Pierre Hardouin

Christophe Velours

Charles Bou-Nader

Nadine Assrir

Soumaya Laalami

Harald Putzer

Dominique Durand

Béatrice Golinelli-Pimpaneau

Dissociation of the dimer of the intrinsically disordered domain of RNase Y upon antibody binding

Keywords: Intrinsically Disordered Proteins (IDP) & Aggregates, Protein Structure & Conformation, X-Ray Crystallography, X-Ray & Neutron Scattering & Diffraction, Optical Spectroscopy (CD & UV-VIS)

Although RNase Y acts as the key enzyme initiating mRNA decay in Bacillus subtilis and likely in many other Gram positive bacteria, its three-dimensional structure remains unknown. An antibody, belonging to the rare IgG2b, lx isotype, was raised against a 12-residue conserved peptide from the N-terminal non-catalytic domain of Bacillus subtilis RNase Y (BsRNaseY) that is predicted to be intrinsically disordered. Here we show that this domain can be produced as a stand-alone protein called Nter-BsRNaseY that undergoes conformational changes between monomeric and dimeric forms. Circular dichroism and Size Exclusion Chromatography coupled with Multi-Angle Light Scattering or with Small Angle X-ray Scattering (SEC-MALS and SEC-SAXS) indicate that the Nter-BsRNaseY dimer displays an elongated form and a high content of a-helices, in agreement with the existence of a central coiled-coil structure appended with flexible ends, and that the monomeric state of Nter-BsRNaseY is favored upon binding the Fab (fragment antigen binding) of the antibody. The dissociation constants of the IgG/BsRNaseY, IgG/Nter-BsRNaseY and IgG/peptide complexes indicate that the affinity of the IgG for Nter-BsRNaseY is in the nM range and suggest that the peptide is less accessible in BsRNaseY than in Nter-BsRNaseY. The crystal structure of the Fab in complex with the peptide antigen shows that the peptide adopts an elongated U-shaped conformation, in which the unique hydrophobic residue of the peptide, Leu6, is completely buried. The peptide/Fab complex may mimic the interaction of a microdomain of the Nterminal domain of BsRNaseY with one of its cellular partners within the degradosome complex. Altogether, our results suggest that BsRNaseY may become accessible for protein interaction upon dissociation of its N-terminal domain into the monomeric form.

Introduction

The processing and degradation of mRNA is a key element in the control of prokaryotic gene expression (1)(2)(3). In E. coli, RNase E has a pivotal role in these processes (4) but Bacillus subtilis and many other Gram positive bacteria have a very different set of enzymes directing mRNA metabolism (5)[START_REF] Lehnik-Habrink | RNA degradation in Bacillus subtilis: an interplay of essential endo-and exoribonucleases[END_REF][START_REF] Durand | sRNA and mRNA turnover in Gram-positive bacteria[END_REF]. Indeed, in B. subtilis, a novel endoribonuclease called RNase Y (BsRNaseY) shares extensive functional homologies with RNase E despite the absence of sequence similarity [START_REF] Shahbabian | RNase Y, a novel endoribonuclease, initiates riboswitch turnover in Bacillus subtilis[END_REF]. Although not essential for cell viability, the crucial role of RNase Y in initiating mRNA cleavage is highlighted by deletion experiments of the rny gene, which led to defects in cell growth, cell morphology and hypersensitivity to antibiotics [START_REF] Figaro | Bacillus subtilis mutants with knockouts of the genes encoding ribonucleases RNase Y and RNase J1 are viable, with major defects in cell morphology, sporulation, and competence[END_REF]. The abundance of a large number of transcripts is affected by decreasing RNase Y expression [START_REF] Shahbabian | RNase Y, a novel endoribonuclease, initiates riboswitch turnover in Bacillus subtilis[END_REF][START_REF] Yao | Initiation of decay of Bacillus subtilis rpsO mRNA by endoribonuclease RNase Y[END_REF][START_REF] Bruscella | RNase Y is responsible for uncoupling the expression of translation factor IF3 from that of the ribosomal proteins L35 and L20 in Bacillus subtilis[END_REF][START_REF] Lehnik-Habrink | RNase Y in Bacillus subtilis: a Natively disordered protein that is the functional equivalent of RNase E from Escherichia coli[END_REF][START_REF] Durand | Three essential ribonucleases-RNase Y, J1, and III-control the abundance of a majority of Bacillus subtilis mRNAs[END_REF][START_REF] Laalami | Bacillus subtilis RNase Y activity in vivo analysed by tiling microarrays[END_REF].

Consequently, the half-life of mRNA is increased two-fold when RNase Y is depleted [START_REF] Shahbabian | RNase Y, a novel endoribonuclease, initiates riboswitch turnover in Bacillus subtilis[END_REF].

The three-dimensional structure of BsRNaseY is unknown but a structure prediction analysis revealed its probable architecture [START_REF] Lehnik-Habrink | RNase Y in Bacillus subtilis: a Natively disordered protein that is the functional equivalent of RNase E from Escherichia coli[END_REF](Figure S1 in the Supporting Material): a transmembrane (TM) region, anchoring the protein to the membrane and predicted to span residues 6-24 according to the TMpred server, is followed by an intrinsically disordered domain (IDD) (Figure 1), a catalytic domain with KH (residues 210-280) [START_REF] Grishin | KH domain: one motif, two folds[END_REF] and HD motifs (residues 330-430) [START_REF] Aravind | The HD domain defines a new superfamily of metal-dependent phosphohydrolases[END_REF] and a C-terminal region of unknown function. Although the IDD region is predicted to be highly disordered, as detected by disorder predictor programs such as IUPRED [START_REF] Dosztanyi | IUPred: web server for the prediction of intrinsically unstructured regions of proteins based on estimated energy content[END_REF] (Figure S1A), it is also predicted to be organized mainly as a-helices by secondary structure prediction programs and to adopt a flexible coiled-coil-like structure [START_REF] Lehnik-Habrink | RNase Y in Bacillus subtilis: a Natively disordered protein that is the functional equivalent of RNase E from Escherichia coli[END_REF]. The IDD is hypothesized to adopt various conformations to enable the protein to interact with several cellular partners (RNA or proteins), a typical behavior for intrinsically disordered proteins [START_REF] Tompa | On the supertertiary structure of proteins[END_REF][START_REF] Liu | Advantages of proteins being disordered[END_REF][START_REF] Varadi | Computational approaches for inferring the functions of intrinsically disordered proteins[END_REF][START_REF] Basu | A structural perspective of RNA recognition by intrinsically disordered proteins[END_REF][START_REF] Van Der Lee | Classification of intrinsically disordered regions and proteins[END_REF]. Only a few in vitro studies of BsRNaseY have been reported, including activity tests [START_REF] Shahbabian | RNase Y, a novel endoribonuclease, initiates riboswitch turnover in Bacillus subtilis[END_REF] and ultracentrifugation experiments that showed that the main oligomeric form is a dimer [START_REF] Lehnik-Habrink | RNase Y in Bacillus subtilis: a Natively disordered protein that is the functional equivalent of RNase E from Escherichia coli[END_REF]. To get more information about the three-dimensional structure of RNase Y, we raised an antibody against a small region of the IDD. Not only could this antibody stabilize the conformation of RNase Y, which would favor crystallization of the complex, but it could also serve to get insights into the structure/function of the IDD.

We report here the characterization of the interaction between BsRNaseY and an antibody named RY79-90 raised against a 12-residue peptide from the IDD (Figure 1) using ELISA (Enzyme Linked ImmunoSorbent Assay) and Size Exclusion Chromatography coupled with Multi-angle light scattering (SEC-MALS). Our results show that the antibody binds BsRNaseY with high affinity but is more specific for its N-terminal domain. We produced this domain as a stand-alone protein called Nter-BsRNaseY and characterized its structure by circular dichroism (CD) spectroscopy and Size Exclusion Chromatography coupled with Small Angle X-ray Scattering (SEC-SAXS). Nter-BsRNaseY, which displays a fast equilibrium between monomeric and dimeric forms, is bound as a monomer when complexed with the fragment antigen binding (Fab) of RY79-90. We also solved the crystal structure of Fab RY79-90 in complex with the peptide used to raise the antibody. We show that the peptide undergoes an induced folding upon binding antibody RY79-90 and discuss how the detailed interactions of the peptide/Fab complex observed in the crystal structure could mimic the binding of a microdomain of the N-terminal domain of BsRNaseY and a cellular partner.

MATERIALS & METHODS

Antibody selection

The sequence of B. subtilis RNase Y was analyzed by the Abmart Company (Shanghai, China) using the SEAL TM algorithm designed for identifying protein fragments most susceptible to yield monoclonal antibodies. Six antigen peptides, five of them contained within the IDD of B. subtilis RNase Y, were selected for immunization. All antibodies were tested for their sensitivity and specificity to detect RNase Y in total B. subtilis cell extracts or the purified protein by Western blotting as described [START_REF] Jamalli | Control of expression of the RNases J1 and J2 in Bacillus subtilis[END_REF]. Briefly, cell extracts, prepared by ultrasonic disruption as reported [START_REF] Putzer | Independent genes for two threonyl-tRNA synthetases in Bacillus subtilis[END_REF], or BsRNAseY purified by Ni-NTA chromatography were separated on a 10% SDS-PAGE gel. Monoclonal antibodies were used at a final concentration of 2 µg/ml and the signal was quantified by chemiluminescence (ECL GE Healthcare) using a horseradish peroxidase (HRP) secondary antibody conjugate (GE Healthcare). Finally, peptide 79-90 (ERRNELQKQENR) was chosen for large scale production of monoclonal antibody in mice ascitic fluid.

Production, purification and characterization of the Fab fragment

The IgG (IgG2b, lx) was produced from mice ascitic fluid, purified by Protein A affinity chromatography by Abmart Company and stored in 50 mM Tris, 100 mM Glycine pH 9, 0.5% ProClin 300 (Sigma-Aldrich). The Fab was generated by papain digestion of the antibody at 37°C in 20 mM sodium phosphate, 274 mM NaCl, 5.4 mM KCl pH 7.4, 1.25 mM EDTA, 1.25 mM 2-mercaptoethanol using a 0.5 % papain: antibody ratio (w/w) for 5 hours. Undigested IgG and Fc fragment were removed by HiTrapQ anion exchange chromatography. The Fab was dialyzed against 200 mM NaCl, 5% glycerol and concentrated to 6.5 mg/ml.

Fab sequence determination

Sequencing of the variable heavy chain was performed by ProMab Biotechnologies Inc, Richmond, CA (USA). Total RNA was extracted from 100,000 hybridoma cells. Using PCR methods and a mixture of constant region sequences, the scFv region was amplified for cloning and sequencing. The sequence of the variable light chain could not be determined by this method, probably because of its very rare Vlx isotype, as identified during structure solving by X-ray crystallography (see below). Mass spectrometry analysis of the lower mass protein band of Fab after migration on a SDS-PAGE gel under reducing conditions confirmed that the amino acid sequences of the Fab RY79-90 light chain and Fab 13F6-1-2 (25) differ only in 3 amino acids.

Expression of Nter-BsRNaseY and BsRNaseY

The coding sequence of the rny gene from K26 to K520 (thus lacking the N-terminal transmembrane domain) was PCR-amplified from chromosomal DNA of strain SSB1002, a wild-type B. subtilis laboratory stock strain derived from strain 168, using oligonucleotides HP1182 (5'TGATTCACTCATATGAAAACCATTGCCGAAGCGAAAATTGCG3') and

HP1116

(5'AGCTAGGATCCTTAGTGATGATGATGATGGTGTTTTGCATACTCTACGGCTCGA GTC3'). The 1.5 kb PCR fragment bearing the sequence coding for the hexahistidine tag at its 3' end was inserted as an NdeI-BamHI fragment into the T7 expression vector pKYB1 plasmid (New England Biolabs) cleaved with NdeI and BglII. Recombinant plasmids from E. coli JM109 were then transformed into E. coli BL21(DE3) star codon⁺ competent cells. One colony was used to inoculate 200 ml of Luria Broth medium supplemented with kanamycine (25 µg/ml) and chloramphenicol (30 µg/ml).

For the Nter-BsRNaseY construct, the sequence corresponding to amino acids V24 to N192 of BsRNaseY appended with a C-terminal hexahistidine tag was PCR-amplified from chromosomal DNA of strain SSB1002 using the Sequence Ligation Independent Cloning method [START_REF] Li | SLIC: a method for sequence-and ligationindependent cloning[END_REF]. The PCR fragment was inserted as an NdeI-BamHI fragment into the T7 expression vector pET22b (New England Biolabs). The recombinant plasmid was then transformed into E. coli BL21(DE3) star codon⁺ competent cells. One colony was used to inoculate 200 ml of Luria Broth medium supplemented with ampicillin (50 µg/ml) and chloramphenicol (30 µg/ml).

Precultures (0.5% v/v) grown overnight at 37 • C at 180 rpm were used to inoculate 2 liters of LB medium supplemented with the same antibiotics. Cultures were grown at 37°C to an OD 600 of 0.8. Expression was then induced by addition of isopropyl-b-D-thiogalactopyranoside (IPTG) to a final concentration of 0.1 mM and incubation was extended overnight at 23°C. After centrifugation for 10 min at 5000 rpm and 4°C, cells were stored at -80°C or used immediately.

Purification of Nter-BsRNaseY and BsRNaseY

Cells were resuspended in 1.5 ml/g of 20 mM HEPES pH 7.5, 500 mM NaCl, 10% glycerol (buffer A) containing 20 µl/ml cOmplete ™ Protease Inhibitor Cocktail (Roche), 10 mM βmercaptoethanol, 2 µg/ml of RNase A (Thermo Fisher), 1 mg/ml lysozyme, 0.25% Tween 20, and disrupted by sonication. Cells debris were removed by ultracentrifugation at 100,000g for 1h at 4°C. The supernatant was loaded onto a 15 mL Ni-NTA Superflow column (Qiagen) equilibrated in buffer A containing 20 mM imidazole, 0.25% Tween 20. The column was washed with 70 mM imidazole and the proteins were eluted with 200 mM imidazole in buffer A. Fractions containing the proteins were then loaded at 2.5 mg/ml (BsRNaseY) or 4.5 mg/ml (Nter-BsRNaseY) on a Superdex 200 26/60 PG column (GE Healthcare) in buffer A. Protein elution was followed by the optical density at 230 nm because of the rarity or absence of aromatic residues in the BsRNaseY and NterBsRNaseY sequences. After concentration to 5 mg/ml using Amicon concentrators (30 kD cutoff, Milllipore), proteins were aliquoted, frozen in liquid nitrogen and stored at -80°C.

Gel shift assay for monitoring interaction between Fab RY79-90 and BsRNaseY

Interaction between Fab RY79-90 and BsRNaseY was assessed on a 4% native PAGE gel.

BsRNaseY at fixed concentration (5 µM) was mixed with Fab at concentrations ranging from 0.1 µM to 5 µM in 20 mM HEPES pH 8.0, 100 mM NaCl, 10% glycerol. After 30 min incubation at 20°C, the mixture was loaded on a native gel (prepared with Tris-glycine pH 8.9 and prerun at 4°C in TBE buffer 0.5x at 100 V for 20 min). The migration was performed at 4°C in TBE 0.5x at 150 V for 150 min.

Determination of the dissociation constants for the IgG/BsRNaseY, IgG/Nter-BsRNaseY, and IgG/peptide complexes The K d s were estimated by three independent measurements using a competitive ELISA assay, as described ( 27) (See Supplementary Methods).

Determination of the absolute molar masses of Fab, Nter-BsRNaseY and the Fab/Nter-

BsRNaseY complex by SEC-MALS

SEC-MALS experiments were performed using an HPLC system (Shimadzu) equipped with static light scattering (mini DAWN TREOS, Wyatt Technology), refractive index (Optilab T-rEX, Wyatt Technology) and UV detectors (SPD-M20A, Shimadzu). 100 µl of Nter-BsRNaseY, Fab RY79-90 and the Nter-BsRNaseY/Fab RY79-90 mixture were loaded onto a Superdex 200 10/300 GL increase column (GE Healthcare) equilibrated in 40 mM MES pH 6.8, 150 mM NaCl buffer at a flow rate of 0.5 ml.min -1 . A 1:2 ratio of Nter-BsRNaseY/Fab RY79-90 was chosen to ensure that all Nter-BsRNaseY was bound in the complex, given that free Nter-BsRNaseY eluted near the Nter-BsRNaseY/Fab complex, which could affect the molar mass determination of the complex. The absolute molar masses of the proteins were calculated using version 6.1 of the ASTRA software (Wyatt Technology) and a refractive index increment (dn/dc) value of 0.183 ml.g -1 . The hydrodynamic radii were determined using the QELS module (Wyatt technology).

Determination of the dissociation constant of the Nter-BsRNaseY dimer/monomer equilibrium using SEC-MALS

The dissociation constant (K d ) of the Nter-BsRNaseY dimer was calculated using SEC-MALS.

Nter-BsRNaseY (100 µl) at a concentration ranging from 30 to 945 µM was loaded onto a Superdex 200 10/300GL increase column equilibrated with 40 mM MES buffer pH 6.8, 200 mM NaCl. Data were fitted by nonlinear regression to the equation:

! " = M % ( 8 M ( + K + --+ . + 8 M ( -+ 4 M ( )
as described previously [START_REF] Proença | A two-component, multimeric endolysin encoded by a single gene[END_REF][START_REF] Loiseau | The UbiK protein is an accessory factor necessary for bacterial ubiquinone (UQ) biosynthesis and forms a complex with the UQ biogenesis factor UbiJ[END_REF]. M w is the weight-averaged molar mass measured, M N is the theoretical molar mass of Nter-BsRNaseY (20,665 Da), [M] T is the molar concentration of the monomer determined by the refractive index detector. The K d for the dimer-monomer equilibrium was calculated using the program CurveExpert Professional 2.6.3 (Hyams Development).

Circular dichroism (CD) spectroscopy

The CD spectrum of Nter-BsRNaseY (485 µM in 40 mM MES pH 6.8, 200 mM NaCl) was recorded at 25°C in a 0.01 mm path-length quartz cuve (121.QS, Hellma) over the 185-260 nm spectral range using a JASCO J-810 CD spectropolarimeter equipped with a PFD 423S/L Peltier temperature controller. The spectrum consisted of an average of five consecutive scans acquired at a speed of 50 nm/min, with a time response of 1s for a 1.0 nm band-width. The spectrum was corrected by subtracting the contribution of the buffer measured under the same conditions. Data recorded in mdeg were converted as Δε (M -1 .cm -1 ). Secondary structure estimates were derived from the normalized spectra using the CDSSTR, SELCON3, CONTIN/LL algorithms of the DICHROWEB server or K2D3 (30 , 31).

Determination of the stoichiometry of the Nter-BsRNaseY/Fab RY79-90 complex by SEC-

MALS

Protein stoichiometry analysis of the Nter-BsRNaseY/Fab RY79-90 complex (at 1:2 ratio) was performed by SEC-MALS using the 'UV extinction from RI peak' method [START_REF] Proença | A two-component, multimeric endolysin encoded by a single gene[END_REF] and the protein conjugate method (29) (See Supplementary Methods). In the first method, the UV 280 extinction coefficients of the peaks are extracted from the SEC-MALS data and compared to the predicted extinction coefficients for the complex at different ratios, which are calculated from the amino acid sequence.

SEC-SAXS data collection and analysis

SAXS data on Nter-BsRNaseY were collected on beam line SWING of the SOLEIL synchrotron on a sample eluting from an analytical SEC-column being directly loaded into the SAXS flow-through capillary cell [START_REF] David | Combined sampler robot and high-performance liquid chromatography: a fully automated system for biological small-angle X-ray scattering experiments at the Synchrotron SOLEIL SWING beamline[END_REF]. Nter-BsRNaseY was injected at 20 mg/ml onto a BioSEC3 column (Agilent) equilibrated in 20 mM HEPES pH 7.5, 150 mM NaCl, 5% glycerol.

The flow rate was 200 µl/min, the frame duration 0.75 s and the dead time between frames 0.01 s. The scattering of the elution buffer before void volume was recorded and subtracted from all protein scattering curves. The scattered intensities were displayed on an absolute scale using the scattering by water. The analysis of the full set of SAXS curves measured during the elution was first performed using FOXTROT [START_REF] David | Combined sampler robot and high-performance liquid chromatography: a fully automated system for biological small-angle X-ray scattering experiments at the Synchrotron SOLEIL SWING beamline[END_REF], the SWING in-house software, and then the US-SOMO HPLC module [START_REF] Brookes | US-SOMO HPLC-SAXS module: dealing with capillary fouling and extraction of pure component patterns from poorly resolved SEC-SAXS data[END_REF]. This program provides for each frame the values of the scattering intensity I(0) and of the radius of gyration R g by applying the Guinier analysis together with a calculation of the approximate molar mass using the Rambo & Tainer approach [START_REF] Rambo | Accurate assessment of mass, models and resolution by small-angle scattering[END_REF]. Identical frames under the elution peak were finally selected using the CorMap program [START_REF] Franke | Correlation Map, a goodness-offit test for one-dimensional X-ray scattering spectra[END_REF] and averaged for further analysis. The distance distribution function P(r) was calculated from the averaged scattering curve using the GNOM program from the ATSAS suite [START_REF] Svergun | Determination of the regularization parameter in indirecttransform methods using perceptual criteria[END_REF].

SAXS modelling

In order to obtain an atomic representation of the protein, we submitted the sequence of Nter-BsRNaseY to the fully automated protein structure homology-modelling server SWISS-MODEL (https://swissmodel.expasy.org/). Interestingly, the program provided a dimer model with a long coiled-coil between residues 61 and 134. The missing N-and C-terminal parts of each monomer were subsequently modeled using DADIMODO (37) [START_REF] Mareuil | A simple genetic algorithm for the optimization of multidomain protein homology models driven by NMR residual dipolar coupling and small angle X-ray scattering data[END_REF], which kept rigid the coiled-coil dimer and refined the position of the terminal parts against SAXS data. This modelling, which was repeated ten times, showed that the N-and C-terminal parts of each monomer are very flexible and that the dimer of Nter-BsRNaseY is able to adopt a great variety of conformations. In these conditions, an approach in terms of conformation ensembles using for example the EOM program (39) would give little additional information and was not explored.

Crystallization of the Fab79-90/peptide complex

Fab79-90 (5 mg/ml) was mixed with a slight excess of the peptide ligand (1 Fab:1.2 peptide ratio) to ensure full occupancy of the Fab combining site. Crystals were grown in 16 % PEG 4000, 0.2 M MgCl 2 , 0.1 M Tris-HCl pH 8.5, then flash cooled in a nitrogen stream at 100K in the same solution containing 20% glycerol.

X-Ray Data Collection and Structure Determination

Diffraction data were recorded on PX1 beamline at the SOLEIL synchrotron. Data were processed with XDS (40) (Table S1). Molecular replacement with PHASER (41) using various antibodies of known structure identified antibody Fab13F6-1-2 (PDB code 2QHR) [START_REF] Lee | Complex of a protective antibody with its Ebola virus GP peptide epitope: unusual features of a V lambda x light chain[END_REF], which belongs to the IgG2a, lx class, as the best model for both the Fv and the CL-CH1 dimer. The atomic model was refined by alternating cycles of model reconstruction with COOT (42) and refinement with BUSTER [START_REF] Bricogne | BUSTER version 2.10[END_REF]. The final refinement statistics are given in Table S1. Figures

were drawn with PYMOL. ThrL51 in CDRL2, which lies in the disallowed region of the Ramachandran plot, belongs to a g turn, as commonly observed in all antibody structures.

RESULTS AND DISCUSSION

RY79-90 is an IgG2b, lx antibody induced against a 12-residue conserved peptide of the

RNase Y IDD

After having identified peptides from BsRNaseY most susceptible to be immunogen, six mouse monoclonal antibodies were produced. Among them, one antibody induced against the 12residue peptide (ERRNELQKQENR) covering residues 79 to 90 was selected for its high affinity for BsRNaseY, as judged by Western-blot analysis (Fig. 1B), and the fact that it belongs to a conserved region of the IDD of BsRNaseY (Fig. 1A). The antibody, named RY79-90, has a heavy chain belonging to the IgG2b isotype and utilizes a rare Vlx light chain that is observed in only 0.5% of antibody sequences (44) (Figure S2). The amino acid sequences of Vlx chains are only 30-33% identical to that of other Vl or Vk light chains. They generally display very few somatic mutations [START_REF] Lee | Complex of a protective antibody with its Ebola virus GP peptide epitope: unusual features of a V lambda x light chain[END_REF][START_REF] Maier | A V lambda x-bearing monoclonal antibody with similar specificity and sequence to encephalitogenic T cell receptors[END_REF][START_REF] Li | The structure of a polyQ-anti-polyQ complex reveals binding according to a linear lattice model[END_REF] and no junctional diversity caused by the rearrangement of the Vlx and Jl2 segments [START_REF] Sanchez | Mouse V lambda x gene sequence generates no junctional diversity and is conserved in mammalian species[END_REF]. A characteristic of the Vlx class is that the complementary determining region (CDR) L3, which is entirely encoded by the Vlx gene fragment, is four amino acids longer than in all other antibody light chains (Figure S2). It has been suggested

that this unusual light chain confers specificity to the antibodies that contain it [START_REF] Galin | Murine V lambda x and V lambda x-containing antibodies bind human myelin basic protein[END_REF]. As shown below, CDRL3 is a major determinant of antigen recognition. To our knowledge, there are only two known crystal structures of Fab or Fv (variable fragment) bearing a lx light chain:

Fab 13F6-1-2 (PDB code 2QHR) [START_REF] Lee | Complex of a protective antibody with its Ebola virus GP peptide epitope: unusual features of a V lambda x light chain[END_REF] and Fv MW1 (PDB code 2OTU and 2OTW)(46), both solved in complex with their peptide antigen.

Sequence comparison of the light chains of Fab RY79-90, Fab 13F6-1-2 and the Vlx germline gene indicates that Fab RY79-90 underwent almost no maturation affinity, as the only mutated amino acid was Glu at position L80 instead of Pro in the germline sequence (Figure S2).

Interestingly, the heavy chain isotype distribution of Vlx-bearing antibodies in normal mouse serum, upon immunization with myelin basic protein, showed a bias towards g2a and 2b

isotypes [START_REF] Galin | Murine V lambda x and V lambda x-containing antibodies bind human myelin basic protein[END_REF]. The heavy chain of Fab RY79-90 belongs also to the IgG2b isotype. VH comes from a combination of the V segment of the Igh-VGAM3.8 VH9 family, a D segment that could not be identified because it is too short and the JH3 segment (Figure S2). There are 10 amino acid mutations compared to the germline gene in the V segment, including one in CDRH1 and two in CDRH2. CDRH3 is exceptionally short, with only three amino acids.

IgG RY79-90 has a higher affinity for the inducing peptide and Nter-BsRNaseY than for

BsRNaseY

In order to find out if the conformation of the peptide in interaction with the antibody is representative of its conformation within BsRNaseY in solution, the affinity of IgG RY79-90 for the inducing peptide, Nter-BsRNaseY and BsRNaseY, both lacking the N-terminal transmembrane a helix, was determined by a competitive ELISA test [START_REF] Friguet | Measurements of the true affinity constant in solution of antigen-antibody complexes by enzyme-linked immunosorbent assay[END_REF]. The dissociation constants for the complexes of the IgG with BsRNaseY, Nter-BsRNaseY and the inducing peptide were 122.6 ± 6.6 nM, 8.9 ± 0.75 nM and 0.36 ± 0.04 nM, respectively (Figure S3).

Thus, the affinity of IgG RY79-90 for the peptide is 14 times stronger than for Nter-BsRNase Y and 350 times stronger than for BsRNaseY. The peptide, which seems less accessible in BsRNaseY than in Nter-BsRNaseY, appears as a main determinant of the interaction of BsRNaseY with the antibody. To further study structural aspects of the antibody/Nter-BsRNaseY interaction, a Fab fragment of IgG RY79-90 was generated by papain digestion and purified. The interaction of Fab RY79-90 and BsRNase Y was assessed on a native PAGE gel (Figure 1C).

Nter-BsRNaseY displays a fast equilibrium between monomeric and dimeric forms

The retention volume of a macromolecule on a SEC column depends on its hydrodynamic radius. Thus, SEC alone allows an estimation of its molar mass only if it is globular. Yet, combining SEC with multi-angle light scattering, UV detector and refractometer (SEC-MALS) provides a useful tool for accurate determination of absolute molar mass and oligomeric state of all proteins in native conditions [START_REF] Folta-Stogniew | Oligomeric states of proteins determined by sizeexclusion chromatography coupled with light scattering, absorbance, and refractive index detectors[END_REF][START_REF] Sahin | Size-exclusion chromatography with multi-angle light scattering for elucidating protein aggregation mechanisms[END_REF]. Moreover, the QELS detector allows the calculation of the hydrodynamic radius independently of the retention volume determined by SEC. In this technique, which does not rely on the calibration of standards and assumptions of the globular state of the protein, the on-line measurement of the intensity of the Rayleigh scattering as a function of the angle as well as the differential refractive index of the eluting peak in SEC allow molar mass calculation at any point in the elution chromatogram and characterization of each population in a mixed sample [START_REF] Trathnigg | Determination of MWD and chemical composition of polymers by chromatographic techniques[END_REF]. Therefore, SEC-MALS was used to analyze Nter-BsRNaseY at different concentrations in solution and determine its absolute molar mass (Figure 2A). The chromatogram shows a single specie eluting around 12.5 ml on a Superdex 200 gel filtration column (Figure 2A &2B, blue curve) whose absolute molar mass depends on protein concentration and is intermediate between the expected value for a monomer (theoretical mass of 20.7 kDa) and a dimer (theoretical mass of 41.3 kDa). This feature indicates that the monomer and another oligomeric form of Nter-BsRNaseY are in a fast equilibrium, as shown in the case of other proteins (51 , 52). MALDI-MS analysis confirmed that the peak around 12.5 ml corresponds to Nter-BsRNaseY and not to a protein contaminant (data not shown). At high concentrations, the molar mass of Nter-BsRNaseY is 40.0 kDa, very close to the theoretical molar mass of the dimer (Table S2). The dissociation constant for the homodimer of Nter-BsRNaseY was then determined by SEC-MALS to be 1.3 µM by monitoring the weight-averaged molar mass as a function of protein concentration (Figure 2A, insert). The existence of a dimeric form of Nter-BsRNaseY is in agreement with bacterial-2-hybrid experiments with the isolated domains, which suggested that the N-terminal domain of BsRNaseY is a main contributor to the dimerization of BsRNaseY [START_REF] Lehnik-Habrink | RNase Y in Bacillus subtilis: a Natively disordered protein that is the functional equivalent of RNase E from Escherichia coli[END_REF].

The Nter-BsRNaseY dimer is probably organized as a central coiled coil appended with disordered regions

The secondary structure of Nter-BsRNaseY was analyzed by far-UV circular dichroism (CD) spectroscopy (Figure 3A). The CD spectrum clearly shows that the dimer at 485 µM is not completely disordered but contains a high content of a-helices. A qualitative analysis using the CDSSTR, CONTIN/LL and K2D3 algorithms indicates a content of 50-60% of a-helices and 30-40% disordered regions. Similar CD spectra and a-helices content were obtained for a concentration of Nter-BsRNaseY between 3 and 485 µM (data not shown), in agreement with the K d value of 1.3 µM for the dimer-monomer equilibrium determined by SEC-MALS, which indicates that the secondary structure of Nter-BsRNaseY is unchanged at all these concentrations. Thus, the CD analysis indicates that a large region of the Nter-BsRNaseY dimer is probably organized as a coiled-coil structure, as predicted [START_REF] Lehnik-Habrink | RNase Y in Bacillus subtilis: a Natively disordered protein that is the functional equivalent of RNase E from Escherichia coli[END_REF].

In-line SEC-SAXS [START_REF] Carter | Prion Protein-Antibody Complexes Characterized by Chromatography-Coupled Small-Angle X-Ray Scattering[END_REF][START_REF] Jeffries | Preparing monodisperse macromolecular samples for successful biological small-angle X-ray and neutron-scattering experiments[END_REF] was used to further analyze the structure of Nter-BsRNaseY in solution. In a SEC-SAXS experiment, SAXS measurements are continuously performed and analyzed immediately after SEC separation while the solution is eluted (see Materials and Methods). Figure 3B shows the evolution during elution of the gyration radius R g and of the molar mass extracted from each SAXS curve. A large plateau is observed for both parameters.

The molar mass keeps constant at a value of about 40 kDa before decreasing at the end of the elution. This is in perfect agreement with the presence of dimers that dissociate at low concentration.

The resulting SAXS curve averaged on the plateau gives values for R g and the maximal extension D max of 61.3 and 215 Å, respectively, much higher than those expected for a compact protein of n = 2 x 176 residues (R g ~ 3n 1/3 = 21 Å and D max ~2.6Rg = 55Å). Therefore, the D max and R g values indicate a highly elongated shape for Nter-BsRNaseY. Moreover, the distance distribution function P(r) and the scattering curve, shown in dimensionless Kratky representation, are strongly different from the dimensionless curves characteristic of a fully unstructured protein (such as protein II-1ng about the same size ( 55)) (Figure S4). Coupling this qualitative information with the strong propensity of Nter-BsRNaseY to form coiled-coil and with the high a-helices content detected by circular dichroism spectrsocopy, we constructed atomic models (see Materials and Methods) that integrate all experimental results and that provide in particular a very good fit to the experimental SAXS curve (Figure 3C).

These models, which display a central coiled-coil appended with flexible ends, are only representative but suggest the possible conformation of free Nter-BsRNaseY in solution.

One Fab binds to one Nter-BsRNaseY monomer

The SEC-MALS profile of Fab RY-79-90 displays two main peaks (Table S2 & Figure 2B, black curve). The major species, eluting at 14.7 ml, exhibits a molar mass of 46.0 kDa corresponding to the theoretical mass of the Fab (46.1 kDa). The Nter-BsRNaseY/Fab mixture leads to the appearance of a new peak eluting at 12.1 ml with a molar mass of 69.7 kDa (Table S2 & Figure 2B, red curve). SEC-MALS data analysis using the "UV extinction from RI peak" method in the ASTRA software (28) indicated an experimental extinction coefficient for the Nter-BsRNaseY/Fab RY79-90 complex of 1.09 ml.mg -1 cm -1 , close to the theoretical extinction coefficient for a complex with a stoichiometry of 1 Fab per one Nter-BsRNaseY monomer (1.103 ml.mg -1 cm -1 ). Moreover, given that the extinction coefficients for Nter-BsRNaseY and Fab RY79-90 are significantly different, the protein conjugate method could also be used to determine the diffusion signal corresponding to each protein [START_REF] Loiseau | The UbiK protein is an accessory factor necessary for bacterial ubiquinone (UQ) biosynthesis and forms a complex with the UQ biogenesis factor UbiJ[END_REF]. The deduced molecular weights of Fab and Nter-BsRNaseY in the complex were 48.2 kDa and 22.0 kDa, respectively, confirming the 1:1 stoichiometry (total mass of 70.2 kDa). Thus, the two calculations are in agreement and indicate that the Fab stabilizes the monomeric state of Nter-BsRNaseY, as a result of steric hindrance or because it competes with the dimerization interface of Nter-BsRNaseY. In fact, the dissociation of the Nter-BsRNaseY dimer upon Fab binding results from the higher affinity of the Nter-BsRNaseY monomer for IgG RY79-90 (nM) than for himself (µM).

Overview of the Fab RY79-90 structure

Co-crystallization trials of Fab RY79-90 with Nter-BsRNaseY remained unsuccessful. Yet, the crystal structure of Fab RY79-90 in complex with the inducing peptide was solved at 2.6 Å resolution by molecular replacement using Fab 13F6-1-2 as model (Table S1). It contains two molecules in the asymmetric unit (Figure S5A). The Fab displays a large elbow angle of 138.5° and 133.1° between the Fv fragment and the CL/CH dimer, for molecules LH and AB, respectively, in agreement with the light chain belonging to the l-class [START_REF] Stanfield | Antibody elbow angles are influenced by their light chain class[END_REF]. The peptidebinding pocket of Fab RY79-90 is a shallow cleft, roughly 50 Å by 44 Å wide and 16 Å deep, at the upper part of the VL/VH interface (Figure S5B and Figure 4A). The peptide exhibits a standard type 1 b-turn implicating an intramolecular hydrogen bond between the Asn4 O and Gln7 N atoms (Figure 4B), and thus adopts a U-shape extended conformation. The Fab surface area buried upon peptide binding is 50.1 and 69.3 Å 2 for molecules LH and AB, respectively.

All CDRs except L2 contribute to the combining site (Figure 4A).

Antibody-peptide interactions

9 out of the 12 residues of the peptide are in contact with the antibody and display clear electron density, whereas electron density was ill-defined for Arg3, Gln9 and Arg12 that were oriented towards the solvent (Figure 4C). The high specificity and binding affinity of the antibody/peptide interaction is achieved through numerous side-chain interactions.

Interestingly, despite its very short length, CDRH3 is involved in several main interactions with the peptide, including a bidentate ionic interaction between ArgH94 (CDRH3) and Glu5. Two additional ionic interactions occur between the side chains of AspH52 (CDRH2) and Arg2, and GluL34 (CDRL1) and Lys8. Hydrogen bonds are formed between the side chains of ThrH96 (CDRH3) and Glu5, AsnH35 (CDRH1) and Asn4, ThrL93 (CDRL3) and Gln7, ThrL32 (CDRL1) and Glu10. There are additional hydrogen bonds between the side chain of Asn11 and the main chains oxygen atoms of GlyL91 (CDRL3) and oxygen and nitrogen atoms of ThrL32 (CDRL1). Interestingly, the only hydrophobic residue of the peptide, Leu6, is completely buried at the bottom of the combining site. Its side chain is stacked between GlyH95 (CDRH3) and TyrL96 (CDRL3). Other peptide side chains are involved in stacking interactions: Arg2 with TyrH53 of CDRH2, Glu5 with TyrH32 (CDRH1) and Asn11 with TyrL31 (CDRL1). A contact between the two molecules in the asymmetric unit is formed by an ionic interaction involving the side chains of GluC1 and ArgD3 from the two different peptides.

Comparison of Fab RY79-90 to Fab 13F6-1-2 and Fv MW1

Considering the rarity of Vlx usage, it is interesting to know whether the other monoclonal antibodies having Vlx light chains share similar antigen specificity. In addition to the structure of Fab reported in this study, two other crystal structures of antibodies Fab or Fv fragments bearing a Vlx light chain have been determined, also in complex with a peptide. The structure of Fab 13F6-1-2, an antibody specific for Ebola viral surface glycoprotein GP was solved in complex with the 11-residue peptide VEQHHRRTDND [START_REF] Lee | Complex of a protective antibody with its Ebola virus GP peptide epitope: unusual features of a V lambda x light chain[END_REF]. Its CDRH3 loop is very long compared to Fab RY79-90, which explains that its peptide binds more on the surface of the combining site compared to the peptide of Fab RY79-90, which is much more buried (Figure 5A). Antibody MW1 was directed against a poly-glutamine sequence relevant for Huntington's disease. The structure of Fv MW1 solved in complex with the GQ 10 G peptide, shows that the latter adopts an extended coil-like structure and is also less buried in the antibody combining site than the BsRNaseY peptide (46) (Figure 5B).

CONCLUSION

Areas of residual structure in unfolded proteins can be critical to forming the initial site of interaction with a binding partner [START_REF] Uversky | Natively unfolded proteins: a point where biology waits for physics[END_REF], and delineation of those sites is required for the design of drugs and antibodies that interfere with binding. Moreover, several monoclonal antibodies combining sites have been shown to act as stable molds of in vivo conformation of important segments of intrinsically disordered proteins [START_REF] Skrabana | Monoclonal antibody MN423 as a stable mold facilitates structure determination of disordered tau protein[END_REF]. Here, we used an antibody directed against a peptide of the IDD of BsRNaseY as a tool to question whether this region could become ordered upon binding a cellular partner. The peptide used to generate RY79-90 lies in a conserved, mostly positively charged region of Nter-BsRNaseY that could be a binding site for RNA or protein partners contributing to the degradosome complex (Figure 1). Similar to E. coli RNase E [START_REF] Callaghan | Studies of the RNA degradosome-organizing domain of the Escherichia coli ribonuclease RNase E[END_REF][START_REF] Marcaida | The RNA degradosome: life in the fast lane of adaptive molecular evolution[END_REF], only microdomains of the intrinsically unstructured region of RNase Y are thought to interact with protein partners to form the B. subtilis degradosome complex [START_REF] Cascante-Estepa | Localization of Components of the RNA-Degrading Machine in Bacillus subtilis[END_REF][START_REF] Cho | The Structure and Function of the Gram-Positive Bacterial RNA Degradosome[END_REF].

If the conserved peptide used to induce RY79-90 belongs to such microdomain, the peptide/Fab RY79-90 interaction could mimic the interaction between the microdomain and a cellular protein partner.

Our studies revealed that the fold adopted by the epitope of the intrinsically disordered region upon binding RY79-90 is a U-shaped extended conformation. Leu6, the unique hydrophobic residue of the peptide, is sandwiched between CDRL3 and CDRH3 at the bottom of the combining site, along with the formation of a b-turn due to hydrogen bonds between Asn4 and Gln7. Interestingly, such a b-turn structure has previously been observed in the complex of a peptide from the intrinsically disordered Tat protein with an Fab fragment, with the burying of a tryptophan residue central to the epitope [START_REF] Serrière | Fab'-induced folding of antigenic N-terminal peptides from intrinsically disordered HIV-1 Tat revealed by X-ray crystallography[END_REF]. In fact, intrinsically disordered proteins have generally the propensity to expose their few hydrophobic residues, probably to favor interactions with their biological partner.

The concept of conformation specific antibodies that recognize a compact fold of the antigen requires rigidity of the antibody-combining site. We did not solve the structure of Fab RY-79-90 alone but, as its CDRH3 loop is extremely short, it is probable that no large conformational changes of the antibody would occur upon peptide binding as these are usually caused by the inherent high flexibility of CDRH3 [START_REF] Sela-Culang | A systematic comparison of free and bound antibodies reveals binding-related conformational changes[END_REF]. In addition, the fact that Fab RY79-90 favors dissociation of the Nter-BsRNaseY dimer suggests that Fab interacts with the same binding region in the N-terminal domain of the BsRNaseY monomer that would also interact in vivo with cellular partners. This is also likely because a large region of Nter-BsRNaseY appears to be organized as a coiled-coil, a structural element often involved in oligomerization and protein interaction [START_REF] Woolfson | Coiled-Coil Design: Updated and Upgraded[END_REF]. The dissociation of the BsRNaseY dimer upon binding the degradosome partners would likely expose the inducing peptide for interaction. Therefore, Fab RY79-90 likely acts as a molecular imprint of a specific structural state of Nter-BsRNaseY and the Fab RY79-90/peptide complex might mimic the interaction of a microdomain of the N-terminal domain of BsRNaseY with one of its cellular partners within the degradosome complex. In future studies, Fab RY79-90 could be used as a tool to induce a specific fold to disordered Nter-BsRNaseY. The colors for Fab RY79-90 are as in Figure 4 and the CDR loops of Fv MW1 are in grey, with the peptide shown as an orange cartoon.

Figure 1 : 4 :

 14 Figure 1: Characterization of a monoclonal antibody induced against a 12-residue peptide

Figure 2 :

 2 Figure 2: SEC-MALS analysis of Nter-BsRNaseY and the Nter-BsRNaseY/Fab RY79-90

Figure 3 :

 3 Figure 3: Nter-BsRNaseY structure. A Circular Dichroism Spectrum of Nter-BsRNaseY.

Figure 4 :A

 4 Figure 4: Recognition of the inducing peptide by Fab RY79-90.

Figure 5 :

 5 Figure 5: Comparison of structures of antibodies with vlx light chain in complex with

Figure 1 Figure 2 Figure 3 Figure 4

 1234 Figure 1

Figure 5 ABFigure S5 :

 5S5 Figure 5

Table S2 : Biophysical parameters of Nter-BsRNaseY, Fab RY79-90 and the Nter- BsRNaseY/Fab RY79-90 complex determined by SEC-MALS

 S2

	Parameter	Fab RY79-90	Nter-BsRNaseY* Nter-BsRNaseY/
				Fab RY79-90
	Molar mass (kDa)	46.0 ± 0.3	40.0 ± 1.1	69.7 ± 0.4
	Rh (nm)	2.6 ± 0.3	4.7 ± 0.3

* Nter-BsRNaseY injected at 945 µM.

Acknowledgements.

This work was supported by ANR (ANR-11-BSV8-026) and the French State Program 'Investissements d'Avenir' (Grants "LABEX DYNAMO", ANR-11-LABX-0011), and has benefited from the facilities and expertise of the Macromolecular Interaction Platform of I2BC.

We thank SOLEIL for provision of synchrotron radiation facilities, Luc Pardanaud for help in hybridoma cell culture, Pierre Legrand for assistance in using beamline PROXIMA 1 (proposal 20150780 for Proxima beamlines), Agnès Thomas-Collignon for the cloning of the Nter-BsRNaseY construct.

PDB accession numbers

Coordinates and structure factors have been deposited in the Protein Data Bank under accession code 6F7T.

Author contributions

BGP conceived and coordinated the study and wrote the major part of the paper. PH performed all biochemical and crystallization assays. SL performed the gene cloning for BsRNaseY production. SL and HP selected the antibody. BGP solved and refined the X-ray structure. CV performed and analyzed the SEC-MALS experiments, CV and NA performed and analyzed the CD experiments, CBN collected the SAXS curves and CBN and DD analyzed the data. All authors designed the experiments, reviewed the results and participated to the final version of the manuscript.

Supporting Citation

Reference 69 appears in the Supporting Material.

Supplementary Methods

Determination of the dissociation constants for the IgG/BsRNaseY, IgG/Nter-BsRNaseY, and IgG/peptide complexes The antibody at constant concentration is first incubated in solution with the antigen at various concentrations until the equilibrium is reached. Then, the amount of unbound antibody is measured by a classical indirect ELISA test (1).

The following protocol was used to establish calibration curves and ensure that the antibody concentration used is in the linear range of the ELISA response and that only a small fraction (≤ 10%) of the total free antibody in solution is retained on the plate. Two 96-well plates were coated in triplicate with 0.5-4 nM BsRNaseY, 5-40 nM Nter-BsRNaseY or 2-16 nM peptide in PBS buffer pH 7.4 and incubated overnight at 4°C. Then, the plates were saturated with 250 µL of 3% BSA, 3% Milk in PBS pH 7.4, 0.05% Tween 20 at 37°C for three hours. After three washes with 20 mM HEPES pH 7.5, 150 mM NaCl, 10% glycerol, 0.05% Tween 20, antibody at various concentrations (0.5 to 64 nM) was dispensed into the wells of the first plate. After 10 min (Nter-BsRNaseY and peptide) or 30 min (BsRNaseY) incubation at 25°C, each well content was transferred to the second plate and incubated in the same conditions. The wells were then washed three times and 100 µL of peroxidase-labeled anti-mouse IgG antibody (Sigma Aldrich®) at a dilution of 1:4000 in 20 mM Hepes pH 7.5, 150 mM NaCl, 10% glycerol and 10 mg/mL BSA was added to each well. After 1 hour incubation at 25°C, plates were washed five times and 100 µL of tetramethylbenzidine (Sigma Aldrich®) was added to each well. After incubation for 10 min in the dark, the reaction was stopped by addition of 1 M sulfuric acid. Absorbance at 450 nm, which should be proportional to the concentration of free antibody, was measured.

Second, the concentration of free antibody in a series of solutions preincubated with various concentrations of the different antigens was determined in triplicate. The binding equilibria were carried out in 20 mM Hepes 7.5, 150 mM NaCl, 10% glycerol, 10 mg/mL BSA overnight at 25°C with the antibody at a fixed concentration (7.6 nM, 1.5 nM and 3.8 nM for the complexes with BsRNaseY, Nter-BsRNaseY or the peptide, respectively), and 31.25, 62.5, 125, 250, 500, 1000 nM BsRNaseY, 3.125, 6.25, 12.5, 25, 50 et 100 nM Nter-BsRNaseY or 0.031, 0.063, 0.125, 0.250, 0.5, 1 nM peptide. Antibody-antigen mixtures were transferred into a 96well plate previously coated overnight with 2 nM BsRNase Y, 40 nM Nter-BsRNaseY or 16 nM peptide in PBS buffer pH 7.4, saturated and washed as described above. After incubation at 25°C for 30 min for the antibody-BsRNaseY complex or 10 min for the antibody-Nter-BsRNaseY complex and antibody-peptide complexes, the concentration of free antibody was determined by the indirect ELISA test described above. The results were plotted using a Scatchard plot as described (1), the slope being equal to -1/K d and the y-axis intercept point to 1/K d.

Determination of the stoichiometry of the Nter-BsRNaseY/Fab RY79-90 complex by SEC-

MALS

Protein stoichiometry analysis of the Nter-BsRNaseY/Fab RY79-90 complex was performed using two methods. The first method, based on the multi-signal detection system of SEC-MALS, uses the 'UV extinction from RI peak' method included in the ASTRA software and the dn/dc value to determine the experimental UV extinction coefficients (2). These values are then compared with the theoretical values predicted from the protein amino acid sequences. The extinction coefficient of Nter-BsRNaseY is negligible due to the rarity of aromatic amino acids in the sequence and that of Fab RY79-90 was predicted from the protein amino acid sequences using the ProtParam tool [http://web.expasy.org/protparam/] to be 1.601 ml.mg -1 cm -1 . The theoretical extinction coefficients for the Nter-BsRNaseY/Fab RY79-90 complex (1 2 ) at 3 different ratios (1:1, 2:1 and 1:1) were obtained using the following equation:

where n N and n F are the fractions of the two partners (Nter-BsRNaseY and Fab RY79-90, respectively) in the complex and M N and M F their molar masses. As e N and e F are significantly different, the protein conjugate method was also used to determine the molecular weights of Nter-BsRNaseY and Fab RY79-90 in the complex and confirm the stoichiometry (3).

Supplementary Figures


Fabrice M Schmidt

A Rodriguez-Villa

Jean-François Agassant

Michel Bellet

F Agassant

Analytical models for the inflation of a polymeric tube

Keywords: elongation flows, tube free inflation, viscoelastic fluid, volumic approach, thin shell assumption, thickness-averaged problem, analytical problem, analytical model, blow moulding process

published or not. The documents may come

Analytical models for the inflation of a polymeric tube

Introduction

Elongation flows occur in many industrial processing operations and particularly in the blow moulding process of Polethylene bottles [START_REF] Rosato | Blow Molding Handbook[END_REF]. In a previous paper [START_REF] Schmidt | Viscoelastic simulation of P.E.T. stretch/blow molding[END_REF], numerical finite element simulations of this process were presented using various constitutive equation. The capability of setting up analytical models for the inflation of a tube allows us to validate finite element simulations, to analyse the influence of processing and rheological parameters, and to compare the solutions calculated using a volumic approach and a thin shell assumption.

In the literature, a few works refer to the development of analytical models for polymeric tube free inflation. [START_REF] Chung | A general elongation experiment: inflation and extension of a viscoelastic tube[END_REF] have investigated the simultaneous inflation and extension of a cylindrical tube made of a Lodge's rubberlike liquid material [START_REF] Lodge | Elastic Liquids[END_REF] as a method for conducting elongational flow experiments. In this approach, the analytical development is facilitated because the axial displacement is prescribed so that the elongation rate remains constant. Ryan andDutta (1982, 1984) have developed quasianalytical models of free and confined tube inflation using viscoelastic differential constitutive equations. The resulting non-linear equation is solved using an iterative scheme based on a Runge-Kutta method where the components of the stress tensor are computed using Simpson's first rule. In a very similar approach, [START_REF] Schmidt | Viscoelastic simulation of P.E.T. stretch/blow molding[END_REF] have developed a simultaneous extension and inflation model of a tube for a Maxwell constitutive equation. Note that in all the previous approaches, the contribution of inertia effects is assumed much smaller than those of viscous and elastic effects. This assumption is justified in the case of elongational flows. Now, if inertia effects are strongly dominant (which is not verified for polymers), [START_REF] Khayat | Uniaxial and biaxial unsteady inflations of a viscoelastic material[END_REF] have found a simple analytical solution for the inner radius.

In the following sections, we will present the equations to be solved using both a thin shell approximation and an exact volumic formulation. Applications to Newtonian and Lodge's viscoelastic constitutive equations will be carried out.

Kinematic assumptions

The radial inflation of a tube (inner radius R(t)), thickness e(t), constant length L 0 is considered (figure 1). A differential inflation pressure P is applied to the inner surface of the tube (i.e., at r = R(t)) and the upper and lower ends of the tube are assumed to move within two fixed horizontal planes (perfectly sliding contact conditions). Hence, the upper and the lower plane can be seen as symmetry planes. The Cauchy stress tensor σ is diagonal at any point of the tube and during all the inflation process. The geometry and boundary conditions being axisymmetric, the following conditions apply to the components of the velocity field u:

u = u r (r, t), u θ = 0, u Z = 0 , (1)
∂u r ∂θ = 0.

(2)

Mass balance and strain rate tensor

The material is assumed to be incompressible. Using Eqs (1) and ( 2), the continuity equations reduces to:

∂(ru r ) ∂r = 0. ( 3
)
Figure 1. Free inflation of a cylindrical tube.

Integrating this first-order differential equation, we get:

u r = A(t) r , (4)
where A only depends on time (A is uniform throughout the tube thickness at each time step). Introducing u r = dR/dt at r = R, we get:

A(t) = R dR dt (5)
and thus:

u r = R r dR dt . ( 6
)
The rate of strain tensor is:

ε = ⎛ ⎜ ⎜ ⎜ ⎝ -R r 2 dR dt 0 0 0 R r 2 dR dt 0 0 0 0 ⎞ ⎟ ⎟ ⎟ ⎠ . (7)
Now, if Eq. ( 3) is integrated along r between the inner radius R and the outer radius (R + e), we obtain:

R+e R ∂(ru r ) ∂r dr = 0 ⇔ (R + e) d(R + e) dt -R dR dt = 0. ( 8
)
Further time integration of this equation from t ′ = 0 to t ′ = t yields the global volume conservation of the tube:

πL 0 (R + e) 2 -R 2 = πL 0 (R 0 + e 0 ) 2 -R 2 0 , (9)
where R 0 and e 0 are, respectively, the initial radius and thickness of the tube. The global volume conservation can be transformed into a more convenient shape using the average radius R = R + (e/2) of the tube:

2π ReL 0 = 2π R 0 e 0 L 0 . (10)

Dynamic equilibrium equations

In a previous paper [START_REF] Schmidt | Viscoelastic simulation of P.E.T. stretch/blow molding[END_REF], it has been shown that inertia effects remain much lower than viscous and elastic effects in the case of a biaxial deformation of a polymer tube. Thus, the stress balance equations reduce to:

∂σ rr ∂r + σ rr -σ θθ r = 0. ( 11
)
The stress balance equation in the r-direction (Eq. ( 11)) may be rewritten as follows:

1 r

∂(rσ rr ) ∂r = σ θθ r ⇔ ∂(rσ rr ) ∂r = σ θθ . ( 12
)
The imposed pressure yields the following boundary conditions:

r = R: σ rr = -P , (13)
r = R + e: σ rr = 0. ( 14
)
Integrating Eq. ( 12) from inner to outer radius with boundary conditions ( 13) and ( 14) leads to:

R+e R σ θθ dr = P R. ( 15
)
Let us define the thickness averaged orthoradial stress σθθ = (1/e) R+e R σ θθ dr. Equation ( 15) can then be rewritten:

σθθ e = P R ⇔ σθθ = P R e (16)
which is the average orthoradial stress in the tube independent on the constitutive equation. If the average radius R is used:

σθθ = P R e - P 2 . ( 17
)

Newtonian behaviour

Introducing the Newtonian constitutive equation leads to:

ε = ⎛ ⎜ ⎜ ⎜ ⎝ -p -2η A r 2 0 0 0 -p + 2η A r 2 0 0 0 -p ⎞ ⎟ ⎟ ⎟ ⎠ , ( 18
)
where p is the hydrostatic pressure and η the viscosity of the material. The stress balance equation leads to a constant hydrostatic pressure at each time step. Using the boundary conditions ( 13) and ( 14), we can express p and A as follows:

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ A = P (R + e) 2 R 2 2η((R + e) 2 -R 2 ) , p = - P R 2 ((R + e) 2 -R 2 ) . ( 19
)
It is to note that A do not depend of dR/dt (Eq. ( 5)). The stress components are:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ σ rr = P (R + e) 2 R 2 ((R + e) 2 -R 2 ) 1 (R + e) 2 - 1 R 2 , σ θθ = P R 2 ((R + e) 2 -R 2 ) 1 (R + e) 2 + 1 R 2 , σ zz = P R 2 ((R + e) 2 -R 2 ) . (20)
Integrating Eq. ( 11) along r using Eq. ( 21) and boundary conditions ( 13) and ( 14), we find

P + 2η R+e R ∂u r ∂r - u r r dr r = 0 ⇔ P 2η + R+e R ∂ ∂r u r r dr = 0. ( 21
)
Introducing u r = dr/dt, Eq. ( 21) becomes:

P 2η + d dt ln R + e R = 0. ( 22
)
And finally:

R + e R 2 = R 0 + e 0 R 0 2 e -P η t . (23)
Using the volume balance (Eq. ( 9)) and the initial shape factor δ 0 = e 0 /R 0 , the inner radius R is determined from Eq. ( 23):

R R 0 = (1 + δ 0 ) 2 -1 (1 + δ 0 ) 2 exp -P η t -1 . ( 24
)
The denominator has to remain positive:

(1 + δ 0 ) 2 e (-P η t) > 1 ⇔ t < t 1 , (25)
t 1 = 2t c ln(1 + δ 0 ) is the maximum blowing time which depends on a characteristic blowing time t c = η/ P and on the initial shape factor δ 0 . Introducing Eq. ( 25) into Eq. ( 24) leads to:

R R 0 = (1 + δ 0 ) 2 -1 (1 + δ 0 ) 2(1-t/t 1 ) -1 ∀t ∈ [0, t 1 [, (26)
which gives the evolution of the inner radius R versus time, for a Newtonian tube.

Viscoelastic behaviour

The simplest constitutive model of integral type which accounts for both viscous and elastic phenomena is Lodge's model, which is mathematically equivalent to Maxwell's differential model which will be used further on. This model is suitable for large deformations (Agasant et al., 1991). It is written

σ = -qI + t -∞ η λ 2 e -t-t ′ λ C -1 (t ′ ) dt ′ , ( 27
)
where q is an arbitrary pressure; λ is a constant relaxation time and C -1 (t ′ ) is the Finger strain tensor which accounts for large deformation (see Eq. ( 31)). One can take advantage of the axisymmetric tube growth and use the Lagrangian coordinate transformation r → X:

X = π r 2 -R 2 . ( 28
)
From Eq. ( 6) it can be deduced that X is a constant for each material point during the inflation process (dX/dt = 0). When r is equated to the outer radius, R + e, X is equal to the surface of the section of the tube, whereas X = 0 for the material points located at the inner radius R. We have the following relationships:

∀t ′ ∈ [0, t] r ′ = r 2 -R 2 + R ′2 , ∀t 0 r ′ = r 0 , R ′ = R 0 . ( 29
)
The components of the gradient tensor F are:

F (t ′ ) = ⎛ ⎜ ⎜ ⎜ ⎝ r r ′ 0 0 0 r ′ r 0 0 0 1 ⎞ ⎟ ⎟ ⎟ ⎠ (30)
and so we have:

C -1 = F -1 • T F -1 = ⎛ ⎜ ⎜ ⎜ ⎝ r ′ r 2 0 0 0 r r ′ 2 0 0 0 1 ⎞ ⎟ ⎟ ⎟ ⎠ . (31)
Finally, using Eqs ( 28) and ( 29), we obtain:

t t ′ 0 C -1 = ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ X/π+R ′2 X/π+R 2 0 0 0 X/π+R 2 X/π+R ′2 0 0 0 1 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ , ( 32
)
∀t 0 and t ′ 0 C -1 = ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ X/π+R 2 0 X/π+R 2 0 0 0 X/π+R 2 X/π+R 2 0 0 0 0 1 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ . ( 33
)
In order to simplify the calculation, dimensionless variables are introduced:

σ = λ σ η , t = t λ , R = R R 0 , X = X πR 2 0 , S = S πR 2 0 = R 0 + e 0 R 0 2 -1, (34)
S is the dimensionless surface of the transverse section of the tube. Using Eq. ( 27), the components of σ are:

σ rr = -q + e -t X + R 2 X + 1 + t 0 e t ′ X + R ′2 d t′ , ( 35
) σθθ = -q + e -t X + R 2 1 X + 1 + t 0 e t ′ d t′ X + R ′ 2 , ( 36
)
σ zz = -q + 1. ( 37
)
The stress balance equation in the z-direction reduces to:

∂ q ∂z = 0 ⇔ q = q X, t . (38)
In addition, if we integrate the stress balance Eq. ( 11) using boundary conditions ( 13), ( 14) and the previous dimensionless variables, we get:

De v + 1 2 θ 0 σ rr -σ θθ X ′ + R 2 d X ′ = 0, ( 39
)
where De v = λ( P /η) is the Deborah number. The non-linear form of Eq. ( 39) excludes any simple analytical solution for R except for some limiting cases (see Section 7). A numerical algorithm has thus been developed to solve Eq. ( 39), details can be found in [START_REF] Schmidt | Etude experimentale et modélisation du procédé d'injection-souflage bi-orientation en cycle froid de bouteilles en PET[END_REF]. At each time step, the dimensionless radius R is determined from Eq. ( 39) using a quasi-Newton iterative procedure. The iterative scheme was stopped when successive values of R differed by less than 10 -3 percent. Simpson's first rule was used to evaluate the integrals.

Then, the components of the stress tensor are deduced.

Thin shell approximation

In this paragraph we propose an analytical solution for thin tubes, i.e. the thickness of which is much smaller than the radius δ 0 = e 0 /R 0 ≪ 1. It is clear that, in that particular geometry, no shear stress develop during the inflation process since the geometry remains axisymmetrical, i.e., the radial normal vector remains normal during inflation (see ( 18) and ( 31)). In that context thin shell approximation leads to neglect the radial stress σ rr , which is O( P ), in front of the orthoradial stress σ θθ , which is O( P (R/e)). Hence, a plane stress state will be considered, and the thickness-averaged problem is solved.

Mean strain-rate tensor

In order to develop a thin shell approach, we average the volumic strain-rate tensor ε over the tube's thickness. We define an average strain-rate tensor ε:

ε = 1 e R+e R ε dr. ( 44
)
From Eq. ( 7) we get:

ε = ⎛ ⎜ ⎜ ⎜ ⎝ -1 R dR dt 0 0 0 1 R dR dt 0 0 0 0 ⎞ ⎟ ⎟ ⎟ ⎠ . ( 45
)

Evolution of the tube radius. Maxwell's behaviour

We introduce the upper convected Maxwell model which is equivalent to the integral Lodge equation ( 27): It is well known that these equations are suitable for large deformations.

σ = -qI + τ , ( 46
) τ + λ δτ δt = 2η ε, ( 47
)
where q is the average pseudo-pressure and τ , the average extra-stress tensor. It should be noted that λ = 0 yields the Newtonian model, q then being the hydrostatic pressure and τ the deviatoric stress tensor. Also note that the equation has been written directly with the average tensors. As stress and rate of strain tensors are diagonal, the upper convected derivative of the tensor is:

δτ δt = d τ dt -2 ε τ . (48)
We are considering here a plane stress state. Pseudo-pressure and orthoradial stress components are solution of the following differential equations:

q + λ d q dt -2λ q εrr = 2η εrr , (49)
σ θθ + q + λ d( σ θθ + q) dt -2λ σ θθ + q εθθ = 2η εθθ . ( 50
)
Introducing average strain-rate tensor ε (Eq. ( 45)), equilibrium equation (Eq. ( 17)) (in which we use the membrane assumption: δ0 = e 0 /R 0 ≪ 1 (where δ0 /(1 + δ 0 /2) ∼ = δ 0 if δ 0 ≪ 1), and mass conservation (Eq. ( 10)) lead to a system of two differential equations with pseudo-pressure q and membrane radius R as unknowns. Note that the imposed internal pressure P has been assumed an arbitrary function of time, and that q and R are only functions of time

2R 0 e 0 q + λ d q dt = -P + λ d( P ) dt R 2 , (51)
4R 0 e 0 dR dt

(λ q + η) = P + λ d( P ) dt R 3 . ( 52
)
Up to our knowledge this system of equations does not have general analytical solutions. Nevertheless, for particular prescribed pressure curves P (t), these solutions do exist.

Initial condition on the imposed pressure curve

The above development is valid for any arbitrary imposed pressure P = P (t). Extra-stress tensor τ in the membrane is assumed to be zero before blowing (say time t = 0). Due to the plane stress state approximation, the pseudo-pressure's value at this moment is zero too ( q(0) = τrr (0) -σrr (0) = 0), and thus the value of orthoradial stress σθθ must also be zero ( σθθ (0) = -q(0) + τθθ (0)). Equilibrium equation ( 17) shows that the imposed pressure must then also start from zero:

P (t = 0) = 0. ( 53
)
Note that in the case of a Newtonian behaviour (λ = 0), it is not necessary to impose a pressure at blowing start up.

Abrupt step approximation

For comparison's sake with the classical problem of a Newtonian membrane inflating under a constant pressure load P (t) = P 0 , we are led to assume an abrupt increase of pressure from the initial value P = 0 to a constant regime value P 0 (figure 2). Let us suppose that this increase takes place from time t =t to time t = 0, interval t being arbitrary small, in particular considerably smaller than relaxation time λ. The imposed pressure is thus:

P (t) = P 0 1 + t t if -t t 0, ( 54
)
P (t) = P 0 if t > 0. ( 55
)
Figure 2. Abrupt increase of pressure load.

Abrupt step solutions

Let us consider the first "fictitious" part of the pressure curve, from t =t to t = 0. Initial conditions are:

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ R(t = -t) = R -t , e(t = -t) = e -t = R 0 e 0 R -t by volume conservation, q(t = -t) = 0. ( 56
)
Introducing Eqs ( 54), ( 51) and ( 52) lead to:

2R -t e -t q + λ d q dt = -P 0 1 + t t + λ t R 2 , ( 57
) 4R -t e -t dR dt (λ q + η) = P 0 1 + t t + λ t R 3 . ( 58
)
The time interval during which this pressure increase takes place has been chosen arbitrary small with respect to the relaxation time λ of the material:

λ t ≫ 1, (59)
λ t ≫ t t = O(1). ( 60
)
Thus, Eqs ( 57) and ( 58), together with volume conservation (Eq. ( 10)), can be approximated by:

2R 0 e 0 q + λ d q dt = -P 0 λ t R 2 , ( 61
)
4R 0 e 0 dR dt

(λ q + η) = P 0 λ t R 3 . ( 62
)
At time t =t, taking into account boundary conditions (56), one obtains:

d q dt -t = - P 0 R 2 -t 2R 0 e 0 t , (63)
dR dt -t = - P 0 λR 3 -t 4ηR 0 e 0 t . (64)
One introduces then a Taylor development of q and R from t =t to t = 0:

q0 = q-t + d q dt -t t, (65)
R 0 = R -t + dR dt -t t, (66)
where q0 is the pseudo-pressure at t = 0. Introducing Eqs ( 63) and ( 64) in this Taylor development leads to an algebraic system of equations, with q0 and R -t as unknowns:

q0 = - R 2 -t A , (67)
1 B R 3 -t + R -t -R 0 = 0, (68)
with A = 2R 0 e 0 /P 0 , B = 2ηA/λ. Constant B is always positive, and Eq. ( 68) has a single real solution, given by

R -t = 3 BR 0 2 + B 2 R 2 0 4 + B 3 27 + 3 BR 0 2 - B 2 R 2 0 4 + B 3 27 . ( 69
)
The expression of q0 , necessary as an initial condition to the second part of the inflation (constant pressure regime), is then found by replacing this R -t value in Eq. ( 67)

Constant load solutions

Let us consider the second of the imposed pressure curve (Eq. ( 55)). Initial conditions are:

R(0) = R 0 , (70)
e(0) = e 0 , (71)
q(0) = q0 , (72)
where q0 has been calculated in the previous paragraph.

Newtonian fluids

As said earlier, the abrupt step approximation is not required for the Newtonian case (for λ = 0 Eq. ( 68) leads to R -t = R 0 ), and thus q0 = -P 0 R 0 /(2e 0 ). Equation ( 52) gives the mean radius evolution for a Newtonian tube:

1

R 3 dR dt = P 0 4ηR 0 e 0 . ( 73
)
This equation is integrated from t ′ = 0 to t ′ = t and yields:

R(t) R 0 = 1 1 -t t 1N , (74)
where

t 1N = 2ηe 0 P 0 R 0 ( 75
)
is the Newtonian limit inflating time for a membrane. It is interesting to compare this result with the limit volumic blowing time t 1 (Eq. ( 25) of Section 5) when the shape factor δ 0 becomes small: lim

δ 0 →0 t 1 = 2η P 0 δ 0 + O δ 2 0 = t 1N . (76)

Viscoelastic fluids

Let us now consider the more general case of a Maxwell fluid submitted to a constant pressure load. Equations ( 51), ( 52) are simplified:

2R 0 e 0 q + λ d q dt = -P 0 R 2 , ( 77
)
4R 0 e 0 dR dt

(λ q + η) = P 0 R 3 . ( 78
)
Including

ξ = λ q + η P 0 R 3 4R 0 e 0 dR dt (79) leads to: 2 R dR dt = 4ηR 0 e 0 λP 0 1 R 3 dR dt - 1 ξ dξ dt - 1 λ , (80)
4R 0 e 0 dR dt

ξ = P 0 R 3 . ( 81
)
Integrating Eq. ( 80) from t ′ = 0 to t ′ = t, one gets:

ln R R 0 2 = K 1 R 2 0 - 1 R 2 -ln ξ ξ 0 - t λ , (82)
where K = 2ηR 0 e 0 /(λP 0 ) and ξ 0 = λ q0 + η. Exponentiating Eq. ( 82) and performing variable separation one gets:

P 0 R 0 e 0 e t/λ dt = 4J 0 R -5 e -K/R 2 dR ( 83
)
with J 0 = ξ 0 R 2 0 e K/R 2 0 . Equation ( 83) can be integrated as a function of time:

1 + K R 2 e -K/R 2 = 1 + K R 2 0 e -K/R 2 0 + K 2 λP 0 2R 0 e 0 J 0 e t/λ -1 . ( 84
)
This equation does not provide explicit R value at a given time step but the blowing time may be calculated explicitly. Equation (84) leads finally to:

t (R) λ = ln ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 1 + De s + R 0 R 2 e 1 De 1- R 0 R 2 -De s -1 2 t 1N R 0 dR dt 0 ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ , (85)
where De s is the Deborah number defined here as De s = λ/t 1N ; we need to known the derivative of R at time t = 0. This value is directly linked to the known value (Eq. ( 62)):

dR dt 0 = λP 0 R 2 0 4e 0 (λ q0 + η) . ( 86
)
As in the Newtonian case, a limit value of inflation time may be obtained (R → +∞ in Eq. ( 85)):

t 1M = λ ln ⎧ ⎨ ⎩ 1 + De s e 1/De -De s -1 2 t 1N R 0 dR dt 0 ⎫ ⎬ ⎭ . ( 87
)

Comparison of volumic and thin shell solutions

We compare the given by both models. Viscosity is η = 50 kPa s. Prescribed pressure is 10 kPa. We have tested three values of δ 0 : δ 0 = 0.01, δ 0 = 0.1, δ 0 = 0.5, for two different fluids (λ = 0 (Newtonian), λ = 0.01 s (viscoelastic)). In the Newtonian case, figure 3 shows the evolution of the dimensionless mean radius R/R 0 versus time. For δ 0 = 0.01 and δ 0 = 0.1 volumic and thin shell solutions are superimposed; even at δ 0 = 0.5, the difference remains limited.

In the Newtonian case we draw the orthoradial stress distribution across the thickness of the tube, at midinflation, for the same values of δ 0 (figure 4). It clearly appears that the difference between volumic and thin shell orthoradial stresses becomes significant even for δ 0 = 0.1. So, in term of kinematic evolution, the thin shell assumption is correct but not in term of stress prediction.

These phenomena are slightly emphasised in the viscoelastic case. Difference between shell and volumic approaches remains of the same order of magnitude for this small value of the Deborah number (figures 5, 6).

It is to note that if we compare the volumic approach with a thin shell solution but using, as this is classically done, the inner radius instead of the mean radius the difference would be more important (Agasant et al., 1996).

The limit viscoelastic blowing time (Eq. ( 87)) as a function of the Maxwell relaxation time λ is presented on figure 7. When increasing λ the limit blowing time decreases and even becomes negative. This may be related to unattainable blowing conditions equivalent to unattainable stretching conditions encountered in fibre spinning and in cast film (Agasant et al., 1996;[START_REF] Rodriquez-Villa | Etude théorique et expérimentale de l'extrusion-soufflage de corps creux en polymères[END_REF]. The unattainable blowing condition become more drastic when the shape factor of the tube δ 0 decreases.

Conclusion

Analytical models for the free inflation of a polymeric tube have been developed for volumic approach and thin shell approximation. They have been compared for Newtonian and Maxwell fluids.

They point out that the thin shell approximation (with an average radius R) remains valid in term of kinematic evolution up to high values of the initial shape factor (δ 0 = 0.5). On the other hand, the difference between volumic and thin shell orthoradial stresses becomes significant for lower values of the initial shape factor. Stresses are significantly underestimated when using a thin shell approximation. It is to notice that more sophisticated finite strain shell theories should reduce these discrepancies.

A limit blowing time may be derived in the Newtonian case for both volumic case and thin shell approaches. The asymptotic value is the same when the initial tube shape factor tends to zero. In the viscoelastic case, a limit viscoelastic blowing time may be derived when using the thin shell approximation. This limit blowing time decreases with the material characteristic relaxation time and may becomes negative for values of the relaxation time exceeding a critical value, which is physically impossible. This could be related to unattainable blowing conditions which are also encountered in other stretching processing (fibre spinning, film drawing). It is to notice that for the viscoelastic volumic approach, this limit blowing times is equivalent to be thin shell one (see figure 5) when convergence may be obtained.

These analytical models provide useful reference solutions for both volumic and thin shell finite element numerical models.

In addition, comparisons between the analytical and finite element simulations allows us to validate the numerical models.

Figure 3 .

 3 Figure 3. Comparison of radii evolutions for a Newtonian fluid.

Figure 4 .

 4 Figure 4. Comparison of orthoradial stress distribution across the tube thickness at mid-inflation (Newtonian).

Figure 5 .

 5 Figure 5. Comparison of radii evolutions for a viscoelastic fluid λ = 0.01 s.

Figure 6 .

 6 Figure 6. Comparison of orthoradial stress distribution across the tube thickness at mid-inflation (viscoelastic fluid λ = 0.01 s).

Figure 7 .

 7 Figure 7. Limit blowing times of viscoelastic membranes.


S Monteix

F Schmidt

Y Le Maoult

R Ben Yedder

R W Diraddo

D Laroche

Experimental study and numerical simulation of preform or sheet exposed to infrared radiative heating

Keywords: Contour integration method, Control-volume method, Heating stage, Infrared camera, Stretch blow moulding, Thermoforming, View factor computation

Introduction

In both the injection stretch blow moulding and thermoforming processes, a tube-or sheet-shaped thermoplastic preform (PET, PVC, PP, etc.) must be heated before forming. In order to increase the temperature of the preform above the glass transition, infrared ovens are used [START_REF] Lindenlaub | Thermoformage, Techniques de l'Inge Ânieur[END_REF]. Infrared heating permits a rapid heatingwith heat ¯ux(proportionalatthefourth power of the source temperature) occurring even on the inner surface of the preform far from the incident radiative source.

The ®nal thickness distribution of the part is dependent on the initial temperature distribution inside the preform. It is very important to investigate parameters which determine the heating stage. They depend both on the geometrical and spectral interactions between lamps and irradiated plastic materials.

As a result of these interactions, the ratio between electric power and absorbed energy in the PET tubular preform is approximately 20%. Optimisation of the infrared oven is then necessary to allow for the control of temperature distribution and also minimisation of the energy costs. In this way, the following methodology has been applied on the heating of PET sheets: characterisation of infrared heaters; measurement of the surface temperature distribution; development of a three-dimensional control-volume model of the heating stage.

Characterisation of the heat source

The spectral properties of the infrared heaters have been already determined in the previous papers [START_REF] Le Maoult | Measurement and calculation of preform infrared heating[END_REF][START_REF] Monteix | Profiles infrared radiative heating in blow molding and thermoforming[END_REF]. The infrared lamps used are sketched in Fig. 1. They are composed of a coiled tungsten ®lament, contained in a quartz tubular enclosure and a re¯ector in order to increase the heat ¯ux ef®ciency to the product.

The lamps are called L300 and L400 with a nominal electric power of 300 and 400 W, respectively. The L300 is used with a cylindrical re¯ector made of polished aluminium. The L400 is commonly plugged into an industrial oven and ®tted with a re¯ector made of a ceramic coating on the backside of the quartz tube as shown in Fig. 2. With a characteristic electrical power in range and a characteristic resistivity, the tungsten ®lament reaches steady state for the L300 at 300 W and a temperature of about 2300 K, and L400 at 400 W and a temperature of about 1700 K.

An experimental set-up (see Fig. 3) using a thermopile detector (remote sensing heat ¯ux) was developed in order to measure the spatial directivity of the lamp.

The ¯ux sensor moves a half circle around the ®lament in order to measure the spatial directivity of each heater. Then, it intercepts the radiation in each direction described by the angle y. The thermopile presents a bandwidth de®ned between 8 and 14 mm. In this spectral range, the quartz tube presents a full absorption of the ®lament radiation [START_REF] Le Maoult | Measurement and calculation of preform infrared heating[END_REF]. We assume that the heaters directivity is the same as that of the quartz tube because of their concentric geometry (the cylindrical ®lament in the quartz tube).

The normalised intensity is plotted versus the angle y as shown in Fig. 4. For all the measurements, the solid angle of the thermopile is centred in the middle of the lamp ®lament. In comparison with a Lambertian source (diffuse radiation), the lamps L400 and the L300 used without re¯ector are similar to an isotropic source (except some border's effects). As expected, we observe that the lamp L300 ®tted with a specular re¯ector exhibits a strong directionality in front of the re¯ector.

Temperature distribution measurement

3.1. Experimental procedure Fig. 5 shows the experimental set-up which has been developed in order to measure the surface temperature evolution when a PET sheet is heated using the infrared lamp.

An 880 LW AGEMA infrared camera (8±12 mm bandwidth) is used to evaluate in a remote sensing, the spatial and transient temperature distributions. The surface dimension of the sheet is 20 cm  20 cm and the thickness is 1.5 mm. The distance between heaters and sheets is 5 cm. The frequency of analysis is 25 frames/s and the device is connected to a 12 bit data acquisition board drive with a real time software (Addelie 1 ).

The camera's objective is oriented normally to the sheet's surface. However, due to the spectral absorbance of the PET [START_REF] Hartwig | Simulation of the stretch-blow molding process and characterization of the relevant material behavior[END_REF], care is required for the surface temperature measurement [START_REF] Lallemand | Mesure de tempe Ârature dans les milieux semi-transparents[END_REF]. To do this, the spectral absorbance of the PET must be characterised prior to determining the bandwidth where the material is assumed to be opaque (see Fig. 6). In this case, only the radiation emitted from the material surface is intercepted by the camera's sensor, although all the other radiation from the bulk or the sheet environment are absorbed before reaching the viewed surface [START_REF] Haji | Radiation pyrometry on semitransparent sheets. I: gray media[END_REF]. Previous work deals with PET transmittivity [START_REF] Shelby | Effects of infrared lamp temperature and other variables on the reheat rate of PET[END_REF], in the spectral range of the heaters intensity, de®ned by Wien's law for a temperature of 2300 K. In the bandwidth of the AGEMA camera (8±12 mm), there is no available data. Then in order to observe the PET behaviour, we have assessed re¯ectivity and transmittivity coef®cients using the infrared camera and a laboratory blackbody.

Results

Temperatures

Measurements have been processed for 0.7 and 1.5 mm PET sheet thicknesses. Tables 1 and2 show a very low average transmittivity on this spectral band. The average emissivity is then calculated from the re¯ectivity measurement.

Taking into account these characteristics, temperatures are measured during heating and cooling stages from the back and front surfaces of the sheet. It takes 3 s to bring the front face in front of the camera during which a cooling stage due to natural convection occurs.

During this transfer, a relative difference less than 0.7% appears in temperature measurements similar to a cooling with a transfer coef®cient, h 10 W m À2 K À1 . In Fig. 7, two infrared thermograms are presented for a heating time of 35 s. Fig. 7a shows the surface temperature distribution measured on the back face of the sheet for the heat source L400. The surface temperature distribution is relatively isotropic due to the presence of the diffuse re¯ector. These results con®rm that the Lambertian source assumption (diffuse radiation) are valid for the lamp L400. As expected, the surface temperature distribution for the lamp L300 (specular re¯ector) is strongly anisotropic (see Fig. 7b).

Thermograms

In this part, thermograms for 1.5 mm PET sheet's thicknesses are examined. Thermograms recorded from the front and back faces with the same experimental heating conditions provide (Fig. 8) the measurement of the temperature gradient reached across the thickness after a heating time of about 35 s. This gradient is related to the location of the measured point on the sheet surface in association with the temperature level at this point. The highest gradient (about 8 8C) is reached at the sheet centre (the same as the heater's centre) corresponding to the highest temperature level of about 85 8C.

The temperature versus time is presented in Fig. 9 for different z-values (see Fig. 7) on the sheet. The dotted vertical line indicates the end of the heating. These measurements during the heating and cooling stages reveal a delay between the heater switch off and the record of the maximum temperature level from the back face. This delay (about 8 s) is less important than the time of the diffusion phenomenon in the semi-in®nite sheet submitted to a surface temperature T s . In this case, the time t diff required to reach half of the initial temperature difference (between T s and the temperature at a depth e) is equal to t diff e 2 =0:92a [START_REF] Agassant | La mise en forme des matie Áres plastiques, Technique et Documentation[END_REF].

The PET diffusivity a 1:4 Â 10 7 m 2 s À1 which corresponds to t diff equal to 17 s over the sheet thickness of 1.5 mm. This discrepancy compared to 8 s reveals the relative in¯uence of the natural convection which tends to accelerate the equilibrium of temperature levels across the sheet thickness. In contrast, the spatial temperature distribution after the same equilibrating time presents a larger difference on the z-axis. This means that the one-dimensional radiative transfer throughout the thickness is more in¯uent than the conductive transfer in any direction of the space.

Control-volume formulation

A three-dimensional control-volume model has been developed for computing the radiative heat transfer during the infrared heating stage. The sheet-shaped domain sketched in Fig. 10 is discretised into cubic elements called control volumes [START_REF]Numerical Heat Transfer and Fluid Flow[END_REF]. The temperature balance (Eq. ( 1)) including radiative transfer (thermal power absorbed by the polymeric sheet from the infrared heaters) is integrated over each control volume and over the time from t to t Dt. V DxDyDz is the control volume, qc the Fourier's ¯ux (q c ÀkrT), qr the internal radiative heat ¯ux. The unknown temperatures are computed at the cell centres. The heat conductivity k, the speci®c mass r as well as the heat capacity C p are assumed to be constant in agreement with previous work [START_REF] Esser | Infrared radiation in the processing of plastics: precise adjustmentÐthe key to productivity[END_REF]:

Z Dt Z V rC p @T @t dV dt À Z Dt Z V rq c qr dV dt (1)

Amount of incident radiation

Previous works [START_REF] Lebaudy | Etude et simulation de la re Âpartition des tempe Âratures dans un cylindre creux de PET soumis a un rayonnement infra-rouge[END_REF][START_REF] Denis | Etude expe Ârimentale et mode Âlisation du chauffage par rayonnement infrarouge de polyme Áres semi-cristallins: influences des centres diffusants[END_REF] dealt PET sheet's heating with a radiant heat ¯ux uniform all around the radiated face or assumed an arbitrary intensity without consideration about heaters dimension. Here, the speci®c lamp geometry is considered, taking into account from a view factor computation [START_REF] Rammohan | Efficient evaluation of diffuse view factors for radiation[END_REF]. We assume a constant ®lament temperature and an uniform source temperature. Thus, the amount of incident radiation reaching each front elements of the irradiated sheet is performed with the contour integration method (Eq. ( 2)) using Stoke's theorem [START_REF] Rammohan | Efficient evaluation of diffuse view factors for radiation[END_REF]:

F H3r s 1 2pS H G H G rs ln r dr H dr r s ( 2
)
where the subscripts H refers to the heater and r s to the sheet front element under consideration; S H the heater surface, r the distance between two elemental linear elements issued from the discretisation of the entire heater border G H and G r s for each surface of control element's border reaching the irradiated sheet's face. This contour integration method was chosen between different numerical view factor method (Monte-Carlo or geometric method). This method allows a good accuracy at a very low computational time and a simple way of use. An efficient numerical method based on Gaussian quadrature has been applied in order to perform the result of this method.

In order to take into account the participation of the diffuse re¯ector in the amount of the incident radiation, a coef®cient of ef®ciency k rf is introduced. k rf is calculated assuming a strip surface for both the heater and its re¯ector and considering all the ¯ux emitted from this equivalent surface equal to the ¯ux emitted from the entire ®lament surface [START_REF] Harmand | Contribution a la mode Âlisation des e Âchanges radiatifs entre un champ d'e Âmetteurs infrarouges et un corps en de Âplacement[END_REF], i.e. S fil pdL fil where d represents the coiled tungsten diameter equal to 4 mm and L ®l the length about 64 mm. Using these assumptions, we have k rf p.

Thus, the amount of incident radiation f(r s ) (Eq. ( 3)) on each surface element is

fr s fx; y 0; z k rf F HÀr s S H S r s Z Dl e l T fil pL 0 l T fil dl (3)
where e(l) is the spectral tungsten emissivity, T fil the filament temperature, L 0 l the blackbody intensity and l a given wavelength between 0.2 and 10 mm.

Internal radiative heat flux

In the ®rst approximation, internal radiative transfer is assumed to be one-dimensional across the sheet thickness. The PET bulk temperature (%400 K) is very low in comparison to the source temperature T s 1700 K. So, the assumption of ``cold material'' is convenient. This leads to express the transmitted ¯ux q r (Eq. ( 4)) across the thickness (y-axis) of the sheet, for an homogenous amorphous PET, using the Beer±Lambert's law in the y-direction q r f l x; y; z f l x; y 0; z e ÀA l y (4)

where A(l) is the spectral absorption coefficient of the material which is given in the below equation

Al À ln tl L ( 5
)
where L is the thickness of the PET film used to obtain an intrinsic transmission coefficient t(l) and, consequently, an accurate A(l) dependence with wavelength. The internal heat source S p on the principal p control volume is given in the following equation:

S p À Z Dt Z V r 3 Á fydV dt À Z Dt Z S q r dS dt ( 6
)
where S are the surfaces of the control volume. We assume the incident flux direction is the same as that of the normal vector of S surfaces (see Fig. 11). Finally, the S p formulation is expressed by

S p fx; y 0; zt y s =L l À t y n =L l ( 7
)
For the next step, the integration of Eq. ( 1) over all the control volumes leads to a linear equations system, which is solved iteratively using the line-by-line Gauss Seidel method in the y-direction of range to obtain a three-dimensional system resolved by a tridiagonalisation method using the Thomas algorithm.

Validation of the numerical model

The numerical model is validated for the analytical case of a semi-in®nite partially transparent solid [START_REF] Wriendt | [END_REF], subjected to a uniform incident heat ¯ux (q HH 0 ). Its value is evaluated using Eq. ( 3) considering the entire spectrum and all the irradiated sheet surface (Eq. ( 8))

q HH 0 k rf S H S R F HÀR e moy sT 4 fil (8)
In these specific conditions, the temperature T(y, t) across the thickness (y-axis) versus time t is computed using Laplace's transforms in order to solve the one-dimensional energy equation with the term source per unit volume, q HHH 0 q HH 0 A e ÀAy and considering adiabatic borders. The constant A corresponds to an average absorption coefficient (Eq. ( 9)), evaluated from Eq. ( 5) with an average PET's transmittivity [START_REF] Siegel | Thermal Radiation Heat Transfer[END_REF]:

t R Dl t l M 0 l T fil dl R Dl M 0 l T fil dl (9)
The solution is given in the below equation

Ty; t T ini À q HHH 0 kA e ÀAy q HHH 0 at p i erfc y 2 at p q HHH 0 2kA e A 2 atAy erfc A at p y 2 at p q HHH 0 2kA e A 2 atÀAy erfc A at p À y 2 at p ( 10
)
where T ini is the initial temperature. Analytical and numerical solutions are compared for a 100 mm  100 mm sheet of 200 mm thickness. Its front surface is subjected to an incident radiation (9) with a ®lament temperature of about 1700 K, corresponding to an average tungsten emissivity of about 0.26 and an absorption coef®cient of 2660 m À1 for a wavelength delimited between 0.2 and 10 mm. The view factor in this case between the strip heater and sheet is evaluated as 0.53.

In Fig. 12, comparisons for the temperature versus thickness are shown. The agreement is good with an error of less than 2%.

PET sheet's heating simulation

After the validation of the model, heating of thin sheets is now performed. The predicted temperature using numerical simulations are compared to experimental temperature measurements in the case of a PET sheet of 1.5 mm thickness. The heating time is 35 s corresponding to typical heating times for industrial processes. Boundary conditions are given below:

À k @T @y y0 Àk @T @y ye hT S À T I ; @T @x x0;l @T @z z0;H 0 ( 11
)
An uniform convection coefficient is applied to the back and front faces (estimated to be 10 W m À2 K À1 ) [START_REF] Sacadura | Initiation aux Transferts Thermiques[END_REF].

The numerical modelling leads to the following temperature distribution through the sheet thickness, in particular, at the sheet centre (see Fig. 13).

During the heating time, a constant temperature gradient across the thickness equal to 7 8C appears. Then, as soon as the heating is switched off, the temperature difference between the front and back faces rapidly decreases. Five seconds are suf®cient to observe the establishment of an equilibrium temperature level all across the thickness. It provides a con-®rmation of the temperature increase, recorded experimentally from the back face as the natural cooling starts. The comparison between experimental and numerical results for the temperature pro®le versus z-axis is plotted in Fig. 14.

Points localised near the sheet centre and front of the heater (in a circle of 2 cm diameter), the difference between the two temperature results is less than 2%. For the other part of the sheet, the discrepancy is more important. This observation is valid for both front and back faces of the sheet. This difference is ®rst due to the incident ¯ux distribution on the irradiated face, more than the assumption, permitting the ¯ux absorption only normal to the incident surface. Several reasons could explain this inaccurate incident front conditions. In particular, the numerical geometry of the strip surface assumed to be equivalent to the radiative cylindrical heater surface. Previous workers [START_REF] Turner | Numerical and experimental analyses of the radiant heat flux produced by quartz heating systems[END_REF] have shown that the quartz tube effect is not negligible denoting a refraction of the radiative beam from the tungsten ®lament centred in the quartz tube. This will lead to an increase of the dispersion angle of the radiant heat ¯ux.

In Fig. 15, the temperature pro®le versus time in the zdirection is plotted. We observe good agreement between experimental and numerical results near the centre of the sheet. Comparisons during the cooling stage exhibits a faster temperature decrease, in particular, at the sheet top extremity. We suppose this discrepancy is due to the one-dimensional description of internal radiative heat transfer. One-dimensional modelling leads to minimised radiative source. It is more signi-®cant when the heat ¯ux arrived on the sheet surface with a consequent angle higher near the vertical sheet extremities.

Conclusion

Preliminary investigations of the infrared heater properties (spatial and spectral) ®tted with diffuse re¯ectors allow to consider them as an isotropic radiative source. The interaction between these lamps and PET sheet has been observed using an infrared camera. The temperature distribution from the back to the front face of the sheet has allowed an estimation of temperature gradient when the heating is switched off. The heating stage of the PET sheet, submitted to an industrial infrared heater, is taking account with the one-dimensional Beer±Lambert's law computed with a control volume model. Experimental and numerical results lead to conclude that the temperature distribution through the sheet thickness (1.5 mm) attain rapidly an equilibrate temperature during the cooling stage. However, due to the poor conductivity of the PET, the non-uniform temperature distribution over the irradiated faces of the sheet still appear after the cooling stage.

Considering an ef®ciency re¯ector coef®cient, the comparison between experimental and numerical results presents discrepancies higher from the centre of the sheet to the extremities due to the one-dimensional radiative heat transfer model.

Further developments will permit to take better into account the real interaction between PET preform and infrared heaters and the oven's environment: defining the reflector efficiency; observing transfer convection; applying this methodology to the tubular PET preform heating.

Fig. 1 .

 1 Fig. 1. Infrared heaters.

Fig. 2 .

 2 Fig. 2. Industrial heater (L400).

Fig. 3 .

 3 Fig. 3. Experimental set-up for infrared heaters characterisation.

Fig. 4 .

 4 Fig. 4. Infrared heater's directivity.

Fig. 5 .

 5 Fig. 5. Experimental heating set-up.

Fig. 6 .

 6 Fig. 6. The PET spectral absorbance.

Fig. 7 .

 7 Fig. 7. (a) Diffuse reflector for thermogram for L400. (b) Specular reflector for thermogram for L300.

Fig. 8 .

 8 Fig. 8. Measurement of the temperature gradient after heating stops.

Fig. 9 .

 9 Fig. 9. Transient evolution of the temperature.

Fig. 10 .

 10 Fig. 10. The sheet shape mesh.

Fig. 11 .

 11 Fig. 11. Radiative internal transfer.

Fig. 12 .

 12 Fig. 12. Comparison between numerical and analytical temperatures versus thickness.

Fig. 13 .

 13 Fig. 13. Numerical temperature distribution through the thickness.

Fig. 14 .

 14 Fig. 14. Gradient temperature in the z-direction.

Table 1

 1 PETs transmittance (8, 12 mm; blackbody temperature 150 C)

	Thickness (mm)	t sheet
	0.7 1.5	5 Â 10 À3 2 Â 10 À3

Table 2

 2

	Normal PET reflectance (8, 12 mm)
	Blackbody temperature (8C)	r n
	70	0.06
	110	0.07
	150	0.07

Acknowledgements

This work is supported by Perrier-Vittel Company. Special thanks goes to G. Denis, A. Contal and J.-P. Arcens.