Abed Bounemoura

Optimal stability and instability for near-linear Hamiltonians

In this paper, we will prove a very general result of stability for perturbations of linear integrable Hamiltonian systems, and we will construct an example of instability showing that both our result and our example are optimal. Moreover, in the same spirit as the notion of KAM stable integrable Hamiltonians, we will introduce a notion of effectively stable integrable Hamiltonians, conjecture a characterization of these Hamiltonians and show that our result prove this conjecture in the linear case.

Introduction and results

1. Let n ≥ 2 be an integer, T n = R n /Z n and B = B R be an open ball in R n of radius R > 1 with respect to the supremum norm. We shall consider a near-integrable Hamiltonian of the form H(θ, I) = h(I) + f (θ, I) |f | ≤ ε < < 1 where (θ, I) ∈ T n × B are action-angle coordinates for the integrable part h and f is a small perturbation in some suitable topology defined by a norm | . |. For simplicity, we shall restrict ourself to the analytic case, that is we assume that h and f are bounded and real-analytic on D = T n × B, so that they have holomorphic extensions to some neighbourhood In the absence of perturbation, that is when ε is zero, all solutions (θ(t), I(t)) of the corresponding Hamiltonian system are quasi-periodic and the action variables I(t) are integrals of motion, but in general this is no longer the case after perturbation. Basically, assuming some non-degeneracy condition on the integrable part h and some regularity on h and f , KAM theory is concerned with the persistence of many quasi-periodic solutions ( [START_REF] Kolmogorov | On the preservation of conditionally periodic motions for a small change in Hamilton's function[END_REF]), while Nekhoroshev theory deals with the variation of the action components of all solutions ( [START_REF] Nekhoroshev | An exponential estimate of the time of stability of nearly integrable Hamiltonian systems[END_REF]).

V σ (D) = {(θ, I) ∈ (C n /Z n ) × C n | |I(θ)| < σ,
In the realm of KAM theory, one can introduce a notion of "KAM-stable" integrable Hamiltonian h for which any sufficiently small perturbation possesses a set of positive Lebesgue measure of quasi-periodic solutions closed to the unperturbed ones, and such that the measure m(ε) of the complement of this set satisfy lim ε→0 m(ε) = 0. These KAM stable Hamiltonians have been characterized: they are exactly Rüssmann non-degenerate Hamiltonians, that is functions h : B → R such that ∇h(B) is not contained in any hyperplane of R n . We refer to the nice survey [START_REF] Sevryuk | The classical KAM theory at the dawn of the Twenty-First Century[END_REF] for precise results and references.

It is tempting to define a notion of "Nekhoroshev-stable" Hamiltonians and to try to characterize them, but first one has to come up with a precise definition. It is easy to see from the perturbative character of the Hamiltonian that for all solutions (θ(t), I(t)) with initial condition (θ 0 , I 0 ), one has

lim ε→0 sup 0≤|t|<ε -1 |I(t) -I 0 | = 0.
Without further hypothesis on h, this cannot be improved. Indeed, following [START_REF]An exponential estimate of the time of stability of nearly integrable Hamiltonian systems II[END_REF] and [START_REF] Niederman | Hamiltonian stability and subanalytic geometry[END_REF], if the restriction of h to some affine hyperplane, whose direction is generated by integer vectors, has a non-isolated critical point, then there exist δ > 0 and an arbitrarily small perturbation of size ε such that sup

0≤t≤ε -1 |I(t) -I 0 | ≥ δ.
A simple example of this type will be given below. This prompts us to introduce the following two definitions.

Definition 1. An integrable Hamiltonian h : B → R is rationally steep if its restriction to any affine hyperplane of the form I 0 + Λ, with I 0 ∈ B and Λ a linear subspace of R n generated by integer vectors, has only isolated critical points.

Definition 2. An integrable Hamiltonian h : B → R is effectively stable if for any f :

T n × B → R with |f | ≤ ε, all solutions (θ(t), I(t)) of the Hamiltonian system H = h + f starting at (θ 0 , I 0 ) satisfy lim ε→0 sup 0≤|t|≤ε -1 |I(t) -I 0 | = 0.
Hence an integrable Hamiltonian is effectively stable if one can ensure that the time of stability T (ε), that is the maximal time during which the variation V (ε) of the action of all solutions satisfy lim ε→0 V (ε) = 0, is at least 1/ε. Then how large one can choose T (ε) will depend on more specific properties of the integrable Hamiltonian.

The following conjecture was essentially made in [START_REF]Stabilité et instabilité des systèmes hamiltoniens presqueintégrables[END_REF].

Conjecture 1. Effectively stable Hamiltonians are exactly rationally steep Hamiltonians.

As we already explained, if a Hamiltonian is effectively stable, then it has to be rationally steep, so that only the converse statement of this conjecture is of interest. In this paper we shall prove this conjecture in the specific case where the integrable Hamiltonian is linear. This is an important case, not only because of its own interest, but also because it is a crucial step in showing the conjecture in its full generality (this point will be briefly discussed at the end of the paper).

2. So from now on we shall restrict to linear integrable Hamiltonians. Given a vector ω ∈ R n \ {0}, we let l(I) = ω.I be the linear Hamiltonian with frequency ω and we consider

H(θ, I) = l(I) + f (θ, I) |f | ≤ ε < < 1. ( * )
Without loss of generality, we may assume that the system has been rescaled so that |ω| = 1, therefore, reordering the components of ω if necessary, we can write

ω = (1, α 1 , . . . , α n-1 ) = (1, α), |α i | < 1, i ∈ {1, . . . , n -1}.
Such "degenerate" integrable systems are never KAM stable, but the problem of effective stability is of different nature as we will show. First, one can easily see that l being non-rationally steep is equivalent to ω being resonant, that is there exists k ∈ Z n \ {0} such that k.ω = 0. In this case, the Hamiltonian

H(θ, I) = ω.I + f (θ), f (θ) = -ε sin(k.θ),
gives rise to a system which can be easily integrated:

θ = ω İ = εk cos(k.θ) =⇒ θ(t) = θ 0 + tω [Z n ] I(t) = I 0 + εk cos(k.θ 0 ).
So any solution starting at (θ 0 , I 0 ) with k.θ 0 = 0 satisfy sup

0≤t≤ε -1 |I(t) -I 0 | = |k| ≥ 1.
Hence non-rationally steep linear Hamiltonians are indeed not effectively stable, and our conjecture in this case states that linear Hamiltonians with a non-resonant frequency are effectively stable. However, the only case for which results are known is when the frequency satisfies a classical Diophantine condition, and there much stronger stability properties hold true in the sense that the time T (ε) is at least exponentially large with respect to the inverse of the size of the perturbation. These results will be recalled below.

In this paper, we shall prove a more general stability result and we will construct an example showing that it is the best one can obtain.

3. Let us state precisely our results. We shall denote by

| . | Z the distance to the integer lattice Z, that is |x| Z = min p∈Z |x -p| for x ∈ R. The frequency vector ω = (1, α 1 , . . . , α n-1 ) = (1, α) being non-resonant, the function Ψ = Ψ ω given by Ψ(K) = max |k.α| -1 Z | k ∈ Z n-1 , 0 < |k| ≤ K , K ∈ N *
is well-defined. It is obviously strictly increasing on N * , hence we can extend it (keeping the same notation) as a strictly increasing continuous function defined on [1, +∞). Then let us also define two additional functions

Λ(x) = xΨ(x), ∆(x) = Λ -1 (x), x ≥ 1,
which are also strictly increasing and continuous.

Theorem 1.1. Let H be as in ( * ), with ω non-resonant. Then there exist positive constants ε 0 , c, c 1 and c 2 depending only on n, R, σ and ω such that if ε ≤ ε 0 , all solutions (θ(t), I(t)) of H with I 0 ∈ B R/2 satisfy the estimates

|I(t) -I 0 | ≤ c 1 δ, |t| ≤ δε -1 exp c 2 ∆(cε -1 ) . for any ∆ cε -1 -1 ≤ c 1 δ < R/2.
Depending on the growth of the function ∆, the exponential factor in the time of stability below might not be very large but choosing δ = ε b with b > 0 arbitrarily small, thanks to the factor δε -1 the time of stability is at least 1/ε and this proves our conjecture in the linear case.

Corollary 1.2. For linear integrable Hamiltonians, rationally steep Hamiltonians are effectively stable.

We shall give an elementary proof of Theorem 1.1 in the case n = 2, using only one rational approximation and a one-phase averaging, in the same spirit as in [START_REF] Lochak | Canonical perturbation theory via simultaneous approximation[END_REF]. This method of proof can be easily extended for any n ≥ 2, but in general it fails to give the best result.

Hence for any n ≥ 2, we shall use more classical techniques, namely general resonant normal forms as in [START_REF] Pöschel | Nekhoroshev estimates for quasi-convex Hamiltonian systems[END_REF] and [START_REF] Delshams | Effective stability and KAM theory[END_REF], and so our proof will not be essentially new. In the special case where the frequency ω is Diophantine, that is when there exist γ > 0 and τ ≥ n -1 such that for all k ∈ Z n-1 \ {0}, |k.α| Z ≥ γ|k| -τ , then, in Theorem 1.1, we can choose

Ψ(x) = γ -1 x τ , Λ(x) = γ -1 x 1+τ , ∆(x) = (γx) 1 1+τ
and our result gives

|I(t) -I 0 | ≤ c 1 δ, |t| ≤ δε -1 exp c 2 (cγε -1 ) 1 1+τ . for any (c -1 γ -1 ε) 1 1+τ ≤ c 1 δ < R/2.
In this case, this is exactly the result obtained in [START_REF] Delshams | Effective stability and KAM theory[END_REF]. In [START_REF] Pöschel | Nekhoroshev estimates for quasi-convex Hamiltonian systems[END_REF], there is a similar but less flexible result, since there one can only choose δ ∼ 1. Let us also note that this Diophantine case was first considered in [START_REF] Fassò | Lie series method for vector fields and Hamiltonian perturbation theory[END_REF], and the result there was even more flexible since one has the freedom to choose δ equals, up to a multiplicative constant, to ε, but the proof is restricted to the Diophantine case and fails without this assumption.

In any cases, our next theorem shows precisely that the time of stability achieved in Theorem 1.1 is the best possible.

Theorem 1.3. For any non-resonant vector ω ∈ R n , there exists a sequence (f j ) j∈N * of analytic functions on V σ (D), with |f j | σ = ε j → 0 when j → +∞, such that the system H j = l + f j has orbits which satisfy

|I(t) -I 0 | = |t|ε j exp -2σ∆(cε -1 j ) .
for a constant c depending only on R, σ and ω.

This result says that for a specific arbitrarily small perturbation and for some solutions of the perturbed system, the inequality of Theorem 1.1 are in fact equality (up to constants depending only on n, R, σ and ω), so that Theorem 1.1 cannot be improved. Conversely, Theorem 1.1 implies that the above theorem is the best possible. The construction of the example of instability in Theorem 1.3 is elementary, and it will follow very naturally from our proof of the stability result in the case n = 2. It is also inspired by an example given in [START_REF] Sevryuk | The classical KAM theory at the dawn of the Twenty-First Century[END_REF].

Therefore, as far as one is interested in exponential stability (that is, T (ε) is exponentially large with respect to ε -1 ), then a polynomial growth of the function Ψ, which is nothing but a Diophantine condition, is both sufficient and necessary. To obtain a non-trivial polynomial time of stability such as ε -r , r > 1, then an exponential growth of the function Ψ is sufficient and necessary.

4. The plan of the paper is the following: the proof of the above theorems will be given in the next section, and further comments on the results are given in the last section. Throughout the paper, in order to avoid cumbersome expressions, we will replace positive constants depending only on n, R, σ and ω with a dot. More precisely, an assertion of the form "there exists a constant c > 0 depending on the above parameters such that u < cv" will be simply replaced with "u <• v", when the context is clear.

Proof of the results

1. We shall start by giving an elementary proof of Theorem 1.1 in the special case n = 2.

Given a vector v = (1, p/q) ∈ R 2 , where p/q is a non-zero rational number in its lowest term, we let l v be the linear Hamiltonian with frequency v. The normal form result that we shall need, which is due to Lochak-Neishtadt ( [START_REF] Lochak | Estimates of stability time for nearly integrable systems with a quasiconvex Hamiltonian[END_REF], see also [START_REF]On Nekhoroshev's estimate at an elliptic equilibrium, Internat[END_REF]), is just a one-phase averaging.

Lemma 2.1. Consider a Hamiltonian

H = l v + f v defined on V σ (D) with |f v | σ <• ε,
and assume that for some K > 1, Kqε <• 1.

(1)

Then there exists an analytic symplectic transformation

Φ : V σ/2 (D) → V σ (D)
with |Φ -Id| σ/2 <• qε such that

H ′ = H • Φ = l v + g + f ′ v
with {g, l v } = 0, and the estimates

|g| σ/2 <• ε, |f ′ v | σ/2 <• εe -•K holds true.
This is a so-called resonant normal form, that is the perturbation f v has been reduced, up to an exponentially small term f ′ v , to its resonant part g which satisfies {g, l v }. Now a simple observation is that if the denominator q is large, then so is the size of any integer vector which is orthogonal to v. In such a case, the resonant term g itself is exponentially small. Let us state this as a lemma which complements the result above.

Lemma 2.2. Under the previous hypotheses, assume also that q > K. Then

g = ḡ + g ′ , ḡ = T n g, |ḡ| σ/2 <• ε, |g ′ | σ/4 <• εe -•K .
Proof. Expanding g as a Fourier series g(θ, I) = k∈Z n ĝk (I)e 2iπk.θ , we can write g = ḡ + g ′ with g ′ (θ, I) = k∈Z n \{0} ĝk (I)e 2iπk.θ . Now the condition that {g, l v } = 0 is easily seen to be equivalent to ĝk

(I) = 0 if k.v = 0, that is g ′ (θ, I) = k.v=0, k =0 ĝk (I)e 2iπk.θ . Now take k = (k 1 , k 2 ) ∈ Z 2 \ {0} such that k.v = 0, that is qk 1 + pk 2 = 0.
First one can see that neither k 1 nor k 2 is zero, otherwise this would imply that either p or q is zero. Therefore q divides pk 2 , but as q and p are relatively prime, then q divides k 2 and hence |k| ≥ |k 2 | ≥ q > K. This means that

|g ′ | σ/2 ≤ |g ′ >K | σ/2 , g ′ >K (θ, I) = |k|>K ĝk (I)e 2iπk.θ .
It is now a classical estimate that

|g ′ >K | σ/4 <• |g| σ/2 e -•K <• εe -•K
with implicit constants depending only on σ.

Part of the arguments given above works only for n = 2. However, for any n ≥ 2, some other simple arguments along these lines can be used (but as we already said, this does not give the best result in general). Now we can prove Theorem 1.1 for n = 2.

Proof of Theorem 1.1, n = 2. Here ω = (1, α) ∈ R 2 with |α| < 1. The size of the perturbation ε > 0 being given, we choose K such that

KΨ(K) =• ε -1 , K = ∆(•ε -1 )
for some implicit constant to be chosen below. First, assuming ε is sufficiently small, that is ε <• 1, we can ensure that K > 1 and we can apply Dirichlet's box principle to approximate α by a rational number with denominator q < Ψ(K): we obtain a vector

v = (1, p/q) ∈ R 2 , p ∈ Z \ {0}, such that |ω -v| ≤ q -1 Ψ(K) -1 , 1 < q < Ψ(K).
Moreover, from the definition of Ψ,

Ψ(q) -1 ≤ |qα -p| < Ψ(K) -1 , so that q > K. Hence |ω -v| ≤ K -1 Ψ(K) -1 =• ε.

Now our Hamiltonian H can be written as

H = l + f = l v + f v , f v = l -l v + f, with |f v | σ <• ε.
Moreover, since q < Ψ(K), then qε <• K -1 , and choosing properly our implicit constant in the definition of K, we can ensure that condition (1) is met and hence we can apply Lemma 2.1: there exists an analytic symplectic transformation

Φ : V σ/2 (D) → V σ (D) with |Φ -Id| σ/2 <• qε <• K -1 such that H ′ = H • Φ = l v + g + f ′ v
with {g, l v } = 0, and the estimates

|g| σ/2 <• ε, |f ′ v | σ/2 <• εe -•K
holds true. Moreover, since q > K, Lemma 2.2 can also be applied and therefore

H ′ = H • Φ = l v + ḡ + g ′ + f ′ v = l v + ḡ + f
with ḡ integrable and

| f | σ/4 ≤ |g ′ | σ/4 + |f ′ v | σ/4 <• εe -•K .
Now let us write (θ, I) = Φ(θ ′ , I ′ ), and take any δ such that K -1 <• δ < R/2. Since l v + ḡ is integrable, the mean value theorem gives

|I ′ (t) -I ′ 0 | <• δ, |t| ≤ δε -1 e •K .
Now coming back to the original coordinates and using |Φ -

Id| σ/2 <• K -1 , this gives |I(t) -I 0 | <• δ, |t| ≤ δε -1 e •K ,
which, recalling that K = ∆(•ε -1 ), is our statement. This completes the proof.

2. Now for any number of degrees of freedom, the strategy we explained above can be easily extended and a result of stability can be obtained. However, it requires to compare simultaneous and linear Diophantine approximation, and it seems to us that, unless the frequency is badly approximable, it cannot give the best result from a quantitative point of view. Therefore we shall use different arguments to prove Theorem 1.1 for any n ≥ 2, following the "dual" approach. Given any sub-lattice Λ of Z n , we shall say that a function g defined on V σ (D) is Λ-resonant if its Fourier expansion is of the form g(θ, I) = k∈Λ ĝk (I)e 2iπk.θ , (θ, I) ∈ V σ (D).

Then, given any K ≥ 1 and any λ > 0, we shall say that a vector

w ∈ R n is (λ, K)-non resonant modulo Λ if ∀k ∈ Z n \ Λ, |k| ≤ K, |k.w| ≥ λ.
Finally, given such a vector w, let us write l w (I) = w.I. The normal form result that we shall use here is due to ).

Lemma 2.3. Consider a Hamiltonian H = l w + f defined on V σ (D), with |f | σ <• ε and assume that Kλ -1 ε <• 1.

(2)

Then there exists an analytic symplectic transformation

Φ : V σ/2 (D) → V σ (D) with |Φ -Id| σ/2 <• λ -1 ε such that H ′ = H • Φ = l w + g + f ′
where g is Λ-resonant, and with the estimates

|g| σ/2 <• ε, |f ′ | σ/2 <• εe -•K .
The above normal form result strictly contains Lemma 2.1 (the latter corresponds to the case where Λ has rank n -1, and we can choose λ = q -1 for any K ≥ 1), as a consequence its proof is more involved. A similar normal form result is contained in [START_REF] Pöschel | Nekhoroshev estimates for quasi-convex Hamiltonian systems[END_REF], but for our purpose it is weaker since the distance to the identity of the normalizing transformation Φ is bigger (it is only of order Kλ -1 ε).

The proof of Theorem 1.1 is now straightforward.

Proof of Theorem 1.1. Here ω = (1, α 1 , . . . , α n-1 ), and this frequency is by definition (Ψ(K) -1 , K)-non-resonant modulo {0}. The size of the perturbation ε > 0 being given, we choose K such that

KΨ(K) =• ε -1 , K = ∆(•ε -1 )
with a suitable implicit constant so that the condition (2) is fulfilled. Then the requirement that K ≥ 1 gives a threshold ε <• 1. Therefore we can apply Lemma 2.3: there exists an analytic symplectic transformation

Φ : V σ/2 (D) → V σ (D)
and therefore |f 2 j | σ ≤ ε j /2 as well. Hence |f j | σ ≤ ε j , and ε j → 0 when j → +∞. Now we can write the Hamiltonian H j (θ, I) = h(I) + f j (θ, I) = v j .I + ε j q -1 j exp(-2σq j ) cos(k j .θ)

and as k j .v j = 0, the associated system is easily integrated:

θ(t) = θ 0 + tv j [Z n ] I(t) = I 0 -tk j q -1 j ε j exp(-2σq j ) cos(k j .θ 0 ).
Choosing any solution with initial condition (θ 0 , I 0 ) satisfying k j .θ 0 = 0, cos(k j .θ 0 ) = 1 and using the fact that |k j |q -1 j = 1, we obtain

|I(t) -I 0 | = |t|ε j exp(-2σq j ).
Recalling that q j = ∆(cε -1 j ) this gives

|I(t) -I 0 | = |t|ε j exp -2σ∆(cε -1 j )
and this concludes the proof.

Further comments

Let us conclude this paper by several remarks.

1. First, let us mention that a perturbation of a linear Hamiltonian system, as we have considered, also describes the dynamics in the neighbourhood of an invariant, linearly stable, isotropic and reducible torus carrying a quasiperiodic motion. Our Theorem 1.1 directly applies to the case where the torus is of full dimension (that is, when the torus is Lagrangian) and gives a non-trivial result of stability. In the case of intermediate dimensions, our result should apply also but there one has to work not only with angle-action coordinates but also with Cartesian coordinates. Anyway, in the case where the torus has dimension zero, that is for an elliptic fixed point, Theorem 1.1 is useless since in this situation the perturbation has a peculiar form and classical Birkhoff normal form estimates apply and give a much better result of stability. Now Theorem 1.3 also applies in this setting, except for elliptic fixed point since in this case there is no proper angle-action coordinates to work with (and in fact, for n = 2, Diophantine and even Brjuno elliptic fixed points are always Lyapounov stable, see [START_REF] Rüssmann | Stability of elliptic fixed points of analytic areapreserving mappings under the bruno condition[END_REF] and [START_REF] Fayad | Herman's last geometric theorem[END_REF]). When the dimension of the torus is at least one, our result applies and in the Diophantine case, without any further assumptions, results of exponential stability are the best one can expect. Assuming further conditions on the Birkhoff invariants, it was shown in [START_REF] Morbidelli | Superexponential stability of KAM tori[END_REF] (see also [START_REF] Bounemoura | Generic super-exponential stability of invariant tori, Ergodic Theory and Dynamical Systems[END_REF] for more general results) that super-exponential stability hold true. Here our result shows that some further conditions are indeed necessary.

2. Then, let us discuss the case where the system is assumed to be less regular than analytic. Our proof of Theorem 1.1 in the case n = 2 only uses a one-phase averaging. Such a result being available in the Gevrey category ( [START_REF] Marco | Stability and instability for Gevrey quasi-convex near-integrable Hamiltonian systems[END_REF]) and in the finitely differentiable category ( [START_REF]Nekhoroshev theory for finitely differentiable quasi-convex Hamiltonians[END_REF]), the proof immediately extends in this case (of course, in finite differentiability, only polynomial stability hold true). Now for any n ≥ 2, we have used a more elaborate normal form which is not known if the system is not analytic. However, such a normal form can certainly be proved using smoothing techniques so that Theorem 1.1 should hold in any regularity. Obviously, the proof of Theorem 1.3 is independent of the regularity and the extension is straightforward.

3. Finally, let us explain how our result can be used to prove in general that effectively stable Hamiltonians are exactly rationally steep Hamiltonians. As we have explained, our result deals with the case where the Hamiltonian is linear. In the non-linear case, first one has to use not only integrable normal forms but also more general resonant forms, which are also well-known.

Then, the condition of being rationally steep roughly says that resonances "do not accumulate" so that any solution, when evolving (of course, if they do not evolve there is nothing to prove), will necessarily encounters nonresonant points, arbitrarily close and in any direction, and therefore the inductive scheme introduced in [START_REF]Prevalence of exponential stability among nearly integrable Hamiltonian systems[END_REF] (see also [START_REF] Bounemoura | Generic Nekhoroshev theory without small divisors[END_REF] and [START_REF]Effective stability for gevrey and finitely differentiable prevalent hamiltonians[END_REF] for refinements), together with some other geometric arguments, should give the desired result.

  d(I, B) < σ} for some σ > 0, where I denotes the imaginary part and d is the distance associated to the supremum norm. Then the norm of the perturbation |f | = |f | σ is defined by |f | σ = |f | C 0 (Vσ(D)) = sup z∈Vσ(D) |f (z)|. * abed@impa.br, IMPA, Estrada Dona Castorina 110, Rio de Janeiro, Brasil, 22460-320 1

Acknowledgments. Thanks to Jean-Christophe Yoccoz, Bassam Fayad, Stéphane Fischler, Pierre Lochak and special thanks to Laurent Niederman and Jean-Pierre Marco. This work has been done during a stay at Penn State University, the author is grateful to Vadim Kaloshin for his very kind invitation. Finally, the author thanks the CNPq for financial support.

with |Φ -Id| σ/2 <• Ψ(K)ε =• K -1 such that

where g is {0}-resonant, that is g = ḡ, and with the estimates

Now, as before, let us write (θ, I) = Φ(θ ′ , I ′ ), and take any δ such that K -1 <• δ < R/2. Since l + ḡ is integrable, the mean value theorem gives

Now coming back to the original coordinates and using |Φ -

which ends the proof, since K = ∆(•ε -1 ).

3. Now let show that our stability result is optimal by proving Theorem 1.3.

Proof of Theorem 1.3. Given ω = (1, α 1 , . . . , α n-1 ), let us pick one component, say α 1 , and let us denote by (p j /q j ) j∈N the sequence of the convergents of α 1 . The vector ω being non-resonant, α 1 is irrational and this means that the sequence (q j ) j∈N is strictly increasing. From the classical estimates (q j + q j+1 ) -1 < |q j α 1 -p j | < q -1 j+1 , j ∈ N, we deduce that q j+1 < Ψ(q j ) < q j + q j+1 < 2q j+1 . Now the perturbation f j will be of the form

First, we choose f 1 j (I) = v j .I -ω.I, where v j = (1, p j /q j , α 2 , . . . , α n-1 ). As |α 1 -p j q -1 j | < (q j q j+1 ) -1 < 2(q j Ψ(q j )) -1 , if we set ε j = c(q j Ψ(q j )) -1 , q j = ∆(cε -1 j ) with a suitable constant c depending on α 1 , R and σ, we obtain |f 1 j | σ < ε j /2. Then, if we define k j = (p j , -q j , 0, . . . , 0) ∈ Z n , we choose f 2 j (θ) = ε j µ j cos(k j .θ), µ j = q -1 j exp(-2σq j ).

For θ ∈ C n with |I(θ)| ≤ σ, we have | cos(k j .θ)| ≤ exp(2σ|k j |) = exp(2σq j ) ≤ q j /2 exp(2σq j )


Peter Hofbauer

Frans J M Smulders

Miroslav Vodnansky

Peter Paulsen
email: peter.paulsen@vetmeduni.ac.at

Waleed R El-Ghareeb

A note on meat quality traits of pheasants ()

Keywords: Pheasant meat, Proximate composition, Drip loss, pH, Cooking loss, Colour

Meat yield, proximate composition, pH and drip loss of breast and thigh muscles were studied in 29 hunted male and 32 slaughtered female pheasants. In the breast muscles of additional 14 hunted male pheasants, colour, cooking loss and shear force were measured. Weight of muscles of hunted male pheasants was higher than that of slaughtered females, but the percentages of breast and thigh muscles relative to the dressed carcass did not differ. Average protein concentrations in lean muscles were above 22%, and average fat was below 1.2%. In breast muscles, pH 24 was lower than in thigh (ca. 5.6 vs. 6.0), and, consequently, drip loss was higher (2.2-3.0% vs. 1.0-1.5%). Colour of breast muscles was characterised by L*a*b* values of about 55, 4 and 8-9, respectively. The shear force of breast muscles was about 30 N/cm 2 .

Introduction

Meat from wild-living game is a highly valued food. Recent studies have dealt with hygiene and food safety implica-tions of the production of meat from game birds, with respect to "Good Hygiene Practice" [START_REF] Paulsen | Influence of storage conditions and shotshell wounding on the hygienic condition of hunted, uneviscerated pheasant (Phasianus colchicus)[END_REF][START_REF] El-Ghareeb | Microbiological condition and shelf life of meat from hunted game birds[END_REF]). However, also sensory quality traits have to be considered, because they will substantially influence the consumers attitude to select and buy such meat or products thereof. Basically, colour/appearance, tenderness, juiciness and flavour/odour are the most important sensory attributes for consumers regarding meat [START_REF] Smulders | Muscle" and "Meat Quality"; biological basis, processing, preparation[END_REF][START_REF] Risvik | Sensory properties and preferences[END_REF]). Among those, colour and drip loss (water-holding capacity; related to juiciness) are the first and more prolonged sensations among the aforementioned sensory quality characteristics [START_REF] Lawrie | Lawrie's Meat Science[END_REF]. This view has been corroborated by consumer behavioural science [START_REF] Becker | Consumer perception of fresh meat quality: a framework for analysis[END_REF]. In particular, water-holding capacity has several implications for shelf life of fresh meat as well as for processing technology [START_REF] Hofmann | What is quality ? Definition, measurement and evaluation of meat quality[END_REF], and is influenced by other factors, as pH [START_REF] Lawrie | Lawrie's Meat Science[END_REF].

Pheasants contribute substantially to the hunting bags in European countries, e.g. 79% of ca. 19 million game birds shot in 2004 in the UK [START_REF] Pacec | The economic and environmental impact of sporting shooting[END_REF], with significant benefits for local economy. Even in small countries, pheasants are among the most relevant small game species, e.g. 61% of a total of ca. 265,000 hunted game birds in Austria in 2008 (Anonymous 2010). Hunted pheasants are either wild-born feral or were bred in aviaries and then released. Farming of pheasants is done not only for restocking/release in hunting but also for slaughter and meat production [START_REF] Golze | Fasanenproduktion zur Fleischgewinnung und zum Auswildern[END_REF].

A number of studies have reported the meat yield and proximate chemical composition of meat from pheasants [START_REF] Petkov | Chemical composition of pheasant meat[END_REF][START_REF] Richter | Zusammensetzung und Qualitätsparameter von Perlhühnern, Fasanen, Tauben, Cairina und Kaninchen[END_REF][START_REF] Tucak | The influence of keeping pheasants in captivity vs. nature on the biological value of meat and its use in human nutrition[END_REF]. Although these data are valuable to characterise the nutritional quality of meat, they will affect the consumers attitude only to a limited extent ("credential quality cues" according to [START_REF] Becker | Consumer perception of fresh meat quality: a framework for analysis[END_REF]. For fresh meat, it is conceivable that sensory perceptions will "override" those credential quality cues. Comparatively few studies have dealt with instrumental measurement of sensory quality of pheasant meat (e.g. [START_REF] Richter | Zusammensetzung und Qualitätsparameter von Perlhühnern, Fasanen, Tauben, Cairina und Kaninchen[END_REF][START_REF] Kuzniacka | Effect of age and sex of pheasants (Phasianus colchicus L.) on selected physical properties and chemical composition of meat[END_REF]. Until recently, the lack of harmonisation in methodology [START_REF] Honikel | Reference methods for the assessment of physical characteristics of meat[END_REF] made comparisons of results to other studies on the same or other species difficult or impossible.

The present study was, therefore, conducted to determine chemical and physical indicators for colour, tenderness and juiciness as important factors for sensory meat quality [START_REF] Smulders | Muscle" and "Meat Quality"; biological basis, processing, preparation[END_REF][START_REF] Risvik | Sensory properties and preferences[END_REF] according to the protocols proposed by [START_REF] Honikel | Reference methods for the assessment of physical characteristics of meat[END_REF]. Also, the yield of the highly valued meat portions from pheasants (breast and thigh) and edible by-products was determined. Hunted (shot) pheasants were included in this study as well as slaughtered pheasants.

Materials and methods

Hunted animals (n=43, all male) were collected from three drive hunts in Lower Austria in autumn/winter where they had been killed by lead shots of 2.5-3 mm diameter. Slaughtered specimens (n=32, all female) were randomly picked from two batches of animals which had been slaughtered in a pheasantry in the Czech Republic (i.e. animals were stunned by a blow on the occipital region of the head and killed by subsequent bleeding under the provisions of Council Directive 93/119/EC (1993)). Carcasses were delivered in uneviscerated condition. All specimens were cooled to 0-2°C internal temperature within 12 h post mortem (p.m.).

Meat yield

Meat yield was determined in 29 carcasses of hunted male and 32 carcasses of slaughtered female pheasants. Animals were eviscerated 24 h p.m., and processed as described earlier [START_REF] El-Ghareeb | Microbiological condition and shelf life of meat from hunted game birds[END_REF]. Weight of dressed carcass (i.e. eviscerated carcass; wings, distal legs, neck, head, crop, feathers and skin removed) and edible by-products was recorded.

Physical characteristics

From 29 carcasses of hunted male and 32 carcasses of slaughtered female pheasants, the right breast muscle (M. pectoralis superficialis) and right thigh were removed 24 h p.m. Measurement of pH was done in the breast muscle and in the M. iliotibialis lateralis (Testo 230; Testo Lenzkirch, Germany). Left breast and thigh were packed in a plastic bag (PA/PE film, 90 µm thickness) and stored for 7 days at 0-2°C for determination of drip loss. Breast and all thigh muscles were used for chemical analysis further on.

Due to the limited size of the muscles, colour, cooking loss and shear force measurements had to be done in breast muscles of separate 14 hunted male pheasants. Right breast muscles were removed from the carcass and tested 24 h p.m., and left breast muscles were removed at 96 h p.m. (storage at 0-2°C). Colour (L*a*b* values) was measured at the muscle surface with a Phyma Codec 400 (Phymacom, Giesshübl, Austria). The mean value of three measurements was recorded. For cooking loss, muscles were weighed, put into plastic bags and heated in a water bath set at 72°C until a core temperature of 70°C was reached. Then, the bags were cooled under running tap water, the meat taken out, gently wiped dry and re-weighed. Cooking loss was calculated as the percentage of weight loss of the meat cut. For shear force (tenderness) measurement, the cooked meat was cut into longitudinal samples with a quadratic cross section of 1 cm 2 . These primatic samples were tested with an Instron 4411 with attached Warner-Bratzler device. The shear blade was 1.2 mm thick and had a rectangular hole (width: 11 mm; height: 15 mm), crosshead speed was 100 mm/min. Per muscle, three subsamples were tested and the mean value of three measurements was recorded.

Chemical analysis

Visible fat and tendons were removed from the meat cuts. Samples were ground and dry matter, crude protein, fat and ash were determined according to the German standards method book (Amtliche Sammlung von Untersuchungsverfahren nach §64 LFBG: L 06. 00-3, 2004; L 06.00-7, 2007; L 06.00-6, 1980; L06.00-4, 2007, respectively).

Statistical analysis

Data are presented as ranges or arithmetic mean ± standard deviation. Traits of breast vs. thigh muscle (separately for hunted males and slaughtered females) and colour/cooking loss/shear force of breast muscle only at 24 vs. 96 h p.m. were compared by Student's t test (SPSS V.14.0, Chicago, USA). Traits of hunted males vs. slaughtered females were not compared.

Results and discussion

Limitations of the data set

All hunted (shot) animals were male, whereas all slaughtered ones were female. This reflected the usual procedure in that particular pheasantry, but also complicates a comparison between these two groups as any difference between the two study groups could be effected by sex, age, feeding, rearing and mode of killing or a combination of these factors. Also, due to the small sizes of muscles, not all of the analyses described below could be done in all animals.

Meat yield

Weights of the male hunted birds (whole bird) were 1336.7± 119.4 g, and those of female slaughtered pheasants were 912.5±142.1 g. These data agree well with other reports, e.g. average weights of 1232±147 g for hunted-males [START_REF] Tucak | The influence of keeping pheasants in captivity vs. nature on the biological value of meat and its use in human nutrition[END_REF]) and 907±90 g [START_REF] Richter | Zusammensetzung und Qualitätsparameter von Perlhühnern, Fasanen, Tauben, Cairina und Kaninchen[END_REF]) or 970±157 g [START_REF] Tucak | The influence of keeping pheasants in captivity vs. nature on the biological value of meat and its use in human nutrition[END_REF]) for slaughtered females.

As in other gallinaceous game birds (e.g. chukar partridge, Özek et al. 2003), female pheasants have a 15-40% lower body weight than males of the same age and keeping condition (farm/aviary or free range), see [START_REF] Richter | Zusammensetzung und Qualitätsparameter von Perlhühnern, Fasanen, Tauben, Cairina und Kaninchen[END_REF], [START_REF] Tucak | The influence of keeping pheasants in captivity vs. nature on the biological value of meat and its use in human nutrition[END_REF] and [START_REF] Golze | Fasanenproduktion zur Fleischgewinnung und zum Auswildern[END_REF]. Also breed, age, feeding regime and keeping condition can have a substantial influence on body weight [START_REF] Tucak | The influence of keeping pheasants in captivity vs. nature on the biological value of meat and its use in human nutrition[END_REF][START_REF] Golze | Fasanenproduktion zur Fleischgewinnung und zum Auswildern[END_REF].

Weights of dressed carcasses were 875.4±86.6 and 555.3±88.8 g for hunted male and slaughtered female pheasants, respectively, or 65.5±1.8 and 60.9±3.7% of the weight of the whole birds. This is somewhat lower than the yields reported by [START_REF] Richter | Zusammensetzung und Qualitätsparameter von Perlhühnern, Fasanen, Tauben, Cairina und Kaninchen[END_REF], with 67.7% and 64.5 % and Golze (2010), with ca. 66-68% (cold carcass weight, sex not specified).

Edible viscera (gizzard, heart, liver), breast and thigh muscles amounted up to 75.0±9.9, 270.8±27.7, 264.0± 22.4 g, respectively, for hunted male birds, and were higher in weight than those of slaughtered females (70.4±8.5, 169.2±28.7, 166.8±23.9 g). These data agree well with those reported by [START_REF] Tucak | The influence of keeping pheasants in captivity vs. nature on the biological value of meat and its use in human nutrition[END_REF] and [START_REF] Richter | Zusammensetzung und Qualitätsparameter von Perlhühnern, Fasanen, Tauben, Cairina und Kaninchen[END_REF]. However, weight of the major muscles (breast and thigh muscles) related to the weight of the dressed carcass in hunted male vs. slaughtered female pheasants was similar (61.2±2.8 vs. 61.5±2.5%). A number of studies indicate that sex has no influence on the percentage of muscle weight to carcass weight [START_REF] Kuzniacka | Effect of age and sex of pheasants (Phasianus colchicus L.) on selected physical properties and chemical composition of meat[END_REF][START_REF] Tucak | The influence of keeping pheasants in captivity vs. nature on the biological value of meat and its use in human nutrition[END_REF][START_REF] Golze | Fasanenproduktion zur Fleischgewinnung und zum Auswildern[END_REF], whereas the origin of the animals (free range vs. pheasantry) will affect the relation breast to leg muscles [START_REF] Golze | Fasanenproduktion zur Fleischgewinnung und zum Auswildern[END_REF].

Chemical composition of muscles

In both study groups, protein concentrations in breast muscle were significantly higher than in thigh muscles (Table 1), similar to the findings reported by Tucak et al. 2008). Higher protein contents were associated with lower contents of moisture and fat. Several studies indicate that the protein content of meat from male and female pheasants will not differ [START_REF] Kuzniacka | Effect of age and sex of pheasants (Phasianus colchicus L.) on selected physical properties and chemical composition of meat[END_REF][START_REF] Tucak | The influence of keeping pheasants in captivity vs. nature on the biological value of meat and its use in human nutrition[END_REF][START_REF] Golze | Fasanenproduktion zur Fleischgewinnung und zum Auswildern[END_REF], but that gross composition of wild and farmed pheasant may differ [START_REF] Tucak | The influence of keeping pheasants in captivity vs. nature on the biological value of meat and its use in human nutrition[END_REF]. Data on fat content depend on the mode of dressing. For lean breast muscle, [START_REF] Tucak | The influence of keeping pheasants in captivity vs. nature on the biological value of meat and its use in human nutrition[END_REF] and Golze (2010) reported 0.6-1% fat.

Average ash concentration was in the range of 1.30-1.40%, and only slightly higher than that reported in literature (ca.

1.1 and 1.2%; [START_REF] Tucak | The influence of keeping pheasants in captivity vs. nature on the biological value of meat and its use in human nutrition[END_REF][START_REF] Richter | Zusammensetzung und Qualitätsparameter von Perlhühnern, Fasanen, Tauben, Cairina und Kaninchen[END_REF]. In summary, our data agree well with those reported from other geographical areas.

Physical indicators

In both study groups, pH measured 24 h p.m. was ca. 0.4 units higher in thigh than in breast muscles, which is in accordance with previous studies [START_REF] Richter | Zusammensetzung und Qualitätsparameter von Perlhühnern, Fasanen, Tauben, Cairina und Kaninchen[END_REF][START_REF] Kuzniacka | Effect of age and sex of pheasants (Phasianus colchicus L.) on selected physical properties and chemical composition of meat[END_REF][START_REF] Paulsen | Influence of storage conditions and shotshell wounding on the hygienic condition of hunted, uneviscerated pheasant (Phasianus colchicus)[END_REF]. A different pattern of myofibres in the studied muscles can account for that: breast muscles of pheasants are predominantly (>70%) composed of fast-twich, glycolytic fibres (IIB type), whereas thigh muscles have higher percentages of other, glycolyticoxidative (IIA) or oxidative fibre types [START_REF] Kiessling | Muscle structure and function in the goose, quail, pheasant, guinea hen and chicken[END_REF]). This difference is common for gallinaceous birds, and reflects the ability for rapid take-off, at the expense of short flight distances [START_REF] Pyörnilä | Fibre types in breast and leg muscles of hand-reared and wild grey partridge (Perdix Perdix)[END_REF]. Also, ante mortem stress can have an influence, especially in muscles rich in IIB fibres [START_REF] Lawrie | Lawrie's Meat Science[END_REF], but in view on the cited studies on pheasants, it is not clear how to weigh stress caused by drive-hunting vs. that due to ante mortem manipulation and slaughter.

As can be expected, drip loss was higher in the muscles with lower pH [START_REF] Hofmann | What is quality ? Definition, measurement and evaluation of meat quality[END_REF]).

An average shear force of 30 N/cm 2 in the breast muscle of pheasant (Table 2) is similar to that reported for broiler and turkey breast [START_REF] Werner | Comparative study of the quality of broiler and turkey meat[END_REF]), but due to subtle differences in methodology, this comparison must be done with caution. No reference on instrumental measurement of pheasant meat could be found, but there are reports on assessment of tenderness by tester panels [START_REF] Golze | Fasanenproduktion zur Fleischgewinnung und zum Auswildern[END_REF]. For other wildlife species, as roe deer and chamois, and using an identical methodology, similar shear force values were reported by [START_REF] Winkelmayer | Qualität des Rückenmuskels-Qualitätsparameter des Rückenmuskels von Rehen aus dem Voralpengebiet in Österreich[END_REF] and [START_REF] Hofbauer | Saisonale Unterschiede von Gemsenfleisch[END_REF].

Colour of the breast muscles (Table 2) was characterised by an average lightness (L*) of 54.2, which increased to 56.6. L* values in the range of 51-60 have also been reported by other authors for pheasant [START_REF] Golze | Fasanenproduktion zur Fleischgewinnung und zum Auswildern[END_REF]) and chicken [START_REF] Werner | Comparative study of the quality of broiler and turkey meat[END_REF]), whereas [START_REF] Kuzniacka | Effect of age and sex of pheasants (Phasianus colchicus L.) on selected physical properties and chemical composition of meat[END_REF] report for pheasant lower L* values of 43-47.

Averages for redness (a*) and yellowness (b*) in stored breast muscles were 4.0 and 9.2, respectively. Our data are quite similar to a* and b* values of 2.98 and 6.65 for the M. pectoralis superficialis of broilers [START_REF] Werner | Comparative study of the quality of broiler and turkey meat[END_REF]), which also is dominated by IIB fibres [START_REF] Barnard | Fibre type composition in chicken skeletal muscles and their changes in muscular dystrophy[END_REF]. Quite different results are reported for this muscle in 12-20week-old pheasants by [START_REF] Kuzniacka | Effect of age and sex of pheasants (Phasianus colchicus L.) on selected physical properties and chemical composition of meat[END_REF], with a* and b* values of 14-15 and 1-2.5, respectively. As the methodology of measurement is not well described, differences in sample preparation of meat (measurement on surface or of a meat cut, fresh or stored meat) could account for this. Also, the concentration of heme pigments and their oxidative status have to be considered when evaluating L*a*b* values of muscles [START_REF] Mancini | Current research in meat color[END_REF].

When comparing results obtained 24 and 96 h p.m., an increase of lightness and yellowness was observed, similar to results reported for broiler and turkey by [START_REF] Werner | Comparative study of the quality of broiler and turkey meat[END_REF].

Breast muscles from pheasants clearly differ in their colour from muscles of wild ruminants, which are darker and have a higher red component, e.g. roe deer and chamois are characterised by L* values in the range of 35-45, a* values of 15-16 and b* values of 11 [START_REF] Winkelmayer | Qualität des Rückenmuskels-Qualitätsparameter des Rückenmuskels von Rehen aus dem Voralpengebiet in Österreich[END_REF][START_REF] Hofbauer | Saisonale Unterschiede von Gemsenfleisch[END_REF]). Again, this can be explained by the dominance of "white" IIB fibres, which is associated with higher L* and lower a* values [START_REF] Lawrie | Lawrie's Meat Science[END_REF].

Table 1

 1 Physical and chemical characteristics of breast and thigh muscles from pheasants (mean ± SD) Data refer to a mix of all thigh muscles for chemical composition and drip loss, and to the M. iliotibialis lateralis for pH

		Hunted male pheasants (n=29)	Slaughtered female pheasants (n=32)
		Breast	Thigh c	Breast	Thigh c
	Moisture d	71.83±1.11a	75.28±1.03a	71.85±1.81	74.20±0.70
	Crude protein d	25.66±1.27a	22.60±2.03a	25.03±1.08b	23.56±0.42b
	Fat d	0.35±0.25a	1.16±0.55a	0.52±0.20	0.84±1.05
	Ash d	1.39±0.15	1.32±0.08	1.30±0.05b	1.39±0.03b
	pH 24	5.66±0.08a	6.03±0.20a	5.55±0.16b	5.93±0.25b
	Drip loss e	3.03±0.57a	1.49±0.36a	2.19±1.37b	1.00±0.56b
	a,b within row and within group (hunted male or slaughtered female), a common letter indicates a significant (p<0.05) difference between breast
	and thigh
	d g/100 g
	e Drip loss within 7 days at 0-2°C, g/100 g

c

Table 2

 2 Selected physical characteristics of breast muscles from hunted male pheasants (n=14), as assessed 24 and 96 h post mortem (p.

	m.)

a Indicates that results for 24 and 96 h p.m. differ significantly, p<0.05

Eur J Wildl Res (2010) 56:809-813

Acknowledgement This work was supported by the "Verein Grünes Kreuz", Austria.


Michel ; Dubois

Sah Cer

Luc ; Demoulin

Caroline ; Charlier

G Singh

S B Godefroy

K Campbell

C T Elliott

Ph Delahaut

Development of ELISAs for detecting domoic acid, okadaic acid, and saxitoxin and their applicability for the detection of marine toxins in samples collected in Belgium

Keywords: Immunoassays, In-house validation, Screening -ELISA Additives/Contaminants: Phycotoxins enzyme immunoassay, domoic acid, okadaic acid, saxitoxin, validation

Development of ELISAs for detecting domoic acid, okadaic acid, and saxitoxin and their applicability for the detection of marine toxins in samples collected in Belgium

Introduction

Over the past two decades, the frequency, intensity, and geographic distribution of harmful algal blooms has increased, along with the number of toxic compounds found in the marine food chain. Different explanations for this trend have been given, such as increased scientific awareness of toxic algal species, increased utilisation of coastal waters for aquaculture, transfer of shellfish stocks from one area to another, cultural eutrophication due to domestic, industrial, and agricultural wastes, increased mobility of humic substances and trace metals from soil due to deforestation and/or acid rain, and unusual climate conditions [START_REF] Hallegraeff | Manual on harmful Marine microalgae[END_REF].

Phycotoxins produced by algae pose a significant potential threat to human health. A variety of symptoms are reported to arise after consumption of contaminated shellfish, including stomach cramps, abdominal pain, diarrhoea, headaches, memory loss, paralysis, and in some extreme cases even death. In addition to the human health risk, the accumulation of these toxins in shellfish can also have severe consequences for the shellfish industry.

Monitoring the various toxin groups is therefore essential. Several families are responsible for poisonings, the symptoms and severity of which depend on the toxin family. Paralytic shellfish poisoning (PSP) is associated with a wide number of derivatives of saxitoxin (SAX). Diarrhetic shellfish poisoning (DSP) is caused by a group of polyether toxins including okadaic acid (OA), the dinophysistoxins (DTX-1, -2, -3...), pectenotoxin, and yessotoxin. Neurotoxic Because of the threat to human health, many countries (including European Union Member States) have enacted regulations on the maximum permitted levels of various phycotoxins. In the European Union, Regulation (EC) No 853/2004 lays down specific hygiene rules for food of animal origin. For live bivalve molluscs, this regulation stipulates that, in addition to meeting microbiological criteria and having organoleptic characteristics associated with freshness and viability, live bivalve molluscs must not contain marine biotoxins in total quantity measured in the whole body (or any edible part separately) exceeding the following limits: 800 ug kg -1 for the PSP toxins, 20 mg kg -1 domoic acid equivalents for the ASP toxins, 160 ug kg -1 okadaic acid equivalents for okadaic acid, dinophysistoxins, and pectenotoxins together. For yessotoxins and azaspiracids, the maximum permissible level is 1 mg kg -1 yessotoxin equivalents and 160 ug kg -1 azaspiracid equivalents.

The method officially recognised by regulatory authorities for the detection of the aforementioned toxins is the mouse bioassay. Alternative methods can be used, provided they are no less effective and ensure the same level of public health protection. A series of instrumental methods [START_REF] Hess | LC-UV and LC-MS methods for the determination of domoic acid[END_REF] [START_REF] Kawatsu | Determination of domoic acid in Japanese mussels by enzyme immunoassay[END_REF][START_REF] Kawatsu | Development and application of an enzyme immunoassay based on a monoclonal antibody against gonyautoxin components of paralytic shellfish poisoning toxins[END_REF][START_REF] Morton | Determination of okadaic acid content of dinoflagellate cells: a comparison of the HPLC-fluorescent method and two monoclonal antibody ELISA test kits[END_REF][START_REF] Osada | Determination of domoic acid by two different versions of a competitive enzyme-linked immunosorbent assay (ELISA)[END_REF][START_REF] Smith | A competitive enzyme-linked immunoassay for domoic acid determination in human body fluids[END_REF], 1995;[START_REF] Vale | Comparaison between HPLC and a commercial immunoassay kit for detection of okadaic acid and esters in Portuguese bivalves[END_REF][START_REF] Renz | Detection of saxitoxin by an indirect competitive ELISA[END_REF][START_REF] Usleber | Direct enzyme immunoassay in microtitration plate and test strip format for the detection of saxitoxin in shellfish[END_REF][START_REF] Traynor | Immunobiosensor detection of domoic acid as a screening test in bivalve mollusks: Comparison with LC based analysis[END_REF][START_REF] Llamas | Development of a novel immunobiosensor method for the rapid detection of okadaic acid contamination in shellfish extracts[END_REF]Fonfria et al 2007), and functional assays have been developed for the determination of phycotoxins in shellfish. This paper describes the development of rapid screening immunoassays for detecting domoic acid, okadaic acid, and saxitoxin. Polyclonal antibodies were raised against each of these compounds and used to develop the corresponding ELISAs, which were then used over a period exceeding 4 months for the analysis of 110 real samples collected in Belgium. temperature for 72 h, then at 4°C for another 12 h. The reaction mixture was dialysed against 1 mmol L -1 acetic acid solution for 6 h, then against deionised water for 18 h. After dialysis the SAX-BSA was precipitated overnight at 4°C with aluminium potassium sulfate (10%, 800 uL) and NaOH (1 mol L -1 , 500 uL). The resulting precipitate was washed six times with cold PBS, pH 7.2.

Materials and methods

Reagents

Preparation of enzyme-labelled drugs.

The two phycotoxins (domoic acid and okadaic acid) were conjugated to horseradish peroxidase (HRP) enzyme label by the same method as for the immunogens.

Saxitoxin was coupled to HRP by the periodate reaction [START_REF] Kawatsu | Development and application of an enzyme immunoassay based on a monoclonal antibody against gonyautoxin components of paralytic shellfish poisoning toxins[END_REF]; HRP (4 mg) was added to a sodium periodate solution (0.1 mol L -1 , 0.2 mL) and the solution was gently mixed for 20 min at room temperature. The mixture was dialysed overnight at 4°C against sodium acetate buffer, pH 4.4, and the pH of the final solution was adjusted with carbonate buffer (0.2 mol L -1 , pH 9.6, 20 µL). The activated enzyme was added to saxitoxin as an acetate salt (80 ug) in acetic acid (0.1 mol L -1 , 250 uL) and the pH was adjusted to 7.5 with sodium carbonate buffer (0.5 mol L -1 , pH 9.6).

After 45 min at room temperature, NaBH 4 (4 mg mL -1 , 0.1 mL) was added. The mixture was incubated for 15 min at 4°C and finally dialysed against PBS (0.15 mol L -1 ). The saxitoxin conjugate was stored at -20°C. Immunogen emulsions were injected subcutaneously into four sites on the animal. Rabbits were immunised every 28 days with 200 ug immunogen, and blood samples were taken from the marginal vein of the ear 10 days after each immunisation (from the third immunisation onward). Test bleeds were monitored every 28 days for the presence of antibodies. When titres stabilised, the rabbits were exsanguinated and the resulting antisera stored at -20°C.

Immunisation of rabbits.

Assessment of antibodies and enzyme conjugates.

All antibodies and enzyme conjugates were initially assessed by means of ELISA chequerboards to determine optimum working dilutions.

Specificity of antibodies.

Specificity means the ability of a method to distinguish between the analyte being measured and other substances. Sixty blank shellfish samples (20 mussel, 20 oyster, and 20 scallop samples) were analysed in blind duplicate.

The specificities of the three antibodies were assessed by determining the extent to which each one cross-reacted, in the ELISA, with analytes having a chemical structure similar to that used to raise the antibody. The degree of antibody specificity within the amnesic, diarrhetic, and paralytic poison families was estimated by means of the competitive ELISAs described in section 2.7. Crossreactivity was calculated by means of the formula: ID 50 of compound used to raise antibody X 100 Ninety-six-well Nunc Maxisorp microtitre plates (Roskilde, Denmark) were coated beforehand with purified sheep antibody directed against rabbit IgG (CER, Marloie, Belgium). The wells were filled with 50 µL working standard prepared in assay buffer (the standard was chosen according to the residue to be detected), 50 µL diluted sample, or 50 µL assay buffer (Bo). The ELISA buffer composition was as follows: 0.15 M NaCl, 0.056 M Na 2 HPO 4 .2 H 2 O, 0.009 M NaH 2 PO 4 .2 H 2 O, 0.2% gelatin, 0.05% Tween 20, 0.01% 8-anilino-1naphthalenesulfonic acid ammonium salt, and 0.0028 M ascorbic acid. Two additional wells were filled with 150 uL assay buffer and were dedicated to nonspecific binding (NSB). The peroxidase conjugate and the antiserum (diluted to its optimal concentration in assay buffer), were then dispensed into the wells (the conjugate and antiserum were chosen according to the residue to be detected). The diluted antibody was added to each well except for the two wells corresponding to NSB. The microtitre plate was incubated at 4°C for 2 hours (OA and DA ELISAs) or overnight (SAX ELISA).

The wells were then emptied and washed three times with washing buffer (0.15 M NaCl, 0.05% Tween 20). A 150-µL aliquot of chromogen mixture [(3, 3', 5, 5'-tetramethylbenzidine) (TMB) and enzyme substrate H 2 0 2 ; v/v)] was added to each well. The enzyme reaction was stopped by addition of 6 M sulfuric acid Tissues were homogenised in a blender (3 min), which was cleaned and rinsed with methanol and distilled water after each sample. One gram of tissue homogenate was vortexed for 1 min in the presence of 5 ml ethanol-water (9/1).

After centrifugation (10 min at 3000 g), the supernatant was decanted and the tissues extracted again with a new aliquot (3 mL) of the same ethanol-water mixture. The final volume of the extract was made up to 10.0 mL and filtered through a 25-mm filter with a 0.45-um-pore-size nylon filter membrane (Whatman). The extract was stored at 4°C and diluted in ELISA buffer prior to ELISA-based detection and quantification.

Determination of detection capability.

The detection capability of an analytical procedure is defined in Decision 2002/657/EC as the smallest amount of a substance that can be detected, identified, and/or quantified in a sample with an error probability of β. Twenty blanks of known non-contaminated mussels and twenty blank mussel samples fortified at 150 ug kg -1 with domoic acid, 50 ug kg -1 with okadaic acid, or 2 ug kg -1 with saxitoxin, were prepared "blind to the analyst" and analysed in duplicate. The ELISA plates were then processed as previously described.

Results and Discussion

ELISA assays for the various toxin classes were developed. Emphasis was placed on keeping the methods and all reagents as similar as possible in the three assay procedures.

The performance characteristics sensitivity, specificity, and detection capability (CCβ) of the three ELISAs in screening for domoic acid, okadaic acid, and saxitoxin in shellfish were determined in order to validate the ELISAs and to

Antibody sensitivity.

All three of the shellfish toxin immunogens resulted in the production of antibodies. Sera exhibiting antibody titres were assessed for their sensitivity and specificity. Each rabbit serum was tested with the corresponding peroxidase conjugate.

The assay sensitivity was calculated as 50% inhibition of deviation (ID 50), the concentration of shellfish toxin necessary to cause 50% antibody binding. The competitive ELISA format described in previously section was used to determine the assay sensitivity. A set of shellfish toxin standards (in the range of 0.005-10 ng mL -1 ) was used. The average optical density of zero standard wells (Bo) represents 100% activity and the remaining standards were normalised with respect to the optical density of Bo.

Several parameters were tested to determine the optimal conditions for each ELISA, notably incubation temperature and incubation time. These parameters were evaluated by comparing the antibody sensitivity (ID 50) while taking into account the practical requirements of potential users. First, the results obtained at two incubation temperatures, 4 and 37°C were compared. The sensitivity was substantially better (no data presented) when the incubation temperature was

Antibody specificity.

The cross-reactivities in assay buffer of the anti-OA, anti-DA, and anti-SAX antibodies are given in Table 1. The ELISAs were performed with increasing amounts of competitor. When no ID 50 could be determined within the tested concentration range because the binding calculated in percent was superior to 50% for the highest concentration, the ID used in the cross-reactivity formula was the ID observed at that concentration and the cross-reactivity was recorded as inferior to the result of the calculation.

The anti-DA antibody showed 100% cross-reactivity with domoic acid and no significant cross-reactivity with any other shellfish toxin. As the total domoic acid concentration in the certified reference material (CRM) MUS-1 was 2), it would appear that the antiserum cross-reacts well with all major isomers of domoic acid and that this ELISA provides a measurement of total domoic acid (the summed concentrations of the various domoic acid isomers).

The anti-OA antibody was tested for cross-reactivity towards two OA analogues of the same toxin family: the dinophysistoxins DTX-1 and DTX-2. It showed 78% cross-reactivity towards DTX-1 and 2.6% cross-reactivity towards DTX-2, as compared to OA. An ELISA developed for DSP detection should detect okadaic acid and the dinophysistoxins equally, but no such ELISA is available.

Detection of DTX-3 requires an extra base hydrolysis step in the extract

preparation procedure to remove the acyl group and convert the toxin to DTX-1.

Commercially available ELISAs for this group include the DSP-Check ® ELISA test kit from UBE Industries, Tokyo, Japan and the Rougier Bio-Tech ® ELISA test kit. The former is used worldwide to screen for OA and DTX-1 at a claimed detection limit of 20 ng g -1 . The latter exploits a monoclonal anti-OA antibody and an anti-idiotypic antibody that competes with OA for binding sites on the anti-OA antibody. This antibody exhibits a much higher sensitivity (10-20 fold) for OA than for either DTX-1 or DTX-2, whereas DTX-3 does not cross-react at all.

The anti-SAX antiserum showed 100% cross-reactivity with saxitoxin, 26.2% with GTX 5 (gonyautoxin-5), 19.2% with decarbamoyl saxitoxin, and 5.6% with GTX 2/3 (gonyautoxin-2 and gonyautoxin-3). It showed very low cross- reactivity with neo-saxitoxin (1.4%). It is a well-recognised problem that all PSP antibodies fail to recognise all members of the toxin family and thereby underestimate the total toxin content. This notably concerns the PSPs, as anti-SAX antibodies show poor cross-reactivity with neo-saxitoxin and vice versa [START_REF] Usleber | Effect of heterologous paralytic shellfish poisoning toxin-enzyme conjugates on the Page 26 of 36 http://mc.manuscriptcentral.com/tfac Email: fac@tandf.co.uk Food Additives and Contaminants cross-reactivity of a saxitoxin enzyme immunoassay[END_REF][START_REF] Huang | Direct Competitive Enzyme-Linked Immunosorbent Assay for Saxitoxin and Neosaxitoxin[END_REF]. The commercial ELISA test kits have only been developed for SAX. However, they are not totally specific for SAX and some reactions are induced to decarbamoyl-SAX (dcSAX) and neoSAX. [START_REF] Usleber | Comparison of enzyme immunoassay and mouse bioassay for determining paralytic shellfish poisoning toxins in shellfish[END_REF] have concluded that ELISA results correlated well with mouse bioassay results when analysing scallops. However, [START_REF] Kasuga | Evaluation of Enzyme-Linked Immunosorbent Assay (ELISA) Kit for Paralytic Shellfish Poisoning Toxins[END_REF] have concluded that the mouse assay can not be replaced with ELISA for the purpose of screening inshore shellfish samples, as unpredictable cross-reactions have occurred, as well as underestimations of toxicity of some naturally contaminated shellfish samples, harvested in the sea near Japan. The predominance of GTX's and the lack of SAX in shellfish samples resulted in the underestimation of toxin content by the ELISA kit in comparison with the PSP mouse bioassay. The major advantages of ELISA are simplicity, and rapidity of the test procedure, and a higher sensitivity compared with other methods. However, quantitative agreement between ELISA and mouse bioassay depends on the antibody specificity and on the toxin profile of the shellfish. Thus both over-and under-estimation of total toxicity may occur.

In order to avoid the poor cross-reactivity of saxitoxin antibodies with neosaxitoxin and vice versa, one could use two or more ELISAs based on antibody Another approach [START_REF] Garthwaite | Integrated enzyme-linked immunosorbent assay screening system for amnesic, neurotoxic, diarrhetic, and paralytic shellfish poisoning toxins found in New Zealand[END_REF] is to develop a competitive enzyme immunoassay based on a gonyautoxin 2/3 (GNTX2/3)-specific monoclonal antibody and a saxitoxinhorse radish peroxidase conjugate. GNTX2/3, dc-GNTX2/3, C1/2, GNTX1/4, STX and neoSTX were detectable at concentrations lower than the regulatory limit of 80 ug/100 g of shellfish tissue.

As far as screening purposes are concerned, ELISAs offer major advantages compared with the mouse bioassay, which is criticized in Europe because of animal welfare. Considering the presence of cross-reactions with lower binding specificity and the potential lack of response to other toxins than saxitoxin within the PSP group, the practical application of these assays will probably remain limited, unless acceptable performance characteristics can be demonstrated in formal collaborative studies according to AOAC International or ISO accepted procedures. The smallest content of the substance that may be detected by each ELISA was assessed by determining the detection capability (CCβ) of the assay for the shellfish toxin to be detected. To evaluate the performance of our ELISAs, three matrices were chosen: mussel, oyster, and scallop. For each matrix, known negative samples were unfortified (blank) and fortified with phycotoxins at various concentrations, extracted, and analysed by means of the optimised ELISAs. The detection capability was determined for each toxin in each matrix.

Detection capability.

In the mussel, oyster and scallop matrix, the respective CCβ values for DA, OA, were 150 and 50 ug kg -1 . For the saxitoxin, the respective CCβ values for mussel, oyster and scallop matrix were 1, 2 and 5 ug kg -1 .

The results presented in Figures 2,3, 4 demonstrate that the blank and spiked oyster samples met the CCβ criterion: all blank oyster samples gave a response below the CCβ, while at least 19 out of 20 spiked samples gave a signal above the CCβ and were thus classified as suspect (β=5%).

In practice, the means of the absorbance values obtained for each sample were divided by the absorbance of the zero standard (Bo) and multiplied by 100. For each group of 20 samples, the mean ratio B/Bo (%) +/-2 standard deviations was calculated (Figures 2,3,4). These data show that there was no B/Bo (%) range overlap between the blank and spiked samples.

For example, in the case of domoic acid in oyster (Figure 2), a 91% threshold is the lower limit of the blank range and 89% is the upper limit of the spiked sample range (cut-off). In the routine application of such a qualitative method a sample was considered suspect if the binding percentage was below or equal the cut-off. Thus the assays developed could be used in highly reliable manner to differentiate between samples free from domoic acid or okadaic acid or saxitoxin from those with these phycotoxins above the CCβ values.

Extraction, recovery.

The present method has the advantage that a single solvent extraction system adequately extracts the ASP, DSP, and PSP toxins. Ethanol is a preferred solvent because of its low toxicity and volatility. In most assays it causes lesser matrix effects than methanol. The extraction efficiency was evaluated by comparing the calibration curves constructed before and after extraction of blank matrices spiked with a known amount of phycotoxin. An extraction efficiency above 90% was obtained for domoic acid, okadaic acid, and saxitoxin, indicating that the proposed extraction is suitable for use in screening for both aqueous and lipophilic toxins. Matrix effects of alcohol extracts in the ELISA were eliminated by dilution of the shellfish extract with PBS until the standard curves obtained for toxins dissolved in ELISA-buffer-diluted shellfish extract coincided with those obtained for toxins dissolved in ELISA buffer.

Toxin recovery was determined by spiking shellfish samples with the target toxin and also by analysing the following CRMs: CRM-ASP-Mus-C and CRM- mg kg -1 domoic acid, as well as several related compounds (isomers). The sum of the concentration of domoic acid and C5'-epidomoic acid is certified at 44±3 mg kg -1 . In CRM-DSP-Mus-b, a CRM for okadaic acid and DTX-1, the toxin levels are 10.1 mg kg -1 for okadaic acid and 1.3 mg kg -1 for DTX-1.

The results for recovery of DA, OA and saxitoxin added to the extracts of mussels, oysters and scallop are shown in table 2 . DA was detected in CRM-ASP-Mus-C with a mean recovery of 105% and coefficient of variation of 9%.

OA was detected in CRM-DSP-Mus-b with a mean recovery of 109% and coefficient of variation of 11%. Saxitoxin added to the mussel was detected with a mean recovery of 112% and coefficient of variation of 8%. over a period exceeding four months. Over this period, the methods showed good robustness. QC samples were analysed to check the performance of the assay. The results for negative QC samples and for positive QC samples spiked at the CCβ concentration were very satisfactory. Reagent blanks were always found compliant and the spiked reagent blanks were always determined as suspect. This confirms that the detection capability set during the initial validation was an acceptable threshold value. Table 3 gives an overview of the contaminated samples. It was concluded that saxitoxin residues were present in one sample (mussel) at the concentration of 2 ug kg -1 and okadaic acid in one sample (mussel) at the CCβ concentration. Domoic acid was detected and quantified in 9 samples (concentration above the CCβ). The incidence of domoic-acid-contaminated samples was 0% for mussel, 0% for oyster, and 15% for scallop. The domoic acid content was above the 20 mg kg -1 limit (the EU-adopted maximum level) in 5 samples (8%).

Analysis of real samples from the

Since the present validated ELISAs for domoic acid, okadaic acid, and saxitoxin in shellfish are screening methods, suspect samples need to be confirmed by mass spectrometry. The contaminated samples were analysed by LC-MS/MS to confirm their contaminated status and to avoid overestimation due to interference. Analysis was done on a Quattro Premier (Waters) mass spectrometer in the ESI mode, using traces 312>266 and 312>161 for quantification and confirmation of domoic acid [START_REF] Holland | Amnesic Shellfish Poisoning Toxins in Shellfish: estimation of uncertainty of measurement for liquid chromatography/tandem mass spectrometry method[END_REF] More rapid analyses of commercial shellfish samples would reduce the time the product must be held in storage and allow the prompt issue of the official documentation on shellfish toxin content required by some export markets. The reported methods have been validated according to the criteria laid down by the EU in Commission Decision 2002/657/EC and were used for more than 4 months for the analysis of the 110 real samples collected in Belgium. The domoic acid content was above the 20 mg kg -1 limit (EU-adopted maximum level) in 5 scallop samples (8%).

  NSP) is the result of exposure to a group of polyethers called brevetoxins, and amnesic shellfish poisoning (ASP) is caused by domoic acid (DA).

  , chemicals, and apparatus. Domoic acid, okadaic acid, saxitoxin, and other phycotoxin standards and certified reference materials (CRM-ASP-Mus-C, CRM-DSP-Mus-b) were purchased from the National Research Council of Canada's Institute for Marine Biosciences (Halifax, Canada). Human serum albumin (HSA), bovine serum albumin (BSA), 1-ethyl-3-(3dimethylaminopropyl) carbodiimide, N-hydroxysuccinimide, and complete Freund's adjuvant were obtained from Sigma (St Louis, MO, USA). New Zealand female rabbits were used for immunisation. Nunc Maxisorp surface 96well polystyrene microtitre plates were purchased from Nunc (Roskilde, All solvents were of LC or analytical reagent grade. Mussel, oyster, and scallop samples were obtained from local supermarkets. A Multiskan EX Thermo Fisher Scientific microplate reader (Zellik, Belgium) was used to read the absorbance in the ELISA-plate wells at 450 nm. Preparation of immunogens. Domoic acid and okadaic acid were conjugated with HSA by means of carbodiimide and N-hydroxysuccinimide. To 50 µL dimethyl sulfoxide (DMSO) containing 1 mg domoic acid or okadaic acid, 20 µL 1-ethyl-3-(3dimethylaminopropyl) carbodiimide hydrochloride solution (25 mg mL -1 in DMSO) and 10 uL N-hydroxysuccinimide solution (30 mg mL -1 in DMSO) were added.After stirring for 1.5 h at 25°C, 0.25 mL of HSA solution (20 mg mL -1 in 0.085 M borate buffer containing 0.06 M NaCl, pH 8.2) was added to the mixture. After incubation for 1.5 h at 25°C, the reaction mixture was loaded onto a PD-10 column (Pharmacia Biotech, Uppsala, Sweden) equilibrated with phosphate buffered saline (PBS). After elution, the protein-containing portion (3.5 mL) was collected. The toxin-protein conjugates were then stored at -30°C until used.The saxitoxin immunogen was prepared according to Renz V and Terplan G (1988). Typically, BSA (2 mg) in sodium acetate buffer (0.1 mol L -1 , pH 4.4, 0.5 mL) was mixed with saxitoxin diacetate (150 ug) in the presence of formaldehyde (37% W/W, 40 uL). The reaction was left to proceed at room

10 (

 10 50 µL). The absorbance of the solution at 450 nm was read within 30 min. The average optical density (OD 450 ) of the Bo wells, containing all constituents except the phycotoxin competitor, was taken as 100% activity. Percent inhibition was calculated as: Sample extraction.

(

  OD 450 test -OD 450 NSB ) (OD 450 Bo -OD 450 NSB )

  The detection capability CCβ (cut-off) was calculated as the concentration at which less than 5% of false-compliant results remained. If 19 samples among 20 fortified samples were declared positive, then CCβ = level of fortification.[START_REF] Morton | Determination of okadaic acid content of dinoflagellate cells: a comparison of the HPLC-fluorescent method and two monoclonal antibody ELISA test kits[END_REF] or less of the fortified samples were declared positive, then CCβ > level of fortification. If all the fortified samples were declared positive then CCβ < level of fortification. The method was repeated by gradually increasing the level of fortification until CCβ was determined. This validation procedure was repeated for the other two matrices of interest (oyster and scallop).

  of using them to screen shellfish for low levels of phycotoxins. The three methods were validated according to the criteria of European Decision 2002/657/EC (Commission 2002) for a screening method.

  , the results obtained at two incubation times, overnight or 2 h, were compared. The best incubation times were 2 hours for domoic acid and okadaic acid and overnight for saxitoxin. The best incubation temperature was 4°C for the three ELISAs. Under these conditions, the respective mid-point of the DA, OA, and SAX calibration curve (ID 50) obtained were 1.8 , 1.3 and 0.04 ng ml -1 .Representative standard curves for domoic acid, okadaic acid and saxitoxin, obtained with the optimised ELISA are shown respectively in Figures 1. It

  appears that inhibition of binding of domoic acid was linear in the range 0.1 -10 ng mL -1 . It was also reproducible in this range, its standard deviation being no more than ± 5%.

  members of the same toxin family.[START_REF] Chu | Screening of paralytic shellfish poisoning toxins in naturally occurring samples with three different direct competitive enzyme-linked immunosorbent assays[END_REF] has demonstrated the feasibility of this approach, finding a very high correlation between the results mouse PSP bioassays and those obtained by summing the results of SAX and neo-SAX ELISAs. The ease and speed of ELISAs makes it relatively simple to conduct several ELISAs in parallel. The use of two ELISAs to cover a particular toxin group requires only a little additional work.

  -b. CRM-ASP-Mus-C is a mussel tissue homogenate containing 41±2

  Belgian market. One hundred and ten samples (42 mussels, 61 scallops and 7 oysters) were collected in Belgium between October 2007 and January 2008. One hundred and three of these samples were collected by the Belgian Veterinary Inspectors for Public Health and seven by scientific personnel from different shops located in the south of Belgium (Walloon region). Samples were frozen and kept at approximately -20°C until analysis. The 110 real samples were analysed with the three validated ELISAs developed for screening shellfish for domoic acid, okadaic acid, and saxitoxin. The analyses were performed on 11 different days

Food

  ) and traces 804>255 and 804>113 for quantification and confirmation of okadaic acid (Fux Page 20 of 36 http://mc.manuscriptcentral.com/tfac Email: fac@tandf.co.uk 2007). These analyses confirmed the contaminant identities and concentrations determined by the ELISAs.In addition, the ELISAs applied to mussel, oyster, and scallop samples successfully detected domoic acid and okadaic acid when they were shown to be present by LC-MS/MS. Hence, the occurrence of false negatives in the ELISAs for domoic acid and okadaic acid appears unlikely.ConclusionsThis work has led to the development of three quick, sensitive ELISAs for the detection of domoic acid, okadaic acid, and saxitoxin respectively in shellfish.The proposed technique requires little expensive apparatus or equipment. A major advantage of ELISAs over other methods is the ability to process relatively large numbers of samples over a short lapse of time. A single technician can prepare and extract up to 20 samples in approximately 4 hours.

Figure 1 :

 1 Figure 1: ELISA standard curves for domoic acid (DA), okadaic acid (OA) and

Figure 2 :

 2 Figure 2: Detection capability (CCβ): graphic representation of 20 blanks

Figure 3 :

 3 Figure 3: Detection capability (CCβ): graphic representation of 20 blanks

Figure 4 :Food

 4 Figure 4: Detection capability (CCβ): graphic representation of 20 blanks

  Figure 2:

Figure 4 :

 4 Figure 4:

Table 1 :

 1 Cross-reactivities of polyclonal antibodies raised against domoic acid (DA), okadaic acid (OA), and saxitoxin (SAX) towards various marine toxins.

	Cross-reactivity (%)

Table 2 :

 2 Recovery or trueness, coefficient of variation (CV) obtained in analyses of domoic acid (DA), okadaic acid (OA), and saxitoxin (SAX) in mussel, oyster, and scallop.

	Spiked sample	Mussel	Oyster	Scallop
	or CRM	Recovery or Trueness	Recovery or Trueness	Recovery or Trueness
		(CV)	(CV)	(CV)
	DA spiked sample	95%	108%	96%
	(5 mg kg -1 )	(11)	(13)	(10)
	OA spiked sample	86%	85%	101%
	(0.16 mg kg -1 )	(7)	(6)	(9)
	SAX spiked sample	112%	116%	106%
	(0.005 mg kg -1 )	(8)	(10)	(11)
	CRM-ASP-Mus-C	105%	-	-
	(44 mg kg -1 )	(9)
	CRM-DSP-Mus-b	109%	-	-
	(11.4 mg kg -1 )	(11)

Table 3 :

 3 Domoic acid (DA), okadaic acid (OA), and saxitoxin (SAX) contamination in real samples of mussel, oyster, and scallop collected in Belgium.

	Domoic acid	Okadaic acid	Saxitoxin
	ELISA	ELISA	ELISA

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Acknowledgements

This work was funded by the SPF-Santé Publique, project Alteval 1 (RT 06/5).

We also thank the FAVV (Dr. Van Den Brande) for the collection of field samples.