Jean-François Mayol

Corinne Loeuillet

Francis Hérodin

Didier Wion
email: didier.wion@ujf-grenoble.fr

The characterization of normal stem cells and cancer stem cells uses the same paradigm. These cells are isolated by a Fluorescent-Activated Cell Sorting step and their stemness is assayed following implantation into animals. However, differences exist between these two kinds of stem cells. Therefore, the translation of the experimental procedures used for normal stem cell isolation into the cancer stem cell research field is a potential source of artefacts. In addition, normal stem cell therapy has for objective the regeneration of a tissue, while cancer stem cell-centred therapy seeks the destruction of the cancer tissue. Taking these differences into account is critical for anticipating problems that might arise in cancer stem cell-centred therapy and for upgrading the cancer stem cell paradigm accordingly.

The cancer stem cell model.

Accumulating evidence indicates that tumours are organized in a hierarchical pattern like normal tissues, with a small population of stem cells named "cancer stem cells" responsible for both development and maintenance of cancer tissue, and a majority of cancer cells with limited proliferative potential which form the bulk of the tumor (for reviews see references (1- 3) . In the cancer stem cell experimental paradigm, cancer stem cells are individually defined by their ability to serially regenerate tumours following implantation in animals. This property illustrates their unlimited proliferation potential, their ability to self-renewed and their capacity to generate a progeny of differentiated cancer cells with limited proliferative potential. Cancer stem cells share several distinctive properties with normal stem cell i.e. selfrenewal and extensive proliferation potentials, and capability to differentiate. However, the term cancer stem cell is not synonymous with "transformed stem cell", since in many cases the nature of the cell to first acquire fully malignant properties, i.e. stem cell, progenitor or differentiated cell, remains unknown [START_REF] Clarke | Cancer stem cells--perspectives on current status and future directions: AACR Workshop on cancer stem cells[END_REF] . In other words, the term "cancer stem cell" defines a function, and does not refer to the identity of the initial transformed cell that originated the tumor.

The concept of cancer stem cell is of paramount importance. If cancer stem cells represent the only cell population with tumorigenic potential, then one would hypothesize that these cells are not only involved in the maintenance of the primary tumour mass but may also play a critical role in metastasis and relapse [START_REF] Quintana | Efficient tumour formation by single human melanoma cells[END_REF][START_REF] Balic | Most early disseminated cancer cells detected in bone marrow of breast cancer patients have a putative breast cancer stem cell phenotype[END_REF][START_REF] Li | Beyond tumorigenesis: cancer stem cells in metastasis[END_REF][START_REF] Brabletz | Opinion: migrating cancer stem cells -an integrated concept of malignant tumour progression[END_REF] . Regarding metastasis, although the cancer stem cell phenotype is not always sufficient to establish metastases, a subpopulation of migrating cancer stem cells involved in tumour metastasis has already been detected in human breast and pancreatic cancers [START_REF] Quintana | Efficient tumour formation by single human melanoma cells[END_REF][START_REF] Balic | Most early disseminated cancer cells detected in bone marrow of breast cancer patients have a putative breast cancer stem cell phenotype[END_REF] . Concerning relapse, the mechanisms involved in the therapeutic resistance of cancer stem cells include for example the preferential activation of the DNA damage checkpoint response, and increased Wnt/β-catenin and Notch signalling [START_REF] Hermann | Distinct populations of cancer stem cells determine tumor growth and metastatic activity in human pancreatic cancer[END_REF] . Drug pumps are also preferentially expressed in cancer stem cells [START_REF] Eyler | Survival of the fittest: cancer stem cells in therapeutic resistance and angiogenesis[END_REF] . Notably, one of these drug transporters involved in chemoresistance, named ABCG2, is the determinant of the Side Population phenotype (SP) which is used for characterizing some normal and cancer stem cells [START_REF] Goodell | Isolation and functional properties of murine hematopoietic stem cells that are replicating in vivo[END_REF][START_REF] Kondo | Persistence of a small subpopulation of cancer stem-like cells in the C6 glioma cell line[END_REF] . Hence, in the cancer stem cell model, a therapy targeting cancer cells but sparing the cancer stem cell population will induce only temporary regression. Therefore, the isolation and the characterization of cancer stem cell are of crucial concern for both basic cancer research and for designing novel efficient therapies.

The purpose of this paper is to discuss the weaknesses of the assays currently used to isolate and characterize the different subpopulations of cancer stem cells that are important for tumour growth and are not efficiently targeted by current therapies. We show that applying the same experimental paradigms for the isolation of normal and cancer stem cell can be misleading because these two kinds of stem cells have fundamental distinct features.

Moreover, the goals of cancer stem cell-centred therapy and regenerative therapy are opposite. Cancer stem cell therapy seeks to eliminate a tissue whereas regenerative stem cell therapy looks for tissue growth. Taking into account these fundamental differences is critical for the future of cancer stem cell research.

Flow cytometry: the gold standard for stem cell isolation.

Since the introduction of monoclonal antibodies directed against cell surface antigens, flow cytometry is the method of choice for characterizing and sorting normal stem cells. Currently, the multi-parameter analysis based on cell surface antigens expression or on the capability to efflux fluorescent dyes, allows one to assess the phenotype of cells residing at the top of the differentiation pyramid of the haematopoietic system. The conceptual analogy between the hierarchical organization of normal and cancer tissues coincides with experimental analogies in the methods used to isolate normal and cancer stem cells. This similarity is evident when the experimental methods developed for assaying either haematopoietic or cancer stem cell potentials are compared (Fig 1).

Basically, the isolation of normal or cancer stem cells requires a cell sorting step that is followed by an in vivo repopulation/tumorigenicity assay, in which the potential of sorted cells to reconstitute normal or tumoral tissues after implantation in animals is evaluated (Fig. 1). This cell sorting step is based on the detection of cell surface antigens or on the ability of some stem cells to efflux the vital fluorescent dye Hoechst 33342. For example, haematopoietic stem cells (HSC) were first characterized from bone marrow as CD34 + [START_REF] Berenson | Antigen CD34+ marrow cells engraft lethally irradiated baboons[END_REF][START_REF] Osawa | Long-term lymphohematopoietic reconstitution by a single CD34-low/negative hematopoietic stem cell[END_REF] .

Currently, other cell surface markers such as the CD133 epitope recognized by the mAb AC133 are used to identify human HSC and some cancer stem cells [START_REF] Miraglia | A novel five-transmembrane hematopoietic stem cell antigen: isolation, characterization, and molecular cloning[END_REF][START_REF] Wu | Persistence of CD133+ cells in human and mouse glioma cell lines: detailed characterization of GL261 glioma cells with cancer stem cell-like properties[END_REF] . On the other hand, a functional assay based on the ability of stem cell to extrude the fluorescent Hoechst 33342 dye (side-population or SP assay) is also widely used to isolate HSC [START_REF] Goodell | Isolation and functional properties of murine hematopoietic stem cells that are replicating in vivo[END_REF] or cancer stem cells [START_REF] Kondo | Persistence of a small subpopulation of cancer stem-like cells in the C6 glioma cell line[END_REF][START_REF] Hirschmann-Jax | A distinct "side population" of cells with high drug efflux capacity in human tumor cells[END_REF][START_REF] Patrawala | Side population is enriched in tumorigenic, stem-like cancer cells, whereas ABCG2+ and ABCG2cancer cells are similarly tumorigenic[END_REF] . Thus, normal or cancer stem cells characterization requires a Fluorescence Activated Cell Sorter (FACS) analysis step, since even when magnetic-activated cell sorting (MACS) is used, the purity of the sorted cell population needs to be evaluated by FACS. However, even though immunophenotyping and SP assays are the gold standards for identifying normal or cancer stem cells, these two methods are basically different. SP analysis is a functional assay evaluating the capability of cells to extrude a fluorescent dye, while immunophenotyping is a structural assay which detects the expression of cell surface markers.

Fig. 1 Normal stem cell and cancer stem cell assays.

The experimental steps used to characterize cancer stem cells follow the experimental paradigm of normal stem cell characterization. At first, cells are isolated from tissue and then sorted by FACS either on their capacity to efflux Hoechst 33342 (SP phenotype) (A,B), or on the basis of the expression of CD34 for haematopoietic stem cells and CD133 for brain cancer stem cells (immunophenotyping) (C,D). In a haematopoietic stem cell assay (A,C), normal stem cells prospectively isolated by FACS, are assayed for their capability to rescue lethally irradiated mice from radiation-induced bone marrow aplasia, and for their ability to provide permanent long-term engraftment [START_REF] Bissell | Context, tissue plasticity, and cancer: are tumor stem cells also regulated by the microenvironment?[END_REF] . In a cancer stem cell assay (B,D), cancer stem cells prospectively isolated by FACS are implanted into an orthotopic site of immunocompromised mice and are assayed for their capability to serially induce tumor formation.

Sorting cancer stem cells on the basis of a functional test: The SP phenotype

History.

Hoechst 33342 is a fluorescent dye designed by the Hoechst firm in 1974 with the aim of providing new chemotherapeutic agents [START_REF] Loewe | Basic-substituted 2,6-bisbenzimidazole derivates, a novel class of substances with chemotherapeutic activity[END_REF] . This drug is part of a family of fluorescent vital dyes and has an excitation wavelength at around 350 nm. Hoechst 33342 binds to DNA and has been first used to visualize nuclei, to quantify cell DNA content, and for sorting living cells according to the different phases of the cell cycle [START_REF] Arndt-Jovin | Analysis and sorting of living cells according to deoxyribonucleic acid content[END_REF] . Interestingly, this compound emits at two wavelengths of fluorescence, blue and red, when it binds to DNA. This polychromatic fluorescence emission depends on the intracellular concentration of the dye [START_REF] Petersen | Chromatic shifts in the fluorescence emitted by murine thymocytes stained with Hoechst 33342[END_REF] and is critical for characterizing the SP phenotype. In 1996 Goodell, demonstrated that mice HSC have the distinct feature to efflux this dye [START_REF] Goodell | Isolation and functional properties of murine hematopoietic stem cells that are replicating in vivo[END_REF] ; this property is the basis of the SP assay which has been used since in more than one hundred studies to characterize normal or cancer stem cells.

Functional basis.

The property of some stem cells to efflux Hoechst 33342 defines by FACS analysis a minor population of low labelled cells referred to as "SP" cells [START_REF] Goodell | Isolation and functional properties of murine hematopoietic stem cells that are replicating in vivo[END_REF][START_REF] Kondo | Persistence of a small subpopulation of cancer stem-like cells in the C6 glioma cell line[END_REF] . These low-labelled cells are detected as a distinct population in the left lower quadrant of FACS profiles (Fig 1B and2A).

Bcrp1/ABCG2 expression has been shown to be the determinant of the SP phenotype in bone marrow cells [START_REF] Scharenberg | The ABCG2 transporter is an efficient Hoechst 33342 efflux pump and is preferentially expressed by immature human hematopoietic progenitors[END_REF][START_REF] Zhou | Bcrp1 gene expression is required for normal numbers of side population stem cells in mice, and confers relative protection to mitoxantrone in hematopoietic cells in vivo[END_REF][START_REF] Zhou | The ABC transporter Bcrp1/ABCG2 is expressed in a wide variety of stem cells and is a molecular determinant of the side-population phenotype[END_REF] . However, depending on the tissue, other ABC transporters can contribute to the SP phenotype. For example, both Bcrp1/ABCG2 and Mdr1a/b (Abcb1a/1b) play a part in the SP phenotype in the mammary gland [START_REF] Jonker | Contribution of the ABC transporters Bcrp1 and Mdr1a/1b to the side population phenotype in mammary gland and bone marrow of mice[END_REF] . Since a drug transporter such as ABCG2 is the functional basis of Hoechst outflow, a control experiment with verapamil an ABCG2 pump inhibitor, must be included for each experiment. In these control experiments, the presence of verapamil during the assay leads to the disappearance of SP cell population (Fig. 1B and2B).

An important consequence of the involvement of the Bcrp1/ABCG2 drug-transporter in the determination of the SP phenotype is that SP cells are de facto resistant to several drugs such as mitoxantrone, and methotrexate [START_REF] Dean | Tumour stem cells and drug resistance[END_REF] .

Controversies:

Verapamil-sensitive upper SP cells:

Although settings for SP analysis place the cell-sorting gate in the left lower quadrant, a more careful examination of FACS profiles obtained in the presence of verapamil demonstrates the existence of other SP cells located in the left upper quadrant of FACS profile [START_REF] Platet | Fluctuation of the SP/non-SP phenotype in the C6 glioma cell line[END_REF][START_REF] Smalley | The mammary gland "side population": a putative stem/progenitor cell marker?[END_REF] . This upper cell population is evident in cancer cell lines (Fig. 2C). In several cancer stem cell experiments, the addition of verapamil induces a convex-concave transition of FACS profiles (see for example [START_REF] Kondo | Persistence of a small subpopulation of cancer stem-like cells in the C6 glioma cell line[END_REF][START_REF] Platet | Fluctuation of the SP/non-SP phenotype in the C6 glioma cell line[END_REF] ). The fact that this observation is not readily observable with normal low cycling cells such as HSC suggests that this upper SP cell population could correspond to SP cycling cells or SP polyploid cells [START_REF] Platet | Fluctuation of the SP/non-SP phenotype in the C6 glioma cell line[END_REF][START_REF] Smalley | The mammary gland "side population": a putative stem/progenitor cell marker?[END_REF] . Unfortunately, this upper SP was not further characterized and goes unnoticed by most investigators working in field of cancer stem cells (11,26) , illustrating our tendency to use the same experimental paradigm for studying normal and cancer stem cells regardless the fundamental differences existing between these two cell types.

Hoechst 33342 activities:

A recurrent question regarding the use of Hoechst 33342 in SP analysis is the consequence of Hoechst staining on cell viability and differentiation since Hoechst 33342 interacts with DNA.

Indeed, the biological consequence of Hoechst nuclear staining is a particularly relevant point in cancer research where many DNA-labelling agents have important cancer therapeutic effects. In fact, it has long been known that Hoechst 33342 can be cytotoxic depending on the concentration and the cell type used [START_REF] Durand | Cytotoxicity, Mutagenicity and DNA damage by Hoechst 33342[END_REF][START_REF] Fried | Effects of Hoechst 33342 on survival and growth of two tumor cell lines and on hematopoietically normal bone marrow cells[END_REF] . Moreover, it is noteworthy that Hoechst interferes with cancer cell differentiation at concentrations used for SP analysis [START_REF] Adamski | Effects of Hoechst 33342 on C2C12 and PC12 cell differentiation[END_REF][START_REF] Steuer | Differentiation of EC cells in vitro by the fluorescent dye Hoechst 33342[END_REF][START_REF] Zhong | Most MCF7 and SK-OV3 cells were deprived of their stem nature by Hoechst 33342[END_REF] . Unfortunately, these data are largely ignored.

Although the mechanism by which Hoechst 33342 influences cell differentiation has not been specifically addressed, several reports demonstrate that Hoechst 33342 poisoned topoisomerase I with a similar mechanism of action as camptothecin [START_REF] Chen | A new mammalian DNA topoisomerase I poison Hoechst 33342: cytotoxicity and drug resistance in human cell cultures[END_REF] . This point is of particular concern since topoisomerase I inhibitors are known to induce cancer cell differentiation [START_REF] Kondo | Induction of in vitro differentiation of mouse embryonal carcinoma (F9) cells by inhibitors of topoisomerases[END_REF][START_REF] Larsen | Involvement of DNA topoisomerases and DNA topoisomerase inhibitors in the induction of leukemia cell differentiation[END_REF] . Hence, an evaluation of Hoechst 33342 effects on cancer cell differentiation should be included in any SP analysis demonstrating a difference between SP and non-SP cells. Regrettably, these control experiments, which are much more cumbersome than evaluating Hoechst cytotoxicity, are never performed. The fact that Hoechst 33342 disrupts TATA box-binding protein/DNA complexes [START_REF] Zhang | Hoechst 33342 induces apoptosis and alters tata box binding protein/DNA complexes in nuclei from BC3H-1 myocytes[END_REF] also raises concerns on the transcriptomics data that compare SP and non-SP cells.

Functional independence of normal stem cells for SP phenotype.

Bcrp1/ABCG2, the molecular determinant of the SP phenotype, is expressed in a wide variety of stem cells [START_REF] Zhou | The ABC transporter Bcrp1/ABCG2 is expressed in a wide variety of stem cells and is a molecular determinant of the side-population phenotype[END_REF] . However, although a large reduction in the number of SP cells is detected in bcrp1 null mouse, the HSC compartment is not affected in these null mice [START_REF] Zhou | Bcrp1 gene expression is required for normal numbers of side population stem cells in mice, and confers relative protection to mitoxantrone in hematopoietic cells in vivo[END_REF] .

Moreover, the SP phenotype could be dispensable for stem cell function since these mice show no obvious stem cell pathology. This could explain some conflicting results obtained with cancer stem cells when SP phenotype fluctuates independently of stemness [START_REF] Platet | Fluctuation of the SP/non-SP phenotype in the C6 glioma cell line[END_REF] . An important point is that, concerning solid tumors, SP analyses are usually done on cancer cell lines or on primary cancer cells which have been amplified in culture. Therefore, the extrapolation of the data to the behaviour of the cells in vivo can be debatable. The situation is different for leukemic cells and for normal hematopoietic cells which are usually analysed immediately after isolation [START_REF] Goodell | Isolation and functional properties of murine hematopoietic stem cells that are replicating in vivo[END_REF][START_REF] Moshaver | Identification of a small subpopulation of candidate leukemia-initiating cells in the side population of patients with acute myeloid leukemia[END_REF][START_REF] Wulf | A leukemic stem cell with intrinsic drug efflux capacity in acute myeloid leukemia[END_REF] . Notwithstanding these limitations since Bcrp1/ABCG2, which is the functional determinant of the SP phenotype is an ABC transporter involved in drug resistance, the SP phenotype provides a selective advantage when cells are challenged with anticancer drugs such as the topoisomerase I inhibitor topotecan or the antifolate agent methotrexate [START_REF] Noguchi | Functions of the breast cancer resistance protein (BCRP/ABCG2) in chemotherapy[END_REF] . Hence, the SP phenotype defines a population of chemo-resistant cancer cells, and this approach is pertinent from a therapeutic standpoint. The recent finding suggesting that therapeutic blockade of either PI3K or Akt might reduce ABCG2 function in cancer stem cell in vivo is, in this context, highly relevant [START_REF] Bleau | PTEN/PI3K/Akt pathway regulates the side population phenotype and ABCG2 activity in glioma tumor stem-like cells[END_REF] .

Fig.2

The upper SP: how the blind adherence to the academic SP cell gating procedures can hinder a relevant cancer SP cell population.

In a typical SP analysis, cells are labelled with Hoechst 33342 in the presence or absence of an ABCG2 inhibitor such as verapamil. The SP cell population assimilated to cancer stem cells is defined as the population of cells that actively outflow Hoechst by a verapamil sensitive process. Therefore, in FACS profiles, the SP region which corresponds to low labelled cells, is indicated by a trapezoid on the left lower quadrant of FACS profile (A). In the presence of verapamil, cells which are localized in the SP region disappears (B). However, when SP analyses are performed on cancer cell lines, another verapamil sensitive population is readily detectable in the left upper quadrant of FACS profile (C,D). Unfortunately this cell population remains unnoticed in almost all published analyses. In the example shown, which is performed on C6 glioma cells, roughly half of the verapamil-sensitive cells are found in this upper side population. Although these cells have not been fully characterized they probably correspond to cycling cells. Indeed this upper SP is not detectable when SP analysis is performed on low-cycling diploid cells such as bone marrow cells (A,B). This example illustrates the limitations of transferring a technology validated in normal stem cell studies to cancer stem cell studies without considering the special features of cancer cells.

Sorting cancer stem cells on the basis of a structural test. Immunophenotyping cancer stem cells on the basis of cell surface markers.

History.

Cancer stem cells immunophenotyping originates directly from methods used more than 30 years ago to identify the heterogeneous populations of cells within bone marrow [START_REF] Berenson | Antigen CD34+ marrow cells engraft lethally irradiated baboons[END_REF] .

Nowadays, prospective cancer stem cells are isolated by flow cytometry on the basis of cell surface markers expression, just like haematopoietic stem cells are [START_REF] Al-Hajj | Prospective identification of tumorigenic breast cancer cells[END_REF][START_REF] Dalerba | Phenotypic characterization of human colorectal cancer stem cells[END_REF][START_REF] O'brien | A human colon cancer cell capable of initiating tumour growth in immunodeficient mice[END_REF][START_REF] Ricci-Vitiani | Identification and expansion of human colon-cancer-initiating cells[END_REF][START_REF] Singh | Identification of human brain tumour initiating cells[END_REF][START_REF] Zhang | Identification and characterization of ovarian cancer-initiating cells from primary human tumors[END_REF] . For example, the cancer stem cell population has been defined as CD34 + CD38 -in acute myeloid leukaemia [START_REF] Lapidot | A cell initiating human acute myeloid leukaemia after transplantation into SCID mice[END_REF] CD133 + in brain [START_REF] Singh | Identification of human brain tumour initiating cells[END_REF] , melanoma [START_REF] Monzani | Melanoma contains CD133 and ABCG2 positive cells with enhanced tumourigenic potential[END_REF] and lung tumors [START_REF] Eramo | Identification and expansion of the tumorigenic lung cancer stem cell population[END_REF] , CD133+ [START_REF] O'brien | A human colon cancer cell capable of initiating tumour growth in immunodeficient mice[END_REF][START_REF] Ricci-Vitiani | Identification and expansion of human colon-cancer-initiating cells[END_REF] or EpCAM high /CD44 + /CD166 + in colon cancer [START_REF] Dalerba | Phenotypic characterization of human colorectal cancer stem cells[END_REF] , CD44 + /CD24

-/low /Lineage -in breast tumors [START_REF] Al-Hajj | Prospective identification of tumorigenic breast cancer cells[END_REF] , CD44 + /CD117 + in ovarian tumors [START_REF] Zhang | Identification and characterization of ovarian cancer-initiating cells from primary human tumors[END_REF] , and CD44 + / 2  1 hi /CD133 + in prostate [START_REF] Collins | Prospective identification of tumorigenic prostate cancer stem cells[END_REF] . In all cases, the prospective cancer stem cell populations sorted on the basis of cell surface markers are then characterized by their ability to regrow a tumor in xenotransplantation assays. This xenograft assay can be viewed as the parallel of the repopulation assay used in haematopoietic stem cell research. Importantly none of the antigens used for identifying cancer stem cells is exclusively expressed on cancer cells since, for example CD34 and CD133 cell surface antigens are also used to immunophenotype normal stem and progenitor cells in several tissues [START_REF] Shmelkov | CD133 expression is not restricted to stem cells, and both CD133+ and CD133-metastatic colon cancer cells initiate tumors[END_REF][START_REF] Krause | CD34: structure, biology, and clinical utility[END_REF] .

Functional basis.

Contrary to the SP phenotype, which relies on the functionality of a drug transporter involved in multidrug resistance and therefore in cancer chemoresistance, the immunophenotyping assay can target membrane proteins not necessarily involved in the "cancer stem cell" function. For example, AC133 one of the most widely antigens used for cancer stem cell immunophenotyping corresponds to CD133, a protein whose function remains elusive.

However, even if the role of CD133 in cancer is not elucidated, CD133 + cells have nevertheless been reported to have important features such as an increased radioresistance [START_REF] Bao | Glioma stem cells promote radioresistance by preferential activation of the DNA damage response[END_REF][START_REF] Blazek | Daoy medulloblastoma cells that express CD133 are radioresistant relative to CD133-cells, and the CD133+ sector is enlarged by hypoxia[END_REF] and increased expression of VEGF [START_REF] Yao | Glioblastoma stem cells produce vascular endothelial growth factor by activation of a G-protein coupled formylpeptide receptor FPR[END_REF] . On the other hand, other antigens used for immunophenotyping are directly implicated in cancer process. For example, the cell surface marker EpCAM is used for immunophenotyping and is known to up-regulate c-myc and cyclin A/E, and to induce cell proliferation [START_REF] Munz | The carcinoma-associated antigen EpCAM upregulates c-myc and induces cell proliferation[END_REF] .

Controversies.

-Effect of cell sorter optical configuration, inter-laboratory variability, and the need for a standard of measurement.

A large prpoportion of the cell sorters commercialised during the end of the 1990s and the beginning of the 2000s such as the FACS Vantage SE from BD Bioscience have an optical configuration that does not allow the optimal collection of the fluorescences emitted by stained cells [START_REF] Shapiro | Practical Flow Cytometry[END_REF] . These devices are still used nowadays in numerous flow cytometry core laboratories. Since the optics of the current machines such as the LSR II from BD Bioscience with a quartz device are more efficient at capturing the emitted light than the stream-in-air system of the old sorters, the percentages of positive cells acquired in these two cases widely differ. This point becomes critical when negative and positive cell populations form a continuum. In this regard it is noteworthy that FACS analyses cluster cells on the basis of positive, negative, low, or high fluorescent intensity. In a sense, FACS is a highly efficient technology of measurement that has forgotten to define an objective standardised unit of measurement relevant to the parameter studied. The judicious selection of a standard for FACS analysis, which could for example consider the number of epitopes by cell, is therefore critical for ensuring the inter-laboratory reproducibility of experimental results [START_REF] Lee | MIFlowCyt: the minimum information about a Flow Cytometry Experiment[END_REF][START_REF] Schwartz | Quantitative flow cytometry[END_REF] .

-Immunophenotyping and SP analyses: similarity, overlap and discrepancy.

The basic tenet of cancer stem cell research is that cancer stem cells represent a unique population of cancer cells responsible for tumour maintenance and recurrence. Hence, it is critical to characterise these cells as they are priority therapeutic targets. In this context, it could have been expected that cancer stem cell populations characterized by immunophenotyping and SP analysis coincide, and that immunophenotyping using different cancer stem cell markers identifies, for a given neoplasm, a unique population of cancer stem cells. However, several studies report a limited overlap between cancer stem cell populations obtained using different cell sorting procedures. For example, in some glioma cell lines, CD133 + cells are not SP [START_REF] Wu | Side population cells in human cancers[END_REF] . Likewise in DAOY medulloblastoma cells CD133 and SP subsets can be two independent compartments [START_REF] Srivastava | Flow cytometric characterization of the DAOY medulloblastoma cell line for the cancer stem-like phenotype[END_REF] .

Although, these data have been obtained with cultured cell lines and might not be representative of the situation found in vivo, other results obtained by immunophenotyping human colorectal xenograft cells which have never been maintained in culture also show some inconsistencies. Thus, the expression of CD133, which has been described as a colorectal cancer stem cell marker [START_REF] O'brien | A human colon cancer cell capable of initiating tumour growth in immunodeficient mice[END_REF][START_REF] Ricci-Vitiani | Identification and expansion of human colon-cancer-initiating cells[END_REF] , is variable in the cancer stem cell population characterized on the basis of CD44 expression [START_REF] Dalerba | Phenotypic characterization of human colorectal cancer stem cells[END_REF] or in colon metastatic cancer stem cell [START_REF] Shmelkov | CD133 expression is not restricted to stem cells, and both CD133+ and CD133-metastatic colon cancer cells initiate tumors[END_REF] .

Likewise, although brain cancer stem cells were first identified as CD133+ [START_REF] Singh | Identification of human brain tumour initiating cells[END_REF] , several recent reports demonstrate the existence of CD133 -cancer stem cells in brain tumors [START_REF] Beier | CD133(+) and CD133(-) glioblastomaderived cancer stem cells show differential growth characteristics and molecular profiles[END_REF][START_REF] Joo | Clinical and biological implications of CD133-positive and CD133-negative cells in glioblastomas[END_REF][START_REF] Ogden | Identification of A2B5+CD133tumor-initiating cells in adult human gliomas[END_REF][START_REF] Wang | CD133 negative glioma cells form tumors in nude rats and give rise to CD133 positive cells[END_REF] . Hence, the use of a single marker such as CD133 for immunophenotyping cancer stem cells could underscore the heterogeneity of the cancer stem cell phenotype as well as the complexity of cancer disease. This emphasizes the importance of phenotyping cancer stem cells on the basis of cell surface marker repertoires.

-Cancer stem cell marker constancy.

Constancy of antigens expression is critical for the relevance of cancer stem cell immunophenotyping. However, as mentioned earlier, a potential drawback of immunophenotyping is that most of the antigens used do not seem to be directly involved in the maintenance of the cancer stem cell function. This situation contrasts with the principle of SP analysis which targets the functionality of ABCG2/Bcrp1, a protein involved in drug resistance [START_REF] Dean | Tumour stem cells and drug resistance[END_REF] . Hence, inconsistencies may occur in the immunophenotyping of cancer stem cells if the markers used are not under selective pressure. Contrary to normal stem cells which have developed several genomic safeguard mechanisms and reside in a unique well-defined microenvironment (the stem cell niche), cancer stem cells are genetically and epigenetically unstable cells which are challenged by changing microenvironments not only in the primary tumor site, but also during metastatic process, and in cell culture. Hence, as cancer is an evolutionary system [START_REF] Dean | Tumour stem cells and drug resistance[END_REF][START_REF] Merlo | Cancer as an evolutionary and ecological process[END_REF][START_REF] Nowell | The clonal evolution of tumor cell populations[END_REF] , the stability of expression over time of a cancer stem cellassociated antigen may depend on the role of this antigen in the maintenance of the cancer stem cell function. This could explain the inconsistency reported above in the use of CD133

to characterise cancer stem cells in brain and colon cancers since the functional contribution of this antigen to cancer stem cell function remains obscure. Thus, both CD133 + and CD133 - cells with cancer stem cell functions can appear in a tumour during tumour progression and metastasis.

In this regard it is note-worthy that environmental conditions, such as low oxygen tension has been described to up regulate CD133 expression [START_REF] Blazek | Daoy medulloblastoma cells that express CD133 are radioresistant relative to CD133-cells, and the CD133+ sector is enlarged by hypoxia[END_REF][START_REF] Griguer | CD133 is a marker of bioenergetic stress in human glioma[END_REF][START_REF] Platet | Influence of oxygen tension on CD133 phenotype in human glioma cell cultures[END_REF] . This finding suggests that CD133 + cells might be cancer cells challenged to a hypoxic microenvironment. Hence the existence of multiple subsets of "cancer stem cells" with different antigenic profiles, even for tumours of a given tissue, must now be integrated into the cancer stem cell paradigm. The characterization of the functional role of the cell surface markers used in cancer stem cell immunophenotyping is therefore critical. In a sense, sorting cancer stem cells on the basis of a functional test such as the SP assay provides both information and understanding since Bcrp1/ABCG2 function is involved in the drug resistance of tumour stem cells. However, immunophenotyping cancer stem cells on the basis of epitopes whose function is not yet defined provides only information.

FACS in normal stem cell and cancer stem cell medicine: One technical approach for two different therapeutic goals.

FACS analysis of cell surface molecules, by its ability to isolate rare cell types to high purity, had a critical impact first on haematopoietic stem cell research, and then on leukaemia research and diagnosis [START_REF] Bene | Immunophenotyping of acute leukaemias[END_REF][START_REF] Noto | Flow cytometry analysis of acute promyelocytic leukemia: the power of 'surface hematology[END_REF] . However, transferring the concepts and methods developed in the context of HSC research to cancer stem cell entities in solid tumors requires a cautious consideration of the clinical difference existing between the therapeutic purpose of a regenerative therapy and that of a stemicide therapy. For example, regarding normal stem cell regenerative therapy, although LT-HSC have been first described as CD34 + or SP, very rare CD34 -and non-SP subsets of LT-HSCs also exist [START_REF] Engelhardt | CD34(+) or CD34(-): which is the more primitive?[END_REF][START_REF] Morita | Non-side-population hematopoietic stem cells in mouse bone marrow[END_REF] . Therefore, sorting HSC on the basis of CD34 expression or SP phenotype eliminates a functionally minor but relevant stem cell population. Omitting these CD34 -or non-SP LT-HSC, however does not seem to have clinical consequences in the repopulation assay since successful haematopoietic reconstitution can be therapeutically achieved with either CD34 + or SP LT-HSC cells only. Therefore, in normal stem cell regenerative medicine, the therapeutic achievement can be obtained with only one the stem cell subpopulation. Likewise, if the cancer stem cell function is supported in the patient by heterogeneous population of cancer stem cells, the implantation of only one of these cancer cell populations is sufficient to regrow the tumour. This outcome fulfils the experimental goal of the xenograft assay model, which is the regeneration of the tumour in the animal. However, in cancer medicine, the therapeutic goal is the eradication of the natural tumour, not its regeneration. If we consider that at the time of diagnosis or recurrence, the primitive cancer stem cell has generated several cancer stem cell sub-populations differing not only genetically or epigenetically, but also according to their surrounding microenvironment and metastatic potential, then a successful stemicide therapy requires the characterization and the targeting of all the cancer stem cell populations present in the patient.

Unfortunately, in a xenograft assay, cancer cells implanted in mice are challenged to a microenvironment very different from the cancer mass from which they are issued. Hence, the xenograft assay may fail to detect human cancer stem cell populations unable to grow in immunocompromised animal but nevertheless capable of sustaining tumour growth in patient [START_REF] Garcion | Cancer stem cells: Beyond Koch's postulates[END_REF][START_REF] Hill | Identifying cancer stem cells in solid tumors: case not proven[END_REF] . The fundamental difference existing between a regenerative and a stemicide stem cell therapy, and the low sensitivity of the xenograft assay, are two possible explanations for a potential disappointing outcome for cancer stem cell-centred therapies.

Concluding remarks and perspectives.

Cancer research is a relatively old field of investigation, and its experimental paradigms have successively evolved throughout the last century according to the most available frames of reference [START_REF] Garcion | Cancer stem cells: Beyond Koch's postulates[END_REF] . ). This could, in part, explain the recent revival of the cancer stem cell concept whose origin is more than a century old [START_REF] Sell | Stem cell origin of cancer and differentiation therapy[END_REF] . As biology is an experimental science, emergence of new reference frames is often linked to the development of new experimental tools. For example, more than one century ago, the development by Koch and Petri of a solid media enabling the rapid isolation of pure cultures from contaminated specimens was a major breakthrough in the development of medical microbiology [START_REF] Koch | Zur Untersuchung von pathogen Organismen[END_REF][START_REF] Petri | Eine kliene Modification des Koch'shen Platten-verfahrens[END_REF] . Interestingly, FACS now achieves the same goal as was achieved by solid media and the Petri dish, that is, cell cloning. However, we must be aware of the current limitations of the FACS experimental procedures used to isolate cancer stem cells (Table 1).

The determination of the functional role played by the different cell surface antigens used for cancer stem cell immunophenotyping is critical. Cancer is in development at all times, and cancer cells are engaged in an evolutionary process which occurs in a very short period of time as limited to the timescale of the illness [START_REF] Merlo | Cancer as an evolutionary and ecological process[END_REF][START_REF] Nowell | The clonal evolution of tumor cell populations[END_REF] . This ecological and evolutionary process involves genetically and epigenetically unstable cancer cells confronted with changing physiological and non-physiological microenvironments. In this context the expression of cancer stem cell epitopes not directly involved in critical stemness functions could fluctuate.

Therefore, identifying cell surface antigens whose function is mandatory for cancer stemness is of paramount importance.

In the absence of such key markers, FACS analyses need to evolve. More complex analyses, taking into account the intensity of cell labelling (high, medium, low, negative) and involving panels of antigens as recently done for colon and breast cancer stem cells [START_REF] Al-Hajj | Prospective identification of tumorigenic breast cancer cells[END_REF][START_REF] Dalerba | Phenotypic characterization of human colorectal cancer stem cells[END_REF] are required. These analyses could target cell surface proteins involved in critical cancer stem cell functions such as chemokine receptors (CXCR4, CXCR3…) [START_REF] Vandercappellen | The role of CXC chemokines and their receptors in cancer[END_REF] adhesion proteins (L1CAM…) [START_REF] Brabletz | Opinion: migrating cancer stem cells -an integrated concept of malignant tumour progression[END_REF] , or ABC drug transporters [START_REF] Dean | Tumour stem cells and drug resistance[END_REF] . Even if these proteins are also expressed on normal cells, a cancer stem cell specific pattern of expression can be expected to emerge.

Functional in vitro assays evaluating for example, invasion and angiogenic assays in Matrigel [START_REF] Benelli | In vitro models of angiogenesis: the use of Matrigel[END_REF][START_REF] Albini | The "chemoinvasion assay": a tool to study tumor and endothelial cell invasion of basement membranes[END_REF] could also be included for characterizing cancer stem cell subpopulations. The evolution of cancer stem cell analyses toward a combination of such structural and functional multiparametric assays is necessary if we consider the complexity of the cancer stem cell phenotype.

However, cancer stem cell characterization is not limited to a FACS sorting step. The stemness of cells prospectively isolated by FACS must be functionally assayed in xenograft experiments. In that case, the overall sensitivity of the cancer stem cell isolation procedure is limited by the experimental step with the lowest threshold of detection. One of the weak link of the cancer stem cell experimental paradigm resides in the sensitivity of the xenograft assay which needs to be improved as it can underestimate the frequency of cancer stem cells [START_REF] Quintana | Efficient tumour formation by single human melanoma cells[END_REF] .

Notably, it has been recently shown that the use of more highly immunocompromised mice increases the detectable frequency of cancer stem cells [START_REF] Quintana | Efficient tumour formation by single human melanoma cells[END_REF] . Numerous studies have demonstrated that microenvironment can either support or inhibit tumorigenesis [START_REF] Hu | Microenvironmental regulation of cancer development[END_REF][START_REF] Bissell | Context, tissue plasticity, and cancer: are tumor stem cells also regulated by the microenvironment?[END_REF] . Cancer stem cell assays are presently performed by implanting cancer cells in normal orthotopic tissue. A tumourigenicity assay based on the injection of cancer cells in pre-tumoral niches could reveal the existence of new class of cancer stem cells not previously detected by the present classical assays.

In conclusion, the parallel between normal and cancer stem cells is both conceptually and experimentally rewarding. However, the striking similarities between normal and cancer stem cells should not understate the fundamental differences existing between these cells (Table 1).

Taking into account these distinctions is essential for upgrading the experimental paradigm currently used in cancer stem cell research and for the development of therapies which will be able to target cancer stem cells while sparing normal stem cells. FACS, are assayed for their capability to rescue lethally irradiated mice from radiation-induced bone marrow aplasia, and for their ability to provide permanent long-term engraftment [START_REF] Bissell | Context, tissue plasticity, and cancer: are tumor stem cells also regulated by the microenvironment?[END_REF] . In a cancer stem cell assay (B,D), cancer stem cells prospectively isolated by FACS are implanted into an orthotopic site of immunocompromised mice and are assayed for their capability to serially induce tumor formation.

Legends to figures

Fig.2

The upper SP: how the blind adherence to the academic SP cell gating procedures can hinder a relevant cancer SP cell population.

In a typical SP analysis, cells are labelled with Hoechst 33342 in the presence or absence of an ABCG2 inhibitor such as verapamil. The SP cell population assimilated to cancer stem cells is defined as the population of cells that actively outflow Hoechst by a verapamil sensitive process. Therefore, in FACS profiles, the SP region which corresponds to low labelled cells, is indicated by a trapezoid on the left lower quadrant of FACS profile (A). In the presence of verapamil, cells which are localized in the SP region disappears (B). However, when SP analyses are performed on cancer cell lines, another verapamil sensitive population is readily detectable in the left upper quadrant of FACS profile (C,D). Unfortunately this cell population remains unnoticed in almost all published analyses. In the example shown, which is performed on C6 glioma cells, roughly half of the verapamil-sensitive cells are found in this upper side population. Although these cells have not been fully characterized they probably correspond to cycling cells. Indeed this upper SP is not detectable when SP analysis is performed on low-cycling diploid cells such as bone marrow cells (A,B). This example illustrates the limitations of transferring a technology validated in normal stem cell studies to cancer stem cell studies without considering the special features of cancer cells.

Fig. 1

 1 Fig.1

  Recently, stem cell biology became a new frame of reference in medical research as evidenced by the increase in the number of journals covering this field (Stem Cells, first issue 1983; Stem Cell and Development, first issue 1999; Stem Cell Research, first issue 2007; Cell Stem Cell, first issue 2007

Acknowledgements. We thank Dr Richard Foucault for critical reading of the manuscript, Dr JP Issartel for suggesting the use of the term "stemicide", and Pr F Berger for support. DW is supported by the Ligue contre le Cancer Comité Rhône..

Characterization of normal and cancer stem cells: One

experimental paradigm for two kinds of stem cells.


Samy Skander Bahoura

HARNACK TYPE INEQUALITY FOR AN ELLIPTIC EQUATION

We give a sup × inf inequality for an elliptic equation.

INTRODUCTION AND MAIN RESULTS

We are on Riemannian manifold (M, g) of dimension n ≥ 3. In this paper we denote ∆ g = -∇ j (∇ j ) the Laplace-Beltrami operator and N = 2n n-2 . We consider the following equation ( 1)

∆ g u = V u N -1 + u α , u > 0.
Where V is a function and α ∈] n n-2 , n+2 n-2 [. For a, b, A > 0, we consider a sequence (u i , V i ) i of solutions of the previous equation with the following conditions:

0 < a ≤ V i ≤ b < +∞, ||∇V i || ∞ ≤ A.
Here we study some properties of this nonlinear elliptic equation. We try to find some estimates of type sup × inf. We denote by S g the scalar curvature.

There are many existence and compactness results which concern this type of equations, see for example . In particluar in [START_REF] Aubin | Some Nonlinear Problems in Riemannian Geometry[END_REF], we can find some results about the Yamabe equation and the Prescribed scalar curvature equation. Many methods where used to solve these problems, as a variationnal approach and some other topological methods. Note that the problems come from the nonlinearity of the critical Sobolev exponent. We can find in [START_REF] Aubin | Some Nonlinear Problems in Riemannian Geometry[END_REF] some uniform estimates for various equations on the unit sphere or for the Monge-Ampere equation. Note that Tian and Siu proved uniform upper and lower bounds for the sup + inf for the Monge-Ampere equation under some condition on the Chern class, see [START_REF] Aubin | Some Nonlinear Problems in Riemannian Geometry[END_REF]. In the case of the Scalar curvature equation and in dimension 2 Shafrir used the isoperimetric inequality of Alexandrov to prove an inequality of type sup + inf with only L ∞ assumption on the prescribed curvature, see [START_REF] Shafrir | A sup+inf inequality for the equation -∆u = V e u[END_REF]. The result of Shafrir is an extention of a result of Brezis and Merle, see [START_REF] Brezis | Uniform estimates and blow-up bihavior for solutions of -∆u = V e u in two dimensions[END_REF] and later, Brezis-Li-Shafrir proved a sharp sup + inf inequality for the same equation with Lipschitzian assumption on the prescribed scalar curvature, see [START_REF] Brezis | A sup+inf inequality for some nonlinear elliptic equations involving exponential nonlinearities[END_REF]. Li in [START_REF] Yy | Harnack Type Inequality: the Method of Moving Planes[END_REF] extend the previous last result to compact Riemannian surfaces. In the higher dimensional case, we can find in [START_REF] Korevaar | Refined asymptotics for constant scalar curvature metrics with isolated singularities[END_REF] a proof of the sup × inf inequality in the constant case for the scalar curvature equation on open set of R n . We have various estimates in [START_REF] Bahoura | Majorations du type sup u × inf u ≤ c pour l'équation de la courbure scalaire sur un ouvert de R n , n ≥ 3[END_REF] when we consider the nonconstant case. To prove our result, we use a blow-up analysis and the moving-plane method, based on the maximum principle and the Hopf Lemma as showed in [START_REF] Bahoura | Majorations du type sup u × inf u ≤ c pour l'équation de la courbure scalaire sur un ouvert de R n , n ≥ 3[END_REF][START_REF] Brezis | A sup+inf inequality for some nonlinear elliptic equations involving exponential nonlinearities[END_REF][START_REF] Korevaar | Refined asymptotics for constant scalar curvature metrics with isolated singularities[END_REF][START_REF] Yy | Harnack Type Inequality: the Method of Moving Planes[END_REF], and a condition on the scalar curvature is sufficient to prove the estimate.

Our main result is:

Theorem 1.1. Assume S g > 0 on M , then, for every compact K of M , there exist a positive constant c = c(α, a, b, A, K, M, n, g) such that:

sup K u i × inf M u i ≤ c.
Remark: in the case where (M, g) = (Ω ⊂ R n , δ) an open set of the euclidean space with the flat metric, we have the same inequality on compact sets of Ω in this case the scalar curvature S δ ≡ 0, see [START_REF] Bahoura | Majorations du type sup u × inf u ≤ c pour l'équation de la courbure scalaire sur un ouvert de R n , n ≥ 3[END_REF].

If we consider the Green function G of the Laplacian with Dirichlet condition on small balls of M , we can have a positive lower bound for G and we have the following corollary: Corollary 1.2. Assume S g > 0 on M , then, for every compact K of M , there exist a positive constant c ′ = c ′ (α, a, b, A, K, M, n, g) such that:

K u 2n n-2 i dv g ≤ c ′ .

PROOF OF THE THEOREM.

Let us consider x 0 ∈ M , by a conformal change of the metric g = ϕ 4/(n-2) g with ϕ > 0 we can consider an equation of type:

(2)

∆ g u + R g u = V u N -1 + ϕ α+1-N u α + R g ϕ 2-N u, u > 0.
with, Ricci g(x 0 ) = 0.

Here

; R g = n -2 4(n -1) S g and R g = n -2 4(n -1) S g
It is clear see the computations in a previous paper [START_REF] Bahoura | Majorations du type sup u × inf u ≤ c pour l'équation de la courbure scalaire sur un ouvert de R n , n ≥ 3[END_REF], it is sufficient to consider an equation of type:

(3)

∆ gu = V u N -1 + u α + µu, u > 0.
with, Ricci ≡ 0 and µ > 0.

Part I: The metric in polar coordinates.

Let (M, g) a Riemannian manifold. We note g x,ij the local expression of the metric g in the exponential map centered in x.

We are concerning by the polar coordinates expression of the metric. Using Gauss lemma, we can write:

g = ds 2 = dt 2 + g k ij (r, θ)dθ i dθ j = dt 2 + r 2 gk ij (r, θ)dθ i dθ j = g x,ij dx i dx j , in a polar chart with origin x", ]0, ǫ 0 [×U k , with (U k , ψ) a chart of S n-1 .
We can write the element volume:

dV g = r n-1 |g k |drdθ 1 . . . dθ n-1 = [det(g x,ij )]dx 1 . . . dx n , then, dV g = r n-1 [det(g x,ij )][exp x (rθ)]α k (θ)drdθ 1 . . . dθ n-1 , where, α k is such that, dσ Sn-1 = α k (θ)dθ 1 . . . dθ n-1 . (Riemannian volume element of the sphere in the chart (U k , ψ) ).
Then,

|g k | = α k (θ) [det(g x,ij )].
Clearly, we have the following proposition: Proposition 2.1. Let x 0 ∈ M , there exist ǫ 1 > 0 and if we reduce U k , we have:

|∂ r gk ij (x, r, θ)| + |∂ r ∂ θ m gk ij (x, r, θ)| ≤ Cr, ∀ x ∈ B(x 0 , ǫ 1 ) ∀ r ∈ [0, ǫ 1 ], ∀ θ ∈ U k . and, |∂ r |g k |(x, r, θ)| + ∂ r ∂ θ m |g k |(x, r, θ) ≤ Cr, ∀ x ∈ B(x 0 , ǫ 1 ) ∀ r ∈ [0, ǫ 1 ], ∀ θ ∈ U k . Remark: ∂ r [log |g k |]
is a local function of θ, and the restriction of the global function on the sphere S n-1 , ∂ r [log det(g x,ij )]. We will note, J(x, r, θ) = det(g x,ij ).

Part II: The laplacian in polar coordinates

Let's write the laplacian in [0,

ǫ 1 ] × U k , -∆ = ∂ rr + n -1 r ∂ r + ∂ r [log |g k |]∂ r + 1 r 2 |g k | ∂ θ i (g θ i θ j |g k |∂ θ j ).
We have,

-∆ = ∂ rr + n -1 r ∂ r + ∂ r log J(x, r, θ)∂ r + 1 r 2 |g k | ∂ θ i (g θ i θ j |g k |∂ θ j ).
We write the laplacian ( radial and angular decomposition),

-∆ = ∂ rr + n -1 r ∂ r + ∂ r [log J(x, r, θ)]∂ r -∆ Sr(x) ,
where ∆ Sr (x) is the laplacian on the sphere S r (x).

We set

L θ (x, r)(...) = r 2 ∆ Sr (x) (...)[exp x (rθ)],
clearly, this operator is a laplacian on S n-1 for particular metric. We write,

L θ (x, r) = ∆ gx,r, S n-1 ,
and,

∆ = ∂ rr + n -1 r ∂ r + ∂ r [J(x, r, θ)]∂ r - 1 r 2 L θ (x, r). If, u is function on M , then, ū(r, θ) = u[exp x (rθ)]
is the corresponding function in polar coordinates centered in x. We have, ( 4)

-∆u = ∂ rr ū + n -1 r ∂ r ū + ∂ r [J(x, r, θ)]∂ r ū - 1 r 2 L θ (x, r)ū.
Part III: "Blow-up" and "Moving-plane" methods

The "blow-up" analysis Let, (u i ) i a sequence of functions on M such that, (5)

∆ g u i + R gu i = V i u N -1 i + ϕ α-N u α i + R g ϕ 2-N u i , u i > 0, N = 2n n -2 ,
It is sufficient to consider an equation of type:

(6) ∆ gu = V u N -1 + u α + µu, u > 0.
with Ricci ≡ 0 and µ > 0.

We argue by contradiction and we suppose that sup × inf is not bounded. We assume that:

∀ c, R > 0 ∃ u c,R solution of (E) such that: R n-2 sup B(x0,R) u c,R × inf M u c,R ≥ c. (H) Proposition 2.2.
There exist a sequence of points (y i ) i , y i → x 0 and two sequences of positive real number

(l i ) i , (L i ) i , l i → 0, L i → +∞, such that if we consider v i (y) = u i [exp yi (y)] u i (y i ) , we have: i) 0 < v i (y) ≤ β i ≤ 2 (n-2)/2 , β i → 1. ii) v i (y) → 1 1 + |y| 2 (n-2)/2
, uniformly on every compact set of R n .

iii) l

(n-2)/2 i [u i (y i )] × inf M u i → +∞ Proof:
Without loss of generality, we can assume that:

V (x 0 ) = n(n -2).
We use the hypothesis (H). We can take two sequences

R i > 0, R i → 0 and c i → +∞, such that, R i (n-2) sup B(x0,Ri) u i × inf M u i ≥ c i → +∞. Let, x i ∈ B(x 0 , R i ), such that sup B(x0,Ri) u i = u i (x i ) and s i (x) = [R i -d(x, x i )] (n-2)/2 u i (x), x ∈ B(x i , R i ). Then, x i → x 0 .
We have, max B(xi,Ri)

s i (x) = s i (y i ) ≥ s i (x i ) = R i (n-2)/2 u i (x i ) ≥ √ c i → +∞.
Set :

l i = R i -d(y i , x i ), ūi (y) = u i [exp yi (y)], v i (z) = u i [exp yi z/[u i (y i )] 2/(n-2) ] u i (y i ) .
Clearly, y i → x 0 . We obtain:

L i = l i (c i ) 1/2(n-2) [u i (y i )] 2/(n-2) = [s i (y i )] 2/(n-2) c 1/2(n-2) i ≥ c 1/(n-2) i c 1/2(n-2) i = c 1/2(n-2) i → +∞. If |z| ≤ L i , then y = exp yi [z/[u i (y i )] 2/(n-2) ] ∈ B(y i , δ i l i ) with δ i = 1 (c i ) 1/2(n-2) and d(y, y i ) < R i -d(y i , x i ), thus, d(y, x i ) < R i and, s i (y) ≤ s i (y i ), we can write, u i (y)[R i -d(y, y i )] (n-2)/2 ≤ u i (y i )(l i ) (n-2)/2 . But, d(y, y i ) ≤ δ i l i , R i > l i and R i -d(y, y i ) ≥ R i -δ i l i > l i -δ i l i = l i (1 -δ i ), we obtain, 0 < v i (z) = u i (y) u i (y i ) ≤ l i l i (1 -δ i ) (n-2)/2 ≤ 2 (n-2)/2 .
We set,

β i = 1 1 -δ i (n-2)/2 , clearly β i → 1.
The function v i is solution of:

-g jk [exp yi (y)]∂ jk v i -∂ k g jk |g| [exp yi (y)]∂ j v i = 1 [u i (y i )] N -1-α v α i + µ [u i (y i )] 4/(n-2) v i +V i v i N -1 ,
By elliptic estimates and Ascoli, Ladyzenskaya theorems, (v i ) i converge uniformely on each compact to the function v solution on R n of, ( 7)

∆v = n(n -2)v N -1 , v(0) = 1, 0 ≤ v ≤ 1 ≤ 2 (n-2)/2 ,
By using maximum principle, we have v > 0 on R n , the result of Caffarelli-Gidas-Spruck ( see [START_REF] Caffarelli | Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth[END_REF]) give, v(y

) = 1 1 + |y| 2 (n-2)/2
. We have the same properties for v i in the previous paper [START_REF] Bahoura | Majorations du type sup u × inf u ≤ c pour l'équation de la courbure scalaire sur un ouvert de R n , n ≥ 3[END_REF].

Polar coordinates and "moving-plane" method

Let, w i (t, θ) = e (n-2)/2 ūi (e t , θ) = e (n-2)t/2 u i o exp yi (e t θ), et a(y i , t, θ) = log J(y i , e t , θ).

We set δ = (n + 2) -(n -2)α 2 .
Lemma 2.3. The function w i is solution of:

(8) -∂ tt w i -∂ t a∂ t w i -L θ (y i , e t ) + cw i = V i w N -1 i + e δt w α i + µe 2t w i , with, c = c(y i , t, θ) = n -2 2 2 + n -2 2 ∂ t a.
Proof:

We write:

∂ t w i = e nt/2 ∂ r ūi + n -2 2 w i , ∂ tt w i = e (n+2)t/2 ∂ rr ūi + n -1 e t ∂ r ūi + n -2 2 2 w i . ∂ t a = e t ∂ r log J(y i , e t , θ), ∂ t a∂ t w i = e (n+2)t/2 [∂ r log J∂ r ūi ] + n -2 2 ∂ t aw i .
the lemma is proved.

Now we have, ∂ t a = ∂ t b 1 b 1 , b 1 (y i , t, θ) = J(y i , e t , θ) > 0,
We can write,

- 1 √ b 1 ∂ tt ( b 1 w i ) -L θ (y i , e t )w i + [c(t) + b -1/2 1 b 2 (t, θ)]w i = V i w N -1 i + e δt w α i + µe 2t w i , where, b 2 (t, θ) = ∂ tt ( √ b 1 ) = 1 2 √ b 1 ∂ tt b 1 - 1 4(b 1 ) 3/2 (∂ t b 1 ) 2 .
Let, wi = b 1 w i , Lemma 2.4. The function wi is solution of:

-∂ tt wi + ∆ g y i ,e t , S n-1 ( wi ) + 2∇ θ ( wi ).∇ θ log( b 1 ) + (c + b -1/2 1 b 2 -c 2 ) wi = (9) = V i 1 b 1 (N -2)/2 wN-1 i + e δt 1 b 1 (α-1)/2 wα i + µe 2t wi ,
where,

c 2 = [ 1 √ b 1 ∆ g y i ,e t , S n-1 ( √ b 1 ) + |∇ θ log( √ b 1 )| 2 ].
Proof:

We have:

-∂ tt wi -b 1 ∆ g y i ,e t , S n-1 w i + (c + b 2 ) wi = V i 1 b 1 (N -2)/2 wN-1 i + +e δt 1 b 1 (α-1)/2 wα i + µe 2t wi , But, ∆ g y i ,e t , S n-1 ( b 1 w i ) = b 1 ∆ g y i ,e t , S n-1 w i -2∇ θ w i .∇ θ b 1 + w i ∆ g y i ,e t , S n-1 ( b 1 ),
and,

∇ θ ( b 1 w i ) = w i ∇ θ b 1 + b 1 ∇ θ w i , we deduce than, b 1 ∆ g y i ,e t , S n-1 w i = ∆ g y i ,e t , S n-1 ( wi ) + 2∇ θ ( wi ).∇ θ log( b 1 ) -c 2 wi , with c 2 = [ 1 √ b 1 ∆ g y i ,e t , S n-1 ( √ b 1 ) + |∇ θ log( √ b 1 )| 2 ].
The lemma is proved.

The "moving-plane" method:

Let ξ i a real number, and suppose ξ i ≤ t. We set t ξi = 2ξ i -t and wξi i (t, θ) = wi (t ξi , θ).

We have,

-∂ tt wξi i +∆ g y i ,e t ξ i S n-1 ( wi )+2∇ θ ( wξi i ).∇ θ log( b 1 ) wξi i +[c(t ξi )+b -1/2 1 (t ξi , .)b 2 (t ξi )-c ξi 2 ] wξi i = = V i 1 b ξi 1 (N -2)/2 ( wξi i ) N -1 + +e δt 1 b 1 (α-1)/2 wα i + µe 2t wi ,
By using the same arguments than in [START_REF] Bahoura | Majorations du type sup u × inf u ≤ c pour l'équation de la courbure scalaire sur un ouvert de R n , n ≥ 3[END_REF], we have:

Proposition 2.5. We have:

1) wi (λ i , θ) -wi (λ i + 4, θ) ≥ k > 0, ∀ θ ∈ S n-1 .
For all β > 0, there exist c β > 0 such that:

2) 1 c β e (n-2)t/2 ≤ wi (λ i + t, θ) ≤ c β e (n-2)t/2 , ∀ t ≤ β, ∀ θ ∈ S n-1 .
We set, Zi = -∂ tt (...) + ∆ g y i ,e t , S n-1 (...) + 2∇ θ (...).∇ θ log( b 1 ) + (c + b

-1/2 1 b 2 -c 2 )(...)
Remark: In the operator Zi , by using the proposition 3, the coeficient c + b

-1/2 1 b 2 -c 2 satisfies: c + b -1/2 1 b 2 -c 2 ≥ k ′ > 0, pour t << 0,
it is fundamental if we want to apply Hopf maximum principle.

We set δ = (n + 2) -(n -2)α 2 .

Goal:

Like in [START_REF] Bahoura | Majorations du type sup u × inf u ≤ c pour l'équation de la courbure scalaire sur un ouvert de R n , n ≥ 3[END_REF], we have elliptic second order operator. Here it is Zi , the goal is to use the "movingplane" method to have a contradiction. For this, we must have:

Zi ( wξi i -wi ) ≤ 0,
-[(c + b -1/2 1 b 2 -c 2 ) ξi -(c + b -1/2 1 b 2 -c 2 )] wξi i + +V ξi i 1 b ξi 1 (N -2)/2 ( wξi i ) N -1 -V i 1 b 1 (N -2)/2 wN-1 i + +e δt ξ i b ξi 1 (1-α)/2 ( wξi i ) α -e δt b
(1-α)/2 1

( wi ) α + µ(e 2t ξ i wξi i -e 2t wi ) ( * * * 1)

Clearly, we have:

Lemma 2.6. b 1 (y i , t, θ) = 1 - 1 3 Ricci yi (θ, θ)e 2t + . . . , R g (e t θ) = R g (y i )+ < ∇R g (y i )|θ > e t + . . . .
According to proposition 1 and lemma 3, Proposition 2.7.

Zi ( wξi

i -wi ) ≤ Ã(e t -e t ξ )( wξi i ) N -1 ) + (1/2)(e δt ξ i -e δt )( wξi i ) α + +C|e 2t -e 2t ξ i | |∇ θ wξi i | + |∇ 2 θ ( wξi i )| + o(1)[ wξi i + ( wξi i ) N -1 + ( wξi i ) α ] + µ wξi i .
Proof:

We use proposition 1, we have:

a(y i , t, θ) = log J(y i , e t , θ) = log b 1 , |∂ t b 1 (t)| + |∂ tt b 1 (t)| + |∂ tt a(t)| ≤ Ce 2t ,
and,

|∂ θj b 1 | + |∂ θj,θ k b 1 | + ∂ t,θj b 1 | + |∂ t,θj,θ k b 1 | ≤ Ce 2t , then, |∂ t b 1 (t ξi ) -∂ t b 1 (t)| ≤ C ′ |e 2t -e 2t ξ i |, on ] -∞, log ǫ 1 ] × S n-1 , ∀ x ∈ B(x 0 , ǫ 1 ) Locally, ∆ g y i ,e t , S n-1 = L θ (y i , e t ) = - 1 |g k (e t , θ)| ∂ θ l [g θ l θ j (e t , θ) |g k (e t , θ)|∂ θ j ].
Thus, in [0, ǫ 1 ] × U k , we have,

A i =   1 |g k | ∂ θ l (g θ l θ j |g k |∂ θ j ) ξi - 1
|g k | ∂ θ l (g θ l θ j |g k |∂ θ j )   ( wξi i ) then, A i = B i + D i with,
B i = gθ l θ j (e t ξ i , θ) -gθ l θ j (e t , θ) ∂ θ l θ j wξi i (t, θ), and, = e (n-2)[(λ-λi)+(ξi-t)]/2 e [(λ-λi)+(ξi-t)] (∂ θj v i )(e [(λ-λi)+(λ-t)] θ) e (n-2)[(λ-λi)+(λ-t)]/2 v i [e (λ-λi)+(λ-t) θ] ≤ Ci , Also: ∂ θj ,θ l w λ i (t, θ) w λ i = e (n-2)[(λ-λi)+(ξi-t)]/2 e 2[(λ-λi)+(ξi-t)] (∂ θj ,θ l v i )(e [(λ-λi)+(λ-t)] θ) e (n-2)[(λ-λi)+(λ-t)]/2 v i [e (λ-λi)+(λ-t) θ] ≤ Ci .

D i = 1 |g k |(e t ξ i , θ) ∂ θ l [g θ l θ j (e
where Ci tends to 0 and does not depend on λ.

We have, We have:

c(y i , t, θ) = n -2 2 2 + n -2 2 ∂ t a + R g e 2t , ( α
w i (2ξ i -t, θ) = w i [(ξ i -t + ξ i -λ i -2) + (λ i + 2)],
Thus, w i (2ξ i -t, θ) = e [(n-2)(ξi-t+ξi-λi-2)]/2 e n-2 v i [θe 2 e (ξi-t)+(ξi-λi-2) ] ≤ 2 (n-2)/2 e n-2 = c.

1 ) b 2 √ b 1 ∂ 2 ) 1 ∆ 2 ∂ tt a, by proposition 1 ,

 1212121 (t, θ) = ∂ tt ( b 1 ) = 1 2 tt b 1 -1 4(b 1 ) 3/2 (∂ t b 1 ) 2 , (α g y i ,e t , S n-1 ( b 1 ) + |∇ θ log( b 1 )| 2 ], (α 3 )Then,∂ t c(y i , t, θ) = (n -2) |∂ t c 2 | + |∂ t b 1 | + |∂ t b 2 | + |∂ t c| ≤ K 1 e 2t .

  if wξi i -wi ≤ 0. θ,e t ξ i -∇ θ,e t )(w ξi i ).∇ θ,e t ξ i log( b ξi 1 ) + 2∇ θ,e t ( wξi i ).∇ θ,e t ξ i [log( b ξi 1 ) -log b 1 ]+ +2∇ θ,e t w ξi i .(∇ θ,e t ξ i -∇ θ,e t ) log b 1

	We write:
	Zi ( wξi i -wi ) = (∆ g y i ,e t ξ i , S n-1	-∆ g y i ,e t , S n-1	)( wξi i )+
	+2(∇

  t ξ i , θ) |g k |(e t ξ i , θ)] -1 |g k |(e t , θ) ∂ θ l [g θ l θ j (e t , θ) |g k |(e t , θ)] ∂ θ j wξi i (t, θ),we deduce,A i ≤ C k |e 2t -e 2t ξ i | |∇ θ wξi i | + |∇ 2 θ ( wξi i )| , We take C = max{C i , 1 ≤ i ≤ q}and if we use ( * * * 1), we obtain proposition 4.

	We have:
	∂ θj w λ i (t, θ)
	w λ i

We set δ = (n + 2) -(n -2)α 2 . The left right side are denoted Z 1 et Z 2 , we can write:

and,

We can write the part with nonlinear terms as:

Because w ξi i ≤ c, we have:

We obtain for t ≤ t 0 < 0:

and, t ξi ≤ t (ξ i ≤ t), we integrate:

(e δt ξ i -e δt ) ≤ δ e (1-δ)t0 (e t ξ i -e t ). Finaly:

-Zi (( wi ) ξi -( wi )) ≤ ( wξi i ) α [-δ c e (1-δ)t0 +A cN-1-α +B](e t -e t ξ i )+( wi ) ξi (µ/2)(e 2t ξ i -e 2t )

We apply proposition 3. We take t i = log √ l i with l i like in proposition 2. The fact

We define ξ i by:

If we use proposition 4 and the similar technics that in [B2] we can deduce by Hopf maximum principle,

It is in contradiction with proposition 2.

Then we have,


Jin Jack Tan
email: jtan@ensta.fr

Cyril Touzé

IMSIA Benjamin Cotté

DOUBLE POLARISATION IN NONLINEAR VIBRATING PIANO STRINGS

The present work studies the double polarisation phenomenon observed in vibrating piano strings. From the experimental viewpoint, it is known that when a string is given an initial displacement in one transverse direction (e.g. hammer excitation in the vertical plane), the second transverse displacement (e.g. in the horizontal plane) is also excited after a few milliseconds and the amplitude can be of similar order to the first transverse displacement. This phenomenon contributes to a characteristic piano sound feature called the "double decay". The purpose of this study is to investigate the role of nonlinearities in inducing double polarisations. The nonlinear vibrations of the strings are studied with a two-degrees-of-freedom (dofs) system extracted from the Kirchhoff-Carrier string equations. The method of multiple scales is used to study the free vibrations of two polarisations having nearly equal eigenfrequencies and thus presenting a 1:1 internal resonance. For an imperfect string with slightly different eigenfrequencies between the two polarisations, it is found out that depending on the energy of the excitation, an uncoupled transverse mode can develop into a coupled mode where there is energy exchange between the two transverse polarisations. The coupled mode is stable and the string oscillates in an elliptic path. Numerical experiments are also carried out, confirming the findings of the analytical approach.

INTRODUCTION

This study on the double polarisation of piano strings is contained in a larger project which aims at developing a physicallybased time-domain piano sound synthesis model. The work was pioneered by Chabassier [START_REF] Chabassier | Modélisation et simulation numérique d'un piano par modèles physiques[END_REF] who proposed a refined complete model of a grand piano [START_REF] Chabassier | Time domain simulation of a piano. Part 1 : model description[END_REF][START_REF] Chabassier | Modeling and simulation of a grand piano[END_REF]. However a few points still need additional developments. For example, one tonal feature that was not observed in the simulated sound is the double decay, i.e. where the temporal signal exhibits two envelopes with different decay rates. It is understood that one of the contributing factors of double decay is the double polarisation of piano strings, as first reported by Weinreich [START_REF] Weinreich | Coupled piano strings[END_REF]. The double polarisation phenomenon is not modeled in Chabassier's work which may explain the absence of double decay in her modeled piano sound. The double polarisation can be caused by various factors, such as nonlinearities of piano strings, asymmetry and complex boundary conditions, coupling to unison strings etc. [START_REF] Hundley | Factors contributing to the multiple rate of piano tone decay[END_REF][START_REF] Suzuki | Acoustics of pianos[END_REF][START_REF] Burred | The acoustics of the Piano[END_REF][START_REF] Karatsovis | Acoustic Features of Piano Sounds[END_REF][START_REF] Chaigne | Acoustics of pianos: physical modeling, simulations and experiments[END_REF].

In this contribution, we focus on the nonlinearity experienced by piano string vibrating at large amplitudes as a possible cause for explaining the birth of double polarisation. More particularly the two fundamental eigenfrequencies of a string with double polarisation are known to have close values, thus showing a 1:1 internal resonance. It is also known from other studies on nonlinear vibrations [START_REF] Touzé | Asymmetric nonlinear forced vibrations of free-edge circular plates. part I: Theory[END_REF][START_REF] Thomas | A harmonic-based method for computing the stability of periodic oscillations of non-linear structural systems[END_REF] that thanks to a 1:1 internal resonance, energy can be exchanged between vibration modes so that even if the motion is initiated along one polarisation only, the nonlinearity can make this motion unstable, so that eventually a coupled vibration arises with the two polarisations involved. The objective of the present contribution is thus to clearly establish if the nonlinearity can be the cause of this coupling, as well as to highlight the main parameters governing the transfer of energy. The 1:1 internal resonance has already been studied in the case of forced vibrations, see e.g. [START_REF] Nayfeh | Nonlinear Oscillations[END_REF][START_REF] Touzé | Asymmetric nonlinear forced vibrations of free-edge circular plates. part I: Theory[END_REF][START_REF] Thomas | A harmonic-based method for computing the stability of periodic oscillations of non-linear structural systems[END_REF]. Here our interest is in the case of free vibration for which only Manevitch and Manevitch present a detailed investigation [START_REF] Manevitch | Free oscillations in conservative and dissipative symmetric cubic two-degree-offreedom systems with closed natural frequencies[END_REF]. The complete problem will hence be fully revisited and applied to the specific case of strings.

The article is organised as follows. First, a system of nonlinear Kirchhoff-Carrier string equations is presented in section 2 and solved via the multiple scales method in section 3. The results are presented and discussed in section 4. The paper continues with a numerical experiment to demonstrate the double polarisations in section 5 before wrapping up with conclusion in section 6.

KIRCHHOFF-CARRIER EQUATION

The Kirchhoff-Carrier equations for a freely vibrating fixedfixed string with two polarised displacements u1 and u2 read [START_REF] Bilbao | Conservative numerical methods for nonlinear strings[END_REF][START_REF] Hélie | Sound synthesis of a nonlinear string using volterra series[END_REF][START_REF] Thomas | A harmonic-based method for computing the stability of periodic oscillations of non-linear structural systems[END_REF]:

ρA ü1 -(T0 + N ) u 1 = 0, ( 1a
)
ρA ü2 -(T0 + N ) u 2 = 0, ( 1b
)
where N is the axial tension created by the large amplitude motions and the coupling with the transverse motion. It reads:

N = EA 2L L 0 u 1 2 + u 2 2 dx. (2)
In this set of equations, L is the length, A the crosse section area, E Young's modulus, T0 the tension and ρ the density. Among others, underlying assumptions are that the inertia of the longitudinal component is negligible, see e.g. [START_REF] Morse | Theoretical Acoustics[END_REF][START_REF] Bilbao | Conservative numerical methods for nonlinear strings[END_REF] for more details. Following [START_REF] Thomas | A harmonic-based method for computing the stability of periodic oscillations of non-linear structural systems[END_REF], the equations can be made nondimensional for a more general treatment. The solutions of the nondimensional equations can be ex-pressed in modal form:

u1(x, t) = K k=1 Φ k (x)p k (t), u2(x, t) = K k=1 Φ k (x)q k (t), (3)
where Φ k (x) are the mode shapes, and {p k (t), q k (t)} k≥1 the modal coordinates. Restraining only the fundamental mode in the truncation for each polarisation (i.e. K = 1) and using the usual projection (Galerkin) technique, on obtains the following two dofs nonlinear system for the two modal coordinates p, q related to each polarisation:

p + ω 2 1 p + ε Γ1p 3 + C1pq 2 = 0, (4a)
q + ω 2 2 q + ε Γ2q 3 + C2qp 2 = 0, ( 4b
)
where ε is a small parameter arising from the nondimensionalisation [START_REF] Thomas | A harmonic-based method for computing the stability of periodic oscillations of non-linear structural systems[END_REF], ε = EAd 2 /T0L 2 , with d the diameter of the string. For a perfect string, the two eigenfrequencies are equal so that ω2 = ω1 = π [START_REF] Thomas | A harmonic-based method for computing the stability of periodic oscillations of non-linear structural systems[END_REF]. The same holds for the nonlinear coefficients, in a perfect case we have

Γ1 = Γ2 = C1 = C2 = π 4 /2.
However, acknowledging that such perfect string does not exist in reality, a detuning parameter σ1 is introduced so that the two eigenfrequencies of the two modes are possibly slightly different, ω2 = ω1 + εσ1.

Following the same lines, the system will be studied with general and different nonlinear coefficients so as to draw the complete picture for the nonlinear string. In a given experimental case, a procedure would be needed for identifying these coefficients.

MULTIPLE SCALES METHOD

The system in equation ( 4) is solved by the multiple scales method which describes the original system to be function of multiple independent time scales. Introducing the "fast" and "slow" time scales:

T0 = t, (6a) T1 = εt, (6b)
p and q can take the following form,

p(t) = p0(T0, T1) + εp1(T0, T1) + O(ε 2 ), (7a)
q(t) = q0(T0, T1) + εq1(T0, T1) + O(ε 2 ), (7b)
where p0 and q0 can be written as:

p0 = A(T1) exp(iω1T0) + c.c, (8a) q0 = B(T1) exp(iω2T0) + c.c, (8b)
where c.c stands for complex conjugates. A and B are unknown complex functions of T1. Substituting equation ( 7) into (4) and grouping all the resonant terms for 1:1 resonance up to order ε, one obtains the solvability conditions. By writing A and B in polar form:

A(T1) = a(T1) exp(iα(T1)), (9a) B(T1) = b(T1) exp(iβ(T1)), (9b)
the solvability conditions can be broken down into a set of four dynamical equations (two for amplitude a and b, two for the phases α and β):

a = - C1 2ω1 ab 2 sin(γ2 -γ1), (10a)
γ 1 = 3Γ1 ω1 a 2 + C1 ω1 b 2 [2 + cos(γ2 -γ1)] , (10b) b
= C2 2ω2 ba 2 sin(γ2 -γ1), (10c)
γ 2 = 3Γ2 ω2 b 2 + C2 ω2 a 2 [2 + cos(γ2 -γ1)] + 2σ1, (10d)
where

γ1 = 2α, γ2 = 2β + 2σ1T1. (11)
The introduction of equation ( 11) is necessary so that the system is made autonomous (not directly dependent on any time scales).

Uncoupled solutions

Let us first consider the uncoupled solutions. They correspond to the motions of the strings that are either in the horizontal direction, or in the vertical one. The first set of uncoupled solutions is found by setting b = 0, in Eqs. [START_REF] Touzé | Asymmetric nonlinear forced vibrations of free-edge circular plates. part I: Theory[END_REF]. The 4-dofs system then degenerates into a two dofs and reads:

a = 0, (12a)
γ 1 = 3Γ1 ω1 a 2 , (12b)
The equations can be easily integrated to give:

a = Ca, (13a)
α = 3Γ1 2ω1 a 2 T1 + αa, (13b)
where Ca and αa are both integration constants independent of T1. Using this result, the solution p0 can be expressed as:

p0 = 2a cos [ωNLt + αa] , (14)
where

ωNL = ω1 1 + ε 3Γ1 2ω 2 1 a 2 . ( 15
)
The first order solution is thus a periodic orbit where only p is involved in the vibration since setting b = 0 implies q = 0. The nonlinear frequency of oscillation ωNL depends on the amplitude a, a usual feature in nonlinear oscillations. For a positive value of Γ1, which is the case for strings, the nonlinearity is of the hardening type, i.e. the oscillation frequency increases with the amplitude.

A similar exercise can be done for the other uncoupled case, i.e. a = 0 and one would obtain a similar result for q0,

q0 = 2b cos [ωNLt + β b ] (16)
where

ωNL = ω2 1 + ε 3Γ2 2ω 2 2 b 2 (17)
One can notice the similarity between the two solutions, coming from the fact that the uncoupled solutions are ruled out by classical Duffing equations. The nonlinearity is completely governed by coefficients Γ1 and Γ2. Figure 1 shows the backbone curve (amplitude-frequency relationship, Eqs. ( 15)-( 17)) for ε = 0.163, a typical value for a string that has been computed using standard values from [START_REF] Chabassier | Modélisation et simulation numérique d'un piano par modèles physiques[END_REF]. The value Γ = 0 is used as an eyeguide to recall that for linear vibrations, the oscillation frequency is independent from the amplitude. The values Γ = π 4 /2, ω = π are the standard values for perfect strings. Finally Γ = π 4 /4 is plotted as an intermediate case between the linear string and the perfect nonlinear string. The figure allows one to estimate the deviation (in radian frequency) of the oscillations with respect to the linear eigenfrequency, as a function of the vibration amplitude.

Coupled solutions

Let us now investigate the coupled solutions of the conservative system given by Eqs. [START_REF] Touzé | Asymmetric nonlinear forced vibrations of free-edge circular plates. part I: Theory[END_REF]. Stationary oscillations occurs at a given energy level so that coupled solutions can be searched for fixed amplitudes, by imposing a = b = 0. This is also in the line of the uncoupled cases where periodic solutions were found for fixed amplitudes and only phase variations, from which the nonlinear amplitude-frequency relationship were derived. From equation (10a) and (10c), it is obvious that for coupled solutions to exist (i.e. a = 0, b = 0), one must have mandatory sin(γ2-γ1) = 0. This implies in particular that cos(γ2-γ1) = ±1. Interestingly, for each case of the possible value of the cosine, the two polarisations p and q are related in a different manner. For cos(γ2 -γ1) = +1, a simple algebra on the system shows that the solutions p and q are related by the following relationship:

q p = ± b a , (18)
while for cos(γ2 -γ1) = -1, p and q are related by:

q 2 4b 2 + p 2 4a 2 = 1. ( 19
)
These particular forms expressed by the coupled solutions has already been commented by Manevicth and Manevitch [START_REF] Manevitch | Free oscillations in conservative and dissipative symmetric cubic two-degree-offreedom systems with closed natural frequencies[END_REF], who refers to them respectively as "normal mode" (NM) (former case, Eq. ( 18)), and "elliptic mode" (EM) (latter case, and Eq. ( 19)). This peculiar relationship expressed for each time between the solution amplitudes leads to a particular motion which is sketched in Figure 2 in the (p, q) (displacements) plane.

The elliptic mode appears particularly interesting for us as it corresponds to the whirling motion observed in piano strings [START_REF] Chaigne | Acoustics of pianos: physical modeling, simulations and experiments[END_REF]. The nonlinear amplitude-frequency relationships defining the backbone curves for coupled solutions can be found out , where the right-hand sides become constants, so that one can retrieve α, β and subsequently the nonlinear frequency ωNL for the coupled modes as:

ωNL = ω1 1 + ε 3Γ1 2ω 2 1 a 2 + rC1 2ω 2 1 b 2 . ( 20
)
In this equation (and in the remainder of the article), the parameter r is such that r = 3 for normal mode, and r = 1 for elliptic modes respectively. As can be seen in equation ( 20), the nonlinear frequency is influenced by the amplitude of both polarisations a and b. The equation forms a surface on the 3D space defined by (a, b, ωNL). However curved line solutions are awaited instead of a whole surface family. Noting that γ2 -γ1 = nπ from the necessary conditions of existence for coupled solutions, one has thus γ 2 -γ 1 = 0. Using such relation, one can obtain:

rC1 ω1 - 3Γ2 ω2 b 2 + 3Γ1 ω1 - rC2 ω2 a 2 = 2σ1. ( 21
)
This relationship expresses the link between the amplitudes a and b for both normal (r = 3) and elliptic (r = 1) coupled modes. The conjuction of Eqs (20) and (21) define the backbone curves for the coupled modes.

Let us now investigate how the coupled solutions can be related to the uncoupled ones. By setting a = 0 or b = 0 in Eq. ( 21), it can be seen that either of the uncoupled modes can branch into the coupled modes provided the following conditions are met: For uncoupled mode a = 0

a 2 ≥ 2σ1 3Γ 1 ω 1 -rC 2 ω 2 , ( 22
)
For uncoupled mode b = 0 b 2 ≥ 2σ1 rC 1 ω 1 -3Γ 2 ω 2 , (23)
These equations provide a limit value, in terms of amplitudes of the uncoupled modes, for which the coupled solutions can develop. Below these limit value, only uncoupled solutions exist. It must be noted that in certain cases, the RHS of both the equations can have two positive values of which the lower one indicates the bifurcation point where the uncoupled solution branches into the coupled solution while the higher one indicates the point where the coupled solution collapse and enter the uncoupled solution [START_REF] Manevitch | Free oscillations in conservative and dissipative symmetric cubic two-degree-offreedom systems with closed natural frequencies[END_REF].

Stability analysis

To obtain the stability of the coupled modes, Eqs [START_REF] Touzé | Asymmetric nonlinear forced vibrations of free-edge circular plates. part I: Theory[END_REF] can be reduced to a 3-DOF system by taking the difference between the two phases or even further to a 2-DOF system as demonstrated by Manevitch and Manevitch [START_REF] Manevitch | Free oscillations in conservative and dissipative symmetric cubic two-degree-offreedom systems with closed natural frequencies[END_REF]. The advantage of using such an approach is that the coupled solutions are then real fixed points of the 3-or 2-dofs systems, so that the usual tools from dynamical system theory can be used for investigating stability. One can thus construct the Jacobian matrix of the corresponding system and solve for the eigenvalues. Using either system will result in the same following stability criteria for the coupled modes:

ω2Γ1 C2ω1 + Γ2ω1 C1ω2 < 2 for normal modes, ( 24a
) ω2Γ1 C2ω1 + Γ2ω1 C1ω2 > 2 3 for elliptics modes. ( 24b
)
It is interesting to see that the stability of the coupled solutions does not depend on the energy but rather on the physical parameters of the system (i.e. eigenfrequencies and nonlinear constants). It also means that regardless of level of excitation, exhibition of stable normal or elliptic modes are pre-determined.

To conclude the analytical study, let us investigate the stability of uncoupled solutions and demonstrate how uncoupled solutions can become unstable in favour of a coupled one. As noted by Manevitch and Manevitch [START_REF] Manevitch | Free oscillations in conservative and dissipative symmetric cubic two-degree-offreedom systems with closed natural frequencies[END_REF], the stability of the uncoupled solutions is determined by the energy of the system. Using the same approach as for the coupled case does unfortunately not give a useful criteria for the stability of uncoupled solutions, which are found to be always unstable. Furthermore, Manevitch and Manevitch do not give an explicit proof of the stability of uncoupled solutions in [START_REF] Manevitch | Free oscillations in conservative and dissipative symmetric cubic two-degree-offreedom systems with closed natural frequencies[END_REF]. The underlying problem is that when setting either b = 0 or a = 0 in the system, the degeneracy is ill-conditioned so that the phase space shrinks down to a two dofs system where the perturbation brought by the other oscillators are not defined and thus cannot be studied. The solution is found from the forced and damped vibration cases by canceling the damping terms and identifying the external excitation frequency to the nonlinear oscillation frequency ωNL. Using the existence conditions derived in [START_REF] Touzé | Asymmetric nonlinear forced vibrations of free-edge circular plates. part I: Theory[END_REF] from a geometric analysis in phase space, one can obtain the following instability regions for the uncoupled solutions: For uncoupled mode a = 0

ωNL = ω2 + εr C2 2ω2 a 2 (25)
For uncoupled mode b = 0

ωNL = ω1 + εr C1 2ω1 b 2 (26)
where r = 1 and r = 3 define the lower and upper bound of the instability region, that are simply related to the connection with either elliptic mode or normal mode. The instability affecting the uncoupled mode a = 0 originates from the eigenfrequency of the other uncoupled mode, ω2 and vice versa. This is because the existence of another uncoupled solution upsets the stability of the original uncoupled solution. It is also interesting to note that the point where the uncoupled solution changes its stability (either in losing or restoring it) is also the point where the uncoupled solution branches into a coupled solution (or a coupled solution leaves and enters the uncoupled solution). Finally, the stability of the uncoupled solutions are seen to depend on the coupling nonlinear coefficients C1 and C2 only, as can be expected. In this section, several case studies are made to demonstrate the properties of the system. Firstly, a perfect string case is considered (i.e. σ1 = 0, C1 = C2 = Γ1 = Γ2). For this case, the two amplitude-frequency relationships for uncoupled solutions given by Eqs. ( 15) and ( 17) are exactly the same so that the backbones in Fig. 3 collapse on the same curve (black and blue curves, only the blue being visible). For the coupled solutions, Eqs. (21) for the normal mode (r = 3) degenerates, indicating that no normal modes are possible in the perfect case. On the other hand, elliptic modes does however exist, and Eqs. (21) shows that they have same amplitude : a = b. Reporting in Eq. (20), one obtains the backbone curve for the coupled, ellipitc modes in the perfect case as: ωNL = ω1(1 + 2ε Γ 1 ω 1 a 2 ). This shows that coupled solutions have a stronger hardening behaviour than uncoupled ondes, as reported in Fig. 3 with a red line. It must be noted that a 2D representation has been chosen for simplicity, by using the same axis for both amplitudes a and b, whereas the whole solutions should be plotted in a 3D space. In such 3D space (a, b, ωNL), uncoupled solutions are restricted respectively to the planes (a, ωNL) and (b, ωNL), whereas the coupled elliptic solutions is in the plane a = b.

RESULTS AND DISCUSSIONS

Finally, examining the different stability conditions found in the previous section for both coupled and uncoupled modes leads to the conclusion that all the solutions reported in Fig. 3 are stable. This leads to the important conclusion that if the motion is excited on a given polarization, then it will stay on it for every time so that no whirling motion of the string would be observed. The coupled elliptic solutions could be observed only if very specific initial conditions are given to the string so that the motion is initiated along this mode.

Let us now turn to the more realistic case of an imperfect string. The most simple imperfection with a slight detuning between the two eigenfrequencies of the polarisation is investigated, by setting σ1 = 1, and keeping all the nonlinear coupling coefficients equal: C1 = C2 = Γ1 = Γ2. The backbone curves are represented in Fig. 4, where now the two uncoupled solutions (black and blue lines) are different and originates respectively from ω1 and ω2 = ω1 + εσ1. Eqs (21) shows once again that in this case normal modes are not likely to exist. Only elliptic modes are possible. The instability condition for uncoupled solutions, provided by Eq. ( 25)-( 26), shows that:

• The uncoupled solutions with b = 0 are always stable.

• On the other hand, uncoupled solutions with a = 0 can be unstable and branch on an elliptic mode, recovering also the criteria given by Eq. ( 22).

The instability region for uncoupled mode with a = 0 given by Eq. ( 25) has two curves. The one for normal mode (r = 3) is not relevant. Finally only the instability line with r = 1 is meaningful, and is represented as a brown line in Fig. 4. The crossing between the sdof uncoupled solution and the instability limit occurs exactly when condition (22) is fulfilled. From this point, uncoupled solutions are unstable, and the branch of elliptic mode solutions (red line) emerges. Once again a 2D representation has been chosen for simplicity, the reader must however keep in mind that the red line is neither in the plane (a, ωNL), nor in (b, ωNL), but really develop in the full 3D space and is not contained within a plane. In particular the crossing between the coupled solution (red line) and uncoupled (b = 0, blue curve) is only a matter of the representation used but does not exist in the full 3D space. The important conclusion that can be drawn from this study is that as soon as an imperfection is taken into account, an unstable region in the backbone curve for uncoupled modes exist. Once the limit amplitude exceeded, uncoupled solutions are unstable so that even though an initial condition is given for that polarisation, an energy transfer will occur so that eventually the system would settle on the stable elliptic mode. Interestingly, one can notice from Eq. (25) that the smallest the detuning σ1, the smallest the amplitude limit for unstable solutions occurs. Hence in order to observe easily this phenomenon, the detuning need not be zero, but should be as small as possible.

NUMERICAL EXPERIMENTS

The main analytical findings of the previous section is now compared to direct numerical simulations of the original system given by Eqs. (4). More particularly, the mots interesting case of the imperfect string is considered, with a slight detuning between the two eigenfrequencies of the two polarisations. The equations of motion are directly integrated numerically in time with a fourth-order Runge-Kutta method. The experiment is carried out with varying level of initial excitation on the first mode only, mimicking the case of a piano string being struck with a hammer of increasing velocities, and hence exciting the string in only one polarization. The values of the case previously studied in Fig. 4 are selected. The analytical study predicts that the uncoupled solution should be stable as long as a < 0.25. A first case is thus studied below this limit value, with an initial condition in displacement only, with p(t = 0) = 0.4 and q(t = 0) = 1e -4. Note that, from Eqs. ( 9)-( 14), a factor 2 between a (resp. b) and the amplitude solution in time for p(t) (resp. q(t)), is present, so that the limit amplitude for stability for p(t = 0) is 0.5. The direct time integration is represented in Fig. 5. One can observe that below the amplitude where instability occurs, the second coordinate q(t) stays at negligible values around 1e -4. p(t), q(t)

Figure 5: Time responses from direct numerical integration of Eqs. (4) and for varying levels of initial displacements p(t = 0). At the top, p(t = 0) = 0.4, below the instability limit; at the bottom, p(t = 0) = 0.8, above the instability limit. Black line correspond to p(t) and blue line to q(t).

For an initial condition above the instability limit, p(t = 0) = 0.8, one can observe that an energy exchange between the two modes occurs and q(t) reaches values up to 0.6. The energy then gets back and forth between the two oscillators, a typical feature of nonlinear conservative oscillations [START_REF] Nayfeh | Nonlinear Oscillations[END_REF]. This numerical experiment confirms the instability of the uncoupled solution. It also shows that, for a given piano string, and for amplitude of excitation that are above a certain threshold that can be predicted, a motion initiated along a single polarisation can be transformed into a coupled whirling motion, as observed experimentally.

CONCLUSION

A detailed study has been conducted to examine the two polarisations of nonlinear vibrating strings, and to show if the geometric nonlinearity due to large-amplitude vibration can be responsible of the coupling between the two polarizations, even though the motion is initiated along a single direction. An analytical study has been conducted using a two-modes approximation for the Kirchhoff-Carrier equation, and then the method of multiple scales. The frequency-amplitude relationships of the two polarisations and the ways whirling motion can take place are identified. The main finding of the anlytical study reveals that an imperfection is needed in order to make uncoupled solutions unstable. When all the coefficients are equal, in the mathematical case of a perfect string, then the periodic solutions are all stable. When a detuning is considered between the two eigenfrequencies, which is always the case in practice, then uncoupled solutions can become unstable and whirling motions of the string can take place even though the motion is initiated along one polarisation only. These results also clearly demonstrate that the geometric nonlinearity can be a potential cause for the whirling motions observed in real piano strings. Future work will consider experimental validations of these findings. The imperfections brought by the specific boundary conditions of the piano string will also be studied in order to assess their role in the appearance of double polarisation.

ACKNOWLEDGEMENT

The research work presented in this paper has been funded by the european commission within the Initial Training Network (ITN) Marie Curie action project BATWOMAN, under the seventh framework program (EC grant agreement no. 605867).

Figure 1 :

 1 Figure 1: Relationship between a and ωNL for various Γ1 at ω1 = π, ε = 0.163.

Figure 2 :

 2 Figure 2: Illustrations of the two modes of coupled solutions. Image not to scale.

Figure 3 :

 3 Figure 3: Amplitude-frequency relationships for the strings, perfect case with ω1 = ω2 = π (no detuning: σ1 = 0), and equal nonlinear coefficients C1 = C2 = Γ1 = Γ2 = π 4 /2. The black blue lines (overlapped with each other) indicate the two uncoupled modes while the red line indicates the elliptic mode. ε = 0.163.

Figure 4 :

 4 Figure 4: Amplitude-frequency relationships for periodic solutions of the nonlinear string, imperfect case with σ1 = 1, all other coefficients being the same as in Fig. 3: C1 = C2 = Γ1 = Γ2 = π 4 /2, ω1 = π, ε = 0.163. Black : uncoupled solution a = 0, blue uncoupled solution b = 0, red coupled elliptic mode. Dashed lines indicates instability. In brown is given the instability limit predicted by Eq. (25).