Pierre Peterlongo

Jacques Nicolas

Dominique Lavenier

Raoul Vorc'h

Joël Querellou

c-GAMMA: Comparative Genome Analysis of Molecular Markers

Discovery of molecular markers for efficient identification of living organisms remains a challenge of high interest. The diversity of species can now be observed in details with low cost genomic sequences produced by new generation of sequencers. A method, called c-GAMMA, is proposed. It formalizes the design of new markers for such data. It is based on a series of filters on forbidden pairs of words, followed by an optimization step on the discriminative power of candidate markers. First results are presented on a set of microbial genomes. The importance of further developments are stressed to face the huge amounts of data that will soon become available in all kingdoms of life.

Introduction

The decade started with the complete sequencing of the Haemophilus influenzae genome in 1995 [START_REF] Fleischmann | Whole-genome random sequencing and assembly of haemophilus influenzae rd[END_REF]. This period was characterized by the multiplication of sequencing projects for getting a better comprehensive view of the whole tree of life. During this time, an exponential rate of sequencing projects was observed, with a ×2 increasing rate every 20 months [START_REF] Koonin | Genomics of bacteria and archaea: the emerging dynamic view of the prokaryotic world[END_REF]. Comparative analyses of complete genomes from Bacteria, Archaea to Human have a huge impact on all aspects of life sciences and is deeply redesigning the evolution theory in the light of genomics [START_REF] Koonin | Darwinian evolution in the light of genomics[END_REF]. To better understand the driving forces in speciation, the diversity in virulence of pathogens, the diversity in metabolic pathways in various key species, more complete genomes of closely related strains of the same species (or species of the same genus) are needed. This recently triggered a flood of sequencing projects for novel strains of key pathogens (Campylobacter, Haemophilus, Mycobacterium, Streptococcus, etc.), model species (Bacillus, Escherichia), ecological key players (Prochlorococcus, Synechococcus) and species potentially interesting for biotechnology (Pyrococcus, Thermococcus). It appears that for these species the number of sequencing projects is growing exponentially, and the time has come to address specifically comparative genomics at micro-scale evolution (Table 1).

One of the main needs is the design of molecular markers that can achieve a high level of discrimination between different species or strains. The use of molecular markers has become increasingly popular in many fields: phylogenetic reconstruction in microbiology, quality control in food industry, traceability in epizooty and epidemic diseases, barcoding of life, forensics, etc. Each domain of activity has its favourite marker(s) working optimally for a specific purpose. The increasing number of complete genomes of related species available in databases raises the question of rapid determination of additional molecular markers through comparative genomics.

This paper proposes a novel approach to characterize molecular markers within a set of complete genomes of related strains or species targeting PCR (Polymerase Chain Reaction). PCR is one of the most important tools in genetic engineering for amplifying specific DNA fragments defined by flanking pairs of words on both side. These pairs of words are matched by complementary short synthetic nucleotides, called primers. Potential applications include strain quality control, identification, taxonomy and possibly phylogeny. 2 Identification of genome species markers using PCR Let us first explain the way markers are used during PCR. Let s denotes the reverse complement of word s over a four letter alphabet A, T, C, G and v denotes the marker to be selectively amplified. A DNA double helix corresponds to the hybridization of sequence x.u.v.w.y with sequence x.u.v.w.y = y.w.v.u.x. PCR aims at hybridizing of subsequence u.v.w with its complementary strand w.v.u, initiated by two short synthetic nucleotides, the primers, which will match u and w respectively (x,u,v,w and y are words). Thus, the word v corresponding to the marker itself is produced in the context of two fixed words corresponding to primer sequences. Most of the specific sequences that are used as molecular markers come from ubiquitous components of the cell with limited nucleic material such as ribosomes. One of the main resources concerns 16S rRNA and can be found on various dedicated websites including the Ribosomal Database project [START_REF] Cole | The ribosomal database project: improved alignments and new tools for rrna analysis[END_REF]. The last release (April 3, 2009) reports 836 814 16S rRNA annotated and aligned sequences. They mostly come from uncultured microbes, as a result of the standard investigation of microbial diversity by molecular methods and more recently by metagenomics.

The limits of 16s rRNA for species identification have been reached to handle this high number of species. Firstly, there is no linear relationship between 16S rRNA similarity and DNA-DNA hybridization. A consensus was reached, specifying that 16S rRNA similarity levels fewer than 97% between two strains is equivalent to DNA-DNA hybridization level fewer than 70%, and discriminates two different species. However, many different species display 16S rRNA sequences similarity within the range of 98-99%, and in those cases, 16S rRNA cannot be used to establish a strain as a novel species. Other variable molecular markers are frequently used in addition to 16S rRNA like housekeeping genes [START_REF] Stackebrandt | Report of the ad hoc committee for the re-evaluation of the species definition in bacteriology[END_REF]. The major drawback is that none of these additional markers are universal. Secondly, in phylogeny reconstruction, 16S rRNA cannot solve all the problems and for some taxonomic groups, tree topologies are uncertain. The help of additional sequences is required and the current trend is to use a set of sequences corresponding to the concatenation of ribosomal proteins.

Another widely used molecular marker for Eukarya "barcoding" is the 648-bp region of the cytochrome oxidase I (COI). DNA barcoding employs a minimum of 500-bp sequence of COI to help species identification and discovery in large assemblages of life [START_REF] Ratnasingham | Bold: the barcode of life data system[END_REF]. Although well adapted to species identification, COI sequences cannot be used in phylogeny nor in ecotypes identification. Here again, additional molecular markers need to be found in the set of complete genomes currently available for various tasks ranging from quality control in laboratory collections of pico and microeukaryotes to traceability of pathogens, of pests in the environment, etc.

Biologists need help both for choosing markers on a less restrictive set of sequences and for choosing the primers that will select these markers. Since most authors consider short sequences, they rely generally on multiple alignments for a subset of species of interest, so that conserved and non conserved regions become directly visible. Targets are then defined as conserved regions delimiting some highly variable regions and potential primers are then checked with the whole database in order to prune solutions matching elsewhere in the sequence. Nice environments have been developed in this context [START_REF] Ludwig | Arb: a software environment for sequence data[END_REF]. However, the task of finding suitable markers is not fully automated and does not scale to many species or long sequences due to the multiple alignment step.

In [START_REF] Pozhitkov | An algorithm and program for finding sequence specific oligonucleotide probes for species identification[END_REF], A. Pozhitkov and D. Tautz propose a program for simply finding one probe discriminating a given set of sequences from others. Although the algorithm could be widely improved, they use the interesting idea of building a crude index of words of fixed size in order to speed the search for common patterns.

Finding the best primer pairs where each primer is a substring of a given sequence is by itself a complex multicriteria task that has been well described in [START_REF] Kampke | Efficient primer design algorithms[END_REF]. It extends beyond string matching and the use of edit distances since it involves criteria including the proximity between primer melting temperatures, minimization of hybridization effects between forward and reverse primers, and avoidance of hybridization of primers with themselves. It may be solved efficiently using dynamic programming schemes that use extensions of approximate string matching equations.

The large scale design of primers has been tackled in another context: the observation of gene expression for a given organism. The issue is to produce for a subset of genes, or the complete set of genes of some genomes, a set of markers that identify each gene. The technique used in such a case, microarrays, involves an array of spots, each being attached to a primer. The main objective, in this context, is to find a set of primers working at a same temperature (called probes), each one recognizing a unique gene in the given set. Combining suffix-tree indexing, dynamic programming and ad'hoc filters, Kaderali and Schliep [START_REF] Kaderali | Selecting signature oligonucleotides to identify organisms using DNA arrays[END_REF] showed that it is possible to identify organisms but that this technique requires long probes for identifying many species. A recent review of long primers (size greater than 40) identification tools is available in [START_REF] Lemoine | An evaluation of custom microarray applications: the oligonucleotide design challenge[END_REF]. Producing a microarray with many long probes remains an expensive operation. One of the interests of working on species instead of gene expression is that the primer-gene association does not needs to be bijective. The issue becomes the choice of a minimal set of primers, a problem easily reduced to the minimum set covering problem [START_REF] Wang | G-primer: greedy algorithm for selecting minimal primer set[END_REF].

The present study shows yet another variation of primer design problem. The idea of working on whole genomes is kept but restricted to PCR as a low cost identification technique. The genome-based marker design problem consists in determining within a set of genomes, (i) primer pairs conserved over these genomes (ii) usable for PCR amplification for genomes differentiation (iii), associated to at least one homologous flanking region (iv) that can be used for diverse objectives: speciation, strain and species rapid identification, taxonomy, search of variable regions and contextual gene analysis.

To the best of our knowledge, this problem has never been stated before. The closest study we are aware of in terms of constraints to be solved, corresponds to a very different application: the study of the variability of individual genomes in terms of deletions or translocations that can occur in mutants and pathogenic states like cancer. A recent clever experimental protocol, PAMP [START_REF] Liu | A novel approach for determining cancer genomic breakpoints in the presence of normal dna[END_REF], uses multiplex PCR to selectively amplify the variations observed in pathogenic cells. Authors have developed an optimization technique to design primer sets based on simulated annealing and integer programming [START_REF] Bashir | Optimization of primer design for the detection of variable genomic lesions in cancer[END_REF]. This technique can processes sequences up to one Mbp. Although the setting is different, it shares an interesting characteristic with our approach, namely the comparison of ordered pairs of primers. We propose a generic formalization and a model for designing primer pairs for PCR amplification for finding markers able to differentiate genomes. The model relies on four steps (see Figure 1 for an overview):

Model

1. Given a set of sequences and primer parameters, detection of oligonucleotides that could be primers and that theoretically hybridize on each of these sequences on direct or reversecomplementary strand. This detection is based on physico-chemical properties of DNA fragments (Section 3.1); 2. From this set of possible primers, selection of all pairs that respect location properties on each sequence. These properties derive from PCR amplification technical constraints (Section 3.2); 3. Selection of primer pairs that define fragments considered as molecular markers; they permit to differentiate sequences from each other, using a simple length criterion (Section 3.3); 4. Selection of all pairs that further define flanking regions (fragments on the left and right hand sides of primer pairs) sharing homology or being highly variable (Section 3.4).

Primers characteristics

The primary goal of a primer is to hybridize a complementary strand of a DNA sequence at a well defined position. Optimal primers are produced on the basis of an hybridization model taking into account various criteria:

1. G+C content: the G+C percentage of a primer is framed between a minimum and a maximum threshold value, typically between 40% and 60%. 2. Melting temperature: the melting temperature of a primer must be in bounded interval.

The computation is based on the nearest neighbour method [START_REF] Santalucia | A unified view of polymer, dumbbell, and oligonucleotide dna nearest-neighbor thermodynamics[END_REF]. The melting temperature calculation also takes into account the concentration of nucleotides and the concentration of salt.

3. Repeats: primers containing large trains of identical nucleotides or dinucleotides are eliminated. 4. Hairpin loops: primers must not include hairpin loops. The size of the stem or the size of the loop must be lower than a predefined value. 5. Self-complementarity: a primer must not self hybridize during PCR. Thus, primers that form a duplex with their complementary strand are removed. 6. Thermodynamic stability at primer ends: The Gibbs Free Energy (∆G: in units of kcal/mole) values computed on the 5' and 3' ends of the primers are bounded. The ∆G value determines the strength of the hybridization and triggers the decision of considering the position as a potential hybridization site.

If a nucleic sequence possesses all these qualities, it can be considered as a successful primer. The next question is: what are the conditions for this primer to hybridize with a DNA sequence? In other words, given this primer and any portion of the genome, can they hybridize together?

The answer is brought by the calculation of the thermodynamic stability between the two strands. The nearest neighbour method proposed by SantaLucia [START_REF] Santalucia | A unified view of polymer, dumbbell, and oligonucleotide dna nearest-neighbor thermodynamics[END_REF] is used to compute ∆G along the two oligonucleotide sequences, with special care in the 5' and 3' extremities of the primer.

primer pairs for PCR amplification

Interesting primer pairs are those defining a fragment that may be amplified by PCR. Their hybridization positions (called PCPP, for Primers Couple PCR Positions) must respect some distances characteristics and some repartition conditions over the hybridization locations of each of these primers. Figure 2 gives an example of PCPP.

For a non-ambiguous characterization of PCR results and for a given a primer pair, two PCPP could not start or end at the same position. This avoids amplification of alternative sizes of fragment at a same position. In the following, the set C pcr of primer pairs defining at least one PCPP on each sequence is defined. Given a set of sequences S, we dispose of two primers sets: P d containing primers that hybridize on the direct strand of each sequence and P r that contains primers hybridizing the reverse complementary strand of each sequence. pos(s, p) is then defined as the set of positions where the primer p ∈ P d hybridizes on the sequence s ∈ S. pos(s, p ′ ) is defined as the set of positions where the primer p ′ ∈ P r hybridizes on the sequence s reverse-complemented. For the sake of clarity, all positions are reported on the direct strand.

C pcr is defined as the set of pairs c = (p, p ′ ) of primers from P d × P r such that for each s ∈ S:

∃a ∈ pos(s, p) and ∃b ∈ pos(s, p

′ ) min ≤ b -a ≤ max (1)
Moreover, the conditions of uniqueness for fragments starting or ending at a given position can be expressed as follows:

∀ a ′ = a ∈ pos(s, p) a ′ < b ⇒ b -a ′ ≥ limit (2) ∀ b ′ = b ∈ pos(s, p ′ ) a < b ′ ⇒ b ′ -a ≥ limit (3)
Conditions [START_REF] Fleischmann | Whole-genome random sequencing and assembly of haemophilus influenzae rd[END_REF] ensures that the pair of primers defines at least a fragment of length in [min + primers length, max]. Conditions (2) and (3) ensure that the selected pair of primers defines non-ambiguous fragments at given positions. Figure 2 represents hybridization locations respecting conditions from (1) to [START_REF] Koonin | Darwinian evolution in the light of genomics[END_REF].

In order to get rid of amplification of alternative sizes of fragment at a same position, C pcr does not contain pairs of primers with hybridization sites respecting condition (1) and not respecting condition (2) and (3). Formally, ∀(p, p ′ ) ∈ C pcr and ∀s ∈ S

∀(a, b) ∈ pos(s, p) × pos(s, p ′ ) min ≤ b -a ≤ max ⇒ (2) ∧ (3) (4)

Primer pairs for sequence differentiation

Primer pairs in C pcr are potential candidates for PCR amplification. Let C dif f be a subset of C pcr containing all pairs of primers defining inner fragments whose length enable to differentiate sequences.

To do so, lengths(s, c) (s ∈ S and c ∈ C pcr ) is defined as the set of lengths of inner fragments defined by PCPP of c on the sequence s. Additionally, C dif f is defined as the subset of pairs c from C pcr such that ∀s, s ′ ∈ S, s = s ′ , ∃l ∈ lengths(s, c) such that, ∀l ′ ∈ lengths(s ′ , c)

max l l ′ , l ′ l ≥ δ, with δ a fixed parameter. (5)
Informally, condition [START_REF] Stackebrandt | Report of the ad hoc committee for the re-evaluation of the species definition in bacteriology[END_REF] ensures that for each pairs of primers c ∈ C dif f and that for each couple of sequences s, s ′ ∈ S, at least one of the fragments defined by any PCPP of c on s has a length different enough from all fragments defined by PCPP of c on sequence s ′ . This property enables selected fragments to differentiate sequences from each other with a simple length-based test.

Moreover, in order to provide readable PCR results by clearly distinguish amplified fragments, an additional parameter max occ. is applied. In C dif f , couples c whose number of hybridization sites is bigger than max occ. are removed. Formally, ∀c ∈ C dif f , ∀s ∈ S : |lengths(s, c)| ≤ max occ., with |E| denoting the cardinality of the set E (this notation is used in the rest of the paper).

Sequence similarity / variability

At last, sequence composition of fragments defined by PCPP is taken into consideration. Given a PCPP, the internal region (red area on Figure 3) and the two flanking regions (yellow fragments on Figure 3) are considered. Depending on the application, one may want these areas to be homologous or variable.

Bearing in mind that any combination of searched homology is possible, C sim (see Figure 1) is defined as the subset of pairs of primers from C dif f , such that there exists at least on PCPP for these pairs with variable centre fragment and at least one homologous flanking region. Each fragment is considered both on the direct and on the reversed strand.

Methods

This section presents the methods for finding potential primers (Section 4.1). Then, respectively, sections 4.2, 4.3 and 4.4 show how previously defined sets C pcr , C dif f and C sim are detected. Given a set S of n sequences, ideally all potential primers that may hybridize at least once on each sequence should be generated. Such an approach is actually unfeasible by enumerating all the primer configurations. In this case study, considering primers of length 25 would lead to test 4 25 elements, which is unrealistic.

Instead, the following approach is used. To search common primers of length l, all l-mers of each sequence s ∈ S are first considered. In addition, to extend the space search, these l-mers are degenerated in their middle. Practically, 2 nucleotides are modified, leading to generate up to 4 2 l-mers per position.

After this stage, a huge set of l-mers are considered as potential primers. Only those respecting conditions presented in Section 3.1 are selected. More precisely, the selection of primers is achieved through a pipeline of filters. Each stage of the pipeline eliminates the candidates which do not fit specific criteria. For efficiency purpose, the most stringent criteria are first taken into consideration. The implementation is based on a series of functions which does not present algorithmic challenges and are not detailed here.

After this process, a new set of l-mers considered as putative primers is available. From this set only those that hybridize on all different sequences are selected. The whole set of primers is thus checked against S. In that way, a list of hybridizing primers is associated to each sequence. The intersection of these lists, results in the set of primers that hybridize at least once on every sequences.

To speed-up the hybridization test, the sequences are first indexed with a seed-based index technique. The length of the seeds are set to 6, meaning that a primer hybridization will be reported only if the primer and the genome share at least 6 common nucleotides (or, more exactly, two complementary 6-nt words). In that case, a ∆G value is computed as presented in section 3.1. Depending of the ∆G value, the primer is added or not to the primer list associated with the genome.

Methods for detection of primer pairs for PCR amplification (C pcr )

From this point, two sets of potential primers are available: P d and P r that hybridize respectively at least once on each sequence s and once on each reverse complementary sequence s.

In order to verify conditions (1) to ( 4), all possible primer pairs (p, p ′ j ) ∈ P d × P r are checked. On each sequence s, the ordered hybridization locations pos(s, p) and pos(s, p ′ ) are available from previous steps.

In a few strokes, the algorithm works as follows: positions over pos(s, p) and pos(s, p ′ ) are read conjointly as long as condition ( 1) is not fulfilled. In case a pair of hybridization positions (a, b) ∈ pos(s, p) × pos(s, p ′ ) respecting these three conditions is found, then previous positions a ′ on pos(s, p) (resp. next position b ′ on pos(s, p ′ )) is checked in order to validate that condition (2) (resp. ( 3)) is respected. In case of success, the pair (p, p ′ j ) is tagged as a potential pair for PCR, otherwise the pair is rejected (condition (4)) and the reading of its positions is stopped. All pairs of primers respecting conditions (1) to (4) are stored is the set C pcr .

For a pair of primers (p, p ′ j ) ∈ P d × P r , this approach reads all positions in pos(s, p) and in pos(s, p ′ ) leading to a complexity in O(|pos(s, p)| + |pos(s, p ′ )|) that is O(N ) with N the total length of the input genomes. As this computation is done for each possible pair of primers, the overall time complexity of this step is in

O(|P d | × |P r | × N ) that is O(N 3
). In practice, the time complexity is much lower, as confirmed by experimental tests described in Section 5.

Methods for detecting primer pairs for sequence differentiation (C dif f )

Finding the subset C dif f from the set C pcr is straightforward. For each primer pair c ∈ C pcr and each sequence s ∈ S, lengths(s, c) is known (see Section 3.3). Trivially, for each primer pair c in C pcr and for each couple of sequences s, s ′ ∈ S, s = s ′ , c is conserved in C dif f if there exists l ∈ lengths(s, c) that is different enough from all l ′ ∈ lengths(s ′ , c) so that condition ( 5) is respected. Simultaneously, it is trivial to conserve in C dif f only primer pairs for which the number of occurrences of PCPP on each sequence respects the max occ. parameter. Pairs of primers from C dif f that define fragments respecting conditions exposed in Section 3.4 (see also Figure 3) are selected to be included in C sim . Knowing the large amount of work previously done for finding multiple local alignments, we decided to not develop our own algorithm. In this framework, we used MEME [START_REF] Bailey | Fitting a mixture model by expectation maximization to discover motifs in biopolymers[END_REF] which provides an e-value estimation enabling used as a formal criterion for creating set C sim . As stated earlier, this step is highly tunable depending on the biological applications. In this framework, the method is the following: for each primer pair c ∈ C dif f , MEME is applied on all combination of PCPP of c on the set of genomes. The primer pair c is stored in C sim if one of the MEME results provides both:

-an e-value bigger than a fixed threshold for center fragments alignments, -an e-value bellow another threshold for flanking regions alignments.

Results

The method has been implemented in the c-GAMMA tool acting as a pipeline of programs. As a preliminary test c-GAMMA was applied on a set S of height Thermococcales genomes (source GOLD database http://www.genomesonline.org/) of total length N ≈ 16 Mb. Thermoccocales were chosen due to their high interest in biotechnology. Species belonging to this family display thermostable hydrolases of interest. It is therefore important to find molecular markers that can help to identify strains within Thermococcus and Pyrococcus species and insure quality control. The goal of our study was to design couples of primers defining molecular markers both identifiable by PCR and by a sequence homology criterion.

All experiments were computed using a PC Intel dual core 2.40 GHz running under Linux Fedora with 2 GBytes memory.

Primer detection results (P d and P r )

The method exposed in Section 4.1 was applied for generating primers of length 25 that hybridize at least once on each genome (direct and reverse complementary strand) in S. The primers generation was done by testing all 25-mers presents on each genome (direct and reverse complementary strands) and degenerating two central positions on of each of them. Thus ≈ 512 million of 25-mers were tested. Each of these 25-mers was selected for further analysis if classical parameters for PCR amplification were respected. This method generated 2 803 510 primers on the direct strand and 2 796 747 on the reverse complementary strand.

Then only primers that hybridized at least once on each sequence (direct and reverse complementary strands) were conserved. This step conserved 62 247 primers on the direct strands (set P d ) defining 6 309 356 hybridization sites. On the reverse complementary strands, 62 764 primers were conserved (set P i ) having a total of 6 295 992 hybridization sites. Note that, in average, a primer hybridization site is found each ≈ 2.38 positions on each strand and that each primer has ≈ 100 hybridization sites. This step is the most time consuming. It was performed in less than six hours. For obtaining C dif f from C pcr , a set of tests using several distinct parameters was performed. Each test was computed in less than 30 seconds. Results are shown Table 2.

This experiment shows that the max occ. parameter (table (a)) has a strong influence and that most of the primer pairs define between 1 and 5 PCPP per genome. However, even by constraining exactly one occurrence per genome (last line of (a)), one notices that still 18122 pairs respect the parameters.

Moreover, these results show that, fortunately, even while applying very stringent parameters, some primer pairs are found. For instance, while asking for a minimal fragment length difference of δ = 10% and at most 2 fragments occurrences on each genome (last line of (b) of Table 2), 11 primer pairs are found. Figure 4(a) shows the theoretical PCR result that would be obtained on the studied set of genomes thanks to a randomly chosen primer pair respecting such conditions. It is worth mentioning that, as expected, this single PCR result clearly permit to distinguish strains from each other, as for any primer pair respecting the required parameters.

Detection of primer pairs taking into account sequence similarity and variability results (C sim )

The goal here is to show that an approach involving similarity criterion in addition to lengths attributes provides realistic biological results. Thus, we show results of an experimentation run on the set of 24 primer pairs generating one PCPP on each sequences with at least δ = 5% of difference of length between them (bold faced line of (c) of Table 2.). For each of these primer pairs PCPP, MEME was applied both on central fragment and on the two flanking areas (over 1000 bp). We selected primers pairs for which the best alignment had an e-value higher than 1 for the central fragment and lower than 10 -1 for any of the flanking regions. Among this 24 primer pairs, one of them gave satisfying results. Indeed, the couple of primers (CGCAGGATTAGCTACAGCCCCACTC, GGCCAATAATACCCAAAGCGGAGGA), having exactly one PCPP on each genome (see Figure 4(b)) define a central fragment highly variable (best local alignment found has an e-value equal to 4.2e+5) and has a left area containing an homologous motif (shown on Figure 5) of length 98 with a e-value of 1.3e -2 .

Conclusion

This paper proposes a generic model to efficiently (1) detect primers on a set of genomes, and (2) define suitable molecular markers for genomes differentiation. The differentiation occurs at two levels, a simple length criterion, and a more precise criterion on flanking regions homologies and/or variability.

The model is fully implemented within a bioinformatics pipeline called c-GAMMA. Applied on a set of height bacterial genomes (16 Mb), c-GAMMA designed primers for the detection of molecular markers in 12 hours on a standard work station, making possible the genomes differentiation both using length and homology criterion.

These encouraging preliminary results open the way to other experimentations on the huge source of data produced by next generation sequencing machines.

Moreover, methods proposed in this framework mark a step over molecular markers detection. They are highly suitable for further enhancements such as:

-improving the primers generation by producing all oligonucleotides that may hybridize on a genome fragment. Generation is currently achieved through a simple degeneration scheme on the middle part of the fragments. Such an approach will provide more suitable results. However, it will dramatically increase the number of possible primer pairs (|P d | × |P r |) and will raise computational issues for finding hybridization sites; -instead of considering only one primer pair on each sequence, the model may be improved by considering simultaneously several primer pairs to perform multiplex PCR in order to efficiently differentiate close species.

Fig. 1 .

 1 Fig. 1. Overview of the model. A set S of four sequences guides the design of two primers (red and green rectangles) sets P d and Pr, whose pairs generate sets Cpcr, C dif f and Csim.

Fig. 2 .

 2 Fig. 2. A portion of two DNA strands is shown. On each strand, a primer has two hybridization locations at position a ′ and a on the direct strand (red rectangles on the bottom line) and position b and b ′ on the reverse complementary one (green rectangles on the top line). This pair shows hybridization sites (a and b) which respect conditions (1) to (3). The shaded area corresponds to a putative molecular marker. If condition (4) is also respected, (a, b) is a PCPP.

Fig. 3 .

 3 Fig. 3. A pair of primers c = (p, p ′ ) ∈ Csim has one PCPP on each of four genomes. For the sake of clarity only one strand is represented. Inner fragments defined by this PCPP (red) present for instance high variability while left or right flanking regions (yellow) present for example high similarity.

4. 1

 1 Methods for primers detection (P d and P r )

  This checking is done in O(|lengths(s, c)|×|lengths(s ′ , c)|) for each couple of sequences s, s ′ and each primer pair c ∈ C pcr . Thus, for each primer pair, this checking is done in O(n 2 ×|lengths(s, c)|× |lengths(s ′ , c)|) leading to an overall time complexity of O(|C pcr |×n 2 ×|lengths(s, c)|×|lengths(s ′ , c)|). Note that in practice n, |lengths(s, c)| and |lengths(s ′ , c)| are negligible with regard to |C pcr |.

4. 4

 4 Methods for detection of primer pairs taking into account sequences similarity and variability (C sim )

Fig. 4 .

 4 Fig. 4. (a) A randomly chosen theoretical PCR obtained on the studied set of genomes using a pair of primers respecting conditions min = 200, max = 2000, limit = 3500, δ = 1.10 and max occ. = 2. (b) theoretical PCR obtained on a primer pair respecting conditions min = 200, max = 2000, limit = 3500, δ = 1.05 and max occ. = 1 and defining a variable marker region and a homologous flanking region between set of genomes.

Fig. 5 .

 5 Fig. 5. Motif found by MEME on left flanking region of couple of primers CGCAGGATTAGCTACAGCC-CCACTC and GGCCAATAATACCCAAAGCGGAGGA.

Table 1 .

 1 Number

	Phylum	Genus	Species, strains Genomes projects Genomes completed
	Arthropoda	Drosophila		10	10
	Euryarchaeota Methanococcus		7
		Pyrococcus		4
		Thermococcus		5
	Firmicutes	Bacillus	anthracis	8
		Bacillus	cereus	14
		Bacillus	other species	13
		Clostridium	botulinum	7
		Clostridium	other species	29	10
		Lactobacillus		12	11
		Staphylococcus aureus	12	12
		Staphylococcus other species	4
		Streptococcus pneumoniae	17
		Streptococcus other species	30	22
	Spirochaetes Borrelia	burgdorferi	7
		Borrelia	other species	4
	Proteobacteria Burkholderia		45	13
		Campylobacter jejuni	9
		Campylobacter other species	6
		Escherichia	coli	31	10
		Haemophilus	influenzae	13
		Haemophilus	other species	4
		Pseudomonas		16	11
		Rickettsia		13	10
		Salmonella	enterica	23
		Shewanella		10
		Vibrio	cholerae	6
		Vibrio	other species	14
		Yersinia	pestis	13
		Yersinia	other species	7
	Actinobacteria Mycobacterium tuberculosis	5
		Mycobacterium other species	12	12
	Tenericutes	Mycoplasma		14	13
		Ureaplasma	urealyticum	11
		Ureaplasma	other species	5
	Cyanobacteria Prochlorococcus marinus	12	12
		Synechococcus		15	10

of genome projects related to important prokaryotic genera and species (Source: GOLD http://www.genomesonline.org/ and Microbesonline http://www.microbesonline.org/, modified, april 2009)

Table 2 .

 2 5.2 Primer pairs for PCR amplification and sequence differentiation results (C pcrandC dif f )For creating the sets C pcr and C dif f from P d and P i , methods presented in Section 4.2 were sequentially applied. For defining C pcr the parameters were the following: min = 200, max = 2000 and limit = 3500. These parameters facilitate standard PCR procedure used by most diagnostic laboratories.First, set C pcr containing pairs of primers that respect conditions (1) to (4) is selected. This was done on all possible primer pairs in |P d | × |P i | (≈ 3.9 billion pairs in this experimentation). This computation took less than four hours and provides 63877 couples. Quantitative results while varying parameters for finding C dif f from Cpcr. Cpcr contained initially 63877 primer pairs. Tests (a) make variation over the maximal number of occurrences (max occ.) of PCPP of each couple on each genome. Tests (b) (resp. (c)) make variation over the parameter δ (see Section 3.3) using at most 2 (resp. 1) occurrences of PCPP of each couple on each genome.

		(a)			(b)			(c)
	δ max occ. |C dif f |	δ max occ. |C dif f |	δ max occ. |C dif f |
	1	10	63872	1.01	2	1149	1.01	1	137
	1	9	63865	1.02	2	301	1.02	1	68
	1	8	63782	1.03	2	180	1.03	1	41
	1	7	63518	1.04	2	107	1.04	1	36
	1	6	63193	1.05	2	71	1.05	1	24
	1	5	62018	1.06	2	56	1.06	1	23
	1	4	59050	1.07	2	37	1.07	1	17
	1	3	53218	1.08	2	11	1.08	1	0
	1	2	42187	1.09	2	11
	1	1	18122	1.10	2	11


L Einig

J Pety

A Roueff

P Vandame

J Chanussot

M Gerin

Orion-B Consortium

Deep learning denoising by dimension reduction Application to the ORION-B line cubes

1/ Hyperspectral line cubes

I Large bandwidth receivers for millimeter radio-telescopes allows the acquisition of position-position-velocity (x, y, v) data cubes of the interstellar medium over a wide field of view and a broad frequency coverage. [1] I Due to the Doppler effect of the moving gas, these lines do not have a dirac-shaped emission spectrum, but rather an envelope with a Gaussian appearance. I Knowing the emission frequency of a molecule, we can interpret the spectral bands as an image of the emission a given molecule moving at a certain radial velocity.

I These cubes contain much information on the physical and chemical properties of the emitting gas but their large size coupled with inhomogenous signal-to-noise ratio are major challenges for consistent analysis and interpretation.

CO isotopologues emission (9 km/s) Ex. of noise map

I Goals # Increase the signal-to-noise ratio, in particular in the region where it is low. # Reduce the dimension of the cube over the spectral axis.

¥ An autoencoder neural network could do both.

[2]

2/ Challenges

I As distant channels are statistically independant, the redundancy that can be exploited is only local.

13 CO (1-0) emission, 7 and 10 km/s channels ¥ The information shared with a given channel decreases with the distance. As a consequence, the amount of exploitable mutual information (MI) is very limited.

Band pairs MI for 13 CO (1-0) and the Indian Pines dataset

3/ Informed local autoencoder

¥ In order to encompass those challenges, we propose significant improvements of typical autoencoder neural network.

1.

As distant channels are independant, they should not be encoded together. We propose an alternative autoencoder architecture.

Dense and locally dense layers Example of locally dense autoencoder

The maximum distance between two channels encoded together is defined from the data analysis as the characteristic distance between two channels at which the remaining mutual information is of 5%.

2. Moreover, as the major part of the voxels (samples (x, y, v)) are signal free, we want the autoencoder to encourage the sparcity of the denoised cube.

L( y, y) = 1 K K k=1   1 J j=1 w jk J j=1 w jk y jk -y j k σ k 2
Controls the reconstruction of the voxels that are likely to contain signal

+ 1 J j=1 w jk J j=1 w jk y jk σ k q  
Controls the sparcity of the voxels that are likely to be signal-free where σ k is the noise RMS for the k-th pixel and w jk an a priori from another segmentation method indicating whether a pixel is likely to contain signal or not.

4/ Results

I We apply this method on some cubes. For each one, a new network has to be trained.

Example of noisy and denoised channel and the residues for 13 CO (1-0).

I We compare the denoising with that of the Gaussian fit method ROHSA. [3]

I We observe different behaviors, in particular the Gaussian fit denoise whatever the signal level and tends to distort the signal of high intensity. The autoencoder tends to preserve these regimes.

I The distortions of the signal (regions where the residuals are correlated) are quite low for the autoencoder.

5/ Conclusion

Statistical denoising with low redundancy Data analysis based architecture and loss function Denoising of the low level signal No significant dimension reduction


Xiaogang Peng

Long Lin

Louise Stuttgé

Marc Rousseau

Thierry Sauvage

Emilie Voirin

Bertrand Donnio

Mircea V Rastei

Jean-Louis Gallani

Role of lattice defects on the magnetism of gold nanoparticles Irradiated with Neutrons

Keywords: Nanoparticles, Magnetism, Neutrons, Defects

The origin of the unexpected magnetic properties of nanoparticles made of gold, silver, or other diamagnetic metals remains obscure despite a large body of experimental and theoretical studies. Whereas many studies have endeavoured at finding correlations between magnetism and nanoparticle size or ligand coating, none so far has attempted at investigating the role of core crystallinity. We have irradiated gold nanoparticles with energetic neutrons or protons with the aim of nucleating lattice defects in the crystalline cores. Comparison of the magnetic behaviours before and after irradiation demonstrates clearly that the presence of defects in the crystalline lattice of the nanoparticles cores contributes efficiently to their magnetism.

Introduction

The question of why noble-metal nanoparticles sometimes display paramagnetism and/or ferromagnetism is still an open one. In spite of many experimental and theoretical work, this behaviour observed in nanoparticles made from metals which are diamagnetic in the bulk is still debated. Many explanations have been brought forth, very often with experimental support : size effect, ligand effect, composition, quantum effects,... but even though the experiments are trustworthy, there is always somewhere another no less trustworthy experimental evidence which hampers a clear interpretation. [START_REF] Nealon | Magnetism in gold nanoparticles[END_REF] One of the major hurdles when investigating macroscopic samples made from many nanoparticles is the lack of reproducibility, a fact which tends to oveshadow all conclusions drawn from the comparison of different samples : no two synthetic batches are identical and within each batch there is an inherent variability of the nanoparticles sizes and structures. [START_REF]We do not consider here so-called "atomically precise nanoclusters[END_REF] Therefore experiments aiming at comparing say, the role of the ligands or the influence of the size are necessarily conducted on different synthetic batches, which adds supplementary uncontrolled parameters and precludes definitive conclusions to be reached. Most nanoparticles comprise a metal core coated with ligands. The ligands being of molecular nature they are necessarily monodisperse and, in the general case, closed-shell diamagnetic molecules. The core is obviously the most variable ingredient. Its diameter cannot be totally controlled, nor can its exact shape and the number of ligands it bears. Moreover, crystallinity is also poorly -if not at all-controlled. Within the course of our own experiments we came across the hypothesis that maybe one should consider more closely the consequences of the quality of cores crystallinity on the magnetic properties of such nanoparticles. The feasibility of a high level of control on the crystallinity of silver nanoparticles has recently been demonstrated. [START_REF] Lin | Ag Nanocrystals with Nearly Ideal Optical Quality: Synthesis, Growth Mechanism, and Characterizations[END_REF] Subsequent experiments on single crystalline and multi-twinned silver nanoparticles then showed that the presence of lattice defects in the metal cores tend to render the nanoparticles para-or ferromagnetic. [START_REF] Lin | Influence of the Crystallinity of Silver Nanoparticles on Their Magnetic Properties[END_REF] We then devised the procedure described hereafter where energetic particles nucleate lattice defects in the cores, with no chemistry involved. Direct irradiation of nanoparticles has one great advantage : rather than having to compare nanoparticles coming from different synthetic batches, it becomes possible to synthesize one unique batch, which is then split in identical aliquots subjected to various experiments, the initial condition remaining identical. We turned to gold nanoparticles as gold is largely unsensitive to oxidation and also has a large enough neutron scatering cross-section which increases the chances to induce defects. Our approach was conforted by a report on the emergence of magnetic moments in plutonium from -particle decay [START_REF] Mccall | Emergent magnetic moments produced by self-damage in plutonium[END_REF] and another study observing the increase of ferromagnetic properties of palladium nanoparticles (5-10nm diameter) irradiated by swift Au ions. [START_REF] Kulriya | Enhancement of ferromagnetism in Pd nanoparticle by swift heavy ion irradiation[END_REF]

Materials and methods

Gold nanoparticles have been synthesized in one batch according to a published procedure. [START_REF] Peng | A facile synthesis of monodisperse Au nanoparticles and their catalysis of CO oxidation[END_REF] After synthesis the oleylamine ligands were exchanged for dodecanethiol ones following standard procedures. Gold has been chosen for its greater chemical stability as compared to e.g. silver. Mean diameter of the cores was 6.0nm, with a size dispersion of ±0.4nm, each nanoparticle comprising ca. 6700 atoms. An approximate "molecular weight" for the coated nanoparticles would be 1.410 6 g/Mol. Electronic configuration of gold is [Xe]4f 14 5d 10 6s 1 . We synthesized a large unique batch of gold nanoparticles (1 gram) which was magnetically characterized and then split in 6 aliquots. Four of them were used for neutron irradiation, one for proton irradiation and one was kept untouched (labelled PRIST). Proton irradiation has been performed using protons at CEMHTI (pelletron accelerator). Energy was set to 2.9MeV, beam current was 4µA and integral number of proton estimated around 1.5310 [START_REF]For example SRIM[END_REF] protons/cm 2 . A 4µm thick layer of nanoparticles was spread on a water-cooled copper block and covered with a 21µm thick copper foil. This sample thickness was chosen so that the Bragg peak fell outside the sample ensuring a more homogeneous irradiation, at the cost of reduced efficiency. After this copper foil the protons energy was down to 1.5MeV, still high enough to generate defects. In spite of the active cooling the temperature of the sample rose to ca. 140°C. Thermogravimetric analysis of the original synthetic batch showed that the nanoparticles were thermally stable up to this temperature (see ESI). This sample is labelled PROT. Neutron irradiation has been performed at the SINQ source of the Paul Scherrer Institute. Source flux is ~10 14 n/cm 2 /s. Four samples labelled NEUTR1, NEUTR 2, NEUTR 3 and NEUTR 4 have been irradiated with thermal neutrons (0.025eV) and received respectively an integral dose of 7.6910 [START_REF] Meechan | Bildungsenergien von leerstellen in kupfer und gold[END_REF] , 3.1210 16 , 3.1210 17 and 2.5010 [START_REF]For example SRIM[END_REF] neutrons/cm 2 . Average sample weight was 50mg, most of this mass comes from Au atoms. We therefore had ca. 1.510 [START_REF] Beyerlein | Effects of He radiation on cavity distribution and hardness of bulk nanolayered Cu-Nb composites[END_REF] gold atoms subjected to irradiation in each sample. As irradiation led to activation of the samples SQUID magnetometry was performed only after the radioactivity had sufficiently decayed (~1 month). Samples were sealed under vacuum in fused silica tubes. During irradiation temperature of the samples could rise slowly up to ca. 85°C (once again within the stability region) due to environmental conditions in the sample chamber. Magnetometry has been performed with a Quantum Design MPMS SQUID magnetometer. The apparatus has been properly calibrated and checked for artifacts. A small amount of each sample (ca. 20mg) was set into a gelatin holder, placed inside a plastic straw. Magnetic response of an empty gelatin holder placed inside an identical straw has been measured as purely diamagnetic and the resulting small magnetic moment has been considered when analyzing the behaviour of the various NPs. No correction for the diamagnetism of the ligand shell of the NPs was applied to the measured values, all curves given here are raw data. In any case the ligands represent at most 8% of the total mass. Magnetization curves have been recorded at room temperature and at 4K. The total magnetic moment measured by the SQUID has been systematically divided by the total mass of the sample (since the mass of the magnetic fraction of the sample is impossible to know), yielding the (mass) magnetization given in the figures. Depending on each sample and its magnetic history, the first magnetization branch of the magnetization curves is not always seen or, when seen, does not always start at the origin. Given the amorphous nature of the samples (no orientational or positional order) this bears no significance.

Transmission Electron microscopy (TEM) was performed on a TOPCON LaB6 microscope from JEOL.

Results

In order to assess the impact of irradiation on the crystalline structure, electron microscopy has first been performed. As the original gold nanoparticles were a mix of monocrystalline and multi-twinned ones, the outcome of these observations was not expected to be obvious. After thoroughly scanning several microscopy grids we became convinced that the most visible change was that some of the nanoparticles were distorted ot had fused together. Radiation-induced agglomeration and increase of the size has also been reported for irradiated Pd nanoparticles. [START_REF] Kulriya | Enhancement of ferromagnetism in Pd nanoparticle by swift heavy ion irradiation[END_REF] Figure 1 is an example of two typical images of the non-irradiated sample (PRIST), the most magnetic neutron-irradiated sample (NEUTR4) and the proton-irradiated sample (PROT). Even though lattice defects are already present in the original sample (green arrows), nanoparticles subjected to high doses of irradiation show clear signs of fusioning and shape altering. This is quite obvious in the picture of proton-irradiated particles which also displays an increased number of defects. More images for all samples are given in ESI. Sample PROT visibly stands out and probably does not deserve much of a discussion as its structure has been significantly altered, not to say destroyed. Many nanoparticules have fusioned, shapes are irregular, major lattice defects are clearly visible and numerous. On the contrary, neutron-irradiated samples have retained their essential features, the most visible difference being the altered shape of some nanoparticles and their fusion in shapeless clusters. Many nanoparticles resemble much the original ones, it seems difficult to assess the exact nature (screw, edge, vacancy, interstitial...) or number of the lattice defects which are visible. It would also be difficult to unambiguously claim that there are more vacancies or interstitials atoms. We will not comment more on the pictures and turn to the magnetic properties of the samples.

All samples have been magnetically characterized by recording their magnetization vs. applied magnetic field at 300K and 4K. Figure 2 compares the magnetization curves recorded at 300K for the nonirradiated sample (PRIST) and the proton-irradiated one (PROT). Figure 3 The non-irradiated sample displays a quasi-diamagnetic behaviour, with only a sliver of ferromagnetism at very low fields, which we reckon is due to the defects inherently present in the metal cores with any synthesis procedure and not to impurities. Possible SQUID artefacts have been looked for and excluded. A clear deviation from this behaviour can then be seen with the irradiated samples PROT (Fig. 2) and NEUTR2 (Fig. 3). Magnetization curves of samples NEUTR1, NEUTR3 and NEUTR4 do not differ much from that of the non-irradiated sample.

Discussion

We aimed to create lattice defaults such as vacancies, interstitial atoms, dislocation loops or stacking faults using only physical means, i.e. no chemistry involved and, as much as possible, without altering the samples in any other way. Nucleation of defects in irradiated materials is a known fact, [START_REF] Lemaignan | Nuclear Materials and Irradiation Effects[END_REF] it is actually one of the main causes of concern in nuclear plants. [START_REF] Hoffelner | Damage assessment in structural metallic materials for advanced nuclear plants[END_REF] Speaking of nuclear plants, it is interesting to observe that contrary to what we report here, the magnetism of the steel of the reactor pressure vessels is seen to diminish upon neutron irradiation. [START_REF] Wang | Study of the magnetization work of RPV steel in dependence on neutron irradiation[END_REF] This is due to the huge difference in structural damage occuring in such high neutron fluxes and, obviously, to the initially much different magnetic properties of the materials, dia-versus para-magnetic. Nanoparticles submitted to energetic irradiation behave differently than macroscopic samples (as it probably should be expected) because of their limited size and high surface-to-volume ratio. Quoted from reference 11 : "Instead of diffusing into organized defect structures, the generated point defects tend to annihilate at the particle surface, or material is ejected from the particle volume, as a result of the energetic collision. This can drastically alter the shape of individual particles and cause agglomeration of closely spaced particle groupings." The change of shape and sintering of Au nanoparticles under an energetic electron beam has indeed been reported. [START_REF] Yu Chen | Sintering of passivated gold nanoparticles under the electron beam[END_REF] Proton irradiation certainly produces similar effects and in the case of our neutron irradiated nanoparticles some sputtering has indeed taken place, since the inner side of the fused silica vials which contained the sample came out slightly gold coloured (see ESI).

Let us first consider neutrons. Neutrons and solid metals can interact in two ways : radiative capture or scattering. In the case of gold, capture reactions are the most probable event and they indeed occurred since our samples became radioactive after irradiation. [START_REF] Kim | Au nanoparticles have been found to become efficient radio-sensitizers upon neutron irradiation[END_REF] Taking a neutron absorption cross section of approximatively 100 barn, 14 with a source flux of 10 14 n/cm 2 /s, one can expect at the most ~1.510 [START_REF] Yu Chen | Sintering of passivated gold nanoparticles under the electron beam[END_REF] atoms hit (neutron capture) per second. Similarly, the scattering cross section is approximatively 7b resulting in ~10 [START_REF] Briggs | Evolution of Gold Nanoparticles in Radiation Environments[END_REF] atoms hit (neutron scattering) per second. These figures, to be taken with caution, result in a maximum of ~10 14 hits for sample NEUTR1 and ~10 [START_REF] Nordlund | Primary radiation damage: A review of current understanding and models[END_REF] hits for sample NEUTR4. In addition to the absorption and scattering of neutrons by the gold atoms, some scattering is also expected from interactions with hydrogen atoms (cross-section of ca. 80b) in the ligands. This will obviously not result in the nucleation of lattice defects but may induce some damage to the ligands. Other elements (C, S, O) will barely interact with thermal neutrons. Our samples were irradiated with thermal neutrons (0.025eV) and energy formation of a vacancy in gold is estimated at 0.67eV, [START_REF] Meechan | Bildungsenergien von leerstellen in kupfer und gold[END_REF] meaning that direct knock-on between a neutron and a gold atom cannot result in the formation of a stable vacancy. The neutron absorption cross section is more than ten times that of the scattering cross section, making it then the most probable mechanism. Upon neutron capture a 197 Au atom evolves towards an unstable 198 Au atom which decays over 2.6 days into 198 Hg by emission of 960keV electron. The 198 Au then recoils with an energy of 2.85eV, larger than the aforementioned energy for creating a lattice vacancy in gold. This recoiling atom is responsible for the defects appearing in the nanoparticles irradiated with thermal neutrons. Of course, most of the time the recoil does not create a lattice defect but the exact proportion of efficient hits is difficult to estimate. The temperature to which samples were subjected during irradiation (85°C) may have had a two-pronged catalytic effect. On the first hand, temperature may help the sintering of NPs since the fusion temperature of Au nanoparticles becomes close to 300K as their diameter is reduced, this sintering being possibly facilitated by the degradation of the ligands. 16 On the other hand, the temperature much certainly helps the created defects to anneal, hence reduces their number. The important subject of defects annealing will be discussed hereafter.

Protons almost have the same mass as neutrons but are positively charged, therefore Coulomb interactions considerably enhance their interaction with matter. Contrary to neutrons, protons will only be scattered and not absorbed. In large solid samples irradiated with fast particles or ions, the so-called Primary Knock-on Atom (PKA) recoils after the shock and, thanks to its electric neutrality, is able to travel over large distances while disrupting the crystalline lattice and generating defects such as vacancies and dislocation loops. Many of these defects (~70%) either recombine or anneal very rapidly (~ps) yet many stable lattice defects are produced during the course of irradiation, quantified by the DPA (displacement per atom). [START_REF] Nordlund | Primary radiation damage: A review of current understanding and models[END_REF] Estimating the DPA is not an easy task as it depends on many parameters but several models [START_REF] Nordlund | Primary radiation damage: A review of current understanding and models[END_REF] and simulation softwares [START_REF]For example SRIM[END_REF] exist for all kinds of ions and particles. In our case the peak of the Bragg curve is expected to occur at a depth of ca. 10µm in solid gold, experimental conditions were therefore set before this peak, ensuring that the sample was more or less homogeneously irradiated throughout its thickness. Under our experimental conditions the average DPA should be around 0.1, we underline it should only be seen as an approximation. All tables and softwares designed for the assessment of radiation damage are intended for use on massive "macroscopic" materials and we are dealing here with heterogeneous particulate materials and nanostructured materials are expected to behave differently. [START_REF] Järvi | Damage production in nanoparticles under light ion irradiation[END_REF][START_REF] Beyerlein | Effects of He radiation on cavity distribution and hardness of bulk nanolayered Cu-Nb composites[END_REF] It was therefore a bit pointless to try optimizing the irradiation conditions for our systems as the maximum length a collided atom can travel is obviously limited by the diameter of the nanoparticles.

Looking now at the magnetic properties of our neutron-irradiated samples, we reckon that irradiation dose of NEUTR1 was not high enough to produce magnetically significant changes. On the contrary NEUTR4 was too altered (especially the organic ligands) by a very long irradiation, as it received the highest dose of all samples. 21 Sample NEUTR3 is maybe more puzzling since it came out with basically unchanged magnetic properties as compared to NEUTR1, in striking difference with NEUTR2, even though it received ten times more neutrons. A first reason to that could be that due to the intrinsically statistical nature of the experimental process, there is unavoidably some variability from sample to sample, the defects created in the core differing in number and coupling from sample to sample. An illustration of that could probably be the magnetization curves recorded at 4K (ESI), with M(H) of NEUTR1 deviating much more diamagnetism than M(H) of PRIST and NEUTR3. Alternatively, it is also possible that in spite of the longer irradiation times, no so many stable defects were generated in sample NEUTR3 or that their magnetic moments were smaller or less coupled than those created in NEUTR2.

The question of the annealing of the defects is worth considering. Two opposing mechanisms are constantly at play : radioactive decay which introduces defects, and thermal annealing which tends to remove damage. Authors of reference 5 report that thermal annealing of their sample at 300K restores its original magnetic state, the recovery process starting to take effect at temperatures as low as 30K for plutonium. Damage was accumulated for several weeks at 5K but duration of the annealing at various temperatures is not given. While providing useful insight, these experiments cannot be readily compared with ours since our irradiation time was shorter and did not take place at cryogenic temperatures. One last comment regarding the spontaneous annealing of defects has to be made. As our samples became radioactive after irradiation we had to wait several weeks before they could be shipped to us and be handled. This unavoidable lapse of time did not allow us to analyze our results in a similar way -nor as rapidly-as the authors of reference 5. Still, we believe that the same mechanisms are acting and that in our case, because of the higher temperatures (up to 385K), the production of defects will peak at a certain time before decreasing and tending to a plateau. Based on these considerations we suggest a phenomenological model where defects are exponentially created and annealed via an Arrhenius process. We emphasize that this model is purely ad hoc and presented here as a simpler way to explain why the magnetization curve M(H) of some neutron-irradiated samples came out with more drastic changes than others. The evolution rate for the population of defects can be described by the following equation (see ESI for details):

= 𝑒 -𝐴𝑒 ( ) 𝑑𝑡 (1)
which numerical integration results in the curve given in figure 4 :

Figure 4.
Modelling of the number of stable defects vs. time. Red symbols have been placed at the respective irradiation duration of the samples. Y-axis scale is meaningless, this is not a fit. Sumperimposed is the variation of temperature vs. time, at the same timescale (blue curve).

If we make the approximation that the magnetization is proportional to the number of stable defects within a sample we can understand the results returned by the magnetic characterization. Sample NEUTR1 has not been irradiated long enough (ca. 70s), NEUTR2 has been sufficiently irradiated (ca. 300s) but annealing did not proceed efficiently because the temperature did not rise much above 300K. Samples NEUTR3 and NEUTR4, irradiated respectively for ca. 310 3 s and 310 4 s, spent long enough time at 385K for all defects to anneal and therefore returned to their original magnetic state, with the ligands of NEUTR4 being possibly too damaged.

Low temperature behaviour of all samples (ESI) was essentially identical to what was observed at 300K. With a very narrow (~35Oe) opening of the hysteresis cycle and low remnant magnetization, sample NEUTR2 behaves like a soft ferromagnet, the magnetization saturating at a field of ca. 3000Oe. Given the size of the nanoparticles, this saturation value of 0.48emu/g corresponds to ~130µB per nanoparticle, averaged over all the nanoparticles. We underline that there is no reason to calculate a magnetization per gold atom as not all atoms bear a moment, only thoses favorably involved in lattice defects, whose number is unknown. Very probably, some nanoparticles will have a moment higher than the average value while other will have a smaller one or none.The saturation magnetization remains practically constant from 300K to 4K (ESI). Similar observations have been reported for gold and other metals. [START_REF] Nealon | Magnetism in gold nanoparticles[END_REF][START_REF] Shinohara | Surface Ferromagnetism of Pd Fine Particles[END_REF][START_REF] Sampedro | Ferromagnetism in fcc Twinned 2.4 nm Size Pd Nanoparticles[END_REF] This sample therefore consists of a limited (and unknown) number of ferromagnetic nanoparticles. The magnetic moment is localized in a few places and not evenly distributed. [START_REF] Mccall | Emergent magnetic moments produced by self-damage in plutonium[END_REF][START_REF] Lin | Influence of the Crystallinity of Silver Nanoparticles on Their Magnetic Properties[END_REF] Thermal behaviour of the magnetization excludes the possibility of a ferromagnetic impurity being present. It has been possible to fit the magnetization curve recorded at 300K with a Brillouin or Langevin function (ESI) but here again, treating the sample as homogeneous makes little sense.

Sample PROT, which has been irradiated with high energy protons, also behaves differently than the original material even if the change is less spectacular than that of NEUTR2. 21 Nevertheless we obviously cannot link with certainty the observed magnetic properties with these fused nanoparticles which are shapeless and much larger than the original ones. Obviously, the annealing process which is acting for the NEUTR samples is unable to recover the extensive damage suffered by the PROT sample, in spite of the higher temperature (140°C). Moreover, the organic ligands lost their integrity which altered the quality of the interparticle electronic insulation. In any case, point defects such as vacancies or insertions suffice for creating spin imbalances and favoring increased magnetic moments. As will be discussed hereafter, we believe that the appearance of magnetism is mostly due to the presence of lattice vacancies and high-energy protons certainly induce much more damage, as clearly seen in the TEM pictures, than just creating vacancies. In any case, irradiation with energetic protons cannot be directly compared with irradiation with thermal neutrons.

As already mentionned in the introduction, many mechanisms have been proffered for explaining the emergence of para-or ferromagnetism in noble metals nanoparticles. Reasons broadly fall in two categories : some effect related to the ligands attached to the surface, or the peculiar role of some surface atoms which dominate over the core atoms of the nanoparticle as its size decreases. Nevertheless, and this being said, materials may also become magnetic upon the presence of defects [START_REF] Esquinazi | Defect-Induced Magnetism in Solids[END_REF][START_REF] Gubina | Magnetism of Nanosized "Nonmagnetic" Materials; the Role of Defects (Review)[END_REF] . The first material clearly displaying defect-induced magnetism (DIM) was graphite. [START_REF] Kopelevich | Ferromagnetic-and Superconducting-like Behavior of Graphite[END_REF][START_REF] Esquinazi | Ferromagnetism in oriented graphite samples[END_REF] It has also been demonstrated that proton irradiation triggers ferro-or ferri-magnetism in HOPG. [START_REF] Esquinazi | Induced Magnetic Ordering by Proton Irradiation in Graphite[END_REF] The most known cases of DIM are that of oxides, nitrides and semiconductors. Most of the time, lattice vacancies -sites where the material's stoichiometry is not respected because of a missing oxygen or nitrogen atom-are at the origin of the effect. Thin films and nano-objects (e.g. nanoparticles, tubes, ribbons, …) where geometry induces peculiar constraints at edges and surfaces are other examples. [START_REF] Lorite | Detection of Defect-Induced Magnetism in Low-Dimensional ZnO Structures by Magnetophotocurrent[END_REF][START_REF] Sundaresan | Ferromagnetism as a universal feature of inorganic nanoparticles[END_REF][START_REF] Ning | Zhang Defects-Driven Ferromagnetism in Undoped Dilute Magnetic Oxides: A Review[END_REF][START_REF] Zhou | Defect-induced magnetism in SiC[END_REF] Metals may also become magnetic when defects are present. For example, a DFT study has considered the case of stacking faults and twin boundaries in palladium, [START_REF] Alexandre | Magnetism of two-dimensional defects in Pd: Stacking faults, twin boundaries, and surfaces[END_REF] and experiments have indeed reported the onset of ferromagnetism in twinned Pd nanoparticles. [START_REF] Sampedro | Ferromagnetism in fcc Twinned 2.4 nm Size Pd Nanoparticles[END_REF] This is interesting but, as commonly stated, "palladium is close to ferromagnetism" so just a little nudge is enough and the authors indeed suggest that an increase of the density of states at the Fermi level is at the origin of the observed ferromagnetism. Another study mentioned earlier reports the emergence of magnetism in plutonium due to the formation of defects by self-emitted alpha particles, with spinless vacancies inducing magnetic moments on the surrounding lattice. [START_REF] Mccall | Emergent magnetic moments produced by self-damage in plutonium[END_REF] Our hypothesis is that perfect Au nanoparticles would be diamagnetic. We also hypothesize that the atoms around a vacancy acquire a magnetic moment through an increased localization of their 5d electrons. A lower coordination favors narrower bands and higher density of states. An increase of the number of unoccupied 5d states is also possible. In this case an electron is transferred and trapped at a defect site, therefore generating a local magnetic moment.

We underline that we do not claim here that the presence of defects is the sole origin of the magnetic properties of nanoparticles made from noble metals. Indeed, the magnetic moments born by defects can coexist happily with other mechanisms (e.g. surface currents [START_REF] Hernando | Giant magnetic anisotropy at the nanoscale: Overcoming the superparamagnetic limit[END_REF] or persistent currents [START_REF] Gréget | Magnetic Properties of Gold Nanoparticles: A Room-Temperature Quantum Effect[END_REF] ) previously envisioned, and can even reinforce them, contributing to the variegated behaviours that have been reported by many experimenters. The question of how these moments are coupled and interact magnetically is unfortunately not totally clear. Many possibilities exist [START_REF] Nealon | Magnetism in gold nanoparticles[END_REF][START_REF] Lin | Ag Nanocrystals with Nearly Ideal Optical Quality: Synthesis, Growth Mechanism, and Characterizations[END_REF] and analysis of the thermal behaviour of the magnetization should in principle help discriminate. The authors of ref. 5, for example, report that the susceptibility of their sample, at the first order, follows a Curie-Weiss law and deduce that a Kondo-like behaviour is observed. This is not the case here, as the thermal behaviour of our samples is featureless and the magnetization basically does not vary between 4K and 300K.

Conclusions

The question as to why nanoparticles made from gold and other diamagnetic metals become magnetic most probably cannot be answered simply. After having worked on the subject for some time our opinion now is that there are several co-existing mechanisms at play. This study aimed at observing the impact of core lattice defects on the magnetic properties using only physical means and no chemistry. The results presented here prove that lattice defects such as vacancies very certainly contribute to the emergence of para-and ferromagnetism in gold nanoparticles. It would be very interesting to devise a means of performing magnetometry while irradiating the nanoparticles. Performing irradiation and magnetic characterization on the same site could bring more definitive results. Theoretical insight on the parameters that make ferromagnetism stable up to 300K in such systems would also be welcome.

Conflicts of interest

The autors declare no conflict of interest.

Figure 1 .

 1 Figure 1. TEM images of non-irradiated nanoparticles PRIST (a) and neutron-irradiated nanoparticles NEUTR4 (b). Lattice defects are already present in the non-irradiated sample (green arrows indicate a few). Note how the shape of some nanoparticles has become less spherical and that some nanoparticles seem to have fused together after irradiation. High energy protons have had a much more visible effect, as seen on the right-hand picture (c).

  compares those of neutronirradiated sample. Details of the low-field region at 300K and magnetization curves at 4K for all samples are given in ESI.

Figure 2 .

 2 Figure 2. Magnetization vs. applied field of non-irradiated sample PRIST (left) and proton-irradiated sample PROT (right).

Figure 3 .

 3 Figure 3. Magnetization vs. applied field of : neutron-irradiated samples at 300K.

Acknowledgements MVR, BD and JLG acknowledge funding received from EOARD (FA8655-13-1-3001) and ANR (ANR-14-OHRI-008 ; ANR-12-BS10-00301). The microscopy plateform of IPCMS is acknowledged for providing access to the JEOL LaB6 microscope. We thank the CIRCé platform and the CEMHTI pelletron for granting us beamtime on their facilities. The magnetometry platform is acknowledged for access to the SQUID magnetometer. A. Vögele at PSI is thanked for his help. L. Joly is thanked for his help with Blender.

Electronic Supplementary Information

Role of lattice defects on the magnetism of gold nanoparticles Irradiated with Neutrons 1) Thermogravimetric analysis of sample PRIST.

Sample is thermally stable in air up to ~140°C. With an average diameter of 7nm, the Au nanoparticles have a core weighing ca. 3.4710 -18 g and a surface of 1.5410 -16 m 2 . On this surface an estimate number of 900 dodecanethiol ligands are grafted (~0.17nm 2 per ligand), weighing 3.010 -19 g. Supposing that these ligands "burn", the resulting weight loss would represent ca. 8%, which is what is observed on the TGA curve : total weight loss 8.23%.

2) Silica tube which contained sample NEUTR4 after irradiation. Note the sputtering of gold on the inner walls.

5) Tentative fit of sample NEUTR2

Left : Fit to a Langevin function with an average moment m0=3.5µB and a number of moments N=38. Right : Fit to a Brillouin curve with S=5/2 and L=0. Total moment M0=131µB.

6) Phenomenological model

We emphasize that this "model" is purely ad hoc and is presented here as a simpler way to explain why the magnetization curve of some neutron-irradiated samples came out with more drastic changes than others. We make the crude approximation that the magnetization is proportional to the number n of stable defects within a sample. Following the authors of the reference 5, as quoted in the main text, we assume that the vacancies are created with an exponential law :

~ 𝑒 (1) gives the rate of vacancies created vs. time t, 𝜏 ≈ 400𝑠 a characteristic time related to the activity of 198 Au. This rate decreases with time since the number of vacancies that can be created in a given volume is limited by the defect-free volume remaining. Eventually a nmax number of defects will be reached :

Sample temperature T is taken to increase from 300K to 385K in 𝜏 ≈ 120𝑠 (samples are sealed under vacuum which prevents quick thermalization) with a similar law : 𝑇(𝑡) = 300 + 85 1 -𝑒 / (3) We underline that temperature was not controlled, its change over times simply comes from the environmental conditions in the experimental chamber. This increase of temperature with time will favor the annealing of the lattice vacancies, thermally activated following an Arrhenius law :

) the rate of defects annealing, A=610 11 s -1 the usual pre-exponetial factor (here the rate of defects annihilation), Ea =1eV the activation energy for the diffusion of a vacancy, kB the Boltzmann constant. With Ea estimated to be ~1eV, it may seem dubious that 385K would be enough to anneal the defects but authors of ref. 5 (main text) observe that their samples are totally annealed and defectless at 300K (~0.026eV), the annealing process starting to take effect at 30K. Gold is not plutonium but vacancy diffusion energy is roughly the same in most metals (Ea~1.4eV for Pu). Gold temperature of fusion is 1064°C, that of plutonium is 639°C, certainly favoring a quicker thermal annealing. At a given time t a number n(t) of vacancies are present and therefore the number of defects which become annealed each second is :

With these three equations is is possible to built a phenomenological equation which gives the number of defects n in the sample as a function of time t, T being given by equation ( 3):

which can be rewritten :

which is then numerically integrated, yielding the following curve :

Modelization of the number of stable defects vs. time. Red symbols have been placed at the respective irradiation duration of the samples. Y-axis scale is meaningless, this is not a fit, we only wish to illustrate the shape of the curve. Sumperimposed is the variation of temperature vs. time, at the same timescale (blue curve).

The number of defects intially increases with irradiation time but thermal annealing of the lattice vacancies kicks in and reduces the number of defects, until no defects are created anymore. We suggest that this levelling of the effective number of radiation-induced defects, added to the fact that probably not all created defects bear a magnetic moment, allows to understand why a longer irradiation time does not result in a higher magnetization or, at least, why the magnetization does not saturate at some higher value. By pure chance, sample NEUTR2 has been irradiated with a duration close to the optimum for defects creation. NEUTR1 was irradiated not long enough, in samples NEUTR3 and NEUTR4 the annealing process was fully active and overcame the nucleation process.

7) TEM images of proton and neutron irradiated samples. PROT This sample is the one which seems to have suffered the most damage. any case, there are much less "surviving" particles in this sample than in the samples which have been hit with neutrons, e.g. sample NEUTR4 below. Moreover, the lattices defects are not just vacancies NEUTR1 Almost no changes as compared to the original non-irradiated sample.

NEUTR2 NEUTR3 NEUTR4

In spite of the high neutron dose, not all nanoparticles are destroyed (right). Organic ligands probably suffered large damage (sample became unsoluble) but this cannot be assessed with electron microscopy.