Ofelia Maniti

Hong-Rong Piao

Jesus Ayala-Sanmartin
email: jesus.ayala-sanmartin@upmc.fr

Basic cell penetrating peptides induce plasma membrane positive curvature, lipid domain separation and protein redistribution

Keywords: Basic peptides, membrane curvature, membrane domains, membrane rolling, plasma membrane CPP, Cell Penetratin Peptide, GUV, Giant Unilamellar Vesicle, LUV, Large Unilamellar Vesicle, PC, L-α-phosphatidylcholine, PE, L--phosphatidylethanolamine, PMS, Plasma Membrane Sphere, PS, L-α-glycerophosphatidyl-

Basic cell penetrating peptides are tools for molecular cellular internalization of non membrane permeable molecules. Their uptake mechanisms involve energy-dependent and energy-independent pathways such as endocytosis, direct translocation or physical endocytosis. These mechanisms are ruled by both, the peptides physicochemical properties and structure and by the membrane lipids characteristics and organisation. Herein we used plasma membrane spheres and membrane models to study the membrane perturbations induced by three arginine-rich cell penetrating peptides. Nona-arginine (R9) and the amphipathic peptide RWRRWWRRW (RW9) induced positive membrane curvature in the form of buds and membrane tubes. Membranous tubes underwent rolling resulting in formation of multilamellar membrane particles at the surface of the plasma membrane spheres. The amphipathic peptides RW9 and RRWRRWWRRWWRRWRR (RW16) provoked lipid and membrane associated protein domain separation as well as changes in membrane fluidity and cholesterol redistribution. These data suggest that membrane domains separation and the formation of multilamellar membranous particles would be involved in arginine-rich cell penetrating peptides internalization.

Introduction

The interaction of proteins and peptides with membranes usually provokes a biological response leading to activation of intracellular transport mechanisms, protein transduction and antimicrobial defence. Based on arginine-rich peptide properties, cell penetrating peptides (CPPs) have been developed to carry active molecules into the cytoplasm of eukaryotic cells [START_REF] Mae | Cell-penetrating peptides as vectors for peptide, protein and oligonucleotide delivery[END_REF]Langel, 2006, Murriel and[START_REF] Murriel | Influence of protein transduction domains on intracellular delivery of macromolecules[END_REF]. It is generally considered that cellular delivery of basic peptides mainly results from two different molecular pathways: direct plasma membrane translocation and internalization by endocytosis with subsequent endosomal release into the cytoplasm [START_REF] Duchardt | A comprehensive model for the cellular uptake of cationic cell-penetrating peptides[END_REF][START_REF] Lundin | Distinct uptake routes of cell-penetrating peptide conjugates[END_REF]. Moreover, one peptide can use both pathways depending on its extracellular concentration [START_REF] Alves | Relationships between membrane binding, affinity and cell internalization efficacy of a cell-penetrating peptide: penetratin as a case study[END_REF][START_REF] Alves | Membrane interaction and perturbation mechanisms induced by two cationic cell penetrating peptides with distinct charge distribution[END_REF] or on temperature [START_REF] Jiao | Translocation and endocytosis for cell-penetrating peptide internalization[END_REF]. It has also been demonstrated that amphipathic and cationic peptides use different endocytic routes for cellular internalization [START_REF] Lundin | Distinct uptake routes of cell-penetrating peptide conjugates[END_REF]. Recently, it was shown that the formation of membranous particles at the cellular surface is involved in direct arginine-rich peptide penetration into the cytosol [START_REF] Hirose | Transient focal membrane deformation induced by arginine-rich peptides leads to their direct penetration into cells[END_REF], and that this phenomenon is favoured by hydrophobic groups on the peptides. Thus, peptide sequence and structure specificities determine the nature of peptide-membrane association and in fine the cellular uptake mechanisms. Therefore, the characterization of peptide-membrane interactions is essential to understand the membrane changes occurring during intracellular traffic and antimicrobial activity as well as the mechanisms of cellular uptake that will help to develop efficient vectors for future therapeutic delivery of molecules.

As in vivo phenomena linked to peptide-membrane interactions are complex, work on membrane models allowed to study the biophysical mechanisms and the characterization of peptide structure and affinity for different membranes (Eiriksdottir et al., Maniti et al., 2010[START_REF] Rydberg | Effects of tryptophan content and backbone spacing on the uptake efficiency of cell-penetrating peptides[END_REF][START_REF] Takechi | Physicochemical mechanism for the enhanced ability of lipid membrane penetration of polyarginine[END_REF][START_REF] Walrant | Membrane interactions of two arginine-rich peptides with different cell internalization capacities[END_REF]. Several studies showed that basic peptides are able to induce membrane negative curvature resulting in membrane invaginations in the form of vesicles and tubes as well as lipid phase separation [START_REF] Alves | Membrane interaction and perturbation mechanisms induced by two cationic cell penetrating peptides with distinct charge distribution[END_REF][START_REF] Lamaziere | Transduction peptides: structural-functional analyses in model membranes[END_REF][START_REF] Lamaziere | Tubular structures in heterogeneous membranes induced by the cell penetrating peptide penetratin[END_REF][START_REF] Lamaziere | Lipid domain separation, bilayer thickening and pearling induced by the cell penetrating peptide penetratin[END_REF][START_REF] Lamaziere | The homeodomain derived peptide Penetratin induces curvature of fluid membrane domains[END_REF][START_REF] Menger | Migration of poly-L-lysine through a lipid bilayer[END_REF]. These peptide-induced membrane deformations would result from changes in membrane asymmetry leading to changes in line tension and bending energy of membrane domains [START_REF] Lipowsky | The conformation of membranes[END_REF][START_REF] Lipowsky | Domain-induced budding of fluid membranes[END_REF]. Furthermore, insertion of polymers in the membrane increases bilayer asymmetry and produce positive membrane curvature resulting in budding and outward tubulation in Giant Unilamellar Vesicles (GUVs) [START_REF] Tsafrir | Budding and tubulation in highly oblate vesicles by anchored amphiphilic molecules[END_REF]. The induction of positive or negative curvature by association of proteins to the membrane has been well established for the so called curvature sensing proteins containing Bar-like domains [START_REF] Antonny | Membrane deformation by protein coats[END_REF][START_REF] Dawson | Bar domain proteins: a role in tubulation, scission and actin assembly in clathrin-mediated endocytosis[END_REF][START_REF] Saarikangas | Molecular mechanisms of membrane deformation by I-BAR domain proteins[END_REF].

The study of cellular implications of peptide membrane-interactions requires the use of cellular models of the plasma membrane. Giant plasma membrane vesicles (GPMVs) or plasma membrane spheres (PMS) are promising models of biological membranes. These vesicles are void of cellular organelles and nuclear fragments [START_REF] Scott | Plasma membrane vesiculation: a new technique for isolation of plasma membranes[END_REF] and their spherical morphology allows the observation of membrane deformations. Their molecular composition is representative of the plasma membrane. They contain intrinsic and extrinsic membrane proteins [START_REF] Baumgart | Large-scale fluid/fluid phase separation of proteins and lipids in giant plasma membrane vesicles[END_REF]. Their overall lipid composition is typical of the plasma membrane containing phosphatidylcholine (PC), phosphatidylethanolamine (PE), phosphatidylserine (PS) and phosphatidylinositols and, as for the plasma membrane, they are enriched on sphingomyelin (SM) and cholesterol [START_REF] Fridriksson | Quantitative analysis of phospholipids in functionally important membrane domains from RBL-2H3 mast cells using tandem high-resolution mass spectrometry[END_REF][START_REF] Scott | Plasma membrane vesiculation: a new technique for isolation of plasma membranes[END_REF]. Not only the global proportion of lipids is conserved but also the proportion of double bonds in acyl chains [START_REF] Fridriksson | Quantitative analysis of phospholipids in functionally important membrane domains from RBL-2H3 mast cells using tandem high-resolution mass spectrometry[END_REF]. Only two anomalies have been reported: a decrease in phosphatidylinositol-bis-phosphate [START_REF] Keller | PI(4,5)P2 degradation promotes the formation of cytoskeleton-free model membrane systems[END_REF] and the loss of PS asymmetry [START_REF] Baumgart | Large-scale fluid/fluid phase separation of proteins and lipids in giant plasma membrane vesicles[END_REF][START_REF] Keller | PI(4,5)P2 degradation promotes the formation of cytoskeleton-free model membrane systems[END_REF][START_REF] Lingwood | Plasma membranes are poised for activation of raft phase coalescence at physiological temperature[END_REF]. However, partial retention of lipid asymmetry has also been reported [START_REF] Yavin | Translocation and turnover of phospholipid analogs in plasma membrane-derived vesicles from cell cultures[END_REF]. PMS are able to undergo protein and phospholipid liquid ordered (Lo) and liquid disordered (Ld) phase separation [START_REF] Baumgart | Large-scale fluid/fluid phase separation of proteins and lipids in giant plasma membrane vesicles[END_REF][START_REF] Keller | PI(4,5)P2 degradation promotes the formation of cytoskeleton-free model membrane systems[END_REF][START_REF] Levental | Cholesteroldependent phase separation in cell-derived giant plasma-membrane vesicles[END_REF][START_REF] Levental | Raft domains of variable properties and compositions in plasma membrane vesicles[END_REF][START_REF] Lingwood | Plasma membranes are poised for activation of raft phase coalescence at physiological temperature[END_REF], they are depleted of metabolic energy and membrane cytoskeleton elements are absent [START_REF] Baumgart | Large-scale fluid/fluid phase separation of proteins and lipids in giant plasma membrane vesicles[END_REF][START_REF] Lingwood | Plasma membranes are poised for activation of raft phase coalescence at physiological temperature[END_REF]. Recently they have been used to study the penetration of several basic cell penetrating peptides [START_REF] Amand | Binding of cellpenetrating penetratin peptides to plasma membrane vesicles correlates directly with cellular uptake[END_REF][START_REF] Maniti | Metabolic energy-independent mechanism of internalization for the cell penetrating peptide penetratin[END_REF][START_REF] Saalik | Penetration without cells: membrane translocation of cell-penetrating peptides in the model giant plasma membrane vesicles[END_REF] and they allowed us to show peptide capacity to induce negative curvature with the consequent formation of plasma membrane invaginations (physical endocytosis) [START_REF] Maniti | Metabolic energy-independent mechanism of internalization for the cell penetrating peptide penetratin[END_REF].

In the present study we used plasma membrane spheres to characterize the interaction of three different basic peptides with the plasma membrane. Nona-arginine (R9) that penetrate cells efficiently [START_REF] Futaki | Argininerich peptides. An abundant source of membrane-permeable peptides having potential as carriers for intracellular protein delivery[END_REF], Mitchell et al., 2000[START_REF] Wender | The design, synthesis, and evaluation of molecules that enable or enhance cellular uptake: peptoid molecular transporters[END_REF], the amphypathic peptide RWRRWWRRW (RW9) because it has been shown that the addition of hydrophobic groups and tryptophanes to polyarginines increase cell penetration [START_REF] Futaki | Argininerich peptides. An abundant source of membrane-permeable peptides having potential as carriers for intracellular protein delivery[END_REF][START_REF] Hirose | Transient focal membrane deformation induced by arginine-rich peptides leads to their direct penetration into cells[END_REF][START_REF] Rydberg | Effects of tryptophan content and backbone spacing on the uptake efficiency of cell-penetrating peptides[END_REF], and RRWRRWWRRWWRRWRR (RW16) to study the effect of the length of an amphipathic peptide. We show that the association of small basic peptides with the plasma membrane results in the induction of positive curvature with subsequent vesicle bud formation and membrane tube elongation. Moreover, the membrane tubes underwent rolling and formed membranous particles attached to the plasma membrane surface. These effects were accompanied by changes in membrane fluidity, lipid domain separation and membrane-associated protein (annexin 2) redistribution.

Material and Methods

Materials.

Nona-arginine (R9), RWRRWWRRW (RW9) and RRWRRWWRRWWRRWRR (RW16)

where synthesized and purified as described [START_REF] Lamaziere | Nonmetabolic membrane tubulation and permeability induced by bioactive peptides[END_REF]. Laurdan was from Molecular Probes, di-4-ANNEPDHQ (ANE) was obtained from Dr Leslie M. Loew (Connecticut, USA) and pyrene-labelled cholesterol (Py-met-chol) was a kind gift of Dr.

André Lopez (Toulouse, France). Egg yolk L-α-phosphatidylcholine (PC), egg yolk L-phosphatidylethanolamine (PE), brain L-α-glycerophosphatidyl-L-serine (PS) and cholesterol were purchased from Sigma-Aldrich.

Plasma membrane spheres (PMS).

Standard and Annexin 2-GFP transfected MDCK cells were cultured in 75 cm 2 flasks in DMEM in standard conditions [START_REF] Ayala-Sanmartin | Homodimerization of Annexin 2. Role of the N-terminal tail and modulation of membrane aggregation properties[END_REF]. Plasma membrane spheres were obtained by the method described in [START_REF] Lingwood | Plasma membranes are poised for activation of raft phase coalescence at physiological temperature[END_REF] adapted in [START_REF] Maniti | Metabolic energy-independent mechanism of internalization for the cell penetrating peptide penetratin[END_REF]. Briefly, PMS induction was performed by overnight incubation of cells with 1 ml of PMS buffer (1.5 mM CaCl2, 1.5 mM MgCl2, 5 mM Hepes, 1 mg ml -1 glucose in PBS pH 7.4).

After PMS formation, 1 ml of PMS buffer was added, the culture was agitated gently and the supernatant containing the PMS was recovered and centrifuged at 1 000 rpm for 15 min. in a standard Eppendorf centrifuge. The PMS were recovered in 100 µl of PBS and conserved at 4°C for several hours. The experiments presented here were performed with 5 independent PMS preparations.

Peptide-PMS incubations and treatments.

All preparations for microscopic observation were performed on µ-slide 8 well slides (Ibidi) containing 160 µl of freshly prepared PMS. For di-4-ANEPPDHQ labelling, a stock solution (200 mM) in ethanol was diluted in PBS. Then, 20 µl were added for a final 5 µM concentration. Peptides were added in 10 µl for a final 10 µM concentration. For experiments in hypertonic solution, 20 or 40 µl of sucrose (500 mM) were added to attain a final concentration of 50 and 100 mM sucrose.

Confocal microscopy.

All confocal images were acquired with a TCS SP2 laser-scanning spectral system (Leica, Wetzlar, Germany) attached to a Leica DMR inverted microscope. Optical sections were recorded with a 63/1.4 or 100/1.4 immersion objectives. We systematically performed 30 confocal slices for each PMS analysed. Considering that most of the PMS were in the range of 15-20 μm, the acquisition of 30 slices allowed the observation of the PMS surface by steps of 500 to 700 nm. Images stacking and 3D-projections were obtained using ImageJ software (NIH). For di-4-ANEPPDHQ imaging, the samples were excited at 488 nm (Ar ion laser) and the fluorescence emission was collected at 570-590 nm for the liquid ordered (Lo) contribution and at 620-640 nm for the liquid disordered (Ld) contribution. Imaging of Annexin 2-GFP was performed by excitation at 488 nm and the emission light was collected at 500-520 nm.

Quantification of plasma membrane deformations.

The quantification of the different structures observed: tubes, rolled tubes, buds and deformations were performed by direct observation under the microscope. Protrusions were considered as tubes when the length was higher than the diameter. The experiments were performed two to four times. The number of PMS analyzed by direct microscopy was: 1656 for the control PMS, 409 for R9, 1150 for the RW9 and 322 for the RW16 treatments. The number of PMS analyzed by random confocal images were: 88 for the control PMS, 69 for R9, 96 for the RW9 and 69 for the RW16 treatments. The statistical significance was assessed by ANOVA with GraphPad Prism. * P<0.05, ** P<0.01, *** P<0.001.

Large unilamellar vesicles.

The appropriate amounts of lipids in a mixture of chloroform/methanol, 2/1 (v/v), were subjected to solvent evaporation under nitrogen. Four lipid compositions were used: PC/PS (75/25), PC/PS/cholesterol (50/25/25), PC/PS/PE (25/15/60) and PC/PS/PE/Cholesterol (17/12/52/19) by weight. Depending on the fluorescence experiments Laurdan was added at 0.1% weight, or Py-met-chol at 3% weight. Lipid films were hydrated with buffer (40 mM Hepes pH 7, 30 mM KCl, 1 mM EGTA) and vortexed extensively to obtain Multilamellar Vesicles (MLVs) at 1 mg/ml. Large Unilamellar Vesicles (LUV) were prepared by extrusion of MLVs through a polycarbonate filter (pore diameter 100 nm) as described [START_REF] Zibouche | The N-terminal domain of annexin 2 serves as a secondary binding site during membrane bridging[END_REF].

Fluorescence spectrometry.

Fluorescence spectra were acquired with a Cary fluorimeter (Varian) as described (Maniti et al., 2010). Temperature was regulated with a Peltier device and controlled by a thermocouple.

The excitation and emission band-pass were set at 5 nm. The Laurdan emission spectra were recorded from 400 to 650 nm using 365 nm excitation. The Py-met-chol emission spectra were recorded from 365 to 600 nm using 330 nm excitation. Sequential recorded spectra were: first, 250 μl of buffer (10 mM Hepes pH 7.4) containing 5 μg LUVs, and second the LUVs 10 min. after peptides addition. Spectra were obtained by triplicate and the buffer background was subtracted. The spectra were normalized as 100% for the maximal emission intensity. Peptides were added to a 1:25 peptide:lipid weight ratio (1-2 μM final concentration).

Results

3.1.Basic peptides induce membrane budding, outward tubulation and tube rolling.

The interaction with membrane lipids is a crucial step for cell penetrating peptides internalisation. Herein, we analysed peptide-induced membrane deformations using plasma membrane spheres (PMS) (Fig. 1a,b). We studied the efficient cell penetrating peptide R9, a variant with five arginine and 4 tryptophan residues and a longer variant (10 arginine and 6 tryptophane residues). Fig. 1c-f shows that two arginine rich peptides, the fully positive (R9), and the amphipathic (RW9) induced plasma membrane outward budding. The number of buds per PMS and their size were variable ranging from one to more than ten and with a radius from about tenths of μm to several μm (see also supplementary Fig. S1). To characterize accurately the membrane topology of the small buds (external to the PMS), we systematically performed 30 confocal slices for each PMS (see methods). A confocal slice-sequence of a PMS incubated with RW9 showing the conservation of membrane continuity between the PMS and the buds is shown in Movie 1.

Besides the buds, a second type of membrane positive curvature protrusion was observed in the form of long tubes (Fig. 1g-h). Tubes were more frequently observed with the RW9 peptide than with R9. Tube diameters were estimated in the range of ≤0.2 to 1 μm and their length varied from about 5 to more than 100 μm (see also Supplementary Fig. S2). The longer tubes seemed to be associated to PMS containing cellular rests and sometimes it was difficult to discern whether the tube grew from the PMS membrane or the cellular rests. We can assume that cellular rests may act as a reservoir for lipids that supply the growth of very long tubes.

The evolution of tubes and buds is illustrated in Fig. 2. Membrane protrusions (buds and tubes) were observed after 15 min. of peptide incubation. Once the bud or the tube starts to grow at the membrane surface, the process continued for a variable time (from 2 to more than 30 min.) (Fig. 2 and supplementary Figs. S1d,S2c). At the end of the growing process, the protrusions seemed to be stable. In fact we very rarely observed retraction of the buds or tubes for as long as four hours of observation. One example of retraction is shown in Fig. 2d. On the contrary, many of the tubes experimented a rolling process. Fig. 2e and the left part of Fig. 2f show confocal reconstructions of PMS just at the end of the rolling of a tube. Notice the movement of the small circular membranes during rolling. In the right part of Fig. 2f it is shown the PMS at the end of the tube rolling process. Notice that the circular membranes do not move anymore. This phenomenon takes about one second making extremely difficult to capture it by confocal microscopy. One example of such phenomenon occurring during confocal slicing of a PMS is shown in Movie 2.

The frequency of budding and tubulation in more than 3 800 PMS was quantified by microscopy. Not all control PMS were spherically perfect and sometimes bud-like structures were present. Fig. 3 shows that in control and in RW16 treated PMS about 13% of the vesicles presented small protrusions (buds or tubes). When incubated with R9 and RW9 about 21 and 28% of PMS showed protrusions respectively. The statistical analysis revealed that RW9 was the more efficient peptide in provoking protrusions, preferentially tubes (16% of tubes, and 12% of buds), and that R9 was more efficient in provoking bud formation (15% of buds, and 6% of tubes) (Fig. 3). However, the number of tubes was underestimated because as described above, many tubes experimented rolling. Therefore some tubes present in R9 and in RW9 incubated PMS were not counted because we were not sure of the origin of small structures attached to the PMS surface and, for quantification, we considered only unambiguous buds and extended tubes. Figure 3 show that the number of protrusions observed when PMS were incubated with the longer amphipathic peptide RW16 was smaller compared to RW9 indicating that the amphipathic peptide length is important for the induction of membrane positive curvature.

The formation of membrane protrusions may be influenced by membrane tension. Therefore, because RW9 showed the most efficient induction of protrusions, we studied its budding capacity in PMS at lower membrane tension. To reduce membrane tension, prior to peptide addition, the PMS were incubated in a hypertonic sucrose solution (50 and 100 mM in PBS).

As shown in Fig. 4b, RW9 was able to induce membrane budding very efficiently (high number of buds per PMS) indicating that a decrease in membrane tension favours peptide induced budding. At higher sucrose concentrations the PMS changed their morphology spontaneously precluding the analysis of peptide induced budding.

All the precedent experiments were performed with PMS labelled with the fluorescent probe di-4-ANEPPDHQ (ANE). To verify that the peptide induced positive curvature effect was not due to the fluorescent probe insertion in the membrane, we added the RW9 peptide to PMS form cells transfected with the membrane binding protein Annexin 2-GFP (Anx2-GFP). The GFP fluorescence was therefore used as the membrane probe. Fig. 4d shows that RW9 was able to induce plasma membrane budding in the absence of the probe ANE.

Membrane lipid and protein domain separation induced by basic peptides.

When performing the experiments with Annexin 2-GFP derived PMS we noticed that in several cases the membrane distribution of Anx2-GFP was perturbed by RW9. As shown in Fig. 5a,b, and in Movie 3, RW9 induced a redistribution of Anx2-GFP in Anx2-rich and Anx2-poor membrane domains. Considering that annexin 2 is a protein that binds only acidic phosholipids and that it is present in different membrane domains, we investigated whether the protein separation in domains was accompanied by lipid redistribution. We used the capacity of the ANE probe to sense rigid and fluid domains by changes in fluorescence emission intensities at 570-590 and 620-640 nm respectively [START_REF] Jin | Characterization and application of a new optical probe for membrane lipid domains[END_REF]. This probe does not respond to inserted peptides and is thus a good probe for lipid packing [START_REF] Dinic | Laurdan and di-4-ANEPPDHQ do not respond to membrane-inserted peptides and are good probes for lipid packing[END_REF]. In the absence of peptide, small fluid and rigid domains co-exist in the PMS that cannot be easily distinguished and merging of the green (rigid) and red (fluid) fluorescence results in a quite homogeneous yellow colour. With R9 peptide we saw no evident changes in the ratio of fluorescence intensities at 570-590 and 620-640 nm (membranes homogeneously coloured). With RW9 we observed changes in membrane fluidity in some PMS. In Fig. 2b the PMS core is yellowish by combination of green (570-590 nm ordered lipids contribution) and red (620-640 nm fluid lipids contribution), and the bud is greener indicating a more ordered membrane domain. With RW16 (Fig. 5d-e, supplementary Fig. S3 and in Movies 4 and 5), we observed stronger differences in the 570-590 and 620-640 nm contributions indicating an efficient separation of membrane lipid domains by RW16.

To get more insight into the peptide induced lipid domain separation, we prepared large unilamellar vesicles labelled with Laurdan, a probe able to monitor lipid fluidity by spectral changes. The fluorescence intensities at the maximum emission wavelength in the ordered (440 nm) and disordered (490 nm) phases characterize the spectrum. In LUVs composed of PC alone, the peptides did not induce changes in membrane fluidity (not shown). However, and as shown in Fig. 5f, the addition of peptides in LUVs containing PS induced a relative increase of the ordered (440 nm) contribution. For LUVs containing cholesterol (PC/PS/cholesterol) a relative decrease of the fluid membrane contribution (490 nm) was observed (Fig. S4a). This effect indicates a global decrease of membrane fluidity after peptide binding. We tested other lipid compositions such as PC/PS/PE/cholesterol and the membrane rigidification effect (increase in the ordered contribution) was also observed (Fig. S4b).

Notice that in line with the confocal microscopy data, RW16 showed the stronger capacity to reduce membrane fluidity. A second fluorescent probe (Pyrene-labelled cholesterol, Py-metchol) was used to monitor lipid redistribution. Py-met-chol is able to follow the cholesterol enrichment of a domain by the induction of a fluorescent signal (at 475 nm) corresponding to the packing of the probe as oligomers (excimers) (Le [START_REF] Guyader | Changes of the membrane lipid organization characterized by means of a new cholesterol-pyrene probe[END_REF]. We studied the effect of peptides on PC/PS/PE LUVs with and without cholesterol. Fig. 5g and Fig. S4c show that R9 and RW9 induced a small and a moderate reduction of Py-met-chol excimers respectively. The stronger effect was observed with RW16 which decreased the Py-met-chol dimerisation in membranes with and without cholesterol. Overall, these data demonstrate that the basic and, especially the amphipathic peptides, are able to induce lipid and protein redistribution on membranes.

Discussion

Direct interaction of molecules with membrane phospholipids can induce diverse membrane deformations with positive or negative curvature as has been shown for Bar and IBar proteins respectively [START_REF] Antonny | Membrane deformation by protein coats[END_REF][START_REF] Saarikangas | Molecular mechanisms of membrane deformation by I-BAR domain proteins[END_REF]. Penetratin and other cell penetrating peptides and toxins were shown to induce negative curvature resulting in membrane invaginations on giant unilamellar vesicles [START_REF] Lamaziere | Nonmetabolic membrane tubulation and permeability induced by bioactive peptides[END_REF][START_REF] Lamaziere | Transduction peptides: structural-functional analyses in model membranes[END_REF][START_REF] Lamaziere | The homeodomain derived peptide Penetratin induces curvature of fluid membrane domains[END_REF][START_REF] Romer | Shiga toxin induces tubular membrane invaginations for its uptake into cells[END_REF] and in plasma membrane spheres [START_REF] Maniti | Metabolic energy-independent mechanism of internalization for the cell penetrating peptide penetratin[END_REF]. Herein we studied three peptides; one short basic non structured (R9) and two structured as amphipathic helix, one short (RW9) and one longer (RW16) with different penetration efficacy. The peptide-membrane interaction was studied in LUVs of different composition and in PMS. PMS were used as a plasma membrane model. One important modification of the PMS compared to the plasma membrane of cells is that the PS asymmetry is perturbed inducing a high abnormal PS content at the extracellular surface. This fact would also modify the peptide behaviour compared to the plasma membrane. However, compared to LUVs and GUVs, the PMS, as mentioned in the introduction, contain all the membrane proteins absent in other models and the proportion of lipids and their modifications and molecular diversity are mostly conserved making of them, to our knowledge the closer model of a living cell plasma membrane. The data show that R9 and RW9 are able to induce positive curvature in plasma membrane spheres and that both peptides along with RW16 induce lipid domains reorganization and/or membrane fluidity reduction in PMS and/or LUVs. Each peptide showed particular behaviour indicating that there is a structural specificity of peptide activities and some tendencies were observed and quantified as statistically significant. R9 induced preferentially budding, RW9 induced strong membrane budding and tubulation and RW16 showed the highest lipid domain separation activity. Strong changes in tension and geometry of membranes in the form of tubes rolling were also observed. It is interesting to note that recently it was shown that the direct penetration of polyarginine (R12) into cells is preceded by the formation of membrane particles structured in multilamellar stacks of about 1 μm in diameter on the cellular surface [START_REF] Hirose | Transient focal membrane deformation induced by arginine-rich peptides leads to their direct penetration into cells[END_REF], and that the lysine rich peptide S4(13)-PV accumulates also at the cell surface in complex multilamellar structures [START_REF] Cardoso | S4(13)-PV cell-penetrating peptide induces physical and morphological changes in membranemimetic lipid systems and cell membranes: implications for cell internalization[END_REF]. Our data suggest that the rolling of tubes observed could be the mechanism leading to multilamellar structures of about 1 μm that would explain the formation of membranous particles observed by others [START_REF] Hirose | Transient focal membrane deformation induced by arginine-rich peptides leads to their direct penetration into cells[END_REF] and [START_REF] Cardoso | S4(13)-PV cell-penetrating peptide induces physical and morphological changes in membranemimetic lipid systems and cell membranes: implications for cell internalization[END_REF] and involved in cell penetration. Moreover, the amphipathic peptide RW9 was shown to be more efficient than R9 in tube formation, and as pointed in the Hirose's study [START_REF] Hirose | Transient focal membrane deformation induced by arginine-rich peptides leads to their direct penetration into cells[END_REF] membrane particles formation and cell penetration were increased by addition of hydrophobic groups to R12. After formation of multilamellar structures at the cell surface, the molecular mechanism for the final penetration into the cytosol [START_REF] Hirose | Transient focal membrane deformation induced by arginine-rich peptides leads to their direct penetration into cells[END_REF] remains unknown.

The authors also suggested that the induction of membrane phase separation upon particle formation may accelerate the peptide penetration into cells. This hypothesis is strongly supported by the membrane domain separation demonstrated in the present study. Overall, positive curvature induction will favour peptide penetration as supported by the fact that R9 and RW9 which provoke stronger curvature are more efficiently internalized than RW16 [START_REF] Jiao | Translocation and endocytosis for cell-penetrating peptide internalization[END_REF]. Moreover, this point is supported by a recent study showing that the membrane positive curvature inducing peptide EPN18 increase the translocation of Alexa488-labeled R8 [START_REF] Pujals | Curvature engineering: positive membrane curvature induced by epsin N-terminal Peptide boosts internalization of octaarginine[END_REF].

What are the molecular mechanisms involved in these peptides-induced membrane changes?

The electrostatic interaction of arginines with phospholipids forming guanidine hydrogen bonds seems to be the first step [START_REF] Rothbard | Role of membrane potential and hydrogen bonding in the mechanism of translocation of guanidinium-rich peptides into cells[END_REF]. This interaction would be stable for R9 that remains unstructured upon membrane binding. However, for the peptides RW9 and RW16, the folding as an amphipathic α-helix upon membrane binding (Eiriksdottir et al., Maniti et al., 2010[START_REF] Rydberg | Effects of tryptophan content and backbone spacing on the uptake efficiency of cell-penetrating peptides[END_REF][START_REF] Takechi | Physicochemical mechanism for the enhanced ability of lipid membrane penetration of polyarginine[END_REF][START_REF] Walrant | Membrane interactions of two arginine-rich peptides with different cell internalization capacities[END_REF] would induce an hydrophobic type of interaction involving snorkelling [START_REF] Mishra | Interaction of model class A1, class A2, and class Y amphipathic helical peptides with membranes[END_REF] of the small RW9 helix, and deeper insertion into the bilayer for the longer peptide RW16.

How positive curvature is induced by these peptides? Membrane budding can be produced by changes in the volume/membrane surface ratio by heating [START_REF] Dobereiner | Budding and fission of vesicles[END_REF][START_REF] Mui | Influence of transbilayer area asymmetry on the morphology of large unilamellar vesicles[END_REF]. However the differences observed in peptide behaviour cannot be explained by merely heating the samples. The curvature induction mechanism could be based in the increase of membrane bilayer asymmetry. The insertion of molecules in the external leaflet of the bilayer increases its mass and consequently the positive curvature as has been shown for polymers with hydrophobic groups [START_REF] Tsafrir | Budding and tubulation in highly oblate vesicles by anchored amphiphilic molecules[END_REF]. A model of the differential peptides action on the membrane is schematized in Fig. 6. R9 will induce a small increase in asymmetry by its interaction with the phospholipid head groups. The induction of positive curvature by R9 does not need the formation of an amphipathic helix. This capacity to create positive curvature without significant membrane insertion is in agreement with a study showing that coating of proteins at the membrane surface without helix insertion into the bilayer can induce membrane tube formation [START_REF] Stachowiak | Steric confinement of proteins on lipid membranes can drive curvature and tubulation[END_REF]. RW9 folded as an amphipathic α-helix would snorkel into the external leaflet of the bilayer with a stronger increase in the external leaflet phospholipid spacing inducing higher curvature. This stronger curvature will favour the RW9-mediated formation of tubes with shorter diameter than buds. RW16 is also able to form an amphipathic α-helix and to snorkel, but being longer than RW9, it would be able to insert into the bilayer perturbing and/or increasing also the internal leaflet mass and therefore inducing a small membrane asymmetry insufficient to provoke significant budding and tubulation. The stronger insertion of RW16 compared to the RW9 peptide is supported by two facts: first, the tryptophan spectral shift observed after membrane binding of RW16 is stronger than that of RW9 [START_REF] Lamaziere | Nonmetabolic membrane tubulation and permeability induced by bioactive peptides[END_REF] indicating a more hydrophobic environment for RW16, and second, that, on the contrary to RW9, RW16 is able to permeabilize the membranes inducing the formation of pores in living cells [START_REF] Lamaziere | Nonmetabolic membrane tubulation and permeability induced by bioactive peptides[END_REF] and in DOPC and DOPC/DOPG LUVs (Jobin et al.). This membrane permeabilisation would indicate a deeper perturbing interaction of RW16 compared to RW9. The exact mechanism for this behaviour is not known at present but we can speculate that to allow membrane insertion of the amphipathic helix RW16 molecules will associate by their arginines thanks to ionic bridges. This peptide aggregation mediated hypothesis is supported by the fact that RW16 induced membrane permeabilization in LUVs is cooperative (Jobin et al.). Therefore, the differential type of peptide membrane interaction creating different degree of asymmetry would produce weak or strong curvature (buds and/or tubes of different diameter).

From a dynamic point of view, the most remarkable behaviour of membrane tubes was rolling. We suggest that rolling is determined by the capacity of the peptides to aggregate [START_REF] Lamaziere | The homeodomain derived peptide Penetratin induces curvature of fluid membrane domains[END_REF][START_REF] Stachowiak | Steric confinement of proteins on lipid membranes can drive curvature and tubulation[END_REF]. Once the tube is homogeneously covered with the peptide inductor of positive curvature, the tube remains extended. The accumulation of the peptide in one side of the tube will result in a redistribution of the curvature effect.

Positive curvature will concentrate on one side and the relaxation of the positive curvature (increase in negative curvature) in the opposite side of the tube will furnish the driving force for rolling (Fig. 6e).

In precedent studies it was shown that short basic peptides were able to induce negative curvature in GUVs [START_REF] Lamaziere | The homeodomain derived peptide Penetratin induces curvature of fluid membrane domains[END_REF]. Herein, R9 and RW9 showed the capacity to induce positive curvature in PMS. Two factors would be responsible for the observed difference between GUVs and PMS. The first is that lipid complexity is stronger in PMS than in the GUVs. We previously show that the peptides were unable to induce negative curvature in Lo phases rich in sphingomyeline and cholesterol in GUVs [17]. The PMS contain higher contents of these lipids and therefore the capacity to induce negative curvature would be also reduced and the positive curvature by bilayer asymmetry (mass increase of the external leaflet) will be allowed. The second factor would be the presence of proteins in the PMS. The absence of proteins in GUVs would favour the movement and clusterization of the peptides facilitating peptide-lipid reorganization competent for negative curvature. In PMS, the presence of proteins would reduce peptide diffusion and clusterization resulting in preferential membrane asymmetry (by mass increase) and positive curvature effects.

Membrane domain separation was observed by redistribution of the membrane-associated protein annexin 2 as well as by separation of domains with different lipid packing (fluidity).

What is the basis of the observed membrane domain separation? First, the different types of interaction (hydrogen bonding for R9, snorkelling for RW9 and bilayer insertion for RW16) with different affinities for phospholipids would result in specific membrane lipids redistribution (Fig. 6). The second factor is that membrane curvature by itself, may induce lipids redistribution as has been shown in GUV membrane tubes obtained by pulling or by photo-oxidation [START_REF] Roux | Role of curvature and phase transition in lipid sorting and fission of membrane tubules[END_REF][START_REF] Yuan | Lipid bilayer discs and banded tubules: photoinduced lipid sorting in ternary mixtures[END_REF]. How peptide-induced lipids redistribution explain the segregation of annexin 2-rich and poor domains in the inner leaflet of the membrane? Molecular modelling suggested that complementary matching of lipids between the inner and the outer leaflet of the membrane take place [START_REF] Stevens | Complementary matching in domain formation within lipid bilayers[END_REF] and that disturbances of anchored membrane proteins in one leaflet, induce redistribution of proteins in the opposite leaflet [START_REF] Yue | Signal transduction across cellular membranes can be mediated by coupling of the clustering of anchored proteins in both leaflets[END_REF]. It has also been demonstrated that cholera toxinB induced lipid separation in the external leaflet provokes redistribution of the cytosolic lipid bound protein Pal-Pal-mCFP [START_REF] Lingwood | Plasma membranes are poised for activation of raft phase coalescence at physiological temperature[END_REF]. Moreover, it was experimentally shown that changes in one membrane leaflet composition were able to induce domain modification in the other leaflet [START_REF] Collins | Tuning lipid mixtures to induce or suppress domain formation across leaflets of unsupported asymmetric bilayers[END_REF]. Therefore, by coupling between the outer and inner layers, the induced lipids redistribution in one side of the membrane might induce reorganization of lipids domains of different affinity for annexin 2 in the other side of the membrane.

It is known that the membrane geometry and lipids domain separation is a very important factor in cell biology. These processes are involved in most of the membrane related functions especially in vesicular and tubular intracellular transport [START_REF] Ayala | Transport and internal organization of membranes: vesicles, membrane networks and GTP-binding proteins[END_REF][START_REF] Simons | Revitalizing membrane rafts: new tools and insights[END_REF][START_REF] Van Meer | Membrane lipids: where they are and how they behave[END_REF] and in peptide cell penetration by membrane particle formation at the cell surface [START_REF] Hirose | Transient focal membrane deformation induced by arginine-rich peptides leads to their direct penetration into cells[END_REF][START_REF] Palm-Apergi | Do cell-penetrating peptides actually "penetrate" cellular membranes?[END_REF].

Conclusion

In conclusion, basic cell penetrating peptides are able to penetrate cells by different mechanisms: by direct crossing of the plasma membrane, by metabolic endocytosis and by "physical endocytosis" if negative curvature is allowed. In this report we show that basic cell penetrating peptide induce membrane domain reorganization, membrane positive curvature and the formation of multilamellar membranous particles formed by membrane rolling. These properties should have implications for peptide-mediated intracellular delivery by formation of membranous structures at the cell surface.

Figures legends

Fig. 1

 1 Fig. 1 Basic peptides induce membrane protrusions on PMS. Confocal images of PMS labelled with the fluorescent probe di-4-ANEPPDHQ. Peptide-free PMS at 15 (a) and 180 min. (b) after di-4-ANEPPDHQ addition. PMS incubated with R9 (c,d), and RW9 (e,f). At the left of the panels are the projections of 30 confocal slices, and at right a selected slice. Notice that the apparent internal vesicles are buds from the plasma membrane. Some buds are pointed by arrows (see also movie 1). Membrane tubes from PMS after incubation with RW9 (g,h). Notice the different length and diameter of the membrane protrusions. Scale bars 10 μm

Fig. 2

 2 Fig. 2 Evolution of membrane protrusions of PMS after incubation with basic peptides. Confocal projections of PMS labelled with the fluorescent probe di-4-ANEPPDHQ. (a); PMS after incubation with R9 at 2 min. interval (left to right). Notice the internal tube (arrowhead) (see (Maniti et al., 2012)). (b); PMS at 20 and 50 min. intervals (left to right) after RW9 addition. (c); PMS at 20 min. intervals (left to right) after incubation with RW9. The "shadows" of the tube in the central image (arrow) are due to tube mobility. (d); Tubes from one PMS at 90 min. interval (top to bottom) after RW9 addition. Notice that the proximal part of the tube attached to the PMS is thicker than the distal part of the tube. The arrow indicate the retraction of a tube. (e); A rolling tube on a PMS surface after RW9 incubation. (f); PMS incubated with RW9 at 2 min. interval (left to right). Notice in (e) and at the left of (f) the shadows during the movement of a tube at the end of the rolling process (arrow). At right of (f) the rolled tube remains stable. Scale bars 10 μm.

Fig. 3

 3 Fig. 3 Frequency of membrane protrusions induced by basic peptides. In (a), bars represent the percent of spherical PMS (none) and the percent of PMS with buds or tubes (protrusions). In (b), bars represent the percent of PMS with buds or with tubes in the whole PMS population. 3 855 PMS were analyzed. Bars are the mean +/-SEM from 2 to 4 independent experiments. * P<0.05, ** P<0.01, *** P<0.001

Fig. 4

 4 Fig. 4 RW9-induced membrane budding in hypertonic solution and in the absence of lipid fluorescent probe. (a); Confocal images of peptide-free PMS labelled with the fluorescent probe di-4-ANEPPDHQ 15 min. after sucrose addition. (b); PMS after RW9 incubation in 100 mM sucrose. (c); Confocal images of peptide-free PMS from Annexin 2-GFP transfected MDCK cells 90 min. after sample preparation under the microscope. Annexin 2-GFP is associated to the plasma membrane. (d); Annexin 2-GFP-PMS after RW9 incubation. At the left of the panels are the projections of 30 confocal slices, and at right a selected slice. Notice that the apparent internal vesicles are outward buds from the plasma membrane. Some buds are pointed by arrows. Scale bars 10 μm.

Fig. 5

 5 Fig. 5 Basic peptides provoke plasma membrane domain separation. Projections of 30 confocal slices from Anx 2-GFP-PMS incubated with RW9 (a,b). Notice the heterogeneous distribution of Anx 2-GFP compared with the control in Fig. 4c. Annexin 2-GFP rich domains (arrows), Anx 2-GFP poor domains (arrowheads). c and d are projections of three consecutive confocal slices of PMS labelled with the probe di-4-ANEPPDHQ (c-d). The colours code for the preferentially ordered rigid domains (570-590 nm in green) and the preferentially fluid domains (620-640 nm in red). Peptide-free PMS after 180 min. of labelling (c). PMS after RW16 addition (d). Zoom of a confocal slice (e) showing the differences in colour intensities (preferential fluid domains arrows, preferential rigid domain arrowhead). See Movies 4 and 5. Scale bars 10 μm. (f); Normalized Laurdan spectra of PC/PS LUVs (75/25) at 37°C. The increase of the rigid-ordered contribution (440 nm) and the decrease of the fluid domain contribution (490 nm) are indicated by arrows. (g); Normalized Py-met-chol spectra of PC/PS/PE/Py-met-chol LUVs (25/15/60/3) at 23°C. The decrease of probe dimers contribution (475 nm) is indicated by the arrow. LUVs (black lines), LUVs incubated with R9 (red dotted lines), RW9 (green dashed lines) and RW16 (blue dash-dotted lines).

Fig. 6

 6 Fig. 6 Schematic model of membrane morphological changes induced by basic peptide-lipid interaction. (a); Membrane bilayer before peptide interaction. (b); R9 (gray structures) interacts with the phospholipid head groups by guanidine hydrogen bonding creating a small separation between them. (c); The amphipathic α-helix of RW9 (small gray helix) is able to snorkel into the external leaflet creating stronger separation of phospholipids and thus stronger positive curvature. (d); The amphipathic α-helix of RW16 (long gray helix) is able to insert profoundly into the bilayer inducing a small change in membrane curvature. To allow membrane insertion of the amphipathic helix molecules of RW16 will associate by the arginines thanks to ionic salt bridges. Lipid domains separation is a consequence of specific peptide-lipid affinities. (e); Peptide redistribution as the driving force for rolling. Progressive accumulation of peptide in one side of the tube would change the positive curvature redistribution resulting in tube rolling and in formation of membranous multilayer structures at the cell surface. For more details see discussion

Acknowledgments

We thank Philippe Fontanges and Romain Morichon for the confocal facility of the platform Imagerie cellulaire et tissulaire (IFR65 Hôpital Tenon, Paris, France). Dr. André Lopez (Toulouse, France) for the pyrene-labelled cholesterol (Py-met-chol), Antonin Lamazière for peptides and useful comments, and Sandrine Sagan for useful discussions. This study was supported by the Centre National de la Recherche Scientifique (CNRS) and the Université Pierre et Marie Curie.


Ruben Di Battista
email: ruben.di-battista@polytechnique.edu

Iván Bermejo-Moreno

Thibaut Ménard

Stéphane De Chaisemartin

Marc Massot

invariants to extract flow statistics: application to canonical objects and the collision of two droplets

Keywords: DNS, geometrical, post-processing, curvature

teaching and research institutions in France or abroad, or from public or private research centers.

Introduction

Classically two-phase flow modeling has been addressed with reduced-order models because generally, even in modern times, the cost of performing high-fidelity simulations is prohibitive for configuration even mildly more complex than academic benchmark cases. A classical work on the subject is [START_REF] Ishii | Two-Fluid Model and Hydrodynamic Constitutive Relations[END_REF] where the multiphase flow equations are derived through an averaging process performed on the instantaneous phase flow equations. Another important result in the frame of two-phase flow modeling is presented in [START_REF] Baer | A Two-Phase Mixture Theory for the Deflagration-to-Detonation Transition (DDT) in Reactive Granular Materials[END_REF], where the entropy inequality is enforced on the equations system providing a problem that is mathematically well-posed in the context of simulation of solid propellants combustion. [START_REF] Gavrilyuk | A New Form of Governing Equations of Fluids Arising from Hamilton's Principle[END_REF] rediscover the variational approach to derive models for two-phase flow sim-ulation. In [START_REF] Baer | A Two-Phase Mixture Theory for the Deflagration-to-Detonation Transition (DDT) in Reactive Granular Materials[END_REF], the procedure to derive equations does not explicitly highlight the convective part of the problem w.r.t. the dissipative part as it's done in [START_REF] Gavrilyuk | A New Form of Governing Equations of Fluids Arising from Hamilton's Principle[END_REF] and [START_REF] Drui | A Hierarchy of Simple Hyperbolic Two-Fluid Models for Bubbly Flows[END_REF]. In particular, in the work of Drui et al., a hierarchy of two-phase flow models is retrieved applying a variational approach that clearly provides as outcome the convective part of the system. The dissipative part is then retrieved again enforcing the entropy inequality, clearly depicting the roles of the different parts composing the system. In [START_REF] Drui | A Hierarchy of Simple Hyperbolic Two-Fluid Models for Bubbly Flows[END_REF], an energy term responsible for the pulsation of bubbles in the unresolved scale (subscale energy term) is proposed. In these modeling efforts, geometrical configuration of the interface between the two phases is generally described only through the volume fraction, while richer approaches could be beneficial to simulation fidelity. A first work on the matter has been proposed by [START_REF] Pope | The Evolution of Surfaces in Turbulence[END_REF] and [START_REF] Drew | Evolution of Geometric Statistics[END_REF], where governing equations for geometrical parameters are given in the context of flame front propagation. Moreover [START_REF] Essadki | Statistical Modeling of the Gas-Liquid Interface Using Geometrical Variables: Toward a Unified Description of the Disperse and Separated Phase Flows[END_REF] have shown that intrinsic topological relation as the Gauss-Bonnet theorem can be used to rely different regimes in a two-phase flow system (with particular interest to injector systems) allowing the connection between the Surface Density Function (SDF), i.e. a statistical description locally to the resolved interface, to the Number Density Function (NDF) in the disperse-phase region of an injection system. A first example of a reduced model for two-phase flows that contains additional geometrical contributions is reported in [START_REF] Cordesse | Derivation of a Two-Phase Flow Model with Two-Scale Kinematics, Geometric Variables and Surface Tension Using Variational Calculus[END_REF], where the Least Action Principle (LAP) is used to obtain a set of equations that include a macroscopic term and a microscopic (subscale) term linked to surface tension. Most of the times introducing additional richness to the geometrical description of the interface brings unclosed terms that need closure, the work described in this paper is motivated by the need of developing such closure. The characterization of 3D objects in terms of geometrical features in high-fidelity simulations is not new in literature, even though, at the best of our knowledge, we did not found other studies directly applied to injections problems. Mean and Gauss curvatures are not the only way to parameterize the phase space of interest, in facts [START_REF] Lewiner | Efficient Implementation of Marching Cubes' Cases with Topological Guarantees[END_REF] introduces two non-linear functions of the curvatures above mentioned, Curvedness and Shape Index, that are then used in Bermejo-Moreno and Pullin (2008) and [START_REF] Bermejo-Moreno | Geometry of Enstrophy and Dissipation, Grid Resolution Effects and Proximity Issues in Turbulence[END_REF] to provide a throughout analysis and characterization of objects within turbulent flow. In [START_REF] Leung | Geometry and Interaction of Structures in Homogeneous Isotropic Turbulence[END_REF] Minkowski functionals are used instead to perform a similar analysis. [START_REF] Dopazo | Strain, Rotation and Curvature of Non-Material Propagating Iso-Scalar Surfaces in Homogeneous Turbulence[END_REF] track iso-surfaces in homogeneous turbulence also in terms of curvature value. The prominent flow contributions to the flame curvature generation are highlighted in [START_REF] Cifuentes | Analysis of Flame Curvature Evolution in a Turbulent Premixed Bluff Body Burner[END_REF] within the context of the combustion in a bluff body burner.

In this study we propose the statistical characterization of two canonical objects of which analytical expression for the curvatures are available (a sphere and an ellipsoid) in order to assess the robustness and reliability of the computational strategy. We then study the Direct Numerical Simulation (DNS) simulation of two droplets colliding against each other in terms of geometrical properties. In section 2 we present the numerical algorithms needed to compute curvature values no the discrete surface triangulation: we present a way of computing these values as 1-ring average values as proposed by [START_REF] Meyer | Discrete Differential-Geometry Operators for Triangulated 2-Manifolds[END_REF]. In section 3 then we analyze the evolution of the geometrical statistical maps while the simulation progresses in order to catch interesting correlations or patterns that could be useful to formulate closures for unknown terms or to enrich existent CFD models and we furthermore clarify how the algorithms behave in term of intrinsic constraints preservation. Section 4 will be reserved to our final remarks.

Geometrical properties as 1-ring averaged values

In this section we describe the numerical strategies used to estimate the geometrical properties on a triangulated surface. The triangulated surface is obtained applying a contouring procedure [START_REF] Schroeder | Flying Edges: A High-Performance Scalable Isocontouring Algorithm[END_REF]) to the initial level-set that is available on the entire volume. The DNS simulations on which we apply these estimations are performed using the ARCHER code from the CORIA lab [START_REF] Ménard | Coupling Level Set/VOF/Ghost Fluid Methods: Validation and Application to 3D Simulation of the Primary Break-up of a Liquid Jet[END_REF]. The volumetric level-set field is given on a regular uniform rectangular grid as the one shown in the slice of fig. 1. We must note that the quality of the estimations are influenced not only by the quality of the strategy chosen to approximate the topological parameters, but also on the quality of the triangulation. Even though the Marching Cubes (MC) algorithm (with its variants as the Flying Edges (FE) algorithm used here) is a well-established method to obtain isolines of a field in a volume, there exist also more advanced revisions [START_REF] Lewiner | Efficient Implementation of Marching Cubes' Cases with Topological Guarantees[END_REF]) of the MC algorithm that ensure additional topological guarantees that could improve the initial estimates for the needed quantities.

The algorithm presented in this section is based on [START_REF] Meyer | Discrete Differential-Geometry Operators for Triangulated 2-Manifolds[END_REF]. The idea is to approximate the value of the mean curvature H and the Gauss curvature G on a vertex Figure 3: A mixed for a general case where one of the triangles is obtuse. The light gray area is the 1-ring neighbourhood and the dark gray area is actually A mixed of the triangulated surface with the average value computed around the 1-ring neighbourhood of said vertex. The 1-ring neighbourhood of a point x i is the region included in the set of points x j that are directly connected to x i , fig. 2 shows a graphical representation. The other quantities are derived from H and G. This strategy of computing geometrical properties has been implemented in a GPL-licensed library called Mercur(v)e (Di Battista ( 2018)).

In order to compute the mean curvature value, the algorithm leverages a local to the 1-ring neighbourhood discretization of the Laplace-Beltrami (LB) operator whose continuous expression is reported in eq. (1).

LB(x) = 2H(x) n(x) = K(x) (1)
The LB operator can be approximated on a triangulated surface via eq. ( 2).

• K (x i ) = 1 2A mixed 1-ring j (cot α ij + cot β ij ) (x i -x j ) (2) With •
• we denote the discrete estimation of a certain quantity on the mesh averaged around the 1-ring. Once the discrete LB has been computed, the scalar value of the mean curvature is simply the half of the norm of the LB vector:

• H = 1 2 • K (3)
A mixed is the equivalent 1-ring area: it is equal to the Voronoi region area if all the triangles that compose the 1-ring are non-obtuse, otherwise is the region whose perimeter is given by all the circumcenters of the non-obtuse triangles and the midpoints of the edges opposed to the obtuse angles for the triangles that are obtuse. See fig. 3 for clarity. The actual algorithm to compute A mixed is given in [START_REF] Meyer | Discrete Differential-Geometry Operators for Triangulated 2-Manifolds[END_REF],

and it is reported in algorithm 1 for reference. What is interesting is to note that the area for a Voronoi region (i.e. the area of the polygon given by all the circumcenters of the triangles composing the 1-ring, all of them being non-obtuse) is given by eq. ( 4), where the (cot α ij + cot β ij ) term can be reused successively in eq. ( 2), saving up on computational cost.

A voronoi = 1 8 1-ring j (cot α ij + cot β ij ) |x i -x j | 2 (4)
Algorithm 1 Algorithm to compute the mixed area 1: procedure MIXEDAREA(x i ) We compute the A mixed on the 1-ring of x i 2:

A mixed = 0

3:

for T in ONERING(x i ) do For each triangle composing the 1-ring 4:

x i , x ij , x ik ← ANGLES(T )
x ij and x ik are the 1-ring points connected to x i 5:

A ←AREA(T ) 6: if cot x ij > 0 or cot x ik > 0 then 7: A mixed += A/4 8: else if cot x i > 0 then 9: A mixed += A/2 10: else 11: A mixed += 1 8 |x i -x ik | 2 cot x ij + |x i -x ij | 2 cot x ik
In order to compute the value of the Gauss curvature G we can just rely on the definition of G for a geodesic polygon:

Definition 1 given a geodesic polygon close to a point x i of area A, the product of the Gauss curvature times the area of the polygon is equal to 2π minus the defect of the exterior angles k of the polygon

G = 1 A 2π - k k (5)
Since for the triangulation of a surface the 1-ring neighbourhood of a point x i is actually a special case of a geodesic polygon for which the edges are straight lines instead of geodesics, the same relation can be recasted for the internal angles θ j , as shown in eq. ( 6).

• G = 1 A mixed   2π - 1-ring j θ j   (6)
Once the mean and Gauss curvature are calculated, the other typical differential geometry parameters can be retrieved from the values of H and G. For example the principal curvatures k 1 and k 2 are computed as follows:

k 1 (x i ) = • H (x i ) + • ∆ (x) (7) k 2 (x i ) = • H (x i ) - • ∆ (x) (8)
As we will see later in section 3, this strategy allows to provide estimates on the triangulation of 3D objects that are quite smooth w.r.t. to other strategies. A problem that arises, unfortunately, is linked to the fact that H and G are calculated independently one from the other. That means in particular that ∆ = H 2 -G, that in the continuous case is always ≥ 0, in the discrete case can be < 0. This problem is mentioned in [START_REF] Meyer | Discrete Differential-Geometry Operators for Triangulated 2-Manifolds[END_REF] to be an "extremely rare occurrence", while in our computations this happens quite often.

Results

In this section we present some results obtained exploiting the numerical strategies to compute geometrical parameters that are described in section 2. First of all in section 3.1 we benchmark the accuracy and convergence rate of the methods on some canonical objects for which the expressions of the curvatures in Cartesian coordinates are available. These 3D objects are generated using superquadrics primitive from VTK library [START_REF] Schroeder | The Visualization Toolkit: An Object-Oriented Approach to 3D Graphics[END_REF]). We then apply the computation to a DNS and we compute an areaweighted Probability Density Function (PDF) to highlight interesting footprints in the H -G phase space of the topological objects produced at each simulation time step.

Validation on canonical 3D objects

In order to assess the validity and reliability of the algorithm we tested the curvatures computation on a sphere of radius R and on an ellipsoid of semi-axes a, b, c respectively for which analytical expression of curvatures in Cartesian coordinates are available analytically (eqs. ( 9) and (10), [START_REF] Bektas | Curvature of the Ellipsoid with Cartesian Coordinates[END_REF]).

G sphere (x) = 1/R 2 H sphere (x) = 1/R (9) G ell (x) = 1 abc x 2 a 4 + y 2 b 4 + z 2 c 4 2 H ell (x) = x 2 + y 2 + z 2 -a 2 -b 2 -c 2 2 (abc) 2 x 2 a 4 + y 2 b 4 + z 2 c 4 3 2 (10)
The 3D objects are generated internally without any contouring procedure in order to provide high-quality triangulation w.r.t. MC-contoured level-sets (as generally done for full DNS), with the idea of separating effects due to the calculation of curvatures from the ones associated to triangulation quality. Two examples of triangulated objects are shown in fig. 4. In fig. 5a and fig. 5b we display the convergence rate for the two objects at increasing number of points. It is possible to notice that for both objects the algorithm computes the values of the two curvatures with reasonable precision and with a stable convergence rate. It is interesting to note how the convergence plot has a minimum around 10 5 points for both cases. We will show that the limit is related to float arithmetic bounds associated to very small cells that progressively start to appear on the object near the poles -this behaviour is associated to the way VTK library uses to build superquadrics surfaces -, caused by the trigonometric functions (notably the cot(α)) that are used to approximate 1-ring averaged H and G in [START_REF] Meyer | Discrete Differential-Geometry Operators for Triangulated 2-Manifolds[END_REF]. In addition, the error level for H and G for the ellipsoid case (fig. 5b) are different. This is due to the fact that the two parameters are estimated independently, without any guarantee on the coupled committed error. For this same reason, as we introduced in section 2, the algorithm produces some points for which ∆ 0; fig. 7 shows the percentage of outliers respectively for the sphere (7a) and the ellipsoid (7b). In the first case, outliers are a big part of the total points, spanning from ∼ 23% to ∼ 16% of the total, while for the ellipsoid outliers are way less (< 2% for most of the resolutions). This behaviour is linked to the fact that for a sphere, on each point of the surface, ∆ = 0 exactly, hence all the analysis is played along the boundary of the admissible curvature values, and the algorithm is not able to cope with this constraint. Nonetheless in fig. 6 an area-weighted error distribution of the relative er- In order to have a better understanding on the error behaviour and the influence of the cell size, we computed two bi-variate PDFs, with and without area-weighting, that are reported in fig. 8 and fig. 9. The area-weighting means that each sample of the distribution is accounted for in the PDF taking into account its associated cumulative 1-ring area in such a way that small triangles are less important in w.r.t. big triangles. Figure 6 is indeed an integrated version of the PDF shown in figs. 8 and 9.

P(E) =

A P(A, E) dA (11)
The zone for which the error E is negative are the zones of points that do not respect the ∆ ≥ 0 constraint. The bigger contributions to the error are given by points that have a relatively big 1-ring area associated to them and, hence, a less precise approximation of the curvature values. In addition, these "big" 1-rings are associated to outliers, since the associated error is always negative. It is worth to note, in addition, that increasing the resolution of the object and consequently reducing the area of the smallest triangles, triggers the generation of error associated to the loss of precision of floating point arithmetic: as we can see, in fig. 8, that is for a coarsely resolved sphere, the error distribution in the space (A, E) is substantially a one-dimensional sub-manifold, for all the surface areas, while for fig. 9, in the zones with very small triangles, the error is populated in all directions, that is with a different propagation mechanism than the one associated to discretization accuracy. This error associated to floating point precision loss explains the rising, after a certain resolution scale, of the L1-error for both curvatures that is shown in fig. 5a: small cells become too small to compute the curvature estimation and they add up to the global error.

Computation on DNS

Once we assessed the capabilities of our strategy on welltriangulated objects, we chose to perform the curvatures estimation on a representative two-phase flow benchmark case: the collision of two droplets. Two droplets with a radius respectively of 130 µm and 200 µm are configured at the beginning of the simulation with a negative relative velocity one w.r.t. to the other. The two spherical objects, then, start to approach until they collide causing surface deformation and topology changes. The simulation of the droplets collision is a DNS performed using the ARCHER code [START_REF] Ménard | Coupling Level Set/VOF/Ghost Fluid Methods: Validation and Application to 3D Simulation of the Primary Break-up of a Liquid Jet[END_REF]. In order to investigate the influence of the mesh resolution on the topological configurations we used three different mesh refinement levels: a coarse configuration with 128 × 128 × 256 cells, a medium case with 256 × 256 × 512 cells and a fine case with 512 × 512 × 1024 cells. We characterize the footprint of each topological object with an areabased PDF in the phase space H -G, P(H, G): A(H, G) is the area associated to a point in the phase space and it is used as a weighting function for the PDF. This PDF is computed on the data after filtering out outliers for which ∆ < 0

(H- √ G) √ G Area Weighted point density sphere -4 -2 0 2 4 6 •10 -5 0 2 4 6 8 •10 -2 (H- √ G) √ G
Area-weighted point density sphere N being N the total number of points in the surface. The corresponding values of H, G are normalized with the equivalent curvatures defined as in eq. ( 12).

R eq = R 1 + R 2 2 H eq = 1 R eq G eq = 1 R 2 eq ( 12
)
We performed the outliers statistics analysis also in the droplets collision case. In fig. 10 we can appreciate again how outliers are more probably produced when the topological objects resemble a sphere, in facts at the beginning of the simulation, when we have two exact spheres in the domain, the outliers are large in absolute terms, even if for the finest case they are percentually less than the other cases. Once the objects is deformed after the collision, the outliers percentage steeply drops and stays averagely the same along the simulation. It's interesting to highlight how the number of outliers is not influenced by the triangulation refinement level but it's quite exclusively associated to the intrinsic topological configuration of the object: more the object is closer to a sphere, higher the number of outliers. We underline that, at least in this specific case, the outliers distribution is not really improved going from a coarse triangulation to a fine one.

A general remark we can provide is that the topological configurations at the same time-step are strongly influenced by the mesh refinement level: in multiphase flow, differently from what it happens in single-phase flow where a minimal scale can be defined -i.e. the Kolmogorov scale -, we do not have a minimal scale that, once reached with the numerical simulation, allows us to resolve everything with a DNS. That means we are always going to commit some sort of error that lies in the fact we are cutting artificially the resolution (due to computational resources) and for this reason we will be always under-resolving something at a certain scale.

In addition to that the physics is strongly influenced by this (for example in the simulation at hands, producing a different number of satellite droplets depending on the case resolution). We can infer this for example comparing fig. 13b with figs. 14a and 14b: more the case is refined, smoother is the PDF that is extracted from the dataset. This behaviour can be related in this specific case to the precision which the surface tension, that depends on the mean curvature, is computed with. The more the mesh is resolved, the more accurate is the surface tension force, directly influencing the topological configuration.

Conclusions

In this work we presented a framework to exploit geometrical information in order to gain some insight on two-phase flow behavior through the post-processing of DNS. We have shown how the algorithm of [START_REF] Meyer | Discrete Differential-Geometry Operators for Triangulated 2-Manifolds[END_REF] fails to ensure ∆ ≥ 0 for objects that are close in slope to a sphere and in addition it cannot be used for cell sizes that are too small since float arithmetic problems arise. We performed the analysis on a real DNS simulation at three different resolutions extracting topological maps that emphasize the different physical behaviour associated to a different surface tension estimation, or in general different physical reproducibility due to the different resolution of the computational meshes. In [START_REF] Essadki | Statistical Modeling of the Gas-Liquid Interface Using Geometrical Variables: Toward a Unified Description of the Disperse and Separated Phase Flows[END_REF] an averaging kernel that preserves Gauss-Bonnet theorem is presented: studying the in- fluence of that averaging process on the outliers and phasespace statistics would be very interesting in order to achieve a more robust strategy w.r.t. curvature changes during topological changes, since the average would take care of bounding the maximum values of curvature in these pathological cases. A preliminary implementation in Python was already present in the work of Essadki et al., it was used to compute NDFs on DNS cases exploiting the Gauss-Bonnet preserving feature together with the averaging process that mitigates the effect of small or misshaped triangles resulting from the isocontouring MC procedure. We plan to implement the same averaging kernel in Mercur(v)e as an indispensable tool to perform a complete analysis of injection cases [START_REF] Blanchard | Modélisation et simulation multiéchelles de l'atomisation d'une nappe liquide cisaillée[END_REF] and [START_REF] Vaudor | A Consistent Mass and Momentum Flux Computation Method for Two Phase Flows. Application to Atomization Process[END_REF]) in terms of topological maps and number statistics in order to gather enough insight to develop closure terms for the reduced models available nowadays. As an additional future perspective, comparing other algorithm to estimate curvatures on discrete surface would allow to choose the most suitable strategy for the specific interfaces that are encountered in injection systems, in terms of robustness against topological changes and performance.

Figure 1 :Figure 2 :

 12 Figure 1: A central slice at τ = 0 of the volumetric levelset field on a uniform grid. The black lines are the 0-isoline of the level-set function

Figure 4 :

 4 Figure 4: Examples of triangulated objects used in this section

Figure 5 :

 5 Figure 5: Convergence rate for H and G: (5a) for the sphere.(5b) for the ellipsoid

Figure 6 :Figure 7 :-Figure 8 :Figure 9 :Figure 10 :

 678910 Figure 6: Error distribution for the sphere outliers. (6a) coarsest resolution. (6b) finest resolution

Figure 11 :

 11 Figure 11: Beginning of the simulation (τ = 0). The color mapping for the 3D objects is on the Gauss curvature G

Figure 12 :Figure 13 :Figure 14 :

 121314 Figure12: (τ = 0.3 × 10 -3 ). The color mapping for the 3D objects is on the Gauss curvature G

th International Conference on Multiphase Flow,ICMF 2019, Rio de Janeiro, Brazil, May 19 -24, 2019