David Gérard-Varet

Emmanuel Dormy

On the ill-posedness of the Prandtl equation *

. We prove here that it is linearly ill-posed in Sobolev type spaces.

. A numerical study of this instability is also provided.

Introduction

One challenging open problem of fluid dynamics is to understand the inviscid limit of the Navier-Stokes equations

     ∂ t u ν + u ν • ∇u ν + ∇p ν -ν∆u ν = 0, x ∈ Ω, ∇ • u ν = 0, x ∈ Ω, u ν | ∂Ω = 0, (1.1)
in a domain Ω with boundaries, endowed with a no-slip boundary condition. Mathematically, the main difficulty is the lack of uniform bounds on the vorticity field, as the viscosity ν goes to zero. In terms of fluid dynamics, this corresponds to a boundary layer phenomenon near ∂Ω.

A natural approach to describe this boundary layer is to look for a double-scale asymptotics, with a parabolic scaling in the normal direction. Consider the case Ω ⊂ R 2 . At least locally, any point x in a neighborhood of ∂Ω has a unique decomposition x = y n(x) + x(x), x ∈ ∂Ω, where y > 0, x is an arc length parametrization of the boundary, and n is the inward unit normal vector at ∂Ω. The velocity field can be written u ν (t, x) = u ν (t, x, y) τ (x) + v ν (t, x, y) n(x), where (τ , n) is the Frénet frame. It is then natural to consider an approximation of the type:

u ν (t, x, y) ≈ u 0 (t, x, y) + u BL (t, x, y/ √ ν), v ν (t, x, y) ≈ v 0 (t, x, y) + √ ν v BL (t, x, y/ √ ν), (1.2)
where u 0 (t, x, y) = u 0 (t, x, y) τ (x) + v 0 (t, x, y) n(x)

satisfies the Euler equation with the no penetration condition, and

(u BL , v BL ) = (u BL , v BL )(t, x, Y )
describes a boundary layer corrector with typical scale √ ν in the normal direction. It is slightly more convenient to introduce u(t, x, Y ) := u 0 (t, x, 0) + u BL (t, x, Y ), v(t, x, Y ) := Y ∂ y v 0 (t, x, 0) + v BL (t, x, Y ). Indeed, inserting this Ansatz in the Navier-Stokes equations, we get formally

           ∂ t u + u∂ x u + v∂ Y u -∂ 2 Y u = ∂ t u 0 + u 0 ∂ x u 0 | y=0 , Y > 0, ∂ x u + ∂ Y v = 0, Y > 0, (u, v) = 0, Y = 0, lim Y →+∞ u = u 0 | y=0 .
(1.3)

These are the so-called Prandtl equations, derived by Ludwig Prandtl [START_REF] Prandtl | Uber flüssigkeits-bewegung bei sehr kleiner reibung[END_REF]. Note that the curvature of the boundary does not appear explicitly in the system. It is however involved in (1.3) through the Euler field, and through the interval of definition of the arc length parametrization x. Up to our knowledge, all studies deal with one of the three following cases: x ∈ R, x ∈ T, or 0 < x < L, supplemented with a condition on u at x = 0. The first and second choices are convenient to describe phenomena that are local in x. The case x ∈ T may also model the outside of a bounded convex obstacle. Finally, the third configuration is adapted to the spreading of a flow around a thin obstacle, where x = 0 corresponds to the tip of the obstacle.

Although this formal asymptotics is very natural, its validity is not clear. As emphasized by physicists, including Prantdl himself, it may not hold uniformly in space and time. One reason is the so-called boundary layer separation, which is observed for flows around obstacles, see [7]. Nevertheless, the description (1.2) fits with many experiments, upstream from the separation zone. In any case, to understand the relevance and limitations of the Prandtl model is a crucial issue.

From the mathematical point of view, one must address two problems:

1. The well-posedness of the Prandtl equation.

The justification of the expansion (1.2).

These two problems depend crucially on the choice of the underlying functional spaces, especially on the regularity that is required in the tangential variable x. Indeed, the main mathematical difficulty is the lack of control of the x derivatives. For example, v is recovered in (1.3) through the divergence condition, and in terms of x-regularity, behaves broadly like ∂ x u. This loss of one derivative is not balanced by any horizontal diffusion term, so that standard energy estimates do not apply.

Within spaces of functions that are analytic in x ∈ R, Y ∈ R + , Sammartino and Caflish have overcome these problems, justifying locally in time the boundary layer asymptotics [START_REF] Sammartino | Zero viscosity limit for analytic solutions, of the Navier-Stokes equation on a half-space. I. Existence for Euler and Prandtl equations[END_REF][START_REF] Sammartino | Zero viscosity limit for analytic solutions of the Navier-Stokes equation on a half-space. II. Construction of the Navier-Stokes solution[END_REF]. But for more "realistic" functional settings, the way solutions of (1.1) behave is still poorly understood. Various instability mechanisms, that are filtered out in an analytic framework, become a huge source of trouble. For example, when the viscosity is small, the Navier-Stokes equation admits exponentially growing solutions which are both small-scale and isotropic in x, y. Their evolution is lost in the anisotropic Prandtl description. This remark was used by Emmanuel Grenier in [START_REF] Grenier | On the nonlinear instability of Euler and Prandtl equations[END_REF], who relied on the so-called Rayleigh instability for inviscid flows to show that the asymptotics (1.2) does not (always) hold in the Sobolev space H 1 (see [START_REF] Grenier | On the nonlinear instability of Euler and Prandtl equations[END_REF] for a precise statement). However, the relevance of this asymptotics in L p , or its relevance in the absence of Rayleigh instabilities, are still open issues.

Above all, the local in time well-posedness of the Prandtl equation for smooth (say Sobolev) initial data has been so far an open problem. Up to our knowledge, the Cauchy problem has only been solved in two settings: i) x ∈ R, with data that are analytic in x : see [START_REF] Sammartino | Zero viscosity limit for analytic solutions, of the Navier-Stokes equation on a half-space. I. Existence for Euler and Prandtl equations[END_REF][START_REF] Lombardo | Well-posedness of the boundary layer equations[END_REF] for more.

ii) 0 < x < L, with data that are monotonic in y : see [START_REF] Oleinik | Mathematical models in boundary layer theory[END_REF][START_REF] Xin | On the global existence of solutions to the Prandtl's system[END_REF] for more.

One may also cite article [START_REF] Engquist | Blowup of solutions of the unsteady Prandtl's equation[END_REF], in which blow up in time of some smooth solutions is exhibited. Finally, let us mention the interesting work [START_REF] Hong | Singularity formation and instability in the unsteady inviscid and viscous Prandtl equations[END_REF], in which the inviscid version of (1.3) is analyzed (no ∂ 2 Y u in the equation). Interestingly, for a smooth initial data, this equation turns out to have an explicit solution through the method of characteristics. In particular, starting from a smooth data, one recovers locally in time a smooth data. More precisely, there is only a finite loss of x-derivatives, so that the Cauchy problem is (weakly) well-posed. We refer to [START_REF] Hong | Singularity formation and instability in the unsteady inviscid and viscous Prandtl equations[END_REF] for all details. See also papers [START_REF] Grenier | On the derivation of homogeneous hydrostatic equations[END_REF][START_REF] Brenier | Homogeneous hydrostatic flows with convex velocity profiles[END_REF] on the hydrostatic equations, that share some features with Prandtl equations. For more on Prandtl equations, see the review [START_REF]Boundary layer theory and the zero-viscosity limit of the Navier-Stokes equation[END_REF].

On the basis of the inviscid result, it seems reasonable to bet for well-posedness of the Prandtl equation (1.3) in Sobolev type spaces. The aim of this paper is to show that it is actually linearly ill-posed in this framework. As we shall see later on, the reason for illposedness is a strong destabilization mechanism due to two ingredients: viscosity, and critical points in the base velocity profile. In particular, it does not contradict the positive results obtained in the inviscid case and for monotonic data.

Let us now describe our results. We restrict ourselves to (x, Y ) ∈ T × R + , and u 0 = 0. To lighten notations, we write y instead of Y . The Prandtl equation comes down to

       ∂ t u + u∂ x u + v∂ y u -∂ 2 y u = 0, in T × R + . ∂ x u + ∂ y v = 0, in T × R + , (u, v)| y=0 = (0, 0), lim y→+∞ u = 0.
(1.4)

Let u s = u s (t, y) a smooth solution of the heat equation

∂ t u s -∂ 2 y u s = 0, u s | y=0 = 0, u s | t=0 = U s , ( 1.5)
with good decay as y → +∞. Clearly, the shear velocity profile (u s , v s ) = (u s (t, y), 0) satisfies the system (1.4). We consider the linearization around (u s , v s ), that is

       ∂ t u + u s ∂ x u + v∂ y u s -∂ 2 y u = 0, in T × R + . ∂ x u + ∂ y v = 0, in T × R + , (u, v)| y=0 = (0, 0), lim y→+∞ u = 0. (1.6)
We wish to study well-posedness properties of (1.6), for a certain class of velocities u s . In this view, we introduce the following functional spaces:

W s,∞ α (R + ) := f = f (y), e αy f ∈ W s,∞ (R + ) , ∀α, s ≥ 0, with f W s,∞ α := e αy f W s,∞
, and

E α,β := u = u(x, y) = k∈Z ûk (y)e ikx , ûk W 0,∞ α ≤ C α,β e -β|k| , ∀k , ∀α, β > 0, with u E α,β := sup k e β|k| ûk W 0,∞ α .
Note that the functions of E α,β have analytic regularity in x. They have only L ∞ regularity in y, with an exponential weight. More regularity in y could be considered as well. Let α, β > 0. We prove in the appendix the following result: Proposition 1 (Well-posedness in the analytic setting)

Let u s ∈ C 0 R + ; W 1,∞ α (R + )
. There exists ρ > 0 such that: for all T with β -ρT > 0, and all u 0 ∈ E α,β , the linear equation (1.6) has a unique solution

u ∈ C ([0, T ); E α,β-ρT ) , u(t, •) ∈ E α,β-ρt , u| t=0 = u 0 .
In short, the Cauchy problem for (1.6) is locally well-posed in the analytic setting. We shall denote

T (t, s)u 0 := u(t, •)
where u is the solution of (1.6) with u| t=s = u 0 . As the spaces E α,β are dense in the spaces

H m := H m (T x , W 0,∞ α (R + y )), m ≥ 0,
this makes sense to introduce the following notation: for all T ∈ L(E α,β , E α,β ),

T L(H m 1 ,H m 2 ) = sup u 0 ∈E α,β T u 0 H m 2 u 0 H m 1 ,
that belongs to R + ∪ {+∞}. In particular, it is infinite when T does not extend to a bounded operator from H m 1 to H m 2 . The main result of our paper is

Theorem 1 (Ill-posedness in the Sobolev setting) i) Let u s ∈ C 0 R + ; W 4,∞ α (R + ) ∩ C 1 R + ; W 2,∞ α (R + )
. Assume that the initial velocity has a non-degenerate critical point over R + . Then, there exists σ > 0, such that for all δ > 0,

sup 0≤s≤t≤δ e -σ(t-s) √ |∂ x | T (t, s) L(H m ,H m-µ ) = +∞, ∀m ≥ 0, µ ∈ [0, 1/2).
ii) Moreover, one can find solutions u s of (1.5) and σ > 0 such that: for all δ > 0,

sup 0≤s≤t≤δ e -σ(t-s) √ |∂ x | T (t, s) L(H m 1 ,H m 2 ) = +∞, ∀m 1 , m 2 ≥ 0.
This theorem expresses strong linear ill-posedness of the Prandtl equation in the Sobolev framework. It is a consequence of an instability process, that holds at high tangential frequencies. We will show that some perturbations with tangential frequency k 1 grow like e √ kt .

The outline of the paper is as follows. Section 2 gives a formal description of the instability mechanism. It relies on an asymptotic analysis of (1.6), in the high tangential frequency limit. Thanks to this analysis, we show that ill-posedness for the PDE (1.6) comes down to a "spectral condition" for a reduced ODE, namely:

(SC) There exists τ ∈ C with Im τ < 0, and a solution W = W (z) of

(τ -z 2 ) 2 d dz W + i d 3 dz 3 (τ -z 2 )W = 0, (1.7
)

such that lim z→-∞ W = 0, lim z→+∞ W = 1.
This spectral condition is studied in section 3, and shown to be satisfied. On these grounds, we prove theorem 1, cf section 4. We end up the paper with numerical computations, which emphasize that our instability mechanism is effective.

The instability mechanism

In this section, we describe the destabilization of system (1.6), leading to the ill-posedness theorem. As we shall see, it takes place at high tangential frequencies, say O(1/ε), and has a typical time O( √ ε). At this timescale, the time dependence of the base velocity (u s (t, y), 0) will not play an important role. Thus, to understand the instability mechanism, we can consider the simpler equation

     ∂ t u + U s ∂ x u + vU s -∂ 2 y u = 0, in T × R + . ∂ x u + ∂ y v = 0, in T × R + , (u, v)| y=0 = (0, 0). (2.1)
Handling of the real equation, that is with u s instead of U s , will require minor modifications, to be achieved in section 4.

System (1.6) has constant coefficients in t and y, so that we can perform a Fourier analysis: we look for solutions in the form

u(t, y) = e ik(ω(k)t+x) ûk (y), v = k e ik(ω(k)t+x) vk (y), k > 0. (2.2)
As we are interested in high tangential frequencies, we denote ε := 1/k 1, and write ω(ε) instead of ω(k), u ε (y), v ε (y) instead of ûk (y), vk (y). The divergence condition yields v ε (y) = -iu ε (y). Using this relation in the first equation in (2.1), one ends up with

(ω(ε) + U s ) v ε -U s v ε + i εv (3) ε = 0, y > 0, v ε | y=0 = v ε | y=0 = 0.
(2.3) Thus, the high frequency limit ε → 0 in variable x yields a singular perturbation problem in variable y. To investigate this problem, one must first consider the inviscid case ε = 0.

The inviscid case

When ε = 0, one can a priori only retain the impermeability condition. The appropriate problem is

(ω + U s ) v -U s v = 0, y > 0, v| y=0 = 0. (2.4)
This spectral problem, as well as the corresponding evolution equation, have been studied exhaustively in [START_REF] Hong | Singularity formation and instability in the unsteady inviscid and viscous Prandtl equations[END_REF]. Clearly, there are non-trivial solutions if and only if ω belongs to the range of -U s . Moreover, the couples

ω a = -U s (a), v a = H(y -a) (U s -U s (a)) , a > 0
where H is the Heaviside function, satisfy (2.4). Note that the regularity of v a depends on the choice of a. When a is a critical point, it belongs to W 2,∞ α (R + ) with a discontinuous second derivative. Otherwise, it is only in W 1,∞ α (R + ), with a discontinuous first derivative. Luckily enough, the additional boundary condition v a | y=0 = 0 is also satisfied.

The viscous perturbation

When ε is not 0, the inviscid eigenelements ω a , v a do not solve (2.3). All boundary conditions are satisfied, cf the above remark, but the equation is not. First, there is a O(ε) remaining term for y > a. More importantly, v a is not smooth at y = a, whereas a solution of this parabolic equation should be.

Nevertheless, at least if a is a non-degenerate critical point, there is an approximate solution near (ω a , v a ). We shall establish this rigorously in section 4. We just give here a formal expansion. It reads

     ω(ε) ∼ ω a + ε 1/2 τ, v ε (y) ∼ v a + ε 1/2 τ H(y -a) + ε 1/2 V y -a ε 1/4 , ( 2.5)
where τ ∈ C, and V = V (z) quickly tends to zero as z → ±∞. Note that the approximation of v ε has two parts: the "regular" part

v reg ε (y) = H(y -a) U s (y) -U s (a) + ε 1/2 τ
and the "shear layer part"

v sl ε (y) = ε 1/2 V y -a ε 1/4 . For ω(ε) = -U s (a) + ε 1/2 τ , the function v reg ε solves (2.
3) up to O(ε), away from the critical point y = a. However, it has a jump at y = a, together with its second derivative. The role of the shear layer v sl ε , which concentrates near y = a, is to cancel these discontinuities. Still formally, we obtain the system satisfied by the profile V :

           τ + U s (a) z 2 2 V -U s (a) z V + i V (3) = 0, z = 0, [V ] | z=0 = -τ, V | z=0 = 0, V | z=0 = -U (a), lim ±∞ V = 0. (2.6)
Let us point out that this system is a priori overdetermined, as jump and boundary conditions provide too many constraints. This justifies the introduction of the parameter τ in the Ansatz (2.5). As we shall see below, there is a τ for which system (2.6) has a solution. Moreover, Im τ is negative. Hence, back to the Fourier representation (2.2), the k-th mode will grow in time like e √ kt . This is the key of the instability mechanism.

To see how the condition (SC) of the introduction steps in, we need a few rewritings. First, τ + U s (a) z 2 2 satisfies the equation in (2.6). We therefore introduce

Ṽ (z) = V (z) + 1 R + τ + U s (a) z 2 2 which leads to        τ + U s (a) z 2 2 Ṽ -U s (a) z Ṽ + i Ṽ (3) = 0, z ∈ R, lim -∞ Ṽ = 0, Ṽ ∼ +∞ τ + U s (a) z 2 2 .
Then, we introduce W such that

Ṽ = τ + U s (a) z 2 /2 W.
We get:

     τ + U s (a) z 2 /2 2 d dz W + i d 3 dz 3 τ + U s (a) z 2 /2 W = 0, lim -∞ W = 0, lim +∞ W = 1.
Finally, we perform the change of variables

τ = 1 √ 2 |U (a)| 1/2 τ , z = 2 1/4 |U (a)| -1/4 z.
Dropping the tildes leaves us with the reduced ODE

     τ + sign(U s (a)) z 2 2 d dz W + i d 3 dz 3 τ + sign(U s (a)) z 2 W = 0, lim -∞ W = 0, lim +∞ W = 1.
If U s (a) < 0, it is exactly the system in (SC). If on the contrary U s (a) > 0, and if (τ, W ) satisfies the system in (SC), then (τ := -τ , W := W ) satisfies the above system. In both cases, back to the original system (2.6), condition (SC) gives a solution (τ, V ) with Im τ < 0. In particular, this √ ε correction to the eigenvalue is a source of strong instability, leading to ill-posedness.

The proof of Theorem 1, which is based on this formal shear layer phenomenon, is postponed to section 4. In the next paragraph, we focus on condition (SC), and prove that it is satisfied.

The spectral condition (SC)

We need to study the existence of heteroclinic orbits for the ODE (1.7). Note that W = 1 is a solution. Equation (1.7) can be written as a second order equation in X = W :

i(τ -z 2 )X -6i z X + (τ -z 2 ) 2 -6i X = 0. (3.1)
To show that (SC) holds, we proceed in three steps.

Step 1. We consider an auxiliary eigenvalue problem:

Au := 1 z 2 + 1 u + 6z (z 2 + 1) 2 u + 6 (z 2 + 1) 2 u = α u. (3.2)
For its study, we introduce the weighted spaces

L 2 := u ∈ L 2 loc , R (z 2 + 1) 4 |u| 2 < +∞ , H 1 := u ∈ H 1 loc R (z 2 + 1) 4 |u| 2 + R (z 2 + 1) 3 |u | 2 < +∞ → L 2 .
with their obvious Hilbert norms. We see A as an operator from D(A)

:= u ∈ H 1 , Au ∈ L 2 into L 2 .
Our goal is to show that A has a positive eigenvalue.

By standard arguments, the domain D(A) is dense in L 2 . Moreover, for any u in D(A), there is a sequence u n of smooth functions with compact support, such that u n → u in H 1 and Au n → Au in L 2 . Integration by parts and use of this density property give easily that

A is symmetric, i.e. ∀u, v ∈ D(A), (Au | v) L 2 = (Av | u) L 2 (3.3)
and that for λ large enough

((λ -A)u | u) L 2 = λ R (z 2 + 1) 4 |u| 2 -6 R (z 2 + 1) 2 |u| 2 + R (z 2 + 1) 3 |u | 2 ≥ 1 2 u 2 H 1 . (3.4)
Then, the coercivity condition (3.4) allows to apply the Lax-Milgram lemma. It implies the invertibility of λ -A, with

(λ -A) -1 f L 2 ≤ (λ -A) -1 f H 1 ≤ C f L 2 .
Moreover, from (3.3), (λ -A) -1 is selfadjoint, and so is A.

First, let us prove that A has positive spectrum. To do so, we claim that it is enough to find u ∈ D(A) with (Au | u) L 2 > 0. Indeed, suppose a contrario that σ(A) is contained in R -. Then, by the spectral theorem,

∀α > 0, (A -α) -1 = 1 d(α, σ(A)) ≤ α -1 .
We deduce: for all u ∈ D(A)

u 2 L 2 ≤ α -2 (A -α)u 2 L 2
Expanding the scalar products, we obtain

0 ≤ α -2 Au 2 L 2 -2α -1 (Au | u) L 2
In the limit α → +∞, we get (Au | u) L 2 ≤ 0 for all u ∈ D(A). This proves our claim. From there, we simply take u = e -2z 2 . A straightforward computation gives

(Au | u) = 439 512 √ π > 0,
and so σ(A) has a positive subset.

It remains to exhibit a positive eigenvalue inside this positive subset of the spectrum. We remark that the operator A can be split into

A = A 1 + A 2 , A 1 u := 1 z 2 + 1 u + 6z (z 2 + 1) 2 u , A 2 u := 6 (z 2 + 1) 2 u.
On one hand, the operator A 1 is negative, and by Lax-Milgram Lemma, A 1 -λ is invertible for any λ > 0. Thus, σ(A 1 ) ⊂ R -. On the other hand, let u n a sequence with u n and A 1 u n bounded in L 2 . This implies that u n is bounded in H 1 , and so has a convergent subsequence in L 2 loc . Moreover, |u n | 2 is equi-integrable over R. Finally, it implies that A 2 u n has a convergent subsequence in L 2 , which means that A 2 is A 1 -compact. Hence, the essential spectra of A and A 1 are the same, see [START_REF] Kato | Perturbation theory for linear operators[END_REF]. In particular, the positive part of σ(A) is made of isolated eigenvalues with finite multiplicity. Eventually, we state: there exists α > 0, and u in D(A) satisfying (3.2).

Step 2. We wish to convert the eigenelements (α, u) of the previous step into an appropriate solution (τ, X) of (3.1). We set τ = -α 1/2 , and z = α -1/4 z, Y (z) = u(z). Dropping the tildes, we obtain a solution of

(τ -z 2 )Y -6 z Y + (τ -z 2 ) 2 -6 Y = 0. (3.5)
By a classical bootstrap argument, Y is smooth. Moreover, it inherits from u its integrability properties at infinity. Actually, the behaviour of Y can be further specified, as shown in:

Proposition 2 The function Y admits a unique extension, still denoted by Y , that is holomorphic in z and satisfies (3.5) in the simply connected domain

U τ := C \ -i∞, -i|τ | 1/2 ∪ i|τ | 1/2 , +i∞ .
Moreover, in the sectors arg z ∈ (-π/4 + δ, π/4 -δ) and arg z ∈ (3π/4 + δ, 5π/4 -δ), δ > 0, it satisfies the inequality

|Y (z)| ≤ C δ exp(-z 2 /4).
Proof. This proposition follows from the general theory of ODE's with holomorphic coefficients. The existence of a holomorphic solution is well-known, because the coefficient τ -z 2 does not vanish on U τ . As regards the inequality, we rewrite equation (3.1) as the first order system:

d dz Y = zA(z)Y, Y = Y z -1 d dz Y , A(z) = 0 1 6-(τ -z 2 ) 2 z 2 (τ -z 2 ) 6 τ -z 2 -1 z 2 . (3.6)
In particular, A is holomorphic at infinity, with A(∞) = ( 0 1 1 0 ). It has two distinct eigenvalues ±1, with eigenvectors 1 ±1 . Hence, we can apply [2, Theorem 5.1 p163]: in any closed sector S inside which Re z 2 does not cancel, there exists solutions Y ± (depending a priori on S) with the following asymptotic behaviour as |z| → +∞:

Y ± ∼   i≥0 Y i ± z α ± -i   e P ± (z) , ( 3.7)
where α ± is a complex constant, and P ± (z) is a polynomial of degree 2. Moreover, the leading term of P ± is ± z 2 2 . Following the same scheme of proof, we get in the present case:

Y 0 ± = 1 ±1 , α ± = - 1 2 (±τ + 7) , P ± = ±z 2 /2. (3.8)
As our solution Y is integrable over R, it is necessarily proportional to the decaying solution.

The bounds in proposition 2 follow. Now, as Y is defined on U τ , we can perform the complex change of variable:

z := e iπ/8 z, τ := e iπ/4 τ Note that τ has a negative imaginary part. Moreover, for z real, the original variable z belongs to the sectors arg z ∈ (-π/4 + δ, π/4 -δ) or arg z ∈ (3π/4 + δ, 5π/4 -δ), with δ = 1/16. By the proposition 2, the function

X(z) = Y (z) satisfies the estimate |X(z)| ≤ C exp(-1/4 z2 ).
Finally, dropping the tildes yields a solution τ, X of (3.1), where X decays at infinity. This concludes step 1.

Step 3. To deduce from the previous step that (SC) holds, it is enough that R X(z)dz be non zero. If so, one can consider

W (z) := R X(z )dz -1 z -∞ X(z ) dz ,
which clearly satisfies all requirements.

Let us assume a contrario that R X(z)dz = 0. Then, the function

V (z) := (τ -z 2 ) z -∞ X(z ) dz is a solution of τ -z 2 V + 2 z V + i V (3) = 0,
which decays exponentially as z goes to ±∞, together with all its derivatives. Differentiation of the equation gives

τ -z 2 V + 2 V + i V (4) = 0,
Then, we multiply by V , that is the complex conjugate of V , and integrate over R. Simple integrations by parts yield:

R (τ -z 2 )|V | 2 -2 R |V | 2 -i R |V (3) | 2 = 0.
The imaginary part of this identity yields

Im τ R |V | 2 = R |V (3) | 2
which contradicts the fact that Im τ < 0. Thus, the condition (SC) is satisfied.

4 Proof of ill-posedness Theorem 1 will be deduced from the formal analysis of section 2. This analysis was performed on (2.1), in which possible time variations of u s were neglected. To account for the original system (1.6) will require a few modifications, notably in the choice of the approximation (2.5).

We will distinguish between the parts i) and ii) of the theorem.

Ill-posedness for general u s

Let u s satisfying the assumptions of part i). Let a be the non-degenerate critical point of u s | t=0 = U s . For the sake of brevity, we consider the case U " s (a) < 0, the other one being strictly similar. The differential equation

∂ t ∂ y u s (t, a(t)) + ∂ 2 y u s (t, a(t)
) a (t) = 0, a(0) = a defines for small time t < t 0 a non-degenerate critical point a(t) of u s (t, •). Let then τ, W given by condition (SC). We set

V := (τ -z 2 )W -1 R + τ -z 2 .
In the light of section 2, we introduce, for ε > 0 and t < t 0 :

ω(ε, t) := -u s (t, a(t)) + ε 1/2 √ 2 |∂ 2 y u s (t, a(t))| 1/2 τ
as well as the "regular" velocity

v reg ε (t, y) := H(y -a(t)) u s (t, y) -u s (t, a(t)) + ε 1/2 √ 2 |∂ 2 y u s (t, a(t))| 1/2 τ ,
and the shear layer velocity

v sl ε (t, y) := ε 1/2 √ 2 ϕ(y -a(t)) |∂ 2 y u s (t, a(t))| 1/2 V |∂ 2 y u s (t, a(t))| 1/4 (y -a(t)) (2ε) 1/4 ,
where ϕ is a smooth truncation function near 0. We then consider the following velocity field:

u ε (t, x, y) := e iε -1 x U ε (t, y), U ε (t, y) = i e iε -1 R t 0 ω(ε,s)ds ∂ y v reg ε (t, y) + v sl ε (t, y) , v ε (t, x, y) := e iε -1 x V ε (t, y), V ε (t, y) = ε -1 e iε -1 R t 0 ω(ε,s)ds v reg ε (t, y) + v sl ε (t, y) .
In order to have a field that is 2π-periodic in x and growing in time, we take ε := 1 n , with n ∈ N * . One verifies easily that u ε = e iε -1 x U ε (t, y) is analytic in x, and W 2,∞ in t, y. Moreover, we have the bounds

c e σ 0 t √ ε ≤ U ε (t, •) W 2,∞ α ≤ C e σ 0 t √ ε , ( 4.1)
for positive constants c, C and σ 0 that do not depend on ε.

Inserting the expression for u ε , v ε into the linearized Prandtl equation (1.6), we obtain

     ∂ t u ε + u s ∂ x u ε + v ε ∂ y u s -∂ 2 y u ε = r ε , in T × R + . ∂ x u ε + ∂ y v ε = 0, in T × R + , (u, v)| y=0 = (0, 0). (4.2)
The remainder term r ε reads

r ε = e iε -1 x R ε (t, y), with R ε (t, y) = e iε -1 R t 0 ω(ε,s)ds -ε -1 u s (t, y) -u s (t, a(t)) -∂ 2 y u s (t, a(t)) y 2 2 ∂ y v sl ε (t, y) + ε -1 ∂ y u s (t, a(t)) -∂ 2 y u s (t, a(t))y v sl ε (t, y) -i ε ∂ 3 y v reg ε (t, y) + i ∂ t ∂ y v reg ε (t, y) + v sl ε (t, y) + O(ε ∞ ) .
The O(ε ∞ ) gathers terms with derivatives of ϕ: as the shear layer profile V decreases exponentially, and the derivatives of ϕ(• -a) are supported away from a, their contribution is indeed exponentially small. Straightforwardly,

R ε (t, •) W 0,∞ α ≤ Ce σ 0 t √ ε , (4.3)
with the same σ 0 as in (4.1).

We are now in a position to prove part i) of Theorem 1. Let us assume a contrario that for all σ > 0, there exists m ≥ 0, µ ∈ [0, 1/2) and δ > 0 such that sup 0≤s≤t≤δ e -σ(t-s)

√ |∂ x | T (t, s) L(H m ,H m-µ ) < +∞. Let T ε (t, s) : W 0,∞ α (R + ) → W 0,∞ α (R + )
the restriction of T (t, s) to the tangential Fourier mode ε -1 . Namely,

T (t, s) e iε -1 x U 0 = e iε -1 x T ε (t, s)U 0 . Similarly, we denote L ε = e -iε -1 x L e iε -1
x , where L is the linearized Prandtl operator around u s . We have, for all 0 ≤ s ≤ t ≤ δ,

T (t, s) L(W 0,∞ α ) ≤ C ε -µ e σ(t-s) √ ε .
Let U = U (t, y) the solution of ∂ t U + L ε U = 0, that coincides initially with the approximation U ε . On one hand, we get

U (t, •) W 0,∞ α ≤ C ε -µ e σt √ ε U (0, •) W 0,∞ α ≤ C ε -µ e σt √ ε . (4.4)
On the other hand, the difference Ũ = U -U ε satisfies, for all t < δ,

Ũ (t, •) = t 0 T ε (t, s)R ε (s)ds. Estimate (4.3) implies Ũ (t, •) W 0,∞ α ≤ C ε -µ t 0 e σ(t-s) √ ε e σ 0 s √ ε ds ≤ C ε 1/2-µ e σ 0 t √ |ε| ,
as soon as σ < σ 0 . Combining this with the estimate (4.1), we obtain the lower bound

U (t, •) W 0,∞ α ≥ U ε (t, •) W 0,∞ α -Ũ (t, •) W 0,∞ α ≥ c e σ 0 t √ ε -Cε µ-1/2 e σ 0 t √ ε
For ε small enough, we get

U (t, •) W 0,∞ α ≥ c e σ 0 t √ ε ,
which contradicts the upper bound (4.4), as soon as σ < σ 0 and t

µ σ 0 -σ | ln(ε)| √ ε.
This achieves the proof of part i).

Stronger ill-posedness for specific u s

It remains to handle part ii) of Theorem 1. Roughly, we must find some u s for which e -σ √ |∂x|(t-s) T (t, s) fails to be bounded from H m to H m-µ , µ ≥ 0 arbitrary. Using notations of the previous paragraph, the keypoint is to build, for any N , a growing solution U ε,N of

∂ t U ε,N + L ε U ε,N = R ε,N , where R ε,N (t, •) W 0,∞ α ≤ C N ε N + t 2N e σ 0 t √ ε .
Indeed, we can then take N + 1/2 > µ, and conclude along the same lines as above.

So far, we have not managed to improve the approximation of the previous paragraph for general u s . This explains the technical restriction µ ∈ [0, 1/2) of part i). In order to obtain a refined approximation, we consider some special profiles: we assume that u s (0, y) = U s (y), for some exponentially decreasing U s , satisfying in the neighborhood of a > 0:

U s (y) = U s (a) (y -a) 2 2 , U s (a) < 0.
Notice that a is a non-degenerate critical point of U s . For such profiles, the approximation of the previous paragraph reads

U ε (t, y) = i e iε -1 R t 0 ω(ε,s) ds ∂ y v reg ε (t, y) + v sl ε (t, y) , ε = 1 n , n ∈ N * .
Using that U s is quadratic near y = a, one can improve this approximation through an expansion of the type

U ε,N (t, y) = U ε (t, y) + i e iε -1 R t 0 ω(ε,s) ds ∂ y N i=1 ε i v i,reg ε (t, y),
with additional terms v i,reg ε . Let us briefly explain the construction of these extra terms. The error terms due to U ε divide into three categories:

1. Shear layer terms involving derivatives of ϕ. As mentioned before, they are O (ε ∞ ), and require no correction.

2. Terms that come from the replacement of u s by its Taylor expansion in the shear layer equation.They read

R ε,1 := -ε -1 e iε -1 R t 0 ω(ε,s) ds u s (t, y) -u s (t, a(t)) -∂ 2 y u s (t, a(t)) y 2 2 ∂ y v sl ε (t, y), R ε,2 := ε -1 e iε -1 R t 0 ω(ε,s) ds ∂ y u s (t, a(t)) -∂ 2 y u s (t, a(t))y v sl ε (t, y).
We write

|R ε,1 | = ε -1 e σ 0 t √ ε y a(t) (z -a(t)) 2 2 ∂ 3 y u s (t, z) dz ∂ y v sl ε (t, y) ≤ ε -1 e σ 0 t √ ε y a(t) (z -a(t)) 2 2 2N -1 k=0 t k k! ∂ k t ∂ 3 y u s (0, z) dz ∂ y v sl ε (t, y) + O(t 2N )e σ 0 t √ ε ≤ ε -1 e σ 0 t √ ε y a(t) (z -a(t)) 2 2 2N -1 k=0 t k k! ∂ 3+2k y U s (z) dz ∂ y v sl ε (t, y) + O(t 2N )e σ 0 t √ ε .
The second inequality stems from a Taylor expansion of u s with respect to t. As u s satisfies the heat equation, each time derivative can be replaced by two space derivatives, so the third line. Because U s is quadratic in a vicinity of a(t) (for short times), and v sl ε and its derivatives decay exponentially fast, we end up with

|R ε,1 | ≤ C t 2N + ε N e σ 0 t √ ε .
A similar bound holds for R ε,2 . Hence, these remainders do not require correction.

3. Terms that come from the time derivative and the diffusion. We focus here on the time derivative, as the diffusion term is simpler and has smaller amplitude. This is

R ε,3 := i e iε -1 R t 0 ω(ε,s) ds ∂ t ∂ y v reg ε (t, y) + v sl ε (t, y) .
Proceeding as for R ε,2 , that is with Taylor expansions in t, leads to

e iε -1 R t 0 ω(ε,s) ds ∂ t ∂ y v sl ε (t, y) ≤ C t 2N + ε N e σ 0 t √ ε .
As regards the regular part,

e iε -1 R t 0 ω(ε,s) ds ∂ t ∂ y v reg ε (t, y) = e iε -1 R t 0 ω(ε,s) ds H(y -a(t)) ∂ t ∂ y u s = e iε -1 R t 0 ω(ε,s) ds H(y -a(t)) F (t, y) + O t 2N e σ 0 t √ ε
where

F (t, y) := 2N -1 k=0 t k k! ∂ 3+2k y U s (y)
comes again from a Taylor expansion in t. The nice thing about this O(1) term is that it is identically zero in the vicinity of y = a(t) (for short times). As a result, the Heaviside function H(y -a(t)) in front of it does not create any discontinuity, and no extra shear layer term is necessary. One takes care of this source term by the introduction of

v 1,reg ε = H(y -a(t)) u s (t, y) -ω(ε, t) y a(t) F (t, z) (u s (t, z) -ω(ε, t)) 2 dz, so that U 1 ε := -i ε e iε -1 R t 0 ω(ε,s) ds ∂ y v 1,reg ε solves ∂ t U 1 ε + L ε U 1 ε = i e iε -1 R t 0 ω(ε,s) ds H(y -a(t))F (t, y) + O(ε).
Proceeding recursively, we obtain an approximation as accurate as we want. This ends the proof of the theorem.

Numerical study

In this last section, we present numerical illustrations of the instability process.

Numerical test of (SC)

To check (SC) numerically, it is more convenient to reformulate it with an Evans function.

We know from Step 2, section 3, that there are solutions Y ± (z) of (3.6) satisfying (3.7)-(3.8). Back to the ODE (3.1), this yields independent solutions X + (τ, •) and X -(τ, •) respectively growing and decaying as z goes to +∞. Furthermore, the following asymptotics holds:

X ± (τ, z) ∼ z ± iτ 2λ -7 2 exp(± 1 2 λz 2 ), ∂ z X ± (τ, z) ∼ ±λz ± iτ 2λ -3 2 exp(± 1 2 λz 2 ), with λ = 1-i √ 2 . Thus, the functions W -(τ, z) := +∞ z X -(τ, s) ds, W + (τ, z) := z 0 X + (τ, s) ds, W 0 (τ, z) = 1,
seen as functions of z, form a basis of solutions of (1.7). They are respectively decaying, growing and constant at +∞. As (1.7) is preserved by the change of variable z → -z, the functions W -(τ, -z), W + (τ, -z), W 0 form a basis as well. They are respectively decaying, growing and constant at -∞. The existence of the heteroclinic orbit is the same as the existence of some constants A and B such that

1 = AW -(-z, τ ) + BW -(z, τ )
for all z, or equivalently

1 0 0 ∈ Vect W -(τ,0) ∂ z W -(τ,0) ∂ 2 z W -(τ,0) , W -(τ,0) -∂ z W -(τ,0) ∂ 2 z W -(τ,0) 0 5e+06 1e+07 1.5e+07 2e+07
√ ε (ω num (ε) + U s (a)
), seen as a function of the tangential frequency k = ε -1 . Top: real part. Bottom: imaginary part. As expected from the theory, both approximately converge to -0.9, as k goes to infinity.

We have discretized this equation in space using finite differences on a stretched grid, and in time through a Crank-Nicholson scheme. Starting from initial random data (i.e. with random values at each gridpoint), we have computed its time evolution for values of k = ε -1 ranging from 1 to 3. if we take for τ the value -0.706-0.706i found in the previous subsection. Moreover, with this value of τ , one can compute directly the solution V of the shear layer equation. After proper rescaling, this allows for comparison between the "numerical" and "theoretical" eigenmodes. More precisely, using the notations of (2.5), one can compare the functions

v th out (y) := 1 √ ε v reg ε (y) v reg ε (∞) - v a (y) v a (∞) and v num out (y) = 1 √ ε v num ε (y) v num ε (∞) - v a (y) v a (∞)
which should both describe the correction to the inviscid eigenmode outside the shear layer.

As regards the shear layer, one can compare Illustrations of these comparisons are given in figures 2 and 3. Again, we obtain an excellent agreement. This confirms that the instability mechanism we have described is indeed effective, and moreover dominates the linear dynamics (1.6).

satisfies

U L ∞ (W 0,∞ α ) ≤ C U 0 W 0,∞ α + F L 1 (W 0,∞ α
) .

( for some ρ. Now, the equation satisfied by U k is

∂ t U k -∂ 2 y U k = i k U s y 0 U k (t, z) dz -U s U k .
Using (5.1), we get

U k (t, •) W 0,∞ α ≤ C U k (0, •) W 0,∞ α + C s k t 0 U k (s, •) W 0,∞ α ds
where C s depends on u s . We conclude by the Gronwall lemma.

Figure 1 :

 1 Figure 1: The correction to the inviscid eigenvalue. Plot of 1√ ε (ω num (ε) + U s (a)), seen as a function of the tangential frequency k = ε -1 . Top: real part. Bottom: imaginary part. As expected from the theory, both approximately converge to -0.9, as k goes to infinity.

10 7 . 1 √ 2

 712 For sufficiently large times, one observes that the numerical solution V num ε behaves likeV num ε (θ, y) ≈ e iω num (ε)θ v num ε ∆θ, y) -V num ε (θ, y) ∆θ V num ε (t, y)gets independent of θ and y. Computations show a relation of the typeω num (ε) ∼ -U s (a) + √ ε (-0.92 -0.91i),see figure1. Here a = is as usual the critical point of U s . This relation is in very good agreement with the theoretical prediction.ω th (ε) := -U s (a) + ε 1/2 |U " s (a)| 1/2 τ √ 2 ≈ -U s (a) + √ ε (-0.92 -0.92i),

  v th in (z) := 1 √ ε v sl ε (ε 1/4 (z + a)) v reg ε (∞) and v num in (z) := 1 √ ε v num ε (ε 1/4 (z + a)) v num ε (+∞)-v num out (ε 1/4 (z + a)).

Figure 2 :

 2 Figure 2: Plots of v th out (dashed line) and v num out (full line), seen as functions of y, at ε = 10 -7 . Left and right figures correspond respectively to the real and imaginary parts. They match, as expected, outside the shear layer.

  Details are left to the reader. This estimate allows to prove proposition 1. Indeed, by decomposing u(t, x, y) =

							5.1)
	This estimate follows directly from the representation formula
				t
	U (t, y) =	R +	S(t, y, z)U 0 (z) dz +	0	R +	S(t -s, y, z)F (s, z) dz ds
	where the heat kernel in the half plane S(t, y, z) reads
	S(t, y, z) := G(t, y -z) -G(t, y + z), G(t, y) :=	1 4πt √	exp(-y 2 /4t).

k∈Z e ikx U k (t, y),

the well-posedness is an easy consequence of the a priori estimate

U k (t, •) W 0,α ≤ C e ρkt U k (0, •) W 0,α

Acknowledgements

The first author would like to thank Guy Métivier, Jeffrey Rauch, and Mikael de la Salle for fruitful discussions. He acknowledges grant ANR-08-JCJC-0104-01.

This last condition is easily seen to be equivalent to W -(τ, 0) = 0, and ∂ 2 z W -(τ, 0) = 0.

Hence, we must find τ with Im τ < 0 such that +∞ 0 X -(τ, s) ds = 0, and ∂ z X -(τ, 0) = 0.

Moreover, we know from Step 3, section 3 that for Im τ < 0, the constraint +∞ 0 X = 0 is satisfied. Finally, the condition (SC) comes down to:

To check this, and get a value for τ , one can use a shooting method. For any τ and any z 0 1, one can start from the approximation

and integrate backwards (3.1) using a Runge-Kutta scheme. This gives access to the function ∂ z X -(τ, z), for any τ and any z ≤ z 0 . Then, a Newton-Raphson procedure allows to find a zero in {Im τ < 0} for the function τ → ∂ z X -(τ, 0). Using such a procedure, we have found τ ≈ -0.706 -0.706 i.

Note that this value is proportional to 1 + i, as expected from the analysis.

Simulation of the instability mechanism

To observe the instability mechanism described in section 2, we have performed direct simulations of system (2.1). More precisely, we have considered the velocity u s (t, y) = U s (y) := 2y exp(-y 2 ) (already studied in [START_REF] Hong | Singularity formation and instability in the unsteady inviscid and viscous Prandtl equations[END_REF] in the inviscid case), and solutions of the type

The profiles V ε = V ε (θ, y) satisfy the singular perturbation problem

One more differentiation gives the parabolic like equation

fulfilled with the boundary conditions

Appendix : Well-posedness in the analytic setting

We start from a simple estimate on the heat equation:


Pierre Lescuyer

Mireille Chevallet

Sylvie Luche

Thierry Rabilloud
email: thierry.rabilloud@cea.fr

Targeted proteomics

Keywords: proteomics, organelle, nuclei, mitochondria

The unit describes the strategy used for targeted proteomics. It relies heavily on previously published units dealing with organelle preparation, protein solubilization and proteomics techniques. A specific commentary for organelle proteomics is provided. Specific protocols for the isolation of nuclei are also provided.

Organelle proteomics

Introduction

With the increasing development of proteomics studies, it becomes more and more obvious that our current proteomics technologies are , by far, not able to cope with the enormous complexity of biological samples. There are therefore two lines of improvement. The first one is to increase the resolving power of the proteomics technologies. While this is in some sense the ideal way, it is also the most difficult and the one for which the results are most difficult to anticipate. the second way is to reduce the complexity of the sample to be analyzed, so that it gets closer to the current performances of the proteomics toolbox. For eukaryotic cells, this relies mainly on the compartimentalization of the cells into various structures such as organelles (i.e. membrane bound subcellular structures) or other components (e.g. nuclear pores, nucleoles etc…), each structure being by far less complex than the complete cell. Thus, a current trend in proteomics is to restrict the analysis to selected subcellular components rather than to complete cells. In addition to providing a better analysis depth by adjusting the complexity of the sample to the performances of the methods, this approach also provides biologically valuable informations such as subcellular localization for proteins.

Targeted proteomics relies on a series of steps, in which the first and obvious ones lead the preparation of the subcellular structure(s) with sufficient purity and reproducibility. This topic has been dealt with in detail in units 4.3 and 22.5 and will therefore not be repeated here. As the cited units deal rather with the isolation of membrane-derived organelles, protocols will be provided for isolation of nuclei. The subsequent steps in targeted proteomics deal with the proteomic analysis per se. This has also been amply detailed in unit 22 (especially units 22.1, 22.2 and 22.4) , with specific aspects also described in units 10 (Electrophoresis) and 16 (mass spectrometry). This will be consequently not re-detailed here. However, a synthetic background discussion will deal with the specific aspects of organelle proteomics compared to the more general aspects discussed in the units mentioned above.

Strategic planning.

In the case of targeted proteomics, the strategic planning will depend whether a single or several components isolated from the biological source is/are to be studied. For a single component, specific isolation procedures can be used which may result in damages or gross contamination of other structures. This cannot be tolerated when several components are to be analyzed. However, one component only-oriented procedures usually result in a higher purity for the target component (see below the example of nuclei).

Basic protocol 1) Isolate the component(s) of interest from the biological source. For all organelles except nuclei, follow the protocols described in units 4.3 and 22.5. For the direct isolation of nuclei, follow alternate protocol 1 or 2. For the simultaneous isolation of nuclei and other organelles, follow alternate protocol 3.

The basic rationale for the isolation of subcellular components is first to lyse the cell under conditions which do not damage the components of interest. This is why nuclei, which are limited by a lamina, can be isolated with non ionic detergents (e.g. alternate protocols 1 and 2), while membrane-limited organelles cannot. This means in turn that the use of detergents in nuclei isolation will dramatically lower the extent of contamination by organelles such as lysosomes or mitochondria. Once the cell has been opened, the components of interest are purified generally by differential centrifugation, using differences in the sedimentation velocity and/or buoyant density of the various subcellular compoents. It must be stressed, however, that there is often a significant overlap in these parameters for various subcellular components. As a result, significant cross-contamination is often experienced, for example between mitochondria and lysosomes. It is also often difficult to separate the total microsomes into the plasma membrane, reticulum and Golgi components. For such delicate final purification, an often interesting track to follow is the use of free-flow electrophoresis (Unit 22.5). This requires however a special apparatus. Furthermore, the price to pay for a high purity preparation is a low yield, and therefore an important sample consumption.

Last but not least, many subcellular components are, one way or another, linked to the cytoskeleton. As this latter component is very strongly assembled, contamination of subcellular structures with cytoskeleton is commonplace and very difficult to remove.

2) Extract the proteins as described in unit 22.4

3) Analyze the proteins by the required proteomics method, e.g. those described in units 22.1 and 22.2. the choice of the method will depend on several, often mutually exclusive, considerations, As an example, membrane proteins are poorly analyzed by techniques based on two-dimensional electrophoresis. However, alternative techniques which perform adequately with membrane proteins, such as those based on SDS electrophoresis followed by LC/MS-MS, are poorly quantitative and offer poor performance in the analysis of posttranslational modifications. The final choice of the experimental proteomics approach will then depend on the aspect that is judged as most important in the study to be carried out.

Alternate protocol 1: direct isolation of nuclei from cultured cells

This protocol is intended for the rapid and direct isolation of nuclei from cultured cells. It is based on the use of nonionic detergents such as Triton X100 to dissolve the lipid bilayers present in the cells, including of course the plasma membrane but also all the inner membrane-bound inner organelles (e.g. Golgi, reticulum, mitochondria and so on) which are thus irreversibly lost. It also dissolves the reticulum-derived cisternae forming the outer nuclear membrane, usually resulting in pure nuclei.

Materials and solutions list cell washing solution. cell lysis solution nuclei wash solution nuclei storage solution Protocol.

Harvest the cells. Collect by centrifugation at 1000g for 5 minutes at 4°C. Wash them twice in ice-cold cell washing solution by resuspension and recentrifugation as above. Estimate the volume of the packed cell pellet.

Resuspend the cell pellet in 10 times its volume of ice-cold cell lysis solution. Incubate on ice for 20 minutes with vortexing for 20 seconds every 5 minutes. Collect the nuclei by centrifugation at 1000g for 5 minutes at 4°C. Resuspend once in an identical volume of icecold cell lysis solution. Incubate on ice for 5 mionutes and recentrifuge as above. resuspend in an identical volume of ice-cold nuclei wash solution, let stand on ice for 5 minutes and recentrifuge as above. Repeat this rinse once. After the last centrifugation, estimate the nuclei pellet volume and resuspend thoroughly in the same volume of storage solution. Dispense in aliquots and store at -80°c, or better in liquid nitrogen.

Alternate protocol 2: direct isolation of nuclei from tissues solutions: same as in alternate protocol 1. protocol: cut the tissue of interest in small pieces with scissors. Wash the pieces with cell washing solution. Homogeneize the tissue pieces in 10 times their columes of cell lysis solution with the help of a Dounce glass-glass homogeneizer . Ten strokes are usually sufficient. Collect and wash the nuclei as described in the basic protocol Alternate protocol 3: cleaning nuclei isolated by other procedures.

the basic protocol and alternate protocol 1 use direct disruption of the cells in a buffer containing Triton X100. As this results in the irreversible loss of most other organelles, these protocols cannot be used when other organelles than the nuclei are also to be collected. In this case, cell disruption protocols respecting the integrity of the organelles are used (see unit 4.3). These protocols generally provide a nuclei-rich pellet at the end of the first low speed centrifugation. This pellet contains nuclei, unbroken cells and contaminating organelles (e.g. mitochondria or membrane sheets). Cleaning of the nuclei is therefore advisable.

Solutions: same as in alternate protocol 1.

protocol: resuspend the nuclei pellet in 10 times its volume with the help of a Dounce glassglass homogeneizer . Five strokes are usually sufficient. Collect and wash the nuclei as described in the basic protocol Support protocol: evaluation of nuclei purity. because of the use of a detergent, the nuclei are generally devoid of contamination by other organelles. The specific enzymes test described in unit 4.3 does not provide an accurate estimate for purity. The most likely contaminants for these nuclei preparation are cytoskeletal proteins and ribosomes. Although these are difficult to quantitate, a good idea of nuclei purity can be provided by microscopy examination using the following protocol.

Add one drop of microscopy stain solution to one drop of nuclei suspension on a microscope slide. Cover with a coverslip and leave at room temperature for 10 minutes before examination on a microsope. Magnification should be at least 200x, and phase contrast affords additional information.

Nuclei appear in light green. One or two pink-colored nucleolae should be visible inside the nuclei. Contaminatiing cytoplasm results in pink material (i.e. ribosome-containing) outside the nuclei. Cytoskeleton is visible under phase contrast as refringent filaments.

Reagents and solutions: cell washing solution: 10 mM phosphate buffer pH 7.5, 0.125M NaCl cell lysis solution: HEPES-NaOH 10 mM pH 7.5, dithiothreitol 1mM, spermidine (trihydrochloride) 1mM, spermine (tetrahydrochloride) 0.25mM, EDTA 0.5 mM, Triton X 100 0.1% (w/v). Add 30µM of 3,4 dichloroisocoumarin (Fluka, predissolved at 30mM in dimethylformamide and kept as stock at -20°C) just before use. nuclei wash solution: HEPES-NaOH 10 mM pH 7.5, dithiothreitol 1mM, sucrose 0.2M, MgCl2 2mM. nuclei storage solution: HEPES-NaOH 10 mM pH 7.5, dithiothreitol 1mM, glycerol 25% (v/v), MgCl2 5mM microscopy stain solution: dissolve 150 mg of methyl green and 250 mg of pyronin Y in 2.5 ml of 95% ethanol. Add 50mM acetate buffer pH 4.8 to 100ml. Add then 40g of sucrose and by mixing. Keep at 4°C in tightly stoppered flasks (to avoid evaporation of the acetic acid)

Background information:

As mentioned above, the rationale in targeted proteomics is to analyze a defined subset of cellular proteins in order to adapt the complexity of the sample to the performances of the analysis tool. Although this trend appeared some time ago (Wilkins et al. 1995[START_REF] Gygi | Evaluation of two-dimensional gel electrophoresis-based proteome analysis technology[END_REF], it becomes more and more apparent that the proteomics toolbox cannot face the complexity of the real biological samples. This holds true for all the methods described to date, whether they are based on 2D gels [START_REF] Gygi | Evaluation of two-dimensional gel electrophoresis-based proteome analysis technology[END_REF] or not [START_REF] Wu | A method for the comprehensive proteomic analysis of membrane proteins[END_REF]. From this evidence, the idea of analyzing only cellular subsets of proteins became obvious a few years ago [START_REF] Rabilloud | Two-dimensional electrophoresis of human placental mitochondria and protein identification by mass spectrometry: toward a human mitochondrial proteome[END_REF][START_REF] Jung | Proteomics meets cell biology: the establishment of subcellular proteomes[END_REF]. These subsets can be obtained from biochemical fractionation of the proteins of the complete sample [START_REF] Fountoulakis | Two-dimensional map of Haemophilus influenzae following protein enrichment by heparin chromatography[END_REF][START_REF] Zuo | Comprehensive analysis of complex proteomes using microscale solution isoelectrofocusing prior to narrow pH range two-dimensional electrophoresis[END_REF], but the performances of the classical biochemical fractionation become limiting very soon. Another approach, which is represented by organelle proteomics, takes advantage of the existing compartimentalization of the eukaryotic cells. In addition to providing samples of reduced complexity for proteomics analysis, organelle proteomics provides cell biology-relevant informations, e.g. the presence of "new" or "unexpected" proteins in a given organelle. This can give rise in turn to important observations in terms of cell biology [START_REF] Gagnon | Endoplasmic reticulum-mediated phagocytosis is a mechanism of entry into macrophages[END_REF]. The relevance of organelle proteomics depends in turn on the purity and reproducibility of the organelle preparation, and these parameters obviously play a role at different levels for modulating the quality of the final proteomics analysis.

Critical parameters

The first, obvious level at which organelle purity plays its role is of course the accuracy of the analysis. A low purity will of course results in contaminating proteins being present in the organelle map. This can lead to misinterpretation of the results, especially when unexpected proteins are present in the map. Another important consequence of organelle purity is the depth of the resulting proteomics analysis. A good example of this phenomenon is provided by the analysis of mitochondria [START_REF] Zischka | Improved proteome analysis of Saccharomyces cerevisiae mitochondria by free-flow electrophoresis[END_REF]. Even for organelle proteomics, the limiting factor is the capacity of the proteomics toolbox. Consequently, when organelle of low purity are analyzed, contaminating proteins occupy a substantial fraction of the analysis space (2D gels or peptide MS peaks) and prevent minor, bona fide organelle proteins from being analyzed. This problem gradually disappears when the organelle purity rises, allowing the analysis of minor proteins with pure organelles [START_REF] Zischka | Improved proteome analysis of Saccharomyces cerevisiae mitochondria by free-flow electrophoresis[END_REF], which is always a challenge in proteomics.

The second critical point is of course the reproducibility of the organelle preparation. While a poor reproducibility is a minor problem for cartographic proteomics purposes (i.e. what is the protein composition of a given sample), it becomes a major problem when comparative, physiology-oriented studies are carried out, as in clinical proteomics. This reproducibility problem can be really tricky to solve, as the changes induced by the pathology studied or upon the the physiological process of interest can interfere with the organelle preparation. As an example, any change involving a major change in the cytoskeleton, a major player and problem in organelle preparation, will alter the purity of the final organelle preparation. This will result in turn into changes in the proteomics map when comparing the different physiological situations of interest. However, these changes are artefactual and, although linked to a change in the cell physiology, are not really related to the organelle of interest. The robustness of the protocol used for organelle preparation is therefore quite critical in such transversal studies, and rather difficult to establish, as a protocol performing well with control samples can fail insidiously with other samples used in the study.

Another major caveat in targeted proteomics is sample consumption, as examplified by proteomics of nuclear proteins. Nuclei represent 10% of the total cell protein mass, and the isolation procedures have a yield close to 50%. This means in turn that 5% of the initial sample is used at the end, and 95% of the protein content is lost. While such losses can be dealt with in some experimental models, they are usually not tolerable when the sample supply is limited, e.g. in clinical proteomics.

These problems lead in turn to the choice of the technology to be used for targeted proteomics. This is obviously linked to the aims of the study to be performed. 2D gel-based technologies are the best choice when quantitative changes and/or changes in post translational modifications are important in the design of the study. As mentioned above, this is seldom the case in current targeted proteomics. Moreover, these technologies have usually a high sample consumption and a strong chemical bias against some classes of proteins, such as membrane proteins. Conversely, peptide separation-based approaches (e.g. in [START_REF] Washburn | Large-scale analysis of the yeast proteome by multidimensional protein identification technology[END_REF], do not have the same drawbacks as 2D gels and are therefore highly complementary. Their interest in targeted proteomics is linked to their low sample consumption [START_REF] Pflieger | Systematic identification of mitochondrial proteins by LC-MS/MS[END_REF], and to the fact that they do not have the same bias against classes of proteins. Membrane proteins present in the sample are therefore more easily analyzed [START_REF] Pflieger | Systematic identification of mitochondrial proteins by LC-MS/MS[END_REF]. These positive features are shared both by methods relying only on peptide separation [START_REF] Washburn | Large-scale analysis of the yeast proteome by multidimensional protein identification technology[END_REF], and by methods using a first step of protein separation by SDS electrophoresis and then a peptide separation step (e.g. in [START_REF] Rout | The yeast nuclear pore complex: composition, architecture, and transport mechanism[END_REF][START_REF] Bell | Proteomics characterization of abundant Golgi membrane proteins[END_REF][START_REF] Pflieger | Systematic identification of mitochondrial proteins by LC-MS/MS[END_REF], which seem easier to implement that the peptide-only approaches [START_REF] Pflieger | Systematic identification of mitochondrial proteins by LC-MS/MS[END_REF].

Anticipated results

Keeping in mind these caveats linked to targeted proteomics, the benefits of targeted proteomics compared to global proteomics can be questioned. However, targeted proteomics brings invaluable information which is out of reach by global proteomics methods.

The first type of information, on a structural point of view, is the identification of the components of the component of interest. Recent examples can be found in organelle proteomics on the mitochondria [START_REF] Taylor | An alternative strategy to determine the mitochondrial proteome using sucrose gradient fractionation and 1D PAGE on highly purified human heart mitochondria[END_REF][START_REF] Lescuyer | Progress in the definition of a reference human mitochondrial proteome[END_REF], on the golgi [START_REF] Bell | Proteomics characterization of abundant Golgi membrane proteins[END_REF] or on the peroxisome [START_REF] Yi | Approaching complete peroxisome characterization by gas-phase fractionation[END_REF]. On the side of non-orrganelle subcellular components, examples can be found on the nuclear pore [START_REF] Rout | The yeast nuclear pore complex: composition, architecture, and transport mechanism[END_REF] or on the nucleolus (Andersen et al. 2002[START_REF] Scherl | Functional proteomic analysis of human nucleolus[END_REF]. These studies also point out the fact that the proteomics part is even not the most difficult part. the validation of the data, i.e. the confirmation of the assignment made to the organelle via the proteomics approach, is much more difficult and time-consuming [START_REF] Rout | The yeast nuclear pore complex: composition, architecture, and transport mechanism[END_REF]. In some occasions, the proteomics approach points to unexpected but verified localizations of proteins. Such events can be the entry point to more cell physiology-oriented studies, as exemplified in [START_REF] Gagnon | Endoplasmic reticulum-mediated phagocytosis is a mechanism of entry into macrophages[END_REF].

The second stage of targeted proteomics lies in more dynamic studies, in which changes in the organelle proteome occurring during various physiological conditions are investigated. Examples of this approach can be found even before the word proteomics even existed [START_REF] Howe | Cytoplasmic and nuclear protein synthesis in preimplantation mouse embryos[END_REF]Solter 1980, Rabilloud et al. 1991). These approaches are however difficult to use because of the sample consumption, as outlined above.

As a conclusion, these questions in the best choice of a subtype of proteomics approach for targeted proteomics must not mask the fact that the real bottleneck in the field is clearly the reproducible preparation of pure subcellular structures.

Literature cited


V Lullien-Pellerin
email: valerie.lullien-pellerin@inrae.fr

Both genetic and environmental conditions affect wheat grain texture: Consequences for grain fractionation and flour properties

Keywords: hardness, milling, puroindoline, vitreousness

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

Introduction

The first step of wheat grain processing, i.e. milling, corresponds to successive grinding and sieving operations [START_REF] Posner | Wheat flour milling[END_REF] with the aim of separating the starchy endosperm from the peripheral tissues (the outer layers and the germ). During this process, the mechanical properties of grain tissues play a major role in the grain breaking behavior and in the fate of the different tissues and consequently in the properties of the resulting product [START_REF] Morris | Wheat. In: Cereal grain quality[END_REF][START_REF] Symes | Influence of a gene causing hardness on the milling and baking quality of two wheats[END_REF].

Direct or indirect methods have been used to evaluate the mechanical resistance of the grain [START_REF] Pasha | Grain hardness: a major determinant of wheat quality[END_REF]. One of the indirect methods used is based on the measurement of the percentage of particles passing through a sieve of defined aperture (75 µm) after grinding the grain, resulting in a particle size index (PSI) that is used to classify wheat in two classes [START_REF] Williams | Attempts at standardization of hardness testing of wheat 1. The grinding sieving (particle size index) method[END_REF]) "soft" and "hard" depending on the PSI obtained. The PSI method was progressively replaced by a near-infrared reflectance spectroscopy (NIRS), which measures scattering of the analyzed ground grain sample at specific wavelengths in comparison with wheat grains with contrasted known PSI values and is used to classify wheat samples in the different classes [START_REF] Norris | A definition of wheat hardness using near infrared reflectance spectroscopy[END_REF][START_REF] Saurer | Use of infra-red reflectance measurement for determination of protein and water content and grain hardness in wheat[END_REF]. A more direct measurement of grain mechanical resistance uses the Single Kernel Characterization System (SKCS) developed by [START_REF] Martin | Development of single kernel wheat characterization system[END_REF] to evaluate the crushing force required to break the grains using a hardness index (HI). Further mechanical resistance of specific grain tissues was also developed to specifically characterize their resistance to breaking at a given temperature and water content. In particular, endosperm bricks or cylinders [START_REF] Haddad | Rapid General Method for Appraising the Rheological Properties of the Starchy Endosperm of Cereal Grains[END_REF]Morris et al., 2011a) were found to be able to correctly evaluate the physical grain resistance to breaking and to be related to the previous hardness classification. Similar measurements of the mechanical properties of tissue samples with well-defined dimensions were also developed for the grain outer layers [START_REF] Mabille | Mechanical properties of seed coats[END_REF] and further refined to characterize almost all component tissues [START_REF] Antoine | Individual Contribution of Grain Outer Layers and Their Cell Wall Structure to the Mechanical Properties of Wheat Bran[END_REF].

A major locus called Ha that controls grain hardness was identified on the short arm of chromosome 5D in common wheat (Triticum aestivum) using chromosome substitution lines [START_REF] Law | The study of grain-protein control in wheat using whole chromosome substitution lines[END_REF][START_REF] Mattern | Locations of genes for kernel properties in the wheat variety "Cheyenne" using chromosome substitution lines[END_REF] and its emergence recorded during wheat selection [START_REF] Chantret | Molecular Basis of Evolutionary Events That Shaped the Hardness Locus in Diploid and Polyploid Wheat Species (Triticum and Aegilops)[END_REF][START_REF] Charles | Sixty Million Years in Evolution of Soft Grain Trait in Grasses: Emergence of the Softness Locus in the Common Ancestor of Pooideae and Ehrhartoideae, after their Divergence from Panicoideae[END_REF]. It was found to encode two specific low molecular weight and tryptophan and cysteine enriched proteins, called puroindolines (PINs) A and B [START_REF] Bhave | Molecular genetics of puroindolines and related genes: regulation of expression, membrane binding properties and applications[END_REF]. These proteins were suspected to play a role in the starch-protein adhesion in the starchy endosperm [START_REF] Chichti | Changes in the starchprotein interface depending on common wheat grain hardness revealed using atomic force microscopy[END_REF][START_REF] Pauly | Wheat (Triticum aestivum L. and T. turgidum L. ssp. durum) kernel hardness: I. Current view on the role of puroindolines and polar lipids[END_REF][START_REF] Turnbull | Wheat endosperm texture[END_REF]. The presence of the wild-type alleles, Pina-D1a and Pinb-D1a, of the genes encoding PINA and PINB results in a soft phenotype whereas deletion or mutations of one of the genes results in a hard phenotype. As durum wheat does not contain the D genome, its grain mechanical properties are the hardest.

A number of different sequences of PIN genes have been recorded but their occurrence depends on wheat selection in different countries. The most common mutation of Pina-D1 in the USA is a null mutation (Pina-D1b) that results in the absence of PINA from the endosperm [START_REF] Morris | Registration of hard kernel puroindoline allele near-isogenic line hexaploid wheat genetic stocks[END_REF]. In Europe, the most frequent mutation occurs in Pinb-D1 gene (allele Pinb-D1b) and corresponds to a Gly to Ser substitution in position 46 [START_REF] Huang | Development of SNP assays for genotyping the puroindoline b gene for grain hardness in wheat using pyrosequencing[END_REF]. The two other relatively frequent mutated alleles, Pinb-D1c and Pinb-D1d, correspond, respectively, to a Leu to Pro change in position 60 [START_REF] Lillemo | A leucine to proline mutation in puroindoline b is frequently present in hard wheats from Northern Europe[END_REF] or to a Trp to Arg change in position 44 [START_REF] Huang | Development of SNP assays for genotyping the puroindoline b gene for grain hardness in wheat using pyrosequencing[END_REF].

A clear relationship between the presence of both puroindoline wild-type alleles and a soft grain phenotype was further reinforced by complementation of corresponding null [START_REF] Wanjugi | The role of puroindoline A and B individually and in combination on grain hardness and starch association[END_REF] or mutated alleles [START_REF] Beecher | Expression of wild-type pinB sequence in transgenic wheat complements a hard phenotype[END_REF]. Moreover, the introduction of PIN genes in other cereals that lack the corresponding genes was found to reduce grain hardness [START_REF] Krishnamurthy | Expression of wheat puroindoline genes in transgenic rice enhances grain softness[END_REF]Morris et al., 2011b;[START_REF] Zhang | Seedspecific expression of the wheat puroindoline genes improves maize wet milling yields[END_REF]. Grain mechanical properties were also found to be affected by endosperm porosity (Dobraszcyk et al., 2002;[START_REF] Haddad | Rheological behaviour of wheat endosperm-proposal for classification based on the rheological characteristics of endosperm test samples[END_REF], which can be evaluated by visual observation of a white endosperm (mealiness), whereas non-porous grains are glassy in appearance. Different methods for more accurate evaluation of vitreousness have been developed in recent years using either transmitted light [START_REF] Neethirajan | Classification of vitreousness in durum wheat using soft x-rays and transmitted light images[END_REF][START_REF] Venora | Quality assessment of durum wheat storage centres in Sicily: evaluation of vitreous and shrunken kernels using an image analysis system[END_REF][START_REF] Xie | Detecting vitreous wheat kernels using reflectance and transmittance image analysis[END_REF], soft X-ray, dual energy X-ray or light reflectance [START_REF] Neethirajan | Classification of vitreousness in durum wheat using soft x-rays and transmitted light images[END_REF][START_REF] Neethirajan | Dual energy X-ray image analysis for classifying vitreousness in durum wheat[END_REF][START_REF] Xie | Detecting vitreous wheat kernels using reflectance and transmittance image analysis[END_REF] or near infrared hyper-spectral image analysis [START_REF] Gorretta | Determining vitreousness of durum wheat kernels using near infrared hyperspectral imaging[END_REF][START_REF] Serranti | The development of a hyperspectral imaging method for the detection of Fusarium-damaged, yellow berry and vitreous Italian durum wheat kernels[END_REF]. All these methods require a well characterized reference sample set for classification of the analyzed sample that consequently needs similar characteristics such as grain geometry, tissue thickness, and color. Recently, a clear relationship between grain porosity and vitreousness was demonstrated through the study of light transmission through wheat grain longitudinal cross sections of different thickness and was shown to strictly follow a Beer-Lambert law [START_REF] Chichti | A wheat grain quantitative evaluation of vitreousness by light transmission analysis[END_REF]. This enabled direct quantification of endosperm vitreousness.

Environmental conditions during wheat grain growth were suspected to play a role in grain porosity despite the possible influence of the genotype background [START_REF] Sharp | Wheat and flour studies, IX. Density of wheat as influenced by freezing, stage of development, and moisture content[END_REF][START_REF] Stenvert | The influence of the physical structure of the protein matrix on wheat hardness[END_REF]. In particular, the influence of environmental conditions on protein content and its relationship with vitreousness was pinpointed in durum wheats as it was controversially discussed in the literature [START_REF] Sieber | Vitreosity, its stability and relationship to protein content in durum wheat[END_REF] and further reinvestigated in durum or common wheats [START_REF] Oury | Relationships between wheat grain physical characteristics studied through near-isogenic lines with distinct puroindoline-b allele[END_REF][START_REF] Fu | Kernel vitreousness and protein content: Relationship, interaction and synergistic effects on durum wheat quality[END_REF]. The relationship between vitreousness and the protein content was shown to be very weak at least in soft common wheat and to depend on the protein content value in hard common wheats [START_REF] Oury | Relationships between wheat grain physical characteristics studied through near-isogenic lines with distinct puroindoline-b allele[END_REF], as well as in durum wheats [START_REF] Fu | Kernel vitreousness and protein content: Relationship, interaction and synergistic effects on durum wheat quality[END_REF].

To clarify the respective roles of puroindoline genes and of environmental conditions in determining the microstructure and mechanical characteristics of wheat grains, as well as their milling behavior, only wheat lines or cultivars whose PIN alleles were known or controlled were grown in different conditions.

Environmental conditions affect grain porosity as revealed by measurements of vitreousness

Near isogenic lines (NILs) of common wheat differing only in the puroindoline allele encoding PINB, were grown along with a control cultivar in three consecutive years (2007,2008,2009) at seven distinct sites in France (ranging from 45°46'N to 50°28'N and from 1°33'E to 3°12'E) with two different nitrogen supplies applied (one adjusted to obtain a high grain yield of around 9t/ha depending on the measured soil N content, and an identical supply with an additional 50 kg/ha at flowering), and their grain characteristics were measured [START_REF] Oury | Relationships between wheat grain physical characteristics studied through near-isogenic lines with distinct puroindoline-b allele[END_REF]. The different NILs carried either the wild-type allele encoding PINB or one of the mutated alleles (Pinb-D1b or Pinb-D1d). The wheat microstructure was evaluated by visual estimation of vitreousness on grain sections (n=500 for each wheat sample) with a Pohl grain cutter, as described in [START_REF] Lasme | A study of puroindoline b gene involvement in the milling behaviour of hard-type common wheat[END_REF].

Analysis of variance including the following factors: year, location and nitrogen supply, revealed a significant effect of all of the factors on vitreousness with a high level of explanation of the model (adjusted r 2 =0.67). The wheat grain samples collected (n=304) showed higher vitreousness in 2008 than in the two other years of cultivation (Figure 1A). But marked differences in vitreousness between the different cultivation sites were also observed in each sample in each year, as shown for one of them in Figure 1B. A higher vitreousness level was also observed with the increased supply of nitrogen (Figure 1C).

Figure 1

Concerning the mechanical characteristics measured by NIRS hardness or SKCS HI, only a year effect was found to be highly significant. When measured using NIRS, neither the location nor the nitrogen supply appeared to explain variability of hardness (adjusted r 2 =0.06) and when measured by SKCS HI, location and nitrogen supply were only significant at the 5% level with a quite low level of explanation of the model (adjusted r 2 =0.18).

In conclusion, environmental conditions clearly affect the grain microstructure as measured by the level of vitreousness.

Pinb-D1 allele appears to affect both grain hardness and porosity

Comparisons were made between the common wheat grain characteristics depending on the Pinb-D1 allele carried [START_REF] Oury | Relationships between wheat grain physical characteristics studied through near-isogenic lines with distinct puroindoline-b allele[END_REF] and are summarized in Table 1 with their average mean values. Significant differences were found in the NIRS and SKCS mean values between grains from NILs (NIL1) carrying the wild-type allele, Pinb-D1a, or the mutated allele, Pinb-D1b; the latter presenting 3-fold higher average NIRS or SKCS values. This result was expected as NIRS and SKCS measurements were carried out with the aim of dividing wheat grain samples into soft or hard classes. But surprisingly, the mean vitreousness level was also found to differ between NILs carrying either the wild-type or the mutated allele of Pinb-D1.

Near isogenic lines carrying the wild-type allele Pinb-D1a, and thus displaying a soft phenotype were found to present 1.5-fold lower mean vitreousness values than the lines carrying the mutated allele, Pinb-D1b, classified as a hard phenotype. Therefore, the presence of Pinb-D1a increases wheat grain porosity regardless of the environmental conditions. A similar decrease in both PSI or SKCS grain hardness and vitreousness due the presence of wild-type alleles of the puroindolines was also reported by [START_REF] Heinze | Puroindoline genes introduced into durum wheat reduce milling energy and change milling behavior similar to soft common wheats[END_REF] in a study of durum wheat in which these genes were translocated (Morris et al., 2011b) in comparison with the non-translocated line. In both cases, the differences in vitreousness cannot be linked with differences in protein content as frequently reported. Indeed, the mean protein content in common wheat grains carrying either Pinb-D1a or Pinb-D1b was (11.8-11.9%, Oury et al., 2015) whereas it was around 16% (15.8-16.2 %) for durum wheat grains carrying or not the puroindoline genes.

Table 1

Comparison of near-isogenic lines carrying either the mutated allele, Pinb-D1b or Pinb-D1d (NIL2) which both lead to a hard phenotype, revealed a highly significant difference (p value <0.001) in average NIRS values, wheat grains carrying Pinb-D1b displaying higher grain hardness measured by NIRS than grains carrying the other mutation Pinb-D1d (Table 1).

Similarly, grains carrying Pinb-D1b also displayed higher SKCS values but with lower significance while average vitreousness values did not differ significantly. These results reflect the fact that NIRS and SKCS measurements do not evaluate the same hardness characteristics. As no significant differences in vitreousness were observed between wheat samples carrying Pinb-D1b and Pinb-D1d, the impact on the required breaking energy, as measured with SKCS was weakly significant. However, differences in the particle size between grounded samples were observed.

Relationships between NIRS hardness, SKCS HI and vitreousness

Relationships between indirect hardness measurements (NIRS hardness) based on the size of the particles after grinding, or direct evaluation of the grain mechanical resistance (SKCS hardness) and vitreousness were studied as a function of the allele, wild-type or mutated, of puroindoline B (Fig. 3).

Figure 3

Figure 3 shows that, regardless of the level of vitreousness, NIRS hardness efficiently distinguish between common wheat near-isogenic lines carrying either the wild-type alleles or the mutated alleles encoding PINB. Only very weak relationships (r 2 <0.3 for common wheat carrying the wild-type or mutated alleles encoding PINB) were found between NIRS hardness and vitreousness measurements (Fig. 3A). By contrast, a strong relationship was found between SKCS HI values and vitreousness (r 2 = 0.73 for soft near-isogenic lines and r 2 = 0.75 for hard genotypes), as illustrated in Fig. 3B. Thus, vitreousness increases the energy required to break the grains. Fig. 3B also shows that the presence of both wild-type alleles encoding PINA and PINB reduces grain mechanical resistance. It also clearly demonstrates that SKCS values above 25 and below 45 can be obtained either with vitreous soft wheat grains or hard wheat grains with low vitreousness. Interestingly, vitreousness values higher than 60% were only found in wheat grains carrying the mutated alleles of Pinb-D1.

Both puroindoline allele encoding PINB and vitreousness affect milling behavior

The milling behavior of common wheat near-isogenic lines carrying the wild-type allele of puroindoline b gene or the mutated alleles and displaying different vitreousness was studied [START_REF] Oury | Bread wheat milling behavior: effects of genetic and environmental factors, and modeling using grain mechanical resistance traits[END_REF]. Results were analyzed in each type of allele but also depending on vitreousness while taking its effect on grain mechanical resistance into account. For this analysis, the grain sample population within each near-isogenic line was divided into two groups according to half the maximum vitreousness value. Significant differences in grain or semolina breaking were found between groups and are reported in Figure 4. Vitreousness was shown to significantly reduce the breaking flour yield at milling whatever the genetic background (Fig. 4A and4B). Reduction of semolina at the reducing step was also studied through the percentage of sizing flour produced from this fraction. Sizing flour production was shown to be reduced only in the case of wheat grains from near-isogenic lines carrying the mutated alleles and displaying a high level of vitreousness (Fig. 4C and4D). Thus, if vitreousness appeared to mainly reduce the ability to produce flour in the case of hard wheat grains (Fig. 4B and4D), in the case of soft wheat grains that carry the wild-type alleles of both puroindoline genes, the negative effect of high vitreousness on break flour production (Fig. 4A) appeared to be balanced by a positive effect on the production of sizing flour (Fig. 4C). As a consequence, total flour appeared to both depend on the puroindoline allele of puroindoline b gene carried and on the level of vitreousness.

Figure 4

Analysis of total flour production from French common wheat cultivars (n=197) carrying different alleles encoding PINB (Pinb-D1a, Pinb-D1b, Pinb-D1c or Pinb-D1d), together with those from the near-isogenic lines (n=72), carrying either Pinb-D1a, Pinb-D1b, or Pinb-D1d, revealed that the highest yield was obtained using grains displaying SKCS hardness values ranging between 30 and 60, which corresponds to either soft vitreous grains or hard grains with a low level of vitreousness (Oury et al., 2017 and Fig. 5).

Figure 5 6. Starch damage and separation between the starchy endosperm and the envelopes

The involvement of both puroindoline genes and vitreousness in the energy required for breaking grains was shown to impact the yield of total flour through the size of the resulting particles, but was also shown to potentially impact flour composition.

Due to the role of puroindolines in reducing the cohesion between the protein network and the starch particles, particle size distribution of flours obtained from near-isogenic lines (Greffeuille et al., 2006a) or common wheat cultivars carrying both of the wild-type alleles encoding PINA and PINB was found to present a bi-modal curve with a first population of particles around the starch granule size (25 µm) and a second population around the flour particle size (150 µm). Conversely, common wheat cultivars or near-isogenic lines carrying the mutated alleles encoding PINB led to a mono-modal distribution around the flour particle size (150 µm). Consequently, less damage to starch was observed in flours made from grains carrying both wild-types alleles encoding PINA and PINB [START_REF] Mayer-Laigle | DRY biorefineries: Multiscale modeling studies and innovative processing[END_REF].

As vitreousness also impacts mechanical resistance of the starchy endosperm, differences in the level of starch damage between vitreous or non-vitreous grains in common wheat nearisogenic lines carrying the same type of puroindoline b allele have also been reported [START_REF] Mayer-Laigle | DRY biorefineries: Multiscale modeling studies and innovative processing[END_REF]. Mealy and vitreous grains carrying both wild-type alleles encoding PINA and PINB displayed a mean starch damage level of 1.9% and 2.2%, respectively, whereas mealy and vitreous grains carrying the mutated allele encoding PINB displayed a mean starch damage level of 3.4% and 5.2%, respectively.

Similarly, the particle size distribution of flours obtained after grain milling of durum wheat (c.v. Svevo) was found to display a mono-modal form whereas flours from the same cultivar into which both wild-type puroindoline genes were introduced (Svevo-Pin) presented a bi-modal curve [START_REF] Heinze | Puroindoline genes introduced into durum wheat reduce milling energy and change milling behavior similar to soft common wheats[END_REF]. Consequently, differences in the starch damage level in milling flours from durum wheat (c.v. Svevo) and Svevo-Pin were 8.3% and 1.9%, respectively. A similar decrease in flour particle size and reduced starch damage due to the introduction of puroindoline genes into durum wheat (c.v. Svevo) has also been observed by other authors [START_REF] Murray | Effect of Soft Kernel Texture on the Milling Properties of Soft Durum Wheat[END_REF][START_REF] Quayson | Empirical rheology and pasting properties of soft-textured durum wheat (Triticum turgidum ssp. durum) and hardtextured common wheat (T. aestivum)[END_REF].

These differences in the level of starch damage may therefore alter the water absorption properties of flours [START_REF] Morrison | Properties of damaged starch granules. IV Composition of ball milled wheat starches and of fractions obtained on hydration[END_REF] when used to make food products.

The involvement of both vitreousness (=1/porosity) and of the presence of PINA and PINB, which have been suggested to play a role in reducing adhesion between starch granules and the protein matrix, in the level of starch damage was clearly shown to match results from numerical models [START_REF] Chichti | Bottom-up model for understanding the effects of wheat endosperm microstructure on its mechanical strength[END_REF]. In these models, voids reflect porosity and different level of adhesion forces were applied between the two main elements of the starchy endosperm (starch and proteins) to mimic the role of puroindolines. [START_REF] Greffeuille | Differences in the aleurone layer fate between hard and soft common wheats at grain milling[END_REF] first pinpointed a different distribution of the aleurone cellular content between cultivars of soft and hard common wheat carrying either wild-type or mutated puroindoline alleles. The higher amount of the aleurone layer cellular content, monitored through phytic acid concentration known to be present in the aleurone globoids [START_REF] Morrison | Histochemistry and fine structure of developing wheat aleurone cells[END_REF], was found in milling flours made from the cultivars carrying the mutated alleles of puroindoline b. This content negatively correlated with the maximum tensile strain of the outer layers (Greffeuille et al., 2006b). This difference in behavior of the mechanical properties of the outer layers was further confirmed using common wheat nearisogenic lines differing only in the wild-type or mutated allele encoding PINB [START_REF] Greffeuille | Mechanical properties of outer layers from near-isogenic lines of common wheat differing in hardness[END_REF].

Recently, [START_REF] Heinze | Puroindoline genes introduced into durum wheat reduce milling energy and change milling behavior similar to soft common wheats[END_REF] clearly showed that differences in the aleurone cellular layer behavior at milling were linked with the presence or absence of both wild-type puroindoline genes. Indeed, with durum wheat (c.v. Svevo), in which both puroindoline genes are absent, the concentration (or proportion of total content) of phytic acid in the flours after milling was significantly 2-3 fold higher than in flours obtained from durum wheat grains into which both wild-type puroindoline genes (Svevo-Pin) were introduced. These differences in the cellular content of the aleurone layer, more precisely the total phytic acid content, in the flours surely impact their technological, nutritional and safety properties. Indeed, the cellular content of the aleurone layer is rich in beneficial micronutrients but may also contain detrimental hydrolytic enzymes [START_REF] Antoine | Nutritional interest of the wheat seed aleurone layer[END_REF] or mycotoxins [START_REF] Rios | Assessment of dehulling efficiency to reduce deoxynivalenol and Fusarium level in durum wheat grains[END_REF]. Furthermore, phytic acid is known to be a complexing agent for minerals and is thus recognized as an anti-nutrient even if it has also been found to display beneficial antioxidant properties due to its iron binding properties [START_REF] Kumar | Dietary roles of phytate and phytase in human nutrition.: a review[END_REF][START_REF] Urbano | The role of phytic acid in legumes: antinutrient or beneficial function?[END_REF]. It has also been suspected to interact with proteins, thereby reducing their digestibility [START_REF] Bye | Dual Effects of Sodium Phytate on the Structural Stability and Solubility of Proteins[END_REF][START_REF] Kumar | Dietary roles of phytate and phytase in human nutrition.: a review[END_REF].

Similarly, the phytic acid content present in the aleurone layer cells has been shown to be higher in durum (c.v. Svevo) wheat flour samples in comparison with corresponding wheat samples (Svevo-Pin) into which both wild-type puroindoline genes were introduced [START_REF] Heinze | Puroindoline genes introduced into durum wheat reduce milling energy and change milling behavior similar to soft common wheats[END_REF]. It was also shown that conversely starch content in bran from durum wheat obtained after milling, as well as proportion of the total starch content, was significantly lower in comparison with the starch content in the bran fraction originating from corresponding Svevo-Pin sample [START_REF] Heinze | Puroindoline genes introduced into durum wheat reduce milling energy and change milling behavior similar to soft common wheats[END_REF]. This suggests a greater loss of starchy endosperm, whose starch is a molecular marker, in the bran fractions milling from grains carrying both wild-type puroindolines. Therefore, separation between the starchy endosperm in grains carrying both wild-type puroindoline genes appeared to preferentially occur in the subaleuronic area, whereas in durum wheat, which does not contain the puroindoline genes and in common wheat grains carrying a mutated puroindoline allele, this separation appeared to occur closer to the aleurone layer.

Conclusions

Our review clearly demonstrates that the milling behavior (breaking energy, generated particle size, tissue distribution) of wheat grain depends on both the type of puroindoline genes (wild-type or mutated) whereas grain vitreousness was determined by environmental growth conditions. Grain vitreousness values were found to depend on the type of Pinb-D1 gene present (from lowest to highest vitreousness grains carrying both wild type puroindoline genes, or carrying the mutated allele Pinb-D1d, or PinbD1b in common wheat, or no puroindoline genes as in durum wheat). SKCS HI values was found to display a linear relationship with vitreousness and was found to differ depending on the presence of wild-type or mutated alleles encoding PINB. The vitreousness level of the former was found to reach a threshold of around 60% corresponding to SKCS values between 35 and 40, therefore presence of PINA and PINB increases grain porosity and lower the breaking energy. The highest total flour yield was obtained at SKCS values ranging between 30 and 60 which correspond to either vitreous grains carrying the wild-type alleles of both puroindolines or grains carrying a mutated allele encoding PINB and displaying low vitreousness. Indeed a negative effect of high vitreousness on break flour production for soft wheats appears to be balanced by a positive effect on the sizing flour production. As both vitreousness and puroindoline alleles have an impact on grain breaking resistance, they consequently also affect the particle size of the milling products and the level of starch damage. Differences in flour and bran composition depending on the presence of wild-type or mutated alleles encoding PINB were also reported and was suggested to result from differences in the grain rupture location. These differences in composition will affect the milling fraction properties. samples in France. We also acknowledge the following projects that helped fund this work: NIRS was measured using a Percon NIRS apparatus (Inframatic 8620) according to AACC method 39-70A. SKCS values were obtained with a Perten SKCS 4100 according to AACC method 55-31 as described in [START_REF] Oury | Relationships between wheat grain physical characteristics studied through near-isogenic lines with distinct puroindoline-b allele[END_REF]. Vitreousness was estimated through visual observation using a Pohl grain cutter according to [START_REF] Lasme | A study of puroindoline b gene involvement in the milling behaviour of hard-type common wheat[END_REF]

Figure 2 Figure 2

 22 Figure2

"

  Valeur Meunière I" between INRA (French National Institute of Research for Agriculture Paris), AFSA (now UFS, Paris), ANMF (National Association of French Flour-Milling, Paris), ARVALIS-Institut du Végétal (Paris), IRTAC (Paris), Danone Vitapole (Palaiseau), Chopin Technologies (Villeneuve la Garenne), and ULICE (Riom); "Valeur Meunière II involving UFS, ANMF, ARVALIS-Institut du végétal, Bühler, Chopin Technologies, ENILIA-ENSMIC (Paris, now in Surgères), IRTAC, INRA and Lu France; and FSOV (Fonds de Soutien à l'ObtentionVégétale, 2007-2010).

Figure captions

  Figure captions

Figure 1 .

 1 Figure 1. Boxplots to compare data obtained from grain vitreousness evaluation of the different samples: in different years of cultivation (A), in 7 different growth sites in 2007 with two different nitrogen input for one of the near-isogenic line containing Pina-D1a-Pinb-D1b (B) and with two different level of nitrogen used for wheat growth (C). For (A) and (C) the number (n) of samples is 304 and for (B), n=14. The graph summarizes the distribution of

Figure 2 .

 2 Figure 2. Boxplots to compare data from SKCS HI measurement for different samples. Number of samples n are 34, 36, 43, 75 and 55 for, respectively, durum wheat (c.v. Svevo), same durum wheat in which both wild-type alleles encoding PINA and PINB were introduced by Morris et al. (2011b), common wheat near-isogenic line carrying the wild-type allele of gene encoding PINB, near-isogenic line carrying the mutated allele Pinb-D1b leading to the following substitution in the amino acid sequence: G46S, near isogenic line carrying the mutated allele Pinb-D1d leading to the following substitution in the amino acid sequence: W44R. See Figure 1 for boxplot description.

Figure 3 .

 3 Figure 3. Relationships between NIRS hardness (A) or SKCS hardness (B) and vitreousness data for common wheat near-isogenic lines carrying the wild-type allele Pinb-D1a (soft) or the mutated allele, Pinb-D1b or Pinb-D1d (hard).

Figure 4 .

 4 Figure 4. Percentages of total breaking flour yield, obtained with a LabMill (Chopin technologies, Villeneuve-la-Garenne, Fr., Dubat et al., 2015), depending on the class of vitreousness within near-isogenic lines carrying the wild-type allele Pinb-D1a (A) or the mutated alleles Pinb-D1b or Pinb-D1d (B) encoding PINB. Percentage of sizing flour produced from semolina reflecting the ability of wheat particles to be reduced into flour depending on vitreousness and on the wild-type (C) or the mutated allele (D) of the gene encoding PINB. Grains within each near-isogenic line was divided into two groups (high and low) according to half of the maximal vitreousness value (V.) obtained for soft and hard nearisogenic lines, which was equal to 30 % and 40 %, respectively. See Figure 1 for boxplot description.

Figure 5 .

 5 Figure 5. Boxplot to compare total flour yield depending on grain SKCS values (0-90) from French common wheat cultivars (n=197) carrying different alleles encoding PINB (Pinb-D1a, Pinb-D1b, Pinb-D1c or Pinb-D1d) and near-isogenic lines (n=72), carrying either Pinb-D1a, Pinb-D1b, or Pinb-D1d. Each defined class of SKCS values represents between 17 and 20 % of the total wheat samples. See Figure 1 for boxplot description.

Fig. 1

 1 Fig. 1

Fig. 2

 2 Fig. 2

Fig. 3

 3 Fig. 3

Fig. 5

 5 Fig. 5

Table 1 : NIRS and SKCS hardness and vitreousness average values from two different types of wheat near-isogenic lines differing by the Pinb-D1 allele.

 1

		Grains from wheat near-isogenic lines	NIRS SKCS Vitreousness
		carrying :			(% )
	NIL1	Pina-D1a-Pinb-D1a	20.2	18.9	29.2
		Pina-D1a-Pinb-D1b	61.6	52.3	41.5
		Significance	***	***	***
	NIL2	Pina-D1a-Pinb-D1b	73.3	51.1	44.9
		Pina-D1a-Pinb-D1d	67.1	46.9	42.1
		Significance	***	(*)	ns

  on grains from two different types of wheat near-isogenic lines encoding PINB conferring respectively a soft or a hard phenotype or both a hard phenotype due to different Pinb-D1 alleles. Tukey's test was used for comparison of means. Number of observed samples were equal to 164 for NIRS and vitreousness data and to 64 for SKCS measurements for NIL1, and equal to 86 and 64 respectively for NIL2. ***pvalue<0.001, (*) pvalue<0.1.

		Different growth
	Wheat samples with	conditions
	defined puroindoline genes
	NIRS and SKCS hardness +
	vitreousness characterisation
	Milling
	behavior
	Flour &
	Bran characterisation

Acknowledgments

We are grateful to the PhD students who contributed to these studies (C. Antoine, E. Chichti, V. Greffeuille, K. Heinze, P. Lasme). We thank F-X. Oury (INRA, UMR GEDEC, Clermont-Ferrand) and C. Michelet (UFS, Paris) for their gift of near-isogenic grains differing in Pinb-D1 alleles. Special thanks to F-X. Oury for his careful reading and comments on this manuscript. We would like to thank Dr Craig F. Morris (USDA-ARS, Pullman, WA, USA) and their colleagues for the gift of durum wheat in which both wild-type puroindoline alleles were introduced and for growing them in the USA, and A. Massi (Syngeta) for the gift of Svevo durum wheat, as well as J. C. Dusautoir (INRA, retired) for cultivating the wheat