Eric Blayo
email: eric.blayo@imag.fr

David Cherel

Antoine Rousseau

Towards optimized Schwarz methods for the Navier-Stokes equations

Keywords: Domain decomposition, Optimized boundary conditions, Navier Stokes equations, Schwarz Alternating Method PACS 65N55, 76D05

This paper presents a study of optimized Schwarz domain decomposition methods for Navier-Stokes equations. Once discretized in time, optimal transparent boundary conditions are derived for the resulting Stokes equations, and a series of local approximations for these nonlocal conditions are proposed. Their convergence properties are studied, and numerical simulations are conducted on the test case of the driven cavity with two subdomains. It is shown that conditions involving one or two degrees of freedom can improve the convergence properties of the original algorithm.

Introduction

A number of domain decomposition approaches have already been proposed for the numerical resolution of Navier-Stokes equations on parallel computers. These approaches can be barely classified into a few main families: algebraic methods working on the discrete matrix system (e.g. [START_REF] Kumar | Purely algebraic domain decomposition methods for the incompressible Navier-Stokes equations[END_REF][START_REF] Ronquist | A domain decomposition solver for the steady Navier-Stokes equations[END_REF][START_REF] Brakkee | Domain decomposition for the incompressible Navier-Stokes equations: solving subdomain problems accurately and inaccurately[END_REF]), variational methods minimizing a misfit at the interfaces between subdomains (e.g. [START_REF] Gunzburger | An optimization-based domain decomposition method for the Navier-Stokes equations[END_REF][START_REF] Kong | An optimization-based domain decomposition method for numerical simulation of the incompressible Navier-Stokes flows[END_REF]), and Schwarz alternating methods. This last class of methods has the advantage of being quite non intrusive, since it does not require modifications in the original computation code except for the boundary conditions at the interface between subdomains. Moreover the Schwarz formulation is a natural framework for coupling different models, which makes it particularly relevant for a very large number of applications. This aspect is our main motivation for the present work, which can be seen as a step in the direction of coupling different models in fluid dynamics.

A number of previous works deal with Schwarz-type algorithms for Stokes, Navier-Stokes and Oseen (i.e. linearized Navier-Stokes) systems. They study either Dirichlet-Dirichlet [START_REF] Strikwerda | A domain decomposition method for incompressible flow[END_REF], Dirichlet-Neumann [START_REF] Xu | On non overlapping domain decomposition methods for the incompressible Navier-Stokes equations[END_REF], Neumann-Neumann [START_REF] Pavarino | Balancing Neumann-Neumann methods for incompressible Stokes equations[END_REF], or Robin-Robin [START_REF] Otto | A nonoverlapping domain decomposition method for the Oseen equations[END_REF][START_REF] Müller | A nonoverlapping domain decomposition method for the nonstationary Navier-Stokes problem[END_REF] algorithms. See also [START_REF] Toselli | Domain decomposition methods -Algorithms and theory[END_REF]. In this paper, our objective is to derive more general optimized conditions, leading to an efficient convergence. To achieve this, we will follow the well-known strategy (e.g. [START_REF] Japhet | The best interface conditions for domain decomposition methods: absorbing boundary conditions[END_REF][START_REF] Gander | Optimized Schwarz methods[END_REF]) which consists first in deriving exact transparent operators (which are pseudo-differential, hence unusable for practical applications) and second in approximating them by differential operators. Note that, using a different approach based on Smith factorization, [START_REF] Dolean | Deriving a new domain decomposition method for the Stokes equations using the Smith factorization[END_REF] have also proposed an efficient domain decomposition algorithm for Stokes and Oseen systems. The non dimensional Navier-Stokes system in a domain Ω with boundary ∂Ω reads:

     ∂v ∂t + v∇v - 1 Re ∆v + ∇p = 0 in Ω div v = 0 in Ω v = g on ∂Ω ( 1
)
where Re is the Reynolds number. In the present work, we will not consider waveform relaxation (or, in other words, global-in-time) domain decomposition methods, but will start with a time discretization of the Navier-Stokes system:

         v n+1 -v n ∆t + v n ∇v n - 1 Re ∆v n+1 + ∇p n+1 = 0 in Ω (2a) div v n+1 = 0 in Ω (2b) v n+1 = g n+1 on ∂Ω. (2c)
Then one has to solve at each time step a Stokes problem:

     1 ∆t v - 1 Re ∆v + ∇p = f in Ω div v = 0 in Ω v = g on ∂Ω (3)
For the sake of simplicity, we will consider the 2-D x -z case in this work. This paper is structured as follows. In Section 2, we will recall the framework of Schwarz algorithms for domain decomposition, and derive transparent boundary conditions following the methodology described in the seminal papers [START_REF] Engquist | Absorbing boundary conditions for the numerical simulation of waves[END_REF][START_REF] Halpern | Artificial boundary conditions for incompressible viscous flows[END_REF][START_REF] Halpern | Artificial boundary conditions for incompletely parabolic perturbations of hyperbolic systems[END_REF]. A series of approximations for these transparent conditions will be proposed in Section 3, and their convergence properties will be studied. Then some numerical simulations on the well known test case of the driven cavity decomposed into two subdomains will be described in Section 4.

Schwarz algorithm and theoretical aspects

In this section, we set the domain decomposition method (DDM) framework of this study, namely the iterative Schwarz algorithm (see [START_REF] Schwarz | Über einen Grenzübergang durch alternierendes Verfahren[END_REF][START_REF] Lions | On the Schwarz alternating method I[END_REF][START_REF] Lions | On the Schwarz alternating method III: a variant for nonoverlapping subdomains[END_REF]). We introduce the analytical expression of the errors (evolving with the iterative process), and provide a generic expression for the convergence rate. We draw the reader's attention to the fact that no specific interface condition is introduced in this section (the operators S j below remain undefined). Naturally, these conditions are required for the actual implementation of the algorithm, and will be fully determined in Section 3.

DDM framework and equations for the errors

The computational domain Ω is split in two subdomains: Ω = Ω 1 ∪ Ω 2 , with possible overlapping. Their internal boundaries, denoted Γ 1 and Γ 2 , are defined by Γ j = ∂Ω j ∩ Ωj j = 1, 2 where j = 3 -j

In the particular case of non overlapping subdomains, Γ 1 = Γ 2 = Γ . On each of the two subdomains Ω j , we look for the solution of the following system of equations:

           1 ∆t v m j - 1 Re ∆v m j + ∇p m j = f in Ω j (4a) div v m j = 0 in Ω j (4b) v m j = g on ∂Ω j \Γ j (4c) IBC on Γ j , (4d)
where IBC stands for interface boundary conditions. Although we postpone the complete definition of these conditions to Section 3, we will use the generic conditions of the form

-σ m j • n j -S j v m j = -σ m-1 j • n j -S j v m-1 j on Γ j , (5)
where σ m j = 1/Re∇v m j -p m j Id is the stress tensor1 in the domain Ω j , n j is the outward normal vector and S j is a generic operator, to be defined later on. These conditions, thanks to the convergence properties analyzed in Section 3.2 below, ensure that both velocities and fluxes are continuous through the interfaces Γ 1 and Γ 2 at the end of the iterative process. In the system of equations ( 4), the integer m denotes the iteration index in the Schwarz algorithm. Since the right-hand side f does not depend on m, the DDM error is solution of the following system:

                 1 ∆t e m j - 1 Re ∆e m j + ∇q m j = 0 in Ω j (6a) div e m j = 0 in Ω j (6b) e m j = 0 on ∂Ω j \Γ j (6c) -σ m j • n j -S j e m j = -σ m-1 j • n j -S j e m-1 j on Γ j , (6d)
where e j = (e u j , e w j ) = v |Ω j -v j and q j = p |Ω j -p j denote the velocity and pressure errors in Ω j , and where σ m j deals now with the error fields.

Analytical expression of the error in infinite domains

In order to compute approximate solutions of equations ( 6), we assume that both domains (Ω j ) j=1,2 are semi-infinite in x, and infinite in z: Ω 1 = (-∞, L) × R and Ω 2 = (0, +∞) × R, with L ≥ 0. The system of equations ( 6) becomes

                 1 ∆t e m j - 1 Re ∆e m j + ∇q m j = 0 in Ω j (7a) div e m j = 0 in Ω j (7b) e m j → 0 as (x, z) → ±∞ (7c) -σ m j • n j -S j e m j = -σ m-1 j • n j -S j e m-1 j on Γ j , (7d)
with

Γ 1 = {L} × R and Γ 2 = {0} × R.
We now proceed to a Fourier transform in the vertical direction, denoting by k the frequency variable associated to the space variable z, and f the Fourier transform of any function f . The system of equations ( 7) becomes:

                         1 ∆t êm j - 1 Re d 2 êm j dx 2 -k 2 êm j +    dq m j dx ik qm j    = 0 in Ω j (8a) dê u,m j dx + ikê w,m j = 0 in Ω j (8b) êm j → 0 when |x| → ∞ (8c) -σm j • n j -Ŝj êm j = -σm-1 j • n j -Ŝj êm-1 j on Γ j . (8d)
As in [START_REF] Halpern | Artificial boundary conditions for incompletely parabolic perturbations of hyperbolic systems[END_REF] and [START_REF] Martin | Schwarz waveform relaxation algorithms for the linear viscous equatorial shallow water equations[END_REF], we look for solutions of this system under the generic form:

E m j := (ê m j , qm j ) T (x) = Φ m j e ξx .
Introducing this expression in (8), we have the following system:

   1 ∆t + k 2 -ξ 2 Re 0 ξ 0 1 ∆t + k 2 -ξ 2 Re ik ξ ik 0    Φ m j = 0. (9)
A necessary and sufficient condition to obtain a non-zero solution to system (9) is to have a singular matrix, which leads to four possible values for ξ:

ξ 1 = λ, ξ 2 = |k|, ξ 3 = -λ and ξ 4 = -|k|,
where

λ = Re ∆t + k 2 . ( 10
)
The solutions of (9) are thus linear combinations of four modes

E m j = 4 l=1 γ m j,l Φ l e ξ l x ,
where (Φ l ) 1≤l≤4 are the eigenvectors (corresponding to eigenvalue 0) associated with each of the ξ l :

Φ 1 =   ik -λ 0   , Φ 2 =   -|k| -ik 1/∆t   , Φ 3 =   ik λ 0   and Φ 4 =   |k| -ik 1/∆t   .
Thanks to the condition (8c), since ξ 1 , ξ 2 > 0 and ξ 3 , ξ 4 < 0, only two modes are retained in the expression of E m j in each of the subdomains. We have:

E m 1 (x) = 2 l=1 γ m 1,l Φ l e ξ l x and E m 2 (x) = 4 l=3 γ m 2,l Φ l e ξ l x . (11)
Equipped with these expressions for the errors E m j , we now want to make use of interface conditions (8d) to study the corresponding convergence rates.

Theoretical convergence rates

We first simplify the expressions of errors by expressing the pressure error term thanks to the vertical component of equation (8a)

qm j = i k 1 ∆t + k 2 Re êm j - 1 Re d 2 êm j dx 2 • (0, 1) t .
The interface boundary conditions (8d) on Γ 1 (x = L) and Γ 2 (x = 0) now reads:

M d 2 êm 1 dx 2 + N dê m 1 dx + P êm 1 -Ŝ1 êm 1 = M d 2 êm-1 2 dx 2 + N dê m-1 2 dx + P êm-1 2 -Ŝ1 êm-1 2 (12)
-M d 2 êm 2 dx 2 -N dê m 2 dx -P êm 2 -Ŝ2 êm 2 = -M d 2 êm-1 1 dx 2 -N dê m-1 1 dx -P êm-1 1 -Ŝ2 êm-1 1 , (13)
with the matrices M , N and P defined as follows

M = - 1 Re 0 i k 0 0 , N = - 1 Re I 2 and P = - 1 Re 0 -i k λ 2 0 0 .
Thanks to [START_REF] Hoang | Space-time domain decomposition methods for mixed formulations of flow and transport problems in porous media[END_REF]

with

M 12 = ik -|k| -λ -ik , e ξ 12 x = e ξ 1 x 0 0 e ξ 2 x and γ m 12 = γ m 1,1 γ m 1,2 , M 34 = ik |k| λ -ik , e ξ 34 x = e ξ 3 x 0 0 e ξ 4 x and γ m 34 = γ m 2,3 γ m 2,4
.

Then ( 12) and ( 13) can be rewritten as Ĥ11 γ m 12 = Ĥ12 γ m-1

34 and Ĥ22 γ m 34 = Ĥ21 γ m-1 12 where Ĥ11 = M M 12 ξ 2 1 0 0 ξ 2 2 + N M 12 ξ 1 0 0 ξ 2 + P M 12 -Ŝ1 M 12 e ξ 12 L Ĥ12 = M M 34 ξ 2 3 0 0 ξ 2 4 + N M 34 ξ 3 0 0 ξ 4 + P M 34 -Ŝ1 M 34 e ξ 34 L Ĥ22 = -M M 34 ξ 2 3 0 0 ξ 2 4 -N M 34 ξ 3 0 0 ξ 4 -P M 34 -Ŝ2 M 34 Ĥ21 = -M M 12 ξ 2 1 0 0 ξ 2 2 -N M 12 ξ 1 0 0 ξ 2 -P M 12 -Ŝ2 M 12 .
We thus obtain the following recursive property for the errors:

γ m 12 = Ĥ-1 11 Ĥ12 Ĥ-1 22 Ĥ21 γ m-2 12 ( 15
)
which means that the growth rate of the error is directly related to the spectral radius of the amplification matrix

Ĥ = Ĥ-1 11 Ĥ12 Ĥ-1 22 Ĥ21 . ( 16
)
More precisely, the convergence rate of the Schwarz algorithm is equal to the square root of this radius in case of an additive version (as written in ( 12)-( 13)-( 15)), and to the radius itself in case of a multiplicative version (since index m -1 is then replaced by m in the r.h.s. of [START_REF] Japhet | The optimized order 2 method: application to convectiondiffusion problems[END_REF], which means that m -2 is replaced by m -1 in ( 15)). In order to ensure the convergence of the Schwarz algorithm for all possible frequencies k ∈ [k min , k max ], we have then to choose appropriately the operator S j such that (e.g. [START_REF] Gander | Optimized Schwarz methods[END_REF][START_REF] Japhet | The best interface conditions for domain decomposition methods: absorbing boundary conditions[END_REF][START_REF] Japhet | The optimized order 2 method: application to convectiondiffusion problems[END_REF])

ρ( Ĥ) = max k∈[k min ,k max ] max µ∈Sp( Ĥ) |µ(k)| < 1. (17)
The objective of Section 3 below is to specify the operator S i that should be used in numerical simulations in order to ensure the well-posedness of (8) and to cancel (or minimize) the spectral radius of the amplification matrix Ĥ.

Transparent boundary conditions and their approximations

This section focuses on the interface boundary conditions that should be used in Equation (8d). More specifically, we will look for operators S i that will be both efficient in term of convergence rate, and numerically tractable.

Exact transparent conditions

It is easy to check that the right-hand sides of equations ( 12) and ( 13), thus of (8d), cancel for

Ŝ * 1 = M M 34 ξ 2 3 0 0 ξ 2 4 M -1 34 + N M 34 ξ 3 0 0 ξ 4 M -1 34 + P Ŝ * 2 = -M M 12 ξ 2 1 0 0 ξ 2 2 M -1 12 -N M 12 ξ 1 0 0 ξ 2 M -1 12 -P
In that case, êm j is the solution of the homogeneous system (8), hence zero, so that the boundary operators S * j are perfectly transparent. We can simplify the preceding expressions, which become:

Ŝ * 1 = 1 Re    λ |k| (λ + |k|) - ik |k| λ ik |k| λ λ + |k|    , Ŝ * 2 = 1 Re    λ |k| (λ + |k|) ik |k| λ - ik |k| λ λ + |k|    (18)
An inverse Fourier transform of these operators would lead to non-local pseudodifferential operators, which would not be numerically tractable. In the sequel, we thus want to approximate them by local operators S app j , that will be called absorbing operators.

Approximate boundary conditions

As indicated above, our present objective is to derive approximations of the S * i s that are easy to implement and that lead to a rapid convergence of the Schwarz algorithm. "Easy" means that the S * i s must be local low-order operators, therefore limited to first-order and second-order derivatives. The matrix coefficients in the approximations of the Ŝ * i s will then be degree two polynomials in ik.

Empirical description of the operators Ŝ * i

As a first step in this direction, we are going to describe simple properties of Ŝ * 1 and Ŝ * 2 that will be useful to design relevant approximations.

Sign, symmetry and parity Given the expressions [START_REF] Martin | Schwarz waveform relaxation algorithms for the linear viscous equatorial shallow water equations[END_REF], obvious observations are that (i) the anti-diagonal terms are opposite and are odd functions of k, and that (ii) the diagonal terms are positive and even functions of k. Preserving these properties when building the approximations leads to the following general forms:

Ŝapp 1 = a 0 -a 2 k 2 -a 1 ik a 1 ik b 0 -b 2 k 2 and Ŝapp 2 = a 0 -a 2 k 2 a 1 ik -a 1 ik b 0 -b 2 k 2 (19)
where the a i and b i coefficients are real numbers to be determined, such that

a 0 -a 2 k 2 ≥ 0 and b 0 -b 2 k 2 ≥ 0.
Orders of magnitude The quantities involved in ( 18) are λ = Re ∆t + k 2 and k. We have thus to compare Re/∆t to k 2 to get insights into the orders of magnitude of the different terms in the Ŝ * i s. The spatial frequencies k that are resolved by the numerical model lie in the interval π H , π ∆z [START_REF] Gander | Optimized Schwarz methods[END_REF]. The problem being nondimensional, H is equal to 1 and 1/∆z is equal to N , the number of grid cells in the z-direction. This implies that [k min , k max ] = [π, N π]. Considering a "standard" non-dimensional Navier-Stokes numerical model, the CFL ratio ∆z/∆t remains almost constant, independently of N , and typically of the order of 10 0 -10 2 . Let denote this ratio by V grid . We have then 1/∆t = V grid /∆z = N V grid , which leads to Re/∆t

k 2 ∈ Re/∆t k 2 max , Re/∆t k 2 min = Re V grid π 2 1 N , Re V grid π 2 N
In actual applications, Re is of the order of 10 2 -10 4 , and N may vary from a few tens to a few thousands. This implies that Re/∆t = N Re V grid is always much larger than k 2 min = π 2 , and is also in most cases larger than k 2 max = N 2 π 2 , except for low Reynolds numbers and relatively high resolution. This means that ε = |k| ∆t/Re is in most cases a small parameter. We have then

λ = Re ∆t (1 + ε 2 ) = Re ∆t 1 + ε 2 2 + O(ε 4 ) (20)
which leads to

Ŝ1 , Ŝ2 = 1 Re |k|( 1 ε 2 + 1 ε + 1 + ε 2 ) ∓ik( 1 ε + ε 2 ) ±ik( 1 ε + ε 2 ) |k|( 1 ε + 1 + ε 2 ) + o(ε) (21)
This Taylor expansion highlights the fact that the upper-left coefficient in the Ŝ * i s is often clearly dominant, since it is one order of magnitude larger than the other coefficients. This observation is illustrated in Figure 1, where the modules of the matrix coefficients are plotted for a few values of Re, V grid and N , corresponding to different behaviors (cf Table 1). The future approximations Ŝapp

= a 0 0 0 0 Ŝ-2,2 i = a 0 -a 2 k 2 0 0 0 a 0 ≥ 0 a 0 -a 2 k 2 ≥ 0 on [k min , kmax]
expansion at order 1/ε degree 0 degree 1 degree 2 polynomial polynomial polynomial

Ŝ-1,0 i = a 0 0 0 b 0 Ŝ-1,1 i = a 0 ∓a 1 ik ±a 1 ik b 0 Ŝ-1,2 i = a 0 -a 2 k 2 ∓a 1 ik ±a 1 ik b 0 -b 2 k 2 a 0 , b 0 ≥ 0 a 0 , b 0 ≥ 0 a 0 -a 2 k 2 , b 0 -b 2 k 2 ≥ 0 on [k min , kmax]

A series of approximations

As indicated above, a general admissible form for the approximate operators Ŝapp i is given by [START_REF] Müller | A nonoverlapping domain decomposition method for the nonstationary Navier-Stokes problem[END_REF]. Rather than looking globally for optimal values of the five coefficients a i and b i , we can propose a series of approximations of increasing complexity, based on the Taylor expansion [START_REF] Otto | A nonoverlapping domain decomposition method for the Oseen equations[END_REF].

-At lowest order in ε, i.e. 1/ε 2 , the upper-left coefficient is the only non zero element in Ŝapp 2).

-At order -1 in ε, the expansion [START_REF] Otto | A nonoverlapping domain decomposition method for the Oseen equations[END_REF] indicates that the four elements of Ŝ * i must be taken into account. This means that the Ŝapp This series of approximate operators is summarized in Table 2.

Theoretical convergence performances

Our goal is to design efficient interface operators, that will lead to a fast convergence of the Schwarz algorithm. As seen in section 2.3, the convergence rate is given by the spectral radius of the amplification matrix [START_REF] Lions | On the Schwarz alternating method I[END_REF]. The analytical expressions of the eigenvalues of this 2 × 2 matrix can be determined for the different approximate boundary operators, as functions of the coefficients a i and b i . However these expressions, that we obtained using a symbolic computation software, are quite long and complicated. Therefore we will not give their detailed expression (except in the very simple case of approximation Ŝ-2,0 i with no overlap), but rather comment on the smallest values of the spectral radius ρ defined in [START_REF] Lions | On the Schwarz alternating method III: a variant for nonoverlapping subdomains[END_REF] that can be obtained after minimization w.r.t. the a i and b i coefficients.

Approximations at order -2 in ε For the approximation Ŝ-2,0 i = a 0 0 0 0 , and in the non overlapping case, the eigenvalues are 1 and

λ (λ 3 -k 2 |k|) + a 0 Re |k| (k 2 -λ 2 ) -λ (λ 3 -k 2 |k|) + a 0 Re |k| (k 2 -λ 2 ) 2 (22)
The eigenvalues for the approximation Ŝ-2,2 i are of course the same, but replacing a 0 by a 0 -a 2 k 2 . An important result here is that one of the eigenvalues is equal to 1, which indicates that the Schwarz algorithm will not converge in this case where the subdomains do not overlap. The explanation for this is actually quite simple. As a matter of fact, when using approximation Ŝ-2,0 i , the conditions occurring in the Schwarz algorithm at the common interface are

       - 1 Re ∂u m 1 ∂x + p m 1 -a 0 u m 1 = - 1 Re ∂u m-1 2 ∂x + p m-1 2 -a 0 u m-1 2 (23a) - 1 Re ∂w m 1 ∂x = - 1 Re ∂w m-1 2 ∂x (23b)
and

       1 Re ∂u m 2 ∂x -p m 2 -a 0 u m 2 = 1 Re ∂u m-1 1 ∂x -p m-1 1 -a 0 u m-1 1 (24a) 1 Re ∂w m 2 ∂x = 1 Re ∂w m-1 1 ∂x (24b)
Due to the change of sign between (23a) and (24a), the first condition will lead to a change from one iteration to the other. But, conversely, the second conditions (23b) and (24b) are similar for both subdomains. Therefore, in the case of a non overlapping configuration, the quantity ∂w/∂x does not evolve, which prevents the algorithm from converging. If the approximation Ŝ-2,2 i is used instead of Ŝ-2,0 i , the conditions (23b) and (24b) remain unchanged, hence the non convergence. In summary, these approximations Ŝ-2,0 i and Ŝ-2,2 i cannot be used without an overlap.

In the case of overlapping subdomains (δ > 0), the spectral radius can however become less than 1. As can be observed on Figure 2 for the approximation Ŝ-2,0 i , the minimal value of ρ( Ĥ) is always smaller than 0.5, so that we can expect a fast convergence of the Schwarz algorithm. Figure 2 is plotted for a 0 ≥ 0, consistently with the constraint expressed in Table 2. However we have also verified that choosing a 0 < 0 leads to poorer results, as will be showed for instance in Figure 3. Among our three sets of parameter values, the more favorable one is case #2, but this cannot be directly evidenced by the analytical expression of ρ( Ĥ) since its dependency with respect to a 0 , δ and the parameters listed in Table 1 is way too complicated. Note that increasing the overlap from δ = ∆x to δ = 2∆x slightly changes the optimal value of a 0 , and significantly improves the spectral radius at a low computational cost. It is interesting to compare the performance of these interface conditions w.r.t. simple Dirichlet conditions (v m j , p m j ) = (v m-1 j , p m-1 j

) on Γ j . In this latter case, the recursive formula for the errors ( 15) is replaced by

γ m 12 = Ĥdir γ m-2 12 with Ĥdir = e -ξ 12 δ M -1 12 M 34 e ξ 34 δ M -1 34 M 12 (25)
The spectral radius of Ĥdir can be easily computed, and we can see ( Fig. 2 Illustration of the theoretical convergence rate for approximation S -2,0 : spectral radius ρ( Ĥ) as a function of a 0 for δ = 0 (blue curves), δ = ∆x (green curves) and δ = 2∆x (red curves). The related configurations are detailed in Table 1.

Results obtained with the approximation Ŝ-2,2 i are displayed in Figure 3. For the sake of clarity, we only consider case #1 here, but the other cases were also performed, leading to similar conclusions. First of all, note that the horizontal section a 2 = 0 of course exactly corresponds to the spectral radius displayed in Figure 2 (left panel) for the approximation Ŝ-2,0 i . Moreover, there is clearly an "area of convergence" in the (a 0 , a 2 ) plane, where ρ( Ĥ) < 1. This area is fully consistent with the empirical constraint a 0 -a 2 k 2 ≥ 0 for k ∈ [k min , k max ] proposed in Section 3.2.1. As a matter of fact, this constraint corresponds to the intersection of the two half-planes a 0 -a 2 k 2 min ≥ 0 and a 0 -a 2 k 2 max ≥ 0, and the area of convergence matches almost exactly this intersection for δ = ∆x and δ = 2∆x. Finally, it appears that adding the second degree term a 2 k 2 in the polynomial approximation of the leading coefficient of Ŝi has almost no influence on the convergence properties within the area of convergence. As a matter of fact, the isolines of ρ( Ĥ) in this region are almost parallel to the vertical axis. Actually, its minimal value is 0.3048, obtained for (a opt 0 ; a opt 2 ) = (230.6; 3.6 10 -4 ), while the corresponding value of the convergence rate for approximation Ŝ-2,0 i was 0.3059, obtained for almost the same value a opt 0 = 229.9. This extremely weak influence of the a 2 term is observed in the 3 cases (see Table 3). We will come back in section 4 to the convergence rate observed in actual numerical simulations. ). As for the preceding approximations at order -2, we obtained these expressions using a symbolic computation software. As expected, approximation at order -1 performs clearly better than approximation at order -2. In the nonoverlapping case, 1 is no longer an eigenvalue, and the convergence rate may become smaller than 1.

-For approximation S -1,0 , the Schwarz algorithm without overlap converges for a wide range of values of a 0 and b 0 , including the quadrant a 0 , b 0 > 0, as can be seen in Figure 4 (note that, if a 0 and b 0 both become large enough, the interface conditions reduce to a simple Dirichlet-Dirichlet case, as can be seen in equations ( 12)-( 13)). The optimal spectral radius reached with this approximation S -1,0 is rather efficient: ρ opt = 0.733, 0.381 and 0.595 for the three test cases respectively (Table 3). Note that the ratio a opt 0 /b opt 0 between the optimal coefficients is fully consistent with the theoretical optimal ratio Ŝ * 11 (k)/ Ŝ * 22 (k) = λ/|k| = 1 + Re/(∆t k 2 ), which is approximately O(10 1 -10 2 ), O(10 0 -10 1 ) and O(10 0 -10 1 ) respectively for the 3 cases (Figure 4 and Appendix).

-Adding a small overlap clearly enhances the convergence, which becomes extremely fast in our first two test cases.

-The addition of the extra diagonal terms ±a 1 ik (approximation Ŝ-1,1 i ) slightly improves the spectral radius, at most by 10-15% in the most favorable case (Table 3). Note that, once again, the ratio a opt 0 /a opt 1 observed in our test cases is generally fully consistent with the corresponding theoretical ratio

k Ŝ * 11 (k)/ Ŝ * 12 (k) = λ + |k|, which is O(10 3 ), O( 10
2
) and O( 103 ) respectively for the 3 cases (see Appendix).

-Considering Ŝ-1,2 i , i.e. adding the a 2 and b 2 terms, makes the optimization much more complex, since one has now 5 parameters to consider. We did not Fig. 4 Illustration of the theoretical convergence rate for approximation S -1,0 . Top line: spectral radius ρ( Ĥ) as a function of a 0 (x-axis) and b 0 (y-axis) for δ = 0. Middle line: same configuration with δ = ∆x. Bottom line: same configuration with δ = 2∆x. The 3 columns correspond to the 3 configurations detailed in Table 1. perform a full optimization, but our attempts to add those two degrees of freedom in the vicinity of values a 0 , b 0 and a 1 that are optimal for Ŝ-1,1 i did not lead to any significant decrease in ρ opt (not shown). This seems to confirm the very weak influence of the a 2 term in approximation S -2,2 w.r.t. S -2,0 that we observed previously.

Discussion From the preceding investigations, we can already draw some first conclusions:

-Approximations Ŝ-k,1 i and Ŝ-k,2 i bring very few improvement w.r.t. Ŝ-k,0 i (k = -2, -1). On the opposite, Ŝ-1,0 i leads to much better results than Ŝ-2,0 i . The approximation Ŝ-1,0 i appears thus as a good overall compromise.

-If ones does not want the subdomains to overlap, then Ŝ-1,p i must be used since Ŝ-2,p i cannot lead to convergence. On the opposite, a small overlap, if affordable, generally accelerates very much the convergence.

-The different forms of matrix Ŝapp i were proposed using a Taylor expansion that is valid if ε is small. However, even if this is not the case, these matrix forms can still be used and their coefficients can be optimized in order to minimize ρ( Ĥ) efficiently. Test case #3 is an illustration of such a situation.

Numerical simulations

The objective of this section is to provide a numerical illustration of the theoretical results presented in Section 3. We first set the numerical framework (time-space finite differences for the lid-driven cavity, see [START_REF] Bruneau | The 2D lid-driven cavity problem revisited[END_REF] for details), and present in the sequel the impact of our optimized boundary conditions on the domain decomposition method used for the driven cavity.

Numerical framework

Driven cavity

The driven cavity problem is well known, since it has been investigated by many authors (see for instance [START_REF] Bruneau | The 2D lid-driven cavity problem revisited[END_REF] and the numerous references herein). The computational domain is a vertical 2-D square domain Ω = (0, 1) × (0, 1), in which the incompressible Navier-Stokes equations (1) are solved, with no-slip boundary conditions on 3 sides of the cavity and a driving velocity on the top of the cavity:

v(0, z, t) = v(1, z, t) = v(x, 0, t) = 0, 0 ≤ x, z ≤ 1, t > 0 v(x, 1, t) = (1, 0), 0 ≤ x ≤ 1, t > 0.
The simulation is started from rest: v(x, z, 0) = 0, 0 ≤ x, z ≤ 1.

Discretization method Discretization in time is performed using a first order Euler scheme, leading to the semi-discrete system [START_REF] Bruneau | The 2D lid-driven cavity problem revisited[END_REF]. Discretization in space is performed on a uniform staggered grid (see Figure 5) using a third order scheme for the nonlinear term, as described in [START_REF] Bruneau | The 2D lid-driven cavity problem revisited[END_REF]. Numerical experiments that will be presented in the following were performed at Re = 10 4 , with ∆x = ∆z = 1/256, ∆t = ∆z/10 which corresponds to the case # 1 in Section 3 and Table 1. For those values of the parameters, the stationary flow exhibits a large primary clockwise vortex (see Figure 5), with secondary vortices in the corners (counter clockwise or clockwise, depending on the order in the well-known vortices cascade). Our numerical model was validated by comparison to corresponding results presented in the literature (e.g. [START_REF] Bruneau | The 2D lid-driven cavity problem revisited[END_REF][START_REF] Erturk | Discussions on driven cavity flow[END_REF]).

Optimized boundary conditions discretized

With the discretization method introduced in Section 4.1, we now illustrate the efficiency of the Schwarz iterative method equipped with the interface conditions introduced in Section 3. For the sake of succinctness, we only consider the test case #1, with Dirichlet-Dirichlet, S -2,0 and S -1,0 boundary conditions. While the Dirichlet-Dirichlet conditions are quite classical and do not require any additional numerical description, let us give a few details regarding the way interface conditions S -2,0 (namely ( 23) and ( 24)) are discretized (processing conditions S -1,0 is similar). In particular for equation (23a) we consider the following 2 :

-1 Re

u 1 I+1/2,j -u 1 I-1/2,j ∆x + p 1 I,j -a 0 u 1 I+1/2,j (26)
= -1 Re

u 2 I+3/2,j -u 2 I+1/2,j ∆x + p 2 I+1,j -a 0 u 2 I+1/2,j + A,
where:

• i = I + 1/2 holds for the interface location, where both u 1 I+1/2,j and u 2 I+1/2,j are defined,

• A = -∆x f I+1/2,j + D 2 z u 1 I+1/2,j + D 2 z u 2 I+1/2,j 2 Re + u 1 I+1/2,j + u 2 I+1/2,j 2 ∆t , • D 2 z u i,j denotes u i,j+1 -2 u i,j + u i,j-1 ∆z 2 .
It is clear that A is of order ∆x, so that equation ( 26) is a consistant approximation of equation (23a). The A term, which may look useless at first glance, is chosen so that equation ( 26) is consistent (after convergence) with the monodomain approximation of the first equation ( 3) on the whole interface, namely:

u I+1/2,j ∆t - D 2 x u I+1/2,j + D 2 z u I+1/2,j Re + D x p I+1/2,j = f I+1/2,j , ∀j, (27)
where D x p i+1/2,j = p i+1,k -p i,j ∆x and D 2 x u I+1/2,j = u I+3/2,j -2 u I+1/2,j + u I-1/2,j ∆x 2 .

Numerical results

Our numerical results were obtained thanks to the Schwarz method (in the multiplicative case) implemented with the aforementioned interface boundary conditions in the case of two subdomains. We illustrate the convergence of the iterative method on one single time step, at time t = 1. 3As expected (see discussion in Section 3), none of the Dirichlet-Dirichlet nor S -k,0 interface conditions makes the DDM algorithm converge when δ = 0 (no overlap). For δ > 0 (overlapping domains), the numerical results are almost identical to the theoretical values as far as the Dirichlet-Dirichlet condition is concerned (compare the first line in Table 4 with the first line in Table 3). Moreover the numerical results obtained with S -2,0 are in very good accordance with theoretical results:

-the shape of the convergence rate with respect to coefficient a 0 (Figure 6) is very similar to the theoretical one (Figure 2, left panel), -the optimal value a * 0 is of the same order of magnitude in both numerical and theoretical cases.

As could be expected, the exact values of a * 0 and of the corresponding numerical convergence rates are not identical (compare the second line in Table 3 with the second line in Table 4), but this can be explained by the hypotheses that were made to build the theoretical analysis (infinite domain) and by the discretization of continuous equations. As expected also from the theoretical results, the obtained convergence rates compare very favorably to the ones obtained with Dirichlet-Dirichlet interface conditions, which supports the use of S -2,0 conditions rather than Dirichlet-Dirichlet conditions as soon as subdomains overlap. Numerical results obtained with S -1,0 are also rather consistent with theoretical ones. Although we did not extensively explore all possible values for (a 0 , b 0 ), sections in the (a 0 , b 0 ) plane (Figure 7) indicate a V-shaped behavior of ρ( Ĥ) as a function of a 0 , and a more complex shape as a function of b 0 . These shapes are consistent with the theoretical behavior plotted in Figure 4 (second row, left column), although less contrasted. Fig. 6 Numerical convergence rate for approximation S -2,0 for δ = 0 (blue), δ = ∆x (green) and δ = 2∆x (red). The related configuration is Test Case #1 in Table 1.

Remark 1 Note that another possible use of the Schwarz domain decomposition algorithm is no longer as an iterative solver, but as a preconditioner for a Krylov method, such as GMRES for instance. It has indeed been noticed in many papers (e.g. [START_REF] Gander | Optimized Schwarz methods[END_REF]) that the convergence is more efficient that way. We thus implemented a version of our test case in such a way, similar to case #1 except that N = 64. Our results are summarized in Table 5 (to compare both methods, we transformed the number of subdomain resolutions in GMRES into an equivalent convergence rate).

Consistently with previous studies, we observed that using the Dirichlet-Dirichlet Schwarz method as a preconditioner for GMRES leads to a much more efficient convergence rate than solving directly with the Dirichlet-Dirichlet classical algorithm (6): ρ = 0.24 instead of ρ = 0.87. More surprisingly, using the Schwarz method with S -2,0 conditions as a preconditioner does not improve the convergence rate any longer. Although the hierarchy in the performances of interface conditions is generally conserved when skipping from using Schwarz as a solver to using Schwarz as a preconditioner, an example of such a similar result was obtained recently in the context of an advection-diffusion problem [START_REF] Hoang | Space-time domain decomposition methods for mixed formulations of flow and transport problems in porous media[END_REF]. This result should thus be studied more carefully, in particular in terms of matrix spectra, and should be complemented by additional results with other interface conditions and other physical configurations. This is however out of the scope of the present paper, and will be the subject of future investigations.

Table 5 Comparison of the convergence rate of the Schwarz method, either used as an iterative solver or as a preconditioner for a GMRES solver (δ = ∆x).

Dir-Dir S -2,0 Schwarz as an iterative solver 0.87 0.41 Schwarz as a preconditioner for GMRES 0.24 0.50

Concluding remarks

In order to complement the preceding study, we would like to point out two additional aspects that have not been discussed yet.

Two-sided conditions

Up to now, the values of the coefficients a i and b i are identical in Ŝapp However it is possible to switch from such so called one-sided conditions to twosided ones, in which these coefficients may have different values in Ŝapp The two-sided spectral radius is plotted in Figure 8. Its optimal value ρ opt (a 0 , a 0 ) is 30-35% smaller than the corresponding optimal one-sided spectral radius ρ opt (a 0 ) (located on the diagonal of each plot), for each of our three test cases (see Appendix). This improvement is quite significant. Fig. 8 Approximation Ŝ-2,0 in the two-sided case: spectral radius ρ( Ĥ) as a function of a 0 (x-axis) and a 0 (y-axis) for δ = ∆x in the case #1. The 3 panels correspond to the 3 configurations detailed in Table 1.

However this two-sided approach doubles the number of degrees of freedom (dof) in the Ŝapp i operators, which makes the optimization harder. The question is then to determine which approximation leads to the best convergence properties for a given number of dof. If we limit the optimization to 2 dof, which is reasonable for the optimization in practical cases, we have then to compare two-sided approximation Ŝ-2,0 (where dof are a 0 and a 0 ) with one-sided approximation Ŝ-1,0 (where dof are a 0 and b 0 ). For our test cases, there is no fully clear conclusion, since one-sided Ŝ-1,0 leads to much better spectral radii than two-sided approximation Ŝ-2,0 in cases #1 and #2, but not in case #3 (see Appendix). However, considering that Ŝ-1,0 does not require any overlap to converge, this latter approximation probably remains the best overall compromise.

Symmetric tensor

As mentioned previously, the stress tensor that we used in Section 2.1 to derive the general form of the interface conditions was the non symmetric form σ = 1/Re∇v -pId. However we could also have chosen the symmetric one σ = 1/(2Re) ∇v + ∇v T -pId. Such a choice modifies the bottom coefficients of matrices introduced in Section 2.3, which become Consequently the expressions of matrices Ĥij and of the spectral radius ρ( Ĥ) are also modified, as well as the expression of corresponding transparent operators (which are no longer antisymmetric). However, in our opinion, there is no theoretical reason indicating that working with the symmetric form of the stress tensor should lead to a better convergence of the DD algorithm than working with the non symmetric form. This is confirmed by our results, since we checked this alternative symmetric approach in the simple case of approximation Ŝ-2,0 , and obtained optimal spectral radii which differ by less than 0.01 from their corresponding "non symmetric" values.

Extension to more than two subdomains

As seen previously, we restricted the present study to a decomposition into two subdomains. Increasing the number of subdomains is out of the scope this paper. However it is well known that the performances of previous (so called one-level) methods are not scalable when the number of subdomains becomes large, due to a lack of a global exchange of information. This can be fixed by an additional solve of a coarse problem, the size of which is of the order of the number of subdomains. Such a so called two-level approach was proposed first in [START_REF] Nicolaides | Deflation of conjugate gradients with applications to boundary value problems[END_REF], and has been further developed in many papers, including very recent ones (see e.g. [START_REF] Efendiev | Robust domain decomposition preconditioners for abstract symmetric positive definite bilinear forms[END_REF][START_REF] Spillane | Abstract robust coarse spaces for systems of pdes via generalized eigenproblems in the overlaps[END_REF][START_REF] Willems | Robust multilevel methods for general symmetric positive definite operators[END_REF] and references herein).

Synthesis

This study leads to the conclusion that S -1,0 i operators are good candidates to provide efficient interface conditions for the Schwarz DDM algorithm applied to the Stokes problem [START_REF] Dolean | Deriving a new domain decomposition method for the Stokes equations using the Smith factorization[END_REF]. They allow for a rapid convergence, even without overlap, in all our test cases, and rely only on two free parameters. They compare favorably to the simple Dirichlet-Dirichlet, but also to the simpler approximation S -2,0 i . Moreover, considering one or two additional parameters in approximations S -1,1 i and S -1,2 i is probably not worth given the additional difficulty in the optimization of these parameters and the relatively small improvement in the convergence rate. Note also that two-sided S -2,0 i operators are a good alternative. This study was performed in the 2-D (x, z) case. Extending it to the 3-D case would be a direct extension of the present work, which would make the derivation of the approximations more complex, but would not require new methodological tools. Conversely, considering the time variable would represent a significant new step. Considering waveform relaxation DDM (also called global-in-time methods) would indeed require coming back to Navier-Stokes equations, dealing with the nonlinear advection term and including the time variable in all previous derivations. However it would allow for the use of different space and time steps in the different subdomains, which is an interesting practical feature. Finally let us remind that the Schwarz formulation is a natural framework for coupling different models, which is an important motivation for the present work, in the prospect of coupling different models in fluid dynamics.

Table 1 Fig. 1

 11 Fig. 1 Comparison of the modules of the coefficients in Ŝ * 1 and Ŝ * 2 as a function of k, with additional zooms away from k min (corresponding to the dashed rectangles). Thick solid line: | Ŝi (1, 1)|; dashed line: | Ŝi (1, 2)| = | Ŝi (2, 1)|; thin solid line: | Ŝi (2, 2)|.

i.

  It can be represented either by a constant positive value a 0 (therefore avoiding second order derivatives in the approximate boundary conditions), or by a degree two polynomial a 0 -a 2 k 2 , positive for k ∈ [k min , k max ]. The two corresponding approximations of Ŝ * i are denoted Ŝ-2

  i s are given either by the full expression (19) further denoted Ŝ-1,2 i , or by its simplified version Ŝ-1,1 i where a 2 = b 2 = 0 if one wants to avoid second order derivatives, or by an even more simplified version Ŝ-1,0 i where a 1 = a 2 = b 2 = 0 if one only allows degree zero polynomials.

Fig. 3

 3 Fig.3Illustration of the theoretical convergence rate for approximation S -2,2 in the case #1. Spectral radius ρ( Ĥ) as a function of a 0 (x-axis) and a 2 (y-axis) for δ = ∆x (left) and δ = 2∆x (right). The dashed lines correspond to a 0 -a 2 k 2 min = 0 and a 0 -a 2 k 2 max = 0.

Fig. 5

 5 Fig. 5 Left: a staggered cell. Right: typical solution of the lid-driven cavity problem at Re = 10 4 .

  .04 --5 --4.96 --4.92

Fig. 7

 7 Fig.7Numerical convergence rate for approximation S -1,0 . The related configuration is Test Case #1 in Table1. Top: convergence as a function of a 0 (with b 0 = -5.03). Bottom: convergence as a function of b 0 (with a 0 = 243.5). The additional zoom corresponds to the dashed rectangle.

1 and Ŝapp 2 .

 12 We have tested this alternative for the simplest case Ŝ-2,0 , i.e. considering Ŝ

Table 2

 2 Summary of the different approximations of Ŝ * 1 and Ŝ *

	2
	expansion at order 1/ε 2
	degree 0	degree 2
	polynomial	polynomial
	Ŝ-2,0 i

Table 3

 3

	, first

Table 3

 3 Optimal value of ρ( Ĥ) as a function of the overlap for the different approximations. The optimal values for coefficients a 0 , b 0 and a 1 are indicated in Appendix.

			Case 1			Case 2			Case 3
		δ = 0	∆x	2∆x	δ = 0	∆x	2∆x	δ = 0	∆x	2∆x
	Dir-Dir	1.	.978	.955	1.	.951	.864	1.	.999	.998
	Ŝ-2,0 i Ŝ-2,2 i Ŝ-1,0 i Ŝ-1,1 i	1. 1. .733 .650	.306 .305 .067 .066	.221 .220 .038 .038	1. 1. .381 .332	.142 .140 .040 .034	.081 .081 .008 .008	1. 1. .595 .592	.462 .461 .394 .356	.386 .386 .265 .244

Table 4

 4 Optimal value of ρ( Ĥ) as a function of the overlap in our numerical experiments. The optimal values for coefficient a 0 are indicated in the last line.

		δ = 0	∆x	2∆x
	Dir-Dir	1.	.950	922
	Ŝ-2,0	1.	.514	.425
	a * 0	N/A	243.5 300.6

We consider here the non symmetric form of the stress tensor. The alternative consisting in using its symmetric form will be discussed in Section 5.

For the sake of readability, the Schwarz indices m and m -1 have been removed, but u 1 should be read as u 1,m , and u 2 as u 2,m-1 as in (23a). In the other case (u 1 = u 1,m-1 and u 2 = u 2,m ), we would obtain the discretized version of (24a).

The time steps from t = 0 to t = 1 have been previously computed with the monodomain algorithm in order to avoid starting from a zero initial solution, which could hide numerical evidences.

Acknowledgements This work was supported by the ANR through contract ANR-11-MONU-005 (COMODO) and by the French national programme LEFE/INSU (project CoCoA).

Appendix

The optimal values of the spectral radius ρ opt and of the parameters a i and b i for the different test cases and for different sizes of the overlap δ are summarized in the following


Huy Quang

Meddy El Lê

Evelyne Alaoui

Bérénice Véricel

Laurent Ségrestin

Michel Soulère

Michel Guichardant

Philippe Lagarde

Catherine Moulin

Calzada

Quang Huy Lê

Meddy El

Evelyne Véricel

Bérénice Ségrestin

Laurent Soulère

Michel Lagarde

Philippe Moulin

Dr Catherine Calzada
email: 33472438524catherine.calzada@insa-lyon.fr

Glycoxidized HDL, HDL enriched with oxidized phospholipids and HDL from diabetic patients inhibit platelet function

Keywords: Anti-aggregatory effects of glycoxidized HDL in type 2 diabetes High density lipoproteins, platelets, hydroxy-octadecadienoic acid, oxidized phospholipids, diabetes AA, arachidonic acid, BHT, butylated hydroxytoluene, CE, cholesteryl esters, cPLA 2 , cytosolic phospholipase A 2, DHA, docosahexaenoic acid, DMA, dimethylacetal, EPA, eicosapentaenoic acid, HDL, high-density lipoproteins, HEDE, hydroxy-eicosadienoic acid, HETE, hydroxy-eicosatetraenoic acid, HODE, hydroxy-octadecadienoic acid, LA, linoleic acid, LDL, low-density lipoproteins, MDA, malondialdehyde, NaBH 4 , sodium borohydride, PC, phosphatidylcholine, PL, phospholipids, PRP, platelet-rich plasma, PUFA, polyunsaturated fatty acids, SR-BI, scavenger receptor BI, TAG, triacylglycerols, TBA, thiobarbituric acid, TLC, thin-layer chromatography, TxA 2 , thromboxane

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

INTRODUCTION

There is growing evidence that HDL composition determines its functional properties besides the levels of HDL cholesterol (1). HDL are highly heterogeneous particles consisting of two hundred individual molecular lipid species and a hundred of proteins (2,3). Amongst lipids, the presence of oxidizable polyunsaturated fatty acids (PUFAs) substrates renders these particles sensitive to oxidative attack. In particular, HDL are the major carriers of primary and terminal end-products of nonenzymatic lipid peroxidation such as lipid hydroperoxides (4) and isoprostanes (5) in human blood plasma. HDL also remove seeding molecules from LDL such as hydroperoxy-octadecadienoic and hydroperoxy-eicosatetraenoic acids [START_REF] Navab | Normal high density lipoprotein inhibits three steps in the formation of mildly oxidized low density lipoprotein: Steps 2 and 3[END_REF]. Oxidatively modified HDL have been detected in vivo [START_REF] Nakano | Immunochemical detection of circulating oxidized high-density lipoprotein with antioxidized apolipoprotein A-I monoclonal antibody[END_REF] in the intima of atherosclerotic plaques in human abdominal aortae [START_REF] Nakajima | Localization of oxidized HDL in atheromatous plaques and oxidized HDL binding sites on human aortic endothelial cells[END_REF] and in patients with type 2 diabetes (T2D) [START_REF] Ueda | Establishment and evaluation of 2 monoclonal antibodies against oxidized apolipoprotein A-I (apoA-I) and its application to determine blood oxidized apoA-I levels[END_REF], however the molecular mechanisms involved were not investigated compared to the extensive studies on oxidized LDL in atherogenesis. Functional properties of oxidized HDL in T2D, a disease associated with increased risk for atherothrombosis and oxidative stress, have been investigated but gave contradictory results. Most studies showed that HDL might lose their protective properties in T2D [START_REF] Nobécourt | Defective antioxidative activity of small dense HDL3 particles in T2D: relationship to elevated oxidative stress and hyperglycaemia[END_REF][START_REF] Sorrentino | Endothelial-vasoprotective effects of high-density lipoprotein are impaired in patients with T2D mellitus but are improved after extended-release niacin therapy[END_REF] while other studies provided evidence for beneficial effects such as increased cholesterol efflux from macrophages in patients with T2D [START_REF] Low | Advanced glycation end-products (AGEs) and functionality of reverse cholesterol transport in patients with T2D and in mouse models[END_REF][START_REF] De Vries | Increased cholesterol efflux from cultured fibroblasts to plasma from hypertriglyceridemic T2D patients: roles of pre [beta]-HDL, phospholipid transfer protein and cholesterol esterification[END_REF]. Studies on the effects of oxidatively modified HDL on platelet functions are few and contradictory, reporting either a stimulation [START_REF] Assinger | Oxidation by hypochlorite converts protective HDL into a potent platelet agonist[END_REF] or inhibition [START_REF] Valiyaveettil | Oxidized highdensity lipoprotein inhibits platelet activation and aggregation via scavenger receptor BI[END_REF] of platelet aggregation by in vitro oxidized HDL. To clarify this issue in the frame of T2D, we undertook two approaches. On one hand, HDL from control subjects were modified in vitro by glycoxidation in order to mimic changes occurring in HDL particles of T2D patients, and their effects on platelet activation were determined and compared to those induced by unmodified HDL. On the other hand, HDL from T2D patients were compared to HDL from healthy control subjects as far as platelet aggregation was concerned. We hypothesized that some elective lipid peroxides in HDL phospholipids (PL) might exert anti-aggregating properties.

SUBJECTS AND METHODS

Study subjects

Eight T2D patients (5 men and 3 women, aged 66 ± 3.5 years) and eight healthy subjects (6 men and 2 women, aged 25 ± 2.9 years) were included. The patients had poorly controlled diabetes (fasting glycemia : 8.7 ± 1.4 mmol/L ; glycated hemoglobin HbA 1C : 8.3 ± 0.6%, 67 mmol/mol). They had mild hypertriglyceridemia (triglycerides: 1.9 ± 0.4 mmol/L), normal LDL-cholesterol (2.4 ± 0.2 mmol/L) and low HDL-cholesterol (1.1 ± 0.1 mmol/L). The patients were recruited at the Department of Endocrinology and Metabolic Diseases, Cardiovascular Hospital, Lyon Bron. The protocol was approved by the Local Ethics Committee (CPP Sud-Est IV, Hospices Civils de Lyon) and the study was conducted in accordance with the principles of the Helsinki declaration. Written informed consent was obtained from each of the participants.

Isolation of HDL by ultracentrifugation

Blood was collected on EDTA and HDL were immediately isolated from plasma by potassium bromide stepwise ultracentrifugation [START_REF] Havel | The distribution and chemical composition of ultracentrifugally separated lipoproteins in human serum[END_REF]. HDL were extensively dialyzed against PBS (pH 7.35) in the presence of 1 mmol/L EDTA. The concentration of proteins was estimated using a Lowry assay [START_REF] Lowry | Protein measurement with the Folin phenol reagent[END_REF].

Preparation of HDL modified by glycoxidation

Glycoxidized HDL consisted of HDL incubated with 50 mmol/L glucose for 5 days at 37°C, dialyzed in PBS to remove excess glucose, and treated with 5 μmol/L CuCl 2 for 1 day at 37°C. Control HDL were prepared by incubating native HDL in the presence of butylated hydroxytoluene (BHT) (5µmol/L) and EDTA (1mmol/L) for 6 days at 37°C. All HDL were finally dialyzed against EDTAfree PBS just before their interaction with platelets.

Platelet isolation

Venous blood was collected on citrate-phosphate-dextrose (19.6 mmol/L citric acid, 89.4 mmol/L sodium citrate, 16.1 mmol/L NaH 2 PO 4 , 128.7 mmol/L dextrose, pH 5.6) from healthy volunteers who had not ingested any aspirin or anti-inflammatory drugs in the previous ten days. Platelet-rich plasma (PRP) was prepared by centrifugation of the blood at 200g for 15 min at 20°C. Platelets were isolated as previously described [START_REF] Lagarde | A simple and efficient method for platelet isolation from their plasma[END_REF].

Synthesis of 1-palmitoyl, 2-(13-hydroxy-octadecadienoyl)-sn-glycero

Synthesis of 1-palmitoyl,2-(15(S)-hydroxy-eicosadienoyl)-sn-glycero-3-phosphocholine

Firstly, 1-palmitoyl,2-eicosadienoyl-sn-glycero-3-phosphocholine PC(16:0/20:2n-6) was chemically synthesized. To a solution of 1-palmitoyl-2-hydroxy-sn-glycero-3-phosphocholine (0.2 mmol), eicosa-

In vitro enrichment of HDL with oxidized phospholipids

To prepare HDL enriched with oxidized PL, native HDL were incubated with dried PC(16:0/13-HODE) for 24 hours at 37°C in a shaking bath. Unbound PL were removed by dialysis in PBS and using PD-10 desalting columns. Similar procedure was applied to prepare HDL enriched with PC(16:0/18:2).

Characterization of HDL

Fatty acid compositions of lipid classes

Fatty acid methyl esters and fatty dimethylacetals were analyzed by gas chromatography as detailed in the Supplemental Methods.

Quantification of monohydroxylated fatty acids

Hydroxylated fatty acids were separated and quantified by reverse-phase HPLC according to the amount of appropriate internal standards, PC(16:0/15-HEDE) and 15-HEDE. For the detailed procedure, please refer to Supplemental Methods.

Stereochemical analysis of hydroxylated fatty acids

Optical isomers of HODE and HETE were separated by chiral phase HPLC as detailed in the Supplemental Methods.

Malondialdehyde (MDA) determination

Overall lipid peroxidation was assessed by quantitation of thiobarbituric acid (TBA)-MDA adducts by reverse-phase HPLC with fluorimetric detection. For details, please refer to Supplemental Methods.

Vitamin E determination

Tocopherol isomers were separated by reverse-phase HPLC and measured fluorimetrically containing tocol as an internal standard. For details, please refer to Supplemental Methods.

Platelet aggregation

Aggregation was measured in isolated platelets in a Chrono-log dual-channel aggregometer (Coulter, Margency, France) according to the method of Born [START_REF] Born | Aggregation of blood platelets by adenosine diphosphate and its reversal[END_REF]. Platelet suspensions were pre-incubated for 5 min at 37°C in the presence or absence of different preparations of HDL and then stimulated with threshold concentrations of collagen (Nycomed, Linz, Austria) with continuous stirring at 1000 rpm.

The threshold concentration of collagen was defined as the concentration of collagen that induced approximately a 60% increase in light transmission. The extent of platelet aggregation was expressed in terms of percentage of change in light transmission 4 min after the addition of collagen.

Platelet p38 MAPK and cytosolic phospholipase A 2 activation

Chemiluminescent Western blotting detection of p38 MAPK, phospho-p38 MAPK and phospho-cPLA 2 was performed as described in the Supplemental Methods.

Determination of intracellular Ca 2+ concentrations

Intracellular calcium concentrations were measured fluorimetrically in Fura-2-loaded platelets as described in the Supplemental Methods.

Statistical analysis

Results are expressed as the means ± SEM. Comparisons between groups were performed using paired Student t-test.

RESULTS

Characterization of glycoxidized HDL compared to control HDL

Fatty acid composition of phospholipids and cholesteryl esters (CE)

Following in vitro glycoxidation, proportions of the main PUFAs, linoleic (LA, 18:2n-6) and arachidonic (AA, 20:4n-6) acids, significantly decreased in PL and CE of glycoxidized HDL compared to control HDL (Table ). In PL from glycoxidized HDL, the proportions of LA, AA, eicosapentaenoic (EPA, 20:5n-3) and docosahexaenoic (DHA, 22:6n-3) acids decreased by 54%, 91%, 56% and 91% respectively. The proportions of 16:0 dimethylacetals (DMA), 18:0 DMA and 18:1 n-9

DMA issued from the alkenyl residue of alkenyl,acyl-glycero-phosphoethanolamine (ethanolamine plasmalogens) decreased in PL from glycoxidized HDL compared to those from control HDL. By contrast, proportions of palmitic (16:0), stearic (18:0) and oleic (18:1n-9) acids increased in modified HDL. In CE, glycoxidation of HDL also led to decreased proportions of LA, AA and EPA, by 53%, 92% and 86% respectively, and increased proportions of palmitic, stearic and oleic acids.

Vitamin E concentration

The concentrations of alpha-tocopherol strongly decreased in glycoxidized HDL compared to control HDL (0 in glycoxidized HDL vs. 2.04 ± 0.28 nmol/mg HDL protein in control HDL, n=5, P<0.001).

Gamma-tocopherol concentrations decreased by 75% in glycoxidized HDL (0.08 ± 0.01 in glycoxidized HDL vs. 0.32 ± 0.02 nmol/mg HDL protein in control HDL, n=5, P<0.001).

MDA concentration

The concentrations of MDA, a marker of overall lipid peroxidation, were more than 10-fold higher in glycoxidized HDL compared to control HDL (1.93 ± 0.32 nmol/mg protein in glycoxidized HDL vs. 0.18 ± 0.13 nmol/mg protein in control HDL, n=4, P<0.01).

Concentrations of hydroxylated fatty acids in lipid classes

The concentrations of the stable primary products of PUFA peroxidation, 13-HODE and 9-HODE, derived from LA, and 15-HETE, derived from AA, were assessed in HDL. In control HDL, most hydroxylated fatty acids were present in CE, followed by PL and TAG, while most of them were present in PL of glycoxidized HDL (Figure 1). HODE concentrations increased strongly in PL of glycoxidized HDL compared to those of unmodified HDL (Figures 1A and1B). to increase in CE and TAG from glycoxidized HDL compared to control HDL (Figure 1C).

Effects of control and glycoxidized HDL on platelet activation

Compared to platelets incubated with collagen, pre-incubation of platelets with glycoxidized HDL for 5 min at 37°C resulted in a dose-dependent inhibition of collagen-induced platelet aggregation with near complete inhibition at 100 µg/mL (Figure 1D). Control HDL had no effect at 25 µg/mL but significantly inhibited collagen-induced platelet aggregation at concentrations greater than or equal to 50 µg/mL. Because scavenger receptor SR-BI binds modified HDL and is expressed in platelets [START_REF] Imachi | Expression of human scavenger receptor B1 on and in human platelets[END_REF], its involvement in the mechanism of action of HDL on these cells was sought. Pre-incubation of platelets with anti-SRBI blocking antibody for 5 min at 37°C fully prevented the inhibitory effects of glycoxidized HDL on collagen-induced platelet aggregation (Figure 1D). Pre-incubation of platelets with anti-SRBI also prevented the inhibition of collagen-induced platelet aggregation in presence of control HDL (aggregation rate: 65 % in platelets incubated with anti-SRBI antibody and 100 µg/ml control HDL vs. 12% in platelets incubated with 100 µg/ml control HDL). Anti-SRBI blocking antibody had no effect on collagen-induced platelet aggregation, and pre-incubation with non-immune isotype control antibody did not reduce the inhibitory effect of glycoxidized HDL on platelet aggregation (data not shown).

To determine the effects of glycoxidized HDL on key enzymes involved in the release of AA from membrane PL, phosphorylation of p38 MAPK and cPLA 2 was determined in platelet suspensions incubated in the absence or presence of HDL for 5 min and further stimulated with collagen for 4 min.

As shown in Figures 2A and2B, addition of collagen to platelets resulted in increased amounts of phosphorylated p38 MAPK and cPLA 2 . Pre-incubation of collagen-stimulated platelets with glycoxidized HDL fully prevented the collagen-induced increased phosphorylation of both enzymes.

Since Ca 2+ is a key second messenger downstream of most signaling pathways and is essential for the translocation of cPLA 2 to membranes, the effects of control and glycoxidized HDL on the kinetics of collagen-induced Ca 2+ release were determined (Figure 2C). Addition of collagen to Fura2-AM loaded platelets resulted in a rapid and transient increase in intracellular Ca 2+ levels. Glycoxidized HDL inhibited collagen-induced intracellular Ca 2+ increase in platelets by 47% while control HDL decreased it by 32%. Pre-incubation of platelets with anti-SRBI blocking antibody for 5 min at 37°C alleviated the inhibitory effects of glycoxidized HDL on phosphorylation of p38 MAPK and cPLA 2 , and Ca 2+ mobilization (data not shown).

Effects of HDL

Effects of glycoxidized HDL obtained from patients with T2D on platelet aggregation

The concentrations of hydroxylated fatty acids were assessed in HDL PL from poorly controlled T2D patients and control subjects. 13-HODE and 9-HODE concentrations were two-fold higher in HDL PL from T2D patients compared to those in control subjects. 15-HETE concentration was 2.4-fold higher in patient HDL PL compared to its concentration in control HDL (Figure 4A). To determine whether hydroxylated fatty acid products originated from enzymatic or non enzymatic lipid peroxidation, chiral phase HPLC was carried out on each isomer. Concerning hydroxylated linoleic acid metabolites, 9(R)-HODE and 9(S)-HODE enantiomers were equally present in HDL from T2D patients (51% 9(S)-HODE and 49% 9(R)-HODE) suggesting that the great majority of 9-HODE originated from non-enzymatic lipid peroxidation. 13-HODE consisted of 78% S isomer and 22% R isomer indicating the involvement of enzymatic lipid peroxidation, presumably by 15-/6-lipoxygenase, together with nonenzymatic lipid peroxidation in the formation of 13-HODE. 15-HETE comprised 84% S and 16% R isomer suggesting again the involvement of 15-lipoxygenase in the oxidation of AA. By comparison, unmodified HDL from control healthy subjects contained equal ratios of 9(R/S)-HODE, 60% S isomer and 40% R isomer for 13-HODE and 15-HETE, suggesting that the lipoxygenation was a minor process compared to what occurred in T2D.

The pre-incubation of platelets with HDL from T2D patients resulted in an inhibition of collageninduced platelet aggregation compared with control platelets incubated with collagen (Figure 4B). The inhibitory effect of patients HDL on collagen-induced platelet aggregation was stronger than the one induced by control HDL because 50 µg/mL patients HDL inhibited collagen-induced platelet aggregation by 63% whereas 50 µg/mL control HDL inhibited it by 28%. To establish the role of oxidized PL in the anti-aggregating effects of HDL, HDL were enriched with PC(16:0/13-HODE) to obtain similar amounts of hydroxylated fatty acid as those found in T2D HDL (Figure 4C). PC(16:0/13-HODE) enriched HDL inhibited collagen-induced platelet aggregation to a similar extent as patient HDL (Figure 4D).

DISCUSSION

The present results indicate that HDL modified by glycoxidation inhibited platelet aggregation via SR-BI in a dose-dependent manner and displayed higher anti-aggregatory potency than control HDL.

Regarding the platelet signaling cascade involved in the mechanism of action of HDL, we present new data showing that in vitro glycoxidized HDL prevented both the collagen-induced increased phosphorylation of p38 MAPK, the stress kinase responsible for the phosphorylation of cPLA 2 [START_REF] Börsch-Haubold | Phosphorylation and activation of cytosolic phospholipase A2 by 38-kDa mitogen-activated protein kinase in collagen-stimulated human platelets[END_REF] and that of cPLA 2 , the key enzyme involved in the release of AA from membrane PL (25) which constitutes a rate-limiting step in the biosynthesis of biologically active eicosanoids. Supporting our in vitro results, we show for the first time that HDL from T2D patients also inhibited platelet aggregation via SR-BI at concentrations as low as 50 µg/mL. Our results are in line with our previous results reporting anti-aggregatory properties of severely oxidized HDL isolated from patients with abetalipoproteinemia, an orphan metabolic disease characterized by the absence of apolipoprotein Bcontaining lipoproteins [START_REF] Calzada | Inhibitory effects of in vivo oxidized high-density lipoproteins on platelet aggregation: evidence from patients with abetalipoproteinemia[END_REF], and extend them in T2D, a common and growing disease known to be associated with chronic oxidative stress [START_REF] Robertson | Beta-cell glucose toxicity, lipotoxicity, and chronic oxidative stress in type 2 diabetes[END_REF]. T2D patients, with [START_REF] Ferroni | Platelet activation in type 2 diabetes mellitus[END_REF] or without cardiovascular complications [START_REF] Véricel | Diabetic patients without vascular complications display enhanced basal platelet activation and decreased antioxidant status[END_REF], show platelet hyperactivation evidenced by increased platelet adhesion and aggregation as well as increased thromboxane A 2 production. Oxidatively modified lipoproteins present in plasma from T2D patients may represent important contributing factors modulating platelet activation, as shown with oxidized LDL from T2D patients which activate platelets [START_REF] Colas | LDL from obese patients with the metabolic syndrome show increased lipid peroxidation and activate platelets[END_REF]. Decreased plasma levels of HDL in T2D patients lead to lower levels of oxidized HDL, and it is likely that the anti-aggregatory effects of HDL from T2D may be reduced and might indirectly contribute to platelet hyperactivation in T2D. So far, few studies have investigated the effects of oxidized HDL on platelet function. While it has been shown that hypochlorite-oxidized HDL may stimulate platelet aggregation via CD36 [START_REF] Assinger | Specific binding of hypochlorite-oxidized HDL to platelet CD36 triggers proinflammatory and procoagulant effects[END_REF], HDL oxidatively modified by copper sulfate or myeloperoxidase strongly inhibit agonist-induced platelet activation and aggregation via SR-BI [START_REF] Valiyaveettil | Oxidized highdensity lipoprotein inhibits platelet activation and aggregation via scavenger receptor BI[END_REF]. Besides effects of HDL on platelet function, oxidative modification of HDL (as a result of in vitro oxidation or diabetes) may decrease cholesterol efflux capacity [START_REF] Nagano | High density lipoprotein loses its effect to stimulate efflux of cholesterol from foam cells after oxidative modification[END_REF][START_REF] Cavallero | Abnormal reverse cholesterol transport in controlled type II diabetic patients. Studies on fasting and postprandial LpA-I particles[END_REF], impair their capacity to protect LDL from oxidation [START_REF] Nobécourt | Defective antioxidative activity of small dense HDL3 particles in T2D: relationship to elevated oxidative stress and hyperglycaemia[END_REF] and alter their anti-inflammatory properties [START_REF] Morgantini | Anti-inflammatory and antioxidant properties of HDLs are impaired in type 2 diabetes[END_REF]. Nevertheless, a few studies reported beneficial effects of HDL modified by oxidative tyrosylation on the efflux of cholesterol in fibroblasts and macrophages [START_REF] Francis | Oxidative tyrosylation of high density lipoprotein by peroxidase enhances cholesterol removal from cultured fibroblasts and macrophage foam cells[END_REF].

Altogether, these discrepant results might be related to differences in the extent and type of HDL oxidation which may determine the binding of HDL to either SR-BI or CD36 platelet receptors and the capacity of HDL to inhibit or stimulate platelet aggregation by binding either to SR-BI or CD36 respectively [START_REF] Van Der Stoep | High-density lipoprotein as a modulator of platelet and coagulation responses[END_REF]. Our results are also in agreement with studies demonstrating that copper-oxidized HDL showed better binding to platelet SR-BI than native HDL [START_REF] Valiyaveettil | Oxidized highdensity lipoprotein inhibits platelet activation and aggregation via scavenger receptor BI[END_REF], and showing that oxidized HDL were more effective competitors than native HDL in cultured endothelial cells [START_REF] Nakajima | Localization of oxidized HDL in atheromatous plaques and oxidized HDL binding sites on human aortic endothelial cells[END_REF]. In addition, it has been reported that the abundance of human SR-BI was reduced on the surface of platelets from patients with atherosclerotic disease and that its expression level correlated negatively with platelet aggregation [START_REF] Imachi | Expression of human scavenger receptor B1 on and in human platelets[END_REF], which might impair the anti-aggregatory properties of HDL in T2D patients.

We also establish the contribution of PL esterified with oxidized LA, especially PC(16:0/13-HODE), in the anti-aggregatory properties of glycoxidized HDL. First, HODEs and 15-HETE were strongly increased in PL from in vitro glycoxidized HDL. It was associated with decreased proportions of LA and AA in this class of lipids, increased MDA levels and vitamin E consumption. Increased concentrations of HODE and 15-HETE, originating from both enzymatic and non enzymatic lipid peroxidation, were also observed in HDL PL from T2D patients compared to control HDL. It should be underlined that the concentrations of HODEs and HETE in HDL PL were negatively associated with collagen-induced platelet aggregation (r = -0.65, P=0.004). This confirms PL esterified with oxidized LA as important biologically active components supporting protective effects of HDL against platelet aggregation. Second, in vitro enrichment of HDL in phosphatidylcholine carrying out 13(S)-HODE inhibited platelet aggregation whereas enrichment of HDL with non-oxidized phosphatidylcholine had no significant effect on platelet aggregation. Similar increases of 13-HODE, 9-HODE and 15-HETE have been recently described in HDL total lipids from patients with diabetes compared with those from patients without diabetes [START_REF] Morgantini | HDL lipid composition is profoundly altered in patients with type 2 diabetes and atherosclerotic vascular disease[END_REF] but this is the first evidence of increased content of hydroxylated fatty acids in the phospholipid class of HDL from T2D patients correlated with atheroprotective activities of HDL. Moreover, the association of higher levels of oxidized PL in HDL and lower response to aggregation are in line with the Cardiovascular Risk in Young Finns study suggesting that an elevated cardiovascular risk profile was associated with lower oxidized HDL lipids levels [START_REF] Kresanov | The associations of oxidized high-density lipoprotein lipids with risk factors for atherosclerosis: the Cardiovascular Risk in Young Finns Study[END_REF]. Due to their localization at the surface of HDL particles, it is conceivable that PL are more accessible targets of free radicals compared to CE and TAG localized in the core of HDL, and are more likely to interfere with platelets following SR-BI interaction. In addition, the structure of lipoproteins in the physiological conditions might increase the preferential oxidation of linoleate into HODE over cholesterol as shown in human plasma [START_REF] Yoshida | Relative susceptibilities of linoleates and cholesterol to oxidation assessed by total hydroxyoctadecadienoic acid and 7-hydroxycholesterol[END_REF]. A limitation of our study is that lipidomic analyses were limited to HODE and HETE species, including their stereo-isomers. The contribution of other oxidized lipid species should be explored in order to decipher the molecules involved in the protective role of glycoxidized HDL on platelet aggregation, although inhibition of platelet aggregation by exogenous PC(16:0/13-HODE) was very similar to that observed by endogenous ones in glycoxidized HDL as well as in HDL from DT2 patients.

We conclude from our experiments that, as far as platelet aggregation is concerned, the ability of HDL to decrease platelet aggregation is not impaired in glycoxidized HDL, but even increased compared to control HDL. Supporting our in vitro results, we also show that glycoxidized HDL from T2D patients retain their anti-aggregating properties. We show that the HDL content of hydroxylated fatty acids esterified in PL could contribute to the inhibitory effects of glycoxidized HDL on platelet aggregation and associated signaling pathways through SR-BI (Figure 5). Our consistent findings challenge the concept of the systematic detrimental effects of oxidized lipids on atherothrombosis.

-3-phosphocholine 1 -

 1 palmitoyl,2-(13(S)-hydroperoxy-octadecadienoyl)-sn-glycero-3-phosphocholine (PC(16:0/13(S)-HpODE) was enzymatically synthesized from 1-palmitoyl,2-linoleoyl-sn-glycero-3-phosphocholine (PLPC or PC(16:0/18:2), Cayman Chemicals, Ann Arbor, MI, USA) following incubation of PC(16:0/18:2) with soybean 15-lipoxygenase (type V) in the presence of deoxycholate for 30 min under a continuous flux of oxygen[START_REF] Januel | Phospholipidhydroperoxide glutathione peroxidase (GPx-4) localization in resting platelets, and compartmental change during platelet activation[END_REF]. Phosphatidylcholine hydroperoxide formed was reduced by sodium borohydride (NaBH 4 ) into phosphatidylcholine hydroxide. 1-palmitoyl,2-(13(S)-hydroxyoctadecadienoyl)-sn-glycero-3-phosphocholine (PC(16:0/13-HODE)) concentration was determined spectrophotometrically by UV absorbance at 235 nm and its purity was checked by HPLC.

  13-HODE concentrations tended to increase (1.8-fold) and 9-HODE concentrations significantly increased in CE from glycoxidized HDL. In TAG, 13-HODE concentrations increased in glycoxidized HDL (2.7-fold) and 9-HODE concentrations tended to increase (1.8-fold) compared to control HDL. 15-HETE concentration increased in PL from glycoxidized HDL (8.5-fold) while 15-HETE concentration tended

enriched with 1 -

 1 palmitoyl,2-(13-hydroxy-octadecadienoyl)-sn-glycero-3phosphocholine on platelet aggregationTo determine the role of oxidized LA-containing PL for anti-aggregating effects of HDL, HDL were pre-loaded with two different concentrations of PC(16:0/13-HODE) prepared by 15-lipoxygenation of LA esterified to phosphatidylcholine, followed by chemical reduction with NaBH 4 (Figure3A). As expected, higher concentrations of 13-HODE, close to those found in glycoxidized HDL, were present in enriched HDL compared to control HDL, while concentrations of 9-HODE were similar in control and enriched HDL, confirming the absence of non enzymatic lipid peroxidation during the preparation of enriched HDL particles. As shown in Figure3B, HDL enriched with PC(16:0/13-HODE) inhibited collagen-induced platelet aggregation. The more HDL were enriched with PC(16:0/13-HODE), the more platelet aggregation was inhibited. HDL enriched with native PC(16:0/18:2) had no significant effect on collagen-induced platelet aggregation compared to platelets incubated with control HDL and stimulated with collagen (data not shown).

Figure 1 .

 1 Figure 1. Hydroxylated fatty acids in control and glycoxidized HDL. Effects of respective HDL

Figure 2 .

 2 Figure 2. Effects of glycoxidized HDL on platelet p38 MAPK and cPLA 2 phosphorylation, and on collagen-induced increases of calcium concentrations in platelets. Phosphorylation levels of p38

Figure 3 .

 3 Figure 3. Effects of HDL enriched with oxidized phosphatidylcholine on collagen-induced

Figure 4 .

 4 Figure 4. Hydroxylated fatty acids in HDL phospholipids from control healthy subjects and

Figure 5 .

 5 Figure 5. Summary diagram for the effects of glycoxidized HDL on platelet signaling pathways.

Table . Fatty acid composition of phospholipids and cholesteryl esters in control and glycoxidized HDL.

 .

	Fatty acid	Phospholipids	Cholesteryl esters
	(mol %)	Control HDL	Glycoxidized HDL	Control HDL	Glycoxidized HDL
	16:0	34.6 ± 2.1	48.9 ± 2.6 ***	14.9 ± 1	28.7 ± 2.7 ***
	18:0	18.7 ± 1.2	26 ± 1.5 ***	2.6 ± 0.4	7.2 ± 2.5 *
	18:1n-9	8 ± 0.5	10.6 ± 0.7 *	19.4 ± 1	27.2 ± 1.9 **
	18:2n-6	14.8 ± 1.6	6.9 ± 0.8 ***	44.5 ± 1.3	20 ± 3.4 ***
	20:4n-6	8.8 ± 0.8	0.8 ± 0.2 ***	7.3 ± 0.5	0.6 ± 0.3 ***
	20:5n-3	0.9 ± 0.1	0.4 ± 0.2 *	0.7 ± 0.2	0.1 ± 0.1 *
	22:6n-3	3.2 ± 0.2	0.3 ± 0.1 ***	1.9 ± 0.4	1.3 ± 0.5
	DMA sum	2.1 ± 0.8	0.53 ± 0.5 ***
	Results, expressed as mol % of main fatty acids, are means ± SEM of 5 independent experiments. *, P
	< 0.05; **, P < 0.01; ***, P < 0.001 vs. control HDL. Dimethylacetal (DMA) sum corresponds to the
	sum of 16:0 DMA, 18:0 DMA and 18:1 n-9 DMA.

ACKNOWLEDGEMENTS

The authors thank Inserm for financial support. Quang Huy Lê is funded by the French Ministry of Education and Research for his PhD thesis. Meddy El Alaoui thanks the "Région Rhône-Alpes" for a PhD grant. Catherine Calzada is supported by the CNRS. The authors are grateful to all participants for their cooperation and to the nurses at Hospices Civils de Lyon for expert blood drawing. phospho-p38 MAPK and phospho-cPLA 2 were visualized by enhanced chemiluminescence, and bands were quantified by densitometry.

Determination of intracellular Ca 2+ concentrations

PRP was acidified to pH 6.4 with citric acid and incubated with 1 μmol/L Fura 2-AM for 45 min at 37°C in a water bath protected from light (5). Platelets were then isolated and suspended in Tyrode-HEPES buffer, and left at room temperature for at least 1 hour in the dark. Fura 2-AM loaded platelets were pre-incubated with 50 µg/ml control or glycoxidized HDL for 5 min then stimulated with collagen. The external Ca 2+ concentration in platelet suspensions was adjusted to 1 mmol/L using CaCl 2 . Platelets were excited alternately at 340 and 380 nm and fluorescence emission was recorded at 510 nm. Intracellular Ca 2+ concentrations were calculated from the ratio of fluorescence emission to excitation.

Supplemental Methods

Fatty acid compositions of lipid classes

Following the addition of appropriate internal standards (1,2-diheptadecanoyl-sn-glycero-3phosphocholine, 1,2-diheptadecanoyl-sn-glycero-3-phosphoethanolamine, 1,2,3-triheptadecanoyl-snglycerol and heptadecanoyl cholesteryl ester) into HDL preparations and extraction with ethanol/chloroform (1:2, v/v) in the presence of BHT (50 μmol/L), lipid classes were separated by thin-layer chromatography (TLC) with hexane/diethylether/acetic acid (80:20:1, by vol.) to separate total PL, cholesteryl esters (CE) and triacylglycerols (TAG) (1). Corresponding silica zones were scraped off and treated with trifluoride boron/methanol (1.3 mol/L, 10%) for 90 min at 100°C. The derivatized fatty acid methyl esters and fatty dimethylacetals were extracted twice with isooctane and separated by gas chromatography using an HP 6890 gas chromatograph equipped with a SP 2380 capillary column (0.25μm, 30m × 0.25mm, Supelco, Bellefonte, PA, USA) and a flame ionization detector.

Quantification of monohydroxylated fatty acids

Following lipid extraction in the presence of appropriate standards (PC(16:0/15-HEDE) and 15-HEDE) and separation of lipid classes by TLC as previously described, dried extracts were reduced by NaBH 4 . Ester bonds were hydrolyzed with 0.5 mol/L potassium hydroxide for 30 min at 60°C followed by an acidification to pH 3 using acetic acid. Non esterified hydroxylated fatty acids and fatty acids were first extracted on an Oasis Sep-Pak cartridge column (Waters, Milford, MA) or were extracted with hexane. Then, non-esterified hydroxylated fatty acids were separated by TLC with hexane/diethylether/acetic acid (60:40:1, by vol.). The spots were scraped off and then extracted with methanol, separated by reverse-phase HPLC on X Bridge C 18 column (3.5 μm, 4.6 × 150 mm, Waters, Milford, MA) using a gradient solvent of acetonitrile and water (pH 3) and measured at 235 nm (2).

Stereochemical analysis of hydroxylated fatty acids

Hydroxylated fatty acids isomers (13(R,S)-HODE, 9(R,S)-HODE and 15(R,S)-HETE) were firstly separated by reverse-phase HPLC as previously described, collected, evaporated and transmethylated with diazomethane for 15 min at room temperature to obtain methyl esters of hydroxylated fatty acids.

Each fraction was injected to chiral phase HPLC on a CHIRALCEL ®OD-H column (5 µm, 250 x 4.6 mm, Daicel), eluted isocratically with hexane/2-propanol/acetic acid (85:15:0.1 by vol ) at a flow rate of 1 mL/min at 25°C. Hydroxylated fatty acids were measured at 235 nm with a diode array detector.

Malondialdehyde (MDA) determination

HDL samples were mixed with thiobarbituric acid (TBA) (10 mmol/L), acetic acid and BHT (5 mmol/L) and the mixture was heated at 95°C for 60 min. The TBA-MDA adducts were extracted with ethyl acetate, separated onto a Nucleosil C 18 column (5μm, 4.6 × 250 mm, Macherey-Nagel, Hoerdt, France) by reverse-phase HPLC and measured fluorimetrically (excitation 515 nm, emission 553 nm) (3).

Vitamin E determination

HDL samples (1 vol.), containing tocol as an internal standard, were extracted twice with hexane (4 vol.) following the addition of ethanol (1 vol.). Tocopherol isomers were separated by reverse-phase HPLC onto a Nucleosil C 18 column (5μm, 4 × 150 mm) and measured fluorimetrically (excitation 295nm, emission 340nm) (4).

Platelet p38 MAPK and cytosolic phospholipase A 2 activation

Following platelet lysis, proteins were denatured, electrophoresed in 12% bis-Tris and transferred to nitrocellulose membranes. The membranes were incubated with either 1:2500 anti-p38 MAPK or antiphospho-p38 MAPK, or anti-phospho-cytosolic phospholipase A 2 (cPLA 2 ) polyclonal antibodies, washed, and incubated with 1:5000 goat anti-rabbit horseradish peroxidase conjugate. P38 MAPK,