Agnel Praveen

Hélène Valadie

N Srinivasan

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

Local structural differences in homologous proteins: Specificities in different SCOP classes

Keywords: amino acid; local structures, backbone conformation, structural alphabet, Protein Blocks, protein folds, indel, structure comparison, Protein Data Bank

The constant increase in the number of solved protein structures is of great help in understanding the basic principles behind protein folding and evolution. 3-D structural knowledge is valuable in designing and developing methods for comparison, modelling and prediction of protein structures. These approaches for structure analysis can be directly implicated in studying protein function and for drug design.

The backbone of a protein structure favours certain local conformations which include α-helices, β-strands and turns. Libraries of limited number of local conformations (Structural Alphabets) were developed in the past to obtain a useful categorization of backbone conformation. Protein Block (PB) is one such Structural Alphabet that gave a reasonable structure approximation of 0.42Å. In this study, we use PB description of local structures to analyse conformations that are preferred sites for structural variations and insertions, among group of related folds. This knowledge can be utilized in improving tools for structure comparison that work by analysing local structure similarities.

Conformational differences between homologous proteins are known to occur often in the regions comprising turns and loops. Interestingly, these differences are found to have specific preferences depending upon the structural classes of proteins. Such class-specific preferences are mainly seen in the all-β class with changes involving short helical conformations and hairpin turns. A test carried out on a benchmark dataset also indicates that the use of knowledge on the class specific variations can improve the performance of a PB based structure comparison approach. The preference for the indel sites also seem to be confined to a few backbone conformations involving β-turns and helix C-caps. These are mainly associated with short loops joining the regular secondary structures that mediate a reversal in the chain direction. Rare β-turns of type I' and II' are also identified as preferred sites for insertions.

Introduction

The three dimensional structure of protein provides precise details on its functional properties like ligand binding or catalysis [START_REF] Baker | Protein structure prediction and structural genomics[END_REF][START_REF] Byers | Acyl carrier protein: structure-function relationships in a conserved multifunctional protein family[END_REF]. Protein structures can also serve as specific drug targets and structure based drug design has been quite successful. The functional properties can be studied by comparing related structures. The analysis of similarities (or variations) in protein structural features among related proteins, demands efficient means of comparing protein folds. Structural divergence occurs less rapidly than sequence divergence and structure based alignments are quite reliable when the proteins have distant relationships [START_REF] Choi | Evolution of protein structural classes and protein sequence families[END_REF][START_REF] Chothia | Lesk AM relation between the divergence of sequence and structure in proteins[END_REF][START_REF] Flores | Comparison of conformational characteristics in structurally similar protein pairs[END_REF][START_REF] Goldstein | The structure of protein evolution and the evolution of protein structure[END_REF][START_REF] Grishin | Fold change in evolution of protein structures[END_REF][START_REF] Salemme | Structural convergence during protein evolution[END_REF][START_REF] Thornton | Protein folds, functions and evolution[END_REF].

Most of the structure comparison methods consider protein folds as rigid bodies and quantify the structural similarity based on an average of atomic distances calculated using backbone coordinates. However, certain regions of a protein structure can be prone to variations, which arise due to structural flexibility or evolutionarily acquired changes. These variations can be either restricted to local regions in the backbone or involve large movements that alter the conformational state of the protein. Unlike the conformational alteration caused by large flexible movements, the local backbone changes are not likely to be affected by the nature of the global fold. Hence the preferences associated with the variations in the backbone conformations can be extracted as a general feature.

The evolutionary information has been used to explore the preferences in amino acid replacements based on empirical approaches [START_REF] Dayhoff | A model of evolutionary change in proteins. Atlas of protein sequence and structure[END_REF][START_REF] Gonnet | Exhaustive matching of the entire protein sequence database[END_REF][START_REF] Jones | The rapid generation of mutation data matrices from protein sequences[END_REF]. Structural contexts of amino acid substitutions involving secondary structures and solvent accessibility have also been studied [START_REF] Goldman | Assessing the impact of secondary structure and solvent accessibility on protein evolution[END_REF][START_REF] Luthy | Secondary structure-based profiles: use of structureconserving scoring tables in searching protein sequence databases for structural similarities[END_REF][START_REF] Overington | Tertiary structural constraints on protein evolutionary diversity: templates, key residues and structure prediction[END_REF][START_REF] Thorne | Combining protein evolution and secondary structure[END_REF][START_REF] Topham | Fragment ranking in modelling of protein structure. Conformationally constrained environmental amino acid substitution tables[END_REF][START_REF] Wako | Use of amino acid environment-dependent substitution tables and conformational propensities in structure prediction from aligned sequences of homologous proteins. II. Secondary structures[END_REF][START_REF] Wako | Use of amino acid environment-dependent substitution tables and conformational propensities in structure prediction from aligned sequences of homologous proteins. I. Solvent accessibility classes[END_REF][START_REF] Przytycka | A protein taxonomy based on secondary structure[END_REF]. Nevertheless, the precise local structural changes that occur need to be understood. Apart from local conformational changes, insertions and deletions (indels) seem to play a major role in protein evolution [START_REF] Grishin | Fold change in evolution of protein structures[END_REF][START_REF] Panchenko | Evolutionary plasticity of protein families: coupling between sequence and structure variation[END_REF][START_REF] Castillo-Davis | The functional genomic distribution of protein divergence in two animal phyla: coevolution, genomic conflict, and constraint[END_REF][START_REF] Petrov | Mutational equilibrium model of genome size evolution[END_REF][START_REF] Sandhya | Length variations amongst protein domain superfamilies and consequences on structure and function[END_REF]. The studies on indels in the context of secondary structures suggested that the loops are more tolerant to indels than regular secondary structural regions and a significant percent of indels are disordered [START_REF] Grishin | Fold change in evolution of protein structures[END_REF][START_REF] Aravind | Trends in protein evolution inferred from sequence and structure analysis[END_REF][START_REF] Jiang | Insertions and the emergence of novel protein structure: a structurebased phylogenetic study of insertions[END_REF][START_REF] Shortle | The emerging role of insertions and deletions in protein engineering[END_REF][START_REF] Sondek | Accommodation of single amino acid insertions by the native state of staphylococcal nuclease[END_REF][START_REF] Taylor | Occurrence and consequences of coding sequence insertions and deletions in Mammalian genomes[END_REF][START_REF] Pascarella | Analysis of insertions/deletions in protein structures[END_REF][START_REF] Kim | Systematic analysis of short internal indels and their impact on protein folding[END_REF]. The inserted regions prefer to be short [START_REF] Pascarella | Analysis of insertions/deletions in protein structures[END_REF] and hydrophobic amino acids were found to be less frequent in the inserted region [START_REF] Chang | Empirical analysis of protein insertions and deletions determining parameters for the correct placement of gaps in protein sequence alignments[END_REF]. A more detailed analysis of the effect of insertions on the flanking regions has also been carried out and insertions were found to break regular secondary structures or cause an alteration in the tertiary structure [START_REF] Zhang | Impact of indels on the flanking regions in structural domains[END_REF].

To study the preferences in the local conformational variations among homologous proteins, a good understanding of the frequent backbone conformations is necessary. The local backbone conformation of a protein chain is usually described in terms of α-helix and βstrand. More than 50% of the backbone is assigned to the coil state which reflects irregularity in the backbone. Later, more precise and comprehensive studies led to the identification of other repeating conformations [START_REF] Offmann | Local Protein Structures[END_REF]. The most important of them are the β-turns which cover about 25%-30% of the residues [START_REF] Bornot | Protein beta-turn assignments[END_REF][START_REF] Chou | Beta-turns in proteins[END_REF][START_REF] Lewis | Folding of polypeptide chains in proteins: a proposed mechanism for folding[END_REF][START_REF] Richardson | The anatomy and taxonomy of protein structure[END_REF][START_REF] Yang | Free energy determinants of secondary structure formation: III. beta-turns and their role in protein folding[END_REF][START_REF] Shepherd | Prediction of the location and type of beta-turns in proteins using neural networks[END_REF][START_REF] Kountouris | Predicting beta-turns and their types using predicted backbone dihedral angles and secondary structures[END_REF]. Out of the 9 different types of β-turns categorized based on the φ/ψ dihedrals, type I and type II are most common representing 31.6% and 10.4% of all turns (i.e., 10 and 4% of all residues). The type IV turns are comprised of those which could not be assigned to other types as per standard definitions and this has the maximum representation of about 43% [START_REF] Hutchinson | PROMOTIF--a program to identify and analyze structural motifs in proteins[END_REF][START_REF] Kabsch | Dictionary of protein secondary structure: pattern recognition of hydrogen-bonded and geometrical features[END_REF].

A more precise and different view of the favorable backbone conformations is provided by Structural Alphabets (SAs). SAs represent a library of limited number of local backbone conformations that are used to approximate the fold of a complete protein chain [START_REF] De Brevern | Bayesian probabilistic approach for predicting backbone structures in terms of protein blocks[END_REF][START_REF] Jones | Using known substructures in protein model building and crystallography[END_REF][START_REF] Kolodny | Small libraries of protein fragments model native protein structures accurately[END_REF][START_REF] Levitt | Accurate modeling of protein conformation by automatic segment matching[END_REF][START_REF] Micheletti | Recurrent oligomers in proteins: an optimal scheme reconciling accurate and concise backbone representations in automated folding and design studies[END_REF][START_REF] Rooman | Automatic definition of recurrent local structure motifs in proteins[END_REF][START_REF] Schuchhardt | Local structural motifs of protein backbones are classified by self-organizing neural networks[END_REF][START_REF] Unger | A 3D building blocks approach to analyzing and predicting structure of proteins[END_REF][START_REF] Sander | Local protein structure prediction using discriminative models[END_REF][START_REF] Thangudu | Analycys: a database for conservation and conformation of disulphide bonds in homologous protein domains[END_REF]. A SA consisting of 16 prototypes called Protein Blocks (PBs) was developed in our laboratory [START_REF] De Brevern | Bayesian probabilistic approach for predicting backbone structures in terms of protein blocks[END_REF][START_REF] De Brevern | New assessment of a structural alphabet[END_REF]. Each PB represents a pentapeptide backbone conformation described as a series of φ, ψ dihedrals and each PB is labeled by a character alphabet ranging from a to p (Figure 1). This SA gives a reasonable approximation of local protein 3D structures with a root mean square deviation (rmsd) of about 0.42 Å [START_REF] De Brevern | New assessment of a structural alphabet[END_REF]. PB description has been used in several bioinformatics approaches including modeling and structure prediction [START_REF] De Brevern | Bayesian probabilistic approach for predicting backbone structures in terms of protein blocks[END_REF][START_REF] De Brevern | Local backbone structure prediction of proteins[END_REF][START_REF] Etchebest | A structural alphabet for local protein structures: improved prediction methods[END_REF][START_REF] Zimmermann | LOCUSTRA: accurate prediction of local protein structure using a two-layer support vector machine approach[END_REF][START_REF] Dong | Analysis and prediction of protein local structure based on structure alphabets[END_REF][START_REF] Benros | Analyzing the sequence-structure relationship of a library of local structural prototypes[END_REF][START_REF] De Brevern | Pinning strategy": a novel approach for predicting the backbone structure in terms of protein blocks from sequence[END_REF][START_REF] Li | Fragment-based local statistical potentials derived by combining an alphabet of protein local structures with secondary structures and solvent accessibilities[END_REF][START_REF] Tyagi | Protein short loop prediction in terms of a structural alphabet[END_REF][START_REF] Chen | Protein local 3D structure prediction by Super Granule Support Vector Machines (Super GSVM)[END_REF][START_REF] Dudev | Discovering structural motifs using a structural alphabet: application to magnesium-binding sites[END_REF][START_REF] Faure | Analysis of protein contacts into Protein Units[END_REF][START_REF] Thomas | Prediction of peptide structure: how far are we?[END_REF][START_REF] Tyagi | Protein structure mining using a structural alphabet[END_REF][START_REF] Zuo | Using reduced amino acid composition to predict defensin family and subfamily: Integrating similarity measure and structural alphabet[END_REF][START_REF] Joseph | A short survey on Protein Blocks[END_REF][START_REF] Joseph | Local Structure Alphabets[END_REF][START_REF] Wu | A structural-alphabet-based strategy for finding structural motifs across protein families[END_REF]. Figure 2 shows practical examples on the association of different PBs with regular secondary structures and Table 1 summarizes this relationship using PROMOTIF [START_REF] Hutchinson | PROMOTIF--a program to identify and analyze structural motifs in proteins[END_REF] based secondary structure assignment.

As in the case of the study of amino acid substitutions that occur during the course of evolution, the preferred local structural changes could be analysed with the help of PBs. This idea was extended to the comparison of protein structures. Approximation of protein structures in terms of SA helps to transform 3D information in 1D. Thus the 3D superposition of protein structures can be carried out with an alignment of sequences encoded in terms of SAs [START_REF] Tyagi | Protein structure mining using a structural alphabet[END_REF][START_REF] Tyagi | Protein Block Expert (PBE): a web-based protein structure analysis server using a structural alphabet[END_REF]. A specialized PB substitution matrix (SM) was developed for this purpose [START_REF] Tyagi | A substitution matrix for structural alphabet based on structural alignment of homologous proteins and its applications[END_REF].

The PB based structure alignment approach performed better than many of the other available tools for structure comparison [START_REF] Tyagi | Protein structure mining using a structural alphabet[END_REF][START_REF] Joseph | Improvement of protein structure comparison using a structural alphabet[END_REF].

In this study we analyse the preferences for the conservation of local backbone conformations with the help of Protein Block abstraction. Initially, we analyse the pattern of PB substitutions and the effect of solvent accessibility on this. Here, we restrict our analysis to the equivalent structural regions found among families of related folds. This knowledge can be utilized in the improvement of structure comparison tools that works based on the similarities in the local backbone or fragment conformations. As the secondary structure content and topology varies between structural classes of proteins (as defined by SCOP [START_REF] Murzin | SCOP: a structural classification of proteins database for the investigation of sequences and structures[END_REF]), we check whether there are class-specific specificities for changes in local pentapeptide conformations. In that case we also verify the use of class specific PB substitution matrices in improving the alignment of structures represented in terms of PB sequences. The preferred local backbone conformations associated with the sites of insertions were studied.

Throughout the study, we associate the PB description of backbone conformation with different secondary structure assignments, to present a different view of the results.

Methods

Protein Blocks. Protein Blocks (PBs) are a set of 16 prototypes of main chain conformations that are 5 residues long. The pentapeptide backbone conformation is described in terms of the φ, ψ dihedral angles. The 16 prototypes are labeled from a to p (Figure 1). They were generated using an unsupervised classifier related to Kohonen Maps [START_REF] Kohonen | Self-Organizing Maps[END_REF] and hidden Markov model. Protein Blocks renders a reasonable approximation of local structures in proteins [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) of 0.42 Å [START_REF] De Brevern | New assessment of a structural alphabet[END_REF]. The assignment of PBs [START_REF] De Brevern | New assessment of a structural alphabet[END_REF] has been carried out using an in-house Python software similar to the one used in iPBA web server [START_REF] Gelly | iPBA: a tool for protein structure comparison using sequence alignment strategies[END_REF].

Figure 2 highlights the correspondence between PBs and regular secondary structures assigned by DSSP (Dictionary of Secondary Structure of Proteins) [START_REF] Kabsch | Dictionary of protein secondary structure: pattern recognition of hydrogen-bonded and geometrical features[END_REF]. The PBs m and d are prototypes for the central region of α-helix and β-strand, respectively. PBs a through c primarily represent the N-cap of β-strand while e and f correspond to C-caps. These N and C caps could also include regions in the loop leading to or arising from a secondary structural element. The PBs p, a, f, h, g and i are often seen in the region of transition between secondary structural elements. Figure 2A-C presents some examples highlighting the association of the PB structures with respect to the secondary structure definition while Table 1 gives a detailed list of this relationships extracted from a subset of PALI (Phylogeny and ALIgnment of homologous protein structures) [START_REF] Balaji | PALI-a database of Phylogeny and ALIgnment of homologous protein structures[END_REF] dataset generated using a sequence identity cut-off of 40%. Figure 2 also highlights some of the frequently occurring PB-PB transitions. PBs g through j are largely associated with coils, PBs k and l are frequent in the N cap of α-helix and n to p in C-caps.

Dataset. The dataset of protein structure alignments used in the study is the recent version of PALI dataset V 2.8a [START_REF] Balaji | PALI-a database of Phylogeny and ALIgnment of homologous protein structures[END_REF][START_REF] Gowri | Integration of related sequences with protein three-dimensional structural families in an updated version of PALI database[END_REF][START_REF] Sujatha | PALI: a database of alignments and phylogeny of homologous protein structures[END_REF]. It consists of 1,922 domain families comprising of 231,000 domain pairs aligned using MUSTANG [START_REF] Konagurthu | MUSTANG: a multiple structural alignment algorithm[END_REF]. The domains are classified based on SCOP definitions [START_REF] Murzin | SCOP: a structural classification of proteins database for the investigation of sequences and structures[END_REF]. SCOP classifies domain structures into four major classes. All-α class consists of proteins with mainly α-helical content while all-β proteins are composed of mainly strand conformation. α/β contains both helical and strand conformations that are mixed in the structure, while they are segregated in the case of α+β class. PB Substitution matrix. Domain pairs in the PALI database that are solved at resolution better than 2Å and share sequence identity less than 40%, were only used for obtaining the substitution frequencies. This corresponds to 5,223 domain alignment pairs from 476 families. The pairwise structural alignments were first represented as PB sequence alignments. The PB pairs occurring in the structurally conserved regions (within 3 Å) were counted for calculating the substitution frequencies. As in our previous work [START_REF] Tyagi | Protein Block Expert (PBE): a web-based protein structure analysis server using a structural alphabet[END_REF], the method presented by Johnson et al. [START_REF] Johnson | A structural basis for sequence comparisons. An evaluation of scoring methodologies[END_REF] was adopted for calculating log odd scores from raw frequencies:

                              M i M j j i M i j i M j j i j i e j i N N N N S 1 1 , 1 , 1 , , , log (1)
where S i,j is the substitution weight and N i,j is the raw substitution frequency between PB i and PB j, M is the total number of different PBs (i.e., [START_REF] Thorne | Combining protein evolution and secondary structure[END_REF].

Structural superposition based on PBs. Protein structures to be aligned were first represented as PB sequences. These sequences have been aligned using Smith-Waterman dynamic programming algorithm [START_REF] Smith | Identification of common molecular subsequences[END_REF], based on the PB substitution scores. Gap penalty of -5.0 was used for alignment [START_REF] Tyagi | Protein structure mining using a structural alphabet[END_REF]. Profit version 3.1 [START_REF] Martinez | Obligatory steps in protein folding and the conformational diversity of the transition state[END_REF] was used to obtain a least squares fit of two protein structures based on the PB sequence alignment. The amino acid sequence alignment corresponding to the PB alignment was given as input for Profit for reading the aligned pairs of residues. The fit was performed on the aligned residue pairs and the Root Mean Square deviation (rmsd) was calculated.

Test Dataset for alignments. The gain in the quality of superposition (quantified as the difference in rmsd of superimposition) obtained using the class specific PB substitution matrices was checked on a smaller dataset. From each SCOP superfamily in the PALI dataset (with two or more families), two families were randomly chosen and from each of these families, a domain pair with sequence identity less than 40%, was chosen. It represents 1,050 domains (comprising of 188,760 residues) from 263 families.

Clustering based on substitution data.

To compare the PB substitution patterns, pairwise correlation coefficients were calculated based on the substitution scores associated with each PB. These values were deducted from 1 to get a distance matrix for hierarchical clustering. The hclust module of 'R' software (http://www.r-project.org/) was used for clustering the PBs based on the distance matrix.

Secondary Structure Assignment. The secondary structure types associated with the PBs were identified with the help of assignments made by DSSP [START_REF] Kabsch | Dictionary of protein secondary structure: pattern recognition of hydrogen-bonded and geometrical features[END_REF], SEGNO [START_REF] Cubellis | Secondary structure assignment that accurately reflects physical and evolutionary characteristics[END_REF] and PROMOTIF [START_REF] Hutchinson | PROMOTIF--a program to identify and analyze structural motifs in proteins[END_REF].

PB accessibility.

A PB is considered solvent accessible if at least 3 residues (out of 5) that it corresponds to, are accessible to the solvent. NACCESS [86] was used for calculating the accessibility of each residue. Different cut-offs of 7%, 15% and 25% for relative solvent accessibility, were used to identify buried residues.

Locating indels. The structural alignments of domain pairs sharing less than 80%

sequence identity cut-off were extracted from PALI. If a continuous stretch of gaps of length n is flanked by aligned regions (each aligned residue pair within 3 Å) that are at least 3 residues long, then that position is considered as a point of insertion/deletion.

Z value:

A likelihood score was computed to identify significant members of a distribution. This was used to identify the local conformation prone to insertions. The preferred series of two PBs (di-PBs) binding the insert site are extracted from the observed distribution of di-PBs. The background frequency of occurrence of di-PBs in the dataset was considered as the expected distribution. Z values were computed based on the deviation from the expected distribution. The di-PBs with Z values greater than 2 were considered as the preferred sites for insertions.

Results

The extent of conservation of local backbone conformations were identified in terms of PBs.

The local structures undergoing subtle conformational differences and those which are preferred as insert sites, were looked into. Pairwise structural alignments from the PALI dataset were used as a reference to study such preferences among related structures in a family.

Local Structure Substitutions

The changes in local backbone conformation were deduced by looking at PB replacements among homologous structures. The reliable alignment regions (residue pairs within 3Å) are only considered for calculating the replacement frequencies. The scores for substituting each PB with the 16 PBs, were calculated from the raw substitution frequencies (see Methods).

Figure 3A shows the substitution preferences associated with each PB. Surprisingly, the PBs associated with the N and C caps of helix and strand do not show highly preferred substitutions with the central helix PB m and central strand PB d respectively. This reflects the preference for conservation of the central or most favoured conformation of these regular structural elements. The PB p, usually found in the C-cap of helices and/or at the N-cap of -strands, favours substitutions with PBs g and i. The PB pairs (p, g) and (p, i) share similar (φ,ψ) dihedrals along the 5 residue stretch (see Figure 3B which compares the dihedral angles associated with these PBs). The substitution (p, g) is dominated by changes in conformation of 3. 10 helices and β-turns and a relatively fewer conversions to α-helix and coil (Table 1, Supplementary data1 & 2). These turns are mainly characterized by β-turns of type I and IV.

On the other hand, (p,i) substitution involves variations in turns (β-turns type I, II and IV) and the substitutions between them and coils. These two substitutions mainly involve the region of helix-helix, strand-strand and helix-strand transitions (Supplementary data 1). PB b which is largely seen in the N cap of -strands, favour replacement with PB i which is frequently seen in the region of strand-strand transitions (Figure 3C). This change is associated with variation in turns and bends, mainly involving transitions between β turns of types I, & IV with types II and IV.

It is expected that the preference for PB substitution is dependent on the extent of structural similarity between PBs. Nonetheless, often the structurally closest PBs are not the ones with the best substitution preference (Figures 3D&E). For instance, the substitution of PB f and PB h is not high preferred (Figure 3E), even though they are very close in terms of the dihedral angle distribution. The preference for replacement can be dependent on the local structural environment. This is also true in the case of substitutions (k, l) and (c,d), which are not highly favoured even though they are structurally closest. PB j, which is usually seen in coils, favours replacement with h (Figure 3A, Supplementary data 3). PB k associated with N-cap of helices, also show preferred substitution with the loop PB h. These two changes are characterized by variations in β-turns and 3. 10 helices (Supplementary data 1). The replacement of h and i which are largely seen in the strand-strand transitions, with central -helix PB m is strongly disfavoured. The more obvious case involving substitutions between helix and strand associated PBs, are not preferred (Figure 3A).

Hence many of the preferred variations in the backbone conformation, corresponds to changes in β-turns. The clustering based on the substitution pattern of each PB (Figure 3E) highlights differences with respect to the association based on PB conformation similarity (Figure 3D). The PBs associated with the helical conformation, i.e. l (N-terminus), m As mentioned in the Methods section, the local conformational changes discussed above were identified using a dataset of domain pairs sharing less than 40% sequence identity. To check whether the nature of backbone conformational changes has significant differences depending on the extent of structure relatedness, we compared the substitution patterns obtained from datasets filtered at different sequence identity cut-offs like 60%, 80%

and finally a dataset with all domain pairs (no filtering, Supplementary data 4). No significant differences were observed with respect to the original dataset (filtered at 40% sequence identity), the PB substitutions had correlation scores close to 1.

PB substitution and accessibility

Each PB was first classified into accessible and buried (see Methods) and the occurrence frequency was calculated. The PB g is dominated by short helical conformations (including 3. 10 helices) and turns, while PB i is very frequent in turns (Table 1). The relative increase in exposure with increase in the threshold for burial also shows a similar trend. The strand associated PBs have a relatively lower increase in the percentage of exposure.

It is interesting to find out whether the substitution patterns vary with solvent accessibility of the local structures. To apprehend it, a substitution matrix was generated for the PBs categorized as exposed and buried (Supplementary data 5). Apart from a few exceptions, the distribution of scores for substitutions between exposed PBs and between buried PBs was largely similar to the general distribution (Figure 3A). Substitution (k, i) is preferred in the buried regions than exposed. Most of the substitutions involving the replacement of an exposed PB by a buried PB of another kind are not favoured. The substitutions (p, g) and (h, j) are exceptions.

Clustering exposed and buried PBs based on the substitution patterns suggests that PBs associate differently depending on their accessibility (Figures 4C andD). The exposed PB (Figure 4C) cluster in a way similar to the general preferences (Figure 3A). In the buried region, the PBs b and i cluster with the loop PBs and not with the strand associated PBs. The substitution patterns associated with the central helix conformation m is not highly similar to the substitutions in the immediate C-terminus (PB n), unlike the exposed regions.

Class specific PB substitutions

The For the all-α class (Figure 5A), the PBs mainly occurring in helix N-terminus, is associated with loop PB h which is largely found in β turns and strand C terminus. For the all-β class (Figure 5B), the group of loop associated PBs cluster is closer to the helix PBs than those which correspond to the strand.

The PBs in the α/β class (Figure 5C) associate in a similar fashion as that of the global distribution, except that the PBs a and c which mark the beginning of strands, cluster closely with the other strand PBs and the helix N cap PB l associates with loop PBs. The clustering in the α+β class (Figure 5D) is closest to the general distribution (Figure 3D).

Preferred substitutions in each class

Thus variations in the substitution preferences of local structure conformations are seen across SCOP classes. Comparison of these class-specific substitution scores with the global matrix (see Methods) highlights a few differences (Figure 6).

It was seen that substitutions involving strand associated PBs and helix associated

PBs have a higher score in the all-α and all-β classes respectively (Figures 6A and6B).

Indeed, they have lower background frequencies or lack sufficient substitution information in these respective classes. Nevertheless, the observed probabilities of changes between strands associated PBs with the central conformation d was low in the all-α class. Similarly, in the all-β class, the substitutions involving central helix conformation m and other helix associated

PBs have low probabilities of occurrence (Supplementary data 7). More class specific preferences for the change in local conformations were evident in the all-α and all-β classes (Figure 6). The substitution patterns associated with each PB was compared with that of the general preferences (Figure 3A) and the cases where the correlation was less than 0.95 were looked into.

In the all-α class, two substitutions (a, e) and (g, j) were found to be more favourable when compared to the global preferences (Figures 7A&B).

Sites of indels

The sites of insertion/deletion events were analysed using PBs. The frequencies of the two PBs (di-PBs) that bind the site of indels, were calculated (see Methods). Preferred sites of insertions were identified using Z-values. The local structural regions where indels occur show some preferences (Table 2 & Figure 8). The length of the insert also affects the preferences for the insert site. However, certain di-PBs like 'p-a' and 'j-a' are the preferred sites for insertions of different lengths.

The preferences for the site of insertions, has variations across different SCOP classes. A few class specific preferences could be found for the all-α and all-β classes, especially for short inserts of length less than 4 (Table 2). Perhaps, many of the preferred sites for insertions/deletions are class-independent. β-turns and the C-capping region of αhelices are largely found as indel sites. These preferred sites are associated with loops that mediate the reversal in the direction of the backbone. Across the different SCOP classes, the two major PB bounds for insertions, are 'h-i' and 'p-a'. The di-PB 'p-a' characterizes helixhelix and helix-strand transitions (Figures 8A andD). This local fold is characteristic of the C-cap motif of α-helices. Both short and long insertions are found associated with this site. In the all-β class, this site is preferred for single residue insertions with an association with beta turn of type I (Figure 8B). These di-PB 'hi' on the other hand, mainly characterizes region of strand-strand transitions (Figures 8B to 8D). Long insertions are found to occur at this site.

The local structural region involving 'hi' is dominated by beta turn of type I' (Figures 8B to 8D).

Single residue insertions are also preferred in the immediate C-terminus of the regular secondary structural elements. Though short insertions are also frequent in helices ('mm') and strands ('dd'), the occurrences are not significantly higher than the background.

Discussion

The precise description of local structures in terms of PBs presents a better view of the preferred local structural differences that occur among homologous proteins. The changes are highly constrained with preferences that are not necessarily correlated with the extent of structural similarity of PBs. -turns are associated with a significant majority of the conformational variations. This involves both variations within a type of -turn and exchanges with other types. Conformational flipping between β-turns has been studied for several years, especially inter-conversions between type I and type II turns and between type I' and II' [START_REF] Martinez | Obligatory steps in protein folding and the conformational diversity of the transition state[END_REF][START_REF] Gunasekaran | Conformational interconversions in peptide beta-turns: analysis of turns in proteins and computational estimates of barriers[END_REF]. Many of these inter-conversions are noted to be associated with functional interaction and dynamics [START_REF] Nicholson | Flexibility and function in HIV-1 protease[END_REF][START_REF] Srinivasan | The T-to-R transformation in hemoglobin: a reevaluation[END_REF]. Fairly low energy barriers are proposed for these changes and flipping of the central peptide unit (linking C-αs of residues i+1 and i+2) is suggested as a mechanism for these changes [START_REF] Gunasekaran | Conformational interconversions in peptide beta-turns: analysis of turns in proteins and computational estimates of barriers[END_REF][START_REF] Hayward | Peptide-plane flipping in proteins[END_REF]. Preferred changes from type I or II to type IV are also seen based on the PB substitution preferences. Replacements between turns and 3. 10 helices also seem to be favoured. In fact, the conformation of 3. 10 helix has similarities with type I β-turn [START_REF] Hutchinson | A revised set of potentials for beta-turn formation in proteins[END_REF]. As the substitution frequencies are calculated from the structurally similar regions, the larger variations are less evident.

Variations in the patterns of local structural changes are observed across different SCOP classes (Figure 5). Specific conformational changes are also preferred in certain SCOP classes (Figure 6). This is most evident in the case of all-β class, where the preferred local structure substitutions are found associated with short helical regions and β-turns. The preferred substitutions involving central helix PB m is rather unexpected. Short helices dominate the helical conformations found in the all-β class (Supplementary data 8). About As they are uncharacterized, they encompass a wide range of conformations.

Using class specific PB substitution matrices for structural alignment

The knowledge on the substitution preferences observed in different SCOP classes could be utilized to improve structural comparisons based on PB sequence alignment [START_REF] Tyagi | Protein structure mining using a structural alphabet[END_REF][START_REF] Tyagi | Protein Block Expert (PBE): a web-based protein structure analysis server using a structural alphabet[END_REF][START_REF] Tyagi | A substitution matrix for structural alphabet based on structural alignment of homologous proteins and its applications[END_REF]. PB based structural alignment method, iPBA, was shown to perform better than other established methods like DALI [START_REF] Holm | Protein structure comparison by alignment of distance matrices[END_REF], MUSTANG [START_REF] Konagurthu | MUSTANG: a multiple structural alignment algorithm[END_REF], VAST [START_REF] Gibrat | Surprising similarities in structure comparison[END_REF], CE [START_REF] Shindyalov | Protein structure alignment by incremental combinatorial extension (CE) of the optimal path[END_REF] and GANGSTA+ [START_REF] Guerler | Novel protein folds and their nonsequential structural analogs[END_REF]. About 82% of the alignments had better quality when compared to DALI in benchmark tests. Comparable performance could be observed with respect to TMALIGN [START_REF] Zhang | TM-align: a protein structure alignment algorithm based on the TMscore[END_REF] and FATCAT [START_REF] Ye | Flexible structure alignment by chaining aligned fragment pairs allowing twists[END_REF].

The substitution matrices generated from the class-specific datasets are adapted for the background PB composition and observed changes. As seen above, specific domain families were found to contribute a significant portion of PB changes, favoured in a specific class. To avoid this bias resulting from non-uniform distribution of different family sizes, the raw frequencies counted from a family was normalized by the family size. As the substitution matrices are generated using the frequencies from the conserved regions of superposition, it is logical to compare the local alignments obtained using the class specific matrices with respect to the global matrix. The structural alignment pairs in the test dataset were used for this assessment.

As seen on Figure 9, a gain in the alignment quality is achieved in the all-α, all-β and α/β classes, with the use of class specific SMs. With the use of all-α class-specific SM for aligning domains in this class, 50.1% and 30.2% of the structural alignments had better and same rmsd values respectively, when compared to those generated using the general SM. For the all-β class, 38.1% of the alignments were better while 26.8% had poor rmsd. For the α/β class 43.3% and 28.8% alignments gave positive and negative results. The α+β class did not show any improvement with the use of specific SM. This suggests that the class specific substitution information could be useful in aligning the structurally similar regions. The negative cases with a lower alignment quality when compared to those generated with the global SM, need to be analysed in detail.

Hot-spots for insertions

The relative frequency of occurrence of insertions is similar across different SCOP classes. The distribution of insertion of different lengths in the classes follows similar pattern (Supplementary data 9). However, single residue insertions have a relatively low frequency in the all-β class. The preferred sites of insertions are highly specific in terms of local conformation. Though some class-specific insert sites are observed, the different SCOP classes share many insert sites. Helix C-caps and hairpin turns mainly constitute the sites favourable for occurrence of indels (Table 2).

Helix capping motifs have been widely studied since many years and exploring the amino acid preferences associated with these motifs, has been a main area of interest [START_REF] Aurora | Helix capping[END_REF][START_REF] Chakrabartty | Helix capping propensities in peptides parallel those in proteins[END_REF]100,101,102]. The dihedral angle distribution of the di-PB 'pa' is close to that observed in the Schellman motif and the α L type caps [START_REF] Aurora | Helix capping[END_REF]. These motifs are stabilized by a specific pattern of backbone hydrogen bonds. Apart from the helix caps, beta turns of types I', II' and I are largely seen to characterize the site of indels. It is interesting to note that the turns of types I' and II' are quite rare, with an occurrence frequency of only about 3% [START_REF] Shepherd | Prediction of the location and type of beta-turns in proteins using neural networks[END_REF].

Hence the preferred insertion sites are largely confined to a few specific conformations.

Both helix caps and beta turns have been implicated in structural stability and protein folding [START_REF] Lewis | Folding of polypeptide chains in proteins: a proposed mechanism for folding[END_REF][START_REF] Yang | Free energy determinants of secondary structure formation: III. beta-turns and their role in protein folding[END_REF]103,104,105,106,107]. These β-turn types associated with indel sites (Table 2) are characterized by short hairpin loops. The conformation of helix C-caps pertaining to the indel sites are also confined to short loops that forms the region of transition with another helix or strand (Figure 8) [START_REF] Aurora | Helix capping[END_REF]. These local folds thus restrict the orientation of the flanking secondary structural elements to an antiparallel conformation. The preferred conformation of insert regions is also reported to be shared among turns and coils and most of the indels are likely to be tolerated as extensions of the local conformation [START_REF] Pascarella | Analysis of insertions/deletions in protein structures[END_REF].

The use of dataset specific substitution information has been implicated in the improvement of amino acid sequence alignment [108,109,110,111,112]. Similar strategy can be adopted in the case of PB based structural alignment too [START_REF] Tyagi | Protein structure mining using a structural alphabet[END_REF][START_REF] Tyagi | Protein Block Expert (PBE): a web-based protein structure analysis server using a structural alphabet[END_REF][START_REF] Tyagi | A substitution matrix for structural alphabet based on structural alignment of homologous proteins and its applications[END_REF]. Class-specific PB substitution matrices have been shown to be useful in improving the quality of alignments pertaining to the class. The nature of specific local structures that act as the hot spots of indels, can be also used to develop specialized gap penalties for structural alignment based on PBs. This strategy has already been reported to improve the quality of alignments generated [START_REF] Chang | Empirical analysis of protein insertions and deletions determining parameters for the correct placement of gaps in protein sequence alignments[END_REF]113]. The PB bounds (di-PBs) that act as sites for insertions/deletions of different lengths are listed. To obtain a better picture of the local fold, the two PBs that are seen on both sides of the indel site were also analysed. The most frequent series are listed and their occurrence frequencies are given in parentheses. PROMOTIF [START_REF] Hutchinson | PROMOTIF--a program to identify and analyze structural motifs in proteins[END_REF] was used for assignment of the local fold corresponding to these frequent PB series. Those regions assigned as coils and are usually found as capping motifs, are labelled as 'caps'. The following are the local fold definitions implied by the PROMOTIF assignment abbreviations: (see also Table 1). HPX:Y -β-hairpins, X and Y indicate the number of residues in loop, based on two different rules [START_REF] Hutchinson | PROMOTIF--a program to identify and analyze structural motifs in proteins[END_REF], GTCLA -Classic γ-turns (φ=75.0±40,ψ=-64.0±40).

Conclusion

(

  central) and n, o and p (C-terminus) have similar preferences for substitution. PB k which is also frequent in the N-cap of helices has patterns of substitution similar to the loop associated PBs (j,h). On the other hand, the PBs mainly occurring at the N-terminus of strands cluster separately from the rest of strand associated PBs.It should be noted that there are significant variations in the substitution preferences, among the helix associated PBs and those associated with the strands. The PBs associated with the central region of helix and its immediate C-terminus, i.e., PBs m and n are found to group closely. Similar relationship is observed in case of strand associated PBs d, e and f.

  Figure4Agives the ratio of the percentage of accessible PBs to buried. PB d found at the central strand regions, has the highest tendency to get buried (Figures 4A&B). The helix associated PBs has a higher preference for solvent exposure than that of the strand associated PBs. The PBs associated with the C-terminus of helices (n, o and p), have a greater tendency to get exposed when compared to the N-cap. On the other hand, both the N and C caps of strands have similar preferences for exposure. The loop associated PBs has variable preferences, with g and i being more accessible than h and j.

  distribution of domain structures in different SCOP classes is based on the secondary structure content and topology. As a result, the background distribution of PBs also varies between the SCOP classes. For instance, the all-α class has very low percentage of strand associated PBs while all-β has a low percentage of helix associated PBs (Supplementary data 6). The PB substitution scores observed in the different SCOP classes were compared to the scores observed in the global distribution. The PB substitution patterns show variations across different SCOP classes. Clustering PBs based on the substitution patterns reflect different behaviours in each structural class.

  Both the substitutions are usually associated with changes in β-turn type II, II' and type IV conformations. The substitutions that are preferred in the all-β class occur in the region of strandstrand transitions (Figures 7C&D).These substitutions can be grouped into the following categories. (i) Those which involve transition between central helix conformation (PB m) and those frequently associated with strands (PBs d and e). This change is usually characterized by changes in short helical regions found in this class. (ii) Those usually associated with beta turns. This includes PB changes (b,g), (c,i), (l,n) and (o,l) in the regions which are mainly characterized by hairpin beta turns. . (iii) Those associated with transitions between central helix and C-terminal PBs. The substitutions (o,m) and (p,m) belong to this category.

69 .

 69 2% of the PB m series occurring in this class are of length 3 or lesser. They are often seen in the region of transition between beta strands. Preferred substitutions with the PBs seen in the N-cap of strands (a & c), usually occur in such regions. Other structural elements associated with preferred local structural differences in the all-β class, are the β-hairpins. This local fold has a very high frequency of occurrence in the all-β class. It is interesting to see that the type IV β-turns are the predominant ones with class specific conformational changes.

LegendsFigure 1 .Figure 2 .

 12 Figure 1. PBs series of φ,ψ backbone dihedral angles. For each PB the series of 8 dihedral angles (ψ i-

Figure 3 .

 3 Figure 3. PB substitutions. (A) The variation in substitution score in the PB substitution matrix is highlighted using a colour-code, as shown. (B) The series of dihedral angles (ψ i-2 , φ i-1 ,ψ i-1 , φ i ,ψ i , φ i+1 ,ψ i+1 , φ i+2 ), associated with the PB substitutions (p, g) and (p,i) and (C) (b,i). These represent some of the preferred local conformational changes (D) Hierarchical clustering of PBs based on the similarity of dihedral angles, measured in terms of angular rmsd. The PBs frequently associated with helices are in red, those found often with beta strands are in blue and the rest are in green (E) Clustering of PBs based on the substitution pattern associated with each PB (see Methods).

Figure 4 .

 4 Figure 4. Clustering PBs based on substitution patterns. (A) Distribution of accessible and buried PBs classified based on different accessibility cut-offs of 7%,15% and 25%. Ratio of frequency of exposed PBs to that of buried, plotted for each of the 16 PBs (B) Hierarchical clustering of PBs classified as exposed (B) and buried (C) at an accessibility cut-off of 15%. The clustering is based on the correlation of substitution scores.

Figure 5 .

 5 Figure 5. PB relationship in each SCOP class derived based on the substitution pattern. (A-D) Hierarchical clustering of PBs based on substitution patterns specific for each SCOP class. The clusters correspond to relationships observed in all-α (A), all-β (B), α/β (C) and α+β (D) classes.

Figure 6 .

 6 Figure 6. Comparison of class-specific PB substitution scores with the global distribution (global substitution matrix). The differences in the PB substitution scores specific for the all-α (A), all-β (B), α/β (C) and α+β (D) classes, with respect to the global matrix, are plotted. The correlation coefficients obtained by performing row-wise comparisons (class-specific PB substitution patterns vs Global) are also indicated adjacent to the difference matrices.

Figure 7 .

 7 Figure 7. PB substitutions highly preferred in certain SCOP classes. The cases where the classspecific substitution scores associated with each PB (each row in the substitution matrix) has a correlation less than 0.95 when compared to the global matrix, were looked into. The absolute differences (class specific vs Global) of substitution patterns (respective rows) were plotted as a boxplot, to identify outliers. Substitution scores lying outside a 1.5 inter-quartile range (IQR), were considered as outliers or significantly different from the global substitutions. For the all-α class, (A) the plots are generated for PBs a and g. (B) highlights examples of backbone conformations corresponding to substitutions detected as outliers. Similarly, boxplots were generated for the all-β class (C) and the examples of significantly different substitutions are shown (D).

Figure 8 .

 8 Figure 8. Preferred local structure for indel events. The di-PBs that bind the site of insertions are shown in the context of secondary structure definition. Parts of four domain structures (A-D) are used to highlight the indel sites.

Figure 9 .

 9 Figure 9. Percentage gain in alignments with better rmsd. Alignment obtained by using class specific PB substitution matrices were compared with that of the global matrix. The percentage of alignments in the dataset with better rmsd is plotted. The performance of each class specific SM in each class is highlighted using different colours.

Table 1 .

 1 Association of PB with secondary structures. The percentage of different secondary structures (assigned by PROMOTIF) found associated with each PB is given. Only the secondary structures with percentage occurrence greater than 0.5% are given. The PBs are listed in the beginning of each row and the secondary structure type is given as header for each column. 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 (φ=-79.0±40,ψ=69.0±40).

		H	G	E	BTI BTII BTIV	BTVIII	BTI' BTII' C	GTINV	AG AC
	a			25.4		14.4	17.0	2.2	1.8	29.5 1.5	4.0
	b			18.1 13.2		14.6	8.7	1.2	35.8 2.3	2.0
	c		0.7	58.3 6.1		6.2	1.9	21.2 2.2
	d			80.4			0.8		14.4 1.2
	e			62.5		12.5	11.3		10.3
	f			38.0 11.6		10.3	3.6	31.2 2.3
	g	6.2	12.8 13.8 17.1 10.1	16.9	3.6	16.4 1.7
	h		1.6	27.2		24.4	31.7		2.1	9.8
	i			7.2		35.1	38.6		15.0
	j	8.6	2..9	10.0 3.2	3.8	22.9		9.1	32.5 1.7
	k	37.1 11.1		23.5 2.3	18.0		5.5
	l	49.1 13.0		13.5 2.3	14.0	1.9	4.3
	m 90.4 2.6		2.5		1.7		2.3
	n	66.3 6.4		6.7		10.3		7.1
	o	20.6 5.0		15.5 5.2	20.0		1.5	29.4
	p	8.3	10.8 1.4	16.7 3.1	14.7	0.8	0.9	38.5 1.2

-67.91, 121.55; 77.85, 10.42

Table 2 .

 2 Preferred indel sites in different SCOP classes.

, φ i-1 ,ψ i-1 , φ i ,ψ i , φ i+1 ,ψ i+1 , φ i+2 ), numbered from 1 to 8, are plotted. i indicates the position of an amino acid in the protein.

Acknowledgments

These works were supported by grants from the French Ministry of Research, University Paris Diderot, Sorbonne Paris Cité, 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 number 3903-E. AdB and NS also acknowledge to CEFIPRA for collaborative grant (number 3903-E).


F Laudarin

C Desceliers

G Bonnet
email: guy.bonnet@univ-paris-est.fr

P Argoul

A non-parametric probabilistic model for soil-structure interaction

Keywords: soil-structure interaction, uncertainties, internal forces, seismic loads

The paper investigates the effect of soil-structure interaction on the dynamic response of structures. A non-parametric probabilistic formulation for the modelling of an uncertain soil impedance is used to account for the usual lack of information on soil properties. Such a probabilistic model introduces the physical coupling stemming from the soil heterogeneity around the foundation. Considering this effect, even a symmetrical building displays a torsional motion when submitted to earthquake loading. The study focuses on a multi-story building modeled by using equivalent Timoshenko beam models which have different mass distributions. The probability density functions of the maximal internal forces and moments in a given building are estimated by Monte Carlo simulations. Some results on the stochastic modal analysis of the structure are also given.

The soil-structure interaction is often taken into account under the assumption of a rigidly moving foundation. Then, the relation between the degrees of freedom of the foundation and the forces and moments applied by the soil on the foundation are then modeled by a soil impedance matrix that can be computed by integration of the three-dimensional equations of elasticity within a half-space soil domain or estimated by simplified methods. In [START_REF] Gazetas | Foundation vibrations (chapter 15)[END_REF], for instance, an easy-to-use engineering model is built to obtain an estimation of the impedance from a compilation of complete computations. It involves simple analytical equations for evaluating the impedance matrix, taking into account the shape of the foundation and the nature of the soil. The ground around and below the foundation is however neither homogeneous nor isotropic and few informations are usually available about the real soil properties. This usual lack of information on dynamic soil properties induces uncertainties on the soil-structure interaction. Such uncertainties have to be taken into account in order to improve the ability of the numerical model for prediction. Obviously, the definition of the seismic loadings include also many uncertainties. In this paper, we restrict the study to uncertainties coming from the foundation soils.

Among the different methods that are available in the literature, the probabilistic approach is the most suitable for modeling these uncertainties. A parametric probabilistic approach would consist in modeling the uncertain parameters of soil impedance model by random variables or random fields. Such an approach has been proposed by [START_REF] Toubalem | Vertical excitation of stochastic soilstructure interaction systems[END_REF] for a Winkler [START_REF] Winkler | Die Lehre von der elasticitaet und festigkeit[END_REF] foundation model. It has been shown that a coupling between translation and torsion of the foundation could then occur even if the foundation has symmetry geometrical properties. Another example of parametric probabilistic formulation of the uncertainties on the soil interaction parameters can be found in [START_REF] Fenton | Three-Dimensional Probabilistic Foundation Settlement[END_REF] where the uncertain mechanical properties of the soil are modeled by random fields. Nevertheless, it is obvious that a parametric probabilistic approach cannot be used to model the uncertainties that are not related to parameters of the model. This is the case if mechanical simplifications are assumed for the construction of the soil impedance model. These simplifications are usually due to a high complexity level of the real physical soil impedance or it can also be due to a lack of experimental measurements on the properties of the soil and of the foundation. Nevertheless, a non-parametric probabilistic formulation allows to quantify these uncertainties. It consists in modeling the uncertain soil impedance by a random matrix for which the probability distribution is explicitly constructed by using the information theory [START_REF] Jaynes | Information theory and statistical mechanics[END_REF][START_REF] Jaynes | Information theory and statistical mechanics[END_REF][START_REF] Shannon | A mathematical theory of communication Bell System[END_REF][START_REF] Kapur | Entropy optimization principles with applications[END_REF]. In such an approach, the parameters of the soil impedance are not modeled by random variables or random fields anymore. Such a construction has been introduced in [START_REF] Soize | A nonparametric model of random uncertainties for reduced matrix models in structural dynamics[END_REF][START_REF] Soize | Maximum entropy approach for modeling random uncertainties in transient elastodynamics[END_REF] for the generalized mass, stiffness and damping matrices of a mechanical dynamical system and extended in [START_REF] Desceliers | Mixed nonparametric-parametric probabilistic model for earthquake reliability of an inelastic reinforced concrete frame structure[END_REF]for modelling uncertainties in the case of nonlinear dynnamics of building submitted to seismic loads. In this paper, we extend this non-parametric probabilistic formulation to the uncertainties related to the soil impedance matrix modeling, along the lines introduced by [START_REF] Cottereau | Construction of a probabilistic model for impedance matrices[END_REF]. The information theory is used to construct the probabilistic model of the random soil impedance matrix, but contrarily to [START_REF] Cottereau | Construction of a probabilistic model for impedance matrices[END_REF], a simplified version of the impedance is used, which is nearer to the usual engineering practice.

In section 2, an engineering model for the soil structure interaction is presented. Then, two simplified mechanical models of multi-story buildings with a rigid foundation are considered. These models involve Timoshenko beams. A widely used beam model in engineering is a "lumped mass model" (referred to as model 1 in the paper) made up of masses that are located at the top ends of weightless Timoshenko beam elements. A more sophisticated lumped mass model allowing torsional effects to be taken into account can be found in [START_REF] Alame | Dynamic models for elastic coupled lateral and torsional response of asymmetric multistorey buildings[END_REF]. In this paper, we present a new Timoshenko beam model with a homogeneous mass distribution (referred to as model 2 in the paper). Then, a non-parametric probabilistic model of the uncertain soil impedance is constructed. The random dynamic model for the soil structure interaction is then constructed by replacing the soil impedance matrix by a random matrix for which the probability density function is constructed by using the maximum entropy principle [START_REF] Jaynes | Information theory and statistical mechanics[END_REF][START_REF] Jaynes | Information theory and statistical mechanics[END_REF][START_REF] Shannon | A mathematical theory of communication Bell System[END_REF][START_REF] Kapur | Entropy optimization principles with applications[END_REF] with the available information (that is related with the algebraic properties of the impedance matrix: its mean value defined by the engineering model and algebraic conditions for the existence of a second order random solution of the random model). Then, some computational results are presented for a multi-story building. The probability density functions of the maximal internal forces and moments inside a building are estimated by the use of the Monte Carlo method [START_REF] Hammersley | Monte Carlo methods Chapman and Hall[END_REF].

2 Deterministic model of the structure and of the soil-structure interaction 2.1 Impedance of the soil-structure interaction In this section, the deterministic model (which will be considered hereafter as the mean model in the probability sense) used for modeling a multi-story building with a rigid foundation and with soil-structure interaction under seismic load is presented. Hereafter, the foundation is considered as a rigid body. Consequently, the impedance in the frequency domain is a complex symmetric 6×6 matrix [Z(ω)] = [K S ]+iω [D S ] where [K S ] and [D S ] are positive-definite symmetric matrices that characterize the stiffness and the damping of the soil interaction. From a general point of view, these matrices are frequency dependent [START_REF] Gazetas | Foundation vibrations (chapter 15)[END_REF]. However, in the following, we use the approximation that these matrices are frequency independent. Such an approximation is indeed very often used by engineers.

Mass matrix of the foundation and added mass matrix of the soil-structure interaction

The inertia of the foundation is defined by a positive-definite symmetric 6 × 6 matrix [M f ] which is block diagonal. The first 3 × 3 block is the diagonal matrix m f [I 3 ] where m f is the mass of the foundation and [I 3 ] is the identity matrix of 4 3 . The last 3 × 3 block is the inertia matrix of the foundation with respect to the axes (Oe 1 ), (Oe 2 ) and (Oe 3 ). In addition, let the positivedefinite symmetric 6 × 6 matrix [M a ] be the added mass matrix corresponding to the mass of the soil activated by the foundation due to soil-structure interaction. We then introduce the positive-definite symmetric 6 × 6 matrix [M S ] which is written as

[M S ] = [M f ] + [M a ].

Model 1 for the structure: lumped mass beam model

The occurrence of an earthquake is modeled by introducing a far free-field ground motion. This free-field motion is characterized by an accelerogram a g (t) = (a 1 (t), a 2 (t), a 3 (t)) in which a 1 (t), a 2 (t) and a 3 (t) are the acceleration components of the free-field ground motion along axes (Oe 1 ), (Oe 2 ) and (Oe 3 ). The displacement of the foundation is then described with respect to a noninertial frame that moves with the free-field soil acceleration.

It can be considered that the constitutive equations of a multi-story building are asymptotically equivalent to the constitutive equations of a Timoshenko beam or a flexure beam [START_REF] Clough | Dynamics of structure[END_REF][START_REF] Boutin | Homogenisation of periodic discrete medium: Application to dynamics of framed structures[END_REF]. Consequently, Timoshenko beam models are widely used in order to compute the dynamical response of multi-story buildings.

A commonly used model is the lumped mass beam model (see Figure 1) for which the ceilings are connected by weightless Timoshenko beams and each ceiling is represented as a rigid body. Let u 0 (t) ∈ 4 3 and θ 0 (t) ∈ 4 3 be the displacement vector and the rotation vector of the center of the foundation. The displacement vector of the node located at the center of the n-th ceiling is denoted by u n (t) = (u n,1 (t), u n,2 (t), u n,3 (t)) and the rotation vector of the n-th ceiling is denoted by θ n (t) = (θ n,1 (t), θ n,2 (t), θ n,3 (t)). The building (with its foundation) is therefore modeled by N sto + 1 rigid bodies (N sto ceilings and one foundation) and the number of degrees of freedom of the beam model is N DOF = 6 (N sto + 1). Then, the effective stiffness parameters of the beam element are obtained from the properties of constitutive elements.

The reduced lumped mass beam model is built by using the modal representation; it means that the displacement along the structure K(t) = (u 0 (t), θ 0 (t), . . . , u Nsto (t), θ Nsto (t)) ∈ 4 NDOF is given by:

K(t) ≃ N α=1 q α (t) Φ α , (1)
where q α (t) are the generalized coordinates of K(t) within the basis of eigenvectors Φ α = (φ 0,α , . . . , φ Nsto,α ) of the building with its foundation coupled with the soil. More details on the construction of this reduced lumped mass beam model can be found in Appendix A. Let the vector q(t) = (q 1 (t), . . . , q N (t)) be the vector of the generalized coordinates. Then, it can be shown that

([M ] + [M S ]) q(t) + [D S ] q(t) + ([K ] + [K S ]) q(t) = f (t) + f S (t) (2)
with the initial conditions q(0) = 0 and q(0) = 0

where the positive-definite symmetric

N × N real matrices [M ], [M S ], [D S ], [K S ]
, the positive symmetric N × N real matrix [K ] and the vectors of the generalized external forces f (t) and f S (t) are explicitly given in Appendix A.

2.4 Model 2 for the structure: Timoshenko beam model with a homogeneous mass distribution

Hereinafter, an alternative Timoshenko beam model is presented. The building is modeled by only one equivalent Timoshenko beam with a homogeneously distributed mass. The foundation is connected to the lower part of the beam. Since the inertia of the Timoshenko beam is taken into account, the Timoshenko beam dynamic equations are used. The kinematics is then described by the displacement field

u G (x 1 , t) = (u 1 (x 1 , t), u 2 (x 1 , t), u 3 (x 1 , t))
of the center G(x 1 ) of the cross section located at position x 1 and by the rotation of the cross section θ(x

1 , t) = (θ 1 (x 1 , t), θ 2 (x 1 , t), θ 3 (x 1 , t))
. The beam section displacement field u is a rigid displacement field that can be written as

u(x 1 , x 2 , x 3 , t) = u G (x 1 , t)+θ(x 1 , t)×(0, x 2 , x 3 ).
The boundary value problem of the beam model is then written as

ρS ü1 -S E u ′′ 1 = -ρ S a 1 (4) ρS ü2 -Gk 2 S(u ′′ 2 -θ ′ 3 ) = -ρ S a 2 (5)
ρS ü3 -G k 3 S (u ′′ 3 + θ ′ 2 ) = -ρ S a 3 (6)
ρI 1 θ1 -G I 1 θ ′′ 1 = 0 ( 7
)
ρI 2 θ2 + G k 3 S (u ′ 3 + θ 2 ) -E I 2 θ ′′ 2 = 0 ( 8
)
ρI 3 θ3 -G k 2 S (u ′ 2 -θ 3 ) -E I 3 θ ′′ 3 = 0 (9)
These equations involve the inertia parameters and the stiffness parameters of the equivalent beam. The mass per unit length is ρ S and the mass moment of inertia per unit length with respect to the axes of R g are ρ I 1 , ρ I 2 and ρ I 3 . The axial stiffness is SE, the shearing stiffnesses are G k 2 S and G k 3 S, the torsion stiffness is G I 1 and the bending stiffnesses are E I 2 and E I 3 . The internal force f int (x 1 , t) and the internal moment m int (x 1 , t) are written as

f int (x 1 , t) =   S E u ′ 1 G k 2 S (u ′ 2 -θ 3 ) G k 3 S (u ′ 3 + θ 2 )   , m int (x 1 , t) =   G I 1 θ ′ 1 E I 2 θ ′ 2 E I 3 θ ′ 3   ( 10
)
Since the top of the building (x 1 = h) is free, we then have the boundary conditions

f int (h, t) = 0 and m int (h, t) = 0 (11)
The lower part of the building is connected with the rigid foundation which is coupled with the soil. We then have

[M S ] ü0 (t) + [D S ] u0 (t) + [K S ] u 0 (t) = f 0 (t) -[M S ] =(t) (12)
where u 0 (t) = (u G (0, t), θ(0, t)) and f 0 (t) = (f int (0, t), m int (0, t)) are the vector of the degrees of freedom and the vector of the internal forces and moments of the beam at x 1 = 0 and where =(t) = (a 1 (t), a 2 (t), a 3 (t), 0, 0, 0). Moreover, at time t = 0, the structure is at rest; so we have the initial conditions

K(x 1 , 0) = 0 and K(x 1 , 0) = 0 for all x 1 ∈ [0, h] . (13)
The reduced Timoshenko beam model with a homogeneous mass distribution is built with the approximation on

K = (u 1 , u 2 , u 3 , θ 1 , θ 2 , θ 3 ) K(x 1 , t) ≃ N α=1 q α (t) Φ α (x 1 ) , (14)
where q α (t) are the generalized coordinates of K(x 1 , t) within the basis of eigenfunctions Φ α for the structure with its foundation coupled with the soil. More details on the construction of this reduced Timoshenko beam model with a homogeneous mass distribution can be found in Appendix B. Let q(t) = (q 1 (t), . . . , q N (t)) be the vector of the generalized coordinates. Then, it can be shown that

([M ] + [M S ]) q(t) + [D S ] q(t) + ([K ] + [K S ]) q(t) = f (t) + f S (t) (15)
with the initial conditions q(0) = 0 and q(0) = 0 [START_REF] Boutin | Homogenisation of periodic discrete medium: Application to dynamics of framed structures[END_REF] where It may be noticed that both reduced models defined by Eqs. ( 2) and ( 15) have the same form, allowing to use a systematic method for building the probabilistic model in the following section.

3 Probabilistic model for the soil-structure interaction and the response of the building

Stochastic soil-structure interaction modeling

The non-parametric probabilistic model (see for instance [START_REF] Soize | A nonparametric model of random uncertainties for reduced matrix models in structural dynamics[END_REF] and [START_REF] Soize | Maximum entropy approach for modeling random uncertainties in transient elastodynamics[END_REF]

p [MS ] ([A]) = 1 M + ([A]) c M (det[A]) λ M -1 exp{- n S -1 + 2 λ M 2 tr([M S ] -1 [A])} , (17)
p [D S ] ([A]) = 1 D + ([A]) c D (det[A]) λ D -1 exp{- n S -1 + 2 λ D 2 tr([D S ] -1 [A])} , (18)
p [KS ] ([A]) = 1 K + ([A]) c K (det[A]) λ K -1 exp{- n S -1 + 2 λ K 2 tr([K S ] -1 [A])} , (19)
where

1 M + ([A]) is equal to 1 if [A] belongs
M S ] βα = Φ β (0) T [M S ] Φ α (0), [D S ] βα = Φ β (0) T [D S ] Φ α (0) and [K S ] βα = Φ β (0) T [K S ] Φ α (0) (for model 2).
Similarly, the components of random vector F S (t) are random variables {F S (t)} β = -Φ 0,β T [M S ] a(t) (for model 1) and

{F S (t)} β = -Φ β (0) T [M S ] a(t) (for model 2).
Consequently, the solution of such a stochastic system of differential equations is a random vector Q(t) such that, [START_REF] Le | Apport de l'analyse continue pour la méthode impact echo. Champs physiques et propagation dans les sols et les structures du génie civil[END_REF] with the initial conditions

([M ] + [M S ]) Q(t) + [D S ] Q(t) + ([K ] + [K S ]) Q(t) = f (t) + F S (t) ,
Q(0) = 0 and Q(0) = 0 . ( 21
)
The random displacement of the structure is given by:

U = N α=1 Q α Φ α , (22)
where Q 1 , . . . , Q N are the random components of the random solution Q and where Φ 1 , . . . , Φ N are the N (deterministic) eigenvectors (for model 1) and eigenfunctions (for model 2) introduced in the previous sections.

Construction of the stochastic response by using the Monte Carlo simulations

The Monte Carlo simulations are used in order to calculate some statistical quantities related to the probabilistic response of the uncertain dynamical system. The method consists in constructing a given number

([M ] + [M S (a)]) Q(t; a) + [D S (a)] Q(t; a) + ([K ] + [K S (a)]) Q(t; a) = f (t) + F S (t; a) , (23)
with the initial conditions

Q(0; a) = 0 and Q(0; a) = 0 , (24)
[M S (a)] βα = Φ β (0) T [M S (a)] Φ α (0) , [D S (a)] βα = Φ β (0) T [D S (a)] Φ α (0) , [K S (a)] βα = Φ β (0) T [K S (a)] Φ α (0) .
Similarly, the random components of vector F S (t; a) are

{F S (t; a)} β = -Φ 0,β T [M S (a)] a(t) , for model 1 , {F S (t; a)} β = -Φ β (0) T [M S (a)] a(t) , for model 2 .
Then samples 7(t; a) (for model 1) or 7(t, x 1 ; a) (for model 2) are given by:

7(a) = N α=1 Q α (a) Φ α , (25)
where Q 1 (a), . . . , Q N (a) are the components of sample vector Q(a). The maximal internal moments and forces over the whole structure and during the seismic load for each model are random variables denoted by

V 1 , V 2 , V 3 and M 1 , M 2 , M 3 . Sample values V 1 (a), V 2 (a), V 3 (a) and M 1 (a), M 2 (a), M 3 (a)
are calculated for each sample 7(a) of the displacement field. Finally, statistical quantities related to the probabilistic response of the uncertain dynamical system are calculated by using usual tools of the statistic with a large enough number N S of samples 7(a 1 ), . . . , 7(a NS ), V 1 (a 1 ), . . . , V 1 (a NS ), V 2 (a 1 ), . . . , V 2 (a NS ), V 3 (a 1 ), . . . , V 3 (a NS ) and M 1 (a 1 ), . . . , M 1 (a NS ), M 2 (a 1 ), . . . , M 2 (a NS ) and M 3 (a 1 ), . . . , M 3 (a NS ).

Numerical results

In this section, the stochastic mechanical beam model is applied to the case of a standard multi-story building.

Description of the building

The building under study is a 8-stories building. Figure 2 shows a schema of the building and its foundation. Every story is 30 m long (e 2 direction), 14 m wide (e 3 direction) and has a height H sto = 2.7 m so that the total height of the building is h = N sto × H sto = 21.6 m. All stories are identical and therefore the stiffness beam parameters are computed by using a finite element computation of a story where the walls, the floor and the ceiling are meshed with Reissner-Mindlin plate finite elements. They are assumed to be made up of an isotropic material for which Young's modulus is 32 GP a, the shearing modulus is 13 GP a and the mass density is 2500 kg.m -3 . For the sake of simplicity the anisotropy induced by the reinforced concrete elements has not been taken into account.

The seismic acceleration due to the earthquake is decribed by using a signal record from the Parkfield earthquake which occurred on September 28, 2004 in the Parkfield area close to Vineyard Canyon in California with a momentmagnitude of 6.0. The 3 components of the seismic accelerogram have been measured in Donna Lee station [17] at 14.5 km from the epicenter. For more information about the 2004 Parkfield event, the reader should refer to [START_REF] Borcherdt R D | Recordings of the 2004 Parkfield Earthquake on the General Earthquake Observation System Array: Implications for the Earthquake Precursors. Fault Rupture, and Coseismic Strain Changes[END_REF]. The P-wave and S-wave velocity were measured in the area by Boore [START_REF] Boore | A compendium of P-and S-wave velocities from surface-to-borehole logging: summary and reanalysis of previously published data and analysis of unpublished Data[END_REF] using layered soil models. The data are available at http://quake.usgs.gov/∼boore/. From the S-wave velocity and P-wave velocity measured in Vineyard Canyon area a shearing modulus of 162 M P a and a Young's modulus of 440 M P a were obtained for the soil properties . Figure 3-a shows the acceleration signal, the Fourier transform modulus and the ridges of the wavelet transform for the transverse component e 2 of the seismic acceleration signal t → a g (t). Ridges are curves in the time frequency plane localizing the concentration of energy. The norm of the complex wavelet transform of the studied signal is maximal on these curves (see [START_REF] Le | Apport de l'analyse continue pour la méthode impact echo. Champs physiques et propagation dans les sols et les structures du génie civil[END_REF] for more details on that technique) . These curves are plotted in gray scale so that the darker the ridges are, the higher is the energy. The process favors the high frequencies components which can be obtained from short durations, which explains that the most energy ridges appear at high frequencies. The first part of the signal ([0, 4]s) contains low frequency energetic components in the horizontal directions which can also be observed on the elastic response spectra (figure 3-b). During the next time period ( [START_REF] Fenton | Three-Dimensional Probabilistic Foundation Settlement[END_REF][START_REF] Cottereau | Construction of a probabilistic model for impedance matrices[END_REF]s) there is a contribution in the range [START_REF] Winkler | Die Lehre von der elasticitaet und festigkeit[END_REF][START_REF] Hammersley | Monte Carlo methods Chapman and Hall[END_REF]Hz. The highest frequencies contribution in the range [START_REF] Hammersley | Monte Carlo methods Chapman and Hall[END_REF][START_REF] Le | Apport de l'analyse continue pour la méthode impact echo. Champs physiques et propagation dans les sols et les structures du génie civil[END_REF]Hz are shorter and are fluctuating, consequently they are never predominant across time.

A global response of the structure can be studied by introducing the dynamic magnification factor [START_REF] Clough | Dynamics of structure[END_REF] H(ω) such that, for all ω in 4,

H(ω) = 1 N ([K ] + [K S ]) [A(ω)] -1 ) F , (26) where [
A(ω)] = -ω 2 ([M ] + [M S ]) -i ω [D S ] + ([K ] + [K S ]
) is the reduced dynamic stiffness of the structure with its foundation coupled with the soil and where • F is the Froebenius norm of matrices. Figure 4 displays the function ω → H(ω). It can be seen from this figure that the mode whose eigenfrequency is 1.9 Hz for the homogeneous mass distribution model, is directly excited at the beginning of the earthquake. In the following stage, two other modes are excited. The wavelet transform presents eventually no relevant energetic frequency contributions after 15s.

Results for the stochastic model

To illustrate the method, the sensitivity of the maximal internal forces and moments with respect to the uncertainties is now studied. As introduced in subsection 3.3, V 1 , V 2 , V 3 and M 1 , M 2 , M 3 are the random maximal values of the internal forces and moments over the whole structure during the application of the seismic loading. They are obtained for both stochastic reduced beam models presented in section 3.2. The Monte Carlo numerical method [START_REF] Hammersley | Monte Carlo methods Chapman and Hall[END_REF] is used in order to calculate the mean values, standard deviations and the probability density functions of these random variables in order to quantify their uncertainties. The values of the dispersion coefficients which have been used are : δ M = 0.2, δ D = 0.5, δ K = 0.5. The values of dispersion coefficients can be identified in solving a statistical inverse problem if experimental measurements are available (see, for instance [START_REF] Desceliers | Determination of the random anisotropic elasticity layer using transient wave propagation in a fluid-solid multilayer: Model and experiments[END_REF]). However, in the case of seismic studies, such a procedure cannot be uses. These coefficients are estimated by the experience of the engineer, due to dispersion on soil data for example. Hereinafter, a sufficiently large amount of samples in the Monte Carlo method have been constructed in order to reach convergence on the statistic estimations [START_REF] Serfling | Approximation theorems of Mathematical Statics[END_REF] of the mean, standard deviation and probability density functions. As an example, the convergence of the mean value with respect to the number of realizations is shown in Figure 5. The stochastic convergence is reached by taking all the eigenvectors up to 500Hz, i.e N = 108.

The graphs of the estimations of probability density functions of V 1 , V 2 , V 3 , M 1 , M 2 and M 3 are shown in figure 6 for which N S ≥ 10000 samples were used to reach convergence. Some internal forces or moments present an important dispersion; the random moment M 3 presents, for instance, a standard deviation of 25% with respect to its mean value. The largest dispersion is given by the random torsional moment M 1 which is completely due to random modeling and could not be computed with a deterministic approach or a parametric modeling of the impedance which would not take into account a spatial variability of the soil properties under the foundation. A substantial effect of the random soil structure interaction is the global rise of the mean internal forces and moments compared to the deterministic values. The effect is stronger with the lumped mass model where the random internal force V 3 and the corresponding random bending moment M 2 respectively increase by 17% and 45%. The beam model with homogeneous mass distribution presents increases of 4% and 18% for the same force and moment.

Conclusions

Soil-structure interaction can modify the dynamic response of a building under earthquakes in the low frequency range. A non-parametric modeling was used to introduce model uncertainties in an easy-to-use engineering model of soil-structure interaction. The model leads to a significant torsion within the building, even if the structure and its foundation are symmetrical. All stochastic computations are performed by using a Monte Carlo simulation which requires solving many times the dynamic equations of the structure . The simplified beam model used for the building mechanical modeling and the simple SSI model allow to perform a sufficient number of such computations . It has been observed that the random soil-structure interaction can produce an increase of internal forces compared to the deterministic results. Then a weak formulation that allows to explicit the relation between this set of matrices is written as: Find K(t) = (u 0 (t), θ 0 (t), . . . , u Nsto (t), θ Nsto (t)) belonging in 4 N DOF such that for all L = (v 0 , Θ 0 , . . . ,

A Reduced lumped mass beam model

v N sto , Θ N sto ) ∈ 4 N DOF , L T [M ] K(t) + v T 0 [M S ] ü0 (t) + v T 0 [D S ] u0 (t) + L T [K] K(t) + v T 0 [K S ] u 0 (t) = -L T [M ] =(t) -v T 0 [M S ] ä0 (t) . (A.1)
Let ω 1 ≤ . . . ≤ ω N be the N first eigenfrequencies which are solutions of the generalized eigenvalue problem: find Φ = (φ 0 , . . . , φ Nsto ) ∈ 4 N DOF and ω > 0 such that, for all L ∈ 4 N DOF ,

L T [K] Φ + v T 0 [K S ] φ 0 -ω 2 L T [M ] Φ -ω 2 v T 0 [M S ] φ 0 = 0 . (A.2)
Let Φ k = (φ 0,k , . . . , φ N sto ,k ) ∈ 4 N DOF be the k-th eigenvector associated with k-th small- est eigenfrequency ω k . Eigenvectors are such that Φ

β [M ] Φα + φ T 0,β [M S ] φ 0,α = δ αβ and Φ β [K] Φα + φ T 0,β [K S ] φ 0,α = ω 2 α δ αβ where δ αβ = 1 if α = β and δ αβ = 0 if α = β. The reduced model is built with the approximation K(t) ≃ N α=1 qα(t) Φα , (A.3)
where qα(t) are the generalized coordinates of K(t) within the basis of eigenvectors Φα. Let the vector q(t) = (q 1 (t), . . . , q N (t)) be the vector of the generalized coordinates. Then, it can be shown that

([M ] + [M S ]) q(t) + [D S ] q(t) + ([K ] + [K S ]) q(t) = f (t) + f S (t) (A.4)
with the initial conditions q(0) = 0 and q(0) = 0 (A.

K = (u 1 , u 2 , u 3 , θ 1 , θ 2 , θ 3 ) ∈ C such that for all L ∈ C, m(K, L) + L(0) T [M S ] K(0, t) + L(0) T [D S ] K(0, t) + k(K, L) + L(0) T [K S ] K(0, t) = f (t, L) -L(0) T [M S ] =(t) (B.1)
where the positive-definite symmetric bilinear form m(•, •), the positive symmetric bilinear form k(•, •) and the linear form f (•) are defined on C × C and C such that where qα(t) are the generalized coordinates of K(x 1 , t) within the basis of eigenfunctions Φα.

m(K, L) = ρ S h 0 u 1 (x 1 ) v 1 (x 1 ) dx 1 + ρ S h 0 u 2 (x 1 ) v 2 (x 1 ) dx 1 + ρ S h 0 u 3 (x 1 ) v 3 (x 1 ) dx 1 + ρ I 1 h 0 u 4 (x 1 ) v 4 (x 1 ) dx 1 + ρ I 2 h 0 u 5 (x 1 ) v 5 (x 1 ) dx 1 + ρ I 3 h 0 u 6 (x 1 ) v 6 (x 1 ) dx 1 (B.2) k(K, L) = G k 2 S h 0 u ′ 2 (x 1 ) v ′ 2 (x 1 ) dx 1 + G k 3 S h 0 u ′ 3 (x 1 ) v ′ 3 (x 1 ) dx 1 + E I 2 h 0 u ′ 5 (x 1 ) v ′ 5 (x 1 ) dx 1 + E I 3 h 0 u ′ 6 (x 1 ) v ′ 6 (x 1 ) dx 1 + G k 2 S h 0 u 6 (x 1 ) v 6 (x 1 ) dx 1 + G k 3 S h 0 u 5 (x 1 ) v 5 (x 1 ) dx 1 + G k 3 S h 0 u 5 (x 1 ) v ′ 3 (x 1 ) dx 1 + G k 3 S h 0 u ′ 3 (x 1 ) v 5 (x 1 ) dx 1 -G k 2 S h 0 u 6 (x 1 ) v ′ 2 (x 1 ) dx 1 -G k 2 S h 0 u ′ 2 (x 1 ) v 6 (x 1 ) dx 1 + S E h 0 u ′ 1 (x 1 ) v ′ 1 (x 1 ) dx 1 + G I 1 h 0 u ′ 4 (x 1 ) v ′ 4 (x 1 ) dx 1 (B.
Let the vector q(t) = (q 1 (t), . . . , q N (t)) be the vector of the generalized coordinates. Then, it can be shown that

([M ] + [M S ]) q(t) + [D S ] q(t) + ([K ] + [K S ]) q(t) = f (t) + f S (t) (B.7)
with the initial conditions q(0) = 0 and q(0) = 0 (B.

  the positive-definite symmetric N × N real matrices [M ], [M S ], [D S ], [K S ], the positive symmetric N × N real matrix [K ] and the vectors of the generalized external forces f (t) and f S (t) are explicitly given in Appendix B.

  ) consists in substituting the deterministic matrices [M S ], [D S ] and [K S ] by random matrices [M S ], [D S ] and [K S ]. The probabilistic model of these random matrices is constructed by using the maximum entropy principle with the available information: the mean values of these random matrices, the algebraic properties of the mass, stiffness and damping random matrices (symmetry, definiteness, etc.), and conditions for the solution to have a finite second order moment. It is shown in the previously quoted papers that the probability density functions p [MS ] , p [DS ] and p [KS ] of those random matrices with respect to the measure d à = 2 nS (nS -1)/4 1≤i≤j≤n [A] ij (with n S = 6) on the set M + of the symmetric positive n S × n S real matrices are then written as

  to M + and is equal to zero if [A] does not belong to M + ; tr(•) is the trace operator; c M , c D and c K are normalization constants; λ M , λ D , λ K are positive real parameters that depend on the statistical fluctuation of random matrices [M S ], [D S ] and [K S ]. Appendix C briefly describes how samples [M S (a)], [D S (a)] and [K S (a)] with probability density functions p [M S ] , p [D S ] and p [K S ] can be numerically computed in practice.3.2 Stochastic reduced model of the building under seismic loadsThe uncertainties on the soil-structure interaction are taken into account by substituting into both reduced models defined by Eqs. (2) or (15), the matrices [M S ], [D S ], [K S ] and vector f S (t) by random matrices [M S ], [D S ], [K S ] and random vector F S (t). The random matrices are obtained by projection of random matrices [M S ], [D S ] and [K S ] (defined in previous section) onto the deterministic modes. Thus [M S ], [D S ] and [K S ] are random definite-positive symmetric N × N matrices whose components are [M S ] βα = Φ T 0,β [M S ] Φ 0,α , [D S ] βα = Φ T 0,β [D S ] Φ 0,α and [K S ] βα = Φ T 0,β [K S ] Φ 0,α (for model 1) and [

For model 1 ,

 1 components of samples [M S (a)], [D S (a)] and [K S (a)] are [M S (a)] βα = Φ T 0,β [M S (a)] Φ 0,α , [D S (a)] βα = Φ T 0,β [D S (a)] Φ 0,α , [K S (a)] βα = Φ T 0,β [K S (a)] Φ 0,α . For model 2, components of samples [M S (a)], [D S (a)] and [K S (a)] are

  In order to exhibit the contribution of matrices [M S ], [D S ] and [K S ] to the lumped mass beam model (model 1), we first introduce matrices [M ] and [K] which are the mass matrix and the stiffness matrix of a lumped mass beam model for which the boundary conditions are the following: (1) the top of the building is free and (2) the foundation is weightless and free. These boundary conditions imply that matrices [M ] and [K] are (not definite) positive symmetric N DOF × N DOF real matrices.

5 )

 5 where the positive-definite symmetric N × N real matrices [M ], [M S ], [D S ], [K S ] and the positive symmetricN × N real matrix [K ] are such that [M ] βα = Φ β [M ] Φα, [M S ] βα = Φ T 0,β [M S ] Φ 0,α , [D S ] βα = Φ T 0,β [D S ] Φ 0,α and [K ] βα = Φ β [K] Φα, [K S ] βα = Φ T 0,β [K S ] Φ 0,α . It should be noted that we also have [M ] βα + [M S ] βα = δ αβ and [K ] βα + [K S ] βα = ω 2 α δ αβ . The components of vectors of the generalized forces f (t) and f S (t) are {f (t)} β = -Φ T β [M ] =(t) and {f S (t)} β = -Φ 0,β T [M S ] =(t). B Reduced Timoshenko beam model A weak formulation for an explicit relation between matrices [M S ], [D S ] and [K S ] and the bilinear forms of the Timoshenko beam model with a homogeneous mass distribution (model 2) is presented hereinafter. Let C be a space of sufficiently regular functions. The weak formulation of the problem is written as: Find

3 )f 0 L(x 1 ) 1 (B. 4 )

 30114 (t, L) = -ρ S h T =(t) dx where for the seek of simplicity, u k , k = 4, 5, 6 are used instead of θ 1 , θ 2 , θ 3 and similarly v k , k = 4, 5, 6 are used for the rotations associated to v k . Let ω 1 ≤ . . . ≤ ω N be the N first eigenfrequencies which are solutions of the generalized eigenvalue problem: find Φ ∈ C and ω > 0 such that, for all L ∈ C, k(Φ, L) + L(0) T [K S ] Φ(0) -ω 2 m(Φ, L) -ω 2 L(0) T [M S ] Φ(0) = 0 (B.5) Let {Φ 1 , . . . , Φ N } be the N eigenfunctions associated with eigenfrequencies ω 1 ≤ . . . ≤ ω N . Eigenfunctions are such that m(Φα, Φ β ) + Φ β (0) T [M S ] Φα(0) = δ αβ and k(Φα, Φ β ) + Φ β (0) T [K S ] Φα(0) = ω 2 α δ αβ where δ αβ = 1 if α = β and δ αβ = 0 if α = β. The reduced model is built with the approximation K(x 1 , t) ≃ N α=1 qα(t) Φα(x 1 ) , (B.6)

8 )Fig. 1 Fig. 2

 812 Fig. 1 Lumped mass model of the building and its foundation

Fig. 4 Fig. 5 Fig. 6

 456 Fig. 4 Graph of the dynamic magnification factor function for the beam models with homogeneous mass (solid line) and the lumped mass model (dashed line). Horizontal axis ω in Hz. Vertical axis: dynamic magnification factor H(ω)

Acknowledgements The authors acknowledge Safia Bedaoui and Tien Minh Nguyen for their help in the use of the wavelet transform software developed at LAMI(ENPC/LCPC), France.

where the positive-definite symmetric N × N real matrices [M ], [M S ], [D S ], [K S ] and the positive symmetric N × N real matrix [K ] are such that

It should be noted that we also have

C Probabilistic model of the random matrices

In this appendix, we present the construction of the probabilistic model of the random matrices involved by the nonparametric probabilistic formulation of the model uncertainties. It has been shown in [START_REF] Soize | A nonparametric model of random uncertainties for reduced matrix models in structural dynamics[END_REF][START_REF] Soize | Maximum entropy approach for modeling random uncertainties in transient elastodynamics[END_REF] that the probabilistic model of random matrices [M S ], [D S ] and [K S ] can be constructed using the information theory with the available information (see section 6.2) yielding

) is equal to 1 if [G] belongs to M + (the set of all the n S × n S symmetric positive matrices) and is equal to zero if [G] does not belong to M + , tr[G] is the trace of matrix [G], a(δ) = (n S +1)/(2δ 2 ), b(δ) = a(δ)(1-δ 2 ), c(δ) = (2π) -n S (S-1)/4 a(δ) n S a(δ) / n+1 j=1 Γ (α j (δ)) in which α j (δ) = a(δ) + (1 -j)/2 and where Γ is the Gamma function.

Then, it has then shown in [9, ?]

where

where V M,j , V D,j and V K,j are positivevalued Gamma random variables whose probability density function p V M,j , p V D,j , p V K,j are written as

v α j (δ K ) e -v Γ (α j (δ K ) .