Erick J Dufourc
email: e.dufourc@cbmn.u-bordeaux.fr

Prof Michèle Auger

Bicelles and Nanodiscs for Biophysical Chemistry*

Keywords: Bicelles, nanodiscs, NMR, membrane peptides & proteins, lipids, membrane dynamics DCPC, 1,2-hexanoyl-1-sn nuclear magnetic resonance, TOCSY, Total Correlation Spectroscopy, NOESY, nuclear overhauser effect spectroscopy, NOE, nuclear overhauser effect, PISA Wheel, Image of Membrane Protein Helical Wheels, SLF, separated local field spectroscopy, PISEMA, polarization inversion spin exchange at magic angle, POPC, 1-palmitoyl-2

Highlights

• Bicelles are pure synthetic lipidic discoidal particles of 20-50 nm diameter that may orient parallel or perpendicularly to magnetic fields. They are called "molecular goniometers" and mimic small pieces of biomembranes. • Bicelles are stable on narrow ranges of composition, temperature and hydration, can embed charged lipids, cholesterol, peptides and membrane proteins and are used as platforms to gain information on protein structure and dynamics by ssNMR.

• Bicelle remain aligned for several hours/days outside magnetic fields and relax according to nematic or smectic properties • Nanodiscs are smaller particles (10-20 nm) that may be built with a wide variety of lipids and apolipoproteins (MSP) making the rim of the disc.

• As for Bicelles, nanodiscs can embed a wide variety of membrane proteins and can be used as "molecular bricks".

• The constraints exerted by the MSP proteins forbid any temperature-driven phase transition, providing a membrane liquid-ordered state on large temperature domains.

Introduction

Biological membranes are very important in life as they provide the "barrier" that delimitates the outside from the inside of a cell, often called the plasma membrane. For eukaryotic organisms, cells have in addition many organelles, such as the nucleus made with a double membrane surrounding the aggregated DNA, the endoplasmic reticulum with a quite intricated membrane structure and the mitochondrion with specific in and out membranes. All these "barriers" not only delineate the organelles but provide them with specific functions of signaling, transport and fusion. These membranes are made of a large variety of lipids, sterols and membrane proteins. Both lipids and proteins give to specific membranes their intrinsic dynamic and structural properties. Although it is possible to purify natural membranes and obtain some dynamical information on their components [START_REF] Uzureau | Mechanism of Trypanosoma brucei gambiense resistance to human serum[END_REF][START_REF] Dazzoni | The unprecedented membrane deformation of the human nuclear envelope[END_REF], there is a lot of work being done to simplify systems, by reducing the number of components, for instance by isolating a given membrane protein and reinserting it inside relevant, but simpler, membrane lipid matrices. Structural and dynamic information can then be obtained by a wide variety of spectroscopic techniques such as Atomic Force Microscopy, Fluorescence, Electron Microscopy, X-rays and Neutrons scattering or solidstate Nuclear Magnetic Resonance (ssNMR) [START_REF] Dufourc | Sterols and membrane dynamics[END_REF][START_REF] Loquet | 3D structure determination of amyloid fibrils using solid-state NMR spectroscopy[END_REF][START_REF] Bechinger | The structure, dynamics and orientation of antimicrobial peptides in membranes by multidimensional solid-state NMR spectroscopy[END_REF][START_REF] Vigh | The significance of lipid composition for membrane activity: New concepts and ways of assessing function[END_REF][START_REF] Daury | Tripartite assembly of RND multidrug efflux pumps[END_REF]. Lipid membranes are amphiphilic assemblies that can be easily handled and are well known: liposomes, giant vesicles, micelles. It is however interesting having a platform that can be handled almost at will to orient samples and providing an additional information on the topology of the different partners within the membrane. This helps a lot in deciphering membrane-associated biological mechanisms. Lipids deposited on glass plates, as bilayers, have been used for years, for X-Rays, Neutron scattering and solid-state NMR [START_REF] Bechinger | Orientations of helical peptides in membrane bilayers by solid state NMR spectroscopy[END_REF][START_REF] Fenske | MODELS OF STRATUM-CORNEUM INTERCELLULAR MEMBRANES -H-2 NMR OF MACROSCOPICALLY ORIENTED MULTILAYERS[END_REF][START_REF] Ramamoorthy | 3-DIMENSIONAL SOLID-STATE NMR-SPECTROSCOPY OF A PEPTIDE ORIENTED IN MEMBRANE BILAYERS[END_REF]. Among these systems Bilayer Micelles and nanodiscs have shown an increase of interest in the Biophysical Chemistry community. Bicelles are made of short chain and long chain lipids and were first proposed from the pioneering work of Gabriel and Roberts using detergents to form "vesicles" of different sizes [START_REF] Gabriel | Spontaneous formation of stable unilamellar vesicles[END_REF]. They indeed identified Bicelles without naming them. The word "Bicelle" for bilayer micelle was proposed by several scientists such as, Sanders, Vold and Prosser [START_REF] Sanders | RECONSTITUTION OF MEMBRANE-PROTEINS INTO LIPID-RICH BILAYERED MIXED[END_REF][START_REF] Vold | Magnetically oriented phospholipid bilayered micelles for structural studies of polypeptides. Does the ideal bicelle exist?[END_REF]. Although different morphologies have been proposed for Bicelles, the disc topology is the most popular and has been clearly detected by Electron Microscopy [START_REF] Arnold | Cation modulation of bicelle size and magnetic alignment as revealed by solid-state NMR and electron microscopy[END_REF][START_REF] Aussenac | Toward bicelle stability with ether-linked phospholipids: Temperature, composition, and hydration diagrams by H-2 and P-31 solidstate NMR[END_REF]. [START_REF] Martinez | Lipid Internal Dynamics Probed in Nanodiscs[END_REF][START_REF] Diller | Bicelles: A natural 'molecular goniometer' for structural, dynamical and topological studies of molecules in membranes[END_REF] Multilamellar Vesicle (∝m) One must mention that perforated lamellae have also been proposed for some conditions of concentration and temperature [START_REF] Triba | Reinvestigation by phosphorus NMR of lipid distribution in bicelles[END_REF]. As the disc topology with a diameter of a few tenths of nanometers is generally used to describe Bicelles (Fig. 1) we will refer herein to this morphology [START_REF] Duerr | The Magic of Bicelles Lights Up Membrane Protein Structure[END_REF]. These pure lipid discs have very interesting magnetic properties that turn to be very useful for NMR structural determinations. They are indeed aligned by magnetic fields because lipids possess a weak positive or negative magnetic susceptibility anisotropy, ∆χ, which turns the disc plane perpendicular or parallel to the magnetic field. A change of orientation can also be obtained by using lanthanide ions. Such a property is very useful for structural studies and Bicelles have been termed as "molecular goniometers". In the following, examples of Bicelles use for understanding membrane dynamical properties and structures of peptides and proteins embedded in membranes will be given. However, Bicelles are made of synthetic lipids of specific chain length, the temperature-compositionhydration diagrams where they can be stabilized is restrained and their use may be limited [START_REF] Beaugrand | Lipid Concentration and Molar Ratio Boundaries for the Use of Isotropic Bicelles[END_REF][START_REF] Warschawski | Choosing membrane mimetics for NMR structural studies of transmembrane proteins[END_REF]. As a consequence other platforms have been developed by Sligar and coworkers [START_REF] Bayburt | Reconstitution and imaging of a membrane protein in a nanometer-size phospholipid bilayer[END_REF][START_REF] Bayburt | Assembly of single bacteriorhodopsin trimers in bilayer nanodiscs[END_REF]. They consist of smaller discs of nanometer size stabilized by membrane scaffold proteins (MSP) that play the role of short chain lipids in Bicelles : they constitute the rim of the disc. Examples will also be given of the use of such nanodiscs (Fig. 1). For clarity we will use the term nanodisc for MSP-stabilized lipid discs of nanometer size and Bicelle for pure lipids discs of similar sizes. A comparison of advantages and drawbacks of such platforms will be given at the end of the paper. There are many articles, which the reader may consult, that discuss Bicelles and nanodiscs [START_REF] Klopfer | Beyond detergent micelles: The advantages and applications of nonmicellar and lipid-based membrane mimetics for solution-state NMR[END_REF][START_REF] Puthenveetil | Solution NMR: A powerful tool for structural and functional studies of membrane proteins in reconstituted environments[END_REF], but this review puts emphasis on ssNMR viewpoint that is particularly adapted to compare these nanoplatforms.

Fig.1 Schematic representation of Multilamellar vesicles (Liposomes), Bicelles (Bilayer Micelles), Micelles and Nanodiscs stabilized by MSP proteins (blue circles). Adapted from

Bicelle Temperature-composition-hydration diagrams

Finding the conditions in which discoidal Bicelles can be stabilized and can be used in structural studies is a compulsory step before undertaking spectroscopic measurements.

There is a wide variety of techniques to build "phase" diagrams. X-rays, neutron scattering, [START_REF] Harroun | Comprehensive examination of mesophases formed by DMPC and DHPC mixtures[END_REF][START_REF] Nieh | Structural phase behavior of high-concentration, alignable biomimetic bicelle mixtures[END_REF], electron paramagnetic resonance (EPR) [START_REF] Garber | Magnetically Oriented Phospholipid Bilayers for Spin Label EPR Studies[END_REF], crystallography [START_REF] Faham | Bicelle crystallization A new method for crystallizing membrane proteins yields a monomeric bacteriorhodopsin structure[END_REF], Circular Dichroism [START_REF] Diller | Bicelles: A natural 'molecular goniometer' for structural, dynamical and topological studies of molecules in membranes[END_REF][START_REF] Khemtemourian | Dimerization of Neu/Erb2 transmembrane domain is controlled by membrane curvature[END_REF][START_REF] Marcotte | Interaction of antimicrobial peptides from Australian amphibians with lipid membranes[END_REF] and NMR [START_REF] Aussenac | Toward bicelle stability with ether-linked phospholipids: Temperature, composition, and hydration diagrams by H-2 and P-31 solidstate NMR[END_REF][START_REF] Triba | Reinvestigation by phosphorus NMR of lipid distribution in bicelles[END_REF][START_REF] Aussenac | Phase diagrams of acyl chain ester-vs. ether-linked lipid bicelles: Towards the bicelle stability[END_REF][START_REF] Bian | Phase separation in short-chain lecithin/gel-state long chain lecithin aggregates[END_REF][START_REF] Raffard | Temperaturecomposition diagram of dimyristoylphosphatidylcholine-dicaproylphosphatidylcholine "bicelles" self-orienting in the magnetic field. A solid state H-2 and P-31 NMR study[END_REF][START_REF] Sternin | Temperature dependence of DMPC/DHPC mixing in a bicellar solution and its structural implications[END_REF] have been mainly used to delineate the boundaries. Interestingly molecular dynamics including coarse grain approaches have been also used [START_REF] Koshiyama | Bicelle-to-Vesicle Transition of a Binary Phospholipid Mixture Guided by Controlled Local Lipid Compositions: A Molecular Dynamics Simulation Study[END_REF]. One must however mention that the wording "phase" is not meant in the thermodynamic meaning, because no transition enthalpy has been measured in tentative reporting of Bicelle diagrams. The system DMPC/DCPC (i.e., C14/C6 PC lipids) has been widely studied. The Fig. 2 shows how ssNMR can be used to build temperature-compositionhydration diagrams. Slow tumbling large liposomes give powder NMR spectra, isotropic micellar phases to NMR isotropic lines and bicelar discs, oriented spectra with one or 2 lines depending on the choice of the reporter nucleus (spin ½ or 1). The great advantage of NMR is the capability to distinguish Bicelles that are oriented with their plane parallel or perpendicular to the magnetic field due to the fact that spectra scale according to the (3cos 2 β-1) law, β being the angle between the normal to the membrane plane and the magnetic field direction. As can be seen on Fig. 2, Bicelles containing saturated lipid chains only (C14/C6) have a large compositional range but a small temperature span (maximum 10°C), they are oriented perpendicular to the field and are stable even with up to 95% hydration. Conversely, Bicelles made with biphenyl moieties in the long chain component (C14BP/C6) display a reduced composition span but exist in a 60°C temperature range. They are oriented with the normal to the bilayer parallel to the field direction and can be hydrated as much as their saturated counterparts. There are also some examples of diagrams on C16/C7 PC lipids [START_REF] Diller | Bicelles: A natural 'molecular goniometer' for structural, dynamical and topological studies of molecules in membranes[END_REF][START_REF] Koshiyama | Bicelle-to-Vesicle Transition of a Binary Phospholipid Mixture Guided by Controlled Local Lipid Compositions: A Molecular Dynamics Simulation Study[END_REF][START_REF] Yang | Small-Angle X-ray Scattering Studies on the Structure of Mixed DPPC/diC(7)PC Micelles in Aqueous Solutions[END_REF], the temperature span is even smaller (few degrees). Adapted from [START_REF] Diller | Bicelles: A natural 'molecular goniometer' for structural, dynamical and topological studies of molecules in membranes[END_REF][START_REF] Loudet | Biphenyl bicelle disks align perpendicular to magnetic fields on large temperature scales: A study combining synthesis, solid-state NMR, TEM, and SAXS[END_REF].

Interestingly Bicelles made of PCs with ether-linked hydrophobic chains have also been investigated and show little difference in "phase diagrams" with their ester-linked counterparts [START_REF] Aussenac | Toward bicelle stability with ether-linked phospholipids: Temperature, composition, and hydration diagrams by H-2 and P-31 solidstate NMR[END_REF][START_REF] Aussenac | Phase diagrams of acyl chain ester-vs. ether-linked lipid bicelles: Towards the bicelle stability[END_REF][START_REF] Ottiger | Bicelle-based liquid crystals for NMR-measurement of dipolar couplings at acidic and basic pH values[END_REF][START_REF] Park | Three-dimensional structure of the transmembrane domain of Vpu from HIV-1 in aligned phospholipid bicelles[END_REF]. The clear advantage is their hydrolysis-resistant character. Most of Bicelle work has been performed with saturated chains lipids such as DMPC, DPPC, DCPC or DHPC. It is however possible to use unsaturated lipids and obtain stable discoidal micelles [START_REF] Minto | A H-2 solid-state NMR spectroscopic investigation of biomimetic bicelles containing cholesterol and polyunsaturated phosphatidylcholine[END_REF][START_REF] Cho | Lipid Domains in Bicelles Containing Unsaturated Lipids and Cholesterol[END_REF][START_REF] Schmidt | Liquid disordered-liquid ordered phase coexistence in bicelles containing unsaturated lipids and cholesterol[END_REF]. Isotropic Bicelles, i.e., fast tumbling and non-magnetically orientable Bicelles have also been reported with native E Coli lipids [START_REF] Liebau | Fast-tumbling bicelles constructed from native Escherichia coli lipids[END_REF][START_REF] Arnold | Studying Natural Structural Protein Fibers By Solid-State Nuclear Magnetic Resonance[END_REF]. For these systems, unfortunately no phase diagram is reported.

B 0

NMR OF LIPIDS 3. Doping Bicelles with sterols, charged lipids or counterions.

When studying the structure and topology of peptides and proteins in interaction or embedded in membranes it is important to have a model membrane that is representative enough of natural membranes, the best being nonetheless to work on natural membranes [START_REF] Uzureau | Mechanism of Trypanosoma brucei gambiense resistance to human serum[END_REF][START_REF] Dazzoni | The unprecedented membrane deformation of the human nuclear envelope[END_REF][START_REF] Grelard | Hepatitis B subvirus particles display both a fluid bilayer membrane and a strong resistance to freeze drying: a study by solid-state NMR, light scattering, and cryo-electron microscopy/tomography[END_REF]. As seen in Fig. 3, cholesterol may be inserted in saturated chain PC Bicelles or in Biphenyl PC Bicelles. The sterol displays the same properties as in liposomes : same orientation parallel to the bilayer normal and same dynamics [START_REF] Loudet | Biphenyl phosphatidylcholine: A promoter of liposome deformation and bicelle collective orientation by magnetic fields[END_REF]. Moreover it clearly partitions in the flat bilayer part of the disc. The well-known promotion of domain formation (liquid-ordered phases) in liposomes is also detected in Bicelles, provided the disc size in large enough [START_REF] Schmidt | Liquid disordered-liquid ordered phase coexistence in bicelles containing unsaturated lipids and cholesterol[END_REF]. Unsaturated lipids can also be added to a cholesterol containing Bicelle [START_REF] Cho | Lipid Domains in Bicelles Containing Unsaturated Lipids and Cholesterol[END_REF][START_REF] Triba | Effects of lipid chain length and unsaturation on bicelles stability. A phosphorus NMR study[END_REF]. Adapted from [START_REF] Loudet | Biphenyl phosphatidylcholine: A promoter of liposome deformation and bicelle collective orientation by magnetic fields[END_REF][START_REF] Aussenac | Detailed structure and dynamics of bicelle phospholipids using selectively deuterated and perdeuterated labels. 2-H NMR and molecular mechanics study[END_REF] Charges at the interface are very important for Bicelle morphology and magnetic field orientation issues. It has been shown that increasing the concentration of cations such as K + , Na + , Ca 2+ and Mg 2+ leads to an increase of disc size and to better nematic-like alignment in magnetic fields [START_REF] Arnold | Cation modulation of bicelle size and magnetic alignment as revealed by solid-state NMR and electron microscopy[END_REF]. The optimal salt concentration for optimal alignment ranges in between 50-200 mM. The most striking effect of charges at the interface is obtained with paramagnetic lanthanides ions with large positive ∆χ (Eu 3+ , Er 3+ , Yb 3+ , Tm 3+ , Dy 3+ ) [START_REF] Tiburu | Development of magnetically aligned phospholipid bilayers in mixtures of palmitoylstearoylphosphatidylcholine and dihexanoylphosphatidylcholine by solid-state NMR spectroscopy[END_REF][START_REF] Ma | Lanthanide ions bind specifically to an added "EF-hand" and orient a membrane protein in micelles for solution NMR spectroscopy[END_REF][START_REF] Prosser | Magnetically Aligned Membrane Model Systems with Positive Order Parameter: Switching the Sign of Szz with Paramagnetic Ions[END_REF]. This results in tilting the disc plane by 90° in a way very similar to that produced by the biphenyl rings of TBBPC (vide supra). Use of paramagnetic lanthanides allows therefore changing the Bicelle orientation with respect to the magnetic field direction.

Bicelles internal dynamics

Membrane dynamics describes complex motional processes occurring at very different time and distance scales : from picoseconds and atomic bonds to seconds and micron-millimeter dimensions [START_REF] Dufourc | Sterols and membrane dynamics[END_REF][START_REF] Aussenac | Detailed structure and dynamics of bicelle phospholipids using selectively deuterated and perdeuterated labels. 2-H NMR and molecular mechanics study[END_REF][START_REF] Dufourc | Dynamics of phosphate headgroups in biomembranes -Comprehensive analysis using P-31 Nuclear-Magnetic-Resonance lineshape and relaxation time measurements[END_REF]. ssNMR has been shown to be the best technique to probe all of these motions by combining several relaxation time measurements and line shape analysis. For Bicelles, the local motions that occur at pico-nanosecond timescales are globally conserved compared to liposome data. This is reflected by an order parameter profile for the long chain lipids that partition on the disc plane very similar to that measured in liposomes : very weak ordering near the center of the bilayer and quite rigid near the lipid glycerol backbone region. This is well reflected in Fig. 3d where the rate of spin lattice relaxation (1/T1Z) was measured on a fully perdeuterated DMPC molecule : two very dynamic regions (low rate of relaxation) for chains and head groups are separated by a very rigid zone (the glycerol backbone). The variance, compared with liposomes, is the slightly more dynamic character of Bicelle long chain lipids, understandable by the presence of the small chain lipids known to promote disorder [START_REF] Aussenac | Detailed structure and dynamics of bicelle phospholipids using selectively deuterated and perdeuterated labels. 2-H NMR and molecular mechanics study[END_REF]. The other variance by comparing with liposomes or large unilamellar vesicles is the absence of collective motions, i.e., membrane undulations linked to hydrodynamic modes of bilayer deformation occurring at microsecond to second time scales [START_REF] Dufourc | Sterols and membrane dynamics[END_REF][START_REF] Dufourc | Dynamics of phosphate headgroups in biomembranes -Comprehensive analysis using P-31 Nuclear-Magnetic-Resonance lineshape and relaxation time measurements[END_REF]. This results for instance in a higher spectral resolution when detecting NMR wide-line spectra. Because Bicelle internal dynamics are very similar to that of liposomes they offer a proper mimic of natural membranes.

Weak orienting media: structure of soluble proteins by liquid-state NMR

One of the first uses of saturated Bicelles was to provide a weak orienting medium for structural biology studies by liquid state NMR. Thanks to the pioneering work of Sanders [START_REF] Sanders | Characterization of magnetically orientable bilayers in mixtures of dihexanoylphosphatidylcholine and dimyristoylphosphatidylcholine by solidstate NMR[END_REF], Bax and coworkers [START_REF] Ottiger | Bicelle-based liquid crystals for NMR-measurement of dipolar couplings at acidic and basic pH values[END_REF][START_REF] Lorieau | Liquid crystalline phase of G-tetrad DNA for NMR study of detergent-solubilized proteins[END_REF][START_REF] Wu | Overall structure and sugar dynamics of a DNA dodecamer from homoand heteronuclear dipolar couplings and P-31 chemical shift anisotropy[END_REF][START_REF] Tjandra | The NMR structure of a DNA dodecamer in an aqueous dilute liquid crystalline phase[END_REF][START_REF] Tjandra | Direct Measurement of Distances and Angles in Biomolecules by NMR in a Dilute Liquid Crystalline Medium[END_REF] one could obtain additional distance constraints using diluted Bicelles (3% w/v): indeed at this low concentration the sample takes the form of a nematic-like medium where discs orient with their plane parallel to the magnetic field direction and are spaced from each other by 30 nm in average. They have a negligible effect on the free diffusion of soluble proteins or DNA molecules. Very interestingly proteins/DNA exhibits a residual degree of alignment that is proportional to the Bicelle concentration.

Residual dipolar couplings, D, can be measured, in addition to the indirect spin-spin couplings J. As the dipolar interaction, also named direct spin-spin coupling depends on nucleus distances through space and orientation with respect to magnetic field, spatial constraints can be obtained in addition to classical measurement of NOE effects.

Peptide 3D ssNMR structure in Bicelles

Conversely to soluble proteins or DNA molecules, amphipathic peptides or membrane proteins may strongly interact or partition entirely into the membrane. The use of liquidstate NMR no longer leads to resolved spectra due to the slow rotational diffusion of the 30-50 nm-sized discs. If, in addition, Bicelles are concentrated, i.e., below 95% hydration, the proton spectrum becomes very broad (Fig. 4a) and forbids the use of solution-state NMR techniques. Wide-line ssNMR can then be used to obtain topology information [START_REF] Ouellet | Insights on the interactions of synthetic amphipathic peptides with model membranes as revealed by P-31 and H-2 solid-state NMR and infrared spectroscopies[END_REF][START_REF] Marcotte | Bicelles as model membranes for solid-and solution-state NMR studies of membrane peptides and proteins[END_REF][START_REF] Marcotte | Interaction of the neuropeptide Metenkephalin with zwitterionic and negatively charged bicelles as viewed by P-31 and H-2 solid-state NMR[END_REF] (vide infra) or the Bicelle sample can be moderately spun at the magic angle (54.7°) to resolve individual isotropic chemical shifts [START_REF] Carlotti | Towards high-resolution 1H-NMR in biological membranes: magic angle spinning of bicelles[END_REF] in proton NMR (Fig. 4b). By comparison with liposomes, which could also be submitted to magic angle sample spinning, the Bicelle structure allows attaining sharper lines due to slightly greater membrane dynamics (vide supra) that provide pre-averaging of the strong 1 H-1 H interactions in the solid state. The MAS rate can therefore be moderate (10 kHz). It is interesting mentioning here that highresolution proton NMR spectra can be obtained on hydrated membrane proteins in the form of aggregates or with Alzheimer's fibrils but with spinning the sample at very high speeds such as 110 kHz [START_REF] Loquet | 3D structure determination of amyloid fibrils using solid-state NMR spectroscopy[END_REF][START_REF] Stanek | NMR Spectroscopic Assignment of Backbone and Side-Chain Protons in Fully Protonated Proteins: Microcrystals, Sedimented Assemblies, and Amyloid Fibrils[END_REF]. As an example of a collaboration with prof Michèle Auger, moderate speed-magic angle sample spinning of DMPS-doped Bicelles containing the Methionin Enkephalin neuropeptide (Menk) allowed obtaining a well-resolved 1 H-NMR spectrum of the lipid and the peptide (Fig. 4b). The use of classical multidimensional liquid state NMR sequences (TOCSY, NOESY) combined with molecular modeling, allowed finding both the 3D structure of Menk and its localization at the membrane interface (Fig. 4c). This structure confirms earlier works that determined the structure of a parent enkephalin, Leucine Enkephalin, using transferred NOE experiments, which are much more difficult to set up [START_REF] Milon | Transferred Nuclear Overhauser Effect Analyses of Membrane-bound Enkephalin Analogs by H-1 Nuclear Magnetic Resonance -Correlation between Activities and Membrane-bound conformations[END_REF].

Structural topology of membrane proteins in oriented Bicelles

The determination of membrane protein structure and topology in a membrane environment, is very difficult. Most NMR experiments of insoluble membrane proteins have been performed by dissolving the membrane in detergent systems that places membrane proteins in an isotropic environment and allows the use of liquid-state NMR multidimensional approaches. As mentioned earlier, use of ultra-high speed magic angle sample spinning allows resolving the structure of insoluble membrane proteins under the form of hydrated aggregates. The topology with respect to the membrane is however inaccessible due to the lack of membrane. On the other hand and although convenient from the experimental viewpoint, the main drawback of the detergent approach resides in possible protein misfolding and the inability to control biological function in such media. Membrane protein topology and structure in lipid membranes have been addressed by many people [68,69] but one must remark the pioneering work of the Opella and Bechinger groups, [START_REF] Bechinger | The structure, dynamics and orientation of antimicrobial peptides in membranes by multidimensional solid-state NMR spectroscopy[END_REF][START_REF] Opella | NMR and membrane proteins[END_REF][71][72] who developed sample alignment experiments that became standards for topology studies of proteins in membranes. Such experiments with macroscopically oriented membrane samples in between glass plates that were placed at defined orientations in the magnetic field offered access to orientational constraints for the calculation of atomicresolution protein structures and topologies with respect to the membrane plane. For instance it is now very easy to determine whether a protein helix, 15 N-labelled in the NH backbone, is perpendicular or parallel to the membrane: 15 N-NMR chemical shifts around 200 ppm reflect a perpendicular orientation whereas resonances at ca. 70 ppm stand for helices lying flat on the membrane [START_REF] Marassi | Correlation of the structural and functional domains in the membrane protein Vpu from HIV-1[END_REF].

In recent years the versatility of Bicelles and the fact that they could be aligned by the magnetic field itself without the need of glass plates has afforded great progresses in the field. Fig. 5 synopsizes this approach by showing 1D and 2D solid-state NMR experiments of uniformly 15 N-labelled Pf1 coat protein embedded in biphenyl lipid bilayers Bicelles that are magnetically oriented with their plane normal to the field direction. The 1D spectrum displays very nicely resolved lines compared to a similar sample sandwiched in-between glass plates or even with a sample made of saturated DMPC/DCPC Bicelles that were flipped by 90° after addition of ytterbium(III) chloride, YbCl3 [START_REF] Park | Solid-state NMR spectroscopy of a membrane protein in biphenyl phospholipid bicelles with the bilayer normal parallel to the magnetic field[END_REF]. The 15 N chemical shifts are detected in between 160 and 220 ppm a range characteristic of tilted trans-membranous helices. In the 2D separated local field (SLF) experiment shown in Fig. 5 (bottom), almost complete spectral resolution is achieved: each amide site leads to a correlation characterized by a unique spot at the intersection of the 1 H- 15 N dipolar coupling and the 15 N chemical shift frequencies. The PISA wheel patterns (dotted ellipse) corresponding to helical wheel projections are formed by the resonances from residues in the transmembrane helix part of the protein. From the breadth of the ellipses a ca. 20-30° tilt can be inferred. Adapted from [START_REF] Park | Solid-state NMR spectroscopy of a membrane protein in biphenyl phospholipid bicelles with the bilayer normal parallel to the magnetic field[END_REF][START_REF] Dufourc | Membrane Interacting Peptides: From Killers to Helpers[END_REF].

Persistence of Bicelle magnetic orientation

The versatility of the Bicelle sample for NMR measurements has been shown above.

Peptides and proteins can be embedded in the bilayer part of the disc and the entire system may be aligned parallel or perpendicular to the magnetic field. This has led us to call this system a "molecular goniometer", which is a slight misuse of language as intermediate orientations cannot yet be reached. Although very interesting such aligned samples thus appear to be restricted to NMR experiments: the magnetic field orients the sample and is used for NMR measurements. The liquid crystal nature of such samples, nematic or smectic, calls for more investigations outside magnetic fields. As shown in Fig. 6 such systems show a persistence of their orientation outside the magnetic field after having been magnetized. Small Angle X-Rays Scattering has been performed on DMPC/DCPC (Fig. 6A), TBBPC/DCPC (Fig. 6B) and DMPC/DCPC/Eu3+ (Fig. 6C) Bicelles after being magnetized for 15 minutes in a 11.7 T magnetic field (z-axis) [START_REF] Loudet-Courreges | Unprecedented Observation of Days-Long Remnant Orientation of Phospholipid Bicelles: A Small-Angle X-ray Scattering and Theoretical Study[END_REF]. The z-axis of the X-rays set-up correspond to the z-axis of the magnetic field. On the Bicelle system made with saturated lipids, 2 spots are detected along the x-direction indicating a macroscopic orientation of the Bicelle plane perpendicular to the field, i.e., nematic orientation. The macroscopic alignment of Bicelle discs is lost after ca. 1 hour as shown by a circular distribution of the scattering pattern. The biphenyl Bicelles system show two spots aligned along the z-axis, i.e., with the disc normal parallel to the magnetic field (smectic order); orientation is such conditions lasts for several days. An isotropic distribution, i.e., a complete return to random orientation, is only obtained after 10 days outside the field. When doping the DMPC Bicelle with Europium ions, the Bicelles are flipped by 90° as seen with two spots along the z-axis (smectic order). They remain well aligned for several hours. When following their orientation/relaxation kinetics one realizes that the orientation mechanism is governed by the disc size: the smaller discs, DMPC/DCPC Bicelles, or DMPC/DCPC/Eu Bicelles, have similar and fast orientation kinetics: it takes about 5 min to align them (Fig. 6 rigth). Conversely, larger discs, TBBPC/DCPC Bicelles, need to stay 15 min in the magnetic field to be aligned. Relaxation outside the field of Bicelles with smectic orientation (DMPC/DCPC/Eu Bicelles or TBBPC/DCPC Bicelles) was very slow, whereas the DMPC/DCPC Bicelles with nematic orientation relaxed back to isotropic orientation after about one hour. Such a remnant long-lived orientation of Bicelles outside the magnetic field was attributed to the higher degree of organization of the smectic phases with respect to nematic ones and was demonstrated by the analysis of anisotropic scattering pattern [START_REF] Loudet-Courreges | Unprecedented Observation of Days-Long Remnant Orientation of Phospholipid Bicelles: A Small-Angle X-ray Scattering and Theoretical Study[END_REF]. Such a long lasting orientation outside the magnetic field, after having been magnetized for a few minutes, is unprecedented and represents an excellent tool for structural studies by X-rays or Neutrons diffraction.

Nanodiscs made of lipids and helical proteins

Bicellar discs are very nice systems for structural biology but remain fragile to handle. If one wishes to embed membrane proteins or peptides into Bicelles, the latter may destabilize the discs as they may compete with the small chain lipids that compensate for membrane edge effect (half torus). This is what indeed happened with Apolipoprotein-like protein (which are amphipathic along the helix edge): Bicelles without short chain lipids are stabilized. They are called nanodiscs. Recent works [START_REF] Hagn | Optimized Phospholipid Bilayer Nanodiscs Facilitate High-Resolution Structure Determination of Membrane Proteins[END_REF] have shown that such 10 nm diameter nanodiscs are good tools to reconstitute membrane proteins in a native environment and solve their 3D structure by spectroscopic techniques. They originated from apolipoprotein A1 (Apo A1), which solubilizes lipids and cholesterol, to deliver them to the liver as high-density lipoprotein (HDL) particles [START_REF] Duong | Characterization of nascent HDL particles and microparticles formed by ABCA1-mediated efflux of cellular lipids to apoA-I[END_REF]. As sketched in Fig. 7a,b they stabilize a piece of lipid bilayer containing or not cholesterol and are well imaged by cryo-electron microscopy. The Apo A1 is also called membrane scaffolding protein (MSP) in the nanodisc community and stabilizes the lipid disc by direct van der Walls interaction of its hydrophobic face with lipid hydrophobic chains. The nanodisc is of small size (10 nm), compared to Bicelles but efforts are being made to design longer MSP proteins to form larger discs, with a diameter of ca. 20 nm or more [START_REF] Mclean | Nanodiscs: A Controlled Bilayer Surface for the Study of Membrane Proteins[END_REF]. Adapted from [START_REF] Daury | Tripartite assembly of RND multidrug efflux pumps[END_REF][START_REF] Martinez | Lipid Internal Dynamics Probed in Nanodiscs[END_REF].

A very nice example of use of nanodiscs has been recently published by Lambert and coworkers [START_REF] Daury | Tripartite assembly of RND multidrug efflux pumps[END_REF] with the 3-component multimeric multidrug efflux system of Gram-negative bacteria. The intermembrane complex is composed of an inner membrane transporter, an outer membrane channel and a periplasmic adaptor protein. It is hypothesized that they form channels within the periplasm facilitating the exit of the drug through the two bacterial membranes. The inner and outer membrane transporter have been reconstituted in MSPnanodiscs and single-particle analysis by electron microscopy has demonstrated that the inner and outer membrane protein components are linked together via the periplasmic adaptor protein (Fig. 7c,d). The 2D electron microscopy projection of all three complexes (Fig. 7d) clearly pictures the essential building block role of the periplasmic adaptor protein in extending the channel over 30 nm (periplasmic space between the inner and outer membranes). It is also interesting mentioning that polymer-encased lipid nanodiscs have been proposed by Ramamoorthy and coworkers: the MSP protein is replaced by polymers [START_REF] Knowles | Membrane Proteins Solubilized Intact in Lipid Containing Nanoparticles Bounded by Styrene Maleic Acid Copolymer[END_REF][START_REF] Ravula | Magnetic Alignment of Polymer Nanodiscs Probed by Solid-State NMR Spectroscopy[END_REF].

Nanodiscs internal dynamics

Although there is a general consensus concerning the driving forces in nanodisc assembly with Apo A1/MSP proteins little is known about the internal dynamics of lipids and the possible dynamic interplay between lipids and proteins in relation with membrane protein activity. Incorporation of fluorescent probes into membranes [START_REF] Nakano | Static and Dynamic Properties of Phospholipid Bilayer Nanodiscs[END_REF] suggested a decrease in

50 nm a) b) c) d)
membrane fluidity in nanodiscs, by comparison to that of vesicles,. However, the use of fluorescent probes might disturb the lipid cooperative behavior [START_REF] Leung | Link between Fluorescent Probe Partitioning and Molecular Order of Liquid Ordered-Liquid Disordered Membranes[END_REF], and direct methods should be preferred. The internal lipid dynamics and their thermotropism has recently been reported by 2 H NMR spectroscopy of chain perdeuterated DMPC molecules embedded in nanodiscs. Such a method allows direct measurement of chain ordering and is very sensitive to the lipid phase nature and chain melting during thermal transitions. The insert shows a representation of the cholesterol containing nanodisc as already described in Fig. 7. Adapted from [START_REF] Martinez | Lipid Internal Dynamics Probed in Nanodiscs[END_REF].

It has been shown (Fig. 8) that chain ordering in the fluid phase is very similar to that of DMPC in large micrometer-size liposomes, although one observes a 10-15% increase in chain ordering. This has been attributed to the constraining role of MSP proteins that limit lateral and rotational diffusion. The well-known ordering effect of cholesterol in natural and synthetic liposomes [START_REF] Dufourc | Sterols and membrane dynamics[END_REF][START_REF] Beck | Plant sterols in "rafts": a better way to regulate membrane thermal shocks[END_REF][START_REF] Aussenac | Cholesterol dynamics in membranes of raft composition: A molecular point of view from H-2 and P-31 solid-state NMR[END_REF][START_REF] Dufourc | Structural and dynamical details of cholesterol lipid intearaction as revealed by deurerium NMR[END_REF] is also observed in 10 nm diameter nanodiscs. The most striking effect is however the observation of fluid phase spectra at temperatures for which one usually observes gel phases with liposomes (Fig. 8). The lamellar gel-to lamellar fluid (Lβ' -Lα) phase transitions, which are well observed by the temperature variation of the first spectral moment in liposomes, are almost completely abolished in the nanodisc. The small size (10 nm) of the nanodisc indicates that there are ca. 70 lipids by leaflet, a value that is very (too) small to define a cooperative unit of lipids and therefore a phase transition.

Cooperative units for liposomes are in the range of several thousand lipids and lead to very sharp transitions (1-2 °C). Those observed for small unilamellar vesicles obtained by sonication (20 nm diameter) lead to very broad transitions (10-20 °C) and are composed of cooperative units of a few hundred of lipids [START_REF] Sugar | COOPERATIVITY AND CLASSIFICATION OF PHASE-TRANSITIONS -APPLICATION TO ONE-COMPONENT AND 2-COMPONENT PHOSPHOLIPID-MEMBRANES[END_REF][START_REF] Gruenewald | CURVATURE INFLUENCE ON THE COOPERATIVITY AND THE PHASE-TRANSITION ENTHALPY OF LECITHIN VESICLES[END_REF][START_REF] Marsh | COOPERATIVITY OF PHASE-TRANSITION IN SINGLE-BILAYER AND MULTI-BILAYER LIPID VESICLES[END_REF]. It thus appears that nanodiscs at low temperatures are maintained in some kind of fluid state due to the constraints produced by the MSP proteins. Interestingly, this result differs from that obtained with larger nanodiscs (ca. 20 nm) from which phase behavior is observed. This reminds also the observations on large Bicelles were solid-ordered and liquid-disordered phases have been observed by NMR [START_REF] Schmidt | Liquid disordered-liquid ordered phase coexistence in bicelles containing unsaturated lipids and cholesterol[END_REF].

Conclusion

Bicelles and Nanodiscs turn to be very interesting platforms primarily for the study of membrane peptides and proteins. They are also used for the structural study of soluble proteins and nucleic acids, as well as possible platforms for drug targeting. Whereas Bicelles are membranous discs entirely made of lipids, Nanodiscs require the presence of MSP proteins or polymers for stability. In both cases increasing the size of these nanoplatforms is a clear requirement to have enough lipids in the disc plane to correctly mimic membrane thermotropism and dynamics. The internal dynamics of Bicelles and Nanodiscs resembles that of liposomes with the variance that lipid order slightly decreases in Bicelles and slightly increases in Nanodiscs on comparing with liposomes. Very interestingly, Bicelles can be manipulated with magnetic fields to be oriented parallel of perpendicular to the field direction, offering the access to "molecular goniometers" for experimentalists. In addition, Bicelles show a remnant alignment after having been magnetized and taken out of the field. In some case this alignment persisting outside the magnetic field lasts for days, offering a new tool for crystallography. Such properties are not yet studied with Nanodiscs made with MSP proteins and deserve to be considered.

Although the structure and dynamics of Bicelles and nanodiscs can be compared, much more information on the composition of nanodiscs is urgently needed. Such a lack is not surprising since nanodiscs were first invented by mimicking natural HDLs while Bicelles were discovered by examining lipid-detergent mixtures. Composition-temperature-hydration diagrams for nanodiscs are needed. The ability to accommodate controlled amounts of cholesterol and sphingolipids would be of primary interest: so-called lipid rafts could be isolated.

The question of phase transitions is also open. While MSP proteins seem to prevent temperature-induced gel-fluid transitions in small nanodiscs, such transitions are difficult to observe in Bicelles because temperature variation modifies the morphology of the pure lipid assembly: very small micelles are observed when the temperature is below Tc. Here again, it would be good to stabilize large nanodisks or Bicelles with enough lipids (a few thousand) to have a cooperative unit capable of producing thermodynamic phase changes.

A review of the literature shows that Bicelles have been considered over the last two decades as the ultimate membrane mimetics for the study of the structure and dynamics of biomolecules. However, the situation is now changing due to the relative fragility of Bicelles; nanodiscs made from MSP or polymers surrounding natural lipids appear to be becoming the fashionable new membrane platform that is robust and easy to manipulate.

Fig. 2 .

 2 Fig. 2. Diagrams with variable composition, temperature and hydration as determined by solid-state wide-line NMR of TBBPC/DCPC or DMPC/DCPC mixtures. (a) Temperaturecomposition and temperature hydration diagrams of Bicelle systems being macroscopically oriented by magnetic fields (B0), either with their bilayer normal, n, perpendicular (B⊥) to the field (DMPC/DCPC) or parallel (B//, TBBPC/DCPC). Mole fractions X are expressed in percent. q represents the long chain-to-short chain molar ratio. Lines are drawn as eye guides to delineate domains. (b) Phosphorus-31 and nitrogen-14 solid-sate wide-line NMR spectra characteristic of phosphocholine lipids embedded in different structures (unoriented lamellar (slow tumbling liposomes), L, isotropic, I, Bicelles with normal oriented parallel (B//) or perpendicular (B⊥) to the field. On the right, molecular structures of DMPC and TBPPC are shown. Adapted from[START_REF] Diller | Bicelles: A natural 'molecular goniometer' for structural, dynamical and topological studies of molecules in membranes[END_REF][START_REF] Loudet | Biphenyl bicelle disks align perpendicular to magnetic fields on large temperature scales: A study combining synthesis, solid-state NMR, TEM, and SAXS[END_REF].

Fig. 3 .

 3 Fig. 3. Cholesterol-containing Bicelles and Bicelle internal dynamics. Solid-state deuterium NMR spectra of 10 mol% deuterium-labeled cholesterol (schematic structure) embedded in DMPC liposomes (a), DMPC/DCPC Bicelles (b), and TBBPC/DCPC Bicelles (c), at 35 °C. Bicelle sketches show the orientation of discs with respect to magnetic field. d) Deuterium spinlattice relaxation rates 1/T1Z, as a function of labeled carbon position, for entirely perdeuterated DMPC incorporated in Bicelles, at 35°C: the higher the rate, the less mobile the carbon deuterium bond. On the right hand side a DMPC molecule is sketched showing the carbon numbering. Adapted from[START_REF] Loudet | Biphenyl phosphatidylcholine: A promoter of liposome deformation and bicelle collective orientation by magnetic fields[END_REF][START_REF] Aussenac | Detailed structure and dynamics of bicelle phospholipids using selectively deuterated and perdeuterated labels. 2-H NMR and molecular mechanics study[END_REF] Charges at the interface are very important for Bicelle morphology and magnetic field orientation issues. It has been shown that increasing the concentration of cations such as K + , Na + , Ca 2+ and Mg 2+ leads to an increase of disc size and to better nematic-like alignment in magnetic fields[START_REF] Arnold | Cation modulation of bicelle size and magnetic alignment as revealed by solid-state NMR and electron microscopy[END_REF]. The optimal salt concentration for optimal alignment ranges in between 50-200 mM. The most striking effect of charges at the interface is obtained with paramagnetic lanthanides ions with large positive ∆χ (Eu 3+ , Er 3+ , Yb 3+ , Tm 3+ , Dy 3+ )[START_REF] Tiburu | Development of magnetically aligned phospholipid bilayers in mixtures of palmitoylstearoylphosphatidylcholine and dihexanoylphosphatidylcholine by solid-state NMR spectroscopy[END_REF][START_REF] Ma | Lanthanide ions bind specifically to an added "EF-hand" and orient a membrane protein in micelles for solution NMR spectroscopy[END_REF][START_REF] Prosser | Magnetically Aligned Membrane Model Systems with Positive Order Parameter: Switching the Sign of Szz with Paramagnetic Ions[END_REF].

Fig. 4 .

 4 Fig. 4. Peptide 3D structure embedded in Bicelles. (a) Static 1 H-NMR spectrum of DMPC/DCPC Bicelles containing the neuropeptide Menk (lipid-to-peptide molar ratio of 25), at 25 °C. (b) Same as (a) with a magic angle sample spinning speed of 10 kHz. A sketch of the MAS set up together with an expansion of the amide resonances are also shown in (b). (c)Position of Menk at the bilayer interface as reconstructed from 2D-NMR constraints and using molecular dynamics. Color code: peptide in pink; lipids: oxygen atoms of fatty acid chains in red, phosphorus in gold, nitrogen in blue, carbon skeleton in light blue, water molecules are not represented for clarity. Adapted from[START_REF] Diller | Bicelles: A natural 'molecular goniometer' for structural, dynamical and topological studies of molecules in membranes[END_REF]

Fig. 5 .

 5 Fig. 5. Topology of membrane proteins in oriented Bicelles.One-dimensional15 N chemical shift spectrum (middle) and two-dimensional15 N chemical shift/ 1 H-15 N dipolar coupling separated local field (SLF) solid-state NMR map (bottom), at 40°C. The Pf1 coat protein is uniformly15 N-labelled and embedded in Bicelle membranes aligned with their bilayer normal parallel to the magnetic field (17.6 T) direction. Bicelles are made with TBBPC/DCPC (sketched on top, left), the molar ratio of lipid bilayers to protein (L/P) is 50. The ideal PISA wheel is superimposed on the 2D-map and leads to the orientation of the transmembrane helix as sketched on top (right). Adapted from[START_REF] Park | Solid-state NMR spectroscopy of a membrane protein in biphenyl phospholipid bicelles with the bilayer normal parallel to the magnetic field[END_REF][START_REF] Dufourc | Membrane Interacting Peptides: From Killers to Helpers[END_REF].

Fig. 6 .

 6 Fig. 6. Remnant orientation of Bicelles. Left panel: SAXS pattern of (A) DMPC/DCPC Bicelles, (B) TBBPC/DCPC Bicelles, and (C) DMPC/DCPC-Eu Bicelles, which were subjected to an 11.7 T magnetic field (B0) during different durations. The graphs show the variation of the integrated scattered intensity as a function of the azimuthal angle θ. Right panel: Schematic representation of kinetics of orientation and relaxation as a function of Bicelles size and ordering. Adapted from[START_REF] Loudet-Courreges | Unprecedented Observation of Days-Long Remnant Orientation of Phospholipid Bicelles: A Small-Angle X-ray Scattering and Theoretical Study[END_REF]

Fig. 7 .

 7 Fig.7. Nanodiscs for structural biology. a) cartoons representing lipid nanodiscs: the MSP protein helices are represented as blue circles and lipid acyl chains are indicated in black and grey, cholesterol is displayed in orange. b) Electron micrographs of 1,2-dimyristoyl-snglycero-3-phosphocholine (DMPC) MSP nanodiscs (scale bar: 10 nm): the left column shows top views of the nanodiscs, whereas the right column shows side views of nanodiscs. c) cartoon representing the assembly of a multimeric (blue, green and orange) membrane protein complex after insertion and stabilization into nanodiscs: the integral membrane proteins are reconstituted into a small lipid bilayer wrapped by MSPs (purple) forming the nanodisc. Lipids are red/yellow. d) isocontour map obtained from cryo-electron microscopy experiments on the multimeric membrane protein complex stabilized into nanodiscs (c).Adapted from[START_REF] Daury | Tripartite assembly of RND multidrug efflux pumps[END_REF][START_REF] Martinez | Lipid Internal Dynamics Probed in Nanodiscs[END_REF].

Fig. 8 .

 8 Fig. 8. Nanodisc and liposome thermotropism and internal chain ordering. Thermal variation of the first spectral moment (M1) of perdeuterated DMPC, as determined from solid-state 2 H NMR spectra for liposomes (red) and nanodiscs (blue) in the presence (dashed lines) and absence (solid line) of 7.5 mol% cholesterol. M1 values near 140 kHz correspond to a chain order parameter near 1 (chains fully extended, full rigidity) and near 0 to entire disorder as in liquids. M1=70 kHz represents a chain ordering of 0.5. The insert shows a representation of the cholesterol containing nanodisc as already described in Fig.7. Adapted from[START_REF] Martinez | Lipid Internal Dynamics Probed in Nanodiscs[END_REF].

N-1 H NMR PISEMA Breadth of ellipsoid = tilt

Acknowledgments CNRS (Institute of Chemistry), University of Bordeaux and Bordeaux-INP are acknowledged for support. The IR-NMR THC (FRCNRS 3040), the Nouvelle-Aquitaine Region, CPER (Contrat de Plan Etat-Region) and the IdEx-Bordeaux are thanked for financial support to buy and maintain NMR spectrometers.


Teko Ekoué

Osseyi Elolo

D Claire

Ako Komla
email: komla.ako@univ-grenoble-alpes.fr

The transitioning feature between uncooked and cooked cowpea seeds studied by the mechanical compression test

Keywords: Cowpeas, texture, gelatinization, yielding, stiffness, cooking time

Mechanical tests can allow for the mimicking of the textural attributes of foods as perceived by humans. Here we report on the use of the mechanical compression test to study the reactions underlying the cooking of the cowpea seeds (beans)until doneness. The creep test was applied to deal with the time-dependence of the strain, ε(t), at a constant force and to derive the elastic, Ke and viscosity, Kv indices for the different heating times. The results, ε(t), were fitted using mostly a stretched exponential with a linear asymptotic(time) function. The flow rate (ω)and the strain (ε) are the constants of the asymptotic function. Both viscoelasticity indices decreased strongly before the first 30 min (from 562 N to 14 N for Ke and from 5.4×10 6 N.s to 0.11×10 6 N.s for Kv), but from 30 to 75 min ± 3 min, Ke remained almost constant (~14 N) while Kv increases (from 0.11×10 6 to 0.24×10 6 N.s). Both viscoelasticity indices decreased strongly again after 75 min ± 3 min (from 14 N and 0.24×10 6 N.s for Ke and Kv respectively). The heating time-dependence of the viscoelasticity indices

Introduction

Cooking is a practice that requires knowledge and technical skills for controlling the food transformation particularly its texture and flavor until the food is considered as ready for consumption. Many vegetables such as fruits, legumes, roots and seeds, are edible uncooked or cooked. Therefore, it is difficult to know fundamentally whether those vegetables are cooked with only the processing time and texture [START_REF] Kouemene | Evaluation of Temperature and Mechanical Properties of Beans during Cooking Process[END_REF]. The appreciation of the food in terms of cooked or uncooked depends on an individual's sensory perception and expectation as the vocabulary can vary with the people's level of training, culture and their own eating habits. However, at any rate the applied cooking method must provide energy to the food sufficiently to activate physical and/or chemical reactions of the food's elementary compounds [START_REF] Coffigniez | Multiresponse modeling of reaction-diffusion to explain alpha-galactoside behavior during the soaking-cooking process in cowpea[END_REF]. Thus, food is cooked when the reactions underlying the cooking of the foods are done [START_REF] Coffigniez | Multiresponse modeling of reaction-diffusion to explain alpha-galactoside behavior during the soaking-cooking process in cowpea[END_REF]. Thermodynamic, i.e. the states and transition states of the food compounds, with kinetics, i.e. the speed of the reactions, drive the cooking process [START_REF] Coffigniez | Multiresponse modeling of reaction-diffusion to explain alpha-galactoside behavior during the soaking-cooking process in cowpea[END_REF][START_REF] Phillips | Effects of Pretreatment on Functional and Nutritional Properties of Cowpea Meal[END_REF]. The difficulty is to find the operational textural variables in phase with both the thermodynamic and kinetic variables of the reactions by which food engineers can tell quantitatively the onset and the end of the cooking reaction processes, or at least can identify the transitioning period between uncooked and cooked state [START_REF] Kouemene | Evaluation of Temperature and Mechanical Properties of Beans during Cooking Process[END_REF].

In this work, the transitioning of the cowpea seeds to the cooked state was investigated.

Cowpea (Vigna unguiculata) is a legume of multiple variety seeds which consists of a testa with a hilum and micropyle, two cotyledons and an embryonic axis [START_REF] Swanson | Seed Microstructure: Review Of Water Imbibition In Legumes[END_REF].

The bean constitutes of moisture (5 -15 %), proteins (20 -26 %), carbohydrates, starch (40 -70 %), fibre (3 -5 %), and fat (1 -3 %) (Adebooye and Singh, 2007;[START_REF] Coffigniez | Impact of soaking process on the microstructure of cowpea seeds in relation to solid losses and water absorption[END_REF]Khattab et al., 2009). Cowpea bean contains B-vitamins namely, thiamine (~0.9 %) and riboflavin (~0.15 %) [START_REF] Edijala | Effects of processing on the thiamin, riboflavin and protein contents of cowpeas (Vigna unguiculata (L) Walp). I. Soaking, cooking and wet milling processes[END_REF][START_REF] Phillips | Effects of Pretreatment on Functional and Nutritional Properties of Cowpea Meal[END_REF] and also some anti-nutritional components such as, tannins, and phytic acids which can precipitate proteins (Adebooye and Singh, 2007;Khattab and Arntfield, 2009). During the cooking of the beans in hot water, the beans uptake water before starch gelatinization and protein denaturation occur. The flavor and color of the beans subsequently change with time but also with the number of the beans and the cooking media composition [START_REF] Phillips | Effects of Pretreatment on Functional and Nutritional Properties of Cowpea Meal[END_REF]. We hypothesis that the kinetics and all the reactions related to gelatinization of starch and denaturation of proteins would predominantly drive the cooking kinetics of cowpea seeds, given that protein and starch alone represent 60 % to 76 % of the total weight of the cowpea seeds. The water (absorption) activity can lead to substantial change of the beans' texture but it is not alone considered as cooking reaction [START_REF] Borges | Effect of Water Activity on the Mechanical Properties of Selected Legumes and Nuts[END_REF], although it can be kinetically activated by heat [START_REF] Coffigniez | Impact of soaking process on the microstructure of cowpea seeds in relation to solid losses and water absorption[END_REF]. Thus, cowpeas beans may be considered cooked when they are gelatinized and/or denatured (Adebooye and Singh, 2007;Adebooye and Singh, 2008;[START_REF] Coffigniez | Multiresponse modeling of reaction-diffusion to explain alpha-galactoside behavior during the soaking-cooking process in cowpea[END_REF]Khattab et al., 2009).

The evolution of the mechanical properties of a system made with similar composition may be affected by those transitions of the beans' compound states. For instance, people in their own homes use essentially the finger test as mechanical test to assess the cooking levels [START_REF] Voisey | Modernization Of Texture Instrumentation[END_REF]. The finger test is used to evaluate how a product behaves mechanically when it is pressed between the fingers. A raw cowpea bean is shown in Fig. 1A.The state of cowpea bean after simmering at 95 °C during 60 min and 90 min is shown in Fig. 1B and1C respectively. Fig. 1D shows how the cowpea bean that has been simmered at 95 °C during 90 min (shown in Fig. 1C) looks like after being crushed between the fingers. The beans are considered well-cooked when they can easily be crushed between the fingers, and this is the way Khattab et al. (2009) have defined the cooking time (Khattab et al., 2009). Adebooye et al. (2007) pressed the beans between two glass plates after every 5 min and the time when the seed yielded was considered to be the cooking time (Adebooye and Singh, 2007). According to [START_REF] Voisey | Modernization Of Texture Instrumentation[END_REF], the mechanical property is the most critical criteria for assessing the texture of food, because this has the strongest influence on the acceptance of food by people and is relevant to controlling mouth feeling [START_REF] Ishihara | Compression Test Of Food Gels On Artificial Tongue And Its Comparison With Human Test[END_REF][START_REF] Mohsenin | Characterization And Failure In Solid Foods With Particular Reference To Fruits And Vegetables[END_REF].

Texture analyser is used for this purpose to have a better control of the applied forces and precision of the food's mechanical responses, to recreate the conditions of consumer interactions with the food and to correlate those results to specific sensory texture attributes.

In this work a mechanical compression test with a force precision of 10mN was used in order to detect the state transition from uncooked to cooked period of the cowpea seeds when they undergo cooking processes. The aim of this study was to find out how the mechanical properties constants of cowpeas seeds evolve between the beginning and the end of the reactions underlying its cooking.

Materials and Methods

Materials

The cowpea seeds were purchased from a local supermarket in Grenoble, France and used as such without additional treatment. A selection of the seeds based on visual appreciation criteria of the seed's surface homogeneity and cleanness was carried out and damaged seeds removed prior to the cooking method. More than 100 seeds were weighed separately using a precision Sartorius balance of 1.0 mg and the seed weight average was 0.200 g with the standard deviation of 0.023 g. The minimum and the maximum seed weight was 0.150 g and 0.249 g respectively.

Physical treatment: Soaking and heating to 95 °C

Four seeds were distributed in four 20 ml containers and filled with 2 g of deionized water to represent 1:10 (w/w). The containers were sealed and immediately heated to 95 °C in a water bath for 10, 30, 60, 75 and 90 min ± 3 min. The cooking time was selected based on the findings by [START_REF] Coffigniez | Impact of soaking process on the microstructure of cowpea seeds in relation to solid losses and water absorption[END_REF][START_REF] Edijala | Effects of processing on the thiamin, riboflavin and protein contents of cowpeas (Vigna unguiculata (L) Walp). I. Soaking, cooking and wet milling processes[END_REF]Khattab et al., 2009) who reported that at T ≥ 95 °C, the cooking time will vary from 35 to 90 min which depends on the type of seeds and the imprecision of the finger test. Additionally, the loss of solid matter is pronounced with the heating time. Therefore only the seeds that looked intact after they had been heated were selected for the study like the ones shown earlier in Fig. 1 (A,B,C). The initial weight (wi) of each seed was recorded as well as the final weight (wf) after heating. The amount in gram of water uptake per gram of cowpea seed termed RCS was determined as:

( ) i i f CS w w w R - = 1.
RCS, water uptake ratio calculations were carried out using four seeds per heating time in quadruplicates. The temperature in the containers reached 60 °C and 95 °C in less than 2 and 5 min respectively.

Compression test

The test of compression was performed using the HAAKE MARS (Fig. 2a), Modular

Advanced Rheometer System, Thermo Scientific instrument with a parallel geometry of 20 mm of diameter (TiLL11030P20). The measurement steps were managed with the HAAKE RheoWin Job manager software 4.85.0002. The zero gap was determined and the temperature was set to 25 °C, then the geometry was raised to 10 mm. A cowpea seed was placed for each measurement between the geometry and the plate then a first step program was run to bring the geometry onto the surface of the seed (Fig. 2b). The instrument geometry stopped going down when it measured a force of 25 mN. The force was reset to 0 at the current height, geometry position, or before the start of the compression test, the geometry position was taken as the initial height (h0) of the seed. The height position of the geometry (Fig. 2c) during the compression (ht) was recorded and the deformation (εt) at time t was calculated as:

( ) 0 0 h h h t t - = ε 2.
Three determinations of the initial height are shown in Fig. 3a. The forces detected by the instrument fluctuated below 10 mN when the geometry ran freely, but at the moment it touched the seed the force increased sharply and stopped at 25 mN, then the force was immediately reset to zero, before running the compression test program. Two types of programs were edited to study the compression of the seed. The first program (insert of Fig. 3b) was applied to investigate the deformation of the seeds over 1h at a constant load force.

The second was a multi-step loading program that consisted of a step-by-step increment of the force during the compression (Fig. 3b). This type of step-loading creep test was used by [START_REF] Mittal | Rheological Characterization Of Apple Cortex[END_REF] to characterize the rheological property of the apple cortex [START_REF] Mittal | Rheological Characterization Of Apple Cortex[END_REF]. The compression tests were repeated four times using four seeds per heating time to give the strain plot that is the average of these four tests and its error bar is ± the standard deviation.

Data analyses

Over the past three decades, some authors have investigated the uniaxial compressibility of food products in the characterization of food texture and recently Kiani Deh Kiani et al.

(2009) have presumably applied the compression test at ambient temperature to determine the elasticity of red bean grains as a function of their moisture content, i.e. not in cooking conditions [START_REF] Finney | Influence Of Variety And Time Upon The Resistance Of Potatoes To Mechanical Damage[END_REF][START_REF] Kiani Deh Kiani | Determination of Poisson's ratio and Young's modulus of red bean grains[END_REF][START_REF] Mohsenin | Characterization And Failure In Solid Foods With Particular Reference To Fruits And Vegetables[END_REF]. The timedependence of strain with constant stress also termed creep function on the heating time is a more practical way to follow the solid-gel transition of the seeds, because the strain is timeindependent for the real solid material, but time-dependent for the gel [START_REF] Voisey | Modernization Of Texture Instrumentation[END_REF]). In the gel state and under constant stress, the seeds can behave mechanically as a solid in a short time but as a fluid in a long time. The critical time, i.e. when the behavior changes from solidlike to fluid-like, is referred to as the yield time and it has been shown that this time for polymer and colloidal gel systems is stress-dependent [START_REF] Sprakel | Stress Enhancement in the Delayed Yielding of Colloidal Gels[END_REF]. Cowpea seeds are heterogeneous and the area on which the force is applied is unknown. Moreover, the contact area between the probe and the seed expands under large deformation compression forces. The factors affecting the determination of Young's modulus as well as the elastic bulk modulus when agricultural products are subjected to compressive forces are documented [START_REF] Finney | Influence Of Variety And Time Upon The Resistance Of Potatoes To Mechanical Damage[END_REF][START_REF] Hamann | Analysis Of Compression, Tension, And, Torsion For Testing Food gel Fracture Properties[END_REF][START_REF] Hammerle | The Determination Of Poisson's Ratio By Compression Tests Of Cylindrical Specimens[END_REF][START_REF] Mohsenin | Characterization And Failure In Solid Foods With Particular Reference To Fruits And Vegetables[END_REF].

Owing to those factors, only the force and the deformation are recorded here to account for the seed viscoelasticity when evaluating the cooking levels of the seeds. Since this implies that the seeds have identical dimension to make meaningful comparison between samples of different heating times [START_REF] Voisey | Modernization Of Texture Instrumentation[END_REF], the analysis is based on the fact that the variation of the seed apparent area (S) and Young's modulus (E) depend strictly on the heating time.

Consequently, the product ExS yields a quantity termed elasticity index, Ke, in Newton unit (N), because its evolution results from E. The Ke evolution is strictly the result of the heating, which could make it possible to distinguish the uncooked and cooked cowpea seeds. The

Poisson's ratio is odd in large deformations of food products [START_REF] Kiani Deh Kiani | Determination of Poisson's ratio and Young's modulus of red bean grains[END_REF][START_REF] Voisey | Modernization Of Texture Instrumentation[END_REF].

The compression test at 1 N of a cowpea seed soaked in demineralized water and heated for 10 min is presented in Fig. 4. The height of the seed decreased sharply at first exponentially then asymptotically over the critical time. The insert of the Fig. 4 shows the force profile with a dash line to represent the time it takes for the instrument to load the force. The time evolution of the height during the loading stage is the result of the coupling between the loading activity and the viscoelastic response of the sample. This stage could be affected by the loading speed capacity of the instrument, therefore one should keep in mind that it is not only the seed response but it could also be an experimental imperfection. However, this loading phase is fundamental for understanding the behavior of the sample afterward. For a perfectly elastic material for instance, the loading stage would not be critical, because the elastic constant is time-independent. However, viscoelastic materials above yield stress have demonstrated time-dependent mechanical behavior [START_REF] Kiani Deh Kiani | Determination of Poisson's ratio and Young's modulus of red bean grains[END_REF][START_REF] Mittal | Rheological Characterization Of Apple Cortex[END_REF][START_REF] Nussinovitch | Characterization Of Gellan Gels By Uniaxial Compression, Stress Relaxation and Creep[END_REF]. Given that the current sample subsequently exhibited flow behavior, it cannot be excluded that the force-loading rate influences the sample's microstructure and consequently the flow property. The experimental condition could have been difficult, if the samples have experienced severe aging in the time interval of the measurement because aging could affect the mechanical behavior [START_REF] Kouemene | Evaluation of Temperature and Mechanical Properties of Beans during Cooking Process[END_REF][START_REF] Landrum | Delayed yield in colloidal gels: Creep, flow, and re-entrant solid regimes[END_REF]. %, ω = 6.35×10 -6 s -1 .

The time necessary for a material to undergo phenomena of disorganization and to start to flow depends on several factors that excite the curiosity of many authors [START_REF] Coussot | Yield stress fluid flows: A review of experimental data[END_REF][START_REF] Sprakel | Stress Enhancement in the Delayed Yielding of Colloidal Gels[END_REF]. The deformation here showed a good coefficient of linear correlation with the forces during the force-loading period, but its consistency was highly time-dependent (Fig. 5). The linear response domain of the cowpea seed seemed to increase with increasing loading rate. Elsewhere, Kiani Deh [START_REF] Kiani Deh Kiani | Determination of Poisson's ratio and Young's modulus of red bean grains[END_REF] found that the Young's modulus of red bean seed increased with increasing loading rate [START_REF] Kiani Deh Kiani | Determination of Poisson's ratio and Young's modulus of red bean grains[END_REF]. With the exception of the imperfection experimental instrument [START_REF] Voisey | Modernization Of Texture Instrumentation[END_REF] the results of time-dependent disorders or phenomena of changing of state which occurs in food materials as soon as they have been strained [START_REF] Coussot | Yield stress fluid flows: A review of experimental data[END_REF]. The forces and the times it takes to load the forces are indicated in the legend. The seeds respond elastically in the beginning of the loading force. The higher the force that is loading, the faster the loading process and the higher the elastic deformation domain. This observation reflects the impact of the loading rate on the compliance of the samples.

These disorder phenomena convert part of the working energy to entropy energy, the conversion time depends on the kinetic of the reactions mechanisms [START_REF] Landrum | Delayed yield in colloidal gels: Creep, flow, and re-entrant solid regimes[END_REF].The material exhibits apparent elastic property if the increase of the load rate compensates the loss of energy. If the loading is done faster than the relaxation time of the material, then the material tends to behave like a perfect solid. Interestingly, the samples in this study exhibit a constant flow rate defined by the slope of the asymptote in Fig. 4. A similar creep curve was found with gellan gel [START_REF] Nussinovitch | Characterization Of Gellan Gels By Uniaxial Compression, Stress Relaxation and Creep[END_REF]. This means that, if a linear relationship between the applied force and the flow rate is verified, then a quantity in N.s unit reflecting the viscosity of the material could be determined. This quantity is referred to as the viscosity index, Kv. performing the compression test at different constant forces. However, the result was noisy which was likely due to the fact that a new seed prepared in the same condition was loaded onto the rheometer when the applied force was changed. The three plots of the Fig. 5 clearly demonstrate the impact of the physical characteristic of the seeds on the reproducibility of the data, given that the elastic index, Ke, corresponding to the three seeds of identical preparation yielded 16.42 N, 42.9 N, and 29.17 N with an average of 29.5 ± 13 N. We believe that the native composition and the dimensions of the seeds could be the main causes of the noise as these affect the axial deformation. Indeed, the error on the deformation, dε, can be expressed as the sum of errors due to:-i) the applied forces (F), dεF, as instrumental imperfection; ii) elastic modulus (E), dεE, as the variation of the seed composition; and (iii) the area (S) over which the applied forces are distributed, dεS as the seed dimensions. If the elastic modulus of the seeds is assumed identical and the errors on the applied force are neglected then the error on the deformation is geometrical as

ε ε ε ⋅       = = S dS d d S 3.
Although the precise values of the elastic modulus is somewhat uncertain, the creep behavior for mostly all the samples is quite clear. Given that mostly all the samples tested in this study displayed the same creep behavior, we tried the model of Eq.( 4) to sort out some characteristic parameters for the discussion on the mechanical properties of these complex systems. The strain was calculated and then the plot was fitted to Eq.( 4) to obtain the values of the fit parameters.

( )

ε ω ε ε ε β + ⋅ + ⋅ - = - t e t N t 0 4. with ( ) α c t t t N = 5.
The fit function, Eq. ( 4), is a combination of the first period where the response of the material to the loading force is elastic-like, and the second period is the flow (creeping) at constant force. The tc in the Nt , Eq. ( 5), gives the characteristic compliance time during the loading phase. The flow of the material takes place in this case following disorder events occurring during compliance. The α in Nt is time structural factor of disordering events that occur in the material. The events can occur from time to time instead of being continuous as the internal dynamics are very weak to consider that they occur freely. The parameter ε0 defines the initial condition of the deformation. For the flow part of the function, ω and ε are the constants of the regression function when applicable and they mean the flow rate and the axial deformation of the seed respectively. The constant ε is represented by the intersection between the y-axis and the function ωt β + ε at t0, so ε is somehow the onset of the elastic response to the applied constant load. The adjusting parameter β can take 1 for all the analysed measurement, but if necessary, a value between 0.95 and 1 was tested to have the best adequacy with the measurement. Unless otherwise indicated, the value of β is 1. For the current samples, the Fig. 4 shows the adjustment and the regression function of the flow part.

Both functions are plotted in solid and dashed lines respectively. The values of the adjustment parameters are also indicated in the legend. The flow rate, ω, of the seed was found to be 6.35×10 -6 s -1 . The current seed felt like hard chewing gum when pressed between the fingers implying that the seed in this state is not yet soft to be eaten (Khattab et al., 2009).

Results and discussion

Deformation under incremental compressive forces

The average of the strain from the measurement of the 4 seeds after 10 min in water at 95 °C is shown in Fig. 6a. The standard deviation per the average strain yields a value δ that fluctuate between 10 % and 50 %. The median of δ values yields 30 %, i.e. 50 % of δ values are below 30 % and 50 % of δ values are above 30 %.This fluctuation was also observed for all the other quantities that were determined (i.e. viscosity indices, elasticity indices, flow rate, etc.). Therefore, unless otherwise indicated, considering a quantity Ψ of the mechanical tests, the standard deviation of the average can be taken as 30 % of the average in this present study. For the sake of clarity, only the averages of the strain-time plots are displayed in Fig. It is noteworthy mentioning that the time delay between the forces was not changed (increased for instance) to see the impact of this on the behavior of the samples. This will be tested in another study.

The compression test was carried out in an unconfined state, so the seeds expanded transversally by the flow behavior but not as an incompressible fluid. There are a number of factors that can make the compression to stop. These are: i) when locally the concentration increases and the compressibility of the seeds decreases as the volume decreases towards the total volume of the uncompressed elementary constituents. The deformation of the seeds will be stopped by the exclusion effect of the elementary constituents, i.e. limit of the samples compressibility; and ii) when the geometry reaches the zero gap. The measurements did not show any remarkable tendency towards stability which means that the maximum degree of compaction was not attained for all the mechanical tests. The time between the application of load and the moment when the sample yields which is the yield or critical time, generally depends on the applied force because yielding mechanism are an activated energy process.

The critical time and stress relationships vary from one system to another. It depends on the competition and the cooperative or synergistic effects of the many yielding events that can take place in the seed at any instant. The thermal motion of small molecules and reptation of biopolymers [START_REF] De Gennes | Dynamics of Entangled Polymer Solution[END_REF], the balance between the dynamic of physical bonds rupture and reconstruction [START_REF] Sprakel | Stress Enhancement in the Delayed Yielding of Colloidal Gels[END_REF] are some of the fundamental yielding events of which the kinetics determine the critical time of the creep behavior of the samples.

The critical time, tc, and α of the fit function are related to the time behavior of the sample under yielding. The values of α fluctuate between 0.5 and 1 when the values of tc fluctuate between 14 s and 70 s with the change of the applied forces. This means that a clear correlation between the parameter α or tc with the applied forces was not observed in the present study. Thus, it was not possible to correlate afterward the critical time with the heating time of the seeds and this implies the complexity of characterizing the yield time of heterogeneous condensed materials under mechanical stress. For instance, it is demonstrated that the relaxation time of polymer and colloidal systems exhibit an exponential decay dependence on the applied stress [START_REF] Sprakel | Stress Enhancement in the Delayed Yielding of Colloidal Gels[END_REF], but for a mixture of polymer and colloidal system or for heterogeneous systems in general this relationship is not fundamentally confirmed. Cowpea seed systems consist of a tightly wrapped packs of starch granule cells in a protein matrix around which there are hydrocolloids and low presence of fat [START_REF] Coffigniez | Impact of soaking process on the microstructure of cowpea seeds in relation to solid losses and water absorption[END_REF][START_REF] Swanson | Seed Microstructure: Review Of Water Imbibition In Legumes[END_REF]. The compression test of this system has shown yielding despite of the applied forces (between 100 mN to 20 N) demonstrating the weakness of the energy needed to activate the yielding mechanisms. The absorption of water by the seeds which causes the microstructure to swell and the intercellular spaces to increase is pronounced by heat, capillary effects, and by the pectin and protein based substances through the migration process in order to balance the osmotic pressure [START_REF] Coffigniez | Impact of soaking process on the microstructure of cowpea seeds in relation to solid losses and water absorption[END_REF].

Although the appearance of the whole seed seemed to be preserved, some of the walls of the interfacial cells could have been damaged and water may continue to diffuse into the whole seed until the osmotic pressure is balanced even at room temperature. The dynamic of the system to reach equilibrium would have been enhanced by a local increase of temperature due to the effects of stress. The lubrication of the interstitial spaces between the seeds cells and their large clusters due to the presence of lipid substances in the intercellular spaces would be also enhanced by the locally induced increase of temperature by stress. It is noteworthy to mention that after 10 min in water at 95 °C the seeds have taken on average about 0.4 ± 0.3 g of water per gram of seeds.

Compressive force and deformation relationships upon heating time

It is shown earlier (in Fig. 5) that the instantaneous responses of the samples are elastic before the elastic response of the sample give way to flow response when the imposed forces are completely loaded. Therefore, the intersection between the (asymptotic) flow function and time 0 axis is taken here as the instantaneous elastic deformation, ε, and assigned to the applied force. Fig. 7 shows the results for different heating times at 95 °C and time 0 for the uncooked (native cowpea) beans , which is denoted as NC. The force as the functions of deformation increases quite linearly as shown by the full line through the data of 90 min and deviates from linearity with higher slope. The compression tests of alginate gels have shown practically similar trend, which the authors have described by a concave upward function using a power-law model, where the pre-factor was referred to as the gel stiffness and the power as the degree of concavity that reflects the deviation from linearity. When the power equal 1, then the power-law model reduces to Hooke's law and the pre-factor coincides with the modulus of elasticity [START_REF] Mancini | Uniaxial Compression And Stress Relaxation Tests On Alginate Gels[END_REF]. Given that the slope is characteristic of the seeds elasticity, the increase of this means strengthening of the seed.

However, under unconfined compressive force, this trend may reflect the compaction effects requiring additional force to strain the sample rather than strain hardening effects [START_REF] Mancini | Uniaxial Compression And Stress Relaxation Tests On Alginate Gels[END_REF]. Furthermore, the resistance to stretch the seed coat could explain the rising of the force-deformation function. At higher compressive forces, the deformation amplitude decreases from the native seed with increasing heating time. However, the way it decreases seems to depend on the applied forces. At a weak or strong applied forces, i.e. 0.1 N or 20 N, the deformation of the seed evolves without a significant discriminative signature between the cooking levels. For the curve of 20 N, the change of the slope is weak between A and B and is absent between B and C. For the curve of 0.1 N, the deformations are too weak to provide relevant information on the periods of the cooking reactions. Therefore, the sort by cooking-level of the seed is weakly precise with both extreme forces. However, for intermediate compressive forces, the test shows a remarkable transitional phase that is characterized by a plateau between level A and C. The controlling compressive forces show a clear-cut between uncooked and cooked seeds as long as the compression test is done with moderate applied forces.

Compressive deformation rate and the seed viscosity index

The deformation rates (ω) of the seeds are plotted as function of the applied forces to the different heating times. Four graphs are displayed in Fig. 9 to illustrate the force-dependence of ω. These graphs are; the raw seed (NC, Fig. 9a) and the seed after heating time of 10 min (Fig. 9b), 60 min (Fig. 9c) and 90 min (Fig. 9d) at 95 °C. The graph of the native seed shows no deformation rate for the lowest compressive forces where as ω starts increasing from roughly 1 N (Fig. 9a). The rise of ω from a stress threshold could characterize the heated seeds too, although it is not seen in the figures 9 b, c, and d, which is mainly attributed to the fact that the yield stress is lower than the forces applied in this case. Indeed, the overall seeds exhibit time-dependent yielding behavior which energy barrier stem from the physical bonds that provide to the seeds the resistance against flow under their own weight. The fact that ω decreases first for the seeds of Fig. 9 b, c, d suggests that free liquid exuded out of the seeds cell interstices. Such expulsion leads to compaction as a result of which ω decreases to a minimum that reflects presumably the transition between the liquid removal and a jamming flow regime. In the jamming regime, ω increases from the minimum around 30 ± 10×10 -6 s -1 to a maximum around 75 ± 10×10 -6 s -1 with increasing applied forces (N) for the heated seeds but the maximum for the native seeds was not reached. The ω of the native seeds is 10 to 20 times lower than that of the heated seeds. The deformation rate is assumed to evolve with the applied forces linearly in the jamming flow regime. This assumption made it possible for the viscosity index, Kv, to be calculated. and Singh, 2008;Biliaderis et al., 1980;[START_REF] Kong | Compression Properties Of Fish-Meat Gel As Affected By Gelatinization Of Added Starch[END_REF]; (ii) denaturation with aggregation of protein [START_REF] Tolkach | Reaction kinetic pathway of reversible and irreversible thermal denaturation of b-lactoglobulin[END_REF]; and (iii) hydration of carbohydrate like pectin, significantly take place [START_REF] Coffigniez | Impact of soaking process on the microstructure of cowpea seeds in relation to solid losses and water absorption[END_REF]. At the end of these processes, after 80 min, the Kv decreases continuously because of additional absorption of water (Fig. 9e)

until the seeds burst.

Seeds stiffness and flow rate as function of the heating time

The relation between ε and the applied forces were interrogated to derive the elastic During the first 10 min, the elasticity index, Ke, of the seeds decreases steeply from 560 N to 19 N, a loss of 97 % of their initial stiffness owing to the flow of water into the seeds macrostructure. The water uptake by the seeds during the first 10 min was on average 38 % of the seeds weight. As a result of this absorption of water by the seed, the flow rate of the seeds significantly increased. When the heating time was increased from 10 min to 30 min, the seeds lost 36 % of their stiffness and the seeds flow rate ω increased weakly. However the amount of water uptake by the seeds in this second period increased by 50 % with almost a constant rate. The decrease of the stiffness during the heating period from 10 min to 30 min is not a result of the absorption of water alone but also a consequence of a collection of mechanisms with opposing effects that prevent both the catastrophic decrease of the stiffness and thickness of the seeds. Thus, we suggest that the gelatinization of starch and denaturation of protein processes in this heating period strengthen the seeds. These processes define the onset of the cooking rather than the water absorption alone. Therefore, it could be hypothesised that the cooking of the seeds starts between 10 min and 30 min at 95 °C. When the heating time was between 30 min and 80 min, the Ke remained practically constant at 14 N, although the absorption of water by the seeds was shown to slightly increase. We could say that the cooking reactions contribute to the stability of the Ke until the end of the reactions somewhere between 75 min and 90 min. Beyond 75 min the stiffness and the thickness of the seeds reduces catastrophically. Heating time longer than 90 min destroyed the seeds and their internal contents were expelled in the cooking media [START_REF] Edijala | Effects of processing on the thiamin, riboflavin and protein contents of cowpeas (Vigna unguiculata (L) Walp). I. Soaking, cooking and wet milling processes[END_REF].

Conclusion

Cowpea is rich in protein (20 % -26 % w/w) and starch (40 % -70% w/w), it contains also vitamin, fat and hydrocolloids but in weak proportion. Cowpea seeds are multiple varieties of species of which the size, weight, seeds coat texture and composition are the specific attributes. Seeds without visual defects were selected to investigate the seeds cooking time using the mechanical compression test given that the mechanical properties of food are correlated with the physical and chemical changes of the food. However: i) how the seeds will behave when they are subjected to mechanical test; ii) how their mechanical properties are correlated with the seeds transformations; and iii) whether this correlation will allow independently food engineers to visualize the transitioning of the seeds to the cooked state cannot be predicted. The seeds are cooked when the physical and chemical reactions underlying the seeds cooking are done. Therefore, we think that the activation of both protein denaturation and starch gelatinization reactions are the starting point of the seed cooking, although before these reactions happen the seeds uptake water during cooking at 95°C. We found that cooking of the seeds starts between 10 min and 30 min because the seeds stiffness (elasticity index) and thickness (viscosity index) decrease weakly during this period of time.

Before this period both mechanical constants decreased strongly, for instance a loss of 97 % of the elasticity index was observed during the first 10 min of heating of cowpea seeds. The time the seeds are cooked was found to be between 75 min and 90 min, because in this period it was observed that both mechanical constants decrease sharply. Furthermore just before, i.e. between 30 min and 75 min, the stiffness was practically constant, and the thickness was increasing or the flow rate was decreasing. To get the stiffness (elasticity index) and the thickness (viscosity index), a stretched exponential plus a linear asymptotic time function was used to fit the creep behavior of the sample where the strain (ε) and the flow rate (ω) were the fit functions parameters. These fit parameters correlate respectively with the applied forces within the linear regime. Viscoelasticity indices correlate with the cooking reaction transitioning.

This work allow to investigate the influence of the cooking medium (salt, pH, sugar, etc.) and other physical seed treatment (from harvest to cooking) on the cooking time and the way the legumes behave mechanically when the undergo cooking reactions in order to achieve commercially acceptable new foods textures with reduction of the cooking energy.

Acknowledgment

The authors declare no conflicts of interest. This The forces and the times it takes to load the forces are indicated in the legend. The seeds respond elastically in the beginning of the loading force. The higher the force that is loading, the faster the loading process and the higher the elastic deformation domain. This observation reflects the impact of the loading rate on the compliance of the samples.

Figure captions

Fig. 1 :

 1 Fig.1: A) Picture of a native black eye cowpea seed; and the seed after having been simmered

Fig. 2 :

 2 Fig.2: Picture of the experimental instrument (a) to show the onset of the compressive test (b)

Fig. 3

 3 Fig.3: a) Initial height position detection test repeated 3 times (open circle, square and

Fig. 4 :

 4 Fig. 4: Time-dependence of deformation εt of a cowpea seed after heating the seed at 95 °C

Fig. 5 :

 5 Fig. 5: Deformation dependence of loading forces on seeds after heating at 95 °C for 10 min.

  6b. The plots were analysed using the fit function (full line), the dash lines show the linear regression function of the flow part of the samples and the points where they meet the time 0 axis. The time delay between the forces (dotted lines) was decided according to the previous tests on the samples of the same series and the compression test program was adjusted to fit with the samples behavior.

Fig. 6

 6 Fig.6: a) Time-dependence of deformation εt of a cowpea seed after heating the seed at 95°C

Fig. 7 :

 7 Fig.7: Applied forces as function of the deformation ε for the native cowpea seed (NC) and

Fig. 8

 8 Fig.8shows for each applied force the deformation values as a function of the heating times.

Fig. 8 :

 8 Fig.8: The deformation of the Fig. 7 is plotted as a function of the heating time at 95 °C when

Fig. 9 :

 9 Fig.9: Force-dependence of deformation, ω, of native cowpea (a), heating times in the figures

  coefficient according to Hooke's law termed elasticity index, Ke in N unit. The Ke is the constant of proportionality between the applied forces and the deformation in the elastic like domain, i.e. before the deformation deviates from linearity. The Ke values are plotted as function of the heating time (result of native seeds is represented by time zero) on the left axis of Fig. 10a in a log-lin scale. The deformation rates, ω, at 0.1 N are plotted on the same figure (Fig. 10a) as function of the heating time using the right axis. The cooking level A, B, C are presented on the top axis of the heating time. The Fig.10b shows the evolution of the water uptake by the seeds.

Fig. 10 :

 10 Fig.10: (a) Evolution of the elasticity constant, Ke on the left axis, and the deformation rate

  work was financially supported by the educational ministry of TOGO. The Laboratoire Rhéologie et Procédés (LRP) is part of the LabEx Tec 21 (Investissements d'Avenir -grant agreement n°ANR-11-LABX-0030) and of the PolyNat Carnot Institut Investissements d'Avenir -grant agreement n°ANR-11-CARN-030-01).

Figures

  Figures

Fig. 2

 2 Fig. 2

  Fig. 3a

  Fig. 3b

Fig. 5

 5 Fig. 5

  Fig. 6a

  Fig. 6b

  Fig. 10b

Fig. 1 :

 1 Fig.1: A) Picture of a native black eye cowpea seed; and the seed after having been simmered

Fig. 2 :Fig. 5 :

 25 Fig.2: Picture of the experimental instrument (a) to show the onset of the compressive test (b)

Fig. 6

 6 Fig.6: a) Time-dependence of deformation εt of a cowpea seed after heating the seed at

Fig. 7 :Fig. 8 :

 78 Fig.7: Applied forces as function of the deformation ε for the native cowpea seed (NC) and

Fig. 10 :

 10 Fig.10: (a) Evolution of the elasticity constant, Ke on the left axis, and the deformation rate

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	Biliaderis, C.G., Maurice, T.J., Vose, J.R., 1980. Starch gelatinization phenomena studied by
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