B Boutin

N Raymond

SOME REMARKS ABOUT FLOWS OF HILBERT-SCHMIDT OPERATORS

This paper deals with bracket flows of Hilbert-Schmidt operators. We establish elementary convergence results for such flows and discuss some of their consequences.

Introduction and results

1.1. Context. Consider a real separable Hilbert space H equipped with the Hilbertian inner product •, • and the associated norm • . The set of bounded operators from H to H will be denoted by L(H) and the associated norm by • . Let (e n ) n∈N be a Hilbertian basis of H and denote by D(H) the subset of bounded diagonal operators with respect to that basis:

T ∈ D(H) ⇔ ∀n ∈ N, ∃λ n ∈ R, T e n = λ n e n .
The purpose of this paper is to analyze the structural and convergence properties of some flows of Hilbert-Schmidt operators for which D(H) turns out to be an attractive Hilbert-Schmidt subspace. We obtain an alternative convergence proof that applies for example to the flows (in infinite dimension) of Brockett, Toda and Wegner. The present paper is mainly motivated by the work by Bach and Bru [START_REF] Bach | Rigorous foundations of the Brockett-Wegner flow for operators[END_REF] where they tackle in particular the example of the Brockett flow in infinite dimension. We extend one of their results to a wider class of flows of Hilbert-Schmidt operators. We may notice here that such flows of operators have deep applications in mathematical physics. For instance, as recently proved by Bach and Bru in [START_REF] Bach | Diagonalizing Quadratic Bosonic Operators by Non-Autonomous Flow Equation[END_REF], they can be used to diagonalize unbounded operators in boson quantum field theory. This paper also aims at being a modest review of the methods and insights appearing in finite dimension with the idea to extend them in infinite dimension. These methods are explained, for instance, in the seminal papers [START_REF] Chu | Isospectral flows and abstract matrix factorizations[END_REF] and [START_REF] Brockett | Least squares matching problems[END_REF] (see also [START_REF] Brockett | Dynamical systems that sort lists, diagonalize matrices, and solve linear programming problems[END_REF], and [START_REF] Wegner | Flow-equations for Hamiltonians[END_REF] for the Wegner flow), but also in the nice survey [START_REF] Chu | Linear algebra algorithms as dynamical systems[END_REF]. Their relations with numerical algorithms such as the QR or LU decompositions are very deep and the idea of interpreting discrete algorithms as samplings of continuous flows has now a long story (see, for instance, [START_REF] Rutishauser | Der Quotienten-Differenzen-Algorithmus[END_REF][START_REF] Rutishauser | Ein infinitesimales Analogon zum Quotienten-Differenzen-Algorithmus[END_REF] and [START_REF] Watkins | On Rutishauser's approach to self-similar flows[END_REF]). Commenting this idea would lead us far beyond the scope of this paper, but we will briefly discuss it when describing our numerical illustrations. 1.2. Results. Let us now describe the results of this paper. For that purpose, we introduce some general notations. Notation 1.1. Below are the notations that we will constantly use in the paper.

(a) We denote L 2 (H) the set of the Hilbert-Schmidt operators on H and the associated norm is • HS . In other words, H ∈ L 2 (H) means that, for any Hilbertian basis (e n ) n∈N , we have n≥0 He n 2 < +∞ , and this last quantity is independent from the Hilbert basis and is denoted by H 2 HS and we have H ≤ H HS . We also recall that the Hilbert-Schmidt operators are compact and that, if (λ n (H)) n∈N is the non-increasing sequence of the eigenvalues of H, we have

H 2 HS = +∞ n=0 |λ n (H)| 2 .
(b) We will denote by S(H), A(H) and U(H) the sets of the bounded symmetric, bounded skew-symmetric and unitary operators respectively. We also introduce S 2 (H) = S(H) ∩ L 2 (H) and A 2 (H) = A(H) ∩ L 2 (H). They are Hilbert spaces equipped with the scalar product (A, B) → Tr(A B). (c) In the sequel we will use a fixed Hilbert basis (e n ) n∈N and for any H ∈ L 2 (H) and any integers i, j ∈ N, we will denote h i,j (t) = H(t)e i , e j .

1.2.1. An elementary convergence result. We will consider infinite systems determined by matrices that are balanced in the sense of the following definition.

Definition 1.2. A family of real-valued functions

(g i,j ) i,j∈N defined on R is said to be balanced if ∀i, j ∈ N, ∃c i,j ∈ L ∞ (R, R + ), ∀t ∈ R, |g i,j (t)| ≤ c i,j (t)|g j,i (t)| .
It is said to be pointwise bounded if, for all t ∈ R, sup i,j |g i,j (t)| < +∞.

Let us now state one of the main results of this paper. It is a general convergence result for infinite systems of differential equations. As we will see, many different classical situations may be reduced to this result.

Theorem 1.3. Consider H ∈ C 1 (R, L 2 (H)) such that i) the function t → H(t) HS is bounded, ii) the family (h i,j ) i,j∈N is balanced, iii)
there exists a pointwise bounded and balanced family of measurable functions (g i,j ) i,j∈N defined on R such that

(1.1) ∀t ∈ R, h i,i (t) = +∞ j=0 g i,j (t)|h i,j (t)| 2 .
Suppose that there exists T > 0 and a sign sequence

( ) ∈N ∈ {-1, 1} N , such that (1.2) ∀t ≥ T, ∀k, ∈ N, k ≥ , g ,k (t) ≥ 0 .
Then we have the integrability property

(1.3) ∀ ∈ N, +∞ j=0 +∞ T |g ,j (t)||h ,j (t)| 2 dt < +∞ ,
and any diagonal term h , (t) converges at infinity to a real value denoted by h , (∞).

Under the additional semi-uniform lower bound condition

(1.4) ∀ ∈ N, ∃c > 0, ∀j = , ∀t ≥ T, |g ,j (t)| ≥ c ,
we have

(1.5) j = +∞ T |h ,j (t)| 2 dt = +∞ T H(t)e -h , (t)e 2 dt < +∞ .
Suppose moreover that, for any ∈ N, the function t → H (t)e is bounded, then for any ∈ N, H(t)e converges to h , (∞)e .

Remark 1.4. Theorem 1.3 remains true if, instead of the assumption (1.2), we only assume that there exists a one-to-one integer mapping function φ : N → N, a time T > 0 and a sign sequence

∈ {-1, 1} ( ∈ N) such that ∀t ≥ T, ∀k ∈ N, k / ∈ {φ(i), 0 ≤ i ≤ -1}, g φ( ),k (t) ≥ 0 .
We will apply Theorem 1.3 to "bracket" flows. Let us state a proposition that describes the fundamental property of such flows.

Proposition 1.5. Let G : S(H) → A(H) be a locally Lipschitzian function and let us consider the following Cauchy problem

(1.6) H = [H, G(H)] , t ≥ 0 , H(0) = H 0 ∈ S 2 (H) .
There exists a unique global solution H ∈ C 1 ([0, +∞), S 2 (H)) unitarily equivalent to the initial data H 0 . More precisely there exists U ∈ C 1 ([0, +∞), U(H)) such that for any t ≥ 0 one has

H(t) = U (t) H 0 U (t).
Let us now provide two corollaries of Theorem 1.3.

First let us introduce two convenient definitions.

Definition 1.6. Given a Hilbertian basis (e n ) n∈N , we introduce i. the symmetric family of symmetric Hilbert-Schmidt operators (E i,j ) (i,j)∈N 2 defined by

E i,j (•) = 1 √ 2 (e * j (•)e i + e * i (•)e j ) if i = j e * i (•)e i if i = j .
ii. the skew-symmetric family of skew-symmetric Hilbert-Schmidt operators (E ± i,j ) (i,j)∈N 2 defined by

E ± i,j (•) = 1 √ 2 (e * j (•)e i -e * i (•)e j ) .
Since the families (E i,j ) i≤j and (E ± i,j ) i<j are Hilbertian bases of S 2 (H) and A 2 (H) respectively, this motivates the following definition. Definition 1.7. We say that G ∈ L (S 2 (H), A 2 (H)) is diagonalizable when there exists a Hilbertian basis (e n ) n∈N and a skew-symmetric family (g i,j ) such that G(E i,j ) = g i,j E ± i,j . In this case, we say that (g i,j ) is the matrix-eigenvalue of G. Note that the matrixeigenvalue of G is balanced and pointwise bounded (G is bounded).

Corollary 1.8. Let G ∈ L (S 2 (H), A 2 (H)) be a diagonalizable map such that its matrix-eigenvalue satisfies (1.2) and (1.4). Then the Cauchy problem (1.6) admits a unique global solution H ∈ C 1 ([0, +∞), S 2 (H)) which weakly converges in S 2 (H) to a diagonal Hilbert-Schmidt operator H ∞ . Moreover, if α is a diagonal term of H ∞ with multiplicity m, then α is an eigenvalue of H 0 of multiplicity at least m.

Then we can apply our theorem to the finite dimensional case and even get an explicit convergence rate. Note that this presentation does not involves a Lyapunov function and relies on the a priori convergence of H(t). This is a way to avoid, as far as possible, the considerations related to the evolution of the corresponding unitary matrices (U (t)) t≥0 in U(H) (that is not compact in infinite dimension).

Corollary 1.9. Consider a finite dimensional Hilbert space H of dimension d. Under the assumptions of Corollary 1.8, the solution of (1.6) converges to a diagonal operator H ∞ = diag (α ) 0≤ ≤d-1 in L(H) and the α are exactly the eigenvalues (with multiplicity) of H 0 . If, in addition, the eigenvalues of H 0 are simple, then there exist T, C > 0 such that, for all t ≥ T , we have

H(t) -H ∞ ≤ Ce -γt , where γ = inf T -|g i,j (α i -α j )| > 0 with T -= {(i, j) ∈ {0, . . . , d -1} 2 : i < j and g i,j (α i -α j ) < 0} .
Note that, in Corollary 1.9, we have not to exclude special initial conditions to get an exponential convergence (see [START_REF] Brockett | Dynamical systems that sort lists, diagonalize matrices, and solve linear programming problems[END_REF]Theorem 3]).

The following proposition states that, even in infinite dimension, we may find exponentially fast eigenvalues of a generic Hilbert-Schmidt operator thanks to a bracket flow. Note that, in this case, we get a stronger convergence as the one established by Bach and Bru in [START_REF] Bach | Rigorous foundations of the Brockett-Wegner flow for operators[END_REF]Theorem 7] for the Brockett flow and that we may also consider non symmetric operators. Proposition 1.10. Assume that H 0 ∈ L 2 (H) (not necessarily symmetric) is diagonalizable with eigenvalues (λ j ) j≥1 with decreasing real parts. Let us also assume that we may find P ∈ L(H) such that P H 0 P -1 = Λ where Λ = diag(λ j ) j∈N is a diagonal operator and such that the minors of P , P J = ( P e i , e j ) 0≤i,j≤J are invertible for all J ∈ N.

We use the choice

G(H) = H --(H -) .
Then, we have, for all ∈ N, Note in particular that, in the symmetric case, H(t) converges to a diagonal operator H ∞ that has exactly the same eigenvalues as H 0 (even in infinite dimension). This property is slightly stronger than the last one stated in Corollary 1.8: for the Toda flow, in infinite dimension, no eigenvalue is lost in the limit.

H(t)e -
1.2.2. Organization of the paper. Section 2 is devoted to the proof of Theorem 1.3, Proposition 1.5 and Corollary 1.8. We also provide examples of applications of our results (to the Brockett, Toda and Wegner flows) in infinite dimension. In Section 3, we discuss the case of finite dimension to the light of our results and discuss the exponential convergence of the flows (Corollary 1.9 and Proposition 1.10).

Existence and global convergence results

2.1. Proof of the main theorem. This section is devoted to the proof of Theorem 1.3. We start by stating two elementary lemmas.

Lemma 2.1. Consider h ∈ C 1 (R + ) a real-valued bounded function, F ∈ L 1 (R + ) and G a nonnegative measurable function defined on R + such that ∀t ≥ 0, h (t) = F (t) + G(t).
Then, the function G is integrable and h converges to a finite limit at infinity.

Proof. Observe that, due to the sign of G, one gets for all x ≥ 0

x 0 |G(t)| dt = x 0 G(t) dt = h(x) -h(0) - x 0 F (t) dt ≤ 2 sup R + |h| + +∞ 0 |F (t)| dt .
Therefore G is integrable and, for all x ≥ 0

h(x) = h(0) + x 0 (F (t) + G(t)) dt ,
which converges as x tend to infinity.

Lemma 2.2. Let w be a function defined on R + with nonnegative values. Suppose that w is Lipschitz continuous with Lipschitz constant C > 0. Then, for any x ≥ 0, one has

w(x) 2 ≤ 2C +∞ x w(y) dy ≤ +∞.
Suppose moreover that w is integrable, then lim

x→+∞ w(x) = 0.
Proof. Since w is Lipschitzian, we have, for all x, y ∈ R,

w(x) ≤ C|x -y| + w(y) ,
so that, integrating with respect to y between x and x + w(x) C , we get

w(x) 2 C ≤ w(x) 2 2C + x+ w(x) C
x w(y) dy , and the conclusion follows.

We can now complete the proof of Theorem 1.3.

2.1.1.

Convergence. Consider first Equation (1.1) with i = 0. Thanks to the sign property (1.2) we have

0 h 0,0 (t) = j∈N |g 0,j (t)||h 0,j (t)| 2 .
The boundedness of t → H(t) HS implies the boundedness of t → h 0,0 (t) and thus Lemma 2.1 implies the existence of a finite limit h 0,0 (∞) and the integrability property

+∞ T j∈N |g 0,j (t)||h 0,j (t)| 2 dt < +∞ .
Then, the rest of the proof is by induction on ∈ N. Let ≥ 1 be an integer and assume that, for any integers 0 ≤ k ≤ -1, we have

+∞ T j∈N |g k,j (t)||h k,j (t)| 2 dt < +∞ .
Equation (1.1) with i = and the sign assumption (1.2) give

h , (t) = k≤ -1 g ,k (t)|h ,k (t)| 2 + j≥ |g ,j (t)||h ,j (t)| 2 .
Again, t → h , (t) is bounded. Let us now prove the integrability of the first sum in the right hand side. Since the families (g i,j ) and (h i,j ) are balanced, we get the following

∃C ≥ 0, ∀t ≥ T, ∀k ≤ -1, |g ,k (t)||h ,k (t)| 2 ≤ C |g k, (t)||h k, (t)| 2 .
Therefore we have (2.1)

+∞ T k≤ -1 |g ,k (t)||h ,k (t)| 2 dt ≤ C k≤ -1 +∞ T |g k, (t)||h k, (t)| 2 dt ≤ C k≤ -1 +∞ T j∈N |g k,j (t)||h k,j (t)| 2 dt < +∞ .
Finally, Lemma 2.1 applies and we get the existence of a finite limit h , (∞) together with the integrability property:

+∞ T j≥ |g ,j (t)||h ,j (t)| 2 dt < +∞ ,
and then, thanks to (2.1), we find

+∞ T j∈N |g ,j (t)||h ,j (t)| 2 dt < +∞ .
This ends the proof of the integrability property and of the convergence of diagonal terms.

2.1.2. Stronger convergence. Consider now the additional assumption (1.4). For any fixed ∈ N, we have

j = +∞ T |h ,j (t)| 2 dt ≤ 1 c j = +∞ T |g ,j (t)||h ,j (t)| 2 dt < +∞ .
For any t ≥ T , we let

w (t) = j = |h ,j (t)| 2 dt = K (t) 2 with K (t) = H(t)e -h , (t)e .
For t ≥ s ≥ T , the Cauchy-Schwarz inequality implies that

|w (t) -w (s)| ≤ K (t) -K (s) K (t) + K (s) .
On one hand, we have

K (t) + K (s) ≤ 2 sup σ≥T H(σ) HS .
On the other hand, we write

K (t) -K (s) ≤ H(t)
∀H ∈ S(H) , F (H) = [H, G(H)] .
Note indeed that, for all H ∈ S(H), by using that G is valued in A(H), we have

F (H) = (HG(H) -G(H)H) = G(H) H -H G(H) = F (H) .
Moreover F is locally Lipschitzian since G is. By using the Cauchy-Lipschitz theorem in the Banach space (S(H),

(2.3) U = U G(H) , U (0) = Id ,
where H is the local solution of (1.6) on [0, T max ). Since the equation is linear, we know that there exists a unique solution U defined on [0, T max ). By using that G(H(t)) ∈ A(H), it is easy to check that, on [0, T max ), we have U U = U U = Id. Then we notice that

H(t) = k≤ h k, (t)E k, , G(H(t)) = k< g k, h k, (t)E ± k, .
We get

H(t)e i = +∞ j=0 h i,j (t)e j , G(H(t))e i = - ∞ j=0 g i,j h i,j (t)e j ,
and thus

h i,i (t) = -2 +∞ j=0 g i,j |h i,j (t)| 2 .
Moreover we notice that

H (t) ≤ 2 H(t) G(H(t)) ≤ 2 G H(t) 2 HS = 2 G H 0 2 HS .
Therefore, thanks to the assumptions on (g i,j ), we may apply Theorem 1.3.

Then we let α = h , (+∞) and we notice that

+∞ =0 H(t)e 2 = H(t) 2 HS = H 0 2 HS ,
and thus, with the Fatou lemma, we get

sup ∈N |α | 2 ≤ +∞ =0 α 2 ≤ lim inf t→+∞ H(t) 2 HS = H 0 2 HS .
We introduce the Hilbert-Schmidt operator

H ∞ := diag (α ) defined by ∀x ∈ H , H ∞ (x) = +∞ =0
α x e , where x = +∞ =0

x e .

In particular, we get that, for all ∈ N, H(t)e converges to H ∞ e . Let us now consider x ∈ H and ε > 0. There exists L ∈ N such that x -

L =0
x e ≤ ε. Since H(t) ≤ H(t) HS = H 0 HS and H ∞ ≤ H 0 HS , we get that, for all t ≥ 0,

(H(t) -H ∞ )x ≤ (H(t) -H ∞ ) L =0
x e + 2ε H 0 HS , and, for t large enough, it follows that

(H(t) -H ∞ )x ≤ 2ε H 0 HS + ε .
Therefore H(t) strongly converges to H ∞ and thus H(t) weakly converges to H ∞ in S 2 (H).

Let us consider an eigenvalue α of H ∞ , with multiplicity m and associated with the eigenvectors e 1 , . . . , e m . Then, for all ε > 0, there exists T > 0 such that, for all j ∈ {1, . . . , m}, H(T )e j -αe j ≤ ε . or equivalently H 0 f j -αf j ≤ ε , with f j = U (T )e j . Therefore, the spectral theorem implies that

∀ε > 0 , range 1 [α-ε,α+ε] (H 0 ) ≥ m .
This proves that α is an eigenvalue of H 0 with multiplicity at least m.

∈ R N + , we take G 1 (H) = [H, A]. Since L 2 (H) is an ideal of L(H), we have G 1 : S 2 (H) → A 2 (H) and G 1 ≤ 2 A . Moreover G 1 is diagonalizable and g i,j = a j -a i . 2.3.2. Toda's choice. If H ∈ S 2 (H), we introduce H -= 1≤i≤j h i,j e * i e j and G 2 (H) = H --(H -) .
By definition, we have

G 2 : S 2 (H) → A 2 (H) and G ≤ 2 H -≤ 2 H HS .
In addition, G 2 is diagonalizable with g i,j = -1 for i < j.

h i,i (t) = 2 +∞ j=0 (h i,i (t) -h j,j (t))|h ij (t)| 2 .
If, for large times, the diagonal terms become distinct (in the sense of (1.2) and (1.4)), then, by Theorem 1.3, H(t) converges to a diagonal matrix H ∞ . Actually, in this case, the convergence is exponential in finite dimension (see Section 3.1.3 where the linearization of the same kind of system is explained).

Exponential convergence

3.1.

A stronger convergence in finite dimension. This section is devoted to the proof of Corollary 1.9.

3.1.1. Reminder of the stable manifold theorem. Let us recall a classical theorem.

Theorem 3.1. Consider F ∈ C 1 (R d , R d
) such that F (0) = 0 and the differential dF 0 is diagonalizable with no eigenvalues with zero real parts. Denote E -and E + respectively the stable and unstable subspaces of dF 0 and k = dim(E -) so that d-k = dim(E + ). Let us consider the following Cauchy problem

(3.1) H (t) = F (H(t)) , t ∈ R , H(0) = H 0 ,
and suppose, for any H 0 ∈ R d , there exists a unique global solution

H ∈ C 1 (R, R d ).
Then there exists V a neighborhood of 0 in R d and two C 1 -manifolds in V, say W -and W + , with respective dimensions k and d -k, tangent at 0 to E -and E + respectively, and such that for all H 0 ∈ V, the solution of (3.1) satisfies:

(a) lim t→+∞ H(t) = 0 ⇔ H 0 ∈ W -, (b) lim t→-∞ H(t) = 0 ⇔ H 0 ∈ W + .
More precisely, the manifold W -(stable at +∞) is the graph of a C 1 -map denoted ϕ -defined over E -∩ V and satisfying ϕ -(0) = 0 et dϕ -(0) = 0. The same result holds for W + .

The above result is a classical consequence of a fixed point argument and the convergence to such hyperbolic equilibrium point is exponential in the stable manifold, with a convergence rate similar to the one of the linearized Cauchy problem in the tangent space E -, i.e. of order at worst O(e -γt ) where

-γ = max ({ (sp(dF 0 ))} ∩ R -) < 0 .
3.1.2. Convergence to the diagonal matrices: a naive estimate. From Corollary 1.8, we know that H(t) converges to H ∞ = diag(α ) where the α are exactly the eigenvalues of H 0 since H is of finite dimension. This convergence analysis is then reduced to a small neighborhood of H ∞ . The differential equation reads

H = F (H) , F (H) = [H, G(H)] ,
and, since G and H ∞ are diagonal in the canonical bases, we have G(H ∞ ) = 0. We deduce that the differential of F at H ∞ is given by

∀H ∈ S(H) , dF H∞ (H) = [H ∞ , G(H)] .
Thus dF H∞ is an endomorphism of the space of symmetric matrices and it is diagonalized in the basis E i,j since a straightforward computation gives

dF H∞ (E i,j ) = g i,j (α i -α j )E i,j .
Since 0 is an eigenvalue of dF H∞ of multiplicity d, we cannot directly apply the stable manifold theorem. Nevertheless, we can elementarily get the following proposition.

Proposition 3.2. Let us introduce

T = {(i, j) ∈ {0, . . . , d -1} 2 : i < j} ,
and assume that -γ := max

T g i,j (α i -α j ) < 0 .
Then, for all ε > 0, there exist T, C > 0 such that, for all t ≥ T ,

H(t) -diag(H(t)) HS ≤ Ce -γt .
Proof. Let us define

R(H) = F (H) -F (H ∞ ) -dF H∞ (H -H ∞ ) = [H -H ∞ , G(H -diag(H))] ,
and

∆(t) = H(t) -diag(H(t)) .
Let us now consider ε > 0 and T > 0 such that, for all t ≥ T , we have

H(t) -H ∞ HS ≤ ε , R(H(t)) HS ≤ 2ε G ∆(t) HS .
For all H ∈ S(H), we notice that diag (dF H∞ (H)) is zero. Therefore we have

∆ (t) = dF H∞ (∆(t)) + R(H(t)) -diag (R(H(t))) .
The Duhamel formula reads, for t ≥ T ,

∆(t) = e (t-T )dF H∞ ∆(T ) + t T e (t-s)dF H∞ {R(H(s)) -diag (R(H(s)))} ds ,
where ∆ and R(H) -diag (R(H)) ∈ Span (E i,j ) i<j . Since (E i,j ) is orthonormalized for the HS-norm, we deduce that ∆(t) HS ≤ e -γ(t-T ) ∆(T ) HS + From the Gronwall lemma, we get, for t ≥ T , e γt ∆(t) HS ≤ e γT ∆(T ) HS e 2ε G (t-T ) , and thus ∆(t) HS ≤ e (-γ+2ε G )(t-T ) ∆(T ) HS .

We deduce that the same convergence holds for the diagonal terms.

Let us now eliminate the dependence in ε in the exponential. We easily have

H (t) = F (H(t)) = [H(t), G(H(t))] = [H(t), G(∆(t))] ,
so that for all t ≥ T

H (t) HS ≤ 2 G H 0 HS ∆(t) HS ≤ 2 G H 0 HS Ce (-γ+ )t .
It follows that

H(t) -H ∞ HS ≤ +∞ t H (s) HS ds ≤ 2 G H 0 HS C -γ + e (-γ+ )t .
We get therefore the existence of a constant D that depends on and H 0 HS only such that for all t ≥ T ,

H(t) -H ∞ HS ≤ De (-γ+ )t .
Coming back to the previous proof with De (-γ+ )t in place of , we get then from the Gronwall lemma e γt ∆(t) HS ≤ e γT ∆(T ) HS e 2D G t T e (-γ+ )s ds , that may be bounded independently from t ≥ T . Therefore we may omit the term in exponential convergence rate.

Remark 3.3. In particular, this proposition may be used as follows. Let us consider ε such that 0 < ε < inf T |α i -α j | and assume that H 0 -H ∞ HS < ε which may be insured by using the flow up to the time T . Then, there exists a permutation Π such that H0 -

H∞ HS = O(ε) , H0 = Π H 0 Π , H∞ = diag(α ) ,
where (α ) ∈{0,...,d} denotes the sequence of the eigenvalues of H 0 in the opposite order with respect to g i,j . Then we have

H(t) -diag( H(t)) HS ≤ Ce -γt .
In other words, up to a permutation and as soon as we are close enough to the limit, we can insure an exponential convergence (with rate γ) to the diagonal matrices.

3.1.3. Proof of Corollary 1.9. It is actually more convenient to work on the side of the unitary transform U . We recall that, from (2.3), U satisfies

U = U G(U H 0 U ) = F(U ) .
Let us first notice that U converges.

Proposition 3.4. The function t → U (t) converges as t tends to +∞. Moreover, the limit U ∞ belongs to the finite set

{Π diag( )Π , Π ∈ S n , ∈ {±1}} . Proof. The orthogonal group is compact since the dimension is finite. If U converges to U ∞ , then U ∞ is still unitary and H ∞ = U ∞ H 0 U ∞ . There exists an orthogonal transformation Q 0 such that Q 0 H 0 Q 0 = diag(α )
where the α are in non-increasing order. Thus, there exists a permutation Π such that

H ∞ = (U ∞ Q 0 Π )H ∞ (ΠQ 0 U ∞ ) ,
Thus, since H ∞ is diagonal with simple coefficients, ΠQ 0 U ∞ is itself diagonal and thus, we get (ΠQ 0 U ∞ ) 2 = Id. Now we know that H(t) tends to H ∞ so that U (t) goes to 0 as t goes to +∞. Since the orthogonal group is compact and that F is continuous with a finite number of zeros, we deduce that U must converge to a zero of F.

We can notice that the proof of Proposition 3.4 relies on the convergence of H(t) when t goes to +∞. Without this a priori convergence, we would not get the convergence of U (t) for any given H 0 . As in [4, Theorem 3], one should exclude from the initial conditions the union of the stable manifolds of all the stationary points that are not attractive. With the compactness of U(H), this would imply the convergence of U (t) towards the (unique) attractive point. Actually such restrictions on the initial conditions are not necessary to get the convergence as explained in the following lines.

Let us now parametrize, locally near U ∞ , the orthogonal group U(H) by its Lie algebra A(H). By the local inversion theorem, it is well known that exp : A(H) → U(H) is a local smooth diffeomorphism between a neighborhood of 0 and a neighborhood of Id. We recall that

(d exp(A)) -1 (H) = exp(-A) +∞ k=0 (adA) k (k + 1)! H .
For t ≥ T , we may write U (t) = U ∞ exp(A(t)). Thus we have

U = U ∞ e A(t) G(e -A(t) H ∞ e A(t) ) .
We get

A (t) = e A(t) G(e -A(t) H ∞ e A(t) )e -A(t) + R 1 (t) , with R 1 (t) = O( A(t) 2 ). We infer that A (t) = G([H ∞ , A]) + R 2 (t) , with R 2 (t) = O( A(t) 2 ). Now A → G([H ∞ , A]
) is an endomorphism of the space of skew-symmetric matrices A(H) and it is diagonalized in the basis E ± i,j with i < j with eigenvalues g i,j (α i -α j ). By assumption these eigenvalues are not zero. We may apply the stable manifold theorem. Since A(t) goes to 0, this means, by definition, that A(t) belongs to the stable manifold near 0. Therefore it goes exponentially to zero, with a convergence rate a priori of order O(e -γt ), with γ = inf T -|g i,j (α i -α j )| > 0. Of course, the convergence may be stronger, depending on the stable manifold on which A(t) is evolving. Then H(t) inherits this convergence rate.

Abstract factorization of Hilbert-Schmidt operators.

Actually it is possible to get a quantitative convergence to the eigenvalues of non self-adjoint operators in infinite dimension. Note that, in Theorem 1.3, we do not have such a quantification, even for symmetric operators.

H (t) = [H(t), G(H(t))] , H(0) = H 0 ∈ L 2 (H) .
We also introduce the equations

g 1 (t) = g 1 (t)G(H(t)) , g 1 (0) = Id , g 2 (t) = (H(t) -G(H(t)))g 2 (t) , g 2 (0) = Id .
Then we have the following lemma (coming from a straightforward adaptation of the theorems in [START_REF] Chu | Isospectral flows and abstract matrix factorizations[END_REF] that are actually true for bounded operators).

Lemma 3.5. The solutions H, g 1 and g 2 are global and of class C 1 . Then, we have

g 1 ∈ C 1 (R, U(H)) and (3.2) H(t) = g 1 (t) -1 H 0 g 1 (t) = g 2 (t)H 0 g -1 2 (t) and thus we have H ∈ C 1 (R, L 2 (H)).
Moreover we have the relations

(3.3)
e tH 0 = g 1 (t)g 2 (t) , e tH(t) = g 2 (t)g 1 (t) .

Proof. Let us just give some insights of the proof. By easy calculations, we get (g 1 g 1 ) = 0, (g 1 Hg 1 ) = 0, and (g -1 2 Hg 2 ) = 0, which ensures both the first algebraic properties and the global character of the solutions in the considered spaces. The first exponential factorization follows from the uniqueness result for the following Cauchy problem Z (t) = H 0 Z(t), t ∈ R, Z(0) = Id , of which both e tH 0 and g 1 (t)g 2 (t) are solutions. The second exponential factorization then easily follows as a consequence. Thus we may write e tH 0 e = j=0 α ,j (t)g 1 (t)e j .

We get P -1 e tΛ f = j=0 α ,j (t)g 1 (t)e j , f = P e .

We have (3.4) j=0 α ,j (t)g 1 (t)e j = P -1 e tΛ f = J j=0 p ,j e tλ j v j + +∞ j=J+1 p ,j e tλ j v j ,

where v j = P -1 e j . We may also write

(3.5) A J   e 0 . . . e J   = P J e tΛ J   g 1 v 0 . . . g 1 v J   + O(e t (λ J+1 ) ) .
and

A J = P J e tΛ J   g 1 v 0 . . . g 1 v J   [e 0 .
. . e J ] + O(e t (λ J+1 ) ) .

By using the simplicity of the eigenvalues, we get, by using a Neumann series and the fact that P J is invertible, that A J is invertible when t is large enough. In addition, we get

(3.6) A -1 J = O e -t (λ J ) .
From (3.4), we get j=0 α ,j (t)H 0 g 1 (t)e j = J j=0 p ,j λ j e tλ j v j + O e t (λ J+1 ) , so that, with (3.2), j=0 α ,j (t)g 1 (t)H(t)e j = J j=0 p ,j λ j e tλ j v j + O e t (λ J+1 ) , which may be written as

A J   H(t)e 0 . . . H(t)e J   = P J Λ J e tΛ J   g 1 v 0 . . . g 1 v J   + O(e t (λ J+1 ) ) .
or, with (3.6),   H(t)e 0 . . .

H(t)e J   = A -1 J P J Λ J e tΛ J   g 1 v 0 . . . g 1 v J   + O(e t (λ J+1 -λ J ) ) .
From (3.5), we get

(3.7)   g 1 v 0 . . . g 1 v J   = e -tΛ J P -1 J A J   e 0 . . . e J   + O(e t (λ J+1 -λ J ) ) , and thus (3.8)
  H(t)e 0 . . .

H(t)e J   = A -1 J P J Λ J P -1 J A J   e 0 . . . e J   + O(e t (λ J+1 -λ J ) ) .
Therefore the space spanned by e 0 , . . . , e J is invariant by H(t) up to an error of order O(e t (λ J+1 -λ J ) ). In view of (3.8), we get the conclusion by induction on J ≥ 0.

In other words, we approximate the diagonal terms of H(t) one by one by moving J from 0 to +∞.

3.3.

Numerics. In the following numerical illustrations, we consider the finite dimensional example of flows evolving in matrices of M d (R) with d = 5. As an initial data, let fix

H 0 = Q diag( 2 ) 1≤ ≤d Q , where Q = exp(B 5
) is a unitary matrix defined through the following choice of B d ∈ A d (R) skew-symmetric:

B d = (b i,j ), b i,j = 1, for i > j .
The numerical solver to compute any of the below approximate solutions is an adaptive 4th-order Runge-Kutta scheme.

choice A = diag(d -) 0≤ ≤d-1 ∈ D(R d
) with non-increasing diagonal elements. With these data, we observe the convergence to the limit H ∞ = diag([25, 16, 9, 4, 1]), see Figure 1. The diagonal terms are sorted in a descending order, according to the coefficients in A. The effective exponential rate of convergence γ = 3 coincides, for sufficiently large times, with the worst expected one (Corollary 1.9). The matrix of g i,j (α i -α j ) is there: .

      0 - 9
Let change the initial data to the following one H0

= Q diag( 2 ) 1≤ ≤d Q , where Q = 1 (0) (0) exp(B 4 )
. Then H0 takes the above block form (3.9) 1 (0) (0) K ,

where K ∈ S 4 (R) has spectrum {4, 9, 16, 25}. Then we observe the convergence to the limit H ∞ = diag([1, 25, 16, 9, 4]), with an exponential rate γ = 5. Namely, the matrix of g i,j (α i -α j ) is then:

     
0 24 30 24 12 24 0 -9 -32 -63 30 -9 0 -7 -24 24 -32 -7 0 -5 12 -63 -24 -5 0

      .
In fact, in that case the exact flow evolves in the linear subspace described in (3.9) and the positive coefficients of the above matrix concern precisely the evolution of the flow transversally to that subspace. It is only because of the very specific form of the initial data that the numerical flow evolves in the same way, elsewhere numerical error due to the finite precision would have induced a convergence of the numerical solution to the totally sorted limit with rate γ = 3.

3.3.2. Toda's choice. For the same test case, the Toda flow gives similar results, see Figure 2. Once again, the effective exponential rate of convergence is γ = 3 and coincides, for sufficiently large times with the theoretical one. Actually the matrix of g i,j (α i -α j ) is there: Under assumptions of Proposition 1.10, exp(H 0 ) is invertible and diagonalizable with eigenvalues (e λ j ) j≥1 with decreasing moduli. This is a sufficient condition the convergence of the QR algorithm applied to e H 0 . We already get g 1 (t) ∈ U(H) and, for that flow, the Cauchy-Lipschitz theorem ensures that g 2 takes value in the subset of upper triangular matrices. At t = 1, e H 0 = g 1 (1)g 2 (1) is therefore nothing but a QR factorization of e H 0 . Thus e H(1) = g 2 (1)g 1 (1) corresponds to the first iteration of the discrete algorithm and, since the differential equation is autonomous, that handling iterates along integer times so that the sampling sequence (e H(n) ) mimics the QR algorithm applied to e H 0 . In fact, the diagonal coefficients of g 2 are not all positive (even if they would be, up to a product with a diag(±1) matrix, that corresponds to a change in the initial data for g 1 and g 2 depending on H 0 ), and the two algorithms slightly differ.

      0 - 9
3.3.4. Wegner's choice. Consider now the above initial data for the Cauchy problem associated to Wegner's flow. The solution converges to H ∞ = diag( [START_REF] Brockett | Dynamical systems that sort lists, diagonalize matrices, and solve linear programming problems[END_REF][START_REF] Watkins | On Rutishauser's approach to self-similar flows[END_REF]16,25,[START_REF] Bach | Rigorous foundations of the Brockett-Wegner flow for operators[END_REF]). The exponential rate to the limit is γ = -9 that matches the expected value. Indeed a simple calculation gives, for E ∈ S(H)

dF H∞ (E) = [H ∞ , [E, H ∞ ]] ,
so that for i < j dF H∞ (E i,j ) = -(α i -α j ) 2 E i,j .

Therefore, at the limit H ∞ , one has the following "matrix-eigenvalue": In that case, for any possible diagonal limit H ∞ , the corresponding eigenvalues of dF H∞ are all negative, except 0 that is an eigenvalue of multiplicity d.

  e , e j e j = λ e + O e -tδ , δ = min 0≤j≤ (λ j -λ j+1 ) .

2. 3 .

 3 Examples. Let us now provide examples of applications of our convergence results. 2.3.1. Brockett's choice. For a given A = diag(a ) ∈ D(H) with a non-increasing sequence (a ) ∈N

  t-s)γ R(H(s)) -diag (R(H(s))) HS ds , so that e γt ∆(t) HS ≤ e γT ∆(T ) HS + 2ε G t T e sγ ∆(s) HS ds .

3. 2 . 1 .

 21 Chu-Norris decomposition for Hilbert-Schmidt operators. Let us consider an application G : L(H) → A(H) of class C 1 and the following Cauchy problem:

j=0α

  ,j (t)e j .

Figure 1 .

 1 Figure 1. Brockett flow -Convergence of diagonal entries (left), exponential convergence of H(t) (right).

Figure 2 .

 2 Figure 2. Toda flow -Convergence of diagonal entries (left), exponential convergence of H(t) (right).

Figure 3 .

 3 Figure 3. Toda flow -Convergence of extradiagonal entries.

On Figure 3 ,

 3 we present the numerical counterpart of Proposition 1.10. For any 1 ≤ ≤ d-1, we compute the norm of the residual column (h j, -α δ j, ) ≤j≤d along the time. The thick dashed curves corresponds to the numerical computations and the thin continuous ones corresponds to the expected rate of convergence, namely (α -α +1 ) ∈ {9, 7, 5, 3} successively, because of the precise distribution of the eigenvalues of H 0 in this example.

Figure 4 .

 4 Figure 4. Wegner flow -Convergence of diagonal entries (left), exponential convergence of H(t) (right).

3. 3 . 3 .

 33 Relation with the QR algorithm. The exponential relations in Lemma 3.5 are the keystone for the connection of the Toda flow to the well-known QR algorithm.

  • ) with the function F , we get the local existence of a solution of the Cauchy problem (1.6), on a maximal time interval [0, T max ), with T max > 0. 2.2.2. Global existence. Let us now show that T max = +∞ and that, we have, for all t ∈ [0, +∞), H(t) ∈ L 2 (H) and H(t) HS = H 0 HS . Let us consider the following Cauchy problem:

  From this, we infer that, for all t ∈ [0, T max ), H(t) ∈ L 2 (H) and H(t) HS = H 0 HS . Therefore, on [0, T max ), the function H is bounded in L 2 (H) ⊂ L(H). Thus we get T max = +∞. 2.2.3. Linear case. Let us now discuss the proof of Corollary 1.8. First we notice that, since L 2 (H) is an ideal of L(H), the function F defined in (2.2) sends the Hilbert space S 2 (H) into itself. Thus, we may apply the Cauchy-Lipschitz theorem in this space and the global existence is ensured by the investigation in the previous section. Let us reformulate (1.6) in the Hilbert basis (e n ) n∈N . We easily have

Then an easy computation shows that the derivative of t → U (t)H(t)U (t) is zero and thus

∀t ∈ [0, T max ) , H(t) = U (t) H 0 U (t) . h i,i (t) = 2 G(H(t))e i , H(t)e i .

  2.3.3. Wegner's choice. Let us finally discuss a non-linear example. We take G 3 (H) = [H, diag(H)]. Since G 3 sends the symmetric operators in the skew symmetric operators, we get that G 3 generates a flow that preserves the Hilbert-Schmidt norm. Thus the flow is global. Explicitly we have H = [H, [H, diagH]]. In particular, we get

  3.2.2. Proof of Proposition 1.10. It is easy to see that g 2 (t)e =


Youssef Dewer

Marie-Anne Pottier

Lisa Lalouette

Annick Maria

Matthieu Dacher

Luc P Belzunces

Guillaume Kairo

David Renault

Martine Maïbèche

David Siaussat
email: david.siaussat@upmc.fr

Martine Maibeche

Behavioral and metabolic effects of sublethal doses of two insecticides, chlorpyrifos and methomyl, in the Egyptian cotton leafworm, Spodoptera littoralis (Boisduval) (Lepidoptera: Noctuidae)

Keywords: Spodoptera littoralis, Crop pest, Insecticide, Sublethal doses, Methomyl, Chlorpyrifos, Olfaction, Insect behavior

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

Introduction

Study of the effects of sublethal doses of neurotoxic chemicals has been poorly investigated until now, and their mechanisms of action are thus only partially understood. Yet, for several years, the policy of many countries has been to reduce the use of pesticides [START_REF] Bellinder | Percentage-driven government mandates for pesticide reduction: the Swedish model[END_REF]. Furthermore, it is well known that many biotic (e.g., plant uptake, microbial, and plant degradation) and abiotic (e.g., wind, volatilization, chemical degradation, and dripping) processes can spatially and temporally change the pesticide doses that an insect is exposed to in the field. It is therefore urgent to determine if environmentally relevant sublethal pesticide concentrations could have a positive or negative effect on non-target and target species. In a context of sustainable development, it also Responsible editor: Philippe Garrigues Youssef Dewer and Marie-Anne Pottier contributed equally to this work.

Electronic supplementary material

The online version of this article (doi:10.1007/s11356-015-5710-1) contains supplementary material, which is available to authorized users. seems important to find new ways to assess the exposure of terrestrial ecosystems to low doses of pesticides and, in particular, to find new biomarkers (molecular to behavioral ones) for the survey of organisms living in these contaminated environments.

Some studies have reported positive effects of pesticide sublethal doses-also called hormetic effects-on population growth and reproductive physiology in various insect species, including in the cotton leafworm Spodoptera littoralis (review in [START_REF] Cutler | Insects, insecticides and hormesis: evidence and considerations for study[END_REF][START_REF] Tricoire-Leignel | Pest insect olfaction in an insecticide-contaminated environment: infodisruption or hormesis effect[END_REF]). However, deleterious impacts of low doses were also reported on fecundity, longevity, and behavior [START_REF] Ripper | Effects of pesticides on balance of arthropod populations[END_REF][START_REF] Haynes | Sublethal effects of neurotoxic insecticides in insect behavior[END_REF][START_REF] Stark | Population level effects of pesticides and other toxicants on arthropods[END_REF][START_REF] Dewer | Effectiveness and safety of some essential oils of aromatic plants on the growth and silk production of the silkworm Bombyx mori[END_REF]. While the effects of high doses were easy to determine because they are usually associated with the mortality of the targeted species, effects of low and sublethal doses were more difficult to observe and are thus relatively less well studied [START_REF] Cutler | Insects, insecticides and hormesis: evidence and considerations for study[END_REF]).

Among a variety of insecticides, organophosphorus and carbamate compounds are widely used worldwide. Chlorpyrifos and methomyl are currently the two most used insecticides belonging to these pesticide families. Both of them cause severe cholinergic poisoning in insects by inhibiting the enzyme acetylcholinesterase (AChE) that hydrolyzes the neurotransmitter acetylcholine. Overstimulation of the nervous system leads to the insect's death. Chlorpyrifos has a well-known environmental persistence and can be found in all environmental compartments [START_REF] Delpuech | Sublethal effects of the insecticide chlorpyrifos on the sex pheromonal communication of Trichogramma brassicae[END_REF]. This pesticide induces an irreversible inhibition of AChE that can be long lasting even at low doses [START_REF] Paudyal | Organophosphorus poisoning[END_REF][START_REF] Carr | Inhibition and aging of channel catfish brain acetylcholinesterase following exposure to two phosphorothionate insecticides and their active metabolites[END_REF]. Indeed, one study highlighted that a short term application (i.e., 4 h) of chlorpyrifos at sublethal doses continued to inhibit AChE for several days afterward [START_REF] Carr | Inhibition and aging of channel catfish brain acetylcholinesterase following exposure to two phosphorothionate insecticides and their active metabolites[END_REF]. This effect was also reported for non-targeted species such as wolf spiders (Lycisidae) for which the AChE inhibition was observed 8 days after a 24-or 48-h exposure to chlorpyrifos [START_REF] Van Erp | Sublethal responses of wolf spiders (Lycosidae) to organophosphorus insecticides[END_REF]. Exposure of the parasitoid Aphytis melinus to median lethal concentration (LC50) of chlorpyrifos reduced longevity and depressed progeny production [START_REF] Rosenheim | Sublethal effects of pesticides on the parasitoid Aphytis melinus (Hymenoptera: Aphelinidae)[END_REF]. As other carbamate insecticides, methomyl induces a reversible AChE inhibition. Contrary to chlorpyrifos, this product has a low persistence in the environment and a high toxicity not only for pest insects but also for birds, aquatic organisms, and beneficial arthropods such as bees [START_REF] Van Scoy | Environmental fate and toxicology of methomyl[END_REF]). Its use is often recommended against pests resistant to organophosphorus pesticides (WHO 1983). Few studies have demonstrated effects of methomyl at sublethal doses in insects: a stimulation of the fecundity was reported in a susceptible strain of the diamondback moth Plutella xylostella [START_REF] Nemoto | Factors inducing resurgence in the diamondback moth after application of methomyl[END_REF], and a transitory growth reduction was observed in larvae of the fall armyworm Spodoptera frugiperda [START_REF] Ross | Inhibition of larval growth in Spodoptera frugiperda by sublethal dietary concentrations of insecticides[END_REF].

These two insecticides were currently used to protect crops against the polyphagous larval stages of S. littoralis [START_REF] Miles | Evidence for negative cross resistance to insecticides in field collected Spodoptera littoralis (Boisd.) from Lebanon in laboratory bioassays[END_REF][START_REF] Riskallah | Reduced sensitivity of cholinesterase as a factor of resistance in leptophos selected strain in the Egyptian cotton leafworm[END_REF]. This noctuid moth is a worldwide economically important pest of cotton, vegetables, and ornamental crops. Pesticide-resistant populations cause severe problems in various countries, and the larvae cause high levels of damages [START_REF] Smagghe | Action of brassinosteroids in the cotton leafworm Spodoptera littoralis[END_REF]. In this nocturnal insect, olfaction is the main sensory modality to communicate with mating partners, localize trophic patches, and oviposition sites, making this species a main pest insect model to decipher the functioning of its olfactory system, from genes to behavior [START_REF] Bigot | Peripheral regulation by ecdysteroids of olfactory responsiveness in male Egyptian cotton leaf worms, Spodoptera littoralis[END_REF][START_REF] Pottier | Cytochrome P450s and cytochrome P450 reductase in the olfactory organ of the cotton leafworm Spodoptera littoralis[END_REF][START_REF] Party | Changes in odor background affect the locomotory response to pheromone in moths[END_REF]. Several studies have focused on the impact of sublethal doses of various pesticides on olfaction using pest insects as models [START_REF] Tricoire-Leignel | Pest insect olfaction in an insecticide-contaminated environment: infodisruption or hormesis effect[END_REF]) in order to determinate if these products could act as info-disruptors by modifying the chemical communication system and thus decrease reproduction chances in target insects [START_REF] Lurling | Info-disruption: pollution and the transfer of chemical information between organisms[END_REF], or if they could act as hormetic factors by enhancing reproduction [START_REF] Cutler | Insects, insecticides and hormesis: evidence and considerations for study[END_REF]. However, none of these studies were conducted on S. littoralis. In addition, the effect of methomyl at sublethal doses on olfaction has not been studied yet in any insect species, whereas few data were available for chlorpyrifos. It has been shown that Trichogramma brassicae males exposed to chlorpyrifos at LD 20 or LD 0.1 were less arrested by female sexual pheromones (review in [START_REF] Desneux | The sublethal effects of pesticides on beneficial arthropods[END_REF]), whereas at LD 20, it caused an increase in host searching in the parasitoid Leptopilina heterotoma (review in [START_REF] Desneux | The sublethal effects of pesticides on beneficial arthropods[END_REF]). These studies were restricted on adults.

In this context, the first step of this study was to determine the sublethal doses of methomyl and chlorpyrifos in S. littoralis larvae. To evaluate if these doses have indeed an effect on the insect physiology, we use a metabolomic approach to compare the levels of several hemolymphatic markers between control and treated larvae. In the second step, we examined the effects of sublethal doses of these two compounds on a vital olfactory-induced behavior (i.e., the attraction of larvae by food odors) and measured in the head of the corresponding larvae the AChE activity.

Materials and methods

Insect rearing and insecticide treatments

Spodoptera littoralis larvae were reared on a semi-artificial diet [START_REF] Hinks | Biosystematics of the genus Euxoa (Lepidoptera: Noctuidae). V. Rearing procedures and life cycles of 36 species[END_REF] at 24 °C, 60-70 % relative humidity, and a 16:8 light/dark cycle until emergence. Topical applications of insecticides were performed on fourth larval instars using a micro-applicator (Hamilton 10 μL syringe and Hamilton dispenser). A range of concentrations of chlorpyrifos (45395, SIGMA PESTANAL, France) diluted in hexane, or of methomyl (36159, SIGMA PESTANAL, France) diluted in dimethyl succinate, were applied to the larval cuticle between the fifth and sixth segment (counting starting from the first head segment). Control insects were treated using hexane or dimethyl succinate, respectively. Forty insects were exposed to each concentration of both compounds. Percentage of mortality was determined 24 h after pesticide application and LD 50 calculated by probit analysis. For further metabolic fingerprinting and behavioral analyses, insects were exposed as described previously but using two chosen sublethal doses for each pesticide, i.e., 1/10 and 1/100 of the LD 50 (noted LD 50 1/10 and LD 50 1/100 afterward). These doses did not induce any mortality.

Metabolic fingerprinting

The hemolymph of larvae was sampled as described in [START_REF] Bigot | Peripheral regulation by ecdysteroids of olfactory responsiveness in male Egyptian cotton leaf worms, Spodoptera littoralis[END_REF]. Hemolymph from four larvae (40±10 μL) was suspended in ethanol (400 μl, purity 99.9 %, MERCK, France). Ten replicates were prepared for each experimental condition. Samples were prepared as described in [START_REF] Khodayari | Deciphering the metabolic changes associated with diapause syndrome and cold acclimation in the two-spotted spider mite Tetranychus urticae[END_REF], with minor modifications. The samples were homogenized in 600 μL of methanol-chloroform (2:1) using a bead-beating device (Retsch ™ MM301, RetschGbmH, Haan, Germany). Four hundred microliters of ice-cold ultrapure water was added, and then aliquots (300 μL) of the upper aqueous phase containing polar metabolites were transferred to micro-tubes and vacuum-dried. Following derivatization (see [START_REF] Khodayari | Deciphering the metabolic changes associated with diapause syndrome and cold acclimation in the two-spotted spider mite Tetranychus urticae[END_REF] for the detailed experimental procedure), metabolites were analyzed by gas chromatography-mass spectrometry (GC-MS) (Figs. 1 and2), which included a CTC CombiPal autosampler (GERSTEL GmbH & Co.KG, Mülheim an der Ruhr, Germany), a Trace GC Ultra chromatograph, and a Trace DSQII quadruple mass spectrometer (Thermo Fischer Scientific Inc., Waltham, MA, USA) [START_REF] Khodayari | Deciphering the metabolic changes associated with diapause syndrome and cold acclimation in the two-spotted spider mite Tetranychus urticae[END_REF]. Fiftyseven metabolites were analyzed (see supplementary data). Peaks were accurately annotated using both mass spectra (two specific ions) and retention times. Calibration curves were set using samples consisting of 58 pure references. Metabolite levels were quantified using XCalibur v2.0.7 software (Thermo Fisher Scientific Inc., Waltham, MA, USA).

Behavioral experiments

The ability of larvae to orient toward food odor was tested using a four-choice olfactometer (Analytical Research Systems, Inc, Florida, USA). Each internal odor source adaptors (OSA) with insect isolation trap was connected to the four cardinal corners of the arena in order to send the airflow (with or without odor) to the larvae placed on a fine mesh grid in the center of the arena. The air flow was generated using an air compressor (OLFM-4C-ADS, Analytical Research Systems, Inc, Florida, USA), pulsed in each OSA at 200 ml/min and regulated with a flow-meter. A vacuum flow evacuated odors through the grid in the center of the arena.

For feeding stimulation, a piece of semi-artificial diet was heated for 10 s in a microwave in order to increase the number of volatiles emitted by the food and then placed immediately in one of the four OSAs before air flow activation. The three other OSAs without food corresponded to air flow controls. The arena was cleaned regularly with 10 % TFD4 (Dutscher, France) to eliminate odor residues. The location of the OSA containing the food was changed between each test in order to prevent any innate preferences for particular parts of the arena. Analyses were performed under red light conditions (T=22± 1 °C, RH=70±10 %) and recorded over a 1-h period with a CC infrared light-sensitive camera (QuickCam ® Pro 9000).

For each experiment, ten larvae (fourth instar) were isolated, food deprived for 24 h, and then introduced into the central starting zone (noted SZ) of the arena (see Figs. 3 and4). Larvae were considered as non-activated when staying in SZ for the duration of the test. Movements of the activated larvae were tracked individually in the four quadrants corresponding to the four airflow stimulations. Each quadrant was divided in two zones: a pre-zone (noted Z) and a target zone (noted C). The quadrant corresponding to the odor stimulation was noted (C0 and Z0), then the three control quadrants with air flow alone were noted (C1 and Z1; C2 and Z2; C3 and Z3) following the reverse rotation clockwise (see Figs. 3 and4). Several parameters were then measured: the time spent in the nine designated sectors of the arena, the proportion of larvae reaching the target, and the speed of each larva. As the insect isolation trap connected to the OSA, insects entering the target zone were not able to move out of this zone.

AChE extraction and assays

AChE was extracted from the head removed from anesthetized adults of S. littoralis. For each sample, 5 heads were weighted, and extraction buffer (10 mM NaCl and 40 mM sodium phosphate pH 7.4 containing 1 % (w/v) Triton X-100, 2 μg/mL antipain, leupeptin, and pepstatin A, 25 units/mL aprotinin, and 0.1 mg/mL soybean trypsin inhibitor as protease inhibitors[1]) was added to obtain 10 % (w/v) extract. Tissues were homogenized using a high-speed homogenizer Tissue Lyser II (Qiagen ® ), for 5 periods of 10 s with oscillation frequencies of 30 Hz at 30 s intervals. This procedure was performed twice with 10 min interval before the extraction product was centrifuged for 20 min at 15,000g. Supernatants were removed for analysis. All of the extraction procedure and sample conservation were performed at 4 °C. For each sample, AChE assay was performed in triplicate with 10 μl of enzyme extract. The final concentrations of the reagents in the reaction medium were 0.3 mM AcSCh.I, 1.5 mM DTNB, and 100 mM sodium phosphate buffer at pH 7.0, following the method of [START_REF] Ellman | A new and rapid colorimetric determination of acetylcholinesterase activity[END_REF] modified by B e l z u n c e s e t a l . ( 1 9 8 8 ) . A C h E a c t i v i t y w a s spectrophotometrically measured at 412 nm for 15 min using a TECAN infinite ® F500 plate reader.

Statistical analyses

Lethal doses for 50 % of the population (LD 50 ) were computed using probit analyses with an α risk of 0.05 (MINITAB Inc., State College, PA, USA). Comparisons of behavioral experiments were analyzed using a Kruskal-Wallis one-way analysis of variance. For metabolic variations and AChE activity, we performed a one-way ANOVA to compare the treatments (i.e., the control group and the two treated groups) for each pesticide. We used the ImPerm package in R 3.2 to compute the p values through a permutation procedure, which allows us to use ANOVA without the need of having normal data [START_REF] Venables | Modern applied statistics with S-Plus, p176 WHO[END_REF]. These statistical analyses were conducted using Minitab ™ 13 statistical software (MINITAB Inc., State College, PA, USA), STATISTICA (StatSoft Enterprise, USA), and R software.

Results

LD determination

The chlorpyrifos and methomyl doses that yielded 50 % mortality in fourth larval instar S. littoralis following dorsal topical application were 0.31±0.061 μg and 0.057±0.0053 μg, respectively.

Sublethal dose effects on hemolymphatic metabolite concentrations

LD 50 1/10 and LD 50 1/100 of chlorpyrifos and methomyl were used to determine their potential systemic effects Twenty-four hours following dorsal topical application of chlorpyrifos at LC 50 1/10 and LC 50 1/100 , and hemolymphs of four larvae were collected, pooled, and analyzed by GC-MS. Metabolite contents (nmoles μL -1 of hemolymph) were expressed as means ± SE (N = 10 replicates for each experimental condition). Single asterisk indicates a significant difference (p<0.05) between control and pesticide treated, double and triple asterisk indicates more significant differences (p<0.01; p<0.0001, respectively). Dotted squares highlight metabolites that present significant differences between control and treated larvae and specific to chlorpyrifos treatment on larvae. Fifty-seven metabolites were quantified in the hemolymph of the larvae 24 h after application of the two pesticides (Tables S1,S2, S3) and compared to the corresponding controls (Figs. 1 and2). Among them, twentyseven metabolites were not detected or not quantified because they were under the threshold detection level (Table S3). Compared to the controls, the levels of some of the 30 quantifiable circulating metabolites were clearly significantly altered by the pesticide applications (Figs. 1 and2) and this with two different patterns. A first set of metabolites was identified because their circulating level was modified after treatment at LD 50 1/10 and/or LD 50 1/100 , whichever pesticide used (Figs. 1 and2). This includes significant increases in phosphoric acid, glycerate, glycine, succinate, citrate, glucose, and trehalose in treated larvae. The second group of metabolites presented alterations specific to either methomyl or chlorpyrifos application. Indeed, whereas chlorpyrifos exposure induced significant circulating level changes for only three polyols (i.e., xylitol, sorbitol, and glycerol-3-phosphate; Fig. 1), methomyl exposure altered a larger number of metabolites including several amino acids (i.e., valine, leucine, isoleuscine, proline, serine), two sugars (i.e., ribose and arabinose), one polyol (i.e., arabitol), and fumarate, which is an intermediate of the citric acid cycle (Fig. 2).

Effects of sublethal doses on the larval behavior

The proportion of time spent by the larvae in the different pre-defined zones of the four-choice olfactometer arena was calculated for each larva 24 h after application of the two sublethal doses for both pesticide or of the corresponding solvent for the controls (Figs. 3 and4). Compared to the Twenty-four hours following dorsal topical application of methomyl at LC 50 1/10 and LC 50 1/100 in fourth larval instar, the pooled hemolymph of four larvae was collected and analyzed by GC-MS. Metabolite contents (nmoles μL -1 of hemolymph) were expressed as means±SE (N=10 replicates for each experimental condition). Single asterisk indicates a significant difference (p<0.05) between control and pesticide treated, double and triple asterisk indicates more significant differences (p<0.01; p<0.0001, respectively). Dotted squares highlight metabolites that present significant differences between control and treated larvae and specific to methomyl treatment control insects (Fig. 3a), chlorpyrifos at LD 50 1/10 and LD 50 1/100 did not alter the larval odorant-induced behaviors (Fig. 3b,c) since (i) the proportions of time spent in the odorant zones (C0+Z0) were largely higher than and significantly different to the three other zones (C1+Z1, C2+Z2, and C3+Z3), and (ii) the times spent in Z0 and C0 odorant zones were similar for the control and the treated insects (Fig. 3d). However, a different response was observed for larvae exposed to methomyl at LD 50 1/10 (Fig. 4). Indeed, larvae exposed to this dose spent more time than the controls in zones without odor stimulation (Fig 4c), whereas no differences were observed for larvae exposed to LD 50 1/100 (Fig. 4a,b). Moreover, LD 50 1/10 -treated larvae spent less time in odorant (C0+Z0) zones compared both to the control and to the LD 50 1/100 -treated larvae (Fig 4d). As the locomotor speed of larvae in all experimental conditions showed no significant differences for both pesticides (Figs. 3 and4e), we can conclude that methomyl, at least at the higher dose of LD 50 1/10 , disrupts the attraction of treaded larvae by food odors, whereas chlorpyrifos at the two chosen doses did not alter the attraction of larvae by food odors.

Effects of sublethal doses on the AChE activity

AChE activity was determined in the head of larvae exposed to the two sublethal doses of both pesticide or to the corresponding solvent for the controls, 24 h after pesticide applications (Fig. 5). The results showed that larvae exposed to sublethal doses of methomyl and chlorpyrifos have an AChE activity similar to that of controls (no significant difference).

Discussion

Our results highlight for the first time the effects of sublethal doses of methomyl and chlorpyrifos on S. littoralis larvae, both at the systemic and behavioral levels. Few data were available to date on the effect of these two insecticides on species from the Spodoptera genus. Moreover, all these studies were focused on lethal dose effects [START_REF] Klein | Comparative toxicity of several insecticides to eggs, larvae and adults of the Egyptian cottonworm, Spodoptera littoralis, in laboratory trials[END_REF] , and the percentage of time spent by pesticide-treated and control (hexane) larvae in designated quadrants of the arena were calculated (mean±SEM). Single asterisk indicates a significant difference (p<0.05) between odorant target and other zones, double and triple asterisk indicates more significant differences (p<0.01; p<0.0001, respectively) S. littoralis [START_REF] Miles | Evidence for negative cross resistance to insecticides in field collected Spodoptera littoralis (Boisd.) from Lebanon in laboratory bioassays[END_REF]. Resistance to methomyl in S. exigua [START_REF] Byrne | An insensitive acetylcholinesterase confers resistance to methomyl in the beet armyworm Spodoptera exigua (Lepidoptera: Noctuidae)[END_REF], S. frugiperda (Rios-Diez et al. 2011), S. litura [START_REF] Tong | Field resistance of Spodoptera litura (Lepidoptera: Noctuidae) to organophosphates, pyrethroids, carbamates and four newer chemistry insecticides in Hunan, China[END_REF][START_REF] Saleem | Resistance to selected organochlorine, organophosphate, carbamate and pyrethroid, in Spodoptera litura (Lepidoptera: Noctuidae) from Pakistan[END_REF], and S. littoralis [START_REF] Riskallah | Reduced sensitivity of cholinesterase as a factor of resistance in leptophos selected strain in the Egyptian cotton leafworm[END_REF] was observed.

The toxicological approach we used allowed first the determination of the median lethal doses which induced 50 % of mortality (LD 50 ) for the two molecules. Despite the lack of previous toxicological data for chlorpyrifos and methomyl in S. littoralis, the LD 50 values we found here for both products were consistent with those previously reported for the related species S. exigua and Helicoverpa zea [START_REF] Pérez | Assessment of insecticide resistance in five insect pests attacking field and vegetable crops in Nicaragua[END_REF].

We then observed a disruption in metabolic homeostasis, as depicted by hemolymphatic alterations of several metabolite concentrations (Fig. 6). Chlorpyrifos exposure specifically induced a decrease of two polyols (xylitol and sorbitol) and an increase of glycerol-3-phosphate concentrations. Polyols and more particularly xylitol and sorbitol are involved in the pentose phosphate pathway [START_REF] Kanfer | Formation of l-lyxonic and lxylonic acids from l-ascorbic acid in rat kidney[END_REF][START_REF] Bánhegyia | Ascorbate metabolism and its regulation in animals[END_REF][START_REF] Jeffery | Enzyme relationships in a sorbitol pathway that bypasses glycolysis and pentose phosphates in glucose metabolism[END_REF]. Glycerol-3-phosphate is involved in glycolysis and triacylglycerol production (Alves-Bezerra and Gondim 2012). Thus, our results highlighted that Fig. 5 AChE activity in S. littoralis fourth instar larvae exposed to methomyl or chlorpyrifos sublethal doses. Twenty-four hours following dorsal topical application of methomyl or chlorpyrifos at LC 50 1/10 and LC 50 1/100 in fourth larval instar, AChE activity in head of larvae was determined. (mean±SEM) energy production was affected in S. littoralis after chlorpyrifos sublethal dose exposure and suggested a shift between the pentose phosphate pathways and pathways using glycerol-3phosphate as intermediates. The energetic alteration observed here in S. littoralis is consistent with the reported effects of chlorpyrifos in several other taxons, such as Caenorhabditis elegans, Mytilus galloprovincialis, and Daphnia magna [START_REF] Jones | Potential new method of mixture effects testing using metabolomics and Caenorhabditis elegans[END_REF][START_REF] Vandenbrouck | Mixtures of similarly acting compounds in Daphnia magna: from gene to metabolite and beyond[END_REF]. In the freshwater carp, Cyprinus Carpio, a non-target vertebrate, chlorpyrifos exposure induced a significant increase in circulating concentration of several metabolites involved in energy metabolism, such as glucose [START_REF] Kokushi | Effects of chlorpyrifos on the metabolome of the freshwater carp[END_REF]. This augmentation was associated with the need to provide energy by gluconeogenesis after insecticide exposure in order to respond to higher metabolic demands [START_REF] Ramesh | Haematological and biochemical responses in a freshwater fish Cyprinus carpio exposed to chlorpyrifos[END_REF].

Methomyl exposure at sublethal doses induced a significant hemolymphatic concentration increase of several amino acids (i.e., alanine, phenylalanine, Threonine, valine, leucine, isoleucine, proline, and serine), two sugars (i.e., ribose and arabinose), five polyols (i.e., arabitol, adonitol, glycerol, inositol, and glycerol-3-phosphate), and fumarate-an intermediate of the citric acid cycle. Interestingly, this kind of amino acid increase has been previously reported with other carbamate insecticides, i.e., carbofuran, propoxur, and carbaryl. In the earthworm Metaphire posthuma and in several model fishes exposed to carbofuran, this increase has been associated with disruption of amino acid metabolism [START_REF] Begum | Carbofuran insecticide induced biochemical alterations in liver and muscle tissues of the fish Clarias batrachus (linn) and recovery response[END_REF][START_REF] Mudiam | Gas chromatography-mass spectrometry based metabolomic approach for optimization and toxicity evaluation of earthworm sub-lethal responses to carbofuran[END_REF]). In the rat, metabolomic profiling of serum and urine after sublethal exposure to propoxur also demonstrated an alteration of metabolites involved not only in amino acid metabolism but also in energy (Liang et al. 2012a, b). In the earthworm Eisenia fetida, a decrease in phenylalanine, tyrosine lysine, alanine, valine, and leucine has been proposed as a potential metabolic response indicator to carbaryl sublethal exposure, since no significant change in metabolic profile was observed after exposure to sublethal doses of chlorpyrifos [START_REF] Mckelvie | Metabolic responses of Eisenia fetida after Environ Sci Pollut Res sub-lethal exposure to organic contaminants with different toxic modes of action[END_REF]. Our results in S. littoralis are thus consistent with the literature available in other taxons and confirm that these metabolic pathways can be used as biomarkers of carbamate exposure.

Previous data and our current findings showed that disruption of energy production seems to be a common denominator in the metabolic responses of all species exposed to methomyl and chlorpyrifos, whatever the dose used. It is therefore not so surprising to observe here variations of metabolites involved in energetic pathways, as revealed by the increases in glycerol-3-phosphate (glycolysis and triacylglycerol production), glycerate (a glycolysis intermediate), citrate, and succinate (citric acid cycle) and glucose levels. Modifications in succinate and citrate levels can be due to a possible disruption in function of the citrate synthase or of the succinate dehydrogenase enzymes [START_REF] Tripathi | Effects of habitats and pesticides on aerobic capacity and survival of soil fauna[END_REF][START_REF] Naqvi | Bioaccumulative potential and toxicity of endosulfan insecticide to non-target animals[END_REF]. The concomitant increase of glucose, glycerate, and glycerol-3-phosphate highlighted a glycolysis disruption probably also due to enzymatic dysfunction. Interestingly, glycerol-3-phosphate appeared in bumblebee to be the optimal oxidative substrates for maintaining the high rates of oxidative metabolism of the mitochondria [START_REF] Syromyatnikov | Isolation and properties of flight muscle mitochondria of the bumblebee Bombus terrestris (L.)[END_REF].

We also observed an increase of phosphoric acid and trehalose concentrations after exposure to sublethal doses of chlorpyrifos and methomyl. Phosphoric acid is crucial for numerous cellular functions such as DNA and membrane lipid synthesis, generation of high-energy phosphate esters, and intracellular signaling. In Drosophila melanogaster, an increase in hemolymphatic phosphate has been associated with a reduction of life span and impaired functioning of Malpighian tubules [START_REF] Bergwitz | Genetic determinants of phosphate response in Drosophila[END_REF]. A metabolomic study in earthworm has highlighted an increase of phosphoric acid after exposure to carbofuran at sublethal dose [START_REF] Mudiam | Gas chromatography-mass spectrometry based metabolomic approach for optimization and toxicity evaluation of earthworm sub-lethal responses to carbofuran[END_REF]. Although it is difficult to determine the role of this metabolite in these contexts, it appeared to be a biomarker of exposure to two pesticides. Trehalose levels, the prominent Bblood sugar^in most insect species [START_REF] Thompson | Trehalose-the insect 'blood' sugar[END_REF], have already been found to increase in Bombyx mori and S. litura following exposure to sublethal doses of various pesticides [START_REF] Nath | Shifts in glycogen metabolism in hemolymph and fat body of the silkworm[END_REF][START_REF] Etebari | Biochemical changes in hemolymph of silkworm larval due to pyriproxyfen residue[END_REF][START_REF] Zhu | RNA interference of NADPH-cytochrome P450 reductase results in reduced insecticide resistance in the Bed Bug, Cimex lectularius[END_REF], suggesting that this sugar serves a protective role for proteins and cellular membranes from inactivation and/or denaturation [START_REF] Elbein | New insights on trehalose: a multifunctional molecule[END_REF].

Few studies have been previously performed on the effect of these two pesticides on insect behavior. For chlorpyrifos, alterations of odorant-driven behaviors in the two parasitoid species T. brassicae and L. heterotoma were already reported but for higher doses (i.e., low lethal doses LD 20 or LD 0.1 [START_REF] Desneux | The sublethal effects of pesticides on beneficial arthropods[END_REF]. Moreover, the impact of sublethal doses of methomyl on such olfactory-driven behavior has not been studied before. High doses of methomyl were only described to have feeding cessation effects [START_REF] Hannig | Feeding cessation effects of chlorantraniliprole, a new anthranilic diamide insecticide, in comparison with several insecticides in distinct chemical classes and mode-of-action groups[END_REF]). We showed here evidence for disruption of larval behavior after methomyl exposure, whereas no modification was observed with chlorpyrifos. We can hypothesize that this behavioral alteration could be due to a modification in the detection of the odorant source at the peripheral olfactory level (i.e., the antennae) and/or in the perception of the stimulus at the olfactory central level (antennal lobes) and/or higher central nervous system structures integrating chemical cues. Although we do not have data on the toxicity of the two pesticides on the AChE activity at the time of the pesticide exposure, our results showed that 24 h after treatment, the AChE activity of treated and control larvae is similar. Our data did not allow us to establish if there is a reversibility of AChE inhibition or if sublethal doses cannot induce AChE inhibition in our model. But, at the time of the behavioral experiment, no cholinergic poisoning by disruption of AChE was observed unlike other studies [START_REF] Hassani | Effects of sublethal doses of fipronil on the behavior of the honey-bee (Apis mellifera)[END_REF][START_REF] Benzidane | Toxicity of thiamethoxam on cockroach locomotor activity associated to its metabolite clothianidin[END_REF][START_REF] Paudyal | Organophosphorus poisoning[END_REF][START_REF] Carr | Inhibition and aging of channel catfish brain acetylcholinesterase following exposure to two phosphorothionate insecticides and their active metabolites[END_REF].

This behavioral disruption could also be due to an indirect consequence of metabolic disruption. Indeed, blocking or inhibiting neurotransmission can induce stress or/and metabolic modulation in neuronal tissues or tissues under neuronal control such as, for example, muscles. Bendahou and coworkers highlighted, for example, biochemical effects of sublethal doses of two neurotoxic pesticides (cypermethrin and fenitrothion) in hemolymph, head, and thoracic muscles of honeybees. [START_REF] Kokushi | Effects of chlorpyrifos on the metabolome of the freshwater carp[END_REF] correlated the high increase of metabolic ammonia levels after exposure to high concentration of chlorpyrifos to a severe convulsion in muscle caused by the inhibition of acetylcholinesterase activity. [START_REF] Moffat | Chronic exposure to neonicotinoids increases neuronal vulnerability to mitochondrial dysfunction in the bumblebee (Bombus terrestris)[END_REF] highlighted that the poor navigation and foraging observed in bumblebee (Bombus terrestris) colonies following a chronic exposure of neonicotinoid can be explained by a mitochondrial dysfunction in the brain of these insects, which may affect more globally the cellular metabolism. Although the mechanisms are not well understood or studied, all these results suggest that low or sublethal doses of neurotoxic pesticides can probably disrupt the metabolism of many tissues under AChE control including the nervous system itself. However, since our metabolic profiling results were obtained with hemolymph of larvae, it is difficult to determine the tissue(s) responsible for this metabolic disruption.

As we have performed topical applications directly on larval cuticle, both pesticides could have escape from midgut detoxification processes, leading to the observed metabolic disruption. But, interestingly, despite these two insecticides sharing the same mode of action by inhibiting both AChE, sublethal doses of chlorpyrifos did not induce any behavioral disruption in our assay conditions. These specific responses may not only be due to differences in the efficacy of these substances, but may also be linked to the different detoxification capacities of the insect toward these compounds, especially at the hemolymph-brain barrier level. As in vertebrates and for other biological barriers (e.g., digestive epithelium), this barrier provides xenobiotic chemoprotection [START_REF] Mayer | Evolutionary conservation of vertebrate blood-brain barrier chemoprotective mechanisms in Drosophila[END_REF][START_REF] Limmer | The Drosophila blood-brain barrier: development and function of a glial endothelium[END_REF]) and involved specific detoxification enzymes. We can hypothesize that the behavioral effects observed here can be due to lower defense capabilities of this hemolymph-brain barrier against methomyl compared to chlorpyrifos.

The metabolic disruption and the info-disruption of the chemical communication that we reported here in a model pest moth participate to a better understanding of the subtle effects of environmentally relevant sublethal pesticide concentrations on a target species. Further studies could allow to test if the effects observed here were observable in field conditions and to validate the metabolic markers identified here in an integrated pest management or ecotoxicological context.

Fig. 1

 1 Fig.1Metabolite contents of hemolymph from chlorpyrifos-treated S. littoralis fourth instar larvae. Twenty-four hours following dorsal topical application of chlorpyrifos at LC 50 1/10 and LC 50 1/100 , and hemolymphs of four larvae were collected, pooled, and analyzed by GC-MS. Metabolite contents (nmoles μL -1 of hemolymph) were expressed as means ± SE (N = 10 replicates for each experimental

Fig. 2

 2 Fig.2Metabolite contents of hemolymph from methomyl-treated S. littoralis fourth instar larvae. Twenty-four hours following dorsal topical application of methomyl at LC 50 1/10 and LC 50 1/100 in fourth larval instar, the pooled hemolymph of four larvae was collected and analyzed by GC-MS. Metabolite contents (nmoles μL -1 of hemolymph) were expressed as means±SE (N=10 replicates for each experimental

Fig. 3

 3 Fig. 3 Behavioral responses of S. littoralis fourth instar larvae exposed to chlorpyrifos sublethal doses. Twenty-four hours following dorsal topical application of chlorpyrifos at LC 50 1/10 and LC 50 1/100

Fig. 4

 4 Fig.4Behavioral responses of S. littoralis fourth instar larvae exposed to methomyl sublethal doses. Twenty-four hours following dorsal topical application of methomyl at LC 50 1/10 and LC 50 1/100 in fourth larval instar, and percentage of time larvae spent in designated quadrants of

Fig. 6

 6 Fig.6Schematic representation of the metabolic pathway disrupted by an exposure to methomyl or chlorpyrifos sublethal doses in S. littoralis fourth instar larvae. Metabolites presenting a hemolymphatic disruption compared to control were labeled M when the disruption was observed following an exposure to sublethal doses of methomyl, C for chlorpyrifos, and B when it was observed for both pesticides

Acknowledgments We thank Dr. Thomas Chertemps for his assistance in the statistical analyses. Dr. Lisa Lalouette was supported by a DIM ASTREA (Région Ile de France) post-doctoral fellowship. Dr. Youssef Dewer was supported by a BMairie de Paris^post-doctoral fellowship: Environ Sci Pollut Res Call 2010, Imhotep exchange program grant (Campus France): Call 2011, and the French Embassy in Egypt post-doctoral fellowship 2012. Dr Marie-Anne Pottier and all experiments were supported by an BEMER-GENCE^grant from the University Pierre and Marie Curie (UPMC-Paris 6) and PHEROTOX grant from ANR Bioadapt.