Huy Duong Bui

Andrei Constantinescu

Hubert Maigre
email: maigre@lms.polytechnique.fr

Numerical identification of linear cracks in 2D elastodynamics using the instantaneous reciprocity gap

This paper considers the identification problem of a linear crack in a body of finite extension within the framework of linear two-dimensional elastodynamics. In a series of prior papers in electricity, elasticity or acoustics, it has been proved using the reciprocity gap that three different series of adjoint wave fields determine in closed-form solution the normal of the plane of the crack, the position of the plane and finally the complete crack extension. The work developed next within the framework of linear elastodynamics defines a novel instantaneous reciprocity gap as the instantaneous work done by the adjoint tractions on the crack opening displacement. This quantity is then used to identify linear cracks in a two-dimensional problem. It is shown using a numerical example that a unique family of planar shear waves permits the identification of the normal, position and a convex hull of a linear crack through simple interpretations of the instantaneous reciprocity gap. This method is more general in the sense that it applies to three-dimensional problems as well.

Introduction

The analysis of the diffraction of elastodynamic waves by a crack in a solid body in order to reconstruct the crack is a classical inverse problem. The problem is of interest to areas such as nondestructive testing or earthquake engineering.

The classical investigations of this inverse problem found in the literature (see, for example, [START_REF] Alves | Inverse scattering for elastic plane waves Inverse[END_REF]) are subject to different limitations: unbounded domain, domain with known Green function, source in the interior of the body, source far away from the crack in order to consider plane waves, etc.

In recent years a series of theoretical results has been obtained on the dynamic problem or on similar time-dependent crack identification problems using the optimal control technique [START_REF] Grasselli | Identifying a spatial body force in linear elastodynamics via traction measurements[END_REF], or by constructing an appropriate adjoint field and applying the method of the reciprocity gap [START_REF] Bui | Diffraction acoustique inverse de fissure plane: solution explicite pour un solide borné[END_REF].

The papers presenting planar crack identification using the reciprocity gap [START_REF] Andrieux | Sur l'identification de fissures planes via le concept d'écart à la réciprocité en élasticité[END_REF][START_REF] Andrieux | Reciprocity principle and crack identification Inverse[END_REF] constructed generally three families of adjoint waves which permitted us to identify in order the normal of the plane, its position and finally the complete extension of the crack.

The purpose of this paper is not the proposition of three new families of adjoint waves suited to identify the planar crack in the elastodynamic case. Its novelty consists of the definition of an instantaneous reciprocity gap, which is the instantaneous mechanical work done by the virtual tractions given by the adjoined field and the crack opening displacement. A simple interpretation of this result permits the determination of the normal and the position of the crack plane, as well as of a 'convex approximation' of the crack.

The paper starts with an introduction of the direct and the inverse problems in a general three-dimensional framework. The next section presents the reciprocity gap in the case of the general elastodynamic and the definition of the instantaneous reciprocity gap. Then a numerical example for the identification of a linear crack in a two-dimensional body is given. The identification results are obtained by a simple interpretation of the instantaneous reciprocity gap. The proposed technique is not dependent on the two-dimensional character of the problem and can therefore be easily extended to three-dimensional problems. The final section shows that the specific boundary condition occurring during slip faults in earthquakes is compatible with the present method and indicates as such a field of applications for this technique.

The direct and the inverse problem

Let us consider a homogeneous elastic body , containing a crack C in its interior. We shall suppose that the crack C is included in a plane . The exterior boundary of will be denoted by ∂ .

For each time instant t ∈ [0, ∞), we shall consider that the vector field of displacements u and the tensor fields of strains and respectively stresses , σ satisfy the following system of equations:

div σ = ρ ü σ = Cε ε= 1 2 (∇ + ∇ T )u (1)
which is equivalent to the hyperbolic elastodynamic equation:

div C∇u = ρ ü (2)
where C denotes the fourth-order tensor of elastic moduli. C is supposed to be positive definite and to respect the classical symmetry relations:

C ij kl = C klij = C ij lk . ( 3
)
We assume that both displacements and tractions are known on the exterior boundary ∂ for each time instant t ∈ [0, ∞):

u| ∂ = ξ σ• n| ∂ = φ ( 4
)
where by σ = C∇u we have denoted the tensor field of stresses. The two lips of the cracks + C and - C are supposed to be stress free:

σ • n| ± C = 0. ( 5
)
These boundary conditions are completed by a series of initial conditions on the displacement and the velocity fields.

We remark that in the most practical case, we shall have a free surface φ = 0, meaning that tractions have not to be measured and that displacements ξ are measured in a finite number of locations generally using accelerometers. This is, for example, the case of the free surface of the earth when modelling earthquakes. A more precise discussion of the boundary conditions and their importance with respect to earthquake problems is given in section 5.

The instantaneous reciprocity gap

The variational principle associated with equation (2) permits after a series of integrations by parts and application of the boundary condition for the solution to define the adjoint problem:

div C∇w = ρ v (6)
and the reciprocity gap:

RB = ∞ 0 C [[u]] • σ[w] • n ds dt (7) = ∞ 0 ∂ {u • σ[w] • n -w • σ[u] • n} ds dt (8) + \ C [u • ∂ t w -∂ t u • w] ∞ 0 dv. ( 9
)
In the definition of the reciprocity gap, u is a solution of the direct problem (2) and directly related to the measurements, whether w is a solution of the adjoint problem ( 6) and arbitrarily fixed.

The instantaneous reciprocity gap will be simply defined by the following expression:

RB(t) = C [[u]] • σ[w] • n ds (10) = ∂ {u • σ[w] • n -w • σ[u] • n} ds. ( 11
)
Defined as such, the instantaneous reciprocity gap actually measures the instantaneous virtual mechanical work done on the crack tips by the real displacements on the adjoint traction field.

If the adjoint field is an adjoined wavefront, it follows that the instantaneous reciprocity gap is zero, unless the adjoint wavefront has reached the crack. As a consequence one can determine the arrival time of the adjoint wavefront at the crack from boundary measurements. The precise knowledge of the propagation velocity and the direction of the adjoint wavefront permits us then to determine the location of the crack. A series of techniques to practically exploit this observation will be clarified in the next section using a two-dimensional numerical example.

Analysis of the numerical results

The example considered here corresponds to the body presented in figure 1. The numerical computations have been done in two steps.

In the first step we have been simulating a direct elastodynamic problem, and obtained a set of 'artificial' displacement measurements. The loading is an initial slip on the crack and is similar to those obtained in earthquakes. In the second step, these measurements have been convoluted with a series of adjoined fields and the results have been interpreted in order to identify the crack.

The direct computations and part of the identification procedure have been programmed using the Cast3M finite element programming language. The complete mesh had 800 nodes and 1500 linear triangular elements. The crack lips have been represented by 10 linear elements. A second-order algorithm has been used for the numerical integration direct wave propagation problem.

The loading of the direct problem has been given a tangential shear vector on the crack lips, which will be completely released after the initial moment t = 0, i.e.:

τ (t) = αY (-t)t ( 12
)
where α is constant, Y is the unit step function and t is the tangent vector on C . All exterior surfaces are considered as stress free:

σ • n| ∂ = 0 i.e. φ = 0. ( 13
)
The boundary displacement has been collected in the direct problem and considered in the identification part as a measurement. These loading conditions slightly open the crack and lead to a large tangential slip when compared to the overlapping of the crack faces. From a practical point of view, it can be shown that this loading type is similar to that of an earthquake, owing to the large amount of slip, but this will not be detailed here. However, in the next section, we shall discuss in detail the effects of nonoverlapping contact boundary conditions on the instantaneous reciprocity gap.

The adjoint wave fields were constructed as 36 planar share wave fields. All directions of propagation of the wave fields were passing through the centre of the body and were equally spaced every 2π/36 radians.

Moreover, the initial time for the adjoined waves was chosen such that all reach the centre of the body at the same instant.

w(x, t) = kY t - 1 c x • p -τ k (14)
where x and t denote the space point and time instant when the displacement w is computed. p, k are the direction of propagation of the wave and respectively the direction of shear and are for each wave orthogonal: p • k = 0. τ is a time parameter chosen such that the shear wave is outside the body at the initial instant t = O. c denotes the velocity of the propagation of the shear waves and equals

c = E ρ(1 + ν)
for an isotropic elastic body with Young modulus E, Poisson coefficient ν and a mass density ρ.

The stress state corresponding to the adjoined wave is a travelling Dirac impulse. It therefore does not produce any mechanical work with the displacement jump [[u]] on the crack unless the adjoined wave field has 'arrived' at the crack at time t p . Therefore the instantaneous reciprocity gap vanishes for t < t p and has a nonzero value after this instant (see figure 3). This is a consequence of the fact that the wavefront has to arrive at the crack in order to produce virtual work on it.

Next, using the previous remark on the interpretation of the instantaneous reciprocity gap and the fact that the adjoined waves are plane, we will be able to identify certain characteristics of the crack:

• normal of the crack

From simple geometric reasoning one can deduce that the adjoint wave having its wavefront parallel to the crack tips will interact with the crack at a later instant when compared to the wave coming from the same half-space with respect to the crack. Figure 2 reproduces the value of the instantaneous reciprocity gap at each time step (x-axis) for each adjoined wave ordered by the incoming angle (y-axis). In this plot one can remark that the angles of the adjoined wave for which the instantaneous reciprocity stays initially the longest period at the zero value, meaning no interference with the crack, are 110 • and 290 • . They are 180 • apart, as they only indicate the direction of the normal to the crack.

• position of the crack plane

Assuming that the normal of the plane is known, let us analyse the instantaneous reciprocity gap obtained from the adjoined waves propagating in the direction of the normal and denote t f , t b as the moments when the adjoint waves begin their interaction with the crack. If R represents the distance from the starting point of the adjoint waves to the centre of the body, we have

c(t f + t b ) = 2R
where v is the propagation speed of the wave. Then the distance of the crack to the centre of the body is given by either

d = R -t f c or d = R + t b c.
The results of this analysis are represented in figure 4 and compared with the initial position of the crack. Another way to analyse the results of the instantaneous reciprocity gap is the following one. On the rays, characterizing the direction of propagation of the adjoint waves we represent at each time instant, in the real position of the wavefront, a segment representing the wavefront. The value of the instantaneous reciprocity gap is represented by a colour code (see figures 5 and 6). We recall that the evolution of the values of the instantaneous reciprocity gap on each line is similar to the graphs displayed in figure 3. The plot of the wave position is stopped when the reciprocity is larger than a small predefined value. The coloured domain represented

-4 -2 2 4 -4 -2 2 4 0. 0.5 1.

RB

Figure 5. The position of the real crack and the 'convex hull' obtained using the front of the adjoint wave and the instantaneous reciprocity gap. Each line segment represents the position of the wavefront and the colour code is the normalized value of the instantaneous reciprocity gap at this position.

-4 -2 defines the region where the interaction between the adjoined wave and the displacement jump on the crack is negligible. As a consequence this domain defines together with the fronts of the adjoined wave at each time location, a sort of 'convex hull' of the crack. This representation is compared in figures 5 and 6 with the position of the real crack.

RB

The small errors at the end of the crack have the length order of one finite element. This is due to the fact that the displacement jump is nonzero only at the first node after the crack tip. As such the virtual work measured by the reciprocity is nonzero only after a penetration of the adjoint wave of more than one element in the crack region.

One can remark that both techniques have given a very good match of the identified position and extension of the crack with the real crack.

Discussion of boundary conditions in earthquake problems

The preceding permits us to identify in a simple way the displacement jump on an internal crack using the instantaneous reciprocity gap and a well-chosen adjoined wave. We shall analyse next the specific crack boundary conditions in the case of an earthquake in order to show that the same method can be applied in the identification of fault slips.

Let us first recall that the complete expression of the reciprocity gap obtained without using the information about a stress-free crack surface is the following:

RB = ∞ 0 C [[u]] • σ[w] • n -w • [[σ[u]]] • n ds dt. ( 15
)
In the case of an earthquake the boundary conditions on the crack corresponding to (5) have a slightly modified form. Let us consider an orthogonal system of coordinates on the fault C given by the tangent and normal unit vectors (t 1 , t 2 , n). We shall consider the tangent and normal components of the displacement and traction vectors on the fault:

u = u 1 t 1 + u 2 t 2 + u n n (16) τ = σ • n = τ 1 t 1 + τ 2 t 2 + τ n n. ( 17
)
The crack C represents in this case the geological fault and is prestressed before the earthquake, for t < 0:

[[u 1 t 1 + u 2 t 2 ]] = 0 [ [u n ]] = 0 (18) [[τ 1 t 1 + τ 2 t 2 ]] = 0 [ [τ n ]] = 0. (19)
The earthquake actually releases the tangential shear stress in the slip of the fault, which implies for t 0 under the hypothesis of a frictionless slip:

[[u 1 t 1 + u 2 t 2 ]] = 0 [ [u n ]] = 0 (20) τ ± 1 t 1 + τ ± 2 t 2 = 0 [ [ τ n ]] = 0. ( 21
)
The following expression for the reciprocity gap reduces then to the following expression:

RB = ∞ 0 C [[u 1 t 1 + u 2 t 2 ]] • (τ 1 [w]t 1 + τ 2 [w]t 2 ) ds dt (22)
and as such measures the mechanical work of the adjoint stresses on the tangential slip on the crack.

If the slip involves friction then there is a continuity of the tangential traction vector on the fault:

[[τ 1 t 1 + τ 2 t 2 ]] = 0 ( 23
)
and we obtain the same expression for the reciprocity gap RB.

The specific boundary conditions in the case of fault slip in an earthquake reduce the expression of the reciprocity gap to the jump of the tangential displacement. Reviewing that the assumption of the preceding identification method still applies in this case. A further numerical study, taking into account no interpenetration and friction on the crack during the measurement simulations should validate the robustness of the identification method in this case.

Conclusion

The previous results show that a simple direct interpretation of the instantaneous reciprocity gap leads to very precise identification of linear cracks in two-dimensional elastodynamic problems. The present technique can easily be applied to three or higher dimensional spaces. Moreover, it has also been shown that the method should also apply to problems with contact boundary conditions, such as those corresponding to earthquake problems. A complete analysis of this case is the subject of future work.

The numerical results are very promising and one can expect that other types of convex defects can also be identified using the same technique in the near future. As an illustration of this fact we presented in figure 6 the identification of two cracks in the same configuration as those discussed in this work.

Figure 1 .

 1 Figure 1. The geometry of the body and the position of the 'real' crack.

Figure 2 .

 2 Figure 2. A plane colour plot representation of the time evolution (x-axis) of the instantaneous reciprocity gap for different angles (y-axis). The angles with the maximum noninterference of the adjoint wave with the crack define the direction of the normal (up to 180 • ).

Figure 3 .Figure 4 .

 34 Figure 3. Time evolution of the computed reciprocity gap computed using two adjoined fields arriving at opposite directions called the backward (at 290 • ) and forward directions (at 110 • ). The arrival times of the adjoint wavefronts on the crack are denoted by t b and t f respectively.

Figure 6 .

 6 Figure 6. The 'convex hull' obtained for two cracks with the technique of the instantaneous reciprocity gap. Each line segment represents the position of the wavefront and the colour code is the normalized value of the instantaneous reciprocity gap at this position.


John F Beckmann

Manon Bonneau

Hongli Chen

Mark Hochstrasser

Denis Poinsot

Mylene Weill

Mathieu Sicard
email: mathieu.sicard@umontpellier.fr

Sylvain Charlat
email: sylvain.charlat@univ-lyon1.fr

Hervé Merçot

Mylène Weill

The toxin-antidote model of cytoplasmic incompatibility: genetics and evolutionary implications

Keywords: Wolbachia, Cytoplasmic Incompatibility, Deubiquitylase, Nuclease, Toxin-antidote 20 systems, Selfish genetic elements

come   L'archive ouverte pluridisciplinaire

A c c e p t e d M a n u s c r i p t

Every living organism is a chimera of different evolutionary lineages living in more or less tight association. Arthropods are emblematic of that rule and often carry bacteria within their own cells, transmitted from mothers to offspring with the egg cytoplasm. These may offer benefits, such as vital nutrients to hosts feeding exclusively on sap or blood [1,[START_REF] Kirkness | Genome sequences of the human body louse and its primary endosymbiont provide insights into the permanent parasitic lifestyle[END_REF], and thus become essential parts of a new whole, but also often colonize host populations through 40 selfish strategies, maximizing their own fitness regardless of possible detrimental effects to hosts [START_REF] Hurst | Reproductive Parasitism : Maternally lnherited Symbionts in a Biparental World[END_REF][START_REF] Charlat | Evolutionary consequences of Wolbachia infections[END_REF]. Cytoplasmic incompatibility (CI, Figure 1, box 1) is one such strategy, which has likely contributed in large part to the radiative success of Wolbachia bacteria, now present in about half of all arthropod species [START_REF] Werren | Wolbachia: master manipulators of invertebrate biology[END_REF][START_REF] Weinert | The incidence of bacterial endosymbionts in terrestrial arthropods[END_REF][START_REF] Bailly-Bechet | How Long Does Wolbachia Remain on Board?[END_REF].

45

CI genetics

While Wolbachia and CI were both discovered a long time ago in Culex pipiens mosquitoes [START_REF] Hertig | Studies on rickettsia-like microorganisms in insects[END_REF][START_REF] Marshall | The british mosquitoes[END_REF][START_REF] Roubaud | Phénomènes d'amixie dans les intercroisements de Culicides du groupe pipiens[END_REF][START_REF] Laven | Crossing experiments with Culex strains[END_REF], the causal link between the two was only made decades later [START_REF] Yen | New hypothesis of the cause of cytoplasmic incompatibility in Culex pipiens L[END_REF]13]. By that time, early models had clarified the invasion dynamics of CI [14], that were later extended [15] and calibrated with empirical data [START_REF] Hoffmann | Unidirectional incompatibility between populations of 455 Drosophila simulans[END_REF][START_REF] Turelli | Cytoplasmic incompatibility in Drosophila simulans: dynamics and parameter estimates from natural populations[END_REF], but only in the 1990s was a formal mechanistic model 50 proposed [START_REF] Werren | Biology of Wolbachia[END_REF][START_REF] Hurst | The evolution of cytoplasmic incompatibility or when spite can be successful[END_REF][START_REF] Breeuwer | Microorganisms associated with chromosome destruction and reproductive isolation between two insect species[END_REF]. The fact that sperm from Wolbachia-carrying males kills uninfected but not infected embryos upon fertilization is compatible with a toxin-antidote model (hereafter TA model) (Figure 2, Key Figure). The toxin factor, deposited in maturing sperm, would kill the embryos unless they are rescued by the antidote. Although concurrent explanations were also proposed [START_REF] Kose | Organization of Wolbachia pipientis in the Drosophila fertilized egg and embryo revealed by an anti-Wolbachia monoclonal antibody[END_REF][START_REF] Callaini | Wolbachia-induced delay of paternal chromatin condensation does not prevent maternal chromosomes from entering anaphase in incompatible 470 crosses of Drosophila simulans[END_REF], the discovery of Wolbachia strains capable of rescuing CI without inducing 55 it further supported the notion that this phenomenon should involve two distinct factors [START_REF] Merçot | Wolbachia of the third kind was overlooked[END_REF]24]. The observations of independent effects of distinct Wolbachia strains, either in the context of multiple infections or mutual incompatibility between different Wolbachia strains, A c c e p t e d M a n u s c r i p t further suggested the toxin and antidote should interact specifically, in a lock-and-key manner [START_REF] Poinsot | On the mechanism of Wolbachia-induced cytoplasmic incompatibility: Confronting the models with the facts[END_REF]. This framework generated a set of testable predictions that fueled the experimental 60 quest to identify the CI genes, which was recently achieved [26][START_REF] Beckmann | A Wolbachia deubiquitylating enzyme induces cytoplasmic incompatibility[END_REF][START_REF] Lepage | Prophage WO genes recapitulate and enhance Wolbachiainduced cytoplasmic incompatibility[END_REF][START_REF] Shropshire | One prophage WO gene rescues cytoplasmic 485 incompatibility in Drosophila melanogaster[END_REF][START_REF] Bonneau | Culex pipiens crossing type diversity is governed by an amplified and polymorphic operon of Wolbachia[END_REF].

The first evidence pointing to the two genes later established as genuine CI factors came from a sperm proteomic study based on the rationale that the hypothetical CI toxin should be present in infected males' mature sperm, even though the bacterium itself is not [26]. Inspired by earlier proteomic analyses of Drosophila sperm [START_REF] Karr | Fruit flies and the sperm proteome[END_REF]32], this approach finally 65 pinpointed the candidate CI genes in the mosquito Culex pipiens, where the high penetrance of CI associated with the wPip Wolbachia strain predicted a toxin protein detectable by mass spectrometry. Here serendipity also played an essential role: the first CI protein identified in sperm in this study was later revealed not as the toxin gene product, as predicted, but as the antidote, the presence of which was not expected under the most parsimonious CI model 70 [START_REF] Poinsot | On the mechanism of Wolbachia-induced cytoplasmic incompatibility: Confronting the models with the facts[END_REF]. Nevertheless, its synteny and co-transcription with another Wolbachia gene, later revealed as the CI toxin, was noted, as was the similarity of this locus with typical toxinantidote systems, usually composed of two genes, the first of which, the antidote, is expressed at higher levels [START_REF] Yamaguchi | Toxin-Antitoxin Systems in Bacteria and Archaea[END_REF]. Proximity of these putative CI genes to prophages in the wPip genome was also pointed out at that time [26]. 75

The next major steps toward the demonstration that these genes and their homologs in Drosophila are responsible for the induction and rescue of CI came from a combination of approaches and model systems [START_REF] Beckmann | A Wolbachia deubiquitylating enzyme induces cytoplasmic incompatibility[END_REF][START_REF] Lepage | Prophage WO genes recapitulate and enhance Wolbachiainduced cytoplasmic incompatibility[END_REF]. In line with the TA model, biochemical analysis and transgenic expression of the putative wPip CI genes in yeast revealed that they encode a toxic deubiquitylase (DUB) and an inhibitor of this toxicity (DUBs are enzymes that specifically 80 remove ubiquitin from ubiquitin-modified proteins). When transgenically expressed in uninfected Drosophila males crossed with uninfected females, these factors recapitulate CI A c c e p t e d M a n u s c r i p t induction during the first embryonic mitosis [START_REF] Beckmann | A Wolbachia deubiquitylating enzyme induces cytoplasmic incompatibility[END_REF]. Importantly, the two proteins were also found to bind tightly to one another in a cognate-specific manner [START_REF] Beckmann | A Wolbachia deubiquitylating enzyme induces cytoplasmic incompatibility[END_REF] consistent with prior lock-and-key predictions [START_REF] Poinsot | On the mechanism of Wolbachia-induced cytoplasmic incompatibility: Confronting the models with the facts[END_REF]. At the same time, independent experiments involving the 85 homologs of these genes from the wMel Wolbachia strain, naturally infecting D. melanogaster, confirmed and complemented these results: their dual expression in uninfected Drosophila males induces a CI-like phenotype that, most importantly, is rescued by the presence of wMel bacteria in females [START_REF] Lepage | Prophage WO genes recapitulate and enhance Wolbachiainduced cytoplasmic incompatibility[END_REF].

Before discussing the many questions raised by these results, a brief note is needed to 90 avoid ambiguities stemming from different CI gene nomenclatures co-existing in the literature [START_REF] Beckmann | A Wolbachia deubiquitylating enzyme induces cytoplasmic incompatibility[END_REF][START_REF] Lepage | Prophage WO genes recapitulate and enhance Wolbachiainduced cytoplasmic incompatibility[END_REF] (Figure 3). One proposal is that the operon inducing a CI-like phenotype when expressed in Drosophila should be called "cid", for "CI-inducing DUB" [START_REF] Beckmann | A Wolbachia deubiquitylating enzyme induces cytoplasmic incompatibility[END_REF]. This functionbased name was chosen to explicitly distinguish cid from cin, short for "CI-inducing nuclease", a paralogous operon in the wPip genome, encoding a nuclease and showing a similar TA-like 95 behavior in yeast, as well as polycistronic transcription [26,[START_REF] Beckmann | A Wolbachia deubiquitylating enzyme induces cytoplasmic incompatibility[END_REF]. Within each operon, the first gene, encoding the putative antidote, is labeled A (e.g., cidA or cinA) and the second, encoding the putative toxin, is labeled B. A contrasting nomenclature proposes that all the different CI-associated genes should be more neutrally noted as "CI factors" (cif), and that the different paralogs should be distinguished on the basis of phylogeny [START_REF] Lepage | Prophage WO genes recapitulate and enhance Wolbachiainduced cytoplasmic incompatibility[END_REF][START_REF] Lindsey | Evolutionary Genetics of Cytoplasmic Incompatibility Genes cifA and cifB in Prophage WO of Wolbachia[END_REF]. The "operon" 100 designation is also questioned by these authors, primarily because the two transcripts can be measured at different levels, which could be indicative of different promoters [START_REF] Lindsey | Evolutionary Genetics of Cytoplasmic Incompatibility Genes cifA and cifB in Prophage WO of Wolbachia[END_REF]. Yet, no such distinct promoters have been identified and different expression levels of the two genes are not uncommon within bacterial operons [START_REF] Güell | Bacterial transcriptomics: What is beyond the RNA horiz-ome?[END_REF]. In our view, the "operon" designation is thus appropriate in the present case. Regardless of this semantic debate, the two adjacent 105 genes within each potential CI locus are labeled A and B in both nomenclature proposals.

A c c e p t e d M a n u s c r i p t

Thus, cidA and cidB are synonymous with cifA and cifB (clade 1), respectively, and these constitute the major CI genes identified to date, recapitulating the phenotype in Drosophila.

We see pros and cons in both nomenclatures and suggest they should be merged into a single system to avoid further confusion. The "cif" term (for CI factor) seems appropriate to 110 designate CI genes in general and we will use it in that sense. To denote their different functional categories and evolutionary histories, we will distinguish cid (the DUB operon), from cin (the nuclease operon) and use this dichotomy to designate specific cif genes from any Wolbachia strain. Some operons have been predicted to carry both functions; we suggest these should be called "cnd" (Figure 3) [START_REF] Beckmann | A Wolbachia deubiquitylating enzyme induces cytoplasmic incompatibility[END_REF]36]. When necessary, we will append the 115 Wolbachia strain name as a superscript. In our opinion, this system merges positives of both previously proposed nomenclatures and fairly acknowledges the concomitant discovery of these gene pairs in two separate studies [START_REF] Beckmann | A Wolbachia deubiquitylating enzyme induces cytoplasmic incompatibility[END_REF][START_REF] Lepage | Prophage WO genes recapitulate and enhance Wolbachiainduced cytoplasmic incompatibility[END_REF].

Following publication of the first two conclusive reports on cif genes, two major unsettled points remained [START_REF] Lepage | Prophage WO genes recapitulate and enhance Wolbachiainduced cytoplasmic incompatibility[END_REF][START_REF] Lindsey | Evolutionary Genetics of Cytoplasmic Incompatibility Genes cifA and cifB in Prophage WO of Wolbachia[END_REF]. First, the antidote activity of the CidA protein, although 120 demonstrated for both CidA wPip and CinA wPip in yeast, was not established in Drosophila. A more recent report has now clarified this point: if expressed in sufficient amount during oogenesis, CidA wMel was found to restore the viability of uninfected embryos fertilized by wMel-carrying Drosophila males [START_REF] Shropshire | One prophage WO gene rescues cytoplasmic 485 incompatibility in Drosophila melanogaster[END_REF]. A second major question, still not fully settled as of this writing, is whether CidB can act alone as the CI toxin or needs some interaction with CidA to 125 express toxicity. Transgenic expression in yeast indicates that the DUB activity of CidB wPip has in itself a toxic effect, which is inhibited by co-expression of CidA wPip . However, Drosophila lines expressing CidB wPip alone were never obtained, making it impossible to directly assess its effect in this context [START_REF] Beckmann | A Wolbachia deubiquitylating enzyme induces cytoplasmic incompatibility[END_REF]. A simple hypothesis, compatible with the idea that CidB alone is indeed the CI toxin, is that CidB wPip can exert deleterious effects outside of the first embryonic 130 A c c e p t e d M a n u s c r i p t mitosis, where CI is usually first realized, and therefore requires co-expression of CidA, the antidote, in any tissue where CidB is present, and at sufficiently high dosage. Such a model would also help explain the paradoxical observation that first turned the spotlight toward the CI operon: the high dosage of CidA protein in mature sperm [26]. If this interpretation is correct, CI would result from removal or inactivation of the paternal CidA protein just before, 135 or just after, fertilization.

At first sight, experiments involving transgenic expression of the cid wMel genes in Drosophila argue against a toxic effect of CidB wMel alone: lines expressing this protein were viable and the CI phenotype was recapitulated only by males expressing both CidB wMel and CidA wMel [START_REF] Lepage | Prophage WO genes recapitulate and enhance Wolbachiainduced cytoplasmic incompatibility[END_REF]. This result has been considered by some authors as arguing for a "two-by-one" 140 model, in which CI induction would somehow rely on both CidA and CidB proteins, while rescue would rely on CidA only [START_REF] Shropshire | One prophage WO gene rescues cytoplasmic 485 incompatibility in Drosophila melanogaster[END_REF]. However, differences in the toxicity of CidB wMel and CidB wPip may reconcile these experimental results with a simple TA model. Although the wMel Wolbachia strain may induce very high CI in a permissive host background (e.g., upon artificial transfer into Drosophila simulans) [START_REF] Poinsot | Wolbachia transfer from Drosophila melanogaster into D. 500 simulans: Host effect and cytoplasmic incompatibility relationships[END_REF], it is well established that it has a low penetrance in its 145 natural host [START_REF] Merçot | Wolbachia infections in Drosophila melanogaster and D. simulans: polymorphism and levels of cytoplasmic incompatibility[END_REF], possibly as a result of past coevolution [START_REF] Turelli | Evolution of incompatibility-inducing microbes and their hosts[END_REF]. A low toxicity of CidB wMel in D. melanogaster would explain why flies expressing this protein alone can survive. But why is CI not expressed in such transgenic lines? One possibility is that CidB wMel is toxic enough to kill those maturing sperm cells where it is most highly expressed. It would follow that the surviving mature sperm would be precisely those where CidB wMel is not sufficiently expressed 150 to induce either sperm or embryonic death. This model could be tested by assessing if D. melanogaster males expressing CidB wMel alone show a reduced sperm production.

Notably, the hypothesis that cidB may be deleterious in various cell types and developmental stages does not imply that the Wolbachia themselves express this gene in A c c e p t e d M a n u s c r i p t many tissues of their native hosts. In other words, the pattern of expression of cidB in 155 transgenic insects says nothing about its natural expression in Wolbachia-infected hosts. The fact that viable uninfected offspring can often be obtained through imperfect Wolbachia transmission or antibiotic curing actually suggests that in a natural context, the CI toxin is not expressed in all tissues. At present, one cannot exclude a "two-by-one" model, where both CidA and CidB proteins would be required to produce a toxic effect, but this would imply 160 drastically different effects of CidA in sperm (where it would contribute to toxicity) and in the eggs (where it is known to act as an antidote). In our view, and as discussed above, a model where CidB and CidA act respectively as toxin and antidote is more likely correct and can be reconciled with the data in hand. As we shall now discuss, comparative genomics of cif genes, both at micro and macro-evolutionary scales, further support this conclusion [START_REF] Bonneau | Culex pipiens crossing type diversity is governed by an amplified and polymorphic operon of Wolbachia[END_REF][START_REF] Lindsey | Evolutionary Genetics of Cytoplasmic Incompatibility Genes cifA and cifB in Prophage WO of Wolbachia[END_REF]. 165

CI evolution

Although C. pipiens provided the original CI model system, the simple pattern presented in Figure 1, which best illustrates the invasiveness of CI, is never seen in natural populations of this species. Wherever they come from, C. pipiens mosquitoes are always infected by 170 Wolbachia (while uninfected specimens can occasionally be found in a cryptic species [START_REF] Rasgon | Wolbachia and cytoplasmic incompatibility in the California Culex pipiens mosquito species complex: parameter estimates and infection dynamics in natural populations[END_REF][START_REF] Dumas | Molecular data reveal a cryptic species within the Culex pipiens mosquito complex[END_REF]). As a consequence, infected males never encounter uninfected females in nature and CI is only expressed in its most elaborate form: infected males and females can only produce offspring if they carry "compatible infections." Indeed, C. pipiens has long been known for harboring a large diversity of cross-incompatible mosquito lines [START_REF] Laven | Crossing experiments with Culex strains[END_REF], which are now known to 175 carry closely related yet incompatible Wolbachia strains [START_REF] Atyame | Wolbachia divergence and the evolution of cytoplasmic incompatibility in Culex pipiens[END_REF]43]. These are said to be "bidirectionally incompatible" if both directions of crosses result in embryo death. By A c c e p t e d M a n u s c r i p t contrast, crosses between "unidirectionally incompatible" strains produce effective CI in only one direction, producing a pattern similar to that illustrated in Figure 1, although all individuals are infected. 180 How can this be? That different Wolbachia strains may harbor different compatibility types is easily explained in the framework of the TA model, especially in its lock-and-key formulation: incompatible Wolbachia strains will carry incompatible locks and keys. In this regard, C. pipiens is not unique: Drosophila simulans, among many other species, also carries several distinct Wolbachia strains, each with its own compatibility type [START_REF] Merçot | Wolbachia infections in Drosophila melanogaster and D. simulans: polymorphism and levels of cytoplasmic incompatibility[END_REF]. In this fruit fly, 185 compatibility relationships between lines can, at least in theory, be parsimoniously explained by variations at a single TA locus [START_REF] Nor | On the genetic architecture of cytoplasmic incompatibility: inference from phenotypic data[END_REF] (although some D. simulans Wolbachia genomes appear to include multiple cif paralogs [45]). In contrast, incompatibility patterns are so complex in C. pipiens that they cannot be explained with a single TA pair per Wolbachia genome [START_REF] Atyame | Wolbachia divergence and the evolution of cytoplasmic incompatibility in Culex pipiens[END_REF][START_REF] Atyame | Multiple Wolbachia determinants control the evolution of A c c e p t e d M a n u s c r i p t 47[END_REF]. Specifically, compatibility relationships are not all transitive in this system: two strains may be 190 mutually compatible although they harbor distinct compatibility patterns with other strains [START_REF] Nor | On the genetic architecture of cytoplasmic incompatibility: inference from phenotypic data[END_REF]. Focusing on compatibility relationships among 19 wild-type C. pipiens lines [43], theoretical analyses grounded in the TA framework concluded that at least five TA pairs may co-occur in one wPip genome [START_REF] Nor | On the genetic architecture of cytoplasmic incompatibility: inference from phenotypic data[END_REF]. Now that the CI genes have been identified, the time has come to test such theoretical predictions. 195

Although all infections from C. pipiens represent a monophyletic group of close

Wolbachia relatives, fine-scale phylogenetic markers allow one to distinguish five clades within which crosses are most often compatible and between which they are most often incompatible [START_REF] Atyame | Wolbachia divergence and the evolution of cytoplasmic incompatibility in Culex pipiens[END_REF]47]. Based on this phylogenetic and phenotypic diversity, Bonneau et al. [START_REF] Bonneau | Culex pipiens crossing type diversity is governed by an amplified and polymorphic operon of Wolbachia[END_REF] selected multiple mosquito lines collected worldwide to assess molecular variation of the cif 200 genes and test their explanatory power with regard to compatibility patterns. Under the A c c e p t e d M a n u s c r i p t hypothesis that these genes underlie CI diversity in Culex, they should be present in several distinct copies within each Wolbachia genome, and strains harboring different CI patterns should carry different cif repertoires. The cid data fully matched these predictions: in the Culex populations studied, cidA and cidB show tremendous variation in both sequence and 205 copy number, resulting in large part from duplication and recombination events, possibly mediated by the prophage region where they occur. By contrast, the cinA and cinB genes were found to be monomorphic in the wPip strains analyzed, indicating that incompatibilities in Culex are the result of cid but not cin variations.

While mutually incompatible strains should harbor different cid repertoires, as was 210 indeed observed, the TA model does not predict that mutually compatible strains should carry exactly the same cid alleles. First, mutations may occur outside of the toxin / antidote interaction sites, which should not affect compatibility patterns and would thus be neutral as far as CI is concerned. Second, the TA model does not demand a strict one-to-one specificity of toxin-antidote interactions: some antidotes may inhibit more than one toxin. Both of these 215 explanations may contribute to explain why a number of mutually compatible strains harbor different cid repertoires. Nevertheless, these strains happen to always share a common cidA variant that may represent a super-antidote, matching several distinct toxins [START_REF] Bonneau | Culex pipiens crossing type diversity is governed by an amplified and polymorphic operon of Wolbachia[END_REF]. Expressing these different Cid variants in an experimentally flexible in vivo system such as yeast, together with biochemical studies, should clarify this issue. 220

What does the cid polymorphism tell us about the evolutionary process of CI diversification? In other words, can cid molecular variations reveal how bidirectionally incompatible Wolbachia evolved? Past theoretical work has highlighted how much the genetic architecture of CI should affect this process [START_REF] Turelli | Evolution of incompatibility-inducing microbes and their hosts[END_REF][START_REF] Nor | On the genetic architecture of cytoplasmic incompatibility: inference from phenotypic data[END_REF][START_REF] Charlat | On the mod resc model and the evolution of Wolbachia compatibility types[END_REF][START_REF] Charlat | Exploring the evolution of Wolbachia compatibility types: a simulation approach[END_REF]. The experimental confirmation that CI genes can occur in multiple copies [START_REF] Bonneau | Culex pipiens crossing type diversity is governed by an amplified and polymorphic operon of Wolbachia[END_REF] greatly simplifies the problem 225 A c c e p t e d M a n u s c r i p t from a theoretical standpoint: following duplication of a TA pair, redundancy between loci may allow new antidotes to emerge without compromising self-compatibility. This first step could be followed by the occurrence of matching mutations of the toxin, producing two distinct CI operons in a genome [START_REF] Nor | On the genetic architecture of cytoplasmic incompatibility: inference from phenotypic data[END_REF]. The possibility that some antidotes may inhibit more than one toxin opens yet other possible scenario, where either side may diversify first if the 230 process goes through a broad spectrum phase before further restriction of specificity, as has been suggested in other TA systems [START_REF] Aakre | Evolving New Protein-Protein Interaction Specificity through 530 Promiscuous Intermediates[END_REF].

The absence of polymorphism in the cin operon (the nuclease paralog of cid) rules out this locus as a driver of CI diversity in Culex. However, could this operon or other cid paralogs operate in other species? Comparative genomics among Wolbachia lineages indicate that 235 some CI-inducing strains do not contain cid genes but only the cin paralogs [START_REF] Lepage | Prophage WO genes recapitulate and enhance Wolbachiainduced cytoplasmic incompatibility[END_REF][START_REF] Lindsey | Evolutionary Genetics of Cytoplasmic Incompatibility Genes cifA and cifB in Prophage WO of Wolbachia[END_REF]. Moreover, neither cid nor cin-related paralogs were found in a close relative of wMel that does not induce CI [START_REF] Lindsey | Evolutionary Genetics of Cytoplasmic Incompatibility Genes cifA and cifB in Prophage WO of Wolbachia[END_REF][START_REF] Sutton | Comparative genome analysis of Wolbachia strain wAu[END_REF], or in nematode Wolbachia strains that have become obligate mutualists [26,[START_REF] Lindsey | Evolutionary Genetics of Cytoplasmic Incompatibility Genes cifA and cifB in Prophage WO of Wolbachia[END_REF]. These results support the involvement of both cid and cin operons in CI induction by Wolbachia, with evidence still emerging. Further, these two loci seem sufficient so far to 240 explain all CI cases associated with Wolbachia. This conclusion does not hold when symbiont lineages distant from Wolbachia are considered. Notably, Cardinium bacteria can induce CI but do not carry identifiable cid or cin genes, suggesting convergent evolution of CI [START_REF] Lindsey | Evolutionary Genetics of Cytoplasmic Incompatibility Genes cifA and cifB in Prophage WO of Wolbachia[END_REF]52]. However, a recent study suggested that TA-like systems showing a putative DUB activity can be found in a diverse array of 245 endosymbionts, not only in Cardinium (albeit in a lineage where CI itself was not demonstrated), but also in other intracellular symbionts, such as Rickettsia and Spiroplasma, that are also known as manipulators of host reproduction [36]. Also notable is the presence of cin genes in non-CI but parthenogenesis-inducing Wolbachia strains [START_REF] Lindsey | Evolutionary Genetics of Cytoplasmic Incompatibility Genes cifA and cifB in Prophage WO of Wolbachia[END_REF]. Although these A c c e p t e d M a n u s c r i p t results do not provide direct evidence that distant cif homologs are involved in CI or other 250 forms of reproductive manipulation, they make this hypothesis worth exploring. Finally, the discovery of CI associated with a non-Wolbachia Alphaproteobacterium [START_REF] Takano | Unique clade of alphaproteobacterial endosymbionts induces complete cytoplasmic incompatibility in the coconut beetle[END_REF] provides another system where cif gene relatives should be sought.

Cui bono? Levels of selection and the origin of CI 255

From Cicero to the detective Columbo, asking "Cui bono?" (Who benefits?) has been a useful avenue of criminal and sociological investigation. This question is also relevant in a Darwinian framework and has prompted novel explanations of evolutionary oddities such as altruism and selfish genetic elements [START_REF] Burt | Genes in Conflict: The Biology of Selfish Genetic Elements[END_REF][START_REF] Hamilton | The evolution of altruistic behavior[END_REF][START_REF] Hickey | Selfish DNA: a sexually-transmitted nuclear parasite[END_REF]. Applying this question to CI led early theorists to highlight the benefit it provides to Wolbachia itself, rather than its host, suggesting that the 260 bacterium represents the right level of selection to understand how this phenomenon came to be [START_REF] Hurst | The evolution of cytoplasmic incompatibility or when spite can be successful[END_REF]. Remarkably, the same early study was visionary in suggesting that the CI genes may be associated with mobile genetic elements: the discovery that they sit in a prophage supported this hypothesis [26- [START_REF] Lepage | Prophage WO genes recapitulate and enhance Wolbachiainduced cytoplasmic incompatibility[END_REF][START_REF] Lindsey | Evolutionary Genetics of Cytoplasmic Incompatibility Genes cifA and cifB in Prophage WO of Wolbachia[END_REF][START_REF] Bordenstein | Eukaryotic association module in phage WO genomes from Wolbachia[END_REF]. In our view, this finding also makes it relevant to revisit the question of the adaptive significance of CI, and ask whether this phenomenon may 265 have evolved in the first place for the benefit not of Wolbachia, but of the bacterium's own intragenomic parasites.

The hypothesis that TA systems, including restriction-modification systems or bacteriocins, constitute fundamentally selfish genetic elements has received ample support in the field of microbial evolution [58][59][60][61][62][START_REF] Kusano | Restriction-modification systems as genomic parasites in competition for specific sequences[END_REF]. In brief, the idea is that TA systems make themselves 270 addictive as soon as they enter a cell: the toxin molecule is typically more stable than the antidote, so that removing the source of both results in cell death. Although this property is not necessarily invasive (an element killing a host once it is lost increases its effective A c c e p t e d M a n u s c r i p t transmission rate but will not, by itself, increase its frequency), it can promote invasion under certain circumstances, especially if the TA system is part of a horizontally transmitted mobile 275 genetic element [START_REF] Rankin | The coevolution of toxin and antitoxin genes drives the dynamics of bacterial addiction complexes and intragenomic conflict[END_REF].

With regard to Wolbachia, it seems reasonable to assume that the nearly universal positioning of the CI genes within prophages [START_REF] Bordenstein | Eukaryotic association module in phage WO genomes from Wolbachia[END_REF][START_REF] Klasson | Genome Evolution of Wolbachia Strain wPip from the Culex pipiens Group[END_REF] is not mere chance, but rather has some adaptive significance. So, who benefits from the CI genes, and more specifically, from their association with prophages? This particular location is not a priori adaptive at the Wolbachia 280 level, but it may well be at the phage level: horizontally transmitted phages should more readily invade Wolbachia populations if they carry a TA system. The occurrence of distant cif relatives in several other bacterial lineages, where they sit in plasmids rather than prophages, argues against a purely viral origin of this gene family [36], and so does their relatedness to typical eukaryotic sequences [26,36,[START_REF] Bordenstein | Eukaryotic association module in phage WO genomes from Wolbachia[END_REF]. However, the association of cif genes with phages in 285 the Wolbachia lineage opens the possibility that they first invaded this clade as a phage adaptation, and only later became "CI genes."

As previously pointed out [START_REF] Poinsot | Infection par Wolbachia chez Drosophila simulans : etude des interactions hote-symbiote[END_REF], the elimination of Wolbachia from maturing sperm establishes conditions for the evolution of such a selfish phage TA system toward genuine CI.

If one simply assumes that the toxin is exported outside of the bacterium and can exert its 290 deleterious effects on the host (eukaryotic) cell, then CI could easily arise. In particular, the paternal pronucleus, threatened by a stable toxin, would need fresh antidote from an infected egg to be able to participate to the first embryonic division. Such a situation would in turn make the bacterium invasive through CI, generating a convergence in the evolutionary interests of the phage and its bacterial host. 295

A c c e p t e d M a n u s c r i p t

Concluding remarks and future perspectives

Recent research has provided a firm answer to the first of many critical questions regarding the molecular biology of CI (Outstanding Questions Box). There is indeed a general agreement that the principal Wolbachia CI genes have been found [26][START_REF] Beckmann | A Wolbachia deubiquitylating enzyme induces cytoplasmic incompatibility[END_REF][START_REF] Lepage | Prophage WO genes recapitulate and enhance Wolbachiainduced cytoplasmic incompatibility[END_REF][START_REF] Shropshire | One prophage WO gene rescues cytoplasmic 485 incompatibility in Drosophila melanogaster[END_REF], and our present analysis 300 concurs with past predictions that they encode toxins and antidotes interacting in a lock-andkey fashion [START_REF] Poinsot | On the mechanism of Wolbachia-induced cytoplasmic incompatibility: Confronting the models with the facts[END_REF]. Nevertheless, significant revisions of the basic TA model are needed to account for all observations. First, we suggest that CifB toxins may exert their deleterious effects not only in incompatible crosses during the first embryonic mitosis, but also in maturing sperm and other cell types, in which the presence of CifA antidotes would also be 305 required. This hypothesis would explain why dual transgenic expression of the two genes in uninfected Drosophila males is required for them to survive (in the case of cid wPip ), or for CI to be transgenically expressed (in the case of cid wMel ) [START_REF] Beckmann | A Wolbachia deubiquitylating enzyme induces cytoplasmic incompatibility[END_REF][START_REF] Lepage | Prophage WO genes recapitulate and enhance Wolbachiainduced cytoplasmic incompatibility[END_REF]. Patterns of molecular variation of the cidA and cidB genes [START_REF] Bonneau | Culex pipiens crossing type diversity is governed by an amplified and polymorphic operon of Wolbachia[END_REF] also suggest that some antidotes may match several toxin products, a conjecture that could be tested by direct assessments of CidA-B interactions in 310 simplified in vivo or in vitro systems. Furthermore, these genetic data confirm the earlier speculation that single Wolbachia strains may carry multiple active CI operons, not only in Culex [START_REF] Atyame | Multiple Wolbachia determinants control the evolution of 565 cytoplasmic incompatibilities in Culex pipiens mosquito populations[END_REF] but also in Drosophila [START_REF] Zabalou | Multiple rescue factors within a Wolbachia strain[END_REF].

While the genetics of CI have been clarified, its molecular mechanisms remain to be worked out (Box 2). Furthermore, the exact nature of the interaction between toxin and 315 antidote proteins remains to be determined, although there are some indications of which residues might be involved [START_REF] Bonneau | Culex pipiens crossing type diversity is governed by an amplified and polymorphic operon of Wolbachia[END_REF]. Clarification of these issues will be needed to design artificial CI systems that may be used in biological control. Comparative genomics will provide a valuable complement to experiments to understand how CI actually works, and to investigate how it evolves. At a microevolutionary scale, further exploration of the Culex system will 320 Building on the observation that the CI genes lie in a mobile genetic element (the WO prophage), we suggested they might have originally been selected as a phage invasive 325 strategy and were later domesticated by the bacterium. In line with this hypothesis, distant homologues of the cid genes are present in other bacterial symbionts, and nearly always in association with phages or plasmids [36]. Pushing the reasoning one step further, one may envisage that the TA operon was initially costly for the phage, and only later became domesticated as an effective invasive strategy. CI would then be a case of a parasitic operon 330 within a parasitic phage within a parasitic symbiont, each relying today on its past inner enemy. The observation that some insects, and many filarial nematodes, cannot live without Wolbachia [69-71] reinforces the idea that such "evolution through addiction" constitutes a never-ending process, producing the Russian dolls that all organisms seem to be.

Box 1: Wolbachia and Cytoplasmic Incompatibility

Perhaps the most crucial aspect of Wolbachia biology is the fact it is transmitted from mothers to offspring through the egg cytoplasm [START_REF] O'neill | Influential Passengers : Inherited Microorganisms and Arthropod Reproduction[END_REF] although horizontal transfer may also occur [START_REF] Bailly-Bechet | How Long Does Wolbachia Remain on Board?[END_REF]45]. Vertical transmission through the female germline will select Wolbachia traits 350 that increase the fitness of infected females, or more technically, the number or the fitness of their infected daughters. CI can be interpreted within such a framework: by protecting infected embryos from the lethal effect of infected males' sperm, Wolbachia increases the relative brood size of infected females. Infected males pay a heavy price for CI (mating with uninfected females drastically reduces their own fertility) but this is costless for Wolbachia 355 because males do not transmit the symbiont to future generations. Notably, only few uninfected females are sterilized through CI when Wolbachia is rare in the host population, so that a low frequency infection has low chances of invasion, unless it combines CI with other traits, such as protection of the host against pathogens [START_REF] Fenton | Solving the Wolbachia Paradox: Modeling the Tripartite Interaction between Host, Wolbachia, and a Natural Enemy[END_REF]. Such protective effects are actually observed [START_REF] Teixeira | The bacterial symbiont Wolbachia induces resistance to RNA 580 viral infections in Drosophila melanogaster[END_REF] and can also block the passage of human pathogens through insect 360 vectors [START_REF] Moreira | A Wolbachia Symbiont in Aedes aegypti Limits Infection with Dengue, Chikungunya, and Plasmodium[END_REF]. The ongoing "World Mosquito Program" makes use of this property: the massive release of CI-inducing mosquitoes allows the spread of the infection, which should reduce overall viral transmission rates [START_REF] Flores | Controlling vector-borne diseases by releasing modified mosquitoes[END_REF] although the implementation and evolutionary outcome of this approach remain uncertain [START_REF] Vavre | Making (good) use of Wolbachia: What the models say[END_REF]78]. This deposition defect could be responsible for improper paternal chromosome condensation in prophase [START_REF] Callaini | Wolbachia-induced delay of paternal chromatin condensation does not prevent maternal chromosomes from entering anaphase in incompatible 470 crosses of Drosophila simulans[END_REF][80][81][82]. How could these features be related to cif genes activities? While 375

bioinformatics predict a number of potential enzymatic properties for the various cif paralogs, the DUB domain of CidB stands at the moment as the strongest CI effector candidate: activesite mutations in CidB eliminate CI in transgenic insects as well as toxicity in yeast models [START_REF] Beckmann | A Wolbachia deubiquitylating enzyme induces cytoplasmic incompatibility[END_REF]. Furthermore, different CidB repertoires induce different levels of CI defects at the cellular level [82]. Although the DUB activity may affect upstream components of the CI causal chain, 380 interference with the ubiquitylation status of key chromatin or cell cycle regulators appears as an obvious hypothesis to link CidB to CI cytology.

However, the observation that some Wolbachia strains may lack cid genes but still induce CI suggests the DUB function is just one tool in a larger cif arsenal [26, [START_REF] Lepage | Prophage WO genes recapitulate and enhance Wolbachiainduced cytoplasmic incompatibility[END_REF][START_REF] Lindsey | Evolutionary Genetics of Cytoplasmic Incompatibility Genes cifA and cifB in Prophage WO of Wolbachia[END_REF]36]. CinB appears as the most likely source of CI in this case, since active-site mutations block CinB 385 toxicity in yeast [START_REF] Beckmann | A Wolbachia deubiquitylating enzyme induces cytoplasmic incompatibility[END_REF]. But why and how would different enzymes generate the same CI phenotype? One hypothesis would be that the DUB and nuclease activities are two upstream components of a common causal chain. Simply put, cutting DNA and cleaving ubiquitin may constitute alternative ways to disturb paternal chromosome condensation. Definitive A c c e p t e d M a n u s c r i p t demonstrations that CinB can induce CI, as well as identification of the CidB and CinB targets 390 will be crucial to resolving these issues.

The fact that CI may be mediated by two distinct effectors, involving DUB or nuclease activities, is also relevant to investigate how CI antitoxins may function. CidA and CinA may inhibit DUB and nuclear activities, respectively, either through distinct pathways or alternatively, through a single mechanism, such as protein sequestration or relocalization. The 395 latter hypothesis may explain why CidA can inhibit the toxicity of CidB without specifically reducing its DUB activity against model substrates [START_REF] Beckmann | A Wolbachia deubiquitylating enzyme induces cytoplasmic incompatibility[END_REF]. Further characterization of the Cid and Cin protein structures and identification of the residues involved in the interaction between cognate partners of both operons appear as promising avenues of research to better understand how infected embryos get rescued from CI. 400 A c c e p t e d M a n u s c r i p t In this naming system, the cif (and Cif) terms designate CI genes (and proteins) in general, while genes from specific operon categories are called according to the enzymatic activity of 420 the putative toxin (DUB, nuclease or both). The first and second gene within each operon are denoted A and B, respectively, and the Wolbachia strain is indicated as a superscript when relevant. The structure of several CI operons is shown to illustrate this system, with active-site residues labeled.

  A c c e p t e d M a n u s c r i p t clarify how different Wolbachia strains become mutually incompatible. On a macroevolutionary scale, genomic comparisons may further reveal which cif genes can actually induce CI, and what are the commonalities and specificities between these systems.

  A c c e p t e d M a n u s c r i p tBox 2: CI molecular mechanisms Whilst the CI effectors have been identified, the question of how they induce embryo death or rescue remains largely unanswered, and can be divided in two: how do cif toxins impede paternal chromosomes, and how do cif antidotes impede the toxins? Functional properties of 370 the cifB genes are obviously relevant to the first question, as are characterizations of the CI phenotype at the cytological level. The earliest detected abnormality in CI crosses is improper deposition of maternal H3.3 histones on the paternal genome after protamine removal[START_REF] Landmann | Wolbachia-mediated cytoplasmic incompatibility is associated with impaired histone deposition in the male pronucleus[END_REF].

Figure 1 .

 1 Figure 1. Cytoplasmic incompatibility in its simplest form: infected females are compatible

Figure 2 .

 2 Figure 2. A schematic view of the toxin-antidote model. In immature sperm, Wolbachia (in

Figure 3 .

 3 Figure 3. A nomenclature proposal and schematic view of the putative CI operons structures.

  A c c e p t e d M a n u s c r i p t Yen, J.H. and Barr, a R. (1973) The etiological agent of cytoplasmic incompatibility in A c c e p t e d M a n u s c r i p t A c c e p t e d M a n u s c r i p t

Figure 2 A

 2 Figure 2

Figure 3 A

 3 Figure 3

Caspari, E. and Watson, G.S. (1959) On the evolutionary importance of cytoplasmic sterility in mosquitoes. Evolution (N. Y). 13, 568-570

oogenesis in a parasitic wasp. Proc Natl Acad Sci U S A 98, 6247-6252

Reed, K.M. and Werren, J.H. (1995) Induction of paternal genome loss by the paternalsex-ratio chromosome and cytoplasmic incompatibility bacteria (Wolbachia): a

comparative study of early embryonic events. Mol ReprodDev 40,[408][409][410][411][412][413][414][415][416][417][418]

Acknowledgments

This work benefited from funding by the United States Department of Agriculture (grant USDA-1015922 to JB), the National Institutes of Health (grant GM046904 to MH) and the French "Agence Nationale de la Recherche" (CIAWOL grant to MS: ANR-16-CE02-0006-01, 340 HORIZON grant to SC: ANR-17-CE02-0021-02).

A c c e p t e d M a n u s c r i p t


Bilel Achour
email: bilel.achour.etu@univ-lemans.fr

Nadia Aloui
email: nadia.aloui@univ-lemans.fr

Najla Fourati
email: najla.fouratiennouri@lecnam.net

Chouki Zerrouki
email: chouki.zerrouki@lecnam.net

Nourdin Yaakoubi
email: nourdin.yaakoubi@univ-lemans.fr

Modelling and simulation of SAW delay line sensors with COMSOL Multiphysics

Keywords: Finite Element Method (FEM), COMSOL Multiphysics, Surface acoustic wave (SAW), 36°lithium tantalate (LiTaO3)

This study concerns 2D and 3D Finite Element Method (FEM) simulation of surface acoustic wave (SAW) sensors using COMSOL Multiphysics software. SAW device has been designed on piezoelectric substrate; 36° rot lithium tantalate (LiTaO3). Simulations were made on well-known structure to ensure the concordance between 2D and 3D models, and to define a 2D one that can account for and predict the electrical behaviour of SAW transducers for the future optimizations. The results show good agreement between numerical simulation and experimental S21 spectra. Accordingly, we can use the 2D built model for simulations intended to optimize the structure of devices, mainly for increasing their sensitivity.

Introduction

Modeling is of primary interest in many areas. It allows suitable optimization of components or structures before the realization steps, offering thus a substantial gain in both time and money. The field of sensors, and particularly Surface Acoustic Wave ones, is mostly concerned. Indeed, SAW devices can be found in wide domains, including physical sensors (temperature, pressure, torque, strain…) chemand bio-sensors (insecticides, pollutants, proteins, biomarkers…), electronics, telecommunication, signal processing (filtering, modulation, RFID…).

The main quality of SAW sensors is their versatility, as they can function in either liquid or gaseous environments, for the detection of various analytes, such as DNA or analogues [START_REF] Hur | Development of an SH-SAW sensor for the detection of DNA hybridization[END_REF][START_REF] Zerrouki | Biological investigation using shear horizontal surface acoustic wave sensor: small "Click generated" DNA hybridization detection[END_REF], antibodies/antigens [START_REF] Rupp | A shear horizontal surface acoustic wave sensor for the detection of antigen-antibody reactions for medical diagnosis[END_REF], gases [START_REF] Marcu | Surface Acoustic Wave Sensors for Hydrogen and Deuterium Detection[END_REF], pesticides [START_REF] Mazouz | Highly Selective Polypyrrole MIP-Based Gravimetric and Electrochemical Sensors for Picomolar Detection of Glyphosate[END_REF], heavy metals [START_REF] Ramshani | SAW sensor based microfluidic system for the detection of heavy metal compounds in liquid environments[END_REF], etc. Moreover, detection of chemical or biological entities can be done in real time and without labeling. Compared to other chemical and biological sensors, the SAW ones are highly sensitive and can achieve very low limits of detection [START_REF] Richardson | Low Insertion Loss and Highly Sensitive SH-SAW Sensors Based on 36° YX LiTaO3 Through the Incorporation of Filled Microcavities[END_REF][START_REF] Li | Gold nanoparticle-based low limit of detection Love wave biosensor for carcinoembryonic antigens[END_REF], which explains the fact that they are still the subject of constant development. Therefore, simulation becomes a powerful tool to efficiently accompany these developments, to design optimized structure for applications were highly sensitive devices are needed. In this paper, we present a comparison between the numerical simulation and experimental results concerning a delay-line structure. This will permit us to define a model that can account for and predict the electrical behavior of any SAW transducers for the future optimizations.

SciForum

Mol2Net-04, 2018, BIOCHEMPHYS-01 (pages 1-x, type of paper, doi: xxx-xxxx http://sciforum.net/conference/mol2net-4

Results and Discussion

3D Finite Element Method (FEM) simulation was considered to simulate the physical and electrical behavior of SAW delay-line, using COMSOL software. The frequency response is obtained by considering the Delta-Function model whose equations are implemented in COMSOL.

The key required parameter is the SAW velocity, that defined the delay-line operating frequency.

The simulations are run from 80 MHz to 200 MHz with a frequency step of 1 MHz. Calculated and experimental frequency responses (S21 spectra) are plotted in Fig. 1a and Fig. 1b respectively. Numerical simulation results show that the delay-line operates around 104 MHz as expected, but the maximum gain value (-55 dB) is lower than the experimental one (-18 dB). This is due to the fact that only 2 pairs of double fingers for both input and output interdigital transducers (IDTs) were used with COMSOL Multiphysics, while the real SAW device is constituted of 30 pairs fingers for each IDT. The chief benefit of simulations being the time saving, it was not possible to model a 3D response with 30 pairs of fingers, the calculation being very long, without guarantee of calculus convergence. Therefore, we redone 2D and 3D calculations on 2 and 3 pairs of fingers to make sure that the obtained results were comparable. Once ensured of results concordance, we simulated a 2D structure with 30 pairs of fingers. The results of the calculations, which lasted only few hours, are shown in Fig. 1.c. S21 spectra presents a comparable shape as experimental one: the characteristic frequency of SAW is clearly identified (with an insertion loss of -21 dB). Moreover, calculation equally account for fast volume waves around 140 MHz. Based on this calculation approach, several optimized structures will be tested before to manufacture new SAW sensors, sensitive enough to perform chemical and biological applications.

Materials and Methods

A SAW delay line was designed on 36°LiTaO3 piezoelectric substrate. Two IDTs were patterned in Cr/Au (20 nm/80 nm) layers with the same periodicity of λ = 40 μm (i.e. corresponding frequency of 104 MHz). The width of each finger is equal to 5 µm [START_REF] Fourati | Surface Acoustic Love Waves Sensor For Chemical And Electrochemical Detection[END_REF]. A metallized Cr/Au (20 nm/80 nm) sensing area of 80 µm length separates the input and output IDTs (Fig. 2). Extra fine triangle elements and a swept mesh with a distribution are used for meshing the geometry of our SAW delay Line. The meshed model is shown in Figure 3.

Conclusions

In this work, we have used COMSOL Multiphysics software to design and optimize the geometry of a SAW delay-line, and to determine its frequency response (S21 spectra). We have shown that 2D calculation can be used instead of 3D one without any information loss, leading to an important time saving. We observed a good correlation between 2D simulation and experimental results. This is of importance as it permits to envisage modelling of any structure before realization steps. This pad the way for new optimized designs of sensors, including adjunction of suitable materials between input and output IDTs, to confine the SAW energy close to the surface, enhancing thus their sensitivity.

Figure 1 .

 1 Figure 1. (a) Calculated S21 spectrum (3D model, 2 pairs of fingers for input and output IDTs); (b) experimental S21 spectrum; (c) Calculated S21 spectrum (2D model, 30 pairs of fingers for input and output IDTs).

Figure 2 .

 2 Figure 2. SAW sensor geometry.

Figure 3 .

 3 Figure 3. Mesh for simulation.

Acknowledgments

N. Yaakoubi and B. Achour wish to thank "La région Pays de la Loire" for the financial support through the WISE / RCSF project.

Conflicts of Interest

The authors declare no conflict of interest.