Dr Meena Balasubramanian
email: meena.balasubramanian@nhs.net

Kath Smith

Lina Basel-Vanagaite

Michal Feingold

Pamela Brock

Gordan Gowans

Pradeep Vasudevan

Lara Cresswell

E J Taylor

Catharine Harris

Basel-Vanagaite

Feingold Mf

Gowans Gc

Vasudevan Pc

Taylor Ej

Harris Cj

Zackai Eh

Theisen A 13

Rosenfeld Ja

Parker Mj

Balasubramanian Al

Case Series: 2q33.1 Microdeletion Syndrome -Further delineation of the phenotype Running Title: 2q33.1 Microdeletion Syndrome: Expanding the phenotype

Keywords: 2q33.1 microdeletion, SATB2, ligamentous laxity, developmental delay, aCGH

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INTRODUCTION

Previous to molecular characterisation, there were very few case reports of interstitial deletion of 2q reported [START_REF] Sa | Interstitial deletion of the long arm of chromosome 2: del(2)(q31q33)[END_REF][START_REF] Miyazaki | Interstitial deletion 2q32.1----q34 in a child with half normal activity of ribulose 5-phosphate 3-epimerase (RPE)[END_REF]. Recently, molecular cytogenetic techniques such as microarray analysis have allowed for the characterisation of microdeletion 2q33.1 syndrome. Van Special AT-rich sequence binding protein 2 (SATB2; OMIM 608148), a DNA-binding protein that plays an important role in craniofacial patterning and brain development, has been identified as a candidate gene responsible for the craniofacial dysmorphology associated with deletions and translocations at 2q32-q33. This follows the identification of a locus for isolated cleft palate on chromosome 2q32, by [START_REF] Brewer | A locus for isolated cleft palate, located on human chromosome 2q32[END_REF], and the identification of translocations that disrupted SATB2 in individuals with isolated cleft palate [START_REF] Fitzpatrick | Identification of SATB2 as the cleft palate gene on 2q32-q33[END_REF]. Based upon several patients, with most of the features of 2q33.1 microdeletion syndrome and deletions within the gene, [START_REF] Rosenfeld | Small deletions of SATB2 cause some of the clinical features of the 2q33.1 microdeletion syndrome[END_REF] suggested that haploinsufficiency of SATB2 is responsible for several of the clinical features associated with 2q33.1 microdeletion syndrome.

We describe seven additional patients with the 2q33.1 microdeletion syndrome and provide genotype-phenotype correlation for some of the clinical features. We also describe several previously-unreported features in one or more individuals, thus expanding the phenotype. We also provide additional evidence that haploinsufficiency of SATB2 causes some of the clinical features in this condition.

MATERIALS AND METHODS

Array-based Comparative Genomic Hybridisation (aCGH)

aCGH was performed on DNA extracted from peripheral blood from Patient (1) using both a Bluegnome Cytochip 1Mb BAC array version 1.1 and an Oxford Gene Technologies 105K version 2 oligonucleotide array. Both were processed according to manufacturer's instructions, using Promega pooled control DNA as a reference. Slides were scanned using an Axon 4100A microarray scanner and analysed using either Bluefuse Multi v 2.1 or OGT Cytosure v3.0.6 software depending on the array manufacturer.

aCGH was performed on DNA extracted from peripheral blood from Patient (4) using Oxford Gene Technologies 60 mer oligo-array printed in 8x60 K International Standard Cytogenomic Array (ISCA) Consortium configuration, according to manufacturer's instructions, using Promega pooled control DNA as a reference. Slides were scanned using an Agilent microarray scanner (G2539A) and analysed using Agilent CGHAnalytics (v4.0) microarray software.

Oligonucleotide-based microarray analysis was performed on Patients ( 5) and [START_REF] Cocchella | The refinement of the critical region for the 2q31.2q32.3 deletion syndrome indicates candidate genes for mental retardation and speech impairment[END_REF] using a 105K-feature whole-genome microarray (SignatureChip Oligo Solution TM version 1.0, custom-designed by Signature Genomic Laboratories, made by Agilent Technologies, Santa Clara, CA), according to previously described methods [START_REF] Ballif | Identification of a previously unrecognized microdeletion syndrome of 16q11.2q12.2[END_REF].

Oligonucleotide-based microarray analysis was performed on Patients (2), ( 3) and ( 6) using a 135K-feature whole-genome microarray (SignatureChip Oligo Solution TM version 2.0, custom-designed by Signature Genomic Laboratories, made by Roche NimbleGen, Madison, WI), according to previously-described methods [START_REF] Duker | Paternally inherited microdeletion at 15q11.2 confirms a significant role for the SNORD116 C/D box snoRNA cluster in Prader-Willi syndrome[END_REF]. Results were displayed using custom oligonucleotide aCGH analysis software (Genooglyphix; Signature Genomic Laboratories).

Fluorescence in situ hybridisation (FISH)

Deletions were confirmed by metaphase fluorescence in situ hybridization (FISH) using BAC clones from the regions found to be deleted by aCGH, as previously described [START_REF] Traylor | Microdeletion of 6q16.1 encompassing EPHA7 in a child with mild neurological abnormalities and dysmorphic features: case report[END_REF], when whole blood samples were available (Patients (2), ( 4) and ( 6)).

Parental samples, when available, were also assayed using metaphase FISH.

CLINICAL REPORTS

Patient (1)

Patient (1) aged 19 years, is the second child of healthy, non-consanguineous Caucasian parents, with no significant family history. The pregnancy was normal. She was a full-term, normal delivery with a birth-weight of 2.72 kilograms (9 th -25 th centile) and was well immediately after birth. However, she had feeding problems and was bottle-fed with an orthodontic teat in the initial neonatal period. She was investigated for 'failure-to-thrive' as an infant, and no cause was found. She was noted to have breath-holding attacks as an infant, but her EEG was reported as normal. At two years of age, her weight was between the 3 rd -10 th centile, length just below the 50 th centile, and weight: length ratio and body mass index (BMI) both <3 rd centile.

Patient (1) had a normal brain-MRI at four years of age. Also, at this age she was diagnosed with conductive hearing loss and had grommets inserted. She was noted to be dysmorphic, with down-slanting palpebral fissures, prominent nose, slightly low-set ears, and a single palmar crease on the left hand (Figure 1a). Over time, the patient developed a 'marfanoid habitus' with arachnodactyly, bilateral claw toes with mild pes cavus, high-arched palate, and a long face (Figure 1b andc). She developed horizontal striae at the base of her back. She also has ligamentous laxity and bilateral mal-tracking patellae with subluxation in extension but no evidence of scoliosis. She has moderate to severe developmental delay, in particular speech and motor delay. She first walked at four years of age. She managed in mainstream school with one-to-one help. She has a deep, hoarse voice. Microlaryngoscopy showed normal vocal cords.

Homocystinuria was excluded, along with other metabolic investigations, which were reported as normal. She has challenging behaviour including hyperactivity, restlessness and has severe expressive and receptive language deficits. She remains under the Psychiatry services for management of her mood swings. She has significant problems with episodes of sub-luxation of her patellae and ligamentous laxity.

Patient (2)

Patient ( 2) is a five-year-old female, who was the first child of healthy nonconsanguineous Jewish parents (Iraqi and Ashkenazi/Moroccan origin). There is no significant family history. She was born after a normal pregnancy, by spontaneous delivery at 39 weeks of gestation, with a birth weight of 3.425 kg (50 th centile), length of 48.5 cm (25 th centile) and head circumference of 36.7 cm (98 th centile). She had feeding difficulties from birth and, at the age of four months, presented with failure to thrive: her weight was on the 10 th centile and her height was on the 3 rd centile.

Development was reported to be normal until the age of 12 months including, standing with support by the age of eleven months. At the age of eighteen months, psychomotor retardation became evident with generalised hypotonia, delayed first walking (25 months) and delayed language. At the age of 2.5 years, she was diagnosed with global developmental delay of about one year.

Dysmorphic features included sparse, thin hair, high forehead, down-slanting palpebral fissures, tented upper lip, downturned mouth corners, high-arched palate, crowded teeth and inverted nipples. She has mild conductive hearing loss on the left side. She was diagnosed with mild to moderate mental retardation. At five years of age, she started to run, has a few words and attends a special nursery. No behavioural or psychiatric problems were reported by the parents.

All metabolic investigations were normal in this patient, apart from a low serum carnitine level, which responded to supplementation and is likely to represent nutritional carnitine deficiency. She had a normal cardiac echocardiogram. MRI-brain demonstrated dilated Virchow-Robin spaces in centrum semiovale. She had a normal EEG and eye examination. Her bone age was normal at two years of age.

Patient (3)

Patient ( 3) is an eight year, eight month-old male with dysmorphic features, cleft palate, missing teeth, hypospadias, severe language impairment, and hypotonia (Figure 2a).

He required nasogastric tube feeding in early childhood. He had a surgical repair of a cleft palate and hypospadias. Neurodevelopmental evaluation at the age of five years and seven months, using the Wechsler Preschool and Primary Scale of Intelligence (WPPSI), showed a performance IQ of 51 (moderate mental retardation). He made vowel sounds, but no clear words, and was diagnosed with severe receptive and expressive language impairment with oral-motor dysfunction. He depends on non-verbal communication. Central nervous system dysfunction manifested as co-ordination and balance difficulties in gross motor skills and hypotonia. He has a happy disposition, but has some problems in school by inappropriately hugging others and touching their faces.

At the age of eight years and eight months, he was normocephalic with a triangular face, a high forehead and a prominent chin. Other dysmorphic features included hypertelorism, upturned nose, prominent and cupped pinnae, long philtrum, narrow palate, crowded dentition with missing front upper incisors, and slight pectus excavatum (Figure 2b). The hands were small with significant joint laxity to the fingers and normal palmar creases; they also had a "doughy" and soft consistency. The thumbs and great toes were

Patient (4)

Patient ( 4) is a nine year old male, the first child of non-consanguineous, White Caucasian, healthy parents with no significant family history. He was born after an uneventful pregnancy (normal antenatal scans), with no known teratogenic exposure at 35 weeks with a birth weight of 2.2 kilograms (25 th centile). A submucous cleft palate was detected at 18 days of age, which was subsequently repaired. He had initial feeding problems with failure to thrive.

On examination aged 17 months, his head circumference, weight and length were below the 3 rd centile. He was not walking at that age. On examination, he had a prominent forehead with prominent orbits, blue sclerae, small mouth, generalised hypotonia, 'myopathic' face and micrognathia (Figure 3a). He also had crowded teeth with alternating convergent strabismus. He had a heart murmur over the left sternal edge, but an echocardiogram did not reveal any structural anomalies of the heart. FISH for 22q11 deletion and Smith-Magenis syndrome, MRI-brain scan, metabolic tests, muscle biopsy, methylation for Angelman's syndrome and telomere analysis were normal. A review at eight years of age revealed low muscle tone, 'myopathic' face, severe learning difficulties and no speech (Figure 3b).

Patient (5)

Patient ( 5) was born at 38 weeks gestation to consanguineous Mexican parents, with normal birth-weight, length and head circumference. Pregnancy was complicated by gestational diabetes. At birth, he was found to have a cleft palate and remained in the hospital for ten days due to feeding problems. His cleft palate was repaired at one year of age. He was referred for a Genetics evaluation at 37 months, because of developmental delay. He reportedly sat at seven months and walked around two years of age. At 3 years of age, he has no speech and is not toilet-trained. Examination revealed normal weight, height, and head circumference. His lateral hairline protruded onto his forehead. He had a beaked nose with hypoplastic alae nasi and a high nasal bridge. His mouth was small with a short philtrum and micrognathia with long, narrow face and low-set ears. He has mild hearing loss and eye examination showed long lashes and intermittent exotropia. His feet have overlapping toes, prominent heels and tight heel cords and one café-au-lait spot was noted.

Patient (6)

Patient ( 6) is a four year-old male who first came to medical attention around two years of age with speech and language delay. There was no significant family history. He was born at term, following an unremarkable pregnancy, with a birth-weight of 3.4 kg (50 th centile). Prenatal ultrasound was normal, apart from a two-vessel cord. Postnatal complications included a weak cry, pallor and feeding difficulties, which resolved with the use of a special teat.

On review at age four, he only had a 10-word vocabulary, did not join words together, and was able to sign approximately 50 words. He was unable to name colours. His initial words were at fifteen months of age. Motor development was normal. Behavioural concerns included autistic-like features, easy distractibility, noise sensitivity (extremely frightened of loud noises), sensory perception issues (touches mother's hair all the time, puts mouth on the cat) and over-fixation on certain activities. Past medical history is significant for hyperopia in the right eye, esotropia in the left eye, recurrent otitis media (requiring placement of grommets), eczema and a heart murmur. MRI-brain was normal apart from multiple prominent peri-vascular spaces bilaterally and symmetrically.

On examination, he had a rectangular, prominent forehead, narrow and high-arched palate, dental crowding, narrow shoulder girdle, pectus carinatum and fifth toe clinodactyly.

Neurological examination was notable for impaired cognition, non-specific vocalisation with inappropriate bouts of laughter, hyperactivity, excessive drooling, pale fundi, and bilateral Babinski response.

Patient (7)

Patient [START_REF] Cocchella | The refinement of the critical region for the 2q31.2q32.3 deletion syndrome indicates candidate genes for mental retardation and speech impairment[END_REF] was one of twins, born to a 42-year-old mother at 40 weeks gestation, by Caesarean section, because of failure to progress. Her mother was on Enoxaparin and aspirin prenatally, because of a maternal clotting disorder. Birth-weight was 3.061kg (25-50 th centile) and there were no neonatal complications reported. The patient was seen by a Gastroenterologist at six months of age for poor weight gain, at which time reflux was diagnosed and treated. She was also evaluated by a Neurologist for hypotonia. Hearing was normal.

Formal developmental evaluation at three years showed mild delay and receptive language that was much better than expressive language. Her speech included only a few words, despite receiving speech and occupational therapy. She has global developmental delay. She rolled at 6-8 months, sat at 13 months, and walked at 19 months. Her twin sister was developing normally. On follow-up at four years, eleven months of age, she had about 4-5 words, and was using 7-8 signs. She was enrolled in a pre-school class with speech, occupational and physical therapy, as well as a sign language class. She was described as a friendly, social girl. Formal developmental evaluation at this age showed mild global developmental delay with more significant expressive language disorder. Examination showed small, upturned nasal tip, long philtrum, and high palate (Figure 4a andb). She has fair skin and curly hair. Her gait was slightly wide-based. At four years and eleven months, her gait had improved, but was noted to still be slightly abnormal. FISH in the parents of Patients ( 1) and ( 6) showed an apparent de novo origin; all other parents were unavailable for testing.

aCGH results

Oligonucleotide

DISCUSSION

Small interstitial deletions of the distal portion of the long arm of chromosome 2 are not widely reported, the majority being terminal or much larger than the deletion detected in our patients. This case series will help further refine genotype-phenotype co-relation in 2q33.1 microdeletion syndrome, as patients in our cohort only include part of the critical region defined and may provide information on the critical candidate genes for the various phenotypes in this microdeletion syndrome. Figure 6 is a schematic representation that summarises the position of the reported deletions and the genes implicated in contributing to the phenotype. Patient (1) and ( 4) have a proximal deletion breakpoint that is more distal to the others reported and may therefore contribute to redefining the minimal deleted region for this syndrome.

In our case series, all patients have varying clinical features of 2q33.1 microdeletion syndrome (Table 1). Facial features in all reported patients with 2q33 microdeletion syndrome appear to be similar including, down-slanting palpebral fissures, low-set ears and prominent nasal bridge with a beaked nose. Most patients also have high-arched/cleft palate.

Microcephaly or anatomical brain malformations on cranial imaging do not seem to be a consistent feature in reported patients. This would be consistent with our cohort of patients, as they all have similar facial dysmorphism to patients reported in the literature. In our opinion, there seems to be two distinct facial gestalts, one with up-turned nose, 'myopathic' appearance (Patients (3), ( 7 3), ( 4) and [START_REF] Brewer | A locus for isolated cleft palate, located on human chromosome 2q32[END_REF] all have cleft palate while the others have a high-arched palate. All of the patients in our cohort have a high prominent forehead and crowded teeth with impaired dentition, which is also reported. Common manifestations include, significant feeding problems, in most cases requiring naso-gastric feeding and failure-to-thrive. Previous reports have alluded to strabismus and other eye manifestations, which is also seen in Patients ( 4), ( 5) and ( 6) in our series. Neurological features of hypotonia, broad-based gait, moderate to severe developmental delay and significant speech delay with no speech in two patients in this report is also in keeping with previously-published cases.

Rosenfeld et al., in 2009 described three patients with smaller microdeletions within 2q33.1. The results of this study showed that haploinsufficiency of SATB2 causes some of the clinical features associated with this condition. Patient [START_REF] Brewer | A locus for isolated cleft palate, located on human chromosome 2q32[END_REF] represents the fourth reported individual with an intragenic SATB2 deletion. He had a cleft palate repaired, developmental delay and similar facial dysmorphism to other reported patients with the 2q33.1 microdeletion syndrome. He did not have any ectodermal features (Table 1). It is important to note that he has significant speech delay, more obvious than delay in other milestones and no reported behavioural problems, which supports the contribution of other genes to the behavioural phenotype in the 2q33.1 microdeletion syndrome. However, abnormal behaviour was reported in the previous cases with small SATB2 deletions [START_REF] Rosenfeld | Small deletions of SATB2 cause some of the clinical features of the 2q33.1 microdeletion syndrome[END_REF].

Patient [START_REF] Brewer | A locus for isolated cleft palate, located on human chromosome 2q32[END_REF] provides further support that SATB2 haploinsufficiency contributes to the developmental delay, cleft palate, and other dysmorphic features seen in the 2q33.1 microdeletion syndrome.

A characteristic feature reported in some individuals with 2q32q33 deletions is an ectodermal dysplasia-like phenotype, with thin, transparent skin and abnormalities of the hair and teeth (Table 1). The skin is reported to be similar to the Wrinkly Skin Syndrome (WSS) and is suggested to be due to haploinsufficiency for COL3A1 and COL5A2 [START_REF] Van Buggenhout | The del(2)(q32.2q33) deletion syndrome defined by clinical and molecular characterization of four patients[END_REF][START_REF] Rifai | Ectodermal dysplasia-like syndrome with mental retardation due to contiguous gene deletion: further clinical and molecular delineation of del(2q32) syndrome[END_REF]. However, other case reports of 2q32 microdeletions of these genes have alluded to thick hair and eyebrows in some patients [START_REF] Mencarelli | Clinical and molecular characterization of a patient with a 2q31.2-32.3 deletion identified by array-CGH[END_REF][START_REF] Prontera | 2q31.2q32.3 deletion syndrome: report of an adult patient[END_REF]. Furthermore, Patients ( 2) and ( 7) in our series have thin, fine and sparse hair, and their deletions do not include these collagen genes, which indicates that there are other genes in the region contributing to the ectodermal phenotype. For example, SATB2 haploinsufficiency apparently contributes to tooth abnormalities, though other ectodermal features are not observed in patients with deletions limited to SATB2 [START_REF] Rosenfeld | Small deletions of SATB2 cause some of the clinical features of the 2q33.1 microdeletion syndrome[END_REF].

Although orthopaedic problems have been reported in a few individuals with the 2q33.1 microdeletion syndrome, Patient (1) has a Marfanoid habitus, ligamentous laxity, recurrent patellar dislocation and on-going problems with the alignment of her toes. All the other patients in this series have delayed gross motor skills and difficulty with mobility.

Further case reports are required to identify if this is an additional feature in the phenotype of patients with the 2q33.1 microdeletion syndrome. This may be explained by a deletion of a gene outside of the critical region described previously in this syndrome.

The behavioural phenotype in patients with 2q33.1 microdeletion syndrome include: hyperactivity, chaotic behaviour, generally 'happy' personalities, but with bouts of anxiety or aggression. 'Happy disposition' as a younger child seems to be a consistently-reported observation in reported patients and was also seen in the patient reported by Urquhart et al., disorder with frequent awakening also seems to be a feature in previously-reported patients and in Patient (1) in our cohort. In Patient [START_REF] Sa | Interstitial deletion of the long arm of chromosome 2: del(2)(q31q33)[END_REF], there are significant behavioural problems and she remains under the Psychiatry services. Similarly, Patient (3) and ( 6) have a specific behavioural phenotype, including 'autistic-like' features and inappropriate behaviour.

However, the deletions in Patients ( 1) and ( 6) do not overlap, suggesting that multiple genes in this region contribute to the behavioural abnormalities seen in these patients. Candidate genes deleted in Patient (1) include CREB1 (OMIM *123810) and ADAM23 (OMIM *603710), which have been implicated as possible candidate genes in autism [START_REF] Pescucci | Chromosome 2 deletion encompassing the MAP2 gene in a patient with autism and Rett-like features[END_REF][START_REF] Valverde | Modulation of anxiety-like behavior and morphine dependence in CREB-deficient mice[END_REF], and hyperactivity candidates ABI2, CYP20A1 and ALS2

[www.genetics.med.ed.ac.uk/suspects/]. Other candidate genes for the neurologic phenotypes seen in individuals with deletions in this region include ZNF533 (180.01-180.43Mb) and MYO1B ), both of which are involved in neuronal function [START_REF] Mencarelli | Clinical and molecular characterization of a patient with a 2q31.2-32.3 deletion identified by array-CGH[END_REF][START_REF] Rifai | Ectodermal dysplasia-like syndrome with mental retardation due to contiguous gene deletion: further clinical and molecular delineation of del(2q32) syndrome[END_REF]. The GLS gene (191.45-191.53 Mb) was also suggested as a candidate for the behavioural phenotype, as it is involved in glutamate production [START_REF] Mencarelli | Clinical and molecular characterization of a patient with a 2q31.2-32.3 deletion identified by array-CGH[END_REF]. In our cohort, GLS and MYO1B are deleted in Patient (2) (Figure 6). Therefore, while these genes are unlikely to be the critical genes for the phenotype of del 2q33, behavioural abnormalities and significant developmental delays are seen in patients with non-overlapping deletions in this region, so many of these candidate genes may cumulatively contribute to the neurologic phenotype. This region on chromosome 2q has linkage evidence for an autism candidate region, and it is possible that there are multiple genes in the region linked to autism and the behavioural component in this syndrome [START_REF] Rabionet | Analysis of the autism chromosome 2 linkage region: GAD1 and other candidate genes[END_REF][START_REF] Mencarelli | Clinical and molecular characterization of a patient with a 2q31.2-32.3 deletion identified by array-CGH[END_REF][START_REF] Prontera | 2q31.2q32.3 deletion syndrome: report of an adult patient[END_REF][START_REF] Cocchella | The refinement of the critical region for the 2q31.2q32.3 deletion syndrome indicates candidate genes for mental retardation and speech impairment[END_REF].

SATB2 is a DNA-binding protein that regulates chromatin organisation and gene expression. It is expressed in the developing brain and is the regulatory determinant of cortico-cortical connections in the developing cerebral cortex [START_REF] Alcamo | Satb2 regulates callosal projection neuron identity in the developing cerebral cortex[END_REF].

Haploinsufficiency of the SATB2 gene causes cleft palate. It has also been shown to be the important determinant in regulating skeletal development, particularly cranio-facial patterning and osteoblast differentiation [START_REF] Dobreva | SATB2 is a multifunctional determinant of craniofacial patterning and osteoblast differentiation[END_REF]. SATB2 is also a target for SUMOylation (small ubiquitive relative modifier), a reversible modification of the protein that modulates its activity as a transcription factor [START_REF] Dobreva | SUMO modification of a novel MAR-binding protein, SATB2, modulates immunoglobulin mu gene expression[END_REF]. Murine in vivo experiments have suggested that SATB2 haploinsufficiency affects jaw development [START_REF] Britanova | Satb2 haploinsufficiency phenocopies 2q32-q33 deletions, whereas loss suggests a fundamental role in the coordination of jaw development[END_REF]. Nonsense mutation of SATB2 has been described in a patient with cleft palate, osteoporosis, severe learning difficulties, epilepsy and cognitive defects [START_REF] Leoyklang | Heterozygous nonsense mutation SATB2 associated with cleft palate, osteoporosis, and cognitive defects[END_REF]. The presence of cleft/high-arched palate in all of the patients reported is in concordance with the proposal by [START_REF] Fitzpatrick | Identification of SATB2 as the cleft palate gene on 2q32-q33[END_REF] that SATB2 located at 2q32-q33 (199.84-200.03Mb) is a candidate gene for cleft palate. Therefore, there is accumulating evidence of the significance of SATB2 gene in human development and its role in the developing brain. In the 2q33.1 microdeletion syndrome, it is likely that haploinsufficiency of SATB2 gene accounts for the developmental delay, cleft/high-arched palate and possibly even the facial dysmorphism and psychiatric problems. Given the role of SATB2 in osteoblast differentiation, more experiments need to be done to identify if this could account for the skeletal features in the 2q33.1 microdeletion syndrome. Interestingly, another gene in the region, PPIL3 has been implicated in cleft palate and is known to share isomerise activity with EBP, which in humans is known to cause X-linked cleft palate and chondrodysplasia punctata [START_REF] Ikegawa | Novel and recurrent EBP mutations in X-linked dominant chondrodysplasia punctata[END_REF]].

The seven individuals described in this study refine and expand the 2q33.1 microdeletion syndrome phenotype and highlight the psychiatric problems which may need to be managed at an earlier stage. Our results further suggest SATB2 is the candidate gene for most of the clinical phenotype in this condition. However, given the variability of the phenotype, the study of additional patients may help elucidate the phenotypic effects of other gene/s in this region.

ACKOWLEDGMENTS

We thank the families for their participation in this report.

COMPETING INTERESTS

A. Theisen and J.A. Rosenfeld are employees of Signature Genomic Laboratories, a subsidiary of PerkinElmer.

LICENCE STATEMENT

The corresponding author has the right to grant on behalf of all authors and does grant on behalf of all authors, an exclusive licence on a worldwide basis to the BMJ Publishing Group Ltd to permit this article to be published in JMG and any other BMJPGL products and sublicences such use and exploit all subsidiary rights, as set out in the BMJ licence.

(http://group.bmj.com/products/journals/instructions-for-authors/licenceforms).

FIGURE LEGENDS

  Buggenhout et al., 2005 described four patients with an interstitial deletion of chromosome 2q32q33. The authors described similarities in clinical findings, including growth retardation, distinct facial dysmorphism, micrognathia, cleft or high palate and persistent feeding difficulties. All had significant learning difficulties. Since then, further case reports by de Ravel et al., 2009, Urquhart et al., 2010, and Rifai et al., 2010 have described patients with this syndrome. Emerging evidence suggests that psychiatric problems, with a distinctive behavioural phenotype including hyperactivity, chaotic behaviour and happy personality with bouts of anxiety or aggression, are a significant part of the clinical features in some individuals [de Ravel et al., 2009; Urquhart et al., 2010].

  -based aCGH in patients 1-7 revealed deletions overlapping the previously-described critical region for the 2q33.1 microdeletion syndrome [Van Buggenhout et al., 2005] (Figure 5-supplementary material). Deletions ranged in size from 35 kb to 10.4 Mb. The deletion in Patient (1) started just distal to SATB2 (chr2:200,063,027-208,691,924, UCSC hg18 coordinates), approximately 20 kb 5' of the gene. The deletion in Patient (6) ended proximal to SATB2 (chr2:195,531,681-199,625,499), approximately 220 kb 3' of the gene. The smallest deletion was found in Patient (5), whose deletion was entirely within SATB2 (chr2:199,877,238-199,911,975). The remaining patients had deletions that encompassed SATB2 (Patient 2, chr2:190,915,507-201,302,003; Patient 3, chr2:194,947,694-200,850,112; Patient 4, chr2:199,340,284-204,310,511; Patient 7, chr2:199,118,846-201,565,929). No other significant copy number changes were found in any individuals.

  ), similar to Van Buggenhout et al., 2005 Patients 2 and 3), and the other with a prominent nose with down-slanting palpebral fissures (Patient (1), similar to Van Buggenhout et al., 2005 Patient 4). Patient (

  2010 and in Patients (3), (4) and (7) in our cohort. Older children with this syndrome are said to develop bouts of hyperactivity and aggression, which was seen in the patients reported by Van Buggenhout et al., 2005 (Patients 1-3 in their series) and de Ravel et al., 2009. Sleep

Figure

  Figure 1a and b: Facial appearance of Patient (1) at 5 and 10 years of age demonstrating

Figure

  Figure 1c and d: Facial appearance of Patient (1) at 19 years of age and arachnodactyly

Figure 6 :

 6 Figure 6: ArrayCGH results in current cohort and patients in the published literature. Green


Valerij G Bardakov
email: bardakov@math.nsc.ru

Andrei Vesnin
email: vesnin@math.nsc.ru

Bert Wiest
email: bertold.wiest@univ-rennes1.fr

Valeriy G Bardakov

Andrei Vesnin Yu

Sobolev

DYNNIKOV COORDINATES ON VIRTUAL BRAID GROUPS

Keywords: braid, virtual braid, Dynnikov coordinates, faithful invariants Mathematics Subject Classification 2010: 20F36, 57M25

Virtual braid groups

The group of virtual braids V B n , n 2, on n strings was introduced by Kauffman [START_REF] Kauffman | Virtual knot theory[END_REF] as a generalization of the classical braid group B n . The most useful system of generators and defining relations of V B n was introduced by Vershinin in [START_REF] Vershinin | Generalization of braids and homologies[END_REF]. The generators of V B n are σ 1 , . . . , σ n-1 , ρ 1 , . . . , ρ n-1 ,

and the defining relations are the following:

σ i σ i+1 σ i = σ i+1 σ i σ i+1 , σ i σ j = σ j σ i if |i -j| > 1, (1.2)
ρ 2 i = 1, i = 1, . . . , n -1, ρ i ρ i+1 ρ i = ρ i+1 ρ i ρ i+1 , ρ i ρ j = ρ j ρ i if |i -j| > 1, (1.3)
σ i ρ j = ρ j σ i if |i -j| > 1, ρ i ρ i+1 σ i = σ i+1 ρ i ρ i+1 . (1.4)
Thus, the group generated by σ 1 , . . . , σ n-1 with relations (1.2) is the braid group B n ; the group generated by ρ 1 , . . . , ρ n-1 with relations (1.3) is the symmetric 1 group S n ; the relations (1.4) will be referred to as mixed relations. The last presented relation is equivalent to

ρ i+1 ρ i σ i+1 = σ i ρ i+1 ρ i . (1.5)
We remark that the relations

ρ i σ i+1 σ i = σ i+1 σ i ρ i+1 , ρ i+1 σ i σ i+1 = σ i+1 σ i ρ i (1.6)
do not hold in V B n , and these relations will be referred to as forbidden relations.

(Adding these forbidden relations yields the so-called braid permutation group [START_REF] Fenn | The braid-permutation group[END_REF]).

There is a natural epimorphism π : V B n → S n defined by

π(σ i ) = π(ρ i ) = ρ i , i = 1, . . . , n -1.
The kernel ker(π) is called the virtual pure braid group and is denoted by V P n . Generators and relations for V P n are described in [START_REF] Bardakov | The virtual and universal braids[END_REF]. It is easy to see that V B n is a semidirect product: V B n = V P n ⋊ S n .

Coordinates on braid groups

In [2, Ch. 8], an action of the braid group B n on the integer lattice Z 2n by piecewiselinear bijections is defined. For the reader's convenience we recall the definition. For

(a, b, c, d) • σ -1 = (a -b + -(d + + f ) + , d + f -, c -d --(b --f ) -, b -f -), (2.2)
where

e = a -b --c + d + , f = a + b --c -d + . (2.3)
For a given vector (a 1 , b 1 , . . . , a n , b n ) ∈ Z 2n we define the action by

σ ε i ∈ B n , where i = 1, . . . , n -1: (a 1 , b 1 , . . . , a n , b n ) • σ ε i = (a ′ 1 , b ′ 1 , . . . , a ′ n , b ′ n ), (2.4)
where

a ′ k = a k , b ′ k = b k if k = i, i + 1, and
(a ′ i , b ′ i , a ′ i+1 , b ′ i+1 ) = (a i , b i , a i+1 , b i+1 ) • σ, if ε = 1, (a i , b i , a i+1 , b i+1 ) • σ -1 , if ε = -1. (2.5)
For a word w in the alphabet {σ ±1 1 , . . . , σ ±1 n-1 } we define an action by w:

(a 1 , b 1 , . . . a n , b n ) • w = (a 1 , b 1 , . . . a n , b n ), if w = 1, ((a 1 , b 1 , . . . a n , b n ) • σ ε i ) • w ′ , if w = σ ε i w ′ .
(2.6)

It can be shown that the above action by B n on Z 2n is well defined, i.e. if two words w 1 and w 2 present the same element of the braid group B n then

(a 1 , b 1 , . . . , a n , b n ) • w 1 = (a 1 , b 1 , . . . , a n , b n ) • w 2 for any vector (a 1 , b 1 , . . . , a n , b n ) ∈ Z 2n
. By the Dynnikov coordinates of a braid we will mean the vector (0, 1, . . . , 0, 1) • w, where w is a word representing that braid.

Example 2.1. Actions by some elements of B 2 on (0, 1, 0, 1) ∈ Z 2 are as follows:

(0, 1, 0, 1) • σ 1 = (1, 0, 0, 2), (0, 1, 0, 1) • σ -1 1 = (-1, 0, 0, 2), (0, 1, 0, 1) • σ 2 1 = (1, -1, 0, 3), (0, 1, 0, 1) • σ -2 1 = (-1, -1, 0, 3), . . . . . . (0, 1, 0, 1) • σ k 1 = (1, -k + 1, 0, k + 1), (0, 1, 0, 1) • σ -k 1 = (-1, -k + 1, 0, k + 1)
, where k ∈ N. Also, acting by some elements of B 3 on (0, 1, 0, 1, 0, 1) ∈ Z 6 we have (0, 1, 0, 1, 0, 1) • σ 1 σ -1 2 = (1, 0, -2, 0, 0, 3), (0, 1, 0, 1, 0, 1) • σ 1 σ 2 σ 1 = (2, 0, 1, 0, 0, 3). Remark 2.2. It is also shown in [START_REF] Dehornoy | Ordering braids[END_REF] that Dynnikov coordinates are faithful invariants of braids, i.e. if (0, 1, . . . , 0, 1) • w 1 = (0, 1, . . . , 0, 1) • w 2 then w 1 = w 2 in B n ; thus, Dynnikov coordinates are very useful for solving the words problem in B n .

Here is an outline of the proof: there is a bijection between vectors in Z 2n and integer laminations of a sphere with n + 3 punctures P 0 , P 1 , . . . , P n+1 , P ∞ ; under this bijection, our action of B n on Z 2n corresponds to the B n -action on a disk containing punctures P 1 , . . . , P n . The key observation is that this disk is filled by the lamination encoded by the vector (0, 1, . . . , 0, 1) (i.e. cutting this disk along its intersection with the lamination yields only disks and once-punctured disks).

Coordinates on virtual braid groups

Let us define an action by elements of V B n on Z 2n . Consider the actions on Z 4 by σ and σ -1 as defined in (2.1) and (2.2), and define the action by ρ as the following permutation of coordinates:

(a, b, c, d) • ρ = (c, d, a, b). (3.1)
For a given vector (a

1 , b 1 , . . . , a n , b n ) ∈ Z 2n we define the action by ρ i ∈ V B n , i = 1, . . . , n -1: (a 1 , b 1 , . . . , a n , b n ) • ρ i = (a ′ 1 , b ′ 1 , . . . , a ′ n , b ′ n ), (3.2)
where

a ′ k = a k , b ′ k = b k for k = i, i + 1, and (a ′ i , b ′ i , a ′ i+1 , b ′ i+1 ) = (a i , b i , a i+1 , b i+1 ) • ρ. (3.3)
The action by σ ε i ∈ V B n on Z 2n is defined according to (2.4) and (2.5).

Suppose that w is a word in the alphabet {σ ±1 1 , . . . , σ ±1 n-1 , ρ 1 , . . . , ρ n-1 } representing an element of the group V B n . Then we define

(a 1 , b 1 , . . . , a n , b n ) • w = ((a 1 , b 1 , . . . , a n , b n ) • ρ i ) • w ′ , if w = ρ i w ′ , ((a 1 , b 1 , . . . , a n , b n ) • σ ε i ) • w ′ , if w = σ ε i w ′ . (3.4)
To show that the action by V B n on Z 2n is correctly defined we will verify that the defining relations of the group V B n are satisfied. Since ρ i acts by permuting pairs of coordinates, the relations of the group S n are obviously satisfied. The fact that relations of the group B n are satisfied follows from [START_REF] Dehornoy | Ordering braids[END_REF]. So, we need to check only the case of mixed relations, i.e. that for any v ∈ Z 2n the relations

v • (σ i ρ j ) = v • (ρ j σ i ), |i -j| > 1, (3.5) v • (ρ i ρ i+1 σ i ) = v • (σ i+1 ρ i ρ i+1 ) (3.6)
hold. Relations (3.5) hold obiously, because σ i acts non-trivial only on the subvector (a i , b i , a i+1 , b i+1 ) and ρ j acts non-trivially only on the subvector (a j , b j , a j+1 , b j+1 ).

In order to verify (3.6) it is enough to consider the case i = 1 in the group

V B 3 . Denote (x, y, z, t) • σ = (a ′ (x, y, z, t), b ′ (x, y, z, t), c ′ (x, y, z, t), d ′ (x, y, z, t)) From (a 1 , b 1 , a 2 , b 2 , a 3 , b 3 ) • (ρ 1 ρ 2 σ 1 ) = (a 2 , b 2 , a 1 , b 1 , a 3 , b 3 ) • (ρ 2 σ 1 ) = (a 2 , b 2 , a 3 , b 3 , a 1 , b 1 ) • σ 1 = ((a 2 , b 2 , a 3 , b 3 ) • σ, a 1 , b 1 )
and

(a 1 , b 1 , a 2 , b 2 , a 3 , b 3 ) • (σ 2 ρ 1 ρ 2 ) = (a 1 , b 1 , (a 2 , b 2 , a 3 , b 3 ) • σ) • ρ 1 ρ 2 = (a ′ (a 2 , b 2 , a 3 , b 3 ), b ′ (a 2 , b 2 , a 3 , b 3 ), a 1 , b 1 , c ′ (a 2 , b 2 , a 3 , b 3 ), d ′ (a 2 , b 2 , a 3 , b 3 )) • ρ 2 = (a ′ (a 2 , b 2 , a 3 , b 3 ), b ′ (a 2 , b 2 , a 3 , b 3 ), c ′ (a 2 , b 2 , a 3 , b 3 ), d ′ (a 2 , b 2 , a 3 , b 3 ), a 1 , b 1 ) = ((a 2 , b 2 , a 3 , b 3 ) • σ, a 1 , b 1 )
we see that (3.6) holds.

Example 3.1. Actions by some elements of V B 2 on (0, 1, 0, 1) ∈ Z 4 are as follows:

(0, 1, 0, 1)

• σ 1 ρ 1 = (0, 2, 1, 0), (0, 1, 0, 1) • σ 1 ρ 1 σ 1 = (3, 0, 0, 2), (0, 1, 0, 1) • σ 1 ρ 1 σ -1 1 = (-2, -1, 1, 3
). Let us demonstrate that the forbidden relations are not satisfied. More exactly, we show that for v = (0, 1, 0, 1, 0, 1) we get

v • (ρ 1 σ 2 σ 1 ) = v • (σ 2 σ 1 ρ 2 ), (3.7) v • (ρ 2 σ 1 σ 2 ) = v • (σ 1 σ 2 ρ 1 ). (3.8)
Indeed, (3.7) holds because (0, 1, 0, 1, 0, 1)•(ρ 1 σ 2 σ 1 ) = (0, 1, 0, 1, 0, 1)•σ 2 σ 1 = (0, 1, 1, 0, 0, 2)•σ 1 = (2, 0, 0, 1, 0, 2), but (0, 1, 0, 1, 0, 1)

• (σ 2 σ 1 ρ 2 ) = (2, 0, 0, 1, 0, 2) • ρ 2 = (2, 0, 0, 2, 0, 1).
Analogously, (3.8) holds because

(0, 1, 0, 1, 0, 1)•(ρ 2 σ 1 σ 2 ) = (0, 1, 0, 1, 0, 1)•σ 1 σ 2 = (1, 0, 0, 2, 0, 1)•σ 2 = (1, 0, 2, 0, 0, 3), but (0, 1, 0, 1, 0, 1) • (σ 1 σ 2 ρ 1 ) = (1, 0, 2, 0, 0, 3) • ρ 1 = (2, 0, 1, 0, 0, 3).
Question 3.2. Is there any relation between our coordinates on V B n and the invariant of virtual braids defined by Manturov in [5]?

Faithfulness of the V B n -action on Z 2n

In this section we will be concerned with the following Question 4.1. Is the V B n -action on Z 2n faithful? In other words, is it true that only the trivial element of V B n acts as the identity on Z 2n ?

In computer experiments, we have tested several billion (10 9 ) random virtual braids with 3, 4 and 5 strands, but failed to find one that would provide a negative answer. The programs used for these tests (written in Scilab) can be obtained from B. Wiest's web page [7].

It should be stressed that nontrivial elements of V B n may very well act trivially on individual vectors. Let us, for instance, look at the V B 3 -action on (0, 1, 0, 1, 0, 1). The actions of ρ 1 and ρ 2 obviously fix this vector, but those of many other braids do, too. Here is one particularly striking example:

Example 4.2. The virtual 3 strand braid

β = σ 1 ρ 2 σ 1 σ -1 2 σ 1 σ 2 σ -1 1 ρ 1 σ 2 ρ 1 σ 1 ρ 2 σ -1 1 ρ 2 σ -1 2 σ -1 1 σ 2 σ -1 1 ρ 2 σ -1
1 acts trivially on the vector (0, 1, 0, 1, 0, 1), and indeed in computer experiments, we found that it acted nontrivially on only about 0.25% of randomly generated vectors of Z 6 with integer coefficients between -100 and 100. In this sense, β is "nearly a negative answer" to Question 4.1. Another one is the virtual braid

σ -1 2 σ 1 ρ 2 σ 2 σ 1 σ -1 2 ρ 2 σ 1 ρ 2 σ 2 ρ 1 σ -1 2 ρ 1 σ -1 1 σ -1 2 ρ 2 σ -1
1 σ 2 which also moves only about 0.6% of random vectors.

We now turn our attention to the case n = 2.

Example 4.3. It is known [5] that β = (σ 2 1 ρ 1 σ -1 1 ρ 1 σ -1 1 ρ 1 ) 2 ∈ V B 2
and the identity element e ∈ V B 2 cannot be distinguished by the Burau representation. Considering actions by these elements on (0, 1, 0, 1) we get (0, 1, 0, 1) • β = (85, 49, -90, -47), while (0, 1, 0, 1)•e = (0, 1, 0, 1). Thus, these elements are distinguished by Dynnikov coordinates.

In fact, in the case n = 2 we have a positive answer to Question 4.1.

Theorem 4.4. The V B 2 -action on Z 4 given by the above formulae is faithful.

Proof. We will, in fact show a stronger result than Theorem 4.4, namely that the only element of V B 2 which acts trivially on a vector (0, x, 0, y), where x and y are different positive integers, is the trivial one. As a simplest vector of this type one can take (0, 2, 0, 1).

Consider the set of symbols S = {0, +, -, +0, -0}. With s ∈ S we associate the following subset of Z: For instance, the vector (-1, -1, 0, 4) belongs to sets associated with (-, -, 0, +) and (-, -0, +0, +), as well as with some other quadruples of symbols.

s = 0 ←→ {x ∈ Z : x = 0}, s = + ←→ {x ∈ Z : x > 0}, s = -←→ {x ∈ Z : x < 0}, s = +0 ←→ {x ∈ Z : x 0}, s = -0 ←→ {x ∈ Z : x 0}. A quadruple (s 1 , s 2 , s 3 , s 4 ),
Let σ, σ -1 , ρ : Z 4 → Z 4 be the transformations defined by (2.1), (2.2), (3.1). Let us apply a sequence of such transformations (without fragments σσ -1 , or σ -1 σ, or ρρ) to the initial vector, say (0, 2, 0, 1). The statement of the theorem will follow from the fact that we trace out a path in the diagram shown in Figure 1, where an arrow with labels β and i.β, where i ∈ {1, 2, . . . , 14} and β ∈ {σ, σ -1 , ρ} from box (s 1 , s 2 , s 3 , s 4 ) to box (t 1 , t 2 , t 3 , t 4 ) means that the image of any element of the subset of Z 4 associated with (s 1 , s 2 , s 3 , s 4 ) under tranformation β belongs to the subset of Z 4 associated with (t 1 , t 2 , t 3 , t 4 ). The numbering of arrows by 1, 2, . . . , 14 is done for the reader's convenience in the further discussion of cases.

It is easy to see from (2.1), (2.2) and (3.1) that for any β ∈ V B 2 there is the following invariant: if (a * , b * , c * , d * ) = (a, b, c, d)•β then b * +d * = b+d. In particular, if the initial vector is taken to be (0, 2, 0, 1) then for any vector in the diagram the sum of its second and fourth coordinates is equal to 3.

Below to simplify expressions we will use the following notations:

(a ′ , b ′ , c ′ , d ′ ) = (a, b, c, d) • σ, (a ′′ , b ′′ , c ′′ , d ′′ ) = (a, b, c, d) • σ -1 .
ρ ? 0,+,0,+ +,0,0,+ ? ? -,0,0,+ ? ?

2.σ 1.σ -1 0,+,+,0 ? By (2.1 ) and (2.2) we have

           a ′ = a + b + + (-a + b -+ c) + , b ′ = d -(a -b --c + d + ) + , c ′ = c + d -+ (a -c + d + ) -, d ′ = b + (a -b --c + d + ) + ,            a ′′ = a -b + -(a + b --c) + , b ′′ = d + (a + b --c -d + ) -, c ′′ = c -d --(-a + c + d + ) -, d ′′ = b -(a + b --c -d + ) -, (4.1)
First of all we remark that the arrows of the diagram related to the action by ρ hold obviously. The other actions will be considered case by case according to the numbering of the arrows.

1. Consider the action by σ -1 on (0, +, 0, +). Each element of (0, +, 0, +) has a form (0, b, 0, d) for some b, d > 0. Since its image (0, b, 0, d) • σ -1 = (-b, 0, 0, b + d) belongs to (-, 0, 0, +), the corresponding arrow of the diagram is proven.

2. Consider the action by σ on (0, +, 0, +). Each element of (0, +, 0, +) has a form (0, b, 0, d) for some b, d > 0. Since its image (0, b, 0, d) • σ = (b, 0, 0, b + d) belongs to (+, 0, 0, +), the corresponding arrow of the diagram is proven.

3. Consider the action by σ -1 on (-, 0, 0, +). Let a < 0, d > 0 then (a, 0, 0, d) • σ -1 = (a, d+(a-d) -, 0, -(a-d) -) = (a, a, 0, -a+d) ∈ (-, -, 0, +) ⊂ (-, -, +0, +).

4. Consider the action by σ on (+, 0, 0, +). Let a, d > 0 then (a, 0, 0, d) • σ = (a, -a, 0, a + d) ∈ (+, -, 0, +) ⊂ (+, -, -0, +). 9. Let us demonstrate that (-, -, +0, +) • σ -1 ∈ (-, -, +0, +). Since a < 0, b < 0, c 0, d > 0, the action is given by formulae

           a ′′ = a -(a + b -c) + , b ′′ = d + (a + b -c -d) -, c ′′ = c -(-a + c + d) -, d ′′ = b -(a + b -c -d) -. Moreover, a + b -c < 0 and a + b -c -d < 0 imply that (a, b, c, d) • σ -1 = (a, a + b -c, c, -a + c + d) ∈ (-, -, +0, +).
10. Let us demonstrate that (+0, +, -, -) • σ -1 ∈ (-, -, +0, +). Since a 0, b > 0, c < 0, d < 0, the action is given by formulae

           a ′′ = a -b -(a -c) + , b ′′ = d + (a -c) -, c ′′ = c -d -(-a + c) -, d ′′ = b -(a -c) -. Moreover, a-c > 0 implies that (a, b, c, d)•σ -1 = (-b+c, d, a-d, b) ∈ (-, -, +, +) ⊂ (-, -, +0, +).
11. Let us demonstrate that (+0, +, -, -)•σ ∈ (+, -, -0, +). Since a 0, b > 0, c < 0, d < 0, the action is given by formulae

           a ′ = a + b + (-a + c) + , b ′ = d -(a -c) + , c ′ = c + d + (a -c) -, d ′ = b + (a -c) + . Moreover, a -c > 0 implies that (a, b, c, d) • σ = (a + b, d -a + c, c + d, b + a -c) ∈ (+, -, -, +) ⊂ (+, -, -0, +).
12. Let us demonstrate that (+, -, -0, +)•σ = (+, -, -0, +). Since a > 0, b < 0, c 0, d > 0, the action is given by formulae

           a ′ = a + (-a + b + c) + , b ′ = d -(a -b -c + d) + , c ′ = c + (a -c + d) -, d ′ = b + (a -b -c + d) + . Moreover, -a + b + c < 0, a -b -c + d > 0, and a -c + d > 0 imply that (a, b, c, d) • σ = (a, -a + b + c, c, a -c + d) ∈ (+, -, -0, +).
13. Let us demonstrate that (-0, +, +, -)•σ ∈ (+, -, -0, +). Since a 0, b > 0, c > 0, d < 0, the action is given by formulae Remark 4.5. Theorem 4.4 allows us to introduce on V B 2 various "coordinate systems". For example, taking (0, 2, 0, 1) as the initial vector, for any β ∈ V B 2 one can define (0, 2, 0, 1) • β as its coordinates. In this sense Theorem 4.4 gives an anolog of Dynnikov coordinates originally defined for braid groups.

           a ′ = a + b + (-a + c) + , b ′ = d -(a -c) + , c ′ = c + d + (a -c) -, d ′ = b + (a -c) + . Moreover, a -c < 0 implies that (a, b, c, d) • σ = (b + c, d, d + a, b) ∈ (+, -, -, +) ⊂ (+, -, -0, +). 14. Let us demonstrate that (-0, +, +, -) • σ -1 = (-, -, +0, +). Since a 0, b > 0, c > 0, d < 0,
Remark 4.6. It is shown in the proof of Theorem 4.4 that the V B 2 -action is faithful on any vector of the form (0, x, 0, y), where x and y are different positive integers. Note that the action on some other vectors of Z 4 can fail to be faithful. For example, (0, 0, 0, 1) • σ = (0, 0, 0, 1) and (0, 1, 0, 1) • ρ = (0, 1, 0, 1).

  x ∈ Z denote x + = max{0, x} and x -= min{x, 0}. Define actions σ, σ -1 : Z 4 → Z 4 on (a, b, c, d) ∈ Z 4 as follows: (a, b, c, d) • σ = (a + b + + (d +e) + , de + , c + d -+ (b -+ e) -, b + e + ), (2.1)

  where s i ∈ S, indicates the set all quadruples (a, b, c, d) ∈ Z 4 such that a, b, c, d belongs to subset of Z associated with symbols s 1 , s 2 , s 3 , s 4 , respectively. For example, (+, +0, 0, -) = {(a, b, c, d) ∈ Z 4 : a > 0, b 0, c = 0, d < 0}.

- 1 Fig. 1 .

 11 Fig. 1. The diagram of actions of V B 2 .

5 .

 5 Consider the action by σ -1 on (0, +, -, 0). Let b > 0, c < 0 then (0, b, c, 0)• σ -1 = (-b -(-c) + , 0, c -(c) -, b -(-c) -) = (-b + c, 0, 0, b) ∈ (-, 0, 0, +).6. Consider the action by σ on (0, +, +, 0). Let b, c > 0 then (0, b, c, 0) • σ = (b + c, 0, 0, b) ∈ (+, 0, 0, +).7. Consider the action by σ on (0, +, -, 0). Let b > 0, c < 0 then (0, b, c, 0)• σ = (b, c, c, bc) ∈ (+, -, -, +) ⊂ (+, -, -0, +).8. Consider the action by σ -1 on (0, +, +, 0). Let b, c > 0 then (0, b, c, 0) • σ -1 = (-b, -c, c, b + c) ∈ (-, -, +, +) ⊂ (-, -, +0, +).

a

  ′′ = ab -(ac) + , b ′′ = d + (ac) -, c ′′ = cd -(-a + c) -, d ′′ = b -(ac) -. (4.2) Moreover, ac < 0 implies that (a, b, c, d) • σ -1 = (ab, d + ac, cd, ba + c) ∈ (-, -, +, +) ⊂ (-, -, +0, +). The proof is completed.

Remark 4 . 7 .

 47 Let us define the norm of a quadruple ||(a, b, c, d)|| = |a|+|b|+|c|+|d|. Obviously, the norm is invariant under the ρ-action. One can easily see from the proof of Theorem 4.4 that all the arrows labelled σ or σ -1 in Figure 1 increase the norm. For example, considering case 13, one gets ||(a, b, c, d) • σ|| = ||(b + c, d, d + a, b)|| = |b + c| + |d| + |d + a| + |b| > |a| + |b| + |c| + |d|, because a 0, b > 0, c > 0, d < 0. However, this property doesn't hold if one takes an arbitrary vector from Z 4 : ||(7, 4, 1, 1)|| = 13, but ||(7, 4, 1, 1) • σ|| = ||(3, 1, 6, 1)|| = 11.

Acknowledgements

Work performed under the auspices of the Russian Foundation for Basic Research by the grant 10-01-00642 and by the French -Russian grant 10-01-91056. The first and the second named authors thank IRMAR, Université de Rennes 1 and Abdus Salam School of Mathematical Sciences, GC University Lahore for their hospitality.


Hideo Aochi
email: h.aochi@brgm.fr

Ariane Ducellier

Fabrice Dupros

Monique Terrier

Jérôme Lambert

Investigation of historical earthquake by seismic wave propagation simulation: Source parameters of the 1887 M6.3 Ligurian, north-western Italy, earthquake

Keywords: séisme ligure de 1887, mouvement du sol, méthode des différences finies, SisFrance 1887 Ligurian earthquake, ground motion, finite difference method, SisFrance

Les simulations numériques haute performance de la propagation des ondes sismiques de la source à la surface du sol sont un outil puissant pour l'étude d'un séisme. Comme la sismologie moderne basée sur les observations instrumentales n'existe que depuis quelques décennies, la simulation numérique est essentielle pour l'étude des séismes historiques ou l'estimation de futurs séismes pour lesquels il n'y a pas de données. Pour la France métropolitaine, le séisme ligure de 1887 est un des séismes les plus destructeurs, dont la position est estimée au nord-ouest de l'Italie, à proximité de la frontière avec la France, au large de la côte méditerranéenne. Nous effectuons des simulations numériques du mouvement du sol, à l'aide d'une méthode de différences finies (FDM) en 3D, basées sur des scénarios potentiels de séismes. En comparant l'extension spatiale des dommages (intensité), nous testons différents scénarios pour ce séisme basés sur des considérations sismotectoniques : (1) une faille normale superficielle proche de la côte, (2) une faille inverse superficielle au large et (3) une faille inverse plus profonde et plus proche de la côte. Les résultats de nos simulations améliorent nos connaissances des phénomènes passés et contribuent à l'évaluation de l'aléa sismique régional. ABSTRACT. High performance numerical simulation of the seismic wave propagation from the earthquake source to the ground motion is a powerful approach to investigate the earthquake and its sequence. As the modern instrumental observational seismology has lasted only since a few decades, the numerical investigation is essential for studying the past historical earthquakes or assessing the future earthquakes for which there are no data. For French Metropolitan, the 1887 Ligurian earthquake is one of the most damaging earthquakes, whose position is estimated in the north-western Italy close to the border to France, off shore of the Mediterranean coast. We carry out numerical simulations of the ground motion, using a 3D Finite Difference Method (FDM), based on the supposed potential earthquake scenarios. Through testing earthquake scenarios and comparing to the spatial pattern of the damaging (intensity), we test different scenarios of this earthquake based on the seismotectonic insights, (1) a shallow normal faulting close to the coastline, (2) a shallow reverse faulting off shore and (3) a deep reverse faulting close to the coastline. Our simulation results improve our knowledge of past phenomena and contribute to the assessment of the regional seismic hazard.

Introduction

The high performance numerical simulations are actually powerful tools for studying the earthquakes and their sequences. The wave propagation in an elastic (sometimes anelastic) medium at large scale can be simulated from the earthquake source to the local ground motion, using for example finite difference scheme (Olsen et al., 1997, Furumura and[START_REF] Furumura | Parallel simulation of strong ground motions during recent and historical damaging earthquake in Tokyo, Japan[END_REF], and many others) and spectral element scheme (e.g. Komatitsch et al., 2004, and some others). Such fully numerical schemes allow us to introduce the complexity in the earthquake source description and/or in the medium property in higher resolution. This helps us to understand better the earthquake mechanism and analyze more quantitatively the sequential ground motion at sites of interest. The recent main damaging earthquakes are studied through such numerical simulations, such as the Mw7.3 1992 Landers, California [START_REF] Olsen | Three-dimensional dynamic simulation of the 1992 Landers earthquake[END_REF], the Mw7.4 1999 Izmit, Turkey [START_REF] Aochi | The 1999 Izmit, Turkey, earthquake: Non-planar fault structure, dynamic rupture process and strong ground motion[END_REF], the Mw6.6 2000 Tottori, Japan [START_REF] Furumura | Visualization of 3D wave propagation from the 2000 Tottori-ken Seibu, Japan, earthquake: Observation and numerical simulation[END_REF], the Mw8.3 2003 Tokachi-oki, Japan [START_REF] Aoi | Three-dimensional finite difference simulation of long-period ground motions for the 2003 Tokachi-oki, Japan, earthquake[END_REF], the Mw7.9 Wenchuan, China [START_REF] Zhang | Numerical simulation of strong ground motion for the M S8.0 Wenchuan earthquake of 12 May 2008[END_REF] and so on. Furthermore the numerical simulations are useful for the historical earthquake (before mid-20th century) for which we do not know well, for example, the 1855 M7 Ansei-Edo, Japan, the 1923 M7.9 Kanto, Japan [START_REF] Furumura | Parallel simulation of strong ground motions during recent and historical damaging earthquake in Tokyo, Japan[END_REF], the 1906 M7.9 San Francisco, California [START_REF] Aagaard | Ground-Motion Modeling of the 1906 San Francisco Earthquake, Part II: Ground-Motion Estimates for the 1906 Earthquake and Scenario Events[END_REF], the 1944 M8 Tonankai, Japan [START_REF] Furumura | An integrated simulation of ground motion and tsunami for the 1944 Tonankai earthquake using high-performance super computers[END_REF]. For the earthquakes instrumentally well recorded, magnitude can be estimated based on the seismic moment (N.m) as moment magnitude Mw, while the historical events are not well constrained (sometimes by geodetic measurement, damage or paleoseismological observation) so that their magnitude are briefly expressed as magnitude M, without precision of the source. We do not aim to cite any more works about the applications for seismic hazard assessment, namely the earthquake scenarios for the future, as there are too many.

The technique to simulate the wave propagation in an elastic medium was well established long time ago, although the improvement is always going on. One of the most popular methods is a finite difference method (FDM), whose framework in seismology is found in [START_REF] Madariaga | Dynamics of an expanding circular fault[END_REF] and already on the textbook of [START_REF] Aki | Quantitative Seismology[END_REF]. Many seismological applications are based on the staggered grid proposed in [START_REF] Madariaga | Dynamics of an expanding circular fault[END_REF] and [START_REF] Virieux | P-SV wave propagation in heterogeneous media: velocity-stress finite-difference method[END_REF], mainly because the scheme is very stable. Later comparing to this fully staggered grid, a partially staggered grid has been proposed [START_REF] Saenger | Modeling the propagation of elastic waves using a modified finite-difference grid[END_REF]. The classical FDM is based on the second order precision in space [START_REF] Madariaga | Dynamics of an expanding circular fault[END_REF][START_REF] Virieux | P-SV wave propagation in heterogeneous media: velocity-stress finite-difference method[END_REF] and then the fourth order precision becomes the most popular from the viewpoint of performance and precision [START_REF] Levander | Fourth-order finite-difference P-SV seismograms[END_REF], although higher order is still possible. The parallel computing became already casual in the mid-1990s [START_REF] Olsen | Simulation of three-dimensional wave propagation in the Salt Lake Basin[END_REF][START_REF] Graves | Simulating seismic wave propagation in 3D elastic media using staggered-grid finite differences[END_REF]. The parallelization is relatively easier than other methods as the computing grids are in most cases structural. Thus The blue rectangle corresponds to the area for which we have originally a 3D structure model. For references, we show the recent earthquakes around Nice (1999 and2001).

the FDM is a very good example for high performance computing. In this paper, we adopt a 3D finite difference method and optimize it by equilibrating the charges on different processors. We demonstrate the numerical simulation for a historical event, the 1887 Liguria earthquake, which damaged the most in France Metropolitan and discuss its source mechanism.

The Liguria region, south-east coast of France along the Mediterranean Sea and close to the border to Italy (Figure 1), is one of the regions of metropolitan France where the necessity of studying seismic hazard and risk is more important. It is known that the region suffered several times from destructive earthquakes in 1494, 1564, 1618, 1644, 1887 after French historical earthquake catalogue (SisFrance, http://www.sisfrance.net ) and there have been previously several studies concerning the regional and local seismic hazard and risk evaluation. For example, the European Risk-UE project (e.g. [START_REF] Mouroux | Presentation of RISK-UE project[END_REF]) evaluated the seismic risk in the city of Nice for an earthquake scenario. In French national project, some earthquake scenarios are proposed according to the interpretation of the regional seismotectonics. In most cases, the reference earthquake is taken as a magnitude M6.3-6.4 which corresponds to the 23th February 1887 Ligurian earthquake (e.g. [START_REF] Salichon | Earthquake scenario in the city of Nice (South-East France): Ground motion simulations[END_REF].

The 1887 event is the largest earthquake in the region. However its source mechanism is not well known. The epicenter of the 1887 Ligurian earthquake is found in north-western Italy, close to the border to France, in the Mediterranean Sea. Significant damage was reported at various towns from north-western Italy and southeastern France (see Figure 2: macroscopic intensity after the database SisFrance). The macroseismic intensity at the epicenter is estimated at 9.5 in the MSK intensity scale (SisFrance, http://www.sisfrance.net ), and the magnitude is estimated at around 6.2 -6.5 [START_REF] Barani | Sensitivity analysis of seismic hazard for Western Liguria (North Western Italy): A first attempt toward the understanding and quantification of hazard uncertainty[END_REF][START_REF] Capponi | The Ligurian earthquake of February 23, 1887[END_REF]Carrozo et al., 1973;[START_REF] Ferrari | The 1887 Ligurian earthquake: A detailed study from contemporary scientific observations[END_REF]. The epicenter is not so fairly determined because it is off-shore, probably at a distance of tenths kilometers from the Ligurian coast (see Figure 1). Table 1 summarizes epicenter location and magnitude proposed in previous studies. The fault mechanism is considered as a thrust fault as reverse mechanisms were observed for recent earthquakes in this area [START_REF] Barani | Sensitivity analysis of seismic hazard for Western Liguria (North Western Italy): A first attempt toward the understanding and quantification of hazard uncertainty[END_REF][START_REF] Baroux | Analyses of the stress field in south-eastern France from earthquake focal mechanisms[END_REF][START_REF] Béthoux | Mécanismes au foyer de séismes en mer Ligure et dans le Sud des Alpes occidentales : résultats et interprétations[END_REF][START_REF] Béthoux | A closing Ligurian Sea?[END_REF]. However, a tsunami with a negative first wave was observed at a large distance along the coast from Genova (Italy) to Cannes (France) (e.g. [START_REF] Eva | The February 23, 1887 tsunami recorded on the Ligurian coast, western Mediterranean[END_REF], which supports the assumption that the earthquake was produced by an off-shore normal fault. Both normal and reverse faultings were used in the previous studies [START_REF] Eva | The February 23, 1887 tsunami recorded on the Ligurian coast, western Mediterranean[END_REF][START_REF] Frisenda | Modellazione numerica 3D di terremoti localizzati in mare: Risultati preliminary[END_REF][START_REF] Frisenda | Numerical modeling of the ground motion produced by the 1887 offshore Western Liguria earthquake[END_REF][START_REF] Pelinovsky | Modelling of tsunami propagation in the vicinity of the French coast of the Mediterranean[END_REF]. Comparing to the detailed regional active fault maps and interpretation of the structural geology [START_REF] Barani | Sensitivity analysis of seismic hazard for Western Liguria (North Western Italy): A first attempt toward the understanding and quantification of hazard uncertainty[END_REF][START_REF] Bigot-Cormier | La surrection du massif cristallin externe de l'Argentera (France-Italie) et ses relations avec la déformation Pliocène de la marge nord-ligure[END_REF][START_REF] Terrier | Identification et classification des failles actives de la région Provence-Alpes-Côte d'Azur -Phase 2 : Analyse et synthèse des connaissances actuelles sous la forme de fiches descriptives des failles[END_REF], both of them seem to be possible. Constraining better the mechanism of this earthquake is important for understanding not only the tectonics and the seismicity in the Liguria region but also the regional seismic hazard and for improving risk analyses. This is the main objective of this study. (after Ferrari, 1991).

Numerical method and numerical performance

We adopt a finite difference scheme for simulating the ground motion from any assumed earthquake source model. The code Ondes3D [START_REF] Dupros | Exploiting intensive multithreading for efficient simulation of 3D seismic wave propagation[END_REF][START_REF] Dupros | Three-dimensional parallel finite-difference modeling of seismic wave propagation on multicore architectures: Application to Nice, France, urban area[END_REF], optimized for parallel computing, is based on the 4th-order staggered-grid framework [START_REF] Madariaga | Dynamics of an expanding circular fault[END_REF][START_REF] Virieux | P-SV wave propagation in heterogeneous media: velocity-stress finite-difference method[END_REF][START_REF] Levander | Fourth-order finite-difference P-SV seismograms[END_REF][START_REF] Olsen | Simulation of three-dimensional wave propagation in the Salt Lake Basin[END_REF]. Any seismic source can be included as a moment distributed on grids [START_REF] Olsen | Simulation of three-dimensional wave propagation in the Salt Lake Basin[END_REF][START_REF] Graves | Simulating seismic wave propagation in 3D elastic media using staggered-grid finite differences[END_REF]. A convolutional perfectly matched layer absorbing condition [START_REF] Komatitsch | An unsplit convolutional Perfectly Matched Layer improved at grazing incidence for the seismic wave equation[END_REF] is introduced for avoiding the artificial wave reflection from the model volume border [START_REF] Ducellier | Implementing a Convolutional Perfectly Matched Layer in a finite-difference code for the simulation of seismic wave propagation in a 3D elastic medium[END_REF]. Our code can treat anelastic medium for introducing a realistic attenuation [START_REF] Day | Memory-efficient simulation of anelastic wave propagation[END_REF]Kristek and Mozco, 2003). However in this study, we limit our calculations to an elastic medium as the Liguria region is presented basically as a horizontally heterogeneous layered structure without any significant basin and anelastic parameter (quality factor Q) is not quantitatively known in this region. The simulations are carried out with a flat ground surface. This is usually enough for regional seismic hazard analyses since we discuss relatively low frequencies.

We use the 3D geological model constructed by compiling different geophysical profiles so as to be consistent with the geological interpretation in the framework of the French national project (QSHA, http://qsha.unice.fr/ ). Although the initial model is constructed for a dimension of about 50 km around the city of Nice, it lacks some extension toward Italy and the source region of the 1887 event. Thus we extend the available 3D model outside to the direction of N60°E supposing a 2D structure along the axis N30°W (e.g. [START_REF] Bertrand | Lithospheric structure of the southern French Alps inferred from broadband analysis[END_REF], briefly perpendicular to the coastline of the region. For the first part of the simulation of the 1887 earthquake, the model volume is therefore given by 100 x 70 x 26 km3. The model parameters used in the simulations are given in Table 2.

The frequency limit in the finite difference scheme is controlled by the grid spacing and the wave velocity in the material. The upper limit is described as

m ax m in / ( ) f v g s    , [1]
where m in v is the minimum velocity in the medium (1400 m/s in the model used in this study), s  is grid size and a constant g which stands for the number of points per wavelength. For 3D calculation with the 4th order scheme, the computation of 5 points per wavelength is usually taken to avoid numerical dispersion (e.g. [START_REF] Levander | Fourth-order finite-difference P-SV seismograms[END_REF]. The calculation up to high frequency requires finer grids. Theoretically Equation ( 1 For our discussion, we focus on values of Peak Ground Velocity (PGV) among various engineering parameters. PGV is numerically a reliable parameter in the current numerical scheme, as it represents relatively low frequencies comparing to, for example, PGA (Peak Ground Acceleration). The correlation between the PGV and damage (or macroseismic intensity) has been discussed mainly in the United States (e.g. Trifunac and Brady, 1975;[START_REF] Wald | Relationshops between peak ground acceleration, peak ground velocity, and Modified Mercalli intensity in California[END_REF] or Japan (e.g. [START_REF] Miyakoshi | Damage ratio functions for buildings using damage data of the Hyogoken Nanbu earthquake[END_REF]. More discussions and applications are found for example in [START_REF] Bommer | The prediction and use of peak ground velocity[END_REF]. Table 2: Model parameters of the numerical simulations in this study. Elastic property of each layer in the model.

Source models for the 1887 Ligurian earthquake

As explained in the previous sections, for the 1887 Ligurian earthquake, two different faulting mechanisms are inferred from seismo-tectonic interpretation. Figure 3 shows a schematic view of these possibilities: reverse faultings, relatively shallower and far from the coast (A) or deeper and close to the coast (B), and normal faulting (C). For all the models, we suppose a magnitude Mw6.3 with a fault dimension of 18 km length and 9 km width. The dip is taken as 30° for the reverse faultings (A and B) and 60° for the normal faulting. The hypocenters are located at the bottom center of each fault model. The hypocenter depth is taken at 7 km depth

dip=30°). B: deeper reverse faulting closer to the coastline with the hypocenter at center bottom (strike=N240°E, dip=30°). C: normal faulting still closer to the coastline with the same hypocenter as Model A (strike=N60°E, dip=60°). Fault dimension is 18 km length by 9 km width.

for A and C (full star in Figure 3), but is shifted to 9 km for B (open star). We first apply a uniform source model (slip distribution) and then introduce a characteristic source model based on [START_REF] Miyake | Source characterization for broadband ground-motion simulation: Kinematic heterogeneous source model and strong motion generation area[END_REF] and [START_REF] Irikura | Recipe for predicting strong ground motion from crustal earthquake scenarios[END_REF]. In all cases, we suppose a very simple, but typical form of the source time function represented by a rectangular pulse of duration (rise time) of 1 second. We choose a rupture velocity of 2.700 km/s. It is possible to introduce sophisticated source time function and spatial heterogeneity, but the simple parameterization is enough for our discussion.

We first compare the homogeneous models A, B and C. The difference in the resultant ground motions is clearly shown in the spatial distribution of horizontal peak ground velocity (PGV) values in Figures 4. It is not easy to compare the PGV distribution and the macroscopic intensity reported at this period. However it is evident that a wide area of high intensity spreads along the coast from Italy to France. For explaining this feature, the source is required geographically to be near the coastline. Indeed, the reverse faulting does not affect significantly the coastline, because the directivity effect of earthquake rupture goes away from the coast. In this meaning, it is model C which can easily generate strong ground motion (e.g. 10 cm/s) on the land. Although model C does not sufficiently supply the strong ground motion (e.g. Figure 2), this fault geometry is necessary to consider for further discussion. For model C, we try to simulate a different hypocenter position, for example as like in model C-unilateral in Figure 5, where the eastern-bottom corner is chosen so as to be the most severe scenario to the west, French Liguria. Although this scenario leads to a wide area of strong ground motion to the west because of the rupture directivity, this instead weakens it to the east. This does not match the wide spread of intensity map, either. Thus, a bilateral rupture scenario should be required in any case.

As long as we consider a homogeneous fault model, the strong ground motion is localized just around the fault and its peak value is not so high so that it is not transported at distance. Then a heterogeneous model is introduced based on the [START_REF] Miyake | Source characterization for broadband ground-motion simulation: Kinematic heterogeneous source model and strong motion generation area[END_REF] and [START_REF] Irikura | Recipe for predicting strong ground motion from crustal earthquake scenarios[END_REF]'s recipe. The asperity (large fault slip) area is 20% of the total rupture area. The number of asperities is assumed to be 2, both of which have the same area and twice the fault slip than the background fault area. Figure 5 also shows this characteristic source model (C-asperities1) and the resultant PGV map. This scenario allows a wide area of strong ground motion along the coast, although the epicenter area is slightly eased.

As the map of intensity (Figure 2) seems to give a slightly larger affected zone at the west of the model than at the east, a last scenario (C-asperities2) is defined with two asperities of different sizes: the western asperity represents 15% of total rupture area and the eastern 5% of total rupture area (this variation is allowed by Irikura and Miyake's recipe). In Figure 5, we also show this characteristic source model and the resultant PGV map. Using two different sizes for the two asperities leaded to eliminate the symmetry which appeared in the previous model and generated more ground motion in the western part than in the eastern part, which can correspond better to the spatial extension of intensity map.

Discussion and summary

Through the simulations performed in this study, preferred source models, explaining the macroseismic intensity observed in the Liguria region, are inferred. However, without more information on the ground motion, it is difficult to improve more the source models proposed here. In fact, the intensity data may not be homogenized between France and Italy for this earthquake and may include any local effect (site and/or vulnerability). It is not completely assured that it indicates the intensity of ground motion. We fixed a magnitude to 6.3 according to the literature, but it is also under debating. If we had more precise ground motion indication for this earthquake, we could constrain more the source parameters, too. About this earthquake, we should not forget that two large aftershocks followed in a day (http://www.sisfrance.net ). It is probable that some historical record were confused between the mainshock and aftershocks, so we need to read it carefully. All these topics should be considered in future works.

The 3D geological model we use in this study remains 2D profile over the whole scale with only three layers and the minimum wave velocity of 1400 m/s. Therefore, our calculation represents the ground motion for the basement at each point. The real ground motion at the ground surface can be amplified by a significant factor. It is expected anyway to improve the geological model so as to include surface layers of at least 700-800 m/s. Our calculation is based on the frequency of 1 Hz. In order to discuss the damages of this era, we will need to refine both the structure and source models until 2-3 Hz.

Regardless of the limits of data and available models, such numerical simulation approaches are useful to discuss the historical earthquakes and their better understanding is important for seismic hazard analysis. In this study, we have tested different scenarios of both reverse and normal faulting, taking into account of tectonic interpretation of this region. Our simulations infer a preference of normal faulting for this earthquake, which differs from some previous studies. All the discussion is still open.

Figure 1 .

 1 Figure 1. Map of the Liguria region. The red rectangle corresponds to the area modeled for the numerical simulation of the 1887 earthquake (brief epicenter shown by star).The blue rectangle corresponds to the area for which we have originally a 3D structure model. For references, we show the recent earthquakes aroundNice (1999 and 2001).

Figure 2 .

 2 Figure 2. Macroseismic intensity map of the 1887 earthquake. This map is provided by French earthquake database SisFrance. (http://www.sisfrance.net )

8ème

  Colloque National AFPS 2011 -Ecole des Ponts ParisTech 5 Layers P-wave velocity Vp [m/s] S-wave velocity Vs [m/s] Density  [kg/m

Figure 3 :

 3 Figure 3 : Projection of fault geometries tested in this study and 3D schematic illustration shown at corner from the same view azimuth. The hypocenters are also marked. A: reverse faulting off-shore with the hypocenter at center bottom (strike=N240°E, dip=30°). B: deeper reverse faulting closer to the coastline with the hypocenter at center bottom (strike=N240°E, dip=30°). C: normal faulting still closer to the coastline with the same hypocenter as Model A (strike=N60°E, dip=60°). Fault dimension is 18 km length by 9 km width.

Figure 4 .

 4 Figure 4. Horizontal PGV (peak ground velocity) in cm/s for models A, B and C. The geometrical map is shown in Figure 3.

Figure 5 .

 5 Figure 5. Horizontal PGV map for the derivative models based on fault geometry model C. ModelC-unilateral: Hypocenter is chosen eastern bottom corner of fault, and fault slip is uniform. ModelC-asperities1: Hypocenter is center bottom. Two asperities of equal dimension (10% each) are given. ModelC-asperities2: Hypocenter is center bottom. Two asperities of large (15%) and small areas (5%) are given.

Table 1 .

 1 Summary of the source parameters of the 1887 earthquake proposed in the literature

			8ème Colloque National AFPS 2011 -Ecole des Ponts ParisTech 4
	Source	Latitude Longitude Intensity (MCS) Magnitude	Depth
	Capponi et al. (1985) 43°52'	8°07'	IX	5.6	15 km
	Carrozo et al. (1973)	43°45'	8°00'	IX	6.2	unknown
	Ferrari (1991)	43°45'	8°00'	IX	6.5	17 km
	Barani et al. (2007)	43,74°	8,13°	IX	6.29	unknown