Nathalie Vergnolle

Christine Comera

Lionel Bueno

C Comkra

Annexin 1 is overexpressed and specifically secreted during experimentally induced colitis in rats

Keywords: colitis, annexin 1, lipocortin 1, secretion, inflammation trinitrobenzenesulfonic acid, PLA,, phospholipase A,, LDH, lactate dehydrogenase, MPO, myeloperoxidase. Enzymes. Phospholipase

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Annexin 1 is overexpressed and specifically secreted during experimentally induced colitis in rats

Glucocorticoids are the most effective anti-inflammatory drugs. They inhibit the expression of various proteins that enhance the production of mediators involved in inflammatory processes 11-51. In the late 1970s, it was also shown that glucocorticoids stimulate both synthesis and secretion of an anti-inflammatory protein, named lipocortin [6]. Later on, this protein was found to be identical to annexin 1 (or lipocortin l), which was purified and cloned in 1986 [7, 81, and belongs to a family of homologous proteins known as annexins [9].

Annexin 1, or some of its fragments, has biological properties identical to those of lipocortin [9-111. First, annexin 1 was shown to be synthethized or released in response to glucocorticoids although this is restricted to certain cells or tissues 19, 12-171. Secondly, it displays extracellular properties reproducing various glucocorticoid actions such as inhibition of the production of pro-inflammatory mediators both in vitro and in vivo [18][19][20][START_REF] Goppelt-Struebe | Enhanced expression of group I1 phospholipase A2 gene in the tissues of endoxin shock rats and its suppression by glucocorticoid[END_REF][START_REF] Cirino | Human recombinant lipocortin 1 has acute local autiinflammatory properties in the rat paw edema test[END_REF][START_REF] Errasfa | A purified lipocortin shares the anti-inflammatory effect of glucocorticosteroids in vivo in mice[END_REF][START_REF] Relton | Lipocortin-I is an endogenous inhibitor of ischemic damage in the rat brain[END_REF][START_REF] Becherucci | Conceivable difference in the anti-inflammatory mechanisms of lipocortin 1 and 5[END_REF][START_REF] Perretti | Modulation of IL-1-induced neutrophil migration by dexamethasone and lipocortin 1[END_REF]. In this regard, in vitro studies showed that annexin 1 inhibits cellular phospholipase A, (PLA,) and, consequently, the production of arachidonic acid, icosanoids, plateletactivating factor, and oxygen free radicals [18][19][20][START_REF] Goppelt-Struebe | Enhanced expression of group I1 phospholipase A2 gene in the tissues of endoxin shock rats and its suppression by glucocorticoid[END_REF]. In vivo studies reported that natural or recombinant annexin 1, and some of its fragments, reduces inflammatory processes, icosanoid production and neutrophil migration when administered either locally or systemically [9-11, 13, 22-26]. In addition, in some models, local or systemic administration of antibodies directed to annexin 1 blocks the anti-inflammatory action of glucocorticoids or annexin 1, suggesting that annexin 1 is a natural mediator involved in glucocorticoid actions 124-261.

However, to exert a physiological extracellular anti-inflammatory action, annexin 1 should be secreted. The occurrence of such a secretion is subject to controversy 191. First, it has been established that annexins are primarily intracellular proteins and are devoid of the signal peptide that is necessary in the conventional protein secretory pathway [9]. Secondly, annexin 1 and other annexins have been identified in extracellular locations, such as on cell surfaces, in cell culture media or in body fluids [8,[12][13][14][15][16][17][START_REF] Pfaffle | Biosynthesis, secretion and extracellular localization of anchorin CI1, a collagen-binding protein of the calpactin family[END_REF][START_REF] Smith | Detection of lipocortin 1 in human lung lavage fluid: lipocortin degradation as a possible proteolytic mechanism in the control of inflammatory mediators and inflammation[END_REF][START_REF] Christmas | Selective secretion of annexin 1, a protein without a signal sequence, by the human prostate gland[END_REF][30][START_REF] Serres | Annexin 1 regulation in human epidermal cells[END_REF]. However, their low extracellular concentration led to the hypothesis that they were non-specifically released from damaged or dead cells rather than actually secreted. Nevertheless, the huge amount of annexin 1 detected in seminal plasma demonstrated a real secretion [START_REF] Christmas | Selective secretion of annexin 1, a protein without a signal sequence, by the human prostate gland[END_REF] and suggested that at least part of the annexins found in extracellular locations were probably secreted by a non-conventional secretory mechanism, as recently described for several other cytosolic proteins 1321.

In addition, other annexins share similar properties with annexin 1. These properties include the inducibility of synthesis or secretion by glucocorticoids [9, 161 and anti-intlammatory properties in vitro or in vivo [19,[START_REF] Comera | Identification and characterization of phospholipase A, inhibitory proteins in human mononuclear cells[END_REF]341. An overexpression of annexin 1 or other annexins has been reported in some pathological processes 19, 10, [START_REF] Elderfield | Lipocortins (annexins) 1, 2, 4 mid 5 are increased in the central nervous system in multiple sclerosis[END_REF][START_REF] Bastian | Localization of annexins in normal and diseased human skin[END_REF][START_REF] Mullens | Up-regulation of lipocortin-1 and its mRNA in reactive astrocytcs in kainate-lesioned rat cerebcllum[END_REF]. Annexin 1 was found to be downregulated in non-perforating Crohn's disease, whereas it was present at control levels in perforating Crohn's disease [START_REF] Sakanoue | Lipocortin-present perforating and lipocortin-absent nonperforating Crohn's disease[END_REF]. Nevertheless, neither the chronology of the appearance of annexin nor the occurrence of a real secretion with respect to inflammation has been well defined.

The aim of this work was therefore to determine whether an experimental colitis in the rat induces local expression and secretion of annexins and, if so, to investigate this phenomenon in relation to the time course of the inflammatory response.

MATERIALS AND METHODS

Animal preparation and colitis induction.

Colitis induc- tion on the proximal colon was adapted from the model of rectocolitis previously described by Moms et al. [START_REF] Morris | Hapten-induced model of chronic inflammation and ulceration in the rat colon[END_REF]. 103 male Wistar rats weighing 225-300 g were used. After an overnight fast, rats were anesthetized by intraperitoneal injection of 0.5 m g k g acepromazine and 100 mg/kg ketamine (Imalgene 1000, RhGne-Merieux). The early effects of trinitrobenzenesulfonic acid (N,ph-SO,H)/ethanol were studied in 31 conscious animals. They were fitted with a polyethylene catheter (0.9 mm in external diameter) inserted into the proximal colon 1 cm from the cecocolonic junction and exteriorized at the back of the neck. Five days later, the animals were fasted overnight and received 0.3 ml N,ph-SO,H (30 mg) in 50% ethanol in the proximal colon, through the catheter. The animals were sacrificed 4, 8, 12, 16, 20 or 24 h after the treatment; four rats were used each time except for the time points 12 h and 16 h where eight and three rats were treated, respectively. The remaining rats were used for the study of later efkcts of N,ph-SO,H/ethanol. They were fasted overnight, anesthetized and received the same N,ph-SO,H/ethanol treatment by direct injection into the proximal colonic lumen. They were sacrificed 1, 3, 6, 9, 12, 15, 18 or 21 days later (eight rats for each time-point). The control animals fitted with a catheter received 0.3 ml saline (0.9% NaCI) and were sacrificed 24 h later while the other control animals were sacrificed h days later the 0.3 ml injection of saline. 1 day after the N,ph-SO ,H/ethanol treatment, no difference was observed between the two modes of administration for all the parameters considered.

Preparation of tissue samples. Animals were sacrificed by cervical dislocation and proximal colons were rapidly collected and rinsed with saline. Colons were cut into equivalent pieces to study either myeloperoxidase (MPO) activity or annexin expression and secretion. MPO activity and annexin expression were determined in all the tissues obtained. Secretion experiments were studied for eight proximal colons from control rats and from animals sacrified at 12 h and on days 1-21 after the N,ph-SO,H/ethanol treatment.

Study of annexin expression. Freshly isolated proximal colons were homogenized using a Polytron for 30 s at 0°C and sonicated for 30 s at 0°C in a lysis buffer, containing 25 mM Tris/HCI, pH 7.4, 5 mM EDTA, 0.25% (by vol.) Nonidet P-40, 1 pg/ml pepstatin, 15 pg/ml benzamidine, 1 pg/ml antipain, 1 pglml leupeptin, 1000 IU/ml aprotinin and 4 mM phenylmethylsulfonyl fluoride. Homogenates were equilibrated in an electrophoresis sample buffer [60 mM Tris/HCI, pH 6.8, 10% (by vol.) glycerol, 5 % (by vol.) 2-mercaptoethanol, 2 % (madvol.) sodium dodecyl sulfate (SDS), 0.01 % (mass/vol.) bromophenol blue], boiled for 5 min, and stored at -80°C. Protein concentrations were measured according to Bradford [START_REF] Bradford | A rapid and sensitive method for the quautification of microgram quantities of protein utilizing the principe of protein-dye binding[END_REF].

Study of annexin secretion. Annexin secretion was studied by washing colonic fragments followed by incubation in cell culture medium. Tissue washes were performed on ice, to block secretory processes, and in the presence of 2 mM EGTA, to remove potential extracellular annexins that could be bound to the cell surface by calcium-dependent interaction. Proximal colonic tissues were cut into small pieces and washed four times for 20min in NaCIlP, (150mM NaCI, 2.5mM KC1, 1 0 m M Na,HPO,, 1.5 mM KHZPO,, pH 7.4) three times in the presence of 2 mM EGTA and one time without EGTA. Thereafter, tissue incubation was performed following two stages, in a culture medium [RPMI 1640 (Eurobio) plus 1 mM glutamine, 0.02 mM EGTA, 20 mM Hepes, pH 7.41. First, colons were incubated for 30 min at 37°C to allow protein secretion. Secondly, samples were adjusted to 2 mM final EGTA, rapidly cooled to O"C, and further incubated for a 15-min period on ice, allowing extracellular annexins to be released into the media. After either washes or incubations, tissues were separated from supernatants, hvmogenized in the lysis buffer, and subsequently equilibrated in the electrophoresis sample buffer as previously described. Supernatants were centrifuged at 13000Xg for 20 s, to eliminate cells or tissue fragments. Supernatant proteins were precipitated by a 10-min incubation on ice with 10% (mass/vol.) trichloroacetic acid (final concentration), centrifuged at 13 OOOXg for 10 min, suspended in the electrophoresis sample buffer, boiled for 5 min, and stored at -80°C. Aliquots of homogenates in the lysis buffer and fresh supernatants were kept at 4°C to measure lactacte dehydrogenase (LDH) activity.

Study of the annexin 1 extracellular form. Two secretion experiments were modified to establish if annexin 1 in the culture medium was in a soluble form. After the 20-s centrifugation at 13000Xg, to eliminate cells or tissue fragments, culture supernatants were supplemented with protease inhibitors (1 pg/ ml pepstatin, 15 pg/ml benzamidinc, 1 pg/ml antipain, 1 yglml leupeptin, 1000 IU/nil aprotinin, and 4 mM phenylmethylsulfonyl fluoride ; final concentrations). Samples were centrifuged for 2 h at 1OOOOOXg. Pellets, containing the vesicular protein fraction, were resuspended in the electrophoresis sample buffer; supernatant soluble proteins were precipitated as described above and resuspended in the electrophoresis sample buffer.

Detection of annexins by SDS/PAGE and immunoblotting. Proteins from homogenate and supernatant samples equili- brated in the sample buffer were separated by SDSlPAGE in 10% polyacrylamide gels according to Laemmli (411, and electrophoretically transferred onto nitrocellulose membranes as described by Towbin et al. [START_REF] Towbin | Electrophoretic transfer of proteins from polyacrylamide gels to nitrocellulose sheets : procedure and some applications[END_REF] in 25 mM Tris/HCI, 192 mM glycine, 0.01 % (by vol.) SDS, 15% methanol. Immunoblottings were performed using different monospecific rabbit polyclonal antibodies. Antibodies were specific to either annexins 1, 3 or 6 [used at 1: 1000, 1 :I00 and 1 :I000 dilutions, respectively, a gift from F. Russo-Marie, Institut National de la SantC et de la Recherche MCdicale (INSERM) U332, Paris, France], to annexin 1 or 2 from (both used at 1 : 1000, provided by L. A. Pradel, Centre National de la Recherche Scientifique UA 11 12, Paris, France), to annexin 1 or 5 (both used at 1 :500, provided by H. Chap, lNSERM U326, Toulouse, France), to annexin 4 (used at 1 :250, from C. E. Creutz, Charlottesville, USA) and to annexin 2 (used at 1 : 1000, a gift from J. C. Cavadore, INSERM U249, Montpellier, France). All the antibodies used were tested for their monospecificity by immunoblotting with pure annexins 1-6 (gifts from F. Russo-Marie) and different cell and tissue homogenates. lmniunoblottings for annexin 1 detection were performed with polyclonal antibodies affinity purified on a human recombinant annexin-I -Sepharose 4B column kindly provided by F. Russo-Marie and J. F. Browning for the recombinant annexin 1 (Biogen Corp.). Incubations were performed in buffer A (25 mM Tris/HCI, pH7.4, 1 5 0 m M NaCI, 0.1% Tween-20) for 1 h with 5 % dried milk, to saturate non-specific binding sites, followed by 1 h in the presence of anti-annexin antibodies.

Nitrocellulose sheets were then washed five times for 5 min in buffer A, incubated for 30 min in the presence of anti-rabbit antibodies made by goat and linked to horseradish-peroxidase (1 : 10000 dilution, Sigma) and washed three times for 5 min in buffer A. Immunolabeled bands were revealed by tluorography, using the ECL reagents (Enhanced Chemiluminescence, Amersham) followed by exposure from 1 min to 1 h to Hyperfilm MP (Amersham). For densitometric quantifications, films were preflashed before exposure for 1 ms with a Sensitize pre-flash unit (Amersham).

Densitometric quantification of annexin 1 secretion. The extracellular and intracellular contents of annexin 1 from secretion studies were assessed by SDSlPAGE and immunoblotting using different dilutions of both homogenates and supernatants and different concentrations of pure human annexin 1 (from F. Russo-Marie). To quantify annexin 1 contents, densitometric scans of the fluorographs were performed with a Shimadzu CS930 chromoscan (Kyoto, Japan) with a linear correlation between densitometric values and the amounts of annexin 1 for 20-100 ng. The percentage of extracellular annexin 1 was calculated by dividing the amount detected in supernatants (38 kDa and 33 kDa) by the sum of the amount detected in supernatants and homogenates multiplied by 100.

Measurement of cellular lysis. Cell viability was determined by spectrophotometric measurement of LDH activities as described by Bergmeyer and Bernt [START_REF] Bergmeyer | Lactate dehydrogenase: UVassay with pyruvate and NADH, in Methods of eniymaric andysis[END_REF], in both fresh supernatants and tissue homogenates from the secretion experiments. The percentage of cellular lysis is equivalent to the percentage of LDH released into supernatants and was calculated by dividing the activity detected in supernatants by the sum of the activities detected in supernatants and in tissue hornogenates multiplied by 100.

Measurement of MPO activity. Proximal colonic tissues

were homogenized twice for 30 s and sonicated three times for 30 s at 0°C in 0.5 % hexadecyltrimethylammonium bromide in 50 mM sodium phosphate, pH 6.0, according to Bradley et al. [START_REF] Bradley | Measurement of cutaneous inflammation: estimation of neutrophil content with an enzyme marker[END_REF]. The homogenates were clarified by centrifugation at 13 OOOXg for 15 min, at 4OC, and were assessed for MPO activity in 3 ml 50 mM sodium phosphate, pH 6.0, containing 16 mM aqueous guaiacol and 5.9 mM H,O, as previously described [START_REF] Smith | Relation of peroxidase activity in gut mucosa to inflammation[END_REF]. The increase in absorbance was measured for 2 min at 470 nm using a Uvikon 860 spectrophotometer (Kontron Instruments). Protein concentrations were determined using the Bio-Rad DC protein assay. MPO activities were expressed in U/mg protein with 1 U hydrolyzing 1 pmol H,O,/min.

Statistics. Statistics were established using a one-way analysis of variance followed by the Student's t test. Differences were considered significant for P values S0.05.

RESULTS

Kinetics of tissue damage and MPO activities.

During the first 24 h following N,ph-SO,H/ethanol treatment, mucosa had a continuous yellow to dark orange colour from 4 h after treatment, and a hyperemia that increased up to 24 h. From 1 to 6 days after N,ph-SO,H/ethanol administration, proximal colons showed a strong inflammation with at least 80% of mucosal necrosis over 1-5 cm, a dark red to brown colour, and apparent hyperemia and ulcers. On day 9, necrotic tissues were dissociated from the bowel wall by slabs and recovered in distal colonic contents. At days 12-21 after N,ph-SO,H/ethanol treatment, no more necrotic tissues were present. The wall thickness increased from day 3 to, at least, day 12, followed by a gradual decrease. However, the colonic diameter remained larger than in the control rats until day 21 after treatment. No bowel perforation was found at any time. MPO activity of the tissues increased from days 1-9 after N,ph-SO,H/ethanol administration, with a factor i n the range 15-26 compared to control values (Fig. 1). From days 12-21, this activity decreased rapidly. All values were significantly different from control values (P < 0.05) except at day 21. Maximal MPO activities corresponded to the cellular stage of inflammation with the presence of numerous leukocytes. treatment with maximal levels at 1-9 days post-treatment (Fig. 2A andB). From 12 h until day 6, colonic annexin 1 was always expressed in two forms: the major one had an apparent molecular mass of 38 kDa and corresponded to the intact protein, the minor one migrated slightly faster at 33 kDa. The two forms of annexin 1 were also detected occasionally at 8 h posttreatment (one animal out of four) and at day 9 (four animals out of eight). All the other colonic homogenates expressed only the 38-kDa form of annexin 1. These two forms were confirmed to be annexin 1 because they were both recognized by monospecific anti-annexin 1 polyclonal antibodies from different sources. Moreover, the 38-kDa protein was recognized by a monoclonal antibody directed to the N-terminal end of human annexin 1 (Zymed; data not shown). Annexin 3 expression was also increased during N,ph-SO,H/ethanol-induced inflammation, especially at days 3-9 post-treatment (Fig. 2B). In contrast, colonic expression of annexins 2 and 6 was similar to that of the control rats throughout the time period after N,ph-SO,H/ethanol administration (Fig. 2B). Annexins 4 and 5 were not detected in any colonic homogenate.

Annexin expression. Annexin 1 expression in the proximal

Annexin secretion. Supernatants from incubated tissues obtained from rats at 12 h to 6 days after N,ph-SO,H/ethanol treatment contained annexin 1 (Fig. 3A, not shown for 12 h). At day 9, four of the eight tissues tested released this protein. Annexin 1 was not found in supernatants of colonic tissues from controls or in treated rats from days 12-21, even after extensive film exposure. All the tissues that expressed the two forms of annexin 1 in homogenates from freshly isolated colons, released the protein in these two forms when incubated in the RPMI medium (Figs 2 and3). Annexins 2, 3 and 6, which are expressed in proximal colons, were not found in supernatants (Fig. 3B). When tissues that expressed the two forms of annexin 1 were washed (three times for 20 min in NaCUP, plus 2 mM EGTA), or washed the same way and incubated in RPMI medium (for 30 min at 37°C followed by 15 rnin at 0°C in the presence of 2 mM EGTA), the whole 33-kDa protein and part of the 38-kDa annexin 1 were recovered in supernatants. The corresponding colonic homogenates contained only the 38-kDa protein (Fig. 4). Extraction from the tissues of both annexin 1 forms required the presence of 2 mM EGTA (data not shown).

To determine if the release of annexin 1 is due to a physiological secretion, the amounts of annexin 1 present in both homogenates and supernatants were quantified by immunoblotting and densitometric analysis for four samples from either control Thereafter, tissues were incubated in RPMI medium for 30 min at 37°C followed by 15 min at 0°C after adding 2 mM final EGTA. The presence of annexin 1 (An I), 2 (An 2), 3 (An 3), and 6 (An 6) is detected by immunoblotting in homogenates (H) and supernatants (S) after RPMI incubation. Equivalent amounts of homogenate proteins were used for each sample. Supernatant volumes loaded represent five times those of the corresponding homogenates. Annexin 1 is the only one of the annexins detected in the colon that is released into the medium. Its release occurs selectively at days 1, 3, 6, and 9 after N,ph-S03H/ethanol treatment. H1, homogenate of freshly isolated colon; H2 and 52, homogenate and pooled supernatants, respectively of colon washes (four washes of 20 min in NaCI/P,: three times with 2 mM final EGTA and subsequently once without EGTA); H3 and S3, homogenate and supernatant, respectively, of colon incubation in RPMI (tissue was washed as previously described, subsequently incubated in RPMI medium for 30 min at 37°C followed by 15 min at 0°C and in the presence of 2 mM final EGTA). Equivalent amounts of proteins were loaded for all homogenates; S2 and S3 volumes loaded represent two and five times more than H2 and H3 volumes, respectively. Migration positions of molecular mass standards are noted on the left (kDa). The whole 33-kDa form and part of the 38-kDa form of annexin 1, which are expressed in the freshly isolated tissue (HI), can be removed in supernatants S2 and S3 in the presence of 2 mM EGTA.

animals or rats after 1, 3, 6 or 9 days of N,ph-SO,H/ethanol colonic administration (Table 1). The results are expressed as amounts of intracellular annexin 1 (present in homogenates) and as percentages of extracellular annexin 1. The latter were compared to the percentage of cell lysis. Whereas all tissues displayed identical cell lysis, which remained inferior to 5.5% at all the times tested, the percentage of extracellular annexin 1 increased up to 24.4% on day 3 (Table 1). Moreover, there was no correlation between the percentage of cell lysis and the percentage of extracellular annexin 1 ( r = 0.109, II = 8). The percentage of extracellular annexin 1 released by rat proximal colons varied with time after N,ph-SO,H/ethanol administration. It increased from days 1-3 (P<O.Ol), and subsequently decreased on days 6 and 9. On days 1-6, N,ph-S0,Wethanol treatment increased annexin 1 intracellular expression up to 25 -36 times compared to initial values. On day 6, it reached a mean value as high as 7.5% of the total protein, which is much more than the previously estimated cellular contents of this protein [8, 461.

The extracellular form of annexin 1. After centrifugation of culture media for 2 h at lOOOOOXg, the pellets contained vesicular proteins, whereas the soluble proteins remained in the supernatants. As detected (Fig. 5) and quantified by immunoblotting, 30% of the total secreted annexin 1 was found in the pellets, in the 38-kDa form. Supernatants contained the two forms of soluble annexin 1 (33 kDa and 38 kDa), which constituted 70% of the total secreted protein.

DISCUSSION

As a main conclusion, our study shows that annexins 1 and 3, but not annexins 2, 4, 5, and 6, are overexpressed during N,ph-SO,H/ethanol-induced rat colitis. Moreover, annexin 1 appears to be selectively secreted during the cellular stage of the colitis. In this model, proximal colons exhibit significantly high levels of MPO activity until day 9, and these high levels corre-Table 1. Time course comparison of the amounts of intracellular annexin 1, extracellular annexin 1 and cellular lysis during secretion experiments. The amounts of intracellular annexin 1 and the percent- ages of extracellular annexin 1 were determined by densitometric quantification after immunoblotting in supernatants and colonic homogenates obtained from secretion experiments. Annexin 1 amounts were calculated as ng/pg total proteins in homogenates. The percentage of extracellular annexin 1 was calculated from the amounts of extracellular (present in supernatants) and intracellular (present in the corresponding homogenates) annexin 1. The percentage of cellular lysis was estimated by LDH activity measurement in homogenates and supernatants. Data are expressed as the meani-SD (n = 4). This experimental colitis increases annexin 1 intracellular expression and release, with no correlation between the percentage of extracellular annexin 1 and the percentage of cellular lysis. spond to the presence of necrotic mucosa. Subsequently, MPO activities decreased rapidly, from day 12 until day 21 after N,ph-SO,H treatment. The activities were significantly different from control values except at day 21. This rapid recovery of normal MPO values after N,ph-SO,H/ethanol administration differs from the findings of several reports [39, 471 although one study reached an identical conclusion [START_REF] Allgayer | Treatment with 16-16'-dimethyl prostaglandin E2 before and after induction of colitis with trinitrohenzenesulfonic acid in rats decreases inflammation[END_REF].

Other investigators previously reported a local increase of the expression of either annexin 1 alone, or various annexins, in pathological conditions [9,10,[START_REF] Elderfield | Lipocortins (annexins) 1, 2, 4 mid 5 are increased in the central nervous system in multiple sclerosis[END_REF][START_REF] Bastian | Localization of annexins in normal and diseased human skin[END_REF][START_REF] Mullens | Up-regulation of lipocortin-1 and its mRNA in reactive astrocytcs in kainate-lesioned rat cerebcllum[END_REF]. For instance, annexin 1 overexpression has been detected in resident keratinocytes from thick skins, or in reactive astrocytes that migrate in kainatelesioned rat cerebellum [36, 371. In addition, annexins 1 and 3 were reported to be abundant in inflammatory leukocytes such as macrophages (annexin 1 ) [13] and neutrophils (annexins 1 and 3) 149, SO]. In our model, the maximal expression of both annexins 1 and 3 occurs during the cellular inflammatory stage, corresponding to maximal MPO activities. This suggests that such overexpressions may be attributed, at least in part, to local leukocyte invasion of inflamed colons, but the cells overexpressing annexin 1 in this colitis remain to be identified. This experimental colitis induces intracellular expression of both annexins 1 and 3, but the concept that annexins can have an intracellular anti-inflammatory activity is a subject of controversy [9]. Among the various hypotheses put forward for the intracellular role of annexins, an action in secretory processes and in signal transduction, including PLA, activation, has been proposed 19, 50, 511. However, the pathophysiological consequence of annexin intracellular overexpression during inflammation is still unknown.

In contrast, the extracellular anti-intlanimatory properties of annexins are well documented [9-111. Our results show that annexin 1 is specifically released into the medium of the incubated colon during the cellular stage of colitis. Two forms of annexin 1 (38 kDa and 33 kDa) were detected in inflamed colonic tissues. Only part of the 38-kDa annexin 1 was extracted from colons, and extensive washes with longer incubation times or higher EGTA concentrations did not improve its release (data not shown). This suggests that part of the 38-kDa protein is extracellular because it can be removed from tissues by chelating calcium ions. The non-extractible part is presumed to remain intracellular. The 33-kDa fragment of annexin 1 was completely removable from tissues by washes in the presence of 2 mM EGTA, indicating an exclusively extracellular location. In agreement with our results, it was shown that annexin 1 has a molecular mass of 38 kDa inside the cells and is partially proteolyzed at its N-terminal end, in the extracellular medium [12, 17, 28, 291. Such a degradation was suggested to downregulate its anti-inflammatory activity 1281. Monoclonal antibodies directed against the N-terminal part of human annexin 1 recognized the 38-kDa rat annexin 1 but not the 33-kDa form, whereas all the polyclonal antibodies tested recognized both forms. This observation strongly suggests that the 33-kDa annexin 1 is a fragment lacking the N-terminus. Moreover, our results support the finding that proteolysis occurs in the extracellular milieu, on the soluble 38-kDa annexin 1. Indeed, after a high-speed centrifugation of culture medium, the 33-kDa form is detected in supernatants, but not in pellets.

Our study supports the view that annexin 1 is also secreted in vivo. The colonic expression of both 33-kDa and 38-kDa annexin 1 in freshly isolated homogenates was restricted to tissues that release annexin 1 in secretion experiments. The fact that extracellular 33-kDa annexin 1 was completely removed from tissues upon washing with ECTA prior to incubation in RPMI, leads to the conclusion that annexin 1 was further secreted into the medium during this in vitro incubation. Moreover, several observations suggest that annexin 1 release in RPMI medium is a physiological secretion by living cells: (a) this release is specific for annexin 1 since we never detected annexins 2, 3 and 6 in the medium; (b) the percentage of cell lysis is identical whether annexin 1 is released or not; (c) the large amounts of annexin 1 found in supernatants, especially on day 3 after N,ph-SO,H treatment, exclude a release by dead or damaged cells.

Several cytoplasmic proteins, devoid of a secretory peptide signal, have been reported to be released and different al teriiative secretory pathways have been proposed. One hypothesis is the post-translational/translocation of proteins across the membranes of several organelles. For example, cytosolic interleukin-1,8 (IL-lj?) is supposed to translocate within recycling endo-somes, and thus the exocytotic fusion of vesicles with the plasma membrane should deliver IL-lP to the milieu in a soluble form (521. Since the secreted annexin 1 in seminal fluid is totally soluble, it could be released by such a pathway. However, in our model, 30% of extracellular annexin 1 is detected in pellets after a high-speed centrifugation, despite the presence of EGTA. This suggests that the protein is either externally bound to vesicles in a Ca2 +-independent manner, or is trapped within them. Indeed, two other secretory pathways, where vesicles containing annexin 1 are released, can be suggested. First, annexin 1 has been localized in late endosomes and has been supposed to mediate inward vesiculation in multivesicular bodies [START_REF] Futter | Annexin 1 is phosphorylated in the multivesicular body during the processing of the epidermal growth factor receptor[END_REF]. In the case of a fusion between multivesicular bodies and the plasma membrane, vesicles containing annexin 1 could be released into the extracellular milieu. Secondly, in another pathway, now referred to as ectocytosis 1541, evagination of the chondrocyte plasma membrane appears to be responsible for the formation of bone matrix vesicles [START_REF] Hale | The mechanism of matrix vesicle formation, 1[END_REF] that are enriched in certain cytoplasmic proteins, including annexin 1 [START_REF] Genge | Identification of phospholipid dependent calcium binding proteins as constituents of matrix vesicles[END_REF]. Such mechanisms can be proposed in our model, but, in order to explain the presence of soluble extracellular annexin 1, vesiculation of the plasma membrane would be followed by a rapid lysis of vesicles.

Using this model, we found that annexin 1 secretion is restricted to the cellular stage of the inflammation, from 12 h to 9 days after N,ph-S03H administration. This supports the argument that annexin 1 secretion is an induced event related to cell activation in the inflamed gut. Other studies have suggested that annexin 1 secretion and its extracellular action do not occur during all the inflammatory period and models. For instance, peritoneal inflammatory macrophages isolated from thioglycolatetreated animals do not secrete annexin 1, even when incubated in the presence of glucocorticoids [13]. Moreover, local antiinflammatory action of annexin 1 was not observed in all the tested models, but it can depend on the major inflammatory mediators involved at the time of its administration 122, 261. For instance, its action in rat paw edema depends on the inflammatory agent used and is not observed during the early stage (1 h after carrageenan administration) that does not involve prostaglandin production but histamine release [START_REF] Cirino | Human recombinant lipocortin 1 has acute local autiinflammatory properties in the rat paw edema test[END_REF].

Taken together, our data indicate that annexin 1 secretion occurs in vivo and in vitro in inflamed proximal colon. Moreover, we show that the major part o f the extracellular annexin l (70%) is in a soluble form. In view of the known anti-inflammatory extracellular properties of this protein, it could be suggested that it is involved in the control of the physiopathology of experimental colitis. The local action of annexin 1 could modulate either the production of icosanoids and other inflammatory mediators, or leucocyte migration. Indeed, both events are induced during the time o f its in situ secretion 147, 481, but whether these are annexin-1-induced effects remains to be demonstrated. Nathalie Vergnolle was the recipient of a grant from the Miniut2re de I'Enseignement Sup*i,e'rieur et de !a Recherche Franpise. The authors thanks M. J. Fargeas, 0. Morteau, H. Chap, and J. Le Bars for their skilful scientific assistance and are very grateful to the fnstitlrt Nutionui de lu Recherche Agrononiique for its financial support.

0 1 3 6 9 12 15 18 21 Fig. 1 .

 1211 Fig. 1. Effect of N,ph-S0,Wethanol treatment on MPO activity in proximal colon tissues: a time course study. MPO activity in colonic homogenates was determined by spectrophotometric measurement and expressed in U/mg protein with 1 U hydrolyzing 1 pmol H202/min. The time labeled 0 corresponds to control rats. Data are expressed as means 2 SD with n = 8 for each time. Compared to controls, values are significantly increased (* for P <0.001 and for P S 0.005) at any time after N,ph-SO,H/ethanol treatment except on day 21,

Fig. 2 .Fig. 3 .

 23 Fig. 2. Immunoblotting analysis of the annexin expression in homogenates of freshly isolated proximal colons at different times after N,ph-S0,Wethanol administration. (A) Early effect of N,ph-SO,H/ethanol on the annexin 1 expression. Colons were recovered at 0 (control rats), 4, 8, 12, 16, 20 and 24 h after treatment. (B) Late effects of N,ph-S0,Wethanol on the expression of annexins 1 (An I), 2 (An 2), 3 (An 3), and 6 (An 6) expression. Colons were isolated at 0 (control rats), 1, 3, 6, 9, 12, 15, 18 and 21 days after treatment. Equivalent amounts of proteins were loaded for each sample. The expressions of annexins 1 and 3 are increased by N,ph-SO,H/ethanol treatment with a maximum from days 1-9 after administration. The expressions of annexins 2 and 6 remain similar at any time.

Fig. 4 .

 4 Fig. 4. Immunoblotting analysis of annexin 1 extraction from one proximal colon isolated 1 day after N,ph-S0,Wethanol administration.H1, homogenate of freshly isolated colon; H2 and 52, homogenate and pooled supernatants, respectively of colon washes (four washes of 20 min in NaCI/P,: three times with 2 mM final EGTA and subsequently once without EGTA); H3 and S3, homogenate and supernatant, respectively, of colon incubation in RPMI (tissue was washed as previously described, subsequently incubated in RPMI medium for 30 min at 37°C followed by 15 min at 0°C and in the presence of 2 mM final EGTA). Equivalent amounts of proteins were loaded for all homogenates; S2 and S3 volumes loaded represent two and five times more than H2 and H3 volumes, respectively. Migration positions of molecular mass standards are noted on the left (kDa). The whole 33-kDa form and part of the 38-kDa form of annexin 1, which are expressed in the freshly isolated tissue (HI), can be removed in supernatants S2 and S3 in the presence of 2 mM EGTA.

*Fig. 5 .

 5 Fig. 5. Immunoblotting analysis of the annexin 1 extracellular form in one proximal colon isolated 1 day after N,ph-S0,Wethanol administration.H and ST, homogenate and total supematant, respectively, of colon incubation in RPMI as previously described. SS, supernatant of colon incubation in RPMI after a 2-h centrifugation at 100000Xg; SP, pellet of colon incubation in RPMI after a 2-h centrifugation at 100000Xg. The volume loaded for each supernatant represents five times the volume of the homogenate. The whole 33-kDa form and the major part of the 38-kDa form of annexin 1 were detected in soluble forms in the supernatant of the colon incubation in RPMI after a highspeed centrifugation; a minor part of the 38-kDa forin was found in the pellet.


V T Renard

O A Tkachenko

V A Tkachenko

T Ota

N Kumada

J-C Portal

Y Hirayama

Boundary-mediated electron-electron interactions in quantum point contacts

An unusual increase of the conductance with temperature is observed in clean quantum point contacts for conductances larger than 2e 2 /h. At the same time a positive magnetoresistance arises at high temperatures. A model accounting for electron-electron interactions mediated by boundaries (scattering on Friedel oscillations) qualitatively describes the observation. It is supported by numerical simulation at zero magnetic field.

Quantum point contacts (QPC) are usually formed when two wide 2D conducting regions are connected by a small constriction. They exhibit a number of intriguing phenomena among which conductance quantization [1,2] is the most emblematic. Nowadays QPCs are very common tools for condensed matter physicists. Examples of recent applications include nuclear spin manipulation, solid state electron optics or precise electron counting [3]. Recently, the puzzling low conductance "0.7 anomaly" [4] has attracted most of the attention since it is related to electron-electron (e-e) interactions. But apart "0.7 anomaly" it is commonly believed that the physics of QPCs is well understood using a single-particle picture (See Ref. 5 and references therein). On the contrary, it is well known that the properties of two-dimensional electron gases (2DEGs) dramatically depend on interactions [START_REF] Altshuler | Electron-electron interaction in disordered systems[END_REF][START_REF] Rudin | [END_REF]8,9,10]. The related corrections to the conductance and tunnelling density of states have their origin in Friedel oscillations (FO) of electron density around impurities. Friedel oscillations are also known to appear at boundaries [11,12,13] and could therefore affect the properties of nano-scale devices. Interestingly, boundary mediated FOs were recently shown to be possibly involved in "0.7 anomaly" physics [14] but this subject is still highly debated [15]. In this context we show that e-e interactions can have a significant influence on transport properties of clean quantum point contacts even in the large conductance regime (G >> 2e 2 /h). We used a combination of relatively low electron densities, high mobility and low series resistance to clearly uncover the effects of e-e interactions. It increases interactions due to reduced screening and ensures that impurity scattering can be disregarded. Starting from the second conductance step, the average slope of the conductance versus gate voltage linearly increases with increasing temperature T . At the same time, the low field magneto-resistance (MR) is non-monotonic at high T and strongly temperature dependent. Some of our findings are present, although not as evident, in previous works [START_REF] Hansen | Proceedings of NANO-7/ECOSS-21[END_REF][START_REF] Pyshkin | [END_REF][START_REF] Cronenwett | [END_REF]19]. They are consistent with a model of e-e interaction mediated by boundaries which is supported by numerical simulation at B = 0 T.

The quantum point contacts were defined employing a split-gates lateral depletion technique [20] on high mobility GaAs quantum wells [START_REF]mobility µ1 ≈ 4 × 10 6 cm 2 /Vs and µ2 ≈ 3 × 10 6 cm 2 /Vs for a density ns1 = 1.5 × 10 11 and ns2 = 1 ×[END_REF]. Needle and square shaped samples of various sizes were used to check the influence of the geometry (see Table I). All samples a have back gate to tune the electron density and some samples have a 0.2 µm-wide center gate to obtain better defined conductance quantization steps when grounded [START_REF] Lee | [END_REF]. Fourterminal resistance measurements were carried out between 350 mK and 10 K with a standard low-frequency technique at small excitation voltage < 20 µV to avoid heating effects. Any obvious temperature dependance (activated parallel conduction, anomalous temperature dependance of the series 2DEG or leakage of the gates) was excluded [23]. More than ten samples showed a qualitatively similar behavior. Figure 1a shows the conductance G of the sample Square1 as the split-gate voltage V sp is varied for different temperatures. The small (∼ 15Ω) 2DEG series resistance was not subtracted. At low T the conductance is quantized to exact integer values of 2 × e 2 /h. Note that the "0.7 anomaly" is observed. Increasing temperature not only shrinks the plateaus but also increases the overall slope of G(V sp ) resulting in the increase of conductance with T . Such a change in the slope of the conductance is not expected from a simple energy averaging, which produces fix temperature independent points in the conductance located at integers of e 2 /h [24]. Regions with alternatively positive or negative temperature dependence should be separated by these fixed points. This is not observed in our experiment. It is however possible to restore this behavior by a phenomenological approach. For each temperature a new effective split-gate voltage can be calculated V * sp = V p + α(T ) × (V sp -V p ) to superpose the curve to the low temperature one (See Fig. 1b). Here, V p is the pinch off voltage and α is the scaling parameter. The strong linear temperature dependence of α points at quantum effects, possibly e-e interactions.

In the Hartee-Fock approximation scattering on the potential created by all the other electrons can be considered as the origin of e-e interactions effects [START_REF] Altshuler | Electron-electron interaction in disordered systems[END_REF][START_REF] Rudin | [END_REF]8,9,12,13]. This potential is connected to the formation of Friedel oscillations of the electron density close to scatterers (defects or boundaries) due to interferences between incoming and reflected electron waves. Spatial variations of their density results in a varying effective potential seen by conducting electrons. In our case the QPC is clean and FOs are created at the boundaries of the constriction. They affect its conductance in two different ways. At low T when the thermal length l T exceeds W the QPC width, FOs located in the 2DEG can reduce the conductance at a plateau via scattering of emitted electrons back to the QPC [11]. As T is increased these oscillations are damped and conductance plateaus reach ideal integer value of 2 × e 2 /h [25]. When l T ≈ W there remain only FOs located inside the QPC. This corresponds to our experimental conditions (l T ≈ 1 µm at T = 300 mK). As T is further increased the transverse oscillations inside the QPC are damped thus effectively widening the constriction (or equivalently shifting 1D energy levels). One has to compensate by applying a lower V sp to obtain a comparable value of conductance (the curves shift to the left). The scaling can be qualitatively understood as follows. Electrostatically defined QPCs have a parabolic cross section potential U = m * ω 2 x 2 /2, m * is the effective mass. The width of the channel can be defined as W = (2/ω) 2E F /m * , where E F is the Fermi level in 2DEG reservoirs. One-dimensional energy levels are written E n = (n + 1/2)hω and in a constant capacitance model W is proportional to

V sp -V p . It follows that ∆E n /E n ∝ ∆ω/ω ∝ ∆W/W ∝ ∆(V sp -V p )/(V sp -V p ).
In analogy to the 2D case where the conductivity σ is renormalized as ∆σ/σ ∼ β ×T /E F in the ballistic regime of interaction [8] we expect that ∆E n /E n ∼ β × T /E F when E n ∼ E F (the interaction-induced shift ∆E n of the closest to Fermi level subband only is considered). Here, β is the interaction parameter. This leads to the rescaling

V sp → V * sp = V p + α(T )(V sp -V p ) with α(T ) ∝ T as measured.
This result is consistant with a numerical non-selfconsistent Hartree-Fock simulation. The principle of the computation is first to calculate the local electron density n(x, y, T, µ) solving the one-particle Shrödinger equation in a "soft wall" electrostatic potential for different chemical potential µ and temperature T . Indeed, instead of calculating the conductance as a function of V sp for a varying potential U 0 (x, y, V sp ) and fixed µ, we chose to fix the electrostatic potential and vary the chemical potential. This greatly simplifies the calculation because the wave-functions are calculated only once. It is valid for small interval of split gate voltage (between two conductance steps). Numerical technique and parameters are described in Ref. 26, 27. In analogy to the 2D case, the interaction correction to the potential is computed as follows δU (x, y, T, µ) = βδn(x, y, T, T * , µ)/D, where β is the interaction parameter, D is the 2D density of states, δn(x, y, T, µ) = n(x, y, T, µ) -n(x, y, T * , µ) and T * a temperature high enough to suppress FOs. The obtained correction which includes only T -dependent terms is then used to solve the transmission problem in the total potential which is now a function of T and µ [START_REF]The computation region was 3 µm long and 0.7 µm wide. The constriction is 250 nm wide in the model. The calculation step along x and y was 5 nm, the energy step was[END_REF]. Contrary to 2D systems containing impurities [8,9] it is not possible to relate the interaction parameter β to the interaction constant F 0 in a simple way. Close to the boundaries the electron density drops to zero. This drop is not completely abrupt and the first Friedel oscillation develops in rarefied weakly-screening electron gas where e-e interactions become particularly strong. This is taken into account in our simulation by the large value of the parameter β = -6. Figures 2a andb show the obtained correction to the electrons density δn(x, y) in the vicinity of the constriction at G ≈ 6 × e 2 /h for T =3.5 K (a) and T =350 mK (b). This clearly illustrates the formation of FOs as T is decreased. Note that pronounced features are demonstrated despite the use of a "soft wall" model. Figures 2c andd compare the experimental conductance of the sample Needle1 to the conductance calculated by the method described above. In both cases the fix temperature independent points are missing and the conductance is on average increasing with temperature. A test of the model consists in neglecting the interaction term of the scattering potential (i.e. setting the interaction parameter β to 0) which restores the dependence for energy averaging as expected (Inset to Fig. 2d). An overall good qualitative agreement is obtained between computer and real experiments which further confirms the qualitative understanding.

In general, the magnetoresistance reveals important informations about scattering, coherent processes and e-e interactions [9]. Figure 3a shows that beside the unusual temperature dependence at B = 0 T the samples present an unexpected field dependence. The data displayed were measured for sample Needle1 at G ≈ 6 × 2e 2 /h (see arrow in Fig. 2c). At high temperature, the magnetoresistance presents a maximum around B = 10 mT and becomes negative at higher fields. At low T the resistance decreases with B at all magnetic fields. The high field slope of the MR is T -dependent. Note that the maximum moves to lower fields in QPCs with larger width (not shown).

A linear negative MR with a slope inversely proportional to the width of the channel is known to appear in QPCs [28]. The increased slope at low temperature that we observe is consistent with the observation at B = 0 T which lead to the conclusion that e-e interactions could in principle reduce the effective width of the channel at low T . As for the positive MR, it cannot be attributed to diffuse boundary scattering which has very different characteristics [29]. In particular it is T -independent and absent in short constrictions. Similarly, the interplay of boundary scattering and electron collisions [30] can be discarded. Finally, commensurability of electron trajectories with the voltage probes can be ruled out since very different Hall bar geometries were tested (Table .

I).

The absence of positive MR at low T confirms that the observed quantum effect is due to e-e interaction and is not an interference effect (i.e. Weak anti-localization) which should increase at low T . Similarly to 2D systems [31], it can most likely be explained by a variation of the parameter β at small magnetic fields which is only visible at high T . Note that simulations with B-independent β did not produce the positive MR. Figure 3b andc demonstrates that the magneto-resistance depends on the electron density and detailed geometry of the point contacts. Carefully adjusting the back-and split-gate voltages, the conductance G(T = 4K, B = 0T) of the sample Needle2 was tuned to the same value (≈ 13 × 2e 2 /h) for two different electron densities. Although qualitatively similar, the effect is found to be more pronounced at the lower density. This is consistent with the Friedel oscillation picture which should have in principle smaller effect at high densities when the system resembles a non-interacting Fermi liquid. Figure 3c compares the results of the sample Needle2 to the effect obtained in the sample Square2 at the same density and conductance. The temperature dependence at B = 0 T is comparable but the field dependence is much more pronounced for the sample Nee-dle2 which has smoother entrances. The geometry dependence therefore appears to be a very interesting tool to study the effect of magnetic field on the measured effect. The positive MR raises interesting questions about the influence of low magnetic field on Friedel oscillations and requires additional theoretical work.

We believe that the presented deviations have not been observed in regular point contacts (for example in Ref. [24]) due to the lower mobility and higher density in these experiments. Indeed, Friedel oscillations are known to depend exponentially on disorder and interaction. However, recent measurements on similar samples show similar phenomenology (see Ref. [START_REF] Hansen | Proceedings of NANO-7/ECOSS-21[END_REF][START_REF] Pyshkin | [END_REF][START_REF] Cronenwett | [END_REF] for the conductance quantization and Ref. 19 for the MR) indicating that our observation is a general effect. It could have particular importance in nano-scale electronics since boundaries dominate transport properties in nano-devices (electrostatically defined quantum dots, rings, Y-junctions etc.).

We are very grateful to I. Gornyi, A. Dmitriev, K. Takashina, Y. Tokura, K. Pyshkin for helpful discussions and K. Muraki for providing some samples. O.A.T. acknowledges UJF for the invitation as "Ma ître de Conférences", the Supercomputing Siberian center and CNRS/IDRIS (project 61778). Part of this work was supported by CNRS/RAS agreement between ISP Novosibirsk and GHMFL Grenoble.

FIG. 1 :

 1 FIG. 1: a) Conductance of the sample Square1 as a function of Vsp (From top to bottom T =4 K, 3 K, 2 K, 1 K and 350 mK). The inset shows SEM images illustrating possible shapes of the point contacts. b) Conductance as a function of the scaled split-gate voltage V * sp for the same temperatures. The inset shows the scaling parameter α.

FIG. 2 :

 2 FIG.2: Calculated correction to the electron density due to Friedel oscillations at G ≈ 6 × 2e 2 /h for T=3.5 K (a) and T=350 mK (b). The scale is m -2 . This clearly illustrate the rise of the interaction potential as T decreases. c) Conductance of the sample Needle1 for different temperature. The vertical arrow corresponds to the voltage at which Fig.3awas recorded. d) Calculated conductance at a similar value of conductance. The inset shows the calculated conductance neglecting e-e interactions.

FIG. 3

 3 FIG. 3: a) Resistance of the sample Needle1 at the gate voltage indicated by an arrow in Fig. 2c. b) Resistance of the sample Needle2 for the electron density ns = 1 × 10 11 cm -2 (dash) and ns = 1.5 × 10 11 cm -2 (solid) for G(T = 4K) ≈ 13×2e 2 /h. c) Resistance of Needle2 (solid) and Square2 (dot) for ns = 1.5 × 10 11 cm -2 and G(T = 4K) ≈ 13 × 2e 2 /h.

TABLE I :

 I Summary of the samples (for the wafer and Hall bar see[START_REF]mobility µ1 ≈ 4 × 10 6 cm 2 /Vs and µ2 ≈ 3 × 10 6 cm 2 /Vs for a density ns1 = 1.5 × 10 11 and ns2 = 1 ×[END_REF]).

	Sample Wafer Hall bar	point contact	center gate
	Square1	1	1	W=0.6 µm; L=0.4 µm	yes
	Square2	1	1	W=0.6 µm; L=1 µm	yes
	Needle1	2	2	W=0.8 µm	no
	Needle2	2	2	W=0.6 µm	no


Long Li

( Fluminance

Time-statistics of wind-driven circulation 1 Governing Equations Location Uncertainty Principles Barotropic Quasi-Geostrophic Model under LU

homogeneous / heterogeneous

d t ✓ + u ? • r✓dt + dB t • r✓ 1 2 r • (ar✓)dt = 0 D t ✓ 4 =
Transport of a random tracer :

( r • = 0 ) d dt Z D 1 2 ✓ 2 = 0 ( r • u ? = 0 ) ( Mémin 2014; Resseguier et al. 2017 ) Long Li (Fluminance INRIA) Stochastic Quasi-Geostrophic Flows 4 / 18
Barotropic Quasi-Geostrophic Model under LU ( Resseguier et al. 2017;Li et al. 2019 ) Evolution of the potential vorticity with source processes :

D t q = S 1 (ru)dt + S 2 (ru)dB t
Evolution of the stream function :

q = /L 2 R + f
Strong incompressible constraints :

u = r ? , dB t = r ? 'dB t , r • u ? = 0
Conservation of total energy :

d dt Z D 1 2 h kr k 2 + ( /L 2 R ) 2 i = 0
Transport of a prognostic passive tracer :

D t q 0 = 0
Conservation of tracer energy :

d dt Z D 1 2 (q 0 ) 2 = 0 h Sketch : Z D dB t • r(q f ) = Z D S 2 (ru) dB t i Long Li (Fluminance INRIA) Stochastic Quasi-Geostrophic Flows 5 / 18

  dX t = u(X t , t)dt + (X t , t)dB t


Covariance operator :

Q(x, y, t, s) = E σ(x, t)dB t σ(y, s)dB s T = δ(t -s)dt Ω σ(x, z, t) σT (y, z, s)dz
Variance tensor (per unit of time) :

a(x, t) = Q(x, t) dt = σσ T (x, t)
Turbulent Kinetic Energy (TKE) :

TKE = 1 2 tr(a) dt (m 2 •s -2 )
Long Li (Fluminance) A Consistent Stochastic Modeling September 12, 2019 5 / 28

Stochastic Reynolds Transport Theorem (SRTT)

We assume that ∇•σ = 0 in the following.

Rate of change of a scalar process θ within a volume transported by the stochastic flow :

d V(t) θ(x, t)dx = V(t) (D t θ + θ ∇• w )dx
Stochastic transport operator :

D t θ = d t θ + (w- 1 2 ∇• a w ) •∇ θdt + σdB t •∇ θ multiplicative noise -∇•( a 2

Conservation Laws

We assume that ∇•w = 0 in the following.

Conservation of extensive scalar :

D t θ = 0 Conservation of tracer energy : [Resseguier, Memin & Chapron, 2017a] d Ω 1 2 θ 2 = Ω θd t θ + 1 2 d θ t = Ω 1 2 θ 2 ∇• (w dt + σdB t )= 0
Energy decomposition of mean and variance fields : Forced-dissipative advection of the potential vorticity (PV) q with source process :

0 = d dt Ω 1 2 (E[θ]) 2 + d dt Ω 1 2 Var[θ] Long Li (Fluminance) A Consistent
D t q = (S 1 + F + D)dt + S 2 dB t S 1 = 1 2 i,j=1,2 ∂ 2 ij (∇ ⊥ a ij • u), S 2 dB t = -tr ∇ ⊥ (σdB t ) T ∇u T
Kinematic relationship between PV and stream function ψ : The mean squared error (MSE) of the ensemble mean forecasts is identical to the average intra-ensemble sample variance (VAR), multiplied by an ensemble-size Ndependent inflation factor :

q = ∇ 2 ψ + f, u = ∇ ⊥ ψ Conservation of kinetic
(q o -Ê[q]) 2 x = N + 1 N Var[q]

  Other UQ metrics[Resseguier, Li, Jouan, Derian, Mémin & Chapron, 2019] : Continuous ranked probability score (CRPS), Energy score, Variogram score.

  energy in the absence of D and F : [Bauer, Heterogeneous Stationary : [Chandramouli, Mémin, Chapron & Heitz 2019b]

	Outline Outline
	Based on snapshot proper orthogonal decomposition (POD) method
	Location Uncertainty (LU) Principles from data Location Uncertainty (LU) Principles
	Stochastic Barotropic Vorticity Equation (SBVE) Stochastic Barotropic Vorticity Equation (SBVE)
	Parametrizations of Noise Parametrizations of Noise
	Long-term Diagnosis of Time-Statistics Long-term Diagnosis of Time-Statistics
	Chandramouli, Chapron, Li & Mémin 2019a] Short-term Verification of Ensemble Forecasts Short-term Verification of Ensemble Forecasts
		d dt Ω	1 2	∇ψ 2 = 0
	Long Li (Fluminance) Long Li (Fluminance) Long Li (Fluminance) Long Li (Fluminance)	A Consistent Stochastic Modeling A Consistent Stochastic Modeling A Consistent Stochastic Modeling A Consistent Stochastic Modeling	September 12, 2019 September 12, 2019 September 12, 2019 September 12, 2019	10 / 28 11 / 28 13 / 28 15 / 28

September 12, 2019

Homogeneous in space Stationary in time : [Resseguier, Memin & Chapron, 2017b] Homogeneous Non-stationary : [Resseguier, Li, Jouan, Derian, Mémin & Chapron, 2019] Based on estimation of the absolute diffusivity spectral density (ADSD).