Rabie Saidi
email: saidi@isima.fr

Mondher Maddouri
email: mondher.maddouri@fst.rnu.tn

Engelbert Mephu Nguifo

Protein sequences classification by means of feature extraction with substitution matrices

Abstract Background

This paper deals with the preprocessing of protein sequences for supervised classification. Motif extraction is one way to address that task. It has been largely used to encode biological sequences into feature vectors to enable using well-known machine-learning classifiers which require this format. However, designing a suitable feature space, for a set of proteins, is not a trivial task. For this purpose, we propose a novel encoding method that uses amino-acid substitution matrices to define similarity between motifs during the extraction step.

Results

In order to demonstrate the efficiency of such approach, we compare several encoding methods using some machine learning classifiers. The experimental results showed that our encoding method outperforms other ones in terms of classification accuracy and number of generated attributes. We also compared the classifiers in term of accuracy. Results indicated that SVM generally outperforms the other classifiers with any encoding method. We showed that SVM, coupled with our encoding method, can be an efficient protein classification system. In addition, we studied the effect of the substitution matrices variation on the quality of our method and hence on the classification quality. We noticed that our method enables good classification accuracies with all the substitution matrices and that the variances of the obtained accuracies using various substitution matrices are slight. However, the number of generated features varies from a substitution matrix to another. Furthermore, the use of already published datasets allowed us to carry out a comparison with several related works. The outcomes of this comparison confirm the efficiency of our encoding method to represent protein sequences in classification tasks.

Background

Analysis and interpretation of biological sequence data is a fundamental task in bioinformatics. Classification and prediction techniques are one way to deal with such task [START_REF] Bhaskar | Machine learning in bioinformatics: A brief survey and recommendations for practitioners[END_REF]. In fact, biologists are often interested in identifying the family to which a lately sequenced protein belongs [START_REF] Bernardes | Structural descriptor database: a new tool for sequence based functional site prediction[END_REF]. This makes it possible to study the evolution of this protein and to discover its biological functions. Furthermore, the study and the prediction of oligomeric proteins (quaternary structures) are very useful in biology and medicine for many reasons [START_REF] Klotz | Human cardiac myosin ATPase and light subunits: A comparative study[END_REF]. Indeed, they often intervene in terms of biomacromolecules functional evolution, reparation of misfolds and defects [START_REF]Price NC: Assembly of multi-subunit structure[END_REF][START_REF] Corrales | Kinetic significance of GroEL 14 •(GroES 7 ) 2 complexes in molecular chaperone activity[END_REF]. They are also involved in many important biological processes such as chromosome replication, signal transduction, folding pathway and metabolism [START_REF] Terry | Determination of protein-protein interactions by matrix-assisted laser desorption/ionization mass spectrometry[END_REF]. Biologists also seek, for instance, to identify active sites in proteins and enzymes [START_REF] Slama | Detection of protein catalytic residues at high precision using local network properties[END_REF], to classify parts of DNA sequences into coding or non-coding zones or to determine the function of the nucleic sequences such as the identification of the promoter sites and the junction sites [START_REF] Nguifo | Prediction of Primate Splice Junction Gene Sequences with a Cooperative Knowledge Acquisition System[END_REF][START_REF] Lemoine | Improving the Efficiency of a User-Driven Learning System with Reconfigurable Hardware. Application to DNA Splicing[END_REF][START_REF] Vignal | Exon prediction in eucaryotic genomes[END_REF].

Alignment is the main technique used by biologists to look for homology among sequences, and hence to classify new sequences into already known families/classes. Since relevant information is represented by strings of characters, this technique generally doesn't enable the use of well-known classification techniques such as decision trees (DT), naïve bayes (NB), support vector machines (SVM) and nearest neighbour (NN) which have proved to be very efficient in real data mining tasks [START_REF] Han | Data Mining: Concepts and Techniques[END_REF].

In fact, those classifiers rely on data described in a relational format.

Meanwhile, different studies have been devoted to motif extraction in biological sequences [START_REF] Huan | Feature Extraction, Construction and Selection: A Data Mining Perspective[END_REF][START_REF] Nevill-Manning | Highly specific protein sequence motifs for genome analysis[END_REF][START_REF] Maddouri | Encoding of primary structures of biological macromolecules within a data mining perspective[END_REF][START_REF] Yongqiang | EXMOTIF: efficient structured motif extraction[END_REF][START_REF] Yongqiang | SMOTIF: efficient structured pattern and profile motif search[END_REF][START_REF] Lopes | Feature selection environment for genomic applications[END_REF]. Motifs extraction methods are generally based on the assumption that the significant regions are better preserved during the evolution because of their importance in terms of structure and/or function of the molecule [START_REF] Nevill-Manning | Highly specific protein sequence motifs for genome analysis[END_REF],

and thus that they appear more frequently than it is expected.

In [START_REF] Maddouri | Encoding of primary structures of biological macromolecules within a data mining perspective[END_REF], authors have shown that motif extraction methods can efficiently contribute to the use of machine learning algorithms for the classification of biological sequences.

In this case, the classification obeys the knowledge discovery in data (KDD) process and hence comprises three major steps:

1. Preprocessing consists of extracting motifs from a set of sequences. These motifs will be used as attributes/features to construct a binary table where each row corresponds to sequence. The presence or the absence of an attribute in a sequence is respectively denoted by 1 or 0. This binary table is called a learning context. It represents the result of the preprocessing step and the new sequence encoding format (figure 1).

2. In the mining step, a classifier is applied to the learning context to generate a classification model.

3. The latter model is used to classify other sequences in the post-processing step. These sequences are also encoded into a relational format using the same features as for the learning context i.e., test context.

In a previous work [START_REF] Saidi | Biological Sequences Encoding for Supervised Classification[END_REF], we proposed a new method to encode protein sequences. It extends an existing method, termed Discriminative Descriptors (DD) [START_REF] Maddouri | Encoding of primary structures of biological macromolecules within a data mining perspective[END_REF], by taking into account the fact that some amino acids have similar properties and thus can be substituted by each other while changing neither the structure nor the function of the protein [START_REF] Henikoff | Amino acid substitution matrices from protein blocks[END_REF]. Hence, there might be several motifs that could be replaced by a single motif. We used amino acids substitution matrices to define such similarity; our encoding method is termed Discriminative Descriptors with Substitution Matrix (DDSM). Preliminary experiments conducted with C4.5 decision tree have shown promising results [START_REF] Saidi | Biological Sequences Encoding for Supervised Classification[END_REF]. This manuscript presents a detailed experimental comparison (in terms of classification accuracy and number of attributes) between several encoding methods using various kinds of classifiers (C4.5 decision tree, NB, SVM and NN) as well as the standard approach based on alignment using Blast [START_REF] Altschul | J: Basic local alignment search tool[END_REF].

Methods

Some existing feature construction methods

The following is a presentation of five existing methods of features construction: the N-Grams (NG), the Active Motifs (AM), the Amino Acid Composition (AAC), the Functional Domain Composition (FDC) and the Discriminative Descriptors (DD).

After this, we re-describe our approach which consists of modifying the DD method by the use of a substitution matrix (DDSM) [START_REF] Saidi | Biological Sequences Encoding for Supervised Classification[END_REF].

N-Grams

The simplest approach is that of the N-Grams, known also as N-Words or length N fenestration [START_REF] Leslie | The spectrum kernel: a string kernel for svm protein classification[END_REF]. The motifs to be built have a predefined length. The N-gram is a subsequence composed of N characters, extracted from a larger sequence. For a given sequence, the set of the N-grams which can be generated is obtained by sliding a window of N characters on the whole sequence. This movement is carried out character by character. With each movement a subsequence of N characters is extracted. This process is repeated for all the analyzed sequences. Then, only the distinct N-grams are kept.

Active Motifs

This method allows extracting the commonly occurring motifs whose lengths are longer than a specified length, called Active Motifs, in a set of biological sequences.

The activity of a motif is the number of matching sequences given an allowed number of mutations [START_REF] Wang | Discovering active motifs in sets of related protein sequences and using them for classification[END_REF]. The motif extraction is based on the construction of a Generalized Suffix Tree (GST) which is an extension of the suffix tree [START_REF] Hui | Combinatorial Pattern Matching[END_REF] and is dedicated to represent a set of n sequences indexed each one by i = 1..n.

Amino Acid Composition

According to the classic definition of this method, the feature set consists of 20 components, representing the 20 native amino acids in proteins. The amino acid composition refers to the occurrence frequency of each of these 20 components in a given protein. Since the information in the primary sequence is greatly reduced by considering the amino acid composition alone, other considerations have been taken into account within several studies such as the sequence-order correlation factors i.e., new features were added to the 20 original which yielded several AAC variants [START_REF] Zhang | Predicting protein structural classes from amino acid composition: Application of fuzzy clustering[END_REF][START_REF] Chou | Predicting protein quaternary structure by pseudo amino acid composition[END_REF][START_REF] Zhang | Classification of protein quaternary structure with support vector machine[END_REF]27,[START_REF] Zhou | An intriguing controversy over protein structural class prediction[END_REF].

Functional Domain Composition

Biological databases, such as PFAM [START_REF] Finn | The Pfam protein families database Nucleic Acids Research[END_REF] and ASTRAL, contain large collections of multiple sequence alignments and Hidden Markov Model (HMM) profiles covering many common protein domains and families [START_REF] Finn | The Pfam protein families database Nucleic Acids Research[END_REF]. Functional domains are determined using computational means, especially HMM profiles, combined with biologist knowledge and other databases information. Since they allow variable length gaps between several components, where each component is a simple motif [START_REF] Yongqiang | EXMOTIF: efficient structured motif extraction[END_REF][START_REF] Yongqiang | SMOTIF: efficient structured pattern and profile motif search[END_REF], functional domains can be considered as structured motifs. But they are more reliable since they obey the expert assessment.

Descriminative Descriptors

Given a set of n sequences, assigned to P families/classes F 1 , F 2 .., F P , this method consists of building substrings called Discriminative Descriptors DD which allow to discriminate a family F i from other families F j , with i = 1..P and i ≠ j [START_REF] Maddouri | Encoding of primary structures of biological macromolecules within a data mining perspective[END_REF]. This method is based on an adaptation of the Karp, Miller and Rosenberg (KMR) algorithm [START_REF] Karp | Rapid Identification of Repeated Patterns in Strings, Trees and Arrays[END_REF]. This algorithm identifies the repeats in character strings, trees or tables. The extracted repeats are then filtered in order to keep only the discriminative and minimal ones.

A substring X is considered to be discriminative between the family F i and the other families F j , with i = 1..P, j = 1..P and i ≠ j if : where α and β are user-specified thresholds between 0 and 1.

1. α ≥ i i F of

Proposed method: Discriminative Descriptors with Substitution Matrix

In the case of protein, the Discriminative Descriptors method neglects the fact that some amino acids have similar properties and that they can be therefore substituted by each other while changing neither the structure nor the function of the protein [START_REF] Henikoff | Amino acid substitution matrices from protein blocks[END_REF]. Indeed, we can find several motifs in the set of the attributes generated by the DD method, which are similar and can derive all from a single motif. In the same way, during the construction of the context (binary table), we are likely to lose information when we denote by 0 the absence of a motif while another one, that can replace it, already exists [START_REF] Saidi | Biological Sequences Encoding for Supervised Classification[END_REF].

As mentioned, the similarity between motifs is based on the similarity between the amino acids which constitute them. Indeed, there are various degrees of similarity between amino acids. Since there are 20 amino acids, the mutations between them are scored by a 20x20 matrix called a substitution matrix [START_REF] Henikoff | Amino acid substitution matrices from protein blocks[END_REF][START_REF] Leslie | The spectrum kernel: a string kernel for svm protein classification[END_REF][START_REF] Malde | The effect of sequence quality on sequence alignment[END_REF].

Terminology

Let M be a set of n motifs, denoted each one by M[p], p = 1.. n. M can be divided into m clusters. Each cluster contains a main motif M* and probably other motifs which can be substituted by M*. The main motif is the one which has the highest probability of mutating to another in its cluster. For a motif M of k amino acids, this probability, noted P m (M), is based on the probability P i (i = 1.. k) that each amino acid M[i] of the motif M does not mutate to any other amino acid. We have:

P m = 1 -∏ = k i 1
P i P i is calculated based on the substitution matrix according to the following formula: x,y) is the substitution score of the amino acid y by the amino acid x as it appears in the substitution matrix. S + (x, y) indicates a positive substitution score. AA j is the amino acid of index j among the 20 amino acids.

P i = S(M[i], M[i]) / ∑ = 20 1 j S + (M[i], AA j ) S(
For our purposes, a motif M substitutes a motif M' if:

1. M and M' have the same length k,

S(M[i], M'

[i]) >= 0 , i = 1.. k,
S m (X, Y) = ∑ = k i 1

S(X[i], Y[i])

It is clear, according to any substitution matrix, that there is only one best motif which can substitute a motif M. It is obviously itself, since the amino acids which constitute it are better substituted by themselves. This proves that the substitution probability of a motif by another one, if they satisfy the substitution conditions, will be between 0 and 1.

Methodology

The encoding method is composed to two parts. First, the number of extracted motifs will obviously be reduced because we will keep only one motif for each cluster of substitutable motifs of the same length. Second, we will modify the context construction rule. Indeed, we will denote by 1 the presence of a motif or of one of its substitutes. The first part can be also divided into two phases: (1) identifying clusters' main motifs and (2) filtering. [START_REF] Bhaskar | Machine learning in bioinformatics: A brief survey and recommendations for practitioners[END_REF] The main motif of a cluster is the one that is the most likely to mutate to another in its cluster. To identify all the main motifs, we sort M in a descending order by motif lengths, and then by P m . For each motif M' of M, we look for the motif M which can substitute M' and that has the highest P m (probability of mutation to another motif). The clustering is based on the computing of the substitution probability between motifs. We can find a motif which belongs to more than one cluster. In this case, it must be the main motif of one of them. (2) The filtering consists of keeping only the main motifs and removing all the other substitutable ones. The result is a smaller set of motifs which can represent the same information as the initial set.

Example

Given a Blosum62 substitution matrix and the following set of motifs (table 1) sorted by their lengths and P m , we assign each motif to a cluster represented by its main motif. We get 5 clusters illustrated by the diagram shown in figure 2.

Experimental environment

NG, AM, DD and DDSM encoding methods are implemented in C language and gathered into a DLL library. The accepted format of the input files is the FASTA format for biological sequences files. The library code that we have implemented generates relational files under various formats such as the ARFF format used by the workbench WEKA [START_REF] Witten | Data Mining: Practical machine learning tools and techniques[END_REF] and the DAT format used by the system DisClass [START_REF] Maddouri | Encoding of primary structures of biological macromolecules within a data mining perspective[END_REF].

Our experiments are divided into 2 parts. In the first one, we make a detailed comparison between NG, AM, DD and DDSM encoding methods to confirm the results obtained in [START_REF] Saidi | Biological Sequences Encoding for Supervised Classification[END_REF]. We perform the sequence classification using DT, SVM, NB and NN algorithms as described in section 1. We also conducted classification experiments using Blast [START_REF] Altschul | J: Basic local alignment search tool[END_REF] coupled with the nearest neighbour algorithm i.e., we assign to a protein query the class with the best hit score. Our method (DDSM) constructs the features using the substitution matrix Blosum62. The choice of this substitution matrix is not based on preliminary experiments, but instead on the fact that it is the most used by alignment tools especially the widespread Blast. We examine three aspects:

1. The effect of each encoding method on the four classifiers to deduce which one is the best in terms of accuracy and number of generated attributes.

2. The comparison of the four classifiers while varying the encoding methods.

The comparison with Blast results.

In the second part, we try to assess the effect of varying the substitution matrices on our method and on the classification quality and hence to determine whether there is a substitution matrix which could be recommended. Then we compare our featureconstruction method with other ones presented in [27, [START_REF] Zhou | An intriguing controversy over protein structural class prediction[END_REF][START_REF] Yu | Classification of protein quaternary structure by functional domain composition[END_REF], which means that we compare with nine related works [27, [START_REF] Zhou | An intriguing controversy over protein structural class prediction[END_REF][START_REF] Yu | Classification of protein quaternary structure by functional domain composition[END_REF][START_REF] Chou | Prediction of protein structural classes from amino acid composition[END_REF]35,[START_REF] Cai | Prediction of protein structural classes by neural network[END_REF]37,[START_REF] Feng | Boosting classifier for predicting protein domain structural class[END_REF][START_REF] Cao | Prediction of protein structural class with Rough Sets[END_REF].

Part 1

To perform our experiments, we use 5 datasets comprising 1604 protein sequences from Swiss-Prot [START_REF] Bairoch | The SWISS-PROT protein sequence database and its supplement TrEMBL in 2000[END_REF] and SCOP [START_REF] Andreeva | SCOP database in 2004: refinements integrate structure and sequence family data[END_REF] described in table 2.

We try to conduct our experiments on various kinds of datasets. These datasets differ from one another in terms of size, number of class, class distribution, complexity and sequence identity percentage. The first dataset DS1 contains 3 distinct and distant protein families. We suppose that classification in this case will be relatively easy since each family will probably have preserved patterns which are different from those of other families [START_REF] Nevill-Manning | Highly specific protein sequence motifs for genome analysis[END_REF]. DS2 represents a bigger dataset comprising two subfamilies of protein sequences belonging to the Rhodopsin Like/Peptide family.

However, the datasets DS3 and DS4 present more difficult classification problems.

DS3 contains seven classes that represent seven categories of quaternary (4D) protein structure with a sequence identity of 25%. The problem here lies in recognizing the 4D structure category from the primary structure. In this case, an important question is to be answered: does the primary structure contain sufficient information to identify the 4D structure? The task relative to DS4 is that of distinguishing between the human Toll-like Receptors (TLR) protein sequences and the non-human ones. The difficulty is due to the structural and functional similarity of the two groups. The choice of this dataset came after Biologists of Pasteur Institute of Tunis (PIT) asked to help them in identifying TLR families especially human ones among the 40 TLR that exist. DS5 consists of 277 domains: 70 all-α domains, 61 all-β domains, 81 α/β domains, and 65 α+β domains from SCOP [START_REF] Andreeva | SCOP database in 2004: refinements integrate structure and sequence family data[END_REF]. This challenging dataset was constructed by Zhou [START_REF] Zhou | An intriguing controversy over protein structural class prediction[END_REF] and has been extensively used to address structural class prediction [27, [START_REF] Zhou | An intriguing controversy over protein structural class prediction[END_REF][START_REF] Chou | Prediction of protein structural classes from amino acid composition[END_REF]35,[START_REF] Cai | Prediction of protein structural classes by neural network[END_REF]37,[START_REF] Feng | Boosting classifier for predicting protein domain structural class[END_REF][START_REF] Cao | Prediction of protein structural class with Rough Sets[END_REF].

Part 2

In this part, we consider again the datasets DS3, DS4 and DS5 since they are considered to be delicate classification tasks and can thus reveal valuable information about the efficiency of the classifiers and the feature-construction methods. We try to investigate the effect of the substitution matrices variation on the quality of our encoding method and hence on the classification quality using C4.5, SVM, NB and NN algorithms. We employ all the substitution matrices used by the standalone version of Blast and belonging to the two well-known families: Blosum [START_REF] Henikoff | Amino acid substitution matrices from protein blocks[END_REF] and Pam [START_REF] Dayhoff | A Model of Evolutionary Change in Proteins[END_REF] i.e., Blosum45, Blosum62 Blosum80, Pam30, Pam70, Pam 250.

Since DS3 is the same dataset as in [START_REF] Yu | Classification of protein quaternary structure by functional domain composition[END_REF], these experiments allow us to compare our encoding method with other related ones presented in that paper, where the nearest neighbour algorithm NN was coupled with each of the following methods: functional domain composition FDC, amino acid composition AAC and Blast alignment tool [START_REF] Altschul | J: Basic local alignment search tool[END_REF], to predict the quaternary structures categories of the proteins. In fact, the investigation of the quaternary structures prediction using computational tools remains a task with important implications for many reasons. First, these structures are involved in many biological processes and have direct link with known diseases like sickle-cell anaemia. Second, the in vitro methods are very slow and costly in spite of being accurate. This comparison allows us to assess whether our featureconstruction method could offer any benefits over the above-mentioned methods quoted in [START_REF] Yu | Classification of protein quaternary structure by functional domain composition[END_REF] while using the same classifier (NN) and learning technique (leaveone-out).

Since prior information on the structure of a protein can provide useful information about its function, many other works similar to [START_REF] Yu | Classification of protein quaternary structure by functional domain composition[END_REF] have investigated this topic [27, 28,34,35,36,37,38,39,43,44,45,46,]. These works often use kinds of amino acid composition or functional domain composition to deal with the prediction of oligomeric proteins or protein structural classes. DS5 represents a challenging dataset that has been extensively used to address structural class prediction [27]. This allows us to compare our method with several works existing in the literature. explained in section 2.3. In part 2, we investigate any potential effect of the substitution matrix variance on the features building and the classification quality, and then we compare it with other classification systems quoted in [27, [START_REF] Zhou | An intriguing controversy over protein structural class prediction[END_REF][START_REF] Yu | Classification of protein quaternary structure by functional domain composition[END_REF].

Part 1 Results

The experimental results vary according to the input data (table 3 andtable 4). The classification of the datasets DS1 and DS2 was relatively easy, as expected. Each family probably has its own motifs which characterize it and distinguish it from the others. This explains the high accuracies reached by all the classifiers with all the encoding methods. But it is notable that the N-Grams encoding gave the best results although it is the simplest method to use. Moreover, since this kind of classification, is easy, it does not require any sophisticated preprocessing and can simply be addressed by using alignment tools; indeed Blast arrived at full accuracy (table 4).

As for DS3, classification represents a real challenge. In fact, it is comprised of 717 sequences unequally distributed into seven classes which represent seven quaternary protein structure categories. It is a question of predicting the 4D structure based only on the primary structure without any complementary information. The AM method could not be used because it generates a great number of attributes (dashes in table 3).

The obtained accuracies with the NG and the DD methods were below the MAV (within 20.9% and 43.2 %) and the result obtained by Blast was acceptable (69.60 %) while the best accuracy reached (79.2%) was obtained with the DDSM method (figure 3 illustrates a sample of ROC curves [START_REF] Zweig | Receiver-operating characteristic (ROC) plots: a fundamental evaluation tool in clinical medicine[END_REF] of the NB classifier based on the DDSM, DD and NG encoding methods with Homotetramer as the positive class from DS3).

The dataset DS4 was not as easy to classify as DS1 and DS2 since the human TLR and the non-human TLR resemble each other in terms of function and structure.

Indeed the two classes share many similar parts, making it difficult to discriminate them. That is why homology based classification (using Blast) didn't reach full accuracy as it did for the two first datasets. The NG and the AM encoding seems to be inefficient since they gave accuracies below the MAV with two classifiers. The DD method outperforms the two previous methods (NG and AM). Since it adopts a discriminating approach to build the attributes, it allowed a better distinction between the human TLR and the non-human TLR. But, to improve classification in the dataset DS4, it is necessary to take into account the phenomenon of mutation and substitution between the amino acids which constitute the protein sequences. Indeed, the DDSM method made it possible to reach the highest precisions with all the classifiers, while reducing the number of generated attributes. Experimental results obtained with DS5 show a good performance for all the encoding methods, though no full accuracy was reached. We can notice that NG performed very well and allowed to improve results with the classifiers C4.5, SVM and NN. Blast allowed also to obtain good accuracy which is due to the high identity percentage within the dataset. But, the best accuracy was obtained with DDSM (≈ 86%).

Moreover, we can notice that SVM generally provided the best accuracies with all the encoding methods, though it is known as a slow classifier. So, we can conclude that the combination (DDSM, SVM) could be an efficient system for the protein sequences classification.

Part 2 results

In this section, we study the effect of the substitution matrices (SM) variation on the classification by applying some of the most often used SMs belonging to the two well-known families: Blosum and Pam [START_REF] Henikoff | Amino acid substitution matrices from protein blocks[END_REF][START_REF] Dayhoff | A Model of Evolutionary Change in Proteins[END_REF]. These SMs are the same used by the standalone version of Blast [START_REF] Altschul | J: Basic local alignment search tool[END_REF].

alignments. Yet it differs from Pam to Blosum. Whereas the Pam matrices have been developed from global alignments of closely related proteins, the Blosum matrices are based on local multiple alignments of more distantly related sequences. This would have an effect on the representation size. Indeed, the number of constructed features varies from a substitution matrix to another. Blosum matrices with low numbers and Pam matrices with higher numbers allow the building of fewer features since they score highly the substitution between amino acids. This would yield larger clusters of substitutable motifs, and hence fewer main motifs i.e., fewer features (see section 2.2.2 and 2.2.3).

However, the variances of accuracies are slight when varying the substitution matrices with the same classifier (table 5, table 6 and table 7). Moreover, no substitution matrix allows obtaining the best accuracy for all the classifiers. We can even notice contradicting results; indeed, in DS3 and DS4, NN algorithm performs worse when coupled with Pam30, while the same matrix allows SVM to reach its best accuracy.

The same phenomenon is noticed in DS5 with the classifiers C4.5 and SVM and the matrix Pam250. If one looks for reduced-size representation, Blosum matrices with low numbers and Pam matrices with higher numbers are recommended.

Since we used the same dataset (DS3) and the same assessment technique (leave-oneout) as in [START_REF] Yu | Classification of protein quaternary structure by functional domain composition[END_REF], we compare our feature-building method (DDSM with default parameter values: α = 0, β = 0, substitution matrix = Blosum62, substitution probability threshold T = 0.9) with the ones studied in [START_REF] Yu | Classification of protein quaternary structure by functional domain composition[END_REF] (FDC, AAC, and Blast coupled each one with the nearest neighbor algorithm NN). Comparative results are reported in table 8. We can notice that the worst results were obtained with the AAC method. Indeed, the obtained results were below the MAV 46.7%. Blast arrived at better results, but the accuracy was not very high. In fact, an analysis of the Protein Data Bank (PDB) [START_REF] Berman | The Protein Data Bank[END_REF], where the protein structures are deposited, reveals that proteins with more than 30% pairwise sequence identity have similar 3D structures [START_REF] Sander | Database of homology derived protein structure and the structural meaning of sequence alignment[END_REF]. But in our case we process a dataset with a sequence identity of 25%. The FDC method seems to be promising since it allowed reaching an accuracy of 75.2 %. But our method was quite better and enabled to reach the highest accuracy rates among the mentioned methods and also coupled with the same classifier i.e., NN algorithm (77%).

If we look for better classification systems we can consider the combinations (DDSM & C4.5) or (DDSM & SVM). In addition higher accuracy can be obtained by using the combination (DDSM & SVM) and the matrix Pam30 which enabled to reach an accuracy of 82% (table 8). This indicates that SVM coupled with our encoding method DDSM represent an efficient system for protein classification.

In the same way, the use of the same dataset (DS5) and the same validation technique (leave-one-out) as in [27,[START_REF] Zhou | An intriguing controversy over protein structural class prediction[END_REF] allowed us to compare our method with these two works as well as six others [START_REF] Chou | Prediction of protein structural classes from amino acid composition[END_REF]35,[START_REF] Cai | Prediction of protein structural classes by neural network[END_REF]37,[START_REF] Feng | Boosting classifier for predicting protein domain structural class[END_REF][START_REF] Cao | Prediction of protein structural class with Rough Sets[END_REF]. In these studies, variants of the amino acid composition AAC have been proposed to encode protein sequences and then coupled with a classifier to predict the protein structural classes. These works are based in the assumption that there is a strong correlation between the AAC and the structural class of a protein. In table 9, we report the results obtained by our method (DDSM with default parameter values: α = 0, β = 0, substitution matrix = Blosum62, substitution probability threshold T = 0.9) coupled with C4.5, SVM, NB and NN as well as the results of the related works (in table 9, AAC x means the AAC variant presented in the paper x). We can claim that our encoding method generally outperforms any AAC encoding method proposed by the above-mentioned works. In [27], authors coupled A ROC graph enables to compare two or more supervised learning algorithms. It depicts relative trade-offs between true positive rates and false positive rates [START_REF] Zweig | Receiver-operating characteristic (ROC) plots: a fundamental evaluation tool in clinical medicine[END_REF].

It is possible to derive a synthetic indicator from the ROC curve, known as the AUC (Area Under Curve -Area Under the Curve). The AUC indicates the probability that the classifier will rank a randomly chosen positive instance higher than a randomly chosen negative instance. There exists a threshold value: if we classify the instances at random, the AUC will be equal to 0.5, so a significant AUC must be superior to this threshold.

Tables

  3. SP(M, M') >= T,where T is a user-specified threshold such that 0 <= T <= 1.We denote by SP(M, M') the substitution probability of the motif M' by the motif M having the same length k. It measures the possibility that M mutates to M':SP(M, M') = S m (M, M') / S m (M, M)S m (X, Y) is the substitution score of the motif Y by the motif X. It is computed according to the following formula:

FiguresFigure 1 -

 1 Figures Figure 1 -Sequence pre-processing based on motif extractionThis figure describes the process of sequence encoding. The extracted motifs are used

Figure 2 -

 2 Figure 2 -Motifs clustering This figure illustrates the set of clusters and main motifs obtained from the data of

Figure 3 -

 3 Figure 3 -ROC curve samples for the NB classifier in the dataset DS3 with the DDSM, DD and NG encoding methods. The positive class is Homotetramer. This figure shows a sample of ROC curves of the NB classifier based on the DDSM,

Table 1 -

 1 Motifs clusteringM is a set of motifs (table 1) sorted by their lengths and P m . The third row shows the cluster main motifs.

	M	LLK	IMK	VMK	GGP	RI	RV	RF	RA	PP
	Pm	0.89	0.87	0.86	0	0.75	0.72	0.72	0.5	0
	Main motif LLK	LLK	LLK	GGP	RI	RI	RI	RV	PP

Table 2 -

 2 Experimental data

	Dataset (source)	Identity percentage (%)	Family/class	Size	Total
			High-potential Iron-Sulfur Protein	19
	DS1 (Swiss-prot)	48	Hydrogenase Nickel Incorporation Protein HypA	20	60
			Hlycine Dehydrogenase	21
	DS2 (Swiss-prot)	48	Chemokine Melanocortin	255 255	510
			Monomer	208
			Homodimer	335
			Homotrimer	40
	DS3 (Swiss-prot)	25	Homotetramer	95	717
			Homopentamer	11
			Homohexamer	23
			Homooctamer	5
	DS4 (Swiss-prot)	28	human TLR Non-human TLR	14 26	40
			All-α domain	70
			All-β domain	61
	DS5 (SCOP)	84	α / β domain	81	277
			α + β domain	65

Table 3 -Machine learning classifiers coupled with encoding methods

 3 Mtr: Metric, Clfr: Classifier, CA: Classification Accuracy (%), NA: Number of Attributes.

				Encoding method
	Data	Mtr	Clfr	NG	AM	DD	DDSM
			C4.5	96.7	95	95	96.7
	DS1	CA	SVM NB	96.7 86.7	93.3 90	96.7 81.7	96.7 80
			NN	63.3	78.3	60	61.7
		NA		4935	2060	4905	2565
			C4.5	99.6	99.4	99.8	99.4
		CA	SVM	100	99.4	100	100
	DS2		NB	100	74.7	100	100
			NN	100	100	100	98.8
		NA		6503	7055	10058	1312
			C4.5	36.4	-	36.7	79.2
	DS3	CA	SVM NB	43.2 43.2	--	43.2 43.1	78.94 59.4
			NN	20.9	-	21.3	77
		NA		7983	-	8403	508
			C4.5	60	57.5	77.5	82.5
	DS4	CA	SVM NB	67.5 57.5	65 40	87.5 92.6	87.5 95
			NN	52.5	60	80	80
		NA		5561	3602	7116	5505
			C4.5	75.5	75.1	67.9	73.3
		CA	SVM	84.1	81.2	82.3	82.3
	DS5		NB	77.3	63.7	84.5	85.9
			NN	80.5	79.4	78	78
		NA		6465	2393	13830	13083

Table 4 -Comparison between Blast and DDSM in term of accuracy (%)

 4

	Dataset	Blast-based	(DDSM & SVM)	Best of DDSM (from table 3)
	DS1	100	96.7	96.7
	DS2	100	100	100
	DS3	69.60	78.94	79.2
	DS4	78.57	87.5	95
	DS5	79.4	82.3	85.9

Table 5 -Experimental results per substitution matrix for DS3

 5

	Accuracy (%)

Acknowledgements

This work was partially supported by the French-Tunisian project CMCU-Utique 05G1412. We would like to thank Jean-Jacques Givry, Alia Bouali, Philippe Fournier-Viger, Shannon Behary and Lamia Nammouchi for English proofreading.

We also would like to thank Sabeur Aridhi for technical help.

Results and discussion

Experimental Techniques

The computations are carried out on a computer with an Intel Centrino 1.6 GHz CPU and 1Go of main memory. Results are shown in the next sub-sections tables. Best accuracies, for each dataset, are shown in bold and results below minimum accepted values results are underlined. The minimum accepted value (MAV) is obtained by assigning all the sequences of a dataset to its biggest class. Hence, we have 35%, 50%, 46.7%, 65% and 29.2% as MAVs respectively for DS1, DS2, DS3, DS4 and DS5. We also show the number of attributes generated by each method.

In the classification process, we use the leave-one-out technique [START_REF] Han | Data Mining: Concepts and Techniques[END_REF] also known as jack-knife test. For each dataset (comprising n instances), only one instance is kept for the test and the remaining part is used for the training. This action is repeated n times.

The leave-one-out is considered to be the most objective test technique compared to the other ones i.e., hold-out, n-cross-validation. Indeed the leave-one-out test allows to obtain the same classification results regardless of the number of runs, which is not the case for the other tests (see the monograph [47] for the mathematical principle and [START_REF] Chou | Review: Prediction of protein structural classes[END_REF] for a comprehensive discussion). For the encoding methods, we use default parameters as in [START_REF] Saidi | Biological Sequences Encoding for Supervised Classification[END_REF]: NG (N=3), AM (min-length = 3, activity = 25%), DD and DDSM (α = 0, β = 0 except for DS3 where β = 1 to reduce the runtime), DDSM (substitution matrix = Blosum62, substitution probability threshold T = 0.9). These parameters can also be specified by users.

We recall that in part 1, we use the following classifiers: C4.5 decision tree, support vector machine SVM, naïve bayes NB and nearest neighbour algorithm NN of the workbench WEKA [START_REF] Witten | Data Mining: Practical machine learning tools and techniques[END_REF]. We generate and test the classification models; then we report the classification accuracy (rate of correctly classified sequences). Moreover, we conducted the leave-one-out test on the same datasets using Blast as already three kinds of AAC with SVM i.e., (AAC & SVM), (pair-coupled AAC & SVM) and (PseAAC & SVM). In the best case, they reached an accuracy of 80.5%, whereas the combinations (DDSM & SVM) and (DDSM & NB) allowed reaching respectively 82.3% and 85.9% of accuracy. To enhance their results, authors in [27] proposed a fusion network that combines the results obtained by the three proposed combinations and they arrived at an accuracy of 87.7%. Although, this result is slightly superior to ours, it does not mean that their encoding method outperforms DDSM. Indeed, the improvement of their results comes from the fusion network classifier and not from the AAC variants they use. Moreover, in most of these related works [27, [START_REF] Zhou | An intriguing controversy over protein structural class prediction[END_REF][START_REF] Chou | Prediction of protein structural classes from amino acid composition[END_REF]35,[START_REF] Cai | Prediction of protein structural classes by neural network[END_REF]37,[START_REF] Feng | Boosting classifier for predicting protein domain structural class[END_REF][START_REF] Cao | Prediction of protein structural class with Rough Sets[END_REF], authors perform a fine-tuning to look for the classifier parameter values allowing to get the best results, whereas we just use the default parameter values of both our encoding method and the classifiers as found in WEKA [START_REF] Witten | Data Mining: Practical machine learning tools and techniques[END_REF]. This fine tuning allowed to reach competitive accuracies which is the case of the combination (AAC & LogitBoost) [START_REF] Feng | Boosting classifier for predicting protein domain structural class[END_REF]. We believe that we can also reach higher accuracies if we perform a fine-tuning of the parameters of our method and the classifiers. But, we chose to just use the default parameter values to make it easier for users who may have no prior knowledge on what these parameters mean or how to specify them.

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

RS participated in the design of the DDSM method, implemented the programs, selected data, carried out the experimental study, performed the statistical analysis and drafted and revised the manuscript. MM conceived the DD method, participated in DDSM design, participated in the data selection and revised the manuscript. EMN conceived the DDSM method, provided assistance in the statistical analysis and revised the manuscript. All authors read and approved the final manuscript. [START_REF] Zhou | An intriguing controversy over protein structural class prediction[END_REF] 79.1 219 AAC [START_REF] Chou | Prediction of protein structural classes from amino acid composition[END_REF] & City-block distance [START_REF] Chou | Prediction of protein structural classes from amino acid composition[END_REF] 59.9 166 AAC [35] & Euclidean distance [35] 55.2 153 AAC [START_REF] Cai | Prediction of protein structural classes by neural network[END_REF] & Neural network [START_REF] Cai | Prediction of protein structural classes by neural network[END_REF] 74.7 206 AAC[37] & SVM [37] 79.4 219 AAC [START_REF] Feng | Boosting classifier for predicting protein domain structural class[END_REF] & LogitBoost [START_REF] Feng | Boosting classifier for predicting protein domain structural class[END_REF] 84.1 233 AAC [START_REF] Cao | Prediction of protein structural class with Rough Sets[END_REF] & Rough Sets [START_REF] Cao | Prediction of protein structural class with Rough Sets[END_REF] 79.4 219

Additional files

Additional file 1 -Experimental data, 328 K Experimental_data.zip comprises all datasets with their classification files.

Additional file 2 -Software

Programs are available upon request from the authors or from www.cril.univartois.fr/~mephu/SeqCod Additional files provided with this submission:

Additional file 1: Experimental_data.zip, 327K http://www.biomedcentral.com/imedia/1206381258352080/supp1.zip


M Martini

M Ericson

G Chanfray

J Marteau

Neutrino and antineutrino quasielastic interactions with nuclei

Keywords: numbers: 25.30.Pt, 13.15.+g, 24.10.Cn

We investigate the interaction of neutrinos and antineutrinos with nuclei. We explore in particular the role played by the multinucleon excitations which can contaminate the quasielastic cross section. For neutrinos the multinucleon term produces a sizable increase of the quasielastic cross section. Part of the effect arises from tensor correlations. For antineutrinos this influence is smaller owing to the axial-vector interference which increases the relative importance of the terms which are not affected by these multinucleon excitations.

The progresses of neutrino experimental physics have allowed the measurement of several partial cross sections in the interaction with nuclei. Most data concern ratios of cross sections [1][2][3][4][5][6][7] but some absolute values are now available [START_REF] Katori | AIP Conf. Proc[END_REF][START_REF] Aguilar-Arevalo | [END_REF][10]. In particular the MiniBooNE collaboration has measured the neutrinos charged current (CC) quasielastic (QE) cross section on 12 C for a neutrino beam with average energy of 788 MeV [10]. In this work ejected nucleons are not detected and the quasielastic cross section is defined as the one for processes in which only a muon is detected in the final state. However it is possible that in the neutrino interaction a pion produced via the excitation of the ∆ resonance escapes detection, for instance because it is reabsorbed in the nucleus, leading to multinucleon emission. In this case it simulates a quasielastic process. The MiniBooNE analysis of the data corrects for this possibility via a Monte-Carlo evaluation of this process. The net effect amounts to a reduction of the observed quasielastic cross section. After application of this correction the quasielastic cross section thus defined still displays an anomaly as compared to a relativistic Fermi gas prediction. The prediction which is sensitive to the cut-off mass of the axial form factor fits the data provided a modified axial form factor is introduced in the calculation, with an increase of the axial cut-off mass from the accepted value M A = 1.03 GeV to the value M A = 1.35 GeV; otherwise the calculated cross section is too small [10].

On the theoretical side Martini et al. [11] have drawn the attention to the existence of additional sources of multinucleon emission which are susceptible to produce an apparent increase of the "quasielastic" cross section. Their evaluation, although approximate, of this contribution shows that it is able to account for this apparent increase. It stresses in particular the role played by the NN tensor correlations in this enhancement. In clarifying this point we will be naturally led to explore the corresponding effect in the case of antineutrinos. We will show that antineutrinos can provide a useful test of the origin of this anomaly, which is the aim of the present work.

In our approach, the same as that of Ref. [11], the neutrino cross section on nuclei is expressed in terms of the nuclear response functions treated in the random phase approximation (RPA). The only nucleon resonance taken into account is the ∆ one. Several responses enter this interaction, as exemplified below in a simplified expression of the charged current cross section where the lepton mass is ignored and the ∆ width is taken to zero. It reads :

∂ 2 σ ∂Ω ∂k ′ = G 2 F cos 2 θ c (k ′ ) 2 2 π 2 cos 2 θ 2 G 2 E ( q 2 µ q 2 ) 2 R N N τ +G 2 A (M ∆ -M ) 2 2 q 2 R N ∆ στ (L) + G 2 A (M ∆ -M ) 2 q 2 ×R ∆∆ στ (L) + G 2 M ω 2 q 2 + G 2 A - q 2 µ q 2 + 2 tan 2 θ 2 × R N N στ (T ) +2R N ∆ στ (T ) +R ∆∆ στ (T ) ±2G A G M k+k ′ M × tan 2 θ 2 R N N στ (T ) + 2R N ∆ στ (T ) + R ∆∆ στ (T ) (1)
where G F is the weak coupling constant, θ c the Cabbibo angle, k and k ′ the initial and final lepton momenta, q µ = k µ -k µ′ = (ω, q) the four momentum transferred to the nucleus, θ the scattering angle, M ∆ (M ) the ∆ (nucleon) mass. The plus (minus) sign in Eq. ( 1) stands for the neutrino (antineutrino) case. The existence of this axialvector interference term is crucial in the present work.

The various responses are related to the imaginary part of the corresponding full polarization propagators :

R(ω, q) = - V π Im[Π(ω, q, q)]. (2)
They are related to the inelastic cross section for a given coupling. For instance for the isospin operator :

R τ = n n| A j=1 τ (j)e i q•xj |0 × n| A k=1 τ (k)e i q•x k |0 * δ(ω -E n + E 0 ). ( 3
)
Similar expressions apply to the other transition operators. An analogous expression applies for a transition to a state with a ∆ excitation which enters the spin-isospin responses. In this case the operators are :

O ∆ στ (L) (j) = (S j • q) T ± j , O ∆ στ (T ) (j) = (S j × q) i T ± j . (4)
In the case of a spin operator the index L or T refers to the direction of the spin, longitudinal or transverse, with respect to the momentum q. In the expression (1) the upper indices, N or ∆, refer to the type of particles, nucleon or ∆, excited by the weak current at the two ends of the RPA chain (see Fig. 1). In the present numerical evaluations we employ, instead of Eq. ( 1), the full expressions given in the Appendix A of our previous publication [11] for which we refer for more details.

We have treated these responses in the ring approximation of the RPA so as to account for collective effects. A limitation of our description is that it does not incorporate final state interactions such as for instance the possibility for a real pion produced by the neutrino to be reabsorbed in the nucleus leading to multinucleon ejection. This fraction of the produced pions would thus be counted in the "quasielastic" events. These events have been subtracted in the MiniBooNE analysis through their Monte Carlo evaluation. Hence our theory can be confronted to their corrected experimental results.

To lowest order the quasielastic cross section is given by the terms in R N N , whether coming from the isovector interaction, R N N τ , or from the isospin spin-transverse one, R N N στ (T ) . The isospin spin-longitudinal one is suppressed, for a vanishing lepton mass, by a cancellation between the space and time components of the axial current [11], which is the reason why it is not explicitly written in Eq.( 1). In the actual calculation its small quasielastic contribution is taken into account. It also contribute to the multinucleon ejection term. In the RPA chain R N ∆ and R ∆∆ also contribute to the quasielastic response as illustrated in Fig. 1. The collective effects produce a mild suppression of the quasielastic response due to the repulsive nature of the residual interaction. The effect that we want to discuss here is of a different nature. It concerns the multinucleon (np -nh) ejection in neutrino interactions other than that due to final state interaction. Here several sources of multinucleon emission enter our description and all type the responses R N N , R N ∆ and R ∆∆ contribute. Examples are given in Fig. 2. One of these multinucleon sources arises from the modification of the ∆ width in the nuclear medium (Fig. 2 (d)). This effect was introduced and parametrized by Oset and Salcedo [12] in the case of real pion or photon absorption. We have used their parametrization of the modified width, although the kinematics of neutrino interaction is different since we are in the space-like region, which could be a source of uncertainty. For the other terms not reducible to a modification of a ∆ width of which some examples are shown in Fig. 2 we have used a parametrization of Delorme and Guichon [START_REF] Delorme | Proceedings of 10 e biennale de physique nucléaire[END_REF]. They exploited a calculation by Shimizu and Faessler [START_REF] Shimizu | [END_REF] of the absorptive part of the p-wave pion-nucleus optical potential at threshold, which writes that the absorption mechanism of pions is a two-nucleon one which means that in the final state two nucleons are ejected. In the many-body language this is a two particletwo hole (2p -2h) excitation. In Ref. [START_REF] Shimizu | [END_REF] the absorption is described by three types of terms (see Fig. 2). The first one (Fig. 2 (a) with pion lines replacing the weak current ones as in the next diagrams) arises from the nucleonnucleon correlations, essentially from the tensor correlations. Another one involves a ∆ excitation (Fig. 2 (c)). The third one is an interference between the nucleon correlation and ∆ terms as in Fig. 2 (b). In these graphs the coupling of the pion to the nucleon involves the pion momentum and is of the spin isospin type, σ • q τ , for the nucleon and a similar expression for ∆ excitation. In the optical potential the pion momentum is expressed by the gradient, the remaining part of the optical potential with the parameter ImC 0 then provides the nuclear 2p -2h bare responses to a probe which couples to the spin and isospin of the nucleon. In principle it is a spin longitudinal coupling but as this term represents a short range effect there is no difference in the bare case (i.e. before the RPA chain) between longitudinal and transverse spin couplings. Therefore it applies to the bare magnetic part of the vector current and to both the spin-transverse and spin-longitudinal parts of the axial current. The pion absorption calculation of Ref. [START_REF] Shimizu | [END_REF] is performed for threshold pion, i.e. for a vanishing three-momentum and an energy ω = m π , which does not correspond to the neutrino situation. Delorme et al. [START_REF] Delorme | Proceedings of 10 e biennale de physique nucléaire[END_REF] have then introduced in each absorption graph the corresponding energy dependence to obtain the bare 2p -2h responses to be inserted in the RPA chain needed for neutrino interaction. However they have completely ignored the momentum dependence. They have left apart the graph (d) of Fig. 2 since it corresponds to a modification of the ∆ width which is taken into account separately through the parametrization of Oset et al. [12]. In a first approach we have used the procedure of Delorme et al. as such in order to evaluate the bare 2p-2h components to be inserted in the RPA chain for the evaluation of the neutrino cross section. In our resulting 2p -2h cross section the modification of the ∆ width is not a dominant effect; it adds a small 2p -2h component (plus a 3p -3h one). It turned out that our overall multinucleon contribution, when added to the genuine quasielastic cross section, was able to account for the anomaly without modification of the axial cut-off mass as already reported in Ref. [11].

(4π/2m π ) ∇ • ImC 0 ρ 2 ∇ [15]. It is known (a) (b) (c)
Since our parametrization of the 2p-2h piece from the extrapolation of pion absorption is questionable, as it ignores in particular any momentum dependence, we have also investigated the effect with our second parametrization [11] of the 2p-2h contribution beyond the one which is reducible to a modification of the Delta width. In the second approach we have used a microscopic calculation of Alberico et al. [16] specifically aimed at the evaluation of the 2p -2h contribution to the isospin spin-transverse response, measured in inclusive (e, e ′ ) scattering. Their basic graphs are similar to those of Shimizu and Faessler. In principle this way of evaluation is definitely more satisfactory since the kinematical variables are correctly incorporated but the results of Ref. [16] are available only for a limited set of energy and momenta. We have extended this range to cover the neutrino one through an approximate extrapolation. We refer to [11] for the details. The result was that the distribution in the energy transfer ω of the differential neutrino cross section is largely modified with a more realistic distribution but once integrated in energy the difference is small. The corresponding neutrino and antineutrino total "quasielastic" cross sections are practically unchanged. Now from the way in which this 2p -2h contribution is built, the corresponding coupling of the weak current to the nucleon or ∆ is a spin isospin one. It is then clear that this 2p-2h term only affects the magnetic and axial responses which enter the neutrino cross section. In the expression (1) these are the terms in G 2 A , G 2 M and the interference term in G A G M . The isovector response (term in R τ ) instead is not affected. This difference is the basis for the test that we propose to help elucidate the origin of the anomaly. The principle is simple. For antineutrinos the interference term in G M G A produces a suppression of the spin isospin response contribution, while it enhances it for neutrinos. In the neutrino case the contribution from the spin isospin terms largely dominates the isovector one. If the suppression of the spin isospin part for antineutrinos by the interference term modifies the balance in such a way that the role of the isovector response in the cross section becomes appreciable, the relative role of the 2p -2h part will be smaller for antineutrinos. We will show below that this is indeed the case. Figure 3 displays the various components to the genuine quasielastic (single nucleon ejection) cross section on 12 C as a function of the energy both for muon neutrinos and antineutrinos. The isovector part is identical in the two cases, as well as the isospin spin-longitudinal one. The isospin spin-transverse one instead changes considerably between neutrinos and antineutrinos due to the interference term with an appreciable reduction in the antineutrino case. As a consequence the isovector relative contribution to the total cross section becomes quite significant for antineutrinos which opens the possibility of an experimental test. Such details are visible in Fig. 4 where the differential cross section dσ dω at E ν =700 MeV is shown in the bare and RPA case, both for neutrinos and antineutrinos. In the same figure the axial-vector interference term is displayed as well as the isovector component illustrating the importance of the latter in the antineutrino case. A remark on the collective nature is in order at this stage. One notices the suppression produced by the RPA which follows from the repulsive character of the residual interaction. It is more pronounced at small energies. The interference term which spreads over all the energy range is therefore less affected by RPA. The antineutrino cross section which peaks at low energies is instead very sensitive to the suppression by the collective RPA effects. Once integrated over the energy transfer the reduction at a typical energy E ν =700 MeV is somewhat larger for antineutrinos than for neutrinos. A comparison between the two is affected by this difference. This could lead to a source of uncertainty in the comparison as the RPA effects which depend on the residual interaction have not been tested in this momentum regime. However we will show below that it does not prevent the comparison to be significant for our purpose.

We now turn to the generalized "quasielastic" cross section which includes the multinucleon contribution. The neutrino and antineutrino genuine and generalized "quasielastic" cross section are plotted in Fig. 5 both in the bare and RPA case. As was already discussed in Ref. [11] the agreement with the MiniBooNE experimental neutrino data [START_REF] Katori | AIP Conf. Proc[END_REF]10] is better when the np -nh com- ponent is added to the genuine QE cross section, whether in the free or in the RPA case. Our prediction for the generalized neutrino "quasielastic" cross section shows only a moderate sensitivity to the collective aspects. For antineutrinos instead the sensitivity to RPA is somewhat larger but it does not hide the important point that the relative importance of the 2p -2h term is smaller for antineutrino. This is illustrated in Fig. 6 which shows the ratio of the multinucleon component to the single nucleon one with and without RPA. In both cases, we find that the ratio for antineutrino is reduced as compared to the neutrino one by a factor 1.7 at E ν =700 MeV. In order to eliminate the uncertainties related to the neutrino energy reconstruction, we give in Table 1 values of quasielastic and multinucleon cross sections on 12 C, as well as their sum, averaged over the respective neutrino [10] and antineutrino [17] MiniBooNE fluxes, so as to provide quantities more in touch with an experimental analysis. We gives these quantities both in RPA and in the free case and various situations are possible. For instance, if our RPA description holds, the ratio of the generalized "quasielastic" cross section, which is the measured cross section, to a theoretical free Fermi gas model is 1.22 for neutrinos and 0.99 for antineutrinos, significantly lower. If the extreme case where RPA effects are totally absent the corresponding ratios are 1.37 for neutrinos and 1.25 for antineutrinos. In all cases the antineutrino numbers are smaller and the difference may be detectable, which offers an experimental test. For neutrinos the fit to the quasielastic data in a relativistic Fermi gas description required an appreciable increase of the axial cut-off mass [10]. For antineutrinos the increase needed to account for the data in the same model should be smaller since the relative role of multinucleon ejection is reduced. This difference offers a possible way to shed light on the origin of the anomaly. Of course the difference which occurs owing to the fact that the target is not pure carbon but CH 2 affects exclusively antineutrino cross section reducing trivially the importance of nuclear effects. It has to be taken into account to reach a significant comparison between neutrino and antineutrino cross sections.

Finally we would like to comment on the absence of final state interactions in our evaluation. In addition to the absorption of a produced pion it also ignores the possibility for an ejected nucleon to interact with the nucleus emitting another nucleon which leads to a final state with two nucleon ejected, the same type of final state as was discussed in this work. For the final sate interaction effect all responses are concerned and not only the spin isospin ones. This has been taken into account by Benhar and Meloni [18] using a spectral function which describes the single particle dynamics in an interacting system, deduced from experimental (e, e ′ p) data. Their conclusion is that this inclusion cannot explain the enhancement of the measured neutrino quasielastic cross section. Although the final sates are the same, (2p -2h), we are dealing in the present work to a different type of correlations. These are ground state correlations, mostly tensor ones which affect only the spin isospin responses, producing an enhancement of the spin isospin sum rule which is reflected in an increase of the "quasielastic" cross section. The rest is due to the ∆ excitation or the interference between the two. These also obviously belong exclusively to the spin isospin sector, which is the basis of our test through a comparison between neutrinos and antineutrinos. A fully realistic calculation should include both the spectral function effect and the ground state correlations or ∆ ones.

In summary we have studied the quasielastic neutrino and antineutrino cross section in the case of the MiniBooNE experiment where multinucleon ejection is not distinguishable from single nucleon production. The "quasielastic" cross section thus defined contains a certain proportion of 2p -2h and 3p -3h excitations. This proportion is large for neutrinos, which may be the interpretation of the increase in the axial cut-off mass needed to describe the data in the relativistic Fermi gas. For antineutrinos we predict a smaller role of the 2p -2h component. The reason is that the vector-axial interference term produces a suppression of the spin isospin response contribution to the cross section leaving a larger role for the isovector response which is not affected be the 2p-2h component. An experimental confirmation of this difference would signal the fact that the excess cross section belongs to the spin-isospin channel, thus displaying the role played by the tensor correlations in neutrino nucleus interactions.

The antineutrino mode which is actively investigated for the general problems of neutrino oscillations and CP violation is also of great relevance for the understanding of the neutrino nucleus interactions.

FIG. 1 :FIG. 2 :

 12 FIG.1:Examples of contributions to the quasielastic cross section. Lowest order contribution from : (a) R NN ; (b) R N∆ and (c) R ∆∆ . The wiggled lines represent the external probe, the full lines correspond to the propagation of a nucleon (or a hole), the double lines to the propagation of a ∆ and the dashed lines to an effective interaction between nucleons and/or ∆s. The dotted lines show which particles are placed on-shell.

FIG. 3 :

 3 FIG.3: Various response contributions to the νµ and νµ-12 C charged current genuine quasielastic cross section. Isovector (τ ) and isospin spin-longitudinal (στ (L)) components are identical for νµ and νµ.

FIG. 4 :

 4 FIG. 4: Differential CC νµ and νµ -12 C cross sections versus the energy transfer.

FIG. 6 :

 6 FIG.6: Ratio of multinucleon component of "quasielastic" cross section on12 C to the single nucleon one for νµ and νµ as a function of neutrino energy.

TABLE I :

 I MiniBooNE flux-integrated CC νµ-12 C and νµ-12 C total cross sections per neutron and per proton respectively in unit of 10 -39 cm 2 . The experimental CCQE νµ-12 C value measured by MiniBooNE is 9.429 × 10 -39 cm 2 with a total normalization error of 10.7 %[10].

	ν		ν
	QE np-nh QE+np-nh QE np-nh QE+np-nh
	bare 7.46 2.77	10.23	2.09 0.52	2.61
	RPA 6.40 2.73	9.13	1.60 0.47	2.07

We thank D. Davesne, T. Ericson, G.T. Garvey and T. Katori for useful discussions.