Emmanuel Chevallier
email: emmanuel.chevallier@mines-paristech.fr

Augustin Chevallier

Jesus Angulo

Jesús Angulo

Computing Histogram of Tensor Images using Orthogonal Series Density Estimation and Riemannian Metrics

Keywords: image histogram, density estimation, volume measure of Riemannian metrics, symmetric-positive-definite matrices

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Introduction

The histogram computation of a scalar image consists in counting the number of pixels at each different intensity value. This extremely simple approximation to the univariate density distribution of image intensities is one of the most important image processing tools to address problems such as contrast enhancement (by histogram linear stretching or using advanced approaches [START_REF] Sapiro | Histogram modification via differential equations[END_REF]), image segmentation (by 1D clustering), texture processing [START_REF] Portilla | A parametric texture model based on joint statistics of complex wavelet coefficients[END_REF], image retrieval [START_REF] Gong | Image indexing and retrieval using color histograms[END_REF], etc. As digital images underlay a quantization step, a more reliable estimation of the image distribution requires techniques such as the classical Parzen window method or a variational approach as the recently proposed in [START_REF] Nikolova | Exact histogram specification for digital images using a variational approach[END_REF]. Computation of histogram of a color image involves in addition the choice of the color space and an appropriate discretization of the colors in the image into a number of bins.

In this paper we are interested in the computation of the histogram of a tensor image, that is, an image where at pixel we have a value in SPD(n): the space of n × n positive definite symmetric matrices. Recently there has been growing interest on processing methods of such images in the computer vision community [START_REF] Jayasumana | Kernel methods on the riemannian manifold of symmetric positive definite matrices[END_REF]. As far as we know, the construction of histogram in SPD(n)-valued images has not been previously addressed. It can be noted that if a normal law density on tensors is assumed, the estimation of the density involves only the computation of the mean tensor as well as a covariance matrix [START_REF] Lenglet | Statistics on the manifold of multivariate normal distributions: Theory and application to diffusion tensor mri processing[END_REF][START_REF] Pennec | Intrinsic statistics on riemannian manifolds: Basic tools for geometric measurements[END_REF]. In our case, we do not assume that the histogram lays in any specific density family. Straightforward approach based on a discretization of SPD(n) into bins, followed by a counting of the tensors belonging to each bin, leads to a histogram estimation which suffers from various problems. A better solution involves the use of the Parzen window technique. We adopt here an alternative approach based on orthogonal series density estimation. An important property of this approach is that it depends on geometry only through the volume element, and not on the geodesic distances. The rest of the paper is organized as follows. The precise definition of image histogram adopted in this paper is given in Section 2. Section 3 introduces the notion of orthogonal series density estimation, including the measure change. A brief remind on SPD(n) space, followed by a discussion on the various measures of SPD(n) are given in Section 4. In particular, by considering SPD(n) as the space of the covariance matrices of multivariate gaussian distribution, we obtain the corresponding density estimation for the measure of the Fisher metric and the Wasserstein metric. Section 5 summarizes the framework for estimating the histograms of tensor valued images and some experimental results on the application of such histogram estimation to DTI image segmentation, texture segmentation and texture recognition. Conclusions in Section 6 end the paper.

Image histogram

In this section we set the theoretical link and notation between images and probabilities. We assimilate the notion of image histogram and the notion of density estimation. Let us consider an image I as the map:

I : Ω → V p → I(p)
We have for instance V = R for grey-scale images or V = R n for multispectral images; Ω is the support space of pixels/voxels, typically a subset of R 2 or R 3 such as a rectangle or a parallelepiped. In the case of tensor images, which is the case addressed in this paper, V ⊆ SPD(n).

We endow Ω with a measure P that reflects its geometry: as the spatial distances are usually evaluated using Euclidean distance, this measure is typically the Lebesgue measure. Indeed the prior importance of an area is often proportional to its Lebesgue measure. After rescaling the measure, the map I can be seen as a random variable. The law of the random variable I contains a synthetic information of the image. Thus, one might be interested in the study of this law. We have access to a finite number of evaluation of I in points p ∈ Ω called pixels. Each I(p) is an evaluation of the random variable I. The set {I(p), p a pixel } is a set of draws of I. We will assume that the set of pixel is uniformly distributed according to the probability P . Formally, if φ : Ω → R is a continuous function, 1 N p φ(p) → φdP when the number of pixel N tends to infinity. If they are not uniformly distributed, we have to take their distribution into account to estimate the law of I. This for instance the case for point cloud images [START_REF] Oudot | Scalar field analysis over point cloud data[END_REF].

The study of law of I is made through this set of draws. We assume that a measure µ is given on the space V . Furthermore, we make the strong assumption that the law of I has a density with respect to µ. The problem now is to estimate this density from the sample set {I(p), p a pixel }. The classical way to proceed is to cut the space V into regular bins and to count the number of draws in each bin. The precision of the estimation depends on the measure of the bins, and on the number of draws. However under some particular circumstances it might be impossible to get such a cutting of the space V in regular bins of same measure. This is often the case is µ is not the Lebesgue measure. If the measures of the bins are not equal, problems appear when the set of draws is not large enough. Let us consider for instance two adjacent bins b 1 and b 2 with µ(b 1 ) much bigger than µ(b 2 ). If a draw falls in b 2 , the obtained density will be much higher than if the draw had fallen in b 1 . This example shows that a small variation in the draw can induce important variations on the values taken by the estimated density.

Orthogonal series density estimation

Let us assume that V is a compact subset of R n , with the measure µ. The measure µ defines a scalar product on the set L 2 (µ) of square integrable functions given by f, g = f gdµ. Let us also assume that {e i } is a Hilbert basis of the functions from V to R, for the previous scalar product. The density of I against the measure µ, denoted f I , can be studied through its expression in this generator system. Indeed we have

f I = ∞ i=-∞ f I , e i e i ,
where f I , e i = f I e i dµ = E (e i (I)) .

Thus f I , e i can easily be approximated by an estimation of the expectation, i.e.,

E (e i (I)) ≈ 1 n n j=1 e i (I(p j ))
where n is the number of pixels of image I. We have then

f I ≈ ∞ i=-∞   1 n n j=1 e i (I(p j ))   e i ,
or in the case of a finite number of basis functions:

f I ≈ N i=-N   1 n n j=1 e i (I(p j ))   e i = fI . (1)
Thus we consider fI as an estimate of the density f I from the I(p j ). We note that using this method, if the {e i } are continuous functions then the estimation is continuous according to the I(p j ). This was not the case using the cutting in bins described previously. The estimation can be easily adapted to a new measure µ ′ if it has a density f µ ′ with respect to µ. The new density f ′ I becomes

f ′ I = f I f µ ′ (2)
Relationship (2) can be easily obtained using the new orthonormal basis system e i / f µ ′ instead of {e i }. Indeed we have

e i f µ ′ e j f µ ′ dµ ′ = e i e j dµ.
Hence, the new system is orthonormal for µ ′ and consequently, we have

f ′ I = N i=-N   1 n n j=1 e i (I(p j )) f µ ′   e i f µ ′ = N i=-N 1 n n j=1 e i (I(p j )) e i f µ ′ = fI f µ ′ . ( 3
)
This property will prove very useful for our work. A deeper presentation of density estimation using orthogonal series can be found in [START_REF] Efromovich | Orthogonal series density estimation[END_REF]. [START_REF] Oudot | Scalar field analysis over point cloud data[END_REF] The space SPD(n) and its parametrization A differentiable manifold is a topological space where the local neighbourhood of each point looks like a vector space. In each point of a manifold we can associate a tangent space composed of tangent vectors to all smooth curves going through the considered point. One can introduce a notion of distance on a manifold by defining a scalar product on each tangent space. If the different scalar product are compatible, that is to say vary smoothly between tangent space, it is possible to define a notion of length of a curve on the manifold. Such a m anifold is called a Riemannian manifold.

We will set here a few notations and describe the structure of the space SPD(n). Let M(n) represents the space of n × n square matrices and GL(n) the subset of M(n) of invertible matrices. By Sym(n), one denotes the space of symmetric matrices. Let A be a matrix in Sym(n). A is positive semidefinite if ∀x ∈ R n , x t Ax ≥ 0 where x t denotes the transpose of x. The set of such matrices A forms a close convex cone whose interior is called the set of symmetric positive definite matrices and is noted SPD(n). If A is in the interior of the cone then we have x t Ax > 0. A complete description of the SPD(n) space can be found in [START_REF] Bhatia | Positive Definite Matrices[END_REF]. Equipped with different scalar products, SPD(n) can be seen as a Riemannian manifold.

Two parametrisations of SPD(n)

In order to compute the histogram of an image valued in SPD(n) matrices, one needs to set the parametrization framework of this space. Indeed there are many different ways to study SPD(n) matrices. The SPD(n) matrices can be seen at first as a subset included in the space M(n). However, this framework is not adapted to histogram computation for the following reason. The random variable I is valued in SPD(n) only. Now, as a vector subspace, the space SPD(n) has a null measure according to the Lebesgue measure of matrices. Thus the random variable I cannot have a density against the Lebesgue measure of matrices M(n).

A way to overcome this problem is to study the SPD(n) space as a subset of the Sym(n) matrices. In that case the space SPD(n) has a non null measure according to the Lebesgue measure of Sym(n). Each matrix can be expressed in the canonical basis of symmetrical matrices, coordinates forms what we will call parametrization 1.

A second approach involves decomposing SPD(n) matrices over rotations and eigenvalues. A symmetrical matrix can be diagonalized in an orthonormal basis. Then each SPD(n) matrix can be represented by its eigenvalues and a rotation matrix. Using any angular representation of the rotation matrix, the SPD(n) matrix can be represented by a set of eigenvalues and a set of angles. This representation forms a convex subset of non null measure of R n(n+1) 2

. We will call this parametrization 2. Now comes the question of the reference measure µ used to compute the histogram. To each parametrization of the SPD(n) space in R n(n+1) 2

is associated a Lebesgue measure. One must be aware that the Lebesgue measure of each parametrization is different. Thus, given two different parametrizations, the computation of histograms against their Lebesgue measure will give different results. Besides Lebesgue measures, other measures can be computed from the two main Riemannian metrics on SPD(n): the Fisher metric and the Wasserstein metric. Indeed any Riemannian metric gives place to a volume measure.

We explain now a key property of the parametrization 2. Given any measure on SPD(n) which is invariant by the action of the rotation, i.e., (M, R) → RM R t , the measure is a product between a measure on the eigenvalues and a the Haar measure of the rotation. Indeed, for all such measures on parametrization 2, it enables us to separate the study of the eigenvalues and the rotations. This result can be found in [START_REF] Robert A Wijsman | Invariant measures on groups and their use in statistics[END_REF], but a for the convenience of the reader, a simple proof is given in Appendix B.

As we explained previously, the change of coordinates between both parametrizations induces a change in volume measure. The modification of the density of the volume measure is expressed by the Jacobian of the change of coordinates [START_REF] Kuriki | Some geometry of the cone of the nonnegative definite matrices and weight of associated χ 2 distribution[END_REF]:

φ : R n(n+1) 2 → SPD(n) (λ 1 , • • • , λ n , θ 1 , • • • , θ n(n-1)
2

) → R θ    λ 1 0 • • • 0 • • • • • • λ n    R t θ
where {λ i } are the n positive eigenvalues ordered in increasing ordering and {θ j } are the n(n -1)/2 angles which parametrize the rotation matrix R θ .

Let us write µ Lebes 1 the Lebesgue measure of the parametrization 1 of SPD(n) and µ Lebes 2 the Lebesgue measure of parametrization 2. The computation of the Jacobian of the coordinate change gives the following relation [START_REF] Kuriki | Some geometry of the cone of the nonnegative definite matrices and weight of associated χ 2 distribution[END_REF]:

dµ Lebes 1 dµ Lebes 2 =   1≤i<j≤n |λ i -λ j |   h Haar (θ 1 , ..., θ n(n-1)
2

) ( 4
)
where h Haar (θ 1 , ..., θ n(n-1)

2

) is the density of the Haar measure on rotation in the parametrization θ j . In the following it will be denoted by h Haar .

Measure for the Fisher and the Wasserstein metrics

We are interested now in the expression of the density of the volume measure of the Fisher metric and the Wasserstein metric in both parametrizations.

The Fisher metric on SPD(n) is the metric induced by the geometry of information [START_REF] Amari | Methods of information geometry[END_REF]. This metric is invariant under the following action of Gl(n) on SPD(n), i.e., a :

Gl(n) × SPD(n) → SPD(n) (G, M ) → GM G t
The Wasserstein metric is another well known metric on SPD(n) induced by the optimal transport of centered Gaussian laws [START_REF] Takatsu | Wasserstein geometry of gaussian measures[END_REF]. Given a Riemannian metric, the density of the volume measure is given by det(G), where G is the matrix of the metric [START_REF] Pennec | Intrinsic statistics on riemannian manifolds: Basic tools for geometric measurements[END_REF]. Given two symmetric matrices U, V ∈ Sym(n) and a matrix W ∈ SPD(n), the scalar product between U and V at the W matrix, take respectively the following forms for the Fisher [START_REF] Pennec | A riemannian framework for tensor computing[END_REF] and the Wasserstein [START_REF] Takatsu | Wasserstein geometry of gaussian measures[END_REF] metric:

U, V Fisher W = tr W -1 2 U W -1 V W -1 2 , U, V Wasserstein W = tr (U W V ) .
From these scalar products we can compute the matrices of the metrics G Fisher W and G Wasserstein W . Then if we express G in the canonical base of symmetric matrices we have:

det(G Fisher W ) = α 1≤i≤n 1 λ n+1 i , det(G Wasserstein W ) = α 1≤i≤n λ i 1≤i<j≤n (λ i + λ j ),
where {λ i } are the eigenvalues of W and α is a multiplicative constant which depends on the dimensionality of the space, i.e α = . These expressions represent the densities of the Fisher and Wasserstein measures with respect to the Lebesgue measure of symmetric matrices µ Lebes 1 .

Using the expression (4) of the density of the Lebesgue measure of symmetric matrices against the Lebesgue measure of the parametrisation 2, we obtain the expressions of the density of the Fisher and Wasserstein measures against the Lebesgue measure of the µ Lebes 2 , given respectively by:

dµ Fisher dµ Lebes 2 = α 1≤i<j≤n |λ i -λ j | 1≤i≤n λ n+1 i h Haar , (5)
dµ Wasserstein dµ Lebes 2 = α 1≤i≤n λ i 1≤i<j≤n (λ i + λ j )   1≤i<j≤n |λ i -λ j |   h Haar .
(6) Despite the fact that we did not find the expression (5) in the literature, this formula might already be known. However, we believe that the case of the density of Wasserstein measure against the Lebesgue has not been previously published elsewhere. Detailed derivations are given in Appendix A.

Experimental results on tensor image histograms

Given a tensor image I : Ω → SPD(n), we perform one of the previous changes of coordinates for each pixel value, that is to say we represent each matrix in one of the parametrizations. The Fourier basis associated with each parametrization forms an orthonormal system with respect to the Lebesgue measure. According to methodology from Section 3, density estimate fI can be evaluated against this basis. The choice of the Fourier basis gives place to an efficient estimation of the image density against the Lebesgue measure of the parametrisation 2 associated to decoupling eigenvalues and rotation angles. We obtain then the histogram of the tensor image according to the Lebesgue measure of the parametrisation 2, i.e., fI;µ Lebes 2 . As explained in Section 3, it is possible to adapt the histogram to any other measure that has a density against the Lebesgue measure by the expression (3). For instance, the histogram estimate according to the Fisher metric measure is obtained by means of (5) by fI;µ Fisher = fI;µ Lebes 2 dµ Fisher dµ Lebes 2 .

Obviously, after choosing the family of the orthogonal functions of the basis, we need to fix the number of components. For the experiments of this paper we have used the basis:

e -i (x) = √ 2 sin(iπx); e 0 (x) = 1; e i (x) = √ 2 cos(iπx); i = 1, • • • , N
We now present applications of histogram construction for tensor images. All what follows is achieved according to the framework described previously. For computational reasons the applications on SPD(n) are here limited to low dimensions, namely n = 2 for structure tensors from 2D texture images, and n = 3 for diffusion tensor images (DTI).

Image segmentation

The first application consists in segmenting tensor images using their histograms. We first compute the complement histogram of the SPD(n) image and then apply a watershed transform [START_REF] Soille | Morphological Image Analysis: Principles and Applications[END_REF] on the histogram. We note that the histogram can be view as a n(n + 1)/2-dimensional grey-level image and the corresponding watershed transformation will provide a partition of I({Ω} in connected sets A i : I(Ω) = ∪A i . In particular, each class i is associated to a regional maximum of the histogram. The segmentation of the image is obtained by assigning a class i to each pixel according to its class on the segmented histogram: Ω = ∪I -1 (A i )

We consider 2D grey-scale texture segmentation using the classical structure tensor representation [START_REF] Knutsson | Representing local structure using tensors[END_REF]. More precisely, given an image composed of different textures, we calculate the structure tensor at each pixel, using a window size that we expect, makes the structure tensor as constant as possible on each texture class of the image. At this point, we have an image where each pixel is a point of SPD(2), parametrized by two eigenvalues and one angle. Then the 3D histogram is computed with respect to one the considered metrics. Fig. 1 shows an example of structure tensor image and its histograms using different measures µ Lebes 2 , µ Fisher and µ Wasserstein . We note that for visualization purposes we have only considered the histogram of the two eigenvalues. It should be also remarked that the contribution concerning the angle by measure h Haar is independent of the metric. Fig. 3 shows the segmentation obtained for a synthetic texture image. As expected, homogenous texture areas are well separated, but the "texture contours" introduce additional texture classes. In particular, this example corresponds to the result for the Wasserstein metric, similar ones are obtained for the other measures. Results given in Fig. 3 correspond to a more complex texture with a comparison of two metrics. From our experiments, we cannot conclude if one of the metrics is better than another for the purpose of texture segmentation. In addition, the obtained segmenta-tion depends also on the regularization window from the structure tensor computation. The second case study deals with the segmentation of DTI images, thus I(Ω) ⊆ SPD(n), a 6-dimensional manifold where each matrix is represented by three eigenvalues and three angles. We can apply the same method as the one we used for tensor structure segmentation. An example of DTI segmentation is shown in Fig. 4. We have compared the various metrics. We have observed for instance that for a global segmentation of the brain into its main areas, the best segmentation results are obtained using only the dominant eigenvalue with respect to its Lebesgue measure, see Fig. 4(b). For this example, we have compared also the performance of segmentation using a histogram obtained by discretization into bins and counting, see Fig. 4(c). As we can observe, this histogram includes an important number of peaks which produces a strong over segmentation into many classes.

Texture recognition

In our last application, we use the structure tensor histogram in order to perform texture recognition. We based ourselves on the Brodatz database [START_REF] Randen | Filtering for texture classification: A comparative study[END_REF] that contains 111 texture images. Each image of the database has been vertically cut into two equal parts. This way, we built a test set and reference set. For each image of the test and the reference set, we computed the histogram of structure tensor. Structure tensor images are computed for different size s of averaging window. For each histogram of the test set, we perform a comparison with all the histograms of the reference set, and select the closest one. The matching is correct if the selected histogram corresponds to the other other half of the image in the reference set. The histogram based on bins gives interesting results in term of performance of classification. However, results are significantly different when using parametrisation 1 or parametrization 2. Indeed the size and the shape of the bins are totally different in the two parametrizations, and we obtain two different estimations of the density. Thus we based the experiment on orthogonal series. The comparison between histograms can be achieved according to several norms. Despite the fact that the L 2 norm might no be the best measure between histograms, it presents a serious advantage in our case. Let c i denote f, e i . An approximation of the L 2 norm can be obtained directly from the coefficients ĉi = 1 n n j=1 e i (I(p j )) without performing any discretization of the density f . Let c 1 i and c 2 i be respectively the coefficient associated with the densities f I 1 and f I 2 . The L 2 norm between these two densities can be approximated by

f I 1 -f I 2 2 L 2 ≈ N i=-N c I 1 i -c I 2 i 2 .
Other norms or divergences requires a preliminary discretization. Results of Brodatz texture recognition are summarized in Table 1. A possible conclusion from this comparison is the stability of relevant results for Fisher metric against the size parameter of the structure tensor.

Conclusion

We started to explore different methods to address the construction of histograms for tensor images. The standard way to build histograms using bins has the advantage of a low computational cost. However, as previously explained, and as different applications show, this method might not always be the most adapted, mainly due to the irregularity induced by the size of the bins. Orthogonal series proved to be an interesting alternative. The choice of the parametrization has also been addressed. Both parametrization 1 and parametrization 2 can be chosen to study histograms. However, parametrization 2 has the following important property : a measure on SPD(n) invariant by the action of rotation RM R t is a product measure between a measure on eigenvalues and a measure on angles. On the other hand, the choice of the orthogonal basis still remains fully open. We chose to use Fourier basis of the different parametrization due to their simplicity. However, other basis might be more adapted to the different measures, in terms of regularity. Good candidates are for instance the eigenfunctions of the Laplacian operator associated to the different metrics.

A Density of Fisher and Wasserstein volume measure

Let us start by considering

U, V f isher W = tr(W -1 2 U W -1 V W -1 2 ) U, V W asserstein W = tr(U W V ) Let E i,j be the canonical basis of M (n). Let E ′ i,i = E i,i and E ′ i,j = E i,j +E j,i √ 2
if i < j The set of matrices E ′ i,j ,i ≤ j, forms a basis of the symmetric matrices. This basis is orthonormal for the Euclidean scalar product on matrices. We can notice that for any rotation R the basis formed by the RE ′ i,j R -1 is also orthogonal for the Euclidean scalar product.

Let G W be the matrix of the metric at a given W = RDR -1 , expressed in an orthonormal basis for the euclidean scalar product. The positive number det(G W ) represents the volume measure associated to the metric G with respect to the volume measure associated to the euclidean scalar product.

We are interested in the calculation of det(G W ) for the Fisher and the Wasserstein metrics.

Let us start with the Fisher metric. We have

RE ′ i,i R -1 , RE ′ j,j R -1 f isher W = δ i,j λ 2 i ,
and for i = j and k = l:

RE ′ i,j R -1 , RE ′ k,l R -1 f isher W = δ i,k δ j,l 2λ i λ j , finally for j = k: RE ′ i,i R -1 , RE ′ j,k R -1 f isher W = 0,
We can now write the matrix G F isher W in the basis formed by the

RE ′ i,j R -1 . G F isher W =           1 λ 2 1 0 . 0 . . 1 λ 2 n 1 2λ i λ j          
The expression of the volume measure is then obtained as

det(G F isher W ) = 1 2 n(n-1) 2 i 1 λ n+1 i
The proceeding is similar for the Wasserstein metric.

RE ′ i,i R -1 , RE ′ j,j R -1 W asserstein W = λ i ,
and for i = j and k = l we have:

RE ′ i,j R -1 , RE ′ k,l R -1 W asserstein W = λ i + λ j 2 ,
finally for j = k:

RE ′ i,i R -1 , RE ′ j,k R -1 W asserstein W = 0
Then we can write: B A measure on SPD(n) invariant under the action of rotations is a product between a measure on eigenvalues and the Haar measure of rotations

G W asserstein W =           λ 1 0 . 0 . . λ n λ i +λ j 2          
Let us set the context. We are working on the space of positive definite matrices which have distinct eigenvalues. We will call this space E. The rotation matrices (O + ) induce a group action on E:

O + × E -→ E (R, M ) -→ RM R t
For every matrix M of E, there exist R ∈ O + such that RM R t =   

λ 1 0 0 0 λ 2 0 0 0 λ 3   ,
and as such, we will note the space of diagonal matrices with positive strictly increasing eigenvalues as E/O + . Furthermore, the following map Furthermore, it is easy to see that A → λ(A) is a measure on E/O + . Since the sets A × B generate the sets of measurable subsets of E/O + × O + (according to Dynkin's Theorem), the image of µ by φ is the product of the two measures λ and µ O + on A and B respectively. It is also worth noting that λ has a density with respect to the Lebesgue measure.

φ : E/O + × O + → E (    λ 1 0 0 0 λ 2 0 0 0 λ 3    , R) -→ R    λ 1 0 0 0 λ 2 0 0 0 λ 3    R T

Figure 1 :

 1 Figure 1: Image (a) is an example of structure tensor image I. (b),(c),(d) represent respectively the densities of µ Lebes 1 ,µ Fisher and µ Wasserstein with respect to µ Lebes 2 as a function of the eigenvalues.

Figure 2 :

 2 Figure 2: Image texture segmentation by watershed partitioning of structure tensor histogram using Wasserstein metric: (a) original image; (b) segmentation.

Figure 3 :

 3 Figure 3: Image texture segmentation by watershed partitioning of structure tensor histogram: (a) original image; (b) segmentation using metric µ Lebes 2 ; (c) segmentation using metric µ Wasserstein .

Figure 4 :

 4 Figure 4: DTI segmentation by watershed partitioning of its histogram: (a) original image; (b) fractional anisotropy; (c) segmentation using a histogram obtained by discretization into bins and counting; (d) segmentation using orthogonal series-based histogram.

µ

  Wasserstein µ Lebes 1 µ Lebes 2 µ Fisher s = a) Three examples from the Brodatz texture dataset. (b) Accuracy (in percentage) of texture recognition in Brodatz database using structure tensor histogram with respect to various measures. Parameter s denotes the size of the window used in structure tensor estimation.

  + λ j )

E 1 AE 1 AE 1 AE 1 A

 1111 is a diffeomorphism. For a matrix M in E we denote M E/O + and M O + the elements of E/O + and O + such thatM = M O + M E/O + M T O + .We seek to show that any measure µ (with a density with respect to the Lebesgue measure) on E invariant by the action of O + can be written as a product measure on E/O + × O + , where the measure on O + is the Haar measure µ O + (the unique measure on O + that is invariant by translations).Consider a measurable subset A of E/O + and let µ A be the measure on O + defined byµ A (B) = (M E/O + )1 B (M O + )dµ(M )We are going now show that this measure on O + is translation invariant, and consequently a multiple of the Haar measure on O + .µ A (RBR T ) = E 1 A (M E/O + )1 RBR T (M O + )dµ(M ) = (M E/O + )1 B (R T M O + R)dµ(M ) = ((R T M R) E/O + )1 B ((R T M R) O + )dµ(M ),by the invariance of µ, we obtainµ A (RBR T ) = (M E/O + )1 B (M O + )dµ(M ) = µ A (B) Therefore, there exists λ(A) ∈ R such that µ A = λ(A)µ O + . If A, B are in E/O + × O + , then µ(φ(A × B)) = E 1 A (M E/O + )1 B (M O + )dµ(M ) = µ A (B) = λ(A)µ O + (B).


Taylor & Francis

Grazia D Raffa

Laura Ciapponi

Giovanni Cenci

Maurizio Gatti

Terminin: A protein complex that mediates epigenetic maintenance of Drosophila telomeres

Introduction

Telomeres of all eukaryotes are nucleoprotein complexes that protect the extremities of linear chromosomes from degradation and fusion, and counterbalance incomplete replication of terminal DNA. When telomeres are not properly capped, they are not recognized as natural chromosome ends but are sensed as double stranded DNA breaks (DSBs), triggering the DNA damage response that arrests the cell cycle until the DNA lesion is repaired. Uncapped telomeres may also undergo inappropriate DNA repair, leading to end-to-end fusions that ultimately result in chromosome breakage. [START_REF] Palm | How shelterin protects mammalian telomeres[END_REF][START_REF] Jain | Telomeric strategies: means to an end[END_REF][START_REF] O'sullivan | Telomeres: protecting chromosomes against genome instability[END_REF] Telomeres also cope with the inability of DNA polymerase to fully replicate the DNA of chromosome termini, often referred to as the end replication problem. In most organisms, this problem is solved by telomerase, a *Correspondence to: Maurizio Gatti; Email: maurizio.gatti@uniroma1.it Submitted: 07/01/11; Revised: 08/24/11; Accepted: 08/25/11 http://dx.doi.org/10.4161/nucl.2.5.17873 in most organisms, telomeres are extended by telomerase and contain GC-rich repeats. Drosophila telomeres are elongated by occasional transposition of specialized retroelements rather than telomerase activity, and are assembled independently of the sequence of the DNA termini. recent work has shown that Drosophila telomeres are capped by a complex, we call terminin, which includes HOAP, HipHop, Moi and ver; these are fast-evolving proteins that prevent telomere fusion, directly interact with each other, and appear to localize and function only at telomeres. with the possible exception of ver that contains an OB fold domain structurally similar to the Stn1 OB fold, none of the terminin proteins is evolutionarily conserved outside the Drosophila species. Human telomeres are protected by the shelterin complex, which comprises six proteins that bind chromosome ends in a sequence-dependent manner. Shelterin subunits are not fast-evolving proteins and are not conserved in flies, but localize and function only at telomeres like the terminin components. Based on these findings, we propose that concomitant with telomerase loss Drosophila rapidly evolved terminin to bind chromosome ends in a sequence-independent fashion, and that terminin is functionally analogous to shelterin.

Terminin

A protein complex that mediates epigenetic maintenance of Drosophila telomeres Grazia D. raffa, [START_REF] Palm | How shelterin protects mammalian telomeres[END_REF] Laura Ciapponi, [START_REF] Palm | How shelterin protects mammalian telomeres[END_REF] Giovanni Cenci [START_REF] Jain | Telomeric strategies: means to an end[END_REF] and Maurizio Gatti Key words: terminin, telomeres, telomere fusion, Drosophila specialized reverse transcriptase that adds short GC-rich repeats to chromosome ends using an internal species-specific RNA template. In Drosophila, telomerase is absent [START_REF] Sasaki | Detection and distribution patterns of telomerase activity in insects[END_REF] and telomeres are elongated by transposition of three specialized non-long-terminal repeat (LTR) retrotransposons, called HeT-A, TART and TAHRE (collectively abbreviated as HTT). These three elements transpose independently of each other and target individual telomeres at rates ranging from 10 -2 to 10 -4 per fly generation. Thus, most Drosophila chromosomes do not terminate with GC-rich repeats but carry HTT arrays of variable length (reviewed in refs. 5-7).

Several studies indicate that Drosophila telomeres are assembled independently of the HTT array and, more generally, independently of the sequence of the DNA termini (reviewed in ref. 8 and 9). In 1938 H.J. Muller observed that following X irradiation of males, terminal deletions could not be recovered. He concluded that chromosome ends are capped by special structures that he called telomeres and that are essential for chromosome stability and transmission. [START_REF] Muller | The remaking of chromosomes[END_REF][START_REF] Muller | An analysis of the process of structural change in chromosomes of Drosophila[END_REF] However, subsequent studies showed that terminal deletions (TDs) can be recovered in several ways. For example, TDs were recovered following irradiation of females carrying a mutation in the mutator gene mu2 and to lesser extent of wild type females. [START_REF] Mason | mu-2: mutator gene in Drosophila that potentiates the induction of terminal deficiencies[END_REF][START_REF] Mason | Germ-line effects of a mutator, mu2, in Drosophila melanogaster[END_REF][START_REF] Dronamraju | Recognition of double strand breaks by a mutator protein (MU2) in Drosophila melanogaster[END_REF] These TDs can be transmitted for many generations without reacquiring HTT elements even if they undergo a progressive loss of terminal DNA due to the end replication problem. [START_REF] Biessmann | Chromosome ends in Drosophila without telomeric DNA sequences[END_REF] It is now clear that TDs that do not end with HTT elements are capped by a neotelomere, which appears to have the same characteristics of the telomeres of intact chromosomes (reviewed in refs. 8 and 9). TDs with neotelomeres have also been recovered from mutational events occurred in the male germline. These events include X-ray induced breaks in the Dp(1;f)1187 mini-chromosome, [START_REF] Tower | Preferential transposition of Drosophila P elements to nearby chromosomal sites[END_REF] the mobilization of a P element located near the telomere, [START_REF] Levis | Viable deletions of a telomere from a Drosophila chromosome[END_REF] breakage of dicentric chromosomes generated by site specific recombination, [START_REF] Ahmad | The transmission of fragmented chromosomes in Drosophila melanogaster[END_REF][START_REF] Titen | Healing of euchromatic chromosome breaks by efficient de novo telomere addition in Drosophila melanogaster[END_REF] and induction of an enzymatic cut in an I-Sce1 site placed within a P element construct inserted near the telomere. [START_REF] Gao | HipHop interacts with HOAP and HP1 to protect Drosophila telomeres in a sequence-independent manner[END_REF] Collectively, these results demonstrate that the HTT elements are not required for fly telomere assembly and that virtually any DNA sequence has the ability to form the nucleoprotein complex that protects the ends of Drosophila chromosomes.

The structural features of the HTT elements, the molecular mechanisms underlying their transposition and their regulation Downloaded by [Institut Pasteur] at 08:42 01 December 2014 anterior TTAGGG duplex, thus creating a telomeric DNA loop (t-loop). A complex of six proteins, called shelterin, specifically associates with the TTAGGG repeats (reviewed in ref. 1). Three of the shelterin subunits directly interact with the TTAGGG repeats; TRF1 and TRF2 bind the TTAGGG duplex, and POT1 binds the 3' overhang. TRF1, TRF2 and POT1 are interconnected by TIN2 and TPP1, and TRF2 interacts with hRap1, a distant homolog of S. cerevisiae Rap1 with no DNA binding ability. [START_REF] Palm | How shelterin protects mammalian telomeres[END_REF] The shelterin subunits share three properties that distinguish them from the non-shelterin telomere-associated proteins: they specifically localize to telomeres; they are abundant at telomeres throughout the cell cycle; and their functions are limited to telomere maintenance. [START_REF] Palm | How shelterin protects mammalian telomeres[END_REF] The CST and shelterin complexes are evolutionarily conserved, even if they vary in composition and architecture in different phyla. The Stn1 and Ten1 subunits of the CST complex are conserved in S. pombe, plants and humans, while shelterin-like elements are found in S. pombe and plants but not in S. cerevisiae. [START_REF] Linger | Conservation of telomere protein complexes: shuffling through evolution[END_REF][START_REF] Lue | Plasticity of telomere maintenance mechanisms in yeast[END_REF][START_REF] Miyake | RPA-like mammalian Ctc1-Stn1-Ten1 complex binds to single-stranded DNA and protects telomeres independently of the Pot1 pathway[END_REF][START_REF] Surovtseva | Conserved telomere maintenance component 1 interacts with STN1 and maintains chromosome ends in higher eukaryotes[END_REF] S. pombe and plants have both shelterin-like and CST-like complexes, both of which are required for telomere protection. The two complexes are present also in humans and are thought to collaborate in telomere protection. However, the human CST complex does not share the shelterin properties and appears to have a relatively minor role in telomere capping. [START_REF] Miyake | RPA-like mammalian Ctc1-Stn1-Ten1 complex binds to single-stranded DNA and protects telomeres independently of the Pot1 pathway[END_REF][START_REF] Surovtseva | Conserved telomere maintenance component 1 interacts with STN1 and maintains chromosome ends in higher eukaryotes[END_REF] The shelterin and CST components of yeast, plant and mammalian telomeres interact with several conserved polypeptides required for telomere function. These polypeptides include many proteins involved in the DNA damage response and in DNA repair such as the ATM and Chk2 kinases, the Ku70/80 heterodimer, the MRE11/RAD50/NBS1 (MRN) complex, Rad51, the ERCC1-XPF and MUS81 endonucleases, the Apollo exonuclease and the RecQ family members WRN and BLM, which are mutated in the Werner and Bloom syndromes, respectively (reviewed in refs. 1, 2, 26 and 30). In addition, yeast and mammalian telomeres are enriched in proteins that are homologous to Drosophila HP1. All non-shelterin and non-CST proteins localize and function not only at telomeres but also elsewhere in the cell. [START_REF] Palm | How shelterin protects mammalian telomeres[END_REF][START_REF] Jain | Telomeric strategies: means to an end[END_REF][START_REF] Linger | Conservation of telomere protein complexes: shuffling through evolution[END_REF][START_REF] Oganesian | Telomeric armor: the layers of end protection[END_REF]

Drosophila Telomeres are Protected by the Terminin Complex

The identification of Drosophila proteins required for telomere protection has mainly relied on the isolation of mutants that display frequent telomeric fusions (TFs) in larval brain cells (Fig. 1). The molecular characterization of the genes specified by these mutants identified ten loci that are required to prevent end-to-end fusion (Table 1). These are Su(var)205 and caravaggio (cav) that encode HP1 and HOAP (HP1/ORC-associated protein), respectively; [START_REF] Fanti | The heterochromatin protein 1 prevents telomere fusions in Drosophila[END_REF][START_REF] Cenci | The Drosophila HOAP protein is required for telomere capping[END_REF] UbcD1 that encodes an E2 enzyme involved in protein ubiquitination; [START_REF] Cenci | UbcD1, a Drosophila ubiquitin-conjugating enzyme required for proper telomere behavior[END_REF] the Drosophila homologs of the ATM, RAD50, MRE11 and NBS1 genes; [START_REF] Bi | Telomere protection without a telomerase: the role of ATM and Mre11 in Drosophila telomere maintenance[END_REF][START_REF] Ciapponi | The Drosophila Mre11/ Rad50 Complex is required to prevent both telomeric fusion and chromosome breakage[END_REF][START_REF] Oikemus | Drosophila atm/telomere fusion is required for telomeric localization of HP1 and telomere position effect[END_REF][START_REF] Silva | ATM is required for telomere maintenance and chromosome stability during Drosophila development[END_REF][START_REF] Song | The Drosophila ATM ortholog, dATM, mediates the response to ionizing radiation and to spontaneous DNA damage during development[END_REF][START_REF] Bi | Drosophila ATM and ATR checkpoint kinases control partially redundant pathways for telomere maintenance[END_REF][START_REF] Ciapponi | The Drosophila Nbs protein functions in multiple pathways for the maintenance of genome stability[END_REF][START_REF] Oikemus | Epigenetic telomere protection by Drosophila DNA damage response pathways[END_REF] without children (woc) that specifies a putative transcription factor; 44 modigliani (moi; also called DTL) that encodes a nonconserved HOAP-binding protein; [START_REF] Raffa | The Drosophila modigliani (moi) gene encodes a HOAP-interacting protein required for telomere protection[END_REF][START_REF] Komonyi | A product of the bicistronic Drosophila melanogaster gene CG31241, which also encodes a trimethylguanosine synthase, plays a role in telomere protection[END_REF] and verrocchio (ver) that specifies an OB-fold containing protein structurally homologous to by small RNAs have been reviewed in recent excellent articles. [START_REF] Mason | Drosophila telomeres: an exception providing new insights[END_REF][START_REF] Frydrychova | Regulation of telomere length in Drosophila[END_REF][START_REF] Pardue | Drosophila telomeres: A variation on the telomerase theme[END_REF] Here, we describe the genes/proteins required for Drosophila telomere capping, with a focus on terminin, a multiprotein complex that evolved after telomerase loss to bind fly telomeres in a sequence-independent fashion.

Telomere Capping Complexes in Organisms with Telomerase

In organisms with telomerase, telomeres associate with capping complexes that specifically bind the telomeric repeats generated by telomerase activity. In S. cerevisiae, telomeres are protected by the Rap1-Rif1-Rif2 complex that associates with the telomeric DNA duplex (dsDNA) through its Rap1 subunit, and by the Cdc13-Stn1-Ten1 complex (CST) that interacts with the 3' single stranded telomere overhang. [START_REF] Shore | Telomere length regulation: coupling DNA end processing to feedback regulation of telomerase[END_REF][START_REF] Gao | RPA-like proteins mediate yeast telomere function[END_REF] The three subunits of the CST complex all contain OB-fold domains and interact with each other to form an RPA-like complex that binds the 3' single stranded overhang via its Cdc13 subunit. [START_REF] Gao | RPA-like proteins mediate yeast telomere function[END_REF][START_REF] Mitton-Fry | Conserved structure for single-stranded telomeric DNA recognition[END_REF][START_REF] Gelinas | Telomere capping proteins are structurally related to RPA with an additional telomerespecific domain[END_REF][START_REF] Sun | Stn1-Ten1 is an Rpa2-Rpa3-like complex at telomeres[END_REF] Human telomeres terminate with a single stranded overhang of tandem TTAGGG repeats, which loops back invading the to the extremities of various types of terminally deleted chromosomes demonstrating that these proteins bind chromosome ends independently of the sequence of terminal DNA. [START_REF] Titen | Healing of euchromatic chromosome breaks by efficient de novo telomere addition in Drosophila melanogaster[END_REF][START_REF] Gao | HipHop interacts with HOAP and HP1 to protect Drosophila telomeres in a sequence-independent manner[END_REF][START_REF] Fanti | The heterochromatin protein 1 prevents telomere fusions in Drosophila[END_REF][START_REF] Cenci | The Drosophila HOAP protein is required for telomere capping[END_REF] It should be noted that despite its direct interaction with HOAP, HipHop and Moi, HP1 should not be considered as a terminin component, because it does not localize exclusively at telomeres and has multiple telomere-unrelated functions (reviewed in refs. 46 and 47).

The structural and functional information on the terminin proteins is still rather limited; the architecture of the complex and the possible roles of its components are not well defined. Biochemical analyses showed that HOAP and HipHop are mutually dependent for their stability, as loss of one protein reduces the amount of the other. [START_REF] Gao | HipHop interacts with HOAP and HP1 to protect Drosophila telomeres in a sequence-independent manner[END_REF] In contrast, HOAP, Moi and Ver do not appear to be interdependent for stability, as loss of one protein does not destabilize the others. However, recruitment of terminin proteins to the telomeres is governed by precise dependencies: localization of Moi and Ver at chromosome ends requires HOAP, while Moi and Ver are mutually dependent for their association with telomeres. HOAP, HipHop and Moi do not exhibit structural features that help define their roles in telomere protection. BLAST searches did not detect HOAP, HipHop or Moi homologs outside the Drosophilidae insects. In addition, these proteins do not appear to contain known functional domains, although HOAP is thought to carry an HMG-like domain. [START_REF] Shareef | Drosophila heterochromatin protein 1 (HP1)/origin recognition complex (ORC) protein is associated with HP1 and ORC and functions in heterochromatin-induced silencing[END_REF] Previous studies showed that HOAP binds double stranded DNA of different sequence, although with different affinities. [START_REF] Shareef | Drosophila heterochromatin protein 1 (HP1)/origin recognition complex (ORC) protein is associated with HP1 and ORC and functions in heterochromatin-induced silencing[END_REF] Ver contains an OB-fold domain that shares structural similarity with the OB fold domain of Rpa2/Stn1 and binds single stranded DNAs of different sequence (reviewed in ref. 32; our unpublished results). Substitution of four critical amino acids in the Ver OB fold abolished the DNA binding ability of the protein (our unpublished results). When this mutated Ver version was expressed in flies, STN1. [START_REF] Raffa | Verrocchio, a Drosophila OB foldcontaining protein, is a component of the terminin telomere-capping complex[END_REF] An additional protein required to prevent telomere fusion, called HP1-HOAP interacting protein (or HipHop), was recently identified among the polypeptides that co-precipitate with HOAP. [START_REF] Gao | HipHop interacts with HOAP and HP1 to protect Drosophila telomeres in a sequence-independent manner[END_REF] Mutations in caravaggio (HOAP), modigliani and verrocchio cause very high frequencies of TFs (~5 per cell), often producing multicentric linear chromosomes that resemble little "trains" of chromosomes. The genes specified by these mutations have been named to reflect this phenotype, as the three Italian trains that are dubbed with the names of these famous painters. The HOAP, Ver and Moi proteins directly interact with each other in GST pulldown assays; HOAP and Moi also bind HP1 but Ver does not. Immunolocalization experiments have shown that HOAP is specifically associated with the telomeres of both mitotic and polytene chromosomes. An analysis of GFP-Moi and Ver-GFP localization on polytene chromosomes showed that these proteins are exclusively enriched at telomeres where they precisely colocalize with HOAP and HP1. However, GFP-Moi and Ver-GFP could not be detected at mitotic chromosome ends, probably due to their very low abundance. [START_REF] Raffa | The Drosophila modigliani (moi) gene encodes a HOAP-interacting protein required for telomere protection[END_REF][START_REF] Raffa | Verrocchio, a Drosophila OB foldcontaining protein, is a component of the terminin telomere-capping complex[END_REF] These results indicate that HOAP, Moi and Ver form a complex that accumulates only at telomeres of both interphase (polytene) and mitotic chromosomes. In addition, available data indicate that HOAP, Moi and Ver function primarily if not exclusively at telomeres. Thus the HOAP-Moi-Ver complex, which has been named terminin (after the name of Rome's train station), has the same properties as human shelterin and is likely to be a functional analog of shelterin. [START_REF] Raffa | The Drosophila modigliani (moi) gene encodes a HOAP-interacting protein required for telomere protection[END_REF][START_REF] Raffa | Verrocchio, a Drosophila OB foldcontaining protein, is a component of the terminin telomere-capping complex[END_REF] HipHop directly interacts with both HOAP and HP1, although it is currently unknown whether it also binds Moi and Ver. HipHop specifically localizes at both mitotic and polytene chromosome telomeres and appears to function only at telomeres. [START_REF] Gao | HipHop interacts with HOAP and HP1 to protect Drosophila telomeres in a sequence-independent manner[END_REF] Thus, HipHop is likely to be an additional terminin component. Remarkably, HOAP, HP1 and HipHop all localize it was recruited at telomeres but was unable to prevent telomere fusion, suggesting that the DNA binding activity of the Ver OB fold is crucial for telomere protection. [START_REF] Raffa | Verrocchio, a Drosophila OB foldcontaining protein, is a component of the terminin telomere-capping complex[END_REF] It is not currently known whether Moi and HipHop can directly bind DNA. However, chromatin immunoprecipitation (ChIP) experiments using TD chromosomes terminating with the white gene sequence revealed that HOAP and HipHop are highly enriched within an 11-kb stretch of telomeric DNA. An analysis of the distribution of these proteins further showed their concentration increases from -11 kb to reach a peak at -1 kb but then drops in the terminal kb of telomeric DNA (Fig. 2). Although these results are rather fragmentary, they permit us to conceive a possible model for the interaction of terminin with telomeric DNA (Fig. 3). We would like to propose that HOAP and HipHop are primarily bound to the telomeric DNA duplex while Ver and Moi are associated with the single stranded overhang, which might span the terminal region of reduced HOAP/HipHop binding.

Although the terminin components are all required to prevent telomere fusion, they do not play identical roles at Drosophila telomeres. Previous work showed that HOAP-depleted telomeres trigger both the DNA damage response (DDR) and the spindle assembly checkpoint (SAC). [START_REF] Cenci | Telomere capping and cellular checkpoints: clues from fruit flies[END_REF][START_REF] Musarò | Unprotected Drosophila melanogaster telomeres activate the spindle assembly checkpoint[END_REF] The SAC appears to be mediated by the BubR1 kinase, which accumulates at the uncapped telomeres in almost all cav mutant cells, but is never targeted to wild type telomeres. In the absence of Moi, Ver or HP1, telomeres have little or no ability to activate the DDR and trigger the SAC. [START_REF] Raffa | The Drosophila modigliani (moi) gene encodes a HOAP-interacting protein required for telomere protection[END_REF][START_REF] Raffa | Verrocchio, a Drosophila OB foldcontaining protein, is a component of the terminin telomere-capping complex[END_REF][START_REF] Cenci | Telomere capping and cellular checkpoints: clues from fruit flies[END_REF][START_REF] Musarò | Unprotected Drosophila melanogaster telomeres activate the spindle assembly checkpoint[END_REF][START_REF] Cenci | Drosophila cell cycle under arrest: uncapped telomeres plead guilty[END_REF] It is currently unknown whether HipHop depletion triggers the DRR and the SAC response. These results suggest that HOAP is crucial for protecting chromosome ends so as to prevent both telomere fusion and checkpoint responses. In contrast, Ver and Moi are not required to prevent checkpoint responses but are essential to hide chromosome ends from the DNA repair machineries that mediate telomere fusion. Interestingly, recent studies have shown that dysfunctional mouse telomeres also recruit BubR1, but it is unclear whether telomereassociated BubR1 can activate the SAC response. [START_REF] Muñoz | TRF1 controls telomere length and mitotic fidelity in epithelial homeostasis[END_REF] The Roles of Drosophila Nonterminin Proteins Required for Telomere Protection Drosophila telomere capping is not only ensured by terminin but also by a number of proteins that do not share the terminin properties; namely, proteins that do not localize or function only at telomeres. To date, we know 7 nonterminin proteins required for telomere protection from fusion events: HP1a, UbcD1, Mre11, Rad50, Nbs, ATM and Woc.

The most characterized nonterminin telomere-capping factor is HP1a. Besides HP1a, the Drosophila genome harbors 5 additional HP1 paralogs (HP1b, HP1c, HP1d, HP1e and Umbrea/HP6), none of which has been unambiguously shown to be required for telomere protection (reviewed in ref. 47; our unpublished results). In polytene chromosomes, HP1a is enriched at the telomeres, the chromocenter, the fourth chromosome and many euchromatic bands. [START_REF] Fanti | The heterochromatin protein 1 prevents telomere fusions in Drosophila[END_REF][START_REF] James | Distribution patterns of HP1, a heterochromatin-associated nonhistone chromosomal protein of Drosophila[END_REF][START_REF] Fanti | Chromosomal distribution of heterochromatin protein 1 (HP1) in Drosophila: a cytological map of euchromatic HP1 binding sites[END_REF] Consistent with this localization pattern, we propose that the very ends of the chromosomes are capped by terminin, which includes HOAP, HipHop (HHop), Moi and ver. HOAP and ver bind double-stranded and single-stranded DNA, respectively, but it is currently unknown whether HipHop and HOAP directly associate with DNA. Moi and ver would be absent from the telomeric DNA duplex proximal to the single stranded overhang; this duplex, however, would be associated with HOAP-HipHop. Drosophila telomeres are also enriched in HP1 and woc, but these proteins are not terminin components because they associate with multiple polytene bands and play functions outside the telomeres. The UbcD1 (Ub) protein is enriched at the telomere region but appears to concentrate near the TAS rather than at the ends of the chromosomes (Cipressa and Cenci unpublished results). The MrN complex and the ATM kinase exhibit a rather uniform distribution along the Drosophila mitotic chromosomes and do not accumulate at polytene chromosome ends. Thus, UbcD1, Mre11, rad50, Nbs and ATM were not included in the telomere region of the scheme, even if these proteins must function at telomeres, as mutations in the genes they specify cause telomeric fusions. components of the Drosophila MRN complex interact with terminin. However, mutations in the rad50, mre11 and nbs genes strongly reduce HOAP accumulation at mitotic telomeres. These mutations also inhibit HOAP and HP1 localization at polytene chromosome ends. [START_REF] Bi | Telomere protection without a telomerase: the role of ATM and Mre11 in Drosophila telomere maintenance[END_REF][START_REF] Ciapponi | The Drosophila Mre11/ Rad50 Complex is required to prevent both telomeric fusion and chromosome breakage[END_REF][START_REF] Bi | Drosophila ATM and ATR checkpoint kinases control partially redundant pathways for telomere maintenance[END_REF][START_REF] Ciapponi | The Drosophila Nbs protein functions in multiple pathways for the maintenance of genome stability[END_REF][START_REF] Oikemus | Epigenetic telomere protection by Drosophila DNA damage response pathways[END_REF] Consistent with these findings, mutations in mre11 prevent Moi localization at polytene chromosome telomeres. [START_REF] Raffa | The Drosophila modigliani (moi) gene encodes a HOAP-interacting protein required for telomere protection[END_REF] Collectively, these results strongly suggest that terminin recruitment to telomeres requires the wild type function of the MRN complex. However, even in the absence of MRN activity, mitotic chromosomes retain the ability to recruit low levels of HOAP [START_REF] Bi | Telomere protection without a telomerase: the role of ATM and Mre11 in Drosophila telomere maintenance[END_REF][START_REF] Ciapponi | The Drosophila Mre11/ Rad50 Complex is required to prevent both telomeric fusion and chromosome breakage[END_REF][START_REF] Bi | Drosophila ATM and ATR checkpoint kinases control partially redundant pathways for telomere maintenance[END_REF][START_REF] Ciapponi | The Drosophila Nbs protein functions in multiple pathways for the maintenance of genome stability[END_REF][START_REF] Oikemus | Epigenetic telomere protection by Drosophila DNA damage response pathways[END_REF] and, presumably, of the other terminin components. These low amounts of terminin are likely to provide a partial protection of telomeres, as suggested by the finding that rad50, mre11 and nbs mutants display much fewer TFs than cav mutants. [START_REF] Cenci | The Drosophila HOAP protein is required for telomere capping[END_REF][START_REF] Bi | Telomere protection without a telomerase: the role of ATM and Mre11 in Drosophila telomere maintenance[END_REF][START_REF] Ciapponi | The Drosophila Mre11/ Rad50 Complex is required to prevent both telomeric fusion and chromosome breakage[END_REF][START_REF] Bi | Drosophila ATM and ATR checkpoint kinases control partially redundant pathways for telomere maintenance[END_REF][START_REF] Ciapponi | The Drosophila Nbs protein functions in multiple pathways for the maintenance of genome stability[END_REF][START_REF] Oikemus | Epigenetic telomere protection by Drosophila DNA damage response pathways[END_REF] Another DNA repair protein involved in Drosophila telomere protection is the ATM kinase. Null mutations in the tefu/atm gene result in lethality at late larval stages and cause both chromosome breakage and TFs in larval brain cells. [START_REF] Bi | Telomere protection without a telomerase: the role of ATM and Mre11 in Drosophila telomere maintenance[END_REF][START_REF] Oikemus | Drosophila atm/telomere fusion is required for telomeric localization of HP1 and telomere position effect[END_REF][START_REF] Silva | ATM is required for telomere maintenance and chromosome stability during Drosophila development[END_REF][START_REF] Song | The Drosophila ATM ortholog, dATM, mediates the response to ionizing radiation and to spontaneous DNA damage during development[END_REF][START_REF] Queiroz-Machado | tef: a mutation that causes telomere fusion and severe genome rearrangements in Drosophila melanogaster[END_REF] tefu mutations do not substantially affect HOAP localization at mitotic telomeres, although they might reduce HOAP accumulation at polytene telomeres. [START_REF] Bi | Telomere protection without a telomerase: the role of ATM and Mre11 in Drosophila telomere maintenance[END_REF][START_REF] Oikemus | Drosophila atm/telomere fusion is required for telomeric localization of HP1 and telomere position effect[END_REF] Thus, loss of Drosophila ATM does not appear to cause telomere fusion by preventing terminin localization at chromosome ends. Null mutants in the ATR-encoding mei-41 gene or in the mus-304 gene that encodes the ATRinteracting protein ATRIP are both viable and their larval brain cells exhibit chromosome aberrations but not TFs. [START_REF] Gatti | Genetic control of chromosome breakage and rejoining in Drosophila melanogaster: spontaneous chromosome aberrations in X-linked mutants defective in DNA metabolism[END_REF][START_REF] Hari | The mei-41 gene of D. melanogaster is a structural and functional homolog of the human ataxia telangiectasia gene[END_REF][START_REF] Brodsky | mus304 encodes a novel DNA damage checkpoint protein required during Drosophila development[END_REF] However, tefu mei-41 and tefu mus-304 double mutants fail to recruit HOAP at telomeres and have significantly higher frequencies of TFs than those observed in tefu single mutants. [START_REF] Bi | Drosophila ATM and ATR checkpoint kinases control partially redundant pathways for telomere maintenance[END_REF][START_REF] Ciapponi | The Drosophila Nbs protein functions in multiple pathways for the maintenance of genome stability[END_REF][START_REF] Oikemus | Epigenetic telomere protection by Drosophila DNA damage response pathways[END_REF] This suggests that ATM and ATR/ATRIP have partially redundant roles in telomere protection and that failure to phosphorylate a common target leads to deprotected telomeres. The nature of this target is currently unknown and it does not appear to be HOAP (ref. 41; our unpublished results).

The mechanism by which the combined action of the MRN complex, ATM and ATR-ATRIP leads to terminin recruitment to telomeres is unclear. The MRN complex plays a central role in detection and repair of DNA double strand breaks and mediates recruitment of ATM at the site of DNA damage (reviewed in ref. 60). Although MRN preferentially associates with the ends of linear DNA molecules in vitro, [START_REF] De Jager | Human Rad50/Mre11 is a flexible complex that can tether DNA ends[END_REF] the Rad50, Mre11 and Nbs protein are uniformly distributed along Drosophila chromosomes (ref. 37; our unpublished results), making it unlikely that MRN directly tethers terminin to telomeres. Thus, it has been hypothesized that interactions of DNA ends with the MRN complex and ATM-ATR result in conformational changes that facilitate terminin recruitment at telomeres. [START_REF] Rong | Telomere capping in Drosophila: dealing with chromosome ends that most resemble DNA breaks[END_REF][START_REF] Ciapponi | The Drosophila Mre11/ Rad50 Complex is required to prevent both telomeric fusion and chromosome breakage[END_REF][START_REF] Cenci | Telomere capping and cellular checkpoints: clues from fruit flies[END_REF] Protecting Drosophila telomeres from fusion also requires the wild type activity of the without children (woc) gene. woc encodes a zinc finger protein that interacts with HP1c and functions both in transcriptional regulation and telomere capping. [START_REF] Raffa | The putative Drosophila transcription factor woc is required to prevent telomeric fusions[END_REF][START_REF] Font-Burgada | Drosophila HP1c isoform interacts with the zincfinger proteins WOC and Relative-of-WOC to regulate gene expression[END_REF][START_REF] Abel | Drosophila HP1c is regulated by an auto-regulatory feedback loop through its binding partner Woc[END_REF] Woc is enriched at polytene chromosome telomeres and co-localizes with all euchromatic bands that associate with the initiating form of Pol II. [START_REF] Raffa | The putative Drosophila transcription factor woc is required to prevent telomeric fusions[END_REF] We do not know whether Woc interacts with terminin. However, Woc localization at telomeres is not affected by HP1a binds diverse proteins involved in a variety of processes including telomere capping, gene silencing, DNA replication and repair, the maintenance of proper chromosome structure, and transcriptional regulation (reviewed in refs. 46, 47, 55 and 56). In Drosophila, HP1a is not only required for telomere protection but it is also involved in the control of telomere length. In stocks heterozygous for lethal mutations in Su(var)205 (which encodes HP1a) the telomeres are much longer than those of wild type flies due to a dramatic elongation of the HTT array. This is a likely consequence of an increased HetA and TART transcription, which has been observed in both Su(var)205 heterozygotes and homozygotes. [START_REF] Savitsky | Heterochromatin protein 1 is involved in control of telomere elongation in Drosophila melanogaster[END_REF][START_REF] Perrini | HP1 controls telomere capping, telomere elongation and telomere silencing by two different mechanisms in Drosophila[END_REF] We never observed telomere elongation in stocks heterozygous for mutations in cav, moi or ver. In addition, our real time RT-PCR experiments did not detect substantial increases in HetA and TART transcription in homozygous cav, moi or ver mutant larvae (our unpublished results). Thus, the extant observations suggest that terminin is not implicated in the control of Drosophila telomere elongation.

effete/UbcD1 was the first Drosophila gene shown to be required for prevention of telomere fusion. [START_REF] Cenci | UbcD1, a Drosophila ubiquitin-conjugating enzyme required for proper telomere behavior[END_REF] UbcD1 is an essential gene that encodes a highly conserved E2 ubiquitin conjugating enzyme implicated in several Drosophila cellular processes. [START_REF] Cenci | UbcD1, a Drosophila ubiquitin-conjugating enzyme required for proper telomere behavior[END_REF] The UbcD1 protein associates with many polytene chromosome bands and is enriched at the telomere region of polytene chromosomes (Cipressa and Cenci G, unpublished observations). This suggests that failure to ubiquitinate one or more telomere proteins leads to fusigenic telomeres. However, we do not know whether UbcD1 interacts with terminin, and the telomere-associated target(s) of UbcD1 remain to be identified. Given that mutations in some of the proteasome components do not cause TFs (our unpublished results), we suspect that UbcD1-mediated ubiquitination of proteins involved in telomere protection is not required for their degradation but is instead a post-translational modification that ensures their proper capping function. Consistent with this idea, polytene chromosomes from UbcD1 mutants and those from wild type controls display comparable telomeric concentrations of both HP1 and HOAP. [START_REF] Cenci | The role of HeT-A and TART retrotransposons in Drosophila telomere capping[END_REF] These results indicate that UbcD1 is required neither for HP1 or HOAP localization at telomeres nor for proteolysis of these proteins. However these findings do not exclude the possibility that either one or both of these proteins are ubiquitinated by UbcD1.

Studies performed in the last few years have shown that several proteins involved in DNA repair are also needed to prevent telomere fusion. Mutants in the Drosophila mre11, rad50 and nbs genes die at late larval stages and exhibit both TFs and chromosome breakage in brain cells. [START_REF] Bi | Telomere protection without a telomerase: the role of ATM and Mre11 in Drosophila telomere maintenance[END_REF][START_REF] Ciapponi | The Drosophila Mre11/ Rad50 Complex is required to prevent both telomeric fusion and chromosome breakage[END_REF][START_REF] Bi | Drosophila ATM and ATR checkpoint kinases control partially redundant pathways for telomere maintenance[END_REF][START_REF] Ciapponi | The Drosophila Nbs protein functions in multiple pathways for the maintenance of genome stability[END_REF][START_REF] Oikemus | Epigenetic telomere protection by Drosophila DNA damage response pathways[END_REF] The Drosophila Mre11, Rad 50 and Nbs proteins are the fly orthologs of human MRE11, RAD50 and NBS1, which form the highly conserved MRN complex involved in both double-strand breaks (DSBs) repair and telomere maintenance. This complex mediates DSB repair by participating in both the homologous recombination (HR) and the nonhomologous end joining (NHEJ) pathways. [START_REF] Lamarche | The MRN complex in double-strand break repair and telomere maintenance[END_REF] In humans, the MRN complex also associates with the TRF2 subunit of shelterin, facilitating telomerase recruitment and participating in detection and signaling of uncapped telomeres (reviewed in ref. 60). It is currently unknown whether the Nucleus volume 2 issue 5

OB fold. [START_REF] Raffa | Verrocchio, a Drosophila OB foldcontaining protein, is a component of the terminin telomere-capping complex[END_REF] We speculate that Drosophila evolved Ver exploiting an Stn1 ancestor similar to the RPA proteins that bind ssDNA with no sequence specificity. The origin of HOAP, Moi and HipHop is unknown, as no conserved proteins that share amino acid sequence homology with these terminin components have been so far identified. It is conceivable that following telomerase loss selective pressure on terminin proteins was much stronger than that exerted on other telomere proteins not specifically involved in capping. Proteins involved in diverse cellular processes are indeed expected to have functional constraints that prevent them from responding to selection with the same high rate of amino acid substitutions as polypeptides that are solely involved in the protection of chromosome ends. Therefore, one would hypothesize that proteins directly and exclusively involved in telomere capping evolved more rapidly than the other telomere-associated proteins. This hypothesis is verified by two findings. First, all Drosophila nonterminin proteins required to prevent telomere fusion have clear mammalian homologs, most of which have been implicated in telomere maintenance (HP1a, ATM, MRE11, RAD50, NBS1). [START_REF] Raffa | The Drosophila modigliani (moi) gene encodes a HOAP-interacting protein required for telomere protection[END_REF][START_REF] Raffa | Verrocchio, a Drosophila OB foldcontaining protein, is a component of the terminin telomere-capping complex[END_REF] Second, HOAP, Moi, Ver and HipHop all exhibit a very high rate of nonsynonymous substitution per nonsynonymous site, and are therefore fast-evolving proteins. In contrast, none of the nonterminin telomere proteins, including HP1, appears to be a fast-evolving polypeptide. [START_REF] Gao | HipHop interacts with HOAP and HP1 to protect Drosophila telomeres in a sequence-independent manner[END_REF][START_REF] Raffa | Verrocchio, a Drosophila OB foldcontaining protein, is a component of the terminin telomere-capping complex[END_REF][START_REF] Schmid | A screen for fast evolving genes from Drosophila[END_REF] Consistent with these findings, sequence analysis of proteins from D. melanogaster and 11 recently sequenced species [START_REF] Clark | Evolution of genes and genomes on the Drosophila phylogeny[END_REF] revealed that the terminin components are substantially more divergent than non-terminin proteins (Fig. 4). In conclusion, the rate of nonsynonymous substitutions in HOAP, Moi, Ver and HipHop is so high and distinctive that we propose to use it as an additional identifying criterion for both the extant terminin components and for terminin proteins that might emerge from future screens.

Recent work has shown that the terminin complex can vary in composition in different Drosophila tissues. In male meiotic and postmeiotic cells, including the sperm nuclei, HipHop is substituted by the product of the ms(3)k81 gene (henceforth referred as k81). [START_REF] Dubruille | Specialization of a Drosophila capping protein essential for the protection of sperm telomeres[END_REF][START_REF] Gao | Paternal imprint essential for the inheritance of telomere identity in Drosophila[END_REF] k81 is a duplication of the hiphop gene, presumably originated through a retroposition mechanism. [START_REF] Loppin | Origin and neofunctionalization of a Drosophila paternal effect gene essential for zygote viability[END_REF] Phylogenic analysis suggests that k81 is a relatively young gene, as it is present only in the melanogaster subgroup of Drosophila species, although an independent hiphop duplication with male-biased expression was also found in D. willistoni. [START_REF] Dubruille | Specialization of a Drosophila capping protein essential for the protection of sperm telomeres[END_REF] K81 associates with the sperm telomeres and is specifically required for telomeric protection of male derived-chromosomes during embryogenesis; in embryos fathered by k81 mutants, the paternal chromosomes display frequent TFs, ultimately leading to embryonic death. [START_REF] Dubruille | Specialization of a Drosophila capping protein essential for the protection of sperm telomeres[END_REF][START_REF] Gao | Paternal imprint essential for the inheritance of telomere identity in Drosophila[END_REF] The relationships between K81 and the other terminin proteins are unclear. In one study, K81 was found to colocalize with HOAP and HP1 in sperm nuclei of wild type males. HP1 and HOAP were absent from the sperm nuclei of k81 mutants, suggesting that K81 is needed for the maintenance of capping complexes at the sperm telomeres. [START_REF] Dubruille | Specialization of a Drosophila capping protein essential for the protection of sperm telomeres[END_REF] In another study, mature sperm of wild type males were found to be devoid of both HP1 and HOAP signals, suggesting that the telomere-capping machinery of the sperm does not include HOAP and HP1. [START_REF] Gao | Paternal imprint essential for the inheritance of telomere identity in Drosophila[END_REF] It would be interesting mutations in Su(var)205, cav, atm or rad50, and mutations in woc do not affect HOAP and HP1 localization at chromosome ends. In addition, woc mutations do not dominantly affect telomere length. [START_REF] Raffa | The putative Drosophila transcription factor woc is required to prevent telomeric fusions[END_REF] These results indicate that the Woc function at telomeres is independent of those played by HOAP, HP1, ATM or Rad50, and that Woc is a transcription factor with telomerecapping properties, as is also the case for yeast Rap1. [START_REF] Shore | Telomeric chromatin: replicating and wrapping up chromosome ends[END_REF] The armitage (armi) and aubergine (aub) genes play a tissuespecific role in Drosophila telomere stability. [START_REF] Khurana | Distinct functions for the Drosophila piRNA pathway in genome maintenance and telomere protection[END_REF] armi and aub mutants are viable but female-sterile, as they cause maternaleffect embryonic lethality. Both genes are involved in the biogenesis of piRNAs; armi encodes an RNA helicase and aub a piRNA-binding Argonaute-like protein (reviewed in ref. 70). In embryos from armi and aub homozygous mothers, there are frequent anaphase bridges that are probably caused by telomeric fusions. Consistent with this interpretation, chromatin immunoprecipitation studies showed that mutation in both genes disrupt telomere binding of HOAP in the embryo. [START_REF] Khurana | Distinct functions for the Drosophila piRNA pathway in genome maintenance and telomere protection[END_REF] Mutations in armi and aub also reduced an embryonic subpopulation of piRNAs that share some homology with telomeric retotransposons, raising the possibility that these small RNAs may facilitate HOAP binding to the HTT array. [START_REF] Khurana | Distinct functions for the Drosophila piRNA pathway in genome maintenance and telomere protection[END_REF] However, these piRNAs or other Armi-and Aub-dependent piRNAs (or the Armi and Aub proteins) must also facilitate HOAP binding to telomeres devoid of the HTT array, as diverse TD chromosomes are regularly transmitted during embryogenesis (see Introduction). It should also be noted that the Armi-and Aub-based mechanism of telomere protection is likely to be restricted to embryogenesis, as homozygous armi and aub individuals generated by heterozygous mothers are viable and do not exhibit TFs (our unpublished results).

Evolution of Drosophila Telomere Proteins

There is a general consensus that the crucial event that led the evolution of Drosophila telomeres was progressive loss of telomerase accompanied by the development of a regulated transposon-based mechanism for telomere elongation. It has also been suggested that the two mechanisms of telomere maintenance might have coexisted for some time. [START_REF] Mason | Drosophila telomeres: an exception providing new insights[END_REF][START_REF] Pardue | Drosophila telomeres: A variation on the telomerase theme[END_REF][START_REF] Louis | Are Drosophila telomeres an exception or the rule?[END_REF] This scenario is not difficult to envisage as in the silkworm Bombyx mori, which retains telomerase, there are two classes of retotransposons that specifically insert into the telomere regions. [START_REF] Takahashi | A new family of site-specific retrotransposons, SART1, is inserted into telomeric repeats of the silkworm, Bombyx mori[END_REF][START_REF] Fujiwara | Telomere-specific non-LTR retrotransposons and telomere maintenance in the silkworm, Bombyx mori[END_REF] It is logical to assume that the transition from a telomerase-driven to a transposondriven telomere elongation mechanism resulted in a divergence of terminal DNA sequences, accompanied by a strong selective pressure toward the evolution of sequence-independent telomerebinding factors. In agreement with this idea, none of the shelterin or the CST components is conserved in flies, and none of the terminin proteins, with the possible exception of Ver, has obvious homologs in yeasts, mammals or plants. [START_REF] Gao | HipHop interacts with HOAP and HP1 to protect Drosophila telomeres in a sequence-independent manner[END_REF][START_REF] Raffa | The Drosophila modigliani (moi) gene encodes a HOAP-interacting protein required for telomere protection[END_REF][START_REF] Raffa | Verrocchio, a Drosophila OB foldcontaining protein, is a component of the terminin telomere-capping complex[END_REF][START_REF] Cenci | The Drosophila HOAP protein is required for telomere capping[END_REF] Thus, we hypothesized that concomitant with telomerase loss, Drosophila lost the shelterin and the CST homologs that bind DNA in a sequence-specific fashion, and evolved terminin to bind chromosome ends independently of the DNA sequence. [START_REF] Raffa | The Drosophila modigliani (moi) gene encodes a HOAP-interacting protein required for telomere protection[END_REF][START_REF] Raffa | Verrocchio, a Drosophila OB foldcontaining protein, is a component of the terminin telomere-capping complex[END_REF] Ver exhibits a very limited amino acid sequence homology with Stn1, but contains an OB fold domain that is structurally similar to the Stn1 Downloaded by [Institut Pasteur] at 08:42 01 December 2014 future work is how terminin is recruited at the telomeres independently of the DNA sequence. We have learned that terminin recruitment requires the function of the MRN complex and the partially redundant activities of the ATM and ATR kinases. However the molecular mechanism underlying terminin recruitment remains elusive.

We have hypothesized that during the transition from a telomerase-based to transposon-based telomere elongation mechanism, Drosophila rapidly evolved terminin to bind chromosome ends independently of the DNA sequence. Consistent with this hypothesis, the Drosophila terminin proteins are not conserved in humans, while the shelterin proteins have no obvious Drosophila homologs. In contrast, Drosophila nonterminin and human nonshelterin telomere proteins are largely conserved from flies to mammals, and many of them play telomere-related functions in both organisms. These findings indicate that the main difference between Drosophila and human telomeres is in the protective complexes that specifically associate with the DNA termini. Thus, apart from the different mechanisms of elongation, Drosophila and human telomeres might not be as different as it is generally thougth. The conservation in humans of the nonterminin telomere proteins further suggests that the identification of additional proteins of this type may lead the discovery of novel components of human telomeres. to determine whether this machinery includes Moi and Ver and other proteins that, like K81, are required for telomere capping in sperm cells. Whatever the composition of the sperm telomere capping complex, the studies on K81 have clearly shown that during early embryogenesis there is substantial remodeling of the telomere structure; during this period K81 is substituted by HipHop and other telomere proteins are possibly recruited. It is also logical to speculate that the K81-HipHop transition period might be crucial for the formation of neotelomeres at the ends of TD chromosomes. It would be interesting to learn whether TD chromosomes, produced in males by means different from irradiation (see introduction), acquire a K81 cap in the sperm or are occasionally capable to escape from fusion events in the embryo and to recruit a normal capping complex during the K81-HipHop transition.

Conclusions and Perspectives

We have described the main characteristics of 15 Drosophila proteins that are necessary to protect telomeres from fusion events (Table 1). In a review published in 2005, we estimated that the Drosophila genome contains at least 40 genes required to prevent telomere fusion. [START_REF] Cenci | The mechanism of telomere protection: a comparison between Drosophila and humans[END_REF] The current results do not alter this conclusion and we still believe that there are many Drosophila telomere-capping genes that await to be discovered. The molecular and genetic analyses of the genes so far identified have shown that Drosophila telomeres are capped by terminin, a complex composed of fast-evolving proteins that specifically bind the telomeres. An important issue that needs to be elucidated by

Figure 1 .

 1 Figure 1. Mutations in genes encoding terminin components display frequent TFs. (A) wild type control metaphase. (B-D) examples of TFs observed in larval brains from ver (B) and moi (C and D) mutants. Mutants in cav, ver and moi show two types of TFs: single TFs (STFs), in which a single telomere associates with either its sister or a nonsister telomere; and double TFs (DTFs), wherein a pair of sister telomeres joins with another pair. STFs and DTFs are likely to be generated during the S-G2 and the G 1 phase, respectively. in wild type cells, the frequency of TFs is less than 0.01/cell, whereas cav, ver and moi mutants exhibit approximately 5 TFs/cell, most of which are DTFs. (B) Metaphase containing 2-2 (arrow) and 4-4 (arrowhead) dicentric chromosomes and a dicentric ring involving both X chromosomes (asterisk), all generated by DTFs. (C) Metaphase showing a 4-4 DTA (asterisk), a 2-2 dicentric ring chromosome (arrowhead) and a 3-3 DTF (diamond); the XL and 3r telomeres exhibit sister union STFs (arrows). (D) Metaphase with a multicentric chromosome (arrow) containing 3-3 and Xr-Xr DTFs and 3 STFs involving both XL telomeres and individual telomeres of chromosomes 2 and 3 (the metaphases shown are from figures in ref. 31 and 32).

Figure 2 .

 2 Figure 2. Distribution of HOAP and HipHop over the terminal DNA of a Drosophila chromosome ending with the white gene sequence (see text and ref. 20 for detailed explanation). Protein localization was determined by chromatin immunoprecipitation using anti-HOAP or anti-HipHop antibodies. Note that HOAP and HipHop exhibit similar distributions with the amount of each protein increasing from -11 to -1 kb from the end of the chromosome to drop in the terminal Kb (adapted from Fig. 6 in ref. 20; with permission of the authors).

Figure 3 .

 3 Figure 3. A tentative model for the molecular organization of Drosophila telomeres.we propose that the very ends of the chromosomes are capped by terminin, which includes HOAP, HipHop (HHop), Moi and ver. HOAP and ver bind double-stranded and single-stranded DNA, respectively, but it is currently unknown whether HipHop and HOAP directly associate with DNA. Moi and ver would be absent from the telomeric DNA duplex proximal to the single stranded overhang; this duplex, however, would be associated with HOAP-HipHop. Drosophila telomeres are also enriched in HP1 and woc, but these proteins are not terminin components because they associate with multiple polytene bands and play functions outside the telomeres. The UbcD1 (Ub) protein is enriched at the telomere region but appears to concentrate near the TAS rather than at the ends of the chromosomes (Cipressa and Cenci unpublished results). The MrN complex and the ATM kinase exhibit a rather uniform distribution along the Drosophila mitotic chromosomes and do not accumulate at polytene chromosome ends. Thus, UbcD1, Mre11, rad50, Nbs and ATM were not included in the telomere region of the scheme, even if these proteins must function at telomeres, as mutations in the genes they specify cause telomeric fusions.

Figure 4 .

 4 Figure 4. identity percentages between the D. melanogaster proteins required for telomere capping and the homologous proteins form 11 Drosophila species (D. mel, D. melanogaster; D. sim, D. simulans; D. sec, D. sechellia; D. yak, D. yakuba; D. ere, D. erecta; D. ana, D. ananassae, D. pse, D. pseudoobscura; D. per, D. persimilis; D. wil, D. willistoni; D. moj, D. mojavensis; D. vir, D. virilis; D. grim, D. grimshawi). The identity percentage (id%; represented by the indicated color) is the percentage of identical matches between two amino acid sequences, calculated using the pairwise alignment eMBOSS Needle software.

  1, * 1 istituto Pasteur-Fondazione Cenci Bolognetti and istituto di Biologia e Patologia Molecolari del CNr; Dipartimento di Biologia e Biotecnologie, Sapienza, Università di roma; roma, italy; 2 Dipartimento di Biologia di Base ed Applicata; Università dell'Aquila; Coppito, L'Aquila, italy

Table 1 .

 1 Drosophila genes required to prevent telomere fusion Mutations in mei-41 or mus-304 do not cause TFs but genetically interact with mutations in tefu, so that mei-41 tefu and mus-304 tefu double mutants exhibit TF frequencies that are much higher than those seen in tefu single mutants. (2) Mutations in armi and aub cause TFs only during embryogenesis.

	Gene name	Protein name	Protein full name	Function outside of telomeres	References
	cav	HOAP	HP1-OrC-Associated Protein	None known	34
	hiphop	HiPHop	HP1-HOAP-interacting protein	None known	20
	moi	Moi	Modigliani	None known	31, 45
	ver	ver	verrocchio	None known	32
	Su(var)205	HP1	Heterochromatin Protein 1	Heterochromatin regulation; transcription factor	33, 46
	eff	UbcD1	Ubiquitin Conjugating enzyme D1	e2 ubiquitin conjugating enzyme	35, 59
	woc	woc	without Children	Transcription factor	44
	mre11	Mre11	Meiotic recombination 11	DNA repair; Component of the MrN complex	36, 37
	rad50	rad50	radiation sensitive 50	DNA repair; Component of the MrN complex	37
	nbs	Nbs	Nijmegen breakage syndrome	DNA repair; Component of the MrN complex	41-43
	tefu	ATM	Ataxia Telangiectasia Mutated	Kinase; DNA damage response	36, 38-41, 61
	mei-41 (1)	ATr	Ataxia Telangiectasia related	Kinase; DNA damage response	41-43
	mus-304 (1)	ATriP	ATr interacting Protein	DNA helicase; DNA damage response	41, 43
	armi (2)	Armi	Armitage	helicase; pirNA biogenesis	69
	aub (2)	Aub	Aubergine	pirNA biogenesis	69
	(1)

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Acknowledgements

We are grateful to John C. Lucchesi for a critical reading of the manuscript. This work was supported in part by a grant from AIRC (Italian Association for Cancer Research) to M.G.


Manuel Núñez-Regueiro

Superconductivity coexistence relation to quantum régimes: application to superconducting domes around quantum critical points

Quantum critical points (QCP) accompanied by superconductivity are ubiquitous in condensed matter physics. In general, the transition temperature ! of an ordered state, e.g. antiferromagnetic, goes to zero under the influence of an external

parameter, e.g. pressure. Superconductivity appears before the disappearance of the ordered state, but reaches its maximum T c when ! = 0. Presently, the implications of the QCP's on superconductivity are a subject of debate. It is propose here that both transition temperatures satisfy the relation ! ! + ! ! = 1, where the tilde indicates normalization to the maximum values. Inspired from the basic postulate of SO(N )

theories of superconductors, it is proposed as an extension from Bilbro-McMillan relation.

The study of phase transitions is central to the understanding of the physical properties of condensed matter [START_REF] Chaikin | Principles of condensed matter physics[END_REF]. Standard phase transitions, at a transition temperature ! ≠ 0, can be treated classically, even those between different quantum states. Quantum phase transitions, where the transition between two different phases takes place at T c = 0 , are qualitatively different, as their critical fluctuations must be treated quantum mechanically [ 2 , 3 ]. Great progress has been made in their understanding, e.g. in the case of magnetic quantum phase transitions [4,[START_REF] Sachdev | Quantum Phase Transitions[END_REF][START_REF] Continentino | Quantum scaling in many-body physics[END_REF]. In a large number of materials including pnictides, heavy fermions, charge or spin density wave materials, the phase transition at T c = 0 as a function of a certain parameter (pressure, doping, etc.) from an ordered, generally bad conductor, state to a conducting disordered phase, is surrounded in theplane by a superconducting region (see Fig. 1). The central point is that there is a region where the ordered and the superconducting state (SC) coexist. For charge density waves (CDW) this coexistence has been studied by several groups [START_REF] Levin | [END_REF]8,9]. In particular, Bilbro and McMillan (BM) [8] studied the interaction of CDW and SC in A15 materials within mean field. Here, both states compete for developing their respective gaps in the same Fermi surface. They obtained the following relation between , the superconducting transition temperature, and ! , the CDW transition temperature,

! !!! ! ! ! ! = ! !"# (1)
where is the fraction of the Fermi surface under the CDW gap and the superconducting transition with no CDW.

Different works have shown that under pressure there are clearly two regimes for the evolution of on its way to the QCP. First a classical BCS at low pressures one and then a quantum fluctuations (QF) one near the QCP. For example, both the

T c n 0 T c Max T 0
one dimensional transition metal trichalcogenide , a well-studied charge density wave (CDW) compound [10,11,12] and Cr metal [13] display, at low pressures, an decrease of following an exponential with pressure BCS tuned variation, consequence of the dependence of ! with the coupling parameter , i.e.

! ~!!/! . A quantum region appears below a crossover value of where QF take control through a power law behavior with an exponent ψ = ½ (See the behavior of !.!! ! ! on Fig. 1).

It is clear that Eq. ( 1) can only be applied in the BCS régime and is that not valid in the QF region. However, at least for CDW and SDW materials, a direct relation between and describing that both states are the result of the same Fermi surface and electron-phonon interaction is expected and needed. In order to have a hint towards it, an analysis of how the BM relation works experimentally is clarifying.

In A-15 compounds, ~ some 10%'s and changes only a few percents in the measured pressure range [8]. Considering then approximately constant in the pressure range, expression (1) becomes,

1 -! ! + ! ! () ≈ ! !"# (2).
We see from Fig. 2 where the variation with pressure of both transition temperatures for the two most studied A-

T 0 T 0 T 0 T c n 0 n 0 V 3 Si Nb 3 Sn n 0 V 3 Si Nb 3 Sn T c Max V 3 Si Nb 3 Sn
However, as described above, the CDW/SDW transition deviates from the exponentially tuned BCS behavior towards a power law dependence of the type

! ∝ ! ! !! ! ! !/!
near to the QCP. It is apparent that the transition temperature does not more follow BCS statistics, but another possibly due to the QF near the QCP. QF accelerate the rhythm of decrease of ! , destabilizing the CDW/SDW and, at the same time, inducing a faster transfer of carriers from the CDW/SDW gap to the SC gap.

Equation ( 1) is thus no longer valid. Phenomenologically, expression (1) adds the variations of the using the functional that is linear with pressure, which in the BCS régime is the logarithm of the transition temperature. The same approach may be taken for the power law régime. Thus, instead of the logarithms, as now ! ! (0) ! = 1 -/ ! , the squares of the transition temperatures should be added and equated to a constant. It is thus expected that the equivalent of BM for the power law region would be a summation of squares of the type

! ! + ! ! = (3)
To advance further in obtaining a BM formula for the quantum régime, it is useful to take into account the SO(5) formulation. In the paper [14] on SO [START_REF] Sachdev | Quantum Phase Transitions[END_REF], it was argued that the simplest way to construct an unified theory of antiferromagnetism and superconductivity for cuprates was to define a five dimensional spinor composed by the three components of the magnetization and the real and imaginary part of the dwave superconducting gap. In this way, it would be possible to recover all the relevant physics of the problem of high temperatures superconducting cuprates [15].

More complicated problems would require higher dimensions spinors to obtain all the physics, i.e. the complete phase diagram with all the possible ground states, e.g. SO (8) for a two chains model of correlated fermions [16]. However, experimentalists can be less ambitious, and just try to describe phenomenologically with the most T x ' s compact expression a restricted region of the phase diagram, e.g. the one near to QCP.

Basically, there are two coexisting phases. The ordered phase should rigorously have several components. However, systems with presumed different number of components, show the same behavior near the QCP. SO [START_REF] Sachdev | Quantum Phase Transitions[END_REF] and SO(4) has been successfully used to explain properties of the Co doped BaFe 2 As 2 system [17], and the Beechgard salt [18] (TMTTF) 2 PF 6 , respectively. Thus, just one component for the ordered phase may be enough for this portion of the phase diagram. A similar reasoning can be used for the superconducting phase, that may also be accounted by one component. The actual formula is suggested by expression (3). It is (Fig. 3a)

! ! (!) ! ! ! !"# ! + ! ! (!) ! ! ! !"# ! = 1 (4)
The procedure to apply this formula is as following : the ! dependence with pressure, concentration, etc. is fitted by a

! = ! !"# !!! ! ! ! ! (5) law, from which the dependence of ! = ! !"# 1 - ! ! ! ! ! ! ! ! ! ( 6
) can be calculated. The normalization to ! ! ! is a way to apply the formula only to the actual region of coexistence of the ordered and superconducting phase (Fig. 3b). Thus, the dependence of ! determines that of ! . The satisfactory application of the formula is also shown for the examples on Fig. 1.

For > ! the dependence of ! is naturally extended to negative values (low-right quadrant of Fig. 3a), and its module is used in the determination of . In the case of magnetic QCP's, this extension can be associated to the crossover or coherence line, below which the system recovers normal Fermi liquid properties. It follows the power law !"! ~ -! !

!" [START_REF] Continentino | Quantum scaling in many-body physics[END_REF], where is the correlation length exponent and z the T c dynamical exponent. As for low dimensions = is often valid, i.e. both ! ≡ ! and T coh should follow the same power law. On the other hand, for CDW-s, a negative ! can be traced to the soft phonon that exists even with values of that hinder the occurrence of the lattice distortion [19,20]. In other words, the interaction, responsible for the ordering, does not obviously disappears when ! = 0, and the extrapolation to negatives values of ! is an approximation to its magnitude.

In fact, already a study of the antiferromagnetic QCP, considering that superconductivity is due to spin fluctuations caused by its criticality, arrives at a formula of similar nature by derivation of the lines of constant coherence lengths [21] [START_REF] Levin | [END_REF]. In the case of a T N ( T coh ) with a critical exponent = 1 ( = 1) , expressions ( 4) and ( 7 [23], ! ! [24] and ( !!! ! ) ! ! [25] show power laws at all pressures.

! ! + !!! ! ! ! ! = ! !"# ! (
Also shown (red circles) is the superconducting dome. The red line is obtained using formula (4), see text and Fig. 3. b) Example of application of formula [4], [START_REF] Sachdev | Quantum Phase Transitions[END_REF] and [START_REF] Continentino | Quantum scaling in many-body physics[END_REF] on CePd 2 Si 2 . The formula for T C automatically uses the fitted value of T N only where there is coexistence between the antiferromagnetic and the superconducting phases (solid curves), although the power law for T N is fitted to all the reported pressure range (dashed curve). For > ! , the expression for T N corresponds, in fact, to that of the coherence line T coh as their power laws often have the same exponents in low dimensions.

  ) are almost identical. It is clear that the type of analysis done on Ref. 21 can be a way for the understanding of superconducting domes around QCP's of different nature. In conclusion, a formula equivalent to the Bilbro-McMillan expression for coexistence between a charge density wave and superconductivity but applicable to the quantum fluctuations controlled regions of the phase diagram is proposed to describe the superconducting domes that appear around quantum critical points. It is shown that it describes phenomenologically the observed behavior for several systems.

Figure 1 !

 1 Figure 1

Figure 2 Left

 2 Figure 2

Figure 3 a

 3 Figure 3