Ph.D Maria A Ciemerych
email: ciemerych@biol.uw.edu.pl

Ph.D Zuzanna ; Maciejewska

Pascal Aude

Ph.D Jacek Z Kubiak
email: jacek.kubiak@univ-rennes1.fr

Aude Pascal

Phosphorylated ERK5/BMK1 transiently accumulates within division spindles in mouse oocytes and preimplantation embryos

Keywords: mouse, oocyte, embryo, meiosis, mitosis, ERK5 mouse, oocyte, embryo, meiosis, mitosis, ERK5

MAP kinases of ERK family play important roles in oocyte maturation, fertilization, and early embryo development. The role of the signaling pathway involving ERK5 MAP kinase during meiotic and mitotic M-phase of the cell cycle is not well known. Here we studied localization of phosphorylated, thus, activated form of ERK5 in mouse maturing oocytes and mitotically dividing early embryos. We show that phosphorylation/dephosphorylation, i.e. activation/inactivation of ERK5 correlates with Mphase progression. Phosphorylated form of ERK5 accumulates in divisi le of bo otic and mitotic cells, and precisely co-local zes w h sp ndle microtubules at metaphase. This localization changes drastically in anapha hen ospho-ERK5 com letely disappears from microtubules and transits to the cytoplasmic granular, vesicle-like structures. In telophase oocytes it becomes incorporated into the midbody. Localization of phospho-ERK5 suggests that it may play important role both in meiotic and mitotic division.

Introduction

MAPK (Mitogen Activated Protein Kinase) signaling pathways are implicated in the regulation of wide range of cellular processes such as proliferation, differentiation, and apoptosis [START_REF] Rubinfeld | The ERK cascade as a prototype of MAPK signaling pathways[END_REF]. Their function has been analyzed in various somatic cells [START_REF] Cude | Regulation of the G2-M cell cycle progression by the ERK5-NFkappaB signaling pathway[END_REF][START_REF] Ranganathan | The MAP kinase ERK5 binds to and phosphorylates p90 RSK[END_REF][START_REF] Cameron | Differential role of MEK5alpha and MEK5beta in BMK1/ERK5 activation[END_REF][START_REF] Roberts | ERK5 is required for VEGF-mediated survival and tubular morphogenesis of primary human microvascular endothelial cells[END_REF][START_REF] Roberts | ERK5 and the regulation of endothelial cell function[END_REF], as well as in oocytes and embryos (e.g. [START_REF] Wang | Entire mitogen activated protein kinase (MAPK) pathway is present in preimplantation mouse embryos[END_REF][START_REF] Verlhac | MAP kinase becomes stably activated at metaphase and is associated with microtubuleorganizing centers during meiotic maturation of mouse oocytes[END_REF]). BMK1/ERK5/MAPK7 (Mitogen-activated protein Kinase 1/Extracellular signal-Regulated Kinase 5/Mitogen Activated Protein Kinase 7; further called by us ERK5) is one of the less characterized MAP kinases of the ERK family. ERK5 mouse knock-out mice proved that the major developmental function of this kinase is related to the placental, cephalic, hindgut, and, importantly, cardiovascular development [START_REF] Yan | Knockout of ERK5 causes multiple defects in placental and embryonic development[END_REF][START_REF] Regan | Erk5 null mice display multiple extraembryonic vascular and embryonic cardiovascular defects[END_REF][START_REF] Hayashi | Targeted deletion of BMK1/ERK5 in adult mice perturbs vascular integrity and leads to endothelial failure[END_REF][START_REF] Sohn | ERK5 MAPK regulates embryonic angiogenesis and acts as a hypoxia-sensitive repressor of vascular endothelial growth factor expression[END_REF][START_REF] Hayashi | Role of the BMK1/ERK5 signaling pathway: lessons from knockout mice[END_REF]. Thus, mouse embryos lacking functional ERK5 gene failed to progress beyond E10.5, but until this developmental stage the cellular proliferation seemed unperturbed. Further analyses suggested, however, that ERK5 is activated at G2-M transition, and thus, involved in the regulation mitotic division [START_REF] Cude | Regulation of the G2-M cell cycle progression by the ERK5-NFkappaB signaling pathway[END_REF][START_REF] Girio | Erk5 is activated and acts as a survival factor in mitosis[END_REF][START_REF] Inesta-Vaquera | ERK5 pathway regulates the phosphorylation of tumour suppressor hDlg during mitosis[END_REF][START_REF] Inesta-Vaquera | Alternative ERK5 regulation by phosphorylation during the cell cycle[END_REF]. Details of the mitotic function of this kinase have been analyzed in somatic cells, but have never been previously reported in oocytes or in embryonic cells.

Among the MAP kinases family members studied so far in oocytes and embryos, the function of only ERK1 and ERK2 is the best known. These two MAP kinases are activated by Mos-MEK1 pathway and are necessary for the establishment of CSF (cytostatic factor) mediated arrest of metaphase II oocytes in mouse [START_REF] Verlhac | MAP kinase becomes stably activated at metaphase and is associated with microtubuleorganizing centers during meiotic maturation of mouse oocytes[END_REF][START_REF] Verlhac | Mos is required for MAP kinase activation and is involved in microtubule organization during meiotic maturation in the mouse[END_REF] as well as in frog [START_REF] Haccard | Induction of metaphase arrest in cleaving Xenopus embryos by MAP kinase[END_REF]. Special attention was paid to ERK1 and ERK2 MAPK pathway since it was shown to be necessary to sustain the meiotic arrest of maturing vertebrate oocytes, establishment of oocyte polarity and impacting at the correct size of the polar body (e.g. [START_REF] Araki | Meiotic abnormalities of c-mos knockout mouse oocytes: activation after first meiosis or entrance into third meiotic metaphase[END_REF][START_REF] Choi | The Mos/mitogen-activated protein kinase (MAPK) pathway regulates the size and degradation of the first polar body in maturing mouse oocytes[END_REF][START_REF] Verlhac | Mos activates MAP kinase in mouse oocytes through two opposite pathways[END_REF][START_REF] Kosako | Mitogen-activated protein kinase kinase is required for the mos-induced metaphase arrest[END_REF][START_REF] Nishiyama | Phosphorylation of Erp1 by p90rsk is required for cytostatic factor arrest in Xenopus laevis eggs[END_REF][START_REF] Inoue | A direct link of the Mos-MAPK pathway to Erp1/Emi2 in meiotic arrest of Xenopus laevis eggs[END_REF]). In-gel MBP kinase assay performed using mouse oocytes and one-cell embryos proved that ERK1 and ERK2 activity is high exclusively in oocytes, but undetectable during mitotic divisions of early mouse embryos [START_REF] Verlhac | MAP kinase becomes stably activated at metaphase and is associated with microtubuleorganizing centers during meiotic maturation of mouse oocytes[END_REF][START_REF] Kalab | Activation of p90rsk during meiotic maturation and first mitosis in mouse oocytes and eggs: MAP kinase-independent and -dependent activation[END_REF]. The same assay showed also that other MBP (myelin basic protein, i.e. MAPK substrate) phosphorylating kinases are active both in oocytes and mitotically dividing embryos [START_REF] Verlhac | MAP kinase becomes stably activated at metaphase and is associated with microtubuleorganizing centers during meiotic maturation of mouse oocytes[END_REF][START_REF] Yan | Molecular cloning of mouse ERK5/BMK1 splice variants and characterization of ERK5 functional domains[END_REF], suggesting potential role of other MAP kinases in regulation of early embryonic mitosis. ERK5 is likely to fulfill this role. The evidence supporting the idea of ERK5 involvement both in meiosis and mitosis comes from analyses of MAP kinases substrates. Phosphorylated form of one of them, i.e. p90 Rsk1 , was detectable in meiotically dividing mouse oocytes, when ERK1 and ERK2 are fully activated, but also during the first mitosis of mouse embryo, when the two ERK kinases remain inactive [ [START_REF] Kalab | Activation of p90rsk during meiotic maturation and first mitosis in mouse oocytes and eggs: MAP kinase-independent and -dependent activation[END_REF], our unpublished data]. Since p90 rsk1 was shown to be phosphorylated by ERK5 in somatic cells [START_REF] Ranganathan | The MAP kinase ERK5 binds to and phosphorylates p90 RSK[END_REF], it is likely that p90 rsk1 is activated by the same pathway during M-phase of oocytes and early embryos. ERK5-encoding mRNA was detected in unfertilized oocytes, and in preimplantation embryos up to blastocyst stage suggesting that the protein could be present at that stages [START_REF] Wang | Entire mitogen activated protein kinase (MAPK) pathway is present in preimplantation mouse embryos[END_REF].The presence of ERK5 protein in oocytes was shown using Western blotting analysis of pig oocytes lacking precise localization analysis [START_REF] Pelech | Antibody Microarray Analyses of Signal Transduction Protein Expression and Phosphorylation during Porcine Oocyte Maturation[END_REF]. Similarly, Wang et al. presented Western blot analysis of preimplantation mouse embryos (3.5 days of development) and non-conclusive immunodetection of active and inactive forms of this antigens [START_REF] Wang | Entire mitogen activated protein kinase (MAPK) pathway is present in preimplantation mouse embryos[END_REF].

In the current paper we show that ERK5 is present in mouse oocytes and early embryo and that its activation is precisely regulated during each of M-phases studied and that its association with both meiotic and mitotic spindle is highly controlled during M-phase progression.

Material and Methods

Ethical issues

All animal studies presented were approved by Local Ethic Committee No. 1 in Warsaw, Poland according to European Union Council Directive 86/609/EEC of 24 November 1986. All animals were raised on the premises.

Collection and culture of oocytes at germinal vesicle stage

Two-to three-month-old F 1 (C57Bl/10 x CBA/H) female mice were injected intraperitoneally with 10 IU pregnant mare serum gonadotrophin (PMSG; Folligon, Intervet, Netherlands) to stimulate the development of ovarian follicles. Forty-eight to fifty-two hours later females were killed by cervical dislocation. Fully grown oocytes arrested at prophase of the first meiotic division [germinal vesicle stage (GV)] were released from ovarian follicles. Oocytes were freed from cumulus cells by pipetting and then cultured for 2 h in M2 medium [medium M16 buffered with HEPES; [START_REF] Fulton | Activation of mammalian oocytes by intracellular injection of calcium[END_REF]] containing bovine serum albumin (BSA; 4 mg/ml). Oocytes were fixed subjected to immunofluorescence at GV stage, at the GVBD (germinal vesicle breakdown) and at selected, subsequent stages of in vitro maturation.

Collection, parthenogenetic activation of ovulated oocytes and in vitro culture of oneand two-cell embryos, blastocysts.

F1 females were superovulated by injection of 10 IU of eCG and 10 IU of hCG (Intervet, Netherlands) 48-52 hours apart. Ovulated oocytes were collected 17-18 hours after hCG injection and cumulus cells were removed with hyaluronidase (Sigma Aldrich, Germany, 300 units/ml PBS). For the parthenogenetic activation the oocytes were exposed to 8% ethanol (POCh, Poland) in M2 medium for 8 minutes at room temperature [START_REF] Cuthbertson | Parthenogenetic activation of mouse oocytes in vitro with ethanol and benzyl alcohol[END_REF], washed carefully and cultured in M2. Only those oocytes that extruded polar bodies and formed pronuclei were used for further analyses. Starting at 14 hours after activation embryos were scored every 10 minutes to determine the timing of the nuclear envelope breakdown (NEBD) and then collected for further analyses at different time after NEBD. Fertilized two-cell embryos were obtained from superovulated and mated females 49 hours after hCG. Embryos were collected and cultured in M2 medium. Starting from 49 hours after hCG embryos were observed every 10 minutes to determine the timing of NEBD and collected for further analyses at different times after NEBD. Blastocysts were obtained from uteri of superovulated and mated females at 90 hours after hCG.

In vitro culture of murine NIH3T3 fibroblasts and C2C12 myoblasts.

NIH3T3 fibroblasts were cultured in Dulbecco's modified Eagle's medium supplemented (DMEM, Invitrogen) with 10 % fetal calf serum (FCS) and antibiotics (Invitrogen) at 37°C in 5 % CO 2 . Media were supplemented with penicillin (100 Units/ml) and streptomycin (100 mg/ml). Cells were fixed 48 hrs after plating and processed for Western blotting and immunodetection of tubulin and phospho-ERK5 epitopes. C2C12 murine myoblasts cell line was obtained from the European Collection of Cell Cultures (ECACC no. 91031101, passage no. 13). Cells were cultured in high-glucose DMEM supplemented with 10% FCS and antibiotics at 37°C in 5 % CO 2 . Cells were fixed and processed for immunodetection of tubulin and phospho-ERK5 epitopes at day 2 after plating, i.e. at the proliferative stage.

Western blotting

Protein samples of NIH3T3 cells were subjected to electrophoresis on 9 % SDS-PAGE gels [START_REF] Laemmli | Cleavage of structural proteins during the assembly of the head of bacteriophage T4[END_REF]. Separated proteins were transferred to nitrocellulose membranes (Hybond C, Amersham Biosciences) and probed with rabbit polyclonal primary antibodies either against ERK5 or Phospho-ERK5 (Upstate) at the dilution of 1:1000. Next, α-tubulin was immunodetected with mouse monoclonal antibody (Sigma Aldrich, Germany). Antigen-antibody complexes were revealed using alkaline phosphatase conjugated anti-rabbit or anti-mouse secondary antibody (diluted 1:20,000, Jackson ImmunoResearch, USA) in combination with Enhanced Chemifluorescence reagent (ECF; Amersham Biosciences). Signal quantification was performed using ImageQuant 5.2 software (Amersham Biosciences).

Immunofluorescence

NIH3T3 cells and C2C12 myoblasts plated and cultured on glass coverslips were fixed in 3.7 % paraformaldehyde in PBS for 10 min at room temperature and permeabilized with 0.1% Triton X100 in PBS for 5 min. Oocytes, embryos were fixed in 3.7 % paraformaldehyde in PBS for 30 min at room temperature and permeabilized with 0.1% Triton X100 in PBS for 5 min. Non-specific binding was blocked by 1 h long treatment with 3% BSA in PBS. Rabbit polyclonal antibodies against phospho-ERK5 (Upstate Biotechnology, USA) were used at the dilution of 1:200 and the incubation was performed overnight at 4°C. Next, α-tubulin was immunodetected with mouse monoclonal antibody (Sigma Aldrich, Germany) followed by FITC-labeled anti-rabbit and TRITC-labeled anti-mouse antibodies (Jackson ImmunoResearch, USA).

Chromatin was visualized either with chromomycin A3 (Sigma Aldrich, Germany) diluted 1:1000 in PBS with 5 mM MgCl2 for 0.5 h at room temperature or propidium iodide present in mounting medium (Vector Laboratories, USA). Samples incubated in chromomycin A3 were mounted in Citifluor (Citifluor Ltd, Great Britain) on glass slides. Immunofluorescence analysis was accompanied by appropriate negative controls, i.e. samples that were not subjected to the primary antibodies, but instead they were incubated in 3% BSA in PBS. All samples were examined with laser scanning confocal microscope (Axiovert 100M, Carl Zeiss Jena, Germany) and processed with Adobe Photoshop 6.0.

Results & Discussion

Since ERK5 is known to be phosphorylated upon M-phase in human cells [START_REF] Cude | Regulation of the G2-M cell cycle progression by the ERK5-NFkappaB signaling pathway[END_REF][START_REF] Inesta-Vaquera | Alternative ERK5 regulation by phosphorylation during the cell cycle[END_REF][START_REF] Zen | ERK5 is a target for gene amplification at 17p11 and promotes cell growth in hepatocellular carcinoma by regulating mitotic entry[END_REF], and its role in mammalian meiosis is virtually unknown, we decided to begin characterization of this MAP kinase in mouse oocytes by follow the dynamics in its phosphorylation state using a specific anti-phospho-ERK5 (P-ERK5) antibody.

In germinal vesicle (GV) stage oocytes (meiotic prophase I) phosphorylated ERK5 is absent from the GV and is detected as a background staining within the cytoplasm (Fig. 1, GV). Following M-phase entry, upon germinal vesicle breakdown (GVBD), the P-ERK5 signal becomes detectable close to condensing bivalents (Fig. 1, GVBD + 1h). This staining increases with the progression of the prometaphase and reflects organization of microtubules forming meiotic spindle around the individualized bivalents. The intensity of the staining is the highest in the vicinity of the chromosome bivalents (Fig. 1,GVBD+3h). In metaphase of the first meiotic division P-ERK5 localizes entirely to the spindle microtubules and the background, i.e. cytoplasmic staining, becomes negligible (Fig. 1, metaphase I). At the time of the first polar body extrusion, at the anaphase I/telophase I transition, P-ERK5 is no more detectable on the spindle microtubules, however, remains clearly positioned next to telophase groups of chromatin and within the midbody, concomitantly with the decrease in the cytoplasmic staining (Fig. 1,telophase I). During the metaphase of the second meiotic division P-ERK5 staining becomes again localized exclusively to the meiotic spindle, as during the metaphase I (Fig. 1, metaphase II). At this stage the cytoplasmic staining is also reduced to the minimum. Thus, the P-ERK5 localization in oocytes undergoing meiotic maturation shows that active form of ERK5 MAP kinase localizes specifically to the meiotic spindle. The affinity of ERK5 to the spindle microtubules manifests very early upon GVBD and progresses during the long, taking about 8 hours, metaphase I. The progressive incorporation of ERK5 to the spindle is manifested by its total disappearance from the cytoplasm when the meiotic spindle is fully assembled both in metaphase I and II. Microtubule-associated localization of other MAP kinases, and some of their activators was reported before. ERK1/ERK2 activating MEK1/2 kinase was shown to be limited to the meiotic spindle, and its active form was shown to be located within the midbody during anaphase [START_REF] Yu | MEK1/2 regulates microtubule organization, spindle pole tethering and asymmetric division during mouse oocyte meiotic maturation[END_REF][START_REF] Sun | MEK1/2 is a critical regulator of microtubule assembly and spindle organization during rat oocyte meiotic maturation[END_REF]. In contrary ERK1/2 MAP kinases were detectable at the spindle poles [START_REF] Verlhac | MAP kinase becomes stably activated at metaphase and is associated with microtubuleorganizing centers during meiotic maturation of mouse oocytes[END_REF]. ERK3 was also recently reported to co-localize to spindle microtubules, however, the activation of this kinase was not investigated [START_REF] Li | ERK3 is required for metaphase-anaphase transition in mouse oocyte meiosis[END_REF]. The localization and activation of ERK5 in mouse oocytes and early embryos was not precisely studied so far, however, its expression was shown in pig oocytes [START_REF] Pelech | Antibody Microarray Analyses of Signal Transduction Protein Expression and Phosphorylation during Porcine Oocyte Maturation[END_REF], and mouse preimplantation embryos [START_REF] Wang | Entire mitogen activated protein kinase (MAPK) pathway is present in preimplantation mouse embryos[END_REF]. However, images of ERK5 and P-ERK5 localization in mouse blastocysts, presented by Wang and colleagues showed high, and uniform signal within the cytoplasm of inner cell mass and trophectoderm cells [START_REF] Wang | Entire mitogen activated protein kinase (MAPK) pathway is present in preimplantation mouse embryos[END_REF], clearly different that in oocytes studied by us. This prompted us to analyze the P-ERK5 localization also in preimplantation mouse embryos.

To verify the specificity of the antibody used we performed Western blotting and immunolocalization analyses of total ERK5 and P-ERK5 in murine somatic NIH3T3 cells. Similarly as presented by others (e.g. [START_REF] Cude | Regulation of the G2-M cell cycle progression by the ERK5-NFkappaB signaling pathway[END_REF][START_REF] Inesta-Vaquera | Alternative ERK5 regulation by phosphorylation during the cell cycle[END_REF]) we detected a clear increase in phosphorylated, up-shifted form of ERK5 in nocodazole-synchronized, i.e. mitotic cells. These result was obtained using both anti-total ERK5 (Fig. 2A,ERK5) and anti-P-ERK5 antibody (Fig. 2B, P-ERK5). Importantly, the phosphorylated form of ERK5 localized not only to the meiotic spindles, as described above, but also to the mitotic spindle, as shown for mouse NIH3T3 cells (Fig. 2B) and in murine C2C12 myoblasts (Fig. 2C). In the interphase NIH3T3 and C2C12 cells a low cytoplasmic staining is detectable, while the nuclei are entirely devoid of P-ERK5 (Fig. 2B and2C, white asterisks). The nuclear localization of P-ERK5 becomes clear during prophase (Fig. 2B andC, prophase). Further, in M-phase, the P-ERK5 staining is restricted to the mitotic spindle only in NIH3T3 cells (Fig. 2B, M-phase), while in murine myoblasts C2C12 P-ERK5 colocalizes not only to the spindle, but also to chromosomes (Fig. 2C, M-phase). Thus, Western blotting and also immunolocalization confirm that the signal detected in murine oocytes corresponds to the phosphorylated form of ERK5.

To perform precise analysis of P-ERK5 during M-phase progression in embryonic cells, we studied the well characterized model of mitotically dividing mouse one-cell embryos. The first mitotic division of mouse embryo can serve, due to its length of approximately 120 minutes, as a useful tool to study the mitotic events [START_REF] Sikora-Polaczek | The first mitosis of the mouse embryo is prolonged by transitional metaphase arrest[END_REF][START_REF] Ciemerych | Control of duration of the first two mitoses in a mouse embryo[END_REF][START_REF] Maciejewska | Spindle assembly checkpoint-related failure perturbs early embryonic divisions and reduces reproductive performance of LT/Sv mice[END_REF][START_REF] Kubiak | On the transition from the meiotic to mitotic cell cycle during early mouse development[END_REF][START_REF] Teperek-Tkacz | Phosphorylation of histone H3 serine 10 in early mouse embryos: Active phosphorylation at late S phase and differential effects of ZM447439 on first two embryonic mitoses[END_REF]. This feature allowed us to precisely analyze the localization, and dynamics of ERK5 activation in M-phase. At metaphase of the first mitotic division of one-cell embryos, similarly as it was shown for oocytes, P-ERK5 perfectly colocalized to the spindle microtubules (Fig. 3A, metaphase). Moreover, the cytoplasmic staining was not detected. As soon as the anaphase started the pattern of P-ERK5 localization changed dramatically. First of all the P-ERK5 signal was lost from the microtubules, and becomes visible as diffused cloud around the spindle (Fig. 3A, early anaphase). This signal dispersed during the progresses to late anaphase and beginning of cytokinesis (Fig. 3A, late anaphase). Concomitantly, the cytoplasmic staining increased (Fig. 3A, early and late anaphase). Thus, following the metaphaseanaphase transition P-ERK5 is no longer present on spindle microtubules, but seems to disperse around the anaphase spindle and further within the cytoplasm. Similar localization of P-ERK5 within the mitotic spindle was observed in two cell stage blastomeres (Fig. 3B), and at further stages of preimplantation development (Fig. 3C).

These results show that: 1) ERK5 is phosphorylated upon M-phase entry in meiosis and mitosis; 2) in both cases it localizes to the metaphase spindle microtubules; 3) and disappears from the spindle in early anaphase. Such localization suggests that ERK5 plays a similar function in spindle assembly/function in oocytes, embryos and somatic cells. ERK5/ BMK1/MAPK7 belongs to the subfamily of the MAP kinase family in mammals, and it has some characteristic features: a large C-terminus and a unique loop 12 sequence [START_REF] Lee | Primary structure of BMK1: a new mammalian map kinase[END_REF]. Mammalian ERK5 was shown to be activated by oxidative stress, hyperosmolarity and several growth factors, including epidermal growth factor and nerve growth factor [START_REF] Lee | Primary structure of BMK1: a new mammalian map kinase[END_REF][START_REF] Abe | Big mitogen-activated protein kinase 1 (BMK1) is a redox-sensitive kinase[END_REF][START_REF] Chao | MEKK3 directly regulates MEK5 activity as part of the big mitogen-activated protein kinase 1 (BMK1) signaling pathway[END_REF][START_REF] Zhou | Components of a new human protein kinase signal transduction pathway[END_REF][START_REF] Kato | Bmk1/Erk5 is required for cell proliferation induced by epidermal growth factor[END_REF][START_REF] Kamakura | Activation of the protein kinase ERK5/BMK1 by receptor tyrosine kinases. Identification and characterization of a signaling pathway to the nucleus[END_REF]. The mitotic function of ERK5 was suggested by the results of the experiments focusing at both non-transformed cells such as fibroblasts and transformed cell lines such as HeLa cells. In HeLa cells expressing of constitutively active MEK5, and as a result activation of ERK5, leads to the increase in the number of mitotic cells [START_REF] Cude | Regulation of the G2-M cell cycle progression by the ERK5-NFkappaB signaling pathway[END_REF]. On the other hand the use of the dominant negative form of ERK5, or siRNA suppressing ERK5 expression reduced mitotic index of analyzed cells [START_REF] Cude | Regulation of the G2-M cell cycle progression by the ERK5-NFkappaB signaling pathway[END_REF]. Moreover, siRNA depletion of ERK5 decreases cell survival of mitotically dividing cells [START_REF] Girio | Erk5 is activated and acts as a survival factor in mitosis[END_REF]. However, the fact that mouse embryos lacking functional ERK5 gene are able to survive until approximately E10.5 strongly suggests that ERK5 function can be replaced by another enzyme. Other MAP kinases, like ERK1 and ERK2, are candidates for such function. Despite that localization of ERK1 and ERK2 does not precisely reflect the P-ERK5 position within the mitotic spindle, it is well known that these kinases play a crucial function in meiotic spindle organization and function [START_REF] Verlhac | MAP kinase becomes stably activated at metaphase and is associated with microtubuleorganizing centers during meiotic maturation of mouse oocytes[END_REF]. Analyses of oocytes of Mos kinase, an upstream ERK1/2 activating kinase, deficient mice, proved that ERK1/2 regulate microtubule dynamics [START_REF] Verlhac | Mos is required for MAP kinase activation and is involved in microtubule organization during meiotic maturation in the mouse[END_REF][START_REF] Araki | Meiotic abnormalities of c-mos knockout mouse oocytes: activation after first meiosis or entrance into third meiotic metaphase[END_REF][START_REF] Choi | The Mos/mitogen-activated protein kinase (MAPK) pathway regulates the size and degradation of the first polar body in maturing mouse oocytes[END_REF]. Recent discoveries of ERK3 localization and function in mouse oocytes identify a new candidate, which together with ERK1/2 and ERK5, might be involved in the regulation of spindle dynamics [START_REF] Li | ERK3 is required for metaphase-anaphase transition in mouse oocyte meiosis[END_REF]. ERK3 was shown to be crucial for the successful mouse development, but again, similarly as it was described for ERK5 knock-out mice, ERK3 deficiency did not prohibit mitotic divisions of vast majority of embryonic tissues [START_REF] Klinger | Loss of Erk3 function in mice leads to intrauterine growth restriction, pulmonary immaturity, and neonatal lethality[END_REF]. This kinase, however, similarly do ERK1/2 and ERK5, was shown to be phosphorylated during M-phase entry and dephosphorylated at metaphase/anaphase transition [START_REF] Tanguay | C-terminal domain phosphorylation of ERK3 controlled by Cdk1 and Cdc14 regulates its stability in mitosis[END_REF]. The ERK4-deficient mice also did not reveal any significant mitotic phenotype, moreover, these animals were not only viable but also fertile [START_REF] Rousseau | Targeted Inactivation of Mapk4 in Mice Reveals Specific Non-Redundant Functions of Erk3/Erk4 Subfamily MAP Kinases[END_REF]. Thus, only the perturbation of the activity ERK1/2 MAP kinases pathway, as shown by Mos knock-out mice analyses, has fatal consequences for the mouse fertility impacting at the oocyte maturation [START_REF] Hashimoto | Parthenogenetic activation of oocytes in c-mos-deficient mice[END_REF][START_REF] Colledge | Disruption of c-mos causes parthenogenetic development of unfertilized mouse eggs[END_REF]. Mos knockout does not influence the mouse embryo development and animal viability probably due to ERK1 and ERK2 activation via Ras/Raf pathway [START_REF] Yamauchi | The Ras/Raf signaling pathway is required for progression of mouse embryos through the two-cell stage[END_REF]. However, it is possible that other MAPK, such as ERK5, may also act to rescue Mos deficiency following fertilization. Further studies will be necessary to solve this issue.

Legend to Figures

Fig. 1.

P-ERK5 localization in maturing mouse oocytes.

Oocytes were analyzed at GV stage, and than during the formation of the metaphase plate and spindle of the first meiotic division (1 and 3 h after GVBD), at metaphase I (5 h after GVBD), telophase I (8 h after GVBD) and at metaphase II (9 h after GVBD). Top rowimmunolocalization of P-ERK (green) and chromatin visualization (blue), overlay; bottom row -immunolocalization of P-ERK (green).

Fig. 2 .

 2 Fig. 2. P-ERK5 in murine NIH3T3 fibroblasts and C2C12 myoblasts. A. Immunoblotting of ERK5 and P-ERK5 in lysates of asynchronously growing and nocodazole synchronized NIH3T3 fibroblasts. Note the presence of upper, slow-migrating, i.e. phosphorylated form of ERK5 in nocodazole treated cells. Immunodetection of αtubulin was performed to show the equal loading of the analyzed samples; B. Immunolocalization of P-ERK5 in asynchronously growing NIH3T3 fibroblasts' C. Immunolocalization of P-ERK5 in asynchronously growing C2C12 myoblasts. NIH3T3 and C2C12 cells were fixed in interphase, prophase and metaphase. Top rowimmunolocalization of P-ERK5 (green), α-tubulin (red), and visualization of chromatin (blue), overlay; middle row -immunolocalization of α-tubulin (red); bottom rowimmunolocalization of P-ERK5 (green). Note the P-ERK accumulation in prophase nuclei and spindle and chromosome localization in metaphase. White stars indicate interphase nuclei lacking P-ERK5 staining.

Fig. 3 P

 3 Fig. 3 P-ERK5 localization in preimplantation mouse embryos. A. One-cell mouse parthenogenetic embryos fixed at metaphase and anaphase of first mitotic division. B. Two-cell mouse embryo fixed at metaphase of the second mitotic division. C. Mitotically dividing inner cell mass in blastocyst. Top rowimmunolocalization of P-ERK5 (green), α-tubulin (red), and visualization of chromatin (blue), overlay; middle row -immunolocalization of α-tubulin (red); bottom row -immunolocalization of P-ERK5 (green).

Figure 1

 1 Figure 1Click here to download high resolution image

Figure 2

 2 Figure 2 Click here to download high resolution image

Figure 3

 3 Figure 3 Click here to download high resolution image

Acknowledgements

JZK was supported by a grant from ARC (4900) and LCC.


Introduction: The German Sänger II Project

From Times Online February 5, 2008

Hypersonic jet could offer day trips Down Under

Day trips to Australia came a step closer to reality today when a British firm unveiled plans for a hypersonic passenger jet which could fly from Europe to Sydney in less than five hours.

Reaction Engines of Oxfordshire says that its A2 plane could be in service within 25 years, carrying 300 passengers at a top speed of almost 4,000 mph, five times the speed of sound and twice the speed of Concorde.

The jet may be more environmentally friendly than other plains because it uses liquid hydrogen for power rather than fossil fuel.

ODE

Project LAPCAT Reading Engines, UK

• Conclusion

Conclusions (hypersonic aircraft problem)

• Detailed model of high complexity for instationary heating

• Reduction of temperature of TPS due to optimal control

• Challenging problem: ODE -PDE state -constrained optimal control

• Lack of theory: nonlinear, state -constrained, ODE-PDE coupling

• Method of lines and SQP methods seem to be at their limits • First optimize, then discretize hardly applicable

• First discretize, then optimize with reliable verification of necessary conditions, but with limitations in time and storage

• Motivation from hypersonic flight path optimization


F Bernelli-Zazzera

F Topputo

Incorporating low-thrust propulsion into the invariant manifolds technique for space trajectories design

 Space trajectory design using non-Keplerian orbits • Transfers to libration point orbits Aims  Improve existing methods • Modify "invariant manifolds technique" to include low-thrust propulsion

• How can we design non-Keplerian orbits for space missions?  Exploit knowledge of the phase space (e.g., equilibrium points, periodic orbits, invariant manifolds, etc.)

The restricted three-body problem (1)

• Two primaries act on a third body (of negligible mass)

• The third body does not affect the motion of the primaries • The two primaries move in circular orbits around their center of mass (circular restricted three-body problem)

The motion of the primaries is given: 6

The restricted three-body problem (2)

• the problem is restricted to the motion of the third body • Newton equation written in inertial frame:

The restricted three-body problem (3)

• Equations written in a dimensionless rotating (synodic) frame make the dynamical system autonomous Conventions:

G = 1 n = 1 m 1 = 1-μ x1 = -μ m 2 = μ x2 = 1-μ μ = m 2 /(m 1 +m 2 )
Mass parameter of the R3BP

Equilibrium points (1)

• The differential system has 5 equilibrium points (libration or Lagrangian points).

• The "nature" of the points is given by their linear behaviour.

x P 1 P 2 L 3 L 1 L 2 L 4 L 5 y   x f x   Ax x   i        i i i       6 / 5 2 4 / 3 1 2 / 1
L4, L5:

Stable (center x center x center) 2 6 / 5 1 4 / 3 2 1       i i       L1, L2, L3: Unstable (saddle x center x center)       0385 . 0 0  < m m if
Note: equilibrium is relative to the synodic system! • Collinear points L1 and L2 are very important for space applications  associated to low energy values (high values of C)  have "fast" stable and unstable dynamics (strong instability)

• Saddle part (stable and unstable eigenvales)  Hyperbolic orbit near L1 and L2

• Center part (complex conjugate eigenvales)  Periodic orbits around L1 and L2

• Invariant manifolds of the points

 W s Li , stable manifold  W u
Li , unstable manifold

• The invariant manifolds of the points have interesting dynamical properties that can be exploited for space applications

Equilibrium points (2)

• Nonlinear invariant manifolds are defined in the phase space as 1D invariant subsets such that

W s Li  {x  R 6 |x(t) → L i , t → +} W u Li  {x  R 6 |x(t) → L i , t → -}
Invariant manifolds of equilibrium points (1)

Sun-Jupiter system -W s L2 (blue) -W u L2 (red)
• The two 1D manifolds extend for more than 3 AU far from the equilibrium point. 13

• If a point lies on the stable (unstable) manifold, in the phase space sense, it will move toward (away) the libration point • Possibility to have free transfers in the solar system

W s,u L1 W s,u L2
Invariant manifolds of equilibrium points (2)

• Large, 3D periodic orbits around collinear Lagrangian points • ω = ν, forced by the higher order (nonlinear) terms • Suitable for many missions (large out-of-plane excursions) • Obtained through semi-analytical methods

Halo orbits (1)

Family of halos around L1 of the Sun-Earth system Richardson's method to construct halo orbits: 1. Find a third-order analytical first guess solution 2. Differential correction under the nonlinear dynamics The R3BP has the following symmetry  a trajectory that crosses perpendicularly the y=0 plane twice is a periodic orbit

Given the analytic initial guess, it is corrected through

Halo orbits (2) 17

• In a neighborhood of the equilibrium points there are also invariant manifolds associated to the periodic orbits!

• These manifolds are more complicated "objects" than the simple manifolds associated to the equilibrium points

Invariant manifolds of Halo orbits (1)

Invariant manifolds of Halo orbits (2)

• Periodic orbits have stable and unstable manifolds too!

• Viewed as equilibrium points on a Poincaré section

• They are given by the stable and unstable eigenvalues and eigenvectors of the monodromy matrix M associated to the periodic orbit ) ( Can we further reduce the propellant mass using low-thrust, high-I sp systems? Can we further improve low-energy solutions?

Introduction of Low-Thrust Propulsion (2)

The effect of low-thrust systems is no longer modeled as instantaneous velocity changes ! Need to re-formulate existing "invariant manifolds technique" to incorporate low-thrust propulsion ... 30

Controlled Restricted Three-Body Problem ?

The guidance law is not given, but rather it is an unknown of the optimal control problem ...

Present thrust magnitude:

Variations of spacecraft mass, m, are considered through last eq.

The system is singular when m → 0

Outline 1 .

 1 Introduction to non-Keplerian orbits 2. Introduction to the restricted three body problem 3. Introduction to equilibrium points, Halo orbits and invariant manifolds 4. Dynamical system theory in space trajectory design  transfers to Lagrange point orbits 5. Introduction of low-thrust propulsion: controlled RTBP 6. Definition and manipulation of attainable sets 7. End-to-end trajectory optimization: controlled RFBP 8. Results and comparisons  low-thrust, stable-manifold transfers to halo orbits 9. Final remarks Non-Keplerian Orbits in Space Trajectory Design • Solution of n-body problems, n ≥ 3  Unique features  Less propellant  Accurate solutions -Non-integrable systems -Chaotic dynamics -Long transfer times -No systematic approach

  has only one integral of motion: the Jacobi integral • The problem is not integrable! Qualitative studies only • Allowed and forbidden regions are established (Hill's curves)The restricted three-body problem (4)• Given x 0 , then C 0 = C(x 0 ) and the allowed region will be (x,y,z) such that Ω(x,y,z) ≥ C 0 /2 9• C = -2E • The higher C, the lower energy! • For C<3, the S/C is allowed to move in the whole configuration spaceThe restricted three-body problem (5)



  initial condition to generate a periodic orbit (given by the Richardson's method)  integrate the dynamical system (of order 42) for 0≤t≤T to obtain x=x(t) and M=M(t)  generic point of the orbit  stable eigenvector of M  initial condition to integrate backward to get the stable manifold of x i 21 • Complete phase portrait  asymptotic orbits  periodic orbits  transit orbits (inside the tubes)  non-transit orbits Asymptotic orbits are useful for the trajectory design of libration point missions Dynamical System Theory in Space Trajectory Design • The phase space structure is exploited in a "natural" way. • Equilibrium points, periodic orbits, invariant manifolds, etc. are used in place of orbital elements. Dynamical System Theory in Space Trajectory Design Typical case studies  Transfers to halo orbits  Low energy transfers to the Moon • The concept of stable manifold is exploited for a transfer to Halo orbits  place the S/C on the stable manifold associated to that orbit  Wait (the S/C wraps on the final orbit)  perform an orbit injection maneuver to stabilize in a halo system • ∆v ≃ 3200 m/s (from LEO) • Single-impulse transfer • ∆v = ? • No single-impulse transfer 27 Dynamical system theory allows us to define solutions with  Unique features (e.g., libration point orbits)  Less propellant mass (e.g., low energy transfers) Invariant manifolds technique is used to reduce ∆v Rocket equation:

First

  associated to an admissible set of initial conditions X 32 Attainable sets • Contain a set of tangential low-thrust orbits associated to admissible initial conditions • Tangential thrust is considered in this stage (preliminary solution later used as first guess of optimal control problem) • Attainable sets are manipulated in the same fashion as invariant manifolds Trajectory Optimization (1)  Improve preliminary solutions found with attainable sets  Eliminate possible discontinuities  More refined dynamical model (controlled, bicircular RFBP):  Includes perturbation of the Sun in the EM model  Sun-spacecraft distance  Normalized Sun parameters: • Sun-EM distance r = 3.8881 10 2 • mass of the Sun m s = 3.2890 10 5 • angular velocity relative to EM system w s Objective function (propellant mass minimization):  Path constraints (saturation of thrusters):  Boundary conditions: • depart at periapsis of parking orbit • perigee and point on final halo orbit  Solution by Direct Transcription and Multiple Shooting:  time domain split into N-1 intervals  each interval is split into M-1 segments  control discretized over the M nodes, using spline interpolation  nonuniform time grid 36 multiple shooting (RKF78 + cubic spline)• end-to-end trajectory transcription • non-uniform grid implementation 37 multiple shooting segments (i.e., ~5100 var., 3300 constr.) • The signature of the preliminary solution is clearly visible

Examplevideo by Shane Ross, Virginia Tech

F. Bernelli-Zazzera, F. Topputo Politecnico di Milano, Dipartimento di Ingegneria Aerospaziale G. Mingotti Universität Paderborn, Institut für Mathematik

Final Remarks

• A method to include low-thrust orbit pieces in the "standard" invariant manifolds frame has been derived

• Low-thrust attainable sets are manipulated in the same fashion as invariant manifolds

• Preliminary solutions are optimized in a four-body frame (end-toend optimization)

• Solutions reduce the propellant mass (due to higher I sp )

• Presented solutions are not achievable neither with patched-


E Huot

I Herlin

G Korotaev

ASSIMILATION OF SST SATELLITE IMAGES FOR ESTIMATION OF OCEAN CIRCULATION VELOCITY

INTRODUCTION

The objective of this study is to compute the surface circulation velocity from oceanographic satellite acquisitions. The problem of apparent motion estimation from a sequence of images has been addressed by the image processing community for a long time. It can be tackled by solving a PDEs system composed of a conservation equation and a regularity hypothesis. On the other hand, data assimilation provides a mathematical solution to combine optimally observations and models; oceanographers use circulation models and data assimilation to estimate and forecast ocean' state.

We recently proposed [START_REF] Korotaev | Retrieving Ocean Surface Current by 4-D Variational Assimilation of Sea Surface Temperature Images[END_REF] a new method for velocity estimation using the data assimilation framework: the available satellite images constitute observations of Sea Surface Temperature (SST) to be assimilated within a dedicated Image Model (IM). The major drawback of this approach was the lack of physical meaning of the evolution equations used in the IM. To take into account the physic of underlying processes, it is possible to consider estimated velocities as pseudo-observations and, further assimilated in a physical circulation model [START_REF] Herlin | Estimation of a motion field on satellite images from a simplified ocean circulation model[END_REF].

In this paper we propose to define a new Image Model taking into account the physical knowledge of ocean processes. This so-called Extended Image Model (EIM) is based on both image evolution properties and some physical equations expressing the evolution of the ocean state. Variational data assimilation is used to assimilate SST observations into the EIM and estimate ocean surface velocity.

STATE OF THE ART

Ocean surface velocity can be estimated by image processing techniques applied to sequences of satellite images. Classical methods [START_REF] Barron | Performance of optical flow techniques[END_REF] rely on a conservation equation designed for rigid or fluid motion [START_REF] Béréziat | A generalized optical flow constraint and its physical interpretation[END_REF][START_REF] Corpetti | Dense estimation of fluid flows[END_REF][START_REF] Vigan | Sea surface velocities from Sea Surface Temperature image sequences[END_REF]. But the conservation equation relies on the spatial and temporal derivatives of the image sequence, missing data on the acquisition damage estimation results. This is crucial with SST data that can be significantly occluded by clouds. Recently, the image processing community has been interested in data assimilation approaches [START_REF] Herlin | Estimation of a motion field on satellite images from a simplified ocean circulation model[END_REF]7,[START_REF] Papadakis | Image assimilation for motion estimation of atmospheric layers with shallow-water model[END_REF], which provides an optimal way to deal with missing data. According to the same idea, we introduced [START_REF] Korotaev | Retrieving Ocean Surface Current by 4-D Variational Assimilation of Sea Surface Temperature Images[END_REF] an Image Model, expressed in the image space, that expresses a link between SST and surface velocity:

8 > > > > > > < > > > > > > : ∂T ∂t + ∇T • v = KT ΔT ∂u ∂t =0 ∂v ∂t =0, (1)
where the first equation is a simplification of the advection-diffusion formalism governing the transport of temperature of an uncompressible fluid; KT stands for the temperature diffusivity parameter. The two other equations constrain the evolution of the two velocity components v =( u, v) through an hypothesis of a constant velocity field. SST satellite data are assimilated into the IM in order to estimate de surface velocity.

The major drawback of this approach is that the evolution equations of velocity have no physical interpretation.

EXTENDED IMAGE MODEL

To define an Extended Image Model with a physical meaning, it is necessary to address ocean modelizations done by oceanographers and numericians. As a first step, we are interested in shallow water models which link the 2D velocity of the merged layer of the ocean associated to the thickness h of this layer. The EIM definition is based on the same transport equation of temperature as the IM but the velocity evolution equations come from the shallow water model:

8 > > > > > > > > > > > > > < > > > > > > > > > > > > > : ∂T ∂t + ∇T • v = KT ΔT du dt -fv = g ′ ∂h ∂x + K v Δu dv dt + fu = g ′ ∂h ∂y + K v Δv ∂h ∂t + ∂(uh) ∂x + ∂(vh) ∂y =0, (2)
where f is the Coriolis parameter; Kv the viscosity coefficient; g ′ = g(ρ0 -ρ1)/ρ0 the reduced gravity, with ρ0 corresponding to the reference density and ρ1 to the average density of the mixed layer.

VARIATIONAL DATA ASSIMILATION

Estimation of surface velocity is performed using a variational 4DVAR data assimilation scheme. Satellite SST acquisition is an observation of the state variable T , and v is estimated minimizing the discrepancy between the model's outputs and the observations. This is formulated as:

8 > > > > < > > > > : ∂X ∂t + F (X)=εm X(t0)=X0 + ε b Y = H(X)+εo (3)
where the first equation is the evolution model ( 2), with state variable X =( T, u, v, h); the second equation is the initial condition; and the third equation links the observations Y to the state variables X through the observation operator H. ε m, ε b , εo stands for errors on model, initialisation and observations respectively. In order to deal with a simpler problem, we are considering several assumptions: the model is consider to be perfect ε m =0;wehaven't any knowledge on the background ε b =0; and observation errors are uncorrelated.

The data assimilation procedure consists in minimizing the energy functional J, that depends upon the control X0:

J(X 0 )= 1 2 Z Ω,t (H(X) -Y ) T R -1 (H(X) -Y )dxdydt (4)
Minimization is performed using a quasi-Newtown involving the adjoint of system (3).

CONCLUSION

This paper presents a velocity estimation method applied to oceanographic image sequences. It is based on variational assimilation of image information into a dedicated image model. The method has two major advantages: it can deal with missing data due to cloud occlusion, and it takes into account some physical knowledge of the ocean circulation.


Mehrdad Yaghoobi
email: yaghoobi@ieee.org

Nam ‡ Sangnam
email: sangnam.nam@inria.fr

Remi Gribonval
email: remi.gribonval@inria.fr

Mike E Davies
email: mike.davies@ed.ac.uk

ANALYSIS OPERATOR LEARNING FOR OVERCOMPLETE COSPARSE REPRESENTATIONS

We consider the problem of learning a low-dimensional signal model from a collection of training samples. The mainstream approach would be to learn an overcomplete dictionary to provide good approximations of the training samples using sparse synthesis coefficients. This famous sparse model has a less well known counterpart, in analysis form, called the cosparse analysis model. In this new model, signals are characterized by their parsimony in a transformed domain using an overcomplete analysis operator. We propose to learn an analysis operator from a training corpus using a constrained optimization program based on L1 optimization. We derive a practical learning algorithm, based on projected subgradients, and demonstrate its ability to robustly recover a ground truth analysis operator, provided the training set is of sufficient size. A local optimality condition is derived, providing preliminary theoretical support for the well-posedness of the learning problem under appropriate conditions.

INTRODUCTION

Sparse signal models, associated to redundant signal dictionaries, are widely used in all areas of signal processing. Traditionally, sparsity is considered using a synthesis model, where high-dimensional and complex data vectors y ∈ R m are approximated using linear combinations of few elements or atoms from an overcomplete collection called dictionary:

y ≈ ∑ x k φ k = Φx, (1)
where Φ ∈ R m×q , m ≤ q, is the dictionary and its columns φ k are the atoms. A plethora of algorithms have been derived that provide sparse representations of a given input in this synthesis approach, or more generally solve linear inverse problems of the type y = Φx where x is sparse. One of the most celebrated approaches is based on 1 minimization min x x 1 s. t. y = Φx.

(2)

Dictionary Learning

Many classes of natural data (audio, images) are associated to known dictionaries (Gabor, wavelets, ...) which provide somewhat accurate approximations of any signal in the class using sparse expansions. When no specific dictionary is available, techniques have been developed to learn a dictionary from a collection of training samples, {y i } i∈I [START_REF] Aharon | K-SVD: an algorithm for designining of overcomplete dictionaries for sparse representation[END_REF][START_REF] Lewicki | Learning overcomplete representations[END_REF][START_REF] Mairal | Online learning for matrix factorization and sparse coding[END_REF][START_REF] Olshausen | Sparse coding with an overcomplete basis set: a strategy employed by V1?[END_REF][START_REF] Plumbley | Dictionary learning for L1-exact sparse coding[END_REF][START_REF] Yaghoobi | Dictionary learning for sparse approximations with the majorization method[END_REF][START_REF] Yaghoobi | Dictionary learning for sparse representations: A pareto curve root finding approach[END_REF]. There has been much practical success in this direction, and some success guarantees have been achieved [START_REF] Geng | On the local correctness of l1 minimization for dictionary learning[END_REF][START_REF] Gribonval | Dictionary identificationsparse matrix-factorisation via 1 minimisation[END_REF] for the learning problem formulated in the form of the following optimization problem, This work is supported by EU FP7, FET-Open grant number 225913 and EPSRC grant EP/F039697/1. MED acknowledges support of his position from the Scottish Funding Council and their support of the Joint Research Institute with the Heriot-Watt University as a component part of the Edinburgh Research Partnership. min X,Φ∈D

X 1 s. t. Y = ΦX, (3)
where Y ∈ R m×l = [y i ] i∈I , X ∈ R q×l and D are respectively the training signal matrix, the coefficient matrix and an admissible set of dictionaries.

Cosparsity and Cosparse Analysis Model

However, the famous sparse synthesis model has a less well-known counterpart, in analysis form, which has received much less attention in terms of algorithms, and virtually no attention at all in terms of model learning. The main contributions of the current paper are to explore this new terrain.

In this new model, called the cosparse analysis model [START_REF] Nam | Cosparse analysis modeling-uniqueness and algorithms[END_REF], signals are characterized by their parsimony in a transformed domain, using a given overcomplete analysis operator Ω ∈ R n×m :

z = Ωy (4)
where z has a minimal representation and Ω is the analysis operator. In this setting, the concept of sparsity in (4) is slightly different to the standard definition of sparsity in [START_REF] Aharon | K-SVD: an algorithm for designining of overcomplete dictionaries for sparse representation[END_REF], as the number of zero elements in z, p = nz 0 has a more important role in analyzing the model (4). It has thus been named cosparsity [START_REF] Davies | Sparse analysis models and algorithms[END_REF][START_REF] Nam | Cosparse analysis modeling-uniqueness and algorithms[END_REF], to prevent any possible confusions. In the context of linear inverse problems, it has been shown that the cosparse analysis model can lead to uniqueness guarantees that mimic that of the sparse model [START_REF] Davies | Sparse analysis models and algorithms[END_REF][START_REF] Nam | Cosparse analysis modeling-uniqueness and algorithms[END_REF], and new cosparse recovery algorithms have been designed [START_REF] Nam | Cosparse analysis modeling-uniqueness and algorithms[END_REF] .

Analysis Operator Learning

When a set of samples Y = [y i ] i∈I , is given, a question is how can we choose a suitable Ω, which provides the highest cosparsity for Y? This is the central problem considered in this paper. Specifically, our main contributions are

• a constrained optimization program for Analysis Operator Learning (AOL) based on 1 minimization, in section 2; • a practical learning algorithm, based on projected subgradient, in section 3; • empirical results suggesting that the approach works reasonably well, even when the algorithm is initialized with a random analysis operator, provided the size of the training set is sufficient, in section 4; • a local optimality condition for testing the optimality of an analysis operator with respect to the proposed program, in sections 5 and 6.

CONSTRAINED ANALYSIS OPERATOR LEARNING

The standard approach for many similar model adaptation problems, is to define a relevant optimization problem such that its optimal solution promotes maximal sparsity of Z := ΩY. A convex sparsity promoting penalty f (Ω) is the sum of absolute values of Z, i.e. f (Ω) = ΩY 1 . Unconstrained minimization of f (Ω) has some trivial solutions. A solution for such a minimization problem is Ω = 0! A suggestion to exclude such trivial solutions is to restrict the solution set to an admissible set C . We here start by investigating the problem of using a closed admissible set. AOL can now be formulated as, min

Ω ΩY 1 s. t. Ω ∈ C (5)
We here initially propose some constraints for the problem (5) and explain why some of them can not individually exclude uninteresting solutions. We finally propose a combined constraint C , which is the Uniform Normalized Tight Frame (UNTF). The proposed constraint is smooth and differentiable on its boundary.

Row norm constraints are insufficient

The first constraint is on the norms of rows of Ω, i.e. ω i = c for the i th row. By applying this row norm constraint on Ω, we find the best ω * ∈ R m , i.e. minimizer of ω T Y 1 , which repeats to generate [1,m] , and thus Rank{Ω * 1 } = 1. Such a solution is not interesting to us as we are looking for a full rank overcomplete operator. If we want to use this constraint individually and still get a reasonable solution, we need to change the objective of (5) to consider the interrelation of the rows of Ω [START_REF] Rubinstein | Training analysis operators[END_REF], which is out of the scope of this paper.

Ω, i.e. Ω * 1 := [ω i = ω] T i∈

Row norm + full rank constraints are insufficient

A full rank fixed row norm constraint C F on Ω provides solutions with very small condition numbers. To illustrate this, denote P C F the orthogonal projection onto C F , and consider A and ε respectively a random Gaussian matrix and a very small constant. The projection P C F {εA + Ω * 1 } has a low objective value in [START_REF] Elad | A generalized uncertainty principle and sparse representation in pairs of bases[END_REF]. We thus need a geometrical constraint on Ω to not allow ω i 's get arbi- trary close to each other, i.e. ω i , ω j ≈ ω i . ω j .

Tight frame constraints are insufficient

In a complete setting m = n, an orthonormality constraint can resolve the ill-conditioning of the problem. The rows of Ω are geometrically as separated as possible. Letting n > m, the orthonormality constraint is not further applicable. An alternative is the orthonormality constraint in the ambient space, ∀i = j, ω i ⊥ω j and ω i 2 = ω j 2 = 1, where ω i and ω j ∈ R n are respectively the i th and j th columns of Ω. The admissible set of this constraint is the set of tight frames in R n×m , i.e. Ω T Ω = I, where I is the identity operator in R m . The admissible set C = {Ω ∈ R n×m : Ω T Ω = I} is smooth and differentiable on its boundary.

Although the tight frame constraint could have seemed appropriate to avoid "trivial" solutions to (5), preliminary empirical and theoretical investigations indicate that when the training set is full rank, the analysis operator minimizing (5) using this constraint is always an orthonormal basis completed by zero columns [START_REF] Davies | A theorem on analysis operator learning[END_REF].

Proposed constraint: Uniform Normalized Tight Frame

This motivates us to apply an extra constraint. Here, we choose to combine the unit row norm and the tight frame constraints, yielding the UNTF constraint set. UNTF frames have many interesting properties, but to the authors' knowledge, there is no analytical method to find the projection of a point onto this set. Many attempts have been done to find such an (approximate) projection, see for example [START_REF] Tropp | Designing structural tight frames via an alternating projection method[END_REF] for an alternating projection based method.

Algorithms ?

Such a constraint C is not convex, hence the optimization problem ( 5) is non-convex. In section 3 we propose a projected subgradient solvers, which is likely to find a local minimum. When the ground truth analysis operator Ω 0 used to generate the training corpus Y is a local minimum of the optimization problem, we expect to identify it using the projected subgradient algorithm, provided we start Algorithm 1 Projected Subgradient Method for Analysis Operator Learning

1: initialization: k = 1, K max , Ω [0] = 0, Ω [1] = Ω in , γ, ε 1 2: while ε ≤ Ω [k] -Ω [k-1]
F and k ≤ K max do 3:

Ω G = ∂ f (Ω [k] ) 4
:

Ω [k+1] = P UN P T F Ω [k] -γΩ G 5: k = k + 1 6: end while 7: output: Ω out = Ω [k-1]
from a point close enough to Ω 0 . However, this can only happen if there are not "too many" spurious local minima. The experimental results, in section 4 will show that this seems to be the case, and the underlying analysis operator is reliably recovered even when the algorithm is started far from it, when the size of training set I is large enough.

PROJECTED SUBGRADIENT ALGORITHM FOR AOL

Subgradient methods have often been used to minimize convex objectives, when the solution is sought only with a few significant figures. These methods are generally slow to find exact solutions, as they converge linearly. In the AOL, we also need to solve (5) to find a solution with a reasonable precision. As the problem is constrained, we use the projected subgradient method. The subgradient of the objective is simply ∂ f (Ω) = Ysgn(Ω T Y) T , where sgn is the extended sign function defined as follows,

{sgn(A)} i j =sgn(A i j ) sgn(a) =    1 a > 0, [-1, 1] a = 0, -1 a < 0. (6)
In the projected subgradient methods, we have to choose a value in the set of subgradients. We randomly choose a value in [-1, 1], when the corresponding element is zero. Projection of an operator, with non-zero rows, onto the space of fixed row norm frames is easy and can be done by normalizing each row to have c norm, we use P UN to denote this projection. If a row is zero, any normalized vector has the same distance to the zero vector, and we thus choose a normalized random vector. The projection can be found by,

P UN {Ω} = [P UN {ω i }] i , (7)
P UN {ω} := ω ω 2 ω 2 = 0 ν otherwise,
where ν is a random vector on the unit sphere. Projection of a full rank matrix onto the tight frame manifold is also easy and can be done using a singular value decomposition of the linear operator [START_REF] Tropp | Designing structural tight frames via an alternating projection method[END_REF]. Let A ∈ R n×m be the given point and A = U Σ V T be a singular value decomposition of A and I n×m be a diagonal matrix with identity on the main diagonal. The projection of A can be found using,

P TF (A) = U I n×m V T . ( 8
)
As mentioned in section 1, a point on the intersection of the uniformly normalized set and the set of tight-frames can often be found by alternatingly projecting onto these sets. Note that, there is no guarantee for convergence to an UNTF using this method, but this technique practically works very well [START_REF] Tropp | Designing structural tight frames via an alternating projection method[END_REF]. As the projected subgradient continuously changes the current point, which needs to be projected onto the UNTF's, we only use a single pair of projections at each iteration of the algorithm. In practice we found that the solutions converge to UNTF's. A pseudocode of this algorithm is presented in Algorithm 1. As the projection onto the admissible set is not exact, although we have not observed instability by choosing a small gradient step size, convergence of this algorithm needs to be investigated in the future work.

EMPIRICAL EVIDENCE

In this section we present some simulation results to empirically show the local optimality of synthetic generative operators. We also empirically demonstrate the convergence of the proposed algorithm and the global recovery of the operator, when the size of the training set is large.

A random operator Ω 0 -∈ R 24×16 was generated using i.i.d. zero mean, unit variance normal random variables 1 . The generative analysis operator Ω 0 is made by alternatingly projecting Ω 0 -onto the sets of UN's and T F's. A set of training samples was generated, with different cosparsities, by randomly selecting a normal vector in the orthogonal complement space of a randomly selected p rows of Ω 0 . Such a vector y i has (at least) p zero components in Ω 0 y i , and it thus is p cosparse. To initialize the proposed algorithm, we used a linear model to generate the initial Ω by combining the generative operator Ω 0 and a normalized random matrix N, i.e. Ω in = Ω 0 + λ N, and then alternatingly projecting onto UN and T F. It is clear that when λ is zero, we actually initialize Ω with the generative model Ω 0 and when λ → ∞, the initial Ω in will be random.

1 Ω 0 -is not necessarily a UNTF and needs to be projected onto the set of UNTF's. In the first experiment, we chose a set of size l = 768 of such training corpus. The projected subgradient method was iterated 50000 times. To check the local optimality of the operator and the size of basin of attraction, we chose λ = 0, 1, and 10. The average percentage of operator (rows) recovery, i.e. the maximum 2 distance of any recovered row and the closest row of the generative operator, is not more than √ .001, for different cosparsity and 100 trials, are plotted in Figure 1. We practically observe that the operator is the local optimum even when the cosparsity of the signal is as low as 3. We also see that the average recovery reduces by starting from a point far from the the actual generative operator. When λ becomes large, we do not further degrade the average recovery as the initial point is close to random, projected onto the admissible set. We also repeated the simulations with a (pseudo-) random admissible Ω in . The result is shown in Figure 2. The average recovery is very similar to the previous experiment, when λ is large, which confirms our expectation.

We now investigate the role of l on the average operator recovery by some simulations. We kept using previous experiment settings and repeated the simulations for two new training sets, with populations of l = 384 and 1536, which are 1/2 and 2 times of the population in the previous experiments. We show the average operator recovery for λ = 0, 1, 10 in Figure 3. The simulation results show not only that Ω 0 can be locally identified with even less cosparse signals, smaller p, but the basin of attraction is also extended and now the generative operator can be recovered by starting from a distant initial point, even using 2 cosparse signals.

The promising results in robust recovery of the generating operator encourage us to investigate the proposed framework by characterizing the local optima of (5) in a general setting. This helps us to check, if a given Ω 0 is a local minimum, and thus locally identifiable using a gradient based solver. We briefly introduce such a qualification for a local optimality in the next section. It is derived using the convexity of the objective and the smoothness of the boundary of C . The local optimality, for the UNTF admissible set, is also investigated as a special case and a straightforward qualification is derived. The result has a flavor similar to those obtained in the context of dictionary learning for the synthesis model [START_REF] Gribonval | Dictionary identificationsparse matrix-factorisation via 1 minimisation[END_REF], as well as in hybrid synthesis/analysis framework [START_REF] Jaillet | An L1 criterion for dictionary learning by subspace identification[END_REF].

LOCAL OPTIMALITY OF AN OPERATOR

The set of matrices in R n×m equipped with the Hilbert-Schmidt inner-product, defined by A, B := tr{A T B}, is an inner-product space. The 1 function can now be reformulated as A 1 = A, sgn(A) , where sgn(•) is the element-wise sign operator.

The set of zeros of Z n×l = ΩY, has an important role in variational analysis of the 1 objective. Let Λ be the index set of non-zero elements of Z and Λ be Λ's complement. Note that the i th column of Z, z i , is found by Ω y i . y i is cosparse, when z i has some zero elements. The cosparsity of the matrix Y can similarly be found by |Λ|. We define X Λ as the matrix which has the components of X on the support Λ, and the rest is zero.

Theorem 1 (Optimality Condition) Let C be a compact set and T C (Ω) be the tangent cone at a boundary point Ω. If Z = ΩY, the necessary and sufficient condition for a boundary point Ω being a local optimum of ( 5) is to satisfy the following inequality for any non-zero

Θ ∈ T C (Ω), | ΘY, sgn(Z) | < (ΘY) Λ 1 , ( 9
)
where Λ is the index set of zeros of Z.

A complete proof of this theorem will be shown in [START_REF] Davies | A theorem on analysis operator learning[END_REF]. We here present a sketch of the proof. The objective of ( 5) is convex and C is smooth on boundary. Using the variational analysis, a boundary point Ω with this setting is a local optimum, if the directional derivative of the objective in Ω is positive for any (non-zero) direction in the tangent space of C at Ω, see [START_REF] Rockafellar | Variational Analysis[END_REF]Theorem 13.24 and Example 13.25]. This theorem can be seen as an extension of Theorem 1 in [START_REF] Jaillet | An L1 criterion for dictionary learning by subspace identification[END_REF], where it is derived in a vector space, under unit norm constraint. The new theorem is more general, and we only need to check [START_REF] Gross | Recovering low-rank matrices from few coefficients in any basis[END_REF] for any non-zero vectors in T C (Ω). Unfortunately this is not an easy task in general. In the next section we reformulate [START_REF] Elad | A generalized uncertainty principle and sparse representation in pairs of bases[END_REF], with a UNTF admissible set. The tangent space of C can be analytically derived. We thus give an explicit qualification for the optimality.

UNIFORM NORMALIZED TIGHT FRAME LEARNING

The set of UNTF's in R n×m , m < n, was defined in the introduction as C = {∀Ω ∈ R n×m : Ω T Ω = I, ∀i ω i 2 = c}, where c = m n and ω i is the i th row of Ω. Note that C is now a manifold and any admissible point is thus on the boundary of C . For a given training Y m×l , l m, the operator learning problem (5) can now be reformulated as,

min Ω ΩY 1 s. t. Ω T Ω = I ∀i ω i 2 = c. (10)
To check the optimality of an admissible point Ω 0 , we can slightly deviate Ω 0 in a direction ∆ in the tangent space of C at Ω 0 , i.e. Ω = Ω 0 + ∆. The tangent space of C is found by letting the directional derivative

2 of h(Ω) = Ω T Ω -I and h(ω i ) = ω i -c, for each i, be zero, ∂ h(Ω)(∆) = ∆ T Ω + Ω T ∆ = 0, (11)
and

∂ h(ω i )(δ i ) = 2ω i δ T i = 0, (12)
which is simply ω i , δ i = 0. We can now rewrite (10) using the new variable ∆,

min ∆ (Ω 0 + ∆)Y 1 s. t. ∆ T Ω 0 + Ω T 0 ∆ = 0 ∀i ω 0 i , δ i = 0, ( 13
)
where δ i is the i th row of ∆. (13) is a convex problem with linear constraints. If the new problem has a single solution ∆ = 0, Ω 0 2 For the definition of directional derivative, see for example [START_REF] Dattorro | Convex Optimization and Euclidean Distance Geometry[END_REF].

would then be a local minimum for [START_REF] Jaillet | An L1 criterion for dictionary learning by subspace identification[END_REF]. To find a qualification that guarantees 0 to be the only solution of (13), we reparametrize the problem. Let Z 0 := Ω 0 Y and ∆ Z := Z -Z 0 = ∆Y. As Y is full rank, we can define Θ := Y T (YY T ) -1 and find ∆, respectively δ i , using,

∆ = ∆ Z Θ , δ i = δ i Z Θ.
To reformulate [START_REF] Nam | Cosparse analysis modeling-uniqueness and algorithms[END_REF] and use a higher dimension problem based on ∆ Z , we should consider the fact that each row of Z, z i , can only live in the subspace spanned by rows of Y. This means that z i does not have any component in the null space of Y. We can now introduce an extra constraint to consider this fact by ∆ Z (I -ΘY) = 0. If we define P := I -ΘY, the problem can be reformulated as, min

∆ Z Z 0 + ∆ Z 1 s. t. Θ T ∆ T Z Z 0 Θ + Θ T Z T 0 ∆ Z Θ = 0 ∀i Z i 0 Θ, δ i Z Θ = 0 ∆ Z P = 0. ( 14
)
Let a vector to matrix operator "vect{•}" be defined such that,

{vect{Z}} k = Z i j , i = k mod l 1 ≤ k ≤ nl j = k -1 l + 1,
and the corresponding inverse operator be "mat{•}". If η := vect{∆ Z } and z 0 := vect{Z 0 }, the constraint of ( 14) is linear and can be presented as Φη = 0. Appendix A explains how to derive Φ. The last reformulation is to rewrite (14) in a vector form,

min η z 0 + η 1 s. t. Φη = 0. (15)
Note that the solution of this problem can be mapped to the matrix form using "mat" operator. ( 15) is the formulation which has been used to show the optimality of a vector z 0 for the problem (2), i.e. an admissible z 0 is the optimal solution of (2) iff η = 0 is the only solution of [START_REF] Plumbley | Dictionary learning for L1-exact sparse coding[END_REF]. Such a qualification has been derived as follows:

η λ 1 < η λ 1 , ∀η = 0 ∈ N Φ , (16)
is the necessary and sufficient condition for the optimality of z 0 , where λ := sup{z 0 } and N Φ is the null-space of Φ [START_REF] Elad | A generalized uncertainty principle and sparse representation in pairs of bases[END_REF][START_REF] Gribonval | Sparse representations in unions of bases[END_REF].

Remark 1 The extra constraint ∆ Z P = 0 in ( 14) can be interpreted as the orthogonality of each row of ∆ Z to the null-space of Y. Therefore the size of this constraint is actually n × (lm), which is smaller than n × l, the size that is related to a full rank P.

Remark 2

The derived local optimality condition is a deterministic qualification. How to check this qualification, is another important question which we do not answer here in this paper. The authors admit that such a test is not easy, but it might be possible to deliver a probabilistic statement using the new technique recently introduced in the matrix completion context [START_REF] Gross | Recovering low-rank matrices from few coefficients in any basis[END_REF].

Remark 3 For any 1 < p < m , we can generate an (at least) p cosparse signal with a given cosparsity pattern. It means that, we can choose a signal in the orthogonal complement of the space spanned by the rows of Ω corresponding to the p cosparsity pattern. Note that if ( 16) is valid for any |λ | = p, it is valid for any larger index sets including λ . This helps us not to worry about checking [START_REF] Rockafellar | Variational Analysis[END_REF] for the situations that by choosing a vector in the orthogonal complement space of the selected p columns of Ω, other elements also vanish.

CONCLUSIONS

This paper presented a new framework for low-dimensional signal model adaptation. The linear model, which is here called the analysis operator, can be used to sparsify a classes of signals. The new framework helps to apply various constraints to the operators. A simple, but efficient, algorithm based on the projected subgradient technique, was also presented to recover such operators. The algorithm relies on the projection onto the constraint set. We used the algorithm to first practically show the local optimality of the operator for the proposed optimization problem, which it shows identifiability of the operator using gradient based methods. We then tested the algorithm to recover the synthetic random operators in another experiment, when the algorithm is fed with a point in the neighborhood of the true analysis operator. We demonstrated that the operator is usually recovered when the training corpus is large enough. Even when we do not know a neighborhood of the generative operator, we practically showed that there is still a good chance to recover the operator, when the training signals are enough cosparse.

In the second half of the paper, the local identifiability of such an operator is investigated and a necessary and sufficient qualification was presented. An example of such constraints, i.e. UNTF, which was empirically shown to be a reasonable constraint for this problem, was investigated in more detail and a more sensible qualification for the identifiability of operators, was derived. The qualification is deterministic but difficult to check. Checking such a qualification is left for future work.

APPENDIX A. DERIVING A LINEAR REPRESENTATION OF THE CONSTRAINTS

Let the subscript • i j be the element located in the i th row and j th column of the operand. To show linearity of the constraint, we only need to represent each constraint as a weighted sum of δ Z i j . Al- though the correct notation is based on the vector subindexing of vector η, we keep using the matrix form of subindexing as it is easier to understand for the readers. A mapping between two types of subindexing is easy, as η = vect{∆ Z }. (14) has three constraints, where the last two are clearly linear, and they can be represented as,

∑ k δ Z i k P k j = 0, ∀i ∈ [1, n], j ∈ [1, l] ∑ k δ Z i k Q ki = 0, ∀i ∈ [1, n], (17)
where Q := ΘΘ T Z T 0 . Let A := Z 0 Θ. The first constraint can now be reformulated as, Θ T ∆ T Z A + A T ∆ Z Θ = 0. To derive this equation in a similar form to (17), we reformulate ∆ Z Θ, then left-multiply the result with A T .

{Θ T ∆ T Z } i j = 1≤q≤l ∑ q δ Z j q Θ qi (18) {(Θ T ∆ T Z ) A} i j = 1≤k≤n ∑ k 1≤q≤l ∑ q δ Z k q Θ qi A k j = 1≤k≤n ∑ k 1≤q≤l ∑ q Θ qi A k j δ Z k q (19)
We now reformulate the first constraint as, 1≤k≤n

∑ k 1≤q≤l ∑ q Θ qi A k j + Θ q j A ki δ Z k q = 0 , ∀i, j ∈ [1, m] (20)
We can generate Φ (n l+m 2 +n)×n l using the weight of ( 17) and ( 20), corresponding to the vector η, and make the linear presentation as Φ η = 0.

Figure 1 :

 1 Figure 1: The average percentage of operator recovery for different λ 's, where λ controls how far is the starting point Ω in form Ω 0 . The x-axis presents the cosparsity of the synthetic data.

Figure 2 :

 2 Figure 2: The average percentage of operator recovery with the random starting point. The x-axis presents the cosparsity of the signals.

Figure 3 :

 3 Figure 3: The average percentage of operator recovery with different training set population size l. The x-axis presents the cosparsity of the signals.