M I Isaev
email: isaev.m.i@gmail.com

Exponential instability in the inverse scattering problem on the energy interval

 (up to the value of the exponent).

Introdution

We consider the Schrödinger equation

-∆ψ + v(x)ψ = Eψ, x ∈ R 3 , (1.1)
where v is real-valued, v ∈ L ∞ (R 3 ), v(x) = O(|x| -3-ǫ ), |x| → ∞, for some ǫ > 0.

(1.2)

Under conditions (1.2), for any k ∈ R 3 \ 0 equation (1.1) with E = k 2 has a unique continuous solution ψ + (x, k) with asymptotics of the form

ψ + (x, k) = e ikx -2π 2 e i|k||x| |x| f k |k| , x |x| , |k| + o 1 |x| as |x| → ∞ uniformly in x |x| , (1.3)
where f (k/|k|, ω, |k|) with fixed k is a continuous function of ω ∈ S 2 .

The function f (θ, ω, s) arising in (1.3) is refered to as the scattering amplitude for the potential v for equation (1.1). (For more information on direct scattering for equation (1.1), under condition (1.2), see, for example, [START_REF] Faddeev | The inverse problem in the quantum theory of scattering[END_REF] and [START_REF] Newton | Inverse Schrodinger scattering in three dimensions[END_REF].)

It is well known that for equation (1.1), under conditions (1.2), the scattering amplitude f in its high-energy limit uniquely determines v on R 3 , where

v(p) = (2π) -3 R 3 e ipx v(x)dx, p ∈ R 3 , (1.4)
via the Born formula. As a mathematical theorem this result goes back to [START_REF] Faddeev | Uniqueness of the solution of the inverse scattering problem[END_REF] (see, for example, Section 2.1 of [START_REF] Newton | Inverse Schrodinger scattering in three dimensions[END_REF] and Theorem 1.1 of [START_REF] Novikov | On determination of the Fourier transform of a potential from the scattering amplitude[END_REF] for details). We consider the following inverse problem for equation (1.1).

Problem 1.1 Given f on the energy interval I, find v.

In [START_REF] Henkin | The ∂-equation in the multidimensional inverse scattering problem[END_REF] it was shown that for equation (1.1), under the conditions (1.2), for any E > 0 and δ > 0 the scattering amplitude f (θ, ω, s) on {(θ, ω, s)

∈ S 2 × S 2 × R + , E ≤ s 2 ≤ E + δ} uniquely determines v(p) on {p ∈ R 3 | |p| ≤ 2 √ E}.
This determination is based on solving linear integral equations and on an analytic continuation. This result of [START_REF] Henkin | The ∂-equation in the multidimensional inverse scattering problem[END_REF] was improved in [START_REF] Novikov | On determination of the Fourier transform of a potential from the scattering amplitude[END_REF]. On the other hand, if v satisfies (1.2) and, in addition, is compactly supported or exponentially decaying at infinity, then v(p)

on {p ∈ R 3 | |p| ≤ 2 √ E} uniquely determines v(p) on {p ∈ R 3 | |p| > 2 √
E} by an analytic continuation and, therefore, uniquely determines v on R 3 .

In the case of fixed energy and potential v, satisfying (1.2) and, in addition, being compactly supported or exponentially decaying at infinity, global uniqueness theorems and precise reconstructions were given for the first time in [START_REF] Novikov | Multidimensional inverse spectral problem for the equation -∆ψ + (v(x) -Eu(x))ψ = 0 Funkt[END_REF], [START_REF] Novikov | The inverse scattering problem at fixed energy for the three-dimensional Schrodinger equation with an exponentially decreasing potential[END_REF].

An approximate but numerically efficient method for finding potential v from the scattering amplitude f in the case of fixed energy was devoloped in [START_REF] Novikov | The ∂-approach to approximate inverse scattering at fixed energy in three dimensions[END_REF]. Related numerical implementation was given in [START_REF] Alexeenko | Solution of three-dimensional acoustical inverse scattering problem,II: modified Novikov algorithm[END_REF].

Global stability estimates for Problem 1.1 were given by Stefanov in [START_REF] Stefanov | Stability of the inverse problem in potential scattering at fixed energy[END_REF] (at fixed energy for compactly supported potentials), see Theorem 2.1 in Section 2 of the present paper. In [START_REF] Stefanov | Stability of the inverse problem in potential scattering at fixed energy[END_REF], using a special norm for the scattering amplitude f , it was shown that the stability estimates for Problem 1.1 follow from the Alessandrini stability estimates of [START_REF] Alessandrini | Stable determination of conductivity by boundary measurements[END_REF] for the Gel'fand-Calderon inverse problem of finding potential v in bounded domain from the Direchlet-to-Neumann map. The Alessandrini stability estimates were recently improved by Novikov in [START_REF] Novikov | New global stability estimates for the Gel'fand-Calderon inverse problem[END_REF].

In the case of fixed energy, the Mandache results of [START_REF] Mandache | Exponential instability in an inverse problem for the Schrödinger equation Inverse Problems[END_REF] show that logarithmic stability estimates of Alessandrini of [START_REF] Alessandrini | Stable determination of conductivity by boundary measurements[END_REF] and especially of Novikov of [START_REF] Novikov | New global stability estimates for the Gel'fand-Calderon inverse problem[END_REF] are optimal (up to the value of the exponent). In [START_REF] Isaev | Exponential instability in the Gel'fand inverse problem on the energy intervals[END_REF] studies of Mandache were extended to the case of Direchletto-Neumann map given on the energy intervals. Note also that Mandache-type instability estimates for the elliptic inverse problem concerning the determination of inclusions in a conductor by different kinds of boundary measurements and the inverse obstacle acoustic scattering problems were given in [START_REF] Cristo | Examples of exponential instability for inverse inclusion and scattering problems Inverse Problems[END_REF].

In the present work we apply to Problem 1.1 the approach of [START_REF] Mandache | Exponential instability in an inverse problem for the Schrödinger equation Inverse Problems[END_REF], [START_REF] Isaev | Exponential instability in the Gel'fand inverse problem on the energy intervals[END_REF] and show that the Stefanov logarithmic stability estimates of [START_REF] Stefanov | Stability of the inverse problem in potential scattering at fixed energy[END_REF] are optimal (up to the value of the exponent). The Stefanov stability estimates and our instability result for Problem 1.1 are presented and discussed in Section 2. In Section 3 we prove some basic analytic properties of the scattering amplitude. Finally, in Section 5 we prove the main result, using a ball packing and covering by ball arguments.

Stability and instability estimates

In what follows we suppose supp v(x) ⊂ D = B(0, 1),

where B(x, r) is the open ball of radius r centered at x. We consider the orthonormal basis of the spherical harmonics in L 2 (S 2 ) = L 2 (∂D):

{Y p j : j ≥ 0; 1 ≤ p ≤ 2j + 1}. (2.2)
The notation (a j 1 p 1 j 2 p 2 ) stands for a multiple sequence. We will drop the subscript

0 ≤ j 1 , 1 ≤ p 1 ≤ 2j 1 + 1, 0 ≤ j 2 , 1 ≤ p 2 ≤ 2j 2 + 1. (2.3) We expand function f (θ, ω, s) in the basis {Y p 1 j 1 × Y p 2 j 2 }: f (θ, ω, s) = j 1 ,p 1 ,j 2 ,p 2 a j 1 p 1 j 2 p 2 (s)Y p 1 j 1 (θ)Y p 2 j 2 (ω). (2.4)
As in [START_REF] Stefanov | Stability of the inverse problem in potential scattering at fixed energy[END_REF] we use the norm

||f (•, •, s)|| σ 1 ,σ 2 = j 1 ,p 1 ,j 2 ,p 2 2j 1 + 1 es 2j 1 +2σ 1 2j 2 + 1 es 2j 2 +2σ 2 |a j 1 p 1 j 2 p 2 (s)| 2 1/2 . (2.5)
If a function f is the scattering amplitude for some potential v ∈ L ∞ (D) supported in B(0, ρ), where 0 < ρ < 1, then

|a j 1 p 1 j 2 p 2 (s)| ≤ C(s, ||v|| L ∞ (D) ) esρ 2j 1 + 1 j 1 +3/2 esρ 2j 2 + 1 j 2 +3/2 (2.6) and, therefore, ||f (•, •, s)|| σ 1 ,σ 2 < ∞, see estimates of Proposition 2.2 of [17]. Theorem 2.1 (see [17]). Let v 1 , v 2 be real-valued potentials such that v i ∈ L ∞ (D) ∩ H q (R 3 ), supp v i ⊂ B(0, ρ), ||v i || L ∞ (D) ≤ N for i = 1,
2 and some N > 0, q > 3/2 and 0 < ρ < 1. Let f 1 and f 2 denote the scattering amplitudes for v 1 and v 2 , respectively, in the framework of equation (1.1) with E = s 2 , s > 0, then

||v 1 -v 2 || L ∞ (D) ≤ c(N, ρ)φ δ (||f 1 (•, •, s) -f 2 (•, •, s)|| 3/2,-1/2 ), (2.7)
where φ δ (t) = (-ln t) -δ for some fixed δ, where, in particular, 0 < δ < 1, and for sufficiently small t > 0.

The main result of the present work is the following theorem.

Theorem 2.2. For the interval I = [s 1 , s 2 ], such that s 1 > 0, and for any m > 0, α > 2m and any real σ 1 , σ 2 there are constants β > 0 and N > 0, such that for any

v 0 ∈ C m (D) with ||v 0 || L ∞ (D) ≤ N, supp v 0 ⊂ B(0, 1/2
) and any ǫ ∈ (0, N), there are real-valued potentials

v 1 , v 2 ∈ C m (D), also supported in B(0, 1/2), such that sup s∈I (||f 1 (•, •, s) -f 2 (•, •, s)|| σ 1 ,σ 2 ) ≤ exp -ǫ -1 α , ||v 1 -v 2 || L ∞ (D) ≥ ǫ, ||v i -v 0 || L ∞ (D) ≤ ǫ, i = 1, 2, ||v i -v 0 || C m (D) ≤ β, i = 1, 2, (2.8)
where f 1 , f 2 are the scattering amplitudes for v 1 , v 2 , respectively, for equation (1.1).

Remark 2.1. In the case of fixed energy s 1 = s 2 we can replace the condition α > 2m in Theorem 2.2 by α > 5m/3. Remark 2.2. We can allow β to be arbitrarily small in Theorem 2.2 if we require ǫ ≤ ǫ 0 and replace the right-hand side in the first inequality in (2.8) by exp(-cǫ -1 α ), with ǫ 0 > 0 and c > 0 depending on β. Remark 2.3. Note that Theorem 2.2 and Remark 2.1 imply, in particular, that for any real σ 1 and σ 2 the estimate

||v 1 -v 2 || L ∞ (D) ≤ c(N, ρ, m, I) sup s∈I φ δ (||f 1 (•, •, s) -f 2 (•, •, s)|| σ 1 ,σ 2 )
(2.9)

can not hold with δ > 2m in the case of the scattering amplitude given on the energy interval and with δ > 5m/3 in the case of fixed energy. Apparently, proceeding from results of [START_REF] Novikov | New global stability estimates for the Gel'fand-Calderon inverse problem[END_REF], it is not difficult to improve estimate (2.7) for

δ = m + o(m) as m → ∞.
(2.11)

Some basic analytic properties of the scattering amplitude

Consider the solution ψ + (x, k) of equation 1.1, see formula (1.3). We have that

ψ + (x, k) = e ikx µ + (x, θ, s), (3.1)
where θ ∈ S 2 , k = sθ and µ + (x, θ, s) solves the equation

µ + (x, θ, s) = 1 - R 3 G + (x, y, s)e -isθ(x-y) v(y)µ + (y, θ, s)dy, (3.2)
where

G + (x, y, s) = e is|x-y| 4π|x -y| . (3.3)
We suppose that condition (2.1) holds and, in addition, for some h > 0 we have that

|Im s| ≤ h, (3.4
)

c 1 (h, D)||v|| L ∞ (D) ≤ 1/2, (3.5)
where D = B(0, 1),

c 1 (h, D) = sup x∈D D e 2h|x-y| 4π|x -y| dy. (3.6)
Then, in particular, e -isθ(x-y) e is|x-y| ≤ e 2h|x-y| .

(3.7) Solving (3.2) by the method of succesive approximations in L ∞ (D), we obtain that

|µ + (x, θ, s)| ≤ 1 1 -c 1 ||v|| L ∞ (D) , θ ∈ S 2 , x ∈ D. (3.8)
Lemma 3.1. Let a j 1 p 1 j 2 p 2 (s) denote coefficients f (s, θ, ω) in the basis of the spherical harmonics {Y p 1 j 1 ×Y p 2 j 2 }, where f is the scattering amplitude for potential v ∈ L ∞ (D) such that conditions (2.1) and (3.5) hold for some h > 0,

f (θ, ω, s) = j 1 ,p 1 ,j 2 ,p 2 a j 1 p 1 j 2 p 2 (s)Y p 1 j 1 (θ)Y p 2 j 2 (ω). (3.9)
Then a j 1 p 1 j 2 p 2 (s) is holomorphic function in W h = {s | |Im s| ≤ h} and

|a j 1 p 1 j 2 p 2 (s)| ≤ c 2 (h, D)s ∈ W h . (3.10)
Proof of Lemma 3.1. We start with the well-known formula

f (θ, ω, s) = 1 (2π) 3 R 3 e is(θ-ω)x v(x)µ + (x, θ, s)dx. (3.11) Note that, since θ, ω ∈ S 2 , |e is(θ-ω)x | ≤ e 2|Im s||x| . (3.12)
Combining it with (2.1), (3.5), (3.8) and (3.11) we obtain that

|f (θ, ω, s)| ≤ c2 (h, D) for s ∈ W h . (3.13)
Using also that

a j 1 p 1 j 2 p 2 (s) = S 2 ×S 2 f (θ, ω, s)Y p 1 j 1 (θ)Y p 2 j 2 (ω)dθdω (3.14)
we obtain the result of Lemma 3.1.

A fat metric space and a thin metric space

Definition 4.1. Let (X, dist) be a metric space and ǫ > 0. We say that a set Y ⊂ X is an ǫ-net for X 1 ⊂ X if for any x ∈ X 1 there is y ∈ Y such that dist(x, y) ≤ ǫ. We call ǫ-entropy of the set X 1 the number

H ǫ (X 1 ) := log 2 min{|Y | : Y is an ǫ-net fot X 1 }. A set Z ⊂ X is called ǫ-discrete if for any distinct z 1 , z 2 ∈ Z, we have dist(z 1 , z 2 ) ≥ ǫ.
We call ǫ-capacity of the set X 1 the number C ǫ := log 2 max{|Z| : Z ⊂ X 1 and Z is ǫ-discrete}.

The use of ǫ-entropy and ǫ-capacity to derive properties of mappings between metric spaces goes back to Vitushkin and Kolmogorov (see [START_REF] Kolmogorov | ǫ-entropy and ǫ-capacity in functional spaces Usp[END_REF] and references therein). One notable application was Hilbert's 13th problem (about representing a function of several variables as a composition of functions of a smaller number of variables). In essence, Lemma 4.1 and Lemma 4.2 are parts of the Theorem XIV and the Theorem XVII in [START_REF] Kolmogorov | ǫ-entropy and ǫ-capacity in functional spaces Usp[END_REF]. Lemma 4.1. Let d ≥ 2 и m > 0. For ǫ, β > 0, consider the real metric space

X mǫβ = {v ∈ C m (R d ) | supp v ⊂ B(0, 1/2), ||v|| L ∞ (R d ) ≤ ǫ, ||v|| C m (R d ) ≤ β}
with the metric induced by L ∞ . Then there is µ > 0 such that for any β > 0 and ǫ ∈ (0, µβ), there is an ǫ-discrete set Z ⊂ X mǫβ with at least exp 2 -d-1 (µβ/ǫ) d/m elements.

W I,γ = { a + b 2 + a -b 2 cos z | |Im z| ≤ γ}. (4.1)
Then there is a constant ν = ν(C, γ) > 0 such that for any δ ∈ (0, e -1 ) there is a δ-net for the space of functions on I with L ∞ -norm, having holomorphic continuation to W I,γ with module bounded above on W I,γ by the constant C, with at most exp(ν(ln δ -1 ) 2 ) elements.

Remark 4.1. In the case of a = b, taking

Y = δ 2 Z [-C, C] + i • δ 2 Z [-C, C], (4.2)
we get δ-net with at most exp(ν ln δ -1 ) elements. Lemma 4.1 and Lemma 4.2 were also formulated and proved in [START_REF] Mandache | Exponential instability in an inverse problem for the Schrödinger equation Inverse Problems[END_REF] and [START_REF] Isaev | Exponential instability in the Gel'fand inverse problem on the energy intervals[END_REF], respectively. For the interval I = [s 1 , s 2 ] such that s 1 > 0 and real σ 1 , σ 2 we introduce the Banach space

X I,σ 1 ,σ 2 = a j 1 p 1 j 2 p 2 (s) | a j 1 p 1 j 2 p 2 (s) X I,σ 1 ,σ 2 < ∞ , (4.3)
where

a j 1 p 1 j 2 p 2 (s) X I,σ 1 ,σ 2 = sup s∈I j 1 ,p 1 ,j 2 ,p 2 2j 1 + 1 es j 1 +σ 1 2j 2 + 1 es j 2 +σ 2 |a j 1 p 1 j 2 p 2 (s)| . (4.4)
We consider the scattering amplitude f for some potential v ∈ L ∞ (D) supported in B(0, ρ), where 0 < ρ < 1. We identify in the sequel the scattering amplitude f (s, θ, ω) with its matrix a j 1 p 1 j 2 p 2 (s) in the basis of the spherical harmonics

{Y p 1 j 1 × Y p 2 j 2 } . We have that sup s∈I ||f (•, •, s)|| σ 1 ,σ 2 ≤ c 3 a j 1 p 1 j 2 p 2 (s) X I,σ 1 ,σ 2 , (4.5)
where σ1 -σ 2 = σ2 -σ 2 = 3 and c 3 = c 3 (I) > 1. We obtain (4.5) from definitions (2.5), (

and by taking c 3 > 1 in a such a way that

j 1 ,p 1 ,j 2 ,p 2 2j 1 + 1 es -3 2j 2 + 1 es -3 < c 3 . (4.6)
For h > 0 we denote by A h the set of the matrices, corresponding to the scattering amplitudes for the potentials v ∈ L ∞ (D) supported in B(0, 1/2) such that condition (3.5) holds.

Lemma 4.3. For any h > 0 and any real σ 1 , σ 2 , the set A h belongs to X I,σ 1 ,σ 2 . In addition, there is a constant η = η(I, h, σ 1 , σ 2 ) > 0 such that for any δ ∈ (0, e -1 ) there is a δ-net Y for A h in X I,σ 1 ,σ 2 with at most exp η (ln δ -1 ) 6 (1 + ln ln δ -1 ) 2 elements.

Proof of Lemma 4.3. We can suppose that σ 1 , σ 2 ≥ 0 as the assertion is stronger in this case. If a function f is the scattering amplitude for some potential v ∈ L ∞ (D) supported in B(0, 1/2), we have from (2.6) that

2j 1 + 1 es j 1 +σ 1 2j 2 + 1 es j 2 +σ 2 |a j 1 p 1 j 2 p 2 (s)| ≤ c 4 (2j 1 + 1) σ 1 (2j 2 + 1) σ 2 2 j 1 +j 2 , ( 4.7)
where c 4 = c 4 (I, h) > 0. Hence, for any positive σ 1 and σ 2 , a j 1 p 1 j 2 p 2 (s)

X I,σ 1 ,σ 2 ≤ sup j 1 ,j 2 c 4 (2j 1 + 1) σ 1 (2j 2 + 1) σ 2 2 j 1 +j 2 < ∞ (4.8)
and so the first assertion of the Lemma 4.3 is proved.

Let l δ,σ 1 ,σ 2 be the smallest natural number such that c 4 (2l + 1) σ 1 +σ 2 2 -l < δ for any l ≥ l δ,σ 1 ,σ 2 . Taking natural logarithm we have that ln c 4 -(σ 1 + σ 2 ) ln(2l + 1) + l ln 2 > ln δ -1 for l ≥ l δ,σ 1 ,σ 2 .

(4.9)

Using ln δ -1 > 1, we get that

l δ,σ 1 ,σ 2 ≤ C ′ ln δ -1 , (4.10)
where the constant C ′ depends only on h, σ 1 , σ 2 and I = [s 1 , s 2 ]. We take W I = W I,γ of (4.1), where the constant γ > 0 is such that W I ⊂ {s | |Im s| ≤ h}. If max(j 1 , j 2 ) ≤ l δ,σ 1 ,σ 2 , then we denote by Y j 1 p 1 j 2 p 2 some δ j 1 p 1 j 2 p 2 -net from Lemma 4.2 with the constant C = c 2 , where the constant c 2 is from Lemma 3.1 and

δ j 1 p 1 j 2 p 2 = es 1 2j 1 + 1 j 1 +σ 1 es 1 2j 2 + 1 j 2 +σ 2 δ. (4.11)
Otherwise we take Y j 1 p 1 j 2 p 2 = {0}. We set

Y = a j 1 p 1 j 2 p 2 (s) | a j 1 p 1 j 2 p 2 (s) ∈ Y j 1 p 1 j 2 p 2 . (4.12)
For any a j 1 p 1 j 2 p 2 (s) ∈ A h there is an element b j 1 p 1 j 2 p 2 (s) ∈ Y such that

2j 1 + 1 es j 1 +σ 1 2j 2 + 1 es j 2 +σ 2 |a j 1 p 1 j 2 p 2 (s) -b j 1 p 1 j 2 p 2 (s)| ≤ ≤ 2j 1 + 1 es j 1 +σ 1 2j 2 + 1 es j 2 +σ 2 δ j 1 p 1 j 2 p 2 ≤ δ (4.13)
in the case of max(j 1 , j 2 ) ≤ l δ,σ 1 ,σ 2 and

2j 1 + 1 es j 1 +σ 1 2j 2 + 1 es j 2 +σ 2 |a j 1 p 1 j 2 p 2 (s) -b j 1 p 1 j 2 p 2 (s)| ≤ ≤ c 4 (2j 1 + 1) σ 1 (2j 2 + 1) σ 2 2 j 1 +j 2 ≤ c 4 (2 max(j 1 , j 2 ) + 1) σ 1 +σ 2 2 max(j 1 ,j 2 ) < δ, (4.14)
otherwise.

It remains to count the elements of Y . We recall that |Y j 1 p 1 j 2 p 2 | = 1 in the case of max(j 1 , j 2 ) > l δ,σ 1 ,σ 2 . Using again the fact that ln δ -1 ≥ 1 and (4.10) we get in the case of max(j 1 , j 2 ) ≤ l δ,σ 1 ,σ 2 : 4 , where n δ,σ 1 ,σ 2 is the number of four-tuples (j 1 , p 1 , j 2 , p 2 ) with max(j 1 , j 2 ) ≤ l δ,σ 1 ,σ 2 . Taking η to be big enough we get that

|Y j 1 p 1 j 2 p 2 | ≤ exp(ν(ln δ -1 j 1 p 1 j 2 p 2 ) 2 ) ≤ exp ν ′ ln δ -1 2 1 + ln ln δ -1 2 . (4.15) We have that n δ,σ 1 ,σ 2 ≤ l 2 δ,σ 1 ,σ 2 (2l δ,σ 1 ,σ 2 + 1) 2 ≤ (2l δ,σ 1 ,σ 2 + 1)
|Y | ≤ exp ν ′ ln δ -1 2 1 + ln ln δ -1 2 n δ,σ 1 ,σ 2 ≤ exp ν ′ ln δ -1 2 1 + ln ln δ -1 2 (1 + 2C ′ ln δ -1 ) 4 ≤ exp η ln δ -1 6 1 + ln ln δ -1 2 . (4.16)
Remark 4.2. In the case of s 1 = s 2 , taking into account Remark 4.1 and using it in (4.15) and (4.16), we get δ-net Y with at most exp η (ln δ -1 )

5 (1 + ln ln δ -1 ) elements.

Proof of Theorem 2.2

We take N such that condition (3.5) holds for any ||v|| L ∞ (D) ≤ 2N for some h > 0. By Lemma 4.1, the set v 0 + X mǫβ has an ǫ-discrete subset v 0 + Z. Since ǫ ∈ (0, N) we have that the set Y constructed in Lemma 4.3 is also δ-net for the set of the matrices, corresponding to the scattering amplitudes for the potentials v ∈ v 0 + X mǫβ . We take δ such that 2c 3 δ = exp -ǫ -1 α , see (4.5). Note that inequalities of (2. (5.5)

This completes the proof of Theorem 2.2.

In the case of fixed energy s 1 = s 2 , using Remark 4.2 in (5.3), we can replace the condition α > 2m in Theorem 2.2 by α > 5m/3.

Lemma 4 . 2 .

 42 For the interval I = [a, b] and γ > 0 consider the ellipse W I,γ ∈ C:

2 ≤≤.( 5 . 2 ) 1 α 6 1 + ln ln(2c 3 ) + ǫ - 1 α 2 ≤ exp η α ǫ - 3 m. ( 5 . 3 )

 25216312353 8) follow from |v 0 + Z| > |Y |, (5.1)where the set Y is constructed in Lemma 4.3 with σ1 = σ 1 + 3 and σ2 = σ 2 + 3. In fact, if|v 0 + Z| > |Y |, then there are two potentials v 1 , v 2 ∈ v 0 + Z with the matrices a j 1 p 1 j 2 p 2 (s)and b j 1 p 1 j 2 p 2 (s) , corresponding to the scattering amplitudes for them, being in the same X I,σ 1 ,σ 2 -ball radius δ centered at a point of Y . Hence, using (4.5) we get thatsup s∈I ||f 1 (•, •, s) -f 2 (•, •, s)|| σ 1 ,σ 2 ≤ c 3 a j 1 p 1 j 2 p 2 (s) -b j 1 p 1 j 2 p 2 (s) X I,σ 1 ,σ 2c 3 δ = exp -ǫ -1 α It remains to find β such that (5.1) is fullfiled. By Lemma 4.3 for some η α = η α (I, σ 1 , σ 2 , α) > 0 |Y | ≤ exp η ln(2c 3 ) + ǫ -Now we take β > µ -1 max N, η m/3 α 2 2m . (5.4) This fulfils requirement ǫ < µβ in Lemma 4.1, which gives |v 0 + Z| = |Z| ≥ exp 2 -4 (µβ/ǫ) 3/m (5.4) > > exp 2 -4 (η m/3 α 2 2m /ǫ)

  Thus Theorem 2.2 and Remark 2.1 show optimality of the Stefanov logarithmic stability result (up to the value of the exponent).

	Remark 2.4. A disadvantage of estimate (2.7) is that
	δ < 1 even if m is very great.	(2.10)

Acknowledgments

This work was fulfilled under the direction of R.G.Novikov in the framework of an internship at Ecole Polytechnique.


Pierre Affaticati

Olivier Mignen

Florence Jambou

Marie-Claude Potier

Isabelle Klingel-Schmitt

Jeril Degrouard

Stéphane Peineau

Elodie Gouadon

Graham L Collingridge

Roland Liblau

Thierry Capiod

Sylvia Cohen-Kaminsky
email: sylvia.cohen-kaminsky@u-psud.fr

Sustained calcium signalling and caspase-3 activation involve NMDA receptor in thymocytes in contact with dendritic cells Running title: NMDAR calcium signalling in thymocytes

Keywords: Calcium signalling / thymocyte / NMDAR APC, antigen-presenting cell, Ca 2+ , calcium, CRAC, Ca 2+ release-activated Ca 2+, DC, dendritic cell, DP, CD4 + CD8 + double-positive, GluR, glutamate receptor, iGluR, ionotropic glutamate receptor, IS, immunological synpase, mGluR, metabotropic glutamate receptor, NMDA, N-methyl-D-aspartate, NMDAR, NMDA receptor, PSD, post-synaptic density, siRNA, small interfering RNA, .TCR, T cell receptor, VGLUT, vesicular glutamate transporter

L-glutamate, the major excitatory neurotransmitter, also plays a role in non-neuronal tissues and modulates immune responses. Whether NMDA receptor (NMDAR) signalling is involved in T cell development is unknown. Here we show that mouse thymocytes expressed an array of glutamate receptors, including NMDARs subunits. Sustained calcium signals, and caspase-3 activation in thymocytes were induced by interaction with antigen-pulsed DCs and were inhibited by NMDAR antagonists MK801 and memantine. NMDAR was transiently activated, triggered the sustained calcium signal, and was corecruited with the PDZ-domain adaptor PSD-95 to thymocyte-DC contact zones. While T cell receptor (TCR) activation was sufficient for relocalization of NMDAR and PSD-95 at the contact zone, NMDAR could be activated only in a synaptic context. In these T-DC contacts, thymocyte activation occurred in the absence of exogenous glutamate, indicating that DCs could be a physiological source of glutamate. DCs expressed glutamate, glutamate-specific vesicular VGLUT transporters and were capable of fast glutamate release through a Ca 2+ -dependent mechanism. We propose that glutamate released by DCs could elicit focal responses through NMDAR-signalling in T cells undergoing apoptosis. Thus, synapses between T and DCs could provide a functional platform for coupling TCR activation and NMDAR signalling which might reflect on T cell development and modulation of the immune response.

INTRODUCTION

L-glutamate is the major excitatory neurotransmitter in the nervous system. The N-methyl-Daspartate (NMDA) receptor belongs to the ionotropic glutamate receptors (iGluRs), mediates fast synaptic responses [START_REF] Mayer | Structure and function of glutamate receptor ion channels[END_REF] , and is involved in neuronal synaptic transmission and plasticity by fine-tuning Ca 2+ signalling [START_REF] Riedel | Glutamate receptor function in learning and memory[END_REF] . There is now evidence to suggest that glutamate acts as a signalling molecule in non-neuronal tissues [START_REF] Nedergaard | Beyond the role of glutamate as a neurotransmitter[END_REF][START_REF] Skerry | Glutamate signalling in non-neuronal tissues[END_REF] , with an emerging role as an immune modulator [START_REF] Pacheco | Role of glutamate on T-cell mediated immunity[END_REF] . Metabotropic G-proteincoupled GluRs (mGluRs) are involved in T-cell activation [START_REF] Pacheco | Group I metabotropic glutamate receptors mediate a dual role of glutamate in T cell activation[END_REF] , and in the inhibition of activationinduced cell death [START_REF] Chiocchetti | Group I mGlu receptor stimulation inhibits activation-induced cell death of human T lymphocytes[END_REF] . iGluRs are involved in cell cycle progression, regulating activation and proliferation, chemotactic migration and integrin-mediated adhesion in T cells [START_REF] Boldyrev | Emerging evidence for a similar role of glutamate receptors in the nervous and immune systems[END_REF][START_REF] Ganor | Human T cells express a functional ionotropic glutamate receptor GluR3, and glutamate by itself triggers integrinmediated adhesion to laminin and fibronectin and chemotactic migration[END_REF][START_REF] Lombardi | Glutamate modulation of human lymphocyte growth: in vitro studies[END_REF] , with some indications for a role of the NMDAR in Ca 2+ signalling [START_REF] Lombardi | Characterization of ionotropic glutamate receptors in human lymphocytes[END_REF][START_REF] Miglio | Human T lymphocytes express N-methyl-Daspartate receptors functionally active in controlling T cell activation[END_REF] .

Increases in [Ca 2+ ]i are key signals in T-cell activation following TCR engagement. T lymphocytes are believed to use store operated calcium (SOC) channel entry as the main mode of calcium influx [START_REF] Lewis | Calcium signaling mechanisms in T lymphocytes[END_REF] . The SOC channels in lymphocytes are known as CRAC channels, whose molecular identity has been recently elucidated, with Orai1 as the pore forming subunit and STIM1 as the sensor of stored calcium [START_REF] Vig | CRACM1 is a plasma membrane protein essential for store-operated Ca2+ entry[END_REF] . Orai1 and STIM1 are recruited to the immunological synapse [START_REF] Lioudyno | Orai1 and STIM1 move to the immunological synapse and are up-regulated during T cell activation[END_REF] . Although CRAC plays a major role in T cells [START_REF] Lewis | Calcium signaling mechanisms in T lymphocytes[END_REF] , it is likely that their complex Ca 2+ signalling also involves purinergic receptors, voltage-gated Ca 2+ channels, and modulation by Ca 2+ -dependent voltage-activated K + channels [START_REF] Vig | CRACM1 is a plasma membrane protein essential for store-operated Ca2+ entry[END_REF][START_REF] Kotturi | Roles of CRAC and Cav-like channels in T cells: more than one gatekeeper?[END_REF] . Elucidating the relationships between all these routes of cytosolic calcium modulation remains a challenge for the future. Until recently, most of the studies exploring Ca 2+ entries in T cells, especially glutamate receptor entry, were not carried out under physiological stimulation by antigen-presenting cells.

The immunological synapse (IS) is a structure which mediates information exchange that typically forms in the contact zone between an antigen-presenting cell (APC) and a T cell bearing a specific TCR. It was first suggested to facilitate the directed secretion of cytokines between T cells and APCs [START_REF] Norcross | A synaptic basis for T-lymphocyte activation[END_REF] . Cytokines may be released directly into the synapse, having an immediate effect on clustered receptors in the contact zone [START_REF] Huse | T cells use two directionally distinct pathways for cytokine secretion[END_REF] . The synapse formed between T cells and dendritic cells (DCs) displays cell-to-cell adhesion, stability and close apposition of membranes [START_REF] Brossard | Multifocal structure of the T cell -dendritic cell synapse[END_REF] , that may help to optimize soluble mediator concentration at the T cell-APC interface, thereby limiting effects on bystander cells. Neuronal synapses and ISs are often compared due to their anatomical similarity, although ISs form rapidly and are transient [START_REF] Dustin | Neural and immunological synaptic relations[END_REF] . To what extent these synapses are functionally similar is not known.

Circulating T cells are thought to be exposed to glutamate in the plasma, glutamate-rich peripheral organs [START_REF] Nedergaard | Beyond the role of glutamate as a neurotransmitter[END_REF][START_REF] Skerry | Glutamate signalling in non-neuronal tissues[END_REF] , and in the brain, facilitating crosstalk with the nervous system [START_REF] Steinman | Elaborate interactions between the immune and nervous systems[END_REF] . It is therefore likely that GluRs are not permanently stimulated in T cells, and we postulated that their activation occurs as a result of IS formation. In analogy to neurons, glutamate should therefore be rapidly released, with faster kinetics than those reported through the cystine/glutamate antiporter (X c -system) [START_REF] Pacheco | Glutamate released by dendritic cells as a novel modulator of T cell activation[END_REF] . Indeed, the possibility that glutamate released by APCs at ISs mediates communication with T cells through GluR-mediated Ca 2+ signalling has never been investigated.

RESULTS

NMDARs are expressed in thymocytes

More than 30 different GluRs are expressed in the central nervous system (CNS) [START_REF] Mayer | Structure and function of glutamate receptor ion channels[END_REF] . Since only mGluRs have been detected in thymocytes [START_REF] Storto | Expression of metabotropic glutamate receptors in murine thymocytes and thymic stromal cells[END_REF] , we used neurotransmission-dedicated oligoarrays to analyze their GluR repertoire. Thymocytes expressed most known GluRs (Supplementary Figure S1a), including NMDAR GluN1, GluN2A and GluN2B subunits (IUPHAR nomenclature of subunits, previously referred to as NR1, NR2A and NR2B subunits [START_REF] Collingridge | A nomenclature for ligandgated ion channels[END_REF] (Figure 1a). These findings were confirmed by PCR (Figure 1a), and confocal microscopy, using well-characterized antibodies (Figure 1b, Supplementary Figure S1c). To study NMDAR function, we focused on GluN1, the obligate subunit of NMDARs which was expressed in thymocytes at a level similar to what was observed in the brain (Figure 1a, Supplementary table S1). FACS analysis, using antibodies that detect NR1 as a band at the expected molecular weight by Western blotting in brain microsomal preparations, indicated that 97.3% of thymocytes expressed NR1, with higher expression in CD4 + or CD8 + cells as compared to DP cells (Figure 1c).

Calcium signalling and apoptosis in thymocytes in contact with DCs, involves NMDAR

To explore NMDAR function in thymocytes, we focused on Ca 2+ and apoptosis signalling. Indeed, both are associated with NMDAR activity in neurones. Glutamate and NMDA, at doses of 1 to 500 µM, did not induce Ca 2+ signalling nor apoptosis in freshly isolated thymocytes (data not shown).

Thus, we investigated whether a thymocyte-DC synaptic context could provide the environment required for coupling Ca 2+ signalling to apoptosis.

To set-up an in vitro system of T cell-DC contacts, we used HNT-TCR-transgenic mice in which most T cells express the same TCR directed to the HA126-138 peptide. In vivo administration of the antigenic HA126-138 peptide in HNT-TCR transgenic mice was shown to induce massive apoptosis, mostly of CD4 + CD8 + double-positive (DP) thymocytes [START_REF] Liblau | Intravenous injection of soluble antigen induces thymic and peripheral T-cells apoptosis[END_REF] . Thus we cocultured thymocytes from HNT-TCR-transgenic mice and HA-peptide pulsed DCs, to enhance the probability to observe antigen-dependent synaptic contacts between thymocytes and DCs. To assess the presence of functional synaptic contacts, we monitored DC capacity to induce thymocyte apoptosis by following the expression of three markers. The first one, Nur77 is an immediate early gene required for the induction of apoptosis in negative selection [START_REF] Woronicz | Requirement for the orphan steroid receptor Nur77 in apoptosis of T-cell hybridomas[END_REF] , and a specific marker of clonal deletion in vivo [START_REF] Cho | Cutting edge: identification of the targets of clonal deletion in an unmanipulated thymus[END_REF] . The second, CD69 is an early T-cell activation marker up-regulated after TCR engagement, and involved in negative selection [START_REF] Merkenschlager | How many thymocytes audition for selection?[END_REF] and was thus used to discriminate thymocytes that have formed synapses with DCs. To analyze apoptosis we used activated caspase-3 in living cells, allowing early detection of apoptosis and avoiding by-stander effects occurring at later time points.

These three markers of thymocyte activation and clonal deletion expressed during negative selection in vivo (Nur77, CD69, and activated caspase-3), that we measured in our own in vivo experiments, were also upregulated in our in vitro model, assessing the presence of functional synapses (Supplementary Figure S2a). In contrast to thymocytes, peripheral CD4 + T cells in contact with DCs showed increased CD69 expression, without caspase-3 activation (Supplementary Figure S2b) and proliferated as expected in response to HA peptide (data not shown).

Then we used this in vitro system to monitor the calcium signal elicited in single thymocytes by contact with antigen-pulsed DCs, in the absence of exogenous glutamate. Antigen-specific contacts of thymocytes with DCs, in a glutamate-free medium, resulted in rapid and sustained increases in [Ca 2+ ] i in T cells (Figure 2a andb). Most thymocytes established long-lasting contacts, 90 % of which resulted in a Ca 2+ peak (ΔR/R=2.93±0.08, n=119), followed by a sustained high-level plateau lasting at least 10 minutes (Figure 2b and Supplementary Video S1). A few unstable contacts with no measurable Ca 2+ signals were observed in the absence of HA-peptide (Figure 2b). Interestingly, contacts with purified peripheral splenic CD4 + T cells resulted in a Ca 2+ signal with a peak similar to that for thymocytes (ΔR/R=3.72±0.19, n=32), but rapidly decreasing (50 % decrease from the peak value in 91±13 s, n=30) to a plateau maintained at a basal level of about one fifth of the peak value (Figure 2a andc, Supplementary Videos S1 and S2).

Thus, thymocyte-DC contact in the absence of exogenous glutamate results in sustained calcium signalling with a high plateau and apoptosis signalling. These results also point to a potential relationship between the sustained shape of the Ca 2+ signal and T cell fate.

NMDAR triggers the sustained calcium response in thymocytes and is involved in thymocyte apoptosis

The primary mechanism of Ca 2+ mobilization in T cells is the release of Ca 2+ from intracellular stores followed by Ca 2+ influx through Ca 2+ release-activated Ca 2+ (CRAC) channels [START_REF] Lewis | Calcium signaling mechanisms in T lymphocytes[END_REF] and memantine respectively) to control cells, but with a plateau close to basal levels (50% decrease in the peak value after 148±16 s, n=18 and 275±37 s, n=14, respectively) (Figure 2d and Supplementary Video S3).

These blockers also significantly affected the percentages of activated caspase-3-expressing cells (27±0.7% inhibition with MK801, n=24 and 30±0.85% with memantine, n=15) (Figure 3a), indicating a role for NMDARs in apoptosis. Of note, the effect on caspase-3 expression might be underestimated since only 50% of the thymocytes engaged in a synapse were sensitive to the drugs, as measured in Ca 2+ signalling experiments, as mentioned above. Activation of Nur77 expression, which is Ca 2+ -dependent [START_REF] Winoto | Nuclear hormone receptors in T lymphocytes[END_REF] , was affected by MK801 as measured by FACS analysis 4 hours after thymocyte-DC contact (Figure 3b andc), suggesting that Nur77 could be an early target of the Ca 2+ signal mediated by NMDAR. To rule out the possibility that NMDAR blockers had non-specific effects on CRAC or other off-target effects on channels involved in T-cell activation, we monitored the Ca 2+ signal by FACS on a large thymocyte population, activated by biotinylated anti-CD3 and anti-CD28 antibodies followed by streptavidin cross-linking, in the absence of glutamate. About 60% of thymocytes responded to TCR stimulation by a Ca 2+ signal that was unaffected by MK801

or memantine (Supplementary Figure S3), indicating that these drugs had no major effect on Ca 2+ signalling resulting from TCR triggering.

These results (NMDA insensitivity of freshly isolated thymocytes and MK801/memantine inhibition of Ca 2+ and apoptosis signalling in DC contacting thymocytes) suggest that NMDA channels are activated only during or after synapse formation. To further investigate the involvement of NMDA channels in this process, we ask if 1) NMDARs are activated only after synapse formation and 2) the sustained Ca 2+ response required a transient NMDAR activation. Using the T cell-DC contact model, we firstly incubated freshly dissociated T cells with MK801, NMDA and D-serine. In these conditions MK801 will definitely block all activable NMDARs present at the membrane. We then washed the thymocytes and, in presence of DCs, recorded the Ca 2+ signalling following synaptic contact. The sustained Ca 2+ responses recorded (n=47, 95% of cells with a sustained signal) (Figure 3d), were similar to the responses recorded in absence of T cell pre-treatment, suggesting that NMDARs are active only after synaptic contact even if their agonists are present. In the same conditions of pretreatment of thymocytes with MK801, NMDA and D-serine, addition of MK801 in the bath solution before synapse formation, revealed a sensitivity of the Ca 2+ plateau to this NMDAR blocker (n=22, 55% of cells with a transient signal) (Figure 3d). Using the same protocol, 200 seconds after the beginning of Ca 2+

signalling due to T cell-DC contacts, we applied MK801. MK801 application after synapse formation failed to inhibit Ca 2+ response (n=20, 90% of cells with a sustained signal) (Figure 3e).

Together these results suggest that NMDARs are necessary for the induction of the sustained Ca 2+ response but are only transiently activated at the beginning of the synaptic response. Furthermore these receptors are also involved in the induction of the thymocyte apoptosis after contact with DCs Thymocyte-DC interaction induces GluN1 and PSD-95 clustering at the contact zone.

Transient activation of NMDA channels at the synapse suggest that an important trafficking of these receptors could occur in thymocytes. The TCR signalosome includes PDZ domain-containing adaptors, initially described at neuronal synapses but increasingly thought to be involved in ISs [START_REF] Round | Dlgh1 coordinates actin polymerization, synaptic T cell receptor and lipid raft aggregation, and effector function in T cells[END_REF] .

PSD-95, a prototypical PDZ domain protein, accumulates in the active zones of neuronal synapses, forming so-called postsynaptic densities (PSDs). It is the first-line signalling adaptor binding directly to the C-terminus of NMDAR subunits [START_REF] Kornau | Domain interaction between NMDA receptor subunits and the postsynaptic density protein PSD-95[END_REF] , involved in stabilization and functional recruitment of NMDARs at the excitatory synapse [START_REF] Kim | PDZ domain proteins of synapses[END_REF] . Staining for GluN1 and PSD-95 indicated that both were distributed over the entire resting thymocytes, but were colocalized in the contact zone in T cell-DC synapses (Figure 4a, and Supplementary Figure S4). When T cells were stimulated with anti-CD3/CD28-coated beads in the absence of glutamate, PSD-95 was nonetheless colocalized with GluN1 in the contact zone (Figure 4a) indicating that the relocalization of NMDAR and PSD-95 to the synapse depends on TCR stimulation. Interestingly, TCR stimulation using anti-CD3/CD28 coated beads, in the presence or absence of NMDA (or glutamate itself), did not recapitulate the effect of glutamate observed in a synaptic context. The Ca 2+ signal was not sustained with a high level plateau (Figure 4b andc), no significant apoptosis was induced (data not shown) and no NMDA current were recorded (data not shown), indicating that in addition to glutamate, one or several other signals provided by the DC, are necessary to activate NMDARs.

We conclude that intense trafficking of NMDAR may occur in thymocytes upon contact with DCs.

The TCR triggering is sufficient to induce GluN1 and PSD-95 clustering at the synapse, while the NMDAR is activated only in a complete synaptic context.

DCs are capable of fast glutamate release

Since all the Ca 2+ signalling experiments in T cells contacting DCs were performed in the absence of exogenous glutamate, we wondered which cell was the physiological source of glutamate in T cell-DC synapses. Immunostaining and confocal microscopy with a specific antibody to glutamate indicated that DCs were probably the principal source of glutamate in T cell-DC synapses (Figure 5a). Moreover, DCs express the glutamate-specific vesicular transporters VGLUT1 and VGLUT2, which confer a glutamatergic phenotype to neurons [START_REF] Takamori | Identification of a vesicular glutamate transporter that defines a glutamatergic phenotype in neurons[END_REF] and VGLUT3 (Figure 5b and Supplementary Figure S5b). Vesicular structures of 100 nm size, with a characteristic electron-dense membrane were observed in DCs, opposite the contact zone and often located in a polarized cytoplasmic region containing mitochondria (Figure 5c). Immunogold labeling of VGLUT2 showed clusters of particles delineating discrete vesicles, opposite the contact zone in DCs (Figure 5d and Supplementary Figure S5d). Purified DCs express mRNAs encoding components involved in neurotransmitter exocytosis via SNARE complex formation (synaptophysin, Snap-23, VAMP-2) or in the regulation of vesicle docking (Munc-18), as well as synaptotagmin-I (Supplementary Figure S5c ande), a Ca 2+ sensor for glutamate release [START_REF] Sudhof | The synaptic vesicle cycle[END_REF] . Double immunogold labelling showed that VGLUT2 and synaptotagmin-I were expressed in the same vesicular structures (Figure 5e), suggesting the presence in DCs of a compartment competent for both glutamate accumulation and exocytosis. The question of glutamate production by immune cells was previously addressed, by measuring glutamate accumulation starting from 24 hours in supernatants of monocyte-derived DCs and in T-DC cocultures [START_REF] Pacheco | Glutamate released by dendritic cells as a novel modulator of T cell activation[END_REF] . Instead, we focused on direct monitoring of glutamate release, in real time, by freshly isolated DCs. We used the L-glutamate dehydrogenase-linked assay, a fluorescence-based assay allowing dynamic quantification of glutamate release in non-excitable cells, to monitor in real time rapid glutamate release from DCs [START_REF] Innocenti | Imaging extracellular waves of glutamate during calcium signaling in cultured astrocytes[END_REF] . In the presence of external Ca 2+ , ionomycin (Figure 5f) and SDF-1α (a physiological stimulus identified for astrocytes [START_REF] Bezzi | CXCR4-activated astrocyte glutamate release via TNFalpha: amplification by microglia triggers neurotoxicity[END_REF][START_REF] Montes | Calcium responses elicited in human T cells and dendritic cells by cell-cell interaction and soluble ligands[END_REF] (Figure 5g) triggered both [Ca 2+ ] i increases and glutamate release in DCs with a similar time course.

We conclude that DCs exhibit features required for regulated glutamate exocytosis and are capable of fast glutamate release in a Ca 2+ -dependent manner.

DISCUSSION

In previous studies, GluRs were potentially involved in immune regulation [START_REF] Pacheco | Role of glutamate on T-cell mediated immunity[END_REF] , T cells were thought to be exposed to glutamate in the CNS, in the bloodstream or in certain peripheral organs [START_REF] Ganor | Human T cells express a functional ionotropic glutamate receptor GluR3, and glutamate by itself triggers integrinmediated adhesion to laminin and fibronectin and chemotactic migration[END_REF] and monocyte derived DCs have been pointed out as a source of slow glutamate release accumulating in T-DC long term-cocultures [START_REF] Pacheco | Glutamate released by dendritic cells as a novel modulator of T cell activation[END_REF] . As T cells communicate through ISs, which are structurally similar to neuronal synapses [START_REF] Dustin | Neural and immunological synaptic relations[END_REF] S2b). The correlation between T-cell fate and the shape of the Ca 2+ signal suggests that the elevated plateau phase is intimately linked to thymocyte apoptosis. This opens new challenges to identify the switches controlling pro-survival vs pro-apoptosis outcome of NMDAR stimulation.

Dendritic cells, a site of glutamate mobilization for exocytosis

A recent study of T cells in coculture with monocyte-derived DCs pulsed with a superantigen, showed glutamate production in the milieu, detectable from day 1 (in the 5 µM range) and accumulating in the coculture supernatant with time. This glutamate production involved a slow release mechanism, mediated by the cystine/glutamate antiporter X c -system, and glutamate was suggested to activate mGluRs in T cells [START_REF] Pacheco | Glutamate released by dendritic cells as a novel modulator of T cell activation[END_REF] . Here, we demonstrated in real time a rapid and substantial release of glutamate by DCs, consistent with previous reports of synaptic polarization and Ca 2+ -regulated release of pre-assembled vesicular cytokine stores for DCs [START_REF] Borg | NK cell activation by dendritic cells (DCs) requires the formation of a synapse leading to IL-12 polarization in DCs[END_REF] . We used several approaches to suggest that DCs are competent for both glutamate accumulation and release. 1)

Immunofluorescence studies showed specific, strong punctate staining for glutamate and VGLUT vesicular transporters, and staining for the calcium sensor synaptotagmin-I and VAMP-2, a SNARE component of the vesicular machinery for exocytosis. 2) Electron microscopy showed that vesiclelike structures were present in the T cell-DC contact zones. 3) Electron microscopy and double staining with gold particles suggested that DCs are competent for both the glutamate accumulation mediated by VGLUT vesicular transporters and for glutamate release through a calcium-and SNARE-dependent mechanism. 4) Finally, an enzymatic-linked assay demonstrated ionomycin and SDF-1α-induced L-glutamate release by DCs. Experimental studies of the mechanism of glutamate release were beyond the scope of this study. It remains to define the precise SNARE-dependent mechanism of exocytosis, using specific drugs or toxins, and to determine the type of Ca 2+ entry involved in exocytosis signalling; mechanisms similar to those reported for astrocytes might be expected [START_REF] Bezzi | CXCR4-activated astrocyte glutamate release via TNFalpha: amplification by microglia triggers neurotoxicity[END_REF] . In our hands, riluzole and bafilomycin inhibited the calcium signal and caspase-3 induction in thymocytes contacting DCs (P.A and SC-K unpuplished data). However, this approach is not entirely appropriate as these drugs as well as specific toxins would target the exocytosis of compounds other than glutamate, such as chemokines. Indeed, as in astrocytes [START_REF] Bezzi | CXCR4-activated astrocyte glutamate release via TNFalpha: amplification by microglia triggers neurotoxicity[END_REF] we have shown that SDF-1α, a physiological stimulus mobilizing intracellular Ca 2+ , triggers the release of glutamate from DCs. How SDF-1 operates in the minimal system of T-DC synapses for glutamate release remains to be determined. In addition, it is difficult to demonstrate glutamate release in the synaptic cleft directly because of the dynamic nature of the synapse (Supplementary videos S1-S3). Ca 2+ transients in DCs interacting with T cells may be involved in triggering glutamate exocytosis, but they have rarely been observed in our study (data not shown), rather suggesting that local Ca 2+ fluxes in DCs might be involved in glutamate release. Since thymic epithelial cells, did not release measurable glutamate in real-time (P.A., T.C. and S.C-K, unpublished data), DCs may be specialized for rapid glutamate delivery to T cells. Glutamate should be considered an "immunotransmitter" as suggested for cytokines and chemokines [START_REF] Trautmann | Chemokines as immunotransmitters?[END_REF] .

PSD-95, a signalling adaptor indicative of the glutamatergic phenotype of the IS

In neuronal synapses, PSDs, which contain an intricate network of scaffolding PDZ-domain proteins, are thought to participate in synapse genesis and to organize Ca 2+ sources, sensors and effectors, and a number of signalling proteins [START_REF] Augustine | Local calcium signaling in neurons[END_REF] . Several studies have described a role for PDZdomain proteins, e.g. CARMA1 and hDlg/Dlgh1/SAP97 have been implicated in T-cell activation [START_REF] Jun | Identifying the MAGUK protein Carma-1 as a central regulator of humoral immune responses and atopy by genome-wide mouse mutagenesis[END_REF] , and TCR activation-induced synapse assembly [START_REF] Round | Dlgh1 coordinates actin polymerization, synaptic T cell receptor and lipid raft aggregation, and effector function in T cells[END_REF] , respectively. Our findings add to the current knowledge the possibility of PSD-95-mediated connections between NMDARs and TCRs. While the role of GluRs in communication between thymocytes and DCs was not entirely unexpected, we were surprised to find that an event associated with TCR triggering alone was sufficient to promote the clustering of both NMDAR and PSD-95 in the contact zone. We suggest that TCR signalling triggers the modeling of the nascent IS into a functionally competent synapse for glutamate signalling. Hence, our data indicated that NMDAR can be activated only in the context of a synapse (Figure 4a). An additional signal from the DC may be needed to activate NMDARs and/or the timing of exposure to glutamate relative to TCR activation may be critical to activate NMDARs.

Thus, we propose that the IS may provide a functional platform for coupling detection of TCR ligation and GluR activation with subsequent focal GluR signalling. PSD-95 may well be a primary adaptor, connecting TCR and NMDAR signalling. A challenge will be to understand the precise molecular mechanism of the interconnection between TCR and NMDAR signalling. Furthermore, a transient activation of NMDARs at the synapse would suggest that an important trafficking of these receptors could occur in thymocytes. Although our data indicate that PSD95 might be involved, the mechanism of NMDAR exocytosis at the synapse as demonstrated in neurones [START_REF] Collingridge | Receptor trafficking and synaptic plasticity[END_REF] remains to be solved.

Conclusions and clinical consequences

For years, glutamate signalling was thought to take place primarily in the CNS; it is now known to control key peripheral physiological functions, such as insulin secretion, keratinocyte differentiation, and remodeling of bone mass. Accordingly, therapeutic applications are being developed for diabetes, psoriasis, and osteoporosis 4 . Our findings provide insights into an additional site of peripheral glutamate signalling and possible involvement of NMDARs in a major physiolological function of the thymus, the negative selection of unwanted and potentially harmful developing T cells, preventing their export to the periphery [START_REF] Palmer | Negative selection--clearing out the bad apples from the T-cell repertoire[END_REF] . There is still a long road to solve the complexity of the cooperation of different GluRs in T-DC synapses, as defined in neuronal synapses.

Understanding the precise role of glutamate signalling in a diversity of immune responses and in determining T-cell fate by modulating the proliferation/differentiation balance, the effector/regulatory and effector/memory transitions, might reflect on the physiology of the adaptive immune response.

Our findings together with previous data [START_REF] Pacheco | Role of glutamate on T-cell mediated immunity[END_REF] raise potential applications for drug-mediated immunomodulation through interactions with L-glutamate signalling. Drugs targeting peripheral GluRs without affecting the CNS are under development. Conversely, the impact on immune competence of drugs used in the CNS should also be examined carefully. The idea that neurodegenerative diseases involving defects in glutamatergic transmission are often associated with immune dysregulation is emerging, but has not yet been fully explored. It would be very interesting to evaluate precisely the immune status of patients with such diseases.

MATERIAL AND METHODS

Mice

The generation and genotyping of HNT-TCR transgenic mice specific for the HA 126-138 (HNTNGVTAACSHE) peptide presented by MHCclass II I-A d have been described elsewhere [START_REF] Liblau | Intravenous injection of soluble antigen induces thymic and peripheral T-cells apoptosis[END_REF] . All animals had a B10.D2 (H2 d ) genetic background and were used at the age of 6 to 10 weeks. All animal experiments conformed to European Community guidelines of animal protection and local legal and ethical requirements. All experiments were approved by local authorities and direction des Services Veterinaires des Hauts-De-Seine (authorization for animal experimentation to SC-K, N°92-131). T cells and DCs were isolated, purified and cocultured using standard methods as described in Supplemental online Material and Methods.

Antibodies and reagents

All antibodies were proved to be specific and are widely used in the neuronal system. Antibodies against NR2A and NR2B were purchased from Upstate Biotechnology. Antibodies against NR1 (M68), VGLUT1, VGLUT2 and VGLUT3 used in flow cytometry or confocal microscopy were from Synaptic Systems. Anti-NR1 antibody M68, widely used in neurones, recognized a band at the expected molecular weight (116KD) in western blotting experiments of thymocytes extracts immunoprecipitated by anti-PSD-95 (data not shown). Biotinylated anti-CD3 (145-2C11) and anti-CD28 (37.51) were from Pharmingen. Antibodies against PSD-95 and glutamate were from Zymed and Sigma, respectively. All inhibitors and antagonists specific for GluRs were from Tocris.

DNA Chip analysis of the GluR repertoire

DNA chip analysis was performed on Neurotrans NT280M oligochips from GeneScore (www.genescore.fr), registered on GEO (Gene Expression Omnibus) (N° GPL4746), already used for studies in the CNS and containing oligonucleotides for 280 genes involved in neurotransmission

(2 oligonucleotides for each gene). [START_REF] Potier | Developement of microarrays to study gene expression in tissue and single cells: analysis of neural transmission[END_REF] Modifications are available in Supplemental online Material and Methods.

.

Single-cell Ca 2+ video imaging

Single-cell Ca 2+ video imaging on T cell-DC conjugates was carried out using a modified version of a previously described method. [START_REF] Delon | Antigen-dependent and -independent Ca2+ responses triggered in T cells by dendritic cells compared with B cells[END_REF] Purified DCs were plated on glass video imaging in isolated thymocytes and DCs, a modification of this protocol was used and is provided in Supplemental online Material and Methods.

Extracellular glutamate imaging

We monitored glutamate efflux in continuous culture, using a well characterized enzymatic assay. [START_REF] Innocenti | Imaging extracellular waves of glutamate during calcium signaling in cultured astrocytes[END_REF] Purified DCs (5 x 10 5 ) were incubated for 20 minutes on glass coverslips treated with polylysine, to allow adhesion. DCs were washed and bathed in the recording solution (mammalian saline as above, supplemented with NAD (1mM), and L-glutamate dehydrogenase (GDH) at 50 IU/ml (Sigma).

Glutamate released from the cells was immediately oxidized to α-ketoglutarate by GDH, with the formation of NADH and fluorescence emission at 450 nm. Controls without L-GDH indicated that detectable levels of NADH are not produced by DCs, either spontaneously or after induction by ionomycin. Agents were added directly to the culture dish.

Flow cytometry and confocal microscopy

Flow cytometry (four color flow cytometry, activated caspase-3 detection and calcium flux studies)

and confocal microscopy on T cell-DC conjugates were performed using standard procedures as described in Supplemental online material. The controls consisted of mouse isotype controls (as control for the monoclonal M68 anti-NR1, anti-Nur77 and anti-synaptotagmine antibodies), and purified rabbit IgG as control for anti-NR2A, NR2B, GluR2/3, PSD-95, KA2, glutamate, VGLUT antibodies. The same settings and exposure time were used for image acquisition by confocal microscopy of the specific staining and the controls.

Electron microscopy

Morphological analysis and post-embedding immunocytochemistry and immunogold labelling of T cell-DC conjugates was performed using classical techniques as described in Supplemental online Material and Methods. Mouse isotype controls and purified rabbit IgG were used as controls for immunogold staining, with the same exposure time and magnification as specific staining.

Statistical analysis

Data are expressed as means ± SEM. Statistical significance was assessed by the non-parametric Mann-Whitney test, using Prism, version 5.0, software (GraphPad Software, San Diego, CA). The significance level was set at P equals .05. The Kolmogorov-Smirnov test (Cellquest Software) was used to compare FACS profiles. (e) Representative traces of the Ca 2+ response in thymocytes in contact with HA-pulsed DCs MK801 (100 µM) was applied after the beginning of Ca 2+ response (t=208 s).

TITLES AND LEGENDS TO FIGURES

  delta T culture dishes (Bioptechs) coated with polylysine and maintained at 37°C in recording solution (mammalian saline: 116 mM NaCl, 5.6 mM KCl, 1.2 mM MgCl 2 , 2 mM CaCl 2 , 5 mM NaHCO 3 , 1 mM NaH 2 PO 4 , 20 mM HEPES, pH 7.3, supplemented with 2 g/l glucose). Thymocytes or CD4 + T cells were loaded for 30 minutes with 4 µM Fura-2/AM (Molecular Probes) by incubation for 30 minutes at 37°C in culture medium. When indicated T cells were preincubated for 10 minutes with the NMDAR antagonists and then dispensed on top of the DCs maintained at 37°C using a temperature-controlled dish, that was fixed above a warmed epifluorescence 40 x oil objective (Bioptechs) and a PTR 200 perfusion temperature regulator (ALA Scientific Instruments, Westbury, NY, USA). Successive fluorescence and DIC images were recorded for 20 minutes. Light for excitation was supplied by a high pressure 100 W xenon arc lamp and the 340 and 380 nm wavelengths were selected with a monochromator (Cairn Research Ltd). Fluorescence images were collected with a Sensicam QE CCD camera (PCO Computer Optics GmbH), digitized, and integrated in real time by an image processor (Metafluor©). The light transmission images were taken every 6 s, alternating with the fluorescent images. The fluorescent signals were analyzed offline, using an image processor (Metamorph©) to account for cell movements during the 20minute recordings. Background fluorescence was subtracted from the corresponding fluorescent images. To allow comparison between different labs, results are expressed as (ΔR/R), where R is the ratio (R) between fluorescence signals at 340 and 380 nm obtained before the addition of any agent, and ΔR the difference between the ratios measured during a response and R. For single cell Ca 2+

Figure 1 :

 1 Figure 1: T cells express a large array of i-and m-GluRs (a) Oligoarray (Neurotrans TM ) analysis

Figure 2 :

 2 Figure 2: NMDAR modulates Ca 2+ signalling in thymocytes in contact with DCs, in the

Figure 3 :

 3 Figure 3: NMDAR triggers a sustained calcium signal, is transiently activated and is involved

Figure 4 :

 4 Figure 4: TCR triggering is sufficient to induce NMDAR relocalization at the thymocyte-DC

Figure 5 :

 5 Figure 5: DCs may be the physiological source of glutamate in T cell-DC synapses

  with DCs[START_REF] Richie | Imaging synapse formation during thymocyte selection: inability of CD3zeta to form a stable central accumulation during negative selection[END_REF] . Our findings indicate that DCs are capable of fast calciumdependent glutamate release, while thymocytes express NMDARs, which modulate TCR-dependent Ca 2+ signalling and apoptosis only in a synaptic context, revealing a novel aspect of IS functioning.

Since glutamate mediates apoptosis of neuronal cells through NMDARs and Ca 2+ signalling

[START_REF] Arundine | Molecular mechanisms of calcium-dependent neurodegeneration in excitotoxicity[END_REF]

, we investigated thymocyte development, a fundamental process in which developing T cells that recognize self-antigens with high affinity are eliminated by Ca 2+ -dependent apoptosis following their close interaction

a Ca 2+ entry in T cell signalling at the IS Until

  , several questions are left unanswered. Does the IS use glutamate signalling? Is that signalling linked to Ca 2+ signals governing T-cell fate? How important is synapse structure in glutamatergic communication between T cells and DCs? To what extent are immunological and neuronal synapses functionally similar? recently, most of the studies exploring Ca 2+ entries in T cells were not carried out on T cells contacting APCs. We observed NR1 expression in the majority of resting thymocytes, which responded to DC contact with an immediate calcium signal, with a sustained plateau sensitive to NMDAR blockers. Our data suggest that this sustained Ca 2+ signal that occurs in thymocytes contacting Our observation that sustained Ca 2+ signalling and apoptosis occurred only in the context of IS and not in solitary thymocytes is in line with the need for a tight control of T-cell activation by antigen-specific contact with DC, to ensure proper thymic negative selection. Indeed, the IS is the physiological structure dedicated to communication in the immune system and represents a logical strategy to provide protection against permanent exposure of thymocytes to glutamate. Such a strategy could also make sense in activation of peripheral T cells which are

	The NMDAR,

DCs is probably not due to a sustained activity of the NMDAR. Similarly to the mechanism involved in synaptic placticity (Long Term Depression (LTD) and Long Term Potentiation (LTP)) of nervous synapses

[START_REF] Collingridge | Receptor trafficking and synaptic plasticity[END_REF]

there might be a transient activation of the NMDAR, which could act as a trigger of a sustained Ca 2+ response carried by other effectors. However, no detectable NMDAR signal was observed in resting thymocytes and in thymocytes stimulated with anti-CD3/CD28 coated beads (ie not engaged in a full synapse with DCs). It is physiologically relevant that the contact with DC may be necessary to activate NMDARs. exposed to serum glutamate concentrations. The sustained, high-level Ca 2+ plateau observed in this study was characteristic of thymocytes and correlated with antigen-dependent induction of apoptosis. It was not observed in naïve peripheral CD4 + splenic T cells that displayed glutamate signalling-dependent proliferation (P.A. and S.C-K unpublished data) but no apoptosis (Supplementary Figure

ACKNOWLEDGMENTS

We thank A. Trautmann and B. Malissen for helpful discussions and critical reading of the manuscript. We thank J.F. Renaud and A. Lombet for continuous support and N. Kerlero de Rosbo for reading the manuscript. We thank B. Lucas for helpful discussions. We thank C. Rücker-Martin and V. Nicolas for advice and assistance on confocal microscopy, L. Dauphinot for expertise in

CONFLICT OF INTEREST

The authors declare no conflict of interest