Camilla Björklund
email: camilla.bjorklund@ped.gu.se

Ulla Runesson Kempe

Framework for analysing children's ways of experiencing numbers

Keywords: Arithmetic skills, early childhood education, mathematical development, variation theory of learning

This study contributes to the field of arithmetic learning a framework based on a conjecture that learning is a change in ways of experiencing numbersdecisive for what is possible to do with numbers in arithmetic tasks. 309 task-based interviews with 5-6-year-old children were used to develop the framework. To validate its use for studies on arithmetic skills development we selected 7 children who between two interview occasions developed from scoring 0 in the first interview to >75% correct answers in the second interview. Based on the results of these children's changing ways of experiencing numbers, we conclude that arithmetic skills development can be found empirically related to changes in the children's ways of experiencing numbers, which validates the framework as being a useful tool to follow and describe children's arithmetic skills over time.

Background and aim

Early arithmetic development has raised an increasing interest during the last decades, providing the mathematics education field with observations of children's counting strategies (Carpenter, Moser & Romberg, 1982) and learning trajectories [START_REF] Clements | Learning and teaching early math: The learning trajectories approach[END_REF]Cross, Woods & Schweingruber, 2009). Multiple studies within this knowledge area have proven that learning arithmetic is complex but most children do in fact learn to solve simple arithmetic problems during their preschool years. Nevertheless, the development of arithmetic skills is a challenge to study, not least because arithmetic computation includes knowledge and awareness of several aspects of numbers and principles [START_REF] Baroody | Curricular approaches to connecting subtraction to addition and fostering fluency with basic differences in grade 1[END_REF]. Even if children have strategies to solve a task, their strategies may not be based on conceptual understanding of relationship between and within numbers and arithmetical principles, but rather on procedural knowledge (see [START_REF] Fuson | Research on whole number addition and subtraction[END_REF]. Relying on procedural knowledge alone has not shown to facilitate further learning and may even induce mathematics difficulties when encountering more advanced arithmetic tasks [START_REF] Gray | An analysis of diverging approaches to simple arithmetic: Preference and its consequences[END_REF][START_REF] Gray | Duality, ambiguity, and flexibility: A "proceptual" view of simple arithmetic[END_REF][START_REF] Neuman | The origin of arithmetic skills: A phenomenographic approach[END_REF]. Of educational and scientific interest is thereby to be able to interpret the way a child understands numbers and what s/he can do with numbers in arithmetic tasks.

Our approach to learning and development is experiential, based on Variation theory of learning [START_REF] Marton | Necessary conditions of learning[END_REF], meaning that how a child acts when trying to solve an arithmetic task depends on the child's way of experiencing numbers in the task. Variation theory explains learning failures in a specific way and spells out the conditions of learning; when learners fail to learn what was intended, they have not discerned aspects that are necessary to discern. So, the very core idea of Variation theory is that discernment is a necessary condition of learning. Furthermore, the way a learner is experiencing numbers is expressed in the way the learner is acting in solving arithmetic tasks. This approach adds to the field of research an alternative approach to understanding development and learning as changes in ways of experiencing numbers in arithmetic tasks, that is what is discerned by the learner and what is yet to be discerned, rather than learning to use more sophisticated strategies.

The aim of this paper is to describe a framework of children's ways of experiencing numbers. To validate its use for studying arithmetic skills, we used empirical data of children's acts and utterances in interviews from a smaller sample of the whole data set and discuss to what extent these children's increased arithmetical skills can be explained in terms of different ways of experiencing numbers.

The study

This particular study is part of a larger project (FASETT) in which a pedagogical program to enhance children's number knowledge and arithmetic skills was developed. To assess this project we needed a framework to consolidate our conjecture of how children develop arithmetic skills that would allow us to study children's development and find theoretically and empirically sound explanations for their learning and possible impact of implemented teaching programs.

103 preschool children participated in researcher-designed interviews based on addition and subtraction tasks (number range 1-10). The interviews were conducted at three occasions: in the beginning of the last preschool year (5-yearolds), in the end of the last preschool year (8 months later) and after another year in pre-primary school (3 x 103 children = 309 interviews). Each interview was video-recorded, with the legal representatives' permission, to ensure repeated analyses. Ethical clearance has been provided to collect personal data for the longitudinal investigations of the children. Furthermore, we chose one group of children (n=7) who made substantial improvements in their interview scores between interview 1 and 2 (from 0 to >75% correct answers on arithmetic tasks) for thorough analyses of their answers in the interviews.

Developing a theoretically informed framework

The interview data, consisting of 3 x 103 interviews (0% falling-off), was used as empirical basis for developing a framework of the outcome space of different ways of experiencing numbers. The children's acts and utterances regarding eight items were the unit of analyses. Several researchers participated in this thorough work, both individually and collectively to fine-tune the characteristics for each category. Six different ways of experiencing numbers have been found. They are distinct in that each category forms a qualitatively different way of seeing and thus understanding what is possible to do with numbers in an arithmetic task. The framework is generated from the analysis of answers to each task, not as a general expression of a specific child. The ways of experiencing numbers are thus analysed as unique observations of each item in the interviews, which gives us a sample of approximately 2400 observations (309 interviews x 8 items).

Validating the framework

The purpose of this paper is furthermore to validate the framework as a tool for studying children's arithmetic skills development. We thereby compared a group of children (n=7) who in the first interview scored low concerning correct answers (0 of 8 possible) and high in the second interview (>75%), eight months later. Our concern is to what extent the improvement can be explained by their different ways of experiencing numbers. These children's ways of experiencing numbers are, in line with the framework, analysed as unique observations of each item in the interviews, which gives us a sample of 56 observations (7 children x 8 items).

The framework

The result from analysing the interviews constitutes six distinct ways of experiencing numbers (Table 1, left column). These are described in terms of criteria for inclusion (Table 1, middle column).

Ways of experiencing numbers

Criteria for category inclusion Strategies enacted when experiencing numbers in respective ways

As Furthermore, the way to experience numbers is conjectured to be related to what is possible to do with numbers, which means that how children use numbers and express their understanding in words and in gestures are keys to interpreting their ways of experiencing numbers (Table 1, right column). In the following these different ways of experiencing numbers and the implications for solving arithmetic tasks will be described with examples from the interviews.

Numbers as words

Children expressing their experiencing numbers as words say random numbers, either solely as an answer to a task, or as a sequence of words (not in an orderly fashion). This category includes instances when a child repeats one of the number words appearing in the task. Thus, numbers are words used in certain contexts but there is nothing you can do with numbers to solve the task at hand.

Task:

Seven marbles are hidden in two hands. How many could be in each hand? Child:

Five, eight, nine [taps her finger at the right hand of the interviewer, once for each number word], ten [points at both hands], six [points at the right hand] and seven [points at the left hand]. (PFÅ09-1)

Number words are said as an answer to a question with a numerical content, but as shown in the excerpt above, there is hardly a numerical reasoning guiding the child's way of using numbers. Number words are in this sense experienced as a type of word in a certain context. They refer to quantitative situations, but the child does not know what numbers they refer to [START_REF] Wynn | Children's acquisition of the number words and the counting system[END_REF].

Numbers as names

Children seeing numbers as names have been observed by [START_REF] Neuman | The origin of arithmetic skills: A phenomenographic approach[END_REF], and described as a procedure of 'word tagging' by [START_REF] Brissiaud | A tool for number construction: Finger symbol sets[END_REF]. The child considers number words as names given to a counted item and the last uttered word denotes only that single object.

Task:

You have ten candies and eat six of them. How many are left? Child:

[shows all the fingers on the left hand and the thumb on the right raised] Ate six. This is six. And then I took the thumb away. So, it's five. (ASJ10-1)

Numbers as names means that each item within a group is given a specific number name. Taking away one (the thumb) of the six fingers and answering "five" to this specific task, is a reasonable answer if the thumb is seen as "the six". Such named items cannot be added to or subtracted from anything in a true sense, since they do not have a cardinal meaning. What characterizes this way of experiencing numbers is thus the ordinal aspect, which means that for example when fingers are used to count on, each finger is given a specific number name and this name cannot be altered because the other fingers are given their unique number names. The lack of cardinal meaning of these number names is also expressed when children answer with the following number word in the counting sequence. This makes sense to the child since the order of the number words implies that the addition of a quantity will include (at least) the next word in the sequence (the n th item).

Numbers as extent

Experiencing numbers as extent means that children have an awareness of numbers in that they use quantities as approximate [START_REF] Neuman | The origin of arithmetic skills: A phenomenographic approach[END_REF].

Task: Seven marbles are divided in two closed hands, how may the marbles be divided? Child:

[points on the closed right hand] One in that one [points at the closed left hand] five in that one. Interviewer: How did you figure that out? Child:

[points at the left hand] Eight in that one and two in that one [points at the right hand]. (PLS05-1) To experience numbers as extent are in a sense cardinal: children give plausible approximations of smaller and larger numbers that together are close to the correct answer. The lack of ordinality means that children do not make any attempts to count in order to determine the quantity of a set. This also makes it impossible to add or subtract in other ways than giving an approximate estimation of more or less than the given numbers in the task. The answer may though be quite close to the correct answer.

Numbers as countables

Experiencing numbers as countables is strongly influenced by the ordinal aspect of numbers, as children count to make a number, often using their fingers to count on. Numbers are thus seen as added units of 'ones' which may be difficult to coordinate in an arithmetic task.

Task:

You have two shells and your friend five. How many do you have together? Child:

One, two [points at his little and ring finger] wait, I have to start over, 1, 2, 3, 4, 5 [pointing at fingers starting from long finger, index, thumb and thumb, index on the other hand. Then starts again from the little finger] 1, 2, 3, 4, 5. Five! (AVT09-1)

This child knows how to count fingers as representations for the items mentioned in the task. However, he does not seem to discern any structure that would help him solve the task in an easier wayhe is able to create parts but has trouble coordinating them (c.f. [START_REF] Steffe | Children's counting in arithmetical problem solving[END_REF]. Numbers as countables further implies that numbers are not comprehended as a composite set, they are created by adding 'ones'. This becomes visible when using the strategy "counting all". Some children do however experience small sets by subitizing for example three fingers (that will not be counted), but when the quantity exceeds the subitizing range the numbers have to be counted in ones.

Numbers as structure

A structural approach in arithmetic problem solving is assumed to promote children in developing their conceptual knowledge of numbers [START_REF] Davydov | The psychological characteristics of the formation of elementary mathematical operations in children[END_REF][START_REF] Lüken | Young Children's Structure Sense[END_REF][START_REF] Schmittau | Vygotskian theory and mathematics education: Resolving the conceptualprocedural dichotomy[END_REF].

Task: You have two shells and your friend five. How many do you have together? Child:

[puts her left hand with all fingers unfolded on the table, then the right hand with thumb and index finger unfolded, looks at her hand shortly] Seven! [with a confident smile] (HNV02-2)

The child knows the finger pattern for five and two, but needs to create the whole by seeing the parts simultaneously both as parts and as a whole, related to each other. This means that some parts may need to be counted, while other parts are experienced for example as finger patterns, but the parts and the whole are experienced as related to each other and are thus possible to handle as a triad (see [START_REF] Baroody | Curricular approaches to connecting subtraction to addition and fostering fluency with basic differences in grade 1[END_REF]. The difference, compared to experiencing numbers as countables, is the way children experience the parts as composite sets that relate to another set and/or the whole. This opens up for a different way of handling arithmetic tasks, for example in "seeing the five in the eight". This has also shown to be a key feature of some young children's spontaneous ways of successfully solving arithmetic problems [START_REF] Björklund | Structuring versus counting: critical ways of using fingers in subtraction[END_REF].

Numbers as known facts

The last category of experiencing numbers reflects an advanced understanding of number relations.

Either, the number relations are known to the child and s/he does not need to calculate, or s/he has an advanced way of reasoning and using retrieved facts to come up with the answer.

Task: Seven marbles are divided in two closed hands, how may the marbles be divided? Child:

Four in that one and three in that one. 'Cause, if you have three and one more it makes four. You have three in that one [points at the left closed hand] and three in that one and one more, makes seven [points at the right closed hand]. (HRM01-2)

In this example the child knows how three, four (seen as 'three and one more') and seven are related and is able to retrieve from memory known facts ('three and three makes six') that he can operate with to find a plausible solution. The difference between numbers as structure and numbers as known facts lies in the child's approach to the arithmetic task. The former is focused on structuring numbers in that the missing addend will be possible to discern (it appears in the structure), while this latter category means that the child sees number relations instantly and is often able to describe how s/he retrieves the answer from known number relations.

Validating the framework

The categories outlined above derives from the 309 interviews and 8 tasks given in each interview.

Our theoretical conjecture has a pedagogical purpose, which directs our attention towards the extent to which the framework can be used to understand children's developing arithmetic skills. Thus, we analysed the target group who made substantial improvements during an eight months period. In Table 2 we can see that children in the first interview are with only one exception found in the three first categories: Numbers as words, Numbers as names and Numbers as extent. From earlier research we know that a simultaneous awareness of numbers' cardinal and ordinal aspects are important for the child's ability to operate with numbers. Without these aspects coordinated in the arithmetic problem solving process it is not possible to use any counting strategies or arithmetic principles [START_REF] Davydov | The psychological characteristics of the formation of elementary mathematical operations in children[END_REF][START_REF] Fuson | An analysis of the counting-on solution procedure in addition[END_REF]. Experiencing numbers in these ways do not include this crucial aspect of arithmetic skills and the children's low achievement on the arithmetic tasks in interview 1 is thereby no surprise.

The second interview includes most observations within the categories Numbers as countables, Numbers as structure and Numbers as known facts. Children's way of experiencing numbers change, in this group, towards ways that allow them to handle numbers in prosperous ways in arithmetic problem solving. However, there are differences in these categories concerning what they enable the children to do with numbers. In some addition tasks the children who experience numbers as countables manage to solve the problems by 'counting all' and thus working with added 'ones'. This way of experiencing numbers stand in bright contrast to numbers as structure or as known facts, in which the parts to be added or taken away already from the beginning are seen as related to each other as composite sets. Because of this latter relational approach to a task, the child usually avoids cumbersome strategies such as working with one unit at a time. In other words, they can handle arithmetic tasks in a more prosperous way when numbers are experienced as structure.

Discussion

The framework was developed first through qualitative interpretations of a large sample of children's different ways of experiencing numbers in simple arithmetic tasks. In this study, we then used the six categories derived from the large data, to test the extent to which it was possible to use the framework to explain differences in arithmetic skills. The results suggest a confirmation of basic principles of Variation theory that failing to solve a task is due to the learner not being able to discern necessary aspects within the task and the strategies children use reflect what aspects they discern. Consequently, the children who score 0 on their initial interview have not discerned necessary aspects of numbers. This is also shown in our definition of the different ways of experiencing numbers. The validation of the framework further shows a hierarchical structure of the categories: when more necessary aspects are discerned there is also an advancement in what the child can do with the numbers in the tasks.

When analysing children's ways of experiencing numbers, their strategies for solving arithmetic tasks become possible to understand, what works and why? Their ways of experiencing numbers change in most cases from primitive ways (that do not enable them to operate with arithmetic tasks) to quite advanced ways of experiencing numbers (that allow them to solve arithmetic tasks as partpart-whole relations). The framework opens for interpreting how new experiences may advance children's arithmetic skills that does not depend on correct answers, or learnt strategies, instead it offers an alternative: what it means to learn about numbers and how to do simple addition and subtraction.

The analysis of the changes in ways of experiencing numbers seems to be solid enough to explain why some children have difficulties solving arithmetic tasks. Since our target group contained children who scored 0 points and improved to almost perfect score, we would suggest that changes in ways of experiencing numbers are possible over time and the ways to experience numbers have significant impact for children's possibilities to solve arithmetic tasks. We would argue that more attention should be directed towards what children are able to discern in arithmetic situations, than strategies to come up with a correct answer. However, more research on a larger sample is needed to confirm our statement.

Table 1 . Overview of the criteria for category inclusion and empirical expressions

 1

	As names	-Numbers are ordered in a sequence	-Ascribing number names to objects
		-Number words describe "the n th "	-Saying the following number word
		object, no cardinal meaning	in the counting sequence as an answer
		-Names cannot be added or subtracted	to an arithmetic task
		from other names, because they lack	-Static way of representing a number
		cardinal meaning	with fingers
	As extent	-Numbers are approximate with a	-No attempts to count single units
		cardinal meaning	-Numbers are used to give a plausible
		-Some sense of plausible quantities	description or comparison of quantities
		related to the task	(however not exact)
	As	-Strong influence of the ordinal aspect	-Counting all parts and the whole
	countables	of number	-Exceeding the subitizing range
		-Added units of 'ones'	imposes counting to determine a set
	As structure -Numbers are composite sets	-Using finger patterns to structure
		-The part-part-whole relation is	numbers' parts and whole
		constituted in the act	-Operating on the relation between
			parts and/or the whole
	As known	-Numbers are instantly recognized as	-Giving an instant (correct) answer
	number facts	a part-whole relation	-Retrieving from known facts

words -Number words are used in quantitative contexts but without numerical meaning -Random number words are used solely or as a sequence, or -Number words from the given task are repeated

  Table2shows the number of observations within each category in the first and the second interview.

	words	names	extent	countables	structure	known facts
	10 / 0	11 / 0	10 / 6	1 / 9	0 / 22	0 / 16

Table 2 : Occurrences of ways of experiencing numbers, interview 1 / interview 2

 2

Acknowledgment

This study was funded by the Swedish Research Council (grant no. 721-2014-1791).


Conception Paul

Hélène Delpech

Delphine Haouzi

Samir Hamamah

Claude Sardet
email: claude.sardet@inserm.fr

Eric Fabbrizio
email: eric.fabbrizio@inserm.fr

H El Ene Delpech

Coprs inactivation leads to a derepression of LINE1 transposons in spermatocytes

Keywords: coprs, Miwi, piRNA, spermatocyte, teratozoospermia, testis

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Repression of retrotransposons is essential for genome integrity during germ cell development and is tightly controlled through epigenetic mechanisms. In primordial germ cells, protein arginine N-methyltransferase (Prmt5) is involved in retrotransposon repression by methylating Piwi proteins, which is part of the piRNA pathway. Here, we show that in mice, genetic inactivation of coprs (which is highly expressed in testis and encodes a histone-binding protein required for the targeting of Prmt5 activity) affects the maturation of spermatogonia to spermatids. Mass spectrometry analysis revealed the presence of Miwi in testis protein lysates immunoprecipitated with an anti-Coprs antibody. The observed deregulation of Miwi and pachytene pre-piRNAs levels and the derepression of LINE1 repetitive sequences observed in coprs-/-mice suggest that Coprs is implicated in genome surveillance mechanisms.

Spermatogenesis is a multistep process that takes place in the seminiferous tubules whereby mature spermatozoa are continuously produced during adulthood reproduction lifetime. Specifically, primordial germ cells (PGCs) generate spermatogonia that either self-renew to maintain a pool of stem cells or undergo differentiation. After two meiotic divisions, secondary spermatocytes are produced. Then, during spermiogenesis, they differentiate into elongating spermatids and finally spermatozoa [START_REF] Dym | Spermatogonial stem cells of the testis[END_REF][START_REF] Fang | Purification and functional characterization of SET8, a nucleosomal histone H4lysine 20-specific methyltransferase[END_REF][START_REF] Kimmins | Chromatin remodelling and epigenetic features of germ cells[END_REF][START_REF] Lacham-Kaplan | In vivo and in vitro differentiation of male germ cells in the mouse[END_REF][START_REF] Mclaren | Germ and somatic cell lineages in the developing gonad[END_REF]. Importantly, these changes are associated with chromatin compaction, transcription arrest, and specific epigenetic modifications that contribute to the production of mature sperm. Concomitantly, to maintain genome integrity, PGCs need to repress repetitive DNA elements because their spread can lead to inherited diseases. These repetitive sequences [for instance, long interspersed nuclear elements 1 (LINEs), short interspersed nuclear element (SINEs), and intracisternal A-particle (IAP) sequences] derive mostly from transposable elements and represent about 40% of the human genome [START_REF] Kimmins | Chromatin remodelling and epigenetic features of germ cells[END_REF][START_REF] Russell | Patterns of mutational sensitivity to chemicals in poststem-cell stages of mouse spermatogenesis[END_REF][START_REF] Steger | Haploid spermatids exhibit translationally repressed mRNAs[END_REF][START_REF] Unhavaithaya | MILI, a PIWI-interacting RNA-binding protein, is required for germ line stem cell self-renewal and appears to positively regulate translation[END_REF]. Thus, their repression is challenging and is tightly controlled in germ cells by the Piwi-interacting RNA (piRNA) pathway and through epigenetic mechanisms [START_REF] Chuma | piRNA and spermatogenesis in mice[END_REF][START_REF] Pillai | piRNAs and their involvement in male germline development in mice[END_REF]. In the mouse, this pathway involves the three Piwi proteins Miwi, Mili, and Miwi2 (PIWIL1, PIWIL2, and PIWIL4 in humans, respectively) that show different developmental expression patterns in testis [START_REF] Bao | Conditional inactivation of Miwi2 reveals that MIWI2 is only essential for prospermatogonial development in mice[END_REF][START_REF] Reuter | Miwi catalysis is required for piRNA amplification-independent LINE1 transposon silencing[END_REF]. Mili is expressed in fetal germ cells up to the round spermatid stage, while Miwi2 expression is restricted to fetal and perinatal germ cells. In adult testes, Miwi is expressed in pachytene spermatocytes up to round spermatid testes [START_REF] Vourekas | Mili and Miwi target RNA repertoire reveals piRNA biogenesis and function of Miwi in spermiogenesis[END_REF]. Mili and Miwi bind to both piRNAs in pachytene spermatocytes and postmeiotic spermatids [START_REF] Zheng | Blockade of pachytene piRNA biogenesis reveals a novel requirement for maintaining post-meiotic germline genome integrity[END_REF]. Deletion of the Miwi gene arrests cells at the round spermatid stage, and Mili deficiency leads to male sterility [START_REF] Zheng | Blockade of pachytene piRNA biogenesis reveals a novel requirement for maintaining post-meiotic germline genome integrity[END_REF]. Chromatin-modifying enzymes also participate in the regulation of these repetitive elements. For instance, protein arginine N-methyltransferase 5 (Prmt5) is implicated in the piRNA pathway in PGCs via a Prmt5-dependent methylation of Piwi proteins. In somatic cells, Prmt5 modulates cell proliferation and differentiation, for instance, during myogenesis and adipogenesis, by interacting with various proteins, including Coprs (previously termed Copr5), a histone-binding protein [START_REF] Karkhanis | Versatility of PRMT5-induced methylation in growth control and development[END_REF][START_REF] Leblanc | Protein arginine methyltransferase 5 (Prmt5) promotes gene expression of peroxisome proliferator-activated receptor gamma2 (PPARgamma2) and its target genes during adipogenesis[END_REF][START_REF] Dacwag | Distinct protein arginine methyltransferases promote ATP-dependent chromatin remodeling function at different stages of skeletal muscle differentiation[END_REF][START_REF] Bandyopadhyay | HOXA9 methylation by PRMT5 is essential for endothelial cell expression of leukocyte adhesion molecules[END_REF][START_REF] Huang | Type II arginine methyltransferase PRMT5 regulates gene expression of inhibitors of differentiation/DNA binding Id2 and Id4 during glial cell differentiation[END_REF][START_REF] Migliori | Symmetric dimethylation of H3R2 is a newly identified histone mark that supports euchromatin maintenance[END_REF][START_REF] Lacroix | The histone-binding protein COPR5 is required for nuclear functions of the protein arginine methyltransferase PRMT5[END_REF][START_REF] Paul | The Wnttarget gene Dlk-1 is regulated by the Prmt5-associated factor Copr5 during adipogenic conversion[END_REF][START_REF] Kirino | Role of arginine methylation in the Piwi-interacting RNA pathway[END_REF][START_REF] Vagin | Proteomic analysis of murine Piwi proteins reveals a role for arginine methylation in specifying interaction with Tudor family members[END_REF]. In addition, Prmt5-methylated arginine residues can be bound by proteins harboring a Tudor domain. In PGCs, many of these proteins (e.g., Piwi proteins) are essential for male fertility [START_REF] Siomi | How does the royal family of Tudor rule the PIWI-interacting RNA pathway?[END_REF]. It was suggested that Prmt5 maintains DNA integrity during global epigenetic reprogramming and that its subcellular localization affects the piRNA pathway, at least in part, and promotes transposon silencing [START_REF] Kim | PRMT5 protects genomic integrity during global DNA demethylation in primordial germ cells and preimplantation embryos[END_REF]. In addition, conditional loss of Prmt5 in early mouse PGCs causes complete male and female sterility that was attributed to global impairment of DNA demethylation in the genome [START_REF] Kim | PRMT5 protects genomic integrity during global DNA demethylation in primordial germ cells and preimplantation embryos[END_REF].

Here, we show that in coprs-/-mice, maturation of spermatogonia to spermatids is affected, although animals are fertile. Moreover, the mRNA levels of Ccna1 (cyclin A1), Prm1 (sperm protamine P1), Miwi, pachytene piRNA precursors, and LINE1 are deregulated in coprs-/testes, suggesting that Coprs contributes to genome surveillance mechanisms via the Piwi-piRNA pathway.

Materials and methods

Flow cytometry

Extraction of cells in testis was performed essentially as described in Mays-Hoopes et al. [START_REF] Mays-Hoopes | Preparation of spermatogonia, spermatocytes, and round spermatids for analysis of gene expression using fluorescence-activated cell sorting[END_REF]. Testes from wild-type (WT n = 3) and knockout (KO n = 4) mice were dissected and decapsulated to release the tubules. Seminiferous tubules were incubated with 0.25 mgÁmL À1 collagenase type IV (Sigma) at 37 °C under rapid agitation for 5 min and washed to release Leydig cells and interstitial cells. Dispersed tubules were allowed to settle and washed twice to remove peritubular cells. Washed tubules were then incubated with 0.5% trypsin/EDTA (Gibco) and 1 lgÁmL À1 DNase RQ1 (Promega) at 37 °C for 5 min. Trypsin digestion was stopped by adding DMEM with 10% FBS. Suspensions were washed and disaggregated into single-cell suspensions by trituration before filtration through a 50-lm cell strainer. For analysis, cells were fixed in 0.4 M citrate buffer (pH 4.5) overnight and resuspended in 1 mL of cold propidium iodide (PI) staining solution (10 mM Tris/HCl (pH 8.0), 1 mM NaCl, 0.1% Nonidet P-40, 50 lgÁmL À1 PI, 10 lgÁmL À1 RNase A), vortexed, and incubated on ice for 10 min to lyse the plasma membrane and stain nuclear DNA. DNA content was assessed on a FACSCalibur II (Becton Dickinson) equipped with the CEL- LQUEST software.

Western blotting

Anti-Prmt5 (Millipore) and anti-Miwi (Abcam) antibodies were used according to the manufacturer's instructions. The anti-Coprs antibody (AGRO-BIO, Clermont-Ferrand, France) was against the last 20 amino acids of Coprs C terminus [START_REF] Lacroix | The histone-binding protein COPR5 is required for nuclear functions of the protein arginine methyltransferase PRMT5[END_REF].

Mass spectrometry analysis

In-gel digestion of bands excised from Colloidal Bluestained gel was done before LC/MS/MS analysis that was performed at the Taplin Mass Spectrometry Facility (Harvard Medical School, Boston).

Immunohistochemistry (IHC)

Tissues were fixed in Bouin's fixative and embedded in paraffin. Then, 4-lm-thick sections were cut and processed for IHC staining. IHC was performed using the same anti-Coprs and anti-Miwi (Abcam) antibodies employed for western blotting, followed by a biotinylated secondary antibody coupled to the streptavidin-peroxidase complex (ABC Vectastain Kit; Vector Laboratories). Revelation was performed with the peroxidase substrate DAB (3,3 0 -diaminobenzidine) from Vector Laboratories.

RNA isolation, cDNA synthesis, and RT-qPCR amplification from mice

Total semen RNA was isolated with the TRIzol Reagent (Life Technologies) according to the manufacturer's instructions. Briefly, 800 lL of TRIzol and 200 lL of chloroform were added to 200 lL of sperm. The mixture was mixed for 15 s and left at room temperature for 5 min. After centrifugation at 12 000 g at 4 °C for 15 min, supernatants were transferred to fresh tubes containing 1 volume equivalent of 70% ethanol. Then, total RNA was purified using the miR-Neasy Serum/Plasma Kit (Qiagen) according to the manufacturer's recommendations. RNA was quantified with a NanoDrop ND-1000 spectrophotometer (NanoDrop Technologies Inc., DE, USA). RNA isolation from mouse testes and RT-qPCR were performed as described [START_REF] Bandyopadhyay | HOXA9 methylation by PRMT5 is essential for endothelial cell expression of leukocyte adhesion molecules[END_REF]. Briefly, testes were lysed in TRIzol reagent (Invitrogen), and total RNA was isolated according to the manufacturer's recommendations. cDNA was synthesized from 1 lg of total RNA using random hexamers and SuperScript III Reverse Transcriptase (Invitrogen). Real-time qPCR was performed on a LightCycler 480 SW 1.5 apparatus (Roche) with Platinum Taq DNA Polymerase (Invitrogen) in the presence and the SYBR Green Mix.

Oligonucleotide sequences

Oct4 f CAATGAGAACCTTCAGGAGATATGC, r TCA ATGCTAGTTCGCTTTCTCTTC;

Amh f TAGTCCTACATCTGGCTGAAGTGATATG, r CCAGGTGGAGGCTCTTGGA;

Ccna1 f CACTTCCTGCTGGATTTCAAC, r CGATGA ATCTCCTCTGCATAC;

Prm1 f GCCGCTCATACACCATAAGG, r CAAGAT GTGGCGAGATGC;

Prm2 f GCAGAAGATCCCGAAGGAG, r CTCCAGG CAGAATGGACAG;

H1t2 f AGGGAAGAGAAGGGACAGGA, r CTTGGA GCCCATATGGAAAA;

Hils1 f GTCCCAAGCCAGAGTGAGAG, r CTTGAA GCGCCAGGTGTTAT;

Miwi f ATGATCGTGGGCAT, r AGGCCACTGCTGT CATA;

pre-piR1 f GTTAGCGAAGGACATTATTCTAACC, r TGACATGAACACAGGTGCTCAGAT;

pre-piR2 f CTATGCTTATGATGGCATTGGAGAG, r TTCCAGTTCAACAGGGACACGGGAC;

pre-piR3 f GTTCTCACTTTATCAGCTCTCAAG, r TGAGAGTGGCATCTAAATGTTTAG;

Line1 f GGAGGGACATTTCATTCTCATCA, r GCTG CTCTTGTATTTGGAGCATAGA;

SineB1 f GTGGCGCACGCCTTTAATC, r GACAGG GTTTCTCTGTGTAG;

IAP f ATTGTTCCCTCACTGGCAAA, r ATTGTTCC CTCACTGGCAAA;

Actin f AGAAGAGCTATGAGCTGCCT, r TCATCG TACTCCTGCTTGCT;

hCOPRS f TCCGCCTCCACTGATACCC, r GGCCCG CCCACCTACAA;

hHPRT f TGACACTGGCAAAACAATGCA, r GGTC CTTTTCACCAGCAAGCT.

Mice and animal care

Animal experiments were approved by the Ethics Committee of the Languedoc-Roussillon Region (France).

Patient's characterization

Nine patients were recruited from our IVF/ICSI program at the ART-PGD Department of Montpellier University Hospital, France, after signature of the written informed consent. Each patient was classified according to the fraction (%) of typical spermatozoon forms (TF) evaluated using one of three classical methods [START_REF] Kruger | Sperm morphologic features as a prognostic factor in in vitro fertilization[END_REF][START_REF] Bisson | Ultrastructural study of anomalies of the acrosome in spermatozoa with irregular heads[END_REF][START_REF] Eustache | Inter-individual variability in the morphological assessment of human sperm: effect of the level of experience and the use of standard methods[END_REF]. The study methodologies were approved by the local ethics committee and conformed to the standards set by the Declaration of Helsinki.

Results

Coprs is localized in spermatogonia and its genetic inactivation leads to male germ cell accumulation at the pachytene stage Our previous studies [START_REF] Lacroix | The histone-binding protein COPR5 is required for nuclear functions of the protein arginine methyltransferase PRMT5[END_REF] and the available Gene Expression Omnibus (GEO) data on Coprs expression profile in various human tissues confirmed that it was strongly expressed in testis (Fig. S1). To investigate Coprs role in testis, we used coprs KO mouse [START_REF] Paul | The Wnttarget gene Dlk-1 is regulated by the Prmt5-associated factor Copr5 during adipogenic conversion[END_REF]. Morphological analysis of 8-and 36-week-old coprs KO males did not show testicular atrophy or reduced body weight (data not shown). IHC analysis of paraffin-embedded testis sections prepared from coprs KO animals and control littermates showed that Coprs was expressed only in germ cells and was mostly localized in the nucleus, although the anti-Coprs antibody stained also faintly the cytoplasmic compartment (Fig. 1A; note the absence of expression in coprs KO testis). A similar nuclear localization was previously observed in different culture cell types [START_REF] Lacroix | The histone-binding protein COPR5 is required for nuclear functions of the protein arginine methyltransferase PRMT5[END_REF]. As we and others previously reported that Coprs is implicated in cell differentiation [START_REF] Leblanc | Protein arginine methyltransferase 5 (Prmt5) promotes gene expression of peroxisome proliferator-activated receptor gamma2 (PPARgamma2) and its target genes during adipogenesis[END_REF][START_REF] Dacwag | Distinct protein arginine methyltransferases promote ATP-dependent chromatin remodeling function at different stages of skeletal muscle differentiation[END_REF][START_REF] Bandyopadhyay | HOXA9 methylation by PRMT5 is essential for endothelial cell expression of leukocyte adhesion molecules[END_REF][START_REF] Paul | The Wnttarget gene Dlk-1 is regulated by the Prmt5-associated factor Copr5 during adipogenic conversion[END_REF], we analyzed germ cell maturation and the distribution of the different cell types in WT and KO testes by flow cytometry. The proportion of germ cells that progressed through meiosis in the leptotene and zygotene stages was reduced in KO mice, and cells accumulated at the pachytene stage (Fig. 1B). Moreover, the proportion of round and elongated spermatids was lower in coprs KO than WT testes (Fig. 1B). These data suggest that Coprs absence perturbs spermatogenesis between the end of meiosis and spermiogenesis.

To further document at what stage coprs inactivation impaired the maturation of spermatogonia, we measured by RT-qPCR the expression level of various differentiation markers in testes of 8-week-old mice.

The mRNA levels of Oct4 and Amh, two markers of undifferentiated spermatogonia and Sertoli cells, respectively, were not significantly different between WT and KO samples (Fig. 1C). This suggested that the initial pool of undifferentiated spermatogonia was comparable in WT and KO animals. In agreement with the flow cytometry data, the mRNA levels of the premeiotic marker Ccna1 were downregulated, whereas the level of the histone variant H1t2, which is usually expressed in pachytene spermatocytes and persists in early spermatids, was upregulated in coprs KO testes (Fig. 1C). Similarly, Prm1 mRNA level was significantly increased in coprs KO testis cells (Fig. 1C). In contrast, spermatid-specific linker histone Hils1 and Prm2 levels did not vary significantly between WT and KO in elongated spermatids (Fig. 1C). Consequently, the Prm1/Prm2 ratio, an indicator of chromatin compaction [START_REF] Aoki | Protamine levels vary between individual sperm cells of infertile human males and correlate with viability and DNA integrity[END_REF], was significantly higher in coprs KO testes than in controls (1.34 in coprs KO mice vs 0.92 in WT animals), suggesting that Coprs deficiency results in perturbations of chromatin compaction.

These data indicated that coprs is implicated in spermatocyte maturation.

Coprs gene ablation leads to deregulation of the piRNA pathway and derepression of LINE1 repetitive elements

To explore how Coprs regulates spermatocyte maturation, we next characterized Coprs protein interactors in testis by proteomics. After immunoprecipitation of Coprs in cell lysates prepared from testes of WT and coprs KO mice, and immunoprecipitate separation by SDS/PAGE and Colloidal Blue staining, we excised bands that were present in WT but not in KO extracts for mass spectrometry analysis (Fig. 2). Although Prmt5 and histone H4 are well-characterized direct partners of Coprs [START_REF] Lacroix | The histone-binding protein COPR5 is required for nuclear functions of the protein arginine methyltransferase PRMT5[END_REF], we did not identify out of the background any specific band in the range corresponding to Prmt5 molecular weight. Conversely and as expected, histones were differentially enriched in Coprs immunoprecipitates, validating the efficiency of the immunopurification procedure. Besides histones, Miwi was the most differentially enriched protein in Coprs immunoprecipitates, suggesting that Coprs is part of a Piwi complex (Fig. 2). Of note, this analysis identified also Stk31 (Tdrd8), a Tudor domain containing protein that interacts with Miwi and Mili upon Prmt5-dependent methylation, but not critical for male germ cell development. IHC analyses of testis tissue sections showed that Miwi staining was lower in samples from coprs KO mice than WT littermates (Fig. 3A). Similarly, Miwi protein (western blotting) and mRNA (RT-qPCR) levels were significantly reduced in testes from coprs KO mice compared with WT littermates (Fig. 3B,C).

As Miwi and piRNAs associate in a functional complex that controls genome integrity, we evaluated whether piRNAs expression was altered upon coprs gene ablation. RT-qPCR analysis showed that pachytene piRNA precursors (pre-piRNAs) 1, 2, and 3 were less abundant in coprs KO than WT testes (Fig. 4A). Although 75% of piRNAs expressed at the pachytene stage are not yet annotated, approximately 17% of them play a key role in inhibiting retrotransposon expression [START_REF] Castaneda | Reduced pachytene piRNAs and translation underlie spermiogenic arrest in Maelstrom mutant mice[END_REF]. Therefore, we used the RNA expression levels of the most abundant retrotransposons, namely LINE1, SINE B1, and IAP, as readout of potential dysfunction in the piRNA pathway. The transcript level of LINE1, but not of SINE B1 and IAP, was upregulated in coprs KO compared to WT testis samples (Fig. 4B).

Altogether, these results suggest that Coprs is involved in the piRNA pathway and that its deficiency in male germ cells perturbs LINE1-related genome surveillance mechanisms.

Coprs and teratozoospermia

To evaluate whether the molecular alterations identified in coprs KO testes influenced reproduction, we first measured the mating efficiency of coprs KO males and control littermates. Several independent crosses of coprs heterozygous animals identified a non-Mendelian inheritance of coprs-floxed allele in males (Fig. 5A). Matings of coprs KO males with WT females were successful without noticeable reduction of litter size (data not shown), indicating that spermatocyte maturation impairment in these mice was not severe enough to impair fertility. Considering that several knockout mice with reduced sperm count or altered spermatogenesis, but normal fertility, have been described previously [START_REF] Schurmann | Reduced sperm count and normal fertility in male mice with targeted disruption of the ADP-ribosylation factor-like 4 (Arl4) gene[END_REF], we evaluated the number, motility, and morphology of spermatozoa in 11-week-old mice. We did not find any significant difference between WT and coprs KO mice (Fig. 5B,C). However, probing GEO data for biological situation of differential Coprs expression, low level of Coprs mRNAs correlated strongly with teratozoospermia, a human pathology linked to male infertility (Fig. 6A). Therefore, we conducted a pilot experiment to evaluate Coprs mRNA level by RT-PCR in patients with teratozoospermia associated with fertility defects and confirmed that it was downregulated compared with patients without teratozoospermia (Fig. 6B,C).

Discussion

Here, we show in mouse that Coprs is expressed in spermatogonia and that in its absence, maturation of spermatogonia during meiosis and spermiogenesis is slightly impaired. This leads to a significant accumulation of spermatocytes at the pachytene stage and a decrease of round/elongated spermatids. Many changes occur within the cell nucleus during spermatocyte maturation, including the programmed replacement of histones by a set of basic nuclear proteins such as histone variants and protamines to increase chromatin compaction, an essential event for male fertility. Consistent with an accumulation of cells in pachytene, the mRNA level of H1t2, a histone variant that is specifically expressed in pachytene spermatocytes, was increased in coprs KO cells. Moreover, the Prm1/Prm2 ratio was increased in KO spermatocytes, a parameter that was reported as being associated with male infertility in humans [START_REF] Aoki | Sperm protamine 1/protamine 2 ratios are related to in vitro fertilization pregnancy rates and predictive of fertilization ability[END_REF][START_REF] Carrell | The human sperm epigenome and its potential role in embryonic development[END_REF][START_REF] Chevaillier | Anomalous protein complement of sperm nuclei in some infertile men[END_REF]. Our proteomic analysis identified Miwi as a potential Coprs interactor, although at this stage our data do not address whether this interaction is direct or through the presence of these two proteins within a common complex. Moreover, both Miwi protein and mRNA levels were significantly decreased in coprs KO mice, as well as the expression level of pachytene piRNAs. Altogether, these results suggest that the Miwi-piRNA pathway is altered in coprs KO testis. How Coprs controls Miwi and piRNA levels remains unknown, and future studies will address this point. It can be hypothesized that the stability of Miwi protein or Miwi complexes requires Coprs. In addition, the reduced Miwi level in coprs KO testes, resulting de facto in a lower Miwi slicer activity, might impact on piRNAs that target active retrotransposons, such as LINE1, thus weakening the maintenance of transposon silencing [START_REF] Reuter | Miwi catalysis is required for piRNA amplification-independent LINE1 transposon silencing[END_REF].

In contrast to Miwi KO mice in which spermatogenesis is arrested at the round spermatid stage with no retrotransposon upregulation [START_REF] Zheng | Blockade of pachytene piRNA biogenesis reveals a novel requirement for maintaining post-meiotic germline genome integrity[END_REF], coprs KO testes present only a slower spermatogenesis progression but showed a significant derepression of LINE1 expression. This suggests that the low but detectable level of Miwi present in coprs KO animals is sufficient to ensure spermatogenesis and that Coprs could contribute to the control of LINE1 expression and genome stability beyond its impact on Miwi expression. A similar mild phenotype, that is, normal fertility and viability associated with a derepression of LINE1 transcripts), was described in mice deficient for Exd1 which encodes for a partner of Miwi2 piRNA biogenesis factor TDRD12 [START_REF] Yang | PIWI slicing and EXD1 drive biogenesis of nuclear piRNAs from cytosolic targets of the mouse piRNA pathway[END_REF]. Importantly, the functional consequences of Exd1 loss in testis, that is, massive derepression of LINE1 elements and an arrest in spermatogenesis, are revealed only in the Exd1 À/À ;Tdrd12 +/À genetic background [START_REF] Pandey | Exonuclease domain-containing 1 enhances MIWI2 piRNA biogenesis via its interaction with TDRD12[END_REF]. As well, we can make the assumption that a stronger impact of Coprs depletion on spermatogenesis might be observed by breeding coprs mice with animals in which other Piwi-associated protein-encoding genes have been invalidated. However, we cannot exclude on the basis of the non-Mendelian inheritance of the coprs-floxed allele that the more severe coprs KO phenotypes might affect early embryo viability. This could explain how coprs absence led to an apparent mild phenotype compared with that of Miwi and Miwi-associated proteins. Thus, it is possible that KO mice upon F2 and/or the following generations might affect more strongly the reproduction parameters and give rise to a phenotype reminiscent of human teratozoospermia, a pathology associated with low Coprs expression level. Although this conclusion requires confirmation in a larger cohort of patients, these data support the hypothesis that Coprs is a candidate diagnostic marker for teratozoospermia.

Fig. 1 .

 1 Fig. 1. In mice, Coprs is localized in spermatogonia and its absence delays spermatid maturation. (A) Left panel: IHC staining of WT (n = 2) and KO (n = 2) testis sections with an anti-Coprs antibody is presented. Bar = 1 mm. Right panel: images at higher resolution. Bar = 50 lm. (B) Flow cytometry analysis (left panels) and histograms (right panel) showing the distribution of the different mouse testis cell types. Sc: Sertoli cells; Rs/Re: round/elongated spermatids; SpB: type B spermatogonia; SpA: type A spermatogonia; L, Z, and P: leptotene, zygotene, and pachytene spermatocytes. Results are expressed as the percentage of each cell type relative to all counted cells and are the mean AE SEM (n = 3 WT; n = 4 KO). *P < 0.1 and **P < 0.05 (Student's t-test). (C) RT-qPCR analysis of the expression of the indicated genes in WT and KO testes. Expression was normalized to actin expression, and values are expressed in arbitrary units (a.u.) and are the mean AE SEM of RNA samples independently prepared (n = 3 WT; n = 4 KO). *P < 0.05 (Student's t-test).

Fig. 2 .

 2 Fig. 2. Identification of Miwi as a major protein immunoprecipitated with Coprs. Protein extracts from WT and coprs KO testes were immunoprecipitated with an anti-Coprs antibody. Bands (*) present only in the WT lane were excised from the Colloidal Blue-stained gel (left) and analyzed by mass spectrometry (right).

Fig. 3 .

 3 Fig. 3. coprs KO affects Miwi expression level. (A) IHC analysis of WT and KO mouse testis sections with or without (negative control, -) an anti-Miwi antibody; bar = 50 lm. (B) Left panel: western blot detection of Miwi expression in whole-cell extracts from WT and KO testes with anti-Miwi and anti-Gapdh antibodies; right panel: quantification of Miwi expression with ImageJ after normalization to Gapdh expression. Values are expressed in arbitrary units (a.u.) and are the mean AE SEM of two independent experiments. (C) Miwi expression was assessed by qPCR. Data were normalized to actin level and expressed in arbitrary units (a.u.) (mean AE SEM of three independent mice/group). **P < 0.05 (Student's t-test).

Fig. 4 .Fig. 5 .

 45 Fig. 4. coprs KO deregulates the expression of pre-piRNAs and LINE1. (A) The expression profile of the indicated pachytene piRNAs precursors (prepiR1, prepiR2, and prepiR3) was assessed by qPCR and normalized to actin level. Values are expressed in arbitrary units (a.u.) and are the mean AE SEM of three independent mice/group. (B) The RNA profile of the indicated transposons was assessed by qPCR and analyzed as in A); *P < 0.1; **P < 0.05; ***P < 0.001 (Student's t-test).

CFig. 6 .

 6 Fig. 6. Low level of Coprs mRNA and teratozoospermia. (A) Transcription profiling of human sperm from individuals with normospermia (C: normally fertile) and teratozoospermia (T) (GEO data: E-GEOD-6872). (B) Coprs expression in semen mRNA samples from controls (C, n = 3)and patients with fertility problems (T, n = 6) was analyzed by RT-qPCR. Data normalized to HPRT RNA and expressed in arbitrary units (a.u.) represent the mean AE SD; *P < 0.1 (Student's t-test). (C) Clinical classification of each patient according to the % of typical spermatozoon forms (TF) evaluated using one of three classical methods, as indicated[START_REF] Schurmann | Reduced sperm count and normal fertility in male mice with targeted disruption of the ADP-ribosylation factor-like 4 (Arl4) gene[END_REF][START_REF] Aoki | Sperm protamine 1/protamine 2 ratios are related to in vitro fertilization pregnancy rates and predictive of fertilization ability[END_REF][START_REF] Carrell | The human sperm epigenome and its potential role in embryonic development[END_REF].

FEBS Open Bio 9 (2019) 159-168 ª 2018 The Authors. Published by FEBS Press and John Wiley & Sons Ltd.

FEBS Open Bio 9 (2019) 159-168 ª 2018 The Authors. Published by FEBS Press and John Wiley & Sons Ltd.

Acknowledgements

We would like to thank especially the RHEM facility for tissue sections. This work was performed thanks to the institutional support of the French CNRS. This work was supported by grants from the Ligue Contre le Cancer (CS) and the Association pour la Recherche contre le Cancer (EF).

Conflict of interest

The authors declare no conflict of interest.

Authors contributions

PC executed and analyzed FACS experiments; HD carried out immunohistochemistry experiments; DH and SH provided human samples and critical discussions of the manuscript; SC provided critical discussions of the manuscript; and FE carried out PCR data analysis, designed the study, and drafted the manuscript.

Supporting information

Additional supporting information may be found online in the Supporting Information section at the end of the article.


N Ouisse

Bouhaddi

Keywords: Quasi-periodic structures, band diagram, frequency stop-band, forced response, metamaterials

published or not. The documents may come

Numerical investigations and experimental measurements on the structural dynamic behaviour of quasi-periodic meta-materials

S Timorian, M.

Numerical investigations and experimental measurements on the structural dynamic behaviour of quasi-periodic meta-materials S. Timorian 1,2 , M. Ouisse 2 , N. Bouhaddi 2 , S. De Rosa 1 and F. Franco 1

Introduction

Recent literature review shows the design of metamaterials as one of the central topics in the vibroacoustic analysis of periodic media. There are interesting subjects in the literature linked to the current investigation which is intended mainly to the design of new architectured metamaterials (i.e. lightweight, easy-to-manufacture, and low frequency stop band properties, noise and vibration reduction) [START_REF] Matlack | Composite 3dprinted metastructures for low-frequency and broadband vibration absorption[END_REF], [START_REF] Claeys | Design and validation of metamaterials for multiple structural stop bands in waveguides[END_REF].

Periodic structures create stop bands effect due to the geometrical and/or material impedance mismatches that can result in great vibroacoustic performances. In terms of material properties, there are various cases that show in-plane and out-of-plane elastic properties. For example the dynamical behaviour of a 2D periodic waveguide, which exhibits in-plane elastic properties (Young's and shear modulus) compared to out-ofplane ones, are described in terms of elastic wave propagation in [START_REF] Ouisse | A piezo-shunted kirigami auxetic lattice for adaptive elastic wave filtering[END_REF][START_REF] Collet | Floquet-bloch decomposition for the computation of dispersion of two-dimensional periodic, damped mechanical systems[END_REF]. The possibility of designing such smart materials or structures, that can partially reduce mechanical waves on certain frequency ranges, is addressed in [START_REF] Billon | Mechanics and band gaps in hierarchical auxetic rectangular perforated composite metamaterials[END_REF][START_REF] Madeo | Modeling phononic crystals via the weighted relaxed micromorphic model with free and gradient micro-inertia[END_REF][START_REF] Martinsson | Vibrations of lattice structures and phononic band gaps[END_REF]. Geometrical discontinuity also plays a central role in creating stop band effects; as an example, sandwich beams with periodic auxetic core, exhibiting impedance mismatch generated by varying elastic and mass properties of the core, are able to produce stop band effects [START_REF] Ruzzene | Wave propagation in sandwich plates with periodic auxetic core[END_REF][START_REF] Qing-Tian | Wave propagation in sandwich panel with auxetic core[END_REF].More complex shapes like star shaped honey-combs can be designed to provide frequency stop band behaviour in low frequency regimes [START_REF] Meng | Band gap analysis of star-shaped honeycombs with varied poisson's ratio[END_REF].

The quasi-periodicity concept which is investigated in this paper can be interpreted as a certain degree of irregularity introduced in a periodic pattern. According to the literature, there are numerous examples of disordered periodicities and/or uncertainties in real structures, like bridges with column spans, and array of fuel tanks interconnected with each other by flexural links. The issues of the non-perfect periodicity in real structures can also be attributed to errors in manufacturing processes [START_REF] Carta | Transmission and localisation in ordered and randomly-perturbed structured flexural systems[END_REF]. The presence of defects and imperfections in geometric and constitutive properties of the structures is generally referred as disorder [START_REF] Chen | Study on band gaps of elastic waves propagating in onedimensional disordered phononic crystals[END_REF]. Numerical solutions of these types of disordered systems can lead to the need of full stochastic approaches [START_REF] Ichchou | Stochastic wave finite element for random periodic media through first-order perturbation[END_REF]. When the irregularity is localized in space (a 'defect in the periodic arrangement), phenomenon of Anderson localization may occur, evidencing that the vibration propagation in a structure is not entirely regular and that could be impeded by the irregularities, giving rise on the average to an exponential decay of vibration level [START_REF] Pierre | Localization of vibrations by structural irregularity[END_REF][START_REF] Hodges | Vibration isolation from irregularity in a nearly periodic structure: theory and measurements[END_REF]. They also demonstrates an example of a string with regular and irregular spacing of added masses in order to use it as a sort of passive vibration control.

On the other hand, the imperfections can be engineered and used as design parameters to tailor the dynamic behaviour. Among others, sequences of impedance mismatches built on numerical series like Fibonacci, Thu-Morse or Rudin Shapiro can be considered as design templates for the engineered irregularity [START_REF] Velasco | Elastic waves in quasiperiodic structures[END_REF].

Hou et al., [START_REF] Hou | Acoustic wave propagating in one-dimensional fibonacci binary composite systems[END_REF], stated that robust quasi-periodic design may offer new vibroacoustic properties to the structures. This robust design refers to the investigation of the transmission properties and the frequency spectra of a Fibonacci composite material with different thickness ratio of two layers. Similarly Aynaou et al. investigated theory of acoustic wave propagation on 1D phononic band gap structures made of slender tube loops pasted together with slender tubes of finite length following Fibonacci sequence [START_REF] Aynaou | Propagation and localization of acoustic waves in fibonacci phononic circuits[END_REF]. In another example similar topological configuration of Fibonacci and Thue-Morse sequences are investigated with an experimental observation of the phononic scattering band structure in 1D quasi-periodic systems [START_REF] King | Acoustic band gaps in periodically and quasiperiodically modulated waveguides[END_REF]. A 1D bar is designed using Fibonacci sequence for stop band distribution and prestress effects by [START_REF] Gei | Wave propagation in quasiperiodic structures: stop/pass band distribution and prestress effects[END_REF][START_REF] Pisano | Fibonacci's liber abaci: a translation into modern english of the book of calculation[END_REF].

The current work is based on the analysis of quasi-periodic 2D structures. A deterministic approach is taken into account to introduce a two directional Thue-Morse sequence for creating a quasi-periodic meta-material to improve the vibroacoustic performances.

In the literature there are various examples of 1D quasi-periodic systems for the dynamic analysis; on the contrary, 2D quasi-periodic systems are very rare. Some sequences are intrinsically ready to be used for 2D cases as the Thue-Morse, while others like Fibonacci are well adapted to the 1D cases.

In this paper, two dimensional lattices are built with the conventional finite element method (FEM) to comply with the Thue-Morse sequence in order to explore the opportunities offered in terms of reduction of the forced response in some frequency bands [START_REF] Timorian | Investigation for the analysis of the vibrations of quasiperiodic structures[END_REF][START_REF] Timorian | Band diagram and forced response analysis of periodic and quasi-periodic panels[END_REF]. Specifically, the work presents modified versions of star-shaped concave unit cells.

The tailored quasi-periodicity is defined by invoking a bi-directional Thue-Morse Morphism sequence on the meta-material. The geometrical impedance mismatch results in asymmetry and follows a combination of two different star-shape elements by variation of different corner angles. The main target of the study is the design of structural stop band/filters in order to isolate maximum vibration level in some frequency ranges.

The contents of this paper is structured as follows. In section 2, quasi-periodic structure is presented. Section 3 introduces the Wave Finite Element Models: dispersion analysis. On the contrary, in section 4, experimental results are validated. Section 5 describes the structural dynamic analysis using Finite Element models. A closing section summarise the conclusion of this work.

Quasi-periodic structures

In this paper the focus is on the application to 2D domains of the Thue-Morse sequence for introducing irregularity in both directions. In this section a possible approach is presented to analyse the vibration performances of quasi-periodic 2D structures.

Quasi-periodic structure section contains three sub-sections. The first one is dedicated to the cells description, followed by the second sub-section that introduces the Thue-Morse Morphism sequence, while the third sub-section concludes with the periodic sequence.

Cells

The waveguide is a star-shaped lattice. The meta-material is made of two different unit cells (A and B). Fig. 1 shows the geometry of the two cells inspired from [START_REF] Chang | Elastic wave propagation of twodimensional metamaterials composed of auxetic star-shaped honeycomb structures[END_REF].

The difference between the two unit cells is in the angles α, which defines the corner opening in the concave shape.

Unit cells A and cell B have a side of 22 mm, and the internal core with star shape is 20 × 20mm 2 . The thickness of the base plate is 3 mm and the wall thickness of the cells is 1mm. Cell A has an angle α A = 30 • and cell cell B has an angle α B = 20 • . The angles are variables of the sequences according to the generating order of the developed

Thue-Morse Morphism sequence

The Thue-Morse sequence is based on binary arrays. Starting from a 1D example, the sequence can be derived by a function s from the set of binary sequences to itself by replacing every A in a 1D sequence with AB and every B with BA [START_REF] Berstel | Combinatorics on words: Christoffel words and repetitions in words[END_REF]:

s(x, y) = s(x)s(y). (1)
Eq.1 defines a map s for all strings x, y. Defining the Thue-Morse morphism s(A) = AB, leads to the relations for increasing the quasi-periodicity pattern in one direction. The associated sequence starts with A then AB, ABBA, ABBABAAB, and so on, [START_REF] Allouche | The ubiquitous prouhet-thue-morse sequence[END_REF].

Considering the sequence in 2D, the first order starts from a 2 × 2 matrix containing [A, B; B, A] which is equivalent to the second order s 2 (A). In the next steps, second order matrix is translated to the right, left, and diagonal directions in order to built s 3 (A) and so on. A loop is generated to translate the same order to the right and left with the opposite entries and similar entries to the diagonal direction to fulfil the pattern of the Thue-Morse sequence. The orders are detailed as follows:

s(A) = AB s 2 (A) = s(s(A)) = AB | BA s 3 (A) = s(s(s(A)) = ABBA | BAAB

Wave and finite element models: dispersion analysis

This section is divided into two subparts. The first one is related to the analysis of the behaviour of infinite periodic 2D lattices by computing the dispersion curves of unit cells, for full cell-A and full cell-B configurations, depicted in Fig. 4. The second is dedicated to other infinite periodic 2D lattices by computing the dispersion curve of alternated (mixed periodic) A-B depicted in Fig. 6, that consists of periodic lattice.

Eigenfrequency analysis is used to compute the band diagram of periodic structures with first 25 Bloch modes for single cells and first 120 Bloch modes for the second case with combined (mixed periodic) A-B [START_REF] Brillouin | Wave propagation in periodic structures: electric filters and crystal lattices[END_REF][START_REF] Mace | Modelling wave propagation in two dimensional structures using finite element analysis[END_REF]. Floquet Bloch (FB) method is used; the periodic boundary conditions are written as

u Rx = e -jk x D u Lx , (2)
u Ry = e -jk y D u Ly ,

where u Rx and u Lx are the right and left side displacement along the x axis, and u Ry and u Ly are the displacement along the y axis, respectively. D is the length of reciprocal or unitary repeated element, k x and k y are the wavenumbers in the x and y directions. The first complete stop bands is observed in the ranges of 3000 -4500 Hz, and the second one is between 5000 -5800 Hz. There is a slight narrow band-gap from 5800 Hz to 6000 Hz which disappears rapidly. The third one starts from 7000 Hz to 8000 Hz which is quite wider and the fourth one with ∆ f = 2 kHz. There is also a very narrow one after 11000 Hz just before the next wider stop band with the ranges of 12000 -13800 Hz.

The overall results of the combined (mixed periodic) A-B show some kind of mix between the dispersion diagram of single cells shown in Fig. 8. Cell-A provides interesting dynamics properties with a few wide band gaps. Cell-B has more band gaps, but with narrow frequency width. The combination of the two in a periodic formation provides a large number of gaps, the largest ones being almost as wide as cell-A, while keeping a number of gaps comparable to cell-B. In the next section, FRFs will be computed on finite structures, in order to first, validate the applicability of the gaps for real structures, and second to check whether or not a quasi-periodic arrangement

Experimental measurements

In this section, the dynamical response of the lattice is validated with experimental tests. The quasi-periodic panel is manufactured by laser cutting as shown in Fig. 10.

The material properties of this test-article are already responded in Tab. 1.

White noise excitation point

Clamped boundary Observed response (output) Observed response (input) Shaker

Finite element analysis

In this section, two periodic and one quasi-periodic lattices are considered. The first two periodic lattices are modelled as finite meta-materials made of cell A and cell B in two separate lattices. The aim is to verify whether those stop bands predicted in the previous section are maintained for this specific finite element configuration i.e. with a finite domain. Secondly the quasi-periodic meta-material in Fig. 15 is compared with a combined cell A and cell B (periodic lattice), and in the last subsection an FRF of a meta-structure (filter junction) with a pure quasi-periodic lattice is described in details.

Point load on z direction

Fixed boundary conditions

Observed response point

Point load on transverse direction

Frequency response of meta-materials

The numerical models are clamped on the right edges and a harmonic point force is applied in the region close to the fixed boundary condition in Fig. 15 with a frequency ranging from 0 -35kHz. The model is meshed by 1061 free tetrahedral and 840 biquadratic triangles element 6 node. Frequency range of 35 kHz is selected for all the FRFs to check how the various cases behave in the higher frequencies.

The response is observed in dark blue spot location in Fig. 15. Fig. 16 shows an FRF for the three types of lattices. Two of them are fully periodic with cell A and cell B and the third one is quasi-periodic lattice. It can be seen from the plots that the predicted band gaps in the dispersion diagram of periodic lattices (Fig. 8) have similarities with the FRFs of the finite counter parts. There are basically three full band gaps, which are associated with the resonant and Bragg type band gaps. As there is no theoretical definition too generic to classify the types of band gaps in this dispersion diagram, however at least the first are resonant band gaps. Fig. 16 provides four band gaps marked as (BG1, BG2, BG3, and BG4). In BG1 the response are almost the same for all the cases, A, B and quasi-periodic arrangement (maybe even random arrangement of A and B) provides similar results.

BG2 has almost the same width for the 3 cases. It is the largest one but possesses a localised mode in the middle for the case A. The quasi-periodic is as large as case B but starts lower in frequency.

In BG3 the band gap is visible only in case A, very narrow on quasi-periodic but not visible as in case B. BG4 almost same width for case A and quasi-periodic, the quasiperiodic fall starts before than case A, no band gap for case B. In conclusion quasiperiodic can be seen as a good compromise between case A and case B.

I.

Operative deformed mode at frequency 2.54 kHz II.

Operative deformed mode at frequency 6 kHz III.

Operative deformed mode at frequency 7.3 kHz IV.

Operative deformed mode at frequency 19 kHz A comparison between the numerical models and the measurements is made in Fig. 22. Two band gaps can be seen in the plot for both numerical and experimental cases.

The first band gap between 2.5 and 5 kHz is almost similar in terms of width for both cases but with a low fall for the experimental part. The second band gap which starts from 8.1 kHz and continues to higher frequencies over 10.9 is similar too, for both cases. As it can be seen the second band gap in the experimental measurement has again low fall compared to the numerical one. The different depth of numerical and experimental band gaps can be function of many concurrent parameters. For example, for very response amplitude in the experimental measurements, the measured data are strongly influenced by the fundamental noise. On the contrary, the computed reduction of the structural response is simply a function of the numerical approximations. In the last section the structural dynamic behaviour of quasi-periodic lattice is computed by FE analysis and compared with the experimental results. The results show an acceptable agreement. In another case study, the FE model of quasi-periodic lattice is embedded between two bare panels which is used as a junction filter. The induced vibration energy transfers through the junction, and it acts as a meta-material filtering property to the elastic waves due to the impedance mismatch in the star-shaped unit cells. The results of FRF in terms of RMS of velocity amplitude gives some elements of novelty to reduce maximum unwanted vibration from the host structure.

The star-shaped unit cells sequences can thus be considered as a viable starting point to the optimisation of the final configurations for designing structures with desired frequency stop-bands.

Figure 1 :

 1 Figure 1: Meso-scale unit cell A and B preliminary design stage

Figure 2 :

 2 Figure 2: 2D mesh of cell-A and cell-B

s 4 (Figure 3 :

 43 Figure 3: Quasi-periodic lattice combining unit cells A and B: following Thue-Morse sequence in two directions

Fig. 5 Figure 4 :Figure 5 :

 545 Fig. 5 shows another periodic lattice with alternated unit cells A and B. As for the previous cases, the whole geometry includes 64 unit cells, in which 32 of them are cells A and 32 cells B.

Figure 6 :Figure 7 :Figure 8 :

 678 Figure 6: FE model of the combined (mixed periodic) A-B with periodic conditions

Figure 9 :

 9 Figure 9: Band diagram of combined A-B lattice

  178

Figure 10 :

 10 Figure 10: Meta-material prototype under experimental test

Figure 11 :

 11 Figure 11: Manufactured meta-material with laser cutting

Fig. 12 Figure 12 :

 1212 Fig. 12 shows a measured FRF which is measured between 0 -100 Hz for the low frequency ranges. On this figure, the eigenfrequencies of the numerical model are shown using vertical lines. The first 3 modes are very well correlated in terms of frequency, which means that the low frequency behaviour of the structure is well captured by the FE model. In a second step, measurements are performed up to 10.9 kHz. The corresponding FRFs are shown in Fig. 13.

Figure 13 :

 13 Figure 13: Velocity response measured in the input point

Figure 14 :

 14 Figure 14: Velocity response measured in the output point

Figure 15 :

 15 Figure 15: Quasi-periodic lattice, numerical model setup for FRF analysis

Figure 16 :

 16 Figure 16: FRF of the two periodic and a quasi-periodic lattices

  . The two lattices are similar in terms of in-plane dimension and volume fraction.The responses provided in Fig.17in terms of velocity amplitude show similar dynamic effects with a slight difference. Only small difference can be observed as an higher depth around 15-dB for quasi-periodic one in blue line compared to strictly periodic in red line. The second difference is due to the wider frequency stop bands around 11 -19 kHz. The quasi-periodic curve has lower peaks starting from 16 kHz and continues up to higher frequencies around 25 kHz. In conclusion both designs have almost similar dynamic effects with a slight difference. The generated combination of quasi-periodicity in the lattice could give reduced response in the attenuation level of frequency range 3.5 -5 kHz compared with the regular periodic combinations. It also has a similar band gap wide in the medium frequency range 11 -19 kHz, with lower attenuation level and peaks after 25 kHz. lattice combined A,B Thue-Morse sequence Strictly periodic lattice combined A,B alternative sequence

Figure 17 :

 17 Figure 17: FRF of the periodic and quasi-periodic lattices

Figure 18 :Figure 19 :

 1819 Figure 18: Geometrical model of the meta-structure

Figure 20 :

 20 Figure 20: Operative modes of the meta-structure in four frequencies[START_REF] Billon | Mechanics and band gaps in hierarchical auxetic rectangular perforated composite metamaterials[END_REF].

Figure 21 :

 21 Figure 21: Iso-surface of three selected response frequencies

Figure 22 :

 22 Figure 22: Comparison of numerical and experimental FRFs

Table 1 :

 1 Material properties

	Material	Polymer
	Modulus of elasticity(Pa) 1.68 × 10 9
	Density (kg • m -3 )	818
	Poisson's ratio	0.38
	Structural loss factor	2.5 %

Acknowledgements

This investigation is carried out in the framework of the VIPER project (VIbroacoustic of PERiodic media). The project has received funding from the European Union's Horizon 2020 research and innovation program under Marie Curie grant agreement No 675441 and EUR EIPHI (ANR 17-EURE-0002) project and special thanks to the S.mart platform for the manufacturing process .