Agnel Praveen

Narayanaswamy Srinivasan

Alexandre G De Brevern
email: alexandre.debrevern@univ-paris-diderot.fr

Cis-trans peptide variations in structurally similar proteins

Keywords: protein folds, cis peptides,  dihedral, cis-trans isomerization, structural alignment, structural alphabet, Protein Blocks, Protein Data Bank

Cis-trans peptide variations in structurally similar

proteins.

Introduction

The large majority of peptide bonds in protein structures are constrained to a planar conformation [START_REF] Ramachandran | Conformation of polypeptides and proteins[END_REF][START_REF] Pauling | The structure of proteins; two hydrogen-bonded helical configurations of the polypeptide chain[END_REF]) (trans) due to the partial double bond nature of the peptide linkage. This has been used as a strict dihedral angle restraint (ω~180°) for protein structure refinement in X-ray crystallography [START_REF] Priestle | Improved dihedral-angle restraints for protein structure refinement[END_REF].

However, several studies have identified deformations in planarity that has been tolerated in protein structures [START_REF] Macarthur | Influence of proline residues on protein conformation[END_REF][START_REF] Weiss | Peptide bonds revisited[END_REF].

A very small percent of all peptides are observed to occur in cis conformation [START_REF] Ramachandran | An explanation for the rare occurrence of cis peptide units in proteins and polypeptides[END_REF]. Significant portion of about 5% of all Xaa-Pro peptides (Xaa being the amino acid preceding the Proline residue) were found in this conformation with the ω dihedral close to 0 o [START_REF] Stewart | Occurrence and role of cis peptide bonds in protein structures[END_REF][START_REF] Macarthur | Influence of proline residues on protein conformation[END_REF][START_REF] Weiss | Peptide bonds revisited[END_REF]. On the other hand, only about 0.03% of the Xaa-nonPro peptides are reported to be in the cis form [START_REF] Weiss | Peptide bonds revisited[END_REF]. This behaviour is due to the imide bond character of the peptide joining a proline residue, which limits the steric cost to have ω close to 0°. Due to the strict constraints used during structure determination and refinement, many of the cis peptide bonds may have gone unreported, especially in low resolution structures [START_REF] Weiss | Peptide bonds revisited[END_REF]).

The Cα i -Cα i+1 distance in cis conformation is nearly 1Å less than that seen in trans and hence there can be a strong influence of structure resolution on the cis bond content [START_REF] Stewart | Occurrence and role of cis peptide bonds in protein structures[END_REF]; [START_REF] Weiss | Peptide bonds revisited[END_REF].

Cis bonds are preserved in protein structures for both structural and functional reasons [START_REF] Brauer | A conserved cis peptide bond is necessary for the activity of Bowman-Birk inhibitor protein[END_REF][START_REF] Lummis | Cistrans isomerization at a proline opens the pore of a neurotransmitter-gated ion channel[END_REF][START_REF] Wu | A cis-prolyl peptide bond isomerization dominates the folding of the alpha subunit of Trp synthase, a TIM barrel protein[END_REF]. Xaa-nonPro peptides in cis conformation are observed near the active sites and are shown to be important for the protein activity [START_REF] Weiss | Peptide bonds revisited[END_REF]Stoddard and Pietrokovski 1998;[START_REF] Jabs | Non-proline cis peptide bonds in proteins[END_REF][START_REF] Guan | Structural mechanism governing cis and trans isomeric states and an intramolecular switch for cis/trans isomerization of a non-proline peptide bond observed in crystal structures of scorpion toxins[END_REF]. Amino acid preferences associated with the cis conformation have also been studied with the help of high resolution structures. Aromatic residues such as Tyr, Trp and other amino acids like Gly, Ala and Pro are more frequent at the position Xaa, before the Pro [START_REF] Pal | Cis peptide bonds in proteins: residues involved, their conformations, interactions and locations[END_REF][START_REF] Weiss | Peptide bonds revisited[END_REF].

The inter-conversion between the cis and trans isomers is a slow reaction and is catalysed in cell by peptidyl cis-trans isomerases [START_REF] Fischer | Regulation of peptide bond cis/trans isomerization by enzyme catalysis and its implication in physiological processes[END_REF]. An activation barrier of about 20 kcal/mol has been proposed for this change [START_REF] Grathwohl | NMR studies of the rates of proline cis-trans isomerization in oligopeptides[END_REF][START_REF] Scherer | Barriers to rotation of secondary amide peptide bonds[END_REF]. Unlike the Xaa-Pro cis bond, the non-prolyl peptides require a considerable additional cost (~2 kcal/mol) to be stabilized in this conformation. Several studies on cis-trans isomerisation underline its importance in protein folding kinetics and function [START_REF] Andreotti | Native state proline isomerization: an intrinsic molecular switch[END_REF]. Proline isomerisation is shown to guide the unfolding and refolding paths of proteins [START_REF] Schmid | Proline isomerization during refolding of ribonuclease A is accelerated by the presence of folding intermediates[END_REF][START_REF] Jakob | Energetic coupling between native-state prolyl isomerization and conformational protein folding[END_REF]. Mutations destabilizing the cis conformation are also shown to reduce the folding rates [START_REF] Odefey | Non-prolyl cis-trans peptide bond isomerization as a rate-determining step in protein unfolding and refolding[END_REF][START_REF] Guan | Structural mechanism governing cis and trans isomeric states and an intramolecular switch for cis/trans isomerization of a non-proline peptide bond observed in crystal structures of scorpion toxins[END_REF][START_REF] Brandts | Consideration of the Possibility that the slow step in protein denaturation reactions is due to cis-trans isomerism of proline residues[END_REF].

Cis-trans conversion is implicated in cell signalling [START_REF] Sarkar | Proline cis-trans isomerization controls autoinhibition of a signaling protein[END_REF][START_REF] Wulf | Phosphorylation-specific prolyl isomerization: is there an underlying theme?[END_REF], neuro-degeneration [START_REF] Pastorino | The prolyl isomerase Pin1 regulates amyloid precursor protein processing and amyloid-beta production[END_REF], amyloid formation [START_REF] Eakin | A native to amyloidogenic transition regulated by a backbone trigger[END_REF], channel gating [START_REF] Lummis | Cistrans isomerization at a proline opens the pore of a neurotransmitter-gated ion channel[END_REF], phage infection [START_REF] Eckert | Prolyl isomerization as a molecular timer in phage infection[END_REF], structural dynamics [START_REF] Grochulski | Two conformational states of Candida rugosa lipase[END_REF][START_REF] Guan | Structural mechanism governing cis and trans isomeric states and an intramolecular switch for cis/trans isomerization of a non-proline peptide bond observed in crystal structures of scorpion toxins[END_REF]) etc. This isomerisation has been recognized as a molecular timer that modulates the amplitude and duration of different cellular processes [START_REF] Lu | Prolyl cis-trans isomerization as a molecular timer[END_REF][START_REF] Nicholson | Prolyl cis-trans Isomerization as a molecular timer in Crk signaling[END_REF]). Hence, it has been also considered as a possible target for therapeutic studies [START_REF] Lu | Prolyl cis-trans isomerization as a molecular timer[END_REF]).

The values of ω are associated with the other backbone dihedrals and a strict correlation was observed with the adjacent φ angle [START_REF] Esposito | Correlation between omega and psi dihedral angles in protein structures[END_REF]. A strong association between the side chain and backbone conformation has also been noted for the cis conformation [START_REF] Pal | Cis peptide bonds in proteins: residues involved, their conformations, interactions and locations[END_REF]. Local structural changes associated with cis-trans isomerisation have been studied based on the Cα distances [START_REF] Reimer | Local structural changes caused by peptidyl-prolyl cis/trans isomerization in the native state of proteins[END_REF]. Except for the position X (Xaa-Pro) the distances between consecutive Cαs did not show much variations. However significant changes were observed in the distances between Cα i and Cα i+2, where i stands for the residue at position X. β-turns of type VI has the characteristic backbone φ,ψ angles for the cis conformation, at positions i+1 and i+2. β-turns of types VIa-1 (φ i+1 ,ψ i+1 ,φ i+2 ,ψ i+2 : -60°,120°,-90°,0°) and VIb-1 (φ i+1 ,ψ i+1 ,φ i+2 ,ψ i+2 : -135°,135°(& >100°),-75°,160°) are predominant among the different sub-classes of type VI turns [START_REF] Pal | Cis peptide bonds in proteins: residues involved, their conformations, interactions and locations[END_REF].

The preferences for both amino acid types and backbone conformations associated with cis peptides have been analysed but specific and comprehensive studies on the conservation of cis conformation have not been carried out. In this study, we analyse the deformations in the planarity of peptide bonds in protein structures with special emphasis on the variations between cis and trans conformers in similar folds. Using protein family alignments involving high resolution structures, we look at the amino acid preferences and the local conformations associated with such cis-trans conversion. A Structural Alphabet namely Protein Blocks (PBs) was used for local structure description [START_REF] De Brevern | Bayesian probabilistic approach for predicting backbone structures in terms of protein blocks[END_REF][START_REF] Etchebest | A structural alphabet for local protein structures: improved prediction methods[END_REF][START_REF] Joseph | A short survey on Protein Blocks[END_REF]. This helps to obtain a precise categorization of pentapeptide backbone conformations into 16 frequent and regular conformations. Only the conserved regions of the alignment are considered for the study.

Methods

Dataset. A set of high quality protein structures solved by X-ray crystallography, with resolution better than 1.6Å and R-factor less than 0.25 is extracted from the PDB. The SCOP domains (version 1.75) corresponding to these structures were identified and all those domains belonging to the same SCOP family were aligned. This resulted in multiple structural alignments of 775 families. The conservation of ω dihedral angles was studied by analysing well-aligned (less than 30% gaps) columns in the alignment.

Protein Blocks. Protein Blocks (PBs) correspond to a set of 16 prototypes of pentapeptide conformations, described based on the  and  dihedral angles (Supplementary data 1). They were obtained by grouping different pentapeptide conformations observed in protein structures using an unsupervised classifier similar to Kohonen Maps [START_REF] Kohonen | Self-Organizing Maps[END_REF] and hidden Markov models [START_REF] Rabiner | A tutorial on hidden Markov models and selected application in speech recognition[END_REF]). This structural alphabet allows a reasonable approximation of local backbone conformations [START_REF] De Brevern | Bayesian probabilistic approach for predicting backbone structures in terms of protein blocks[END_REF] with an average root mean square deviation (rmsd) evaluated at 0.42 Å (de Brevern 2005; [START_REF] Joseph | A short survey on Protein Blocks[END_REF]). PBs (de Brevern 2005) have been assigned using a in-house Python software as in the previous studies [START_REF] Gelly | iPBA: a tool for protein structure comparison using sequence alignment strategies[END_REF][START_REF] Joseph | Improvement of protein structure comparison using a structural alphabet[END_REF].

Secondary Structure Assignment. The secondary structure types associated with the PBs were identified with the help of assignments made by PROMOTIF [START_REF] Hutchinson | PROMOTIF--a program to identify and analyze structural motifs in proteins[END_REF].

Multiple Structural Alignment. A structural alignment approach had been developed

based on the representation of protein 3D structures as PB sequences [START_REF] Gelly | iPBA: a tool for protein structure comparison using sequence alignment strategies[END_REF][START_REF] Joseph | A short survey on Protein Blocks[END_REF][START_REF] Joseph | Improvement of protein structure comparison using a structural alphabet[END_REF][START_REF] Tyagi | Protein structure mining using a structural alphabet[END_REF]. The protein structural backbone was transformed into PB sequence and these sequences were aligned to obtain a comparison of the 3D structures. Dynamic programming algorithm was used to generate PB sequence alignments [START_REF] Joseph | Improvement of protein structure comparison using a structural alphabet[END_REF]. Pairwise structure comparison based on PB representation (iPBA) outperformed other popular structural alignments tools in several benchmark tests [START_REF] Joseph | Improvement of protein structure comparison using a structural alphabet[END_REF][START_REF] Tyagi | Protein structure mining using a structural alphabet[END_REF][START_REF] Gelly | iPBA: a tool for protein structure comparison using sequence alignment strategies[END_REF].

The PB based pairwise alignment was extended to perform multiple structure comparisons. A progressive alignment strategy similar to that used in CLUSTALW [START_REF] Thompson | CLUSTAL W: improving the sensitivity of progressive multiple sequence alignment through sequence weighting, positionspecific gap penalties and weight matrix choice[END_REF], was adopted for multiple PB sequence alignments. The various different parameters and gap penalties were optimized based on the alignment quality. This method performed better than established tools like MUSTANG [START_REF] Konagurthu | MUSTANG: a multiple structural alignment algorithm[END_REF], MULTIPROT [START_REF] Shatsky | A method for simultaneous alignment of multiple protein structures[END_REF], SALIGN [START_REF] Madhusudhan | Alignment of multiple protein structures based on sequence and structure features[END_REF], MATT [START_REF] Menke | Matt: local flexibility aids protein multiple structure alignment[END_REF] and MAMMOTH [START_REF] Lupyan | A new progressive-iterative algorithm for multiple structure alignment[END_REF]) (Joseph et al, submitted).

Results

The amino acid and conformational preferences associated with cis-trans peptide variations were studied using a dataset of alignment of families of high quality structures.

Initially, the preferences associated with the cis conformation, were studied. Then the sequence and structural features associated with cis-trans changes were analysed. PB representation was used to have a reasonable categorization of the backbone conformations associated with such changes.

Figure 1 gives the distribution of the values of ω angles observed in the databank. In the alignment databank, 94% of the peptides have  angles that varies at most 10 o from planarity (180 o ). Only about 0.31% of all peptides are found to have cis conformation with ω close to 0 o . The distribution of ω angles was also studied in a larger dataset of structures solved at resolutions under 2.5Å. Only 0.23% of the peptide conformations are in cis form.

Cis peptide preferences

Throughout this work, we indicate the two amino acids that form the peptide, as Xaa1 -Xaa2. The preference for each amino acid to occur at these two positions of a cis peptide was calculated. At the position Xaa1, aromatic amino acids have a higher preference compared to others. About 12.6% of the position Xaa1 is occupied by Gly (Figure 2A), while 87.6% of the position Xaa2 is occupied by Pro (Figure 2B). About 0.62% and 0.67% of all Tryptophans and Tyrosines are in the first position Xaa1 of a cis-bond (Figure 2C). Apart from the aromatic amino acids (W,Y, F), Gly, Ala, Asn, Pro and Cys also prefer to occupy the position Xaa1 (> 0.3%). Aromatic amino acids are reported to stabilize Pro in the cis conformation by forming pi interactions [START_REF] Wu | Local control of peptide conformation: stabilization of cis proline peptide bonds by aromatic proline interactions[END_REF].

5.4% of all prolines occur as Xaa2 in the cis-conformation (Figure 2D). All other amino acids are under-represented at this position, Tyr has a relatively higher occurance of about 0.12%. For 61% of the cis peptides, the solvent accessibility at the second position was higher than the first reflecting a greater exposure at this position (Supplementary data 2). About 26% of the cis peptides occur in β turns of type IV (Figure 3). Other β turns of types VIa1, VIa2 and VIb (that are associated with a cis-Pro [START_REF] Hutchinson | PROMOTIF--a program to identify and analyze structural motifs in proteins[END_REF]) also dominate the cis conformation. The conformation at the first position (Figure 3A) is represented by β turns VIa1 and VIb with an occurrences of 13.1% and 16.2% respectively.

The coil and extended (β strand) conformations share about 14% each, at this position, which is quite low compared to the background frequencies. At the second postion (Figure 3B), β turns VIa1 and VIb contribute to 12.2% and 21.1% of the cis conformations. The strand conformation is less frequent (4.4%) at this position. The preferred conformations were also studied in terms of Protein Blocks (PBs). PBs generally associated with β strands, af, are preferred in both positions (Figure 3C & D), PB b, a specific PB often located after long loop at the entrance of -strand, has a higher preference for the 2 nd position. Loop associated PBs, g and j are preferred in the first position while PBs h and k are preferred in the 2 nd . Note that the PBs g and h are frequently found in succession, similarly j and k share common dihedrals. Supplementary data 3 gives the dihedral angles associated with these PBs.

Cis-Trans Variations

The multiple structural alignments involving high resolution structures were used to study the conservation of cis conformation in structurally related proteins. The conserved regions in the alignment (< 30% gaps) were only considered for analysis. The frequency of changes in the ω dihedral was also computed from the alignments (Figure 4). 65% of the ω angles are highly conserved with a maximum variation of only 10 o . On the other hand, the cis conformation is not well conserved; about 36% of the cis peptides prefer exchange with the trans conformer.

The change from cis to trans is indicated by the change Xaa1-Xaa2 to Xaa1'-Xaa2', where Xaa1 and Xaa2 are amino acids linked by the peptide in cis conformation and Xaa1' and Xaa2' represents amino acids linked by the peptide in trans conformation after the change. It was quite striking to note that nearly 65% of the positions that were associated with changes in the cis conformation, are marked with gaps in the alignment.

Based on the residue occupancies at the two positions, cis peptides can have different tendencies to go to the trans state. To study the relationship of amino acid types with the stability of the cis conformation, we calculated the percentage of residues undergoing cistrans variations. Figure 5 gives the percentage propensity of each amino acid (in cis peptides) to go to the trans conformation in a related protein. At positions Xaa1 and Xaa2, proline has about 31.8% and 33.8% chance of undergoing the change (Figure 5A &B). At position Xaa1, isoleucine and glutamine have the highest tendency for isomerisation (64.3% and 61.6%).

Other amino acids like glycine also show high preference of about 47.5%, for conversion (Figure 5A). Trp in the position Xaa1 prefers to conserve the cis conformation (only 11.5% change). At position Xaa2, the very low occurrences of non-Pro residues restrict a reliable quantification of the preferences (Figure 5B). Only the changes with at least 10 instances are considered for analysis. Lys and Gly shows higher preferences (47.5% and 55.5% respectively) for the change, at the second position Xaa2. As mentioned above, only 33.8% of Pro undergoes cis-trans changes.

Figure 5C & D gives the preferences in amino acid substitutions associated with cistrans variation. Only the conserved alignment positions (no gaps at both positions) are considered for studying the substitution preferences. At the first position (Xaa1 to Xaa1') many of the amino acids prefer replacement with non-polar amino acids like Gly, Ala, Leu and Ile, for undergoing the change. Substitutions with Glu and Pro are also preferred. On the other hand, at the second position (Xaa2 to Xaa2'), most of the amino acids prefer to be conserved (Figure 5C). Pro, which is dominant at this position, is not highly conserved (only 13.3%) upon the cis-trans conversion. Some amino acids like Thr, Asp, Gln and His prefer substitutions with Gly upon conversion. Leu and Ile are preferably replaced with Asp or Phe for the cis-trans change.

The local conformational changes associated with cis-trans changes was also studied (Figure 6). At the first position (Figure 6A), β turns of types VIII and IV are more prone to cis-trans change (61% and 41%). However it has to be noted that only a few cis peptides occur in type VIII β turns (Figure 3A). β turn VIb and the coil states also have conversion propensities above 30% (38.5% and 34.9% respectively). At the second position (Figure 6B), the coil and strand conformations have conversion frequencies close to 40%. The β turn VIb also show a relatively higher preference (38.3%).

The propensities were also studied in terms of PB changes. At the first position (Figure 6C), PBs l and p have higher propensities for cis-trans change (75.9% and 69.7% respectively). PB l is largely found in the N-cap of helices while p is frequent at the C terminus (and at the N terminus of strands). PB i mainly associated with coils and has a high dominance of type IV β turns, has a propensity of about 50% for conversion. For the second position on the other hand (Figure 6D), PBs j and a have a relatively higher preferences with frequencies 54.4% and 46.0% respectively.

On the conserved regions of alignment (no gaps at both positions), the PB changes associated with cis-trans change was analysed (Figure 7). At both positions a preference for substitution with a strand associated conformation (PBs b, c and d), was observed. At the first position (Figure 7A), loop associated PBs i and j, prefer replacement with PB c which is frequent at the N terminus of strands. PB g, that represents a significant portion of 3 10 helices, prefers substitution with central helix conformation (PB m). This preference is observed at the second position also. At the second position (Figure 7B), PBs i and j, have greater preferences to get replaced by strand associated PBs, b and d. Apart from g, PBs h, k and p also prefer change with PB m, upon cis-trans change.

These substitution preferences were compared with the favourable PB changes (substitution with different PBs) observed in all the aligned regions in the dataset. Figure 7C & D gives the distribution of differences in the substitution preferences. It can be seen that many of the preferred changes associated with cis-trans variation differ highly from the general preferences for PB substitutions. Also, the largely expected changes involving the replacement of central helix conformation (PB m) with the capping regions (l and n) are under-represented.

Discussion

The cis peptides are maintained in protein structures for both structural and functional reasons. Hence these are expected to be well-conserved across the proteins sharing the same structure and function. In this study we have looked into the amino acid and conformational preferences associated with the cis state and how these preferences influence the replacement with a trans conformer in the structurally equivalent regions. Multiple structural alignments of different SCOP [START_REF] Murzin | SCOP: a structural classification of proteins database for the investigation of sequences and structures[END_REF] families involving high resolution crystal structures were used for analysis. SCOP domain families where at least one instance of cis-trans change, were extracted (181 families). Nearly 80% of the domain families correspond to enzymes and the rest are also dominated by ligand binding activity (Supplementary data 4). A few cases among these were looked into.

The structure of Chitinase B holds a non-Pro cis bond between Glu 144 and Tyr 145 (numbering taken from the PDB file) in the active site at the centre of the TIM barrel (Figure 8A) [START_REF] Kolstad | Structure of the D140N mutant of chitinase B from Serratia marcescens at 1.45 A resolution[END_REF]. These two residues are involved in direct contact with the substrate [START_REF] Van Aalten | Structural insights into the catalytic mechanism of a family 18 exo-chitinase[END_REF] and are also in the vicinity of the dimer interface (Figure 8A). This cis conformation is not conserved in Imaginal Disc Growth Factor -2 (IDGF2) [START_REF] Varela | Crystal structure of imaginal disc growth factor-2. A member of a new family of growth-promoting glycoproteins from Drosophila melanogaster[END_REF] which shares a similar TIM barrel fold with chitinase B. However the absence of many active site residues preclude the chitinase activity, the protein is reported to hold the ability to bind carbohydrates but in a different conformation (Figure 8B). Comparison of the two structures (Figure 7A &B) showed that the local PB change at the site of cis bond is from PB series ce to ae, involving variation in β turn type IV conformation. By comparing IDGF2 protein with related folds, Valera and co-workers suggested that the protein may have evolved from chitinases to acquire the new function as growth factors [START_REF] Varela | Crystal structure of imaginal disc growth factor-2. A member of a new family of growth-promoting glycoproteins from Drosophila melanogaster[END_REF].

Another set of proteins that have the same fold and bind similar substrates, are the Glycinamide ribonucleotide transformylases and Glycinamide ribonucleotide synthetase enzymes. The transformylase catalyses the formylation of glycinamide ribonucleotide (GAR) using Mg(2+)ATP and formate while the synthetase uses phosphoribosylamine, glycine, and ATP to form glycinamide ribonucleotide. Both enzymes are important in the purine biosynthesis pathway. A cis bond between Ala and Pro is found in the vicinity of the active site of the synthetase [START_REF] Wang | X-ray crystal structure of glycinamide ribonucleotide synthetase from Escherichia coli[END_REF]) (Figure 8C). This region is at the C terminus of a β strand and is associated with the PBs eb. However the cis peptide is not conserved in the transformylase enzyme [START_REF] Thoden | PurT-encoded glycinamide ribonucleotide transformylase. Accommodation of adenosine nucleotide analogs within the active site[END_REF] (Figure 8D). The cis conformation is replaced by trans with a change in the strand conformation (PB change eb to df ). This part is in direct contact with the substrate glycinamide ribonucleotide (GAR). Though both enzymes bind ATP or its derivatives and GAR, and they also have similar binding sites, the catalysis mechanisms are different. The conformation of bound ATP (or ADP) is also different in the two crystal structures (Figure 8C andD). Hence the cis-trans peptide changes can be implicated in the evolution of new function.

Variation in cis-conformation is also observed while comparing the structures of beta keto acyl carrier protein synthase III [START_REF] Qiu | Refined structures of beta-ketoacyl-acyl carrier protein synthase III[END_REF]) and HMG (3-hydroxy-3-methylglutaryl) -CoA synthase [START_REF] Theisen | 3hydroxy-3-methylglutaryl-CoA synthase intermediate complex observed in "realtime[END_REF]), which have similar folds. Changes in CoA binding sites are also observed (Supplementary data 5) between the two structures. Hence the changes in the cis conformation are generally associated with the acquiring of a different function with variations in substrate binding modes.

Apart from ligand binding and catalysis, cis peptides are also involved in intra and inter protein interactions. The structure of Hypoxic Response Protein I (HRP1) [START_REF] Sharpe | The structure and unusual protein chemistry of hypoxic response protein 1, a latency antigen and highly expressed member of the DosR regulon in Mycobacterium tuberculosis[END_REF]) has a cystathionine beta synthase (CBS) domain that is reported to lack AMP binding activity. A novel protein with similar fold from Thermoplasma acidophilum [START_REF] Proudfoot | Biochemical and structural characterization of a novel family of cystathionine beta-synthase domain proteins fused to a Zn ribbon-like domain[END_REF]) has an additional N terminal Zn ++ ribbon domain (Figure 9A). A cis proline is observed at the interface between this domain and CBS and this directly takes part in the interaction.

The absence of this cis peptide loop in HRP1 reflects its functional relevance. It is interesting to analyse the structural or functional importance of a cis-trans variation with conserved amino acids. In the structure of hydroxynitrile lyase (from Almond), a cis Proline is observed at a Glycosylation site (Figure 9B), when bound to sugar moiety. Though the Proline is conserved at the second position, the structure of related fold, pyranose 2 oxidase [START_REF] Bannwarth | Reaction geometry and thermostable variant of pyranose 2-oxidase from the white-rot fungus Peniophora sp[END_REF], has a trans conformation at this position which is likely to be due to the absence of the glycosylation site.

Non Pro cis-trans isomerisation was observed in the structures of myrosinase bound to different substrate analogs (Figure 10A) [START_REF] Bourderioux | The glucosinolate-myrosinase system. New insights into enzyme-substrate interactions by use of simplified inhibitors[END_REF][START_REF] Burmeister | High resolution X-ray crystallography shows that ascorbate is a cofactor for myrosinase and substitutes for the function of the catalytic base[END_REF].

The B-factors associated with this region were not higher compared to the rest of the protein, reflecting a reliable conformational change dependant on the nature of ligand. A similar case was also observed in the structures of quinone reductase 2 where a cis conformation was adopted upon substrate binding (Figure 10B) [START_REF] Calamini | Kinetic, thermodynamic and Xray structural insights into the interaction of melatonin and analogues with quinone reductase 2[END_REF]). In the absence of the substrate (melatonin), the loop conformation changes from β turn VIb to type I in the trans state.

Conclusion

Specific studies have shown that cis-trans isomerisation has an important role in the protein folding process. Due to the high energy cost, cis peptides accounts for only a negligible percent of the total population. But the ones that are observed in protein structures are seen to be essential for the structural integrity and function. Here we show that the change between cis and trans conformations are efficiently utilized for the emergence of new function among similar protein folds. Many of the non-Pro cis peptides are more prone to such changes than the Xaa-Pro peptides. The local conformational changes associated with the cis-trans variations are significantly different from those generally observed in similar structures. [START_REF] Burmeister | High resolution X-ray crystallography shows that ascorbate is a cofactor for myrosinase and substitutes for the function of the catalytic base[END_REF]. The cis peptide (WA) is highlighted. (B) Comparison of Quinone reductase structures [START_REF] Calamini | Kinetic, thermodynamic and Xray structural insights into the interaction of melatonin and analogues with quinone reductase 2[END_REF], one with the cis peptide (IP) (yellow) bound to Melatonin and the other (green) unbound at this site.

Figure 1 .

 1 Figure 1. Distribution of ω angles. (A) The percentage distribution of ω angles is plotted. The plot on the right (B) zooms the distribution with occurrence less than 2%.

Figure 2 .

 2 Figure 2. Amino acid propensity for cis -conformation. Percentage of each amino acid in the cisconformation at the positions Xaa1 (A) and Xaa2 (B) that forms the peptide bond. For (C) and (D), the percentage is calculated with respect to individual amino acid occurrence frequency.

Figure 3 .

 3 Figure 3. Local conformational preferences for cis -conformation. The local backbone conformation was analysed in terms of secondary structures and PBs. PROMOTIF (Hutchinson and Thornton 1996) was used for assignment of the local fold corresponding to the cis peptides. Percentage of each type of local fold associated with the first (A) and second (B) positions, are given. Abbreviation of PROMOTIF assignments: BTX -β-turns, X is the type of β-turn, AG -Antiparallel strands, G1 type β-bulge, where the first residue is in the left handed helical conformation (usually Glycine), AC -Antiparallel strands, Classic type beta bulge, one extra residue forms the bulge, GTINV -Inverse γturns, HPX:Yβ-hairpins, X and Y indicate the number of residues in loop, based on two different rules (Hutchinson and Thornton 1996). (C) and (D) indicates the percentage of each PB in the cisconformation at the positions corresponding to Xaa1 and Xaa2. For (C) and (D), the percentage is calculated with respect to individual PB occurrence frequency.

Figure 4 .

 4 Figure 4. Conservation of ω angles represented by plotting the substitutions observed in ω angles (divided into 10 o bins).

Figure 5 .

 5 Figure 5. Conservation of cis state: Amino acid propensity for cis-trans change. Percentage of each amino acid that participate in cis-trans change at the positions Xaa1 (A) and Xaa2 (B). The percentage is calculated with respect to individual amino acid occurrence frequency in cis conformation. (C) and (D) gives the percentage substitution of each amino acid (x axis) (cis) by all the 20 amino acids (trans), associated with cis-trans conversion. The legend gives the substitution percentages; each column adds up to 100%.

Figure 6 .

 6 Figure 6. Local conformational changes associated with cistrans conversion. The local backbone conformation was analysed in terms of secondary structures and PBs. PROMOTIF (Hutchinson and Thornton 1996) was used for assignment of the local fold. Percentage of each type of local fold involved in cis-trans conversion at the first (A) and second (B) positions, are given. Abbreviation of PROMOTIF assignments: BTX -β-turns, X is the type of β-turn, AG -Antiparallel strands, G1 type β-bulge, where the first residue is in the left handed helical conformation (usually Glycine), AC -Antiparallel strands, Classic type beta bulge, one extra residue forms the bulge, GTINV -Inverse γturns, HPX:Yβ-hairpins, X and Y indicate the number of residues in loop, based on two different rules (Hutchinson and Thornton 1996). (C) and (D) indicates the percentage of each PB involved in cis-trans change at the positions corresponding to Xaa1 and Xaa2. The percentage is calculated with respect to individual occurrence frequency in cis conformation.

Figure 7 .

 7 Figure 7. Preferences for PBs at different positions for cis-trans change. (A) and (B) gives the percentage substitution of each PB (x axis) (cis) with all the 16 PBs (trans), associated with cis-trans conversion. The legend gives the substitution percentages, each column adds up to 100%. The percentages of non-similar (except the conserved ones) PB changes associated with cis-trans change were compared with the general preferences observed in the databank. The differences in these preferences (cis-trans -general) are plotted in (C) and (D).

Figure 8 .

 8 Figure 8. Evolution of new functions with cis-trans conversions. (A) The structure of Chitinase B (PDB ID: 1GOI) (Kolstad et al. 2002) (yellow) highlighting the binding site of N-acetyl Dglucosamine (NAG) (orange) (van Aalten et al. 2001). The residues involved in cis peptide (EY) are also indicated (red). The other monomer is shown in cyan. (B) The structure of Imaginal Disc Growth factor -2 (IDGF2) (green) has a similar fold but lack the cis peptide (Varela et al. 2002). The sugar moiety (NAG and mannose) bound to the structure is also shown. The residues structurally equivalent to the cis peptide are also labelled. (C) The structure of Glycinamide Ribonucleotide Synthetase from E-coli (Wang et al. 1998) (yellow) that holds a cis peptide (AP) at the active site. The ADP binding site is also highlighted, ANP -Phosphoaminophosponic acid Adenylate ester. (D) The structure of Glycinamide Ribonucleotide Transformylase (Thoden et al. 2002) (green) bound to ATP and Glycinamide Ribonucleotide (GAR).

Figure 9 .

 9 Figure 9. Structural modifications associated with cis-trans change. (A) Comparison of structures of Cystathione beta synthase (CBS) domain containing protein (Proudfoot et al. 2008) (yellow, PDB ID: 1PVM) and Hypoxic Response Protein 1 (HRP1) (Sharpe et al. 2008) (green). The location of the cis peptide (KP) in 1PVM (yellow), is highlighted. (B) Comparison of structures of Hydroxynitrile Lyase (Dreveny et al. 2001) (yellow) and Pyranose 2 oxidase[START_REF] Bannwarth | Reaction geometry and thermostable variant of pyranose 2-oxidase from the white-rot fungus Peniophora sp[END_REF]) (green). The sugar moiety bound to HRP1 is also highlighted.

Figure 10 .

 10 Figure 10. Role of cis-trans isomerisation in ligand binding. (A) Comparison of structures of Myrisinases bound to substrate analogs (inhibitors) S-benzyl-phenylacetothiohydroximate-O-sulphate (SEH) (Bourderioux et al. 2005) (yellow) and Nojirimycine Tetrazole (NTZ) (Burmeister et al. 2000). The cis peptide (WA) is highlighted. (B) Comparison of Quinone reductase structures (Calamini et al. 2008), one with the cis peptide (IP) (yellow) bound to Melatonin and the other (green) unbound at this site.

Acknowledgments

These works were supported by grants from the French Ministry of Research, University of Paris Diderot -Paris 7, French National Institute for Blood Transfusion (INTS), French Institute for Health and Medical Research (INSERM) and Indian Department of Biotechnology. APJ is supported by CEFIPRA/IFCPAR number 3903-E. NS and AdB also acknowledge to CEFIPRA/IFCPAR for collaborative grant (number 3903-E).


Nicolas Raymond

Semiclassical 3D Neumann Laplacian with variable magnetic field: a toy model

In this paper we investigate the semiclassical behavior of the lowest eigenvalues of a model Schrödinger operator with variable magnetic field. This work aims at proving an accurate asymptotic expansion for these eigenvalues, the corresponding upper bound being already proved in the general case. The present work also aims at establishing localization estimates for the attached eigenfunctions.

1 Motivation and main results

Motivation

In this paper we are interested in a toy model of Schrödinger operator with variable magnetic field (with Dirichlet and Neumann conditions on the boundary) in the bounded region Ω 0 = {(x, y, z) ∈ R 3 : |x| ≤ x 0 , |y| ≤ y 0 and 0 < z ≤ z 0 }, where x 0 , y 0 , z 0 > 0. We will need the following notation for the part of the boundary carrying the Dirichlet condition:

∂ Dir Ω 0 = {(x, y, z) ∈ Ω 0 : |x| = x 0 or |y| = y 0 or z = z 0 }.
Definition of the operator For h > 0, α ≥ 0 and θ ∈ 0, π 2 , we consider the self-adjoint operator:

L(h, α, θ) = h 2 D 2 y + h 2 D 2 z + (hD x + z cos θ -y sin θ + αz(x 2 + y 2 )) 2 , (1.1)
with domain:

Dom(L(h, α, θ)) = {ψ ∈ L 2 (Ω 0 ) : L(h, α, θ)ψ ∈ L 2 (Ω 0 ), ψ = 0 on ∂ Dir Ω 0 and ∂ z ψ = 0 on z = 0}.

1

We will denote by (λ(h), u h ) an eigenpair (we omit the dependence on α and θ). The vector potential is expressed as:

A(x, y, z) = (V θ (y, z) + αz(x 2 + y 2 ), 0, 0) = (A, 0, 0) where V θ (y, z) = z cos θ -y sin θ.

(1.

2)

The associated magnetic field is given by: ∇ × A = β = (0, cos θ + α(x 2 + y 2 ), sin θ -2αyz).

(1.3)

Constant magnetic field case (α = 0) Let us examine the important case when α = 0:

L(h, 0, θ) = h 2 D 2 y + h 2 D 2 z + (hD x + V θ (y, z)) 2 ,
viewed as an operator on L 2 (R 3 + ). We perform the rescaling:

x = h 1/2 r, y = h 1/2 s, z = h 1/2 t (1.4)
and the operator becomes (after division by h):

H(θ) = D 2 s + D 2 t + (D r + V θ (s, t)) 2 .
Making a Fourier transform in the variable r denoted by F, we get:

FH(θ)F -1 = D 2 s + D 2 t + (τ + V θ (s, t)) 2 .
(1.5)

Then, we use a change of coordinates:

U θ (τ, s, t) = τ , ŝ, t = τ, s - τ sin θ , t (1.6)
and we obtain:

H Neu (θ) = U θ FH(θ)F -1 U -1 θ = D 2 ŝ + D 2 t + V θ (ŝ, t) 2 .
Notation 1.1 We denote by Q θ the quadratic form associated with H Neu (θ).

The operator H Neu (θ) We can consider H Neu (θ) as an operator acting on L 2 (R 2 + ) ; we denote by H Neu (θ) this realization. The bottom of its spectrum is denoted by σ(θ). In [START_REF] Fournais | Spectral methods in surface superconductivity[END_REF] (see the references therein, in particular [START_REF] Helffer | Magnetic bottles for the Neumann problem: the case of dimension 3[END_REF][START_REF] Lu | Surface nucleation of superconductivity in 3-dimensions[END_REF]), it is proved that σ is analytic and strictly increasing on 0, π 2 , that σ(0) ∈ (0, 1), σ π 2 = 1 and σ ess (H Neu (θ)) = [1, +∞[. Therefore, σ(θ) is a simple eigenvalue. Notation 1.2 We denote by u θ the positive and L 2 -normalized eigenfunction (which is in the Schwartz class, see [START_REF] Raymond | On the semiclassical 3D Neumann Laplacian with variable magnetic field[END_REF]) associated with σ(θ).

Let us also recall that the lower bound of the essential spectrum is related, through the Persson's theorem (see [START_REF] Persson | Bounds for the discrete part of the spectrum of a semi-bounded Schrödinger operator[END_REF]), to the following estimate:

q Neu θ (χ R u) ≥ (1 -ε(R)) χ R u , ∀u ∈ Dom(q Neu θ ),
where q Neu θ is the quadratic form associated with H Neu (θ), where χ R is a cutoff function away from the ball B(0, R) and ε(R) is tending to zero when R tends to infinity. Moreover, if we consider the Dirichlet realization H Dir (θ), we have:

q Dir θ (u) ≥ u 2 , ∀u ∈ Dom(q Dir θ ). (1.7)
Let us finally mention that an accurate analysis of the eigenpairs of H Neu (θ) is done in [START_REF] Bonnaillie | Discrete spectrum of a model schrödinger operator on the half-plane with neumann conditions[END_REF] in the regime θ → 0.

Motivations Let us explain why we are led to consider this model.

The reason comes from the theory of superconductivity. As much in 2D as in 3D, the magnetic Laplacian appears in this theory when studying the third critical field H C 3 (which distinguish between normal and superconducting states) after the linearization of the Ginzburg-Landau functional (see for instance [START_REF] Lu | Eigenvalue problems of Ginzburg-Landau operator in bounded domains[END_REF][START_REF] Lu | Surface nucleation of superconductivity in 3-dimensions[END_REF] and also the book of Fournais and Helffer [START_REF] Fournais | Spectral methods in surface superconductivity[END_REF]). It turns out that H C 3 can be related to the lowest eigenvalue of the magnetic Laplacian in the regime h → 0. Many properties of the eigenpairs are investigated when the magnetic field is constant (see [START_REF] Bauman | Stable nucleation for the Ginzburg-Landau system with an applied magnetic field[END_REF][START_REF] Bernoff | Onset of superconductivity in decreasing fields for general domains[END_REF][START_REF] Pino | Boundary concentration for eigenvalue problems related to the onset of superconductivity[END_REF][START_REF] Helffer | Magnetic bottles in connection with superconductivity[END_REF][START_REF] Fournais | Accurate eigenvalue asymptotics for the magnetic Neumann Laplacian[END_REF] for the 2D and [START_REF] Helffer | Magnetic bottles for the Neumann problem: curvature effects in the case of dimension 3 (general case)[END_REF] for the 3D), but less are known when the magnetic field is variable (see [START_REF] Lu | Eigenvalue problems of Ginzburg-Landau operator in bounded domains[END_REF][START_REF] Raymond | Sharp asymptotics for the Neumann Laplacian with variable magnetic field: case of dimension 2[END_REF] for the 2D and [START_REF] Lu | Surface nucleation of superconductivity in 3-dimensions[END_REF][START_REF] Raymond | On the semiclassical 3D Neumann Laplacian with variable magnetic field[END_REF] for the 3D). When the magnetic vanishes, we can refer to [START_REF] Montgomery | Hearing the zero locus of a magnetic field[END_REF][START_REF] Helffer | Semiclassical analysis for the ground state energy of a Schrödinger operator with magnetic wells[END_REF][START_REF] Pan | Schrödinger operators with non-degenerately vanishing magnetic fields in bounded domains[END_REF][START_REF] Helffer | Spectral gaps for periodic Schrödinger operators with hypersurface magnetic wells: analysis near the bottom[END_REF][START_REF] Dombrowski | Semiclassical analysis with vanishing magnetic fields[END_REF].

Let us introduce the fundamental invariant in the case of variable magnetic field and our generic assumptions. We let:

β(x, y) = σ(θ(x, y)) β(x, y, 0) ,
where θ(x, y) is the angle of β(x, y, 0) with the boundary z = 0:

β(x, y, 0) sin θ(x, y) = β(x, y, 0) • ν(x, y),
where ν(x, y) is the inward normal at (x, y, 0). It is proved in [START_REF] Lu | Surface nucleation of superconductivity in 3-dimensions[END_REF] that the semiclassical asymptotics of the lowest eigenvalue is:

λ 1 (h) = min(inf z=0 β, inf Ω 0 β )h + o(h).
In this paper (like in [START_REF] Raymond | On the semiclassical 3D Neumann Laplacian with variable magnetic field[END_REF]), we are interested in the case when the following generic assumptions are satisfied:

inf z=0 β < inf Ω 0 β (1.8)
β admits a unique and non degenerate minimum.

(1.9)

Under these assumptions, a three terms upper bound is proved for λ 1 (h) in [START_REF] Raymond | On the semiclassical 3D Neumann Laplacian with variable magnetic field[END_REF] and the corresponding lower bound, for a general domain, is still an open problem. The aim of this paper is to establish such a lower bound for an example. For α > 0, the toy operator (1.1) is the simplest example of a generic Schrödinger operator with variable magnetic field satisfying Assumptions (1.8) and (1.9). Let us explain in which sense. In the general case studied in [START_REF] Raymond | On the semiclassical 3D Neumann Laplacian with variable magnetic field[END_REF] and after using Taylor formulas near (0, 0, 0) at the order 3, we are reduced, in the case of a flat metrics, to an operator in the form:

h 2 D 2 z + (hD y + A (3) y ) 2 + (hD x + A (3) x ) 2 .
It turns out that the terms A

y and the terms of order 2 of A

x are not crucial in order to satisfy the generic assumptions: The most important come from the terms of order 3 in A

(3)

x . Then, we cancel a few more terms to get a diagonal matrix for the Hessian of β. Thus, this model is the one with the less terms and so that the Hessian matrix of β is diagonal. Let us now verify the assumptions. Using the computations of [START_REF] Raymond | On the semiclassical 3D Neumann Laplacian with variable magnetic field[END_REF], we have the Taylor expansion:

β(x, y) = σ(θ) + αC(θ)(x 2 + y 2 ) + O(|x| 3 + |y| 3 ).
(1.10) with:

C(θ) = cos θσ(θ) -sin θσ (θ).
Moreover, it is proved in [5, Proposition 5.1] that C(θ) > 0, for θ ∈ 0, π 2 . Thus, Assumption (1.9) is verified if x 0 , y 0 and z 0 are fixed small enough. Using σ(θ) < 1 when θ ∈ 0, π 2 and β(0, 0, 0) = 1, we get Assumption (1.8). Finally, we can briefly compare to what is done in [START_REF] Helffer | Magnetic bottles for the Neumann problem: curvature effects in the case of dimension 3 (general case)[END_REF] in the case of the constant magnetic field. In their paper, the authors establish many localizations properties through Agmon estimates and reduce the general operator to simplest models. By doing this, they refine the localization properties and are led to a normal form involving an harmonic oscillator (see [START_REF] Helffer | Magnetic bottles for the Neumann problem: curvature effects in the case of dimension 3 (general case)[END_REF]Subsection 13.8]). This will be the main philosophy of this paper, even if we already start from a simplest operator.

Remark on the function β Using the explicit expression of the magnetic field, we have:

β(x, y) = βrad (R), R = α(x 2 + y 2 )
and an easy computation gives:

βrad (R) = β rad (R) σ arctan sin θ cos θ + R , with β rad (R) = (cos θ + R) 2 + sin 2 θ.
The computations of [5, Proposition 5.1] imply that βrad is strictly increasing and

∂ R βrad (R = 0) = C(θ) > 0.
Consequently, β admits a unique and non degenerate minimum on R 3 + and tends to infinity far from 0. This is easy to see that:

inf R 3 + β = cos θ.
We deduce that, as long as σ(θ) < cos θ, the generic assumptions of [START_REF] Raymond | On the semiclassical 3D Neumann Laplacian with variable magnetic field[END_REF] are satisfied with

Ω 0 = R 3 + .
1.2 Recall and improvement of existing results and statement of the main theorem

Let us recall and improve the main result of [START_REF] Raymond | On the semiclassical 3D Neumann Laplacian with variable magnetic field[END_REF] in our simplified case. We perform the scaling (1.4) and, after division by h, L(h, α, θ) becomes:

L resc (h, α, θ) = D 2 s + D 2 t + (D r + t cos θ -s sin θ + hαt(r 2 + s 2 )
). Using the Fourier transform F (see (1.5)) and the translation U θ (see (1.6)), we have:

U θ FL resc (h, α, θ)F -1 U -1 θ = D 2 ŝ +D 2 t + V θ (ŝ, t) + hα t D τ - D ŝ sin θ 2 + ŝ + τ sin θ 2 2
. This operator will be shortly denoted by L Normal (h) and the corresponding quadratic form q Normal h . We write:

L Normal (h) = H Neu (θ) + hH 1 + h 2 H 2 ,
where:

H 1 = α t D τ - D ŝ sin θ 2 V θ + V θ D τ - D ŝ sin θ 2 + 2V θ ŝ + τ sin θ 2 , H 2 = α 2 t2 D τ - D ŝ sin θ 2 + ŝ + τ sin θ 2 2 ≥ 0.
We look for quasi-eigenpairs in the form:

µ ∼ j≥0 µ j h j and ψ ∼ j≥0 ψ j h j .
We solve the following problem in the sense of formal series:

L Normal (h)ψ ∼ µψ.
Term in h 0 We solve:

H(θ)ψ 0 = µ 0 ψ 0 .
We take: µ 0 = σ(θ) and:

ψ 0 (τ , ŝ, t) = u θ (ŝ, t)f 0 (τ ),
f 0 being to be determined.

Term in h Then, we must solve:

(H(θ) -σ(θ))ψ 1 = (µ 1 -H 1 )ψ 0 .
We apply the Fredholm alternative and we write:

(µ 1 -H 1 )ψ 0 , u θ ŝ, t = 0.
Let us introduce a fundamental operator involved in the asymptotics. We let:

S θ (D τ , τ ) = 2 tV θ u 2 θ dŝd t H harm + 2 sin θ tV θ u 2 θ dŝd t τ + d(θ),
where

H harm = D 2 τ + τ 2 sin 2 θ and d(θ) = sin -2 θ t(D 2 ŝ V θ + V θ D 2 ŝ )u θ , u θ + 2 tŝ 2 V θ u 2 θ dŝd t.
We recall the important fact that (see [24, Formula (2.31)]):

2 tV θ u 2 θ dsdt = C(θ) > 0,
so that S θ (D τ , τ ) can be viewed as the harmonic oscillator up to dilation and translation. We denote ν n (S θ (D τ , τ )) the n-th eigenvalue of S θ (D τ , τ ). The compatibility equation rewrites:

S θ (D τ , τ )f 0 = µ 1 f 0 and we take µ 1 = ν n (S θ (D τ , τ )
) and for f 0 the corresponding L 2 -normalized eigenfunction. Then, we can write the solution ψ 1 in the form:

ψ 1 = ψ ⊥ 1 + f 1 (τ )u θ (ŝ, t)
where ψ ⊥ 1 is the unique solution orthogonal to u θ . We notice that it is in the Schwartz class.

Further terms ("Grushin procedure") We perform a recursion to get the next terms (this is related to the so-called Grushin procedure, see [START_REF] Šin | Hypoelliptic differential equations and pseudodifferential operators with operator-valued symbols[END_REF]). We assume that ψ k is determined in the Schwartz class for k = 0, • • • , n -1 and that ψ n is written in the form:

ψ n = ψ ⊥ n + f n (τ )u θ (ŝ, t),
where ψ ⊥ n is determined in the Schwartz class and where f n is to be determined. We also assume that µ k is determined for k = 0, • • • , n. The equation for h n+1 is in the form:

(H(θ) -σ(θ))ψ n+1 = (µ 1 -H 1 )(f n u θ ) + µ n+1 ψ 0 + R n (τ , ŝ, t),
where R n (τ , ŝ, t) is determined and in the Schwartz class. The Fredholm alternative provides:

(S θ (D τ , τ ) -µ 1 )f n = µ n+1 f 0 (τ ) + r n (τ ), with r n = R n , u θ ŝ, t
Applying again the Fredholm alternative, we deduce that µ n+1 = -r n , f 0 τ permits to determine a unique solution f n in the Schwartz class which is orthogonal to f 0 . As a consequence, we can write:

ψ n+1 = ψ ⊥ n+1 + f n+1 (τ )u θ (ŝ, t).
Thus, we have determined (µ j ) j≥0 and (ψ j ) j≥0 . To emphasize the dependence on n, we shall write (µ j,n ) j≥0 and (ψ j,n ) j≥0 . Thanks to the spectral theorem, we deduce:

Theorem 1.
3 For all α > 0, θ ∈ 0, π 2 , there exists a sequence (µ j,n ) j≥0 and there exist positive constants C, h 0 such that for h ∈ (0, h 0 ):

d σ(L(α, θ, h)), h J j=0 µ j,n h j ≤ Ch J+2 and we have µ 0,n = σ(θ), µ 1,n = ν n (S θ (D τ , τ )).
Main result The aim of this paper is to establish the following accurate estimate for λ n (h): Theorem 1.4 For all α > 0, θ ∈ 0, π 2 and n ≥ 1, there exists a sequence (µ j,n ) j≥0 and

ε 0 > 0 s. t. for |x 0 | + |y 0 | + |z 0 | ≤ ε 0 , λ n (h) ∼ h→0 h j≥0 µ j,n h j and we have µ 0,n = σ(θ), µ 1,n = ν n (S θ (D τ , τ )).
Organization of the paper The paper is organized as follows. In Section 2, we recall a first rough lower bound for λ n (h) and provide the corresponding normal Agmon estimates for the eigenfunctions. In particular, we will see that the first eigenfunctions are living in a neighborhood of the boundary of size h 1/2 . In Section 3, we use these rough results to improve the localization of the eigenfunctions with respect to the variables (ŝ, t). In Section 4, we use those localizations estimates to reduce the problem to the study of a normal form (similarly as in [START_REF] Helffer | Magnetic bottles for the Neumann problem: curvature effects in the case of dimension 3 (general case)[END_REF]) involving an harmonic oscillator operating in τ to improve the approximation of the eigenfunctions in order to estimate the spectral gap between the eigenvalues and deduce Theorem 1.4.

First lower bound and localization estimates

In this section, we are concerned by Agmon estimates (with respect to (x, y)) satisfied by an eigenfunction u h associated to λ n (h).

Agmon estimates

We shall prove the following tangential localization: Proposition 2.1 For all δ > 0, there exist C > 0 and h 0 > 0 such that for h ∈ (0, h 0 ):

Ω 0 e δ(x 2 +y 2 )/h 1/4 |u h | 2 dxdydz ≤ C u h 2 , Ω 0 e δ(x 2 +y 2 )/h 1/4 ∇u h 2 dxdydz ≤ Ch -1 u h 2 .
Before starting the proof, let us recall a rough lower bound for λ 1 (h) first obtained in [START_REF] Lu | Surface nucleation of superconductivity in 3-dimensions[END_REF] (see also [START_REF] Fournais | Spectral methods in surface superconductivity[END_REF]Theorem 9.1.1]):

Proposition 2.2 There exist C > 0 and h 0 > 0 such that, for h ∈ (0, h 0 ) :

λ n (h) ≥ σ(θ)h -Ch 5/4 .
Combining the upper and lower bounds of λ n (h), this is standard to deduce the following normal Agmon estimates (see [START_REF] Helffer | Magnetic bottles for the Neumann problem: the case of dimension 3[END_REF]):

Proposition 2.3 There exist δ > 0, C > 0 and h 0 > 0 such that for all h ∈ (0, h 0 ), we have :

Ω 0 e δh -1/2 z (|u h | 2 + h -1 |(ih∇ + A)u h | 2 ) dxdydz ≤ C u h 2 .
Let us now deal with the proof of Proposition 2.1.

Proof of Proposition 2.1 We begin by writing the Agmon identity (see [START_REF] Agmon | Lectures on exponential decay of solutions of second-order elliptic equations: bounds on eigenfunctions of N -body Schrödinger operators[END_REF][START_REF] Agmon | Bounds on exponential decay of eigenfunctions of Schrödinger operators[END_REF]):

λ n (h) exp(Φ)u h 2 = q h (exp(Φ)u h ) -h 2 |∇Φ| exp(Φ)u h 2 , (2.1)
for Φ a Lipschitzian function (depending on h) to be determined. We estimate q h (exp(Φ)u h ) thanks to a localization technique. We use the following partition of unity :

j χ 2 j,h = 1, (2.2)
j ∇χ j,h 2 ≤ Ch -2ρ , (2.3)
where the χ j,h are smooth cutoff functions supported in balls of size h ρ . The IMS formula (cf. [START_REF] Cycon | Schrödinger operators with application to quantum mechanics and global geometry[END_REF]) provides:

q h (exp(Φ)u h ) ≥ j q h (χ j,h exp(Φ)u h ) -Ch 2-2ρ j χ j,h exp(Φ)u h 2 .
We let: φ j,h = χ j,h exp(Φ)u h . We now perform the approximation by the constant magnetic field case to give a lower bound for q h (φ j,h ). We introduce the linear approximation of A in the ball B j :

A lin j (x, y, z) = A(x j , y j , z j )+2αz j x j (x-x j )+(2αz j y j -sin θ)(y-y j )+(cos θ+α(x 2 j +y 2 j ))(z-z j ).

It satisfies:

|A(x, y, z) -A lin j (x, y, z)| ≤ C(|x -x j | 2 + |y -y j | 2 + |z -z j | 2 ).
Then, we have, with the Cauchy-Schwarz inequality, for all ε > 0 :

q h (φ j,h ) ≥ (1 -ε)( hD y φ j,h 2 + hD z φ j,h 2 + (hD x + A lin j )φ j,h 2 ) -ε -1 Ch 4ρ φ j,h 2 .
For the balls intersecting the boundary, we have:

hD y φ j,h 2 + hD z φ j,h 2 + (hD x + A lin j )φ j,h 2 ≥ β(x j , y j )h φ j,h 2 ≥ h φ j,h 2 σ(θ) (2.4
) and for the other balls:

hD y φ j,h 2 + hD z φ j,h 2 + (hD x +A lin j )φ j,h 2 ≥ β(x j , y j ) h φ j,h 2 ≥ h φ j,h 2 inf Ω 0 β .
(2.5) The optimization of ε provides: ε = h 2ρ-1/2 and the optimization of ρ leads to: ρ = 3 8 . With this choice and (2.1), we get:

j q h (φ j,h ) -Ch 5/4 φ j,h 2 -h 2 φ j,h ∇Φ 2 -λ 1 (h) φ j,h 2 ≤ 0.
We recall the rough upper bound (cf. Theorem 1.3):

λ 1 (h) ≤ σ(θ)h + Ch 2 .
We split the sum into two parts:

j = bnd j + int j .
We choose Φ = δh -1/4 (x 2 + y 2 ), so that, on the one hand, there exists c > 0 (cf. (2.5)) such that:

int j q h (φ j,h ) -Ch 5/4 φ j,h 2 -h 2 φ j,h ∇Φ 2 -λ 1 (h) φ j,h 2 ≥ ch int j φ j,h 2 ≥ 0
and on the other hand (cf. (2.4)):

bnd j q h (φ j,h ) -Ch 5/4 φ j,h 2 -h 2 φ j,h ∇Φ 2 -λ 1 (h) φ j,h 2 ≥ bnd j Ω 0 β(x j , y j )h -σ(θ)h -C(δ)h 5/4 ) |φ j,h | 2 dxdydz.
Therefore, we deduce:

ch int j φ j,h 2 + bnd j Ω 0 β(x j , y j )h -σ(θ)h -C(δ)h 5/4 ) |φ j,h | 2 dxdydz ≤ 0.
By non degeneracy of the minimum of β (see (1.10)), we get the existence of c 1 > 0 and c 2 > 0 such that for all j:

c 2 (x 2 j + y 2 j ) ≥ β(x j , y j ) -σ(θ) ≥ c 1 (x 2 j + y 2 j ).
We now split the sum on the boundary into two parts: the j's such that x 2 j + y 2 j ≥ 2h 1/4 c -1 1 C(δ) and the j's such that:

x 2 j + y 2 j < 2h 1/4 c -1 1 C(δ).
We infer:

c int j Ω 0 |χ j,h exp(Φ)u h | 2 dxdydz + bnd j Ω 0 |χ j,h exp(Φ)u h | 2 dxdydz ≤ C u h 2 .
Corollary 2.4 For all η > 0, we have:

|x|+|y|≥h 1/8-η |x| k |y| l |z| m (|u h | 2 + |D x u h | 2 + |D y u h | 2 + |D z u h | 2 ) dxdydz = O(h ∞ ) u h 2 . z≥h 1/2-η |x| k |y| l |z| m (|u h | 2 + |D x u h | 2 + |D y u h | 2 + |D z u h | 2 ) dxdydz = O(h ∞ ) u h 2 .
Let us consider η > 0 small enough and introduce the cutoff function defined by:

χ h (x, y) = χ 0 h -1/8+η x, h -1/8+η y, h -1/2+η z ,
where χ 0 is a smooth cutoff function being 1 near (0, 0, 0). We can notice, by elliptic regularity, that χ h u h is smooth (as it is supported away from the vertices).

Let us consider N ≥ 1. For n = 1, • • • , N , let us consider u n,h a L 2 -normalized eigenfunction associated with λ n (h) so that u n,h , u m,h = 0 for n = m. We let:

E N (h) = span n=1,••• ,N u n,h .
We notice that Propositions 2.3 and 2.1 hold for the elements of E N (h). As a consequence of Propositions 2.3 and 2.1, we have: Corollary 2.5 We have:

q h (ũ h ) ≤ λ N (h) + O(h ∞ ), with ũh = χ h u h , where u h ∈ E N (h).
Agmon estimates of higher order In the last paragraph we proved Agmon estimates for u h and its first derivatives. We will also need estimates for the higher derivatives. This is the aim of the following proposition (this kind of higher order estimates can be found for instance in [START_REF] Helffer | Semi-classical analysis for the Schrödinger operator and applications[END_REF]): Proposition 2.6 For all ν ∈ N 3 , there exists δ > 0, γ ≥ 0, h 0 > 0 and C > 0 such that, for h ∈ (0, h 0 ):

e δh -1/2 z |D ν ũh | 2 dxdydz ≤ Ch -γ ũh 2 . e δh -1/4 (x 2 +y 2 ) |D ν ũh | 2 dxdydz ≤ Ch -γ ũh 2 ,
where u h ∈ E N (h).

Proof: For |ν| = 1, this is already proved. Let ν such that |ν| ≥ 1 and assume that the inequality is proved for all multi-indices with length smaller than |ν|. We recall that, when u h is an eigenfunction:

L(h)ũ h = λ n (h)ũ h + [L(h), χ h ]u h .
Let us apply D ν to the equation. We get:

L(h)D ν ũh = λ n (h)D ν ũh + D ν [L(h), χ h ]u h + [L(h), D ν ]ũ h .
It follows, for Φ = δh -1/2 z or Φ = δh -1/4 (x 2 + y 2 ):

L(h)e Φ D ν ũh = λ n (h)e Φ D ν ũh + e Φ D ν [L(h), χ h ]u h + e Φ [L(h), D ν ]ũ h + [L(h), e Φ ]D ν ũh .
Then, we take the scalar product with e Φ D ν ũh . From the recursion assumption and up to choose δ smaller to absorbe polynomials, we deduce the existence of γ 1 > 0 and γ 2 > 0 such that:

[L(h), e Φ ]D ν ũh , e Φ D ν ũh = h 2 ∇Φe Φ D ν ũh 2 ≤ Ch -γ 1 ũh 2 , | e Φ [L(h), D ν ]ũ h , e Φ D ν ũh | ≤ Ch -γ 1 ũh 2 , | e Φ D ν [L(h), χ h ]u h , e Φ D ν ũh | ≤ Ch -γ 1 ũh 2 + Ch γ 2 q h (e Φ D ν ũh ).
We infer:

q h (e Φ D ν ũh ) ≤ Ch -γ 1 ũh 2 .
In particular, we get the control of D t D ν , D s D ν and, up to loose again some negative power of h, we get the control of D r D ν . The extension to u h ∈ E N (h) is then standard.

In particular, we infer:

Corollary 2.7 For all η > 0, we have, for all ν ∈ N 3 :

|x|+|y|≥h 1/8-η |x| k |y| l |z| m |D ν ũh | 2 dxdydz = O(h ∞ ) ũh 2 , z≥h 1/2-η |x| k |y| l |z| m |D ν ũh | 2 dxdydz = O(h ∞ ) ũh 2 ,
where u h ∈ E N (h).

Toward a normal form

For u h ∈ E N (h), we let:

w h (r, s, t) = χ resc h (r, s, t)u resc h (r, s, t) = χ 0 (h 3/8+η r, h 3/8+η s, h η t)u h (h 1/2 r, h 1/2 s, h 1/2 t) (2.6) and v h (τ , ŝ, t) = U θ Fw h .
We consider F N (h) the image of E N (h) by these transformations. We can reformulate Corollary 2.5.

Corollary 2.8

With the previous notation, we have the lower bound, for v h ∈ F N (h):

q Normal h (v h ) ≤ λ resc N (h) + O(h ∞ ),
where

λ resc N (h) = h -1 λ N (h).
We can also notice that, when u h is an eigenfunction associated with λ p (h):

L Normal (h)v h = λ resc p (h)v h + r h , (2.7)
where the remainder r h is O(h ∞ ) in the sense of Corollary 2.7.

In the following, we aim at proving localization and approximation estimates for v h rather that u h . Moreover, these approximations will allow us to estimate the energy q Normal h (v h ).

Approximation and refined localizations in the phase space

This section aims at estimating momenta of v h with respect to polynoms in the phase space. Before starting the analysis, let us recall the link (cf. (1.6)) between the variables (τ, s, t) and (τ , ŝ, t):

D τ = D τ + 1 sin θ D s , D ŝ = D s , D t = D t . (3.1)
We will use the following obvious remark:

Remark 3.1 We can notice that if φ is supported in supp(χ h ), we have:

q resc h (φ) ≥ (1 -ε)Q θ (φ) -Ch 1/2-6η ε -1 φ 2 .
Optimizing in ε, we have:

q resc h (φ) ≥ (1 -h 1/4-3η )Q θ (φ) -Ch 1/4-3η φ 2 .
Moreover, when the support of φ avoids the boundary, we have:

Q θ (φ) ≥ φ 2 .

Localizations in ŝ and t

In this subsection, many localizations lemmas with respect to ŝ and t are proved.

Estimates with respect to ŝ and t We begin to prove estimates depending only on the variables ŝ and t.

Lemma 3.2 Let N ≥ 1. For all k, n, there exist h 0 > 0 and C(k, n) > 0 such that, for all h ∈ (0, h 0 ):

tk ŝn+1 v h ≤ C(k, n) v h , (3.2) tk D ŝ(ŝ n v h ) ≤ C(k, n) v h (3.3) tk D t(ŝ n v h ) ≤ C(k, n) v h ., (3.4
)

for v h ∈ F N (h).
Proof: We prove the estimates when v h is the image of an eigenfunction associated to λ p (h) with p = 1, • • • , N .

Case n = 0 Let us analyze the case n = 0. (3.4) follows from the normal Agmon estimates. We have:

q Normal h ( tk v h ) ≤ λ resc p tk v h 2 + | [D 2 t , tk ]w h , tk v h | + O(h ∞ ) v h 2 .
The normal Agmon estimates provide:

| [D 2 t , tk ]v h , tk v h | ≤ C v h 2
and thus:

q Normal h ( tk v h ) ≤ C v h 2 .
We deduce (3.3). We also have:

tk (-ŝ sin θ + t cos θ + R h )v h 2 ≤ C v h 2 .
We use the basic lower bound:

tk (-ŝ sin θ + t cos θ + R h )v h 2 ≥ 1 2 tk ŝ sin θv h 2 -2 ( tk+1 cos θ + tk R h )v h 2 .
Moreover, we have (using the support of χ resc h ):

tk R h v h ≤ Ch(h -3/8-η ) 2 tk+1 v h ≤ Ch(h -3/8-η ) 2 v h ,
the last inequality coming from the normal Agmon estimates. Thus, we get:

tk ŝv h 2 ≤ C v h 2 .
Recursion We apply tk ŝn+1 to (2.7), take the scalar product with t k ŝn+1 v h and it follows:

q Normal h ( tk ŝn+1 v h ) ≤ λ resc p (h) tk ŝn+1 v h 2 + C tk-2 ŝn+1 v h tk ŝn+1 v h + C tk-1 D t ŝn v h tk ŝn+1 v h + C tk D ŝ ŝn w h tk ŝn+1 v h + C tk ŝn-1 v h tk ŝn+1 v h + | tk [ŝ n+1 , (-ŝ sin θ + t cos θ + R h ) 2 ]v h , tk ŝn+1 |, where R h = hα t D τ -(sin θ) -1 D ŝ 2 + ŝ + (sin θ) -1 τ 2 . (3.5)
We have:

[ŝ n+1 , (-ŝ sin θ + t cos θ + R h ) 2 ] = [ŝ n+1 , R h ](-ŝ sin θ + t cos θ + R h ) + (-ŝ sin θ + t cos θ + R h )[ŝ n+1 , R h ].
Let us analyze the commutator [ŝ n+1 , R h ]. We can write:

[ŝ n+1 , R h ] = αh t[ŝ n+1 , D τ -(sin θ) -1 D ŝ 2 ]
and:

[ D τ -(sin θ) -1 D ŝ 2 , ŝn+1 ] = (sin θ) -2 n(n + 1)ŝ n-1 + 2i(sin θ) -1 (n + 1)(D τ -(sin θ) -1 D ŝ)ŝ n
we infer:

[ŝ n+1 , (-ŝ sin θ + t cos θ + R h ) 2 ] = αh t(sin θ) -2 n(n + 1)ŝ n-1 + 2iαh t(sin θ) -1 (n + 1)(D τ -(sin θ) -1 D ŝ)ŝ n (V θ + R h ) + (V θ + R h ) αh t(sin θ) -2 n(n + 1)ŝ n-1 + 2iαh t(sin θ) -1 (n + 1)(D τ -(sin θ) -1 D ŝ)ŝ n
After having computed a few more commutators, the terms of [ŝ n+1 , (-ŝ sin θ + t cos θ + R h ) 2 ] are in the form:

tl ŝm , h tl (D τ -(sin θ) -1 D ŝ)ŝ m , h 2 tl (D τ -(sin θ) -1 D ŝ) 3 ŝm , h 2 tl (ŝ + (sin θ) -1 τ ) 2 (D τ + (sin θ) -1 D ŝ)ŝ m
with m ≤ n + 1 and l = 0,

Let us examine for instance the term h 2 tl (ŝ + (sin θ) -1 τ ) 2 (D τ + (sin θ) -1 D ŝ)ŝ m . We have, after the inverse Fourier transform and translation:

h 2 tl (ŝ + (sin θ) -1 τ ) 2 (D τ + (sin θ) -1 D ŝ)ŝ m v h ≤ Ch 2 (h -3/8-η ) 3 tl ŝm v h
where we have used the support of χ resc h (see (2.6)). We get:

| tk [ŝ n+1 , (-ŝ sin θ + t cos θ + R h ) 2 ]v h , tk ŝn+1 v h | ≤ C tk ŝn+1 v h n+1 j=0 k+2 l=0 tl ŝj v h .
We deduce by the recursion assumption:

q Normal h ( tk ŝn+1 v h ) ≤ C v h 2 .
We infer that, for all k:

D t( tk ŝn+1 )v h ≤ C v h and D ŝ( tk ŝn+1 )v h ≤ C v h .
Moreover, we also deduce:

(V θ + R h ) tk ŝn+1 v h ≤ C v h ,
from which we find:

tk ŝn+2 v h ≤ C v h .
We will also need a control of the derivatives with respect to ŝ.

Lemma 3.3 For all m, n, k, there exist h 0 > 0 and C(m, n, k) > 0 such that for h ∈ (0, h 0 ):

tk D m+1 ŝ ŝn v h ≤ C(k, m, n) v h (3.6) tk D m ŝ D t ŝn v h ≤ C(k, m, n) v h , (3.7)
for v h ∈ F N (h).

Proof: For m = 0, this is done in Lemma 3.2.

Let us assume that the assumptions are true for all l ≤ m and all k and n. We apply tk D m+1 ŝ ŝn to (2.7) and take the scalar product with tk D m+1 ŝ ŝn v h . We shall estimate commutators. We get:

| [D 2 t , tk D m+1 ŝ ŝn ]v h , tk D m+1 ŝ ŝn v h | ≤ C tk D tD m+1 ŝ ŝn v h tk-1 D m+1 ŝ ŝn v h + C tk-2 D m+1 ŝ ŝn v h tk D m+1 ŝ ŝn v h ≤ C D tt k D m+1 ŝ ŝn v h tk-1 D m+1 ŝ ŝn v h + C tk-2 D m+1 ŝ ŝn v h tk D m+1 ŝ ŝn v h + C tk-1 D m+1 ŝ ŝn v h 2
and we deduce, with the recursion assumption, for all ε > 0, the existence of C ε > 0 such that:

| [D 2 t , tk D m+1 ŝ ŝn ]v h , tk D m+1 ŝ ŝn v h | ≤ C(k, m, n)εq Normal h ( tk D m+1 ŝ ŝn v h ) + C ε v h 2 .
Moreover, we have in the same way:

| [D 2 ŝ , tk D m+1 ŝ ŝn ]v h , tk D m+1 ŝ ŝn v h | ≤ C(k, m, n)εq Normal h ( tk D m+1 ŝ ŝn v h ) + C ε v h 2 .
Finally, we shall analyze:

[(V θ (ŝ, t) + R h ) 2 , tk D m+1 ŝ ŝn ].
We have:

[(V θ (ŝ, t) + R h ) 2 , D m+1 ŝ ŝn ] = (V θ (ŝ, t) + R h )[(V θ (ŝ, t) + R h ), D m+1 ŝ ŝn ] + [(V θ (ŝ, t) + R h ), D m+1 ŝ ŝn ](V θ (ŝ, t) + R h ) The worst term is h t[(D ŝ -(sin θ) -1 D τ ) 2 , D m+1
ŝ ŝn ] but it can be dealt by using the support of w h (see (2.6) and the proof of Lemma 3.2). We deduce:

|[(V θ (ŝ, t) + R h ) 2 , tk D m+1 ŝ ŝn ]v h , tk D m+1 ŝ ŝn v h | ≤ C(k, m, n) v h 2 .
We infer the existence of C(k, m, n) > 0 such that:

(1 -C(k, m, n)ε)q Normal h ( tk D m+1 ŝ ŝn ) ≤ C(k, m, n) v h 2 .
and the conclusion follows by choosing ε small enough..

In the next paragraph, we establish partial Agmon estimates with respect to ŝ and t. Roughly speaking, we can write the previous lemmas with τ v h and D τ v h instead of v h .

Partial estimates involving τ

We now aim at adding τ and D τ in the previous estimates. For instance, let us begin to prove that: Lemma 3.4 For all k ≥ 0, there exist h 0 > 0 and C(k) > 0 such that, for h ∈ (0, h 0 ):

tk τ v h ≤ C( τ v h + v h ), tk D t τ v h ≤ C( τ v h + v h ),
and tk D ŝ τ v h ≤ C( τ v h + v h ), for v h ∈ F N (h).
Proof: For k = 0, we multiply (2.7) by τ and take the scalar product with τ v h . There is only one commutator to analyze:

[(V θ + R h ) 2 , τ ] = [(V θ + R h ), τ ](V θ + R h ) + (V θ + R h )[(V θ + R h ), τ ]
so that:

[(V θ + R h ) 2 , τ ] = [R h , τ ](V θ + R h ) + (V θ + R h )[R h , τ ].
We deduce, thanks to the support of w h :

| [(V θ + R h ) 2 , τ ]v h , τ v h | ≤ C v h τ v h ≤ C( τ v h 2 + v h 2 )
and we infer:

q Normal h (τ v h ) ≤ C( τ v h 2 + v h 2 ).
We get:

D t τ v h ≤ C( τ v h + v h ) and D ŝ τ v h ≤ C( τ v h + v h ).
Recursion We assume that the property is true for all 0 ≤ l ≤ k. We now multiply (2.7) by tk+1 τ and take the scalar product with tk+1 τ v h . We have:

| [D 2 t , tk+1 τ ]v h , tk+1 τ v h | ≤ C( tk D t τ v h + tk-1 τ v h ) t k+1 τ v h ≤ Cε tk+1 τ v h 2 + C ε ( τ v h 2 + v h 2 ).
We have:

| tk+1 [(V θ + R h ) 2 , τ ]v h , tk+1 τ v h | ≤ Cεq Normal h ( tk+1 τ v h ) + C ε tk+1 τ v h 2 .
We deduce:

(

1 -Cε)q Normal h ( tk+1 τ v h ) ≤ (σ(θ) + Cε) tk+1 τ v h 2 + C τ v h 2 + C v h 2
It remains to apply Remark 3.1 and to use σ(θ) < 1 to deduce the control of tk+1 τ v h and thus the one of tk+1 D t and tk+1 D ŝ.

As an easy consequence of the analysis of Lemma 3.4, we have:

Approximation of v h

Let us state the approximation result of this subsection: Proposition 3.9 There exists C > 0 and h 0 > 0 such that, if 0 < h ≤ h 0 and v h ∈ F N (h):

v h -Πv h + V θ v h -V θ Πv h + ∇ ŝ,t (v h -Πv h ) ≤ Ch 1/4-2η v h ,
where Π is the projection on u θ defined by: Πψ(τ , ŝ, t) = ψ, u θ ŝ, tu θ (ŝ, t).

In particular, Π :

F N (h) → Π(F N (h)
) is an isomorphism (sometimes called Feshbach-Grushin projection).

Proof: As usual we start to prove the inequality when v h is the image of an eigenfunction associated with λ p (h), the extension to v h ∈ F N (h) being standard. We want to estimate

(H Neu (θ) -σ(θ))v h .
We have:

(H Neu (θ) -σ(θ))v h ≤ (H Neu (θ) -λ p (h))v h + Ch 1/4 v h .
With the definition of v h and with Corollary 2.7, we have:

(H Neu (θ) -λ p (h))v h ≤ h H 1 v h + h 2 H 2 v h + O(h ∞ ) v h .
Then, we can write:

H 1 v h ≤ C tV θ D τ - D ŝ sin θ 2 v h +C t D τ - D ŝ sin θ 2 V θ v h +C tV θ ŝ + τ sin θ 2 v h
With Lemma 3.2 and the support of u h , we infer:

h H 1 v h ≤ Ch 1/4-2η v h .
In the same way, we get:

h 2 H 2 v h ≤ Ch 1/2-4η v h .
We deduce:

(H Neu (θ) -σ(θ))v h ≤ Ch 1/4-2η v h .

Let us write

v h = v ⊥ h + Πv h We have: (H Neu (θ) -σ(θ))v ⊥ h ≤ Ch 1/4-2η v h .
The resolvent, valued in the form domain, being bounded, the result follows.

Harmonic oscillator and last refinements

In this section, we prove Theorem 1.4. In order to do that, we first prove a localization with respect to τ and then use it to improve the approximation of Proposition 3.9.

Control of v h with respect to τ

This subsection is devoted to the reduction to a model operator involving an harmonic oscillator.

Proposition 4.1 There exist h 0 > 0 and C > 0 such that for all C 0 > 0 and h ∈ (0, h 0 ):

q h (v h ) ≥ (1 -C 0 h) D tv h 2 + D ŝv h 2 + V θ (ŝ, t) + αh tH harm v h 2 - C C 0 h H harm v h , v h -Ch v h 2 , for v h ∈ F N (h). Proof: Let us consider q h (v h ) = D tv h 2 + D ŝv h 2 + V θ (ŝ, t) + αh t {H harm + L(τ , D τ , ŝ, D ŝ)} v h 2 .
where L(τ , D τ , ŝ, D ŝ) = (sin θ) -2 (-2 sin θD ŝD τ + 2 sin θŝτ + D 2 ŝ + ŝ2 ). For all ε > 0, we have:

q h (v h ) ≥ (1 -ε) D tv h 2 + D ŝv h 2 + V θ (ŝ, t) + αh tH harm v h 2 -ε -1 α 2 h 2 tL(τ , D τ , ŝ, D ŝ)v h 2
We take ε = C 0 h. We apply Lemmas 3.3, 3.7 and 3.8 to get:

tL(τ , D τ , ŝ, D ŝ)v h 2 ≤ C( D τ v h 2 + τ v h 2 + v h 2 ).
A model operator From the last proposition, we are led to study the model operator:

M h = D 2 ŝ + D 2 t + (V θ (ŝ, t) + αh tH harm ) 2 .
We can write M h as a direct sum:

M h = n≥1 M n h ,
Then, we get:

hH 1 D τ v h ≤ Ch 1/4-2η v h and: h 2 H 2 D τ v h ≤ Ch 1/2-4η v h .
We deduce:

(H Neu (θ) -σ(θ))D τ v h ≤ Ch 1/4-η v h .
The conclusion is the same as for the proof of Proposition 3.9.

Control with respect to τ The analysis for τ can be done exactly in the same way.

Conclusion: proof of Theorem 1.4 We recall that:

q Normal h (v h ) = D tv h 2 + D ŝv h 2 + V θ (ŝ,
tV θ (ŝ, t)D τ v h , D τ v h = tD τ v h , V θ D τ v h .
We have:

tD τ v h ≤ C v h
and thus, with Proposition 4.3:

tD τ v h , V θ D τ v h = tD τ v h , V θ ΠD τ v h + o(1) v h 2 .
We can write:

tV θ ΠD τ v h ≤ C v h
and it follows:

tD τ v h , V θ ΠD τ v h = ΠD τ v h , tV θ ΠD τ v h + o(1) v h 2 .
We deduce that:

tV θ (ŝ, t)D τ v h , D τ v h = tV θ (ŝ, t)D τ Πv h D τ Πv h + o(1) v h 2 .
Let us consider: tV θ (ŝ, t)τ v h , τ v h = tτ v h , V θ τ v h .

In order to treat this term, we use:

tτ v h ≤ C v h .
Approximations of the other terms Another term we should approximate is:

tV θ (ŝ, t)D ŝv h , D τ v h = tD ŝv h , V θ (ŝ, t)D τ v h .

We can notice that: tD ŝv h ≤ C v h so that:

tD ŝv h , V θ (ŝ, t)D τ v h = tD ŝv h , V θ (ŝ, t)ΠD τ v h + o(1) v h 2 .
We write: tD ŝv h , V θ (ŝ, t)ΠD τ v h = D ŝv h , tV θ (ŝ, t)ΠD τ v h and use: tV θ (ŝ, t)ΠD τ v h ≤ C v h to find:

D ŝv h , tV θ (ŝ, t)ΠD τ v h = D ŝΠv h , tV θ (ŝ, t)ΠD τ v h + o(1) v h 2 .
In the same way, we find:

tŝv h , V θ τ v h = tŝΠv h , V θ τ Πv h + o(1) v h 2 , tŝ 2 v h , V θ v h = tŝ 2 Πv h , V θ Πv h + o(1) v h 2 , tD 2 ŝ v h , V θ v h = tD 2 ŝ Πv h , V θ Πv h + o(1) v h 2 .
We infer:

q Normal h (v h ) ≥ σ(θ) v h 2
+ αh 2 tV θ (ŝ, t)H harm + tV θ L(τ , D τ , ŝ, D ŝ) + tL(τ , D τ , ŝ, D ŝ)V θ (ŝ, t)Πv h , Πv h + o(h) v h 2 .

This implies:

q Normal h (v h ) ≥ σ(θ) v h 2 + αh S θ (D τ , τ )φ h , φ h τ + o(h) v h 2 ,
where φ h = v h , u θ ŝ, t and v h ∈ F N (h). We infer:

h -1 (λ N (h) -σ(θ) + o(h)) v h 2 ≥ α S θ (D τ , τ )φ h , φ h τ .
With Proposition 3.9 and the min-max principle, we deduce that:

λ N (h) ≥ σ(θ) + αhν N (S θ (D τ , τ )) + o(h).

This provides the spectral gap between the lowest eigenvalues. Jointly with Theorem 1.3, we deduce Theorem 1.4.

  t) + αh t {H harm + L(τ , D τ , ŝ, D ŝ)} v hIt remains to approximate v h by Πv h modulo a lower order term.Approximation of the terms involving H harm Let us examine first:

	2
	so that we get:
	q Normal

h (v h ) ≥ H(θ)v h , v h + αh 2 tV θ (ŝ, t)H harm + tV θ L(τ , D τ , ŝ, D ŝ) + tL(τ , D τ , ŝ, D ŝ)V θ (ŝ, t)v h , v h

Acknowledgments The author would like to thank Soeren Fournais for his invitation at the University of Aarhus (Winter 2010) where the first ideas of this paper were discussed. The author is also grateful to Mikael Persson for many useful discussions.

We can now prove the following lemma: Lemma 3.6 For all k, n, there exist h 0 > 0 and C(k, n) > 0 such that, for all h ∈ (0, h 0 ):

)

Proof: The proof is based on a recursion with respect to n. For n = 0, this is a consequence of Lemmas 3.4 and 3.5. We refer to the proof of Lemma 3.2. The only difference appears in the computation of the commutator

] provides an additional term in ŝn+1 but this one is controlled by Lemma 3.2.

From this lemma, we deduce a stronger control with respect to the derivative with respect to ŝ: Lemma 3.7 For all m, n, k, there exist h 0 > 0 and C(m, n, k) > 0 such that for h ∈ (0, h 0 ):

)

Proof: The proof can be done by recursion on m. The case m = 0 comes from the previous lemma. Then, the recursion is the same as for the proof of Lemma 3.3 and uses Lemma 3.3 to control the additional commutators.

By the same arguments (symmetry between τ and D τ ), we have: Lemma 3.8 For all m, n, k, there exist h 0 > 0 and C(m, n, k) > 0 such that for h ∈ (0, h 0 ):

for v h ∈ F N (h).

In the next subsection, we prove that v h behaves like u θ (ŝ, t) with respect to ŝ and t.

with

, so that, we shall analyze:

From [5, Proposition 5.1], we deduce the existence of c > 0 such that, for all γ ≥ 0:

Taking C 0 large enough in Proposition 4.1, we deduce the proposition: Proposition 4.2 There exist C > 0 and h 0 > 0 such that:

and:

Refined approximation of v h

The control of v h with respect to τ provided by Proposition 4.2 permits to improve the approximation of v h .

Proposition 4.3 There exist C > 0, h 0 > 0 and γ > 0 such that, if h ∈ (0, h 0 ):

Proof:

Control with respect to D τ Let us apply D τ to (2.7). We have the existence of γ > 0 such that:

We can write:


Abderrezak Rachedi
email: rachedi@univ-mlv.fr

Lamia Kaddar
email: lamia.kaddar@prism.uvsq.fr

Ahmed Mehaoua
email: ahmed.mehaoua@mi.parisdescartes.fr

EDES-Efficient Dynamic Selective Encryption Framework to Secure Multimedia Traffic in Wireless Sensor Networks

In this paper we propose a new framework able to ensure security, multimedia quality and energy efficiency in Multimedia Wireless Sensor Networks (MWSNs). The energy is an important and limited resource, which has a direct impact on the lifetime of nodes in MWSNs. In addition, the characteristics of MWSNs like the limited bandwidth and non-deterministic channel access have a significant impact on the QoS of multimedia traffic. We propose the Efficient Dynamic Selective Encryption Framework (EDES) in order to reduce the energy consumption and increase the QoS while ensuring a secure multimedia traffic. EDES proposes three security levels (high, medium and low) and the selection of each level depends on the energy and QoS parameters. Moreover, the cross-layer approach is selected for EDES to take into account the different parameters at physical, MAC and upper layers. The capacity metric is proposed to evaluate the possibility to increase or decrease the security level. The simulation results illustrate the importance of the security level adaptation according to the QoS and the energy parameters. EDES increases the lifetime duration of nodes by almost 40% compared to static encryption.

I. INTRODUCTION

The Multimedia Wireless Sensor Networks (MWSNs) have gained the interest of researchers and industrials, because of their large applications. Unlike the Wireless Scalar Sensor Networks (WSSNs), MWSNs offer new network services like video and audio communications. The application fields of MWSN are mainly related to security such as: video surveillance, people or objects tracking, border monitoring, etc [START_REF] Akyildiz | A survey on wireless multimedia sensor networks[END_REF] [START_REF] Boulanouar | Energy-Aware Object Tracking Algorithm using Heterogeneous Wireless Sensor Networks[END_REF]. Therefore, these applications require security services like the end-to-end data confidentiality between the multimedia sensors and the server (sink), the mutual authentication, and the data integrity. However, the characteristics of MWSNs such as: the limited resources (bandwidth, energy, memory and processing), the wireless link and mobility make the proposition of an efficient security solution a real challenge. Many solutions proposed to secure both WSSNs and MWSNs are mainly based on static public or symmetric cryptography algorithms [START_REF] Guerrero-Zapata | The future of security in Wireless Multimedia Sensor Networks : A position paper[END_REF]. In the case of public cryptography the Elliptic Curve Cryptography (ECC) implemented in hardware [START_REF] Gaubatz | Sate of art in ultra-low public key[END_REF] or software [START_REF] Watro | Tinypk: securing sensor networks with public key technology[END_REF] is used. However, in the case of symmetric cryptography, the Tesla and micro Tesla [START_REF] Liu | Tesla: Broadcast authentication for distributed sensor networks[END_REF] are used. Most security solutions proposed for MWSNs do not only encounter the energy constraint but also the QoS constraints. We know that the security cost can directly impact the network performance and QoS. For instance, when the size of packets increases because of the addition of security information like numerical packet signature, the data throughput of data decreases.

Unlike existing works, which focus on security, QoS and energy separately, we introduce in this work the security of the multimedia traffic while taking into account QoS and energy. We propose a new framework called EDES able to adapt the security levels according to QoS and energy parameters. The basic idea of EDES is to improve security in terms of robustness without negatively impacting the QoS and lifetime of sensors. We introduce a new metric called capacity to assess the available resources in terms of throughput, delay, link quality and energy. The cross-layer approach is selected to correctly evaluate the different parameters at different layers. Moreover, the different types of video frames do not have the same importance. The compression exploits temporal and spatial correlations in an image sequence, so there is a dependency between frames created by the inter-frame coding. Inter-frame coding uses motion estimation and compensation between successive video frames. For instance, in the case of MPEG4 codec the frame types P and B need the adjacent frame I and frame P respectively in order to be correctly decoded. In other words, it is not possible to rebuild frames P and B without the data in frames I and P respectively. Therefore, the frame types significance is not the same. That is why, we use in this work the selective encryption process to differentiate between different frames.

The rest of the paper is organized as follows: in section 2, we present the summarization of the existing works related to security and encryption in MWSNs. Section 3 presents our proposed solution Efficient Dynamic Selective Encryption Framework (EDES). The fourth section is dedicated to the results obtained with the simulators and their analysis. Finally, section 5 concludes the paper and presents our future works.

II. RELATED WORKS

Recent works dealing with Quality of Service (QoS) in MWSNs are mainly focused on the QoS maximization and energy consumption minimization. Kandris et al. [START_REF] Kandris | Energy efficient and perceived QoS aware video routing over Wireless Multimedia Sensor Networks[END_REF] proposed a scheme called PEMuR which aims at saving energy and maximizing the QoS rate for an efficient video communication. PEMuR proposes the combined use of an energy aware hierarchical routing protocol with an intelligent video packet scheduling algorithm. This algorithm may cope with limited available channel bandwidth by selectively dropping less significant packets prior to their transmission. Another work proposed by Cobo et al. [START_REF] Cobo | Ant-based routing for wireless multimedia sensor networks using multiple QoS metrics[END_REF] dealt with the QoS routing model for Wireless Multimedia Sensor Networks (WMSN). The proposed solution called AntSensNet is based on the traditional ant-based algorithm and multi-QoS routing metric. However, these solutions did not deal with security issues.

Other proposed solutions dealt with security in the case of low-power computing and communication devices by using a selective encryption concept. The idea of selective encryption is to encrypt chosen units of message while leaving the remaining units unencrypted. This concept was introduced by Spanos and Maples in [START_REF] Spanos | Performance study of a selective encryption scheme for security of networked, real-time video[END_REF]. Meyer and Gadegast [START_REF] Meyer | Security mechanims for multimedia data with example MPEG-I video[END_REF] propose to reduce the amount of encrypted MPEG data in a video sequence while at the same time providing an acceptable level of security. However, these works did not take into account the constraints of Multimedia Wireless Sensor Networks (MWSNs). In addition, main works on selective encryption focused on performance issues and not on security issues. That is why, Lundin and Lindskog have evaluated the security impact of the selective encryption by using the entropy and guesswork [START_REF] Lundin | Security Implications of Selective Encryption[END_REF]. In this work, we combine between opposite QoS and security parameters with energy consumption consideration.

III. EFFICIENT DYNAMIC SELECTIVE ENCRYPTION FRAMEWORK

The Efficient Dynamic Selective Encryption framework (EDES) is based on two main steps. The first step consists in assessing the network performance parameters. We introduce the new capacity function called CAP . This function combines between QoS parameters and residual energy. The QoS parameters consist of the throughput, delay and link quality. In order to assess these parameters, we select the cross-layer approach which is able to deal with MAC, routing and other upper layers parameters. The second step consists in selecting a security level and using or not the selective encryption algorithm. The security level is selected according to the CAP function. The capacity function allows to evaluate the resources availability at the transmitter camera sensor node and it is based on four parameters: the throughput, the delay, the link quality index (LQI) and the residual energy. The throughput and the delay are the most important parameters for multimedia traffic. We know that this kind of traffic needs an important throughput and minimum delay. The strategy of CAP function is summarized as follows:

• Maximize the residual energy of the camera sensor (RE) • Maximize the throughput • Maximize the Link Quality Index (LQI)

• Minimize the delay The CAP function between two nodes i and j can be expressed according to the chosen QoS metrics:

CAP i,j = k c k × f k (x k ij ) (1)
where x k ij is the value of metric k relatively to the link between two nodes i and j, c k is the preference weight of metric k with k c k = 1, and f k (:) is a normalized function. x k ij presents the following QoS parameters: Throughput, Delay, Link quality and Residual energy. The choice of the weights c i depends on the application and the type of traffic (the delay is more important than the loss rate for streaming). CAP function introduces four formalized functions f k (:). The normalized function is introduced to express different characteristics of different units with a comparable numerical representation. The most commonly used normalized functions are the sigmoidal (S-shaped) functions. Indeed, sigmoidal functions are well-known functions often used to describe QoS perception [START_REF] Badia | An economic model for the radio resource management in multimedia wireless systems[END_REF]. We consider the following analytic expression for the sigmoid form:

f (x) = (x -x m ) ζ 1 + (x -x m ) ζ (2)
where x m > 0 and ζ ≥ 2 are tunable parameters, according to which different users' utilities are differentiated. EDES defines three security levels: low, medium and high. The low security enables the receiver nodes to control the data integrity and the sender authentication by using the Hashedbased Message Authentication Code (HMAC). However, this level does not ensure the data confidentiality. The medium level has the same security services as low level and ensures data confidentiality by using Standard (AES) algorithm. If the network traffic requires confidentiality, EDES chooses whether the medium or the high security level according to the network status. In this work, we did not focus on the cryptographic keys distribution. In MWSNs the camera sensors send the data to the sink node (Many-to-one communication). We assume that the multimedia sensor nodes shared secret keys with the sink node and all known nodes shared the secret group key [START_REF] Rachedi | A secure and resistant architecture against attacks for mobile ad hoc networks[END_REF]. The difference between the high and medium security levels is the percentage of encryption which is higher than %T h high in the case of a high security level. In addition, EDES differentiates between the frame types in order to select the important frame called key-frame (ie. frame I in the case of MPEG-4 codec). The percentage of encryption in the case of I frame is not the same for other frames. In this work, we consider two types of video codecs: MPEG4 and H263. In both codecs, I and PB frames are important frames in MPEG4 Where C P , C B and C P B are coefficients of importance of P, B and PB frames compared to I frames. In the case of H263 the frame P has the same importance as frame I (Enc I = Enc P ). In order to switch between three security levels, we define three CAP thresholds values. The high security level needs a sufficient network performance and residual energy. The medium security level is an intermediate level which enables to ensure the confidentiality with an acceptable resource consumption. Without enough resource availability the minimum security is ensured by the low level. The following equation defines the switching between different security levels:

   if (CAP ≥ %T h high ) Enc I = %T h high if (%T h min ≤ CAP ≤ %T h high ) Enc I = 50% if (CAP ≤ %T h min ) Enc I = 0%
where T h high and T h min can be tuned according to wireless nodes technologies and the desirable security. EDES framework can be summarized into two flowcharts illustrated in figures 1 and 2. Figure 1 shows the set of operations executed by the sender. The CAP function assesses the resource availability in terms of network performance parameters and residual energy. If the network traffic needs Fig. 2. Flowchart of EDES at the Receiver node confidentiality, the low security level is discarded. The specific operations are launched according to the selected security level. In the case of low security level the information related to data integrity and authentication is added by using two HMAC: first one is based on group key (K G ) for the relayed nodes and and the second one is based on the key shared by the sensor camera and the sink node (K ER ). The key group enables the relayed node to check if the packet is generated by authorized network member nodes and the data integrity. This HMAC is changed at each hop in order to check each relayed node. This is a common operation between the different security levels. However, in the case of high and medium security levels, the frame study and encryption percentage of each type of frame is specified. Moreover, information (M) is added to indicate if the packet is encrypted or not. The selective encryption is carried out according to the chosen %Enc I of each frame. Finally the digital packet signature is added for authentication and data integrity.

Figure 2 shows the flowchart at the receiver node (sink or relayed node). When the relayed node receives the packet, it checks the HMAC associated with the packet. If it is correct, the packet is relayed or generated by an authenticated node using K G . If the receiver is the sink node, it checks the HMAC by using the secret shared key (K ER ) before the decryption process. If the operation is succeeds the decryption process is applied.

IV. PERFORMANCE EVALUATION

In order to evaluate the performance of EDES, we simulate the proposed selective encryption algorithms and CAP function by using N S2 [START_REF]NS2 -Network Simulator[END_REF] and M ica2dot characteristics platform [START_REF] Arvinderpal | Energy analysis of public-key cryptography for wireless sensor network[END_REF]. We used two different video codecs: MPEG4 with low quality and H263 with 16kbps rate. The basic difference between both video codecs is that in MPEG4 coding all the important information are in the I-frame, for example the average of I-frames is 7000 bytes, whereas the average size of P-frames is 300 bytes and less than 100 bytes for B-frames. However, with H263 there is only one I-frame (the first one) then the frame's size is distributed between PB-frames and Bframes and the PB-frames are more frequent than the I-frames in MPEG4. Moreover, the simulation parameters are presented in Table I.

A. CAP function performance

The obtained simulation results are plotted in Figure 3 which shows the CAP function behavior with different parameters such as: the residual energy, throughput and link quality. We remark that the high security level is reached when both parameters: throughput and residual energy are high otherwise the medium security level depends on the selected T h min .

B. Evaluation of Security

In order to evaluate the security aspect, we used SecV LC tool [START_REF] Kaddar | SecVLC : Secure Transmission over Multimedia Wireless Ad Hoc Networks with Energy-awareness[END_REF]. We select a video with a high rate of movement as testbed in order to evaluate the performance of EDES in terms of security, energy consumption and network lifetime. MSU Video Quality Measurement Tool (MSU VQMT) [START_REF] Vatolin | MSU Video Quality Measurement Tool[END_REF] is used to analyse the quality of different videos. This application enables to carry out an objective comparison of the different video codecs and to perform filters in the video analysis.

We used M SE (Mean Square Error) as metric to evaluate the image quality. M SE measures the average of "errors" squares. It assesses the quality of an estimator according to its variation. When MSE is low the loss of information for a potential attacker is high. Figure 4 shows M SE versus frames with different security levels. We remark that M SE increases significantly between two frames encrypted with a We used P SN R (Peak Signal-to-Noise Ratio) [START_REF] Huynh-Thu | Scope of validity of PSNR in image/video quality assessment[END_REF] as another metric in order to evaluate the power of signal and power of corrupting the noise which affects the quality of video representation. Typical values for the P SN R in lossy image and video compression are between 30 and 50 dB, where higher is better. Acceptable values for wireless transmission quality loss are considered to be about 20 dB to 25 dB.

Figure 5 shows the average PSNR versus frames with different security levels varying from highest Enc I = 70% to lowest Enc I = 10%. We remark that the PSNR decreases when the percentage of I-frame encryption increases. In the case of Enc I = 70% the PSNR is around 6.68db which is a very low value. However, when Enc I = 33% and 10% the PSNR is 14.08dB and 17.99dB respectively which are high values. Despite the low encryption percentage, the PSNR does not exceed 18db which is lower than 20dB and 25dB.

C. Energy Saving Evaluation

We know that the cost of energy consumption by the cryptography algorithms is inferior to the cost of radio communication [START_REF] Arvinderpal | Energy analysis of public-key cryptography for wireless sensor network[END_REF]. The equations to compute the energy consumption in the case of transmission and reception are as follows:

ET x = P acketsize * 59, 2µJoule ERx = P acketsize * 28, 6µJoule

where 28, 6µJoule is the cost of receiving one byte and 59, 2µJoule is the required cost to transmit one byte. Let's consider the case of a sensor node using 2 × AA batteries. That means that the initial battery of a sensor is 2 × 2700mAh. We know that 1wh = 3600joule, then the initial energy expressed in joule is E Initial = 17820Joule.

Figure 6 illustrates the encryption energy consumption versus time with EDES and the classical encryption algorithm. We remark that the energy consumption of EDES is significantly inferior to and more stable than the classical encryption algorithm. The high variation of energy consumption in the case of the classical algorithm is mainly due to the encryption of all frames because the frames size variation directly impacts the energy consumption and it is not the case with EDES.

In order to evaluate EDES with both video codecs: MPEG4 and H263, we simulate EDES and the obtained results are plotted in figure 7. We remark that using H263 video codec with EDES increases the MWSN lifetime by around 50% compared to MPEG4. We conclude that H263 is more adapted to MWSNs characteristics. Moreover, EDES improves the lifetime by 40% compared to the classical encryption algorithm. The reduction of energy consumption is more important at the source node compared to the destination node, because in MWSNs the destination node is the sink node (server). Therefore, the impact at the receiver node can be neglected in MWSNs.

V. CONCLUSION

In this paper, we proposed a new framework called Efficient Dynamic Selective Encryption Framework (EDES) in order to ensure dynamic security levels while taking into account the network performance and the energy consumption. The innovation of the proposed EDES consists in the combination of QoS parameters and security levels. EDES proposes three security levels (high, medium and low) with the ability to ensure the authentication, the data integrity and the confidentiality. The selection of each level depends on the energy and QoS parameters. The capacity function is proposed to evaluate the possibility to increase or decrease the security level. The assessment of this function is based on the crosslayer approach to take into account the different parameters at physical, MAC and upper layers. The simulation results illustrate that the security is ensured even with a low security level. In addition, EDES increases the lifetime duration of nodes by around 40% compared to the classical encryption algorithm. In our future works, we plan to extend our study to dynamic MWSNs by taking into account a new parameter: mobility.

Fig. 1 .

 1 Fig. 1. Flowchart of EDES at the Sender node

Fig. 3 .

 3 Fig. 3. CAP versus Residual energy

Fig. 4 .

 4 Fig. 4. MSE measure of the encrypted videos

Fig. 6 .

 6 Fig. 6. Encryption Energy Consumption

This research was funded by LIGM (CNRS UMR8049) and partially supported by the Korea Science and Engineering Foundation, under the WCU (World Class University) program.