Wiener Chaoses Guillaume Poly

Guillaume Poly

Regularization

REGULARIZATION ALONG CENTRAL CONVERGENCE ON SECOND AND THIRD

REGULARIZATION ALONG CENTRAL CONVERGENCE

ON SECOND AND THIRD WIENER CHAOSES by

Guillaume Poly

Abstract. -Consider F an element of the second Wiener chaos with variance one. In full generality, we show that, for every integer p ≥ 1, there exists ηp > 0 such that if κ4(F ) < ηp then the Malliavin derivative of F admits a negative moment of order p. This entails that any sequence of random variables in the second Wiener chaos converging in distribution to a nondegenerated Gaussian is getting more regular as its distribution is getting close to the normal law. This substantially generalizes some recent findings contained in [HLN14, HNT + 15, NN + 16] where analogous statements were given with additional assumptions which we are able to remove here. Moreover, we provide a multivariate version of this Theorem.

Our main contribution concerns the case of the third Wiener chaos which is notoriously more delicate as one cannot anymore decompose the random variables into a linear combination of i.i.d. random variables. We still prove that the same phenomenon of regularization along central convergence occurs. Unfortunately, we are not able to provide a statement as strong as the previous one, but we can show that the usual non-degeneracy estimates of the Malliavin derivative given by the Carbery-Wright inequality can be improved by a factor three. Our proof introduces new techniques such that a specific Malliavin gradient enabling us to encode the distribution of the Malliavin derivative by the spectrum of some Gaussian matrix. This allows us to revisit the fourth moment phenomenon in terms of the behavior of its spectral radius.

The case of the second Wiener chaos

Let us consider F an element of the second Wiener chaos Ker(L + 2Id) where L stands for the Ornstein-Uhlenbeck operator. We refer the reader to the book [START_REF] Bakry | Analysis and geometry of Markov diffusion operators[END_REF] for an introduction to the formalism of Wiener chaoses from the point of view of Markov generators. It is a standard fact that one may find {α k } k≥1 a sequence of real numbers in l 2 (N ⋆ ) such that (1)

F = ∞ k=1 α k G 2 k -1 .
Above, {G k } k≥1 stands for an i.i.d. sequence of standard Gaussian random variables which belongs to the first Wiener chaos. The reader may for instance consult [NP12a, 2.7.13] for a corresponding proof. We shall assume that E F 2 = 1, it is then a well known fact that κ 4 (F ) = E(F 4 ) -3 controls the proximity in distribution between F and the standard Gaussian law. Our main goal is to show that, provided that κ 4 (F ) is small enough, the regularity of the distribution f of F increases. The key ingredient is to establish negative moments for the Malliavin derivative of F , namely

(2) ∀p ≥ 1, ∃δ p > 0, s.t. κ 4 (F ) < δ p ⇒ 1 Γ[F, F ] ∈ L p (Ω, F, P).
Actually we will prove a more quantitative version of the previous statement which is given in the Theorem below.

Theorem 1. -Let F ∈ Ker(L + 2Id) which satisfies E(F 2 ) = 1. One has, for every p ≥ 1, (3) κ 4 (F ) < 24 2 p (p + 1)! ⇒ 1 Γ[F, F ] ∈ q< p 2 L q (Ω, F, P).
The latter considerably improves recent findings on this topic for the case of the second Wiener chaos since it completely removes any kind of additional assumptions and only requires the central convergence since the integrability of the inverse of the Malliavin derivative is directly related to the smallness of the fourth cumulant. One can for instance consult [HLN14, Thm 7.2 and Thm 7.3], [HNT + 15, Thm 1.5 : case of the second Wiener chaos], [NN + 16] or else the nice survey [CHZ14, page 377] for related statements, all of them requiring additional assumptions ensuring negative moments for the Malliavin derivative.

Proof. -First of all, using the representation (1), we immediately deduce that

Γ[F, F ] = 4 ∞ k=1 α 2 k G 2 k ;
Computing the Laplace transform of the previous expression gives

2 (4) ∀λ > 0, E (exp (-λΓ[F, F ])) = ∞ k=1 1 1 + 8λα 2 k .
Now, let us introduce the symmetric elementary functions

(5)

S p = i 1 <i 2 <•••<ip α 2 i 1 α 2 i 2 • • • α 2 ip ,
as well as the so-called Newton sums which are in turn expressed in terms of cumulants:

(6) N p = ∞ k=1 α 2p k = 1 2 2p-1 (2p -1)! κ 2p (F ).
It is a standard fact that symmetric elementary functions can be expressed in terms of the Newton sums. In particular, one has the following identities, which are due to Newton and Girard and whose proof may be found for instance in [START_REF] Richter | Ein einfacher beweis der newtonschen und der waringschen formel für die potenzsummen[END_REF].

(7) S p = (-1) p m 1 +2m 2 +•••+pmp=p p i=1 (-N i ) m i m i !i m i
One needs to estimate the denominator of the right hand side of equation (4). A simple expansion gives

∞ k=1 (1 + 8λα 2 k ) = 1 + ∞ p=1 8 p λ p S p .
Now we will isolate in the formula (7) the case m 1 = p, which forces

m 2 = m 3 = • • • = m p = 0. S p = N p 1 p! + (-1) p m 1 <p m 1 +2m 2 +•••+pmp=p p i=1 (-N i ) m i m i !i m i = 1 2 p p! + (-1) p m 1 <p m 1 +2m 2 +•••+pmp=p p i=1 (-N i ) m i m i !i m i .
We notice that the condition m 1 < p implies that there exists i ∈ {2, • • • , p} such that m i > 0. Besides, since for any i ≥ 1 we have α 2

i ≤ 1 we get that for all i ≥ 2 we have N i ≤ N 2 = 1 48 κ 4 (F ) as well as N i ≤ 1 for every i ≥ 2. We have then the rough inequalities

m 1 <p m 1 +2m 2 +•••+pmp=p p i=1 (-N i ) m i m i !i m i ≤ κ 4 (F ) 48 m 1 +2m 2 +•••+pmp=p p i=1 1 m i !i m i ≤ p κ 4 (F ) 48
The last inequality above comes from the fact that

m 1 +2m 2 +•••+pmp=p p i=1 1 m i !i m i ≤ p i=1 p m i =0 1 m i !i m i ≤ p i=1 e 1 i ≤ p
There is certainly room for improvement in the aforementioned inequalities but we are not focused here on establishing sharp bounds. As a result, we have established for all p ≥ 1 the following inequality

(8) S p - 1 2 p p! ≤ p 48 κ 4 (F ).
Hence, as soon as p 48 κ 4 (F ) ≤ 1 p! 1 2 p -1 2 p+1 one has S p ≥ 1 2 p+1 p! . This holds when κ 4 (F ) < 24 2 p (p+1)! and it implies

∀λ > 0, ∞ k=1 (1 + 8λα 2 k ) ≥ 1 + 1 2 p+1 p! 8 p λ p = 1 + 4 p 2 p! λ p .
and thus

(9) ∀λ > 0, E (exp (-λΓ[F, F ])) = 1 1 + 1 2 p! 4 p λ p ≤ √ 2 p! 2 p λ p 2
. Now, take ǫ > 0, one has

P (Γ[F, F ] < ǫ) = P (exp (-λΓ[F, F ]) ≥ exp(-λǫ)) ≤ e λǫ E e -λΓ[F,F ] ≤ √ 2 p! 2 p e λǫ λ p 2
.

Setting λ = 1 ǫ gives that

P (Γ[F, F ] < ǫ) ≤ √ 2 p! 2 p ǫ p 2 ,
and ensures that 1/Γ[F, F ] ∈ L q for every q < p 2 which achieves the proof.

Multivariate case in the second Wiener chaos

In this section, we extend to the multivariate case the content of Theorem 1. We will write it in a sequential way since it is more convenient for us, though it dos not provide a quantitative statement. Consider

F n = (F n,1 , • • • , F n,d ) such that for every i ∈ {1, • • • , d} we have F n,i ∈ Ker(L + 2Id
). Moreover we will also assume that Cov(F n ) = Id d . Then we shall prove the following Theorem:

Theorem 2. -Assume that for every i ∈ {1, • • • , d} we have κ 4 (F n,i ) → 0.
Then, for every q ≥ 1 there exists C q > 0 and n q ≥ 1 such that

(10) E e i d k=1 ξ k F k,n ≤ C q ξ 2 1 + • • • + ξ 2 d q 2 .
Using classical Fourier inversion methods, this shows that the sequence of the joint densities of (F n,1 , • • • , F n,d ) converges towards the Gaussian density in C ∞ topology.

Proof. -Usually, to handle the non-degeneracy of a random vector by Malliavin calculus, one needs to show negative moments for the determinant of its Malliavin matrix. Here, we were not able to prove that det (Γ[F n,i , F n,j ) 1≤i,j≤d admits negative moments for n large enough. Instead of that we will only work somehow with univariate variables.

Using for instance Peccati-Tudor Theorem [START_REF] Peccati | Gaussian limits for vector-valued multiple stochastic integrals[END_REF] , whose an alternate proof may be found in [CNP + 16], we know that for every i

= j, Γ[F n,i , F n,j ] → 0 in L 2 (P). As a result, for every (t 1 , • • • , t d ) ∈ S d-1 the unit sphere of R d we have Var Γ d i=1 t i F n,i , d i=1 t i F n,i = Var   d i,j=1 t i t j Γ [F n,i , F n,j ]   = d i=1 t 4 i Var (Γ[F n,i , F n,i ]) + i =j or k =l t i t j t k t l Cov (Γ[F n,i , F n,j ], Γ[F n,k , F n,l ]) 5
As a result we get (11)

Var Γ d i=1 t i F n,i , d i=1 t i F n,i ≤ max 1≤i≤d Var (Γ[F n,i , F n,i ])+d 2 max i =j Γ[F n,i , F n,j ] 2 .
As the right hand side is independent of (t 1 , • • • , t d ) and since on any Wiener chaos Var(Γ[F n , F n ]) → 0 and κ 4 (F n ) → 0 are equivalent statements, one gets that ( 12) max

(t 1 ,••• ,t d )∈S d-1 κ 4 d i=1 t i F n,i → 0.
Thus, in virtue of Theorem 1, for any q ≥ 1 there exists n q ≥ 1 such that

∀n ≥ n q , ∀(t 1 , • • • , t d ) ∈ S d-1 , 1 Γ d i=1 t i F n,i , d i=1 t i F n,i ∈ L p (Ω, F, P) . Now, let us fix t = (t 1 , • • • , t d ) ∈ S d-1 and let us introduce for convenience the notation F n (t) = d i=1 t i F n,i .
Then, using integrations by parts techniques (see for instance [Nua06, Prop 2.1.4]) and combining the uniformity in the bound (12) with Theorem 1, one can show that there is a constant C p > 0 such that for every φ ∈ C p-1 b (R):

∀t ∈ S d-1 , ∀n ≥ n p , E φ p-1 (F n (t) ≤ C p φ ∞ . Applying this to t ξ = 1 √ ξ 2 1 +•••+ξ 2 d (ξ 1 , • • • , ξ d ) and φ(x) = e ix √ ξ 2 1 +•••+ξ 2 d gives that E e i d k=1 ξ k F k,n = E φ(F n (t ξ ) ξ 2 1 + • • • + ξ 2 d ≤ C p ξ 2 1 + • • • + ξ 2 d p-1 2 .

The case of the third Wiener chaos

Let us consider here F an element of Ker(L+3Id) which satisfies E(F 2 ) = 1. Contrarily to the case of the second Wiener chaos, it is not anymore possible to decompose F as a linear combination of independent random variables. This issue remains for higher order chaoses and mainly explains why it is in general very difficult to extend to these settings some results holding true for the second Wiener chaos. For instance, one still ignores what are the possible limits in distribution of chaotic random variables or order strictly greater than 2. In contrast, we know that every limit in distribution of random variables in the second Wiener chaos may be written as an element of the second Wiener chaos plus some independent Gaussian random variable, see for instance [NP + 12b, Thm 3.1] or else [BKNP15, Thm 1.2] for the multivariate case. Another important instance where the second Wiener chaos provides stronger results is given by [NP + 12b, Thm 3.4], where for a wide class of targets, the convergence in distribution is shown to be equivalent to the convergence of a finite number of moments/cumulants. For sequences lying in Wiener chaoses of order larger than 3, when the target is not Gaussian or else Gamma, we still don 't know whether the convergence in distribution is ensured by the convergence of a finite number of moments/cumulants. Closely related to these topics, one might read [START_REF] Bourguin | Four moments theorems on markov chaos[END_REF] where quantitative fourth moment theorems are given for targets lying in the Pearson family or else [DP + 18] for analogous questions in the Poisson space.

In this section, we investigate whether the phenomenon of regularization along central convergence, which is established for the second Wiener chaos in the last section, still holds for the third chaos. Unfortunately, we are not able to provide a statement as strong as Theorem 1. Nevertheless, we shall prove that provided that κ 4 (F ) is small enough, one can improve by a factor tending to three the usual estimates of the non degeneracy of Γ[F, F ] given by the Carbery-Wright inequality. Concretely, we will establish that Theorem 3. -For every θ < 3 4 , there exits ǫ > 0 such that

κ 4 (F ) < ǫ ⇒ 1 Γ[F, F ] θ ∈ L 1 (Ω, F, P).
This should be compared with the threshold 1 4 which is given by a standard application of Carbery-Wright inequality. It should also be emphasized that we will not provide a quantitative proof of the previous result, since it seemed to us too technical and beyond the scope of this article. Abstract constants proceed from a reasoning by the absurd at the end of the third step of the proof below. Finally let us mention that Carbery-Wright inequality is a key ingredient in a serie of recent papers as it enables one to provide quantitative statements in terms of total variation, see for instance the following highly non exhaustive list [BC19, NP13, BKZ18]. As such, any improvement of the Carbery-Wright result in many applications regarding the current state of this art.

Proof. -Step 0: preliminary material and notations Let us consider (X i ) i≥1 an i.i.d. sequence of standard Gaussian random variables and the underlying probability space (Ω, F, P). Without loss of generality, we shall assume that F = σ(X i ; i ≥ 1). Let us briefly recall that the Malliavin operators on the Wiener space (Ω, F, P) are defined in the following way:

∀f ∈ C 1 b (R d , R), Γ[f (X 1 , • • • , X d ), f (X 1 , • • • , X d )] = 1≤i,j≤d ∂ i f (X 1 , • • • , X d ) 2 ∀f ∈ C 2 b (R d , R), L [f (X 1 , • • • , X d )] = ∆f (X 1 , • • • , X d ) - d i=1 X i ∂ i f (X 1 , • • • , X d ).
Consider {a(i, j, k)} 1≤i,j,k≤N a sequence of real numbers indexed by 1, N 3 which is assumed to be symmetric: ∀σ ∈ S 3 , and every (i 1 , i 2 , i 3 ) ∈ 1, N 3 one has a(i σ(1) , i σ(2) , i σ(3) ) = a(i 1 , i 2 , i 3 ). We will also assume that whenever i 1 = i 2 or i 2 = i 3 or else i 1 = i 3 we have a(i 1 , i 2 , i 3 ) = 0. Then, let us introduce

F := F (X 1 , • • • , X N ) = 1≤i 1 ,i 2 ,i 3 ≤N a(i 1 , i 2 , i 3 )X i 1 X i 2 X i 3 ∈ Ker(L + 3Id).
We will further assume that

E(F 2 ) = 6 1≤i 1 <i 2 <i 3 ≤N a(i 1 , i 2 , i 3 ) 2 = 1.
In the sequel we will work with F which is a homogeneous polynomial of degree 3 into a finite numbers of Gaussian random variables. All the forthcoming estimates will be independent on the number of entries N , and therefore can be extended verbatim to the infinite dimensional setting.

Finally the proof requires here a particular gradient, customarily called the sharp operator, which has been introduced by N. Bouleau, see [BH10, page 135] or [START_REF] Bouleau | Error calculus for finance and physics: The language of Dirichlet forms[END_REF]page 80]. This gradient is particularly convenient since it maintains somehow the Gaussian structure with opposition with the standard choice of the literature to introduce an auxiliary Hilbert space of the form L 2 ([0, T ]). It plays a crucial role in our approach. To do so, we need a copy ( Ω, F, P) of (Ω, F, P) as well as ( Xi ) i≥1 a corresponding i.i.d. sequence of standard Gaussian such that F = σ( Xi ; i ≥ 1). For any m ≥ 1 and any F ∈ C 1 Pol (R m , R), the set of functions of C 1 (R m , R) whose gradient has a polynomial growth, one may define the following sharp operator :

(13) ♯ F (X 1 , • • • , X m ) := m i=1 ∂ i F (X 1 , • • • , X m ) Xi .
Step 1) Encoding the law of Γ[F, F ] by the spectrum of a Gaussian matrix

For the sake of clarity, E denotes the expectation with respect to P and Ê stands for the expectation with respect to P. Relying on the definition (13) and the formula for the square field operator which was recalled previously, one deduces that Γ[F, F ] = Ê(( ♯ F ) 2 ). Moreover, conditionally to (X 1 , • • • , X N ), the random variable ♯ F is a centered Gaussian whose variance is Γ[F, F ]. As a result we get the formula ( 14)

E e -Γ[F,F ] 2 ξ 2 = E Ê e iξ ♯ F
This relates the Laplace transform of the square field operator with the characteristic function of the sharp gradient. On the other hand one may write

♯ F = 1≤i 1 ,i 2 ,i 3 ≤N a(i 1 , i 2 , i 3 ) ♯ (X i 1 X i 2 X i 3 ) = 1≤i 1 ,i 2 ,i 3 ≤N a(i 1 , i 2 , i 3 ) ♯ X i 1 X i 2 X i 3 + X i 1 ♯ X i 2 X i 3 + X i 1 X i 2 ♯ X i 3 = 1≤i 1 ,i 2 ,i 3 ≤N a(i 1 , i 2 , i 3 ) Xi 1 X i 2 X i 3 + X i 1 Xi 2 X i 3 + X i 1 X i 2 Xi 3 = 1≤i 1 ,i 2 ≤N α(i 1 , i 2 )X i 1 X i 2
where, by using symmetry properties of the sequence {a(i, j, k)}, we have

α(i 1 , i 2 ) = 3 1≤k≤N a(i 1 , i 2 , k) Xk .
The matrix  ∈ M N (R) whose entries are (α(i 1 , i 2 )) 1≤i 1 ,i 2 ≤N is then a symmetric matrix with Gaussian entries. We can diagonalize it and find N random variables ( λ1 , • • • , λN ) which correspond to the spectrum of  and are ordered in the following way:

| λ1 | ≥ | λ2 | ≥ • • • ≥ | λN |.
There also exists P a random variable taking values in O N (R), the set of orthonormal matrix,

such that  = P Diag( λ1 , • • • , λN ) t P . As a result, denoting by (Y 1 , • • • , Y N ) = (X 1 , • • • , X N )P , we have 1≤i 1 ,i 2 ≤N α(i 1 , i 2 )X i 1 X i 2 = (X 1 , • • • , X N ) Ât (X 1 , • • • , X N ) = (Y 1 , • • • , Y N )Diag( λ1 , • • • , λN ) t (Y 1 , • • • , Y N ) = N k=1 λk Y 2 k = N k=1 λk (Y 2 k -1),
where the last equality comes from the fact that the trace of  is zero. Let us note that, since P is an orthonormal matrix which is independent of

(X 1 , • • • , X N ), we get that (Y 1 , • • • , Y N ) is a vector of i.i.d. standard Gaussian conditionally to ( X1 , • • • , XN ).
As a result, in virtue of the Fubini Theorem and the conditional independence aforementioned, one gets

E Ê e iξ ♯ F = ÊE e iξ ♯ F = Ê N k=1 E e iξ λk (Y 2 k -1) = Ê   N k=1 1 1 -2i λk ξ  
Note that, √ 1 + ix may be defined unambiguously for every x ∈ R by

√ 1 + ix = (1 + x 2 )
1 4 e i arctan(x)/2 . Coming back to the equation ( 14) we get the following relation between the square field operator and the spectrum of Â:

(15) ∀ξ ∈ R, E e -Γ[F,F ] 2 ξ 2 = Ê     1 det Id -2iξ Â     .
As a matter of fact, the distribution of Γ[F, F ] is fully encoded by the distribution of the spectrum of Â.

Step 2) Bounding the spectral radius of  by the fourth cumulant of F :

In this step we use tail estimates of Gaussian chaoses which have been proved for instance in [L + 06, Thm1 or equation (2)]. We will need them only for the case of the second Wiener chaos. These estimates assert that there exists an absolute constant C > 1 such that, for every N ≥ 1, any symmetric real matrix M ∈ M N and any p ≥ 1:

(16) 1 C √ p Tr(M 2 ) + pρ M ≤ i,j≤N M i,j X i X j p ≤ C √ p Tr(M 2 ) + pρ M
where ρ M stands for the spectral radius of M . Applying this in the case M = Â and considering the 2p-norm with respect to P only gives the following inequality P-almost surely.

E ♯ F 2p ≥ 4 C 2 p | λ1 | 2p p 2p .
Taking now the expectation with respect to Ê gives

ÊE ♯ F 2p = E Ê ♯ F 2p = E (Γ[F, F ] p ) (2p)! 2 p p! ≥ 4 C 2 p p 2p Ê |λ 1 | 2p .
The last equality uses the fact that ♯ F is Gaussian of variance Γ[F, F ] conditionally to F and (2p)! 2 p p! is the moment of order 2p of a standard Gaussian random variable. Note that by Stirling formula one has

(2p)! 2 p p! 1 p 2p ∼ √ 2 2 p e p p p 1 p 2p ∼ √ 2 2 e p 1 p p
As a result, for some universal constant C we have

(17) λ1 2p ≤ C √ p Γ[F, F ] p .
On the other hand, by the triangle inequality and hypercontractivity on Wiener chaoses, we get

Γ[F, F ] p ≤ Γ[F, F ] -3 p + 3 ≤ (p -1) 2 Γ[F, F ] -3 2 + 3 ≤ p 2 Var (Γ[F, F ]) + 3
Besides, a central result of Nourdin-Peccati theory, applied here for the third Wiener chaos case, asserts that

Var (Γ[F, F ]) ≤ 3 κ 4 (F ).
Many proofs may be found in the literature, one can consult for instance [ACP14, Thm3.2] for a proof or else [START_REF] Nourdin | Normal approximations with Malliavin calculus: from Stein's method to universality[END_REF] for an overview of the literature around this type of inequalities. Gathering all theses facts gives, for some universal constant C which may change from line to line, that

λ1 2 ≤ λ1 2p ≤ C √ p p 2 Var (Γ[F, F ]) + √ 3C √ p ≤ C √ pκ 4 (F ) 1 4 + 1 √ p
Optimizing in p then gives, for some absolute constant C > 0 that (18) λ1 2 ≤ Cκ 4 (F )

1 8 .
Step 3) Behavior of Tr( Â2 ):

In this step we shall use the celebrated Nourdin-Peccati bound which is at the heart of the Malliavin-Stein method. It asserts that, as soon as X belongs to some Wiener chaos of any order and has variance 1 we have

(19) d T V (X, N (0, 1)) ≤ 1 √ 3 E(X 4 ) -3.
Let us apply this bound to ♯ F , conditionally to ( X1 , • • • , XN ). Taking into account that E( ♯ F 2 ) = 2Tr( Â2 ), the bound (19) gives, conditionally to F that

E e iξ ♯ F -e -ξ 2 Tr( Â2 ) ≤ d T V ♯ F, N (0, 2Tr( Â2 ) ≤ 1 √ 3Tr( Â2 ) κ 4 ( ♯ F ) = 1 √ 3Tr( Â2 ) 48 N k=1 λ4 k ≤ 4 | λ1 | Tr( Â2 )
Note that Tr( Â2 ) is a quadratic form in Gaussian random variables which satisfies

Ê Tr( Â2 ) = 9 1≤i 1 ,i 2 ,i 3 ≤N a(i 1 , i 2 , i 3 ) 2 = 3 2 .
This is why we have P Tr( Â2 ) = 0 = 0 and the above inequalities are unambiguous. Taking the expectation with respect to Ê gives

E e -ξ 2 Γ[F,F ] 2 -Ê e -ξ 2 Tr( Â2 )
= ÊE e iξ ♯ F -Ê e -ξ 2 Tr( Â2 )

≤ Ê E e iξ ♯ Fe -ξ 2 Tr( Â2 )

≤ 4 Ê   | λ1 | Tr( Â2 )   ≤ 4 Ê | λ1 | √ α + P Tr( Â2 ) ≤ α
However Tr( Â2 ) is polynomial of degree 2 in the Gaussian r.v ( X1 , • • • , XN ) and the Carbery-Wright inequality (see e.g. [START_REF] Carbery | Distributional and lˆq norm inequalities for polynomials over convex bodies in rˆn[END_REF]) gives, for some absolute constant C > 0 which, in the sequel, may change from line to line that

P Tr( Â2 ) ≤ α ≤ C √ α Ê Tr( Â2 ) ≤ c √ α.
Gathering all theses facts leads to the inequalities valid for every ξ ∈ R:

E e -ξ 2 Γ[F,F ] 2 -Ê e -ξ 2 Tr( Â2 ) ≤ C 1 √ α Ê | λ1 | + √ α ≤ C Ê | λ1 | ≤ Cκ 4 (F ) 1 16 ,
where we have used the bound (18) at the end.

Let us now show by the absurd that

(20) ∀ǫ > 0, ∃δ ǫ > 0 s.t. κ 4 (F ) < δ ǫ ⇒ Var Tr Â2 ≤ ǫ.
If it would not be true, for some ǫ > 0 one might find a sequence {F n } n≥1 of random variables in the third chaos such that :

-E(F 2 n ) = 1, -κ 4 (F n ) ≤ 1 n -Var Tr Ân 2 ≥ ǫ.
Up to extracting a subsequence, we may assume that Tr Ân 2 converges in distribution towards Z ∞ and one deduces by passing at the limit that, for every ξ ∈ R:

E e -ξ 2 3 2 = Ê e -ξ 2 Z∞ .
Finally, by injectivity of the Laplace transform of positive random variables one deduces that Z ∞ = 3 2 and thus necessarily Tr Â2 n → 3 2 and we finally get Var Tr Â2 n → 0, which is contradictory. Unfortunately reasoning by the absurd leads to abstract constants and we lose here the ability of providing explicit quantitative statements. One way to bypass this problem is to relate the infinite norm of the difference of the Laplace transforms of Γ[F, F ] and Tr Â2 to some usual probability distance, such as Forter-Mourier. This falls beyond the scope of this article and the statement (20) would be enough for us.

Step 4): Improving Carbery-Wright rate for Γ[F, F ] by a factor 3: Set 0 < θ < 3 4 and chose an integer p large enough such that 2θ

< 1 2 + p 2 -p p 2 + 2p -2 < 3 2 .
The equation (15) gives the following inequality:

E e -Γ[F,F ] 2 ξ 2 = Ê     1 det Id -2iξ Â     ≤ Ê    N k=1 1 1 + 4 λ2 k ξ 2 1 4    .
Let us use again some notations of the last section and set for every q ≥ 1:

Ŝq = 1≤i 1 <i 2 <•••<iq≤N λ2 i 1 λ2 i 2 • • • λ2 iq Nq = N i=1 λ2q i .
As previously, the Newton-Girard inequalities give

Ŝp = N p 1 p! + (-1) p m 1 <p m 1 +2m 2 +•••+pmp=p p i=1 (-Ni ) m i m i !i m i ,
with p being the integer previously chosen. Besides we also have for every choice of (m 1 , • • • , m p ) such that m 1 + 2m 2 + • • • + pm p = p and m 1 < p there exits i ≥ 2 such that m i > 0. Hence, for this index i we have

| Ni | ≤ | λi | 2i-2 N1 which implies that p i=1 | Ni | m i ≤ | λ1 | N i=2 (2i-2)m i N m 1 +•••+mp 1
Let us recall that one has Ê N1 = Tr Â2 = 3 2 as well as the estimate (18). One deduces that, provided that κ 4 (F ) is small enough, then As a matter fact, we have shown that, provided that κ 4 (F ) is small enough, one gets that

Ê Ŝp ≥ 1 2p! Ê N p 1 ≥ 1 2 3 p 2 p p! .
P Ŝp ≤ α ≤ C p α 1 2p ,
where the constant c p is an absolute constant depending only on p. We are now ready for the final argument. Let us firs note that, since for every p ≥ 1, Ŝp ≥ 0, the following inequality holds: (23) ∀q ≥ 1, ∃η q > 0 s.t., κ 4 (F ) ≤ η ⇒ Tr( Â2 ) = N 2 = Ŝ1 ∈ L -q ( Ω, F, P).

Assuming that κ 4 (F ) < η p and applying Hölder inequality gives, for a constant C p only depending on p and changing from line to line:

E    N k=1 1 1 + 4 λ2 k ξ 2 1 4    ≤ C p   1 ξ p 2 ǫ 1 4 + 1 √ ξ E 1 Ŝ p 4 2 1 p P Ŝp < ǫ p-1 p   ≤ C p 1 ξ p 2 ǫ 1 4 + 1 √ ξ ǫ 1 2p p-1 p
When minimizing on ǫ one must have ≤ C p e ξ 2 ǫ 1 ξ 2θ .

Choosing ξ = 1 √ ǫ leads to the desired conclusion.

  On the other hand, the Newton-Girard formulas expresses Ŝp as linear combinations of Tr( Âk ) with k ∈ {2, 4, • • • , 2p} which ensures that Ŝp is a polynomial function of degree 2p of the Gaussian variables { X1 , • • • , XN } and which is positive. One can use the Carbery-Wright inequality to Ŝp which asserts that

k

  One the other hand, relying on the statement (20) established in the previous step, we know that Var(Tr( Â2 )) tends to zero when κ 4 (F ) tends to zero. Besides,Tr( Â2 ) = 9 1≤i 1 ,i 2 ≤N N k=1 a(i 1 , i 2 , k) Xk 2is a positive quadratic Gaussian form which can be diagonalized and written in where {β k } k≥1 is a collection of positive real numbers whose sum is necessarily equal to Ê Tr Â2 = 3 2 and where the Ĝk are independent standard Gaussian in the space Ω, F, P . Combining Theorem 1 with the observation (20) entails that:

.

  To conclude the proof we use again Markov inequality and writeP (Γ[F, F ] < ǫ) = P e -ξ 2 Γ[F,F ] ≥ e -ξ 2 ǫ ≤ e ξ 2 ǫ E e -ξ 2 Γ[F,F ]


Géraldine Sarret
email: geraldine.sarret@ujf-grenoble.fr

Pierre Saumitou-Laprade

Valérie Bert

Olivier Proux

Jean-Louis Hazemann

Agnès Traverse

Matthew A Marcus

Alain Manceau

Footnotes

Forms of Zn Accumulkated in Arabidopsis halleri 1

The chemical forms of zinc in the Zn tolerant and hyperaccumulator Arabidopsis halleri and in the non-tolerant and non-accumulator Arabidopsis lyrata ssp. petraea were determined at the molecular level by combining chemical analyses, extended X-ray absorption spectroscopy (EXAFS), synchrotron-based X-ray microfluorescence (µSXRF) and µEXAFS.

Plants were grown in hydroponics with various Zn concentrations, and A. halleri specimens growing naturally in a contaminated site were also collected. Zn speciation in A. halleri was independent of the origin of the plants (contaminated or non-contaminated) and Zn exposure.

In aerial parts, Zn was predominantly octahedrally coordinated and complexed to malate. A secondary organic species was identified in the bases of the trichomes, which contained elevated Zn concentrations, and in which Zn was tetrahedrally coordinated and complexed to carboxyl and/or hydroxyl functional groups. This species was detected thanks to the good resolution and sensitivity of µSXRF and µEXAFS. In the roots of A. halleri grown in hydroponics, Zn phosphate was the only species detected, and is believed to result from chemical precipitation on the root surface. In the roots of A. halleri grown on the contaminated soil, Zn was distributed in Zn malate, Zn citrate, and Zn phosphate. Zn phosphate was present in both the roots and aerial part of A. lyrata ssp. petraea. This study illustrates the complementarity of bulk and spatially resolved techniques, allowing the identification of (1) the predominant chemical forms of the metal, and (2) the minor forms present in particular cells, both types of information being essential for a better understanding of the bioaccumulation processes.

INTRODUCTION

Metal tolerant plants have the ability to survive and reproduce on soils containing high concentrations of metals in forms that are toxic or inimical to other plants [START_REF] Macnair | Metal tolerance in plants: Evolutionary aspects[END_REF]. Metal hyperaccumulating plants have the additional property of storing large amounts of metals in their aerial parts, more than typically 10000 µg g -1 (d. w.) for Zn (Baker, 1990). This characteristic makes hyperaccumulators highly suitable for phytoremediation, a soft method in which plants are used for the cleanup of metal-polluted soils (Books, 1998, Baker et al., 2000).

The genetics and the biochemical processes involved in metal uptake, transport, and storage by hyperaccumulating plants are still poorly understood, although this basic information is fundamental for the improvement of the technique [START_REF] Van Der Lelie | The binding of zinc, but not cadmium, by phytic acid in roots of crop plants[END_REF]. Zinc is one of the most important metal contaminant in industrialized countries [START_REF] Nriagu | Quantitative assessment of worldwide contamination of air, water and soils by trace metals[END_REF], and numerous studies have been conducted on the species Thalspi caerulescens [START_REF] Vazquez | Localization of zinc and cadmium in Thlaspi caerulescens (brassicaceae), a metallophyte that can hyperaccumulate both metals[END_REF][START_REF] Vazquez | Compartmentation of zinc in roots and leaves of the zinc hyperaccumulator Thlaspi caerulescens J & C Prel[END_REF][START_REF] Pollard | The quantitative genetics of zinc hyperaccumulation in Thlaspi caerulescens[END_REF][START_REF] Lasat | Altered Zn compartmentation in the root symplasm and stimulated Zn absorption into the leaf as mechanisms involved in Zn hyperaccumulation in Thlaspi caerulescens[END_REF][START_REF] Küpper | Cellular compartmentation of zinc in leaves of the hyperaccumulator Thlaspi caerulescens[END_REF][START_REF] Salt | Zinc ligands in the metal hyperaccumulator Thlaspi caerulescens as determined using X-ray absorption absorption spectroscopy[END_REF][START_REF] Frey | Distribution of Zn in functionally different leaf epidermal cells of the hyperaccumulator Thlaspi caerulescens[END_REF], Lasat et al., 2000, Assunçaõ et al., 2001) and, to a lesser extent, on Arabidopsis halleri [START_REF] Macnair | Zinc tolerance and hyperaccumulation are genetically independent characters[END_REF], Bert et al., 2000[START_REF] Küpper | Cellular compartmentation of cadmium and zinc in relation to other elements in the hyperaccumulator Arabidopsis halleri[END_REF], Zhao et al., 2000). This latter species is of particular interest because it is one of the closest relatives to A. thaliana [START_REF] Koch | Molecular systematics of the Brassicaceae: evidence from coding, plastidic MatK and nuclear Chs sequences[END_REF], whose genome is entirely sequenced [START_REF] Kaul | Analysis of the genome sequence of the flowering plant Arabidopsis thaliana[END_REF][START_REF] Meinke | Arabidopsis thaliana: A model plant for genome analysis[END_REF]. This information, together with the huge amount of literature available on A. thaliana, should facilitate our understanding of metal tolerance and hyperaccumulation in A. halleri.

A. halleri is a pseudo-metallophyte, which means that it is found both in polluted and nonpolluted areas. It is known as a Zn hyperaccumulator, but recent studies showed that it can also hyperaccumulate Cd (Dahmani-Muller et al., 2000, Küpper et al., 2000, Bert et al., in press). By analyzing F2 progenies produced by interspecific crosses between A. halleri and the non-tolerant and non-hyperaccumulating Arabidopsis lyrata ssp. petraea, [START_REF] Macnair | Zinc tolerance and hyperaccumulation are genetically independent characters[END_REF] demonstrated that Zn tolerance and Zn hyperaccumulation are two genetically independent characters. Moreover, by comparing Zn tolerance and Zn hyperaccumulation abilities of several populations of A. halleri originating from contaminated and uncontaminated areas, Bert et al. (2000) showed that both characters are constitutive properties of the species, but that populations from uncontaminated sites are slightly less Zn tolerant but exhibit higher Zn accumulation rates than populations from contaminated sites.

Recent studies by scanning electron microscopy and energy dispersive X-ray microanalysis (SEM-EDX) documented the cellular distribution of Zn in the tissues of A. halleri grown in hydroponics [START_REF] Küpper | Cellular compartmentation of cadmium and zinc in relation to other elements in the hyperaccumulator Arabidopsis halleri[END_REF], Zhao et al., 2000). In the leaves, Zn was mostly sequestered in the base of the trichomes and in mesophylle cells. Trichomes are epidermal hairs present at the surface of plant leaves, and their function can be as diverse as the exudation of various molecules, the protection against the wind and sunlight, or the storage of metals [START_REF] Rodriguez | Biology and Chemistry of Plant Trichomes[END_REF]. The chemical form of Zn accumulated in the trichomes and in mesophylle cells of A. halleri was not determined. Another study on A. halleri grown in Zn-containing hydroponics showed a correlation between the concentration of Zn and the concentration of P, citric, and malic acid in the roots [START_REF] Zhao | Zinc hyperaccumulation and cellular distribution in Arabidopsis halleri[END_REF]. The Zn-P correlation was attributed to Zn phosphate precipitates at the root surface. No Zn correlation with P or organic acids was found in the leaves.

In hydroponic studies, the nutrient solution used is generally devoid of Si because this element is not considered essential to plants (Epstein, 1999). However, some Zn-containing silicate aggregates were observed in the cytoplasm and in pinocytotic vesicles of A. halleri leaves grown on polluted soils, suggesting that Zn was transiently present as Zn silicate in the cytoplasm, before being translocated and stored in the vacuoles in an undetermined form [START_REF] Neumann | Silicon and heavy metal tolerance of higher plants[END_REF].

The aim of this study is to address several open questions concerning the mechanisms of Zn tolerance and hyperaccumulation in A. halleri. First, what are the accumulation forms of Zn in the roots and in the aerial parts of A. halleri, and are they specific to this species or common to a non-tolerant and non-hyperaccumulating Arabidopsis species such as A. lyrata ssp. petraea? Second, within A. halleri species, do Zn accumulation forms depend on the origin of the plant (contaminated or non-contaminated)? Third, does the nature of the growing medium (soil vs. hydroponics) and Zn concentration in the nutrient solution influence Zn speciation in the plant?

To address these questions, two populations of Arabidopsis halleri, one originating from a contaminated site (A. h. -C) and one from a non-contaminated site (A. h. -NC), as well as a nontolerant and non-hyperaccumulating species, Arabidopsis lyrata ssp. petraea (A. l.) [START_REF] Macnair | Zinc tolerance and hyperaccumulation are genetically independent characters[END_REF], were grown in hydroponics at various Zn levels. In addition, natural specimens of A. halleri growing in a contaminated soil were collected. The chemical form of Zn in the roots and in the aerial parts of the plants was studied by Zn K-edge extended X-ray absorption fine structure (EXAFS) spectroscopy on powder samples, and results were interpreted in light of elemental and organic acids concentrations. The localization and speciation of Zn in the leaves of A. halleri was also investigated at the micron scale by synchrotron-based X-ray microfluorescence (µSXRF) and µEXAFS spectroscopy.

RESULTS

Elemental and organic acids concentrations

Total concentrations of Zn, P and organic acids in the aerial parts and in the roots of the plants are presented in Table I. For the two populations of A. halleri grown in hydroponics, Zn concentrations increase with Zn exposure. The transfer coefficient ([Zn] aerial parts /[Zn] roots ) is always close to or less than 1, which is unexpected for an hyperaccumulating species. Such a low transfer coefficient was already observed in hydroponic experiments [START_REF] Küpper | Cellular compartmentation of cadmium and zinc in relation to other elements in the hyperaccumulator Arabidopsis halleri[END_REF], and is attributed to the precipitation of Zn phosphates on the root surface. Indeed, Figure 1 shows that P and Zn concentration are clearly correlated in the roots of hydroponic plants (group B on Fig. 1), but not in the other samples (group A on Fig. 1, including the aerial parts of all plants and the roots of A. h. grown on a contaminated soil). Moreover, A. h. grown on a contaminated soil presents a higher transfer coefficient than hydroponic plants (1.5), which is consistent with a chemical precipitation of Zn phosphate on hydroponic roots only. We shall see below that this interpretation is also supported by EXAFS results.

For a given Zn concentration in solution (250 or 100 µM Zn), the population from noncontaminated origin (A. h. -NC) accumulates more Zn in its aerial parts than that from contaminated origin (A. h. -C), which confirms previous observations made at lower Zn concentration (50 µM, Bert et al., 2000). The higher aerial Zn accumulation in A. h. -NC was not accompanied by visible toxicity signs, such as chlorosis or low growth. A. lyrata ssp. petraea (A.

l.) grown in 10 µM Zn exhibits a very low transfer coefficient (0.1), as expected for a nonhyperaccumulating species.

The concentrations of the three organic acids most often inferred to bind metals (citrate, malate and oxalate) [START_REF] Verkleij | Mechanisms of metal tolerance in higher plants[END_REF][START_REF] Streit | Chemical Properties of Metals and the Process of Bioaccumulation in Terrestrial Plants[END_REF], Brooks, 1998) were also measured, and compared to total Zn concentrations (Table I). In the roots, for all, but two, samples, the organic acid/Zn molar ratios are lower than 1 (Table I, Figure 2). Moreover, the sum of the three organic acids/Zn ratio is lower than one for all, but three, samples. Thus, these ligands are not concentrated enough to bind all Zn atoms present in the roots. In the aerial parts, the malate/Zn molar ratio is higher than 1 in all the samples, whereas citrate/Zn and oxalate/Zn ratios are lower than 1. Thus, malate could bind all Zn atoms present in the aerial parts by forming 1:1 complexes (the predominant complex if we consider a solution containing equivalent concentrations of Zn and malate at pH 5.5, which is the pH of the vacuoles), whereas citrate and oxalate could not. However, the malate concentration is not linearly correlated to Zn (Fig. 2).

These results differ from those obtained by [START_REF] Zhao | Zinc hyperaccumulation and cellular distribution in Arabidopsis halleri[END_REF] on A. halleri plants grown in hydroponics, in which malate and citrate were correlated to Zn in the roots, but not in the aerial parts.

Zn Speciation in the bulk samples

The Zn K-edge EXAFS spectra for all plant samples are shown in Fig. 3. The whole set of data was first treated by principal component analysis (PCA) (Ressler et al., 2000 and references therein). This statistical analysis allows the determination of the number of independent components contained in a set of spectra. The number of primary components corresponds to the number of Zn species present in the set of spectra, provided no species has a constant fractional amount ('background' species, [START_REF] Manceau | Quantitative speciation of heavy metals in soils and sediments by synchrotron X-ray techniques[END_REF]. Then, an operation called "target transform" evaluates whether a reference spectrum is a likely principal component of the system. Once all components have been identified, their proportion in the various samples is determined by least-squares fitting of the unknown spectra to the combination of reference spectra previously identified by PCA. This approach is particularly powerful for the analysis of natural samples containing multiple forms of the same metal because the number and nature of these forms cannot be assumed a priori [START_REF] Isaure | Quantitative Zn speciation in a contaminated dredged sediment by µ-PIXE, µ-SXRF, EXAFS spectroscopy and principal component analysis[END_REF]. An important condition for the PCA is that the number of spectra should be greater than the number of unknown species, a condition amply satisfied here.

The number of primary components was evaluated from three criteria: the weight of each component, which is directly related to how much of the signal it represents, the indicator (IND) of each component, which reaches a minimum for the least significant component representing real signal [START_REF] Malinowski | Factor Analysis in Chemistry[END_REF], and the residuals between experimental and reconstructed spectra using one, two, three, or more components. If the system contains two principal components, each spectrum should be well fitted by two components, and adding a third one should not significantly improve the quality of the fit.

In the present study, the weights of the first four components were, in decreasing order, 107, 44, 8, and 6, with IND values of 0.11, 0.04, 0.05 and 0.06, respectively. The spectra were correctly reconstructed with two components, with the normalized sum-square

(NSS = Σ[k 3 χ(k) exp -k 3 χ(k) reconstr. ] 2 / Σ [k 3 χ(k) exp ]
2 ) between 3.7 x 10 -2 and 4.2 x 10 -3 , and the quality of the fits was not much improved with three components (NSS between 2.5 x 10 -2 and 3.5 x 10 -3 ).

Thus, it was concluded from this analysis that two Zn species are significantly present in the set of samples. Note that species representing less than 10% of total Zn are not detected by this method.

The two statistically-significant Zn species were subsequently identified by target transformation using a large library of reference spectra (aqueous Zn 2+ , Zn complexed to organic acids, to amino acids, Zn sorbed on mineral surfaces, Zn minerals, Sarret et al, 1998a[START_REF] Manceau | Quantitative Zn speciation in smelter-contaminated soils by EXAFS spectroscopy[END_REF]. Several references gave satisfactory fits, including Zn malate, Zn histidine, aqueous Zn 2+ , Zn citrate and Zn phytate. Other references, for instance Zn phosphate tetrahydrate or Zn oxalate gave unsatisfactory fits. Among the five compounds retained, the most likely pair of primary components should allow the reproduction of all the plant spectra by linear combinations of these two spectra. Thus, all possible pairs were tested, and Zn malate + Zn phytate was the only one satisfying this condition. Phytate, a myo-inositol kis-hexaphosphate, contains six phosphate groups, which lend the molecule a high affinity for cations (Cosgrove, 1980). In Zn phytate, the metal is four-fold coordinated (R = 1.96 Å), with a second shell modeled by only one P atom at 3.08 Å, which corresponds to a disordered Zn phosphate environment. It is difficult to conclude on the presence of Zn phytate or disordered Zn phosphate mineral in the plant samples, so in the following text and in Table II, the generic term "Zn phosphate" will be used for this species. In this case study, the identification of the two species was facilitated by the fact that some samples were pure end-members, i.e., contained 100% Zn malate or 100% Zn phosphate (Table II).

The percentage of Zn malate and Zn phosphate in each sample was estimated next by leastsquares fitting the unknown spectra with linear combinations of the two references (Table II; Fig. 3 and4). The fits pointed Zn malate as the major species in the aerial parts of the two A. halleri populations and in the roots of A. halleri from the contaminated soil. These results are consistent with malate/Zn ratios (higher than 1) except for the roots of A. halleri grown on soil (Table II). In this latter sample, citrate is well represented (138 µmol g -1 ). The simulation of the EXAFS spectrum by a mixture of Zn malate (29 ± 10%), Zn citrate (39 ± 10%) and Zn phytate (32 ± 10%) gave a satisfactory fit, with a NSS of 5.3 x 10 -2 . As Zn citrate was among the compounds positively identified by the target transformation, its presence in this sample is likely. The occurrence of Zn citrate in the other samples was tested by including Zn citrate as a third component of the simulations, but the proportions determined were always below 5 %, which is within the precision of the method. The fact that the PCA pointed out two instead of three principal components may be due to the fact that Zn citrate is present in only one sample, in which it represents less than 50% of total Zn. The leaves of A. h. -C exposed to 10 µM Zn contained Zn malate plus a minor proportion (33 ± 10% of total Zn, i.e., 4 µmol g -1 ) of Zn phosphate. The fact that this Zn species was undetected at higher Zn concentration indicates that its proportion decreases when Zn increases (Fig. 4A). Zn phosphate was clearly the major Zn species in the roots of all plants grown in hydroponics, and in the aerial parts of A. lyrata ssp. petraea.

Zn structural parameters determined by numerical fits confirmed the results obtained by PCA and linear combinations (Table III; Fig. 5). In the aerial parts of all A. halleri plants, Zn was found to be octahedrally coordinated (d(Zn-O) = 1.99 to 2.03 Å) and surrounded by a nextnearest C shell at 2.80 to 2.87 Å, in agreement with a Zn malate complex (Table III). In the roots of the hydroponic plants, and in the aerial parts of A. lyrata ssp. petraea, the Zn-O distance

Zn Speciation in the trichomes of A. halleri

High Zn concentrations were recently observed in the bases of the trichomes in the leaves of A. halleri [START_REF] Küpper | Cellular compartmentation of cadmium and zinc in relation to other elements in the hyperaccumulator Arabidopsis halleri[END_REF], Zhao et al., 2000). The distribution and speciation of Zn in the leaves of A. h. -C grown on the contaminated soil were investigated at the micron scale using µSXRF and Zn K-edge µEXAFS spectroscopy. Elemental maps of Ca and various metals present in the leaves are presented in Fig. 6. Ca is almost evenly distributed in the leaf, whereas transition metals are concentrated in the bases of the trichomes. For instance, Zn signal is about 10-fold greater in these spots than in the leaf itself (75000 cts/s/I 0 compared to 4000-8000 cts/s/I 0 ).

Considering the thickness of the leaf and of the trichome spots, it corresponds to a Zn concentration at least 100-fold higher. The same elemental distribution was observed in other leaves of different ages. Zn-K-edge µEXAFS spectra in different Zn 'hot spots' were recorded and found to be identical. Figure 7 compares the µEXAFS spectrum of a trichome to the EXAFS spectra of the roots and aerial parts of A. h. -C, together with a selection of Zn references. The trichome spectrum is clearly different from all the others: Its frequency matches that of the roots of A. h. -C grown in hydroponics, but the two spectra clearly have a distinct shape. The trichome spectrum was compared to a large number of Zn organic and mineral references (see previous paragraph), but no good match was obtained. The occurrence of mineral Zn in this highly metalconcentrated zone was ruled out, as EXAFS spectra of inorganic compounds such as zincite or hydrozincite exhibit complex shapes owing to the presence of heavy atoms in the second or higher coordination shells (Fig. 7). The comparison of the radial distribution function (RDF) for the trichome and the bulk plant samples indicates that Zn is tetrahedrally coordinated, as in the roots of the hydroponic plant (Fig. 8). The second shell peak of the trichome is centered at R + ΔR = 2.5 Å, compared to 2.6 Å for the aerial parts (C shell) and 2.8 Å for the roots (P shell). This short distance is suggestive of a C shell (Sarret et al., 1998a). This structural interpretation is strongly supported by the relative position of the modulus and imaginary part of the Fourier transform, whose maxima are superimposed in the case of a Zn-P pair, and opposite for a Zn-C pairs (see arrows in Fig. 8, and Sarret et al., 1998a). Hence, the second coordination shell of Zn in the trichome likely consists of C atoms. These data suggest that in the bases of the trichomes, Zn is four-fold coordinated and complexed to carboxyl and/or hydroxyl groups belonging to organic acid(s). This chemical form differs from the average form (i.e., Zn malate) identified by powder EXAFS, which means that it is quantitatively minor. Despite the high concentration of Zn in the base of the trichomes, these cells account for a minor proportion of the leaf biomass, so they do not represent the major sink of Zn. The combination of µEXAFS and powder EXAFS shows that the metal is distributed as Zn malate in the leaf itself (predominant form), and as a tetrahedral Zn-organic acid(s) complex in the trichomes (minor form).

DISCUSSION

A. halleri is supposed to accumulate Zn in the vacuolar compartment of the leaves [START_REF] Neumann | Silicon and heavy metal tolerance of higher plants[END_REF], similarly to T. caerulescens [START_REF] Frey | Distribution of Zn in functionally different leaf epidermal cells of the hyperaccumulator Thlaspi caerulescens[END_REF][START_REF] Küpper | Cellular compartmentation of zinc in leaves of the hyperaccumulator Thlaspi caerulescens[END_REF][START_REF] Vazquez | Localization of zinc and cadmium in Thlaspi caerulescens (brassicaceae), a metallophyte that can hyperaccumulate both metals[END_REF]. Organic acids, including malate, citrate, and oxalate, which are primarily located in the vacuoles [START_REF] Ryan | Plant vacuoles[END_REF], thus are often inferred to chelate metals. In Thlaspi caerulescens, malate was shown to be the most abundant organic acid in the shoots (164 to 248 µmol g -1 f. w.), followed by citrate, succinate and oxalate [START_REF] Tolra | Zinc hyperaccumulation in Thlaspi caerulescens. II. Influence on organic acids[END_REF].

However, XANES spectroscopy showed that malate was not involved in Zn binding in this species, the chemical forms of Zn being, in decreasing proportion, citrate, aqueous Zn 2+ , histidine, and Zn bound to the cell wall [START_REF] Salt | Zinc ligands in the metal hyperaccumulator Thlaspi caerulescens as determined using X-ray absorption absorption spectroscopy[END_REF]. In the present study, EXAFS and chemical analyses showed that Zn is predominantly complexed to malate in the leaves of the two A. halleri populations. A secondary Zn organic species was identified in the trichomes, in which Zn is tetrahedrally coordinated and complexed to carboxyl and/or hydroxyl functional groups.

The function of the trichomes in metal storage or exudation is still unclear.

Although A. lyrata ssp. petraea has a malate/Zn molar ratio much higher than 1 in its aerial parts, this non-tolerant and non-hyperaccumulating species sequesters Zn as a phosphate species, similarly to various crop species [START_REF] Van Der Lelie | The binding of zinc, but not cadmium, by phytic acid in roots of crop plants[END_REF][START_REF] Sarret | Accumulation forms of Zn and Pb in Phaseolus vulgaris in the presence and absence of EDTA[END_REF]. The fact that malate is not a marker of tolerance and hyperaccumulation is also supported by Shen et al.'s results (1997), who showed that the hyperaccumulator T. caerulescens and the non-tolerant and non-hyperaccumulator T. ochroleucum had constitutively high concentrations of malate in shoots.

Instead, the location of malate (vacuolar or cytoplasmic) and the quantity of Zn transmembrane transporters [START_REF] Lasat | Molecular physiology of zinc transport in the Zn hyperaccumulator Thlaspi caerulescens[END_REF], Pence et al., 2000, Assunçaõ et al., 2001) are probably key factors conditioning Zn hyperaccumulation.

In the roots of hydroponic plants, Zn was speciated as inorganic or organic Zn phosphate. As phosphate precipitates have been observed previously at the root surface of hydroponic plants [START_REF] Küpper | Cellular compartmentation of cadmium and zinc in relation to other elements in the hyperaccumulator Arabidopsis halleri[END_REF], Zhao et al., 2000), the inorganic form is more likely. Although the nutrient solutions were undersaturated with respect to Zn-phosphate solids, chemical precipitation may have been induced by the root activity. This phenomenon would account for the low measured values of the root-to-leaf transfer coefficients (Table I). Zn phosphate was also present in small proportion in the roots of the plant grown on soil. Its location, either at the surface of the roots or inside the cells, is unknown, but the high phosphorus content of the soil (3 to 4 g kg -1 d. w. P 2 O 5 ) tends to favor the first hypothesis.

These results were obtained on freeze-dried and ground plant materials for bulk EXAFS experiments, and on freeze-dried whole leaves for µEXAFS experiments. For bulk EXAFS, grinding is required to obtain homogeneous samples at the scale of the X-ray beam (a few hundreds of micrometers in our experiment). To avoid chemical reactions between different cell compartments during this step, the plant material can be frozen or freeze-dried. This latter conditioning was preferred in order to avoid a possible partial defrosting and mixing of the cell compartments during grinding or sample transfer. However, it is difficult to completely dismiss the possibility of artifacts induced by this preparation. For instance, could Zn malate and Zn phosphate be the products of reactions occurring during the dehydration between Zn 2+ , malate and phosphate ions ? The high affinity of Zn 2+ for malate and phosphate (complexation constant log K = 2.9 for Zn malate [START_REF] Smith | Critical Stability Constants[END_REF]), solubility constant log K S = -32 for Zn phosphate tetrahydrate (MINTEQA2 database)) is a point in favor of the preexistence of the two species in the fresh material. Moreover, these reactions would imply proton exchange, whose possible occurrence at low temperature (-52°C in the freeze-dryer used in this work) is unknown to our knowledge.

In conclusion, the major, and some minor, chemical forms of Zn in the aerial parts and in the roots of A. halleri and A. lyrata ssp. petraea have been elucidated at the molecular-scale by the combination of chemical analyses and EXAFS spectroscopy. However, the role of the genes involved in Zn tolerance and hyperaccumulation on the speciation of Zn is still unknown. In addition, the biochemical processes responsible for Zn absorption, transfer, and storage remain to be clearly delineated.

MATERIALS AND METHODS

Plant origins

Seeds of A. halleri were collected on single mother plants in 1999 at two different sites. Seeds of A. halleri from the polluted site (A. h.-C) were collected in a field contaminated by the atmospheric fallouts of a nearby Zn smelter in Auby (North of France). Seeds of A. halleri of non-contaminated origin (A. h. -NC) were collected in Tatransla Javorina, a conservation area of the High Tatras in Slovakia. A. lyrata ssp. petraea (A. l.) originated from Unhošt, an uncontaminated woodland in the valley of Lodenice in Central Bohemia (Czech Republic).

Plant culture

Seeds were germinated on sand in a greenhouse, and eight weeks after germination seedlings were transferred to 10L polycarbonate vessels (6 plants per vessel) containing a growth medium.

The medium consisted of 0.5 mM Ca(NO 3 ) 2 , 0.2 mM MgSO 4 , 0.5 mM KNO 3 , 0.1 mM K 2 HPO 4 , 0.2 μM CuSO 4 , 2 μM MnCl 2 , 10 µM H 3 BO 3, 0.1 µM MoO 3, 10 µM FeEDDHA, and 0.2 µM ZnSO 4 . The vessels were kept in a controlled-growth chamber (temperature 20°C day / 15°C night; light 16h day / 8h night). The pH of the solution was maintained at 5.0 ± 0.1 using 2morpholinoethanesulphonic (MES) acid buffer (2 mM), which is known to be chemically inert towards metals. After three weeks, the nutrient solutions received ZnSO 4 at the following concentrations: 10, 100 or 250 μM for A. h.-C, 100 or 250 μM for A. h.-NC (the plants grown on 10 µM were accidentally lost), and 10 μM for A. l. (1 vessel containing 6 plants per Zn concentration). The theoretical speciation of Zn in the nutrient solutions was calculated using MINTEQA2 program. Zn speciation was almost constant at the three Zn concentration, with free Zn 2+ as major species (84 to 85 %), and aqueous ZnSO 4 as minor species (15 to 16 %). The saturation indexes for Zn minerals were always negative, so no Zn precipitates should have formed. During the experiment, nutrient solutions were renewed every 8 days. The position under lights in the growth chamber was randomly modified each 4 days. Plants were harvested after 5 weeks of Zn treatment. In parallel to the hydroponic culture, six A. halleri plants growing naturally in the polluted site of Auby were sampled. After harvesting, plant samples were rinsed with deionized water and divided into roots and aerial parts. For each species and each culture condition, the roots and the aerial parts of the six plants were pooled in order to have enough material for the EXAFS and chemical analyses and freeze-dried. To allow a rapid freezing, each sample was placed in a large container, transferred into the freeze-dryer at room temperature, and the container was filled with liquid nitrogen before starting the dehydration. The samples were then ground using a mechanical agate mill. An aliquot was kept for EXAFS, and the rest was divided into six aliquots, three for the analysis of Zn and P, and three for the analysis of organic acids. Some freeze-dried leaves of A. halleri from the contaminated site were kept for µSXRF and µEXAFS analysis.

Chemical Analyses

For Zn and P analysis, plant powders were digested with HNO 3 /HClO 4 (80:20, v/v) and Zn and P concentrations were determined using inductively coupled atomic emission spectrometry (ICP-AES). For the determination of malic, citric and oxalic acid concentrations, the plant powders were placed in a 0.1 N HCl solution and ultrasonicated for 1 hour in order to extract and dissociate the Zn-organic acids complexes. The suspension was then filtered at 0.45 µm, and cations were extracted from the solution using a cationic exchange resin (On Guard H Dionex).

The solution was then neutralized to pH 7 using a 1 N NaOH solution. Organic acids concentrations were measured by ionic chromatography (Dionex DX500). All values are given as mean concentrations over three samples ± standard deviation.

X-ray Absorption Spectroscopy

Zn malate standard was obtained by slow evaporation of a solution containing 10 -2 M Zn(NO 3 ) 2 and 8 10 -2 M Na malate at pH 5.5. Zn citrate was purchased from Alfa. Zn phytate was kindly provided by J. Cotter-Howells. Other Zn standards were presented previously (Sarret et al., 1998a[START_REF] Manceau | Quantitative Zn speciation in smelter-contaminated soils by EXAFS spectroscopy[END_REF][START_REF] Isaure | Quantitative Zn speciation in a contaminated dredged sediment by µ-PIXE, µ-SXRF, EXAFS spectroscopy and principal component analysis[END_REF]. Pressed pellets were prepared from the aerial parts and roots powder. Zn K-edge EXAFS spectra of Zn-rich samples were measured at room temperature on beamline D42 at the Laboratoire du Rayonnement Electromagnétique (LURE, Orsay, France) in transmission mode using ionization chambers, and on beamline BM32 at the European Synchrotron Radiation Facility (ESRF, Grenoble, France) in fluorescence mode using a 30-element solid-state Ge detector (Canberra) for diluted samples ([Zn] < 5000 mg kg -1 ).

Data extraction was performed according to standard methods. The principal component analysis and the least-squares spectral decomposition were performed with our own software, and EXAFS structural parameters (coordination numbers, interatomic distances, and Debye Waller factors) were determined using WinXAS 2.0 [START_REF] Ressler | WinXAS: A new software package not only for the analysis of energydispersive XAS data[END_REF]. For this determination, k 3 -weighted χ(k) functions were Fourier transformed over the 3.5-12 Å -1 range using a Bessel window with a smoothing parameter of 4. Then fits of the first two shells were carried out using Zn-O, Zn-P and Zn-C theoretical scattering functions calculated with FEFF7 [START_REF] Rehr | Theoretical X-ray absorption fine structure standards[END_REF] from the structure of Zn malate dihydrate [START_REF] Reed | The crystal structure of S-Malatodiaquozinc(II) hydrate[END_REF] and hopeite [START_REF] Whitaker | The crystal structure of hopeite, Zn 3 (PO 4 ) 2 , 4H 2 O[END_REF]. Fits were performed both in k and R space to check for consistency.

Microprobe analyses

µSXRF and Zn K-edge µEXAFS measurements on the leaves of A. h.-C grown on the soil were performed on beamline 10.3.2 at the Advanced Light Source (ALS, Berkeley, USA), operating at 1.9 GeV and 200-400 mA. Fragments of freeze-dried leaves were fixed on a kapton tape, mounted on a x-y translation stage, and studied in air at room temperature. The beam was focused using a pair of elliptically bent mirrors in the Kirkpatrick-Baez configuration [START_REF] Kirkpatrick | Formation of optical images by X-rays[END_REF]. The incident beam intensity was measured using two copper paddles forming a miniature ionization chamber, and the fluorescence yield was measured using a sevenelement Ge solid-state detector. For µSXRF, the spot size was 5*5 µm, and the fluorescence yield was normalized by I0 and the dwell time. Four maps of different leaves were recorded. For µEXAFS, the spot size was 15*5 µm. Three µEXAFS scans were performed on a Zn-rich trichome from three different leaves. All spectra were identical. II. III.

ACKNOWLEDGMENTS

  (d(Zn-O) = 1.95 to 1.99 Å) is characteristic of a tetrahedral coordination, and the next-nearest shell consists of P atoms at 3.06 to 3.16 Å as in phosphate compounds. Samples containing several Zn species (roots of A. h. -C grown on soil and aerial parts of A. h. -C grown in the 10 µM solution), have Zn structural parameters intermediate between those of the two (Zn malate and Zn phytate) or three (Zn malate, Zn citrate and Zn phytate) references.

Figure 1 .

 1 Figure 1. Relationship between Zn and P concentrations in the plant samples. Two groups of points can be defined: The first one (group A) represents the aerial parts of all plants and the roots of A. halleri from the contaminated soil. For these samples, Zn and P are not correlated (R2, regression coefficient = 0.07). The second group (group B) represents the roots of all plants, except those of A. halleri from the contaminated soil. A Zn-P correlation clearly exists for these samples (R2 = 0.86).

Figure 2 .

 2 Figure 2. Organic acids content as a function of Zn content in the roots (A) and in the aerial parts (B) of the plants (values given in TableI). The line y = x is shown in each plot.

Figure 3 .

 3 Figure 3. Zn K-edge EXAFS spectra for the plant samples (A. h. -C, Arabidopsis halleri, contaminated origin; A. h. -NC, Arabidopsis halleri, uncontaminated origin; A. l., Arabidopsis lyrata ssp. petraea; R, roots; AP, aerial parts) and for some Zn reference compounds. Solid lines are data and dashed lines are linear combinations of Zn malate and Zn phytate.

Figure 4 .

 4 Figure 4. Concentration of Zn species in the aerial parts (A) and in the roots (B) of the plants calculated from EXAFS fitting percentages and Zn concentrations, as explained in TableII.

Figure 5 .

 5 Figure 5. Radial distribution functions (modulus and imaginary part) for the roots and the aerial parts of A. halleri (contaminated origin) grown in solution containing 250 µM Zn. Solid lines are data and dashed lines are numerical simulations. EXAFS parameters are given in TableIII.

Figure 6 .

 6 Figure 6. µSXRF elemental maps of a leaf fragment (incident beam energy: 9.7 KeV, beam size and pixel size: 5*5 µm, dwell time: 150 ms/pixel). The number of fluorescence-yield counts were normalized by I0 and the dwell time. Metals are concentrated in the bases of the trichomes.

Figure 7 .

 7 Figure 7. Comparison of the µEXAFS spectrum for the trichome to the powder EXAFS spectra for A. h. -C (Arabidopsis halleri, contaminated origin) and for a selection of reference compounds.

Figure 8 .

 8 Figure 8. Radial distribution functions for the trichome and for powder samples (roots and aerial part) of A. h. -C (Arabidopsis halleri, contaminated origin).

Table I .

 I Zn, P, malate, citrate and oxalate concentrations in the roots (R) and in the aerial parts (AP) of the plants. Values are means of three samples ± SD. A. h. -C, Arabidopsis halleri, contaminated origin; A. h. -NC, Arabidopsis halleri, non-contaminated origin; A. l., Arabidopsis lyrata ssp. petraea. a Transfer coefficient = [Zn] aerial parts /[Zn] roots .

							Concentrations (μmol g -1 )
				Zn		P			Malate	Oxalate	Citrate
		[Zn]
	Plants Grown in Hydroponics	solution	R	AP	T a	R	AP	R	AP	R	AP	R	AP
		(μM)
	A. h. -C	250 218 ± 15 160 ± 1 0.7	469 ± 5 67 ± 2	2 ± 1 165 ± 24	5 ± 0 5 ± 0	22 ± 4	31 ± 5
		100 105 ± 15 62 ± 1 0.6	419 ± 57 74 ± 6	6 ± 1 144 ± 28	5 ± 0 16 ± 3	10 ± 1	19 ± 5
		10	17 ± 4	11 ± 2 0.6	181 ± 45 67 ± 12	6 ± 0 113 ± 6	4 ± 0 18 ± 9	28 ± 7	18 ± 1
	A. h. -NC	250 218 ± 30 217 ± 19 1.0	551 ± 54 73 ± 7	5 ± 2 221 ± 35	5 ± 0 10 ± 3	9 ± 1	6 ± 0
		100	74 ± 13 80 ± 3 1.1	239 ± 47 56 ± 7	21 ± 6 111 ± 32	5 ± 1 8 ± 4	56 ± 10 16 ± 5
	A. l.	10	46 ± 2	6 ± 0	0.1	292 ± 2 77 ± 3	10 ± 4 220 ± 22	0 ± 0 0 ± 0	9 ± 1	10 ± 1
	Plant from the Contaminated Soil
	A. h. -C	-	112 ± 11 169 ± 8 1.5	70 ± 5	95 ± 2	36 ± 5 447 ± 26	6 ± 0 34 ± 7	138 ± 24 59 ± 6

We would like to acknowledge the European Synchrotron Radiation Facility (ESRF) in Grenoble, the LURE in Orsay, and the Advanced Light Source (ALS) in Berkeley for the provision of beamtime. We are grateful to Vlastimil Mikolas for help in finding population sites in Slovakia, to Marc Macnair for providing seeds of Arabidopsis lyrata ssp. petraea, to Nicolas Geoffroy for his technical support, and to Jaco Vangronsveld for fruitful discussions.

a This research was supported by the CNRS, "Programme Environnement, Vie et Société" (grant n° 00N55) and by the Nord-Pas-de-Calais Region, "Programme de Recherches Concertées".

LITERATURE CITED

Assunçaõ

AGL, Martins PD, De Folter S, Vooijs R, Schat H, Aarts MGM (2001) Elevated expression of metal transporter genes in three accessions of the metal hyperaccumulator Thlaspi caerulescens. Plant Cell and Environment 24: 217-226 Baker AJM, Walker PL (1990) Ecophysiology of metal uptake by tolerant plants. In AJ Shaw, ed. Heavy Metal Tolerance in Plants: Evolutionary Aspects. CRC Press, Boca Raton, pp 155-177 Baker AJM, McGrath SP, Reeves RD, Smith JAC (2000) Metal hyperaccumulator plants: a review of the ecology and physiology of a biochemical resource for phytoremediation of metal-polluted soils. In N Terry, G Banuelos, J Vangronsveld, eds. Phytoremediation of Contaminated Soil and Water. Lewis Publishers, Boca-Raton, pp 85-107 Bert V, Macnair MR, De Laguerie P, Saumitou-Laprade P, Petit D (2000) Zinc tolerance and accumulation in metallicolous populations of Arabidopsis halleri (Brassicaceae). New Phytol 146: 225-233 Bert V, Bonnin I, Saumitou-Laprade P, de Laguérie P, Petit D (2002) Heavy metal (Zn, Cd, Pb) accumulation in natural populations of Arabidopsis halleri from contaminated and normal habitats, New Phytol, in press Brooks RR (1998) Plants That Hyperaccumulate Heavy Metals. CAB International, Wallingford. Chao GY (1969) Refinement of the crystal structure of parahopeite. Zeits Krist 130: 261-266 Cosgrove DJ (1980) Inositol Phosphate. Their Chemistry, Biochemistry and Physiology. Elsevier, New York. Dahmani-Muller H, Van-Oort F, Gelie B, Balabane M (2000) Strategies of heavy metal uptake by three plant species growing near a metal smelter. Environ Pollut 109: 231-238 Epstein E (1999) Silicon. Annual review of Plant Physiology. Plant Mol Biol 50: 641-664

-NC, Arabidopsis halleri, non-contaminated origin; A. l., Arabidopsis lyrata ssp. petraea. a Proportions of Zn species (in molar % of total Zn) were determined by simulating the plant EXAFS spectra by a linear combination of Zn malate and Zn phytate spectra (Fig. 3). b Concentrations are calculated from the proportion of the two species and the total Zn concentrations (Table I). c The quality of the fit is estimated by the normalized sum-square

). d For this sample, two Zn distributions are given', one including Zn malate and Zn phytate, and one including Zn citrate as well. The error bars correspond to an uncertainty of 10% for the proportion of Zn species, to the standard deviation over three samples for the measured concentrations, and to the combination of both uncertainties for the deduced concentrations of Zn species.

Table III. Zn structural parameters in reference compounds and plant samples. . a The quality of the fit is estimated by the residual Res = Σ⏐k 3 χ(k) exp -k 3 χ(k) fit ⏐/ Σ⏐k 3 χ(k) exp ⏐*100. Estimated errors on R and N are ± 0.01 Å and ± 10% for the first shell, ± 0.03 Å and ± 20% for the second shell, respectively.


Andrii Repula

Mariana Oshima Menegon

Cheng Wu

Paul Van Der Schoot

Eric Grelet
email: eric.grelet@crpp.cnrs.fr

Directing liquid crystalline self-organization of rod-like particles through

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

and highly localized directional attractive interaction between one of the ends of the particles. This has been performed by functionalizing the tips of filamentous viruses by means of regioselectively grafting fluorescent dyes onto them, resulting in a hydrophobic patch whose attraction can be tuned by varying the number of bound dye molecules. We show, in agreement with our computer simulations, that increasing the single tip attraction stabilizes the smectic phase at the expense of the nematic phase, leaving all other liquid crystalline phases invariant. For sufficiently strong tip attraction the nematic state may be suppressed completely to get a direct isotropic liquid-tosmectic phase transition. Our findings provide insights into the rational design of building blocks for functional structures formed at low densities.

There is a considerable interest in the self-organization of fluid dispersions of nanoparticles into hierarchical structures and morphologies. On the one hand, there is a fundamental interest in elucidating the physical principles that govern the self-assembly of colloidal particles [1]. On the other hand, there is also a technological interest in the context of fabricating novel functional materials bottom-up, that is, via self-assembly [2,3]. For both reasons, anisometric building blocks are seen as highly promising systems, because of their versatility in surface functionalization and their ability to form complex architectures, as the liquid crystalline phases [4][5][6][7][8][9][10][11][12][13].

Among the desired organizations relevant in the context of materials science and nanotechnology, layered structures stand out for their outstanding optical and mechanical properties [14][15][16][17]. Such lamellar or smectic phases usually appear at relatively high packing fractions, which tend to render them difficult to handle experimentally. [17][18][19]. It would therefore be appealing to develop methods and approaches to obtain the smectic ordering at lower particle loadings. We have recently shown that the self-organization and phase stability of highly ordered liquid crystalline states of filamentous viruses, including the smectic phases, is dominated by volume exclusion and hence by entropy [20], confirming the role of model colloidal system of these biological rods.

Here, we go beyond relying on a purely hard-core interaction and homogeneous surface functionalization [21,22], and introduce a tunable localized directional attraction between the tips of the virus particles by specifically grafting hydrophobic fluorescent dyes to one of the two ends of our virus-based colloidal rods. We investigate experimentally the impact of this "enthalpic" patch on the self-assembly behavior of the particles and compare our findings with computer simulations. The regioselective functionalization of the tips of the rods into hydrophobic patches gives rise to highly localized attractive interactions, which strongly influence the relative stability and structure of the various liquid crystalline phases.

In particular, we show how an increasing tip attraction stabilizes the smectic A phase at the expense of the nematic and eventually also the isotropic phase, extending the stability of the smectic A phase to relatively low concentrations. We demonstrate in this Letter the efficiency of introducing a single attractive patch in the design of anisotropic building blocks to sensitively control the balance between entropy and enthalpy, and thus to control the self-organization of these particles into the desired architecture.

In our experiments, we made use of mutants of the filamentous bacteriophages M13KE The white beads from different particles interact via a repulsive soft-core potential, while the red ones located at one end of the rods are attractive. and M13C7C, which only differ by the number of cysteine groups available at their proximal end on the P3 proteins (Fig. 1A-B). Both viruses are rod-shaped with a contour length of L = 1 µm, and a diameter of 7 nm. The particles are semi-flexible with a persistence length of L p 3L [START_REF] Barry | Proc. Natl. Acad. Sci. USA[END_REF]. The presence of cysteine residues only at one of the ends allows, after chemical reduction, for their selective bioconjugation with maleimide activated fluorescent compounds (Dylight 550 and 594 Maleimide, ThermoFisher), as described elsewhere [START_REF] De La Cotte | [END_REF]25].

This results in single-tip labeled viruses, whose degree of functionalization, i.e., the average number of fluorescent dyes per virus can be controlled in our experiments from n dyes = 1, 3 to 10 by varying the molar excess during the labeling reaction (See the Supplemental Material [26].)

The dye molecules are partially hydrophobic due to the presence of aromatic rings [21],

implying that the number of grafted dye molecules dictates the size of the hydrophobic patch on the otherwise hydrophilic surface of the virus. It is reasonable to presume that the strength of the attraction between the virus tips increases with the patch area. Whether there is a linear relationship between the number of dyes and the strength of the attraction is contentious, as the cysteine reduction and dye labeling leads to partial unfolding of the P3 tip proteins. This causes hydrophobic moieties of buried amino acids to become exposed to the aqueous solution. Still, it seems reasonable to assume that the number and size of these exposed hydrophobic groups increases with the degree of labeling, as confirmed by our experiments (See the discussion below).

Samples of single-tip functionalized virus suspensions have been prepared by dilution with BisTris-HCl-NaCl buffer, setting the pH at 7 and the ionic strength at 20 mM. These are then studied by optical microscopy [25] and small angle X-ray scattering (SAXS, [20])

(see the Supplemental Material [26]). In our molecular dynamics simulations, we model the chiral filamentous virus particles as achiral overlapping bead-spring chains, where 21 beads are connected via springs of rest length measuring half bead diameter and very large spring constant (Fig. 1C). Therefore, the aspect ratio of the simulated particles is 11, which is smaller by about one order of magnitude than the effective (i.e. accounting for the electrostatic repulsion between the charged viruses [20]) aspect ratio of the experimental particles.

The consequences for the comparison between results from experiments and simulations are discussed below. The beads interact via a steeply repulsive potential. A bending potential has been introduced to mimic the flexibility of the virus particles in order to reproduce the ratio between the persistence and contour lengths of the virus, L p /L ∼ 3. One of the end beads (displayed in red in Fig. 1C) representing the labeled virus patch interacts attractively through a Lennard-Jones potential with the other tip beads, and with a purely repulsive interaction with the other beads forming the rod particles. The strength of the tip attraction u is the depth of the Lennard-Jones potential. Approximately 4600 chains are placed in size-adjustable simulation box initially organized in 8 AAA-stacked bilayers. We performed NPT simulations at various pressures, using the simulation package LAMMPS according to a method described in [START_REF] De Braaf | [END_REF].

We construct the experimental phase diagram for patchy rods as a function of the concentration and the degree of functionalization (Fig. 2A), and compare this with the phase behavior of the pristine viral particles. By comparing the stability limits of the various mesophases, which includes nematic, smectic A, smectic B and columnar phases, we conclude that increasing the number of dye molecules grafted at the tips of the virus particles strongly affects the nematic-smectic A (N-SmA) transition, yet has almost no effect over the other phase transitions. Our main finding is the increased stabilization of the smectic phase, at the expense of the nematic phase, with increasing number of grafted dyes, and concomitant widening phase gap implying that the transition becomes more strongly first order. cysteine groups at the virus tip (see the Supplemental Material [26]). The isotropic liquidto-nematic phase (I-N) transition remains unchanged in both phase diagrams, except for the highest tip attraction where the simulations point at a relatively weak widening of the coexistence range. This suggests that our patchy interaction is rather weak and localized, as rods with stronger attractive interaction, driven by either depletion interaction [START_REF] Lekkerkerker | Colloids and the Depletion Interaction[END_REF][START_REF] Dogic | [END_REF] or by a residual van der Waals interactions between the bodies of the rods [30], exhibit a significant widening of the I-N coexistence range.

The results from our experiments and the simulations diverge at very high packing fractions. We do not find a stable columnar phase in our simulations of rod-like particles.

This could be due to the difficulty of stabilizing the columnar organization in numerical simulations for entropy driven, single-component systems [31]. It is also possible that the columnar phase does not form in suspensions of particles with aspect ratios below 30, as suggested in [32]. Another obvious difference between experiments and simulations is the strongly first order transition between the smectic A and smectic B phases in the latter.

Experimentally, it is second order or weakly first order [20]. An extension of the smectic-B range by increasing the tip attraction that we find experimentally, is lacking in simulations for which there is also an intrinsic difficulty to clearly distinguish between the smectic B from the crystalline phase. The absence of one-to-one correspondence between the mass concentration in the experiments and the volume fraction in the simulations is not really surprising given the crude nature of the interaction potential, the modest aspect ratio of the particles in the simulations, and the overestimation of the size of the attractive bead in the simulations compared to the size of the attractive sites on the virus tip-proteins.

The overall qualitative agreement between experiments and simulations is however manifest. This is true for the dependence on tip attraction of the transitions between isotropic, nematic and smectic A phases (Fig. 2), but turns out to be true as well as for the local ordering displayed in these phases (Fig. 3). For the purpose of direct comparison, we added a tracer amount of body labeled viruses with green fluorescent dyes to our suspensions. The striking feature of the optical texture as seen by fluorescence microscopy is the presence of red colored clusters in the isotropic phase. By varying the depth of focus, we evidence the clusters to have a two-dimensional structure, forming bilayer "lamellae" in which the viruses assemble at their red tips and lie nearly perpendicular to them. We cannot exclude the possibility that some of these clusters are caused by chemical rather than physical cross-linking, during the tip functionalization process.

Similar lamellar structures can be observed in the nematic phase, except that in this case they are oriented perpendicular to the director (defined as the average rod orientation) whereas in the isotropic phase they are randomly oriented (Fig. 3A). Furthermore, the disappearance of the chiral nematic or cholesteric phase in favor of the uniaxial nematic phase upon grafting even a single dye molecule to the virus tip (Fig. 2A), we ascribe to the presence of these lamellae. We argue that they must interfere with the chirality amplification on the mesoscopic scale. In our simulations we observe bilayer clusters similar to those seen experimentally with particles assembled by their attractive tips in both sides, as shown by the snapshots in the isotropic and nematic phases displayed in Fig. 3B.

At increased particle concentration, the lamellar aggregates grow and condense into smec-tic domains in a nematic background, corresponding to the N-SmA coexistence region (See Fig. 3, central images). As expected, the particles are aligned along the director in the two phases, in both experiments and simulations. An example of the single smectic domain is given in Fig. 3A, where the alignment of the rod-like particles is perpendicular to the layer allowing us to rule out any smectic C or other types of tilted smectic.

In contrast to the smectic A and B phases, which we are able to distinguish by means of SAXS measurements (see the Supplemental Material [26]) [20,33], and which do exhibit large single domains, the columnar phase is characterized by finite domain sizes of only a few micrometers width, as shown both in Figs. 2A and3A. The absence of bright red localized signals supports the lack of layered structure and is therefore consistent with the liquid-like order along the columns. The variation of red fluorescence intensity arguably does not reflect strong clustering, but may be interpreted as the result of the integration over the sample thickness of the fluorescence signal coming from domains with different orientations.

As the main effect of the tip patchiness is to widen the smectic stability range, we have characterized this phase by determining the associated molecular field U layer [34]. This unidimensional ordering potential can be obtained by measuring the distribution of longitudinal rod fluctuations with respect to the middle of the layers, from which is deduced the probability P (z) of finding a particle at position z along the director. P (z) is related to the ordering potential via the Boltzmann factor P (z) ∝ exp(-U layer (z)/k B T ). The free energy landscape of both experimental and simulated particles is presented in Fig. 4 and shows the same trends: (i) the magnitude of the ordering potential increases with increasing tip patchiness for a given particle packing fraction (Fig. 4A-B), and (ii) U layer increases with the particle concentration, for both repulsive and attractive tips (Fig. 4C-D). Note in addition that the smectic potential also becomes narrower with increasing density and functionalizing the tips of the viruses. This implies that the amplitude of the fluctuations of the particles around their equilibrium positions in the layers become weaker, and hence that the particle positions become more localized. As the aspect ratio of the particles is smaller in our numerical simulations, we expect lower smectic potentials compared to the experimental ones, as shown in Fig. 4. The reason is that the stability of the smectic A phase of repulsive rod-like particles reduces with decreasing length [35]. Notice that irrespective of the strength of tip attraction, we find the same slope of the ordering potential as a function of the particle concentration, both in the experiments and the simulations. This is to be expected because In all graphs, open black and full red symbols correspond to "raw" repulsive and tip-functionalized rod-like particles, respectively.

the molecular field a test particle experiences in a lyotropic smectic must be proportional to the average density [36]. Even though we have not been able to find a sensible mapping between our experimental and simulation results because of the large disparity between the respective aspect ratios of the particles, our simulations do account for most of the features we observe in our experimental system. This is true both for the phase behavior and ordering potentials, suggesting that our prediction that a tip attraction strength as small as u ≈ 1 -2 k B T is sufficient to fully suppress the nematic phase and promote the smectic organization in dispersions of otherwise mutually repelling rod-like particles is plausible.

This small value is actually not surprising, considering that free energy differences between particles in co-existing liquid crystalline phases of rod-like particles are typically of the order of a thermal energy and often much smaller than that.

In summary, we report on the achievement of tip-functionalized rod-like virus particles exhibiting sticky patches with tunable interaction. We find that the range of stability of the smectic phase of these particles can be enlarged continuously by increasing the strength of the patch attraction. Extending the stability of the smectic phase to lower concentrations happens at the expense of the nematic phase, in which bilayer lamellar aggregates form.

Other phase transitions are, by and large, not affected by the tip functionalization. Our experiments and computer simulations suggest that the reason why only the smectic ordering responds to the tip functionalization, is that it brings together the interacting ends that are otherwise not highly correlated in the other phases. Our findings open up perspectives in the rational design and site-specific post-modifications of particulate building blocks for soft self-assembled materials, showing how the introduction of a single and tiny enthalpic patch is able to steer the structuring of complex fluids.

FIG. 1 .

 1 FIG. 1. (A) Transmission electron microscopy of the filamentous virus used in this work (Scale bar: 200 nm), and (B) schematic representation of pristine (left) and single-tip functionalized viral rod with red dyes (right), resulting in a localized directional attractive interaction. (C) Schematic of the semi-flexible rod-like particles, modeled as a bead-spring chain, used in computer simulations.

FigureFIG. 2 .FIG. 3 .

 23 Figure 2B presents our simulation phase diagram as a function of the strength of the tip attraction, u. The resulting phase behavior shows qualitative agreement with the experimental data: increasing the stickiness of the tips affects mainly the nematic-smectic A

  FIG. 4. Smectic ordering potentials or molecular fields calculated from experimental (A) and simulated (B) distribution of particles, as a function of the particle position, normalized by the smectic layer spacing, for different particle concentrations and volume fractions. In gray we indicate the experimentally measured potential of immobile particles, which is the equivalent for potentials of the point spread function (PSF) of the optical setup. Smectic potential barriers as a function of virus concentration for experiments (C) and as a function of volume fraction for simulations (D).

ACKNOWLEDGMENTS

This project has received funding from the European Union Horizon 2020 research and innovation programme under the Marie Sk lodowska-Curie Grant Agreement No. 641839.