Jacques M Bahi
email: jacques.bahi@univ-fcomte.fr

Christophe Guyeux
email: christophe.guyeux@univ-fcomte.fr

Antoine Perasso

P Pg

Chaos in DNA Evolution

Keywords: Genome evolution models, Mutations, Mathematical topology, Devaney's chaos

 accurately simulates gene mutations over time. First, we demonstrate that the CM model is a truly chaotic one, as defined by Devaney. Then, we show that mutations occurring in gene mutations have the same chaotic dynamic, thus making the use of chaotic models relevant for genome evolution.

Introduction

Codons are not uniformly distributed in the genome. Over time, mutations have introduced some variations in their frequency of apparition. It can be attractive to study the genetic patterns (blocs of more than one nucleotide: dinucleotides, trinucleotides...) that appear and disappear depending on mutation parameters. Mathematical models allow the prediction of such an evolution, in such a way that statistical values observed in current genomes can be recovered from hypotheses on past DNA sequences. A first model for genome evolution was proposed in 1969 by Thomas Jukes and Charles Cantor [START_REF] Jukes | Evolution of Protein Molecules[END_REF]. This first model is very simple, as it supposes that each nucleotide A, C, G, T has the probability m to mutate to any other nucleotide, as described in the following mutation matrix, Cq. This first attempt has been followed up by Motoo Kimura [START_REF] Kimura | A simple method for estimating evolutionary rates of base substitutions through comparative studies of nucleotide sequences[END_REF], who has reasonably considered that transitions (A ÐÑ G and T ÐÑ C) should not have the same mutation rate as transversions (A ÐÑ T , A ÐÑ C, T ÐÑ G, and C ÐÑ G), leading to the following mutation matrix. ‹ ‹ ' This model was refined by Kimura in 1981 (three constant parameters, to make a distinction between natural A ÐÑ T , C ÐÑ G and unnatural transversions), Joseph Felsenstein [START_REF] Felsenstein | A view of population genetics[END_REF], Masami Hasegawa, Hirohisa Kishino, and Taka-Aki Yano [START_REF] Hasegawa | Dating of the human-ape splitting by a molecular clock of mitochondrial dna[END_REF] respectively. The differences between these models are in the number of parameters they use, but all of the latter manipulate constant parameters.

However, they all are rudimentary as they only allow to study nucleotide evolution, not genetic patterns mutations. From 1990 to 1994, Didier Arquès and Christian Michel proposed models based on the RY purine/pyrimidine alphabet [START_REF] Arquès | Periodicities in coding and noncoding regions of the genes[END_REF][START_REF] Arquès | A model of dna sequence evolution[END_REF][START_REF] Arquès | A simulation of the genetic periodicities modulo 2 and 3 with processes of nucleotide insertions and deletions[END_REF][START_REF] Arquès | Identification and simulation of new non-random statistical properties common to different populations of eukaryotic non-coding genes[END_REF][START_REF] Arquès | Identification and simulation of new nonrandom statistical properties common to different eukaryotic gene subpopulations[END_REF][START_REF] Arques | Analytical expression of the purine/pyrimidine autocorrelation function after and before random mutations[END_REF]. These models have been abandoned by their own authors in favor of models over the tA, C, G, T u alphabet. More precisely, in 1998 Didier Arquès, Jean-Paul Fallot, and Christian Michel proposed a first evolutionary model on the tA, C, G, T u alphabet that is based on trinucleotides [START_REF] Arquès | An evolutionary analytical model of a complementary circular code simulating the protein coding genes, the 5' and 3' regions[END_REF]. With such a model, the mutation matrix now has a size of 64 ˆ64 (there are 64 trinucleotides). This model comprises 3 parameters p, q, r that correspond, for a given trinucleotide XY Z, to the probability p of mutation of the first nucleotide X, the mutation probability q of Y , and the probability r that Z mutates. As for the nucleotides based models, this new approach has only taken into account constant parameters. In 2004, Jacques M. Bahi and Christian Michel published novel research work in which the 1998 model was improved by replacing constants parameters by new parameters dependent on time [START_REF] Jacques | A stochastic gene evolution model with time dependent mutations[END_REF]. In this way, it has been possible to simulate a gene evolution that is non-linear. However, the following years, these researchers returned to models embedding constant parameters, probably due to the fact that the 2004 model leads to poor results: only one of the twelve studied cases allows to recover values that are close to reality. For instance, in 2006, Gabriel Frey and Christian Michel proposed a model that uses 6 constant parameters [START_REF] Frey | An analytical model of gene evolution with six mutation parameters: an application to archaeal circular codes[END_REF], whereas in 2007, Christian Michel used a model with 9 constant parameters that generalize those of 1998 and 2006 [START_REF] Christian | An analytical model of gene evolution with 9 mutation parameters: an application to the amino acids coded by the common circular code[END_REF].

Finally, Jacques M. Bahi and Christian Michel have recently introduced in [START_REF] Bahi | Chaotic iterations and topological chaos[END_REF], a last model with 3 constant parameters, but whose evolution matrix evolves over time. In other words, trinucleotides that have to mutate (modifying trinucleotide content without changing their location) are not fixed, but they are randomly picked among a subset of potentially mutable trinucleotides. This model, called "chaotic model" (CM), allows good recovery of various statistical properties detected in the genome. Furthermore, this model fits well with the hypothesis of some primitive genes that have mutated over time.

In this paper, we ask why the CM model yields such good results. Obviously, it is reasonable to assume that not all of the trinucleotides have to mutate each time as, for instance, the stop codons that have very small mutation probabilities. However, such a biological claim is not sufficient to explain the success of the CM model to accurately simulate the dynamics of mutations in genomes. Our proposal is that the dynamics of genomes evolution is indeed chaotic, as defined by the Devaney's formulation [START_REF] Devaney | An Introduction to Chaotic Dynamical Systems[END_REF][START_REF] Banks | On devaney's definition of chaos[END_REF]. This is why linear non-chaotic models of evolution are far from what they attempt to model, leading to a poor accuracy in their prediction. By contrast, we have recently established that discrete dynamical systems in chaotic iterations satisfy Devaney's definition of chaos [START_REF] Jacques | Hash functions using chaotic iterations[END_REF]. Thus the CM model, which is the first mutation model based on chaotic iterations [START_REF] Bahi | Chaotic iterations and topological chaos[END_REF] (considering that the set of trinucleotides that can possibly mutate evolve over time), uses a chaotic dynamical system to describe a chaotic behavior, leading to a model of the same nature as the phenomenon under study. We finally demonstrate that, in contrast to inversions, mutations occurring in genomes have a chaotic dynamics. So at least one type of genomes reorganization process is chaotic, according to the formulation of such a behavior in the mathematical theory of chaos.

The remainder of this research work is organized as follows. In Section 2, the CM model of genomes evolution is recalled and its performances are synthesized. Then, in the next section, basic recalls concerning chaotic iterations and Devaney's chaos are given. Genomics mutations are formalized through a discrete dynamical system and studied in Section 4. In particular, they are proven to be chaotic according to Devaney. Other categories of genomics rearrangements are investigated too, namely transpositions and inversions. This research work ends with a conclusion section, where the contribution is summarized and intended future work is listed.

The CM Model of Genome Evolution

In this section, the CM model is presented, its capability to reasonably approximate mutations into genomes is recalled, and its relationship with chaotic iterations is stated.

Gene mutations shown as chaotic iterations

When considering the model of 2007 with 9 constant parameters [START_REF] Christian | An analytical model of gene evolution with 9 mutation parameters: an application to the amino acids coded by the common circular code[END_REF] that generalizes the models of 1998 and 2006 ( [START_REF] Arquès | An evolutionary analytical model of a complementary circular code simulating the protein coding genes, the 5' and 3' regions[END_REF] and [START_REF] Frey | An analytical model of gene evolution with six mutation parameters: an application to archaeal circular codes[END_REF] respectively), all of the trinucleotides have to mutate at each time. These models do not take into account the low mutability of the stop codons. Furthermore, they do not allow mutation strategies to be applied to certain given codons, while the other codons do not mutate. This is why the model with 3 constant parameters and a chaotic strategy has been proposed in [START_REF] Bahi | Chaotic iterations and topological chaos[END_REF][START_REF] Jacques | A stochastic model of gene evolution with chaotic mutations[END_REF]. In this model, the set of trinucleotides is divided into two subsets at each time t: the first one comprises trinucleotides that can possibly mutate at time t, whereas in the second set, trinucleotides cannot change at the considered time. The trinucleotides that mutate with replacement at time t are randomly picked following a uniform distribution on the set of all possible subsets of trinucleotides (other distribution of probabilities like discrete Poisson process have not been regarded by these authors). Consequently, the size and the constitution of the subset of mutable trinucleotides change at each time t. This subset is denoted by Jptq, and this new model has been called "chaotic model" CM by the authors of [START_REF] Bahi | Chaotic iterations and topological chaos[END_REF][START_REF] Jacques | A stochastic model of gene evolution with chaotic mutations[END_REF], as opposed to the former "standard model" of 1998 [START_REF] Arquès | An evolutionary analytical model of a complementary circular code simulating the protein coding genes, the 5' and 3' regions[END_REF], due to its relationship with the chaotic iterations recalled below. Since the trinucleotides that do not mutate in the chaotic model are not derived from the mutation of other trinucleotides (as, at each iteration, we focus only on the subset of trinucleotides that are allowed to mutate at the considered time), their probabilities of occurrence are constant. Conversely, mutation parameters of the mutable trinucleotides are the same as the 1998 model: p, q, and r with p `q `r " 1, for each of the three nucleotide sites. Let P i ptq the probability of occurrence of the trinucleotide i at time t. Let A ptq be the mutation matrix at time t, whose element pi, jq is P ptq pi Ñ jq: the probability that the i´th trinucleotide (ordered in lexicla order) mutates into the j´th one. For instance, in line 1 and column 2, there is P ptq p1 Ñ 2q " P ptq pAAA Ñ AACq. The previous remarks lead to the following formulation:

$ ' & ' % P 1 i ptq " 0 if i R Jptq, P 1 i ptq " 64 ÿ j"1
pA ptq ´Iq ji P j ptq if i P Jptq.

Obviously, this new model is a generalization of the 1998 version. Indeed, if we suppose that A ptq " A for every t, then denoting Jptq is the set of all the trinucleotides at time t, the latter system can be summarized to its second line, which is exactly the 1998 model. As the number of mutable trinucleotides changes over time, the mutation matrix is not constant, which leads to the fact that the resolution method used in the standard model cannot be applied here. To solve the system, authors of [START_REF] Bahi | Chaotic iterations and topological chaos[END_REF][START_REF] Jacques | A stochastic model of gene evolution with chaotic mutations[END_REF] have considered discrete times small enough to be sure that the mutation matrix does not change between t i and t i`1 , where the length of rt i , t i`1 s is small enough compared to the mutation rate. Let A pkq be the (constant) mutation matrix during the time interval rt k´1 , t k s.

To compute P 1 i pt k´1 q, authors of [START_REF] Bahi | Chaotic iterations and topological chaos[END_REF][START_REF] Jacques | A stochastic model of gene evolution with chaotic mutations[END_REF] have considered that:

d P i pt k´1 q dt " P i pt k q ´Pi pt k´1 q h ,
where h " t k ´tk´1 is supposed small and constant. By putting this formula into the previous system, these authors have finally obtained:

$ ' & ' % P i pt k q " P i pt k´1 q if i R Jpt k q, P i pt k q " h 64 ÿ j"1
pA pkq ´Iq ji P j pt k´1 q `Pi pt k´1 q if i P Jpt k q.

This model has been called the "discrete time chaotic evolution model CM" in [START_REF] Bahi | Chaotic iterations and topological chaos[END_REF][START_REF] Jacques | A stochastic model of gene evolution with chaotic mutations[END_REF]. We will show that this discrete version is, indeed, a gene evolution model that uses chaotic iterations. To understand the interest of this discrete time chaotic evolution model, we must firstly recall the discovery by Michel et al. of a C 3 ´code and its properties [START_REF] Arquès | A complementary circular code in the protein coding genes[END_REF].

Relevance of the CM model

A computation of the frequency of each trinucleotide in the 3 frames of genes, in a large gene population (protein coding region) of both eukariotes and prokaryotes, it was established in 1996 that the distribution of trinucleotides in these frames is not uniform [START_REF] Arquès | A complementary circular code in the protein coding genes[END_REF]. Such a surprising result has led to the definition of 3 subsets of trinucleotides, denoted by X 0 , X 1 , and X 2 . These sets are defined as follows. For each of the 60 trinucleotides different from AAA, T T T , CCC, GGG, computes its frequency in the reading frame R 0 , in the frame R 1 obtained by a shift of 1 nucleotide to the left of R 0 , and in the frame R 2 obtained by a shift of 2 nucleotides. If the considered trinucleotide is more frequent in R 0 (resp. R 1 , R 2 ), put it in X 0 (resp. X 1 , X 2 ). This procedure is repeated, with small variations, until X 0 , X 1 , and X 2 are respectively made-up of 20 trinucleotides. These sets are linked by the following permutation property: X 1 " tPptq, t P X 0 u, X 2 " tPptq, t P X 1 u, where P is defined for all trinucleotide t " n 0 n 1 n 2 by Pptq " n 1 n 2 n 0 . Additionally, if we denote c : N ÝÑ N the complementary function defined on the set of nucleotides N " tA, T, C, Gu by: cpAq " T , cpT q " A, cpCq " G, cpGq " C, and for all words of nucleotides u and v, cpuvq " cpvqcpuq, then we have cpX 0 q " X 0 , cpX 1 q " X 2 , and cpX 2 q " X 2 , which is referred to the "complementarity property". More details about the research context, the constitution of these sets, and their properties (C 3 code, rarity, largest window length, higher frequency of "misplaced" trinucleotides, flexibility) can be found in [START_REF] Bahi | Chaotic iterations and topological chaos[END_REF][START_REF] Jacques | A stochastic model of gene evolution with chaotic mutations[END_REF]. Among other things, it has been proven that X 0 occurs with the highest probability (48.8%) in genes (reading frames 0), whereas X 1 and X 2 occur mainly in the frames 1 and 2 respectively. In other words, X 0 is not pure in the reading frame (its probability is less than 1): it is mixed with X 1 and X 2 . Such a property has been explained by authors of [START_REF] Bahi | Chaotic iterations and topological chaos[END_REF][START_REF] Jacques | A stochastic model of gene evolution with chaotic mutations[END_REF] as follows. Suppose that X 0 represents the set of trinucleotides used to build the gene of the last common ancestor of the considered set of species. Random mutations have introduced noise during evolution, leading to a decreased probability of X 0 [START_REF] Bahi | Chaotic iterations and topological chaos[END_REF][START_REF] Jacques | A stochastic model of gene evolution with chaotic mutations[END_REF]. Another fundamental property is asymmetry in the sense that codes X 1 and X 2 satisfy P pX 1 q ă P pX 2 q. The standard and chaotic models (with particular strategies for the stop codons) can explain both the decreased probability of the code X 0 and the asymmetry between the codes X 1 and X 2 in genes, by the following procedure. Construct the "primitive" genes, i.e., genes before random substitutions, with trinucleotides of the circular code X 0 . Starting from this initial condition, the systems (standard or chaotic) are launched, iterating their processes until a stop condition is checked. By doing so, and for rates chosen carefully, it is possible to be close to the current frequency of each of the three codes X 0 , X 1 , and X 2 in genes. In this situation, CM models largely outperform the standard models, being closer to the observed probabilities for X 0 , X 1 , and X 2 discussed above. In particular, the chaotic model called "CM T AA " with low mutability of the stop codon TAA, matches as much as possible the probability discrepancy between the circular codes X 1 and X 2 observed in reality. For further details, the reader is referred to [START_REF] Bahi | Chaotic iterations and topological chaos[END_REF][START_REF] Jacques | A stochastic model of gene evolution with chaotic mutations[END_REF].

All the properties described before show that the gene mutation prediction is suitable to describe these phenomena. This kind of manifestation of chaos in genomics is somewhat surprising and needs, in our opinion, to be further investigated, determining whether more fundamental reasons can justify the success of chaotic models to well simulate genome evolution. In the following section, we will propose some reasons explaining why chaos is related to genomes. More precisely, we will show that some genome evolution mechanisms, as modeled in the present article, are chaotic according to Devaney. To achieve this goal, we first need to recall the bases of the mathematical theory of chaos.

Basic Remainders

Let us now rigorously introduce the notions of Devaney's chaos and of chaotic iterations, with their respective links.

Devaney's chaotic dynamical systems

Consider a topological space pX , τ q and a continuous function f : X Ñ X .

Definition 1 Function f is said to be topologically transitive if, for any pair of non empty open sets U, V Ă X , there exists k ą 0 such that f k pU q X V ‰ H.

Definition 2

The point x P X is a periodic point for f of period n P N ˚if f n pxq " x.

Definition 3 Function f is said to be regular on pX , τ q if the set of periodic points for f is dense in X : for any point x in X , any neighborhood of x contains at least one periodic point.

Definition 4 Function f is said to be chaotic on pX , τ q if f is regular and topologically transitive.

In cases where the topology τ can be described by a metric d, the chaos property is strongly linked to the notion of "sensitivity", defined on a metric space pX , dq by: Definition 5 Function f has sensitive dependence on initial conditions if there exists δ ą 0 such that, for any x P X and any neighborhood V of x, there exists y P V and n ě 0 such that d pf n pxq, f n pyqq ą δ. Then δ is called the constant of sensitivity of f . Indeed, Banks et al. have proven in [START_REF] Banks | On devaney's definition of chaos[END_REF] that when f is chaotic on pX , dq, then f has the property of sensitive dependence on initial conditions (this property was formerly an element of the definition of chaos). To sum up, quoting Devaney in [START_REF] Devaney | An Introduction to Chaotic Dynamical Systems[END_REF], a chaotic dynamical system "is unpredictable because of the sensitive dependence on initial conditions. It cannot be broken down or simplified into two subsystems which do not interact because of topological transitivity. And in the midst of this random behavior, we nevertheless have an element of regularity". Fundamentally different behaviors are consequently possible and occur in an unpredictable way.

Chaotic Iterations

Definition 6 Let X be a set, N P N ˚, f : X N ÝÑ X N be a function, and S be a sequence of subsets of v1; Nw called a "chaotic strategy". The chaotic iterations are the sequence px n q nPN of elements of X N defined by x 0 P X N and @n P N ˚, @i P v1; Nw,

x n i " # x n´1 i if i R S n `f px n´1 q ˘i if i P S n .
In other words, at the n th iteration, only the components of S n are "iterated". Note that the term "chaotic" in the name of these iterations, has a priori no link with the mathematical theory of chaos, which will be recalled in the next section. However, it has been proven in [START_REF] Bahi | Hash functions using chaotic iterations[END_REF] that, for a large variety of functions, chaotic iterations are indeed really chaotic.

Genomics Mutations as a Discrete Dynamical System

Presentation of the Problem

We now ask whether the evolution of a DNA sequence under evolution can be predicted or not. In this section, we will more specifically focus on the following questions. Firstly, given a genome (or any DNA sequence) G of interest, and a more or less precise idea of mutations that it will probably face in future (for instance, some areas in the genomes are known to mutate more frequently than other ones), is it possible to infer a set of the more probable genomes that can result, in the future, from this original sequence G after mutations? Second, given a sequence known at the current generation (say, at time t n ), is it possible to determine what was the most probable aspect of this sequence in the past (at time t m , m ă n)? Thirdly, given two DNA sequences, the second one being the result of some mutations on the first one, is it possible to find the mutation sequence that has changed the first sequence in the second one (taking into account the fact that a given nucleotide can mutate several times). Obviously, with no information about the mutation rate and history of the considered DNA sequence, this prediction is quite impossible. But what happens if we can follow the DNA sequence over several generations, learning by doing so information about the possible form of its mutations sequence? For instance, following a lineage of Escherichia coli during 40000 generations gives us a lot of informations concerning the behavior of mutations in the genomes of the considered lineage. Is it possible to use this knowledge to predict the genome of this lineage at generation number 45000 ? In other words, knowing the initial DNA sequence G 0 at time t 0 and the 40000 first terms of the mutations sequence, can we predict the DNA sequence at time t 45000 ? With the knowledge of G 0 and the whole mutations sequence S " pS 0 , . . . , S 45000 q, the genome G 45000 can be obtained without prediction, but what happens to our ability to make a prediction when using only the head pS 0 , . . . , S 40000 q of this sequence? This head can be seen as an approximation of the true mutations sequence S, and if the evolution dynamics of the mutations is quite stable through approximations, in the sense where a small perturbation at the origin yields a small perturbation at the end of the process, then this prediction makes sense. To measure the stability of the mutation dynamics through small errors or approximations, and the capability to predict the evolution of genomes under mutations, we must firstly write this mutation operation as a dynamical system, provide an accurate distance that corresponds to the "approximation" quoted below, and measure the effects of our ignorance on the complete mutations sequence on the prediction of genomes evolution.

Formalization of DNA Mutation Evolution

A genome having N nucleotides is formalized here as a sequence of N integers belonging in t1, 2, 3, 4u, where 1 (resp. 2, 3, and 4) refers to the adenine (resp. cytosine, guanine, and thymine). The benefit of using integers t1, 2, 3, 4u instead of tA, C, G, T u is justified by the construction of a metric for the mutation process (see Section 4.3). An evolution under nucleotide mutations of this genome is a sequence of couples of v1, Nw ˆv1, 4w, where we infer that:

• time has been divided into a sequence t 0 , t 1 , . . . , t n , . . . such that at most one mutation can occur between two time intervals,

• the i´th couple of the mutation sequence is equal to pm, nq if and only if the m´th nucleotide of the genome is replaced into the nucleotide n. If the m´th nucleotide was n, then no mutation has occurred at time t i .

Such a sequence will be called "mutations sequence" in the remainder of this document. S N " ď nPN pv1, Nw ˆv1, 4wq n will denote the (infinite) set of all possible mutations (finite) sequences. We introduce the phase space X N " v1, 4w N ˆSN as the set of mutating genomes. It is constituted by couples of points that store the information of a genome and its future evolution: the first coordinate of the couple is the current DNA sequence whereas the second coordinate is the sequence of mutations that will appear in the future (the problem is that this sequence can only be, in the best case, approximate concretely).

Example 1 For instance, the point pp1, 1, 2, 1, 3q, pp2, 2q, p2, 3q, p1, 4qqq P X 5 corresponds to the evolution tAACAG, ACCAG, AGCAG, T GCAGu: the left coordinate p1, 1, 2, 1, 3q means that we start with the sequence AACAG, whereas the second coordinate pp2, 2q, p2, 3q, p1, 4qq explains that:

1. the first mutation p2, 2q is a substitution of the second nucleotide by C, 2. the second mutation p2, 3q is a substitution of the second nucleotide by G, 3. the third and last mutation p1, 4q refers to the substitution of the first nucleotide by T , which is designed here by 4.

Let us now introduce the initial and shift operators i and σ defined respectively by i : S N ÝÑ v1, Nw ˆv1, 4w ps 0 , s 1 , . . .q Þ ÝÑ s 0 and σ : S N ÝÑ S N ps 0 , s 1 , . . .q Þ ÝÑ ps 1 , s 2 , . . .q.

The shift operator corresponds to the so-called symbolic dynamics, a wellstudied mathematical example of chaotic dynamics [START_REF] Formenti | Automates cellulaires et chaos : de la vision topologique à la vision algorithmique[END_REF]. With this material, the mutation operation M can be written as follows:

M : X N ÝÑ X N pG 1 , . . . , G N q, Sq Þ ÝÑ `pG 1 , . . . , G ipSq1´1 , ipSq 2 , G ipSq1`1 , . . . , G N q, σpSq ˘. (1)
In other words, the nucleotide at position ipSq 1 in the genome pG 1 , . . . , G N q is replaced by the nucleotide ipSq 2 , and the first substitution ipSq in the mutation sequence S is removed (as the mutation has already been achieved). Thus the DNA evolution as the generations pass can finally be written as the following discrete dynamical system:

" X 0 " pG 0 , Sq P X N X n`1 " MpX n q. (2)
Example 2 Let us consider Example 1 another time. As stated before, X 0 " pp1,

, 4qqq P X 5 . Then X 1 " MpX 0 q " pp1, 2, 2, 1, 3q, pp2, 3q, p1, 4qqq, X 2 " MpX 1 q " pp1, 3, 2, 1, 3q, pp1, 4qqq, and X 3 " MpX 2 q " pp4, 3, 2, 1, 3q, ∅q. The last DNA sequence, obtained after 3 mutations (3 iterations of the dynamical system), is thus equal to G 3 " X 3 1 " p4, 3, 2, 1, 3q, which is T GCAG.

A Metric for Mutation based Genomes Evolution

A relevant metric must now be introduced in order to measure the correctness of the prediction, and to give consistency to the notion of approximation that has occurred several times in the previous section. This distance must be defined on the set X N , to measure how close is a predicted DNA evolution to the real one. It will be constructed as follows: given X " pX 1 , X 2 q, Y " pY 1 , Y 2 q P X N , the number dpX, Y q:

• has an integer part that computes the differences between the two DNA sequences X 1 (for instance, the predicted or approximated genome) and Y 1 (the real genome), that is, the number of nucleotides that do not correspond in the two genomes.

• has a fractional part that must be as small as the evolution processes X 2 , Y 2 will coincide for a long enough duration. More precisely, the k´th digit of dpX, Y q will be equal to 0 if and only if, after k generations, the same position (nucleotide) will be changed in both X 1 and Y 1 genomes, and the same nucleotide is inserted in each case.

Qualitative properties

Firstly, the topological transitivity property implies indecomposability [START_REF] Ruette | Chaos en dynamique topologique, en particulier sur l'intervalle, mesures d[END_REF].

Definition 9 A dynamical system pX , f q is indecomposable if it is not the union of two closed sets A, B Ă X such that f pAq Ă A, f pBq Ă B.

Hence, taking into account only a small part of a genome in the modeling process, in order to simplify the complexity of the studied dynamics, takes away from us to a global vision of mutations. Moreover, we will prove that genomic mutations are topologically mixing, which is a strong version of transitivity: Definition 10 A discrete dynamical system is said to be topologically mixing if and only if, for any couple of disjoint open sets U, V ‰ ∅, n 0 P N can be found so that @n ě n 0 , f n pU q X V ‰ ∅.

We have the result, Theorem 3 pX N , Mq is topologically mixing. This property is an immediate consequence of the lemma below.

Lemma 1 For all open ball B Ă X N , there exists an integer n P N ˚such that M pnq pBq " X N , where M pnq is the n-th composition of operator M defined in [START_REF] Arques | Analytical expression of the purine/pyrimidine autocorrelation function after and before random mutations[END_REF].

Proof Let B " B ``pG 1 , . . . , G N q , pS 0 , . . .q ˘, ε ˘, k 0 " ´log 10 p|ε|q, and X " ``Ǧ 1 , . . . , ǦN ˘, p Š0 , . . .q ˘P X N .

We define X 1 " `pG 1 , . . . , G N q , pS 0 , . . . , S k0 , p1, Ǧ1 q, . . . , pN, ǦN q, Š0 , . . . ˘. This point is such that X 1 P B and M pk0`Nq pX 1 q " X.

Mutations M satisfy the notion of chaos according to Knudsen too, which is defined by [START_REF] Knudsen | Aspects of noninvertible dynamics and chaos[END_REF]: Definition 11 A discrete dynamical system is chaotic according to Knudsen if:

• it is sensitive to the initial conditions,

• there is a dense orbit.

Let us prove that, Proposition 6 The mutation operator is chaotic according to Knudsen on pX N , dq.

Proof The sensitivity to the initial condition has yet been stated. Let us define a point X on X N having a dense orbit under iterations of M. X N " v1, 4w N ˆSN being the Cartesian product of two countable sets, it is countably infinite too: there exists a bijection σ : N ÝÑ X N . Let G : X N ÝÑ v1, 4w N , pG, Sq Þ ÝÑ G be the first projection. Then X can be defined as follows:

X " pp1, 1, . . . , 1q, p p1, Gpσp0qq 1 q, p2, Gpσp0qq 2 q, . . . , pN, Gpσp0qq N q, σp0q 2 , p1, Gpσp1qq 1 q, p2, Gpσp1qq 2 q, . . . , pN, Gpσp1qq N q, σp1q 2 , . . .qq

This X is such that @Y P X N , Dn Y P N, M nY pY q " X, which is stronger than the required density.

To a certain extent, this notion of chaos is less restrictive than the one of Devaney. More precisely, Devaney's chaos implies Knudsen's chaos in compact spaces [START_REF] Formenti | Automates cellulaires et chaos : de la vision topologique à la vision algorithmique[END_REF].

Discussion

Conclusion of this study of mutations is that they present a chaotic behavior leading to the impossibility to qualify the long term effect of an error in predicting the location and frequency of mutations into genomes. In the worst case scenario, this error will be amplified until having a completely different genome (all the nucleotides are different, as the constant of sensitivity is greater than the length of the genome). However this case is rather marginal, mutations do not occur as frequently as the generations pass, and a mutation implies a change of only one nucleotide, leading to the opinion that, at least in the short term, the general aspect of the genome under consideration still remains under control when only mutations occur.

Inversion and transpositions are another genomics rearrangements that mostly affects more than one nucleotide. Thus an error in the prediction of these operations can potentially more largely impact the genome evolution. To qualify such impact, we first give some definitions useful to formalize inversions.

Definition 12

The complementary function c : v1, 4w ÝÑ v1, 4w is defined by cp1q " 4, cp4q " 1, cp2q " 3, and cp3q " 2.

Then the complement of adenine A is thymine T, and cp2q " 3 means, for instance, that the complement of cytosine is guanine. We can now define the inversion process on a chromosome: Definition 13 Let N P N ˚, and pn 1 , . . . , n N q a chromosome. Inversions have the form: pn 1 , . . . , n i´1 , n i , n i`1 . . . , n j´1 , n j , n j`1 , . . . , n N q ÝÑ pn 1 , . . . , n i´1 , cpn j q, cpn j´1 q . . . , cpn i`1 q, cpn i q, n j`1 , . . . , n N q.

Example 3 For instance, ACCT GT AAT GT T A is a possible inversion of ACCT T T ACT GT T A.

Obviously, it is impossible to map the DNA sequence AAAAAAAA into CCCCCCCC using only inversions, as the complement of A is T . This fact is in contradiction with the property of transitivity, leading to the statement that, Proposition 7 The inversion rearrangement is not chaotic on the set of all genomes of size N.

Let us finally investigate the dynamics of transposition inside genomes. Transposons are DNA sequences that can move into a given genome following a cut and paste mechanism of transposition: pn 1 , . . . , n i´1 , n i , . . . , n j , n j`1 , . . . , n k , n k`1 , . . . , n N q ÝÑ pn 1 , . . . , n i´1 , n j`1 , . . . , n k , n i , . . . n j , n k`1 , . . . , n N q.

Obviously this transposition cannot fit the requirements of transitivity, as the number of adenines, thymines, guanines, and cytosines are preserved. Then, for instance, it is impossible to join a genome with an high rate of thymine, starting transpositions on a genome with a low rate of T . Thus, Proposition 8 Transposition of transposons is not chaotic according to Devaney.

Conclusion

In this document, the three operations of genomics rearrangement that can be modeled by discrete dynamical systems (due to the preservation of the size of the genomes) have been studied using mathematical topology. It has been stated that mutations are chaotic, whereas transpositions and inversions are not. The proposed models lead to the feeling that genome evolution generates moderate chaos, and that this evolution can probably be predicted to a certain extent. This claim will be further investigated in our future work, by making a larger and complete study of all the possible rearrangements into genomes, measure and study their frequency using the related literature, and discussing to what extend this prediction can be realized. In particular, authors will study the set of mutations, transpositions, and inversion strategies, to take into account for the presence of recombination hotspots.

Such requirements lead to the introduction of the following function: @X, Y P X N , dpX, Y q " d G pX 1 , Y 1 q `dS pX 2 , Y 2 q where $ ' ' ' ' & ' ' ' ' % 1 ,

where δ is the discrete metric on R, that is, for x, y P R, δpx, yq " # 1 if x ‰ y, 0 else, and F : R 2 ÝÑ R `is given by F px 1 , x 2 q " |x 1 | `δp0, x 2 q.

Proposition 1 Function d is a metric on X N .

Proof The function d G is clearly a metric on v1, 4w N as being the 1-product metric of the N metric spaces pv1, 4w, δq. We now prove that d S is a metric. Firstly, d S is well defined since for pX 2 , Y 2 q P S N , one gets F pX k 2 ´Y k 2 q ď N `2 for every k P N, implying the convergence of the series in the definition of d S . Coincidence axiom and symmetry being obvious, we only prove the subadditivity of f S . If x 2 , y 2 P R are such that δp0, x 2 ´y2 q " 1 then x 2 ‰ y 2 and for every z 2 P R, either x 2 ‰ z 2 or y 2 ‰ z 2 and so δp0, x 2 ´z2 q `δp0, z 2 ´y2 q ě 1. Consequently for every x, y, z P R 2 , F px ´yq ď F px ´zq `F pz ´yq. The series being convergent for every X 2 , Y 2 P S N , one deduces that d G satisfies the subadditivity property on S N and is a metric on this set. As a consequence, d is a metric on X N . Proof This result will be established by using the sequential characterization of the continuity. Let pX n q nPN a sequence of X N that converges to ℓ P X N . We must prove that MpX n q ÝÑ Mpℓq in pX N , dq. Let ε ą 0. X n ÝÑ ℓ in pX N , dq, then Dn 1 P N, @n ě n 1 , dpX n , ℓq ď 0.09. So all X n for n ě n 1 have the same first coordinate (genome), which is ℓ 1 . Furthermore, consequently to the definition of d S , the first term of each

2 , ℓ 2 q ď 1 10 k0`1 , meaning that the sequences X n 2 , n ě n 2 and ℓ 2 start all with the same k 0 `1 terms. As the operation of M on the second coordinate of points of X N is a shift of one term to the left, we conclude that @n ě n 2 , d G pMpX n q 1 , Mpℓq 1 q " 0 and d S pMpX n q 2 , Mpℓq 2 q ă 1 10 k0 ď ε, and thus MpX n q ÝÑ Mpℓq in pX N , dq, which ends the proof.

As M is continuous, we can thus study the chaotic behavior of the discrete dynamical system of Eq. 2.

Chaotic Behavior of DNA Mutations

We first prove that, Proposition 3 M is topologically transitive on pX N , dq.

Proof Let X " pG, Sq and X " p Ǧ, Šq two points of X N , and ε ą 0. We will find n P N and a point X 1 " pG 1 , S 1 q P B pX, εq, the open ball centered on X with radius ε, such that M n pX 1 q " X. Let k 0 " r´log 10 pεqs. Thus any point of the form pG, pS 0 , S 1 , . . . , S k0 , s k0`1 , s k0`2 , . . .qq, with s k0`1 , s k0`2 , . . . P v1, Nw, is in B pX, εq. Suppose that G and Ǧ have m different nucleotides, in position i 1 , . . . , i m P v1, Nw. Thus the point X 1 " pG, pS 0 , S 1 , . . . , S k0 , pi 1 , Ǧi1 q, . . . , pi m , Ǧim q, Š0 , Š1 , . . .q P B pX, εq is such that M k0`m`1 pX 1 q " X, leading to the transitivity of M.

Remark 1 A stronger result than the topological transitivity has indeed been stated in the proof above. It is called strong transitivity and is defined by: for all X, Y P X and for all neighborhood V of X, it exists n P N and X 1 P V such that M n pX 1 q " Y . Obviously, the strong transitivity implies the transitivity property.

We now prove that, Proposition 4 M is regular on pX N , dq.

Proof Let X P X N and ε ą 0. We have to exhibit a periodic point X 1 P BpX, εq. Let k 0 " r´log 10 pεqs. Suppose that X " pG, pS 0 , S 1 , . . .q, and that the genome M k0 pXq 1 differ from m nucleotides of G in position i 1 , . . . , i m P v1, Nw. Then the point:

X 1 " pG, pS 0 , . . . , S k0 , pi 1 , G i1 q, . . . pi m , G im q, S 0 , . . . , S k0 , pi 1 , G i1 q, . . . pi m , G im q, . . .q is a periodic point in BpX, εq.

Let us finally demonstrate that:

Proposition 5 The mutation operator M has sensitive dependence on initial condition, and its constant of sensitivity is equal to

Proof Let X " pG, Sq P X N and ε ą 0. Let k 0 " r´log 10 pεqs. Consider a finite sequence of nucleotides pn 1 , . . . , n N q P v1, 4w N such that for each i P v1, Nw, n i ‰ `Mk0`N pXq 1 ˘i, and an infinite sequence ps j q jPN such that for every j P N,

Then the point X 1 " pG, pS 0 , . . . , S k0 , p1, n 1 q, . . . pN, n N q, s 0 , s 1 , . . .q P BpX, εq and is such that

. Due to the definition of pn 1 , . . . , n N q and ps j q jPN , the infimum in the latter equality is optimal, and the distance cannot be enlarged systematically for the neighborhood of all points.

The three previous propositions lead to the following result.

Theorem 1 Genome mutations as modeled by our approach have a chaotic behavior according to Devaney.

4.5. Further Investigations 4.5.1. Quantitative properties Genomic mutations possess the instability property:

Definition 7 A dynamical system pX , f q is unstable if for all x P X , the orbit γ x : n P N Þ ÝÑ f n pxq is unstable in the following sense, Dε ą 0, @δ ą 0, Dy P X , Dn P N, s.t. dpx, yq ă δ and d pf n pnq, f n pyqq ě ε.

This property, which is implied by sensitive dependence on initial conditions, leads to the fact that in all neighborhoods of any genome evolution pG, Sq there are points that can be separated with distance bigger than ε in the future through mutations.

Let us now recall another common quantitative measure of disorder.

Definition 8 A function f is said to have the property of expansivity if Dε ą 0, @x ‰ y, Dn P N, dpf n pxq, f n pyqq ě ε.

Then ε is the constant of expansivity of f : an arbitrarily small error on any initial condition is always amplified of ε. Let us prove that, Theorem 2 Mutation operator M is expansive and its constant of expansivity is at least equal to 1.

Proof If X 1 ‰ Y 1 , then dpX, Y q " dpM 0 pXq, M 0 pY qq ě 1.

Or else necessarily X 2 ‰ Y 2 . Let n " mintk P N, X k 2 ‰ Y k 2 u. Then @k ă n, M k pXq 1 " M k pY q 1 and M n pXq 1 ‰ M n pY q 1 , so dpM n pXq, M n pY qq ě 1.


Stephane Sengmany

Mathilde Sitter

Eric Léonel
email: leonel@icmpe.cnrs.fr

Erwan Le Gall

Gervaise Loirand

Thierry Martens

Didier Dubreuil

Florian Dilasser

Morgane Rousselle

Vincent Sauzeau

Stéphane Sengmany

Jacques Lebreton

Muriel Pipelier

Rémy Le Guével

Synthesis and biological evaluation of 3-amino-, 3-alkoxy-and 3-aryloxy-6-(hetero)arylpyridazines as potent antitumor agents

Keywords: Electrosynthesis, Nickel catalysis, Arylpyridazines, Biological evaluation, Cytotoxic activity

come

The pyridazine ring system can currently be considered as a privileged scaffold in medicinal chemistry due to its widespread distribution in active compounds displaying significant and varied biological properties. 1 Pyridazines also represent interesting phenyl surrogates due to possible secondary electrostatic interactions with proteins' receptors and exhibit modified ADME profile. Among these, very important subclasses are 3-amino-6-arylpyridazines, and to a lesser extent, 3-aryloxy-and 3-alkoxy-6-arylpyridazines, which prove to display diverse biological activities including analgesic, 2 anti-inflammatory, [3][4][5][6] anti-hypertensive, [7][8][9][10] and anti-depressant activities. [11][12][13] They have also found applications in the treatment for obesity, [14][15][16][17] neurodegenerative diseases, [18][19][20][21][22][23][24] and were additionally used as GABA-A receptor antagonists. [25][26][27] Interestingly, whereas fused pyridazinic rings are also privileged in numerous potential anticancer drugs acting either as DNA chelators, angiogenesis inhibitors, or protein kinase inhibitors, 28 only few examples of simple 3,6-disubstituted pyridazine derivatives exhibiting cytotoxic properties have been documented yet (Fig. 1). Therefore, in 2002, general screening using the rat aortic ring assay (in vitro functional model of angiogenesis) led to the discovery of a thiadiazole pyridazinic derivative (R 90324) with nanomolar in vitro anti-angiogenic potency. 29 A few years ago, George described the pyrazoline derivatives bearing phenyl pyridazine core as new apoptosis inducers. 30 IC 50 up to 1.67 M are reported against HepG-2, MCF-7 and CaCo-2 cancer cell lines. Although limited, these activities have to be balanced with a remarkable selectivity, normal cell lines remaining safe until 300 M. In 2013, Ahmed described the synthesis and biological evaluation of a series of novel 3,6disubstituted pyridazines based on vatalanib's (PTK787) structure. 31 One of the tested compounds proved five times more active on HT-29 cancer cell lines than the reference drug. More recently, Ahmed reported the synthesis of new pyridazinic analogues of imatinib targeting VEGFR kinase. 32 Biological assays revealed that one compound shows improved cytotoxic activity compared to the reference drug imatinib on colon cancer cell line (HCT-116). It was also demonstrated that the antitumor activity of some compounds is greater in vivo than that of imatinib against Ehrlich's ascites carcinoma (EAC) solid tumors in mice. Fig. 1. Aminopyridazine derivatives described as potential antitumor agents.

In the last years, our group focused on electroreductive cross-couplings of diazinic rings with aryl halides using a sacrificial anode process. 33,34 As such experimental conditions are well mastered in the laboratory, we desired to highlight the efficiency of the electrochemical process for the synthesis of a wide range of compounds embedding a 3,6-subtituted pyridazinic ring. In addition, as this subunit is a common feature of some scaffolds exhibiting promising anti-cancer activities, we were prompted to undertake the biological assessment of this series of compounds against various cancer cell lines. Therefore, we report herein the easy access to various 3-amino-, 3aryloxy-and 3-alkoxy-6-arylpyridazines and their biological evaluation as potent antitumor agents.

The synthesis of 3-amino-, 3-aryloxy-and 3-alkoxy-6-(hetero)arylpyridazines 3 and 5 was undertaken from the corresponding chloropyridazines 1 and 4, in the presence of aryl or heteroaryl halide partners 2 following the general and effective electrochemical reductive cross-coupling process developed in our laboratory (Scheme 1). 33 Scheme 1. Overview of the electrochemical couplings.

Electrolyses were performed in an undivided cell, under galvanostatic mode, using an iron/nickel (64/36) anode and a nickel foam cathode, until consumption of the starting materials. Thus, after a short optimization of the general reaction parameters, it was found that under a constant current intensity of 0.2 A, the cross-coupling between chloropyridazines 1 or 4 (1 equiv.) proceeded better with (hetero)aryl halides 2 (2 equiv.) in N,N-dimethylformamide (DMF) at room temperature, in the presence of NiBr 2 bpy (10 mol%) as a pre-catalyst and tetrabutylammonium bromide as supporting electrolyte.

The synthesis of 3-amino-6-(hetero)arylpyridazines 3 from 3-amino-6-chloropyridazines 1 was examined first. [START_REF] Sengmany | For a method of synthesis of precursors[END_REF] Results are reported in Table 1. Coupling products were prepared using either aryl bromides or iodides [START_REF]Aryl chlorides do not work in the process[END_REF] in limited to good yields (26 to 72%), depending on the substitution pattern of the organic halide. It was also observed that substituted phenyl bromides are more efficient as partners for the coupling reaction when the substituent is electron-withdrawing whereas iodides react better in the presence of electron-donating groups (compounds 3a, 3l, 3q-s, 3w, 3y and 3ac). The position of the substituent has a non-negligible influence on the coupling efficiency as variable yields are obtained with ortho, meta, and para-substituted aryl halides (compounds 3m, 3n and 3o). Both electronic effects and steric hindrance can explain low yields with aryl bromides or heteroaromatic partners substituted at the ortho-position of an electronwithdrawing group like CO 2 Et (compound 3o). [START_REF]from aryl bromides substituted at the ortho position with electron-withdrawing groups[END_REF] The nature of the amino group at the para-position shows notable influence on the coupling yield, as a pyrrolyl group (compound 3ab) leads to worst results than aliphatic amino groups like diethylamino or morpholino groups (compounds 3i and 3n). However, amino groups like piperazines are well tolerated, with some yields close to 70% (compounds 3w, 3x and 3y).

Encouraging biological results obtained with para-amino esters 3g and 3h (see biological part below) prompted us to prepare the amino acid mimic 3g' derived from 3g (Scheme 2). Indeed, the hydrolysis of the ester group prior to biological assessment would likely anticipate potential hydrolysis that often occurs in physiological medium. It was thus considered relevant to compare the in vitro cytotoxic activity of a genuine amino ester to that of the corresponding hydrolysed species.

Scheme 2. Synthesis of amino acid 3g'.

Going further through this work, the synthesis of the oxygenated counterparts 3-alkoxy-and 3-aryloxy-6-(hetero)arylpyridazines 5 from 3-alkoxy-and 3-aryloxy-6-chloropyridazines 4 was undertaken. Results are reported in Table 2. Electrosyntheses worked fairly well and afforded cross-couplings products in 36 to 86% yields. In a general manner, results of cross-couplings in oxygenated series are comparable to those obtained from aminated pyridazinic substrates, whatever the nature of the substituents on the phenyl ring. It can be noted that the coupling is also efficient with heteroaryl halide such as 3-bromothiophene (compound 5h). Due to the possible presence of an aryloxy substituent linked to the pyridazinic moiety, it was envisaged to additionally assess to effect of substituents connected to the O-Ar moiety through the preparation of compounds 5i and 5j embedding para-fluoro and -cyano substituent, respectively.

The antiproliferative activity of these 3-substituted-6-(hetero)aryl-pyridazine derivatives 3 and 5 were then evaluated in vitro in seven human cancer cell lines HuH7 (hepatocellular carcinoma), Caco-2 (colorectal adenocarcinoma), MDA-MB-231 (breast carcinoma), HCT116 (colorectal carcinoma), PC3 (prostate carcinoma), NCI-H727 (lung carcinoid) and HaCaT (skin carcinoma) with roscovitine (cyclin-dependent kinase inhibitor), doxorubicin (DNA topoisomerase inhibitors), Taxol (mitotic inhibitor) and Pi103 (dual PI3Kα/mTOR inhibitor) as the reference compounds (see 2.1. and 2.2. of supplementary data). IC 50 , reported in Table 3, was defined as the drug concentration causing 50% inhibition of cell proliferation (values between parentheses indicate the maximum cell death percentage induced by the pyridazine derivatives).

' (µM) >25 >25 >25 >25 >25 >25 >25 >25
3h (µM) 9 (60%) 4 (80%) 1.5 (70%) 0.5 (90%) 1 (80%) 1.5 (60%) 0.5 (80%) 3 (40%)

3i (µM) >25 >25 >25 >25 >25 30 (40%) >25 >25 3j (µM) >25 >25 >25 >25 >25 >25 - >25 3k (µM) >25 >25 >25 >25 >25 >25 - >25 3l-3v (µM) >25 >25 >25 >25 >25 >25 >25 >25 3w-3ac (µM) 30 (75%) 50 (80%) >25 >25 >25 >25 >25 >25 5a-5j (µM) >25 >25 >25 >25 >25 >25 >25 >25
Among all evaluated 1-amino-6-(hetero)arylpyridazines in series 3, three derivatives 3f, 3g and 3h showed a noticeable proliferation inhibition on selected cancer cells lines with a lower impact on healthy cells (i.e. fibroblast) which shown almost 60% of survival cells, albeit with a similar IC 50 . While 3-N,N-diethylamino-6-paraacetophenone-pyridazine (3f) showed a modest antiproliferative activity with IC 50 ranging between 5 and 25 µM, the two alkyl 4-(6-(diethylamino)pyridazin-3-yl)benzoate derivatives 3g and 3h exhibited significant IC 50 with mean ranges of 0.1 µM for 3g and 3 µM for 3h. Antiproliferative activity of 3g, in which a methyl ester group substitutes the aryl moiety in para position, appeared to be the most efficient with ten-fold increase on CaCo-2, MDA-MB231, HCT116, and HaCaT cell lines, and hundred-fold on HuH7 and PC3, compared to the ethyl ester homologue 3h.

Increasing the electron-withdrawing effect by the introduction of a cyano group (compound 3e) resulted in a total loss of antiproliferative activity (IC 50 > 25 µM). The presence of either electron-donating or electron-withdrawing groups at meta-position of the phenyl did not restore the activity (compounds 3a-3d and 3i, IC 50 > 25 µM). A similar result was observed with a cyclic amino group at 3-position of pyridazines instead of the 3-N,N-dialkylamino residues, as shown with the morpholino derivatives 3l-3v. None of other modifications performed at 3-and 6positions of pyridazines 3w-3ac, using various combinations of cyclic amino residues (substituted piperazine, pyrrolidine) and aryl or hetetoaryl moieties, respectively, were able to restore proliferation inhibition activity.

These data seem to highlight a surprising selective behavior of the alkyl 4-(6-(dialkylamino)pyridazin-3yl)benzoate scaffold that would be ascribed, with caution, to a possible electronic delocalization, occurring from the electron-donating dialkylamino residue to the electron-withdrawing aromatic ring, through the pyridazine moiety. This subtle contribution of a polarized form in the activity of such pyridazine backbone is widened by the antiproliferative inefficiency of all other analogues including compounds in series 5 designed to establish this hypothesis further. In this latter series 5a-5j, the N,N-dialkylamino group at 3-position of the pyridazine derivatives has been replaced by less electron-donating ether linkages or electron-withdrawing groups, giving IC 50 values over 25 µM on all tested cancer cell lines.

More intriguing, ethyl 4-(6-(dimethylamino)pyridazin-3-yl)benzoate 3j and its methyl ester homologue 3k, also appeared inactive under 25 µM in all cancer cell lines. These two analogues 3j and 3k only differ from pyridazines 3h and 3g by N,N-dialkylamino residue, which could counterbalance the electronic hypothesis made previously as the only or main cause of their biological antiproliferative activity. As predictable, the amino acid 3g' did not bring more information due to a probable inability to cross cell membrane at physiological pH.

Thus, these previous results demonstrated that methyl 4-(6-(diethylamino)pyridazin-3-yl)benzoate 3g is able to decrease cancer cells proliferation, albeit with low selectivity between lines. However, the cancer cells, regardless of their origin, appear more sensitive to this new compound than healthy cells (i.e. fibroblasts). The results strongly suggest a cytostatic activity on cell cycle rather than a cytotoxic effect from 3g that increase the interest of such novel series of small molecules. Currently, the molecular mechanism allowing this activity is not identified inasmuch as a slight change of substituent undergo a drastic decrease of their cancer cells antiproliferative effect.

The capacity of the selected molecules 3f, 3j, 3k, 3h and 3g to disrupt the phosphorylation level of p44/42 and Akt, two enzymes involve in the cell cycle process, was then examined. However, instead of focusing the investigation on widely studied HCT-116 tumor cells, the ability of these particular short aminoarylpyridazines to inhibit these key enzymes on human triple-negative breast cancer cell line, one of the most aggressive cancers highly associated with metastasis formation, was assessed. Only, few therapeutic solutions are currently available to prevent oncogenicity of these particular cells and the discovery of simple molecules limiting their proliferation is off great interest to design new therapeutic drugs. The breast cancer MDA-MB-468 cell line is characterized by the ability to hyperactivate p44/42 and Akt-dependent signaling pathways promoting cell proliferation and linked to tumor progression in triple-negative breast cancer. [START_REF] Jang | [END_REF]39 The capacity of the selected molecules 3f, 3j, 3k, 3h and 3g to disrupt the phosphorylation level of p44/42 and Akt at a concentration of 10 -5 M was examined first (Fig. 1A). By western blot analysis (see 2.3. of supplementary data), it was observed that EGF-stimulated MDA-MB-468 cells show potent decreased of P-p44/42 abundance (almost 70% of inhibition) after both 3h and 3g treatments, whereas 3f exhibits less potency (50%) (Fig. 1B). However, 3g seems to maintain a potent effect on P-Akt inhibition (over 70%) compared to 3h and, as observed in proliferation assays, no significant inhibitory effect was recorded with compounds 3j and 3k upon both enzyme abundances. Inhibition of clonogenic outgrowth of human breast cancer cells was then evaluated, showing a very good correlation with their antiproliferative activity. MDA-MB 468-Luc cell line was cultured for 15 days in the presence or not of molecules 3f, 3g, 3h, 3j and 3k. Then, cell cultures were stained with Giemsa to visualize and count the foci of transformed cells (see 2.4. of supplementary data). As expected the two selected compounds 3g and 3h exhibited markedly, whereas 3f, 3j and 3k displayed very poor or no significant effects. Increasing concentrations allowed to establish IC 50 (Fig. 2) and to confirm the higher activity of alkyl 4-(6-(diethylamino)pyridazin-3-yl)benzoate analogues 3g and 3h (IC 50 = 0.015 and 0.12 µM, respectively) compared to their dimethylamino counterparts 3j and 3k, respectively (IC 50 = 26 for 3j and 3.5 µM for 3k). In this experiment, pyridazine 3f lacking ester function also exhibited an unexpected activity with an IC 50 of 0.65 µM. The toxicity of the most active molecules 3g and 3h was further evaluated in vitro on human hepatocytes, HepaRG (see 4. of supplementary data) and in vivo on zebrafish assays (see 5. of supplementary data). Therefore, HepaRG cells were incubated with 2.5 µM solutions of compounds 3g and 3h for 48 h (n = 3) without shown toxicity. Furthermore, zebrafish embryos, incubated for 72 h with both molecules at two different concentrations (0.8 and 2.5 µM), expressed toxicity only at a concentration of 2.5 µM, not observed at 0.8 µM. At this concentration, the embryos remained alive, mobile, and their development was similar to the DMSO reference without visible deformation. However, a reduced size of eyes observed in the presence of 3g at upper concentration of 2.5 µM seemed to indicate a possible effect on this organ, undetectable at 0.8 µM.

In conclusion, several 3-amino-, 3-aryloxy-and alkoxy-6-(hetero)arylpyridazines (series 3 and 5) have been synthesized by an efficient electrochemical reductive cross-coupling process. All the prepared compounds were tested for their in vitro antiproliferative activity against seven cancer cell lines and showed a very sensitive structural effect depending of the nature of amino substituent and aryl ester group at positions 3 and 6 of the pyridazine anchor, respectively. Only N,N-dialkylamino functions, with a preference for N,N-diethylamino vs N,N-dimethylamino, and their exclusive association with a para-benzoate group at position 6, provided noticeable activity. Consequently, the highest activity was found for compound 3g, exhibiting IC 50 values at 10 -1 µM range on several cancer cell lines (MDA-MB231, HCT116, PC3, HaCaT, HuH7 and CaCo-2), albeit with a lack of selectivity. Regarding its ability to inhibit key hyperactivated p44/42 and Akt-dependent signaling pathways, pyridazine 3g proved to be very interesting with a potential tumor-selective mechanism of action. This compound also showed a potent ability to inhibit clonogenicity of human breast cancer cell line, one of the most aggressive cancer, without inducing toxicity under 2.5 µM neither on human hepatocytes in vitro and in vivo on zebrafish thus paving the route for the design of novel short pyridazine derivatives with therapeutic interest.

Fig. 1 .

 1 Fig. 1. (A) Immunoblot analysis of P44/42 and Akt phosphorylation in MDA-MB-468-Luc cells treated with indicated molecules. (B) Corresponding quantification of P44/42 and Akt phosphorylation.

Fig. 2 .

 2 Fig. 2. Focus formation assay. The human breast cancer cell line MDA-MB-468-Luc was treated with indicated molecules (3f, 3g, 3h, 3j and 3k). Foci were photographed and counted after 2 weeks.

Table 1

 1 Synthesis of 3-amino-6-(hetero)arylpyridazines 3 a,b .

	3a (X=I, 44%)	3b (X=Br, 48%)	3c (X=Br, 68%)	3d (X=Br, 65%)
	3e (X=Br, 38%)	3f (X=Br, 28%)	3g (X=Br, 48%)	3h (X=Br, 60%)
	3i (X=Br, 61%)	3j (X=Br, 62%)	3k (X=Br, 55%)	3l (X=I, 62%)
	3m (X=Br, 58%)	3n (X=Br, 70%)	3o (X=Br, 30%)	3p (X=Br, 70%)
	3q (X=I, 58%)	3r (X=I, 64%)	3s (X=I, 46%)	3t (X=Br, 71%)

a Typical reaction conditions: iron/nickel (64/36) rod anode, nickel foam cathode, DMF (50 mL), tetrabutylammonium bromide (1.2 mmol), 1,2-dibromoethane (2.5 mmol), NiBr 2 bpy complex (10 mol%), chloropyridazine 1 (4 mmol), aryl or heteroaryl halide 2 (8 mmol), I = 0.2 A. b Isolated yields.

Table 2

 2 Synthesis of 3-aryloxy-and 3-alkoxy-6-(hetero)arylpyridazines 5 a,b .

	5a (X=Br, 55%)	5b (X=Br, 58%)	5c (X=Br, 63%)	5d (X=Br, 86%)

b Isolated yields.

Table 3

 3 IC 50 for compounds 3 and 5 on cancer cell lines.

					Cell lines
	Compound	HuH7	CaCo-2	MDA-MB-231	HCT116	PC3	NCI-H727 HaCaT	Fibroblast
	DMSO	>25	>25	>25	>25	>25	>25	>25	>25
	Rosco (µM)	20 (90%)	10 (90%)	12 (80%)	9 (100%)	12 (8%)	20 (40%)	12 (80%)	12 (40%)
	Doxo (µM)	0.03 (40%)	0.09 (60%)	0.012 (50%) 0.05 (80%)	0.08 (80%)	0.03 (60%) 0.03 (70%)	0.005 (40%)
	Taxol (µM)	0.012 (50%) 0.02 (60%)	0.06 (60%)	<0.001 (90%) <0.001 (60%) n.d.	0.002 (70%) <0.001 (30%)
	Pi103 (µM)	6 (40%)	0.8 (60%)	8 (40%)	3 (80%)	1.2 (60%)	>25	2 (70%)	>25
	3a-3e (µM)	>25	>25	>25	>25	>25	>25	>25	>25
	3f (µM)	25 (80%)	15 (80%)	5 (60%)	6 (100%)	>25	25 (80%)	20 (80%)	>25
	3g (µM)	0.09 (60%)	0.25 (80%)	0.14 (60%) 0.06 (100%) 0.09 (72%)	0.09 (60%) 0.08 (85%)	0.10 (40%)
	3g

Acknowledgements

The authors are grateful to the CNRS and the Université Paris-Est Créteil for financial support on this work.

A. Supplementary data

Supplementary data associated to this article can be found, in the online version, at https://dx.doi.org/xxxx.


Michaël Berges
email: michael.berges@u-bourgogne.fr

Romain Léger
email: romain.leger@mines-ales.fr

Véronique Person

Vincent Placet
email: vincent.placet@univ-fcomte.fr

Emmanuel Ramasso

Jérôme Rousseau

Xavier Gabrion

Stéphane Corn

Stéphane Fontaine

Patrick Ienny

M Bergès

R Leger

Effect of moisture uptake on flax-reinforced composite laminates: influence on dynamic and quasi-static properties

Keywords: Flax, composites, laminates, fatigue, water uptake

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

Introduction

Because of environmental and sustainable issues, new rules and regulations, many industrial sectors have more and more interest in bio-based materials [START_REF] Shah | Developing plant fibre composites for structural applications by optimising composite parameters: a critical review[END_REF]. Thanks to outstanding research and recent developments of the last ten years, plant fibres reinforced bio-composites are usually dedicated to non structural applications, e.g. where high loading levels are not required. For example, these applications take places in automotive, aerospace, packaging and building industries. High mechanical properties of fibres extracted from plants, such as flax, hemp or sisal make them good candidates to be used as composite reinforcements for lightweight structural applications. Up-to-date critical reviews of literature [START_REF] Shah | Developing plant fibre composites for structural applications by optimising composite parameters: a critical review[END_REF][START_REF] Summerscales | A review of bast fibres and their composites. Part 2 -Composites[END_REF][START_REF] Dittenber | Critical review of recent publications on use of natural composites in infrastructure[END_REF][START_REF] Bhowmick | Mechanical properties of natural fibre-reinforced composites[END_REF][START_REF] Huber | A critical review of all-cellulose composites[END_REF][START_REF] Koronis | Green composites: A review of adequate materials for automotive applications[END_REF][START_REF] Mantia | Green composites: A brief review[END_REF] show that several technological and scientific barriers still remain to be knocked down in order to obtain fully optimised plant fibre composites (PFCs) dedicated to structural applications. One of the main question concerns the long-term durability of such composites. Indeed, for some intended applications, Plant Fibre Composites (PFC) will be heavily exposed to moisture, various temperatures and UV radiation. These environmental loadings can strongly affect their mechanical properties. Their durability will be thus drastically affected. Indeed, for some intended applications, Plant Fibre Composites (PFC) will be heavily exposed to moisture, various temperatures and UV radiation. These environmental loadings can strongly affect their mechanical properties. Their durability will be thus drastically affected.

Most of the polymer composites absorb moisture in humid atmosphere and when immersed in water. Water absorption can lead to a change in the physical, chemical, mechanical and dimensional properties of the matrix (plasticization, hydrolysis, swelling). Water can also degrade fibre-matrix interface. This induces a decrease in the stress transfer efficiency and in the mechanical properties of the composite materials. Contrary to glass or carbon fibres, plant fibres have a high affinity with water. This property lead composite material to high water uptake capacity. Vapour or liquid water can thus be easily absorbed by PFCs through fibres. Consequences are (i) water adsorption into the cell wall through hydrogen bonds (ii) degradation of cellulose macromolecules, , (iii) swelling and (iv) risk of drastic decrease of the mechanical properties of the PFCs [START_REF] Placet | Influence of environmental relative humidity on the tensile and rotational behaviour of hemp fibres[END_REF][START_REF] Davies | Effect of Environmental Relative Humidity and Damage on the Tensile Properties of Flax and Nettle Fibers[END_REF][START_REF] Stamboulis | Effects of environmental conditions on mechanical and physical properties of flax fibers[END_REF][START_REF] Symington | Tensile Testing of Cellulose Based Natural Fibers for Structural Composite Applications[END_REF]. It is therefore of paramount importance to characterize moisture absorption and its influence on the long-term behaviour of PFCs. Despite recent works on fatigue, the influence of moisture absorption and environmental loading has not been yet considered.

Thus, the aims of this paper is to quantify and show the influence of water vapour sorption on a large spectrum of mechanical behaviours of flax fibres reinforced epoxy laminates. Here, targets are quasistatic, fatigue. This work was conducted as part of the scientific collaboration between three French laboratories (DRIVE, FEMTO-ST and C2MA). The laboratories are members of a research consortium called BIOCOMP. This consortium groups several laboratories. These partners have expertise in the field of mechanical and fiber reinforcements, methods for preparation and characterization of fibrous reinforcements, polymers and composites, biochemistry as well as chemistry of interfaces.

Material and methods

Composite materials

Composite bio-based-materials have been fabricated from prepregs flax fibres and epoxy resin. FlaxPreg UD 150 was supplied by Lineo company (France). The flax fabric has an areal weight of 154 g/m 2 . It contains 95% of warp yarns and 5% of weft yarns in weight. The epoxy resin is an Epoxy Huntsman LY 5150. The fibre fraction is approximately 50% in weight before curing.

Composites fabrication

Five plates (400 mm x 400 mm x 2.2 mm) were fabricated using a thermo-compression process under specific pressure and temperature cycle (curing under 130°C/4,5 Bar during 90 minutes). They are numerated P1 to P5. These plates consist in 10 elementary unidirectional plies. Each plate has been characterized in terms of porosity and fibre contents, according to ASTM D 3171-99 standard (table 1).

Figure 1. Scanning Electron Microscopy (SEM) cliché / cross-section of the produced material

A scanning electron microscope (Hitachi S-3000N) has also been used to observe the cross-section areas of the produced materials. Images have been recorded under partial vacuum with back-scattered electrons. A specific attention was dedicated to the study and the distribution of the voids (figure 1). On such image, the warp yarns (white areas) and the voids (black areas) can be easily identified. The presence of weft yarns into the reinforcement structure is also easily observed.

The Table 1 summarize principal characteristics of the produced plates and the way they have been aged and mechanically tested.

Sample preparation

Rectangular specimens (250 mm long and 25 mm large) were sawn with a high-speed diamond cutting machine. This operation allowed to produce unidirectional laminates, first in the fibres direction (UD 0°) and second perpendicularly to the fibres direction (UD 90°). After cleaning and preparing the surface (sanding and cleaning with acetone), aluminum tabs (50 mm long, 25 mm large and 3 mm thick) have been bonded (with Loctite super glue 3 adhesive) to the test specimens dedicated to quasistatic and fatigue tests. Samples were speckled in order to follow strain plane tensor by digital image correlation [START_REF] Christmann | Mechanical behaviour at large strain of polycarbonate nanocomposites during uniaxial tensile test[END_REF].

Hygrothermal conditioning (P2, P3, P4, P5)

After fabrication, samples were stored in a regulated chamber at 21°C and 50% RH before mechanical testing. This hygroscopic state was named "DRY". Samples were also aged in a hygrothermally regulated chamber at 70°C and 85% RH for more than 2 weeks. During this period, relative weight uptake, dimensions, dynamic tensile modulus were monitored. A Mettler-Toledo AT200 balance was used for weight monitoring (precision of 0.001 g) and a caliper with a precision of 10 -2 mm was used to evaluate the swelling. At the end of the two-week monitoring period, an equilibrium state named WVS (Water Vapour Saturated state) has been achieved.

Monotonic tensile tests (P3, P4, P5)

Young modulus, ultimate tensile stress and strain were determined by uniaxial tensile tests on a MTS testing machine (model Criterion C45.105), equipped with a 100 kN load cell. Samples were loaded at a crosshead speed of 1 mm/min. Each set of samples (DRY and WVS) have been tested.

Tensile-tensile fatigue tests (P1, P2)

The tensile-tensile fatigue tests were performed under a triangular waveform loading, at a frequency of 5 Hz. Using a load amplitude control mode, the ratio between minimum and maximum stress (R) was fixed to 0.1. The experimental configuration (loading machine, specimen shape and preparation) was similar to monotonic tensile tests. Fatigue tests were performed in the fibre direction for DRY and WVS specimens. Four levels of maximum stress were applied. The levels were 80%, 65%, 50% and 40% of the quasi-static tensile failure stress (at 2 mm.min -1 ) for the DRY specimens and approximately 75%, 60%, 55% and 45 % for the VWS specimens. At least three replicates were tested at each stress level. Tests were stopped at failure.

Tests were carried out under controlled hygrothermal conditions (T = 21 °C ±3 °C, RH = 50% ±5%).

For WVS specimens, it was necessary to keep the water-saturated state during fatigue loading. Thus, the specimens were covered, after the treatment, with a flexible coating film. This film was made with an aqueous mixture of iron oxide-filled and acrylic resin. The mass of each specimen was measured after hygrothermal treatment but also before and after fatigue loading. The mass variation was checked to be negligible and the water-saturation state has been assumed to be constant during tests. Peak-to-peak and average amplitudes of load and strain sensors were recorded for each cycle.

Results and discussion

Water Vapour absorption behaviour

Figure 2 displays the water uptake of samples from P2, P4 and P5 under humid environment described before. The diffusion kinetic follows a one dimensional Fickian behaviour (solid lines in figure 2) as presented by Scida et al. [START_REF] Scida | Influence of hygrothermal ageing on the damage mechanisms of flax-fibre reinforced epoxy composite[END_REF]. After 6 days, the final water uptake reaches an equilibrium between 3,0 and 3.3 ± 0.2%. The coefficient of diffusion D is estimated at 6.47x10 -6 ± 0.14x10 -6 m 2 .s -1 . It is interesting to notice the different kinetic in the case of P4 specimen. This plate shows a lower maximal mass uptake at 3,0 %. This phenomenon can be explained by its lower void ratio (6.3%) in comparison to other specimens from P2 and P5 (respectively 7.2% and 8.1% -see table 1).

Figure 2.

Water uptake of composite samples and 1D-Fickian kinetic prediction (solid lines) when exposed to a hot humid environment (70°C/85%rh)

Monotonic tensile tests

Results are presented in figure 3 and completed by values presented in table 2. Moduli were evaluated in the strain interval 0.01 to 0.15%, using linear regression. Concerning DRY specimens, the mean values of tensile moduli (approximately 30 GPa in the fibre direction and 4.7 GPa in the transverse direction), tensile strength (approximately 320 MPa for UD 0° and 30 MPa for UD 90° direction) and strain at failure (approximately 1.5% in the fibre direction and 1% in the transverse direction) are in good agreement with data of literature collected on the same type of flax-epoxy laminates [START_REF] Sawi | An investigation of the damage mechanisms and fatigue life diagrams of flax fiber-reinforced polymer laminates[END_REF][START_REF] Poilâne | Polymer reinforced by flax fibres as a viscoelastoplastic material[END_REF]. ). An increase of strains, a decrease of the Young modulus as well as a slight increase of the Poisson's ratio are observed. These evolutions can be attributed to the plasticization of both matrix and fibres by water molecules. WVS specimens exhibit slightly superior tensile strengths to DRY specimens. Munoz and Garcia-Manrique [START_REF] Munoz | Water Absorption Behaviour and Its Effect on the Mechanical Properties of Flax Fibre Reinforced Bioepoxy Composites[END_REF] also observed a higher tensile strength of flax-epoxy composites after aging. They attributed this behaviour to the swelling of flax fibres in the composite material (as a result of water absorption). This phenomenon was assumed to induce a stronger interfacial bonding between the fibre and the matrix. 4. For each studied load level, and particularly for WVS specimens, there is a slight dispersion in the maximum tensile stress. This is due to the variation in cross-section of the specimens. Results show that both set of specimens (DRY and WVS) present a gradual decline in fatigue strength when the number of fatigue cycles increases. For a fixed loading level, dispersion in fatigue life duration is observed. The order of magnitude of this dispersion is in agreement with data collected on flax-epoxy laminates [START_REF] Sawi | An investigation of the damage mechanisms and fatigue life diagrams of flax fiber-reinforced polymer laminates[END_REF][START_REF] Poilâne | Polymer reinforced by flax fibres as a viscoelastoplastic material[END_REF]18]. The fatigue strength can be fitted by the Wöhler law, as a function of number of cycles. The fatigue strength for a high number of cycles (1 million) is approximatively 140 MPa for DRY specimens. This value represents 42 % of the monotonic tensile strength. This range of value corresponds to the observations produced by El Sawi et al. [START_REF] Sawi | An investigation of the damage mechanisms and fatigue life diagrams of flax fiber-reinforced polymer laminates[END_REF] on the same type of unidirectional flax-epoxy laminates. Results also show that the fatigue strength is significantly influenced by water sorption. Concerning WVS specimens, the initial dynamic modulus is 22 GPa, against 33GPa for DRY specimens. Contrary to all expectations, the fatigue strength for a high number of cycles is higher for WVS specimens than for DRY ones. In the case of WVS, the fatigue strength reach 155 MPa (47% of the monotonic tensile strength). This phenomenon is assumed to be due to the plasticization of the laminate.

Conclusions

This study focused on the consequences of water vapour sorption of unidirectional flax-epoxy laminates. Mechanical investigations concerned monotonic and fatigue behaviours. The motivations of this work was to evaluate the durability of bio-based laminates for structural applications. Results show that, when exposed to hygrothermal conditioning at 70°C and 85% RH, the diffusion kinetic of UD flax-epoxy composite follows a one dimensional Fickian behaviour. Water vapour sorption is showed to induce a significant change in the shape of the tensile stress-strain curve and a decrease in dynamic elastic modulus. Contrary to all expectations, water saturation does not degrade the monotonic tensile strength of these flax-epoxy composites. Moreover, an increase in the fatigue strength is clearly observed on wet specimens. Water sorption seems to activate some hardening phenomena. This is a benefit for both static and fatigue strength. This result is of paramount importance for bio-based composites end-users. Future of this work will help to quantify the influence of the exposure time to such hygrothermal conditions (i.e. ageing) on the mechanical behaviours of PFCs. The final purpose is to understand how microstructural mechanisms helps to harden these biobased composites composite when thy are submitted to moisture.

Figure 3 .

 3 Figure 3. Tensile stress-strain curves for 0°UD specimens before and after aging

Figure 4 .

 4 Figure 4. S-N curves of DRY (P1) and WVS (P2) specimens

Table 1 .

 1 Plate thickness, porosity and fibre/matrix volume fractions

	Plates Thickness	Fibre vol.	Matrix	Void vol.	Water	Monotonic	Fatigue
		(mm)	ratio (%)	vol. ratio	ratio (%)	uptake	tests	tests
				(%)
	P1	2.19	45.0	46.4	8.6			X
	P2	2.18	45.4	47.4	7.2	X		X
	P3	2.16	45.6	47.7	6.7	X	X
	P4	2.17	45.7	48.0	6.3	X	X
	P5	2.17	45.8	46.1	8.1	X	X

Table 2 .

 2 Properties obtained in monotonic tensile tests (mean value ±standard deviation)

	Specimens	EL (GPa)	(MPa)	(MPa)	(%)	(%)	LT
							(-)
	DRY (P4, P5)	30.4 ± 2.7	319 ± 20	30.0 ± 1.3 1.59 ± 0.14	0.99 ± 0.16	0.34 ± 0.02
	WVS(P4, P5)	19.6 ± 2.0	320 ± 20	25.9 ± 1.5 2.32 ± 0.14	1.80 ± 0.21	0.37 ± 0.02
	3

.3 Tensile-tensile fatigue tests S

  -N curves for tensile-tensile fatigue tests on DRY and WVS specimens are presented in figure

Acknowledgements

The authors thank the Foundation de Coopération Scientifique Bourgogne Franche-Comté for its financial support through the project Bonus Qualité Recherche « Durabilité des composites biosourcés » (FCS-BFC BQR2014-19).