Adélaïde Doray

Roxane Lemoine

Marc Severin

Stéphanie Chadet

Osbaldo Lopez-Charcas

Audrey Héraud

Christophe Baron

Pierre Besson

Arnaud Monteil

Falsig Stine

Pedersen

PhD Sébastien Roger
email: sebastien.roger@univ-tours.fr

PEDERSEN Stine Falsig

PhD Raphael Rappetti-Mauss
email: raphael.rapetti-mauss@univ-cotedazur.fr

PhD Philippe Chavrier
email: philippe.chavrier@curie.fr

PhD Aubin Penna
email: aubin.penna@univ-poitiers.fr

Stine Falsig Pedersen

THE VOLTAGE-GATED SODIUM CHANNEL BETA4 SUBUNIT MAINTAINS EPITHELIAL PHENOTYPE IN MAMMARY CELLS

Keywords: NaVβ4, epithelial phenotype, β-catenin, Epithelial-to-Mesenchymal Transition, mammary cells NaVβ4, epithelial phenotype, β-catenin, Epithelial-to-Mesenchymal Transition, mammary cells

published or not. The documents may come

Introduction

Voltage-gated sodium channels β (NaVβ) proteins, encoded by SCNxB genes, define a family of four transmembrane proteins with a short C-terminal intracellular domain and a large N-terminal Immunoglobulin (Ig)-like extracellular domain [START_REF] Brackenbury | Na Channel beta Subunits: Overachievers of the Ion Channel Family[END_REF]. These proteins have initially been characterized as auxiliary subunits of sodium channels [START_REF] Messner | The sodium channel from rat brain. Separation and characterization of subunits[END_REF]. Indeed, they were isolated along with pore-forming voltage-gated sodium channel (NaVα) isoforms, which they interact with through covalent or non-covalent associations [START_REF] Mccormick | Molecular determinants of Na+ channel function in the extracellular domain of the beta1 subunit[END_REF][START_REF] Meadows | The intracellular segment of the sodium channel beta 1 subunit is required for its efficient association with the channel alpha subunit[END_REF][START_REF] Chen | Identification of the cysteine residue responsible for disulfide linkage of Na+ channel alpha and beta2 subunits[END_REF][START_REF] Gilchrist | Crystallographic insights into sodium-channel modulation by the beta4 subunit[END_REF]. Hence, NaVβ proteins regulate NaVα membrane trafficking as well as their biophysical (Calhoun and Isom, 2014) and pharmacological properties (Lenkowski et al., 2003, Zhang et al., 2013, Wilson et al., 2011). Along with these roles as sodium channel activity modulators, NaVβ proteins have also been proven to conduct other specific cellular functions in excitable cells (Isom, 2001), such as an important role as cell adhesion molecules (CAMs), allowing for both trans-homophilic and trans-heterophilic cell-cell and cell-matrix adhesions in cells expressing NaVα isoforms (Isom, 2002). Therefore, the expression and roles of NaVβ are best known in excitable cells such as in neurons in which they participate in the regulation of Na + influx but also control neurite outgrowth, axonal fasciculation and interaction with glial cells (O' Malley and Isom, 2015).

The most recently identified NaVβ4 subunit, encoded by the SCN4B gene, is the least characterized member of the family. It was initially demonstrated to be expressed in the nervous system (dorsal root ganglia, spinal cord and restricted areas or nuclei in the brain), in skeletal and cardiac muscle cells (Yu et al., 2003). It shares sequence similarity with the NaVβ2 subunit and engages in covalent interactions with NaVα in the extracellular Ig domain (Yu et al., 2003[START_REF] Gilchrist | Crystallographic insights into sodium-channel modulation by the beta4 subunit[END_REF]. NaVβ4 was demonstrated to control NaVα activity and particularly the generation of resurgent (Grieco et al., 2005, Miyazaki et al., 2014, White et al., 2019) or persistent (Aman et al., 2009, Bant andRaman, 2010) sodium currents in neurons. NaVβ4 was also identified to participate in cell-cell adhesion [START_REF] Shimizu | Parallel homodimer structures of the extracellular domains of the voltage-gated sodium channel beta4 subunit explain its role in cell-cell adhesion[END_REF] and neurite extension [START_REF] Miyazaki | BACE1 modulates filopodia-like protrusions induced by sodium channel beta4 subunit[END_REF]. Evidence is shown that NaVβ4 dysregulation is involved in epilepsy [START_REF] Sheilabi | The Sodium Channel B4-Subunits are Dysregulated in Temporal Lobe Epilepsy Drug-Resistant Patients[END_REF], in Rett Syndrome [START_REF] Oginsky | Hyperexcitability of Mesencephalic Trigeminal Neurons and Reorganization of Ion Channel Expression in a Rett Syndrome Model[END_REF] and mutations in the SCN4B gene have been linked to cardiac arrythmia [START_REF] Li | Mutations of the SCN4B-encoded sodium channel beta4 subunit in familial atrial fibrillation[END_REF], Yang et al., 2019, Xiong et al., 2019[START_REF] Medeiros-Domingo | SCN4B-encoded sodium channel beta4 subunit in congenital long-QT syndrome[END_REF] and sudden death syndromes [START_REF] Tan | Sudden infant death syndrome-associated mutations in the sodium channel beta subunits[END_REF].

Recently, we identified a critical role for NaVβ4 in cancer progression (Bon et al., 2016). Specifically, we have shown that NaVβ4 was strongly expressed in normal epithelial cells and tissues from breast, colon, rectum, lung and prostate but consistently downregulated in cancer samples, to be almost absent in high-grade primary and metastatic tumours (Bon et al., 2016). In mammary cancer cells, reducing NaVβ4 expression potentiated cell migration and invasiveness through the acquisition of a hybrid mesenchymal-amoeboid aggressive phenotype which resulted in an increase in mammary tumour growth and a higher metastatic colonisation. This effect was independent of NaVα channel activity (Bon et al., 2016). Later on, similar results were obtained in cervical cancer cells [START_REF] Sanchez-Sandoval | Contribution of voltage-gated sodium channel beta-subunits to cervical cancer cells metastatic behavior[END_REF] and the preserved expression of SCN4B in papillary thyroid cancer was proposed to be a favourable indicator of recurrence-free survival [START_REF] Gong | Preserved SCN4B expression is an independent indicator of favorable recurrence-free survival in classical papillary thyroid cancer[END_REF]. It was therefore suggested that the SCN4B gene might be considered as a metastasis-suppressor gene (Bon et al., 2016). Nevertheless, the function of NaVβ4 in normal epithelial cells is not known. We therefore hypothesized that NaVβ4 is important for epithelial phenotype. To test this, we explored the consequences of reducing NaVβ4 expression in non-cancer MCF10A mammary cells. We show that knocking-down NaVβ4 induced a loss of epithelial phenotype that is reminiscent to early stages of carcinogenesis.

Results

Bioinformatics gene expression analyses, using The Cancer Genome Atlas (TCGA) and the UCSC Xena browser (https://xenabrowser.net), confirmed initial studies (Bon et al., 2016) indicating that the SCN4B gene, encoding for NaVβ4, is significantly down-regulated in all breast cancer stages compared to adjacent non-tumoral breast tissues (Fig. 1a). A lower expression of SCN4B was even identified in stage IIA compared to stage I (Fig. 1a). In line with these initial results, NaVβ4 protein expression was almost 10 times lower in human mammary cancer MDA-MB-231 compared to noncancer MCF10A cells (Fig. 1b). In order to assess the potential role of NaVβ4 in non-cancer mammary cells we developed, using the CRISPR/Cas9 technique, a cell line derived from the parental MCF10A with a permanent knockdown of NaVβ4, named MCF10A Crβ4, as well as a control cell line MCF10A CTL. The efficiency of NaVβ4 knock-down was verified by western blotting indicating a median reduction of protein expression of 85% (Fig. 1c). Reduction of NaVβ4 had important effects on MCF10A cell morphology, as observed in conventional 2-dimensional cultures (Fig. 1d).

Particularly, MCF10A Crβ4 showed an elongated morphology characterised by an increased cell length as compared with MCF10A CTL (Fig. 1e, 89.9 ± 5.1 vs. 33.1 ± 1.7 µm, respectively). Furthermore, while MCF10A CTL cells showed a tendency for being closely associated in clusters, as epithelial cells do, MCF10A Crβ4 were more scattered and established less intercellular interactions (Fig. 1f). These morphological changes were also identified by epifluorescence imaging in fixed cells labelled to visualize cell nuclei and F-actin network (Fig. 1g). Image analysis demonstrated that MCF10A Crβ4 showed a greater cell surface than MCF10A CTL cells (Fig. 1h, 2,318 ± 123 vs. 1,062 ± 52 µm 2 , respectively). Interestingly, MCF10A Crβ4 also appeared to have a stronger intensity of F-actin labelling (Fig. 1i), characterised by the presence of actin stress fibres (Fig. 1g).

These initial data suggested that the loss of NaVβ4 expression affected epithelial cell phenotype and intercellular organization. We next investigated how the loss of NaVβ4 affected cell polarisation in 3 dimensions, using a cyst forming assay in a 3D extracellular matrix [START_REF] Jensen | The Na(+) /H(+) exchanger NHE1 localizes as clusters to cryptic lamellipodia and accelerates collective epithelial cell migration[END_REF]. We found that, in 3D cultures, MCF10A CTL cells formed regular cysts with a general spheroid morphology (Fig. 2a), characterized by a circularity index (calculated from pictures) approaching 0.9, maintained during the 3 weeks of the experiment (Fig. 2b). MCF10A CTL cysts demonstrated a regular growth that was slightly slowed after 14 days culture (Fig. 2c). After this time, the number of remaining individualized cysts tended to decrease, mostly due to the dispersion of some cells into the extracellular matrix (Fig. 2d). MCF10A Crβ4 cysts also exhibited a generally spheroid shape, albeit demonstrating a lower circularity index (Fig. 2a,b), but their growth was slower compared to MCF10A CTL cysts (Fig. 2c). This could be partly due to a reduced cell proliferation rate. Indeed, loss of NaVβ4 expression slightly reduced MCF10A cell viability after 5 days of culture (Suppl. Fig. 1a) which appeared to be due to a slower DNA synthesis (Suppl. Fig. 1b,c). However, it also appeared that cells evaded from MCF10A Crβ4 cysts to invade the 3D extracellular matrix (Fig 2a,arrows), thus resulting in the complete disorganisation of cysts and in the reduction of the number of individualized cysts with time (Fig. 2a,d). We thus characterized cysts ultrastructure. Cysts were fixed and stained in order to visualise cell nuclei, vimentin expression and F-actin network by fluorescence imaging. MCF10A CTL cysts demonstrated a typical and regular epithelial organisation with compacted cell nuclei uniformly distributed within the structure, the expression of vimentin at the external periphery of the cyst (equivalent to the basal side of the epithelial layer), contiguous cells displaying a cortical submembrane F-actin network, and the appearance of a central lumen (Fig. 2e).

By contrast, MCF10A Crβ4 cysts were completely disorganized, lacked a central lumen and did not show any apico-basal polarisation (Fig. 2e). Cell nuclei appeared to be less compacted, vimentin expression was more intense and diffuse, and cortical F-actin network was disrupted to be more diffuse inside the cytosol. Furthermore, multiple cells detached from the cyst to invade the extracellular matrix. Because this property might require proteolytic activities, we explored the capacity of MCF10A cysts to degrade the fluorogenic substrate DQ-Gelatin when incorporated into the extracellular matrix, after 7 days growing (Fig. 2f). MCF10A CTL cysts demonstrated almost undetectable proteolytic activity. By contrast MCF10A Crβ4 cysts showed a strong proteolytic activity towards the extracellular matrix (green fluorescence) and a strong invasion by cells originating from the cyst (Fig 2f,g). Because cell-cell junctions appeared to be disrupted in MCF10A

Crβ4 cysts, we then analysed the expression of adherens junction proteins β-catenin and E-cadherin.

While MCF10A CTL cysts were characterized by a strong colocalization of β-catenin and E-cadherin at the plasma membrane of cells, MCF10A Crβ4 cysts demonstrated a weaker and diffuse expression of both proteins with no colocalization (Fig. 3a). These observations prompted us to investigate the expression of β-catenin depending on that of NaVβ4. Consistent with the more mesenchymal phenotype, the mRNA level of gene encoding for β-catenin, CTNNB1, was significantly higher in MCF10A Crβ4 compared to MCF10A CTL cells (40% median increase, Fig. 3b). In contrast, βcatenin protein level was down-regulated in MCF10A Crβ4 compared to MCF10A CTL cells (52% median decrease, Fig. 3c). A possible explanation for this apparent discrepancy could be an increased recycling of β-catenin in MCF10A Crβ4 cells. Therefore, we incubated MCF10A CTL and MCF10A

Crβ4 cells with the proteasome inhibitor MG132 for different time durations and assessed levels of β-catenin (Fig. 3d). These led to an increased immunodetection of β-catenin in both cell lines.

Nevertheless, the increase in β-catenin protein expression was significantly higher in MCF10A Crβ4 than in MCF10A CTL cells (Fig. 3e). Similarly, MCF10A cells transfected with specific silencing RNA targeting NaVβ4 (siβ4, Fig. 3f) also displayed a reduced expression of β-catenin compared to scramble siRNA (siCTL, Fig. 3g) and this expression was restored upon treatment with MG132 (Fig. 3h). These results suggested that the loss of the plasma membrane NaVβ4 leads to the disruption of intercellular junctions and to a reduced half-life of β-catenin. Together with the loss of the epithelial morphology in individual cells, the disorganisation and the increased extracellular invasiveness of cysts, the disruption of adherens junctions argued in favour of a transition towards a mesenchymallike phenotype. Therefore, we next examined the expression of genes associated with either epithelial phenotype (CDH1, encoding for E-cadherin) or with mesenchymal phenotype (CDH2, encoding for N-cadherin; SNAI1, SNAI2, TWIST, ZEB1 encoding for transcription factors promoting epithelial-tomesenchymal transition, VIM, encoding for vimentin, and ACTA2, encoding for α-SMA). The loss of NaVβ4 expression was associated with a significant reduction in the epithelial marker CDH1 expression, and an increase in the expression of mesenchymal markers CDH2, SNAI2, TWIST, ZEB1, VIM and ACTA2 (Fig. 4a). Only SNAI1 expression was not modified. These changes observed at the transcriptional level were confirmed at the protein level for E-Cadherin, N-Cadherin, Vimentin and α-SMA (Fig. 4b). Finally, we explored the possibility to rescue the epithelial phenotype in MCF10A

Crβ4 by overexpressing NaVβ4 (Fig. 4c). This partially restored CDH2, VIM and ACTA2 expression (Fig. 4d).

Discussion

Pore-forming NaVα and auxiliary NaVβ proteins of voltage-gated sodium channels were initially characterized in excitable cells in which they are responsible for the triggering and the propagation of action potentials. However, over the past years it has been shown that both NaVα and NaVβ subunits are dysregulated in cancers, in which they have non-excitable roles. Most of the time, their overexpression in carcinoma cells has been associated with cancer progression, and their activity was shown to promote pro-cancerous properties [START_REF] Lopez-Charcas | Pharmacological and nutritional targeting of voltage-gated sodium channels in the treatment of cancers[END_REF]. Pore-forming NaV1.5, NaV1.6 and NaV1.7 appear to be specifically up-regulated in cancer cells and their activity, through an inward sodium current, has been shown to promote invasive properties [START_REF] Roger | Voltage-gated sodium channels and cancer: is excitability their primary role?[END_REF]. It has also been found that some NaVβ subunits are up-regulated in cancer and that they bear important roles in cancer cell biology and in cancer progression [START_REF] Lopez-Charcas | Pharmacological and nutritional targeting of voltage-gated sodium channels in the treatment of cancers[END_REF]. In contrast, the NaVβ4 subunit has recently been identified as being expressed at high levels in normal epithelial tissues, but consistently down-regulated in cancer samples. This was identified in breast, lung, prostate and colorectal cancer (Bon et al., 2016), in cervical cancer (Sanchez-Sandoval and Gomora, 2019) and in papillary thyroid cancer [START_REF] Gong | Preserved SCN4B expression is an independent indicator of favorable recurrence-free survival in classical papillary thyroid cancer[END_REF]. In these studies, the preserved expression of the SCN4B gene was considered as a favourable biomarker of metastasis-free survival suggesting that it could be a metastasis-suppressor gene.

The expression of the SCN4B gene was shown to be lower in highly invasive cancer as compared to weakly invasive or to non-cancer cells (Bon et al., 2016[START_REF] Sanchez-Sandoval | Contribution of voltage-gated sodium channel beta-subunits to cervical cancer cells metastatic behavior[END_REF][START_REF] Roger | Voltage-gated sodium channels potentiate the invasive capacities of human non-small-cell lung cancer cell lines[END_REF]. Both in vitro and in vivo experiments, performed with different human cancer cell lines, demonstrated that reducing NaV 4 expression potentiated cell migration, invasiveness and tumour progression, while overexpressing it had opposite effects (Bon et al., 2016[START_REF] Sanchez-Sandoval | Contribution of voltage-gated sodium channel beta-subunits to cervical cancer cells metastatic behavior[END_REF][START_REF] Diss | Beta-subunits of voltage-gated sodium channels in human prostate cancer: quantitative in vitro and in vivo analyses of mRNA expression[END_REF]. In breast cancer cells, the loss of NaV 4 promoted RhoA activity and the acquisition of a hybrid mesenchymal-amoeboid phenotype associated with highly invasive capacities (Bon et al., 2016). However, the role of NaV 4 in normal epithelial cells is not known, and whether its loss might participate to events of early carcinogenesis has never been characterized.

In this study, we confirmed that SCN4B / NaV 4 is down-regulated in cancer compared to non-cancer cells and tissues. Furthermore, we demonstrated that its repression in non-cancer mammary epithelial cells induced dramatic morphological and functional changes associated with a loss of epithelial polarisation, a disruption of epithelial junctions, the overexpression of genes associated with a mesenchymal phenotype and the acquisition of pro-invasive capacities. Consequently, loss of NaV 4 completely abrogated the establishment of 3D epithelial structures (cysts). The loss of NaV 4 favoured the degradation of β-catenin, thus reducing its half-life and leading to the disruption of adherens junctions. It can be speculated that the NaV 4 subunit, expressed at the plasma membrane of epithelial cells, stabilizes β-catenin in proximity with E-cadherin. Loss of NaV 4 might lead to cytosolic release of β-catenin which could, for one part translocate to the nucleus and induce the expression of EMT-related genes, and for a second part undergo proteasomal degradation. These cellular events might be critical during carcinogenesis.

However, what could trigger the repression of SCN4B during cancer transformation is still unknown.

Interestingly, recent studies pointed out the involvement of several miRNAs that are dysregulated in some cancers, as important regulators of SCN4B expression. In colorectal cancer, the increased expression of miR-424-5p in tumour samples was associated with poor prognosis [START_REF] Dai | miR-424-5p promotes the proliferation and metastasis of colorectal cancer by directly targeting SCN4B[END_REF].

In this study, the authors demonstrated that SCN4B was directly inhibited by miR-424-5p thus promoting colon cancer cell proliferation, migration and invasion [START_REF] Dai | miR-424-5p promotes the proliferation and metastasis of colorectal cancer by directly targeting SCN4B[END_REF]. Another miRNA, miR-3175, has recently been shown to be overexpressed in prostate cancer and to participate in cancer cell growth and invasion. Knocking down miR-3175 in prostate cancer cells increased SCN4B and Ecadherin expression, inhibited N-cadherin expression, and importantly reduced cell proliferation, migration and invasion [START_REF] Huang | Silencing of microRNA-3175 represses cell proliferation and invasion in prostate cancer by targeting the potential tumorsuppressor SCN4B[END_REF]. Together, these recent studies brought new elements on the involvement of miRNA in the promotion of cancer progression through the regulation of SCN4B expression in tumours. As such, exploring the expression of these miRNAs during carcinogenesis might be of importance.

In conclusion, this study demonstrates the critical role played by NaV 4 in the maintenance of an epithelial phenotype in normal cells, and how its repression might contribute to cellular dysplasia and early carcinogenesis. In this context, maintaining its expression level in epithelial cells would be determinant in order to prevent, or delay, tumour transformation. Statistically different at * p<0.05 (Mann-Whitney rank sum test), otherwise no statistical difference. mediated knock-down was assessed by western blotting and stability of cells was followed for a minimal duration of 6 weeks. Mycoplasma contamination tests were performed routinely (Lonza,

MycoAlert™ Mycoplasma Detection Kit).

Small interfering RNA transfection -MCF10A mammary epithelial cells were transfected with siRNA directed against human SCN4B mRNA (siβ4) or scramble siRNA as a control (siCTL), both of which were purchased from ON-TARGETplus siRNA (Horizon Discovery, Cambridge, UK).

Cells were transfected using Pepmute™ siRNA Transfection Reagent (SinaGen laboratories, USA).

Experiments were performed 24h-48h after transfection and efficacy of silencing was assessed by western blotting.

Overexpression of SCN4B gene -MCF10A Crβ4 cells were transfected with 2 µg of a pcDNA3.1(+) plasmid containing the SCN4B gene (pcSCN4B, Synbio Technologies, USA) or with a pcDNA3.1(+) empty vector as a control. Transfection was realized using Lipofectamine 2000 (Invitrogen, France).

RNA extraction, reverse transcription (RT) and quantitative-polymerase chain reaction

(qPCR) -Total RNA was extracted using TRIzol™ Reagent (Invitrogen, France), quantified by measuring absorbance at 260 nm using Nanodrop 2000™ (Thermofisher, France) and reversetranscribed with the PrimeScript™ RT Reagent Kit (Takara Bio Group, France). Quantitative PCR were performed using SYBR qPCR Premix Ex Taq (Takara Bio Group, France) and LightCycler 384 wells (Roche, France). Control gene was HPRT1. All primers sequences and corresponding efficiencies are described in Supplementary Table I.

Cell viability -cell viability was evaluated by the tetrazolium salt assay (MTT) as already described [START_REF] Jelassi | P2X(7) receptor activation enhances SK3 channels-and cystein cathepsin-dependent cancer cells invasiveness[END_REF]. Briefly, cells were seeded in different densities indicated in the figure and grown for 4 days at 37°C and 5% CO2 in their normal culture medium. Cell viability was measured after incubation with 0.5 mg/mL MTT for 60 minutes at 37°C by measuring the absorbance at 540 nm.

Cell proliferation -Cells were seeded at a density of 2x10 5 in wells of a 6-well plate. 72h after seeding, medium was removed and cells were incubated in a fresh culture medium containing 10 µM 5-Ethynyl-2'deoxyuridine (EdU). Cells were then washed in phosphate-buffered saline (PBS), trypsinized and the incorporation of EdU was monitored by flow cytometry (BD FACS Canto, Becton Dickinson, France) using the Click-iT™ Plus EdU Alexa Fluor488 kit (Invitrogen, France).

Cysts production and analyses -

A 40 µL layer of Geltrex® (Sigma-Aldrich, France) was added into the wells of Nunc™ Lab-Tek™ II Chamber Slide™ (Thermo Scientific) mimicking the extracellular matrix. The chamber slide was incubated at 37°C for 20 min. MCF10A CTL or MCF10A Crβ4 cell suspensions were prepared in medium containing 25 µL/mL Geltrex® and seeded at 10,000 cells per 400 µL in the chamber slide wells. The culture medium of cells was replaced every 2 days by 400 µL containing 25 µL/mL Geltrex®. Pictures of cysts were taken in bright field before changing medium (Invitrogen EVOS M7000, Thermofisher, France) in order to monitor growth and circularity of cysts. After 3 weeks of culture, cysts were washed twice in PBS, then fixed in 4% paraformaldehyde (PFA, Invitrogen) for 30 min at room temperature. Cysts were washed three times for 10 min in 100 mM glycine solution. Cysts were permeabilised for 5 min in a 0.5% Triton X-100 solution, then washed three times for 10 min with PBS. Unspecific site blocking was be realized by incubating for 1h in 5% BSA solution. Primary antibodies in 150 µL of 5% BSA solution were added to the wells and incubated at 4°C overnight. Wells were washed twice for 10 min in PBS, then 150 µL of fluorescent secondary antibodies were added for 1h at room temperature. Hoechst 33342

(1/1,000, Invitrogen) was used to visualize cell nuclei and Phalloidin-AF594 to visualize the F-actin network. Wells were finally washed four times with PBS, and micrographs were obtained using confocal microscopy using a 20 x objective (LEICA SP8 STED). Primary antibodies used were dedicated to identify β-catenin (Cell Signalling Technology D10A8, 1/200), E-cadherin (Invitrogen 13-1700, 1/1,000), Vimentin (Abcam ab92547, 1/100). Growth and circularity of cysts were analysed using the ImageJ software.

Western Blotting -Cells were washed with PBS and lysed in presence of a lysis buffer (50 mM Tris, pH7, 100 mM NaCl, 5 mM MgCl2, 10% glycerol, 1 mM EDTA), containing 5% sodium dodecyl sulphate (SDS) and protease inhibitors (S8830, Sigma-Aldrich, France). Western blotting experiments were performed according to standard protocols. Total protein concentrations were determined using the Pierce® BCA Protein Assay Kit Thermoscientific (Fisher Scientific, France).

Protein sample buffer was added and the samples were heated at 95°C for 5 min. Total protein samples were electrophoretically separated by SDS-polyacrylamide gel electrophoresis in 10% gels, and then transferred to polyvinylidene fluoride membranes (Millipore, USA). NaVβ4 proteins were detected using anti-NaVβ4 rabbit polyclonal primary antibodies (1/5,000, HPA017293, Sigma-Aldrich). Other primary antibodies used were: rabbit monoclonal anti-β-catenin (Cell Signalling Technology D10A8, 1/2,000), mouse monoclonal anti-E-cadherin (Cell Signalling Technology, 4A2, 1/2,000), rabbit monoclonal anti-Vimentin (Cell Signalling Technology, D21H3, 1/2,000), rabbit monoclonal anti-αSMA (Cell Signalling Technology, D4K9N, 1/2,000), rabbit monoclonal anti-N-Cadherin (Cell Signalling Technology, D4R1H, 1/2,000). Secondary horseradish peroxidase (HRP)conjugated goat anti-mouse or goat anti-rabbit IgG secondary antibodies were obtained from BioRad (1/5,000) and from Jackson ImmunoResearch (1/10,000), respectively. HSC70 protein was used as a sample loading control using anti-HSC70 mouse primary antibody at 1/10,000 (Santa-Cruz). Also in some conditions, a β-actin-HRP mouse monoclonal antibody (1/1,000, SantaCruz) was used as a control for sample loading. Proteins were detected using electrochemiluminescence-plus kit (Pierce® ECL Western Blotting Substrate, Fisher Scientific, France) and captured on CL-XPosure Films (Thermoscientific, France). Densitometric analyses were performed using ImageJ software, and quantifications of proteins of interest are expressed relatively to that of the control protein used (either HSC70 or β-actin) and to the control condition. Full uncropped blots are shown in Supplementary Figure 2.

Epifluorescence experiments -MCF10A CTL and Crβ4 cells were grown for 48h in Lab-Tek™ chambers. In some cases, chambers were coated with a layer of Geltrex™ containing 25 µg/mL of DQ-Gelatin 488. Cells were fixed in 4% paraformaldehyde for 30 min and then incubated with 5% BSA for 30 min. F-actin was visualized after staining the cells with phalloidin-AF594 (1/200, Cell Signaling Technology, France) for 1 h. Slides were mounted using ProLong® Gold Antifade Mountant with DAPI to visualize cell nuclei (Invitrogen, France). Epifluorescence microscopy was performed with an EVOS M7000 microscope (Thermofisher, France). Images were analysed using the ImageJ software.

Data presentation and statistical analysis -

Figure 1 :

 1 Figure 1: NaVβ4 downregulation induces morphological changes in non-cancer mammary cells

Figure 2 :

 2 Figure 2: NaVβ4 sustains epithelial polarity in 3-dimensional MCF10A cysts.

Figure 3 :

 3 Figure 3: NaVβ4 prevents β-catenin degradation

Figure 4 :

 4 Figure 4: NaVβ4 expression prevents mesenchymal transition in MCF10A epithelial mammary

  Figure 1

  Figure 4

Acknowledgements

This work was supported by the "Ministère de la Recherche et des Technologies", the "Ligue Nationale Contre le Cancer -Interrégion Grand-Ouest -comités 29, 36, 86, 37" to S.R., the Région Centre-Val de Loire (grant "NavMetarget"). S.R. was recipient of a prize "Prix Ruban Rose Avenir 2017" from the charity "le Cancer du sein: parlons-en!". A.D. obtained a PhD fellowship from the University of Tours, and a mobility grant from the Doctoral School ED549. O.L.-C. was recipient of a post-doctoral funding from the "Fondation pour la Recherche Médicale (FRM)" (SPF201909009198). R.L. was funded by a grant from the Région Centre-Val de Loire (grant "CanalEx") to S.R. We thank Mrs Carole Desplanches for secretary and administrative assistance.

Our confocal microscopy data were obtained with the assistance of the IBiSA Electron Microscopy Facility of the University of Tours, and we are grateful to M. Julien Burlaud-Gaillard for his help.

Author contributions

Declaration of Interest

The authors declare no competing interests.

STAR Methods

Bioinformatic analyses -Gene expression data were obtained from The Cancer Genome Atlas (TCGA) and Genotype-Tissue Expression (GTEx) databases using the UCSC Xena Browser (https://xenabrowser.net) 49 . The IlluminaHiSeq (log2-normalized_count+1) files were downloaded from the "TCGA Breast Cancer (TCGA-BRCA)" cohort, in order to compare expressions between adjacent non-tumoral tissues and primary tumour. From the "TCGA TARGET GTEx" cohort, the RSEM norm_count (log2-normalized_count+1) files were downloaded, in order to compare expressions between adjacent non-tumoral tissues, primary tumour and metastases.

Inhibitors and chemicals -The proteasome inhibitor MG132 as well as all chemicals were purchased from Sigma-Aldrich (France). Fluorescent probes DQ™-Gelatin and Hoechst 33342 were purchased from Invitrogen (France), and Phalloidin-AF594 from Cell Signaling Technology (France). ProLong® Gold Antifade Mountant containing DAPI was purchased from Invitrogen, (France).

Cells and cell culture -

The human breast cancer cell line MDA-MB-231 and non-cancer mammary epithelial MCF10A cells were acquired from the American Type Culture Collection (ATCC), through LGC Standards (France), and were grown at 37°C in a humidified 5% CO2 incubator. MDA-MB-231 cells were grown in Dulbecco's modified Eagle's medium (DMEM) supplemented with 5% foetal calf serum (FCS). The immortalized non-cancer mammary epithelial MCF10A cells were cultured in DMEM/Ham's F-12, 1:1 mix containing 5% horse serum (Dutscher, France), 10 µg/mL insulin, 20 ng/mL epidermal growth factor, 0.5 µg/mL hydrocortisone, and 100 ng/mL cholera toxin. A stable MCF10A cell line knocked-down for the expression of the SCN4B gene, encoding for NaVβ4, was generated using the CRISPR/Cas9 technique, as previously described [START_REF] Brisson | P2X7 Receptor Promotes Mouse Mammary Cancer Cell Invasiveness and Tumour Progression, and Is a Target for Anticancer Treatment[END_REF]


Amélie Barbe
email: amelie.barbe@ens-lyon.fr

Paulo Gonc
email: paulo.goncalves@ens-lyon.fr

Marc Sebban
email: marc.sebban@univ-st-etienne.fr

Pierre Borgnat
email: pierre.borgnat@ens-lyon.fr

Rémi Gribonval
email: remi.gribonval@ens-lyon.fr

Titouan Vayer
email: titouan.vayer@ens-lyon.fr

Optimization of the Diffusion Time in Graph Diffused-Wasserstein Distances: Application to Domain Adaptation

Keywords: Optimal transport, graphs, domain adaptation, heat kernel

The use of the heat kernel on graphs has recently given rise to a family of so-called Diffusion-Wasserstein distances which resort to the Optimal Transport theory for comparing attributed graphs. In this paper, we address the open problem of optimizing the diffusion time used in these distances and which plays a key role in several machine learning settings, including graph domain adaptation or graph classification. Inspired from the notion of triplet-based constraints used, e.g., in metric learning, we design a loss function that aims at bringing two graphs closer together while keeping an impostor away, this latter taking the form of a Wasserstein barycenter. After a thorough analysis of the properties of this function, we show on synthetic and real-world data that the resulting Diffusion-Wasserstein distances outperforms the Gromov and Fused-Gromov Wasserstein distances on unsupervised graph domain adaptation tasks. Additionally, we give evidence in such a setting that our method for optimizing the diffusion parameter allows to overcome the limitation of the widely used circular validation strategy.

I. INTRODUCTION

A Domain Adaptation (DA) scenario arises in machine learning when we observe a change of distribution (a.k.a. domain shift) between the training data (the source distribution) and the samples used at test time with the deployed model (the target distribution). To cite a few examples, DA can occur in image processing, when changing the lighting or camera lens while acquiring images, in demography with social mobility of people or in fraud detection, with fraudsters trying to adapt over time to better mimic genuine behaviors. Most of the time, training a new model from the target distribution is not Work supported by the ACADEMICS grant of the IDEXLYON, project of the Université de Lyon, PIA operated by ANR-16-IDEX-0005 desirable for several reasons: (i) the algorithmic complexity required for optimizing from scratch the parameters of a new model; (ii) the lack of target training examples; (iii) the lack (or absence) of supervision (i.e. no labelled target data available), etc. In such a setting, the domain adaptation theory [START_REF] Ben-David | A theory of learning from different domains[END_REF], [START_REF] Redko | Advances in Domain Adaptation Theory[END_REF] suggests to reduce the divergence between the source and the target distributions while learning an efficient model from the labelled source data.

One way to solve DA problems is to use Optimal Transport [START_REF] Peyré | Computational optimal transport[END_REF], [START_REF] Ollivier | Optimal Transport -Theory and Applications, ser[END_REF] (OT). As illustrated in Figure 1, OT provides a natural geometry for comparing and aligning two distributions in the space of probability measures. In the discrete case, when dealing with point clouds, it looks for a coupling matrix (and its corresponding Wasserstein distance) that minimizes the global cost of transporting the individual masses from the source to the target distribution. In [START_REF] Courty | Optimal transport for domain adaptation[END_REF], the authors introduced OTDA, the first DA algorithm based on OT which moves the source on the top of the target by preventing -using a group-sparse regularization -two source data of different labels from being transported on the same target example. Once the alignment is achieved, a model is learned from the labelled source data and applied on the target distribution.

During the past few years, OT has received a tremendous interest from the machine learning community for comparing structured data, like attributed graphs. To address this task, two OT-based distances have been introduced in the literature. The Gromov-Wasserstein distance [START_REF] Peyré | Gromov-wasserstein averaging of kernel and distance matrices[END_REF] (GW) has been originally defined for comparing two distributions that do not necessarily lie in the same feature space. Based on intra-distribution pairwise distances/costs, GW provides a nice framework for computing a distance between two graphs by encoding some structure, like the shortest path between two vertices. In order to consider both the features associated to the nodes and the structure of the graphs, the Fused-Gromov-Wasserstein (FGW) distance has been recently introduced in [START_REF] Vayer | Fused gromov-wasserstein distance for structured objects[END_REF]. FGW acts as a combination of the Wasserstein distance (focusing only on the features) and the GW distance (considering only the structure). Although FGW has been shown to be significantly better than GW in graph classification tasks, it has the shortcoming of not scaling to large graphs due to a complexity in O(m 2 n 2 ) (with m (resp. n) the size of the source (resp. target) graph).

To overcome this limitation, a new family of graph Diffusion Wasserstein (DW) distances have been defined in [START_REF] Barbe | Graph diffusion wasserstein distances[END_REF] exploiting the heat diffusion on attributed graphs. The main advantage of DW is to capture both the feature and structural information in one term based on the Laplacian exponential kernel applied on the features associated to the nodes. This heat kernel is defined as exp(-τ L), where L is the graph Laplacian and τ is the time during which the features diffuse along the structure of the graph. Even though DW has been shown to provide promising results when addressing graph DA tasks [START_REF] Barbe | Graph diffusion wasserstein distances[END_REF], the quality of this family of Wasserstein distances highly depends on the value of the diffusion time τ . Because of the absence of target labels in unsupervised DA, tuning this parameter remains a tricky and unstable task even with the widely used circular validation strategy [START_REF] Bruzzone | Domain adaptation problems: A dasvm classification technique and a circular validation strategy[END_REF].

The main contribution of this paper consists in addressing this challenging task by designing a new optimization problem dedicated to automatically learn the optimal diffusion time. Inspired from the triplet-based constraints used in metric learning [START_REF] Bellet | Metric Learning, ser. Synthesis Lectures on Artificial Intelligence and Machine Learning[END_REF], we design a loss function that aims at bringing the source and target graphs close together while keeping an impostor away. This latter is built from the feature-based Wasserstein barycenter of the two original graphs. We present in this paper a thorough analysis of the properties of this new loss function. We also discuss some strategies to alleviate the additional complexity cost of minimizing it. The experimental results obtained on both synthetic and real datasets give indisputable evidence that our method for optimizing the diffusion parameter allows to outperform the state-of-the-art graph-based Wasserstein distances and overcome the limitation of the circular validation.

The rest of this paper is organized as follows. Section II is dedicated to the preliminary knowledge on OT and the definition of GW, FGW and DW distances. Section III is devoted to the presentation and analysis of our optimization problem aiming at learning the diffusion time. Section IV describes experiments performed on synthetic and real data. We conclude in Section V by opening future promising lines of research.

II. PRELIMINARIES

In this section, we provide the necessary background in OT and recall the definition of the Wasserstein distance (W), as well as those of the Gromov Wasserstein (GW), Fused-Gromov Wasserstein (FGW) and Diffusion Wasserstein (DW) distances.

A. Discrete Optimal Transport and Wasserstein distance

Let us consider two empirical probability measures µ and ν, called source and target distributions, and supported on two sample sets X = {x i } m i=1 and Y = {y j } n j=1 , respectively, lying in some feature space X and with weights a

= (a i ) m i=1 , b = (b j ) n j=1 such that µ = m i=1 a i δ xi and ν = n j=1 b j δ yj where δ is the Dirac measure. If X = R r for some integer r ≥ 1, a matrix representation of X (resp. of Y ) is the matrix X ∈ R m×r (resp. Y ∈ R n×r ) whose rows are x i , 1 ≤ i ≤ m (resp. y j , 1 ≤ j ≤ n). Let M = M (X, Y ) ∈ R m×n + be a cost matrix, where M ij def = [d(x i , y j )] ij
is the cost (w.r.t. to some distance d) of transporting x i on y j . OT [START_REF] Peyré | Computational optimal transport[END_REF], [START_REF] Kantorovich | Mathematical methods of organizing and planning production[END_REF] aims at moving µ on the top of ν in an optimal way with respect to M . Let Π(a, b) be a transportation polytope defined as the set of admissible coupling matrices γ:

Π(a, b) = {γ ∈ R m×n + s.t. γ1 n = a, γ T 1 m = b},
where γ ij is the mass transported from x i to y j and 1 k is the vector of dimension k with all entries equal to one. The p-Wasserstein distance W p p (µ, ν) between the source and target distributions is defined as follows:

W p p (µ, ν) = min γ∈Π(a,b) γ, M p (X, Y ) F , (1)
where ., . F is the Frobenius inner product and M p (X, Y ) := (M ij ) p is the entrywise p-th power of M (X, Y ) with exponent p > 0.

Equation ( 1) is a constrained linear optimisation problem which always has a solution (at least, the naive one of the form γ = ab T which does not take into account the costs of the displacements). Its complexity is polynomial in the sample set sizes (super cubic in practice [START_REF] Pele | Fast and robust earth mover's distances[END_REF] when using a minimum cost flow solver), and thus becomes quickly prohibitive as the number of samples grows. Moreover, the solution of Equation (1) might be unstable because inherently the linear program looks for vertices in the polytope, and therefore the solution can very easily change from one vertex to another by slightly changing the cost function M . To address these limitations, an entropy-regularized version has been introduced in [START_REF] Peyré | Computational optimal transport[END_REF], [START_REF] Cuturi | Sinkhorn distances: Lightspeed computation of optimal transport[END_REF] and is defined as follows:

W p,ε (µ, ν) = min γ∈Π(a,b) γ, M p (X, Y ) F -εE(γ), (2)
where E(γ) =i,j γ ij (log γ ij -1) is the Shannon entropy.

B. Graph-based Wasserstein distances

Let us now consider empirical probability measures supported on graphs, for which we need to extend the notations of the previous section. Following [START_REF] Vayer | Fused gromov-wasserstein distance for structured objects[END_REF], we define an undirected attributed graph as the quadruplet G = (V, E, l f , l s ). V and E are the vertices and edges of the graph respectively. l f : V → X is a function that associates a feature x i to each vertex v i . l s : V → Ω s is a function that associates a structural representation z i to each vertex v i ; z i is further used to define a cost function C : Ω 2 s → R + which measures the structural similarity between two nodes. Additionally, if G is a labeled graph, each vertex is assigned a label from some label space C. Finally, when associated with a probability distribution on the nodes, the undirected graph G can be seen as a probability measure, and thus can be considered in an OT scheme.

In the following, we first recall the definition of the Gromov-Wasserstein distance (GW) [START_REF] Peyré | Gromov-wasserstein averaging of kernel and distance matrices[END_REF]. Unlike W, the source and target distributions are not required for GW to lie in the same feature space. Instead of using a cost matrix between the source and target samples, it considers two intra-distribution pairwise cost matrices. Therefore, GW can by directly applied on graphs, by using, e.g., the shortest-path matrix (or any structural matrix) for the cost functions C's.

Definition II.1. Let G s and G t be two source and target graphs of size m and n respectively with their associated probability distributions µ and ν over the nodes. Let C s ∈ R m×m + and C t ∈ R n×n + be two pairwise cost matrices associated with each graph. The Gromov-Wasserstein (GW) distance between G s and G t is defined as follows:

GW p p (C s , C t , µ, ν) = min γ∈Π(a,b)    i,j,k,l |C s ik -C t jl | p γ ij γ kl    . (3)
Note that for the sake of simplicity, the dependency on C s and C t will be omitted in the rest of the paper. It is worth noting that GW only considers the structural information of the graphs. In order to take also into account the feature information at the node level, the Fused Gromov-Wasserstein distance (FGW) has been recently introduced in [START_REF] Vayer | Fused gromov-wasserstein distance for structured objects[END_REF].

Definition II.2. Let G s and G t be two source and target graphs of size m and n respectively with their associated probability distributions µ and ν over the nodes. Let α ∈ [0, 1]. The Fused Gromov-Wasserstein (FGW) is defined as follows:

FGW p p (µ, ν) = min γ∈Π(a,b) { i,j,k,l (1 -α)M p ij +α|C s ik -C t jl | p γ ij γ kl }. (4)
The hyper-parameter α allows to control the trade-off between the importance of the structure and the features, and has to be tuned according to the task at hand. FGW covers two special cases. When α = 0, we retrieve the Wasserstein distance, that is α = 1, we recover the Gromov-Wasserstein distance, that is FGW p p (µ, ν | α = 1) = GW p p (µ, ν). Roughly speaking, the optimal coupling matrix γ will tend to associate two source and target nodes if both their features and structural representations are similar.

FGW p p (µ, ν | α = 0) = W p p (µ, ν). When
As implied by the four indices involved in the sum of Equation ( 4), FGW has the shortcoming of not scaling well to large graphs due to a complexity in O(m 2 n 2 ). To overcome this drawback, the Diffusion-Wasserstein distance (DW) has been introduced in [START_REF] Barbe | Graph diffusion wasserstein distances[END_REF]. It resorts to the so-called heat kernel which diffuses the features in the graph before computing the transport. Therefore, DW incorporates both the structural and the feature information. This process depends on a diffusion time τ which allows to capture the graph's structure at different scales (see Figure 2 for an illustration on 1D features).

Definition II.3. Let G s and G t be two source and target graphs of size m and n respectively with their associated probability distributions µ and ν over the nodes. Let L s ∈ R m×m and L t ∈ R n×n be the Laplacian matrices of G s and G t . Let X ∈ R m×r , Y ∈ R n×r represent respectively the source and the target sample sets associated to the features (∈ R r ) on their vertices.

Given parameters 0 ≤ τ s , τ t < ∞, consider the diffused sample sets X, Ỹ represented by the matrices

X = exp(-τ s L s )X ∈ R m×r , Ỹ = exp(-τ t L t )Y ∈ R n×r and define M (τ s , τ t ) := M ( X, Ỹ ) ∈ R m×n ,
a cost matrix between features that takes into account the structure of the graphs through diffusion operators. We define the Diffusion Wasserstein distance (DW) between µ and ν as:

DW p p (µ, ν | τ s , τ t ) = min γ∈Π(a,b) γ, M p . (5)
Here again, M p is the entrywise p-th power of M . The underlying distance is implicit in M (•, •). For the sake of concision, the dependency on τ s and τ t will be omitted from the notation DW p p (µ, ν) if not specifically required, and simplified as DW p p (µ, ν | τ ) when the diffusion times are equal. For the same reason, we will omit in the rest of this paper the superscript p used in all the Wasserstein distances.

III. OPTIMIZATION OF THE DIFFUSION TIME τ

In this section, we present the main contribution of this paper. Unlike the circular validation [START_REF] Bruzzone | Domain adaptation problems: A dasvm classification technique and a circular validation strategy[END_REF] that aims at tuning hyper-parameters in unsupervised DA by benefiting from pseudo-target labels, we suggest here to directly optimize the diffusion time τ in a self-supervised way.

A. Circular validation

One peculiarity of unsupervised domain adaptation comes from the absence of target labels. In such a setting, a standard method to tune hyper-parameters is the circular validation [START_REF] Bruzzone | Domain adaptation problems: A dasvm classification technique and a circular validation strategy[END_REF]. The "circular" aspect is due to the fact that the labels go backand-forth between the source and the target data. Let us detail the underlying principle in the context of an OT-based graph domain adaptation task. Given a transport map γ and a set of labels l s ∈ C for the source graph, one can define pseudolabels lt for the target graph by choosing, for each node, the label from which the maximum weight comes from:

lt j = argmax l∈C    1≤i≤m γ ij δ l s i =l    . (6)
Like-wise, pseudo-labels for the source graph can be reinferred in a similar fashion:

ls i = argmax l∈C    1≤j≤n γ ij δ lt j =l    . (7)
It is now possible to define an unsupervised score for ranking the transport maps obtained by different sets of hyperparameters. This score measures the level of agreement between the original and pseudo source labels:

s(γ) = 1 m m i=1 δ l s i = ls i . (8)
It is important to note that while a low score of s(γ) is an evidence that the considered hyper-parameter does not lead to a good model, a high score would not allow us to definitely conclude. Indeed, for two graphs of the same size, any permutation matrix γ would produce a perfect score of 1, that can make the circular validation unstable.

B. A triplet-based loss function to learn τ

To address the limitation of the circular validation, we propose in the following to learn the diffusion time by minimizing a loss function that considers an impostor attributed graph. Inspired from the triplet-based constraints used, e.g., in metric learning [START_REF] Bellet | Metric Learning, ser. Synthesis Lectures on Artificial Intelligence and Machine Learning[END_REF], the impostor facilitates the choice of the diffusion time τ that brings G s and G t close together without suffering from degenerate phenomena.

Definition III.1. Let G s and G t be two source and target graphs of size m and n respectively with their associated probability distributions µ and ν over the nodes. The impostor G 0 with respect to G s and G t is a graph with i = m+n 2 Fig. 3: Illustration of the construction of the impostor (in grey) of two graphs (in red and blue). Features correspond to the 2D coordinates of the nodes. Impostor nodes are supported on the Wasserstein barycenter (eq. 9) of the original graphs' features. Impostor edges are drawn uniformly with probability equal to the average connection probability of the two original graphs.

nodes whose features X 0 are defined as the minimizer of the following Wasserstein barycenter problem:

X 0 = argmin X∈R i×r 1 2 W(X s , X) + W(X t , X) . (9)
The adjacency matrix A 0 of G 0 is sampled according to an Erdös-Rényi model [START_REF] Gilbert | Random Graphs[END_REF], with connection probability p 0 = p s +p t 2

the average connection probability of the two graphs.

The solution of the Wasserstein barycenter problem is difficult in practice and is the subject of a rich literature (see [START_REF] Altschuler | Wasserstein barycenters can be computed in polynomial time in fixed dimension[END_REF]). Our problem is a free-support barycenter problem where the main obstacle is that we have to optimize on the support X of the barycenter [START_REF] Cuturi | Fast computation of wasserstein barycenters[END_REF], [START_REF] Luise | Sinkhorn barycenters with free support via frank-wolfe algorithm[END_REF]. It has been recently proved that this problem can be solved in polynomial time when the number of points of each measure is fixed [START_REF] Ben-David | A theory of learning from different domains[END_REF]. In our case, we chose to rely on the heuristic proposed in [START_REF] Altschuler | Wasserstein barycenters can be computed in polynomial time in fixed dimension[END_REF] which is reasonable as there are only 2 distributions involved and the weights of the barycenter are considered as fixed (uniform in our case). Overall it boils down to iterating over 1) solving two linear OT problems W(X s , X) and W(X t , X) 2) finding the support X which can be done in closed-form as detailed in [START_REF] Cuturi | Fast computation of wasserstein barycenters[END_REF] (Equation 8). The procedure for generating the impostor is illustrated in Figure 3 for two toy graphs with 2D features.

The diffusion parameter can be now defined as the solution of the following optimization problem:

τ * = argmin τ ≥0 {L(τ )} , with (10)
L(τ ) = DW p (G s , G t | τ )-DW p (G s , G 0 | τ ) + DW p (G t , G 0 | τ ) . (11
) Intuitively, like in metric learning, the idea is to learn the parameters of a model (here, a unique diffusion time τ ) that (i) constrains G s and G t to get closer while (ii) preventing a scenario facilitating the bringing together of G s and G t with a "different" graph. However, unlike supervised metric learning where the impostors of a pair of points of the same class can be defined as the training samples of the opposite label in the close neighborhood, the notion of "different" is here ill-defined because of the absence of target labels in DA tasks. By using G 0 as defined above, we generate an impostor sufficiently close in terms of features (defined as the Wasserstein barycenter) and structure (mean structure of G s and G t in terms of vertices) forcing the identification of a parameter τ which aligns as well as possible G s and G t while ensuring a certain "margin" to a different but related graph.

C. Analysis of the triplet-based loss function

In this section, we provide a thorough analysis of our loss function. First, we illustrate the correlation between the minimizer of Eq. ( 11) and the corresponding accuracy on a DA task. Then, after the derivation of some theoretical properties, we give evidence that L(τ ) is very robust to some approximations aiming at reducing its algorithmic complexity.

a) Correlation between L(τ ) and the accuracy in DA:

The coupling matrix γ resulting from the calculation of DW p (G s , G t | τ ) with τ the solution of Problem (11) can be directly used for addressing a graph DA task. Given γ, each target node receives the mass of some source vertices and then can be assigned the majority class among the transported labels. According to this classification rule, one can compute the DA accuracy measuring the level of agreement between the predicted and the expected labels. In Figure 4, we illustrate the correlation between L(τ ) and the DA accuracy from two source and target synthetic community graphs, made of 200 nodes and 3 classes. A community graph is built by (i) assigning a random class to each node, (ii) generating 2D features using a Gaussian distribution with a different center for each class, (iii) connecting all points of the same class that are closer than some radius r, and (iv) linking points of different classes at random with a small connection probability. The two resulting source and target graphs are presented in Figure 4a. In Figure 4b, we plot the loss L(τ ) and the accuracy obtained from a large range of τ .

The most interesting point is that the global minimum of the loss (which is not convex in general with potentially several local minima) corresponds to the parameter τ yielding the maximum accuracy. On the other hand, the global maximum of L(τ ) matches with the worst behavior in DA. Therefore, this correlation between L(τ ) and the accuracy confirms the interest of our loss for addressing graph-based optimal transport tasks with Diffusion Wasserstein distances. b) Theoretical properties: Given the definition of the loss function L(τ ), we can derive the following properties.

Theorem III.2. Let G s and G t be two connected attributed graphs. The loss function L(τ ) of Eq. ( 11) is continuous, negative, and its limits are:

lim τ →0 L(τ ) = lim τ →∞ L(τ ) = 0. ( 12
)
Proof. Continuity stems from continuity of all the functions involved. Then, writing X u τ the distribution of the features of a graph G u diffused for a time τ (u ∈ {0, s, t}), we have:

L(τ ) = DW p (G s , G t | τ ) -DW p (G s , G 0 | τ ) + DW p (G t , G 0 | τ ) = W(X s τ , X t τ ) -(W(X s τ , X 0 τ ) + W(X 0 τ , X t τ )). (13)
W being a distance, negativity follows from triangle inequality. Now, denote X s and X t the features of G s and G t , and X 0 the features of the impostor G 0 . By definition of X 0 as a minimizer:

W(X s , X 0 ) + W(X t , X 0 ) ≤ W(X s , X t ) + W(X t , X t ) ≤ W(X s , X t ), (14)
and by triangle inequality we have:

W(X s , X 0 ) + W(X t , X 0 ) ≥ W(X s , X t ). (15)
By combining Eq.( 14) and ( 15), we get:

L(0) = W(X s , X t ) -(W(X s , X 0 ) + W(X t , X 0 )) = 0. ( 16
)
Finally, from [START_REF] Peyré | Computational optimal transport[END_REF], Remark 9.1, we have that the expected value of the barycenter is the barycenter of the expected values: From [START_REF] Barbe | Graph diffusion wasserstein distances[END_REF], Proposition 1, we have the limit of DW 2 :

E[X 0 ] = 1 2 E[X s ] + E[X t ] . (17)
lim τ →∞ DW 2 (G s , G t ) = E[X s ] -E[X t ] 2 . (18)
Therefore, we know that the limit of L(τ ) exists and is:

lim τ →∞ L(τ ) = E[X s ] -E[X t ] 2 -E[X s ] -E[X 0 ] 2 -E[X t ] -E[X 0 ] 2 = E[X s ] -E[X t ] 2 -1 /2 E[X s ] -E[X t ] 2 -1 /2 E[X s ] -E[X t ] 2 = 0. (19)
Thanks to Theorem. III.2, the proposed triplet loss prevents the trivial values τ = 0 or τ → ∞ from being chosen. c) Robustness of L(τ ) w.r.t. different approximations:

The algorithmic complexity of the loss L(τ ) depends on three main elements: (i) the exponential of the graph Laplacians, (ii) the Wasserstein distance W on the diffused source and target features and (iii) the size of the impostor. Different strategies can be used to simplify the overall complexity, including (but not exhaustively) the entropic regularization [START_REF] Cuturi | Sinkhorn distances: Lightspeed computation of optimal transport[END_REF], the reduction of the size of G 0 , or a Chebychev approximation of the diffusion [START_REF] Marcotte | Fast multiscale diffusion on graphs[END_REF]. In the following, we study the capacity of L(τ ) to resist these approximations and thus to provide a similar global minimum.

Entropic regularization. The entropic regularization of W [START_REF] Cuturi | Sinkhorn distances: Lightspeed computation of optimal transport[END_REF], as defined in Eq. ( 2), allows to overcome the limitations of the original problem due to the super-cubic complexity, the instability and the non uniqueness of the solution. The entropyregularized version is several orders of magnitude faster (an η-approximation is computed in O(n 2 log(n)η -3 ) [START_REF] Altschuler | Near-linear time approximation algorithms for optimal transport via sinkhorn iteration[END_REF]). Using the same graphs as those of Figure 4a, Figure 5a shows the magnitude of the loss L(τ ) for different values of the regularization parameter ε. We can see that whatever the value of ε the shape of the resulting loss L(τ ) does not change much, meaning that the global minimum stays very close to the one corresponding to the maximum accuracy (see the dashed-line in Fig. 4b).

Size of the impostor. While Def. III.1 suggests to set the size of G 0 to m+n 2 , we study here the impact of reducing the number of nodes of G 0 before computing the Wasserstein barycenter. The behavior of the loss L(τ ) is reported in Figure 5b for different reduction ratios (from 2 to 10). Interestingly, we can observe that even though the curves get smoother as the size of the impostor decreases, the global minimum changes very little, pointing to a relatively stable value for τ . Note that the connection probability of G 0 has to adapt to the reduction ratio. For instance, if the size is reduced by a factor of 2, p 0 must be doubled.

Chebychev approximation of the diffusion. Finally, instead of calculating the exact heat kernel applied to the features, one can resort to polynomial approximations [START_REF] Phillips | Interpolation and Approximation by Polynomials, ser. CMS Books in Mathematics[END_REF] of the diffusion. Following [START_REF] Marcotte | Fast multiscale diffusion on graphs[END_REF], we applied a Chebychev approximation of the exponential operator where the truncation order (degree of the final approximating polynomial) offers a trade-off between accuracy and computational speed. An illustration of the effect of various truncation orders on the loss L(τ ) is given in Figure 5c. Once again, the global minimum of L(τ ) appears to be very robust to changes in the degree of the approximation.

IV. EXPERIMENTAL STUDY

In this section, we perform two series of experiments dedicated to compare several optimal transport methods on graph DA tasks. In the first one, we use Contextual Stochastic Bloc Models [START_REF] Deshpande | Contextual stochastic block models[END_REF] to generate synthetic data. The second one concerns the ogbn-arxiv graph [START_REF] Hu | Open graph benchmark: Datasets for machine learning on graphs[END_REF] and aims at classifying papers published in a given year from articles published before. All experiments are written in Python and use the libraries POT [START_REF] Flamary | Pot: Python optimal transport[END_REF] for optimal transport methods, PyGSP [START_REF] Defferrard | Pygsp: Graph signal processing in python[END_REF] for graph generation, and NumPy and SciPy for other computations. They run on a Intel© Core™ i5-5300U CPU @ 2.30GHz×2 processor and 15.5 GiB RAM on a Linux Mint 20 Cinnamon1 .

A. Domain Adaptation on synthetic data a) Synthetic data generation: The process for generating a synthetic graph of size N is the following. For each of the N nodes, a label +1 or -1 is chosen randomly with equal probability. For each node, a 1D feature is generated randomly by sampling a Gaussian distribution N (l, σ), where l is the node's label and σ is considered as a hyper-parameter. Links between nodes are sampled according to a Bernoulli distribution of probability p l,l /N where l and l are the node's labels. Therefore, a graph is described by its size N , its labels l ∈ {+1, -1} N , its attributes X ∈ R N and its adjacency matrix A ∈ {0, 1} N ×N . We will add super-scripts s or t to designate either source or target quantities. The hyper-parameters are the sizes m and n of G s and G t , the bandwidths σ s and σ t , and the connectivity matrices p s +1,+1 b) OT methods compared: We compare the following OT methods that can take advantage of the features and structure of both graphs, and only the source labels:

• The Wasserstein distance (W) [START_REF] Peyré | Computational optimal transport[END_REF] using only the features.

• The Gromov Wasserstein distance (GW) [START_REF] Peyré | Gromov-wasserstein averaging of kernel and distance matrices[END_REF] taking only into account the structure of the graphs. • The Fused Gromov-Wasserstein distance (FGW) [START_REF] Vayer | Fused gromov-wasserstein distance for structured objects[END_REF] using both the feature and the structural information.

• The DA method based on the Wasserstein distance with Laplacian regularisation OT_LAPLACE [START_REF] Flamary | Optimal transport with Laplacian regularization[END_REF].

• The DA method based on the Wasserstein distance with a group-lasso regularizer for the labels (L1L2_GL) [START_REF] Courty | Optimal transport for domain adaptation[END_REF]. • The Diffusion-Wasserstein distance (DW CV) [START_REF] Barbe | Graph diffusion wasserstein distances[END_REF], where τ is tuned using a circular validation criterion. • The Diffusion-Wasserstein distance, where τ is the minimizer of our loss function L(τ ), with (DWL ε ) and without (DWL) entropic regularization.

c) Experimental setup and results: 50 source/target graphs pairs are generated using the CSBM model described above. A transport map γ is learned for each method. Then the labels are transported according to the rule defined in [START_REF] Peyré | Gromov-wasserstein averaging of kernel and distance matrices[END_REF]. Using the ground truth labels, the accuracy is computed for each method. Each method is given 20 trials to find its best hyperparameters, using random search and a circular validation criterion. For FGW, α is linearly sampled in [10 -6 , 1 -10 -6 ]. For DW, τ is logarithmically sampled in [10 -3 , 10 -0.5 ], and for all regularization based methods, the corresponding hyperparameters range logarithmically in the interval The results are reported in Figure 6. The accuracy scores of each method over the 50 graph pairs are plotted as a boxplot, displaying the median performance, the quartiles and the 10th and 90th percentiles as well as the outliers (i.e. out of the 10th and 90th percentiles). We can make the following comments. First, we can note that the Gromov Wasserstein distance is worse than random guessing. This behavior can probably be explained by a coupling matrix that permutes the classes. Second, the approaches that take into account both the feature and the structural information (i.e. FGW and DW-based methods) outperform the competitors. Third, DWL ε and DWL are better than any other method with a more stable behavior for the regularized version of our loss function. Finally, as expected, learning τ yields a significant improvement compared to DW CV based on the circular validation.

B. Domain Adaptation on real data

This second series of experiments concerns the real ogbn-arxiv graph [START_REF] Hu | Open graph benchmark: Datasets for machine learning on graphs[END_REF]. Although originally designed for node classification, we cast the problem as a Domain Adaptation task and we address it using the same methods as in the previous section. Each node of the graph represents a paper published in Arxiv. A link from one node to another indicates that this later is cited by the former. The feature of a node is an embedding of the paper's title and abstract, and it lies in R 128 . Nodes are labelled according to their corresponding subject area among 40 possible labels. Finally, each node is associated with a publication year.

In our setting, the source graph corresponds to the papers published before 2004. Its size is m = 1279. The target graph contains the articles published before 2005 (n = 1666 nodes). This makes the source graph a sub-graph of the target one and therefore, the DA accuracy is measured only on nodes of year 2005:

acc(γ) = 1 387 1666 j=1280 δ lt j =l t j . ( 20
)
For GW, the source and target cost matrices are built from the shortest-distances in the graph. Because the graph is not connected and the solver cannot handle infinite costs between two nodes, infinite values are replaced by twice the longest length path. For FGW, the same cost matrices are used, along with the pairwise Euclidean distance between the features. For the hyper-parameter α, 10 values are sampled uniformly in [0, 1] and the best one is selected using circular validation. For DW, the hyper-parameter τ is either determined by circular validation among 10 logarithmically spaced values in [10 -3 , 10 1 ], or chosen by minimizing L. Finally, the size of the impostor graph G 0 is set to 500 nodes. The test accuracies are reported in Table I as well as the computation times. We can note that DW outperforms the competitors in terms of accuracy and remains much cheaper than FGW from a computational angle. The results also confirm that learning τ by minimizing the triplet loss L yields much better results than the circular validation.

V. CONCLUSION

In this paper, we presented an optimization method aiming at learning the diffusion time involved in the so-called Diffusion Wasserstein Distances. The experimental results give evidence that our strategy (i) allows to overcome the limitation of the widely used circular validation and (ii) outperforms the state-of-the-art OT-based domain adaptation methods. Note that we optimized so far only one τ used afterwards for both source and target graphs. In complex scenarios, the optimization of two diffusion times will be probably necessary. This will require to design a new way for building the impostor and derive new theoretical properties on the resulting loss function. We also plan to use our method on graph classification tasks. Finally, a promising perspective would consist in learning the ground metric which plays a key role in the coupling matrix.

Fig. 1 :

 1 Fig. 1: Illustration of OT on a toy example: (a) two source (red circles) and target (blue crosses) distributions are given. (b) reduction of the domain shift after minimization of the global cost of transportation from the source to the target.

Fig. 2 :

 2 Fig. 2: Illustration of the heat kernel diffusion for three values of τ of a 1D signal (r = 1) centred on two nodes in the Minnesota Read Network graph [14].

  (a) Source (left) and target (right) community graphs (b) L(τ ) and DA accuracy as a function of τ .

Fig. 4 :

 4 Fig. 4: Correlation between L(τ ) and the DA accuracy: (a) source and target community graphs G s and G t ; (b) the global minimum of L(τ ) corresponds to the maximum accuracy reachable in a DA task.

Fig. 5 :

 5 Fig. 5: Behavior of the loss L(τ ) computed from the two source and target Community Graphs of Fig. 4a in three approximation scenarios: (a) effect of the entropic regularization used in the inner DW distance for various values of the regularization parameter ε; (b) effect of downsizing the number of nodes of the impostor; (c) effect of approximating the diffusion process with different Chebychev polynomials. The dashed line represents the exact L(τ ).

  [10 -2 , 10 -0.5 ]. The synthetic data parameters are n = m = 960, σ s = 2 and σ t = 4, source and target connectivity matrices 96

Fig. 6 :

 6 Fig. 6: Median, quartile and decile accuracy of various OT methods on the task of transferring the labels of G s to G t .

TABLE I :

 I Computation time and test accuracy of various OTbased DA methods on the ogbn-arxiv graph restricted to years ≤ 2004 (for the source) and ≤ 2005 (for the target) with 1279 and 1666 nodes respectively.

	Method	Computation time Test accuracy
	DWL ε=0.1	96s	30%
	DWL	319s	28%
	L2L1 GL	870s	24%
	DW CV	130s	23%
	FGW	2100s	22%
	OT LAPLACE	228s	18%
	W	2s	18%
	GW	275s	8%

In case of acceptation, the code and the datasets will be made available.


Pierre-Vincent Dugué

Mohammed El Gibari

Mathieu Halbwax

Stephane Ginestar

Vanessa Avramovic

Jean-Pierre Vilcot

Hong Wu

Pierre-Vincent Dugue
email: pierre-vincent.dugue@etu.univ-nantes.fr

Mohammed El-Gibari

Hongwu Li

A new equivalent circuit scheme for grounded back-to-back GCPW-MS-GCPW transitions fabricated on a thin low-k substrate

établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

A New Equivalent Circuit Scheme for Grounded Back-to-Back GCPW-MS-GCPW Transitions Fabricated on a Thin Low-k Substrate

INTRODUCTION

The evolution in the mobile communications that we have witnessed for 20 years raises a lot of interest in radio frequency (RF) systems. Connected devices and civil/military communications are certainly sectors targeting the increase in the working frequency range [START_REF] Larson | RF and microwave hardware challenges for future radio spectrum access[END_REF]. The amount of requests for devices allowing users to get seamless high data-rate signals processing is increasing day by day. Notably, Monolithic Microwave Integrated Circuits (MMICs) focus interest in reducing their footprint and increasing their operation bandwidth by adjusting the embedded transmission lines and transitions. Bandwidth optimization of many transitions has already been explored on thick film substrates [START_REF] Sain | Impact of Ground via placement in grounded coplanar waveguide interconnects[END_REF][START_REF] Xu | Resonance suppression of grounded coplanar waveguide in submount for 40 Gb/s optoelectronic modules[END_REF][START_REF] Zhou | Development of a broadband coplanar waveguide-to-microstrip transition with vias[END_REF].

It results in the integration of multiple via-holes along the transitions to reduce losses and improve mode conversion. Hereby, we seek to validate a new equivalent circuit model over an ultra-large bandwidth and to demonstrate the feasibility of a microwave transition between a Grounded Coplanar Waveguide (GCPW) line and a Microstrip (MS) line in a thin film configuration as encountered in MMICs. The single via solution that is used for the grounding pads of the GCPW portion is optimized, and the bandwidth is increased up to 110 GHz. Furthermore, this transition design could be a great solution in domains like multiple layers MMICs which need high integration level and in new domains such as microwave photonic devices [START_REF] Lee | Broadband modulation of light by using an electro-optic polymer[END_REF][START_REF] Goldfarb | Modeling via hole grounds in microstrip[END_REF].

METHODS

HFSS Modelling

In order to discern the impact of single via-hole grounding pads on the performance of a GCPW-MS transition, such a structure has been modelled using High Frequency Structure Simulator (HFSS) software. A back to back GCPW-MS-GCPW component has been modelled since it corresponds to the practical case that will be used for characterization. The layout of the component is shown in Figure 1. The overall length of the transition is L = 1 cm. The grounding pad dimensions are: width S = 1 mm and length L1 = 1 mm. The gap between the pad and center line is 25 µm. The MS line is 54 µm wide. The substrate thickness is 20 µm. The bottom (T ground ) and top metallization (T ms ) thicknesses are 200 nm and 900 nm, respectively. More details on materials can be found in Section 2.2 which is relative to the component fabrication. The diameter (D) of the via-holes varies from 100 to 900 µm. The results of HFSS simulation are given in Figures 2(a) and 2(b). Cutoff frequency is determined by either an S 21 value at -3 dB from its value in the low frequency range (-0.5 dB @ 250 MHz here), so at -3.5 dB or an S 11 value below -10 dB. Apart from the inherent propagation losses of the MS line, the resonance peaks of the S 21 curves, due to the parasitic inductance of the via-holes, are the main factor of bandwidth reduction, as shown by Goldfarb and Pucel [START_REF] Goldfarb | Modeling via hole grounds in microstrip[END_REF]. The parasitic inductance L para of a metallized cylindrical via-hole is directly related to its volume and can be expressed as follows:

L para = μ 0 2π • h • ln h + √ r 2 + h 2 r + 3 2 r -r 2 + h 2 (1)
where μ 0 is the vacuum permeability; h is the depth of the via-hole, equal here to the substrate thickness T BCB ; and r is its radius. According to this expression that is valid for dielectric thickness values less than 3% of the free-space wavelength of the working frequency, the via-hole inductance increases when its depth increases, and its radius diminishes. This directly supports the fact that bandwidth limitation is reduced using thin substrates and large via-hole diameters. For example, for T sub = 635 µm, L para = 0.23 nH and 0.47 pH respectively for via-hole diameters of 100 µm and 900 µm. Those values are respectively 405 fH and 44.5 fH when a substrate thickness T sub = 20 µm is used. When via-hole diameter is increased, the S 21 resonance peaks tend to be pushed towards higher frequency range, allowing a bandwidth around 100 GHz to be reached for via-hole radius of 900 µm. The deduced bandwidths are summarized in Table 1 versus the via-hole diameter.

Figure 3 shows the electric field distributions obtained by HFSS simulation for a transition, using 500 µm-diameter via-holes, for two representative frequencies, one corresponding to the major resonant peak at 102.5 GHz (Figure 3(a)) and the other in the transmission band, i.e., outside the resonant peak, here at 40 GHz (Figure 3(b)). It is noticeably observed that, at resonance frequency, the main part of the electric field is concentrated on the ground pads and that little energy is then propagating on the MS line whereas it is completely the reverse situation in the operating bandwidth of the transition.

Fabrication of the GCPW-MS-GCPW Component

The component has been fabricated following the layout shown in Figure 1. The substrate material is BCB (Benzocyclobutene resin). The metallic layer of the bottom ground plane and the via-hole filling are made of copper. The top metallization is made of gold that has been chosen mainly owing to its compatibility with the microwave probing equipment. The mask layout incorporates all the component

Processing Flow

The substrate is made of a photosensitive BCB: Cyclotene TM XUS35078-Type 3 from Dow R [START_REF]CYCLOTENE 4000 series -Kayaku advanced materials[END_REF]. Its permittivity, ε r , is 2.65, and it has been particularly chosen for its low value of dielectric loss tangent (tan δ = 0.0008 @ 20 GHz), for the thicknesses that can be deposited (a wide range from 15 µm to 40 µm for the formulation used) and for the advantages brought by its photosensitive nature. Indeed in a lot of devices, via-holes are opened using etching techniques like RIE (Reactive Ion Etching), DRIE (Deep Reactive Ion Etching), ICP (Inductively Coupled Plasma) [START_REF] Kondo | High-aspect ratio copper-via filling for three dimensional chip stacking[END_REF][START_REF] Lin | Via-filling capability of copper film by CVD[END_REF][START_REF] Wu | High aspect ratio silicon etch: A review[END_REF], or inkjet printing [START_REF] Mckerricher | Fully inkjet printed RF inductors and capacitors using polymer dielectric and silver conductive ink with through vias[END_REF] in the substrate (polymer, metal, silicon), which makes the manufacturing process complicated and expensive. For this demonstration purpose, using photosensitive BCB is simple because it allows opening the via-holes just using a lithography process. The process is presented below (Figure 4). -metallization of the bottom ground plane (200 nm) by sputtering deposition of a Ti/Cu layer (Ti is used as adhesion layer); -deposition of BCB (around 20 µm) by spin-coating; -lithography of BCB defining via-hole; -via-hole filling by electrodeposition of copper; -lithography of top layer metallization; -metallization of top ground pads and MS line (400 nm) by e-beam evaporation of Ti/Au layer (Ti is used as adhesion layer);

Substrate Deposition

The components are fabricated on a 3 silicon wafer. Before dielectric film is deposited, the bottom ground plane is full-wafer deposited using the sputtering deposition of, firstly, a layer of 20 nm-thick titanium that plays the role of adhesion layer and, secondly, a layer of 300 nm of copper. Then, a 27 µm-thick BCB XUS35078 film is deposited by spin-coating (a primary spin coat of adhesion promoter AP3000 was made). This allows to get a final thickness about 20 µm (DOW R recommends to have 6-7 µm thicker film than the desired value at spin-coat level since the post-bake process tends to densify the material and to reduce its thickness). A soft bake at 110 • C was made on a hot plate prior to exposure in order to remove any residual solvent.

Opening of the Via-Holes

The exposure of the via-hole patterns has been performed on a MA6/BA6 Suss MicroTec UV240-UV 400 nm at 500 mJ/cm 2 for 200 s. Development was made by immersion in DS3000 developer @ 35 • C for 7 minutes; it allows a total opening of via-hole patterns (Figure 5

Via-Holes Filling by Electrodeposition

Copper electrodeposition process has been widely used for via-hole filling [START_REF] Josell | Via filling by electrodeposition: Superconformal silver and copper and conformal nickel[END_REF][START_REF] Lühn | Filling of microvia with an aspect ratio of 5 by copper electrodeposition[END_REF]. Patterned BCB sample was dipped jointly with a copper bar inside an electrolytic bath of sulfuric acid and water. The bath is agitated at 200 rpm to maintain a uniform electrolytic behavior by keeping a homogeneous volume concentration of molecules. A voltage of 0.1 V and a current of 4.1 µA are applied on the sample for 7 minutes to achieve the full via filling.

Deposition of the GCPW-MS-GCPW Electrode Structure

Lithography of the top level has been made using AZ1512HS photoresist from Clariant because it offers a good undercut profile, an essential condition for the next lift-off process. A 20 nm-thick Ti layer and a 900 nm-thick Au layer were then full-wafer deposited. E-beam evaporation was used in order to ease the lift-off process. The latter was made using AR 300-76 stripper (Allresist) @ 75 • C for 60 minutes. The AR 300-76 stripper was chosen since it has no effect on the BCB layer contrary to other possible stripper solutions. The result, after lift-off, is shown in Figure 6.

EXPERIMENTAL RESULTS

The characterization has been carried out with a vector network analyzer (VNA) R&S R ZVA 24 associated with ZVA-Z110 heads and GSG 100 Infinity TM probes allowing the measurement of Sparameters over the frequency range from 250 MHz to 110 GHz. An LRRM (Line-Reflect-Reflect-Match) calibration method is performed preliminarily using a Cascade ISS-104-783 calibration substrate. The system is controlled by Wincal Cascade software. Measured results (Figure 7 and Table 2) are in good agreement with those obtained with HFSS electromagnetic simulation (cf. Figure 2). Resonant peaks appear at the same frequency, and the response is damped, as usual, compared to that obtained in simulation. For via-hole diameter above 700 µm, no resonant peak is observed in the measurement frequency band, and the transition bandwidth is superior to 110 GHz.

EQUIVALENT CIRCUIT MODELLING

Wiatr et al. proposed an equivalent circuit for such a GCPW-MS transition (only one via-hole centered in the top ground pad) fabricated on a thick (175 µm) and high permittivity [START_REF] Wu | High aspect ratio silicon etch: A review[END_REF] substrate [START_REF] Wiatr | Coplanar-waveguide-to-microstrip transition model[END_REF]. The proposed circuit was validated over a 30 GHz bandwidth; we used it as a basis and adapted it to extend its validity to thin and low permittivity substrates as well as to a larger bandwidth, 100 GHz typically.

The new equivalent circuit (Figure 8) differs by a residual capacitance C V ia that appears between the superior ground pad of the via-hole and the bottom ground plane and by a resistance R V ia related to the via-hole metallization. The GCPW-MS transition is now modelled by:

-C V ia , R V ia , and L V ia which are the three elementary components representing the via-hole, -C coupl and L coupl are respectively the coupling capacitance and the link inductance between ground pads and microstrip line, and R coupl is the resistance related to L coupl . Table 2. Measurement of the S-parameters of GCPW-MS-GCPW components as function of the via-hole diameter: the cutoff frequency is defined by the lowest of the two values corresponding to S 21 dB ≥ -3.5 dB and S 11 dB ≤ -10 dB.

Via-hole diameter (µm) S 21 dB ≥ -3.5 dB (GHz) S The microstrip section between the two GCPW-MS transitions is represented by a distributed element available in the ADS library, which allows ADS simulation to be easy, accurate, and fast.

Each passive element value has been optimized (Table 3) to fit the ADS simulated and experimental S-parameters values within 10% difference. First, the via-hole capacitance, C via , remains the same whatever the diameter is; the intrinsic capacity related to the via-hole is slightly affected due to the low variation in the dimensions of the ground pad, especially. L via logically decreases with the increase in via-hole diameter, roughly by 200 pH every 200 µm. R via remains unchanged, around 400 Ω. The value of C coupl , the capacitance which appears between the ground pad and the microstrip line, does not change significantly, around 10 fF whatever the diameter is; in fact, the gap between the ground pad and the MS line does not change either, explaining this invariance. L coupl also tends to decrease, same as L via , following pretty much the same tendency. Finally, R coupl shows an increase with diameter increase; this is almost unexplained right now, a behavior like that observed for C coupl would have been awaited.

The comparison between the simulation results obtained using this equivalent circuit on ADS and the measured S-parameters values shows a good agreement on their overall behavior and, especially, on the position of the resonance peaks (Figure 9). In addition, this equivalent circuit predicts bandwidth values closer to the experimental ones than HFSS simulation does. For example, for a via-hole diameter of 700 µm, due to its overestimation of S 11 module, HFSS simulation predicts a reflection-limited bandwidth of 76.4 GHz while it is over 110 GHz according to both measurement and ADS simulation results (Figure 9).

There are of course some minor differences between measurement and ADS simulation results concerning:

• the bandwidth limitation for smaller via-hole diameters, e.g., for via-hole diameter of 300 and 500 µm: the values obtained are 81 GHz and 92 GHz, respectively, by ADS modelling versus 85.75 GHz and 94.75 GHz for experimental ones (cf. S 21 curves on left top graph in Figure 9). • the discrepancy in reflection coefficient: it is more pronounced for larger via-holes diameters (cf.

S 11 curves on right bottom graph in Figure 9), but fortunately, reflection coefficients for viahole diameters of 700 µm and 900 µm always remain below the -10 dB criterion. There is so no consequence on bandwidth value.

The supplementary low amplitude oscillations observed on the experimental S 21 curves can be explained by the experimental positioning of the probes that always leaves a "dead end", i.e., the probes are never positioned at the perfect beginning of the component, resulting in a small echo on the transmission and reflection parameters.

CONCLUSION

We studied the influence of the via-hole diameter on the transmission and reflection parameters of a GCPW-MS transition on thin substrate. According to simulation results from HFSS, the bandwidth of such transitions using grounding pads with a single circular via-hole can exceed 100 GHz. GCPW-MS-GCPW components have been fabricated on a 20 µm-thick BCB substrate and measured over the 250 MHz-110 GHz frequency range; the results show a wider bandwidth than the one predicted by HFSS. Based on the equivalent circuit of the GCPW-MS transition proposed by Wiatr et al., an improved model has been developed to reproduce the experimental measurements up to 110 GHz on low permittivity and thin substrates. ADS simulation results using this equivalent circuit showed good agreement with experimental ones. Even if right now, a parametric model has not be fully set, the optimized value of each element constituting this equivalent circuit gives the trend. For the larger fabricated via-hole diameters, i.e., 700 µm and 900 µm, a bandwidth higher than 110 GHz has been recorded, bringing such a GCPW-MS transition on a thin film of BCB as one exhibiting the largest bandwidth currently reported in the literature.

Figure 1 .

 1 Figure 1. Modelled GCPW-MS-GCPW transition including input and output coplanar-microstrip transitions with grounding via-holes and a microstrip transmission line in between.

Figure 2 .Figure 3 .

 23 Figure 2. Simulated S-parameters issued from HFSS modelling as a function of the via-hole diameter: (a) S 21 and (b) S 11 .

Figure 4 .

 4 Figure 4. Process flow (main steps) for the component fabrication.

  (a)). A profilometer measurement shows complete opening of the via-holes just after development (Figure 5(b)). A post-bake at 250 • C for 30 minutes was then performed to totally harden the BCB material.

Figure 5 .

 5 Figure 5. (a) Optical microscope view of a 500 µm-diameter via-hole pattern under development; (b) Corresponding profilometer measurement prior to the BCB post-bake.

Figure 6 .

 6 Figure 6. Part view of the wafer on which a set of nine components, each of them including transitions with different via-hole diameters, from 100 µm to 900 µm, has been fabricated.

Figure 7 .

 7 Figure 7. Measured S-parameter as function of the via-hole diameter: S 21 and (b) S 11 .

Figure 8 .

 8 Figure 8. Equivalent circuit of the GCPW-MS-GCPW component shown in Figure Lumped and distributed elements are used to model the vias and the MS line, respectively.

Figure 9 .

 9 Figure 9. Comparison between ADS simulation and measurement results of the S-parameters of the GCPW-MS-GCPW component shown in Figure 1.

Table 1 .

 1 HFSS simulation of GCPW-MS-GCPW component as function of the via-hole diameter: the cutoff frequency is defined by the lowest of the two values corresponding to: S 21 dB ≥ -3.5 dB or S 11 dB ≤ -10 dB.

	Via-hole diameter (µm)	S 21 dB ≥ -3.5 dB (GHz) S 11 dB ≤ -10 dB (GHz)	Cutoff frequency (GHz)
	100	47.3	47.3	47.3
	200	58.8	56.6	56.6
	300	70.7	65.5	65.5
	400	76.9	74.6	74.6
	500	84.4	74.7	74.7
	600	85.9	75.5	75.5
	700	94.3	76.4	76.4
	800	95.4	85.5	85.5
	900	104.1	95.1	95.1

variants mentioned above, i.e., components with via-hole diameters from 100 µm up to 900 µm by 100 µm step. This allows a collective fabrication of all components ensuring the different material layers to be identical from one component to another, removing some possible error sources comparing experimental results between variants.

Table 3 .

 3 Component values extracted from ADS after optimization.

	Via-hole diameter 300 µm 500 µm 700 µm 900 µm
	C via	6.04 fF 6.17 fF 6.12 fF 6.51 fF
	L via	950 pH 765 pH 546 pH 352 pH
	R via	408 Ω	388 Ω	406 Ω	401 Ω
	C coupl	10.8 fF 10.5 fF 8.00 fF 13.1 fF
	L coupl	977 pH 643 pH 598 pH 356 pH
	R coupl	1.02 Ω	53.9 Ω	191 Ω	287 Ω

ACKNOWLEDGMENT

The authors would like to thank the support of the French National Research Agency (ANR) and the Directorate General of Armaments (DGA) of the French Armed Forces Ministry under the ADC-Poly project (ANR-17-ASTR-0004). Pierre-Vincent DUGUE got a doctoral fellowship from both DGA (agreement N • 2017 60 0052) and Région des Pays de la Loire (agreement N • 2017-08522). Authors also thank the RENATECH network for its support in device fabrication.