Oscar Hernández-Hernández

Céline Guiraud-Dogan

Géraldine Sicot

Aline Huguet

Sabrina Luilier

Esther Steidl

Stefanie Saenger

Elodie Marciniak

Hélène Obriot

Caroline Chevarin

Annie Nicole

Lucile Revillod

Konstantinos Charizanis

Kuang-Yung Lee

Yasuhiro Suzuki

Takashi Kimura

Tohru Matsuura

Bulmaro Cisneros

Maurice S Swanson

Fabrice Trovero

Bruno Buisson

Jean-Charles Bizot

Michel Hamon

Sandrine Humez

Guillaume Bassez

Friedrich Metzger

Luc Buée

Arnold Munnich

Nicolas Sergeant

Geneviève Gourdon

Mário Gomes- Pereira

PhD. Mario Gomes-Pereira

Synaptic dysfunction in myotonic dystrophy

Keywords: myotonic dystrophy, transgenic mice, synaptic transmission, RAB3A, synapsin I

Myotonic dystrophy type 1 is a complex multisystemic inherited disorder, which displays multiple debilitating neurological manifestations. Despite recent progress in the understanding of the molecular pathogenesis of DM1 in skeletal muscle and heart, the pathways affected in the central nervous system are largely unknown. To address this question, we studied the only transgenic mouse line expressing CTG trinucleotide repeats in the central nervous system. These mice recreate molecular features of RNA toxicity, such as RNA foci accumulation and missplicing. They exhibit relevant behavioral and cognitive phenotypes, deficits in short-term synaptic plasticity, as well as changes in neurochemical levels. In the search for disease intermediates affected by disease mutation, a global proteomics approach revealed RAB3A upregulation and Synapsin I hyperphosphorylation in the central nervous system of transgenic mice, transfected cells and post-mortem brains of myotonic dystrophy type 1 patients. These protein defects were associated with electrophysiological and behavioral deficits in mice, and altered spontaneous neurosecretion in cell culture. Taking advantage of a relevant transgenic mouse of a complex human disease, we found a novel connection between physiological phenotypes and synaptic protein dysregulation, indicative of synaptic dysfunction in myotonic dystrophy type 1 brain pathology.

INTRODUCTION

Myotonic dystrophy (DM) is the most common form of adult muscular dystrophy [START_REF] Harper | Myotonic Dystrophy[END_REF]. DM1 is caused by the expansion of a CTG trinucleotide repeat in 3'UTR of the DM protein kinase (DMPK) gene [START_REF] Brook | Molecular basis of myotonic dystrophy: expansion of a trinucleotide (CTG) repeat at the 3' end of a transcript encoding a protein kinase family member[END_REF][START_REF] Fu | An unstable triplet repeat in a gene related to myotonic muscular dystrophy[END_REF][START_REF] Mahadevan | Myotonic dystrophy mutation: an unstable CTG repeat in the 3' untranslated region of the gene[END_REF]. Repeat number correlates directly with disease severity and inversely with age of onset. The DM1 repeat shows a marked tendency towards further expansion in intergenerational transmissions and somatic tissues [START_REF] Gomes-Pereira | Chemical modifiers of unstable expanded simple sequence repeats: what goes up, could come down[END_REF]. The prevailing model of disease pathogenesis points to a transdominant effect of expanded DMPK transcripts, which accumulate in nuclear foci, interfering with at least two families of alternative splicing regulators: the muscleblindlike (MBNL) and CUGBP/Elav-like (CELF) proteins. MBNL1 loss of function, through sequestration into ribonuclear foci [START_REF] Miller | Recruitment of human muscleblind proteins to (CUG)(n) expansions associated with myotonic dystrophy[END_REF], and CELF1 upregulation [START_REF] Timchenko | RNA CUG repeats sequester CUGBP1 and alter protein levels and activity of CUGBP1[END_REF] disturb a developmentally regulated splicing program, resulting in aberrant expression of embryonic isoforms in adult skeletal muscle and heart [START_REF] Ranum | RNA-Mediated Neuromuscular Disorders[END_REF]. Missplicing explains important DM1 symptoms, such as myotonia [START_REF] Charlet | Loss of the musclespecific chloride channel in type 1 myotonic dystrophy due to misregulated alternative splicing[END_REF][START_REF] Lueck | Chloride channelopathy in myotonic dystrophy resulting from loss of posttranscriptional regulation for CLCN1[END_REF][START_REF] Mankodi | Expanded CUG repeats trigger aberrant splicing of ClC-1 chloride channel pre-mRNA and hyperexcitability of skeletal muscle in myotonic dystrophy[END_REF] and insulin resistance [START_REF] Savkur | Aberrant regulation of insulin receptor alternative splicing is associated with insulin resistance in myotonic dystrophy[END_REF].

Although traditionally considered a muscle disease DM type 1 (DM1) presents many debilitating neurological manifestations. Adult-onset DM1 patients show prevalent hypersomnia and fatigue, as well as visuoconstructive impairment, attention deficits, reduced initiative and apathy (Harper, 2001, Meola and[START_REF] Meola | Cerebral involvement in myotonic dystrophies[END_REF], suggesting executive dysfunction and the involvement of frontal lobes [START_REF] Gaul | Subtle cognitive dysfunction in adult onset myotonic dystrophy type 1 (DM1) and type 2 (DM2)[END_REF][START_REF] Meola | Cerebral involvement in myotonic dystrophies[END_REF][START_REF] Meola | Executive dysfunction and avoidant personality trait in myotonic dystrophy type 1 (DM-1) and in proximal myotonic myopathy (PROMM/DM-2)[END_REF][START_REF] Sistiaga | Cognitive/personality pattern and triplet expansion size in adult myotonic dystrophy type 1 (DM1): CTG repeats, cognition and personality in DM1[END_REF]. Reduced intelligence quotients (IQ) were measured in one third of adult and in the majority of childhood-onset patients (Angeard et al., 2007, Meola and[START_REF] Meola | Cerebral involvement in myotonic dystrophies[END_REF], while moderate to severe mental retardation is a feature of congenital DM1 [START_REF] Harper | Myotonic Dystrophy[END_REF].

Additional signs of personality disorder, such as increased anxiety, depression and anhedonia, were reported in a proportion of DM1 patients [START_REF] Bungener | Psychopathological and emotional deficits in myotonic dystrophy[END_REF][START_REF] Delaporte | Personality patterns in patients with myotonic dystrophy[END_REF][START_REF] Meola | Executive dysfunction and avoidant personality trait in myotonic dystrophy type 1 (DM-1) and in proximal myotonic myopathy (PROMM/DM-2)[END_REF][START_REF] Winblad | Temperament and character in patients with classical myotonic dystrophy type 1 (DM-1)[END_REF]. Region-specific structural abnormalities (including both white and grey matter affection) and metabolic changes, revealed by imaging studies, may contribute to DM1 neuropsychological manifestations [START_REF] Minnerop | The brain in myotonic dystrophy 1 and 2: evidence for a predominant white matter disease[END_REF][START_REF] Romeo | Retrospective study on PET-SPECT imaging in a large cohort of myotonic dystrophy type 1 patients[END_REF][START_REF] Weber | Comparative analysis of brain structure, metabolism, and cognition in myotonic dystrophy 1 and 2[END_REF].

RNA foci accumulate in post-mortem DM1 brains and co-localize with MBNL1 and MBNL2 [START_REF] Jiang | Myotonic dystrophy type 1 is associated with nuclear foci of mutant RNA, sequestration of muscleblind proteins and deregulated alternative splicing in neurons[END_REF]. Nevertheless, the mechanistic links between the genetic mutation and neuropsychological abnormalities remain elusive [START_REF] Meola | Cerebral involvement in myotonic dystrophies[END_REF]. We previously generated transgenic mice expressing DMPK transcripts in multiple tissues under the control of the human gene promoter, within the environment of the human DM1 locus. In contrast to control DM20 lines (overexpressing short 20-CTG tracts), homozygous mice from two independent DM300 expansion lines (carrying 500-600 CTG), expressing enough toxic DMPK transcripts, showed wide RNA foci accumulation in a variety of tissues and developed a multisystemic phenotype [START_REF] Guiraud-Dogan | DM1 CTG expansions affect insulin receptor isoforms expression in various tissues of transgenic mice[END_REF][START_REF] Panaite | Myotonic dystrophy transgenic mice exhibit pathologic abnormalities in diaphragm neuromuscular junctions and phrenic nerves[END_REF][START_REF] Seznec | Mice transgenic for the human myotonic dystrophy region with expanded CTG repeats display muscular and brain abnormalities[END_REF]. Dramatic intergenerational instability in DM300 mice generated DMSXL animals carrying >1000 CTG, which develop a more severe phenotype [START_REF] Gomes-Pereira | CTG trinucleotide repeat "big jumps": large expansions, small mice[END_REF][START_REF] Huguet | Molecular, physiological and motor performance defects in DMSXL mice carrying >1000 CTG repeat from the human DM1 locus[END_REF]. We now used DMSXL mice to characterize RNA toxicity in the brain and to identify disease intermediates and pathways affected by DM1 in the central nervous system (CNS). We gathered electrophysiological, neurochemical and molecular evidence of synaptic protein dysregulation and synaptic dysfunction, which translates into DM1-associated behavior deficits and most certainly mediates neurological symptoms.

MATERIALS AND METHODS

Transgenic mice. The DM transgenic mice used in this study carried 45 kb of human genomic DNA cloned from a DM1 patient as described [START_REF] Gomes-Pereira | CTG trinucleotide repeat "big jumps": large expansions, small mice[END_REF][START_REF] Seznec | Transgenic mice carrying large human genomic sequences with expanded CTG repeat mimic closely the DM CTG repeat intergenerational and somatic instability[END_REF]. Transgenic mice were raised and kept at Centre d'Exploration et de Recherche Fonctionnelle Expérimentale (CERFE, Genopole, Evry, France). Animal housing, care and handling were performed according to the French and European legislations, and the ethical guidelines of the host institution. Genotyping procedures are described in the Supplementary Materials and Methods online. The generation, characterization and genotyping of Mbnl1-and Mbnl2 knock-out mice is described elsewhere [START_REF] Charizanis | Muscleblind-like 2mediated alternative splicing in the developing brain and dysregulation in myotonic dystrophy[END_REF][START_REF] Kanadia | A muscleblind knockout model for myotonic dystrophy[END_REF].

Quantitative real-time RT-PCR quantification of DMPK and Rab3A transcripts.

Total RNA extraction and cDNA synthesis were performed as previously described [START_REF] Gomes-Pereira | CTG trinucleotide repeat "big jumps": large expansions, small mice[END_REF]. DMPK, Dmpk and Rab3A transcripts were amplified in a 7300 Real Time PCR System (Applied Biosystems) using SybrGreen detection and oligonucleotide primer sequences described in Supplementary Table 1. DMPK and Rab3A mRNA levels were expressed relative to 18S transcripts. The quantification of DMPK transcript levels was performed on a cDNA sample synthesized from equal RNA quantities extracted from three individual DMSXL homozygous mice. Rab3A transcript levels were quantified in six individual DMSXL homozygotes and wild-type controls.

Two independent replicate experiments were performed.

Fluorescent in situ hybridization (FISH).

Ribonuclear inclusions were detected with a 5'-FITC-labelled 2'-O-methyl-(CAG) 10 oligonucleotide probe, or a 5'-Cy3-labelled (CAG) 5 PNA probe, as previously described [START_REF] Seznec | Mice transgenic for the human myotonic dystrophy region with expanded CTG repeats display muscular and brain abnormalities[END_REF]. RNA foci quantification was performed using Discovery Automaton (Roche) [START_REF] Bassez | Type 2 myotonic dystrophy can be predicted by the combination of type 2 muscle fiber central nucleation and scattered atrophy[END_REF]. Mouse brain regions of interest were anatomically identified and the percentage of cells containing nuclear foci was calculated based on the observation of >100 cells in each brain region, in each mouse.

Immunofluorescence (IF) combined with FISH. Following the 1X SSC posthybridization wash of the FISH procedure, sections were incubated in primary antibodies overnight at 4°C (antibody references and working dilutions are listed in Supplementary Table 2), washed five times with PBS for 2 min, and then incubated in secondary antibody and 0.001% (m/v) diamidino-2-phenylindole (DAPI) for 15 min at room temperature. Sections were washed five times in PBS prior to mounting.

RT-PCR analysis of alternative splicing.

Total RNA extraction, cDNA synthesis and semi-quantitative RT-PCR analysis were performed as previously described [START_REF] Gomes-Pereira | CTG trinucleotide repeat "big jumps": large expansions, small mice[END_REF], using oligonucleotide primers described in Supplementary Table 1. The inclusion ratio of alternative exons in each individual animal was determined in two replicate PCR amplifications to minimize experimental variation. The percentage of exon inclusion was calculated as [exon inclusion band/(exon inclusion band + exon exclusion band)] x 100.

Behavioral analysis. Since DMSXL mice display muscle deficits [START_REF] Huguet | Molecular, physiological and motor performance defects in DMSXL mice carrying >1000 CTG repeat from the human DM1 locus[END_REF] behavioral tests were carefully selected, designed and interpreted to minimize the confounding effects of muscular impairment on brain functional deficits. All the experiments were performed according to the guidelines of the French Ministry of Agriculture for laboratory animal experimentation (law 87-848; Agreement N°: A-45-234-8) and the mouse genotype was unknown to the researcher. Mouse behavioral testing was conducted in order to reduce animal stress to the minimum. Three groups of animals were studied, in order to limit the number of tests performed on each animal. Group A: Open-field, Y maze, passive avoidance and sucrose intake. Group B:

Marble-burying and Morris water maze. Group C: Saccharine intake. Further details on behavioral phenotyping are described in the Supplementary Materials and Methods online.

Electrophysiological profiling. Input/output (I/O) properties, Paired-Pulse Facilitation (PPF), Long Term Depression (LTD) and Long Term Potentiation (LTP) were assessed on male DMSXL homozygotes and wild-type controls, aged seven months (I/O, PPF and LTD) or four months (LTP). Mice were sacrificed by fast decapitation, without previous anesthesia. Brains were processed in oxygenated buffer and artificial cerebro-spinal fluid as previously described [START_REF] Steidl | The adult rat hippocampal slice revisited with multi-electrode arrays[END_REF]. Extracellular field excitatory post-synaptic potentials (fEPSP) were measured with Multi-Electrode Arrays Two-dimensional protein electrophoresis and mass spectrometry. Ten-monthold female mice were used to compare the global proteomic profile in frontal cortex and brainstem between six homozygotes carrying expanded repeats (~500-600 CTG) and six females carrying short 20-CTG control tracts. Isolectric focusing and mass spectrometry was performed as previously described [START_REF] Sergeant | Truncated beta-amyloid peptide species in pre-clinical Alzheimer's disease as new targets for the vaccination approach[END_REF]. CELF1 and CELF2 phosphorylation was assessed by two-dimensional gel electrophoresis and immunodetection, using antibodies in Supplementary Table 2.

Western blot analysis. Primary antibody references and working dilutions are shown in Supplementary Table 2.

Neurosecretion assays. Neurosecretion assays in cell culture were performed as previously described [START_REF] Lee | Overexpression of BACE1 stimulates spontaneous basal secretion in PC12 cells[END_REF].

Human tissue samples. Autopsy materials were obtained from nine DM1 patients (mean age 60.9 years, range 32-73 years) and six non-DM1 controls (three with no neurologic disease, one with Charcot-Marie-Tooth disease, one with a brain tumor and one with Limb-girdle muscular dystrophy; mean age 68.2 years, range 53-79 years).

Six DM1 patients (i, j, m, n, o) had signs of classical adult-onset DM1, and they all died of complications related to the disease (respiratory failure or heart disease). Patient "DM1 g" developed late onset DM1. Patient "DM1 h" carried a large CTG repeat expansion in blood and brain tissue; her clinical form of DM1 was not fully described by the clinicians, but she presented gait problems and general brain atrophy evidenced by neuroimaging. The clinical form of patient "DM l" was not fully described either; she carried large CTG repeats, presented gait problems and mild cognitive impairment, and imaging assessment revealed general brain atrophy. Patient "DM1 k" exhibited mental retardation and was diagnosed with congenital DM1. He died at the age of 32, of pneumonia. Neuropshychological profiling and neuroimaging was not systematically conducted on the majority of these patients. Further patient details are given in Supplementary Table 3. All experiments using human samples were approved by the Ethics Committee of Asahikawa Medical Center and Okayama University. Written informed consent specimen use for research was obtained from all patients. Statistical analysis. Statistical analyses were performed with JMP 5, Prism 5 and Excel software. When two groups were compared, a two-tailed Student's t-test for statistical significance was performed, unless otherwise stated. The significance level was set at P values less than 0.05 for all statistical analyses. All data are expressed as mean ± SEM, unless otherwise stated.

Hernandez-Hernandez et al.

10

RESULTS

Expression of DM1 CTG expansions results in RNA foci accumulation and missplicing in the CNS of DMSXL mice

The toxic RNA hypothesis proposes that accumulation of CUG-containing transcripts is the initiating pathogenic event in DM1. To assess to what extent DMSXL mice recreate central molecular aspects of DM1, we measured DMPK transgene expression in CNS regions dissected from one-month-old DMSXL homozygous mice and found expanded transcripts in all CNS regions investigated, with some regional differences (Supplementary Fig. 1A): DMPK transcript levels were higher in the hippocampus, thalamus/hypothalamus, cerebellum and brainstem, than in frontal and temporal cortex and in striatum. Overall, the expression levels of the DMPK transgene in frontal cortex and cerebellum were approximately three times higher than those of the endogenous Dmpk gene [START_REF] Huguet | Molecular, physiological and motor performance defects in DMSXL mice carrying >1000 CTG repeat from the human DM1 locus[END_REF]. Expanded DMPK transcripts accumulated in one to ~20 ribonuclear inclusions per cell nucleus in both neurons and astrocytes (Fig. 1A) and co-localized with MBNL1 and MBNL2 (Fig. 1B). Astrocytes showed a significantly higher frequency of foci than neurons overall (62% versus 44%

foci-containing nuclei) (Fig. 1C). Interestingly, foci distribution was not homogeneous throughout DMSXL brains: they were present in 20% nuclei of striatum, up to 70% in brain cortex and ~80% in the dorsal raphe and raphe magnus nuclei of brainstem (Supplementary Fig. 1B). In addition to the accumulation of RNA foci in neurons previously reported in DM1 human brains [START_REF] Jiang | Myotonic dystrophy type 1 is associated with nuclear foci of mutant RNA, sequestration of muscleblind proteins and deregulated alternative splicing in neurons[END_REF], we found nuclear foci in human astrocytes (Supplementary Fig. 1C).

To investigate the trans-dominant effect of DMPK transcripts in the CNS, we studied the alternative splicing of candidate genes. We focused primarily on foci-rich brain regions relevant to disease manifestations: while frontal cortex is involved in executive tasks [START_REF] Robbins | The neuropsychopharmacology of fronto-executive function: monoaminergic modulation[END_REF], brainstem is involved in the control of sleep cycles, respiratory and cardiac function [START_REF] Izac | Basic anatomy and physiology of sleep[END_REF]. Mild region-specific missplicing events were detected in frontal cortex and brainstem of one-month-old DMSXL homozygotes (Fig. 1D, Supplementary Fig. 2A).

The missplicing of Mbnl1 exon 7 and Ldb3 exon 11, two alternative exons specifically regulated by MBNL1 activity [START_REF] Kalsotra | A postnatal switch of CELF and MBNL proteins reprograms alternative splicing in the developing heart[END_REF], confirmed the impact of the sequestration of MBNL proteins (Fig. 1B) on alternative splicing deregulation.

Similarly, mild missplicing of CELF1-dependent exons 15 and 16 of Fxr1 predicted a role of CELF1 in CNS spliceopathy. To confirm this hypothesis we quantified the steady-state levels of CELF proteins and found significant CELF1 and CELF2

upregulation (of about 30%) in DMSXL frontal cortex (Fig. 1E; Supplementary Fig. 1D). In DMSXL brainstem, only CELF1 was significantly upregulated. CELF1 upregulation was associated with mild protein hyperphosphorylation in DMSXL frontal cortex and brainstem (Supplementary Fig. 1E). CELF2 phosphorylation levels did not differ between genotypes in frontal cortex and appeared to decrease in DMSXL brainstem. The upregulation of CELF proteins was more pronounced in human DM1 frontal cortex: CELF1 levels were 70% higher and CELF2 showed a four-fold increase relative to non-DM individuals (Fig. 1F; Supplementary Fig. 1F).

Interestingly, wild-type splicing profiles were region-specific (e.g.

Grin1/Nmdar1, Fig. 1D, Supplementary Fig. 2A). Regional differences were associated with varying steady-state levels of MBNL and CELF proteins between frontal cortex and brainstem in adult mice (Supplementary Fig. 1G), suggesting a determining role of the ratio between antagonistic splicing regulators in the establishment of adult splicing profiles.

We studied the alternative splicing of alternative genes throughout wild-type mouse brain development (Supplementary Fig. 2B), and found that the DMSXL 3A).

However, DMSXL mice showed a significant decrease in exploratory activity, shortly after transfer into the open-field arena. While the total number of rearings did not differ between the two genotypes, the percentage of rearings over the first minute out of the first five minutes spent in the arena was significantly lower in DMSXL mice (Fig. 2A). This result reveals freezing behavior in response to a new and unfamiliar environment and indicates novelty-induced inhibition, suggestive of increased anxiety.

This was confirmed by the assessment of obsessive-compulsive behavior using a marble-burying test. DMSXL mice displayed a statistically significant shift towards a higher number of fully buried marbles (Fig. 2B), indicative of increased anxiety.

We then examined if the expression of expanded DMPK transcripts resulted in memory impairment as observed in DM1 patients. Mouse spatial memory was assessed by the Morris water maze. The platform was located in the same target quadrant during the training period, and removed during the probe trial. DMSXL mice showed impaired spatial memory, illustrated by a significantly lower number of entries in the target quadrant during the probe trial (Fig. 2C). DMSXL swimming speed was not significantly different between genotypes (Supplementary Fig. 3B), excluding biases introduced by impaired motor performance. In the assessment of working memory, a component of the executive function [START_REF] Robbins | The neuropsychopharmacology of fronto-executive function: monoaminergic modulation[END_REF] 2D). Non-spatial long-term memory, assessed by the passive avoidance conditioning test, was not affected in DMSXL mice (Supplementary Fig. 3C).

Anhedonia, previously reported in DM1 patients, was assessed by the consumption of a highly palatable solution of saccharine [START_REF] Schweizer | Chronic mild stress (CMS) in mice: of anhedonia, 'anomalous anxiolysis' and activity[END_REF] , suggesting an anhedonic-like behaviour (Fig. 2E).

DMSXL mice exhibit deficits in short-term synaptic plasticity

In parallel to the behavioral and cognitive phenotyping of DMSXL mice, we have assessed the physiological impact of toxic DMPK transcripts on synaptic function. To this end we performed the electrophysiological profiling of DMSXL hippocampus, a brain region showing foci accumulation and missplicing (Supplementary Fig. 4). We first examined basal synaptic transmission by stimulating hippocampal Schaffer collaterals at increasing intensities and generating input/output (I/O) curves from measures of field excitatory post-synaptic potentials (fEPSP). DMSXL homozygotes exhibited slightly higher fEPSP relative to wild-type controls, but the overall I/O curves were similar between both genotypes (Fig. 3A), indicating no major deficits in basal transmission. We next investigated some short-term plasticity properties through the quantification of paired-pulse facilitation (PPF). DMSXL slices displayed significantly reduced PPF ratios, indicative of presynaptic dysfunction (Fig. 3B). Finally, we measured long-term depression (LTD) and potentiation (LTP), which support some forms of learning and memory. Standard LTD and LTP protocols did not reveal overt abnormalities in DMSXL mice (Figs. 3C and3D). However, early after low frequency stimulation, LTD amplitude was slightly lower in DMSXL than in wild-type slices (36±7% vs. 47±3%), which might suggest mild impairment of the post-synaptic response. The difference between genotypes, however, did not reach statistical significance (P=0.142, repeated measures two-way ANOVA). Depression stabilized after 20 minutes, being similar for both genotypes at the endpoint of the experiment.

In summary, the number of animals studied did not show marked abnormalities in long-term plasticity, but revealed significant deficits in short-term synaptic plasticity.

DMSXL brains show abnormal levels of dopamine and serotonin metabolites

Neurotransmitter dysregulation can cause behavioral and electrophysiological dysfunction. Little is known about neurochemical signaling and metabolism in DM1 brains. To further dissect the neurological phenotype of DMSXL mice and assess whether behavioral and electrophysiological phenotypes were associated with changes in neurotransmitter levels, we measured key neurosignaling molecules in DMSXL brains. Adult DMSXL mice revealed a significant reduction of dopamine (DA) in the frontal cortex, as well as a tendency to decreased levels of its precursor (L-DOPA) and metabolites (DOPA, HVA). A significant decrease of 5-hydroxyindoleacetic acid (5-HIAA, the main serotonin metabolite) was detected in the brainstem of DMSXL mice (Fig. 3E).

Molecular abnormalities in synaptic proteins in the CNS of DMSXL mice and DM1 patients

To identify pathways affected in the CNS that might contribute to the behavioral and electrophysiological phenotypes, we compared the proteomic profiles of adult 4A, Supplementary Figs. 5A and5B). To further test if synaptic protein dysregulation could be mediated by overexpression of short DMPK transcripts and protein, we analysed DM20 mice carrying 20 CTG repeats by western blot. This DMPK overexpressing control line did not show RAB3A upregulation or SYN1 hyperphosphorylation (Fig. 4B), indicating that the defects observed are mediated by the expression of expanded CUG RNA repeats. We extended the analysis to other synaptic proteins to assess the extent of synaptic dysfunction, but found no additional abnormalities (Supplementary Figs. 5C and5D). Our data suggest that, rather than interfering with synaptic proteins in general, the repeat expansion may affect (directly or indirectly) a limited number of synaptic targets.

To address the mechanisms of synaptic protein dysregulation, we first tested if these abnormalities were associated with missplicing of known alternative exons of Rab3A and Syn1 transcripts. RT-PCR analysis revealed that Rab3A and Syn1 alternative splicing was not affected in DMSXL mice at four months (Supplementary Fig. 5E), an age when synaptic protein abnormalities are detected. RAB3A and SYN1 protein changes are not mediated by spliceopathy of the candidate alternative exons studied. Interestingly, RAB3A upregulation was associated with a significant increase of mRNA transcript levels in DMSXL frontal cortex and hippocampus (Fig. 4C).

We then tested if Mbnl1 or Mbnl2 inactivation was sufficient to dysregulate synaptic proteins, through the analysis of knock-out mice [START_REF] Charizanis | Muscleblind-like 2mediated alternative splicing in the developing brain and dysregulation in myotonic dystrophy[END_REF][START_REF] Kanadia | A muscleblind knockout model for myotonic dystrophy[END_REF]. Western blot analysis revealed that Mbnl1 inactivation (but not Mbnl2) resulted in a significant increase of RAB3A protein levels in mouse frontal cortex (Fig. 4D). Neither Mbnl1 nor Mbnl2 loss of function affected SYN1 phosphorylation levels. To test whether SYN1 hyperphosphorylation is mediated by upregulation of CELF proteins, we transfected PC12 cells with expressing vectors encoding CELF1 or CELF2. We studied SYN1 phosphorylation by western blot and found protein hyperphosphorylation in PC12 cells overexpressing CELF1 or CELF2 (Fig. 4E).

This analysis indicated that upregulation of CELF proteins is sufficient to dysregulate SYN1 phosphorylation levels.

To ascertain whether RAB3A and SYN1 dysregulation in mice reflected a pathophysiological event of the human condition, we investigated post-mortem DM1 frontal cortex (Fig. 4F).

Expanded CUG-containing DMPK transcripts affect neuronal exocytosis in culture.

To investigate the functional consequences of DM1 repeat expansions, and in particular altered expression of synaptic proteins on vesicle trafficking, we studied exocytosis in an established transfected cell culture model of neurosecretion. Regulated exocytosis is triggered by extracellular stimulus and exhibits low basal spontaneous secretion in the absence of stimulation [START_REF] Sudhof | The synaptic vesicle cycle[END_REF]. Basal and regulated neurosecretion can be measured in cultured PC12 cells transiently transfected with human growth hormone (hGH) [START_REF] Lee | Overexpression of BACE1 stimulates spontaneous basal secretion in PC12 cells[END_REF][START_REF] Sugita | Human growth hormone co-transfection assay to study molecular mechanisms of neurosecretion in PC12 cells[END_REF]. The percentage of hGH secreted prior and following osmotic stimulation serves as an estimate of basal and regulated neurosecretion, respectively. To assess the effect of CTG repeats on vesicle trafficking, PC12 cells were co-transfected with an hGH-producing plasmid and with DMPK constructs carrying a CTG expansion or no CTG repeats. Expanded CUG-containing RNA accumulated in the nucleus of PC12 cells, co-localized with MBNL1 and MBNL2, and induced mild missplicing. More importantly, transfected PC12 cells exhibited RAB3A upregulation and SYN1 hyperphosphorylation (Supplementary Fig. 6). The neurosecretion assay demonstrated that the expression of toxic RNA repeats enhanced basal neurosecretion in the absence of stimulation, relative to no-repeat control constructs (Fig. 5). The effect was observed in four independent experiments with a significant average basal secretion enhancement of 60±16%, during 15 minutes of incubation. Interestingly, CUG-containing transcripts did not disturb regulated neurosecretion after osmotic stimulation.

DISCUSSION

To explore DM1 neuropathology, we studied DM1 transgenic mice that reproduce key molecular aspects of RNA toxicity and exhibit relevant behavioral phenotypes. DMSXL mice display neurochemical and electrophysiological signs of synaptic dysfunction, which are associated with molecular abnormalities in synaptic proteins, observed not only in mice but also in transfected cells and post-mortem DM1 brain samples. The association between physiological and molecular phenotypes of the synapse indicates synaptic dysfunction in DM1 neuropathophysiology.

The expression of expanded DMPK transcripts in the CNS of transgenic mice results in the accumulation of RNA foci in neurons and astrocytes. RNA toxicity in both cell types indicates that DM1 may be associated not only with neuronal dysfunction, but also with glial abnormalities. Interestingly, brain areas showing the highest foci content (e.g. frontal cortex) did not necessarily express the highest DMPK levels, suggesting that RNA aggregation into foci may depend on region-and/or cell typespecific factors. Like in skeletal [START_REF] Lin | Failure of MBNL1dependent post-natal splicing transitions in myotonic dystrophy[END_REF][START_REF] Orengo | Expanded CTG repeats within the DMPK 3' UTR causes severe skeletal muscle wasting in an inducible mouse model for myotonic dystrophy[END_REF] and cardiac [START_REF] Wang | Elevation of RNA-binding protein CUGBP1 is an early event in an inducible heart-specific mouse model of myotonic dystrophy[END_REF] The missplicing of alternative exons specifically regulated by MBNL or CELF proteins [START_REF] Kalsotra | A postnatal switch of CELF and MBNL proteins reprograms alternative splicing in the developing heart[END_REF] indicated that MBNL loss of function by protein sequestration, and CELF upregulation induce missplicing in the CNS. In support of this view, Mbnl1-or Mbnl2-deficient mice show abnormal splicing in brain [START_REF] Charizanis | Muscleblind-like 2mediated alternative splicing in the developing brain and dysregulation in myotonic dystrophy[END_REF][START_REF] Suenaga | Muscleblind-like 1 knockout mice reveal novel splicing defects in the myotonic dystrophy brain[END_REF].

CELF1 upregulation in CNS appears to be mediated by mild protein hyperphosphorylation, as in DM1 heart [START_REF] Kuyumcu-Martinez | Increased steady-state levels of CUGBP1 in myotonic dystrophy 1 are due to PKC-mediated hyperphosphorylation[END_REF]. It remains to be investigated whether PKC activity is increased in the DM1 brains. In contrast, CELF2 upregulation in DMSXL frontal cortex was not associated with protein hyperphosphorylation, suggesting alternative mechanisms of protein regulation. The marked upregulation of CELF2 in human DM1 frontal cortex predicts a pathogenic role for this protein in DM1 neuropathogenesis. The modulation of CELF2 expression by miRNA species, recently reported in a transgenic mouse model of spinal-bulbar muscular atrophy (SBMA) suggests new possibilities for miRNA-mediated therapies in DM1 [START_REF] Miyazaki | Viral delivery of miR-196a ameliorates the SBMA phenotype via the silencing of CELF2[END_REF] The behavioral phenotyping of DMSXL mice revealed reduced exploratory activity, increased anxiety, spatial memory impairment and anhedonia, which resemble DM1 neurological manifestations. Previous neuropsychological assessment of DM1 patients revealed low scores in the exploratory scale [START_REF] Winblad | Temperament and character in patients with classical myotonic dystrophy type 1 (DM-1)[END_REF], a higher prevalence of anxiety-related behaviors [START_REF] Delaporte | Personality patterns in patients with myotonic dystrophy[END_REF][START_REF] Meola | Executive dysfunction and avoidant personality trait in myotonic dystrophy type 1 (DM-1) and in proximal myotonic myopathy (PROMM/DM-2)[END_REF], visualspatial impairment [START_REF] Modoni | Characterization of the pattern of cognitive impairment in myotonic dystrophy type 1[END_REF]) and anhedonia associated with emotional blunting and depressive symptomatology [START_REF] Bungener | Psychopathological and emotional deficits in myotonic dystrophy[END_REF]. The phenotypic parallel between patients and transgenic mice illustrates the impact of toxic DMPK transcripts on CNS physiology, and corroborates the use of the DMSXL line to recreate DM1 brain pathology. Mild deficits in working memory were also found in DMSXL mice.

Although not statistically significant, the initial difference between the two genotypes in the acquisition trial, during the working memory assessment, might confound the analysis, and the results should be interpreted with some caution. Alternative behavioral tests (e.g. elevated or multiple arm radial mazes) might prove useful to provide definitive evidence of working memory deficits in DMSXL mice.

The behavioral abnormalities of DMSXL mice are associated with deficits in short-term plasticity, as well as changes in neurochemicals, suggesting altered synaptic function and neurotransmission in response to the CTG repeat expansion. The neurochemical data, in particular, provide insight into the neuronal circuits affected by DM1. Decreased dopamine in frontal cortex may account for motivation and reward deficits (Arias-Carrion and Poppel, 2007), while the reduced serotonin metabolism in brainstem may increase susceptibility to depressive-like behaviors [START_REF] Werner | Classical neurotransmitters and neuropeptides involved in major depression: a review[END_REF], thereby contributing to anhedonia. The high foci content in dopaminergic (substantia nigra) and serotonergic (raphe nucleus) brain centers may contribute, at least partially, to the neurochemical deficits of DMSXL mice. In humans, loss of catecholaminergic neurons (dopamine is an abundant catecholamine) and serotonin-containing neurons was previously reported (Ono et al., 1998, Ono et al., 1998). The involvement of dopaminergic and serotonergic pathways in DM1 neuropathology requires further investigation, and may provide insight into future means of therapeutic intervention, such as the modulation of the dopaminergic and serotinergic circuits.

In the search for pathways affected by the DM1 repeat mutation in the CNS, we found RAB3A upregulation and SYN1 hyperphosphorylation, not only in transgenic mice expressing large CTG expansions, but also in post-mortem DM1 brains. Transgenic mice over-expressing short DMPK transcripts did not show abnormal synaptic protein dysregulation, indicating that RAB3A and SYN1 misregulation is specifically associated with expanded transcripts.

Decreased glucose metabolism [START_REF] Fiorelli | Decreased cerebral glucose utilization in myotonic dystrophy[END_REF] and blood perfusion [START_REF] Meola | Reduced cerebral blood flow and impaired visual-spatial function in proximal myotonic myopathy[END_REF] have been previously described in the frontal lobe of DM1 patients. Since abnormal glucose metabolism can alter the protein content of synaptic vesicles [START_REF] Gaspar | Diabetes differentially affects the content of exocytotic proteins in hippocampal and retinal nerve terminals[END_REF], it is reasonable to speculate that changes in the exocytocic machinery in frontal cortex might result (at least partially) from brain hypoperfusion and/or hypometabolism. Despite the possible contribution of altered glucose metabolism to synaptic dysfunction, cells transfected with DM1 repeat expansions displayed RAB3A upregulation and SYN1 hyperphosphorylation in culture, supporting the view that synaptic protein dysregulation is also a direct consequence of CUG RNA toxicity, rather than a simple indirect effect, secondary to general brain dysfunction. To further support this hypothesis, the analysis of complementary animal and cell models of DM1 revealed that RAB3A upregulation is mediated by MBNL1 inactivation, and SYN1 hyperphosphorylation is mediated by upregulation of CELF proteins. We propose that transcriptional dysregulation [START_REF] Osborne | Transcriptional and post-transcriptional impact of toxic RNA in myotonic dystrophy[END_REF] or altered post-transcriptional regulation of mRNA decay [START_REF] Masuda | CUGBP1 and MBNL1 preferentially bind to 3' UTRs and facilitate mRNA decay[END_REF] through loss of function of MBNL proteins could mediate RAB3A upregulation. Hyperphosphorylation of SYN1 may result from dysregulated kinase and/or phosphatase activities, as a result of altered CELF levels. Dysregulation of micro-RNAs, also reported in DM1 skeletal muscle and heart [START_REF] Gambardella | Overexpression of microRNA-206 in the skeletal muscle from myotonic dystrophy type 1 patients[END_REF][START_REF] Perbellini | Dysregulation and cellular mislocalization of specific miRNAs in myotonic dystrophy type 1[END_REF][START_REF] Rau | Misregulation of miR-1 processing is associated with heart defects in myotonic dystrophy[END_REF], may extend to the CNS, opening new avenues for future research.

RAB3A is an abundant synaptic vesicle protein that regulates neurotransmission [START_REF] Sudhof | The synaptic vesicle cycle[END_REF]. Mouse Rab3a inactivation results in increased hippocampal PPF [START_REF] Geppert | The small GTP-binding protein Rab3A regulates a late step in synaptic vesicle fusion[END_REF], while RAB3A overexpression in cell culture activates spontaneous exocytosis [START_REF] Schluter | Localization versus function of Rab3 proteins. Evidence for a common regulatory role in controlling fusion[END_REF], similar to PC12 cells transfected with expanded DMPK constructs. Given the involvement of RAB3A in short-term synaptic plasticity and neurotransmitter release, RAB3A upregulation in DMSXL mice likely contributes to altered PPF and mediates the increase in basal neurosecretion in transfected PC12 cells. In addition RAB3A has been implicated in visual-spatial learning (D 'Adamo et al., 2004). Therefore, altered RAB3A expression may contribute to the cognitive deficits of DMSXL mice and DM1 patients. Synapsins comprise the most abundant proteins in synaptic vesicles. SYN1 serves as a phosphorylation-dependent regulator of neurotransmitter release [START_REF] Rosahl | Short-term synaptic plasticity is altered in mice lacking synapsin I[END_REF]: while non-phosphorylated SYN1 attaches synaptic vesicles to the actin cytoskeleton, stimulation-dependent phosphorylation decreases the affinity for synaptic vesicles and potentiates exocytosis [START_REF] Fdez | Vesicle pools and synapsins: new insights into old enigmas[END_REF]. Consistent with a role of SYN1 in short-term plasticity [START_REF] Fiumara | Phosphorylation of synapsin domain A is required for post-tetanic potentiation[END_REF], chemically-induced hyperphosphorylation of SYN1 in vivo was previously associated with decreased PPF [START_REF] Tallent | In vivo modulation of O-GlcNAc levels regulates hippocampal synaptic plasticity through interplay with phosphorylation[END_REF], like in DMSXL mice.

The marked RAB3A upregulation and abnormal SYN1 hyperphosphorylation in postmortem DM1 brains corroborate their role in DM1 neuropathology. RAB3A and SYN1

variability between patients may be associated with different degrees of disease severity, as previously reported for splicing abnormalities in skeletal muscle and heart [START_REF] Orengo | Expanded CTG repeats within the DMPK 3' UTR causes severe skeletal muscle wasting in an inducible mouse model for myotonic dystrophy[END_REF], Wang et al., 2007). In summary, we found RAB3A upregulation and SYN1 hyperphosphorylation in DM1 transgenic mice, transfected cells and in human DM1 brain samples. These protein defects were associated with electrophysiological and behavioral abnormalities in mice, as well as altered spontaneous neurosecretion in cell culture.

If changes in synaptic proteins reflect greater abnormalities in the dynamics and microstructure of the brain cell membrane, the molecular abnormalities identified might have wider implications and may correlate with the prevalent white matter lesions reported in DM1 brains [START_REF] Minnerop | The brain in myotonic dystrophy 1 and 2: evidence for a predominant white matter disease[END_REF]. Future MRI (magnetic resonance imaging) of DMSXL brain integrity will address this possibility, through the characterization of tissue changes in response to the DM1 mutation. Additional imaging measurements of regional brain blood flow and/or glucose metabolism in transgenic mice by PET (positron emission tomography) and/or SPECT (single-photon emission computed tomography) must be performed to evaluate and localize the functional impact of DM1 in the CNS, and provide insight into the underlying mechanisms.

There is currently debate whether CNS dysfunction in DM1 is neurodegenerative, neurodevelopmental or neurofunctional. Although clinical data are

not enough to answer this difficult question, transgenic mice may provide significant insight. The molecular and electrophysiological abnormalities detected in DMSXL mice are consistent with functional deficits in adult brain. Future analyses of cell loss and the CNS investigation throughout mouse development will be required to address the contribution of neurodegeneration and impaired neurodevelopment, respectively, towards DM1 CNS dysfunction. It remains possible that these processes are not mutually exclusive and that they all participate in DM1 brain pathophysiology."

In the context of the pre-clinical assessment of future therapies, it will be interesting to investigate whether therapeutic schemes targeting the CNS (and in particular RAB3 and SYN1) will be able to reverse the behavioral phenotypes (e.g.

visual-spatial memory impairment, increased anxiety, anhedonia) and/or electrophysiological profiles (e.g. PPF deficits) of DMSXL mice. In cell culture, RAB3A

and SYN1 can modulate miniature endplate currents of neurons [START_REF] Chiappalone | Opposite changes in glutamatergic and GABAergic transmission underlie the diffuse hyperexcitability of synapsin I-deficient cortical networks[END_REF][START_REF] Wang | Impaired activity-dependent plasticity of quantal amplitude at the neuromuscular junction of Rab3A deletion and Rab3A earlybird mutant mice[END_REF]. It is tempting to investigate whether RAB3A knocking-down and/or SYN1 dephosphorylation in primary cultures can rescue the electrophysiological phenotype of DMSXL neurons.

Using molecular and physiological approaches to explore the molecular mechanisms of a complex human disease, we generated evidence of the impact of DM1 on synaptic proteins, vesicle secretion, neurotransmission and synaptic plasticity.

The validation of our molecular findings in post-mortem DM1 brains substantiate a role of synaptic dysfunction in DM1, through changes in proteins involved in the regulation of synaptic vesicle release.

(

  MEA) technology (100 µm distant electrodes) on 350 µm thick hippocampal slices. One of the electrodes stimulated Schaeffer collaterals at the CA3/CA1 interface. The stimulus, consisting of a monopolar biphasic current pulse (negative for 60 µs and then positive for 60 µs), evoked responses (field potentials: fEPSP) in the CA1 region. I/O properties: I/O curves were plotted by measuring the response evoked by stimulation intensities varying between 100 µA and 800 µA. Details on the electrophysiological procedures are included in the Supplementary Materials and Methods online.

  homozygous mice carrying CTG expansions, with those of DM20 control mice overexpressing short DMPK transcripts. The use of DM20 controls excluded the identification of disease intermediates possibly affected by overexpression of DMPK protein. The proteomics analysis suggested altered expression of RAB3A and posttranslational modifications of Synapsin I (SYN1). Western blot quantification confirmed a statistically significant upregulation of RAB3A and hyperphosphorylation of serine residues of SYN1 in DMSXL frontal cortex and hippocampus at four months of age (Fig.

  muscles, CUG toxicity interferes with developmental alternative splicing in the CNS, increasing to a limited extent the abnormal expression of embryonic isoforms of some transcripts in adult DMSXL brains. Splicing abnormalities in the CNS may contribute to DMSXL phenotype. In particular, Grin1/Nmdar1 and Mapt/Tau splicing defects likely participate in synaptic dysfunction detected in DMSXL mice. Further experiments are required to investigate how Grin1/Nmdar1 missplicing influences intracellular localization and function of this receptor, as well as the impact of Mapt/Tau missplicing on DM1 neurofibrillary degeneration.

FIGUREFigure 1 .

 1 FIGURE TITLES AND LEGENDS

Figure 2 .

 2 Figure 2. DMSXL exhibit novelty-induced inhibition, anxiety, spatial and working memory impairment and anhedonia. (A) Assessment of novelty-induced inhibition in DMSXL homozygotes (n=16) and age-matched controls (n=16). The graph represents the percentage of the number of rearings (±SEM) in the first minute out of the first five minutes spent in the first open-field session. (B) Assessment of mouse anxiety levels by the marble-burying test. The graph represents the average number of marbles (±SEM) unburied and buried by DMSXL (n=15) and wild-type mice (n=15). The DMSXL line shows a significant shift towards a higher number of buried marbles (P=0.0087, Fisher's exact test). (C) Spatial memory assessment by the Morris water maze test. The graph represents the number of entries in the target quadrant (±SEM) during the probe trial, for DMSXL (n=15) and wild-type (n=15) mice. (D) Working memory assessment by the Morris water maze test. The graphs represent the average time and distance travelled (±SEM) to reach the platform in acquisition and retention trials for both genotypes (n=15 per genotype). (E) Saccharine consumption test for anhedonia. The graph on the left represents the average volume of saccharine solution (±SEM) drank by DMSXL homozygotes (n=12) and wild-type controls (n=12). The volume of water consumed once saccharine became available did not differ between the two genotypes (graph on the right). *, P<0.05; **, P<0.01.

Figure 3 .

 3 Figure 3. DMSXL mice exhibit deficits in short-term synaptic plasticity and changes in neurochemical levels. (A) Electrophysiological profiling of DMSXL mice and agematched controls. I/O characteristics in the CA1 region. Mean value of fEPSP (±SEM) is expressed as a function of stimulation intensity. (B) PPF in the CA1 region. The mean ratio of the second peak compared to the first one (fEPSP 2 /fEPSP 1 , ±SEM) is expressed

Figure 5 .

 5 Figure 5. Toxic CUG repeats dysregulate neurosecretion in culture. Quantification of the effect of CTG repeat expansion on neurosecretion. The average secretion from hGH-expressing PC12 cells co-transfected with expanded DT960 or no-expansion DMPKS plasmids is plotted as a percent of total hGH content (±SEM). Basal secretion was measured in control medium containing 5.6 mM KCl. Stimulus-dependent secretion was measured in media containing 56 mM or 100 mM KCl.

  spliceopathy in the CNS modified certain transcripts towards embryonic/newborn splicing profiles. For instance, the splicing patterns of Grin1 exon 21 and Mbnl2 exon 7 in DMSXL frontal cortex, as well as App exon 8 and Frx1 exons 15/16 in DMSXL

	brainstem resembled, to a limited extent, the splicing patterns of newborn wild-type
	mice. The effect was less pronounced in other alternative exons, which displayed
	intermediate inclusion ratios in DMSXL mice, between those characteristic of the
	embryonic and adult developmental stages (e.g. Ldb3 exon 11 in frontal cortex and
	brainstem; Grin1 exon 5 in brainstem).

CTG repeat expansions induce mouse behavioral abnormalities

Following the validation of toxic RNA expression, foci accumulation and missplicing in DMSXL brains, we then investigated the impact of expanded DMPK transcripts on mouse behavior and cognition through blinded phenotyping of adult DMSXL homozygotes. We first assessed mouse activity in the open-field test. DMSXL mice displayed overall levels of horizontal (P=0.402, Student's t-test) and vertical activity (P=0.355, Student's t-test) similar to those of wild-type controls, excluding a major effect of muscular deficits on mouse performance in this test (Supplementary Fig.

ACKNOWLEDGEMENTS

We thank Dr. Shuzo Sugita and Dr. Thomas Cooper for providing the phGHCMV5, DMPKs, DT960, CELF1 and CELF2 plasmids, and to Dr. Glenn Morris and Dr. Manuel Hernández for providing the anti-MBNL2 and anti-ß-Actin antibodies, respectively. We are grateful to Amine Bouallague and to the personnel of CERFE (Centre d'Exploration et de Recherche Fonctionelle Expérimentale, Genopole, Evry, France) personnel for attentively caring for the mice. We thank our colleagues at Inserm U781 and the DM1 French Splicing Network for helpful discussions.

FUNDING

This work was supported by ANR (Agence Nationale de Recherche, France; "DM1MICE" project), AFM (Association Française contre les Myopathies, France; "DM Brain" project), Prosensa (The Netherlands), INSERM (Institute National de la Santé et Recherche Médicale, France) and Université Paris Descartes (Paris, France). OHH was partially funded by a post-doctoral fellowship from CONACyT (Consejo Nacional de Ciência y Tecnología, Mexico). GS was awarded a PhD student fellowship from Ministère Français de la Recherche et Technologie.

SUPPLEMENTARY MATERIAL

Supplementary material includes five figures and two tables.


Daniele A Di

Robert Eymard
email: robert.eymard@univ-mlv.fr

Simon Lemaire
email: simon.lemaire@ifpen.fr

Roland Masson
email: roland.masson@ifpen.fr

Hybrid Finite Volume Discretization of Linear Elasticity Models on General Meshes

Keywords: hybrid finite volumes, linear elasticity, general meshes MSC2010: 74S10, 74B05

This paper presents a new discretization scheme for linear elasticity models using only one degree of freedom per face corresponding to the normal component of the displacement. The scheme is based on a piecewise constant gradient construction and a discrete variational formulation for the displacement field. The tangential components of the displacement field are eliminated using a second order linear interpolation. Our main motivation is the coupling of geomechanical models and porous media flows arising in reservoir or CO2 storage simulations. Our scheme guarantees by construction the compatibility condition between the displacement discretization and the usual cell centered finite volume discretization of the Darcy flow model. In addition it applies on general meshes possibly non conforming such as Corner Point Geometries commonly used in reservoir and CO2 storage simulations.

Introduction

The oil production in unconsolidated, highly compacting porous media (such as Ekofisk or Bachaquero) induces a deformation of the pore volume which (i) modifies significantly the production, and (ii) may have severe consequences such as surface subsidence or damage of well equipments. This explains the growing interest in reservoir modeling for simulations coupling the reservoir Darcy multiphase flow with the geomechanical deformation of the porous media [START_REF] Settari | Coupling of Geomechanics and Reservoir simulation models[END_REF]. Similarly, poromechanical models are also used in CO2 storage simulations to predict the over pressure induced by the injection of CO2 in order to assess the mechanical integrity of the storage in the injection phase.

The most commonly used geometry in reservoir and CO2 storage models is the so called Corner Point Geometry or CPG [START_REF] Ponting | Corner Point Geometry in reservoir simulation[END_REF]. Although the CPG discretization is initially a structured hexahedral grid, vertical edges of the cells may typically collapse to account for the erosion of the geological layers and vertices may be dedoubled and slide along the coordlines (i.e. straight lines orthogonal to the geological layers) to model faults. In addition non conforming local grid refinement is used in near well regions. The resulting mesh is unstructured, non conforming, it includes all the degenerate cells obtained from hexahedra by collapsing edges, and hence it is not adapted to conforming finite element discretizations.

The objective of this paper is to introduce a new discretization scheme for linear elasticity equations which should

• apply on general meshes possibly non conforming;

• guarantee the stability of the coupling with Darcy flow models using cell centered finite volume discretization for the Darcy equation [START_REF] Murad | On Stability and Convergence of Finite Element Approximations of Biot's Consolidation Problem[END_REF].

Our discretization is based on the family of Hybrid Finite Volume schemes introduced for diffusion problems on general meshes in [START_REF] Eymard | Discretisation of heterogeneous and anisotropic diffusion problems on general non-conforming meshes, SUSHI: a scheme using stabilisation and hybrid interfaces[END_REF] and also closely related to Mimetic Finite Difference schemes [START_REF] Brezzi | A family of mimetic finite difference methods on polygonal and polyhedral meshes[END_REF] as shown in [START_REF] Droniou | A unified approach to mimetic finite difference, hybrid finite volume and mixed finite volume methods[END_REF]. The degrees of freedom are defined by the displacement vector u σ at the center of gravity of each face σ of the mesh as well as the displacement vector u K at a given point x K of each cell K of the mesh. Following [START_REF] Eymard | Discretisation of heterogeneous and anisotropic diffusion problems on general non-conforming meshes, SUSHI: a scheme using stabilisation and hybrid interfaces[END_REF], a piecewise constant gradient is built and can be readily used to define the discrete variational formulation which mimics the continuous variational formulation for the displacement vector field. In order to stabilize this formulation and to reduce the degrees of freedom, the tangential components of the displacement are interpolated in terms of the neighbouring normal components. Numerical experiments on two dimensional and three dimensional meshes show that the resulting discretization is stable and convergent. In addition, this discretization satisfies by construction the compatibility condition or LBB (see [START_REF] Brezzi | Mixed and Hybrid Finite Element Methods[END_REF], [START_REF] Murad | On Stability and Convergence of Finite Element Approximations of Biot's Consolidation Problem[END_REF]) condition for poroelastic models when coupled with a cell centered finite volume scheme for the Darcy flow equation.

Discretization of Linear Elasticity Models

Let Ω be a d-dimensional polygonal or polyhedral domain (d = 2 or 3) and let us consider the following linear elasticity problem in infinitesimal strain theory:

   div (σ (u)) + f = 0 on Ω , u = u D on ∂ Ω D , σ (u) • n = g on ∂ Ω N , (1)
where u ∈ R d is the unknown displacement field and D and N the two exponents standing respectively for Dirichlet and Neumann boundary conditions. σ (u) is the Cauchy stress tensor and is given by Hooke's law σ

(u) = 2µε(u) + λ tr (ε(u)) 1,
where µ, λ are the Lamé parameters and ε(u) = 1 2 ∇u + ∇u T is the infinitesimal strain tensor.

Hybrid Finite Volume Discretization

The simulation domain Ω is discretized by a set of polygonal or polyhedral control volumes K ∈ K , such that Ω = K∈K K. The set of faces of the mesh is denoted by E and splits into boundary interfaces E ext and inner interfaces E int . Among the boundary interfaces, we denote by E D ext and E N ext the subsets of boundary faces verifying Dirichlet or Neumann conditions. The center of gravity of the face σ is denoted by x σ and its d -1 dimensional measure by |σ |. A point x K is defined inside each cell K of the mesh. The set of faces of each cell K is denoted by E K , and the distance between x K and σ by d K,σ . The cone of base σ ∈ E K and top x K is denoted by K σ .

Brief reminder of the hybrid finite volume discretization of a scalar diffusion problem (see [5]):

We first define the discrete hybrid spaces

V = (v K ∈ R) K∈K , (v σ ∈ R) σ ∈E and V 0 = v ∈ V | v σ = 0 ∀σ ∈ E D ext . V 0 is endowed with the discrete H 1 0,D (Ω ) norm v V 0 = ∑ K∈K ∑ σ ∈E K |σ | d K,σ |v σ -v K | 2 1 2 . (2)
Then, following [START_REF] Eymard | Discretisation of heterogeneous and anisotropic diffusion problems on general non-conforming meshes, SUSHI: a scheme using stabilisation and hybrid interfaces[END_REF], a discrete gradient is defined on each cone K σ which only depends on v K and v σ ′ for σ ′ ∈ E K . This gradient is exact on linear functions and satisfies a weak convergence property. According to [START_REF] Eymard | Discretisation of heterogeneous and anisotropic diffusion problems on general non-conforming meshes, SUSHI: a scheme using stabilisation and hybrid interfaces[END_REF], it can be written

∇ K σ v = ∑ σ ′ ∈E K (v σ ′ -v K )y σ σ ′ K ∀ v ∈ V, (3)
where y σ σ ′ K ∈ R d only depends on the geometry.

Hybrid finite volume discretization of the linear elasticity model:

As we did above, we introduce

V = (v K ∈ R d ) K∈K , (v σ ∈ R d ) σ ∈E as the discrete hybrid space.
With an equivalent definition for V 0 , the discrete norm is now defined as

v 2 V 0 = ∑ d i=1 v i 2 V 0 . Let us also introduce the space W = (w K ∈ R d ) K∈K , (w σ ∈ R d ) σ ∈E D ext , (w n σ ∈ R) σ ∈E int ∪E N ext
and the following projection operator P W :

V → W, v → (v K ) K∈K , (v σ ) σ ∈E D ext , (v σ • n σ ) σ ∈E int ∪E N ext ,
where n σ is a unit vector normal to σ which orientation is fixed once and for all. Let us finally define the space

W 0 = P W (V 0 ) endowed with the discrete norm w W 0 = inf v∈V 0 | P W (v)=w v V 0 .
The main novelty of our discretization lies in the definition of a linear interpolation operator I : W → V. This linear interpolation operator must be second order accurate to preserve the order of approximation of the scheme and interpolant in the sense that P W (I(w)) = w for all w ∈ W. It must also be local in the sense that it computes the displacement vector v σ at a given face σ ∈ E int ∪ E N ext in terms of a given number of normal components v σ ′ • n σ ′ taken on a stencil S σ ⊂ E of neighbouring faces σ ′ of σ (with σ ∈ S σ ). An example of construction of such an interpolator is given in subsection 2.3.

The use of the interpolation operator I will bring two crucial improvements to the discretization: first a drastic reduction of the degrees of freedom and secondly a stabilization of the discretization.

Finally, generalizing the scalar framework to the vectorial case of the linear elasticity model, we introduce a piecewise constant discrete gradient for each cone K σ :

∇ K σ v = ∑ σ ′ ∈E K (v σ ′ -v K ) ⊗ y σ σ ′ K ∀ v ∈ V. ( 4
)

Discrete Variational Formulation

Starting from (1), we deduce a discrete weak formulation of the problem in

W 0 . Setting ε K σ (v) = 1 2 ∇ K σ v + ∇ K σ v T and σ K σ (v) = 2µε K σ (v)+λ tr (ε K σ (v)) 1 for all v ∈ V, we introduce the discrete bilinear form on W × W a D (u, v) = ∑ K∈K ∑ σ ∈E K |K σ | σ K σ (I(u)) : ε K σ (I(v)) . (5)
Then, the discrete variational formulation reads: find u ∈ W such that u σ = u D σ for all σ ∈ E D ext and such that, for all v ∈ W 0 ,

a D (u, v) = ∑ K∈K |K| f K • v K + ∑ σ ∈E N ext |σ | g σ • I(v) σ , (6)
where

u D σ = 1 |σ | σ u D dσ , f K = 1 |K| K f dx and g σ = 1
|σ | σ g dσ are average values. It is important to keep in mind that numerical experiments show that without interpolation, the bilinear form a D on the space V × V leads to an unstable scheme with vanishing eigenvalues on triangular or tetrahedral meshes with mixed Neumann Dirichlet boundary conditions.

Note also that for the solution of the linear system (6), the unknowns u K can easily be eliminated without any fill in, reducing the degrees of freedom to the face normal components of the displacement.

Interpolation of the tangential components of the displacement

Given a face σ ∈ E int ∪ E N ext , for each component i ∈ 1, d of the displacement field u σ , we look for a linear interpolation of the form

ū i σ (x) = d ∑ j=1 α i j σ x j + β i σ . (7)
In order to determine the d(d + 1) coefficients (α i j σ ) i, j∈ 1,d , (β i σ ) i∈ 1,d as linear combinations of normal components u σ ′ • n σ ′ , σ ′ ∈ S σ , we look for a set S σ of d(d + 1) neighbouring faces σ ′ of the face σ , with σ ∈ S σ and such that the system of equations ūσ (x σ ′ ) • n σ ′ = u σ ′ • n σ ′ is non singular. The set S σ is built using the following greedy algorithm:

1. Initialization: for a given number k > d(d + 1), we select the k closest neighbouring faces of the face σ which are sorted from the closest to the farthest using the distance between the face center and x σ :

σ 0 = σ , σ 1 , • • • , σ k-1 .
We set S σ = {σ } and q = 1, l = 0; 2. while q < d(d + 1) and l < k -1:

• l = l + 1; • if the equation ūσ (x σ l ) • n σ l = u σ l • n σ l is linearly independent with the set of equations ūσ (x σ ′ ) • n σ ′ = u σ ′ • n σ ′ for all σ ′ ∈ S σ , then S σ = {σ l } ∪ S σ , q = q + 1; 3. if q < d(d + 1
), the algorithm is rerun with a larger value of k.

Note that since σ ∈ S σ , the interpolation operator satisfies as required the property P W (I(u)) = u for all u ∈ W.

Compatibility condition with cellwise constant pressure for poroelastic models

Let L 2 0 (Ω ) be the subspace of functions of L 2 (Ω ) with vanishing mean values. For the sake of simplicity but without any loss of generality, we consider here ∂ Ω D = ∂ Ω and u D = 0. It is well known (see [START_REF] Murad | On Stability and Convergence of Finite Element Approximations of Biot's Consolidation Problem[END_REF]) that the well-posedness of linear poroelasticity models relies on the well-posedness of the following saddle point problem: find

(u, p) ∈ H 1 0 (Ω ) d × L 2 0 (Ω ) such that a(u, v) + b(v, p) = (f, v) L 2 (Ω ) d for all v ∈ H 1 0 (Ω ) d , b(u, q) = (h, q) L 2 (Ω ) for all q ∈ L 2 0 (Ω ), ( 8
)
where a is the bilinear form of the linear elasticity model and b

(v, p) = -(div v, p) L 2 (Ω ) .
The stability of this saddle point problem results from the coercivity of a and the following LBB condition (see [START_REF] Brezzi | Mixed and Hybrid Finite Element Methods[END_REF]) which guarantees the existence and uniqueness of the solution:

inf p∈L 2 0 (Ω ) sup v∈H 1 0 (Ω ) d b(v,p) v H 1 0 (Ω ) d p L 2 (Ω )
≥ γ > 0.

The following theorem states that the LBB condition holds on the discrete spaces W 0 × M 0 , where M 0 is the space of cellwise constant functions on the mesh K with vanishing mean values endowed with the L 2 (Ω ) norm, and for the discrete divergence operator defined by:

b D (w, p) = -(div D w, p) L 2 (Ω ) = -∑ K∈K p K ∑ σ ∈E K ∩E int |σ | w n σ (n σ • n K,σ ) , (9)
for all (w, p) ∈ W 0 × M 0 , and where n K,σ is the normal to the face σ outward K. It implies that, assuming the coercivity of a D , the coupling of our discretization for the elasticity model with a cell centered finite volume scheme for the Darcy pressure equation will lead to a convergent and stable scheme for the poroelastic model.

Theorem 1. The bilinear form b D defined on W 0 × M 0 satisfies the LBB condition

inf p∈M 0 sup w∈W 0 b D (w, p) w W 0 p L 2 (Ω ) ≥ γ D > 0, ( 10
)
with a constant γ D depending only on usual regularity parameters of the mesh.

Proof. From the continuous LBB condition, for all p ∈ M 0 , there exists a displace-

ment field v ∈ H 1 0 (Ω ) d such that b(v, p) ≥ γ v H 1 0 (Ω ) d p L 2 (Ω ) . Let u be the ele- ment of V 0 such that u K = 1 |K| K v dx for K ∈ K and u σ = 1 |σ | σ v dσ for σ ∈ E . Then, w = P W (u) ∈ W 0 satisfies b D (w, p) = b(v, p). Since it is shown in [9] that u V 0 ≤ κ v H 1 0 (Ω )
d , with a constant κ depending on usual regularity parameters of the mesh, and since we have by definition the inequality w W 0 ≤ u V 0 , the discrete LBB condition holds with γ D = γ κ . ⊓ ⊔

Numerical experiments

In this section, the convergence of the scheme is tested on the linear elasticity model with exact solution

u i = e cos ∑ d j=1 χ i j x j , i = 1, • • • , d. ( 11
)
The right hand side and the Dirichlet boundary conditions are defined by the exact solution. The Lamé parameters λ and µ are set to 1.

The tests have been held using an object oriented C++ implementation which original approach is described in [START_REF] Di Pietro | Lowest order methods for diffusive problems on general meshes: A unified approach to definition and implementation[END_REF]. The relative l 2 error on the displacement and on the gradient of the displacement are plotted function of the number of inner faces. In the computation of these errors, the cellwise constant discrete solution and discrete gradient are used. We first consider the triangular meshes (mesh family 1), the Cartesian grids (mesh family 2), the local grid refinement (mesh family 3) and the Kershaw meshes (mesh family 4) from the FVCA5 2D benchmark. The exact solution is defined by χ = 1 1 2 -1 .

The results presented on figure 1 show the good convergence behaviour of the scheme. The expected order of convergence is reached for all the meshes (for the solution and its gradient) and is even exceeded for the gradient on Cartesian grids. Next, let us consider the Cartesian grids (mesh family A), the randomly distorted grids (mesh family AA), and the tetrahedral meshes (mesh family B) from the FVCA6 3D benchmark. The exact solution is defined by

χ =   1 1 1 2 1 -1 -1 1 2   .
The results exhibited on figure 2 show again the good convergence behaviour of the scheme with the expected order on the three meshes for both the discrete solution and its gradient.

Conclusion

In this paper, a new discretization method has been introduced for linear elasticity using only one degree of freedom per face. It applies to general polygonal and polyhedral meshes possibly non conforming. In addition this discretization satisfies the compatibility condition when coupled with cell centered finite volume schemes for the Darcy equation in poroelastic models.

First numerical experiments in 2D and 3D exhibit the stability and convergence of the scheme. In the near future, further testings will be performed on CPG grids with erosions, local grid refinement and faults, to assess the potential of this scheme for reservoir and CO2 storage simulations.

Fig. 1 l

 1 Fig. 1 l 2 error for the displacement and for the gradient of the displacement function of the number of inner faces for the triangular meshes, the Cartesian grids, the local grid refinement and the Kershaw meshes.

Fig. 2 l

 2 Fig. 2 l 2 error for the displacement and for the gradient of the displacement function of the number of inner faces for the Cartesian grids, the randomly distorted grids and the tetrahedral meshes.


Nadine Abdallah

Audrey Queudet
email: audrey.queudet@univ-nantes.fr

Maryline Chetto

Rafic Hage

Partitioned EDF Scheduling in Multicore systems with Quality of Service constraints

In this paper we study the partitioned EDF scheduling in a homogeneous multiprocessor environment with Quality of Service (QoS) constraints. The system considered here is a real-time multiprocessor system assumed to be powered by rechargeable batteries. We address the issue of how to best partition a set of firm real-time tasks that can occasionally skip one instance according to a predefined QoS threshold. The main goal is to minimize the energy consumption of the system while offering solutions with respect to transient energy starvation situations the system can experiment. The contribution of the paper is twofold. First, we present a schedulability analysis of firm multiprocessor task sets under QoS constraints. Second we propose new partitionning heuristics integrating skips. The evaluation is conducted from several points of view (minimization of the total processor number, maximization of the spare capacity on each processor).

I. INTRODUCTION

The computer science literature generally divides real-time systems in three main categories: soft, hard and firm [START_REF] Liu | Real Time Systems[END_REF]. Soft real-time systems allow some jobs to miss their deadlines in order to improve resource usage or average performance. Hard real-time systems embody guaranteed timing and cannot miss deadlines. A single failure in one timing constraint can cause an intolerable cost (in terms of human lives, equipment damage or economic loss). Firm real-time systems instead allow some of their constraints to be occasionally lost, the performance being then quantified in terms of Quality of Service (QoS).

In recent years, there has been increasing interest to incorporate real-time scheduling techniques that deal with power/energy conservation. Many energy-oriented real-time scheduling techniques have been proposed to reduce energy consumption. Among them, Dynamic Voltage and Frequency Scaling (DVFS) algorithms [START_REF] Chen | Energy-efficient scheduling for real-time systems on dynamic voltage scaling (DVS) platforms[END_REF] reduce energy consumption by changing processor speed and voltage at run-time depending on the needs of the applications running. Another trend is based upon Dynamic Power Management (DPM) policies [START_REF] Benini | A Survey of Design Techniques for System-Level Dynamic Power Management[END_REF] which trade off the performance for the power consumption by selectively placing components into low-power states. All these techniques have been proposed for uniprocessor realtime systems. However, with todays computational demands and as the miniaturization of integrated circuits reaches its physical limits, a valid solution to supply sufficient resources is to use multicore platforms, but actually, much less work has been done for power awareness in multicore systems.

In our research we consider multicore real-time systems with both QoS and energy constraints. The objective is to exploit the flexibility offered by QoS-based real-time tasks to face both processor overload and energy starvation situations, skipping the execution of some task instances. For that purpose, we propose to tackle the problem of the repartition of QoS-constrained tasks over such platforms. Our contribution is twofold. First, we design a schedulability test for firm multiprocessor task sets under QoS constraints. Second, based on this test, we propose new partitioned scheduling heuristics to assign tasks with QoS constraints to processors, so as to minimize the number of processors, thus minimizing the energy consumption. To the best of our knowledge, this paper is the first to introduced QoS contraints into partitioning heuristics.

We rely on previous partitioning results [START_REF] Lupu | Multi-Criteria Evaluation of Partitioning Schemes for Real-Time Systems[END_REF][9] [START_REF] Zapata | Analysis of Real-Time Multiprocessors Scheduling Algorithms[END_REF] to preselect which partitioning heuristic and which task set sorting criterion can best fit both QoS and energy constraints. The performance of each heuristic is analyzed according to the success ratio (i.e. the number of schedulable task sets among the total number of generated task sets). We also study the influence of the task sorting criteria on the schedulability of each heuristic so as to underline the impact of task sorting criterion on the performance of the heuristics chosen.

The remainder of this paper is organized as follows: in Section 2, we describe some background material. Section 3 presents the models and definitions considered in this paper. Section 4, 5 and 6 presents the main contribution which begins with a schedulability analysis under QoS constraints and then relies on the partitioning under QoS contraints to end with performance evaluation. In Section 7, we conclude and give some future lines of investigation.

II. BACKGROUND MATERIAL

A. EDF Scheduling

Earliest Deadline First (EDF) scheduling algorithm [START_REF] Dertouzos | Control Robotics: The Procedural Control of Physical Processes[END_REF] is an algorithm in which jobs with earliest deadlines have higher priority. EDF is an optimal scheduling algorithm on preemptive uniprocessors, in the following sense: if a collection of independent jobs can be scheduled (by any algorithm) such that all the jobs complete by their deadlines, EDF will schedule this collection of jobs such that they all complete by their deadlines.

B. Skip-Over model

We consider a uniprocessor system consisting of firm periodic and preemptable tasks. Tasks are assumed to be independent. Each task is divided into instances where each instance occurs during a single period of the task. The possibility of skipping task instances was introduced by Koren and Shasha [START_REF] Koren | Skip-over: Algorithms and Complexity for Overloaded Real-Time Systems[END_REF]. In their model, a task is characterized by its worst case execution time, its period, its deadline and a skip parameter s (2 ≤ s ≤ ∞). This parameter gives the tolerance of a task to miss deadlines. The higher s is, better is the QoS. It represents the minimum QoS required by a task. Every instance of the task can be red or blue. Red instances must complete before their deadline but blue instances can be skipped or aborted at any time. A task set is deeply-red when all tasks instances are initially activated at the same time and are red. The distance between two consecutive skips must be at least s periods. After missing a deadline, the next s -1 instances must complete before their deadlines. On the contrary, if a blue instance completes within its deadline, the next instance is still blue.

Here are two algorithms based on the Skip-Over model:

• RTO (Red Tasks Only): this algorithm never tries to execute blue instances. Red instances only are scheduled according to EDF. In the deeply-red model, this algorithm is optimal which means that all feasible task sets will be schedulable using RTO. • BWP (Blue When Possible): this algorithm is more flexible in the sense that it schedules red instances according to EDF and tries to schedule blue instances when there are no ready red instances.

C. Partitioning

Partitioning a task set is equivalent to the Bin-Packing problem: how to place n objects of different sizes in m identical boxes. This problem is known to be NP-hard. The only known solution for this kind of problem is to enumerate all possible configurations and verify their correctness one by one. Some heuristics [START_REF] Lupu | Multi-Criteria Evaluation of Partitioning Schemes for Real-Time Systems[END_REF][9] [START_REF] Zapata | Analysis of Real-Time Multiprocessors Scheduling Algorithms[END_REF] have been proposed in the literature in order to solve it. All of them imply a sequential assignment of tasks to processors: a task is assigned on a processor if it verifies the schedulability test after assignment. According to First Fit (FF) a task is assigned to the first possible processor, starting from π 1 (the first processor). Best Fit (BF) assigns a task τ i to the processor which minimizes the remaining processor capacity. According to Worst Fit (WF) a task is assigned to the processor which maximizes the remaining processor capacity. Next Fit (NF) assigns a task τ i to the first possible processor in the range π j , ..., π m (π j being the current processor). The procedure starts from π 1 .

III. MODELS AND DEFINITIONS

A. System and task model

In this paper, we consider π a platform with m identical processors: π = {π 1 , ..., π m }. We refer to the Skip-Over periodic task model. A periodic task τ i is defined by a 4-tuple (C i , P i , D i , s i ) where C i is the worst-case execution time (WCET), P i the period, D i the relative deadline and s i the skip parameter. A task can be instantiated an infinite number of times. The task set τ = {τ 1 , ..., τ n } is composed of n periodic tasks with constrained deadline (D i ≤ P i ). Tasks are assumed to be independent and preemptable. The system is considered deeply-red.

Each task is characterized by:

• an utilization factor:

u i = Ci Pi
• an equivalent utilization factor integrating skips:

u * i = Ci Pi × si-1 si • a density: δ i = Ci min{Di,T i} .
• an equivalent density integrating skips:

δ * i = Ci min{Di,Pi} × si-1 si B. Known results
1) Processor utilization factor: Given a set τ = {τ i (P i , D i , C i )} of n periodic tasks scheduled on a processor, the processor utilization factor is defined as [START_REF] Liu | Scheduling algorithms for multi-programming in a hard real time environment[END_REF]:

U τ = n i=1 C i P i (1)
2) Equivalent processor utilization factor: Given a set τ = {τ i (P i , D i , C i , s i )} of n periodic tasks with implicit deadline that allow skips, the equivalent processor utilization factor integrating skips is defined as [START_REF] Koren | Skip-over: Algorithms and Complexity for Overloaded Real-Time Systems[END_REF]:

U * τ = max L≥0 { i D(i, [0, L]) L } (2)
where

D(i, [0, L]) = L Pi -L Pisi .
3) Load factor: Given a set τ = {τ i (P i , D i , C i )} of n periodic tasks with arbitrary deadlines, the load factor is defined as [START_REF] George | A norm approach for Partitioned EDF Scheduling of Sporadic Task Systems[END_REF]:

Load(τ ) = max{U τ , sup L∈[Dmin,P ) DBF (τ, L) L } (3)
where D min = min{D 1 , ..., D n } and P = lcm{P 1 , ..., P n }.

For a given t, the Demand Bound Function (DBF) represents the upper bound of the workload generated by all tasks with activation times and absolute deadlines in the same interval [0, t]:

DBF (τ, [0, L]) = n i=1 1 + L -D i P i × C i (4)
DBF will be computed for L corresponding to absolute task deadlines between 0 and D min .

4) Exact schedulability test: A necessary and sufficient (i.e. exact) schedulability test for EDF for arbitrary deadlines on uniprocessor systems is given by [START_REF] Liu | Scheduling algorithms for multi-programming in a hard real time environment[END_REF]:

Load(τ ) ≤ 1 (5)
IV. SCHEDULABILITY ANALYSIS UNDER QOS CONSTRAINTS 5) New equivalent load factor: Given a set τ = {τ i (P i , D i , C i , s i )} of n periodic tasks with arbitrary deadlines that allow skips, we define the equivalent load factor as:

Load QoS (τ ) = max{U * τ , sup L∈[Dmin,P ) DBF QoS (τ, L) L } (
DBF QoS (τ, [0, L]) = n i=1 1 + L -D i P i - 1 s i 1 + L -D i P i × C i (7)
As the DBF QoS changes its value only at instants corresponding to absolute deadlines, DBF QoS will be computed for L corresponding to absolute tasks deadlines between 0 and D min . The equivalent processor utilization factor U * τ integrating skips is defined as it previously appears in equation (2).

6) New exact schedulability test: A necessary and sufficient (i.e., exact) schedulability test for EDF for arbitrary deadlines QoS-constrained tasks on uniprocessor systems is given by :

Load QoS (τ ) ≤ 1 (8)
V. PARTITIONING UNDER QOS CONSTRAINTS

A. Identification of suitable heuristics

As mentioned previously, partitioning, which reduces to a bin-packing problem, is known to be NP-Hard. According to previous results in [START_REF] Lupu | Multi-Criteria Evaluation of Partitioning Schemes for Real-Time Systems[END_REF], we will use two heuristics :

• First Fit (FF) because it minimizes the number of processors used (in our case, it will minimize the energy consumption); • Worst Fit (WF) because it maximizes the remaining processor capacity, thus offering more flexibility for recharging the system in case of energy starvation. These heuristics have been adapted in our case. We systematically apply a schedulability test (see equation ( 8)) before assigning a task to a processor. If the test is not verified we continue testing on others processors until we find the good one able to accept the task. If a task is not assigned to any processor, it means that the task set is not schedulable.

B. New task sorting criteria

All the partitioning algorithms proposed in the literature often include task sorting criteria with the purpose of increasing their success ratio (i.e. ratio of schedulable task sets). A task sorting criteria consists in ordering the tasks before assigning them to a processor. It has an influence on the tasks assignment on processors. For instance, sorting tasks in order of decreasing density will force to first assign heaviest tasks on the processors. Based on the observation of previous results in the literature [START_REF] Lupu | Multi-Criteria Evaluation of Partitioning Schemes for Real-Time Systems[END_REF], we retained three criteria: the density, the utilization factor and the period. We adapted them to QoS constraints, analyzing the general influence of the 4 following criteria in increasing/decreasing order: equivalent density δ * i , equivalent utilization factor u * i , period multiplied by skip parameter P i s i and the skip parameter itself s i

VI. PERFORMANCE EVALUATION

In this section, we evaluate the performance of all possible combinations between a series of 2 partitioning heuristics (adapted FF and WF) and 8 task sorting criteria (δ * i , u * i , P i s i , s i considered in increasing/decreasing order).

A. Task generation methodology

In our task generation methodology we consider tasks with constrained deadlines (D i ≤ T i ) which is the hardest assumption to take.

• P i is uniformly chosen from [20, 40]; • C i is set to provide a task utilization factor equal to the one given in input; • D i is uniformly chosen between C i and P i ; • s i is uniformly chosen from [s min , s max ] where s min and s max are input parameters; • m is the number of processors used and is given as a parameter.

We consider m-identical processor platforms. For m processors, we generate a system that contains 2 × m tasks. ∀i ∈ [1, n], δ i ≤ 1, which means that task density doesn't exceed the utilization capacity of a processor. For our simulations, we generated 1000 different task sets.

B. Performance criteria

We evaluate each combination of the previously mentioned parameters according to a performance parameter named Success Ratio which is defined by:

Success ratio = N success N generated (9)
where N success denotes the number of task sets successfully scheduled and N generated the total number of task sets generated. This criteria helps us to determine which combination schedules the largest number of task sets.

C. Experimental results

In this section, we present an initial empirical investigation for QoS-constrained tasks defined according to the Skip-Over model, examining the effectiveness of our heuristics on 4 processors. This section deals with the impact of a task sorting criterion on the success ratio of schedulability tests. In the corresponding graphs, D means "decreasing", I means "increasing" and E means "equivalent". The simulation results depicted in Figure 1 and2 show that, taking all heuristics together, the sorting criteria which maximize the success ratio are: Decreasing Equivalent Utilization and Decreasing Equivalent Density. Figure 1 shows that for task sets with a total utilization factor less than 25% of the platform capacity, all sorting criteria give the same performance. However, for task sets with total density greater than 25% of the platform capacity, Decreasing Equivalent Utilization factor and Decreasing Equivalent Density exhibit the best behavior. Decreasing Equivalent Utilization factor is slightly more efficient than Decreasing Equivalent Density. Figure 2 shows that for task sets with the total utilization factor less than 75% of the platform capacity, all sorting criteria give the same performance. However, for task sets with total density greater than 75% of the platform capacity, Decreasing Equivalent Utilization factor and Decreasing Density exhibit the best behavior. In fact, for a total utilization factor equals to 3.2, at most 99% of task sets are schedulable with First Fit Decreasing Equivalent Utilization and 84% with Worst Fit Decreasing Equivalent Utilization and Density. In Figure 1, we also notice for Decreasing Equivalent Utilization that when skips are comprised in [START_REF] Baruah | Algorithms and complexity concerning the preemptive scheduling of periodic real-time tasks on one processor[END_REF][START_REF] Caccamo | Optimal Scheduling for Fault-Tolerant and Firm Real-Time Systems[END_REF] we have a better success ratio than when skips are comprised in [START_REF] Baruah | Algorithms and complexity concerning the preemptive scheduling of periodic real-time tasks on one processor[END_REF][START_REF] Liu | Real Time Systems[END_REF]. In fact, the skip parameter has also an influence on schedulability: the more we authorize skips higher is the success ratio.

VII. CONCLUSION

While low-power uniprocessor real-time systems have fueled much recent work on energy-aware scheduling, the same issue upon multicore platforms has been somewhat neglected. Motivated by this, we proposed a flexible solution based on QoS-constrained real-time tasks, allowing to skip some task instances in case of either processor overload or energy starvation.

The major contributions of our study are as follows: (i) we provide a schedulability test for multiprocessor task sets under QoS contraints, (ii) we propose and evaluate two new partitionning heuristics (First Fist variant that minimizes the energy consumption, and Worst Fist variant that leaves maximum spare capacity on processors in order to allow the battery to charge). From the simulations, we showed that the best task sorting criterion are equivalent decreasing density and equivalent decreasing utilization. In the future, we want to extend this study to semi-partitioning algorithms, providing a complete analysis with new evaluation criteria such as the number of migrations and preemptions.

Fig. 1 .

 1 Fig. 1. EDF-based sorting criteria with WF heuristic

  [START_REF] Dertouzos | Control Robotics: The Procedural Control of Physical Processes[END_REF] where D min = min{D 1 , ..., D n } and P = lcm{s 1 P 1 , ..., s n P n }. According to the RTO algorithm, the DBF changes. Its new equation is :

ACKNOWLEDGMENT

The work presented in this paper is sponsored by a CE-DRE project, namely GreenEmbedded, which is a bilateral collaboration between University of Nantes and the Lebanese University.