Céline Vebert

Yassin Hamini

Ludovic Héricher

Sylvain Michon

Lionel Aranda

Patrice Berthod

Casting and characterization of Ni-, Fe(

come

Ni-based alloys

For the same carbon and tantalum contents the alloy microstructures vary versus the nature of the base element. Globally they were well predicted by thermodynamic calculations performed here. Among the alloys with equal atomic tantalum content and carbon content it is possible to obtain only tantalum carbides as interdendritic strengthening particles in the chromium-rich Fe-based, {Fe + Ni}-based and Co-based alloys. In contrast this is obviously not the case of the Ni-based alloys which necessarily also contain chromium carbides. With their solidus temperature higher than 1400°C, the best refractoriness is given by the iron-based alloys. Their melting point may increase again by a simple modification of the carbon and tantalum contents. Unfortunately the refractoriness of the other alloys cannot be really improved since thermodynamic calculations showed that they have already the chemical compositions which lead to the maximal solidus temperatures.

Faculté des Sciences et Technologies

Interdendritic carbides of chromium or tantalum are among the main reinforcing phases in superalloys. They allow them being used at high temperature under mechanical stresses. However the presence of such carbides can be detrimental for the refractoriness. In addition the morphologies and volume fractions of these carbides may evolve to better correspond to the thermodynamic equilibrium at the service temperature, with possible consequences on the properties of the alloys. The study aims to better know the fusion temperature ranges, the metallurgical stable states at high temperature of several foundry chromium-rich alloys based on iron+nickel and nickel and reinforced by chromium or tantalum carbides. The set of studied alloys will be also enriched with a TaC-containing cobalt alloy. Thereafter the metallographic observations and quantitative measurements will allow testing the accuracy of thermodynamic calculations in the (Fe and/or Ni)-Cr-C-Ta before using this tool for identifying new chemical compositions allowing to improve refractoriness without too important loss of strengthening carbides.

  of Ni-, Fe(Ni)-and Fe-based 30%Cr-containing alloys strengthened by TaC; comparison with thermodynamic calculations Céline VEBERT, Yassin HAMINI, Ludovic HERICHER, Sylvain MICHON, Lionel ARANDA, Patrice BERTHOD Institut Jean Lamour, Université Henri Poincaré, NancyUniversité BP 70239, Bd des Aiguillettes, 54506 Vandoeuvre-lès-Nancy, France

  solidus and liquidus temperatures measured by DTA and issued from Thermo-Calc calculations Stables microstructures of the alloys with equal atomic contents of Ta and C


Rúben Faria

Milan Paul

Swati Biswas

Eric Vivès

Prisca Boisguerin

Ângela Sousa

Diana Costa

Peptides vs. Polymers: Searching for the Most Efficient Delivery System for Mitochondrial Gene Therapy

come


est destinée au dépôt

  Hafedh Fessi, Jean-Christophe Szelag, Cécile Courivaud, Philippe Nicoud, Didier Aguilera, Olivia Gilbert, Marion Morena, Michel Thomas, Bernard Canaud, Jean-Paul Cristol To cite this version: Hafedh Fessi, Jean-Christophe Szelag, Cécile Courivaud, Philippe Nicoud, Didier Aguilera, et al.. Safety and Efficacy of Short Daily Hemodialysis with Physidia S3 System: Clinical Performance Assessment during the Training Period. Journal of Clinical Medicine, 2022, 11 (8), pp.2123. 10.3390/jcm11082123. hal-03650081


Gabriel Peyré
email: peyre@ceremade.dauphine.fr

Sébastien Bougleux
email: bougleux@ceremade.dauphine.fr

Laurent D Cohen
email: cohen@ceremade.dauphine.fr

Non-local Regularization of Inverse Problems

Introduction

State of the art image denoising methods perform a non-linear filtering that is adaptive to the image content. This adaptivity enables a non-local averaging of image features, thus making use of the relevant information along an edge or a regular texture pattern. This article shows how such adaptive non-local filterings can be extended to handle general inverse problems beyond simple denoising.

Adaptive Non-Local Image Processing. Traditional image processing methods use local computation over a time-frequency or a multi-scale domain [?]. These algorithms use either fixed transforms such as local DCT or wavelets, or fixed regularization spaces such as Sobolev or bounded variations, to perform image restoration.

In order to better respect edges in images, several edge-aware filtering schemes have been proposed, among which Yaroslavsky's filter [?], the bilateral filter [?], Susan filter [?] and Beltrami flow [?]. The non-local means filter [?] goes one step further by averaging pixels that can be arbitrary far away, using a similarity measure based on distance between patches. This non-local averaging shares similarities with patch-based computer graphics synthesis [?,?].

These adaptive filtering methods can be related to adaptive decompositions in dictionaries of orthogonal bases. For instance the bandlets best basis decomposition [?] re-transform the wavelet coefficients of an image in order to better capture edges. The grouplet transform of Mallat [?] does a similar retransformation but makes use of an adaptive geometric flow that is well suited to capture oriented oscillating textures [?].

Regularization and inverse problems. Non-local filtering can be understood as a quadratic regularization based on a non-local graph, as detailed for instance in the geometric diffusion framework of Coifman et al. [?], which has been applied to non-local image denoising by Szlam et al. [?]. Denoising using quadratic penalization on image graphs is studied by Guilboa and Osher for image restoration and segmentation [?].

These quadratic regularizations can be extended to non-smooth energies such as the total variation on graphs. This has been defined over the continuous domain by Gilboa et al. [?] and over the discrete domain by Zhou and Schölkopf [?]. Elmoataz et al. [?] consider a larger class of non-smooth energies involving a p-laplacian for p < 2. Peyré replaces these non-linear flows on graphs by a non-iterative thresholding in a non-local spectral basis [?].

A difficult problem is to extend these graph-based regularizations to solve general inverse problems. The difficulty is that graph-based regularizations are adaptive since the graph depends on the image. To perform denoising, this graph can be directly estimated from the noisy image. To solve some specific inverse problems, the graph can also be estimated from the measurements. For instance, Kindermann et al. [?] and Buades et al. [?] perform image deblurring by using a non-local energy computed from the blurry observation. A similar strategy is used by Buades et al. [?] to perform demosaicing, where the non-local graph is estimated using an image with missing pixels. For inpainting of thin holes, Gilboa and Osher [?] compute a non-local graph directly from the image with missing pixels.

These approaches are different from recent exemplar-based methods introduced to solve some inverse problems, such as super-resolution (see for instance [?,?,?]). Although these methods operate on patches in a manner similar to nonlocal methods, they make use of pairs of exemplar patches where one knows both the low and high resolution version.

Contributions. This paper proposes a new framework to solve general inverse problems using a non-local and non-linear regularization on graphs. Our algorithm is able to efficiently solve for a minimizer of the proposed energy by iteratively computing an adapted graph and a solution of the inverse problem. We show applications to inpainting, super-resolution and compressive sampling where this new framework improves over wavelet and total variation regularization.

Non-local Regularization

Inverse Problems and Regularization

Many image processing problems can be formalized as the recovery of an image f ∈ R n from a set of p n noisy linear measurements

y = Φf + ε ∈ R p .
where ε is an additive noise. The linear operator Φ typically accounts for some blurring, sub-sampling or missing pixels so that the measured data y only captures a small portion of the original image f one wishes to recover.

In oder to solve this ill-posed problem, one needs to have some prior knowledge on the kind of typical images one expects to restore. This prior information should help to recover the missing information. Regularization theory assumes that f has some smoothness, for instance small derivatives (linear Sobolev regularization) or bounded variations (non-linear regularization).

A regularized solution f ⋆ to the inverse problem can be written in variational form as

f ⋆ = argmin g∈R n 1 2 ||y -Φg|| 2 + λJ(g), (1)
where J is small when g is close to the smoothness model. The weight λ needs to be adapted to match the amplitude of the noise ε, which might be a non-trivial task in practical situations. Classical variational priors include Total variation: The bounded variations model imposes that f ⋆ has a small total variation and uses

J tv (g) def. = ||g|| TV def. = |∇ x g|dx. (2)
This prior has been introduced by Rudin, Osher and Fatemi [?] for denoising purpose. It has been extended to solve many inverse problems, see for instance [?].

Sparsity priors: Given a frame (ψ m ) m of R n , one defines a sparsity enforcing prior in this frame as J spars (g)

def. = m | g, ψ m |. (3)
This prior has been introduced by Donoho and Johnstone [?] with the orthogonal wavelet basis for denoising purpose. It has then been used to solve more general inverse problems, see for instance [?] and the references therein. It can also be used in conjunction with redundant frames instead of orthogonal bases, see for instance [?,?].

Graph-based Regularization

Differential operators over graphs. We consider a weighted graph w that links together pixels x, y over the image domain with a weigth w(x, y). This graph allows to compute generalized discrete derivatives using the graph gradient operator

∀ x, ∇ w x f = w(x, y)(f (y) -f (x)) y ∈ R n .
This operator defines, for any pixel x, a gradient vector

∇ w x f ∈ R n , see [?]. The divergence operator div w = (∇ w )
T is the adjoint of the gradient, viewed as an

operator f → ∇ w f ∈ R n×n . For a gradient field F x ∈ R n , the divergence is (div w (F ))(x) = y w(x, y)(F x (y) -F y (x)).
The total-variation energy of an image, according to the graph structure given by w is then defined as

J w (f ) = x ||∇ w x f ||, (4)
where Non-local graph Given an image ∈ R n to process, one wishes to compute an adapted graph w(f ) so that the regularization by J w efficiently removes noise without destroying the salient features of the image. In order to do so, we use a non-local graph inspired by the non-local means filtering [?], which has been used in several recent methods for denoising, see for instance [?,?] This non-local graph is built by comparing patches around each pixel. A patch p x (f ) of size τ × τ (τ being an odd integer) around a pixel position x ∈ {0, . . . ,

√ n -1} 2 is ∀ t ∈ {-(τ -1)/2 + 1, . . . , (τ -1)/2} 2 , p x (f )(t)
def.

= f (x + t).

A patch p x (f ) is handled as a vector of size τ 2 . Color images f of n pixels can be handled using patches of dimension 3τ 2 . The non-local means algorithm [?] filters an image f using the following image-adapted weights w = w(f )

w(x, y) = w(x, y) Z x where w(x, y) = e -||px(f )-py (f )|| 2σ 2 if ||x -y|| δ 2 , 0 otherwise, (5)
where the normalizing constant is Z x def.

= y w(x, y). The parameter δ > 0 restrict the non-locality of the method and also allows to speed-up computation. The parameter σ controls how many patch are taken into account to perform the averaging. It is a difficult parameter to set and ideally it should also be adapted to the noise level |ε|.

In the following, we consider the mapping f → w(f ) as a simple way to adapt a set of non-local weights to the image f to process.

Graph-based regularization of inverse problems. We propose to use this graph total-variation (??) to solve not only the denoising problem but arbitrary inverse problems such as inpainting, super-resolution and compressive sampling.

Our non-local graph regularization framework tries to recover an image f ⋆ from a set of noisy measurements y = Φf +ε using an adapted energy J w(f ) . The graph w(f ) should be adapted to the image f to recover, but unfortunately, one does not have this information since only the noisy observations y are available. In order to cope with such a problem, one performs an optimization over both the image f ⋆ to recover and the optimal graph w(f ⋆ ) as follow

f ⋆ = argmin g∈R n 1 2 ||y -Φg|| 2 + λJ w(g) (g). ( 6
)
It is important to note that the functional prior J w(g) depends non-linearly on the image g being recovered through equation (??).

Proximal Resolution of the Graph Regularization

The optimization of (??) is difficult because the energy J w(g) (g) makes it non-convex. We use an iterative approximate minimization that optimizes successively on the optimal graph and then on the image to recover.

Since J w is a non-smooth functional, gradient descent methods are inefficient to minimize J w . In order to cope with this issue, we use proximal iterations. The resulting algorithm is based on three main building blocks:

The non-local graph adaptation procedure, equation (??), to compute a graph w(f k ) adapted to the current estimate f k of the algorithm. Proximal iterations to solve for (??) when the graph is fixed. Fixed point iterations to compute the proximity operator needed for the proximal iterations.

Proximal iterations. Equation (??) allows to compute a graph w adapted to a current estimate of the algorithm. In order to solve the initial regularization (??), we suppose that the graph w is fixed, and look for a minimizer of

f ⋆ (w) = argmin g∈R n 1 2 ||y -Φg|| 2 + λJ w (g). (7)
This non-smooth convex minimization is difficult to solve because of the rankdefficient matrix Φ that couples the entries of g. In order to perform the optimization, we use iterative projections with a proximal operator.

Proximal iterations replace problem (??) by a series of simpler problems. This strategy has been developed to solve general convex optimizations and has been applied to solve non-smooth convex problems in image processing [?,?].

The proximity operator of a convex functional J : R n → R + is defined as

Prox J (f ) = argmin g∈R n 1 2 ||f -g|| 2 + J(f ). (8)
A proximal iteration step uses the proximity operator to decrease the functional (??) one wishes to minimize

f (k+1) = Prox λ µ Jw f (k) + 1 µ Φ T (y -Φf (k) ) . (9)
It uses a step size µ > 0 that should be set in order to ensure that ||Φ T Φ|| < µ.

If the functional J is lipshitz continuous (which is the case of J w ), then one has the convergence f (k) → f ⋆ (w) a minimizer of (??), see for instance [?].

Computation of the proximity operator. In order to compute the proximity operator Prox Jw for the functional J w , one needs to solve the minimization (??), which is simpler than the original problem (??) since it does not involve anymore the operator Φ.

In order to do so, we use a fixed point algorithm similar to the one of Chambolle [?]. It is based on the computation of a field ∈ R n at each pixels x

g i+1 (x) = g i (x) -ηh(x) 1 + η||h(x)|| with h(x) = ∇ w x div w (g i ) - µ λ f ) . ( 10
)
can then prove that if η is small enough, the iterations (??) satisfy

f -λ div w (g i ) i→+∞ -→ Prox λ µ Jw (f ).
In numerical computation, since the graph w defined by equation (??) is relatively sparse (each pixel is connected to less than δ 2 pixels), g(x) is stored as a sparse vector.

Graph regularization algorithm. The algorithm to minimize approximately (??) is detailed in table ??. It proceeds by iterating the proximal mapping (??). Each computation of this proximity operator requires m inner iterations of the dual gradient descent (??). Since the proximal iterations are robust against imperfect computation of the proximity operator, m is set to a small constant.

1. Initialization: set f (0) = 0 and k ← 0.

Enforcing the constraints: compute f

(k) = f (k) + 1 µ Φ T (y -Φf (k) ). 3. Update the graph: compute the non-local graph w (k) = w( f (k) ) adapted to f (k)
using equation (??). 4. Compute proximal iterations: Set g k) . Perform m steps of the iterations of (??)

(k) 0 = ∇ w (k) f (
g (k) i+1 (x) = g (k) i (x) -ηh(x) 1 + η||h(x)|| with h(x) = ∇ w (k) x " div w (k) (g (k) i ) - µ λ f (k) " .
Set the new estimate

f (k+1) = f (k) -λ div w (k) (g (k)
m ). 5. Stopping criterion: while not converged, set k ← k + 1 and go back to 2.

Numerical Illustration

In the numerical simulations, we consider three different regularizations: The total variation energy J tv , defined in equation (??). An algorithm very similar to the algorithm of table ?? is used for this minimization, excepted that ∇ x is the classical gradient and that the step 3 of algorithm ?? is not needed. The sparsity energy J spars , defined in equation (??), using a redundant tight frame of translation invariant wavelets (ψ m ) m . An algorithm very similar to the one of table ?? is used for this minimization, excepted that the proximal projection is computed with a soft thresholding as detailed in [?] and that the step 2 of algorithm ?? is not needed. The non-local total variation regularization J w in an optimized graph, solved using algorithm ??. For this regularization, the parameter σ of equation (??) is fixed by hand in order to have consistent results for all experiments. The locality parameter δ of equation (??) is fixed to 15 pixels. Both total variation and non-local total variation require approximately the same number of iterations. For these two methods, the number of inner iterations to solve for the proximity operator is set to m = 10. The non-local iteration is computationally more intensive since the computation of the non-local weights (w(x, y)) y requires to explore δ 2 pixels y for each pixel x. In the three applications of sections ??, ?? and ??, we use a low noise level |ε| of .02||y||. For all the proposed methods, the parameter λ is optimized in an oracle manner in order to minimize the PSNR of the recovered image

f ⋆ PSNR(f ⋆ , f ) = -20 log 2 (||f ⋆ -f ||/||f || ∞ ).

Inpainting

Inpainting corresponds to the operation of removing pixels from an image

(Φf )(x) = 0 if x ∈ Ω, f (x) if x / ∈ Ω,
where Ω ⊂ {0, . . . , √ n-1} 2 is the region where the input data has been damaged.

In this case, Φ T = Φ, and one can take a proximity step size µ = 1 so that the proximal iteration (??) becomes

f (k+1) = Prox λJ ( f (k) ) where f (k) (x) = f (x) if x ∈ Ω, f (k) (x) if x / ∈ Ω.
Classical methods for inpainting use partial differential equations that propagate the information from the boundary of Ω to its interior, see for instance [?,?,?,?]. Sparsity promoting prior such as (??) in wavelets frames and local cosine bases have been used to solve inpainting as a inverse problem [?,?].

Our method iteratively updates the optimal graph w(x, y) and iterations of the algorithm update in parallel all the pixels inside Ω. This is similar to the exemplar-based inpainting algorithm of [?] which uses a non-local copying of patches, however in their framework Ω is filed progressively by propagating inward from the boundary. The anisotropic diffusion of Tschumperle and Deriche [?] also progressively builds an adapted operator (parameterized by a tensor field) but they solve a PDE and not a regularization as we do.

Figure ?? shows some numerical examples of inpainting on images where 80% of the pixels have been damaged. The wavelets method performs better than total variation in term of PSNR but tends to introduce some ringing artifact. The non-local total variation perform better in term of PSNR and is visually more pleasing since edge are better reconstructed.

Super-resolution

Super-resolution corresponds to the recovery of a high-definition image from a filtered and sub-sampled image. It is usually applied to a sequence of images in video, see the review papers [?,?]. We consider here a simpler problem of increasing the resolution of a single still image, which corresponds to the inversion of the operator

∀ f ∈ R n , Φf = (f * h) ↓ k and ∀ g ∈ R p , Φ T g = (g ↑ k ) * h
where p = n/k 2 , h ∈ R n is a low-pass filter, ↓ k : R n → R p is the sub-sampling operator by a factor k along each axis and ↑ k : R p → R n corresponds to the insertion of k -1 zeros along horizontal and vertical directions. Figure ?? shows some graphical results of the three tested super-resolution methods. The results are similar to those of inpainting, since our method improves over both wavelets and total variation.

Compressive-sampling

Compressive sensing is a new sampling theory that uses a fixed set of linear measurements together with a non-linear reconstruction [?,?]. The sensing operator computes the projection of the data on a finite set of p vectors

Φf = { f, ϕ i } p-1 i=0 ∈ R p , (11)
where (ϕ i ) p-1 i=0 are the rows of Φ. Compressive sampling theory gives hypotheses on both the input signal f and the sensing vectors (ϕ i ) i for this non-uniform sampling process to be invertible. In particular, the (ϕ i ) i must be incoherent with the orthogonal basis (ψ m ) m used for the sparsity prior, which is the case with high probability if they are drawn randomly from unit normed random vectors. Under the additional condition that f is sparse in an orthogonal basis (ψ m ) m # {m \ f, ψ m = 0} s then the optimization of (??) using the energy (??) leads to a recovery with a small error || f -f || ≈ |ε| if p = O(s log(n/s)). This results extend to approximately sparse signals, such as for instance signals that are highly compressible in an orthogonal basis.

In the numerical tests, we choose the columns of Φ ∈ R p×m to be independent random unit normed vectors. Figure ?? shows examples of compressive sampling reconstructions. The results are slightly above the wavelets method and tend to be visually more pleasing.

Conclusion and Future Work

This paper proposed a new framework for the non-local resolution of linear inverse problems. The variational minimization computes iteratively an adaptive non-local graph that enhances the geometric features of the recovered image. Numerical tests show how this method improves over some state of the art methods for inpainting, super-resolution and compressive sampling. This new method also open interesting questions concerning the optimization of the non-local graph. While this paper proposes to adapt the graph using a patch comparison principle, it is important to understand how this adaptation can be re-casted as a variational minimization.

Fig. 1 .

 1 Fig. 1. Examples of inpainting where Ω occupates 80% of pixels. The original images f are displayed on the left of figure ??.

Fig. 2 .

 2 Fig. 2. Examples of image super-resolution with a down-sampling k = 8. The original images f are displayed on the left of figure ??.

Fig. 3 .

 3 Fig. 3. Examples of compressed sensing reconstruction with p = n/8.

  || • || is the euclidean norm over R n . This energy was proposed by Gilboa et al. [?] in the continuous setting. It is used in the discrete setting by Zhou and Schölkopf [?] and Elmoataz et al. [?] in order to perform denoising.

Table 1 :

 1 Block coordinate descent algorithm to minimize approximately (??).

This work was partially supported by ANR grant SURF-NT05-2 45825.


L Golinskii

S Kupin

AND J Leblond

M Nemaire

ON DISCRETE SPECTRA OF BERGMAN-TOEPLITZ OPERATORS WITH HARMONIC SYMBOLS

Keywords: 2010 Mathematics Subject Classification. Primary: 47B35; Secondary: 47A55, 30H20, 3010 Discrete spectrum of a perturbed operator, Lieb-Thirring type inequality, Blaschke type inequality, Hardy-Toeplitz operator, Bergman-Toeplitz operator

In the present article, we study the discrete spectrum of certain bounded Toeplitz operators with harmonic symbol on a Bergman space. Using the methods of classical perturbaton theory and recent results by Borichev-Golinskii-Kupin and Favorov-Golinskii, we obtain a quantitative result on the distribution of the discrete spectrum of the operator in the unbounded (outer) component of its Fredholm set.

Introduction

Various problems of modern analysis require the study of certain classes of "model" operators. One of the important families is the class of Toeplitz operators and operators related to them. Probably, the most classical objects of this kind are Toeplitz operators on Hardy spaces of analytic functions, see Nikolski [START_REF] Nikolski | Operators, functions, and systems: an easy reading, I. Hardy, Hankel, and Toeplitz[END_REF] for a complete account on the subject. The applications of operators of this class can be found in Nikolski [START_REF] Nikolski | Operators, functions, and systems: an easy reading, II. Model operators and systems[END_REF]. Another "similar" class is the family of Toeplitz operators on Bergman spaces. Their study started in late 80's of the last century, see Zhu [START_REF] Zhu | Operator theory in function spaces[END_REF] for a nice overview of the topic.

We proceed with some definitions. For a function ϕ ∈ L ∞ (T), the Hardy-Toeplitz operator T ϕ : H 2 (T) → H 2 (T) is defined as

T ϕ h = P + (ϕh), h ∈ H 2 (T), (1.1)
where P + is the well-known Riesz othogonal projection from L 2 (T) to H 2 (T), see Garnett [START_REF] Garnett | Bounded analytic functions[END_REF]. The function ϕ is called a symbol of the operator. For the sake of brevity, we call operator T ϕ (1.1) an HT-operator.

The definition of a Bergman-Toeplitz operator T ψ (a BT-operator, for short), is rather similar to the above one. Indeed, let L 2 a (D) be the closed subspace in L 2 (D) of analytic on D functions. Given ψ ∈ L ∞ (D), set

T ψ : L 2 a (D) → L 2 a (D), T ψ h = P+ (ψh), (1.2)
where P+ is the orthogonal projection acting from L 2 (D) to L 2 a (D), see Zhu [START_REF] Zhu | Operator theory in function spaces[END_REF]Chapter 7].

For a function ϕ ∈ L ∞ (T), it is sometimes convenient to consider a harmonic function φ on D, the harmonic extension of ϕ to D, given by

φ(z) = T 1 -|z| 2 |t -z| 2 ϕ(t) m(dt), z ∈ D. (1.3)
Here, m is the probability Lebesgue measure on T.

Certainly, φ ∈ L ∞ (D) and φ ∞ ≤ ϕ ∞ .
Given an HT-operator T ϕ (with its symbol defined on T), we consider its associated BT-operator T φ with symbol φ given by (1.3). Notice that even though we use "similar-looking" notation for an HT-operator T ϕ and a BT-operator T φ, the confusion is not possible since the functions ϕ and φ are defined on T and D, respectively. The domains of definitions of corresponding symbols will be always clear from the context of the discussion.

We shall be concerned with Toeplitz operators T ϕ , T φ having symbols defined as follows. Set

ϕ(t) := ḡ(t) + f (t), t ∈ T, f, g ∈ H ∞ (T).
(1.4)

Clearly, we have

ψ(z) := φ(z) = ḡ(z) + f (z), z ∈ D. (1.5)
It is plain that φ has the non-tangential boundary values on the unit circle

φ(t) = lim r→1- φ(rt), for a. e. t ∈ T,
and these boundary values coincide with ϕ a.e. on T.

Despite the similarity of definitions (1.1), (1.2), the BT-operators exhibit considerably reacher spectral behavior as compared to HT-operators. For instance, the essential spectrum of HT-operator T ϕ is connected, see Widom [START_REF] Widom | On the spectrum of a Toeplitz operator[END_REF], while, in general, the essential spectrum of BT-operator T ψ is not. There are non-trivial compact BT-operators with quite simple (even radial) symbols [START_REF] Zhu | Operator theory in function spaces[END_REF]Sections 7.2,7.3], while a compact HT-operator is necessarily zero [START_REF] Nikolski | Operators, functions, and systems: an easy reading, I. Hardy, Hankel, and Toeplitz[END_REF]Part B,Chapter 4].

Sundberg-Zheng [START_REF] Sundberg | The spectrum and essential spectrum of Toeplitz operators with harmonic symbols[END_REF] showed that there are BT-operators with harmonic symbols having isolated eigenvalues in their spectrum. In subsequent papers, Zhao-Zheng [START_REF] Zhao | The spectrum of Bergman Toeplitz operators with some harmonic symbols[END_REF], Guan-Zhao [START_REF] Guan | Invertibility of Bergman Toeplitz operators with harmonic polynomial symbols[END_REF] and Guo-Zhao-Zheng [START_REF] Guo | The spectral picture of Bergman Toeplitz operators with harmonic polynomial symbols[END_REF] presented a class of BT-operators with harmonic symbols posessing "rather big" discrete spectrum, that is, the set of isolated eigenvalues of finite algebraic multiplicity.

So, in contrast to HT-operators, the notion of the discrete spectrum of a BT-operator with harmonic symbol makes sense. The study of the properties of the discrete spectrum for BT-operators with symbols (1.5) is the core of the present paper. Unlike the articles [START_REF] Guan | Invertibility of Bergman Toeplitz operators with harmonic polynomial symbols[END_REF][START_REF] Guo | The spectral picture of Bergman Toeplitz operators with harmonic polynomial symbols[END_REF][START_REF] Zhao | The spectrum of Bergman Toeplitz operators with some harmonic symbols[END_REF], our results are essentially based on the perturbation techniques from operator theory and function-theoretic results of Borichev-Golinskii-Kupin [START_REF] Borichev | A Blaschke-type condition and its application to complex Jacobi matrices[END_REF][START_REF] Borichev | On zeros of analytic functions satisfying nonradial growth conditions[END_REF] and Favorov-Golinskii [START_REF] Favorov | Blaschke-type conditions on unbounded domains, generalized convexity, and applications in perturbation theory[END_REF].

We also need the definition of the Sobolev space W 1,2 (T) of absolutely continuous functions on the unit circle T with derivative in L 2 :

W 1,2 (T) := {h : T → C, h ∈ AC, h ∈ L 2 (T)}.
For further purposes, we would like to introduce two closely related characteristics of compact sets on the complex plane C. The following definitions are borrowed from Perkal [12, Section 2] and Peller [START_REF] Peller | Spectrum, similarity, and invariant subspaces of Toeplitz operators[END_REF]Section 4], respectively.

Definition 1.1. Let r > 0. A closed set E ⊂ C is called r-convex if C\E = B(x, r) : B(x, r) ⊂ C\E ,
that is, the complement to E can be covered with open disks of a fixed radius r > 0, which lie in that complement.

Definition 1.2. A compact set K ⊂ C is called circularly convex, if there is r > 0 such that for each λ ∈ C\K with dist(λ, K) < r there are points µ ∈ ∂K and ν ∈ C\K so that

|µ -ν| = r, λ ∈ (µ, ν], {ζ : |ν -ζ| < r} ⊂ C\K.
For example, if K is a convex set, or the boundary ∂K is of C 2 -class (without intersections and cusps), then K is a circularly convex set.

Note that the later definition is a bit more stringent than the former one. When K is a (closed) Jordan curve (a rectifiable continuous curve with no self-intersections), one can also see that the above definitions are equivalent.

A short reminder on standard notions and notations from operator theory is given in Subsection 2.1 below. For instance, see (2.3) for the notion of the unbounded (outer) open component of the Fredholm domain F 0 (T ).

The main result of this note is the following theorem.

Theorem 1.3. Let T ϕ be an HT-operator with the symbol ϕ (1.4) from W 1,2 (T), φ its harmonic extension (1.3), and T φ be the BT-operator associated to T ϕ . Assume that the spectrum σ(T ϕ ) is a circularly convex set. Then, for each ε > 0

λ∈σ d (T φ)∩F 0 (Tϕ) dist 3+ε (λ, σ(T ϕ )) ≤ C(ϕ, ε) ϕ 2 2 .
(1.6)

Corollary 1.4. Let q and p be algebraic polynomials, ϕ = q + p be a harmonic polynomial, and assume that the image ϕ(T) is a Jordan curve without cusps. Then (1.6) holds for the discrete spectrum of BT-operator T φ.

Some preliminaries

2.1. Generalities from operator theory. In this section, we recall some well-known notions of the classical operator theory, see Kato [8, Section IV. ∼5].

Let T be a bounded linear operator on a (separable) Hilbert space H. As usual, the resolvent set of T is

ρ(T ) := {λ ∈ C : (T -λ) : H → H is bijective }. (2.1)
It follows that (T -λ) -1 is bounded for λ ∈ ρ(T ). The spectrum of T is defined as

σ(T ) = C\ρ(T ). (2.2)
Furthermore, we say that a bounded operator T is Fredholm, if its kernel and co-kernel are of finite dimension. The essential spectrum of T is defined as

σ ess (T ) = {λ ∈ C : (T -λ) is not Fredholm}.
One can see that σ ess (T ) is a closed subset of σ(T ). One considers also the Fredholm domain of T , F(T ) = C\σ ess (T ). Clearly, ρ(T ) ⊂ F(T ). We represent F(T ) as

F(T ) = ∞ j=0 F j (T ), (2.3)
where F j (T ) are disjoint (open) connected components of the set. We agree that F 0 (T ) stays for the unbounded connected component of F(T ).

The discrete spectrum σ d (T ) of T is the set of all isolated eigenvalues of T of finite algebraic multiplicity. For convenience, we put

σ 0 (T ) := σ d (T ) ∩ F 0 (T ) ⊂ σ d (T ).
(2.4)

Let A 0 , A be bounded operators on a Hilbert space such that A -A 0 is compact. The operators A and A 0 are called compact perturbations of each other. The celebrated Weyl's theorem states that

σ ess (A) = σ ess (A 0 ), (2.5)
see Kato [8, Section IV.5.6].

We shall be interested in the situation when σ 0 (A) is at most countable set, σ 0 (A) = {λ j } j≥1 and it accumulates to the essential spectrum σ ess (A 0 ) only.

2.2.

Reminder on Hilbert-Schmidt operators. In this subsection, we recall briefly the notion of a Hilbert-Schmidt operator and its simplest properties, see Birman-Solomyak [1, Section 11.3].

Let A be a compact operator. The sequence of singular values {s j (A)} j≥1 is defined as

s j (A) = λ j (A * A) 1/2 , s j (A) ≥ 0,
where λ j (A * A) are eigenvalues of the compact operator A * A. Without loss of generality one can suppose that {s j (A)} j≥1 forms a decreasing sequence, and, moreover lim j→+∞ s j (A) = 0.

One says that A ∈ S 2 , the Hilbert-Schmidt class of compact operators, iff

A 2 S 2 := j≥1 s j (A) 2 < ∞.
Equivalently, A ∈ S 2 if and only if {s j (A)} j≥1 ∈ 2 . Alternatively, the S 2 -norm of the operator can be computed as

A 2 S 2 = j,k≥1 |(Ae j , e k )| 2 ,
where {e j } j≥1 is an arbitrary orthonormal basis in the given Hilbert space.

2.3. On the discrete spectrum of a perturbed operator: a result of Favorov-Golinskii. Some useful quantitative bounds for the rate of convergence of the discrete spectrum of a perturbed operator are given in Favorov-Golinskii [4, Section 5]. A special case of [4, Theorem 5.1] (cf. a remark right after its proof and formula (5.8)) looks as follows.

Theorem 2.1 ([4]

). Let A 0 be a bounded linear operator on a Hilbert space, which satisfies the conditions:

(1) The spectrum σ(A 0 ) is an r-convex set.

(

) The resolvent R(z, A 0 ) = (A 0 -z) -1 is subject to the bound R(z, A 0 ) ≤ C(A 0 ) dist p (z, σ(A 0 )) , p > 0, z ∈ F 0 (A 0 ). (2.6) 2
Let B be a Hilbert-Schmidt operator, and

A = A 0 + B. Then for each ε > 0 λ∈σ d (A)∩F 0 (A 0 ) dist 2p+1+ε (λ, σ(A 0 )) ≤ C(σ(A 0 ), p, ε) B 2 2 . (2.7)
If σ ess (A 0 ) does not split the plane, and (2.6) holds for all λ ∈ C\σ(A 0 ), then (2.7) is true for the whole discrete spectrum σ d (A). For the class of (non-selfadjoint) HT-operators A 0 = T ϕ , ϕ in (1.4), the essential spectrum, in general, splits the plane.

Proof of the main result

Let ϕ be as in (1.4). Consider the HT-operator T ϕ (1.1) and the associated BT-operator T φ (1.2). The technical way to compare these operators is to look at their matrices in appropriately chosen bases. Namely, define

e H,n (t) = t n , e B,n (z) = √ n + 1 z n , n ≥ 0. (3.1)
It is plain that the systems {e H,n } n≥0 and {e B,n } n≥0 are the orthonormal bases in H 2 (T) and L 2 a (D), respectively. Set

T ϕ = [(T ϕ e H,i , e H,j ) H 2 (T) ] i,j≥0 , T φ = [(T φe B,i , e B,j ) L 2 a (T) ] i,j≥0 . (3.2)
The operators T ϕ and T φ are unitarily equivalent to original operators T ϕ and T φ, and they both act on 2 (Z + ). So, one can argue on the operators T ϕ and T φ being "close" in a certain sense.

Rewriting relation (1.4) in more detailed form, we have

φ(z) = g(z) + f (z), f (z) = ∞ k=0 f k z k ∈ H ∞ (T), g(z) = ∞ k=0 g k z k ∈ H ∞ (T),
ϕ(e iθ ) = φ(e iθ ) = j∈Z b j e ijθ , b j = f j , j ≥ 0, g -j , j < 0. Finally,

τ k,k+l = k + 1 k + l + 1 b -l , k, l ≥ 0.
The same formula holds for τ k+l,k , k, l ≥ 0, and so

τ i,j = min(i, j) + 1 max(i, j) + 1 b i-j , i, j ≥ 0. (3.5)
Let us estimate the Hilbert-Schmidt norm

T φ -T ϕ 2 S 2 = i,j≥0 |τ i,j -b i-j | 2 = i,j≥0 |b i-j | 2 1 - min(i, j) + 1 max(i, j) + 1 2 . As i,j≥0 a ij = ∞ l=0 ∞ k=0 a k,k+l + ∞ l=1 ∞ k=0 a k+l,k , we have T φ -T ϕ 2 S 2 = ∞ l=0 l 2 |b l | 2 + |b -l | 2 ∞ k=0 1 (k + l + 1) √ k + l + 1 + √ k + 1 2 = l∈Z |l b l | 2 ∞ k=0 1 (k + |l| + 1) k + |l| + 1 + √ k + 1 2 ≤ l∈Z |lb l | 2 ∞ k=0 1 (k + 1) (2 √ k + 1) 2 = l∈Z |lb l | 2 ∞ k=0 1 4 (k + 1) 2 = π 2 4 • 6 ϕ 2 2 = π 2 24 ϕ 2 2 .
as claimed.

Weyl's theorem concerning spectra of compact perturbations, mentioned above, leads to the following Corollary 3.2. Let the symbol ϕ satisfy hypothesis of the above Proposition. Then the essential spectrum of BT-operator T φ is

σ ess (T φ) = σ ess (T ϕ ) = ϕ(T) =: Γ,
and the discrete spectrum σ d (T φ) is at most countable set of eigenvalues of finite algebraic multiplicity with all its accumulation points on Γ.

We go on with the proof the quantitative version of the above Corollary.

Proof of Theorem 1.3. Let A 0 = T ϕ , A = T φ, see (3.2). We only have to ensure that the conditions of Theorem 2.1 are met.

It is clear that ϕ ∈ W 1,2 (T) implies that ϕ ∈ W , the Wiener algebra of absolutely convergent Fourier series. By [START_REF] Peller | Spectrum, similarity, and invariant subspaces of Toeplitz operators[END_REF]Theorem 4], the resolvent (A 0 -z) -1 admits the linear growth, that is, (2.6) holds with p = 1. Next, by Proposition 3.1, the difference A -A 0 is the Hilbert-Schmidt operator with the norm bound (3.4)

(3. 3 ) 3 . 1 . 2 S 2 4 )

 331224 Proposition Assume that the symbol ϕ (1.4) belongs to W 1,2 (T). Then T φ -T ϕ is a Hilbert-Schmidt operator andT φ -T ϕ Proof. The matrix representation of T ϕ is obvious: T ϕ = [b i-j ] i,j≥0 . So let us compute the matrix T φ = [τ i,j ] i,j≥0 in the orthonormal basis {e B,n } n≥0 (3.1). For l, k ≥ 0 τ k,k+l = (T φe B,k+l , e B,k ) L 2 a (D) = 1 π D φ(z)e B,k+l(z) e B,k (z) dxdy =(k + l + 1)(k + 1) π D φ(z)z k+l z k dxdy,or, in polar coordinates, τ k,k+l = (k + l + 1)(k + 1) n r |n|+2k+l+1 e i(n+l)θ drdθ.

  . The proof is complete. 2 B. Verkin Institute for Low Temperature Physics and Engineering, 47 Nauky ave., Kharkiv 61103, Ukraine Email address: golinskii@ilt.kharkov.ua; leonid.golinskii@gmail.com IMB, CNRS, Université de Bordeaux, 351 ave. de la Libération, 33405 Talence Cedex, France Email address: skupin@math.u-bordeaux.fr FACTAS, INRIA Sophia Antipolis-Méditerranée, 2004 route des Lucioles -BP93, 06902 Sophia Antipolis Cedex, France Email address: juliette.leblond@inria.fr IMB, CNRS, Université de Bordeaux, 351 ave. de la Libération, 33405 Talence Cedex, France Email address: Masimba.Nemaire@math.u-bordeaux.fr FACTAS, INRIA Sophia Antipolis-Méditerranée, 2004 route des Lucioles -BP93, 06902 Sophia Antipolis Cedex, France Email address: Masimba.Nemaire@inria.fr

Acknowledgements. SK, JL, MN are partially supported by the project ANR-18-CE40-0035.


Martino Adamo
email: martino.adamo@unito.it

Enrico Büttner

Harald Kellner

Patricia Luis

Sophie Comtet-Marre

Laurent Vallon

Rocio Prego

Martin Hofrichter

Mariangela Girlanda

Pierre Peyret

Roland Marmeisse

Fungal dye-decolorizing peroxidase diversity: roles in either intra-or extracellular processes

Keywords: DyP-type peroxidase, Lignin degradation, Fungi, Sequence capture

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

Introduction

Lignin is a complex plant cell-wall polymer made of different cross-linked phenylpropanoid monomers [START_REF] Schmidt | Wood and tree fungi[END_REF]) that represents about 30% of the plant-derived organic carbon available in the biosphere [START_REF] Boerjan | Lignin biosynthesis[END_REF]. Besides providing structural rigidity in living plants, another important function is to protect cell wall polysaccharides from degradation by hydrolytic enzymes secreted by microorganisms (lignin barrier). Control of lignin degradation represents an important field of research either for the use of this polymer as a source of aromatic molecules or simply for its removal from plant biomass to access cell-wall polysaccharides for biorefinery and biofuel production [START_REF] Duval | A review on lignin-based polymeric, micro-and nano-structured materials[END_REF][START_REF] Chio | Lignin utilization: a review of lignin depolymerization from various aspects[END_REF].

White-rot (WR) fungi, a specific functional group of saprotrophic Basidiomycota, are considered the main group of microorganisms able to perform enzymatic ligninolysis. Phylogenomic studies support the assumption that the WR phenotype is an ancestral character that arose once early in the evolution of the Basidiomycota [START_REF] Floudas | The paleozoic origin of enzymatic lignin decomposition reconstructed from 31 fungal genomes[END_REF]. Subsequently, the WR phenotype was lost several times independently during the course of evolution to give rise to different lineages of either brown-rot (BR) saprotrophic or mutualistic ectomycorrhizal (EM) Basidiomycota that associate to plant roots [START_REF] Kohler | Convergent losses of decay mechanisms and rapid turnover of symbiosis genes in mycorrhizal mutualists[END_REF][START_REF] Martin | Unearthing the roots of ectomycorrhizal symbioses[END_REF][START_REF] Nagy | Genetic bases of fungal white rot wood decay predicted by phylogenomic analysis of correlated gene-phenotype evolution[END_REF][START_REF] Miyauchi | Large-scale genome sequencing of mycorrhizal fungi provides insights into the early evolution of symbiotic traits[END_REF]. Both BR and EM fungi have lost the capacity of performing enzymatic ligninolysis, although being capable of modifying the lignin structure to some extent using oxidizing mechanisms [START_REF] Arantes | Peculiarities of brown-rot fungi and biochemical Fenton reaction with regard to their potential as a model for bioprocessing biomass[END_REF][START_REF] Rineau | The ectomycorrhizal fungus Paxillus involutus converts organic matter in plant litter using a trimmed brownrot mechanism involving Fenton chemistry[END_REF].

From a mechanistic point of view, biochemical and genetic studies have demonstrated that several secreted enzymes belonging to the so-called heme-containing, fungal-specific class II peroxidases (E.C. 1.11.1.-) participate directly or indirectly in lignin decomposition [START_REF] Hammel | Role of fungal peroxidases in biological ligninolysis[END_REF][START_REF] Salame | Inactivation of a Pleurotus ostreatus versatile peroxidaseencoding gene (mnp2) results in reduced lignin degradation[END_REF][START_REF] Ayuso-Fernández | Evolutionary convergence in lignin-degrading enzymes[END_REF]). Among these class II peroxidases, lignin (LiP,E.C. 1.11.1.14) and versatile (VP,E.C. 1.11.1.16) peroxidases interact directly with lignin to cleave non-phenolic ether bonds that link phenylpropanoid monomers. A further class II enzyme, manganese peroxidase (MnP,E.C. 1.11.1.13), but also VPs, oxidize Mn 2+ to Mn 3+ , which is then chelated with organic acids and cleaves as low-molecular mass and diffusible redoxmediator phenolic structures of lignin (Ruiz-Dueñas et al. 2021). Besides biochemical studies, the essential role of class II peroxidases in lignin decomposition was also supported by phylogenomic studies. Diversification of this protein family correlates with diversification of WR fungi and the several transitions from a WR to BR or EM trophic modes almost always went along with a loss or a sharp contraction of class II peroxidase gene family in the fungal genomes [START_REF] Kohler | Convergent losses of decay mechanisms and rapid turnover of symbiosis genes in mycorrhizal mutualists[END_REF][START_REF] Martin | Unearthing the roots of ectomycorrhizal symbioses[END_REF][START_REF] Nagy | Genetic bases of fungal white rot wood decay predicted by phylogenomic analysis of correlated gene-phenotype evolution[END_REF][START_REF] Miyauchi | Large-scale genome sequencing of mycorrhizal fungi provides insights into the early evolution of symbiotic traits[END_REF]Ruiz-Dueñas et al. 2021).

Despite the undoubted contribution of class II peroxidases to lignin decomposition, their exclusive role in this process is debated. Indeed, substantial lignin degradation seemingly occurs in the absence of organisms producing these enzymes, as in the case of the termite gut [START_REF] Li | Lignocellulose pretreatment in a fungus-cultivating termite[END_REF]. On the other hand, other microbial enzymes have been shown to be able to cleave phenolic and/or nonphenolic bonds of artificial lignin substrates (e.g. β-O-4dimers = adlerol) and/or oxidize Mn 2+ to Mn 3+ and therefore behave similarly to either class II LiP/VPs or MnPs [START_REF] Fernández-Fueyo | Description of the first fungal dye-decolorizing peroxidase oxidizing manganese(II)[END_REF]. One such enzyme family is that of "dye-decolorizing peroxidase" (DyP) (pfam no. PF04261, E.C. 1.11.1.19) [START_REF] Celis | Substrate, product, and cofactor: the extraordinarily flexible relationship between the CDE superfamily and heme[END_REF]. These heme-containing peroxidases are unrelated to class II peroxidases and were identified for their unique ability to oxidize high-redox potential anthraquinone dyes [START_REF] Kim | Characteristics of a newly isolated fungus, Geotrichum candidum Dec 1, which decolorizes various dyes[END_REF], a property which makes them good candidates for the remediation of dye-polluted waste-water [START_REF] Scheibner | Novel peroxidases of Marasmius scorodonius degrade β-carotene[END_REF]. DyPs present a ferredoxin-like fold [START_REF] Singh | The multihued palette of dye-decolorizing peroxidases[END_REF], which is unique among peroxidases, and are present in all three domains of life (Bacteria, Archaea, and Eukarya) as opposed to class II peroxidases that are restricted to the Eumycota (true fungi) [START_REF] Zámocký | Molecular evolution of hydrogen peroxide degrading enzymes[END_REF][START_REF] Celis | Substrate, product, and cofactor: the extraordinarily flexible relationship between the CDE superfamily and heme[END_REF]. Global phylogenies of the DyP (sub)family show that, with very few exceptions, fungal sequences are monophyletic and form a well-supported clade that does not include sequences from other taxonomic groups [START_REF] Sugano | DyP-type peroxidases comprise a novel heme peroxidase family[END_REF][START_REF] Zámocký | Independent evolution of four heme peroxidase superfamilies[END_REF].

Different levels of biochemical information exist for at least 13 fungal DyPs (12 from Basidiomycota, one from Ascomycota), including 3D structures (based on protein crystals) for two of them [START_REF] Yoshida | The catalytic mechanism of dye-decolorizing peroxidase DyP may require the swinging movement of an aspartic acid residue[END_REF][START_REF] Strittmatter | First crystal structure of a fungal highredox potential dye-decolorizing peroxidase substrate interaction sites and long-range electron transfer[END_REF], identification of amino acids participating to catalysis through site-directed mutagenesis and biophysical measurements (Linde et al. 2015b) as well as reconstruction and characterization of ancestral fungal sequences [START_REF] Zitare | Mutational and structural analysis of an ancestral fungal dye-decolorizing peroxidase[END_REF].

DyPs can form surface protein radicals (surface-exposed tryptophan radicals) via a long-range electron transfer (LRET) and are hereby able to oxidize phenolic compounds and synthetic dyes. In Pleurotus ostreatus PosDyP4, a Mn 2+ -oxidation activity located on a different surface site, surrounded by acidic amino acids, was identified [START_REF] Fernández-Fueyo | Description of the first fungal dye-decolorizing peroxidase oxidizing manganese(II)[END_REF]. This activity is analogous to that accomplished by MnPs and VPs (Fernández-Fueyo et al. 2018), and [START_REF] Krahe | A dyP-type peroxidase of Pleurotus sapidus with alkene cleaving activity[END_REF] were able to use a closely related DyP of Pleurotus sapidus with Mn 2+ -oxidation activity to cleave alkenes (aryl alkenes (E)-methyl isoeugenol, α-methylstyrene, and trans-anethole) in the presence of Mn 2+ , which could be relevant in flavor production. A further Mn 2+ -oxidizing DyP activity has been recently reported for the first characterized ascomycetous DyP from Xylaria grammica [START_REF] Kim | Characteristics of a newly isolated fungus, Geotrichum candidum Dec 1, which decolorizes various dyes[END_REF].

The roles of DyPs in fungal biology have never been addressed by forward genetic approaches, and they have been proposed to contribute to the detoxification of naturally occurring soil-borne or plant metabolites [START_REF] Valette | Antifungal activities of wood extractives[END_REF], including natural anthraquinone derivatives [START_REF] Sugawara | Degradation of antifungal anthraquinone compounds is a probable physiological role of DyP secreted by Bjerkandera adusta[END_REF]. They have also repeatedly been suspected to participate in lignin decomposition and in the oxidation of lignin-related compounds [START_REF] Liers | DyP-like peroxidases of the jelly fungus Auricularia auricula-judae oxidize nonphenolic lignin model compounds and high-redox potential dyes[END_REF][START_REF] Celis | Substrate, product, and cofactor: the extraordinarily flexible relationship between the CDE superfamily and heme[END_REF][START_REF] Cragg | Lignocellulose degradation mechanisms across the tree of life[END_REF][START_REF] De Gonzalo | Bacterial enzymes involved in lignin degradation[END_REF][START_REF] Catucci | Biochemical features of dye-decolorizing peroxidases: current impact on lignin degradation[END_REF][START_REF] De Eugenio | Characterization of a dye-decolorizing peroxidase from Irpex lacteus expressed in Escherichia coli: an enzyme with wide substrate specificity able to transform lignosulfonates[END_REF], despite the fact that fungal class II peroxidases perform far better on "lignin substrates" than bacterial or fungal DyPs [START_REF] Linde | Comparing ligninolytic capabilities of bacterial and fungal dye-decolorizing peroxidases and class-II peroxidase-catalases[END_REF]. In support of their putative roles in lignin oxidation, fungal DyPs have been also often reported as secreted enzymes, i.e. extracellularly acting biocatalysts. Several of them were purified from fungal culture filtrates, and they have been identified as well in the secretomes of fungi growing on lignin-rich substrates [START_REF] Salvachúa | Lignin depolymerization by fungal secretomes and a microbial sink[END_REF]. From an evolutionary perspective, as for class II peroxidases and other gene families, the fungal DyP (sub)family diversification parallels the evolution of WR fungi and regresses in BR and EM lineages [START_REF] Nagy | Genetic bases of fungal white rot wood decay predicted by phylogenomic analysis of correlated gene-phenotype evolution[END_REF].

In the absence of forward genetic data and in the presence of uncertain biochemical activities, the objective of the present study has been to further evaluate the potential biological functions of fungal DyPs, including their putative contribution to lignin decomposition using phylogenetic and correlative approaches. Two strategies were followed. One was to establish a comprehensive view on the evolution of the DyP (sub)family in the fungal kingdom, using available genomic data, and to confront these data to our current knowledge on DyPs' catalytic mechanisms and evolution of trophic modes in fungi. A second strategy was to isolate new fungal DyP genes expressed in different environments (soils and decomposing wood) in an attempt to characterize potentially divergent DyP sequences distantly related to sequences present in public databases. The latter approach could potentially identify new DyPs with potentially useful catalytic properties for novel applications in biotechnology.

Materials and methods

Species and sequence datasets

Publicly available DyP sequences were extracted from Gen-Bank and the JGI Mycocosm portal (https:// genome. jgi. doe. gov/ progr ams/ fungi/ index. jsf; [START_REF] Grigoriev | MycoCosm portal: gearing up for 1000 fungal genomes[END_REF]) databases as well as from our own unpublished whole-genome sequences. DyP protein sequences were identified in public databases using BlastP searches [START_REF] Altschul | Basic local alignment search tool[END_REF]) using as queries the protein sequences of two biochemically characterized DyPs from Auricularia auricula-judae (accession No. JQ650250) and Bjerkandera adusta (CDN40127.1).

A threshold E-value of 10 -5 was retained for the inclusion of the sequences in the dataset. DyP sequences extraction from the unpublished genomes was performed using HMMER (http:// hmmer. org/). A HMMER profile was first created starting from a Clustal Omega alignment [START_REF] Sievers | Clustal Omega, accurate alignment of very large numbers of sequences[END_REF] of 150 DyP sequences from the public databases. Phylogenetically-related bacterial DyP sequences, all belonging to the DyP clade C [START_REF] Zámocký | Independent evolution of four heme peroxidase superfamilies[END_REF], were identified following blastp searches using as queries different sequences representative of the phylogenetic diversity of the fungal DyPs.

Phylogenetic and network analyses

Only putative full-length protein sequences, starting with methionine and ending with a stop codon, corresponding to CDS longer than 1000 bp and without gaps in the regions where amino acids essential for catalysis have been identified (as described in Linde et al. 2015a), were used for a phylogenetic reconstruction of the fungal DyP (sub)family. In the case of the Aspergillus and Penicillium, genera for which the genomes of numerous species are available, only sequences from Aspergillus oryzae and Penicillium italicum were included in the phylogenetic analysis. Sequences were aligned using MUSCLE [START_REF] Edgar | MUSCLE: multiple sequence alignment with high accuracy and high throughput[END_REF]. ModelFinder [START_REF] Kalyaanamoorthy | ModelFinder: fast model selection for accurate phylogenetic estimates[END_REF], as implemented on the IQ-Tree web Server (http:// iqtree. cibiv. univie. ac. at/; [START_REF] Trifinopoulos | W-IQ-TREE: a fast online phylogenetic tool for maximum likelihood analysis[END_REF])), was used to find the best substitution model (LG + R). IQ-TREE Web Server was also used to perform a maximum likelihood phylogenetic analysis [START_REF] Nguyen | IQ-TREE: a fast and effective stochastic algorithm for estimating maximum-likelihood phylogenies[END_REF]. One thousand bootstrap replicates were performed to assess tree topology.

A fungal species phylogeny that encompassed all examined fully-sequenced basidiomycetous species was obtained by modifying a publicly available phylogenetic tree (#Tr106951, [START_REF] Zhao | A six-gene phylogenetic overview of basidiomycota and allied phyla with estimated divergence times of higher taxa and a phyloproteomics perspective[END_REF], built using an alignment of six genes, downloaded from TreeBase (www. treeb ase. org/). Species whose genomes have not been sequenced were removed from the original six-genes alignment while the corresponding six genes from newly sequenced species were added to it to construct a new species tree following the approach presented in [START_REF] Zhao | A six-gene phylogenetic overview of basidiomycota and allied phyla with estimated divergence times of higher taxa and a phyloproteomics perspective[END_REF]. Species taxonomy was assigned using the MycoBank database (www. mycob ank. org). Both gene and species trees were edited using the online software iTOL (https:// itol. embl. de; [START_REF] Letunic | Interactive tree of life (iTOL) v3: an online tool for the display and annotation of phylogenetic and other trees[END_REF]). We inferred gene loss and duplication along the species phylogenetic tree using a comparative genomics method with NOTUNG 2.9 as described in the original article of [START_REF] Chen | NOTUNG: a program for dating gene duplications and optimizing gene family trees[END_REF].

To visualize similarities between fungal DyP protein sequences, a sequence similarity network (SSN) was 1 3 computed using the EFI-Enzyme Similarity Tool (https:// efi. igb. illin ois. edu/ efi-est/; [START_REF] Gerlt | Enzyme function initiative-enzyme similarity tool (EFI-EST): a web tool for generating protein sequence similarity networks[END_REF]) starting from the complete set of DyP protein sequences. The lowest pairwise alignment score limit for the output file was set to 66. The full-network output file was visualized with Cytoscape 3.5.1 [START_REF] Shannon | Cytoscape: a software environment for integrated models of biomolecular interaction networks[END_REF]. The network was represented with an "organic" layout, and only edges with a pairwise alignment score above 110 were visualized. To further evaluate the potential contribution of agaricomycetous DyP genes to organic matter (e.g. wood) and more specifically, to lignin degradation, we compared the prevalence of the different DyP clades in polyphyletic species groups of Agaricomycota, which differed with respect to their trophic modes ("trophic guilds"). Using the FUNguild [START_REF] Nguyen | FUNGuild: an open annotation tool for parsing fungal community datasets by ecological guild[END_REF]) and experts' opinions, we defined four trophic modes. "White-rot" (WR), sensu stricto encompasses saprotrophic or facultative pathogenic, lignicolous species (living on wood and coarse woody debris), which degrade lignin enzymatically to produce a "white-rot" phenotype in compact wood (where finally whitish cellulose fibers prevail). All of these species possess mostly numerous class-II peroxidase genes (Mn, versatile, and/or lignin peroxidases) in their genomes [START_REF] Floudas | The paleozoic origin of enzymatic lignin decomposition reconstructed from 31 fungal genomes[END_REF]. "Brown-rot" (BR) species, although also lignicolous, produce a brown-rot phenotype that results from a selective degradation of cell-wall polysaccharides (cellulose and xylan/hemicelluloses) along with a nonenzymatic modification of the lignin polymer. "Ectomycorrhizal" (EM) species are mutualistic symbionts of mainly trees (larger woody plants) that have (almost) lost the ability to degrade plant cell walls [START_REF] Shah | Ectomycorrhizal fungi decompose soil organic matter using oxidative mechanisms adapted from saprotrophic ancestors[END_REF]. Finally, a fourth group named "other saprotrophs" (OS) encompasses all other species that are either litter-decomposing saprotrophs (e.g. Agaricus bisporus), coprophilic species (e.g. Coprinopsis cinerea), or endophytic ones (e.g. Sebacina vermifera). Since these OS species do not grow on compact wood, their modes of degrading the different plant polymers do not exactly fit into the white-rot or brown-rot definitions, although several authors classify several of these species as primarily white-rotters [START_REF] Eastwood | The plant cell wall-decomposing machinery underlies the functional diversity of forest fungi[END_REF][START_REF] Nagy | Comparative genomics of early-diverging mushroom-forming fungi provides insights into the origins of lignocellulose decay capabilities[END_REF].

Prediction of sequence features

Prediction of N-terminal signal peptides for eukaryotic secreted proteins was performed with SignalP (https:// servi ces. healt htech. dtu. dk/ servi ce. php? Signa lP-5.0; [START_REF] Nielsen | Predicting secretory proteins with SignalP[END_REF]) with default settings except for the D-cutoff value, which was set at 0.34 for both the noTM (transmembrane domain) and the TM networks. SignalP raw results were reported on the protein phylogenetic tree. The number of putative N-glycosylation sites was predicted using NetNGlyc (https:// servi ces. healt htech. dtu. dk/ servi ce. php? NetNG lyc-1.0) [START_REF] Gupta | Orphan Protein Function and Its Relation to Glycosylation[END_REF]. Other protein features, such as protein length, absolute numbers of specific amino acids, predicted protein isoelectric point (pI), and global protein hydropathy using the "grand average of hydropathy" (GRAVY), were predicted using tools of the sequence manipulation suite (https:// www. bioin forma tics. org/ sms2/; [START_REF] Stothard | The sequence manipulation suite: JavaScript programs for analyzing and formatting protein and DNA sequences[END_REF]).

Principal component analysis (PCA) was performed to cluster DyP protein sequences according to their protein features in the R environment using the R 3.6.3 (R Core Team 2019). Principal components were extracted with the prcomp function of the R base package and visualized with the ggbiplot package v0.55. Kmeans analysis was performed with the kmeans function of the R base package. Variable's (i.e. protein features) contribution to principal components (PC) was determined using the fviz_contrib function from factoextra v1.0.6 R package. Statistical differences were tested using the nonparametric Kruskal-Wallis test and Dunn's post hoc tests with dunn. test package v1.3.4 [START_REF] Dinno | Nonparametric pairwise multiple comparisons in independent groups using Dunn's test[END_REF]. DyP abundance comparisons among different fungal guilds were performed with the Yates-corrected Chi-square test.

Probes design for gene capture by hybridization

From 1267 publicly available fungal DyP DNA-coding sequences, we designed 69 (70 bp long) degenerated probes (Supplemental Table S1) with the KASpOD software (https://g2im. u-clermont1.fr/kaspod/index.php, [START_REF] Parisot | KASpOD-a web service for highly specific and explorative oligonucleotide design[END_REF]). In silico probe coverage evaluation, on the 1267 fungal DyP sequences detects 94.2% of the ascomycetous sequences and 81.5% of the basidiomycetous ones (with four allowed mismatches). Oligonucleotides corresponding to probe sequences were synthesized with two flanking adaptor sequences for their PCR amplification and conversion to biotinylated RNA probes using the T7 RNA polymerase, as described in [START_REF] Bragalini | Solution hybrid selection capture for the recovery of functional full-length eukaryotic cDNAs from complex environmental samples[END_REF].

Environmental samples, RNA extraction, and cDNA synthesis

Grassland and forest soil, as well as deadwood samples from six different contrasted geographic sites in Italy and France, were included in this study (Supplemental Table S2). Four of them had already been described in [START_REF] Adamo | Metabarcoding on both environmental DNA and RNA highlights differences between fungal communities sampled in different habitats[END_REF] and one in [START_REF] Bragalini | Solution hybrid selection capture for the recovery of functional full-length eukaryotic cDNAs from complex environmental samples[END_REF].

RNA from soil samples was extracted from 2 g of material using the RNA Power Soil extraction kit from MOBIO Laboratories (Carlsbad, CA, USA) according to the manufacturer's instructions. Deadwood RNA was extracted from 100 mg of decaying wood following the protocol described in [START_REF] Adamo | RNA extraction from decaying wood for (Meta)transcriptomic analyses[END_REF]. Eukaryotic cDNAs were synthesized from 2 μg of total environmental RNA using the "Mint-2 cDNA synthesis and amplification kit" according to the manufacturer's instructions (Evrogen, Moskow, Russian Federation). First-strand synthesis was initiated at the mRNA poly-A end using a modified poly-dT primer (CDS-4 M). All resulting cDNAs were bordered at their 5′ and 3′ extremities by the same M1 sequence (5′-AAG CAG TGG TAT CAA CGC AGAGT-3′) that can be used for priming cDNA PCR amplification.

Gene capture by hybridization

DyP cDNA sequence capture was performed as already described in [START_REF] Denonfoux | Gene capture coupled to high-throughput sequencing as a strategy for targeted metagenome exploration[END_REF] and [START_REF] Bragalini | Solution hybrid selection capture for the recovery of functional full-length eukaryotic cDNAs from complex environmental samples[END_REF]. Briefly, 2 μg of heat-denatured cDNA were hybridized to 500 ng of a mix of biotinylated RNA probes for 24 h at 65 °C in microcentrifuge tubes. The probes/DyP cDNA hybrids were then captured on streptavidin-coated paramagnetic beads (Dynabeads M-280 Streptavidin, Invitrogen, Waltham, MA USA). After several washing steps to eliminate unbound cDNAs, the captured cDNAs were detached from the beads using 0.1 M NaOH and purified using the "Qiaquick PCR purification kit" (Qiagen, Hilden, Germany). Captured cDNAs were further amplified using "Encyclo Plus PCR kit" and the M1 primer and were purified using the "Qiaquick PCR purification kit." Captured cDNA was then subjected to the second round of capture identical to the first one. The enrichment in DyP sequences along the capture protocol was evaluated via a semi-quantitative PCR using DyP-specific primers [START_REF] Bragalini | Solution hybrid selection capture for the recovery of functional full-length eukaryotic cDNAs from complex environmental samples[END_REF][START_REF] Kellner | Widespread occurrence of expressed fungal secretory peroxidases in forest soils[END_REF].

Sequencing of captured cDNAs and bioinformatics analysis of the sequencing data

Since known fungal DyP CDSs range in size from 1500 to 2000 bp, captured cDNA was the first size fractionated to sequence only putatively full-length cDNAs and to eliminate truncated and contaminating sequences. Fractionation of 1 µg amplified cDNA (between 1400 and 2000 bp) was performed using a "BluePippin" instrument (Sage Science, Beverly, MA, USA). Fractionated cDNA was quantified by fluorimetry using the "Qubit dsDNA HS Assay Kit" and "Qubit Fluorimeter 2.0" (Thermo Fisher Scientific, Waltham, MA USA) and re-amplified by PCR using the M1 primer and "KAPA HiFi Taq polymerase" (KAPA Biosystems, Wilmington, MA USA). The 50 μl PCR mix contained 10 ng of cDNA, 10 μl of 5X KAPA HiFi Fidelity Buffer, 1.5 μl of 10 mM dNTPs, 10 μl of the 10 μM concentrated M1 primer, and 1 U of KAPA HiFi Taq polymerase. After an initial denaturation at 94 °C for 3 min, cDNA fragments were amplified for 25 cycles comprising 20 s at 98 °C, 15 s at 66 °C, and 1 min at 72 °C. Amplification was terminated by a final elongation at 72 °C for 3 min. The "Agencourt® AMPure® XP kit" (Beckman Coulter, Brea, CA USA) was used to perform a final purification and partial size fractionation of the captured cDNA. A 0.5X concentration of beads was used in order to remove small unspecific fragments from the eluted material. Purified cDNA quantity was assessed using the "Qubit dsDNA HS Assay Kit" and "Qubit Fluorimeter 2.0", while DNA purity was assessed by measuring the OD 260/OD 280 and OD 260/OD 230 ratios using a NanoDrop TM 2000 spectrophotometer (Thermo Fisher Scientific, Waltham, MA USA).

All captured cDNA samples were sequenced using the Illumina HiSeq 2000 2 × 250 bp technology (I.G.A. Technologies, Udine, Italy) using the Illumina MiSeq 2 × 250 bp standard protocol for samples preparation.

Adapter sequences were eliminated using Cutadapt (Martin 2011). Sequence quality was evaluated with Trimmomatic [START_REF] Bolger | Trimmomatic: a flexible trimmer for Illumina sequence data[END_REF]; bases with quality lower than 20 were eliminated, and only sequences longer than 60 bases were kept. Trimmed sequences were assembled using IDBA-UD [START_REF] Peng | IDBA-UD: a de novo assembler for single-cell and metagenomic sequencing data with highly uneven depth[END_REF]) and the default parameter. To obtain longer contigs, the resulting contigs were then assembled using CAP3 and default parameters (Huang 1999). For the identification of DyP sequences, a similarity search between known fungal DyP sequences and contigs was performed using DIAMOND v0.9.22.123 [START_REF] Buchfink | Fast and sensitive protein alignment using DIAMOND[END_REF], with the BLASTx command in "sensitive" mode (i.e. a maximal e-value of 1.e -05 and a minimal identity of 50%). Matching cDNA sequences were translated using the Expasy [START_REF] Gasteiger | ExPASy: the proteomics server for in-depth protein knowledge and analysis[END_REF]) translate tool and further analyzed for the presence of the DyP peroxidase domain using ScanProSite and the Prosite database [START_REF] De Castro | ScanProsite: detection of PROSITE signature matches and ProRule-associated functional and structural residues in proteins[END_REF][START_REF] Sigrist | New and continuing developments at PROSITE[END_REF]. Identified DyP proteins were clustered using a 90% identity threshold using CD-HIT v4.7 [START_REF] Fu | CD-HIT: accelerated for clustering the next-generation sequencing data[END_REF]. Sequences are available on NCBI with the accession numbers OM674899-OM674912.

Results

Global distribution of DyP genes in the fungal kingdom (Eumycota)

Among the 227 fungal genomes inspected, 454 DyP-coding genes were identified in 162 species belonging exclusively to the Dikarya (Ascomycota and Basidiomycota; Supplemental Table S3) and not in any other fungal taxonomic groups. In the Ascomycota, 134 DyP genes were identified in the genomes of 84 species, all from the Pezizomycotina and none from the Saccharomycotina and Taphrinomycotina. For Basidiomycota, DyP genes were absent from the 1 3 early-branching clades of Ustilagomycotina (subphylum of smut fungi), and present in the Pucciniomycotina (rust fungi) and Agaricomycotina. In these latter groups, 306 sequences were identified in the genomes of 78 species (34% of inspected species). Predicted numbers of putative DyPencoding genes per genome differed greatly, ranging from 0 to 1 up to 64 genes in the case of the basidiomycetous species Sphaerobolus stellatus (which is remarkable since this fungus contains also 78 genes of unspecific peroxygenases/ UPOs, EC 1.11.2.1; [START_REF] Hofrichter | Fungal peroxygenases: a phylogenetically old superfamily of heme enzymes with promiscuity for oxygen transfer reactions[END_REF]. In Ascomycota, the number of predicted DyP genes never exceeded six per genome (e.g. Paraconiothyrium sporulosum). A detailed list of DyP genes identified in the inspected species is given in Supplemental Table S4.

Evolution of fungal DyP proteins

DyP amino acid sequences were used to compute a sequence similarity network, which individualized seven well-separated clusters of sequences (Fig. 1) using a pairwise alignment score of 110. Following a phylogenetic analysis, each cluster defined one single monophyletic clade, with the exception of cluster VI-1, from which emerged clade VI-2 (Fig. 2). From the taxonomic point of view, while clade VI-1 encompassed sequences from both Basidiomycota and Ascomycota, other clades had a narrower distribution pattern as illustrated for clade II present in the plant pathogens belonging to the class Pucciniomycotina (with the exception of one sequence originating from the Agaricales Termitomyces sp.). When representative sequences of each of the seven fungal DyP clades were individually used in blast searches to identify their closest relatives among bacterial DyP sequences, the same set of bacterial sequences was identified. These bacterial sequences all belonged to group C of DyP sequences as defined by [START_REF] Zámocký | Independent evolution of four heme peroxidase superfamilies[END_REF]. A selection of these bacterial sequences was included in the fungal DyP phylogeny. They all clustered in a single basal clade (Fig. 2).

For each fungal protein sequence, we predicted a number of features that either reflected the global protein structure (GRAVY hydropathy index, isoelectric point (pI), sequence length, number of putative N-glycosylation sites, number of acidic amino acid residues at the putative Mn 2+ oxidation site), which could be related to catalytic activities as in the case of the number of aromatic residues (tyrosine, tryptophan, and phenylalanine), which can participate in long-range electron transfers (LRETs) (Supplemental Table S4). When incorporated in a PCA analysis, these different variables separated several of the proteins according to the clades they belonged to (Fig. 3). The first two axes of the PCA altogether explained 52% of the variation in the dataset. Clade VI-2 was particularly well separated from all other clades with significantly lower average pI values and significantly higher GRAVY indices (Fig. 2), as well as the number of putative N-glycosylation sites per polypeptide (Dunn's post hoc test, p < 0.05) (Supplemental Fig. S1). Furthermore, 69% of the proteins affiliated with clade VI-2 also possessed a predicted N-terminal signal peptide for secretion (Fig. 2). Outside of clade VI-2, only four proteins from Puccinia spp., belonging to clade II, also processed a putative signal peptide. Interestingly, 13 of the 16 purified and characterized fungal DyPs identified in the literature belong to clade VI-2; the three others were affiliated with clade VI-1. Most of these characterized proteins indeed appeared to be extracellular as they were purified from fungal culture filtrates presenting a dye-decolorizing activity (toward Reactive Blue5).

Inspection of amino acid residues present at positions essential for catalysis in the AauDyPI enzyme from A. auricula-judae [START_REF] Strittmatter | First crystal structure of a fungal highredox potential dye-decolorizing peroxidase substrate interaction sites and long-range electron transfer[END_REF]Linde et al. 2015b) highlighted differences between the different clades (Supplemental Fig. S2). Positions in the distal side of the heme pocket are well conserved across all the clades, except in the case of Clade III, where the aspartate residue is replaced by a glycine one (position 168 in AauDyPI), and phenylalanine frequently replaced by valine (position 359). In the case of the proximal side of the heme pocket, while the histidine and aspartate residues, implicated in Fe 3+ ligation as well as in peroxide-binding and cleavage, are well conserved across the clades, the central valine (position 253) residue present in the AauDyPI protein is frequently replaced by isoleucine, methionine, or phenylalanine residues, the latter two being predominant in clades I and IV, and III, respectively (Supplemental Fig. S2). Finally, the radical-forming residues tyrosine and tryptophan, probably involved in LRETs, present at positions 337 and 377 of the AauDyPI protein are conserved in all clades, except in clades VI-1 and VI-2, where they are frequently replaced by phenylalanine, another aromatic amino acid that can participate in LRETs [START_REF] Acebes | Mapping the longrange electron transfer route in ligninolytic peroxidases[END_REF].

DyPs and the evolution of trophic modes in the Agaricomycotina

We constructed a species phylogenetic tree that included 128 Agaricomycotina species and two species belonging to Ustilaginomycotina (as an outgroup) with fully sequenced genomes (Fig. 4), based on the tree published by [START_REF] Zhao | A six-gene phylogenetic overview of basidiomycota and allied phyla with estimated divergence times of higher taxa and a phyloproteomics perspective[END_REF]. According to [START_REF] Zhao | A six-gene phylogenetic overview of basidiomycota and allied phyla with estimated divergence times of higher taxa and a phyloproteomics perspective[END_REF], the topology of this tree reflected the time-dependent divergence between the main agaricomycetous orders. Mapping the occurrence of DyP genes and clusters on this consensus species tree showed that most major agaricomycetous DyP clades (II, III, V, and VI, see Fig. 2) diverged early during the evolution of this taxon, being already present in all the most basal taxa of the Agaricomycotina, such as Cantharellales, Sebacinales, Auriculariales, and Tremellales [START_REF] Hibbett | A higher-level phylogenetic classification of the fungi[END_REF]). The tree also illustrates the recurrent loss, but also duplication, of several of the DyP clades with a differential distribution among the main agaricomycetous orders. For example, current genomic data suggest the loss of DyP genes belonging to clades VI-1 and VI-2 in the ancestor of the Boletales, which seemingly only possess members of clade V.

DyP phylogeny did not match fungal species phylogeny. One single species could host genes belonging to up to three clades (as in the case of the agaric species Gymnopus luxurians or the polypore B. adusta that both possess DyPs from clades VI-1, V, and III), and the presence Fig. 3 Principal component analysis (PCA) separates fungal DyP protein sequences (dots) belonging to different sequence clusters (same color code as in Fig. 2) according to their structural characteristics (vectors: hydropathy index (GRAVY), theoretical isoelectric point (pI), number of predicted N-glycosylation sites (N-glyc), polypeptide length (aa), number of specific amino acid residues (acidic, F, W, Y)) of genes encoding DyPs representative of one clade never excluded the presence of a gene of another clade. We predicted the number of gene loss or gene gain events, using the NOTUNG software, separately for the most sequence-rich clades VI-1 and VI-2. Diversification at the level of the tree leaves (species-level) of clades VI-1 and VI-2 was not systematically coordinated between DyP clades (Fig. 4). To further evaluate the potential contribution of agaricomycetous DyP genes to organic matter conversion, and more specifically, to lignin degradation, we compared the prevalence of the different DyP clades in Agaricomycotina with respect to their trophic modes. The first observation from the cluster analysis is that clade VI-2 (putatively secreted extracellular DyPs) encompassed sequences exclusively from WR and OS species and none from BR or EM ones (Fig. 1). This distribution is specific to clade VI-2. Indeed, all other clades with agaricomycetous sequences (clades II, III, V, and VI-1) contained sequences from WR, OS, BR, and EM species (the latter being absent from clade III). Although clade VI-2 sequences are absent from nearly two-thirds of the species classified as WR, and more WR species have been sequenced compared to BR and EM ones (Fig. 4), Chi 2 tests strongly support (p < 0.01) the higher prevalence of clade VI-2 sequences in WR (or WR and OS) species than in BR and EM ones. On the contrary, we did not find evidence for a higher prevalence of other clades in groups of species presenting a specific trophic mode (WR, OS, BR, or EM; Chi 2 test, p < 0.5).

Diversity of fungal DyPs expressed in the environment

A set of 69 biotinylated degenerated RNA probes, whose sequences spanned the known phylogenetic diversity of fungal DyP sequences, was designed and used to capture homologous cDNAs synthesized from mRNA directly extracted from different environmental matrices (decaying wood, forest and grassland leaf-litter, and soils). DNA recovered after two successive rounds of capture (enrichment) was randomly sequenced, and the reads were successfully assembled to recover 14 full-length or nearly full-length DyP cDNAs. All of these environmental sequences encode different DyP not present in the database. They were clearly affiliated with either clade VI-2 (5 sequences) or clade VI-1 (9 sequences) (Fig. 2). In clade VI-1, three environmental DyP sequences were related to reference sequences from Ascomycota and six to sequences from Basidiomycota. We affiliated "cluster 4" to Basidiomycota, although also close to Ascomycota sequences in the phylogeny. In clade VI-2, three of the five protein sequences had a predicted N-terminal signal peptide.

Discussion

We provide an exhaustive and comprehensive view of DyPencoding genes distribution during the evolution of the Dikarya fungi, using all fungal genomes publicly available at the start of the study. Fungal (Eumycota) DyP coding genes of the clade D, as defined by Zamocky et al. (2015), are present in some, but not all, Dikarya species. Search for the most similar bacterial sequences and their placement in the fungal DyP phylogenetic tree suggest that fungal sequences may have originated from a single horizontal transfer of a bacterial class C gene to an ancestor of the Dikarya more than 400 My ago [START_REF] Taylor | Perithecial ascomycetes from the 400 million year old rhynie chert: an example of ancestral polymorphism[END_REF], followed by multiple independent events of gene loss, duplication and diversification during the course of evolution of this taxonomic group.

The access to a large (more than 450) number of fungal DyP sequences allowed the identification of distinct clades, defined on the basis of sequence similarities that diverged from each other at different periods during the evolution of the Dikarya. In the Agaricomycotina, and to a far lesser extent in the Pezzizomycotina, DyP diversification did not parallel taxa diversification since DyP genes belonging to different (up to three) clades can co-exist in the genome of a single extant fungal species. The four main DyP clades (III, V, VI-1, and VI-2) present in extant Agaricomycotina have thus followed different and apparently independent evolutionary trajectories, being independently eliminated, recruited, or amplified in specific lineages. It seems, for example, that clade VI-2 experienced more frequent episodes of both gene diversification and loss than clade VI-1. Consequently, clade VI-1 is more prevalent in extant agaricomycetous species compared to clade VI-2, but with an average lower number of copies per genome (1.5 versus 3.9). These independent evolutionary trajectories suggest that enzymes from each of the clades may participate in different functions and are under different selective pressures.

DyP clades, which form distinct clusters in network analyses, can also be, to some extent, distinguished from each other based on "global" protein characteristics (hydropathy indices, isoelectric points, absolute numbers of aromatic residues, and abundance of N-glycosylation sites or the presence/absence of an N-terminal signal peptide for secretion). As observed for other enzyme families, including enzymes implicated in biomass degradation [START_REF] Aspeborg | Evolution, substrate specificity and subfamily classification of glycoside hydrolase family 5 (GH5)[END_REF][START_REF] Viborg | A subfamily roadmap of the evolutionarily diverse glycoside hydrolase family 16 (GH16)[END_REF], these observations suggest that the divergence between DyPs could be associated with significant changes in catalytic properties, substrate range, and cellular functions [START_REF] Zallot | Discovery of new enzymatic functions and metabolic pathways using genomic enzymology web tools[END_REF]. It is, however, difficult to speculate on the nature of these changes since current structural (e.g. 3D structure) and functional information on fungal DyPs are fragmentary and with regard to related enzymes belonging exclusively to the phylogenetically-related clades VI-1 and VI-2. Enzymes belonging to this latter clade share a number of very distinctive structural features that clearly distinguish them from those of other clades. The most striking one is the presence of a sequence peptide that suggests secretion through the conventional endoplasmic reticulum pathway. Although SignalP may sometimes fail to predict signal peptides, we are confident that in our case, this should be a marginal problem as the presence of a signal peptide 1 3 was predicted for almost all of the sequences of clade VI-2. This hypothesis is further supported by the presence in clade VI-2 proteins of a higher number of putative N-glycosylation sites, a common feature of endoplasmic reticulum-secreted proteins [START_REF] Medus | N-glycosylation triggers a dual selection pressure in eukaryotic secretory proteins[END_REF]. Although it is tempting to hypothesize that the emergence of clade VI-2 corresponded to a transition from an intracellular to an extracellular status, this hypothesis would need further experimental validation. On the one hand, a majority of the biochemically characterized fungal DyPs have been purified from culture filtrates that precisely belong to clade VI-2 (see Fig. 2). On the other hand, novel DyPs also purified from culture filtrates belong to clade VI-1, from which emerged clade VI-2. Clade VI-1 is the only clade that encompasses both ascomycetous and basidiomycetous sequences, and one of the characterized enzymes was from the wood-rot Ascomycota X. grammica. It could thus be hypothesized that clade VI-1 DyPs are either intracellular or extracellular, secreted through nonconventional secretion systems as proposed for a rhamnosidase of Xylaria polymorpha and other fungal proteins (Nghi et al. 2012;[START_REF] Prudovsky | The non-classical export routes: FGF1 and IL-1α point the way[END_REF]. Acquisition or loss of structural elements favoring secretion of DyPs is also documented in the bacteria, where type A DyPs possess a Tat secretion signal not found in the other bacterial clades such as clade C from which fungal DyPs may derive [START_REF] De Gonzalo | Bacterial enzymes involved in lignin degradation[END_REF]. Secreted type A bacterial DyPs are suspected to be involved in chemical lignin modification [START_REF] Sugano | DyP-Type Peroxidases: Recent Advances and Perspectives[END_REF].

Besides clade VI (VI-1 and VI-2) DyPs, no functional information exists for DyPs belonging to other clades, including their putative cellular localization, although several, but not all, DyPs of rust fungi (Puccinomycotina) possess signal peptides. We suggest that future functional studies should focus on these other clades, not only to precise their cellular localization but also on their catalytic activities and substrate range to understand their cellular functions and evaluate their potential use as catalysts in biotechnology.

Regarding the most debated role of fungal DyPs, i.e. their contribution to lignin degradation, our correlative analyses suggest a facultative contribution of a subset of these enzymes in this process, as suggested in [START_REF] Nagy | Genetic bases of fungal white rot wood decay predicted by phylogenomic analysis of correlated gene-phenotype evolution[END_REF], who considered the fungal DyP family as a whole without splitting it in distinct clades with different evolutionary histories. We show a strong association between clade VI-2 and species of Agaricales classified as whiterotters or litter-decomposers (soil saprotrophs) that, for many of them, can degrade lignin and humic substances [START_REF] Steffen | Mineralisation of 14 C-labelled synthetic lignin and ligninolytic enzyme activities of litter-decomposing basidiomycetous fungi[END_REF][START_REF] Steffen | Degradation of humic acids by the litter-decomposing basidiomycete Collybia dryophila[END_REF]. Intuitively, enzymes active on bulky extracellular polymers need to be secreted as we hypothesized that they are in the case of clade VI-2 DyPs. This strong association contrasts with the complete absence of this clade from the genomes of either brown rot or ectomycorrhizal species sequenced thus far. In the case of the Boletales, a taxonomic group composed of BR and ECM species, the loss of clade VI-2 DyPs could even represent an ancestral characteristic. BR and ECM species can, however, possess DyPs belonging to other clades. If clade VI-2 DyPs participate in lignin degradation/modification, this role may be rather accessory as (i) many WR species lack these enzymes and (ii) species that possess clade VI-2 DyPs also possess class II peroxidases, including MnPs, VPs, and/or LiPs. Besides Agaricales, clade VI-2 DyPs are also present in saprotrophic or plant symbionts/endophytes of the orders Cantharellales/Sebacinales that are not considered WR species [START_REF] Nagy | Comparative genomics of early-diverging mushroom-forming fungi provides insights into the origins of lignocellulose decay capabilities[END_REF]) but possess a rich set of genes encoding plant cell-wall degrading enzymes in their genomes [START_REF] Kohler | Convergent losses of decay mechanisms and rapid turnover of symbiosis genes in mycorrhizal mutualists[END_REF]. Among these species is Sistotrema sp. (Cantharellales), sometimes considered a white-rot fungus although not possessing class-II peroxidases [START_REF] Riley | Extensive sampling of basidiomycete genomes demonstrates inadequacy of the white-rot/ brown-rot paradigm for wood decay fungi[END_REF].

Contribution of clade VI-1 enzymes to lignin degradation by the wood-rot ascomycete X. grammica has also been suggested as one corresponding enzyme purified from culture filtrates of this species possesses a Mn 2+ -oxidizing activity [START_REF] Kim | Characteristics of a newly isolated fungus, Geotrichum candidum Dec 1, which decolorizes various dyes[END_REF]. Incidentally, we observe that several other ascomycetous species possessing clade VI-1 DyPs are associated with decaying wood (e.g. Ascocoryne sp., Diaporthe ampelina, Eutypa lata, Kretzschmaria deusta, Rosellinia necatrix, and Xylaria spp.). These genera and species could be targeted to study the transcription profile of their DyPs when grown on lignocellulosic substrates, their (extra)cellular localization, and substrate specificity. Other ascomycetous species possessing clade VI-2 DyPs were however not associated with rotting wood, this is also the case for all ascomycetous species processing clade I or IV DyPs, the two other clades present in this phylum.

Regarding DyPs expressed in the environment, we successfully assembled 14 complete or nearly complete DyP sequences following sequence capture of cDNAs retrotranscribed from soil and decaying wood RNAs. All DyP sequences were affiliated to either clades VI-1 or VI-2 despite the fact that capture probes also targeted sequences from the other fungal DyP clades. Clades VI-1 and VI-2 sequences represent altogether about two-thirds of all DyP present in databases. The exclusive presence of clades VI-1 or VI-2 DyPs among environmental sequences suggests that fungal species harboring DyPs belonging to these clades are either overrepresented and/or that the corresponding genes are overexpressed in the studied environments. These observations are comforted by the data of [START_REF] Kellner | Widespread occurrence of expressed fungal secretory peroxidases in forest soils[END_REF], who used PCR to obtain partial DyP sequences from different forest soils. Affiliation of these 61 partial sequences to the different clades showed that 81% originated from clades VI-1 (42%) and VI-2 (24%), while a minority of the sequences came from clades I (4 sequences), III (6), and V(1). The absence of environmental sequences that define novel DyP clades may signify that fungal genome sequencing may have already captured the entire phylogenetic diversity of the DyP D subfamily or those additional clades are too divergent to be captured by the designed probes or too rare to be easily identified using this approach. The occurrence of expressed clades VI-1 and VI-2 DyPs, from both basidiomycetous and ascomycetous taxa in organic matter (OM)-rich environments (leaf-litter, uppermost soil) reinforces our hypothesis for their involvement in OM degradation and at the same time should incite us to characterize the catalytic properties of the more enigmatic and somewhat structurally divergent members of clades I to V.

Fig. 1

 1 Fig. 1 Sequence-similarity clustering of DyP protein sequences retrieved from fully sequenced fungal genomes and of biochemically-characterized fungal DyPs. Clustering using the EFI-EST server identified seven separate clusters (I -VI-1 and VI-2). Symbol shape indicates the presence (diamond) or absence (circle) of a predicted signal peptide. Symbol color indicates the trophic mode of the corresponding fungal species; red, white rot; green, brown rot or ectomycorrhizal; grey, other. Symbols with a yellow contour indicate biochemically-characterized enzymes

Fig. 2

 2 Fig. 2 Phylogenetic analysis of the fungal DyP peroxidase family based on the alignment of 301 protein sequences retrieved from fully-sequenced genomes, corresponding to biochemically characterized enzymes, or encoded by environmental cDNAs isolated by gene capture by hybridization. Branch colors correspond to each of the sequence clusters illustrated in Fig. 1. The clade with black branches corresponds to bacterial sequences with the highest similarity values to fungal protein sequences. Grey dots indicate the bootstrap sup-

Fig. 4

 4 Fig. 4 Phylogenetic tree of 128 fully-sequenced Agaricomycotina species and two Ustilaginomicotina (used as outgroup) computed based on a six genes sequence alignment according to Zhao et al. (2017). Species names are colored according to the order they belong to. Colored dots associated with species names indicate their trophic mode, as reported in the legend. The number of predicted

Acknowledgements

We would like to acknowledge Dr. Francesco Venice (University of Turin) for the suggestions in bioinformatics.

The study was performed in the framework of the PeroxiDiv project co-funded by the ANR (Agence Nationale de la Recherche, France; ANR-RF-2015-01) and the DFG (Deutsche Forschungsgemeinschaft, Germany). Martino Adamo was supported by a grant from the Italian Ministry of Education, University and Research (MIUR) and the CMIRA (Coopération et mobilité internationales Rhône-Alpes) program of region Région Auvergne-Rhône-Alpes. We are grateful to the Mésocentre Clermont Auvergne University and AuBi platform for providing computing resources.

Supplementary Information

The online version contains supplementary material available at https:// doi. org/ 10. 1007/ s00253-022-11923-0.

Author contribution RM, MA, HK, MG, PL, and MH conceived and designed the research. MA, EB, and LV conducted experiments. MA, PP, SCM, and RP analyzed data. MA, MG, and RM wrote the manuscript. All authors read and approved the manuscript.

Data availability

Sequences generated during the current study are available in the NCBI-nr repository, with accession numbers OM674899-OM674912. All accession numbers of the sequences analyzed during this study are included in this published article (Supplementary Table S4).

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