Christophe A Girard
email: girard@ipmc.cnrs.frlambeau@ipmc.cnrs.fr

Barbara Seitz Seitz-Polski

Guillaume Dolla

Arnaud Augert

David Vindrieux

David Bernard

Gerard Lambeau

Barbara Seitz-Polski

Gérard Lambeau

Nouveaux rôles physiopathologiques pour le récepteur PLA2R1 dans le cancer et la glomérulonéphrite extramembraneuse

est destinée au dépôt

Ligands de PLA2R1

Notre équipe a découvert PLA2R1 (initialement appelé récepteur de type M) en 1990 à l'aide d'une phospholipase A2 sécrétée (sPLA2) de venin de serpent (donc exogène) appelée OS2 [START_REF] Lambeau | Identification and purification of a very high affinity binding protein for toxic phospholipases A2 in skeletal muscle[END_REF]. Cette dernière se lie à PLA2R1 avec une très haute affinité de 7 pM. On ne connaissait en 1990 que deux sPLA2 de mammifère, IB et IIA, et la question se posait alors de savoir si ces sPLA2 pouvaient être les ligands naturels endogènes de PLA2R1. Suite à une série de résultats en apparence confus car obtenus avec des sPLA2 et du PLA2R1 issus de différentes espèces de mammifères, une première réponse claire a été apportée en 1999, au moins chez la souris, pour ces deux sPLA2 [START_REF] Cupillard | Both group IB and group IIA secreted phospholipases A2 are natural ligands of the mouse 180-kDa M-type receptor[END_REF]. Dans cette espèce, les sPLA2 IB et IIA se lient à PLA2R1 avec une haute affinité de l'ordre du nM. Cependant, les résultats apparaissaient différents chez le rat et l'homme, suggérant des propriétés de liaison de PLA2R1 dépendantes de l'espèce [START_REF] Cupillard | Both group IB and group IIA secreted phospholipases A2 are natural ligands of the mouse 180-kDa M-type receptor[END_REF]. On sait maintenant qu'il

Structure de PLA2R1

PLA2R1 (phospholipase A2 receptor 1) est une glycoprotéine transmembranaire de 180-kDa appartenant à la superfamille des lectines de type C (Figure 1) [START_REF] Lambeau | Receptors for a growing family of secreted phospholipases A2[END_REF]. Il présente une structure complexe composée de 10 domaines extracellulaires (représentant 95 % de la protéine), d'un seul segment transmembranaire et d'une courte queue cytoplasmique d'une quarantaine d'acides aminés contenant un motif consensus d'endocytose NPXY [START_REF] Ancian | The human 180-kDa receptor for secretory phospholipases A2. Molecular cloning, identification of a secreted soluble form, expression, and chromosomal localization[END_REF][START_REF] Zvaritch | Endocytic properties of the M-type 180-kDa receptor for secretory phospholipases A2[END_REF]. Depuis son extrémité amino-terminale extracellulaire, on distingue une région riche en cystéine (CysR), un domaine fibronectine de type II (FNII) et huit domaines lectine de type C distincts (CTLD). Trois autres récepteurs ont la même structure et forment un sous-groupe au sein de la superfamille des lectines de type C : le récepteur mannose du macrophage (MRC1), le récepteur Endo-180 (MRC2) et le récepteur des cellules dendritiques (DEC-205, LY75) (pour revue voir [START_REF] East | The mannose receptor family[END_REF]). Les séquences en acides aminés de ces quatre récepteurs présentent cependant moins de 30 % d'identité, suggérant des fonctions distinctes. Par ailleurs, PLA2R1 existe sous deux formes, transmembranaire et soluble [START_REF] Ancian | The human 180-kDa receptor for secretory phospholipases A2. Molecular cloning, identification of a secreted soluble form, expression, and chromosomal localization[END_REF]. La forme soluble est clivée après le CTLD8, et est dépourvue de segment transmembranaire et un moyen efficace de stopper leur activité biologique, en inhibant leur activité enzymatique et en favorisant leur élimination du milieu extracellulaire. En effet, in vitro, la liaison des sPLA2 au récepteur PLA2R1 soluble entraîne une inhibition de leur activité enzymatique, et leur liaison au récepteur membranaire conduit non seulement à leur inhibition, mais aussi à leur internalisation et leur dégradation dans les lysosomes [START_REF] Ancian | The human 180-kDa receptor for secretory phospholipases A2. Molecular cloning, identification of a secreted soluble form, expression, and chromosomal localization[END_REF][START_REF] Zvaritch | Endocytic properties of the M-type 180-kDa receptor for secretory phospholipases A2[END_REF][START_REF] Rouault | Recombinant production and properties of binding of the full set of mouse secreted phospholipases A2 to the mouse M-type receptor[END_REF]. Inversement, PLA2R1 pourrait relayer la transduction de certaines fonctions biologiques des sPLA2 en activant diverses voies de signalisation intracellulaire [START_REF] Hanasaki | Phospholipase A2 receptor: a regulator of biological functions of secretory phospholipase A2[END_REF] ou, peut-être même, en les adressant dans des compartiments intracellulaires, comme le noyau [START_REF] Fayard | Nuclear location of PLA2-I in proliferative cells[END_REF]. Mais, la démonstration formelle du rôle de PLA2R1 dans ces deux derniers cas reste à apporter.

Rôles physiopathologiques du récepteur PLA2R1

Rôles opposés de PLA2R1 dans l'inflammation Plusieurs sPLA2 (IIA, V et X) et le récepteur PLA2R1 sont exprimés dans les poumons [START_REF] Murakami | Emerging roles of secreted phospholipase A2 enzymes: Lessons from transgenic and knockout mice[END_REF]. Les sPLA2 IIA et X sont présentes dans les lavages broncho-alvéolaires de existe une douzaine de sPLA2 humaines ou de souris (IB, IIA, IIC, IID, IIE, IIF, III, V, X, XIIA, XIIB et otoconin-95) [START_REF] Lambeau | Biochemistry and physiology of mammalian secreted phospholipases A2[END_REF][START_REF] Murakami | Emerging roles of secreted phospholipase A2 enzymes: Lessons from transgenic and knockout mice[END_REF]. À nouveau chez la souris, nous avons montré que sept de ces 12 sPLA2 se lient à PLA2R1, ce qui renforce l'idée que ces dernières sont bien des ligands naturels de PLA2R1 [START_REF] Rouault | Recombinant production and properties of binding of the full set of mouse secreted phospholipases A2 to the mouse M-type receptor[END_REF]. Les propriétés de liaison des sPLA2 sur PLA2R1 restent cependant à caractériser finement dans les autres espèces de mammifères, notamment chez l'homme. On sait aussi que PLA2R1, au moins dans certaines espèces, peut lier du collagène et certains sucres, suggérant que PLA2R1 pourrait avoir d'autres types de ligands endogènes, associés à différentes fonctions de type adhésion ou lectine (Figure 1) [START_REF] Rouault | Recombinant production and properties of binding of the full set of mouse secreted phospholipases A2 to the mouse M-type receptor[END_REF][START_REF] Ancian | Multifunctional activity of the extracellular domain of the M-type (180 kDa) membrane receptor for secretory phospholipases A2[END_REF]. Enfin, les expériences de mutagenèse dirigée ont montré que le domaine fibronectine de type II lie le collagène de type IV [START_REF] Ancian | Multifunctional activity of the extracellular domain of the M-type (180 kDa) membrane receptor for secretory phospholipases A2[END_REF], tandis que les domaines CTLD4-6 lient les sPLA2 et certains sucres, comme le mannose ou le glucose [START_REF] Nicolas | Identification of the binding domain for secretory phospholipases A2 on their M-type 180 kDa membrane receptor[END_REF]. La liaison des sPLA2 sur les domaines CTLD4-6 de PLA2R1 se fait cependant par interaction protéine-protéine [START_REF] Rouault | Recombinant production and properties of binding of the full set of mouse secreted phospholipases A2 to the mouse M-type receptor[END_REF]. L'interaction sPLA2-PLA2R1 pourrait avoir des rôles opposés (Figure 1). Les sPLA2 sont de petites enzymes très stables de 13-à 18-kDa qui hydrolysent les phospholipides pour libérer des acides gras et des lysophospholipides, deux précurseurs de la synthèse de nombreux médiateurs lipidiques [START_REF] Lambeau | Biochemistry and physiology of mammalian secreted phospholipases A2[END_REF][START_REF] Murakami | Emerging roles of secreted phospholipase A2 enzymes: Lessons from transgenic and knockout mice[END_REF][START_REF] Ait-Oufella | Lp-PLA2 et sPLA2 : biomarqueurs cardiovasculaires[END_REF] (➜). Cette propriété est à la base d'un grand nombre de leurs fonctions biologiques [START_REF] Lambeau | Biochemistry and physiology of mammalian secreted phospholipases A2[END_REF][START_REF] Murakami | Emerging roles of secreted phospholipase A2 enzymes: Lessons from transgenic and knockout mice[END_REF]. La liaison sPLA2-PLA2R1 pourrait donc être [START_REF] Augert | The M-type receptor PLA2R regulates senescence through the p53 pathway[END_REF]. Lorsque ces cellules entrent en sénescence, elles expriment davantage de PLA2R1 (Figure 2). La diminution

Entrée en apoptose des cellules tumorales

Ré-expression de PLA2R1 dans les cellules tumorales

Perte de PLA2R1

Stress oxydatif médicaments). La GEM idiopathique est une maladie auto-immune dont l'évolution clinique est variable. Certains patients entrent en rémission spontanée, tandis que d'autres progressent vers une insuffisance rénale terminale, nécessitant le recours à la dialyse ou à la greffe de rein. Le diagnostic de GEM idiopathique repose exclusivement sur l'analyse histologique d'une biopsie rénale. La GEM se caractérise par la présence de dépôts auto-immuns au niveau de la membrane basale glomérulaire (MBG). Ces dépôts, généralement composés d'immunoglobulines de type IgG4 et de protéines du complément, sont associés à une perte de sélectivité de la barrière glomérulaire, ce qui conduit au passage anormal de protéines sanguines dans les urines, expliquant la protéinurie de rang néphrotique retrouvée chez les patients atteints de GEM (Figure 3). L'identification des cibles moléculaires des auto-anticorps a fait l'objet d'intenses recherches. En 1959, Heymann fut le premier à développer un modèle de GEM chez le rat et à montrer que l'antigène principal était dans ce cas la mégaline, une protéine exprimée à la surface des podocytes, les cellules épithéliales essentielles à la filtration glomérulaire [START_REF] Ronco | Pathogenesis of membranous nephropathy: recent advances and future challenges[END_REF][START_REF] Heymann | Production of nephrotic syndrome in rats by Freund's adjuvants and rat kidney suspensions[END_REF]. La mégaline étant absente du podocyte humain, ces découvertes n'étaient cependant pas transposables à l'homme. Il fallut attendre 2002 pour que l'équipe de Pierre Ronco identifie chez l'homme l'endopeptidase neutre (NEP) comme un premier antigène de la GEM [START_REF] Debiec | Antenatal membranous glomerulonephritis due to antineutral endopeptidase antibodies[END_REF][START_REF] Ronco | Allo-immunisation foeto-maternelle anti-CD10[END_REF]. Cependant, cet antigène est impliqué dans une forme rare de GEM néonatale de type allo-immune, mais pas dans les GEM idiopathiques de l'adulte. C'est enfin en 2009 que l'équipe de David Salant, en collaboration avec notre équipe, a mis en évidence qu'environ 70 % des patients atteints de GEM idiopathique avaient des auto-anticorps circulants de type IgG4 dirigés contre PLA2R1 [START_REF] Beck | M-type phospholipase A2 receptor as target antigen in idiopathic membranous nephropathy[END_REF]. Les anticorps anti-PLA2R1 sont absents chez les patients souffrant de GEM secondaires ou d'autres néphropathies glomérulaires. Ces anticorps reconnaissent des épitopes conformationnels de PLA2R1, et sont insensibles à son état de glycosylation. En accord avec l'hypothèse émise 50 ans plus tôt par Heymann, PLA2R1 est bien exprimé à la surface des podocytes chez le sujet sain (Figure 3). Chez les patients, son expression serait augmentée, et il est colocalisé avec les IgG4 qui forment les dépôts autoimmuns [START_REF] Beck | M-type phospholipase A2 receptor as target antigen in idiopathic membranous nephropathy[END_REF][START_REF] Debiec | PLA2R autoantibodies and PLA2R glomerular deposits in membranous nephropathy[END_REF]. Ces résultats suggèrent que les dépôts immuns observés in situ au niveau de la MBG des patients atteints de GEM idiopathique résultent de la reconnaissance par les auto-anticorps anti-PLA2R1 circulants de l'antigène PLA2R1 exprimé à la surface des podocytes. Cette reconnaissance pourrait déclencher l'activation du complément ou d'autres mécanismes d'expression de PLA2R1 par ARN interférence retarde l'entrée en sénescence, tandis que la surexpression de PLA2R1 l'induit de façon prématurée [START_REF] Augert | The M-type receptor PLA2R regulates senescence through the p53 pathway[END_REF]. L'effet pro-sénescent de PLA2R1 dépend de la production de radicaux oxygénés (reactive oxygen species, ROS), de l'induction de dommages à l'ADN et de l'activation du gène suppresseur de tumeurs p53 [START_REF] Augert | The M-type receptor PLA2R regulates senescence through the p53 pathway[END_REF]. Les sPLA2 IB et IIA contrôlent également la sénescence cellulaire [START_REF] Augert | The M-type receptor PLA2R regulates senescence through the p53 pathway[END_REF][START_REF] Kim | Induction of cellular senescence by secretory phospholipase A2 in human dermal fibroblasts through an ROS-mediated p53 pathway[END_REF][START_REF] Bischof | Une re-vue de la sénescence cellulaire[END_REF][START_REF] Augert | Les facteurs sécrétés associés à la sénescence[END_REF]. Néanmoins, la contribution de PLA2R1 dans leurs effets n'est pas clairement établie [START_REF] Augert | The M-type receptor PLA2R regulates senescence through the p53 pathway[END_REF]. PLA2R1 joue également un rôle clé dans l'OIS. En effet, in vitro, la déplétion de PLA2R1 permet l'échappement à l'OIS, ainsi que la transformation de kératinocytes et des HMEC exposés à l'oncogène Ras V12G [START_REF] Vindrieux | PLA2R1 mediates tumor suppression by activating JAK2[END_REF]. In vivo, cela se traduit par l'apparition plus rapide et en plus grand nombre de papillomes chez les souris Pla2r1 -/-dans un modèle de tumorigenèse cutanée induite chimiquement [START_REF] Vindrieux | PLA2R1 mediates tumor suppression by activating JAK2[END_REF]. L'analyse de la base de données Oncomine (https://www.oncomine.org) révèle que l'expression de PLA2R1 diminue dans plusieurs cancers (sein, rein, côlon, tête, cou, etc.) [START_REF] Vindrieux | PLA2R1 mediates tumor suppression by activating JAK2[END_REF]. Chez certains patients atteints de leucémie, et dans des lignées tumorales lymphocytaires (Jurkat, U937) et mammaires (MDAMB-453, Cama1, BT20), cette perte d'expression semble être la conséquence d'un contrôle épigénétique via une hyperméthylation du promoteur [START_REF] Menschikowski | Aberrant methylation of the M-type phospholipase A2 receptor gene in leukemic cells[END_REF]. Lorsque PLA2R1 est réintroduit par une approche virale dans des lignées tumorales mammaires, il induit leur apoptose [START_REF] Augert | PLA2R1 kills cancer cells by inducing mitochondrial stress[END_REF]. Cette réponse pro-apoptotique, plutôt que pro-sénescente, pourrait résulter de l'altération fréquente dans ces cellules des voies p16/Rb et p53, qui participent à l'effet pro-sénescent de PLA2R1 dans les cellules normales. L'effet pro-apoptotique de PLA2R1 ne semble dépendre ni de son domaine cytosolique, ni du domaine de liaison des sPLA2. L'utilisation d'inhibiteurs pharmacologiques suggère que l'effet de PLA2R1 ne dépend pas non plus de la production de médiateurs lipidiques [START_REF] Vindrieux | PLA2R1 mediates tumor suppression by activating JAK2[END_REF][START_REF] Augert | PLA2R1 kills cancer cells by inducing mitochondrial stress[END_REF]. En revanche, il impliquerait l'activation d'une voie de signalisation dépendante de JAK2 (Janus kinase 2) et la production de ROS d'origine mitochondriale [START_REF] Vindrieux | PLA2R1 mediates tumor suppression by activating JAK2[END_REF][START_REF] Augert | PLA2R1 kills cancer cells by inducing mitochondrial stress[END_REF]. L'action de PLA2R1 pourrait ainsi être déterminante à plusieurs étapes clés du processus tumoral. Dans les cellules normales, PLA2R1 pourrait empêcher la formation de tumeurs en participant à leur entrée en sénescence, à la suite d'un stress oncogénique. Dans les cellules cancéreuses, sa ré-expression pourrait freiner la progression tumorale en favorisant leur apoptose. Les travaux de Judith Campisi ont montré que la sénescence cellulaire produit un environnement pro-inflammatoire riche en cytokines [START_REF] Campisi | Aging, cellular senescence, and cancer[END_REF][START_REF] Augert | Les facteurs sécrétés associés à la sénescence[END_REF]. Il serait donc tentant de rapprocher les effets de PLA2R1 décrits ci-dessus sur l'inflammation et la production de cytokines de son rôle pro-sénescent. Cependant, nous n'avons à ce jour aucune donnée suggérant une contribution claire des cytokines dans la réponse de type sénescence induite par PLA2R1.

Rôle de PLA2R1 dans la glomérulonéphrite extra-membraneuse idiopathique

La glomérulonéphrite extra-membraneuse (GEM) est la première cause de syndrome néphrotique chez l'adulte [START_REF] Ronco | Pathogenesis of membranous nephropathy: recent advances and future challenges[END_REF][START_REF] Ponticelli | Glomerular diseases: membranous nephropathy -A modern view[END_REF]. On distingue la GEM primitive ou idiopathique (80 % des cas), dont l'origine est indéterminée, de la GEM secondaire (20 % des cas), qui est associée à une pathologie extra-rénale (lupus, cancers, infections, expositions à certains SYNTHÈSE REVUES cytotoxiques, conduisant au dysfonctionnement des podocytes et au syndrome néphrotique. Plusieurs études indiquent aujourd'hui que le titre des anticorps anti-PLA2R1 est corrélé à l'activité de la maladie (Figure 3) et serait prédictif de la sévérité du pronostic [START_REF] Hofstra | Anti-phospholipase A2 receptor antibodies correlate with clinical status in idiopathic membranous nephropathy[END_REF][START_REF] Kanigicherla | Anti-PLA2R antibodies measured by ELISA predict long-term outcome in a prevalent population of patients with idiopathic membranous nephropathy[END_REF]. Ainsi, il diminue sensiblement chez les patients en rémission spontanée ou induite par un traitement, et augmente lors de rechutes [START_REF] Hofstra | Anti-phospholipase A2 receptor antibodies correlate with clinical status in idiopathic membranous nephropathy[END_REF][START_REF] Beck | Rituximab-induced depletion of anti-PLA2R autoantibodies predicts response in membranous nephropathy[END_REF]. De plus, les taux des anticorps anti-PLA2R1 circulants varient plus rapidement après un traitement immunosuppresseur que la protéinurie des patients, permettant de prédire la réponse au traitement et d'anticiper la rechute [START_REF] Beck | Rituximab-induced depletion of anti-PLA2R autoantibodies predicts response in membranous nephropathy[END_REF]. Ces observations pourraient s'avérer particulièrement intéressantes pour prédire les récidives de GEM chez les patients transplantés [START_REF] Rodriguez | The pathology and clinical features of early recurrent membranous glomerulonephritis[END_REF], bien que l'apport du suivi des anticorps anti-PLA2R1 reste encore discuté [START_REF] Debiec | PLA2R autoantibodies and PLA2R glomerular deposits in membranous nephropathy[END_REF][START_REF] Stahl | PLA2R autoantibodies and recurrent membranous nephropathy after transplantation[END_REF][START_REF] Debiec | Autoantibodies specific for the phospholipase A2 receptor in recurrent and de novo membranous nephropathy[END_REF]. Enfin, certains polymorphismes des gènes PLA2R1 et HLA-DQA1 confèrent un risque accru de développement d'une GEM idiopathique [START_REF] Stanescu | Risk HLA-DQA1 and PLA2R1 alleles in idiopathic membranous nephropathy[END_REF].

Ces découvertes ont déjà permis de développer de nouveaux tests diagnostiques non invasifs permettant une meilleure prise en charge des malades [START_REF] Hofstra | Antiphospholipase A2 receptor antibody titer and subclass in idiopathic membranous nephropathy[END_REF][START_REF] Dahnrich | Development of a standardized ELISA for the determination of autoantibodies against human M-type phospholipase A2 receptor in primary membranous nephropathy[END_REF]. De plus, elles devraient contribuer à une meilleure classification des GEM et à une meilleure stratification des patients à risque. Enfin, elles pourraient ouvrir la voie vers des traitements plus spécifiques de la maladie, ciblant PLA2R1, les auto-anticorps et/ou leurs voies d'activation.

Conclusion et perspectives

PLA2R1 apparaît donc aujourd'hui comme un récepteur multifonctionnel capable d'exercer plusieurs fonctions dans l'inflammation, le cancer et la GEM, une maladie auto-immune. Par ses différents domaines extracellulaires, il serait capable de lier différents types de ligands, comme les sPLA2, le collagène et certains sucres, mais l'importance relative de ces ligands dans ses effets biologiques reste à préciser. Il semble aussi que ses fonctions et ses propriétés de liaison dépendent de l'espèce, et ne sont pas systématiquement conservées de la souris à l'homme et vice versa. Ainsi, chez la souris, PLA2R1 jouerait un rôle dans l'inflammation qui semble associé à la liaison des sPLA2, mais une telle fonction, ainsi que la contribution des sPLA2, restent à valider chez l'homme. Inversement, son effet antitumoral et son rôle dans la GEM idiopathique ont d'abord été mis en évidence chez l'homme. Un rôle équivalent chez la

Titre des IgG4 anti-PLA2R1

Sujet sain

Formation de complexe immuns in situ

Protéinurie

Lésion des podocytes

Patient atteint de GEM

SUMMARY

New physiopathological roles for the PLA2R1 receptor in cancer and membranous nephropathy PLA2R1 is a large transmembrane receptor of 180-kDa that belongs to the superfamily of C-type lectins. It was discovered because of its high affinity for secreted phospholipases A2 (sPLA2), enzymes that play a key role in lipid mediator synthesis. Early PLA2R1 physiological roles include the clearance of sPLA2 from the extracellular medium and/or promotion of their actions. Over the last four years, two independent studies suggested that PLA2R1 plays a role in cancer as a tumor gene suppressor and is the major target antigen of auto-immune antibodies involved in idiopathic membranous nephropathy, a severe human kidney disease. These novel findings shed light on PLA2R1 and pave the way for its use as a reliable biomarker and an attractive therapeutic target in these diseases. ‡

LIENS D'INtÉRÊt

Les auteurs déclarent n'avoir aucun lien d'interêt concernant les données publiées dans cet article.

REMERCIEMENtS

  'asthme.Henderson et al. ont mis en évidence un rôle clé de la sPLA2-X dans un modèle d'asthme chez la souris[START_REF] Henderson | Importance of group X-secreted phospholipase A2 in allergen-induced airway inflammation and remodeling in a mouse asthma model[END_REF]. Dans ce modèle induit par administration intranasale d'ovalbumine, les souris invalidées pour la sPLA2-X produisent moins d'icosanoïdes pro-inflammatoires et ont moins d'asthme. In vitro, la liaison des sPLA2 à PLA2R1 entraîne leur internalisation et leur dégradation dans la voie lysosomale[START_REF] Zvaritch | Endocytic properties of the M-type 180-kDa receptor for secretory phospholipases A2[END_REF][START_REF] Hanasaki | Phospholipase A2 receptor: a regulator of biological functions of secretory phospholipase A2[END_REF]. Ces observations ont conduit Tamaru et al. à étudier le rôle de PLA2R1 dans l'asthme[START_REF] Tamaru | Deficiency of phospholipase A2 receptor exacerbates ovalbumin-induced lung inflammation[END_REF]. Dans le même modèle que celui utilisé par Henderson et al., ils ont observé que les souris invalidées pour PLA2R1 (souris Pla2r1 -/-) présentent une réponse inflammatoire plus intense que les souris sauvages, caractérisée par une infiltration plus importante de cellules inflammatoires dans les voies respiratoires. Les lavages broncho-alvéolaires des souris Pla2r1 -/-contenaient davantage de sPLA2 IB et X, et des concentrations plus élevées de plusieurs icosanoïdes et de cytokines Th2. Par administration intratrachéale de sPLA2 radiomarquée, ces auteurs ont pu mettre en évidence que ces taux plus élevés de sPLA2 résultaient d'une clairance et d'une dégradation réduites de ces sPLA2[START_REF] Tamaru | Deficiency of phospholipase A2 receptor exacerbates ovalbumin-induced lung inflammation[END_REF]. Au niveau pulmonaire, PLA2R1 pourrait donc jouer un rôle protecteur dans la pathogenèse de l'asthme en inhibant l'activité des sPLA2 et en les éliminant du milieu extracellulaire. Contrairement à son rôle anti-inflammatoire au niveau pulmonaire, il semblerait que PLA2R1 joue un rôle pro-inflammatoire dans le choc septique[START_REF] Hanasaki | Resistance to endotoxic shock in phospholipase A2 receptor-deficient mice[END_REF]. L'administration de lipopolysacharide (LPS, un composant de la membrane des bactéries Gram-négatives) chez des souris sauvages induit un choc septique qui provoque la mort des animaux en 24 heures. Chez ces souris, d'importantes quantités de TNFα (tumor necrosis factor α) sont produites dans les poumons, la rate, le rein et l'utérus, des tissus qui expriment la sPLA2 IB. L'expression de PLA2R1 est également augmentée dans les cellules épithéliales des alvéoles pulmonaires et dans les lymphocytes spléniques des souris sauvages exposées au LPS[START_REF] Yokota | Enhanced tissue expression and elevated circulating level of phospholipase A(2) receptor during murine endotoxic shock[END_REF].

Figure 2 .

 2 Figure 2. Rôle antitumoral de PLA2R1. Représentation schématique des mécanismes de sénescence. Différents stimulus induisent l'entrée en sénescence de cellules normales. Dans ces cellules, l'inactivation de PLA2R1 permet un échappement à la sénescence et la transformation maligne. Sa réexpression dans les cellules tumorales induit leur entrée en apoptose.

Figure 3 .

 3 Figure 3. Rôle de PLA2R1 dans la GEM idiopathique. A. Modèle de formation de dépôts immuns in situ conduisant à une protéinurie chez les patients atteints de GEM idiopathique. B. Analyse microscopique d'une biopsie de rein humain montrant la localisation de PLA2R1 (coloration marron) au niveau du glomérule. G : glomérule ; TCP : tubule collecteur principal ; TCD : tubule collecteur distal. C. Corrélation entre la sévérité de la maladie (maladie active : protéinurie > 3,5 g/g de créatine urinaire ; rémission partielle : protéinurie entre 0,5 et 3,5 g/g de créatine urinaire ; rémission totale : protéinurie < 0,5 g/g de créatine urinaire) et le taux circulant d'anticorps (IgG4) anti-PLA2R1 chez les patients atteints de GEM idiopathique.

Figure 1. Structure schématique et fonctions moléculaires possibles de PLA2R1. PLA2R1

	PLA2R1 membranaire			est composé d'une large région extracellulaire,
				d'un seul segment transmem-
	CysR		Adhésion ?	branaire et d'une courte queue
				cytoplasmique contenant un motif
				d'internalisation (NPXY). La région
			Inhibition	extracellulaire comprend : un
			des sPLA2	domaine riche en cystéine (CysR),
				un domaine fibronectine de type
			Fonction lectine ?	II (FNII) qui lie le collagène, et huit domaines lectines de type C
				(CTLD), dont les domaines CTLD4 à
			Autres rôles	-6 (en jaune) qui lient les sPLA2 et
			biologiques ?	certains sucres. PLA2R1 peut exis-
			Milieu	ter sous forme membranaire ou
			extracellulaire	soluble. En liant certaines sPLA2,
				PLA2R1 pourrait, via l'activation
				de diverses voies de signalisation,
	NPXY		Milieu intracellulaire	relayer la transduction de certains de leurs effets biologiques ou,
				inversement, inhiber leur action
	Internalisation et dégradation des ligands	Activation de voies de signalisation	Réponses biologiques : -migration -prolifération -invasion -sénescence/apoptose -lésion des podocytes	via leur internalisation et leur dégradation. En liant d'autres ligands, PLA2R1 pourrait aussi agir indépendamment des sPLA2. BSA :
				bovine serum albumin.

  In vitro, l'application de sPLA2 IB sur des cellules de parenchyme pulmonaire ou des cellules endothéliales vasculaires provoque la sécrétion d'icosanoïdes. Lorsqu'ils sont appliqués sur des macrophages exposés à une endotoxine, ces dérivés lipidiques stimulent la production de cytokines pro-inflammatoires, telles que le TNFα et l'IL-1β (interleukine-1β).

	Ces observations ont conduit Hanasaki et al. à étudier le
	rôle du complexe sPLA2 IB/PLA2R1 dans la pathogenèse
	du choc septique. Ils ont montré que, comme le LPS, la
	sPLA2 IB induit un choc septique lorsqu'elle est admi-
	nistrée à des souris préalablement sensibilisées avec
	des doses sub-léthales de LPS. En revanche, les souris
	Pla2r1 -/-sont plus résistantes que les souris sauvages
	au choc septique, induit aussi bien par le LPS que par
	la sPLA2 IB [17]. De plus, chez ces souris, les concen-
	trations plasmatiques de TNFα et d'IL-1β induites par
	le LPS sont sensiblement réduites. L'ensemble de ces
	résultats suggèrent que dans ce modèle inflammatoire,
	PLA2R1 pourrait cette fois relayer la transduction de
	certains effets des sPLA2 et participer, notamment via
	la production de cytokines pro-inflammatoires, à la
	progression du choc septique.
	Rôles de PLA2R1 dans le cancer
	L'équipe de David Bernard, en collaboration avec notre
	équipe, a été la première à suggérer un rôle antitumoral
	de PLA2R1, lors d'un criblage génétique fonctionnel
	destiné à identifier des gènes contrôlant la sénescence
	cellulaire [19]. La sénescence est un arrêt irréversible
	de la prolifération cellulaire. Elle constitue, au même
	titre que l'apoptose, un puissant
	mécanisme antitumoral [20,
	42, 43]. Plusieurs mécanismes
	provoquent l'entrée en sénes-
	cence : l'épuisement du pouvoir
	réplicatif des cellules (on parle
	alors de sénescence réplicative),
	et l'exposition à un stress onco-
	génique (on parle alors d'on-
	cogene-induced senescence ou
	OIS). Le rôle crucial de PLA2R1
	dans la sénescence réplicative
	a d'abord été mis en évidence
	in vitro, dans des cultures pri-
	maires de fibroblastes pulmo-
	naires humains (WI38) et dans
	des cellules épithéliales mam-
	maires (HMEC)

  souris, notamment dans le cas de la GEM, et la contribution des sPLA2 ou des autres ligands de PLA2R1 dans ces effets, restent à démontrer. Nos travaux futurs devront prendre en compte cet état des lieux. Par exemple, au niveau fonctionnel, la mise en place d'un modèle animal de la GEM pourrait s'avérer délicate chez la souris puisque PLA2R1 est fortement exprimé dans le rein humain au sein des podocytes, mais pas dans le rein de souris, de rat ou de lapin, ce qui suggère une fonction rénale de PLA2R1 spécifique à l'homme. L'étude du rôle antitumoral de PLA2R1 chez la souris pourrait aussi être rendue difficile par les différences de liaison et d'expression observées entre les récepteurs dans les deux espèces. Il s'agira aussi de savoir dans quel contexte les patients atteints de GEM génèrent des auto-anticorps contre PLA2R1, et s'il existe un lien entre la fonction antitumorale de PLA2R1 et son dysfonctionnement dans les podocytes, après liaison des auto-anticorps de patients. Au niveau moléculaire, les travaux à venir auront pour but d'identifier les ligands endogènes du récepteur PLA2R1, notamment chez l'homme, et aussi d'identifier les partenaires moléculaires directs de PLA2R1, chez l'homme et la souris, lesquels pourraient être différents. Enfin, les propriétés de liaison et les effets en apparence complexes de PLA2R1 sont à mettre en regard de ce que l'on sait, ou plutôt de ce que l'on ne sait pas, pour ses paralogues MRC1 (mannose receptor, C type 1), MRC2 et LY75 (lymphocyte antigen 75). Leurs fonctions et leurs ligands sont, là aussi, loin d'être compris, aussi bien chez la souris que l'homme. Nonobstant cette complexité apparente et la place des sPLA2, premiers ligands de PLA2R1, ce dernier apparaît maintenant sous un jour nouveau, comme un acteur clé de la GEM idiopathique, cible directe d'auto-anticorps, et peut-être comme un nouveau gène antitumoral, ouvrant de nouvelles perspectives diagnostiques et thérapeutiques en santé publique. C'est déjà le cas pour la GEM, puisque le dosage des auto-anticorps améliore significativement la prise en charge des patients, et permet de mieux identifier leur maladie et de leur prescrire un traitement plus efficace. ‡

	A
		Podocyte		PLA2R1
				Anticorps
				anti-PLA2R1
	Membrane
	basale
	glomérulaire
	Cellule
	endothéliale
				Chambre
				urinaire
	B	C
			10 000
	TCP	G	8 000
		G	6 000
	TCD		4 000
			2 000
			0
			Maladie	Rémission	Rémission
			active	partielle	totale

  Nous tenons à remercier la fondation ARC pour la recherche sur le cancer, l'association pour la recherche internationale sur le cancer (AICR), et le ministère des Affaires sociales et de la santé (PHRC PRAM-KT) pour leur soutien financier. Nous remercions aussi leDr. Damien Ambrosetti (service d'anatomopathologie du CHU de Nice) pour la figure d'immunomarquage de PLA2R1 dans le glomérule rénal humain et Nadège Parassol-Girard pour ses commentaires. Ce travail a bénéficié d'une aide de l'Agence nationale de la recherche au titre du programme Labex SIGNALIFE « investissements d'avenir » portant la référence ANR-11-LABX-0028-01.

m/s n° 5, vol.30, mai 2014

(➜) Voir la synthèse de H. Ait-Oufella et al., page ??? de ce numéro


Vincent Huin
email: vincent.huin@inserm.fr

Isabelle Strubi-Vuillaume

Kathy Dujardin

Marine Brion

Marie Delliaux

Delphine Dellacherie

Jean-Christophe Cuvellier

Jean-Marie Cuisset

Audrey Riquet

Caroline Moreau

Luc Defebvre

Bernard Sablonnière

David Devos

Expanding the phenotype of SCA19/22: Parkinsonism, cognitive impairment and epilepsy

Keywords: Spinocerebellar ataxia, Parkinson's disease, Epilepsy, Cognitive disorders, KCND3 mutation, Neurogenetics, Neuropsychiatric disorder

come

Introduction

Spinocerebellar ataxia types 19 and 22 (SCA19/22) are, rare inherited neurodegenerative disorders characterized by slowly progressing ataxia, frequent cognitive impairment, and signs of frontal lobe dysfunction [START_REF] Duarri | Mutations in potassium channel kcnd3 cause spinocerebellar ataxia type 19[END_REF][START_REF] Lee | Mutations in KCND3 cause spinocerebellar ataxia type 22[END_REF]. Other neurological features observed in some (but not all) cases include postural head tremor, myoclonus, pyramidal signs and neuropathy [START_REF] Schelhaas | Clinical, psychological, and genetic characteristics of spinocerebellar ataxia type 19 (SCA19)[END_REF]. SCA19 and SCA22 almost certainly correspond to the same disease, which is caused by loss-of-function mutations within the potassium voltage-gated channel subfamily D member 3 (KCND3) gene [START_REF] Schelhaas | SCA19 and SCA22: evidence for one locus with a worldwide distribution[END_REF]. The KCND3 gene encodes the voltage-gated potassium channel Kv4.3, which is known to be important for repolarization of the action potential in excitable cells [START_REF] Dixon | Role of the Kv4.3 K+ channel in ventricular muscle. A molecular correlate for the transient outward current[END_REF]. In SCA19/22, mutations alter the channel's location and/or function. In turn, this alters the excitability of Purkinje neurons and leads to their neurodegeneration [START_REF] Duarri | Spinocerebellar ataxia type 19/22 mutations alter heterocomplex Kv4.3 channel function and gating in a dominant manner[END_REF]. Here, we describe two novel SCA19/22 families and broaden the phenotypic spectrum to encompass mild Parkinsonism and epilepsy.

Methods

Clinical examination

Two unrelated French families displaying ataxia with an autosomal-dominant inheritance pattern were examined in the Department of Neurology at Lille University Medical Center (Fig. 1). All affected family members underwent a detailed clinical evaluation. During examination, Parkinsonism was defined with the classical association of rigidity and akinesia that can be associated with a rest tremor. The symptoms of four deceased family members were also recorded. Repeat expansions in the SCA1, 2, 3, 6, 7, 8, 12, 17 genes and conventional mutations in the SCA13, 14, 21, 28 genes were ruled out. All individuals gave their written informed consent to participation in the study.

Cognitive and behavioral measures

Twelve affected individuals underwent a neuropsychological evaluation. Given that some individuals had trouble reading and writing (due to learning disabilities and ataxia), an adapted testing procedure was mainly focused on non-verbal tests.

Overall cognition was assessed with the Montreal Cognitive Assessment (MoCA).

Raven's colored progressive matrices (PM47) were used to assess abstract reasoning and non-verbal intelligence. Attention and working memory were assessed with the forward and backward Corsi block-tapping test. We used simple and choice reaction time tests to assess vigilance, inhibition and divided attention. Visuospatial learning and memory performance were tested with the 10/36 spatial recall test.

Executive function was analyzed with the Tower of London test, with four difficulty levels: (i) "3N" items, where the goal can be achieved in a minimum of 3 moves, (ii) "5N" items, where the goal can be achieved with a minimum of 5 moves, and where initially no bead can be brought directly to its final destination, (iii) "5Iþ" items, where the goal can be achieved with a minimum of 5moves, and one bead can be immediately moved into its final destination, (iv) "5I-" items, where the goal can be achieved with a minimum of 5 moves, and one bead can be immediately moved into its final destination but will immediately prevent the solution of the problem.

Performance was assessed by the mean number of moves at each difficulty level.

Given the individuals' ataxia, the test was not timed. A 15-item version of the Benton judgment of line orientation (BJLO) test was used to assess visuospatial abilities.

Receptive language abilities were rated with the Token test. The Neuropsychiatric Inventory was administered to determine the presence and severity of behavioral disorders.

Molecular analysis

Genomic DNA was isolated from peripheral blood, according to standard procedures.

Targeted, next-generation sequencing of 24 genes involved in dominant hereditary ataxia was performed in the probands and in one healthy sibling (Data S1).

Variations were confirmed by Sanger sequencing. Repeat expansion in the SCA10 gene was ruled out using a fluorescent repeat-primed PCR assay [START_REF] Cagnoli | Detection of large pathogenic expansions in FRDA1, SCA10, and SCA12 genes using a simple fluorescent repeat-primed PCR assay[END_REF].

Results

Clinical features

The probands were referred to our clinic for gait disorders, poor balance and dysarthria. The family medical history was suggestive of an inherited disease with autosomal-dominant transmission. Table 1A summarizes the clinical characteristics of the 16 affected patients.

Eleven patients (68.8% of the carriers) developed a classic SCA19/22 phenotype, i.e. slowly progressing cerebellar ataxia with predominant gait impairment (Video 1 and 2). Poor balance was the first symptom to be identified. As the disease progressed, the patients developed gait disorders, dysarthria, nystagmus and then dysphagia. The estimated mean age at ataxia onset was 23.1 years (range: 2e66). Mild Parkinsonism was observed in eight patients (50% of the carriers) (III-6, III-9, IV-5, IV-6, IV-9, V-2 in family A and II-2, III-2 in family B).

The phenotype was dominated by epilepsy in five younger patients (31.3%). In family A, the patients IV-10, V-2, V-4 and V-6 suffer from epilepsy with a mean age at epilepsy onset of 5.3 years (range: 3e12), whereas ataxia was mild, delayed or not noticed. The patient IV-10 had a history of tonic-clonic epilepsy between the age of 3 and 6. The patient V-2 had complex partial seizures sometimes complicated by generalized tonic-clonic seizures. His brother (V-4) had complex partial seizures (temporal area) followed by generalized atonic seizures. He experienced one episode of tonic-clonic seizure. The patient V-6 had complex partial seizure corresponding to benign rolandic epilepsy and complicated by generalized atonic seizures. EEG in patients V-2 and V-4 revealed spike wave or polyspikewave discharges in the right frontotemporal area (V-2) or in the right frontal area (V-4). EEG in patient V-6 revealed paroxysmal rhythmic theta waves or biphasic theta waves in the right frontotemporal region. The fifth and last patient (IV-1 in family B) reported frequent myoclonus. The EEG of patients V-4 and V-6 in family A are depicted in supplementary Figure S2.

Brain MRI datasets were available for seven patients and showed isolated atrophy of the vermis in the five oldest individuals (III-9, IV-1, IV-6, IV-7 in family A and II-2 in family B).

Cognition, behavior and neuropsychiatric disorders

The results of the comprehensive neuropsychological test battery are shown in Table 1B. Most patients had experienced learning difficulties, and five (31.3%) had attended special schools. For affected adults, the time spent in formal education ranged from 8 to 14 years (median: 10 years). The estimated intelligence quotient was >80 for ten patients (83.3%) and between 60 and 70 for two patients (16.7%). Behavioral disorders were detected in four patients: slight aggressiveness (n = 2), mild agitation (n = 1), mild depression (n = 3), severe apathy (n = 1), and severe sleep disorders (n = 1). A history of major depressive disorder was reported by five patients, and obsessive-compulsive disorder was reported by one. Two patients were taking medication for anxiety.

Impairments in overall cognition

Genetic testing

Next-generation sequencing revealed the same KCND3 mutation (c.679_681delTTC, p.F227del) (NM_004980.4) in the two families. Sanger sequencing confirmed the segregation of this mutation in all affected individuals and one unaffected individuals (Supplemental Figure S1). The ATTCT pentanucleotide repeat expansion in the intron 9 of ATXN10 (SCA10) was in the normal range with all patients having between 10 and 18 ATTCT repeats.

Discussion

The F227del variation is a recurrent mutation that has been previously reported in different ethnic groups, e.g. in a Han Chinese family and a French family with SCA19/22 diagnosis [START_REF] Lee | Mutations in KCND3 cause spinocerebellar ataxia type 22[END_REF]. This mutation is an in-frame deletion of a highly conserved amino acid residue in KCND3 that reduces the protein length and is absent from controls in Exome Sequencinq Project, 1000 Genomes Project, Exome Aggregation Consortium and the Genome Aggregation Database. In vitro functional studies has been performed in two previous studies [START_REF] Lee | Mutations in KCND3 cause spinocerebellar ataxia type 22[END_REF][START_REF] Dixon | Role of the Kv4.3 K+ channel in ventricular muscle. A molecular correlate for the transient outward current[END_REF] and are supportive of a damaging effect with defect in the trafficking and cell surface expression of Kv4.3 and thus loss of the potassium channel current amplitude.

In the SCA19/22 patients assessed here, Parkinsonism did not have an impact on daily living and did not justify treatment with Ldopa, since this might have worsened balance even more. However, the presence of Parkinsonism might help the clinician to diagnose SCA19/22 because this sign is not frequent in most forms of SCA [START_REF] Rossi | Autosomal dominant cerebellar ataxias: a systematic review of clinical features[END_REF]. Parkinsonism has already been described in SCA2, 17, 21 and also in SCA3 or Machado-Joseph disease that is the most common SCA worldwide.

Parkinsonism in in SCA3, especially the akinetic-rigid syndrome is considered as a subphenotype of the SCA3 and may respond to treatment by levodopa or dopamine agonists. The existence of Parkinsonism features in SCA3 is not fully understood but could involve dysfunction of the midbrain dopaminergic system caused by neurodegeneration of the basal ganglia and especially of the substantia nigra [START_REF] Pedroso | The cerebellum in Parkinson's disease and Parkinsonism in cerebellar disorders[END_REF]. Interestingly, mild cogwheel rigidity was noted in two patients (aged 61 and 77) with an F227del mutation in the French family described by Lee et al. [START_REF] Lee | Mutations in KCND3 cause spinocerebellar ataxia type 22[END_REF]. Moreover, extrapyramidal symptoms are compatible with (i) the known expression pattern for Kv4.3, and (ii) the moderate neurodegeneration of the substantia nigra reported in a neuropathological examination of a SC19/22 patient who suffered from isolated ataxia and mild cognitive impairment [START_REF] Seidel | First patho-anatomical investigation of the brain of a SCA19 patient[END_REF]. Indeed, neurodegeneration in SCA19/22 is not restricted to the cerebellum alone but also involves distinct brainstem nuclei (such as the substantia nigra, the raphe nucleus and the inferior olivary nucleus).

Schelhaas et al. reported that most SCA19 patients had cognitive impairment

(including executive function) and mood disturbance, despite a MMSE score >24.

They also observed a low premorbid IQ (below 105) [START_REF] Schelhaas | Clinical, psychological, and genetic characteristics of spinocerebellar ataxia type 19 (SCA19)[END_REF][START_REF] Schelhaas | SCA19 and SCA22: evidence for one locus with a worldwide distribution[END_REF]. Most of the individuals assessed in the present study also had a rather low IQ. However, when assessed with the MoCA (which is more sensitive to impaired executive function), overall cognitive efficiency was impaired in most individuals. We also observed the frequent impairment of visuospatial function. The performance pattern for visuospatial memory revealed a frequent impairment in the long-term storage of information via episodic memory. Furthermore, mood disturbances and behavioral disorders were frequent in the two families studied here. Overall, this pattern of cognitive deficits and behavioral disorders is close to the cerebellar cognitive affective syndrome as introduced by Schahmann and Sherman [START_REF] Schmahmann | The cerebellar cognitive affective syndrome[END_REF]. This syndrome due to damage to the cerebellum is characterized by the association of deficits in executive function, visuospatial cognition and language with behavioral disorders, namely impulsivity, disinhibition, depression, anxiety and apathy. By consequence, it is possible that cognitive and behavioral symptoms in SCA19/22 patients are associated with cerebellar dysfunction. Although the cognitive impairment in the SCA19/22 patients is mild and does not frequently lead to mental retardation, it must not be neglected; it is nevertheless a frequent feature (in 75% of individuals) and has a major impact on quality of life. It is important to note that most of the individuals studied here did not complete their formal education, and three even had slight impairment in receptive language abilities. Given that the symptoms of SCA were combined with neurologic impairments and (in some cases) epilepsy, there is a clear requirement for extended psychosocial care in these patients.

Epilepsy has been described most frequently in SCA10 with variable frequencies among different families. Many of the SCA10 patients have recurrent generalized tonic-clonic seizures, but combinations of myoclonic, complex partial, and secondary generalized tonic-clonic seizures has been described [START_REF] Lin | Recent progress in spinocerebellar ataxia type-10 (SCA10)[END_REF]. As in SCA10, three SCA19/22 patients have generalized tonic-clonic seizures or myoclonia. However it seems that the epilepsy in SCA19/22 is not strictly similar to SCA10. To the best of our knowledge, there was no description in SCA10 of benign rolandic epilepsy or generalized atonic seizures. Moreover, in SCA10 patients the epilepsy seems generally to develop a few years after the ataxia. Finally, there is also a clear age effect on the form of the epilepsy, which could also interplay. Potassium channel defects are frequently associated with epilepsy [START_REF] Köhling | Potassium Channels in Epilepsy[END_REF], and two SCA19 patients from the original Dutch family had myoclonus [START_REF] Schelhaas | Clinical, psychological, and genetic characteristics of spinocerebellar ataxia type 19 (SCA19)[END_REF]. The benign epilepsy phenotypes and EEG of the two French families are similar to those previously reported in one patient with a de novo KCND3 mutation who had generalized epilepsy with paroxysmal rhythmic theta waves in frontal and parietal regions [START_REF] Smets | First de novo KCND3 mutation causes severe Kv4.3 channel dysfunction leading to early onset cerebellar ataxia, intellectual disability, oral apraxia and epilepsy[END_REF]. Surprisingly, the epilepsy and myoclonus phenotypes recorded to date show male predominance (seven men and only one woman).

  were detected in all but two of the twelve individuals having undergone neuropsychological testing. However, this impairment was severe (MoCA score <21) in only four individuals. Most individuals achieved normal levels of performance in the attention and working memory tests. Although levels of performance in the visuospatial learning and memory tests were normal for immediate recall, most subjects had impairments for delayed recall. Half of the individuals displayed impairments in visuospatial functions. Planning and inhibition abilities were relatively unaffected in most individuals. In two of the younger patients in family A (V-4 and V-6), cognitive impairment had been suspected during clinical examination, and both had learning delay. One patient (I-1 in family B) died before cognitive performance could be assessed. In summary, 12 of the 16 patients (75%) in whom cognitive function had been assessed were found to display mild cognitive impairment.

  These findings suggest that a diagnosis of SCA19/22 should be considered for patients presenting with Parkinsonism, epilepsy, and a family history of neurological disorders. KCND3 should be included in the list of candidate genes for epilepsy, Parkinsonism and cognitive impairment.

	B	COGNITION, BEHAVIOR AND NEUROPSYCHIATRIC DISORDERS Overall cognition Attention and working memory Visuospatial function Language Executive Functions Tower of London 577. Visuospatial learning and memory Cognitive impairment ++ ++ ++ ++ --+ + + + + n.a. -+ + + Education PS PS PS SS SS n.a. SS PS SS NA NA n.a. PS PS SS PS Time in formal education (years) 8 8 n.a. 10 14 n.a. 12 12 11 NA NA n.a. 8 10 11 10 Special education --+ --n.a. -+ + + NA n.a. ---+ Learning delay + + ++ --n.a. -+ + + + n.a. + + -++ IQ (PM47) <70 >80 80 60-70 >80 n.a. >80 >8 0 >8 0 n.a . n.a . n.a. >80 >80 >80 > 80 MoCA (/30) 10 20 17 18 28 n.a. 25 22 24 n.a . n.a . n.a. 28 22 24 23 Corsi forward span 4 4 4 5 4 n.a. 5 4 4 n.a . n.a . n.a. 5 5 6 6 Corsi backward span 3 4 3 4 5 n.a. 5 5 5 n.a . n.a . n.a. 4 5 6 5 Simple reaction time (ms) 749 364 378. 5 247 324. 5 n.a. 233. 5 31 5.5 27 4.5 n.a . n.a . n.a. 344. 5 303 234 285 Go No Go (ms) 657 506 429 594 531. 5 n.a. 355. 5 41 3 42 4 n.a . n.a . n.a. 563. 5 404. 5 356. 5 442. 5 Divided attention (ms) 824. 5 727 502 622 730 n.a. 464 49 7.5 48 6 n.a . n.a . n.a. 752 5 501 473. 10/36 immediate recall (/30) 13 14 11 12 13 n.a. 22 14 17 n.a . n.a . n.a. 9 20 16 19 10/36 delayed recall (/10) 4 4 4 4 4 n.a. 8 4 4 n.a . n.a . n.a. 4 7 5 6 BJLO (/15) 3 9 4 6 14 n.a. 13 15 10 n.a . n.a . n.a. 11 13 6 12 BJLO (/30) 14 22 15 19 29 n.a. 28 30 24 n.a . n.a . n.a. 26 28 19 27 Token test (/36) 23.5 29 33 30 34 n.a. 34 34 33 n.a . n.a . n.a. 32.5 34 33 33 3N items 3.33 3.67 U 3.33 3 n.a. 3 3 3 n.a . n.a . n.a. 10.3 3 3 3.33 3.67 5N items 11 8 U 6.33 8.67 n.a. 5 9.6 7 11. 67 n.a . n.a . n.a. 6.67 5.67 6.67 9.33 5I+ items 5 5.33 U 5 5 n.a. 7.67 8.6 7 5 n.a . n.a . n.a. 6.33 5 6.33 7.67 5I-items 12 8.67 U 9.67 7 n.a. 8.33 8.6 7 7.6 7 n.a . n.a . n.a. 7.67 9.33 9 6.33

Acknowledgement

We thank the patients, their families and the Association "Connaître les Syndromes Cérébelleux" for their participation in this study. We thank Sirine Hamitouche, Alexandre Genet and Kathy Dupont for technical assistance.

Funding

This work was funded by Lille University Medical Center.

Conflicts of interest

None.

Author contributions

Vincent Huin was involved in the analysis and interpretation of data (genetic testing) and drafting the manuscript. Isabelle Strubi-Vuillaume was involved in the analysis and interpretation of data (genetic testing). Kathy Dujardin was involved in the collection, analysis and interpretation of data (neuropsychological tests), and drafting and revision of the manuscript. Marine Brion was involved in data collection

HIGHLIGHTS

• SCA19 and SCA22 (caused by KCND3 gene mutations) are rare forms of inherited ataxia.

• We observed Parkinsonism is a high proportion of individuals with SCA19/22.

• Epilepsy seems to be a feature of SCA19/22.

• We characterized cognitive and behavioral disorders.

• KCND3 may be a candidate gene for epilepsy, Parkinsonism and cognitive disorders.

SUPPORTING INFORMATION Data S1: Next-generation sequencing

Gene panel design: A list of the 24 genes associated with dominant hereditary ataxia was selected and included in an SCA panel after analyzing the literature and the OMIM database. Genes with trinucleotide repeat expansions and those associated with SCA were not included in the panel. The gene panel comprised AFG3L2 (SCA28), ATP1A2 (episodic ataxia), ATP1A3 (episodic ataxia), CACNA1A

(episodic ataxia type 2), CACNB4 (episodic ataxia type 5), CCDC88C (SCA40), (episodic ataxia type 6), SL2A2 (episodic ataxia), SPTBN2 (SCA5), TGM6 (SCA35), TTBK2 (SCA11), and TMEM240 (SCA21). The SCA panel of 24 genes was designed using SureDesign software (v4.5, Agilent Technologies, Santa Clara, CA, USA).

Targeted sequencing: Exon capture of the 24 selected genes and libraries was prepared using the Haloplex kit (Agilent Technologies), according to the manufacturer's instructions. Pooled libraries (n=20) prepared using the SCA panel were sequenced on a MiSeq system (Illumina) with MiSeq Reageant kit v2.

Bioinformatics analysis: Sequence analysis was performed using SeqPilot (v4.


Member, IEEE Marc Jungers
email: marc.jungers@univ-lorraine.fr

Member, IEEE Francesco Ferrante
email: francesco.ferrante@gipsa-lab.fr

Jérôme Lohéac
email: jerome.loheac@univ-lorraine.fr

Dissipativeness and Dissipativation of discrete-time switched linear systems

published or not. The documents may come

Dissipativeness and dissipativation of discrete-time switched linear systems

I. INTRODUCTION

The notion of dissipativeness for dynamical systems has been formalized by Willems in [START_REF] Willems | Dissipative dynamical systems part I: General theory[END_REF] for continuous-time nonlinear systems. When considering a supply rate, a system is called dissipative if there exists a storage function associated with this supply rate. It can be interpreted as the fact that the system absorbs more energy from the external environ than it supplies. This concept extends Lyapunov theory, related to autonomous system, to the framework of open systems and has crucial implications for the interconnection of dynamical systems [START_REF] Willems | Dissipative dynamical systems part I: General theory[END_REF] and small gain theory. A large part of physical systems are dissipative as emphasized in [START_REF]Dissipative dynamical systems[END_REF]. That consolidates also the link between physics and control theory. Independently, Yakubovich [START_REF] Yakubovich | Solution of some matrix inequalities encountered in the automatic control theory[END_REF] and Kalman [START_REF] Kalman | Lyapunov functions for the problem of Lur'e in automatic control[END_REF] obtained what is now known as the Kalman-Yakubovich-Popov (KYP) Lemma for continuous-time time-invariant linear systems, (or positive real lemma). This lemma allows one to establish an equivalent relation between linear matrix inequalities (concerning to the state space representation of linear systems) and a condition in the frequency domain. It was originally obtained to solve the Lur'e problem [START_REF] Lur | On the theory of stability of control systems[END_REF][START_REF] Popov | Absolute stability of nonlinear systems of automatic control[END_REF]. See also [START_REF] Rantzer | On the Kalman-Yakubovich-Lemma[END_REF][START_REF] Brogliato | Dissipative Systems Analysis and Control[END_REF] for a detailed discussion.

The counterpart for discrete-time time-invariant systems has been provided in [START_REF] Hitz | Discrete positive-real functions and their application to system stability[END_REF] and fills the gap between linear matrix inequalities weighted by the matrices of the state space representation and the harmonic responses in the frequency domain.

Dissipativeness of specific classes of systems has been studied (see for instance the references in [START_REF] Brogliato | Dissipative Systems Analysis and Control[END_REF]), yet the notion of passivity encounters a great attention in the literature. Passivity is a particular type of dissipativeness for systems with equal number of inputs and outputs, when considering a supply rate defined as the scalar product between the input and the output.

For switched systems, some preliminary results have been published several years ago, mainly in [START_REF] Pogromsky | On stability and passivity of a class of hybrid systems[END_REF] and [START_REF] Žefran | A notion of passivity for hybrid systems[END_REF]. In the latter, it is shown that as for stability, the notion of passivity for switched system is not intuitive and underlines that switching among passive modes may destroy passivity.

The paper [START_REF] Zhao | A notion of passivity for switched systems with state-dependent switching[END_REF] presented a notion of passivity for switched systems with a multiple storage function approach. First the analysis of passivity is provided for a given switching law and secondly, under suitable conditions (involving localized form of KYP-Lemma and the S-procedure), a min-switching law strategy over the modal storage functions is proposed. In [START_REF]Dissipativity theory for switched systems[END_REF], dissipativity for continuous-time switched systems is investigated by considering multiple storage functions and multiple supply rates, including the concept of cross supply rate related to the link between active and inactive modes.

In [START_REF] Geromel | Passivity of switched linear systems: Analysis and control design[END_REF], on the one hand, the link between passivity analysis of continuous-time switched systems and linear matrix inequalities and on the other hand, the link between the design of switching law to ensure the passivity of the resulting switched system and Yakubovitch-Metzler inequalities are provided.

The framework of discrete-time switched system has been investigated in [START_REF] Bemporad | Passivity analysis and passification of discrete-time hybrid systems[END_REF]: the contribution deals with piecewise affine systems and provides first passivity analysis and also offers sufficient conditions in the form of linear matrix inequalities to design a switched state feedback to ensure closed-loop passivity.

It should be emphasized that in [START_REF] Wang | On passivity of a class of discrete-time switched nonlinear systems[END_REF], passivity of nonlinear discrete-time switched systems switching suitably among passive and not passive modes is studied, but this paper assumes that there exists a common storage function. This is a very restrictive assumption.

Passivity for hybrid systems and more generally for the interconnection of hybrid sub-systems is more delicate and is discussed for instance in [START_REF] Naldi | Passivity-based control for hybrid systems with applications to mechanical systems exhibiting impacts[END_REF][START_REF]Sufficient conditions for passivity and stability of interconnections of hybrid systems using sums of storage functions[END_REF]. First contributions have been obtained when considering only weak notions of passivity: flow-passivity and jump-passivity, that is, when only the flow or jump is passive.

The contribution of this paper is focused on two main topics. The first study deals with the dissipativeness analysis of a discrete-time switched linear systems when considering an arbitrary switching law. The second contribution provides sufficient conditions to design of a min-switching law related to modal storage functions for the dissipativation of discrete-time switched linear systems. In both cases, we provide sufficient conditions in terms of linear matrix inequalities (LMIs) or parameterized ones.

This paper is organized as follows. In Section II, discrete-time switched linear systems are presented and the notion of dissipativeness with time-dependent supply rates and time-dependent storage functions is defined. Several closely related concepts of dissipativeness are discussed. Finally standard results in the frequency domain associated with the Kalman-Yakubovich-Popov (KYP) Lemma are recalled for linear time invariant systems. Section III is focused on the analysis of dissipativeness of discrete-time linear switched systems with arbitrary switching laws. Our main contribution consists of sufficient conditions in the form of linear matrix inequalities (LMIs) solving in a constructive way time-dependent supply rate and storage function for any switching law. Section IV is dedicated to design a min-switching law, which is a state feedback guaranteeing that the resulting system when such a switching law is applied, is dissipative. Discussions about sufficient conditions to ensure stability and stabilization of discrete-time switched systems are also provided. A numerical illustration emphasizes the applicability of the main contribution before concluding remarks.

Notation: Notation is quite standard. R denotes the set of real numbers, Z the set of integer ones, and N the set of nonnegative integer numbers. Moreover N * = N/{0} and N N = {1, • • • , N }. The symbol stands for a symmetric bloc and • is used as a shortcut such that (•) M N = N M N , for suitable matrices N and M , where M is the transpose of the matrix M and M * its Hermitian. The set M ⊂ R N ×N of Metzler matrices is composed of matrices with nonnegative off-diagonal elements

M := Π ∈ R N ×N , π ji ≥ 0, ∀(i, j) ∈ N 2 N , i = j . We define the subclass M d of M such that M d := Π ∈ M, π ii ≥ 0, ∈N N π i = 1, ∀i ∈ N N .
The convex combinations of positive definite matrices P i with weights being elements of the Metzler matrix Π ∈ M d is denoted as (P ) p,i = ∈N N π i P .

II. DESCRIPTION OF THE SYSTEM AND DEFINITIONS

Let us consider a discrete-time switched linear system, with N ∈ N * modes satisfying ∀k ∈ Z:

x k+1 = A σ(k) x k + B σ(k) u k , (1)
z k = C σ(k) x k + D σ(k) u k , (2)
with x k ∈ R n the state of the system, u k ∈ R m the exogenous input and z k ∈ R p the output. The switching law σ : N → N N indicates at each time which mode is active.

Switched system (1)-( 2) is by nature a time-varying system. It is of interest to notice that Willems in his fundamental paper has already considered the framework dissipativeness of nonstationary systems [1, Section 6]. Especially he pointed out that the supply rate, and also the storage function, may be time-varying. Here we adapt to the discrete-time domain the definitions by Willems. Definition 1 (Willems Dissipativeness): A time-varying dynamical system (1)-( 2) with a locally summable supply rate w :

R m × R p × Z → R (3)
is said to be dissipative if there exists a nonnegative function

S : R n × Z → R + , called storage function, such that, for any (k 0 , k 1 , x k0 ) ∈ Z × Z × R n , k 1 ≥ k 0 , and any input u k defined over {k 0 ; • • • ; k 1 }, S(x k0 , k 0 ) + k1 k=k0 w(u k , z k , k) ≥ S(x k1 , k 1 ), (4)
or equivalently,

S(x k+1 , k + 1) -S(x k , k) ≤ w(u k , z k , k), (5)
where x k1 is the state at time k 1 of the solution to system (1)-( 2) when applying the control u k . It is called lossless when the equality occurs in (4).

Several particular cases of dissipativeness can be pointed out, depending on the choice of supply rate, which can be viewed as the input power for the system [START_REF] Willems | Dissipative dynamical systems part I: General theory[END_REF].

-When considering a quadratic supply rate,

w(u k , z k , k) = z k Q(k)z k +2z k L(k)u k +u k R(k)u k , (6)
with

Q(k) = Q(k) ∈ R p×p , R(k) = R(k) ∈ R m×m and L(k) ∈ R p×m , we say that the system is (Q(k), L(k), R(k))-dissipative. -When Q(k) = -I p , L(k) = 0 p×m and R(k) = γ 2 I m ,
with γ a scalar, the system has finite gain. In addition, when p = m, i.e. when there are as many inputs as outputs, we use the following concepts:

-

If w(u k , z k , k) = z k u k , (that is Q(k) = R(k) = 0 m and L(k) = 1 2 I m ) the system is said passive. -If w(u k , z k , k) = z k u k -νu k u k with ν > 0, the system is said input strictly passive. -If w(u k , z k , k) = z k u k -νz k z k with ν > 0, the system
is said output strictly passive. Definition 1 involves a storage function. There are other definitions of dissipativeness for system, among them, Hill and Moylan considered the following definition by using only a condition about the supply rate [START_REF] Hill | The stability of nonlinear dissipative systems[END_REF] for the trajectories starting from the origin, that can be also extended to the time-varying case.

Definition 2 ( Hill & Moylan's Dissipativeness): System (1)-( 2) with a locally summable supply rate w k , given by (3), is said to be dissipative if for any k 1 ≥ k 0 , all admissible input u k defined over {k 0 ; • • • ; k 1 }, any trajectory x k starting from x k0 = 0 satisfies:

k1 k=k0 w(u k , z k , k) ≥ 0. (7)
Under some weak assumptions, which are given below, we can establish a link with the dissipativeness in the sense of Willems and provide necessary and sufficient conditions summarized in Theorem 1 for quadratic supply rates. This is a time-varying discrete-time counterpart of [START_REF] Hill | The stability of nonlinear dissipative systems[END_REF]Theorem 1].

Assumption 1: The state space of the system (1) is reachable from the origin. That is to say that given any

(x f , k f ) ∈ R n × Z, there exist k i = k i (x f , k f ) ∈ Z, k i < k f and admissible control law u k = ũk (x f , k f ) and switching rule σ(k) = σk (x f , k f ), that are defined over k ∈ {k i ; • • • ; k f }, such that the corresponding state of (1) is steered from x ki = 0 to x k f = x f .
Assumption 2: For any y = 0, and k ∈ Z, there exists some u such that the supply rate (3) verifies w(u, y, k) < 0.

Remark 1: Conditions to ensure the satisfaction of Assumption 1 have been provided in [START_REF] Ge | Reachability and controllability of switched linear discrete-time systems[END_REF]. Here we assume those conditions hold. It should be noted that for discrete-time systems, as opposed to continuous-time ones, there may exist a bound related to the duration k f -k i . However this does not have any impact on the results. Assumption 1 concerns a structural property of the combination of the modes and is independent of the nature of the switching law (given, arbitrary or controlled). Assumptions 2 is only related to a suitable choice of matrices Q(k), L(k) and R(k) in the quadratic supply rate [START_REF] Popov | Absolute stability of nonlinear systems of automatic control[END_REF].

Theorem 1: System (1)-( 2) is dissipative, in the sense of Definition 2, with respect to the quadratic supply rate given by ( 6) if and only if there exist a function φ : R n × Z → R, which is quadratic with respect to its first argument, and real functions : R n × Z → R q and W : R n × Z → R q×m (for some integer q) satisfying φ(x, k) > 0, for x = 0 and φ(0, k) = 0 [START_REF] Brogliato | Dissipative Systems Analysis and Control[END_REF] and

∀(x k , u k , k) ∈ R n × R m × Z, solution of the system (1)-(2), (•) [•] -H(k) 0 0 H(k + 1) M σ(k) x k u k = (•) [•] R(k) L(k) Q(k) N σ(k) x k u k -(x k , k) + W (x k , k)u k 2 , ( 9
)
where for any i ∈ N N ,

M i = I n 0 n×m A i B i , N i = 0 m×n I m C i D i , (10)
and

H(k) = 1 2 ∂ 2 φ(x, k) ∂x 2
is half of the Hessian of the quadratic function φ at time k.

Remark 2: It should be mentioned that in the original result in the continuous-time domain [START_REF] Hill | The stability of nonlinear dissipative systems[END_REF]Theorem 1], the function φ is not assumed to be quadratic. This assumption is valid thanks to the result in [START_REF] Trentelman | Storage functions for dissipative linear systems are quadratic state functions[END_REF] stating that for a dissipative linear system associated with a quadratic supply rate, any storage function can be represented as a quadratic function of the state. The time-varying assumption does not affect this result.

Proof: The proof follows the lines of [START_REF] Hill | The stability of nonlinear dissipative systems[END_REF]Theorem 1]. Sufficiency: Suppose that conditions (8)-( 9) are verified, by rearranging the terms, we have

k1 k=k0 w(u k , z k , k) = φ(x k1+1 , k 1 + 1) -φ(x k0 , k 0 ) + k1 k=k0 (x k , k) + W (x k , k)u k 2 .
Taking x k0 = 0, we conclude that (7) holds. Necessity: Let introduce žk = z(x k0 , {u ; σ( )} k0≤ ≤k , k) denoting the value of the output z k at time k related to the trajectory starting at k 0 from x 0 and when the input control and switching law {u ; σ( )} k0≤ ≤k are applied. Assuming that condition (7) holds, we show first that the available storage function φ a (x k , k) defined by

φ a (x 0 , k 0 ) = sup K≥k0, {u k ;σ(k)} k 0 ≤k≤K - K k=k0 w(u k , žk ), (11)
is well defined and solves ( 8)-( 9). Independently of the sequence of input control and switching law {u ; σ( )} k0≤ and by using Assumption 1, we can determine a time k i < k 0 and admissible input and switching laws, defined over {k i , • • • , k 0 -1}, such that x ki = 0 can be steered to x 0 at k = k 0 . We denotes with some abuse of notation, ũk and zk the resulting input and output of this trajectory. Condition [START_REF] Rantzer | On the Kalman-Yakubovich-Lemma[END_REF] implies

- K k=k0 w(u k , z k , k) ≤ k0-1 k=ki w(ũ k , zk , k) < +∞,
where the righthand side is an implicit function of the initial state x 0 . The available storage function φ a defined by ( 11) is finite. In addition, thanks to Assumption 2, φ a is positive definite. By following the lines of [START_REF] Willems | Dissipative dynamical systems part I: General theory[END_REF], the available storage function is a solution to (4). To end the proof, we need to determine (x k , k) and W (x k , k) such that for φ = φ a , conditions (8)-( 9) are satisfied. Introducing

d(x k , u k , k) = φ a (x k+1 , k + 1) -φ a (x k , k) + w(u k , z k , k), we have d(x k , u k , k) ≥ 0 thanks to (5) applied on φ a .
Hence, φ a and w being quadratic functions, and x k+1 and z k being affine in the input u k , we can factorize

d(x k , u k , k) as d(x k , u k , k) = (x k , k) + W (x k , k)u k 2 , with (x k , k) ∈
R q and W (x k , k) ∈ R q×m for some integer q. That leads to conditions (8)-( 9). The framework of Theorem 1 is close to the one in [START_REF] Wang | On passivity of a class of discrete-time switched nonlinear systems[END_REF] related also to switched discrete-time systems but that assumes only a unique supply rate and storage function. It extends furthermore to time-dependency the continuous-time result provided in [19, Theorem 1].

In the next sections, sufficient conditions will be provided to derive Theorem 1 in a constructive way in distinct frameworks. To end this preliminary section, we recall well established results for the dissipativeness of a discrete-time linear time-invariant (LTI) system, namely

x k+1 = Ax k + Bw k , (12)
z k = Cx k + Dw k , (13)
with state x k ∈ R n , exogenous input w k ∈ R m , and output z k ∈ R p . The transfer function of this system is denoted by T (q) = C(qI n -A) -1 B + D, where q ∈ C. Considering system ( 12)-( 13) and quadratic (Q, L, R)-dissipativeness, without lack of generality, it is possible to look for quadratic storage functions. The dissipative inequality ( 5) is then equivalent to the LMI in a symmetric matrix P ∈ R n×n such that

[•] -P 0 0 P I n 0 n×m A B ≤ [•] R L Q 0 m×n I m C D . (14)
If ( 14) is verified, a storage function for system ( 12)-( 13) is given by S(x) = x P x, which is a solution to the time invariant dissipative inequality S(x k+1 ) -S(x k ) ≤ w(u k , z k ). When considering the particular case of passivity, LMI ( 14) reads (see for example [9, Lemma 3]):

A P A -P A P B -C /2 B P B -(D + D )/2 ≤ 0 n+m . ( 15
)
The existence of a matrix P solution to LMI ( 14), without assumption on its definiteness is related to a frequency domain criterion, called circle criterion and mentioned in the following lemma (see the discussion in [START_REF] Rantzer | On the Kalman-Yakubovich-Lemma[END_REF]).

The advantage of the frequency domain criterion in the KYP-Lemma lies in the fact that it can be numerically checked without the computation of the given matrix P , and as a consequence answer to the dissipative issue of a LTI system.

Lemma 1: Suppose that the pair (A, B), of the system ( 12)-( 13), is controllable. The two following conditions are equivalent:

1) For all ω ∈ R, iω not eigenvalues of matrix A,

0 m ≤ I m T (e iω ) * R L Q I m T (e iω ) ; (16)
2) There exists a symmetric matrix P ∈ R n×n such that LMI ( 14) is verified. Lemma 1 calls several comments and remarks. Remark 3: When considering strict inequalities ( 14) and ( 16), the equivalence between the two items is true even if the controllability assumption is relaxed.

Remark 4: In practice, in order to have a positive definite storage function, P in ( 14) will be assumed to be positive definite. That will be assumed in the following.

Remark 5: Feasibility of the LMI ( 14) in the variable P allows a time independent solution of relation [START_REF] Hitz | Discrete positive-real functions and their application to system stability[END_REF], in which the choice of (x k , k) and W (x k , k) is made to reach the equality.

Remark 6: The (Q, L, R)-dissipativeness with P = P > 0 n and C QC ≤ 0 n , guarantees stability of system [START_REF] Zhao | A notion of passivity for switched systems with state-dependent switching[END_REF]. This implication comes from the fact that the left-upper block in matrix inequality ( 14) reads A P A -P < C QC. This is obviously true when considering the particular case of passivity, because C QC = 0 m .

III. DISSIPATIVENESS OF DISCRETE-TIME SWITCHED SYSTEMS WITH ARBITRARY SWITCHING LAW

In this section, we consider the system (1)-( 2), with the assumption that the switching law σ(•) is arbitrary. Sufficient conditions to ensure that this system is dissipative irrespective to the choice of the switching law are provided in Theorem 2. The idea is to take advantage of the time-varying nature of the conditions in Theorem 1 to deal with the switching behaviour of the system (1)-( 2). Nevertheless, due to the restriction of the time-dependency via the switching law, we provide only sufficient conditions. The conservatism of this is still an open issue. The obtained sufficient conditions, based on LMIs, allow to build a switching storage function related to a switching supply rate. This is an extension of [START_REF] Wang | On passivity of a class of discrete-time switched nonlinear systems[END_REF].

Theorem 2: Consider the system (1)-( 2) and N triplets of matrices

(Q i , L i , R i ) ∈ R p×p × R p×m × R m×m with Q i = Q i and R i = R i . If there exist N symmetric positive definite matrices P 1 , • • • , P N such that ∀(i, j) ∈ N 2
N the following LMIs are verified

[•] -P i 0 0 P j M i < [•] R i L i Q i N i , (17)
with M i and N i defined by [START_REF] Pogromsky | On stability and passivity of a class of hybrid systems[END_REF], then the system (1)-( 2) is dissipative with respect to a time-varying, or switching, quadratic supply rate

w(u k , z k , k) = w σ(k) (u k , z k ), with w i (u k , z k ) = z k Q i z k + 2z k L i u k + u k R i u k , ∀i ∈ N N , (18)
and switching storage function

S(x k , k) = x k P σ(k) x k .
Proof: The proof is straightforward by considering (i, j) = (σ(k), σ(k + 1)) and multiplying the LMI (17) at left by ( x k u k ) and at right by its transpose. It results that S(x k , k) = x k P σ(k) x k is positive definite and

x k+1 P σ(k+1) x k+1 -x k P σ(k) x k ≤ w σ(k) (u k , z k ).
LMIs [START_REF] Naldi | Passivity-based control for hybrid systems with applications to mechanical systems exhibiting impacts[END_REF] with i = j recovers condition ( 14) for discrete-time LTI system. As a consequence, each mode i ∈ N N of system (1)-( 2) is a LTI system which is

(Q i , L i , R i )-dissipative with respect to the supply rate w i (u k , z k ) and modal (time independent) storage function S(x k , k) = x k P i x k .
Remark 7: When Theorem 2 applies, thanks to the positive definiteness of matrices

P i , i ∈ N N , if C i Q i C i ≤ 0 n , for all i ∈ N N , then the first diagonal block of LMI (17) reads A i P j A i -P i < 0 n , for any (i, j) ∈ N 2
N , which ensures the stability of the discrete-time switched autonomous system x k+1 = A σ(k) x k under any switching rule σ(•) [START_REF] Daafouz | Stability analysis and control synthesis for switched systems : A switched Lyapunov function approach[END_REF].

When considering passivity as a specific dissipativeness, Theorem 2 applies with LMIs (17) replaced by the LMIs

A i P j A i -P i A i P j B i -C i /2 B i P j B i -(D i + D i )/2 < 0 m+n .

IV. SWITCHING LAW DESIGN FOR DISSIPATIVATION OF DISCRETE-TIME SWITCHED SYSTEMS

This section is devoted to offer sufficient conditions allowing the dissipativation of a discrete-time linear switched system of the form (1)- [START_REF]Dissipative dynamical systems[END_REF], that is to say the existence of a switching rule making the system dissipative. Furthermore, under these sufficient conditions, it is possible to explicitly build and exhibit such a switching rule. The main result is gathered in the following theorem.

Theorem 3: Consider the system (1)-( 2) and N triplets of matrices

(Q i , L i , R i ) ∈ R p×p × R p×m × R m×m with Q i = Q i and R i = R i .
If there exist N symmetric positive definite matrices P i , i ∈ N N and a Metzler matrix Π ∈ M d such that for any i ∈ N N the following matrix inequalities are verified:

[•] -P i 0 0 (P ) p,i M i < [•] R i L i Q i N i , (19)
with M i and N i defined by [START_REF] Pogromsky | On stability and passivity of a class of hybrid systems[END_REF], then system (1)-( 2) is rendered dissipative by applying the min-switching rule related to the multiple storage functions

σ(k) = µ(x k ) ∈ arg min i∈N N x k P i x i . (20)
Proof: Assume that the condition ( 19) is verified. Let us multiply it at left by ( x k u k ) and at right by its transpose, that yields

(A i x k +B i u k ) (P ) p,i (A i x k +B i u k )- x k P i x k ≤ w i (u k , C i x k + D i u k ).
Let us introduce the minimum storage function as S min (x) = min i∈N N x P i x, ∀x ∈ R n . Thus, we have

S min (A i x k + B i u k ) = min j∈N N (•) P j (A i x k + B i u k ), ≤ (•) (P ) p,i (A i x k + B i u k ), ≤ x k P i x k + w i (u k , C i x k + D i u k ),
because the minimum of a set of positive scalars is always less or equal to any of their convex combinations. By selecting the min-switching strategy defined by [START_REF] Ge | Reachability and controllability of switched linear discrete-time systems[END_REF], it yields

S min (A µ(x k ) x k + B µ(x k ) u k ) -S min (x k ) ≤ w µ(x k ) (u k , C µ(x k ) x k + D µ(x k ) u k ). (21)
The positive definiteness of S min (•) is inherited by the positive definiteness of matrices P i , in finite number. The function S min (•) is thus the induced storage function related to the min-switching strategy [START_REF] Ge | Reachability and controllability of switched linear discrete-time systems[END_REF].

It should be noticed that the min-switching rule, when the matrices P i are given, is only a (non-linear) state-feedback and does not depend explicitly on the supply rates w i (u k , z k ) defined by [START_REF]Sufficient conditions for passivity and stability of interconnections of hybrid systems using sums of storage functions[END_REF]. This choice may be done irrespectively with the input u k .

Remark 8: With C i Q i C i ≤ 0 n , the left-upper block of inequality [START_REF] Hill | The stability of nonlinear dissipative systems[END_REF] leads to the Lyapunov-Metzler inequalities A i (P ) p,i A i -P i < 0 n , which is a sufficient condition to stabilize the autonomous system x k+1 = A σ(k) x k via the min-switching strategy [START_REF] Ge | Reachability and controllability of switched linear discrete-time systems[END_REF] (see [START_REF] Geromel | Stability and stabilization of discrete-time switched systems[END_REF] or [START_REF] Fiacchini | Necessary and sufficient condition for stabilizability of discrete-time linear switched systems: a set-theory approach[END_REF][START_REF] Fiacchini | On the stabilizability of discrete-time switched linear systems: novel conditions and comparisons[END_REF] for more details). Less conservative implications may be investigated in future research.

Remark 9: The result in Theorem 3 deals with switched systems having not necessarily

(Q i , L i , R i )-dissipative modes. Nevertheless the (Q i , L i , R i )-dissipativeness of the LTI system x k+1 = π 1/2 ii A i x k + π 1/2 ii B i u k , (22)
y k = C i x k + D i u k , (23)
is a required condition for the feasibility of inequalities [START_REF] Hill | The stability of nonlinear dissipative systems[END_REF], because it can be rewritten as

[•] -P i 0 0 π ii P i M i < [•] R i L i Q i N i - j∈N N ,j =i [•] 0 0 0 π ji P j M i . (24)
One further necessary condition for the feasibility of inequalities [START_REF] Hill | The stability of nonlinear dissipative systems[END_REF]

is that R i +L i D i +D i L i +D i Q i D i ≤ 0 m .
That can be viewed by deriving the lower right block of inequality [START_REF] Hill | The stability of nonlinear dissipative systems[END_REF] and noticing that B i (P ) p,i B i ≤ 0 n .

It is noteworthy that our framework using multiple storage functions leads to a storage function, which apparently does not depend on the switching rule, however a main difference with the approach using a common storage function, as in [START_REF] Wang | On passivity of a class of discrete-time switched nonlinear systems[END_REF] is that S min (x) may not be differentiable with respect to x ∈ R n .

V. NUMERICAL ILLUSTRATION This section provides a numerical example illustrating the contribution in Section IV. Let us consider the following example with n = 3, m = p = 1 and two modes (N = 2), resulting from the discretization, with an unitary sampling period, of exemple [14, Exemple 2].

A 1 B 1 C 1 D 1 =    0.

  

and the quadratic supply rates defined by (Q 1 , L 1 , R 1 ) = (Q 2 , L 2 , R 2 ) = (0, 1/2, 0). The two modes are stable, but are not passive. That can be checked numerically because the frequency domain inequality ( 16) is not satisfied, as shown in Figure 1, with frequency ω ∈ [-π, π] and q = e jω . Nevertheless, there is a solution to inequalities [START_REF] Hill | The stability of nonlinear dissipative systems[END_REF] with Let us consider as exogenous input u k = 4 sin(k). The min-switching strategy σ(k) defined by ( 20) is depicted on Figure 2. This figure emphasizes also that the dissipative inequality ( 21) is verified, as expected.

P 1 = 9

VI. CONCLUSION

The issue of dissipativeness for discrete-time linear switched systems has been investigated in this paper. Starting from the derivation of necessary and sufficient conditions in the time-dependent framework for dissipativeness, we obtain a concept of dissipativeness for a given switching law. Then, we assume that the supply rates are quadratic. That allows to establish our two main contributions. First, we provide sufficient conditions in terms of linear matrix inequalities ensuring that for any switching law, the discrete-time linear switched system is dissipative. The second contribution consists of providing parameterized linear matrix inequalities with parameters belonging to the entries of a Metzler matrix. That guarantees that there exists a min-switching law that renders the discrete-time linear switched system dissipative. Such a min-switching law is then build via multiple storage functions associated to multiple supply rates. These results have been discussed and illustrated in a numerical exemple.

Fig. 1 .

 1 Fig. 1. Frequency domain inequality (16) for mode 1 (blue dashed line), and mode 2 (red line).

Fig. 2 .

 2 Fig. 2. Top: min-switching law σ(k) = µ(x k ) defined by (20); Bottom: current value of the supply rate (circle in red); Smax(x k+1 ) -Smax(x k ) (star in black).

Research funded in part by ANR via project HANDY, number ANR-18-CE40-0010.