Marlène Legrand

Antoine Mousson

Philippe Carl

Léa Rossé

Hélène Justiniano

Jean-Pierre Gies

Daniel Bouvard

Emilie Sick

Denis Dujardin

Philippe Rondé
email: philippe.ronde@unistra.fr

Protein dynamics at invadopodia control invasion-migration transitions in melanoma cells

Keywords: ECM, extracellular matrix, MMP, matrix metalloprotease, FAK, Focal Adhesion Kinase, FA, Focal adhesion Pyk2, cortactin, Tks5, FAK, dynamics, invadopodia, invasion, migration, melanoma

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Introduction

Melanoma is a very aggressive form of skin cancer and patients with spreading melanoma have a very poor prognosis with a 5-year survival rate less than 5% [START_REF] Shain | From melanocytes to melanomas[END_REF] . In melanoma as in many invasive epithelial cancers, such as prostate, head and neck or breast cancer, cancer cells acquire the ability to breach basement membrane using supramolecular complexes called invadopodia.

Invadopodia are sub-cortical protrusions that localize matrix-degrading activity to cell-substrate contact points and represent major hubs where many signalling pathways converge [START_REF] Hoshino | Signaling inputs to invadopodia and podosomes[END_REF] .

Invadopodia are composed of an actin-rich core which include actin activators, nucleators and regulators that is surrounded by adhesion, scaffolding and signaling proteins [START_REF] Murphy | The 'ins' and 'outs' of podosomes and invadopodia: characteristics, formation and function[END_REF] . These structures are enriched in proteolytic enzymes such as matrix metalloproteinases (MMPs), which mediate extracellular matrix (ECM) degradation. Significant efforts have been made to characterize the mechanism of invadopodia formation and the sequence of events that allows the correct assembly and regulation of inavadopodia components start to be elucidated [START_REF] Eddy | Tumor Cell Invadopodia: Invasive Protrusions that Orchestrate Metastasis[END_REF] .

However, one of the key questions that remains to be solved concerns the maturation step which regulates the switch between immature invadopodia devoid of invasive properties to fully mature invadopodia able to degrade the ECM. Several proteins or signalling pathways have been already implicated in this process such as Tks5 5 , NHE-1 6 profilin 1 [START_REF] Valenzuela-Iglesias | Profilin1 regulates invadopodium maturation in human breast cancer cells[END_REF] , talin2 [START_REF] Baster | Talin2 mediates secretion and trafficking of matrix metallopeptidase 9 during invadopodium formation[END_REF] , Mena IV9 , protrudin [START_REF] Pedersen | Protrudin-mediated ER-endosome contact sites promote MT1-MMP exocytosis and cell invasion[END_REF] or the CCL7/CCR3 axis [START_REF] Qi | The chemokine CCL7 regulates invadopodia maturation and MMP-9 mediated collagen degradation in liver-metastatic carcinoma cells[END_REF] . In melanoma the transition from in-situ melanoma (melanoma cells located in their original tissue) to invasive melanoma seems to be correlated with direct contact of melanoma cells with distal keratinocytes of the epidermal layer thereby triggering Noth signaling pathway [START_REF] Golan | Interactions of Melanoma Cells with Distal Keratinocytes Trigger Metastasis via Notch Signaling Inhibition of MITF[END_REF] . Of note, inhibition of Notch signalling or suppression of cell-cell contact reduced Mena IV expression in tumour cells and invadopodia activity [START_REF] Pignatelli | Macrophagedependent tumor cell transendothelial migration is mediated by Notch1/MenaINV-initiated invadopodium formation[END_REF] raising the possibility that in-situ melanoma display already invadopodia structure that mature during their transition into invasive melanoma.

Another aspect of the invadopodia lifetime that is poorly understood concerns the disassembly process. Several lines of evidence seem to indicate that focal adhesion and invadopodia turn-over are regulated by similar pathways. Thus, calpain has been linked to focal adhesion and invadopodia disassembly through cleavage of their respective substrates including talin [START_REF] Franco | Calpain-mediated proteolysis of talin regulates adhesion dynamics[END_REF] , paxillin [START_REF] Cortesio | Calpain-mediated proteolysis of paxillin negatively regulates focal adhesion dynamics and cell migration[END_REF] and cortactin [START_REF] Cortesio | Calpain 2 and PTP1B function in a novel pathway with Src to regulate invadopodia dynamics and breast cancer cell invasion[END_REF] . In line with this, calpain has also been reported to induce podosome disassembly in dendritic cells, through cleavage of important components like WASP, Pyk2 and talin [START_REF] Calle | Inhibition of calpain stabilises podosomes and impairs dendritic cell motility[END_REF] . FA turn-over which tune the migration speed [START_REF] Giannone | Calcium oscillations trigger focal adhesion disassembly in human U87 astrocytoma cells[END_REF][START_REF] Webb | FAK-Src signalling through paxillin, ERK and MLCK regulates adhesion disassembly[END_REF] is also controlled in part by phosphorylation and dephosphorylation processes of key molecules such as FAK and paxillin [START_REF] Deramaudt | FAK phosphorylation at Tyr-925 regulates cross-talk between focal adhesion turnover and cell protrusion[END_REF][START_REF] Hamadi | Regulation of focal adhesion dynamics and disassembly by phosphorylation of FAK at tyrosine 397[END_REF][START_REF] Hildebrand | Paxillin, a tyrosine phosphorylated focal adhesionassociated protein binds to the carboxyl terminal domain of focal adhesion kinase[END_REF] . While FAK is absent from dot-like invadopodia [START_REF] Chan | FAK alters invadopodia and focal adhesion composition and dynamics to regulate breast cancer invasion[END_REF][START_REF] Kolli-Bouhafs | FAK competes for Src to promote migration against invasion in melanoma cells[END_REF][START_REF] Mousson | Inhibiting FAK-Paxillin Interaction Reduces Migration and Invadopodia-Mediated Matrix Degradation in Metastatic Melanoma Cells[END_REF] , it is interesting to note that RhoG modulate the phosphorylation of paxillin during invadopodia disassembly [START_REF] Goicoechea | A RhoG-mediated signaling pathway that modulates invadopodia dynamics in breast cancer cells[END_REF] . Similarly, in Rous sarcoma virus-transformed baby hamster kidney cells, paxillin phosphorylations on tyrosine 31 and 118, also allows invadopodia disassembly [START_REF] Badowski | Paxillin phosphorylation controls invadopodia/podosomes spatiotemporal organization[END_REF] .

In addition, the turn-over of adhesion structures is controlled by the p21 Rho family of small GTPase. For exemple, Rac1 induces the turnover of both focal adhesions and podosomes [START_REF] Manser | Expression of constitutively active alpha-PAK reveals effects of the kinase on actin and focal complexes[END_REF][START_REF] Webb | PAK1 induces podosome formation in A7r5 vascular smooth muscle cells in a PAK-interacting exchange factor-dependent manner[END_REF] .

Controlled focal adhesion turn-over take place notably at the lamellipodia which extend protrusions that mediate cell migration. Recently, Rac1 has also been shown to control invadopodia disassembly via activation of the downstream effectors p21 activated kinase 1 (Pak1) with subsequent phosphorylation of cortactin [START_REF] Moshfegh | A Trio-Rac1-Pak1 signalling axis drives invadopodia disassembly[END_REF] . Because Rac1 is necessary for both focal adhesion and invadopodia turn-over and focal adhesion turn-over is essential for lamelipodia extension, it is generally admitted that invadopodia assembly and cell migration occurs in a sequential manner. Indeed, this is consistent with the fact that degradation activity at specific areas requires the time necessary for invadopodia assembly and maturation which is difficult to reconcile with sustained migration. Thus, a spatio temporal control of FA and invadopodia turnover is necessary for proper invasion capabilities. Recently it was shown that this regulation could be made by differential activation of downstream effectors of Rac1. Indeed, local activation of Rac1 at the leading edge promotes Wave2-dependent signalling for further lamillipodia extension [START_REF] Yamaguchi | Invadopodia and podosomes in tumor invasion[END_REF] . This extension is also under the control of the underneath focal complex turnover driven in part by Dock180 a GEF for Rac1 [START_REF] Deramaudt | FAK phosphorylation at Tyr-925 regulates cross-talk between focal adhesion turnover and cell protrusion[END_REF] . On the other hand, Rac1 through activation of another GEF Trio, promotes Pak1-mediated invadopodia disassembly [START_REF] Moshfegh | A Trio-Rac1-Pak1 signalling axis drives invadopodia disassembly[END_REF] .

In this study, using high-end imaging techniques, we investigated invadopodia maturation and disassembly processes. We show that Pyk2 and cortactin are major components of both focal adhesion and invadopodia. During invadopodia disassembly Pyk2 and cortactin are repeatedly relocated to nearby focal complexes promoting lamelipodia extension. We also found that during invadopodia turn-over, cell migration is reduced which may be due to the dynamic redistribution of invadopodia components to focal adhesion in close proximity.

Finally, we show that a small Pyk2 inhibitor reduced melanoma migration without enhancing invadopodia activity contrarily to a small FAK inhibitors that reduced only the migration process but enhanced matrix degradation via relocation of FA component to invadopodia [START_REF] Kolli-Bouhafs | FAK competes for Src to promote migration against invasion in melanoma cells[END_REF] .

Material and methods

Antibodies

For immunofluorescence, anti-cortactin (05-180) from Millipore; anti-paxillin (AHO0492) and mouse anti-phospho-Y402 Pyk2 (44-625G) from Invitrogen; rabbit anti-phospho-Y402 Pyk2 (3291) from Cell Signalling; phalloidin 568 (A12380), phalloidin 647 (A22287) and Alexa Fluor-conjugated secondary antibodies from Molecular Probes.

Cells culture

Human melanoma cell lines, were purchased from Rockland and cultured in MCDB153/L-15 medium (M7403, Sigma; 11415064, Invitrogen) supplemented with 2% foetal bovine serum, 1,68 mM CaCl2 and 1% penicillin/streptomycin. The A375 melanoma cell line from ATCC was maintained in DMEM (11960044, Lonza) supplemented with 10% foetal bovine serum, 1mM L-glutamine and 1% penicillin/streptomycin antibiotics. Melanoma cells were used between passage 5 and 30.

Cells transfections

FAK-GFP, Src-Y530F-mCherry were obtained as previously described [START_REF] Hamadi | Src activation and translocation from focal adhesions to membrane ruffles contribute to formation of new adhesion sites[END_REF] McNiven, (Mayo Cancer Center, Rochester). All plasmids were verified by sequencing and isolated using the JetStarPlasmid kit (Genomed) following the manufacturer's protocol.

Melanoma cells were transfected with 2 µg of vinculin-mCherry or Pyk2-GFP or Tks5-GFP or Cortactin-dsRed using lipofectamine 2000 (Invitrogen), according to the manufacturer instructions. Transfected cells were incubated for 24h at 37°C before use.

Matrix degradation assay and immunofluorescence

Glass bottom Ibidi dish (81158, Ibidi) were coated with fluorescent-labelled-FITC gelatine or Cy3 gelatine as described previously [START_REF] Mousson | Inhibiting FAK-Paxillin Interaction Reduces Migration and Invadopodia-Mediated Matrix Degradation in Metastatic Melanoma Cells[END_REF] . Then, 3.5.10 4 previously starved melanoma cells were plated on fluorescent gelatine and incubated at 37°C for 24h in medium supplemented with 10% foetal bovine serum with or without PF-573228 or PF-431396. Cells were fixed using 4% paraformaldehyde permeabilized using triton-X-100 at 0,1% and incubated in 2% of bovine serum albumin at room temperature. Cells were then labelled for 1h with anti-cortactin (1/250), anti-phospho-Y402 Pyk2 (1/100), anti-FAK (1/250), anti-phospho-Y118 paxillin (1/250) or anti-paxillin (1/250). After three washes with PBS, cells were incubated with Alexa-488conjugated goat anti-rabbit (1/1000), Alexa-647-conjugated goat anti-rabbit (1/1000), Alexa-405-conjugated goat anti-mouse (1/1000), Alexa-647-conjugated goat anti-mouse (1/1000), Alexa-568-conjugated phalloidin (1/250) or Alexa-647-conjugated phalloidin (1/250) for 1, washed and mounted in Prolong Gold mounting media (Invitrogen). Cells were imaged using Leica TSC SPE confocal microscope (x63 HCX Pl Apo 1.40 NA or x20 HCX Pl Apo 0.7 NA objective, Wetzlar, Germany). Invadopodia were identified as actin and cortactin rich punctate structure. Areas of degradation were identified as "black holes" within the fluorescent gelatine.

Invadopodia and areas of degradations were quantified using ImageJ software. Degradation areas measurements were based on cells displaying degration activity, and the frequency of degradation was based on randomly selected cells.

Wound healing and 2,5D invadopodia assay

Melanoma cells were seeded in 12 well plates containing medium supplemented with 10% foetal bovin serum and transfected with siCtrl or siFAK as described before. After 24h, confluent cell layers were treated with mitomycin C and with PF-573228 or PF-431396 as described before and then wounded with a pipet tip. Images of cell migration were acquired using a Leica DMIRE2 microscope (x10 N PLAN 0.25 NA objective) equipped with a 37°C 5% CO2 control system (Life Imaging Services) with a Leica DC350FX CCD camera piloted by the FW4000 software (1 image every 30 min during 24h). The 2,5D invadopodia assay was performed as previously described (Mousson). Briefly, melanoma cells were added in the upper side of the insert at 5.10 4 cells and incubated 24 hours at 37°C in 5% CO2 incubator. After incubation, inserts were washed with PBS and fixed as described in "Matrix degradation assay and immunofluorescence" section. Cells were imaged using a Leica confocal microscope TSC SPE (#63 HCX Pl Apo 1.40 NA objective, Wetzlar, Germany). 3D images were reconstructed using ImageJ or Imaris. The length of each protrusion containing actin and cortactin staining was calculated using ImageJ.

Live cell imaging, FRAP and TIRF experiments

For live cell imaging, cells were plated on fluorescent gelatin-coated coverslips and mounted in a Ludin Chamber (Life Imaging Services). The cells were then placed at 37°C, 5% CO2 on a iMIC microscope equipped with a multi-LED Lumencor Spectra X. Images were acquired with an Olympus 60x TIRFM (1.45 NA) objective every 5 to 10 min during 24 h and a Hamamatsu Flash 4 V2+ camera (Iwata) piloted by the Live Aquisition software (Till Photonics).

Expressing cells were initially located via the GFP or mCherry signal, and were subsequently followed only via dual phase contrast/fluorescent signal together with the FITC or RITCcoupled gelatin substrate. 10 to 20 different fields were sequentially recorded during each experiment using a Marzhauser automated stage piloted by the iMIC software. Mean fluorescence intensity at focal adhesion, invadopodia and underlying substrate at each time points were calculated. TIRF images were acquired using an iMIC microscope (Till Photonics) equipped with a Cobolt Dual Calypso Laser 491/532 nm and an Olympus 60x TIRFM (1.45 NA) objective. During acquisition, cells plated on plasma-cleaned coverslips, were maintained at 37°C in a 5% CO2 humidified atmosphere using an environmental control system (Life Imaging Services).

Azimuthal TIRF images were acquired every 1 min during 1 hour on an Hamamatsu Flash 4 V2+ camera (Iwata) and analyzed using ImageJ software.

FRAP experiments were done at 37°C, 5% CO2 on a iMIC microscope. To avoid possible artifacts of overexpression, only cells expressing low but detectable amounts of protein were chosen for further analysis. Briefly, fluorescence intensity was measured at low laser power before bleach in selected regions of interest at invadopodia. Photobleaching was done at 100% of the 491 nm line with 4 iterations. Recovery was followed with low laser power at various time intervals until the intensity had reached a steady plateau. For each time point, the intensity of the bleached area was normalized to that of a corresponding unbleached area. Fluorescence during recovery was normalized to the prebleached intensity. FRAP measurements were fitted to a single exponential curve, I(t) = I(0) + k1e -k2t to determine the characteristic time of recovery.

Statistical analysis

Statistical significance was calculated using Student's t test or ANOVA followed by Tukey's test for multiple comparisons. Values were considered statistically significant when pvalue was <0.05. For all figures: *, P < 0.05; **, P < 0.01; ***, P < 0.001.

Results

Actin and Cortactin labelled both active and inactive invadopodia

To elucidate the specific role of invadopodia related-protein during invadopodia maturation, we first plated melanoma cells on fluorescent gelatin and analyzed invadopodia activity. We recently reported that melanoma cells form inactive invadopodia when located in the epidermis that matures into fully degradating invadopodia during the transition from in-situ to metastatic locations [START_REF] Mousson | Inhibiting FAK-Paxillin Interaction Reduces Migration and Invadopodia-Mediated Matrix Degradation in Metastatic Melanoma Cells[END_REF] . Indeed, we found that, although many invadopodia, identified as actin-and cortactin-rich punctate structures, were present in in-situ WM1862 melanoma cells (mean value: 13.87 ± 1.19 invadopodia/cell) none of them displayed degradative properties as reported by in-situ zymography assay (Figure 1A, see quantification of the gelatin signal in green along the yellow arrows 1, 2 and 3). To confirm the identity of these actin-and cortactin-rich structures we tested their capacity to form ventral protrusions when plated on 1µm transwell filters. 3D reconstruction of confocal images indicated the presence of ventral protrusions containing actin and cortactin in both in-situ (WM 1862 and WM 1552C) and malignant cell lines (WM983B and A375). Of note invasive melanoma displayed longer invadopodia compared to non-invasive cells that was reduced in the presence of the metalloprotease inhibitor GM6001 (Figure 1B).

We next analyzed invadopodia activity in metastatic A375 melanoma cells. As previously reported, many invadopodia were identified in these cells using dot-like actin/cortacin colocalisation (mean value: 14.34 ± 0.57 invadopodia/cell) but only a few of them were able to degrade the fluorescent gelatin, the degradation frequency being 20.08 % ± 1.81 (Figure 1C). Moreover, invadopodia-induced gelatin degradation was completely inhibited in the presence of GM6001 (Supplemental Figure 1). Thus altogether, these results indicate that actin and cortactin labelled both active and inactive invadopodia.

Pyk2 localisation at invadopodia correlates with matrix degradation

Recently, Pyk2 was found to localize to invadopodia in invasive breast cancer cells. We thus sought to determine the role of Pyk2 in invasive melanoma cells. Immunofluorescent labelling of endogenous activated Pyk2 (pY402-Pyk2) in A375 cells showed colocalization with cortactin and actin at both ventral areas and peripheral structures resembling to focal adhesion. Of note, the distribution of activated Pyk2 at ventral areas display a more restricted pattern than the total distribution of invadopodia identified by dot-like actin cortacin colocalisation (Figure 2A, see quantification of the signal along the yellow arrows 1 and 2). To test whether Pyk2 is located at active invadopodia, A375 cell were plated on fluorescent ECM and colocalisation analysis of Pyk2 with actin spots at degradation punctia was performed. We found consistent localisation of activated Pyk2 only at actin positive structures associated with areas of degradation (Figure 2B). Next, to clearly identify focal adhesion, immunofluorescent experiments were performed using an anti-paxillin antibody. Activated Pyk2 colocalised with paxillin at focal adhesion whereas paxillin did not colocalised with Pyk2 at ventral degradation areas indicating that Pyk2 contrarily to paxillin localised to both focal adhesion and active invadopodia (Figure 2C). These results suggest that Pyk2 may be a marker of active invadopodia. To test this hypothesis, immunofluorescent labeling of activated Pyk2 was assessed in a panel of melanoma cells ranging from non-invasive in-situ melanoma (WM1552 and WM1862) to invasive (WM115 and WM983A) and metastatic melanoma (WM983B and A375). In in-situ melanoma, displaying immature invadopodia unable to degrade the ECM, Pyk2 localisation was restricted to focal adhesions (Figure 3A). In close opposition, active Pyk2 was observed at both focal adhesion and mature invadopodia, in invasive and metastatic cells.

Of note, immature invadopodia present in invasive cells do not display detectable amount of active Pyk2 (Figure 3A). Quantification of pY402-Pyk2 at active versus inactive invadopodia in this panel of melanoma cells revealed that active Pyk2 is significantly over-expressed only at active invadopodia indicating a major role of Pyk2 in invadopodia maturation processes (Figure 3B, Supplemental Table 1).

Pyk2-GFP and Cortactin-dsRed localize to a subset of focal adhesion and invadopodia

To examine the role of these proteins in the stages of invadopodium maturation and disassembly and their potential relation with focal adhesion in more detail, we next used timelapse videomicroscopy. Previously, we have shown that FAK is not located at invadopodia in melanoma cells [START_REF] Kolli-Bouhafs | FAK competes for Src to promote migration against invasion in melanoma cells[END_REF][START_REF] Mousson | Inhibiting FAK-Paxillin Interaction Reduces Migration and Invadopodia-Mediated Matrix Degradation in Metastatic Melanoma Cells[END_REF] . Indeed, in A375 cells expressing FAK-GFP, FAK colocalized with activated Pyk2 and paxillin at focal adhesions whereas Pyk2 but not FAK colocalised with actin dots at invadopodia (Supplemental Figure 2A). Similarly, in A375 cells expressing vinculin-mCherry, vinculin colocalized with actin and cortactin at focal adhesion but not at invadopodia (Supplemental Figure 2B). On the other hand, both Pyk2 and cortactin localised to a subset of focal adhesions and invadopodia in invasive melanoma expressing either Pyk2-GFP or cortactin-dsRed. Indeed, in these cells, Pyk2-GFP localised to peripheral focal adhesion and to some invadopodia identified by dot-like actin/cortactin colocalisation (Supplemental Figure 3A) whereas cortactin-dsRed localised to some focal adhesion and to invadopodia (Supplemental Figure 3B). This result suggests a dynamic recruitment of Pyk2 at specific stages of invadopodia assembly and further show that cortactin and/or Pyk2 are suitable for invadopodia-focal adhesion interplay investigations.

Loss of Cortactin at invadopodia is time-correlated to the formation of focal adhesioncontaining cortactin during migration

Next, using time-lapse imaging of live A375 cells expressing cortactin-dsRed plated on fluorescent gelatin, we found that cortactin localisation at invadopodia occurs several minutes before gelatin degradation (Figure 4, Video 1) in agreement with previous studies [START_REF] Webb | Dissecting the functional domain requirements of cortactin in invadopodia formation[END_REF] . The increase in cortactin level reached a peak about 200-230 min after the beginning of the invadopodia formation with a second phase that seems more pronounced at t=100 (Figure 4B) which may be related to the established role of cortactin in the regulation of protease secretion [START_REF] Clark | A new role for cortactin in invadopodia: regulation of protease secretion[END_REF] .

Analysing in further details invadopodia turn-over, we observed, intact, actively sliding invadopodia translocation toward the peripheral cell border as the cell migrate forward which end-up into the complete disassembly of the structure (Figure 5, Video 2). It is interesting to note that the disappearance of cortactin at this site is temporally correlated to the appearance of cortactin at nearby nascent adhesion in 71,4% of the cells displaying active invadopodia, suggesting that the formation of these adhesions could be related to the disassembly of invadopodia via exchange of cortactin (Figure 5 A andB).

Disassembly of invadopodia-containing cortactin and Pyk2 trigger nascent adhesion formation

To investigate protein dynamic at invadopodia at high resolution and analyse the relationship between invadopodia and focal adhesion, we performed dual epifluorescence/TIRF microscopy of A375 cells transfected with either dsRed-cortactin, or Pyk2-GFP. Time-lapse acquisition of dsRed-cortactin revealed several dot-like invadopodia localised at the ventral area of the cells that can be visualized in both epifluorescence and TIRF microscopy (Figure 6A, left, Video 3). During the time course of the experiment we observed by TIRF microscopy that, in 62.6 % of the cells analysed, many dot-like invadopodia, gradually changed their morphology to adopt more dispersed structures (arrowheads 1 to 3) resembling to nascent adhesion (Figure 6A). The time-course of these changes is relatively rapid and occurred in 5-6 min which is consistent with the previously reported time of invadopodia disassembly [START_REF] Moshfegh | A Trio-Rac1-Pak1 signalling axis drives invadopodia disassembly[END_REF] . To clearly characterize these structures, we then used cells expressing Pyk2-GFP as we have shown that P-Pyk2 is both a marker of active invadopodia and also located at focal adhesion (Figure 2 and 3). Indeed, we identified both dot-like-invadopdia (arrows) at ventral areas and focal adhesion at peripheral areas (arrowheads) with epifluorescence and TIRF microscopy (Figure 6B, left, Video 4). Time-lapse acquisition in TIRF mode of Pyk2-GFP shows that, in 66.7% of the cells analysed, newly formed fluorescent adhesions were initiated at the dot-like invadopodia location before spreading away in a manner similar to what has been described for nascent adhesion dynamics at lamellipodia (Figure 6B, insets). We also observed that Pyk2 fluorescence at invadopodia decreases while it increases at nearby nascent adhesion suggesting that invadopodia and focal adhesion likely exchange protein component during their turn-over.

To verify whether this mechanism is specific to protein shared by both invadopodia and focal adhesions we performed time-lapse imaging of Tks5-GFP a protein not found in other protrusions and adhesion structures [START_REF] Martino | Cdc42 and Tks5: a minimal and universal molecular signature for functional invadosomes[END_REF][START_REF] Paterson | Invadosomes are coming: new insights into function and disease relevance[END_REF][START_REF] Seals | The adaptor protein Tks5/Fish is required for podosome formation and function, and for the protease-driven invasion of cancer cells[END_REF] . Several Tks5-GFP-containing invadopodia were clearly identified by both epifluorescence and TIRF microscopy (Supplemental Figure 4, Video 5). However, in close opposition to the observations made in cells expressing either cortactin-dsRed or Pyk2-GFP, dynamic analysis of Tks5-GFP over time by TIRF microscopy revealed that, translocation of this protein to other structure could be observed in only 11,5 % of the cells analysed. Protein dynamic at invadopodia and their exchange rate with focal adhesion is dependent on the time-residency of these proteins at invadopodia. We thus assessed these parameters by fluorescence recovery after photobleaching (FRAP) experiments. The GFP, ds-Red or m-Cardinal moiety contained in either Tks5, Pyk2, cortactin or actin constructs present at invadopodia randomly selected at the ventral part of the cells were photobleached with short high-power 491 or 532 nm laser excitation, and the recovery of fluorescence in the bleached regions was followed by time-lapse imaging in TIRF mode over the ensuing 200 s, at invadopodia. Recovery of fluroescence after photobleaching at invadopodia was fast for all the 4 proteins investigated indicating that invadopodia components are continuously renewed during their life-time (Figure 6C). Of note, Pyk2 (t1/2 = 2.30 ± 0.48 s) and to a lesser extend cortactin (t1/2 = 4.36 ± 1.01 s) are substantially faster compared with Tks5 (t1/2 = 5.77 ± 0.90 s) or actin (t1/2 = 8.57 ± 0.51 s) which may be related to the dynamic exchange of Pyk2 and cortactin between invadopodia and focal adhesion.

Migration speed is reduced during invadopodia activity

Because invadopodia and nascent adhesion exchange proteins essential to maintain their structure we hypothesised that cell migration and active invadopodia assembly occurs sequentially. To test this hypothesis we acquired time-lapse images of A375 metastatic melanoma cells known to spontaneously assemble invadopodia. Cells were engineered to express the fluorescent invadopodia marker mCardinal-LifeAct and cultured on fluorescent gelatin. Cell migration speed were quantified by tracking the nucleus path over time before the invadopodia assembly, during their life-time, and after their disassembly (Figure 7). In the beginning of the recording, before invadopodia assembly, the tracked cells migrate relatively fast at a mean value of 0.232 ± 0.021µm/min. However, during the lifetime of active invadopodia the cells migrate significantly less rapidly (0.180 ± 0.013µm/min) before returning close to the initial migration speed (0.222 ± 0.021µm/min) after complete invadopodia disassembly (Figure 7B). This result clearly indicates a negative balance between invadopodia activity and migration. Taken together, our results suggest that the frequency of switching between the invasion state and migration state is, at least in part, controlled by the exchange rate of common proteins at invadopodia and focal adhesion.

The dual FAK/Pyk2 inhibitor PF-431396 reduced both migration and matrix degradation of melanoma cells

Since tumor invasion requires both cell migration and MEC degradation, we investigated whether a pharmacological inhibitor of a protein implicated in both processes would be an efficient strategy to prevent cell invasion. Indeed, several kinase inhibitors targeting migration and/or invasion processes are currently into clinical trials [START_REF] Meirson | Targeting invadopodiamediated breast cancer metastasis by using ABL kinase inhibitors[END_REF][START_REF] Mousson | Targeting Focal Adhesion Kinase Using Inhibitors of Protein-Protein Interactions[END_REF][START_REF] Naser | Endogenous Control Mechanisms of FAK and PYK2 and Their Relevance to Cancer Development[END_REF] . Thus, we compared the effect of the selective FAK inhibitor PF-573228 and the dual FAK/Pyk2 inhibitor PF-431396 on melanoma cell migration and ECM degradation. Using the wound healing assay, we observed that FAK inhibition reduced the migration speed of metastatic melanoma cells by 30-40% compared to control conditions, as already demonstrated in other cancers cell lines (Figure 8 A-C). These results confirmed that FAK expression is necessary for optimal melanoma cell migration independently of melanoma aggressivity. Interestingly, treatment with PF-431396 did not further inhibit cell migration as compared to treatment with PF-573228 suggesting FAK and Pyk2 may have redundant effects on cell migration in these cell lines. We next analysed the effect of FAK inhibition and dual FAK/Pyk2 inhibition on invadopodia properties. Confirming our previous results, we show that FAK inhibition significantly increased invadopodia-induced matrix degradation as compared to control conditions (Figure 8 B-C). In close opposition, dual FAK/Pyk2 inhibition did not modify invadopodia activity. This indicates that Pyk2 inhibition is sufficient to counteract the effect of FAK inhibition-mediated increased invadopodia activity. These results are in line with the localisation of active Pyk2 at invadopodia in these cells shown in Figure 3 and 4 and confirmed previous data establishing that Pyk2 is necessary for invadopodia activity in breast cancer cell line [START_REF] Genna | Pyk2 and FAK differentially regulate invadopodia formation and function in breast cancer cells[END_REF] . Altogether, these results indicate that inhibition of both focal adhesion and invadopodia activity may represent a novel approach to reduce metastatic processes.

Discussion

Using live-cell video microscopy, high resolution TIRF microscopy and FRAP analysis, we have characterized protein dynamics at invadopodia of melanoma cells adhering to a thin 2D matrix. To our knowledge, we are the first to demonstrate that components shared by invadopodia and focal adhesion, such as Pyk2 and cortactin displayed rapid dynamics at invadopodia allowing fast redistribution to nascent adhesions during invadopodia disassembly.

In this study, we observed invadopodia formation and elongation in in-situ and metastatic human melanoma cell lines using 2,5D invasion assays. As invadopodia initially form precursors enriched in F-actin, Arp2/3 and cortactin which then mature to acquire proteolytic activity [START_REF] Artym | Dynamic interactions of cortactin and membrane type 1 matrix metalloproteinase at invadopodia: defining the stages of invadopodia formation and function[END_REF][START_REF] Beaty | Digging a little deeper: the stages of invadopodium formation and maturation[END_REF][START_REF] Sharma | Tks5 and SHIP2 regulate invadopodium maturation, but not initiation, in breast carcinoma cells[END_REF] , we suggest that in-situ melanoma cells assemble invadopodia precursor and that the maturation step occurs during melanoma progression. In agreement, previous studies described the possible involvement of signalling pathways in invadopodia maturation. Notch signalling has been described to initiate melanoma progression by inhibiting MITF function [START_REF] Golan | Interactions of Melanoma Cells with Distal Keratinocytes Trigger Metastasis via Notch Signaling Inhibition of MITF[END_REF] .

Moreover, Notch signalling initiate Mena IV expression which stimulate invadopodia formation and function by enhancing the phosphorylation of Y421 cortactin [START_REF] Weidmann | MenaINV dysregulates cortactin phosphorylation to promote invadopodium maturation[END_REF] . Our results show that another protein, Pyk2 is specifically localized at degradative but not immature invadopodia thereby identifying Pyk2 as a major protein implicated in the maturation process. The mechanism implicated in Pyk2-induced invadopodia activity remains to be determine but recent studies have shown that Pyk2 recruits Src kinase into invadopodia and control cortactin phosphorylation in breast cancer cells [START_REF] Genna | Pyk2 and FAK differentially regulate invadopodia formation and function in breast cancer cells[END_REF] suggesting a possible mechanism for invadopodia maturation. This mechanism may depend on MMP activity as its inhibition reduced both invadopodia length and Pyk2 localisation to invadopodia as reported here.

Depending of the cells studied, it has been reported that the life-time of invadopodia is relatively high, ranging from minutes to hours thereby allowing significant matrix degradation at these sites. Therefore, for efficient ECM degradation leading to dissemination of melanoma cells, high migration speed should be either avoided or redirected toward the degradation area.

We found that, during invadopodia lifetime, melanoma migration speed was reduced confirming previous studies on breast carcinoma [START_REF] Pourfarhangi | ECM Cross-Linking Regulates Invadopodia Dynamics[END_REF] . FA are key regulators of migration speed, and our results raise the possibility that signaling cues originating from invadopodia may regulate FA dynamics. Indeed, as FA and invadopodia share many structural components, a dynamic equilibrium of proteins between FA and invadopodia could represent a mean to regulate the cycle between proteolytic ECM remodeling and cell migration thereby enhancing the invasion process. In support to this hypothesis, we observed fast dynamic of Pyk2 and cortactin at invadopodia together with rapid redistribution of these proteins to newly formed nascent adhesions at lamellipodia during invadopodia disassembly. The fact that this redistribution occurred in 62 to 66% of the cells analyzed suggest that theses exchanges of component take place at both immature and mature invadopodia. This dynamic exchange of proteins is in agreement with previous studies showing that Src is located either at invadopodia or at FA depending on the FAK phosphorylation state of the cells, thereby regulating either invadopodia activity or FA dynamics [START_REF] Chan | FAK alters invadopodia and focal adhesion composition and dynamics to regulate breast cancer invasion[END_REF][START_REF] Kolli-Bouhafs | FAK competes for Src to promote migration against invasion in melanoma cells[END_REF] . Although Pyk2 localization was described as mainly cytosolic, previous studies have found that Pyk2 localised to FA in certain cancer cell lines [START_REF] Litvak | Targeting of PYK2 to focal adhesions as a cellular mechanism for convergence between integrins and G protein-coupled receptor signaling cascades[END_REF][START_REF] Selitrennik | PYK2 integrates growth factor and cytokine receptors signaling and potentiates breast cancer invasion via a positive feedback loop[END_REF] . Moreover, recently it was shown that cortactin is a substrate and interactor of Pyk2 41 and our study revealed the presence of cortactin and Pyk2 at both nascent adhesions and invadopodia suggesting that the shuttling of these molecules between the two structures regulate the cycling between migration and ECM degradation.

However, invadopodia are believed to be distinct from FA based on their molecular components, architecture and spatio-temporal regulation. For exemple, invadopodia contain proteins that regulate actin polymerization, such as the Arp2/3 complex and N-WASP contrarily to FA. Nevertheless, in Src-transformed fibroblasts, invadopodia formation is initiated at adhesion sites in response to the focal generation of phosphatidylinositol-3,4-bisphosphate that triggers the recruitment of a protein complex including Grb2, Tks5 and N-WASP [START_REF] Oikawa | PtdIns(3,4)P2 instigates focal adhesions to generate podosomes[END_REF] . Therefore, approaches to identify key molecules that drive the transition between FAs to invadopodia have been used and led to the discovery that the combination of low PKCα activity with increased PI3K promotes enhanced formation and activity of invadopodia along with changes in the organization of focal adhesions [START_REF] Hoshino | Network analysis of the focal adhesion to invadopodia transition identifies a PI3K-PKCα invasive signaling axis[END_REF] . Furthermore, because adhesion-related protein are often located at or near invadopodia it has become clear that FA may contribute at least in some conditions to the formation of invadopodia [START_REF] Hoshino | Signaling inputs to invadopodia and podosomes[END_REF] . Conversely, our study shows that invadopodia also may contribute to the formation of nascent adhesion. This is in agreement with previous observations showing that, in Src-transformed fibroblasts, rapid rosette disassembly was frequently accompanied by localized, dynamic bursts of lamellipodia and nascent focal adhesions formation [START_REF] Gulvady | Hic-5 regulates Src-induced invadopodia rosette formation and organization[END_REF] . Thus, we propose the existence of a dynamic exchange of protein components between these two adhesion structures aimed at controlling the cycles of migration and ECM degradation for efficient invasion.

Our findings indicate that activated Pyk2 may be a marker of melanoma aggressiveness as its expression was only detected in invasive melanoma. This confirmed a recent study showing that increased expressions of P-Pyk2 were positively correlated with advanced stage, metastasis, and Clark grade in melanoma patients [START_REF] Shen | PYK2 mediates the BRAF inhibitor (vermurafenib)-induced invadopodia formation and metastasis in melanomas[END_REF] . Moreover, altogether our results suggest also the interesting possibility of targeting protein components shared by both invadopodia and focal adhesion to reduce invasion processes. To challenge this hypothesis we tested the dual FAK/Pyk2 inhibitor PF-431396 on ECM degradation and migration assays. We found that, contrarily to the selective FAK inhibitor PF-573228, which reduced cell migration but enhanced ECM degradation, PF-431396 significantly inhibited cell migration without altering ECM degradation. We and others have already shown that inhibiting FAK via either pharmacological inhibitors targeting the kinase domain, siRNA strategies or FAK mutation at Y397, increases ECM degradation via inhibition of the Src binding domain at FA [START_REF] Chan | FAK alters invadopodia and focal adhesion composition and dynamics to regulate breast cancer invasion[END_REF][START_REF] Kolli-Bouhafs | FAK competes for Src to promote migration against invasion in melanoma cells[END_REF] thereby enhancing Src localization to invadopodia targets. Of note, among invadopodia targets, Pyk2 was recently shown to interact with Src thus promoting invadopodia activity [START_REF] Genna | Pyk2 and FAK differentially regulate invadopodia formation and function in breast cancer cells[END_REF] . Therefore, our results imply that inhibiting Pyk2 at invadopodia is able to counteract the effect of Src relocalisation at these sites. On the other hand, it is not clear whether FAK and Pyk2 have redundant or specific effects on FA dynamics and cell migration. As the dual FAK/Pyk2 inhibitor did not further inhibit cell migration as compared to the selective FAK inhibitor, this suggests that FAK and Pyk2 have redundant effect on melanoma cell migration. This is in accordance with previous results

showing that knockdown of FAK or Pyk2 in breast cancer cells displayed similar inhibitory effect on 2D migration [START_REF] Genna | Pyk2 and FAK differentially regulate invadopodia formation and function in breast cancer cells[END_REF] . Thus, altogether, these results show that inhibiting a protein implicated in both invadopodia and FA biological processes have the potential to decrease both cell migration and ECM remodeling thereby reducing efficiently cancer cell invasion.

Author Contributions: M. L, and A. M contributed to the study design, performed data acquisition, analysis and interpretation. L.R. and P. C and H.J performed data acquisition and analysis. E.S and D.D assisted with data analysis and interpretation and supervised the research. D. B. and J-P.G helped with data analysis and interpretation. P.R. conceived, designed and

  . Plasmid expresing vinculin-mCherry, Pyk2-GFP, Tks5-GFP and Cortactin-dsRed were kindly provided by I. Lavelin (Weizmann Institute of Science, Rehovot, Israel), J.A. Girault (Institut du fer à Moulin, Paris), S. Courtneige (Oregon Health & Science University, Portland) and M.A.

Figure 1 :

 1 Figure 1: Actin and Cortactin labelled both active and inactive invadopodia. (A) In-situ melanoma WM1832 cells were plated on FITC-Gelatin (Gray), fixed, stained with Hoechst and labelled for actin (Red) and cortactin (Cyan). Boxed regions and insets depict invadopodia. Graphs indicate fluorescent intensity in arbitrary units (A.U.) of F-actin (red), Cortactin (Cyan) with respect to Gelatin (green) over the indicated line scan in the inset. Note the co-localisation of actin and cortactin in dot-like structure and the absence of degradation area. Scale bar: 10 µm (B) X-Y (top, left) confocal images of in-situ melanoma WM1552 showing invadopodia (insets, white arrows) as identified by dot-like F-actin (Red) and cortactin (Cyan) colocalisation. X-Z (bottom, left) images of invadopodia projecting into a collagen/gelatin matrix plated in the top chamber of a 1-μm transwell filter. Histograms represents mean invadopodia length and whiskers with 10-90 percentile with 54 to 117 invadopodia from 27 to 56 cells analysed per condition. (C) Metastatic melanoma A375 cells were plated on FITC-Gelatin (Gray), fixed, stained with Hoechst and labelled for actin (Red) and cortactin (Cyan). Boxed regions and insets depict invadopodia (White) and degradation areas were identified as black holes on fluorescent gelatin. Graphs indicate fluorescent intensity in arbitrary units (A.U.) of Actin (red), Cortactin (Cyan) with respect to Gelatin (green) over the indicated line scan in the inset. Dotted line indicates inactive invadopodia whereas straight lines indicate active invadopodia. Note that colocalisation of actin and cortactin in dot-like structure label both active and inactive invadopodia. Scale bar: 10 µm. The mean number of invadopodia/cell was calculated for 57 WM1862 and 123 A375 cells.

Figure 2 :

 2 Figure 2: P-Pyk2 is expressed at both focal adhesion and invadopodia. (A) Metastatic melanoma A375 melanoma cells were fixed stained with Dapi and labelled for P-Pyk2 (Cyan)

Figure 4 :

 4 Figure 4: Time-lapse imaging of Cortactin-dsRed revealed biphasique Cortactin appearance at active invadopodia. (A) Dual time-lapse images of Cortactin-dsRed and Cy3-Gelatin were taken every 15 min for 24 h; representative images are shown every 30 min. Insets show magnified views of active invadopodia in the cortactin channel (white arrows) and the gelatin channel (black arrows) Scale bar: 10 µm. (B) Graphs indicate fluorescent intensity in arbitrary units (A.U.) of Cortactin-dsRed (Red), and Cy3-gelatin (black) over the indicated yellow line scan in the inset at t=60, 120 and 390 min. Note high fluorescence intensity of Cortactin-dsRed (left) at invadopodia before degradation (t=60), a second increase in fluorescence intensity after degradation (t=120) and the return to background levels after degradation (t=390). Right graph represents the mean intensity of Cortactin-dsRed and Cy3gelatin fluorescence over time of this example. Bottom graph represents the mean F/Fmax intensity ± SEM of Cortactin-dsRed and F/Fmin intensity ± SEM of FITC-gelatin over time from 30 invadopodia. For each invadopodia, T=0 was defined as the maximal fluorescence intensity of cortactin.

Figure 5 :

 5 Figure 5: Loss of Cortactin at invadopodia is time-correlated to the formation of focal adhesion-containing Cortactin during migration. (A) Dual fluorescent time-lapse images of Cortactin-dsRed and Cy3-Gelatin and bright field images were taken every 15 min for 24 h; representative images are shown. Dot-like cortactin fluorescence appeared at t= 15min (white arrow) and disappeared at t=150 min, whereas matrix degradation appeared at t= 60min (black arrow). Note that the disappearance of cortactin at invadopodia is time-correlated to the appearance of cortactin at focal adhesion (arrowhead) during migration clearly visualised by the movement of the nucleus in the bright field images (yellow arrow) Scale bar: 10 µm. (B) Graphs indicate fluorescent intensity in arbitrary units (A.U.) of Cortactin-dsRed (Red), and Cy3-gelatin (black) over the indicated line scan in the inset at t=15, 60 and 150 min. Note high fluorescence intensity of Cortactin-dsRed (left) at invadopodia before degradation (t=15), and the return to background levels after degradation (t=150). Graph represents the mean intensity of Cortactin-dsRed at invadopodia and focal adhesion and Cy3-gelatin fluorescence at invadopodia over time. Note the concomitant increase and decrease in cortactin fluorescence at focal adhesion and invadopodia respectively.

Figure 6 :Supplemental Figure 4 :Video 1 :

 641 Figure 6: Analysis of invadopodia dynamics and protein dynamics at invadopodia. A375 cells expressing either Cortactin-dsRed (A), Pyk2-GFP (B) were imaged by dual Epifluorescence/TIRF mode at 1 image/10 sec during 1 hour. Insets show magnified views of invadopodia (white arrows). One image in epifluorescence (EPI) is shown for each condition to help visualise the cortical location of invadopodia. Note the transformation of dot-like invadopodia into focal complexes (yellow arrowheads) in cells expressing cortactin (A) and Pyk2 (B). Representative time-lapse images are shown from 27 to 37 cells. Graph represent normalized fluorescence intensity over time at invadopodia and nascent adhesions. Scale bar: 10 µm. (C) Analysis of protein dynamics at invadopodia by FRAP. Left, A typical recovery curve of Tks5-GFP at invadopodia is shown. Middle, Box chart represent the mean t1/2 and

Funding: This work was in part supported by the CNRS, the Université de Strasbourg, La Ligue Contre le Cancer and the NANOTUMOR / Plan Cancer with logistical support from Aviesan and ITMO Cancer and BMSV. AM and ML were recipient of a Fellowship from the Ministère de la Recherche.

supervised the research and wrote the manuscript. All authors critically read and approved the final manuscript.

Conflict of interest.

The authors declare that they have no conflict of interest. actin (Red) and Cortactin (Magenda). Boxed regions and insets depict focal adhesion (top, white) and typical dot-like invadopodia (bottom white). Graphs indicate fluorescent intensity in arbitrary units (A.U.) of P-Pyk2 (Cyan), actin (red) and Cortactin (Magenda) over the indicated line scan in the inset at invadopodia (left) and focal adhesion (right). Note high level of P-Pyk2 in only one of the two invadopodia selected. Note also high level of P-Pyk2 at focal adhesion. Scale bar: 10 µm (B) Metastatic melanoma A375 cells were plated on FITC-Gelatin (Gray), fixed, stained with Hoechst and labelled for actin (Red) and PY407-Pyk2 (Cyan). Boxed regions and insets depict invadopodia (White) and degradation areas were identified as black holes on fluorescent gelatin. Graphs indicate fluorescent intensity in arbitrary units (A.U.) of actin (red), P-Pyk2 (Cyan) with respect to Gelatin (green) over the indicated line scan in the inset. Note that all active invadopodia present high level of both actin and P-Pyk2. Scale bar: 10 µm. (C) Metastatic melanoma A375 cells were plated on FITC-Gelatin (Gray), fixed, stained with Hoechst and labelled for Paxillin (Red) and PY407-Pyk2 (Cyan). Boxed regions and insets depict invadopodia and focal adhesion (White). Degradation areas were identified as black holes on fluorescent gelatin. Graphs indicate fluorescent intensity in arbitrary units (A.U.) of paxillin (red), P-Pyk2 (Cyan) with respect to Gelatin (green) over the indicated line scan in the inset. Note that all active invadopodia (straight lines) present high level of P-Pyk2 but background level of paxillin whereas focal adhesion (dotted line) present high level of both paxillin and P-Pyk2 but no degradation. Scale bar: 10 µm. 1: Active invadopodia displayed high level of P-Pyk2 signal. In-situ invasive and metastatic melanoma cells were plated on FITC-Gelatin, fixed, and labelled for actin, cortactin and P-Pyk2. P-Pyk2 was quantified by normalizing P-Pyk2 signal at invadopodia over P-Pyk2 in the whole cell. Inactive invadopodia were defined as dot-like actin/cortactin with no degradation in the gelatin layer whereas active invadopodia were defined as colocalisation of actin/cortactin with degradation areas in the gelatin layer. The mean value ± SEM of P-Pyk2invadopodia/P-Pyk2cell are presented.

Supplemental Figure 1: A375 melanoma cells treated with GM6001 were plated on FITC-Gelatin (Gray), fixed, and labelled for cortactin (Red) and P-Pyk2 (Cyan). Boxed regions and insets depict invadopodia (White). Note the absence of degradation at the gelatin layer and the absence of p-Pyk2 at invadopodia.

Supplemental


Tong Wu

Tomos David

Wouter J T Bos

Point-vortex statistical mechanics applied to turbulence without vortex stretching

Keywords: Vortex-stretching, condensation, statistical mechanics, turbulence

In turbulent systems with inverse cascades, energy will pile up at large scales if no large-scale sink is present. We observe that in forced-dissipative threedimensional turbulence from which vortex-stretching is removed, such condensation is observed, associated with an inverse cascade of helicity. The large-scale structure of this condensate is characterized by a hyperbolic sine relation between vorticity and velocity, analogous to the sinh relation between vorticity and stream function observed in freely decaying 2D turbulence in periodic domains. We generalize a 2D point-vortex statistical mechanics approach to our 3D system. It is shown that the predictions of this approach are in agreement with observations of both the forceddissipative system, after appropriate averaging, and of the freely decaying system.

I. INTRODUCTION

In two-dimensional (2D) turbulence at high Reynolds numbers, nonlinear interactions yield a transfer of energy towards large scales [START_REF] Kraichnan | Inertial ranges in two-dimensional turbulence[END_REF] . If energy is continuously injected, and in the absence of a large-scale dissipation mechanism, energy eventually accumulates in low wave numbers, leading to a condensate 1 . This condensation phenomenon has been observed numerically and, in 2D periodic square domains, the physical manifestation of this condensation is the generation of a large-scale counter-rotating vortex pair 2,3 . Such large-scale coherent structures in turbulent flows are at the heart of a number of geophysical processes [START_REF] Van Kan | Condensates in thin-layer turbulence[END_REF][START_REF] Xia | Upscale energy transfer in thick turbulent fluid layers[END_REF][START_REF] Bouchet | Random changes of flow topology in two-dimensional and geophysical turbulence[END_REF] .

Large-scale structuring in 2D and quasi-2D systems has been extensively studied using statistical mechanics [START_REF] Bouchet | Statistical mechanics of two-dimensional and geophysical flows[END_REF][START_REF] Esler | Universal statistics of point vortex turbulence[END_REF][START_REF] Gauthier | Giant vortex clusters in a two-dimensional quantum fluid[END_REF] . Two different statistical mechanics approaches are the Lee-Kraichnan equilibrium statistical mechanics [START_REF] Kraichnan | Inertial ranges in two-dimensional turbulence[END_REF][START_REF] Lee | On some statistical properties of hydrodynamical and magnetohydrodynamical fields[END_REF] and the Robert-Sommeria-Miller (RSM) approach [START_REF] Robert | Relaxation towards a statistical equilibrium state in twodimensional perfect fluid dynamics[END_REF][START_REF] Miller | Statistical mechanics of Euler equations in two dimensions[END_REF][START_REF] Eyink | Onsager and the theory of hydrodynamic turbulence[END_REF] , respectively. Kraichnan predicted the shape of energy spectra associated with the absolute equilibrium state of a truncated Fourier system taking into account the two inviscid invariants surviving Galerkin truncation: energy and enstrophy [START_REF] Kraichnan | Inertial ranges in two-dimensional turbulence[END_REF] . However, a weakness of Kraichnan's theory is that it cannot predict structuring and does not take into account the higher-order moments of the vorticity. For instance, an observation that is not captured by such an approach is the following: In freely evolving 2D turbulence in square periodic domains, at long times, vorticity ω and stream function Ψ are locally related by a hyperbolic sine function, corresponding to a steady solution of the 2D Euler-equations. More precisely, observations in numerical simulations are consistent with a functional relation

ω = -c -1 sinh(bΨ), (1)
where c and b are constants (c > 0, b < 0) [START_REF] Montgomery | Relaxation in two dimensions and the "sinh-Poisson[END_REF] . This relation was predicted by applying statistical mechanics to a point-vortex system. Indeed, building upon the ideas of Onsager [START_REF] Onsager | Statistical hydrodynamics[END_REF] , Joyce and Montgomery obtained the above quantitative prediction [START_REF] Joyce | Negative temperature states for the two-dimensional guiding center plasma[END_REF] .

Navier-Stokes turbulence is a dissipative system, whereas the point-vortex dynamics constitute an ideal finite-dimensional Hamiltonian system. The agreement of theory with observations of freely evolving Navier-Stokes turbulence is, therefore, not a trivial fact.

However, point-vortex descriptions have allowed important progress in the understanding of two-dimensional turbulence [START_REF] Esler | Universal statistics of point vortex turbulence[END_REF][START_REF] Pointin | Statistical mechanics of two-dimensional vortices in a bounded container[END_REF][START_REF] Taylor | Interacting vortices and spin-up in twodimensional turbulence[END_REF] .

It is even less straightforward that these approaches should work when a turbulent system is continuously stirred by a body force. In such forced-dissipative systems, inviscid invariants are continuously modified so that the application of equilibrium statistical mechanics is not well founded in principle. Nevertheless, on average such approaches might work. Indeed, recently it was observed that maximum entropy theory could be applied to forced-dissipative geostrophic turbulence [START_REF] David | Eddy-mixing entropy and its maximization in forced-dissipative geostrophic turbulence[END_REF][START_REF] Singh | The climate system and the second law of thermodynamics[END_REF] . Another indication that statistical mechanics might be applied to averaged turbulent systems is the success of the application of statistical mechanics of axisymmetric turbulence [START_REF] Leprovost | Dynamics and thermodynamics of axisymmetric flows: Theory[END_REF][START_REF] Thalabard | Statistical mechanics of the 3d axisymmetric Euler equations in a Taylor-Couette geometry[END_REF][START_REF] Qu | Direct numerical simulation of axisymmetric turbulence[END_REF] to observations of experiments of turbulent flows, which are only axisymmetric on average [START_REF] Monchaux | Properties of steady states in turbulent axisymmetric flows[END_REF] .

In the present work, we continue this line of research and show how point-vortex statistical mechanics can be applied to forced-dissipative and freely decaying three-dimensional (3D) turbulence scenarios, however, without vortex stretching. We have recently established a framework for incompressible turbulence without vortex stretching [START_REF] Bos | Three-dimensional turbulence without vortex stretching[END_REF][START_REF] Wu | Statistical mechanics of the Euler-equations without vortex stretching[END_REF] . In cases where no forcing or dissipation is present, the governing dynamics can be described by:

∂ω ∂t + u • ∇ω = -∇P ω , (2)
where u represents the velocity with ∇ • u = 0, ω = ∇ × u denotes the vorticity, and P ω is a pressure term ensuring the vorticity to remain solenoidal. Indeed, Eq. ( 2) is obtained by removing the vortex-stretching term, ω • ∇u, from the curl of the 3D Euler equation. In previous work [START_REF] Wu | Cascades of enstrophy and helicity in turbulence without vortex stretching[END_REF] , we illustrated that the forced turbulent system without vortex stretching exhibits an inverse helicity cascade (and also a forward enstrophy cascade) bearing similarities to the behavior observed in 2D turbulence. However, different from 2D turbulence, the no-vortex-stretching system does not conserve kinetic energy. This inverse transfer led us to consider the following questions: In the absence of dissipation at large scales, will this system manifest a condensation structure? Can we predict the properties of such a structure using point-vortex theory?

In this paper, we show that, in forced-dissipative turbulence without vortex stretching, condensation indeed occurs at the scale of the box. We will also investigate the system when it is freely decaying from this condensation state. Moreover, we attempt to predict the characteristics of large-scale structures in both the forced-dissipative and freely decaying cases using the principle of point-vortex statistical mechanics.

The rest of this paper is organized as follows. In Sec. II, we predict properties of large-scale structures in 3D turbulence without vortex stretching by using a generalized point-vortex model. In Sec. III, we present the numerical setup. Then, in Sec. IV we report on the assessment of the theoretical results. Finally, Sec. V presents the conclusions.

II. ANALYTICAL CONSIDERATIONS

At large but finite Reynolds numbers, a 2D Navier-Stokes fluid will relax to a quasistatic maximum entropy state where most of the energy is concentrated at large scales.

When this state is attained, a sinh relation between vorticity and stream function emerges (Eq. ( 1)). This was observed numerically [START_REF] Montgomery | Relaxation in two dimensions and the "sinh-Poisson[END_REF][START_REF] Montgomery | Navier-Stokes relaxation to Sinh-Poisson states at finite Reynolds numbers[END_REF] and explained analytically [START_REF] Onsager | Statistical hydrodynamics[END_REF] . Then Joyce and Montgomery used this idea to study the 2D electrostatic guiding center plasma [START_REF] Joyce | Negative temperature states for the two-dimensional guiding center plasma[END_REF][START_REF] Montgomery | Statistical mechanics of "negative temperature" states[END_REF] . The system of interacting point-vortices is equivalent to the system which describes the interaction of long, uniformly charged rods aligned with a uniform magnetic field. Joyce and Montgomery considered N rods of charge +e and N more of charge -e and predicted the emergence of the hyperbolic sine relation.

We investigate whether there is a similar functional relation in our turbulent system without vortex stretching. We generalize the point-vortex model to a 3D system. Thereto we propose that a 3D vorticity field can be considered as a combination of 6N point-vortices, the axes of which are each parallel to one of the three coordinate axes. We suppose that in each direction there are N point-vortices of positive vorticity +1, and N ones of negative vorticity -1. Without loss of generality, we use non-dimensional vorticities +1 and -1 here.

Similar to Ref. [START_REF] Joyce | Negative temperature states for the two-dimensional guiding center plasma[END_REF] , we imagine the total volume V = L 3 subdivided into small cells of volume ∆ ≪ V . The cells are large enough, however, to contain many particles. We call

N + i,x , N + i,y , N + i,z and N - i,x , N - i,y , N - i,z
the number of positive and negative point-vortices in each direction inside cell i. We can consider that one small cell constitutes a basic unit, which means the resultant quantities (such as velocity, vorticity, etc.) of all point vortices in a cell can be considered as the quantity at the center point of the cell. Furthermore, it might seem artificial that all point vortices are aligned with the coordinate axes. However, since all cells contain a large number of vortices, we can without loss of generality assume that three vortices, in the three directions, represent the three components of one point vortex. This would reduce the total number of vortices in the description by a factor of three, which will not make any difference in the asymptotic limit, where all N ± i,a (a indicates x, y, or z) tend to infinity. Then the vorticity field can be written as

         ω x (r) = i (N + i,x -N - i,x )δ (r -r i ) , ω y (r) = i (N + i,y -N - i,y )δ (r -r i ) , ω z (r) = i (N + i,z -N - i,z )δ (r -r i ) , (3)
where r i indicates the position of cell i. The associated velocity field can be obtained using Biot-Savart's law

u(x) = 1 4π R 3 ω(y) × (x -y) |x -y| 3 dy. (4)
Substituting Eq. (3) into Eq. ( 4), yields expressions for the velocity components in each direction

         u x (r) = 1 4π i (N + i,y -N - i,y )(rz-r i,z )-(N + i,z -N - i,z )(ry-r i,y ) |r-r i | 3 , u y (r) = 1 4π i (N + i,z -N - i,z )(rx-r i,x )-(N + i,x -N - i,x )(rz-r i,z ) |r-r i | 3 , u z (r) = 1 4π i (N + i,x -N - i,x )(ry-r i,y )-(N + i,y -N - i,y )(rx-r i,x ) |r-r i | 3 (5)
with r i,x , r i,y , r i,z the x-, y-, and z-component of r i and r x , r y , r z the x-, y-, and z-component of r.

This point-vortex model allows us to define an entropy using Boltzmann's formula. The entropy of this 3D point-vortex system can be written as

S = lnW (6)
with

W = (6N )! i N + i,x !N - i,x !N + i,y !N - i,y !N + i,z !N - i,z ! . ( 7
)
The quantity W indicates the number of microstates associated with the macrostate where there are

N + i,x , N + i,y , N + i,z positive and N - i,x , N - i,y , N - i,z negative point-vortices inside cell i. We assume N + i,x , N + i,y , N + i,z , N - i,x , N - i,y , N - i,z
to be large enough for Stirling's formula to be valid. We then obtain (8) Similar to a 2D system, we assume the final state of 3D turbulence without vortex stretching to be close to the maximum entropy state. We will therefore attempt to determine this state by maximizing the entropy under the constraints imposed by the dynamical equation. Indeed, our system (2) conserves enstrophy and helicity [START_REF] Wu | Statistical mechanics of the Euler-equations without vortex stretching[END_REF] . To proceed, we need therefore to consider formulas of these two invariants expressed as a function of the variables governing the 3D point-vortex system. The total helicity of the system is given by

H = 1 2 u • ωdr = 1 8π i,j 1 |r j -r i | 3 ((N + j,x -N - j,x ) (N + i,y -N - i,y )(r j,z -r i,z ) -(N + i,z -N - i,z )(r j,y -r i,y ) + (N + j,y -N - j,y ) (N + i,z -N - i,z )(r j,x -r i,x ) -(N + i,x -N - i,x )(r j,z -r i,z ) + (N + j,z -N - j,z ) (N + i,x -N - i,x )(r j,y -r i,y ) -(N + i,y -N - i,y )(r j,x -r i,x ) ). ( 9
)
And enstrophy of this 3D point-vortex system can be written as

W = i (1) 2 N + i,x + i (-1) 2 N - i,x + i (1) 2 N + i,y + i (-1) 2 N - i,y + i (1) 2 N + i,z + i (-1) 2 N - i,z . (10)
The maximum entropy state is now obtained by solving the variational problem, δS -γδH -αδW = 0 (11) with γ, α Lagrange multipliers. Similar to what was assumed for 2D systems, we suppose the numbers of positive or negative point-vortices to be independent of each other. Thus, the partial derivative of helicity with respect to

N + j,x is ∂H ∂N + j,x = 1 8π i (N + i,y -N - i,y )(r j,z -r i,z ) -(N + i,z -N - i,z )(r j,y -r i,y ) |r j -r i | 3 = u x (r j ) 2 . ( 12
)
Other derivatives have similar forms. Then, substituting Eq. ( 8), ( 9) and (10) into Eq. ( 11)

and deriving the integrand with respect to N + i,x and N - i,x yields the expressions

lnN + i,x + α + γ 2 u x (r i ) = 0, lnN - i,x + α -γ 2 u x (r i ) = 0. ( 13
)
Results in the other two directions are similar. Hence, the number of point-vortices in cell i with vorticity vectors along the x-direction are

N + i,x = exp(-(α + γ 2 u x (r i ))), N - i,x = exp(-(α - γ 2 u x (r i ))), (14)
at the maximum entropy state. Using Eq. (3), we find

ω x = exp(-α)(exp(- γ 2 u x ) -exp( γ 2 u x )) =2 exp(-α) sinh(- γ 2 u x ). ( 15
)
After applying similar derivations in the other two directions, the vorticity components in the yand z-directions are

ω y = 2 exp(-α) sinh(-γ 2 u y ), ω z = 2 exp(-α) sinh(-γ 2 u z ). ( 16
)
This sinh relation between vorticity and velocity, Eqs. ( 15) and ( 16) are the principal theoretical results of the present investigation.

III. NUMERICAL METHOD AND SETUP

Direct numerical simulations (DNSs) are performed using a standard pseudo-spectral solver with a third-order Adams-Bashforth time integration scheme 31 , which was modified to remove vortex stretching [START_REF] Wu | Statistical mechanics of the Euler-equations without vortex stretching[END_REF] . Our computational domain is a cubic periodic box of size L = 2π. Aliasing errors are removed using the 2/3 rule. DNSs are executed on grids of size 256 3 .

The Biot-Savart operator is applied in Fourier space to the vorticity equation in order to obtain an equation for the velocity field. Then the equation for Fourier coefficients of velocity û, which is solved in our code, is

∂ û ∂t + i k 2 k × F[(u • ∇)ω] = f -νk 2η û ( 17
)
with k the wave vector, k = |k| the wave number, F[•] the Fourier transform, f a forcing term, ν the (hyper)viscosity, and η the hyperviscosity parameter. The physical space equivalent of (17) (Eq. ( 2)) contains a pressure term which allows to impose incompressibility.

Equivalently, in Fourier-space, the incompressibility (∇

• u = 0) is contained in i k 2 k × F[•]
which projects the nonlinearity of the evolution-equation onto a plane perpendicular to the wave-vector.

We consider two cases. The first one is a forced-dissipative system. For the second case, we eliminate the force and allow the system to decay freely. Since we are concentrating on the condensation state without focusing too much on the dissipation range, we reduce the size of the latter by using hyperviscosity and we use therefore the value η = 4 (normal viscous dissipation corresponds to η = 1). The parameter ν represents the hyperviscosity rate which is set to 10 -13 in our simulations. We choose an injection mechanism which keeps the energy constant in a narrow wavenumber range 32,33 29.5 < k < 31.5 around the forcing wavenumber k f = 30.5 at a level E (k) = 10 -3 . A convenient property of this type of forcing, with respect to the present investigation, is that it does not only inject energy in the system, but also helicity [START_REF] Wu | Cascades of enstrophy and helicity in turbulence without vortex stretching[END_REF] .

The initial energy spectrum is chosen as E(k) = 5 × 10 -7 . The initial value of the energy is small enough to allow a clear observation of the energy condensation process. Complex phases of û are set randomly at the initial moment.

IV. NUMERICAL RESULTS

We first show the energy condensation process in the forced system in Sec. IV A. Subsequently, in Sec. IV B, we eliminate the force and allow the system to decay freely from the condensation state. In both cases, we verify our predictions from the point-vortex model by illustrating the emergence of a hyperbolic sine relation between velocity and vorticity.

A. Self-organization in the forced system

We start by forcing the system from a random initial condition, dissipating mainly the small scales. In Fig. 1, we show the time evolution of the helicity flux normalized by its dissipation rate ϵ H , defined as ϵ H = 2νk 2η H(k)dk, during the condensate phase from t = 5 to t = 160. In turbulence governed by the reduced Navier-Stokes equation Eq. ( 17), the flux of helicity is defined as Π since helicity is conserved by the nonlinearity and there is only very weak damping present at scales k < k f , the physical process leading to the build-up at the large scales of the system is associated with the helicity which is transferred to large scales. After arriving at the largest scales of the system, no dissipation mechanism is able to absorb the helicity injected by the forcing. The helicity and its associated energy piles then up, leading to large spectral and k -4/3 , as expected for the present system [START_REF] Wu | Cascades of enstrophy and helicity in turbulence without vortex stretching[END_REF] . In unmodified 3D turbulence, the helicity is transferred in the opposite direction (to small scales). However, dimensional arguments lead in that case also to the possibility of a -7/3 scaling 34 , but this is only observed if the forcing scheme is tuned such that the flow becomes extremely helical [START_REF] Kessar | Non-Kolmogorov cascade of helicitydriven turbulence[END_REF] . Due to the limitation of computing resources, the inertial range is not sufficiently large to clearly display these theoretical scalings. In Fig. 3, we show the time evolution of kinetic energy, helicity, and enstrophy in the three directions, respectively. After t ≈ 60, kinetic energy, helicity, and enstrophy in the y direction become larger than in the other two directions. At this time the condensation process starts. The associated condensate-like structure is visualized in Fig. 4. This structure is constituted of highly anisotropic columnar structures in the y direction. A parallel can here be drawn with 2D turbulence where the condensate-like structure is a pair of counter-rotating vortices.

H (k) = -Σ k Re N (k) • û * (k) dk with N (k) = F[-(u•∇)ω]
After reaching this condensate state, we assess the relation between velocities and vor- ticities as predicted by the point-vortex model in Sec. II. Fig. 5 shows the scatter plot of u y against ω y for the condensation state at t = 160. A tendency of a sinh-relation can be observed, but a considerable noise hinders us from identifying a clear functional relationship. In order to reduce the noise, instead of investigating the relation between velocity and vorticity components directly, we focus on their averaged values

⟨u y ⟩ y (x, z) = 1 M y y u y (x, y, z), ⟨ω y ⟩ y (x, z) = 1 M y y ω y (x, y, z), (18)
where u y and ω y indicate velocity and vorticity in the y-direction, ⟨•⟩ y indicates a space average in the y-direction and M y is the number of grid points in the y-direction (M y = 256 in our simulations). Similar definitions can be defined in the other directions. As sinh is not a linear function, theoretically, we can not derive ⟨ω y ⟩ y = 2 exp(-α) sinh(-γ 2 ⟨u y ⟩ y ) from Eq. ( 15) and ( 16). But if u y is nearly constant along the y-direction, we can get an approximation sinh(-γ 2 u y ) y ≈ sinh(-γ 2 ⟨u y ⟩ y ). An instantaneous visualization of three velocity components for the condensation state at t = 160 is shown in Fig. 6. We observe that u x , u y and u z depend only weakly on the x, y and z directions respectively. Thus, the averaged values are appropriate for investigating the hyperbolic sine relation. Furthermore, the structure of u y is similar to a dipole, while structures of u x and u z are closer to a unidirectional flow. Note that dipoles and unidirectional flows are two condensation states of 2D turbulence caused by bifurcations of the stochastic Navier-Stokes equations [START_REF] Bouchet | Random changes of flow topology in two-dimensional and geophysical turbulence[END_REF] .

The values of the two Lagrange multipliers α and γ can be determined a priori. Firstly, we introduce the averaged quantities associated with helicity and enstrophy, as

H ave = 1 2 y,z ⟨u x ⟩ x (y, z) ⟨w x ⟩ x (y, z) + 1 2
x,z ⟨u y ⟩ y (x, z) ⟨w y ⟩ y (x, z)+ 1 2

x,y ⟨u z ⟩ z (x, y) ⟨w z ⟩ z (x, y),

W ave = 1 2 y,z ⟨w x ⟩ 2 x (y, z) + 1 2 x,z ⟨w y ⟩ 2 y (x, z) + 1 2 x,y ⟨w z ⟩ 2 z (x, y) (19)
respectively. We substitute the relation ⟨ω m ⟩ m = 2 exp(-α) sinh(-( γ 2 ⟨u m ⟩ m )) into Eq. ( 19) where m indicates x, y and z, so that H ave and W ave can be written as functions of averaged velocities ⟨u m ⟩ m . At each moment, exact values of ⟨u m ⟩ m , H ave , and W ave can be obtained from the simulation data. Then the two unknowns α and γ can be calculated from these two equations of H ave and W ave . At t = 160, the analytically predicted curve is ⟨ω m ⟩ m = 1.5207 sinh(0.3779 ⟨u m ⟩ m ) in our simulation.

In Fig. 7, we show scatter plots of averaged components of velocities and vorticities in three directions for the condensation state at t = 160. A hyperbolic sine relation is observed between y-components ⟨u y ⟩ y and ⟨ω y ⟩ y . The yellow lines in Fig. 7 indicate the analytically predicted curve, which overlaps the data points in the direction where most energy is contained. This collapse verifies our analytical prediction of the sinh relation in the y-direction. However, for other directions, we observe a linear instead of a sinh relation between velocities and vorticities, as shown in Fig. 7(a) and (c). The Taylor expansion of sinh(cx) for small x yields cx with c a constant number, which indicates that the form of sinh(cx) resembles a straight line when x varies in a small range around 0. We insist that

V. CONCLUSIONS

In this paper, we have illustrated a large-scale condensation of energy in 3D turbulence without vortex stretching. As we presented in previous work, helicity cascades from small to large scales [START_REF] Wu | Cascades of enstrophy and helicity in turbulence without vortex stretching[END_REF] . During the cascade, energy is carried by helicity towards the smallest wavenumbers. In the absence of friction at large scales, energy accumulates and forms a large-scale structure. And this large-scale structure persists when the force is removed.

This condensate of 3D turbulence without vortex stretching behaves like that of 2D turbulence. We showed analytically a generalized 3D point-vortex model to predict a hy- perbolic sine relation between vorticity and velocity in the direction which contains most energy. In a forced system, after the condensate occurs, the predicted hyperbolic sine relation is observed between averaged vorticity and velocity, which suggests that the statistical mechanics approach can be applied to averaged systems, as also observed in experiments [START_REF] Monchaux | Properties of steady states in turbulent axisymmetric flows[END_REF] .

At the freely decaying phase, the relation between vorticity and velocity very clearly relaxes to the predicted hyperbolic sine function, similar to the sinh relation between vorticity and stream function in freely decaying 2D turbulence.

ACKNOWLEDGEMENT

We acknowledge the China Scholarship Council (CSC) for the financial support. All simulations were carried out using the facilities of the PMCS2I ( École Centrale de Lyon).

For the purpose of Open Access, a CC-BY public copyright licence has been applied by the authors to the present document and will be applied to all subsequent versions up to the Author Accepted Manuscript arising from this submission.

Fig. 1 .

 1 Fig. 1. Flux of helicity normalized by the dissipation rate of helicity during the force-dissipative phase at t = 5, 10, 20, 30, 40, 50, 60, 70, 80, 100, 130, 160. The coloring of the curves evolves in time from dark to light.

Fig. 2 .

 2 Fig. 2. Time evolution of wavenumber spectra of the kinetic energy (a) and helicity (b) during the force-dissipative phase. The time-instants are the same as indicated in Fig. 1.

Fig. 3 .

 3 Fig. 3. Time evolution of integral quantities in three directions during the forced phase. ⟨•⟩ indicates the volume average. (a) Energy. (b) Helicity. (c) Enstrophy.

Fig. 4 .

 4 Fig. 4. Visualizations of condensate-like structure at t = 160. (a) Energy. (b) Helicity.

Fig. 5 .

 5 Fig. 5. Scatter plot of y-components velocities and vorticities at the condensation state at t = 160.

Fig. 6 .

 6 Fig. 6. Velocity components in three directions at the condensation state at t = 160. (a) u x . (b) u y . (c) u z .

Fig. 7 .

 7 Fig. 7. Scatter plots of averaged components of velocities and vorticities in three directions at the condensation state at t = 160. The yellow lines present the analytically predicted curve ⟨ω m ⟩ m = 1.5207 sinh(0.3779 ⟨u m ⟩ m ). (a) X-components. (b) Y -components. (c) Z-components.

Fig. 8 .

 8 Fig. 8. Time evolution of integral quantities contained in three directions during the freely decaying phase. (a) Energy. (b) Helicity. (c) Enstrophy.

Fig. 9 .

 9 Fig. 9. Time evolution of wavenumber spectra of kinetic energy (a) and helicity (b) during the freely decaying phase at t = 0, 1, 2, 4, 8, 16. The insets in (a) and (b) provide a detailed view around the fist modes.

Fig. 10 .

 10 Fig. 10. Visualization of energy during the freely decaying phase. (a) t = 1. (b) t = 2. (c) t = 4. (d) t = 8 (e) t = 16.

Fig. 11 .

 11 Fig. 11. Scatter plots of the components of velocities and vorticities during the freely decaying phase. (a) u y versus ω y at time instants t = 0, 1, 2, 4, 8, 16 from dark to light. (b) x-and zcomponents of velocities and vorticities at t = 16. Black lines in (a) and (b) are the analytically predicted curves at t = 16, given by ω y = 1.511 sinh(0.3718u y ).

Fig. 12 .

 12 Fig. 12. Simulation results of the case where energy in two of the three directions are larger than that in the third direction. (a) Mean square values of velocity components in three directions. (b-d) Scatter plots of averaged components of velocities and vorticities in three directions at t = 126. (b) Relation between ⟨u x ⟩ x and ⟨ω x ⟩ x . (c) Relation between ⟨u y ⟩ y and ⟨ω y ⟩ y . (d) Relation between ⟨u z ⟩ z and ⟨ω z ⟩ z . Yellow lines in (b), (c) and (d) are the analytically predicted curves ⟨ω m ⟩ m = 1.5391 sinh(0.3479 ⟨u m ⟩ m ).

3

  M. Chertkov, C. Connaughton, I. Kolokolov, and V. Lebedev. Dynamics of energy condensation in two-dimensional turbulence. Phys. Rev. Lett., 99:084501, 2007.

S ≈ 6N ln(6N ) -6N + i (-N + i,x lnN + i,x + N + i,x -N - i,x lnN - i,x + N - i,x -N + i,y lnN + i,y + N + i,y -N - i,y lnN - i,y + N - i,y -N + i,z lnN + i,z + N + i,z -N - i,z lnN - i,z + N - i,z )

these functional relations are not direct fits of the sinh relation to the data but obtained evaluating the averaged quantities ⟨u m ⟩ m , H ave and W ave only.

Moreover, we note that the y direction is not necessarily the direction which contains most energy. Further details can be found in Appendix A, where we present results from additional simulations.

B. Freely decaying state

At the condensation state at t = 160, the external force is removed from the system, and we indicate in this section the moment when the removal of force commences as time t = 0.

In Fig. 8, we show the time evolution of kinetic energy, helicity, and enstrophy in the three directions, respectively. During the free-decay phase, most of kinetic energy, helicity, and enstrophy remain concentrated in the y direction. As shown in Fig. 8(a), energy displays an increasing trend during the dissipation phase, especially in the y direction, which is not violating any conservation laws because energy is not conserved in turbulence without vortex stretching [START_REF] Wu | Statistical mechanics of the Euler-equations without vortex stretching[END_REF] . From Fig. 8(b), we can see that the value of helicity exhibits very little change over time since most of the helicity cascades towards small wavenumbers and is only weakly dissipated by the hyperviscosity. Fig. 8(c) shows the decrease of enstrophy.

In Fig. 9(a) and (b), we show the time evolution of energy and helicity spectra, respectively. We observe that the energy and helicity at small scales decay during the dissipation.

The insets in Fig. 9 provide a detailed view around the first modes, revealing that energy and helicity at large scales increase for a duration t ≤ 16 after the external force is removed.

The large-scale structures persist after eliminating the force, as shown in Fig. 10.

In Fig. 11(a), we show scatter plots of y-components of velocities and vorticities, along with the analytically predicted black curves. Here, no-averaged values u p , H, and W are used to calculate Lagrange multipliers α and γ. During the dissipation phase, noise at small scales decreases, and we observe a clear hyperbolic sine relation between u y and ω y . Thus, we can infer that the noise originates from the small-scale motions, and the sinh relation is characteristic of the large-scale coherent structures. Our analytical prediction of the sinh relation in the y direction is confirmed by the overlapping of the predicted curve and data points. For other directions, similar to the forced phase, we observe a linear instead of a sinh relation between velocities and vorticities, as shown in Fig. 11(b).

Appendix A: Supplemental simulations

To assess the robustness of the results presented above, we also executed eight supplemental DNS-runs using the same initial energy spectrum but different initial complex phases of û. We find that the y direction is not necessarily the direction which contains most energy. Furthermore, in one of the eight simulations, energy in two of the three directions is comparable and larger than that in the third direction. In this case the sinh relation is found in the two directions containing larger energy, presented in Fig. 12(b-d).

In all our nine simulations we have thus observed the same type of symmetry breaking where two components of the kinetic energy take the same value, the other being either larger or smaller. No observations are reported where all three components tend to the same value.


Cet ouvrage est la traduction anglaise d'un livre précédemment publié en 2014 à la Oxford University Press de Hong Kong sous le titre He wei Zhongguo : Jiangyu, minzu, wenhua yu lishi 何為中國 : 疆域, 民族, 文化與歷史, lui-même précédé par une version en japonais parue peu avant sous le titre Chūgoku saikō: Sono ryōiki, minzoku, bunka 中国再考:その領 域、民族、文化 (Tōkyō, Iwanami Shoten, 2014).

Il a été traduit et enrichi d'un avant-propos par Michael Gibbs Hill, sinologue du William & Mary College (Virginie), spécialiste de l'histoire intellectuelle des XIX e et XX e siècles chinois. L'ouvrage n'est pas un essai d'un seul tenant, mais une mosaïque d'écrits pour la plupart déjà publiés sous différentes formes (articles, cours ou conférences), notamment dans le magnum opus récent de l'auteur, Here in "China" I dwell (Zhai zi Zhongguo 宅茲中國, Leiden, Brill, 2017). Plusieurs de ces textes ont été rédigés à l'occasion de prises de parole à l'étranger, au Japon, en Corée ou en Thaïlande.

L'objectif de Ge Zhaoguang est défini clairement dans la postface : « What I hope to share with readers in this book is how a Chinese scholar understands "China", "Chinese history", and "Chinese culture". » (p. 149) Ce livre est donc une sorte d'essai-témoignage et c'est ainsi qu'il faut le considérer. En effet, qui s'attendrait, comme pourrait le suggérer le titre, à une sorte de réflexion déconstructiviste sur les fondements historiques de l'entité territoriale chinoise actuelle serait fort déçu, car l'auteur cherche davantage à mettre en avant l'histoire de la construction de la Chine en tant que nation, qu'il considère comme très singulière. Comme il l'explique dans l'introduction, il s'emploie chemin faisant à répondre aux théories déconstructivistes ou post-modernes de la recherche actuelle (notamment américaine) et à démontrer que l'évolution européenne de la formation des États-nation n'est pas applicable à la Chine.

Pour tout dire, la lecture de l'introduction fait craindre le pire pour la suite, tant elle comporte d'affirmations tranchées et discutables. Ge insiste tout particulièrement sur la singularité de la Chine d'une façon qui en devient presque choquante : "As a state, the true nature of China can be understood neither through a simple application of the European concept of empire nor through the use of definitions or theories of the early modern European nation-state. Questions concerning China's territories, nations and peoples, faiths, territorial boundaries, and identities are far more complicated than for any other country in the world." (p. 3) Ailleurs, le propos est si schématique qu'il en devient faux sur le plan historique : « All the way down to the Sui and Tang dynasties, China still maintained an empire that reached across much of East Asia and exercised a substantial degree of control over the various people within its territories. » (p. 4) C'est oublier ici la période de fragmentation territoriale des III e -VI e siècles où précisément aucun empire unique ne gouvernait l'écoumène chinois, à l'exception de la courte unification des Jin entre 280 et 317. Par ailleurs, dire que l'empire chinois atteignait la plus grande partie de l'Asie orientale est très exagéré… Ailleurs encore, Ge évoque le Japon : "since at least the times of the Sui and Tang dynasties, Japan has had a consciousness of itself as occupying a position of parity with China, but it would probably not be until the Yuan dynasty […] that a true orientation toward one's own state began to develop in political, economic, and cultural terms." (p. 6) Il me semble que les souverains japonais n'ont pas attendu l'époque mongole pour se sentir les égaux des monarques chinois, pour preuve le célèbre passage du Suishu 隋書, cité pourtant en note par Ge, où l'empereur du Japon se présente lui-même comme un Fils du Ciel (tianzi 天子) (note 8, p. 155).

Heureusement, la suite de l'ouvrage ne contient pas de raccourcis aussi trompeurs ou d'affirmations aussi discutables. Les cinq chapitres qui suivent sont thématiques et proposent chacun une perspective historique différente sur le sujet : la vision du monde, les frontières et le territoire, la question des peuples, le multiculturalisme, les rapports avec la Corée et le Japon. Il se termine par un chapitre sur la place de la Chine dans le monde actuel. S'agissant au départ de textes indépendants, l'ouvrage foisonne d'exemples et de données ; cela dit, l'auteur revient à plusieurs reprises sur quelques thèmes qui semblent lui être chers. L'une des idées fortes de Ge Zhaoguang est le passage de l'idéologie du « Monde sous le Ciel » (tianxia 天下) à celui de la « Myriade d'États » (wanguo 萬國), pour reprendre les expressions qu'il emploie, c'est-à-dire le moment où l'empire chinois prend conscience qu'il existe de par le monde des États qui lui sont égaux sur le plan politique. Pour Ge ce passage est progressif, mais l'un des moments clés est celui des Song, en particulier lorsque ceux-ci sont contraints d'établir une sorte de relation diplomatique égalitaire avec les Khitans, avec/par la reconnaissance d'une frontière entre les deux États (un processus sur lequel Ge insiste beaucoup). Même si cette idée est convaincante, Ge Zhaoguang n'insiste sans doute pas assez sur le fait que les connaissances sur le monde extérieur n'ont pas conduit à un changement paradigmatique dans les relations étrangères des empires Ming ou Qing. Ce fut la brutalité des puissances occidentales qui causa ce changement, non le savoir.

Un autre thème clé du livre est la question du multiculturalisme de la Chine. On le sait, la République populaire actuelle se définit comme un État pluriethnique (duo minzu guojia 多民 族國家). Ge s'emploie à montrer que ce caractère est ancien et que l'ethnie Han elle-même est issue d'un mélange de peuples et de cultures. Il rappelle de manière judicieuse qu'à la chute de l'empire des Qingempire multiethnique s'il en estla question se posa de savoir si la République devait se fonder uniquement sur la population Han (et donc se limiter à la Chine des dix-huit provinces héritées des Ming) ou si elle devait inclure aussi les autres populations et territoires contrôlés par les Qing (notamment la Mongolie, la Mandchourie, le Xinjiang et le Tibet). La seconde solution fut finalement choisie sous le principe « la République des cinq ethnies » (wuzu gonghe 五族共和) ; les cinq ethnies en question étant les Han, les Mandchous, les Mongols, les Hui et les Tibétains, ouvrant ainsi la voie aux distinctions ethniques faites sur le modèle soviétique qui caractérisent la République populaire fondée en 1949. Ge Zhaoguang revient, dans le chapitre consacré à l'« ethnicité », sur la création de l'Academia Sinica en 1928 (p. 77), fondée, selon lui, non seulement afin de mieux comprendre les territoires et les peuples de la jeune République mais aussi de répondre à l'orientalisme japonais et européens. Les savants chinois de l'époque s'employèrent à montrer que la culture et le peuple chinois étaient indigènes, manière de répliquer aux théories, en vogue dans certains milieux savants européens, qui voulaient que la culture chinoise vînt de l'Ouest, les vrais indigènes étant les Miao. Ge met ainsi en lumière le fait que de nombreuses études furent menées à cette époque pour montrer que tel ou tel peuple appartenait à la « généalogie » du peuple chinois et pour prouver, d'une certaine manière, l'existence de la « grande famille de la nation chinoise » (p. 84). Ainsi, dès sa création, l'Academia Sinica poursuivait un but tout autant politique que scientifique.

De manière plus générale, on sent l'auteur assez embarrassé lorsqu'il s'agit de définir la « culture chinoise ». Tout en mettant en avant les échanges de la Chine avec l'étranger sur les plans économiques, religieux et sur celui des idées, et tout en soutenant que la culture chinoise ne se limite pas à la culture Han, il admet qu'elle en est le coeur. L'unification impériale réalisée sous les Qin et les Han est pour lui fondamentale, et si le noyau culturel formé à cette époque fut remodelé par la suite, notamment par des influences étrangères, il n'en demeure pas moins le « cultural mainstream » de la Chine, pour reprendre une expression qui revient souvent. Ge met particulièrement en lumière l'importance des Song dans la construction de l'identité chinoise. Ils opèrent une sorte de recréation de la culture prenant pour cadre le peuple Han et d'une éthique fondée sur la pensée confucéenne (p. 109). Si l'auteur ne donne pas de définition précise de ce qu'est pour lui la culture Han traditionnelle, il en énumère cinq facettes clés : 1. l'écriture chinoise 2. la structure familiale, celle du clan et de l'État, 3. les trois enseignements (confucianisme, taoïsme, bouddhisme) et leur unité (san jiao he yi 三教合一) 4. les rapports de l'homme et du Ciel (c'est-à-dire la cosmologie corrélative du yin, du yang et des cinq agents) 5. la théorie du Tianxia et les relations internationales fondées sur le tribut.

Au final, ce petit ouvrage atteint son objectif qui est de donner quelques clés afin de comprendre comment un intellectuel chinois d'aujourd'hui se débat (souvent avec peine) entre deux pôles irréconciliables : celui d'une tradition historique locale fortement conservatrice et celui d'une approche hypercritique et déconstructiviste.

Damien Chaussende, CNRS-CRCAO