and columns of the reinforced concrete frame structure is strongly relevant for lateral loads induced by horizontal accelerations. Such a behaviour induces localized plastic hinges. The Takeda nonlinear model [START_REF] Takeda | Reinforced concrete response to simulated earthquakes[END_REF] is used to describe the inelastic constitutive equations within the hinges. The Takeda nonlinear model for a beam is completely defined by a trilinear moment-curvature curve, representing the elastic undamaged, damaged and post yielding stages [START_REF] Takeda | Reinforced concrete response to simulated earthquakes[END_REF]. This trilinear moment-curvature curve is defined by 6 mechanical parameters (see Fig. 3-a): the curvature and moment

m 1 = E 0 Ic 1 , (1)
m 2 = m 1 + E 1 I(c 2 -c 1 ) , (2)
m 3 = m 2 + E 2 I(c 3 -c 2 ) , (3)
where E 0 , E 1 , E 2 are the elastic, cracked and plastic stiffness moduli and I is the moment of inertia of the cross section of the beam ( underlined characters are used to avoid confusion with other notations within the paper). In addition, the hysteretic phenomena are controlled by 3 parameters: h 1 for the reduction of Young's modulus during cyclic loading, h 2 for the pinching effect and h 3 for the strength degradation under cyclic loading (see Fig. 3-b).

Fig. 4 The finite element mesh of the simplified nonlinear mechanical model

Nonlinear mechanical model

For such a structure, the simplified nonlinear mechanical model (see section 1) is built by using the finite element method (see for instance [START_REF] Zienkiewicz | The finite element method[END_REF]) for the reinforced concrete frame structure. The finite element mesh used thereafter is shown in Fig. 3. It is made of 436 beam and plate elements and comprises 2006 degrees of freedom. Let [M ], [D] and [K] be the finite element mass, damping and stiffness matrices of the reinforced concrete frame structure. At time t > 0, the vector of the N degrees of freedom related to the finite element mesh of the structure is denoted by u(t) and belongs to R N . We then have, for t > 0

[M ] ü(t) + [D] u(t) + [K] u(t) = -[M ] üs(t) -g N L ({u(τ )} 0≤τ ≤t , t; w) , (4)
with the initial conditions

u(0) = u(0) = üs(0) = 0 , (5)
where a dot means the derivative with respect to time t and us(t) is the vector of the constrained degrees of freedom at the foundation points of the structure meshing. Stiffness matrix [K] takes into account the linear part of the elastoplastic model (modulus E 0 shown in Fig. 3

-a). Matrices [M ] , [D] ,

[K] define the low strain "linear model" used thereafter to build a basis of eigenvectors. The vector of the "equivalent nonlinear forces" g N L represents the effect of the inelastic constitutive equations of the material at the ends of the beams and columns of the reinforced concrete frame structure. The vector w

= (c 1 , m 1 , c 2 , m 2 , c 3 , m 3 , h 1 , h 2 , h 3
) includes all parameters of the Takeda nonlinear model. It should be noted that the vector of the "nonlinear forces" g N L depends of the parameters of the Takeda model (components of vector w) and of the history {u(τ )} 0≤τ ≤t of u.

Modal formulation of the nonlinear model : the reduced nonlinear model

The formulation of the previous section has shown that taking into account the nonlinear behaviour of the plastic hinges leads to solve a system of nonlinear equations.

From another point of view, the nonparametric model is built from the reduced dynamic model of the structures built on the linear modes. A modal formulation of the nonlinear model, built on the modes related to the associated linear model is therefore presented in this section. Let λ 1 ≤ . . . ≤ λ N be the N eigenvalues which are solutions of the generalized eigenvalue problem: find φ ∈ R N and λ > 0 such that

[K] φ = λ [M ] φ.
Let {φ 1 , . . . , φ N } be the N eigenvectors associated with eigenvalues λ 1 ≤ . . . ≤ λ N . Then, the vector u(t) can be rewritten as u(t) = N α=1 qα(t) φα where qα(t) are the generalized coordinates of vector u(t) within the basis of eigenvectors φα. The reduced nonlinear model is built with the approximation u(t) ≃ u n (t) in which

u n (t) = n α=1 qα(t) φα , ( 6
)
where n is the order of the reduced modal expansion which is chosen such that n < N . This last equation is rewritten as

u n (t) = [Φn] q n (t) , (7)
where the n × N modal matrix [Φn] is such that [Φn] α j = {φα} j and where the vector q n (t) is such that {q n (t)}α = qα(t). Equations ( 4) and ( 5) yield

[M] qn (t)+[D] qn (t)+[K] q n (t) = -[M] qn s (t)-[Φn] T g N L ({[Φn] q n (τ )} 0≤τ ≤t , t; w) , (8)
q n (0) = qn (0) = qn s (0) = 0 , (9)
where [M], [D] and [K] are the generalized mass, damping and stiffness matrices defined as

[M] = [Φn] T [M ] [Φn] , [D] = [Φn] T [D] [Φn] , [K] = [Φn] T [K] [Φn] , (10)
and where

q n s = [M] -1 [Φn] T [M ] us(t).
The second term in the right hand side of Eq. ( 8) represents the " equivalent nonlinear forces" due to the nonlinearities induced by the inelastic behaviour of materials (see section 3). It should be noted that the nonlinear forces are located at the ends of the beams and columns within the frame structure. It is noteworthy that strong nonlinear deformations do appear within these parts of the structure and that a projection onto the low frequency eigenvectors only is not sufficient to represent accurately such a deformation. Then, it is necessary to use a basis which also contains higher frequencies eigenvectors that allow local displacements to be accurately represented even if they are out of the frequency content of the signal related to seismic loading. Thus, a higher number of eigenvectors may be necessary to build the reduced nonlinear model.

For any given symmetric positive-definite matrices [M], [D], [K] and any given vector w, Eqs. ( 8) and ( 9) are solved by using an implicit Newmark scheme. For each time step of the Newmark scheme, the nonlinear equations are solved with a Newton algorithm. Then, using Eq. ( 7), approximation t → u n (t) can be built. Consequently, we can define a mapping f n such that for any values of

[M], [D], [K] and w, displacement u n (t) is given by u n (t) = f n (t; [M], [D], [K], w) . ( 11
)
6 Stochastic model

Mixed nonparametric-parametric probabilistic formulation of the uncertainties

This chapter is devoted to the presentation of the probabilistic approach which is used to account for the uncertainties involved during the process of studying the behaviour of buildings under seismic loading. From a general point of view, the designer prepares a "nominal model" which is the best approximation which takes into account the main features of the structure under study : geometry, structural elements, physical properties, etc, as defined within the previous sections. When a probabilistic approach is used, this "nominal model" can be considered as a "mean model" (in the probabilistic sense), within a probabilistic model whose objective is to take into account uncertainties related to the geometry, the physical parameters,etc.

( 1 presents different probabilistic approaches which account for the scattering of variables around the "mean model". For the usual probabilistic approach or so-called "parametric probabilistic approach", only uncertainties related to the physical parameters (mechanical strength, geometrical properties of the structural elements, etc) are taken into account, while for a "nonparametric probabilistic approach", it is possible to take into account, within a global approach and in the dynamic case, the uncertainties related to the model and to the physical parameters. The "model uncertainties" come from the necessary simplifications involved during the process of modelling : suppression of "non significant" structural elements, discretization procedures, simplified structural elements (beams, plates, etc). As shown in previous works, one of the consequences of these model uncertainties leads to the coupling of the modes which are estimated from the "mean model". The "nonparametric" method is therefore well suited to deal with structures or parts of the structures which behave linearly. The present work combines the parametric and nonparametric approaches to recover the advantages of both methods : uncertainties related to the linear part of the dynamical system are modeled by a "nonparametric" probabilistic approach and uncertainties related to the nonlinear part of the dynamical system are modeled by a "parametric" probabilistic approach. The approach thus developed is called "mixed probabilistic formulation", to recall both components of the method. In the following, both "nonparametric" and "parametric" probabilistic formulations are presented.

Nonparametric probabilistic formulation

Entering into more details, the nonparametric probabilistic approach (see for instance( [START_REF] Shannon | A mathematical theory of communication Bell System[END_REF][START_REF] Soize | Maximum entropy approach for modeling random uncertainties in transient elastodynamics[END_REF][START_REF] Soize | Nonlinear dynamical systems with nonparametric model of random uncertainties Uncertainties in[END_REF]) consists in substituting in Eq. [START_REF] Pinto | Reliability methods in earthquake engineering Progress in Structural Engineering and Materials[END_REF] the matrices

[M], [D], [K] by random matrices [M], [D], [K].
Usually, these random matrices are full as shown in [START_REF] Soize | Random matrix theory for modeling uncertainties in computational mechanics[END_REF]. In the following, it will be assumed that only uncertainties affecting the eigenfrequencies around the frequency content of the excitation signal are relevant for the dynamic computation of the structure. Let φ 1 , . . . , φn d , be the eigenvectors whose eigenfrequencies are relevant for the dynamic computation. Then, matrices [M], [D], [K] can be rewritten as :

[M] = [Mn d ] 0 0 [Mn qs ] , [D] = [Dn d ] 0 0 [Dn qs ] , [K] = [Kn d ] 0 0 [Kn qs ] , (12)
where In the present paper, the nonparametric probabilistic model for the model uncertainties yields random matrices

[M], [D], [K] defined as [M] = [Mn d ] 0 0 [Mn qs ] , [D] = [Dn d ] 0 0 [Dn qs ] , [K] = [Kn d ] 0 0 [Kn qs ] , (13)
where [Mn d ], [Dn d ] and [Kn d ] are random full matrices whose probabilistic model is built in [START_REF] Soize | Random matrix theory for modeling uncertainties in computational mechanics[END_REF] by using the information theory. In other words, it means that the probability density functions which appear in the model maximize the entropy under the constraint to comply to the available information. This information is defined as follows :

( The nonparametric probabilistic model of such random matrices is presented [START_REF] Soize | Random matrix theory for modeling uncertainties in computational mechanics[END_REF] and briefly recalled in Appendix A. It should be noticed that the statistical fluctuation of random matrices [M], [D], [K] in this probabilistic model are only controlled by three dispersion coefficients δ M , δ D and δ K (see also Appendix A). This low number of parameters accounting for the dispersion is one of the advantages of the method.

Parametric probabilistic formulation of the uncertainties for the parameters of the inelastic constitutive equation

The parametric probabilistic approach consists in substituting in Eq. ( 11) the deterministic vector w = (c 1 , m 1 , c 2 , m 2 , c 3 , m 3 , h 1 , h 2 , h 3 ) which contains the physical parameters defining the non-linear behaviour of plastic hinges by a random vector

W = (C 1 , M 1 , C 2 , M 2 , C 3 , M 3 , H 1 , H 2 , H 3 ) in which C 1 , M 1 , C 2 , M 2 , C 3 , M 3 are sta-
tistically dependent R-valued random variables such that, as deduced from Eqs (1) to (3),

M 1 = E 0 IC 1 , M 2 = M 1 + E 1 I(C 2 -C 1 ) , M 3 = M 2 + E 2 I(C 3 -C 2 ) , (14) 0 < C 1 ≤ C 2 ≤ C 3 , 0 ≤ H 1 ≤ 1 , 0 ≤ H 2 ≤ 1 , 0 < H 3 (15)
In addition, the mean values and the standard deviations of C 1 , C 2 , C 3 , H 1 , H 2 , H 3 are assumed to be given and they are denoted by c 1 , c 2 , c 3 , h 1 , h 2 , h 3 (for the mean values) and σ C1 , σ C2 , σ C3 , σ H1 , σ H2 , σ H3 (for the standard deviations). The construction of the probabilistic model of these random variables is presented in [START_REF] Hamza | Analyse probabiliste de la vulnérabilité sismique des btiments existants: application aux structures en portiques en béton armé[END_REF] using the information theory and is briefly recalled in Appendix B.

Random response spectrum

The solution of the stochastic model is a R N -valued stochastic field {U n (t

) : t ∈ [0, T ]} such that U n (t) = f n (t; [M], [D], [K], W) (16)
It means that at each realization of W, [M], [D], [K] is associated a realization of U n (t). Let S j (ξ, ω) be the random normalized acceleration response spectrum of the stochastic transient response {U n } j of the jth degree of freedom in which the damping ratio ξ ∈ [0, 1] and the eigenfrequency ω belongs to the frequency band of analysis [ω min , ωmax]. We then have, for any given damping ratio ξ and radial frequency ω

S j (ξ, ω) = ω 2 g max t∈[0,T ] |X j (t)| ( 17
)
where g is a normalization constant and X j (t) is the R-valued stochastic field indexed by [0, T ] such that

Ẍj (t) + 2ξω Ẋj (t) + ω 2 X j (t) = -{ Ün (t)} j , t ∈ [0, T ], (18)
X j (0) = Ẋj (0) = 0. ( 19
)
The stochastic solver used to estimate the confidence region of S j (ξ, ω) is based on the Monte Carlo method and is presented in Appendix C. 7 Application and numerical results

Parameters of the stochastic model for the application

In this section, the results related to an application concerning the building defined in section 2 are presented. For this application, it is necessary to precise all parameters defining the probabilistic model : mean values and dispersion parameters in the case of the parametric part of the model(parameters related to the nonlinear behaviour), dispersion parameters in the case of the nonparametric part of the model (dispersion parameters on stiffness, damping and mass matrices). The application is performed hereafter with δ M = δ D = δ K = 0.15 for the nonparametric probabilistic formulation and

c 1 = 0.0018m -1 , m 1 = 10.5×10 3 N.m, c 2 = 0.0123m -1 , m 2 = 22×10 3 N.m, c 3 = 0.1225m -1
, m 3 = 24.5 × 10 3 N.m, h 1 = 0.3, h 2 = 0.8, h 3 = 50 and the standard deviation σ C1 = 9×10 -5 m -1 , σ C2 = 5×10 -3 m -1 , σ C3 = 6.13×10 -3 m -1 , σ H1 = 0.06, σ H2 = 0.23, σ H3 = 5 for the parametric probabilistic formulation.

Convergence analysis with respect to the number of modes n

The convergence analysis of the nonlinear reduced model with respect to the parameter n is carried out by using the mapping n → Conv 1 (n) that is defined as

Conv 1 (n) = T 0 t(u n (t)) 2 dt ( 20
)
where t(u n (t)) is the vector of the shear forces due to the displacement Fig. 6 Comparison between the time-step solution and the reduced modal solution. Graphs of the j A th degree of freedom t → {u(t)} j A (solid lines) and t → {u n (t)} j A (dashed lines). Horizontal axis: time t ∈ [0, 3] s (left) and time t ∈ [2.3, 2.8] s (right). Vertical axis : {u(t)} j A (solid lines) and {u n (t)} j A (dashed lines) in m -

u n (t) = f n (t; [M], [D], [K], w).
Shear effort (N) Fig. 7 Comparison between the time-step solution and the reduced modal solution. Graphs of the time evolution of the shear force {t} j A with respect to the displacements {u(t)} j A (solid lines) and {u n (t)} j A (dashed lines). Horizontal axis: displacement {u(t)} j A and {u n (t)} j A between -8 × 10 -3 m and 8 × 10 -3 m (left) and between ×10 -3 m and 6 × 10 -3 m (right). Vertical axis: {t(u(t))} j A (solid lines) and {t(u n (t))} j A (dashed lines) in N and (9) and using Eq. ( 7)) to be compared. Figures 6 show the graphs of t → {u(t)} j A (solid lines) and t → {u n (t)} jA (dashed lines) representing the displacement along direction X of the node A. Figures 7 show the time evolution of the shear forces {t(u(t))} j A and {t(u n (t))} j A at node A with respect to the displacement {u(t)} j A (solid lines) and {u n (t)} jA (dashed lines). It should be noted that, for node A and n = 1400, there is a good matching for displacements and shear forces obtained from the modal response and from the finite element computation taking into account all the structure and discretized time history.

Confidence region of S j (ξ, ω)

In order to buildthe confidence regions associated with a probability level Pc = 95% of the random normalized response spectrum, we choose n d = n B2 where n B2 = 27 is the number of eigenfrequencies that belong to the frequency band B 2 = [0, 100]Hz. The stochastic solver used for this construction is based on the Monte Carlo method with ns = 300 statistical independant realizations of S jA k (ξ, ω) related to the displacement Fig. 8 Uncertainties quantification of the building for a linear stochastic approach (left) and a stochastic nonlinear approach (right). Horizontal axis: frequency in Hz. Vertical axis: confidence region of random acceleration spectrum S j A k (ξ, ω) with the probability level Pc = Pc = 95% for the nodes A k with k = 1, . . . , 4 in the column P 13 (see fig. 1).

of node A k along direction X (see Fig. 1b) and whose coordinates are (x k , y k , z k ) with x k = -6.93 m, y k = 6.3 m and z k ∈ [0, 2.36] m.

The confidence regions of S j (ξ, ω) for j = j A1 , j A2 , j A3 , j A4 , where z 1 = 0.29 m, z 2 = 0.58 m, z 3 = 0.98 m and z 4 = 2.36 m, are shown in Fig. 8 (right) for the solution of the nonlinear model. It can be seen that for the frequency band [0, 10]Hz, the widthes of the confidence regions are nearly constant and similar for every node A k . It means that the uncertainties levels related to the random acceleration response spectrum S jA k (ξ, ω) do not depend of frequency ω and of coordinate z k of node A k . On the other hand, the width of the confidence region in the frequency band [START_REF] Pauley | Seismic design of concrete and masonry buildings[END_REF]100]Hz increases with frequency ω and decreases with coordinate z k . It means that the uncertainties levels increase with ω in frequency band [START_REF] Pauley | Seismic design of concrete and masonry buildings[END_REF]100]Hz and decrease with coordinate z k . Thus, the most important uncertainties levels are reached for nodes close to the base of columns where plastic hinges are located. These plastic hinges support the most important plastic strain.

The confidence regions of S j (ξ, ω) for j = j A1 , j A2 , j A3 , j A4 , j A5 are shown in Fig. 8 (left) for the linear case, i.e. with g N L = 0 (stochastic linearized reduced model). It can be seen that the widthes of the confidence regions are similar for all nodes A k and for all ω in the frequency band of analysis. It means that the uncertainties levels are equivalent for all nodes A k and are nearly frequency independent in the frequency range [0, 10]Hz.

Conclusions

A mixed nonparametric-parametric probabilistic model for the uncertainties in a structure made up of inelastic materials has been presented. The nonparametric formulation allows the model uncertainties due to approximations in the construction of the simplified nonlinear model to be taken into account. The parametric formulation allows uncertainties related to the parameters of the inelastic materials to be taken into account. It should be noted that such a probablistic model does not allow to take into account uncertainties related to the seismic loads, physical nonlinearities other than those already accounted for, geometrical nonlinearities, etc. In addition, the main difficulty for using such a probabilistic model comes from the choice of the value of the dispersion parameters δ M , δ D , δ K . In other engineering fields, these parameters can be identified from experimental data by solving a stochastic inverse problem (see for instance [START_REF] Durand | Structural-acoustic modeling of automotive vehicles in presence of uncertainties and experimental identification and validation[END_REF][START_REF] Soize | Probabilistic model identification of uncertainties in computational models for dynamical systems and experimental validation[END_REF]). Unfortunately, such experimental data are difficult to obtain in seismic engineering. In this case, these parameters must be chosen from expert appraisement. The parametric probabilistic model has been developed and presented in the paper using the information theory. The method developed in the paper allows the confidence regions of the random normalized acceleration response spectrum to be estimated by using a stochastic solver based on the Monte Carlo method. An example of one-storey building has been studied and it has been shown that material nonlinearities increase the uncertainties levels of the random normalized acceleration response spectrum when compared to the stochastic dynamical linear model, especially for frequencies higher than 10 Hz and for nodes located close to the base of the columns. Within the class of uncertainties taken into account by this probabilistic model, it appears that the nonlinear mean model is robust (no increase dispersion with frequency) up to frequencies around 20Hz which is the practical upper band for frequency range in seismic engineering. In addition, the dispersion of sprectra would appear of second order compared to uncertainties associated with seismic excitations.

A. Probabilistic model of random matrices for the nonparametric probabilistic formulation of the model uncertainties

In this appendix, we present the construction of the probabilistic model of the random matrices involved by the nonparametric probabilistic formulation of the model uncertainties. It has been shown in [START_REF] Soize | Random matrix theory for modeling uncertainties in computational mechanics[END_REF] that the probabilistic model of random matrices [Mn d ], [Dn d ] and [Kn d ] can be constructed using the information theory with the available information (see section 6.2) yielding

[Mn d ] = [L M ] T [G M ][L M ] , [Dn d ] = [L D ] T [G D ][L D ] , [Kn d ] = [L K ] T [G K ][L K ] .
[G M ] , p [G D ] , p [G K ] are such that p [G M ] ([G]) = I M + ([G]) c(δ M ) (det[G]) b(δ M ) exp{-a(δ M )tr[G]} , p [G D ] ([G]) = I M + ([G]) c(δ D ) (det[G]) b(δ D ) exp{-a(δ D )tr[G]} , p [G K ] ([G]) = I M + ([G]) c(δ K ) (det[G]) b(δ K ) exp{-a(δ K )tr[G]} , where I M + ([G]) is equal to 1 if [G] belongs to M + and is equal to zero if [G] does not belong to M + , tr[G] is the trace of matrix [G], a(δ) = (n d + 1)/(2δ 2 ), b(δ) = a(δ) × (1 -δ 2 ), c(δ) = (2π) -n d (n d -1)/4 a(δ) n d a(δ) / n+1
j=1 Γ (α j (δ)) in which α j (δ) = a(δ) + (1 -j)/2 and where Γ is the Gamma function.

Then, it has then shown in [START_REF] Soize | Random matrix theory for modeling uncertainties in computational mechanics[END_REF] that random matrices

[G M ], [G D ], [G K ] can be written as [G M ] = [L M ] T [L M ] , [G D ] = [L D ] T [L D ] , [G K ] = [L K ] T [L K ] ,
where (2) for j = j ′ , [L M ] jj ′ , [L D ] jj ′ and [L K ] jj ′ are positive-valued random variables written as [L M ] jj ′ = σ M 2 V M,j , [L D ] jj ′ = σ D 2 V D,j where V M,j , V D,j and V K,j are positivevalued Gamma random variables whose probability density function p V M,j , p V D,j , p V K,j are written as

p V M,j (v) = I R + (v) v α j (δ M ) e -v Γ (α j (δ M ) , p V D,j (v) = I R + (v) v α j (δ D ) e -v Γ (α j (δ D ) , p V K,j (v) = I R + (v) v α j (δ K ) e -v
Γ (α j (δ K ) .

Then, ns statistical independent realizations of random matrices B.Probabilistic model of the random vector for parametric formulation of uncertainties related to the inelastic constitutive equations

In order to satisfy the first inequality in Eq. ( 15), random variables C 1 and C 2 are rewritten as

C 2 = A C 1 , C 3 = A B C 1 , (21)

  . The modulus of the Fourier transform of the acceleration is shown in Fig.2-b. It can be seen that the accelerogram belongs to the frequency band B 1 = [0, 50]Hz.

Fig. 1

 1 Fig. 1 Residential building under consideration (Fig. a), columns distribution (Fig. b) and a typical outline (Fig. c)

Fig. 2 Fig. 3

 23 Fig. 2 Accelerogram of the seismic signal (Fig. a) and modulus of the Fourier transform of the seismic signal (Fig. b)

  [Mn d ], [Dn d ] and [Kn d ] are positive diagonal (n d × n d ) matrices and where [Mn qs ], [Dn qs ] and [Kn qs ] are positive diagonal (nqs × nqs) matrices such that nqs = nn d .

  ) [Mn d ], [Dn d ], [Kn d ] are second order random variables with values in the set M + of all the (n d × n d ) real symmetric positive-definite matrices; (2) the mean values of these random matrices are E{[Mn d ]} = [Mn d ], E{[Dn d ]} = [Dn d ], E{[Kn d ]} = [Kn d ]; (3) the Frobenius norm of inverse matrices [Mn d ] -1 , [Dn d ] -1 , [Kn d ] -1 are secondorder random variables.

Fig. 5

 5 Fig. 5 Convergence analysis of the reduced matrix model. Graph of the mapping n → Conv 1 (n). Horizontal axis: dimension n of the reduced model. Vertical axis: Conv 1 (n)

Figure 5

 5 shows the graph of mapping n → Conv 1 (n). It can be deduced that the reduced nonlinear model is converged with n = 1400 as shown by the plateau which can be seen in this figure. This number is high when compared with the number of eigenfrequenies belonging to the frequency band of analysis B 1 = [0, 50]Hz which is n B1 = 5. This high value of n was discussed in section 5. Figures6 and 7allow the solutions of the nonlinear model by time-step computation (i.e solving Eqs. (4 for the whole structure) and (5)) and of the reduced nonlinear model (i.e solving Eqs.[START_REF]Information theory and statistical mechanics[END_REF]

  in which the (n d × n d ) upper triangular matrices [L M ], [L D ], [L K ] correspond to the Cholesky factorizations [Mn d ] = [L M ] T [L M ], [Dn d ] = [L D ] T [L D ], [Kn d ] = [L K ] T [L K ] and where [G M ], [G D ], [G K ] are random matrices for which the probability density functions p

  [L M ], [L D ] and [L K ] are random upper triangular (n d × n d ) real matrix such that (1) for j < j ′ , [L M ] jj ′ , [L D ] jj ′ and [L K ] jj ′ are real-valued Gaussian random variables with zero mean and variance equal to σ M = δ M / √ n d + 1, σ D = δ D / √ n d + 1 and σ K = δ K / √ n d + 1;

  [Mn d ], [Dn d ] and [Kn d ] are constructed such that[Mn d (θ 1 )] = [L M ] T [G M (θ 1 )] [L M ] , . . . , [Mn d (θn s )] = [L M ] T [G M (θn s )] [L M ] [Dn d (θ 1 )] = [L D ] T [G D (θ 1 )] [L D ] , . . . , [Dn d (θn s )] = [L D ] T [G D (θn s )] [L D ] [Kn d (θ 1 )] = [L K ] T [G K (θ 1 )] [L K ] , . . . , [Kn d (θn s )] = [L K ] T [G K (θn s )] [L K ]

5 ]) σ 2 A = (σ 2 C 2 +c 2 2 )/(σ 2 C 1 +c 2 1 )-(c 2 /c 1 ) 2 and σ 2 B = (σ 2 C 3 +c 2 3 )/(σ 2 C 2

 5222212122322 where A and B are two random variables with values in [1, +∞[ for which the mean values are A = c 2 /c 1 and B = c 3 /c 2 and for which the standard deviations σ A and σ B are such as (see[+c 2 2 )-(c 3 /c 2 ) 2 . It should be noted that σ A and σ B are defined only if inequalities σ C 3 /c 3 > σ C 2 /c 2 > σ C 1 /c 1 hold. Let p C 1 , p A , p B , p H 1 ,p H 2 and p H 3 be the probability density functions of random variables C 1 , A, B, H 1 , H 2 and H 3 . It has been shown in[START_REF] Hamza | Analyse probabiliste de la vulnérabilité sismique des btiments existants: application aux structures en portiques en béton armé[END_REF] thatp C 1 (x) = p 1 (x; c 1 , σ C 1 , 0) , p A (x) = p 1 (x; A, σ A , 1) , p B (x) = p 1 (x; B, σ B , 1) (22) p H 1 (x) = p 2 (x; h 1 , σ H 1 ) , p H 2 (x) = p 2 (x; h 2 , σ H 2 ) , p H 3 (x) = p 1 (x; h 3 , σ H 3 , 0) (23) in which p 1 (x; X, σ, a) = I [a,+∞[ (x) e -3 k=1 {L 1 (X,σ,a)} k x k-1 (24) p 2 (x; X, σ) = I [0,1] (x) e -3 k=1 {L 2 (X,σ)} k x k-1(25) where I D (x) is the indicator function of a given subset D ⊂ R and L 1 (X, σ, a) = arg min

L∈R 3 H 1

 31 (L; X, σ, a) , L 2 (X, σ) = arg min

L∈R 3 H 2 (H 1 ( 1 0 e - 3 k=1

 32113 L; X, σ, a) = {L} 1 + {L} 2 X + {L} 3 (σ 2 + X 2 ) + +∞ a e -3 k=1 {L} k x k-1 dx (27) H 2 (L; X, σ) = {L} 1 + {L} 2 X + {L} 3 (σ 2 + X 2 ) + {L} k x k-1 dx (28)

Table 1 :

 1 Comparison of the different kinds of modelling

	A) Nominal model
	Simplification of the structure :	→ Mean model
	keeping main structural elements,
	beam and plates simplification
	use of mean beam and plate thickness
	mean physical properties
	(B) Alea on parameters
	Random physical properties,	→ Parametric model
	Dispersion on thickness,...
	(C) Alea on the model
	Random stiffness, damping	→ NonParametric model
	and mass matrices
	Coupling induced between
	dynamical modes
	Alea on nonlinear properties (B) + on model and linear properties (C)
	Randomness on all parts of the model → Mixed model
	Table

C. Stochastic solver for the estimation of the confidence region of S j (ξ, ω)

The confidence region associated with a probability level Pc is constructed using quantiles (see for instance [START_REF] Serfling | Approximation theorems of Mathematical Statics[END_REF]). The method is briefly recalled. Let F S j (ξ,ω) (s j ; ξ, ω) = P {S j (ξ, ω) ≤ s j } be the cumulative distribution function of the random variable S j (ξ, ω). For p ∈]0, 1[, the pth quantile of F S j (ξ,ω) is defined as ζ(p) = inf{s j : F S j (ξ,ω) (s j ; ξ, ω) ≥ p}. Then, the upper envelope s + j and the lower envelope s - j of the confidence interval are defined by s + j = ζ((1 + Pc)/2) and s - j = ζ((1 -Pc)/2). Let s 1 j = S j (ξ, ω; θ 1 ), . . . , s ns j = S j (ξ, ω; θn s ) be ns independent realizations of S j (ξ, ω). Let s 1 j = S j (ξ, ω; θ 1 ) ≤ . . . ≤ s ns j = S j (ξ, ω; θn s ) be the order statistics associated with s 1 j , . . . , s ns j . Therefore, we have the following estimation: s + j = s k + j with k + = int(ns(1 + Pc)/2) and s - j = s k - j with k -= int(ns(1 -Pc)/2) in which int(z) is the integer part of the real number z.