Olivier Faugeras

James Maclaurin
email: james.maclaurin@inria.fr.

A representation of the relative entropy with respect to a diffusion process in terms of its infinitesimal-generator

Keywords: relative entropy, Kullback-Leibler, diffusion, martingale formulation

In this paper we derive an integral (with respect to time) representation of the relative entropy (or Kullback-Leibler Divergence) R(µ||P ), where µ and P are measures on C([0, T ]; | d ). The underlying measure P is a weak solution to a Martingale Problem with continuous coefficients. Our representation is in the form of an integral with respect to its infinitesimal generator. This representation is of use in statistical inference (particularly involving medical imaging). Since R(µ||P ) governs the exponential rate of convergence of the empirical measure (according to Sanov's Theorem), this representation is also of use in the numerical and analytical investigation of finite-size effects in systems of interacting diffusions.

Introduction

In this paper we derive an integral representation of the relative entropy R(µ||P ), where µ is a measure on C([0, T ]; | d ) and P governs the solution to a stochastic differential equation (SDE). The relative entropy is used to quantify the distance between two measures. It has considerable applications in statistics, imaging, information theory and communications. It has been used in the long-time analysis of Fokker-Planck equations [START_REF] Plastino | Minimum Kullback entropy approach to the Fokker-Planck equation[END_REF][START_REF] Desvillettes | On the trend to global equilibrium in spatially inhomogeneous entropy-dissipating systems. Part 1: The Linear Fokker-Planck Equation[END_REF], the analysis of dynamical systems [START_REF] Yu | The Kullback-Leibler Rate Metric for Comparing Dynamical Systems[END_REF] and the analysis of spectral density functions [START_REF] Georgiou | Kullback-Leibler approximation of spectral density functions[END_REF]. It has been used in financial mathematics to quantify the difference between martingale measures [START_REF] Fritelli | The minimal entropy martingale measure and the valuation problem in incomplete markets[END_REF][START_REF] Grandits | On the minimal entropy martingale measure[END_REF]. It has also been shown in [START_REF] Miyahara | Minimal Relative Entropy Martingale Measure of Birth and Death Process[END_REF] that the existence problem of the minimal relative entropy martingale measure problem of birth and death processes can be reduced to the problem of solving the Hamilton-Jacobi-Bellman equation; furthermore the minimal entropy martingale measures (MEMMs) for geometric Levy processes are investigated in [START_REF] Miyahara | On the Minimal Entropy Martingale Measures for Geometric Lévy Processes[END_REF].The finiteness of R(µ||P ) has been shown to be equivalent to the invertibility of certain shifts on Wiener Space, when P is the Wiener Measure [START_REF] Ustunel | Entropy, invertibility and variational calculus of the adapted shifts on Wiener Space[END_REF][START_REF] Lassalle | Invertibility of adapted perturbations of the identity on abstract Wiener space[END_REF]. However one of the most frequent uses of the relative entropy is in statistical inference (particularly in medical imaging) [START_REF] Akaike | Likelihood of a model and information criteria[END_REF][START_REF] Do | Wavelet-Based Texture Retrieval Using Generalized Gaussian Density and Kullback-Leibler Distance[END_REF]. For example, in data-fitting, it is a standard technique to select the parameters which minimize the relative entropy of two conditional probability distributions [START_REF] Bozdogan | Akaike's Information Criterion and Recent Developments in Information Complexity[END_REF].

Modelling in medical imaging increasingly involves diffusion process with state space C([0, T ]; | d ), for which the expression R(µ||P ) = E µ [log dµ dP ] or the variational definition in Definition 1 may not always be tractable. Furthermore, it is not always clear that one may simply approximate the relative entropy by successively calculating it for the marginals over increasingly fine time-discretisations, since these expressions may asymptote to infinity (see (4) below).

Another very important application of the relative entropy is in the field of Large Deviations. Sanov's Theorem dictates that the empirical measure induced by independent samples governed by the same probability law P converge towards their limit exponentially fast; and the constant governing the rate of convergence is the relative entropy [START_REF] Dembo | Large deviations techniques[END_REF]. Large Deviations have been applied for example to spin glasses [START_REF] Ben-Arous | Large deviations for Langevin spin glass dynamics[END_REF], neural networks [START_REF] Moynot | Large deviations and mean-field theory for asymmetric random recurrent neural networks[END_REF][START_REF] Faugeras | A large deviation principle for networks of rate neurons with correlated synaptic weights[END_REF][START_REF] Faugeras | Large Deviations of an Ergodic Synchoronous Neural Network with Learning[END_REF] and mean-field models of interacting particles [START_REF] Budhiraja | Large deviation properties of weakly interacting processes via weak convergence methods[END_REF][START_REF] Fischer | On the form of the large deviation rate function for the empirical measures of weakly interacting systems[END_REF] In the mean-field theory of neuroscience in particular, there has been a recent interest in the modelling of 'finite-size-effects' [START_REF] Faugeras | Large Deviations of an Ergodic Synchoronous Neural Network with Learning[END_REF][START_REF] Baladron | Mean Field description of and propagation of chaos in recurrent multipopulation networks of Hodgkin-Huxley and FitzHugh-Nagumo neurons[END_REF]; that is, the deviations from the limiting behaviour for a population of a particular size. Large Deviations provides a mathematically rigorous tool to do this. In this systems, the limiting system is typically the law P of a stochastic process, and therefore the likelihood of the empirical measure of the system being 'near' some measure µ is the relative entropy R(µ||P ). However the numerical calculation of R(µ||P ) is not straightforward: the results of this paper provide an alternative characterization of R(µ||P ) which assists in this calculation.

For example the rate function for the Large Deviation Principle of the interacting particle model of [START_REF] Fischer | On the form of the large deviation rate function for the empirical measures of weakly interacting systems[END_REF] is directly in terms of the relative entropy between two measures on the space of continuous functions (see in particular Theorem 5.2 of this paper). Similarly the rate function in [START_REF] Faugeras | Large Deviations of an Ergodic Synchoronous Neural Network with Learning[END_REF]Theorem 10] may be expressed as a function of the relative entropy. In more detail, the rate function J in [START_REF] Faugeras | Large Deviations of an Ergodic Synchoronous Neural Network with Learning[END_REF]Theorem 10] is of the form J(µ) = lim n→∞

1 |Vn| R(µ Vn ||Ξ Vn ).
Here Ξ is the law of the process in [START_REF] Faugeras | Large Deviations of an Ergodic Synchoronous Neural Network with Learning[END_REF]Equation (31)], i.e. the law of a ) d -indexed stochastic process. µ Vn and Ξ Vn are the marginals over the finite hypercube V n of side length (2n + 1). The results of this paper give a means of evaluating R(µ Vn ||Ξ Vn ) and therefore J(µ).

In this paper we derive a specific integral (with respect to time) representation of the relative entropy R(µ||P ) when P is the law of a diffusion process. The representation is in terms of the infinitesimal generator of P . This P is the same as in [START_REF] Dawson | Large deviations from the mckean-vlasov limit for weakly interacting diffusions[END_REF]Section 4]. The representation makes use of regular conditional probabilities. We expect that in some circumstances, it ought to be more tractable than the standard definition in 1, and thus it might be of practical use in the applications listed above.

Outline of Main Result

Let T be the Banach Space C([0, T ]; | d ) equipped with the norm

X = sup s∈[0,T ] {|X s |}, (1)
where |•| is the standard Euclidean norm over | d . We let (F t ) be the canonical filtration over (T , B(T )).

For some topological space X , we let B(X ) be the Borelian σ-algebra and M(X ) the space of all probability measures on (X , B(X )). Unless otherwise indicated, we endow M(X ) with the topology of weak convergence. Let σ = {t 1 , t 2 , . . . , t m } be a finite set of elements such that t 1 ≥ 0, t m ≤ T and t j < t j+1 . We term σ a partition, and denote the set of all such partitions by J. The set of all partitions of the above form such that t 1 = 0 and t m = T is denoted J * . We define |σ| = sup 1≤j≤m-1 {t j+1t j }. For some t ∈ [0, T ] and σ ∈ J * , we define σ(t) = sup{s ∈ σ|s ≤ t}. The following definition of relative entropy is standard.

Definition 1. Let (Ω, H) be a measurable space, and µ, ν probability measures.

R H (µ||ν) = sup f ∈E {E µ [f ] -log E ν [exp(f )]} ∈ | ∪ ∞,
where E is the set of all bounded functions. If the σ-algebra is clear from the context, we omit the H and write R(µ||ν). If Ω is Polish and H = B(Ω), then we only need to take the supremum over the set of all continuous bounded functions.

Let P ∈ M(T ) be the following law governing a Markov-Feller diffusion process on T . P is stipulated to be a weak solution (with respect to the canonical filtration) of the local martingale problem with infinitesimal generator

L t (f ) = 1 2 1≤j,k≤d a jk (t, x) ∂ 2 f ∂x j x k + 1≤j≤d b j (t, x) ∂f ∂x j , for f (x) in C 2 (| d ), i.
e. the space of twice continuously differentiable functions. The initial condition

(governing P 0 ) is µ I ∈ M(| d ). The coefficients a jk , b j : [0, T ] × | d → | are assumed to be continuous (over [0, T ] × | d
), and the matrix a(t, x) is strictly positive definite for all t and x. P is assumed to be the unique weak solution. We note that the above infinitesimal generator is the same as in [22, p 269] (note particularly Remark 4.4 in this paper). We note that P is the law of the solution Y = (Y j ) to the following stochastic differential equation:

for j ∈ [1, d], dY j t = b j (t, Y )dt + d k=1 a jk (t, Y )dW k .
Here (W k ) are independent Wiener Processes. Our major result is the following. Let µ ∈ M(T ) govern a random variable X ∈ T . For some x ∈ T , we note µ |[0,s],x , the regular conditional probability (rcp) given X r = x r for all r ∈ [0, s]. The marginal of µ |[0,s],x at some time t ≥ s is noted µ t|[0,s],x . Theorem 1. Let (σ (m) ) m∈) + be any series of partitions such that σ (m) ⊆ σ (m+1) and |σ (m) | → 0 as m → ∞. For µ ∈ M(T ),

R (µ||P ) = R F 0 (µ||P ) + sup σ∈J * Γ(σ) = R F 0 (µ||P ) + lim m→∞ Γ σ (m) (2)
where

Γ(σ) = E µ(x) T 0 sup f ∈D ∂ ∂t E µ t|[0,σ(t)],x [f ] -E µ t|[0,σ(t)],x (y) L t f (y) + 1 2 d j,k=1 a jk (t, y) ∂f ∂y j ∂f ∂y k dt . ( 3
)
Here D is the Schwartz Space of compactly supported functions

| d → |, possessing continuous derivatives of all orders. If ∂ ∂t E µ t|[0,σ(t)],
x [f ] does not exist, then we consider it to be ∞.

Our paper has the following format. In Section 3 we make some preliminary definitions, defining the process P against which the relative entropy is taken in this paper. In Section 4 we employ the projective limits approach of [START_REF] Dawson | Large deviations from the mckean-vlasov limit for weakly interacting diffusions[END_REF] to obtain the chief result of this paper: Theorem 1. This gives an explicit integral representation of the relative entropy. In Section 5 we apply the result in Theorem 1 to various corollaries, including the particular case when µ is the solution of a Martingale Problem. We finish by comparing our results to those of [START_REF] Budhiraja | Large deviation properties of weakly interacting processes via weak convergence methods[END_REF] and [START_REF] Fischer | On the form of the large deviation rate function for the empirical measures of weakly interacting systems[END_REF].

Preliminaries

We outline some necessary definitions.

For σ ∈ J of the form σ = {t 1 , t 2 , . . . , t m }, let σ ;j = {t 1 , . . . , t j }. We denote the number of elements in a partition σ by m(σ). We let J s be the set of all partitions lying in [0, s]. For 0 < s < t ≤ T , we let J s;t be the set of all partitions of the form σ ∪ t, where σ ∈ J s .

Let π σ : T → T σ := | d×m(σ) be the natural projection, i.e. such that π σ (x) = (x t 1 , . . . , x t m(σ) ). We similarly define the natural projection π αγ : T γ → T α (for α ⊆ γ ∈ J), and we define π

[s,t] : T → C([s, t]; | d ) to be the natural restriction of x ∈ T to [s, t].
The expectation of some measurable function f with respect to a measure µ is written as

E µ(x) [f (x)], or simply E µ [f ] when the context is clear. For s < t, we write F s,t = π -1 [s,t] B(C([s, t]; | d )) and F σ = π -1
σ B(T σ ). We define F s;t to be the σ-algebra generated by F s and F γ (where γ = [t]). For µ ∈ M(T ), we denote its image laws by

µ σ := µ • π -1 σ ∈ M(T σ ) and µ [s,t] := µ • π -1 [s,t] ∈ M(C([s, t]; | d )). Let µ ∈ M(T ) govern a random variable X = (X s ) ∈ T . For z ∈ | d , we write the rcp given X s = z by µ |s,z . For x ∈ C([0, s]; | d ) or
T , the rcp given that X u = x u for all 0 ≤ u ≤ s is written as µ |[0,s],x . The rcp given that X u = x u for all u ≤ s, and X t = z, is written as µ |s,x;t,z . For σ ∈ J s and z ∈ (| d ) m(σ) , the rcp given that X u = z u for all u ∈ σ is written as µ |σ,z . All of these measures are considered to be in M(C([s, T ]; | d )) (unless indicated otherwise in particular circumstances). The probability laws governing X t (for t ≥ s), for each of these, are respectively µ t|s,z , µ t|[0,s],x and µ t|σ,z . We clearly have µ s|s,z = δ z , for µ s a.e. z, and similarly for the others.

REMARK. See [START_REF] Karatzas | Brownian motion and stochastic calculus[END_REF]Definition 5.3.16] for a definition of a rcp. Technically, if we let µ * |s,z be the r.c.p given X s = z according to this definition, then

µ |s,z = µ * s,z • π -1 [s,T ] and µ t|s,z = µ * s,z • π -1 t .
By [START_REF] Karatzas | Brownian motion and stochastic calculus[END_REF]Theorem 3.18], µ |s,z is well-defined for µ s a.e. z. Similar comments apply to the other rcp's defined above.

In the definition of the relative entropy, we abbreviate R Fσ (µ||P ) by R σ (R||P ). If σ = {t}, we write R t (µ||P ).

4. The Relative Entropy R(•||P ) using Projective Limits In this section we derive an integral representation of the relative entropy R(µ||P ), for arbitrary µ ∈ M(T ). We start with the standard result in Theorem 2, before adapting the projective limits approach of [START_REF] Dawson | Large deviations from the mckean-vlasov limit for weakly interacting diffusions[END_REF] to obtain the central result (Theorem 1).

We begin with a standard decomposition result for the relative entropy [START_REF] Donsker | Asymptotic Evaluation of Certain Markov Process Expectations for Large Time, IV[END_REF].

Lemma 1. Let X be a Polish Space with sub σ-algebras G ⊆ F ⊆ B(X). Let µ and ν be probability measures on (X, F), and their regular conditional probabilities over G be (respectively)

µ ω and ν ω . Then R F (µ||ν) = R G (µ||ν) + E µ(ω) [R F (µ ω ||ν ω )] .
The following Theorem is a straightforward consequence of [25, Theorem 6.6]: we provide an alternative proof using the theory of Large Deviations in Section 6.1 of the Appendix.

Theorem 2. If α, σ ∈ J and α ⊆ σ, then R α (µ||P ) ≤ R σ (µ||P ). Furthermore, R Fs,t (µ||P ) = sup σ∈J∩[s,t] R σ (µ||P ), (4)
R Fs;t (µ||P ) = sup σ∈Js;t R σ (µ||P ). ( 5
)
It suffices for the supremums in (4) to take σ ⊂ Q s,t , where Q s,t is any countable dense subset of [s, t].

Thus we may assume that there exists a sequence σ

(n) ⊂ Q s,t of partitions such that σ (n) ⊆ σ (n+1) , |σ (n) | → 0 as n → ∞ and R Fs,t (µ||P ) = lim n→∞ R σ (n) (µ||P ). (6)
We now provide a technical lemma.

Lemma 2. Let t > s, α, σ ∈ J s , σ ⊂ α and s ∈ σ. Then for µ σ a.e. x, R t (µ |σ,x ||P |s,xs ) = R(µ t|σ,x ||P t|s,xs ). Secondly, E µσ(x) R t (µ |σ,x ||P |s,xs ) ≤ E µα(z) R t (µ |α,z ||P |s,zs ) .
Proof. The first statement is immediate from Definition 1 and the Markovian nature of P . For the second statement, it suffices to prove this in the case that α = σ ∪ u, for some u < s. We note that, using a property of regular conditional probabilities, for µ σ a.e x,

µ t|σ,x = E µ u|σ,x (ω) µ t|α,v(x,ω) , (7)
where v(x, ω) ∈ T α , v(x, ω) u = ω, v(x, ω) r = x r for all r ∈ σ.

We consider A to be the set of all finite disjoint partitions a ⊂ B(| d ) of | d . The expression for the entropy in [START_REF] Dupuis | A Weak Convergence Approach to the Theory of Large Deviations[END_REF]Lemma 1.4.3] yields

E µσ(x) R µ t|σ,x ||P t|s,xs = E µσ(x) sup a∈A A∈a µ t|σ,x (A) log µ t|σ,x (A) P t|s,xs (A) .
Here the summand is considered to be zero if µ t|σ,x (A) = 0, and infinite if µ t|σ,x (A) > 0 and P t|s,xs (A) = 0. Making use of ( 7), we find that

E µσ(x) R µ t|σ,x ||P t|s,xs = E µσ(x) sup a∈A A∈a E µ u|σ,x (w) µ t|α,v(x,w) (A) log µ t|σ,x (A) P t|s,xs (A) ≤ E µσ(x) E µ u|σ,x (ω) sup a∈A A∈a µ t|α,v(x,ω) (A) log µ t|σ,x (A) P t|s,xs (A) = E µα(z) sup a∈A A∈a µ t|α,z (A) log µ t|σ,πσαz (A) P t|s,zs (A) .
We note that, for µ α a.e. z, if µ t|σ,πσαz (A) = 0 in this last expression, then µ t|α,z (A) = 0 and we consider the summand to be zero. To complete the proof of the lemma, it is thus sufficient to prove that for µ α a.e. z sup a∈A A∈a µ t|α,z (A) log µ t|α,z (A) P t|s,zs (A) ≥ sup a∈A A∈a µ t|α,z (A) log µ t|σ,πσαz (A) P t|s,zs (A) .

But, in turn, the above inequality will be true if we can prove that for each partition a such that P t|s,zs (A) > 0 and µ t|σ,πσαz (A) > 0 for all A ∈ a,

A∈a µ t|α,z (A) log µ t|α,z (A) P t|s,zs (A) - A∈a µ t|α,z (A) log µ t|σ,πσαz (A) P t|s,zs (A) ≥ 0.
The left hand side is equal to A∈a µ t|α,z (A) log

µ t|α,z (A)
µ t|σ,πσαz (A) . An application of Jensen's inequality demonstrates that this is greater than or equal to zero.

REMARK. If, contrary to the definition, we briefly consider µ |[0,t],

x to be a probability measure on T , such that µ(A) = 1 where A is the set of all points y such that y s = x s for all s ≤ t, then it may be seen from the definition of R that

R F T µ |[0,t],x ||P |[0,t],x = R F t,T µ |[0,t],x ||P |[0,t],x = R F t,T µ |[0,t],x ||P |t,xt . (8)
We have also made use of the Markov Property of P . This is why our convention, to which we now return, is to

consider µ |[0,t],
x to be a probability measure on

(C([t, T ]; | d ), F t,T ).
This leads us to the following expressions for R(µ||P ).

Lemma 3. Each σ in the supremums below is of the form {t 1 < t 2 < . . . < t m(σ)-1 < t m(σ) } for some integer m(σ).

R (µ||P ) = R 0 (µ||P ) + m(σ)-1 j=1 E µ [0,t j ] (x) R Ft j ,t j+1 µ |[0,t j ],x ||P |t j ,xt j , (9)
R (µ||P ) = R 0 (µ||P ) + sup σ∈J * m(σ)-1 j=1 E µσ ;j (x) R t j+1 µ t j+1 |σ ;j ,x ||P t j+1 |t j ,xt j , (10)
E µ [0,s] (x) R t µ t|[0,s],x ||P t|s,xs = sup σ∈Js E µσ(y) R t µ t|σ,y ||P t|s,ys , (11)
where in this last expression 0 ≤ s < t ≤ T .

Proof. Consider the sub σ-algebra F 0,t m(σ)-1 . We then find, through an application of Lemma 1 and ( 8), that

R (µ||P ) = R F 0,t m(σ)-1 (µ||P ) + E µ [0,t m(σ)-1 ] (x) R Ft m(σ)-1 ,t m(σ) µ |[0,t m(σ)-1 ],x ||P |t m(σ)-1 ,xt m(σ)-1 .
We may continue inductively to obtain the first identity. We use Theorem 2 to prove the second identity. It suffices to take the supremum over J * , because

R σ (µ||P ) ≥ R γ (µ||P ) if γ ⊂ σ. It thus suffices to prove that R σ (µ||P ) = R 0 (µ||P ) + m(σ)-1 j=1 E µσ ;j (x) R t j+1 µ t j+1 |σ ;j ,x ||P t j+1 |t j ,xt j . ( 12
)
But this also follows from repeated application of Lemma 1. To prove the third identity, we firstly note that

R Fs;t (µ||P ) = R 0 (µ||P ) + sup σ∈Js;t m(σ)-1 j=1 E µσ ;j (x) R t j+1 µ t j+1 |σ ;j ,x ||P t j+1 |t j ,xt j . = sup σ∈Js R σ (µ||P ) + E µσ(x) R t µ t|σ,x ||P t|s,xs .
The proof of this is entirely analogous to that of the second identity, except that it makes use of (5) instead of (4). But, after another application of Lemma 1, we also have that

R Fs;t (µ||P ) = R F 0,s (µ||P ) + E µ [0,s] (x) R t (µ t|[0,s],x ||P t|s,xs ) .
On equating these two different expressions for R Fs;t (µ||P ), we obtain

E µ [0,s] (x) R t (µ t|[0,s],x ||P t|s,xs ) = sup σ∈Js R σ (µ||P ) -R F 0,s (µ||P ) +E µσ(x) R t µ t|σ,x ||P t|s,xs . Let (σ (k) ) ⊂ J s , σ (k-1) ⊆ σ (k) be such that lim k→∞ R σ (k) (µ||P ) = R F 0,s (µ||P )
. Such a sequence exists by [START_REF] Georgiou | Kullback-Leibler approximation of spectral density functions[END_REF]. Similarly, let (γ (k) ) ⊆ J s , be a sequence such that E µ γ (k) (x) R t µ t|γ (k) ,x ||P t|s,xs is strictly nondecreasing and asymptotes as k → ∞ to sup σ∈Js E µσ(x) R t µ t|σ,x ||P t|s,xs . Lemma 2 dictates that

E µ σ (k) ∪γ (k) (x) R t µ t|σ (k) ∪γ (k) ,x ||P t|s,xs
asymptotes to the same limit as well. Clearly lim k→∞ R σ (k) ∪γ (k) (µ||P ) = R F 0,s (µ||P ) because of the identity at the start of Theorem 2. This yields the third identity.

Proof of Theorem 1

In this section we work towards the proof of Theorem 1, making use of some results in [START_REF] Dawson | Large deviations from the mckean-vlasov limit for weakly interacting diffusions[END_REF]. However first we require some more definitions.

If K ⊂ | d is compact, let D K be the set of all f ∈ D whose support is contained in K. The corresponding space of real distributions is D ′ , and we denote the action of θ ∈ D ′ by θ, f . If

θ ∈ M(| d ), then clearly θ, f = E θ [f ]. We let C 2,1 0 (| d
) denote the set of all continuous functions, possessing continuous spatial derivatives of first and second order, a continuous time derivative of first order, and of compact support. For f ∈ D and t ∈ [0, T ], we define the random variable

∇ t f : | d → | d such that (∇ t f (y)) i = d
j=1 a ij (t, y) ∂f ∂y j (for x ∈ T , we may also understand ∇ t f (x) := ∇ t f (x t )). Let a ij be the components of the matrix inverse of a ij . For random variables X, Y : T → | d , we define the

inner product (X, Y ) t,x = d i,j=1 X i (x)Y j (x)a ij (t, x t ), with associated norm |X| 2 t,x = (X(x), X(x)) 2 t,x . We note that |∇ t f | 2 t,x = d i,j=1 a ij (t, x t ) ∂f ∂z i (x t ) ∂f ∂z j (x t
). Let M be the space of all continuous maps [0, T ] → M(| d ), equipped with the topology of uniform convergence. For s ∈ [0, T ], ϑ ∈ M and ν ∈ M(| d ) we define n(s, ϑ, ν) ≥ 0 and such that

n(s, ϑ, ν) 2 = sup f ∈D ϑ, f - 1 2 E ν(y) |∇ t f | 2 t,y . ( 13
)
This definition is taken from [22, Eq. (4.7)] -we note that n is convex in ϑ. For γ ∈ M(T ), we may naturally write n(s, γ, ν) := n(s, ω, ν), where ω is the projection of γ onto M, i.e. ω(s) = γ s . It is shown in [START_REF] Dawson | Large deviations from the mckean-vlasov limit for weakly interacting diffusions[END_REF] that this projection is continuous. The following two definitions, lemma and two propositions are all taken (with some small modifications) from [START_REF] Dawson | Large deviations from the mckean-vlasov limit for weakly interacting diffusions[END_REF].

Definition 2. Let I be an interval of the real line. A measure µ ∈ M(T ) is called absolutely continuous if for each compact set K ⊂ | d there exists a neighborhood U of 0 in K and an absolutely continuous function

H K : I → | such that |E µu [f ] -E µv [f ]| ≤ |H K (u) -H K (v)| ,
[h 1 , h 2 ] = t s E νu(x) [(h 1 (u, x), h 2 (u, x)) u,x ] du.
We denote by L 2 s,t,∇ (ν) the closure in L 2 s,t (ν) of the linear subset generated by maps of the form (x, u) → ∇ u f , where f ∈ C 2,1 0 ([s, t], | d ). We note that functions in L 2 s,t,∇ (ν) only need to be defined du ⊗ ν u (dx) almost everywhere.

Recall that n is defined in [START_REF] Bozdogan | Akaike's Information Criterion and Recent Developments in Information Complexity[END_REF], and note that * L t µ t , f := µ t , L t f . Proposition 1. Assume that µ ∈ M(C([r, s]; | d )), such that µ r = δ y for some y ∈ | d and 0 ≤ r < s ≤ T . We have that [START_REF] Dawson | Large deviations from the mckean-vlasov limit for weakly interacting diffusions[END_REF]Eq. 4.9 and Lemma 4.8]

s r n(t, μt - * L t µ t , µ t ) 2 dt = sup f ∈C 2,1 0 (| d ) E µs(x) [f (s, x)] -f (r, y) - s r E µt(x) ∂ ∂t + L t f (t, x) + 1 2 |∇ t f (t, x)| 2 t,x dt . ( 14
)
It clearly suffices to take the supremum over a countable dense subset. Assume now that s r n(t, μt - * L t µ t , µ t ) 2 dt < ∞. Then for Lebesgue a.e. t, μt = * K t µ t , where [START_REF] Dawson | Large deviations from the mckean-vlasov limit for weakly interacting diffusions[END_REF]Lemma 4.8(3)]

K t f (•) = L t f (•) + 1≤j≤d (h µ (t, •)) j ∂f ∂x j (•), (15)
for some h µ ∈ L 2 r,s,∇ (µ) which satisfies [START_REF] Dawson | Large deviations from the mckean-vlasov limit for weakly interacting diffusions[END_REF]Lemma 4.8(4)]

s r n(t, μt - * L t µ t , µ t ) 2 dt = 1 2 s r E µt(x) |h µ (t, x)| 2 t,x dt < ∞. ( 16
)
REMARK. We reach [START_REF] Faugeras | A large deviation principle for networks of rate neurons with correlated synaptic weights[END_REF] from the proof of Lemma 9 in [START_REF] Dawson | Large deviations from the mckean-vlasov limit for weakly interacting diffusions[END_REF]Eq 4.10]. One should note also that [START_REF] Dawson | Large deviations from the mckean-vlasov limit for weakly interacting diffusions[END_REF] write the relative entropy R as L

ν in (their) equation ( 4.10). To reach [START_REF] Faugeras | Large Deviations of an Ergodic Synchoronous Neural Network with Learning[END_REF], we also use the equivalence between (4.7) and (4.8) in [START_REF] Dawson | Large deviations from the mckean-vlasov limit for weakly interacting diffusions[END_REF].

Proposition 2. Assume that µ ∈ M(T ), such that µ r = δ y for some y ∈ | d and 0 ≤ r < s ≤ T . If R Fr,s (µ||P |r,y ) < ∞, then µ is absolutely continuous on [r, s], and [START_REF] Dawson | Large deviations from the mckean-vlasov limit for weakly interacting diffusions[END_REF]Lemma 4.9]

R Fr,s (µ||P |r,y ) ≥ s r n(t, μt - * L t µ t , µ t ) 2 dt. ( 17
)
Here the derivative μt is defined in Lemma 4. For all f ∈ D, [START_REF] Dawson | Large deviations from the mckean-vlasov limit for weakly interacting diffusions[END_REF]Eq (4.35)]

E µs [f ] -log E P s|r,y [exp(f )] ≤ s r n(t, μt - * L t µ t , µ t ) 2 dt. ( 18
)
We are now ready to prove Theorem 1 (the central result).

Proof. Fix a partition σ = {t 1 , . . . , t m }. We may conclude from ( 9) and ( 17) that,

R (µ||P ) ≥ R 0 (µ||P ) + m-1 j=1 E µ [0,t j ] (x) t j+1 t j n(t, μt|[0,t j ],x - * L t µ t|[0,t j ],x , µ t|[0,t j ],x ) 2 dt. ( 19
)
The integrand on the right hand side is measurable with respect to E µ [0,t j ] (x) due to the equivalent expression [START_REF] Dembo | Large deviations techniques[END_REF]. We may infer from ( 18) that, In turn, for each n there exist (f

E µ [0,t j ] (x) t j+1 t j n(t, μt|[0,t j ],x - * L t µ t|[0,t j ],x , µ t|t j ,x ) 2 dt ≥ E µ [0,t j ] (x) sup f ∈D E µ t j+1 |[0,t j ],x [f ] -log E P t j+1 |t j ,x t j [exp(f )] = E µ [0,t j ] (x) sup f ∈C b (| d ) E µ t j+1 |[0,t j ],x [f ] -log E P t j+1 |t j ,x t j [exp(f )] . (20
(m) n ) ∈ D Kn such that we may write Kn f (x)dν(x) = lim m→∞ Kn f (m) n (x)dν(x).
This allows us to conclude that the two supremums are the same. The last expression in [START_REF] Fischer | On the form of the large deviation rate function for the empirical measures of weakly interacting systems[END_REF] is merely

E µ [0,t j ] (x) R t j+1 µ t j+1 |[0,t j ],x ||P t j+1 |t j ,xt j .
By [START_REF] Akaike | Likelihood of a model and information criteria[END_REF], this is greater than or equal to E µσ ;j (y) R t j+1 µ t j+1 |σ ;j ,y ||P t j+1 |t j ,yt j .

We thus obtain the theorem using (10).

Some Corollaries

We state some corollaries of Theorem 1. In the course of this section we make progressively stronger assumptions on the nature of µ, culminating in the elegant expression for R(µ||P ) when µ is a solution of a martingale problem. We finish by comparing our work with that of [START_REF] Budhiraja | Large deviation properties of weakly interacting processes via weak convergence methods[END_REF][START_REF] Fischer | On the form of the large deviation rate function for the empirical measures of weakly interacting systems[END_REF].

Corollary 1. Suppose that µ ∈ M(T ) and R(µ||P ) < ∞. Then for all s and µ a.e. x, µ |[0,s],x is absolutely continuous over [s, T ]. For each s ∈ [0, T ] and µ a.e. x ∈ T , for Lebesgue a.e. t ≥ s

μt|[0,s],x = * K µ t|s,x µ t|[0,s],x (21)
where for some

h µ s,x ∈ L 2 s,T,∇ (µ |[0,s],x ) K µ t|s,x f (y) = L t f (y) + d j=1 h µ,j s,x (t, y) ∂f ∂y j (y). (22)
Furthermore,

R (µ||P ) = R 0 (µ||P ) + 1 2 sup σ∈J * T 0 E µ(w) E µ t|[0,σ(t)],w (z) h µ σ(t),w (t, z) 2 t,z dt. ( 23
)
For any dense countable subset Q 0,T of [0, T ], there exists a series of partitions σ

(n) ⊂ σ (n+1) ∈ Q 0,T , such that as n → ∞, |σ (n) | → 0, and
R (µ||P ) = R 0 (µ||P ) + 1 2 lim n→∞ T 0 E µ(w) E µ t|[0,σ (n) (t)],w (z)
h µ σ (n) (t),w (t, z) 2 t,z dt. (24)
REMARK. It is not immediately clear that we may simplify (23) further (barring further assumptions). The reason for this is that we only know that

E µ |[0,σ(t)],w (z) h µ σ(t),w (t, z) 2 t,z
is measurable (as a function of w), but it has not been proven that h µ σ(t),w (t, z) is measurable (as a function of w).

Proof. Let σ = {0 = t 1 , . . . , t m = T } be an arbitrary partition. For all j < m, we find from Lemma

3 that R Ft j ,t j+1 µ |[0,t j ],x ||P |t j ,xt j < ∞ for µ [0,t j ] a.e. x ∈ C([0, t j ]; | d ).
We thus find that, for all such x, µ |[0,t j ],x is absolutely continuous on [t j , t j+1 ] from Proposition 2. We are then able to obtain ( 21) and ( 22) from Propositions 1 and 2. From ( 16), ( 2) and ( 21) we find that

R (µ||P ) = R 0 (µ||P ) + 1 2 sup σ∈J * E µ(x) T 0 E µ t|[0,σ(t)],x (z) h µ σ(t),x (t, z) 2 t,z dt. (25)
The above integral must be finite (since we are assuming R(µ||P ) is finite). Furthermore

E µ t|[0,σ(t)],x (z) h µ σ(t),x (t, z) 2 t,z
is (t, x) measurable as a consequence of the equivalent form [START_REF] Dembo | Large deviations techniques[END_REF]. This allows us to apply Fubini's Theorem to obtain [START_REF] Karatzas | Brownian motion and stochastic calculus[END_REF]. The last statement on the sequence of maximising partitions follows from Theorem 2.

Corollary 2. Suppose that R(µ||P ) < ∞. Suppose that for all s ∈ Q 0,T (any countable, dense subset of [0, T ]), for µ a.e. x and Lebesgue a.e. t, h µ s,x (t, x t ) = E µ |[0,s],x;t,x t (w) h µ (t, w) for some progressively-measurable random variable

h µ : [0, T ] × T → | d . Then R (µ||P ) = R 0 (µ||P ) + 1 2 T 0 E µ(w) |h µ (t, w)| 2 t,wt dt.
Proof. Let G s,x;t,y be the sub σ-algebra consisting of all B ∈ B(T ) such that for all w ∈ B, w r = x r for all r ≤ s and w t = y. Thus h µ s,x (t, y) = E µ |[0,s],x;t,y (w) h µ (t, w) = E µ [h µ (t, •)|G s,x;t,y ]. By [27, Corollary 2.4], since ∩ s<t G s,x;t,xt = G t,x;t,xt (restricting to s ∈ Q 0,T ), for µ a.e. x,

lim s→t -E µ |[0,s],x;t,x t (w) h µ (t, w) = h µ (t, x), (26)
where s ∈ Q 0,T . By the properties of the regular conditional probability, we find from [START_REF] Donsker | Asymptotic Evaluation of Certain Markov Process Expectations for Large Time, IV[END_REF] that

R (µ||P ) = R 0 (µ||P ) + 1 2 lim n→∞ T 0 E µ(w) E µ |[0,σ (n) (t)],w;t,w t (v) [h µ (t, v)] 2 t,wt dt. ( 27
)
By assumption, the above limit is finite. Thus by Fatou's Lemma, and using the properties of the regular conditional probability,

R (µ||P ) ≥ R 0 (µ||P ) + 1 2 T 0 E µ(w) lim n→∞ E µ |[0,σ (n) (t)],w;t,w t (v) [h µ (t, v)] 2 t,wt
dt.

Through use of [START_REF] Dupuis | A Weak Convergence Approach to the Theory of Large Deviations[END_REF],

R (µ||P ) ≥ R 0 (µ||P ) + 1 2 T 0 E µ(w) |h µ (t, w)| 2 t,wt dt.
Conversely, through an application of Jensen's Inequality to ( 27)

R (µ||P ) ≤ R 0 (µ||P ) + 1 2 lim n→∞ T 0 E µ(w) E µ |[0,σ (n) (t)],w;t,w t (v) |h µ (t, v)| 2 t,wt dt.
A property of the regular conditional probability yields

R (µ||P ) ≤ R 0 (µ||P ) + 1 2 T 0 E µ(w) |h µ (t, w)| 2 t,wt dt.
REMARK. The condition in the above corollary is satisfied when µ is a solution to a Martingale Problem -see Lemma 5.

We may further simplify the expression in Theorem 1 when µ is a solution to the following Martingale Problem. Let {c jk , e j } be progressively-measurable functions [0, T ] × T → |. We suppose that c jk = c kj . For all 1 ≤ j, k ≤ d, c jk (t, x) and e j (t, x) are assumed to be bounded for x ∈ L (where L is compact) and all t ∈ [0, T ]. For f ∈ C 2 0 (| d ) and x ∈ T , let

M u (f )(x) = 1≤j,k≤d c jk (u, x) ∂ 2 f ∂y j ∂y k (x u ) + 1≤j≤d e j (u, x) ∂f ∂y j (x u ).
We assume that for all such f , the following is a continuous martingale (relative to the canonical filtration) under µ f

(X t ) -f (X 0 ) - t 0 M u f (X)du. (28)
The law governing X 0 is stipulated to be ν ∈ M(| d ).

From now on we switch from our earlier convention and we consider µ |[0,s],x to be a measure on T such that, for µ a.e. x ∈ T , µ |[0,s],x (A s,x ) = 1 where A s,x is the set of all X ∈ T satisfying X t = x t for all 0 ≤ t ≤ s. This is a property of a regular conditional probability (see [START_REF] Karatzas | Brownian motion and stochastic calculus[END_REF]Theorem 3.18]). Similarly, µ |s,x;t,y is considered to be a measure on T such that for µ a.e. x ∈ T , µ |s,x;t,y (B s,x;t,y ) = 1, where B s,x;t,y is the set of all X ∈ A s,x such that X t = y. We may apply Fubini's Theorem (since f is compactly supported and bounded) to (28) to find that

µ t|[0,s],x , f -f (x s ) = t s E µ |[0,s],x [M u f ] du.
This ensures that µ |[0,s] is absolutely continuous over [s, T ], and that

μt|[0,s],x , f = E µ |[0,s],x [M t f ] . (29
) If R(µ||P ) < ∞ then R(µ||P ) = R(ν||µ I ) + 1 2 E µ(x) T 0 |b(s, x s ) -e(s, x)| 2 s,xs ds . (30
Proof. It follows from R(µ||P ) < ∞, (21) and ( 22) that for all s and µ a.e. Equation (31) thus follows from Corollary 2. [START_REF] Budhiraja | Large deviation properties of weakly interacting processes via weak convergence methods[END_REF][START_REF] Fischer | On the form of the large deviation rate function for the empirical measures of weakly interacting systems[END_REF] We have already noted in the introduction that one may infer a variational representation of the relative entropy from [START_REF] Budhiraja | Large deviation properties of weakly interacting processes via weak convergence methods[END_REF][START_REF] Fischer | On the form of the large deviation rate function for the empirical measures of weakly interacting systems[END_REF] by assuming that the coefficients of the underlying stochastic process are independent of the empirical measure in these papers. The assumptions in [START_REF] Fischer | On the form of the large deviation rate function for the empirical measures of weakly interacting systems[END_REF] on the underlying process P are both more general and more restrictive than ours. His assumptions are more general insofar as the coefficients of the SDE may depend on the past history of the process and the diffusion coefficient is allowed to be degenerate. However our assumptions are more general insofar as we only require P to be the unique (in the sense of probability law) weak solution of the SDE, whereas [START_REF] Fischer | On the form of the large deviation rate function for the empirical measures of weakly interacting systems[END_REF] requires P to be the unique strong solution of the SDE. Of course when both sets of assumptions are satisfied, one may infer that the expressions for the relative entropy are identical.

Comparison of our Results to those of Fischer et al

Appendix

Proof of Theorem 2

The following is an alternative proof to that of [START_REF] Xanh | Ergodic Theorems for Spatial Processes[END_REF]Theorem 6.6] which employs the theory of Large Deviations. The fact that, if α ⊆ σ, then R α (µ||P ) ≤ R σ (µ||P ), follows from Lemma 1. We prove the first expression (4) in the case s = 0, t = T (the proof of the second identity ( 5) is analogous). and for all closed sets F lim

N →∞ N -1 log Γ N (F ) ≤ -inf x∈F I(x).
If furthermore the set {x : I(x) ≤ α} is compact for all α ≥ 0, we say that I is a good rate function.

We define the following empirical measures.

Definition 5. For x ∈ T N , y ∈ T N σ , let μN (x) = 1 N 1≤j≤N δ x j ∈ M(T ), μN σ (y) = 1 N 1≤j≤N δ y j ∈ M(T σ ).
Clearly μN σ (x σ ) = π σ (μ N (x)). The image law P ⊗N • (μ N ) -1 is denoted by Π N s,t ∈ M(M(T )). Similarly, for σ ∈ J, the image law of P ⊗N

for some γ ∈ J and open O (with respect to the weak topology of M(T γ )).

We may continuously embed M(T ) into the projective limit M(T ) of its marginals, letting ι denote this embedding. That is, for any σ ∈ J, (ι(µ))(σ) = µ σ . We note that ι is continuous because ι -1 (A γ,O ) is open in M(T ), for all A γ,O of the form in (33). We equip M(T ) with the Borelian σ-algebra generated by this topology. The embedding ι is measurable with respect to this σ-algebra because the topology of M(T ) has a countable base. The embedding induces the image laws (Π N • ι -1 ) on M(M(T )). For σ ∈ J, it may be seen that

Π N σ = Π N • ι -1 • (π M σ ) -1 ∈ M(M(T σ ))
, where π M σ (⊗ α µ(α)) = µ(σ). It follows from [START_REF] Dawson | Large deviations from the mckean-vlasov limit for weakly interacting diffusions[END_REF]Thm 3.3] that Π N • ι -1 satisfies a Large Deviation Principle with rate function sup σ∈J R σ (µ||P ). However we note that ι is 1 -1, because any two measures µ, ν ∈ M(T ) such that µ σ = ν σ for all σ ∈ J must be equal. Furthermore ι is continuous. Because of Sanov's Theorem , (Π N ) is exponentially tight (see [14, Defn 1.2.17, Exercise 1.2.19] for a definition of exponential tightness and proof of this statement). These facts mean that we may apply the Inverse Contraction Principle [START_REF] Dembo | Large deviations techniques[END_REF]Thm 4.2.4] to infer Π N satisfies an LDP with the rate function sup σ∈J R σ (µ||P ). Since rate functions are unique [START_REF] Dembo | Large deviations techniques[END_REF]Lemma 4.1.4], we obtain the first identity in conjunction with Sanov's Theorem. The second identity (5) follows similarly.

We may repeat the argument above, while restricting to σ ⊂ Q s,t . We obtain the same conclusion because the σ-algebra generated by (F σ ) σ⊂Qs,t , is the same as F s,t . The last identity follows from the fact that, if α ⊆ σ, then R α (µ||P ) ≤ R σ (µ||P ).

)

  This last step follows by noting that if ν ∈ M(| d ), and f ∈ C b (| d ), and the expectation of f with respect to ν is finite, then there exists a series (K n ) ⊂ | d of compact sets such that | d f (x)dν(x) = lim n→∞ Kn f (x)dν(x).

) Lemma 5 .

 5 If R(µ||P ) < ∞ then for Lebesgue a.e. t ∈ [0, T ] and µ a.e. x ∈ T , a(t, x t ) = c(t, x).

Definition 4 .

 4 A series of probability laws Γ N on some topological space Ω equipped with its Borelian σ-algebra is said to satisfy a strong Large Deviation Principle with rate function I : Ω → | if for all open sets O, lim N →∞ N -1 log Γ N (O) ≥inf x∈O I(x)

σ•

  (μ N σ ) -1 on M(T σ ) is denoted by Π N σ ∈ M(M(T σ )). Since T and T σ are Polish spaces, we have by Sanov's Theorem (see[START_REF] Dembo | Large deviations techniques[END_REF] Theorem 6.2.10]) that Π N satisfies a strong Large Deviation Principle with good rate function R(•||P ). Similarly, Π N σ satisfies a strong Large Deviation Principle on M(T σ ) with good rate function R Fσ (•||P ).We now define the projective limit M (T ). If α, γ ∈ J, α ⊂ γ, then we may define the projectionπ M αγ : M(T γ ) → M(T α ) as π M αγ (ξ) := ξ • π -1 αγ . An element of M(T ) is then a member ⊗ σ ζ(σ) ofthe cartesian product ⊗ σ∈J M(T σ ) satisfying the consistency condition π M αγ (ζ(γ)) = ζ(α) for all α ⊂ γ. The topology on M(T ) is the minimal topology necessary for the natural projection M(T ) → M(T α ) to be continuous for all α ∈ J. That is, it is generated by open sets of the form A γ,O = {⊗ σ ζ(σ) ∈ M(T ) : ζ(γ) ∈ O},

  x, for Lebesgue a.e. t ≥ s

	E µ |s,x;t,x t [c(t, •)] = a(t, x t ).	(32)
	Let us take a countable dense subset Q 0,T of [0, T ]. There thus exists a null set N ⊆ [0, T ] such that for
	every s ∈ Q 0,T , µ a.e. x and every t / ∈ N the above equation holds. We may therefore conclude (30)

using

[START_REF] Revuz | Continuous Martingales and Brownian Motion[END_REF] Corollary 2.4]

and taking s → t -. From (29), we observe that for all s ∈ [0, T ] and µ a.e. x, for Lebesgue a.e. t h µ s,x (t, x t ) = E µ |[0,s],x;t,x t [e(t, •)].

Acknowledgments

This work was supported by INRIA FRM, ERC-NERVI number 227747, European Union Project # FP7-269921 (BrainScales), and Mathemacs # FP7-ICT-2011.9.7

Conflicts of Interest

"The authors declare no conflict of interest".


S Pierre

Katarzyna Tarnowska

Lamia Hachfi

S Coupé

R Simide

Sylvain Couvray

Cédric Garnier

Michel Grimaldi

Simone Richard

S Gaillard

N Prévot-D'alvise

Professor J P Grillasca
email: grillasca@univ-tln.fr

EFFECTS OF WATER TEMPERATURE INCREASE AND HEAVY METALS CONTAMINATION ON WAP65 GENE EXPRESSION IN SEA BASS (Dicentrarchus labrax) LIVER

Keywords: Dicentrarchus labrax, gene expression, hsp70, RNA, wap65, Warm Temperature Acclimationrelated adenosine triphosphate, Cd: cadmium, Co: cobalt, Cu: copper, DPASV: Differential Pulse Anodic Stripping Voltammetry, GST: glutathione Stransferase, HDPE: high-density polyethylene, Hg : mercury, HSP: heat shock protein, MT: metallothionein, Ni: nickel, PAH: polycyclic aromatic hydrocarbon, Pb: lead, PCB: polychlorobiphenyl, PCR: Polymerase Chain Reaction, TBT: tributyltin, UV: ultraviolet, WAP65: Warm temperature Acclimation-related 65kD Protein, Zn: zinc

d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

Changes in marine environmental conditions include both natural events, e.g. temperature fluctuations (long-term such as global warming, and short-term seasonal variations), and anthropogenic actions such as chemical contamination. Among pollutants, metals are of a great importance. Their sources in aquatic environment are various, but significant contributions are man-made. Metals concentrations in marine environments are often relatively high, because their elimination in wastewater treatment plants is often limited [START_REF] Choubert | Influent concentrations and removal performances of metals through various conventional and advanced wastewater treatment processes[END_REF].

Due to environmental changes, multiple dynamic equilibrium regulation mechanisms in marine organisms are aimed at maintaining efficient and optimal behavior in new environmental conditions (reestablishing homeostasis). Therefore, a variety of responses on cellular and molecular levels occurs. Natural changes in environmental conditions such as seasonal changes in water temperature provoke acclimatory responses. In poikilothermic fish, such as the European sea bass, this response involves, among others, regulation of a set of genes that includes myosin, lactate dehydrogenase, Δ 9 -desaturase, warm temperature acclimation-related 65kDa protein gene (wap65) and ATP synthase (in cold temperature) (23 and references therein, 39). Responses and adaptations differ in case of sudden temperature perturbations, which provoke short-term adjustments, exemplified by the production of heat shock proteins (HSPs) [START_REF] Iwama | Heat shock protein expression in fish[END_REF]. Water composition also influences the quality of commercially valuable fish, because of the accumulation of the pollutants inside the body [START_REF] Canli | The relationships between heavy metal (Cd, Cr, Cu, Fe, Pb, Zn) levels and the size of six Mediterranean fish species[END_REF][START_REF] Dallinger | The role of metallothionein in cadmium accumulation of Arctic char (Salvelinus alpinus) from high alpine lakes[END_REF][START_REF] Dugo | Concentration of Cd (II), Cu (II), Pb (II), Se (IV) and Zn (II) in cultured sea bass (Dicentrarchus labrax) tissues from Tyrrhenian Sea and Sicilian Sea by derivative stripping potentiometry[END_REF]. Contamination with metals has a profound effect on gene expression patterns in fish. One of the responses to metal intoxication is the upregulation of metallothioneins (MT), which can mitigate the harmful effects of various metals [START_REF] Kagi | Biochemistry of metallothionein[END_REF].

Moreover, cytochrome P4501A, vitellogenin and heat shock proteins have been examined in several metal toxicity studies on fish [START_REF] Tom | Transcript and protein environmental biomarkers in fish. A review[END_REF].

On the basis of responses to environmental perturbations, different biomarkers have been proposed in order to assess exposure of aquatic organisms to pollutants and to identify certain substances with low concentrations levels (34, for fish reviewed in [START_REF] Van Der Oost | Fish bioaccumulation and biomarkers in environmental risk assessment: a review[END_REF]. There is a strong interest in finding novel genetic markers to monitor environmental perturbations that may affect fish. It is especially important for species with important commercial value like the European sea bass (Dicentrarchus labrax), in order to assure high fish quality. In 2009, the global sea bass capture fisheries production was ~11 933 tons with France accounting for 57% of the production (10). In the past 25 years aquaculture of this species has developed rapidly, reaching a global production amount of ~113 653 tons in 2009.

It has been shown that in fish the expression of the gene coding for WAP65, the "Warm temperature Acclimation-related 65kDa Protein" changes as a result of environmental perturbations. However, its function has yet to be fully characterized. It resembles mammalian hemopexin (in amino acid sequence and functional sites), a gene encoding a plasma glycoprotein that is synthesized in the liver and functions as a scavenger of free heme. In contrast to the single isoform of hemopexin found in mammals, there are two paralogous wap65 genes in fish (wap65-1, wap65-2) (6 and references therein), presumably as a result of the teleost whole genome duplication event [START_REF] Amores | Zebrafish hox clusters and vertebrate genome evolution[END_REF]. These forms probably differ in function. The evolution rate of wap65-2 is much lower than the one of wap65-1 and its tissue distribution restricted (mainly liver). Therefore, it is more similar to mammalian hemopexin. WAP65 in fish is known to play a role in the warm temperature acclimatory response. The protein (or mRNA coding for it) has been found in goldfish (Carassius auratus) and carp (Cyprinus carpio) acclimated to 30°C for at least 1 month [START_REF] Kikuchi | The warm temperature acclimation-related 65-kDa protein, Wap65, in goldfish and its gene expression[END_REF][START_REF] Kinoshita | cDNA cloning characterization of the warm-temperatureacclimationassociated protein Wap65 from carp, Cyprinus carpio[END_REF][START_REF] Watabe | Cold and warmtemperature acclimation induces specific cytosolic protein in goldfish and carp[END_REF]. It also appeared in the muscle tissues of goldfish within 5 days of high temperature exposure and was maintained at high concentrations for at least 9 days [START_REF] Kikuchi | The 65-kDa cytosolic protein associated with warm temperature acclimation in goldfish, Carassius auratus[END_REF]. Increased wap65-1 mRNA levels were also found in liver within four hours after exposure to 30°C, with maximum expression observed after 3 days. After 3 weeks wap65 transcript levels diminished, but remained 10 times higher than before the warm water exposure [START_REF] Kikuchi | The Wap65 gene expression of goldfish (Carassius auratus) in association with warm water temperature as well as bacterial lipopolysaccharide (LPS)[END_REF]. The role of WAP65 in acclimatory processes appears to vary considerably among fish, as significant upregulation of this gene was not recorded in pufferfish (Takifugu rubripes) and medaka (Oryzias latipes) exposed to high water temperature for one month [START_REF] Hirayama | Primary structures and gene organizations of two types of Wap65 from the pufferfish Takifugu rubripes[END_REF][START_REF] Hirayama | The occurrence of two types of hemopexin-like protein in medaka and differences in their affinity to heme[END_REF].

Moreover, it was demonstrated that wap65 expression levels change in reaction to pathogen infections (e.g. [START_REF] Sarropoulou | Evolution of a multifunctional gene: the warm temperature acclimation protein Wap65 in the European sea bass Dicentrarchus labrax[END_REF][START_REF] Sha | The warm temperature acclimation protein Wap65 as an immune response gene: its duplicates are differentially regulated by temperature and bacterial infections[END_REF][START_REF] Shi | Molecular cloning of liver Wap65 cDNA in ayu (Plecoglossus altivelis) and mRNA expression changes following Listonella anguillarum infection[END_REF], during fish development [START_REF] Hirayama | The occurrence of two types of hemopexin-like protein in medaka and differences in their affinity to heme[END_REF][START_REF] Nakaniwa | Genomic sequences encoding two types of medaka hemopexin-like protein Wap65, and their gene expression profiles in embryos[END_REF][START_REF] Sarropoulou | Evolution of a multifunctional gene: the warm temperature acclimation protein Wap65 in the European sea bass Dicentrarchus labrax[END_REF] and in the reaction to intoxication with certain heavy metals and other pollutants [START_REF] Aliza | Identification of Wap65, a human homologue of hemopexin as a copper-inducible gene in swordtail fish, Xiphophorus helleri[END_REF][START_REF] Lie | Gene expression profiling in Atlantic cod (Gadus morhua L.) from two contaminated sites using a custom-made cDNA microarray[END_REF][START_REF] Lie | Effects of environmental relevant doses of pollutants from offshore oil production on Atlantic cod (Gadus morhua)[END_REF]. However, in comparison to the number of studies concerning wap65 expression after water temperature increase, knowledge concerning regulation of this gene after intoxication with pollutants remains scarce.

In the present study we investigated: (i) the delay between the temperature increase and the change in wap65 expression, and (ii) wap65 expression in reaction to heavy metal intoxication.

MATERIALS AND METHODS

Animals and aquariums

All experiments were performed according to the guidelines of the Canadian Council on Animal Care under the appropriate approved animal care protocol described in «the care and use of fish in research, teaching and testing». Fish used for our experiments were collected from a fish farm located on the Mediterranean coast at La Seyne sur Mer (south of France), where they were kept in floating cages in natural environmental conditions and fed once a day ad libitum with commercial fish diet (Le Gouessant, France). One year old sea bass fish (average weight of 80-100 g) were used for the experiments. Experiments were conducted in 150 L aquariums filled with sea water. The sea water was filtered with a mechanical silica filter, sterilized with ultraviolet (UV), filtered with biological filter with nitrifying bacteria and with the microfiltration system with different mesh sizes. Before the experiments, aquariums were decontaminated with hydrochloric acid (0.1%) and rinsed three times with sea water.

Experiment: increased water temperature

Fish were held in aquariums at 12°C for 4 weeks and fed ad libitum. In the first experiment, the aquarium water temperature was increased by 1°C every hour from 12°C to 30°C, and maintained at 30°C for 4 weeks. There were three fish for each experimental condition (12°C control group and 30°C groups acclimated for 1 h, and from 1 to 4 weeks). Temperature was regulated using a series of resistors (Shego, 300 W) connected with the thermostat (Thermostab, Aquael) which resulted in an accuracy of the water temperature between +/-0.5 °C.

Experiment: intoxication with heavy metals

Fish intoxication experiments were performed in water containing four heavy metals: lead (Pb), cadmium (Cd), copper (Cu) and zinc (Zn). Experimental conditions were defined accordingly to the metals levels measured in water pumped from the Bay of Toulon (south of France) used to fill the aquariums [START_REF] Pierre | Contribution à la pisciculture marine. De la nécessité de diversifier et de contrôler la qualité sanitaire[END_REF] and directly in the aquariums (to check the cleaning) and compared to concentrations recorded along transects in the Bay of Toulon (Jean, Université du Sud Toulon-Var, France-personal communication). Sample treatment and analysis was adapted from Louis et al. [START_REF] Louis | Kinetic and equilibrium studies of copper-dissolved organic matter complexation in water column of the stratified Krka River estuary (Croatia)[END_REF]. Water samples were filtered through 0.45 μm pre-cleaned cellulose nitrate filters (Sartorius), and stored at 4 °C in the dark until analysis in pre-cleaned 60 mL HDPE (Nalgene) for trace -metal measurements (preservation by acidification at pH < 2 using suprapur HNO 3 ). Total dissolved metal concentrations were determined in acidified samples by Differential Pulse Anodic Stripping Voltammetry (DPASV). Measurements were carried out with a voltammetric analyser μAutolabI (EcoChemie) controlled by GPES 4.9 software (EcoChemie) coupled with a three -electrode cell of 663 VA Stand (Metrohm). The analytical procedure, fully automated, was based on 3 successive steps : (a) simultaneous measurement of Cd, Cu and Pb by DPASV with standard additions (using Cavro XL 3000 Syringe Pump) of standard solution containing the 3 metals, (b) addition of a 4 M sodium acetate solution (from sodium acetate suprapur, Merck) to increase the pH to 4, (c) measurement of Zn by DPASV with standard additions of Zn standard solution. Deposition time of the DPASV procedures was adapted accordingly to metal levels in samples. Metal levels measured in water sampled in pre-cleaned aquarium were on the level of those obtained on water pumped from the Bay of Toulon used to fill the aquariums, proving the efficiency of aquarium cleaning procedure. In comparison to metal concentrations recorded in various zones of the Bay of Toulon (Jean, Université du Sud Toulon-Var, France-personal communication), these values correspond to the levels obtained in the north of the small Bay of Toulon, largely higher than open-sea values reported for the western Mediterranean Sea [START_REF] Morley | Distribution and behaviour of some dissolved trace metals in the western Mediterranean Sea[END_REF], reflecting the significant contamination of the Bay due to numerous anthropogenic activities and low sea water renewal.

Accordingly to the measured levels, two concentrations of each metal were used: the first (M1), slightly higher than the one found in the natural environment and the second (M2), 100 higher than M1 (Table 1). Metals concentrations measurement in the control systems (i.e. in the absence of fish) showed that concentrations remained stable (i.e. they do not adsorb significantly neither on aquariums' walls nor on filters) either on non-spiked aquariums or with M1 and M2 concentrations. Before the experiment, fish were kept in aquariums at 20 °C for 4 weeks and fed ad libitum. There were 12 fish for each experimental condition (6 fish per aquarium). Four metals were tested, each of them at two concentrations (Table 1). Control without contaminants contained 14 fish. The intoxication experiment lasted for 96 h (acute intoxication).

Tissue collection, RNA extraction, reverse transcription and polymerase chain reaction

Samples of 100 mg tissue were immediately harvested from the liver of freshly scarified fish. They were stored in 1 mL of RNAlater (Ambion, the RNA Company) at 4 °C overnight for complete infiltration in the tissue and then kept at -80 °C until the analysis. Total RNA was extracted using RNeasy Mini Kit (Qiagen) according to the manufacturer's instructions. A DNAse treatment (DNAse I Amp Grade, Invitrogen) was performed after the extraction. RNA quality was assessed by electrophoresis on an agarose gel, showing clear individual 28S and 18S rRNA bands and no smears, confirming integrities of all prepared RNA. The cDNAs were generated following the standard protocol of the SuperScript II RNAse H Reverse Transcriptase kit (Invitrogen) using random Hexa-primers and 1 μg of RNA. The polymerase chain reactions (PCR) were performed using a Mastercycler ep gradient thermocycler (Eppendorf). For the temperature increase experiment PCR started with initial denaturation step of 94 °C for 2 min, followed by 24 cycles of denaturation step at 94 °C for 1 min, primer hybridization step at 54 °C for 1 min and elongation step at 72 °C for 2 min. The final elongation step was performed at 72 °C for 5 min. For the metal intoxication experiment it started with 2 min at 94 °C followed by 26 cycles at 94 °C for 1 min, 50 °C for 1 min and 65 °C for 1 min. A 15 min step at 65 °C ended the reaction.

In temperature increase experiments PCR reactions were performed to simultaneously amplify fragments of three genes : β-actin, wap65-1 and hsp70. β-actin was used to standardize the results of DNA amplification, as its expression is not supposed to differ among individuals and experimental conditions. Expression of heat shock protein (hsp70) was investigated for comparative purposes.

The primers used to amplify β-actin and wap65 genes were identical to those previously used by Pierre et al. [START_REF] Pierre | Cloning of Wap65 sea bass (Dicentrarchus labrax) and sea bream (Sparus aurata) and expression in sea bass tissues[END_REF]. The primers for hsp70 amplification were designed with Primer3 software (37) using the published sequence of this gene of D. labrax (Genebank accession no. : AY423555). The following primers were chosen: HSPdir (5'-GAGAACAAGATCACAATCAC-3') and HSPrev (5'-CTTCTGTTGATGTTCATACTC-3'). Each PCR mixture contained 1 μL of first strand cDNA, 2 μL of each primer (3 primer pairs) (10 pmol/ μL), 0.3 μL of 5U/μL HotMaster Taq DNA Polymerase (5 PRIME), 5 μL of 10× HotMaster Taq buffer (with 25 mM of Mg 2+ ) and 2 μL dNTPs at 5 mM. The volume was brought to 50 μL with RNAse DNAse free H 2 O. For samples derived from the metal intoxication experiments the PCR mixture contained the same proportions of reagents, however only the β-actin and wap65 genes were examined.

Gene expression quantification and statistical analysis

The visualization of amplified cDNA was performed by a 2% TAE agarose gel electrophoresis (UV, coloration with ethidium bromide). The 1 Kb Plus DNA Ladder (Invitrogen) was used to check the size of amplified bands. Gene expression levels were quantified with ImageJ software (36) and standardized with β-actin. The ratios of wap65/β-actin and hsp70/β-actin (in case of temperature increase experiment) were calculated and compared among individuals.

In order to test the significance of differences in gene expression levels among groups of fish subject to different metal concentrations the Mann-Whitney test was performed in R environment [START_REF] Development | R: A language and environment for statistical computing[END_REF].

RESULTS

Wap65 expression after water temperature increase

The wap65 and hsp70 genes expression levels differed among individuals and among experimental conditions. Both genes were expressed in the controls at 12°C. The highest expression levels for these two genes were recorded after 1 h of exposure to 30°C (Fig. 1). These data indicated a 2-2.5-fold increase in either wap65 or hsp70 transcript levels following the temperature increase. The patterns of expression of both genes from 1 to 4 weeks of exposure to 30°C were similar and these levels were close to the ones recorded at 12°C (control group) (Fig. 2).

Wap65 expression after intoxication with heavy metals

After treatment with Pb at the M1 concentration (Table 1), which was slightly higher than the concentrations found in the natural environment, the level of wap65 expression did not significantly change. However, after intoxication with Pb at the concentration M2, which was 100 times higher than M1, wap65 expression fluctuated greatly (either down or up). Wap65 was upregulated after intoxication with Cd (p < 0.05) at M1, but not at M2. After the intoxication with Cd at M1, 70% of individuals demonstrated wap65 expression levels twice as high as the control.

Treatment with Cu at M1 provoked a slight downregulation of wap65 (p < 0.05), which was not observed at M2. The exposure to zinc resulted in an upregulation of wap65 at both concentrations, but there was a high level of variability among individuals. For only two metals, Cu (downregulation) and Zn (upregulation), the tendency in expression pattern was similar between M1 and M2 concentrations (Fig. 3, Fig. 4). 1) in respect to the control sample. Wap65 expression level was standardized with ß-actin expression level. The numbers correspond to the numbers of individuals from each experimental condition included in the data analysis.

DISCUSSION

Expression of wap65 after water temperature increase

In the European sea bass wap65-1 upregulation was recorded as soon as one hour after exposure to 30 °C and its expression returning to baseline after ~1 week. The wap65 expression pattern was similar to that of the heat shock protein, hsp70. Based upon these results, we speculate that in D. labrax, WAP65 is involved in a "shock" response to rapid perturbation, rather than in a more long term acclimatory process. However, there are only a few studies on the expression of wap65 after a heat shock [START_REF] Clark | Characterisation of the warm acclimated protein gene (wap65) in the Antarctic plunderfish (Harpagifer antarcticus)[END_REF][START_REF] Kikuchi | The Wap65 gene expression of goldfish (Carassius auratus) in association with warm water temperature as well as bacterial lipopolysaccharide (LPS)[END_REF]. As for the European sea bass, detailed study of the expression patterns between one hour and one week of warm temperature exposition would help to estimate more exactly the time after warm water exposition, when wap65 is not upregulated anymore.

Our study suggests that WAP65-1 in European sea bass probably has little function in the acclimatory response. However, such function was found for WAP65-1 in goldfish and carp [START_REF] Kikuchi | The warm temperature acclimation-related 65-kDa protein, Wap65, in goldfish and its gene expression[END_REF][START_REF] Kinoshita | cDNA cloning characterization of the warm-temperatureacclimationassociated protein Wap65 from carp, Cyprinus carpio[END_REF][START_REF] Watabe | Cold and warmtemperature acclimation induces specific cytosolic protein in goldfish and carp[END_REF]. It has been suggested that WAP65-1 underwent neofunctionalization, and the form which maintained function resembles the ancestral one, functioning in acclimation to warm temperature or in mediating an immune response. For sea bass, this would correspond to WAP65-2 [START_REF] Hirayama | Primary structures and gene organizations of two types of Wap65 from the pufferfish Takifugu rubripes[END_REF][START_REF] Hirayama | The occurrence of two types of hemopexin-like protein in medaka and differences in their affinity to heme[END_REF][START_REF] Sarropoulou | Evolution of a multifunctional gene: the warm temperature acclimation protein Wap65 in the European sea bass Dicentrarchus labrax[END_REF]. In support for this idea it has been shown that in the channel catfish (Ictalurus punctatus) the transcriptional response of wap65-2 was quite slow, occurring days after warm temperature treatment (acclimatory response) or bacterial infection [START_REF] Sha | The warm temperature acclimation protein Wap65 as an immune response gene: its duplicates are differentially regulated by temperature and bacterial infections[END_REF]. However this may not be generalizable since in the Antarctic plunderfish (Harpagifer antarcticus) after the increase of water temperature (from 0 °C to 6 °C), a strong downregulation of wap65-2 was recorded within 2-8 hours. Recovery to near control levels took place within 48 hours [START_REF] Clark | Characterisation of the warm acclimated protein gene (wap65) in the Antarctic plunderfish (Harpagifer antarcticus)[END_REF].

A clear phylogenetic distinction between ancestral (e.g. Cypriniformes, like goldfish and carp) and modern (e.g. Perciformes, like sea bass) teleosts was found for both wap65-1 and wap65-2 sequences [START_REF] Sarropoulou | Evolution of a multifunctional gene: the warm temperature acclimation protein Wap65 in the European sea bass Dicentrarchus labrax[END_REF]. It is therefore probable that different taxa acclimate to temperature increase in different ways. However, the number of studied species remains too low for detailed conclusions and, for example, in goldfish, both acclimatory reaction and rapid response to increased temperature (after 4 hours) was demonstrated [START_REF] Kikuchi | The warm temperature acclimation-related 65-kDa protein, Wap65, in goldfish and its gene expression[END_REF][START_REF] Kikuchi | The Wap65 gene expression of goldfish (Carassius auratus) in association with warm water temperature as well as bacterial lipopolysaccharide (LPS)[END_REF].

Expression of WAP65 after intoxication with heavy metals

There are only a few studies on the expression of wap65 after intoxication with heavy metals. Exposure to Hg contaminated sediment provoked its upregulation in cod (Gadus morhua) [START_REF] Olsvik | Transcriptional responses in juvenile Atlantic cod (Gadus morhua) after exposure to mercury-contaminated sediments obtained near the wreck of the German WW2 submarine U-864, and from Bergen Harbor, Western Norway[END_REF]. Wap65 upregulation was also seen in swordtail fish (Xiphophorus helleri) after contamination with Cu for 24 h (upregulation was detected at 50 μg/l and was highest at 1000 μg/l after 24 h) (1). We did not observe upregulation of the sea bass wap65 after the intoxication with copper for 96 h (slight downregulation was found at M1), although the concentrations tested (5 μg/l and 500 μg/l) were within the range previously tested [START_REF] Aliza | Identification of Wap65, a human homologue of hemopexin as a copper-inducible gene in swordtail fish, Xiphophorus helleri[END_REF]. If metals intoxication provokes a rapid response, in our experiments, the expression levels could have returned to baseline before our measurements were taken. Therefore, our and the previous results are not necessarily contradictory. The M1 concentrations used were slightly higher than the ones found in the natural environment. The M2 concentrations were 100 times higher than M1 and may mimic spills or other concentrated events. The latter concentrations were used to test the reaction to acute pollution. This acute intoxication did not provoke any mortality, proving some resistance of fish at least to the short-term exposure to these metals at the concentrations examined. As for the concentrations that could be encountered in the natural environment (M1), wap65 is regulated after the contamination with cadmium (upregulation) and probably with copper (slight downregulation) and zinc (high level of variability among individuals with a global tendency of upregulation). In case of copper and zinc the study should be repeated using a higher number of individuals and a range of different concentrations to confirm the recorder patterns.

Although, the response of wap65 to metals intoxication has not been much studied before, there are some available results concerning other pollutants.

Wap65 upregulation was demonstrated in cod experimentally exposed to alkylphenols [START_REF] Lie | Effects of environmental relevant doses of pollutants from offshore oil production on Atlantic cod (Gadus morhua)[END_REF]. Its downregulation was also found in this species from Store Lungegårdsvann, in Bergen, Norway, which is polluted with polycyclic aromatic hydrocarbons (PAHs), tributyltin (TBT), polychlorobiphenyls (PCBs), Cd, and Hg [START_REF] Lie | Gene expression profiling in Atlantic cod (Gadus morhua L.) from two contaminated sites using a custom-made cDNA microarray[END_REF].

Intoxication with metals influences numerous metabolic pathways and physiological processes in fish. In response to a toxic insult, fish may increase their metabolic activity in order to mobilize protective measures against the pollutant involved. Therefore, enzymes involved in energy production, such as 6-phosphofructo-2-kinase (key regulatory enzyme in glycolysis) might be upregulated [START_REF] Hogstrand | Application of genomics and proteomics for study of the integrated response to zinc exposure in a non-model fish species, the rainbow trout[END_REF]. The reaction to heavy metals intoxication may also provoke a general inflammatory reaction, potentially resulting in the expression of glutathione S-transferases (GSTs) [START_REF] Hollis | Cadmium accumulation, gill Cd binding, acclimatation and physiological effects during long term sub-lethal Cd exposure in rainbow trout[END_REF]. WAP65 has been demonstrated to play a role in the inflammatory processes (e.g. [START_REF] Sarropoulou | Evolution of a multifunctional gene: the warm temperature acclimation protein Wap65 in the European sea bass Dicentrarchus labrax[END_REF], so the inflammation provoked by intoxication with some metals could lead to its upregulation. Moreover, some metals, like cadmium (upregulated wap65 in sea bass) bind to fish gills provoking their damage [START_REF] Hollis | Cadmium accumulation, gill Cd binding, acclimatation and physiological effects during long term sub-lethal Cd exposure in rainbow trout[END_REF] and leading to tissue hypoxia. Upregulation of wap65 was detected during hypoxia conditions in goby Gillichthys mirabilis [START_REF] Gracey | Hypoxiainduced gene expression profiling in the euryoxic fish Gillichthys mirabilis[END_REF]. Therefore, the increased wap65 expression in sea bass that we observed may be due to hypoxic stress induced by binding of cadmium to the gills. On the other hand, our data did not reveal an increase in wap65 expression levels after intoxication with copper. This metal can also induce gill damage and it has been proposed as one of the possible explanations for wap65 upregulation after the copper intoxication in the previous studies [START_REF] Aliza | Identification of Wap65, a human homologue of hemopexin as a copper-inducible gene in swordtail fish, Xiphophorus helleri[END_REF].

Various metals like Cd, Cu, nickel (Ni) and cobalt (Co) can affect heme homeostasis (e.g. 41 and references therein). Free iron can promote free radical formation resulting in oxidative stress [START_REF] Olsvik | Transcriptional responses in juvenile Atlantic cod (Gadus morhua) after exposure to mercury-contaminated sediments obtained near the wreck of the German WW2 submarine U-864, and from Bergen Harbor, Western Norway[END_REF]. It has been demonstrated that the intoxication of fish with cadmium induces the expression of genes responsive to oxidative stress [START_REF] Williams | Development of the GENIPOL European flounder (Platichthys flesus) microarray and determination of temporal transcriptional responses to cadmium at low dose[END_REF][START_REF] Williams | Transcriptomic responses of European flounder (Platichthys flesus) to model toxicants[END_REF]. Oxidative stress in fish might be also provoked by copper [START_REF] Gabbianelli | DNA damage induced by copper on erythrocytes of gilthead sea bream Sparus aurata and mollusk Scapharca inaequivalvis[END_REF], lead (41 and references therein) or zinc [START_REF] Franco | Biochemical alterations in juvenile carp (Cyprinus carpio) exposed to zinc: glutathione reductase as a target[END_REF]. The oxidative stress induced by exposure to mercury (Hg) increased transcription of wap65, but also of transferrin and heme oxygenase, genes involved in iron metabolism, suggesting that WAP65 might play a role in these mechanisms [START_REF] Olsvik | Transcriptional responses in juvenile Atlantic cod (Gadus morhua) after exposure to mercury-contaminated sediments obtained near the wreck of the German WW2 submarine U-864, and from Bergen Harbor, Western Norway[END_REF].

The mammalian homologue of WAP65, hemopexin, participates in transporting heme from the plasma to the liver and preventing oxidative stress [START_REF] Dooley | Emergence of the acute-phase protein hemopexin in jawed vertebrates[END_REF][START_REF] Paoli | Crystal structure of hemopexin reveals a novel high-affinity heme site formed between two beta-propeller domains[END_REF][START_REF] Tolosano | Hemopexin: structure, function, and regulation[END_REF]. Induction in hemopexin expression has been shown to increase the rate of the protein binding to heme. The binding of heme-hemopexin-like complexes with the hemopexin receptor located on the cell surface of the liver provokes among others, the induction of metallothionein gene expression (1 and references therein). Increased synthesis of metallothioneins and their binding to heavy metals reduces the amount of freely available metals (in the host) thus decreasing their potential toxicity [START_REF] Dallinger | The role of metallothionein in cadmium accumulation of Arctic char (Salvelinus alpinus) from high alpine lakes[END_REF][START_REF] Klaassen | Metallothionein: an intracellular protein to protect against cadmium toxicity[END_REF]. Therefore, a crucial question in uncovering the function of WAP65 after exposure to heavy metals is whether WAP65 has an activity similar to hemopexin. There are many hypotheses based on the structure of the putative protein concerning the ability of WAP65 (either of the two isoforms) to have a hemopexin-like activity [START_REF] Choi | Expression of Wap65 mRNA and physiological changes with increasing temperature in black porgy, Acanthopagrus schlegeli[END_REF][START_REF] Hirayama | The occurrence of two types of hemopexin-like protein in medaka and differences in their affinity to heme[END_REF][START_REF] Pierre | Cloning of Wap65 sea bass (Dicentrarchus labrax) and sea bream (Sparus aurata) and expression in sea bass tissues[END_REF][START_REF] Sarropoulou | Evolution of a multifunctional gene: the warm temperature acclimation protein Wap65 in the European sea bass Dicentrarchus labrax[END_REF]. Although functional analyses of proteins are still scarce, it has been demonstrated that shark hemopexin binds heme [START_REF] Dooley | Emergence of the acute-phase protein hemopexin in jawed vertebrates[END_REF].

It is possible that WAP65 is involved in maintaining homeostasis, likely by preventing high concentration of free heme and free metals within the body, thereby reducing toxicity [START_REF] Hirayama | The occurrence of two types of hemopexin-like protein in medaka and differences in their affinity to heme[END_REF]. However, functional studies of WAP65 proteins are needed to confirm these hypotheses. Moreover, cadmium, which is the only metal inducing wap65 upregulation, is not the only one influencing heme homeostasis and causing oxidative stress in fish. The lack of significant wap65 upregulation after intoxication with other metals should be confirmed by additional experiments.

WAP65-not only "warm temperature acclimation-related protein"

Numerous studies suggest that WAP65 is not only involved in temperature acclimation, but also acts as a multifunctional agent in several biological protection processes. Taking into account the multitude of functions and the rapidity of reaction to stress, there is some similarity between WAP65 and heat-shock proteins (HSP), which has a general role in cell physiology. Initially heat shock proteins were thought to be linked only with environmental temperature increases, but it is now widely known that these proteins are upregulated in cells that are exposed to a broad range of stressors [START_REF] Iwama | Heat shock protein expression in fish[END_REF]. Similarly, the term ''warm-temperature protein" was assigned to a 65-kDa protein due to its high abundance in various tissues after warm temperature acclimation, however, WAP65 is involved in many other host processes including in development, immune response, as well as mediating certain chemical stress responses (e.g. [START_REF] Hirayama | The occurrence of two types of hemopexin-like protein in medaka and differences in their affinity to heme[END_REF][START_REF] Sarropoulou | Evolution of a multifunctional gene: the warm temperature acclimation protein Wap65 in the European sea bass Dicentrarchus labrax[END_REF][START_REF] Sha | The warm temperature acclimation protein Wap65 as an immune response gene: its duplicates are differentially regulated by temperature and bacterial infections[END_REF].

In the Mediterranean Sea, from where the European sea bass was sampled, the water temperature changes from 12°C in the winter to over 30°C in the summer. Therefore, the experimental temperature range reflects the one encountered by this poikilothermic species in its natural environment. The fact that there is no increased wap65 expression after more than a week of exposure to 30°C (for example during acclimatory reaction to the seasonal change of water temperature), makes wap65 a potential marker for a rapid detection of heat shock. This is particularly important in aquaculture, where fish in cages do not have a possibility to move to deeper, colder water when the surface water temperatures increase.

Due to the reaction to different physical and chemical stimuli wap65 could also be a potential marker to detect stress in fish. However, variability among individuals in expression patterns after exposure to heavy metals was considerable. Therefore, further studies with additional chemical substances and wider concentrations range are needed in order to verify the utility of wap65 in ecotoxicological studies as a possible biomarker of exposure. Some attempts to include wap65 in DNA microarrays, where its expression is analyzed in parallel with many other genes, has already been undertaken [START_REF] Lie | Gene expression profiling in Atlantic cod (Gadus morhua L.) from two contaminated sites using a custom-made cDNA microarray[END_REF][START_REF] Lie | Effects of environmental relevant doses of pollutants from offshore oil production on Atlantic cod (Gadus morhua)[END_REF][START_REF] Olsvik | Transcriptional responses in juvenile Atlantic cod (Gadus morhua) after exposure to mercury-contaminated sediments obtained near the wreck of the German WW2 submarine U-864, and from Bergen Harbor, Western Norway[END_REF]. Understanding the physiological pathways involved in wap65 regulation will lead to a better understand of the function of this protein and the reasons behind the differences in expression patterns seen in response to different pollutants.

Figure 1 .

 1 Figure 1. Examples of wap65 and hsp70 genes expression patterns in sea bass liver tissue after the exposition to increased water temperature (MW-molecular weight, h-hour, w-week).

Figure 2 .

 2 Figure 2. Average levels of wap65 and hsp70 genes expression in sea bass liver tissue after the exposition to increased water temperature. Wap65 and hsp70 expression levels were standardized with ß-actin expression level (hhour, w-week).

Figure 3 .

 3 Figure 3. a) Examples of wap65 gene expression patterns after intoxication with heavy metals at the concentrations M1 and M2 (see Table 1), b) Wap65 expression after intoxication with lead (Pb) at M2 concentration (perturbing effect; either upregulation or downregulation) (MW-molecular weight).

Figure 4 .

 4 Figure 4. Average level of wap65 expression in sea bass liver tissue after intoxication with heavy metals at the concentrations M1 and M2 (see Table1) in respect to the control sample. Wap65 expression level was standardized with ß-actin expression level. The numbers correspond to the numbers of individuals from each experimental condition included in the data analysis.

Table 1 .

 1 Concentrations of metals in the aquariums during metals exposure experiment.

	Metal	Pb	Cd	Cu	Zn
	M1, μg/l	1	0.5	5	10
	M2, μg/l	100	50	500	500

Acknowledgements -We are very grateful to the Conseil Régional Provence-Alpes Côte d'Azur (France) and Serge Carle from Aqualog for the PhD scholarship and the financing of research of Stéphanie Pierre. We acknowledge the marine farm Dentech for their generous gifts. We also thank Conseil Général du Var (France) and Toulon Var Technologie (France) for their partnerships.


Gilles Rougeron

Jason Lambert

Ekaterina Iakovleva

Lionel Lacassagne

Nicolas Dominguez

Implementation of a GPU Accelerated Total Focusing Reconstruction Method within Civa Software

Keywords: Total Focusing Method, FMC, GPU, OpenCL

This paper presents results of a TFM implementation for Full Matrix Capture acquisitions in CIVA, proposed as a post-processing tool for accurate analysis. This implementation has been made on GPU architecture with OpenCL to minimize the processing time and offer computational device flexibility (GPU/CPU). Examples on immersion configurations on isotropic 2D CAD specimen with planar extrusion are proposed to illustrate the performances. Reconstructions on 2D or 3D areas of direct echoes with mode conversion are allowed. Probe scanning can also be taken into account. Reconstruction results and a benchmark explaining the speedup are presented. Further improvements are also reviewed.

INTRODUCTION

Ultrasonic Testing has been greatly transformed and modernized thanks to the introduction of phased array capabilities in the last decades. NDT performances have benefited from this technology mainly because of two factors: i) the increase in testing speed because of large aperture arrays combined with electronic scanning, ii) the possibility to replace or limit complex mechanical scanning by electronic scanning. Now the nature of phased arrays offers much wider possibilities than just sweeping and steering ultrasonic beams, and the step further is to go towards reconstructed data in the specimen framework. First examples of such reconstructed views are the so-called "true scan" or "corrected scan" views which represent the ultrasonic information along ultrasonic paths in the specimen. Other approaches aim at processing the data so that to produce reconstructed images of the defects in the part, like for instance the Total Focusing Method (TFM) [START_REF] Seydel | Ultrasonic synthetic-aperture focusing techniques in ndt[END_REF] which applies a Synthetic Focusing [START_REF] Holmes | Post-processing of the full matrix of ultrasonic transmit receive array data for nondestructive evaluation[END_REF] to phased array data. Such methods offer enhanced spatial resolution and signal to noise ratio but still suffer from increased computation times which limit the industrial deployment. Previous work in order to accelerate computations of TFM with GPUs were limited in the case of [START_REF] Sutcliffe | Real-time full matrix capture for ultrasonic non destructive testing with acceleration of post-processing through graphic hardware[END_REF] to probes in contact with the specimen, and for [START_REF]OpenCL[END_REF] to planar surfaces. This paper presents recent progress made on the acceleration of the TFM processing in CIVA, by implementation on massively parallel hardware such as GPU (Graphical Processing Units) or multi-core CPU for probes in immersion and complex surfaces. In the remainder of this paper, we will recall the basic principle of TFM algorithm. Then we will provide details on parallelized CPU and GPU implementations of an optimized version of the algorithm. We will then show performance benchmark results. Finally after briefly reviewing the industrialization of the algorithm within CIVA software [START_REF]OpenCL[END_REF] by use of OpenCL emerging standard [START_REF]CIVA : State of the art simulation software for Non Destructive Testing[END_REF], we will draw perspectives and conclude.

TFM METHOD

Basic Principle

The TFM is a technique used to post-process the data from Full Matrix Capture (FMC) or Sparse Matric Capture (SMC) to produce a scalar image, I(P), of the inspected region, where the array is focused in transmission and reception at every point P in the image. The intensity of the TFM image I(P) is given by: .

) ( ) ( 1 ,     t N j i jp ip ij T T S P I (1)
where Sij is the acquired temporal signal, Tip and Tjp are the time delays relative to the point P for i'th and j'th elements, respectively. In immersion mode, the transducer is separated from the mechanical part by a coupling liquid medium, which is generally water. For this inspection mode, propagation of ultrasonic waves in two media with different sound velocities results in refraction at the interface. As is shown in Figure 2., for a planar surface, ultrasound beam is refracted at the point I(x' i ; 0) at the liquid/isotropic solid interface given by z = 0. The propagation time T Eip can be expressed as follows:

2 2 2 ' 1 2 2 ' ) ( ) ( c z x x c z x x T i i i i Eip       (2)
where c1 and c2 are the velocity of ultrasonic waves in liquid and solid, respectively, (x i ; z i ) is the position of the i'th element, (x' i ; 0) is the positions of the refraction point I and (x; z) are the coordinates of the point P. The x-coordinate of the refraction point I can be determined using Snell's law which leads, as detailed in (3), to the following nonlinear equation:

2 2 ' 2 ' 2 2 ' 1 ' ) ( . ) ( z x x c x x z x x c x x i i i i i i i        (3)
The solution of equation 3 can be approximated by using some iterative methods in order to find the roots of the associated polynomial of fourth degree. In our case, the selected root finding method is Laguerre's because of its capacity to always converge for any initial guess and with a cubical rate of convergence for simple roots. For each root found, polynomial deflation is applied using Horner's Method to reduce the polynomial degree: in the case of a real root P(X) is divided by (X -xroot), whereas in the case of a complex root P(X) is divided by (X-xroot) (X -xroot_conjugate).

Curved surfaces imply solving higher degree polynomials depending on the surface: from 4 th to 10 th degree for cylindrical surface depending on the type of control, up to 16 th degree for the case of a torical surface..

ALGORITHM IMPLEMENTATION

Optimizations

The following two optimizations are of algorithmic nature and are used both by CPU and GPU implementation.

Time-of-Flight Symmetry

For a given point P and a given element, the time delay between them depends only on their positions which do not change during the computation. Thus, this time delay is the same whether the element acts as a transmitter or as a receiver. That is why, it is possible to say that there is a symmetry regarding the role of each element. Then, the algorithm can be optimized by computing each time delay only once.

Time-of-Flight Interpolation

For homogeneous specimen with smooth front surface time of flights can be computed on a coarser grid and then interpolated for each pixel of the image. Step grid value is set to the wavelength of the type of wave used for inspection.

With previous two optimizations, the algorithm is organized as follows:

1. For each grid point of a coarse grid time of flight with each element is computed. 2. For each pixel and each couple of element in emission (i) and in reception (j) Time of flight t pixel_ij is computed by interpolation of the grid point time of flight in the neighborhood of the pixel, t pixel_ij = t interp_i +t interp_j Pixel intensity is updated by accumulation of the signal S ij amplitude at time t pixel_ij .

CPU Implementation

Parallelization of the algorithm on CPU is straightforward. An OpenMP implementation has been made with: 1. Parallel computation of time of flights on coarser grid (one CPU thread per grid point). 2. Parallel interpolation of time of flights + amplitude summation (one CPU thread per pixel) No synchronization is required as computation and interpolation of times of flight are independent for each grid point / each pixel. Furthermore amplitude summations for each pixel are made in separate threads.

GPU Implementation

The CUDA Toolkit version 4.2 was used to develop codes aimed at CUDA 2.x enabled device. With this programming architecture, computing task will be distributed among the GPU's streaming multiprocessors (SM). To fit to this architecture, the CUDA model subdivides a compute task on two levels to form a grid:

 A thread level with each disposing of its own program counter and of its own registers.  It defines a block as a group of contiguous threads distributed to a single SM where its threads can share its resources (shared memory, registers...).

FIGURE 4. GPU implementation.

This architecture is most efficient with regular computations, where every thread of a block do the same computations and when threads access GPU's global memory sequentially.

The two steps of a TFM optimized algorithm (with symmetry and interpolation of times of flight) have been implemented as followed:

1. Computation of time of flights: a block of threads is assigned to a rectangular group of pixels corresponding to a coarser grid cell, and they simultaneously compute times of flight at each 4 corners of this cell. Results are stored in shared memory to accelerate data access. Threads are synchronized at the end of this step. 2. Time of flight interpolation plus amplitude summation: this step requires careful attention to improve signal data access. As they are stored in global memory, threads should make coalesced access. This way, they can access global memory through a single memory transaction. To reach this goal, threads of a block are divided and interlaced in N work-groups. N contiguous threads work simultaneously on N different pixels for a given couple of element, thus a given signal. As time of flight for neighbor pixels are very similar, access to same signal data are coalesced. Finally amplitude summation over a workgroup for each pixel requires a reduction.

OpenCL Implementation

GPU implementation described in previous paragraph, initially developed with Cuda, has been ported to OpenCL ( [START_REF]OpenCL[END_REF]). The same code then could be executed both on CPU (through Intel and AMD OpenCL implementation) and on GPU (Nvidia and AMD HD Radeon GPU ones).

PERFORMANCE BENCHMARK

In this paragraph performance results are provided comparing the different implementations on different hardware.

Test Cases

Two test cases were used, one on a planar specimen, and the other on a cylindrical one (Figure 3.). Characteristics of both cases are presented below in Table 1.

Hardware and Software

CPU used was: 2x Xeon X5690 (6 cores) @3,47GHz, while GPUs used were an Nvidia GTX580 (1.5 GB), Nvidia Tesla C2070 (6 GB), and an AMD Radeon HD6970 (2 GB).

On CPU, OpenMP implementation was made with Microsoft Visual C++ 2010 compiler. OpenCL implementation was executed through Intel SDK 1.5 and AMD APP SDK 2.7.

On GPU, Cuda implementation was made with Cuda SDK 4.2. OpenCL implementation was executed on Nvidia GPUs through Nvidia OpenCL 1.1. provided with Cuda SDK 4.2. For AMD Radeon HD6970, AMD APP SDK 2.7 was used.

Results

CIVA PERSPECTIVE

OpenCL implementation of TFM algorithm has been integrated in Civa software ( [START_REF]CIVA : State of the art simulation software for Non Destructive Testing[END_REF]) developed by CEA. This language has been chosen as it is an emerging industrial standard. It offers portability among a vast variety of massively parallel architectures (multi-many cores CPU/GPU/FPGA etc…). It provides good to acceptable performances compared with native platforms languages/library : i.e. Cuda on Nvidia, OpenMP on CPU 1 .

This implementation offers the following features:  Specimen can be planar, cylindrical or 2D CAD with planar extension (Figure 4.)  Signal data set with multiple probe positions can be taken into account (Figure 4.)  Reconstructions can be done on 2D or 3D areas (Figure 5.)  Echo with direct mode conversion can be reconstructed (Figure 6.)

FURTHER IMPROVEMENTS

New features shall be added to TFM OpenCL implementation in Civa in order to reach the same level of functionalities as native previous version:

 More specimen: 2D CAD with cylindrical extension shall be taken into account. As they can potentially contain conical or torical surfaces, polynomial of high degrees shall have to be solved.  Reconstruction with skips (corner and indirect echoes) shall be available  With current integration, signal data have to be transferred entirely on GPU memory before launching OpenCL code execution. An out-of-core solution when FMC data exceeds GPU memory shall be provided.

Performances shall be enhanced:  A Newton method for simple surfaces (ex. Planar) could be used instead of Laguerre. It is should be more efficient for such simple geometries. The goal is to reach a level of performance high enough in order to do TFM reconstruction in real-time (with a frame rate of at least 25 images per second.

CONCLUSION

Implementations of an optimized and fully parallelized Total Focusing Reconstruction method, both on CPU and GPU with native languages (OpenMP and Cuda) have been presented. Performance results on two test cases have been showed and analyzed. Industrialization of this fast reconstruction method within CIVA platform through an OpenCL port has been made. Current features of this integration have been presented. TFM on GPU should be available in a CIVA 11.X version.

FIGURE 1 .

 1 FIGURE 1. TFM basic principle.

FIGURE 2 .

 2 FIGURE 2. Evaluating Snell-Descartes law at a planar surface.

FIGURE 3 .

 3 FIGURE 3. CPU implementation.

FIGURE 5 .

 5 FIGURE 5. The two tests cases used for benchmark performance.

FIGURE 6 .

 6 FIGURE 6. Reconstruction over a complex 2D CAD specimen. On the left with one position, on the right with a scanning of ten positions.

FIGURE 7 .

 7 FIGURE 7. Reconstruction over a 3D area.

FIGURE 8 .

 8 FIGURE 8. Reconstruction for different direct echo modes (LL top-left, LT=TL top-right, TT bottom).

TABLE 2 .

 2 PerformancesOn CPU, best performances are obtained with OpenMP implementation. It provides a good scaling. OpenCL implementation is 3 to 5 times slower compared with OpenMP. As this implementation is an adaptation of a Cuda code optimized for GPU architecture, it is not necessarily well suited for a CPU.On GPU, best performances are obtained with Cuda implementation. OpenCL performances are nearly comparable to Cuda.

						(en s)
			CPU				GPU
		OpenMP	OpenCL	Cuda			OpenCL
		Mono	Multi	AMD	Intel	GTX580	C2070	GTX580	C2070 HD6970
	Planar	4.927	0.446	1.570	2.284	0.192	0.227	0.255	0.348	0.381
	Cylindrical	5.431	0.473	1.654	2.276	0.197	0.235	0.266	0.368	0.384

OpenCL is improving on CPU (see latest Intel SDK

with automatic parallelisation and code vectorization)

ACKNOWLEDGMENTS

This work has been partly supported by the French Agence Nationale de la Recherche OPARUS project (ANR-10-COSI-010-01).