Stefan Le

Tai-Peng Tsai
email: ttsai@math.ubc.ca

FINITE AND INFINITE SOLITON AND KINK-SOLITON TRAINS OF NONLINEAR SCHR ÖDINGER EQUATIONS

Keywords: . 2010 Mathematics Subject Classification. 35Q55(35C08, 35Q51) soliton, multi-soliton, kink-soliton train, infinite soliton train, nonlinear Schrödinger equations

We will first review known results on multi-solitons of dispersive partial differential equations, which are special solutions behaving like the sum of many weakly-interacting solitary waves. We will then describe our recent joint work with Dong Li on nonlinear Schrödinger equations: Assuming the composing solitons have sufficiently large relative speeds, we prove the existence and uniqueness of a soliton train which is a multi-soliton composed of infinitely many solitons. In the 1D case, we can add to the infinite train an additional half-kink, which is a solution with a non-zero background at minus infinity.

Introduction

The nonlinear Schrödinger equation (nls) iu t + ∆u + f (u) = 0,

u(t = 0) = u 0 ,
where u is a complex valued function on R × R d and the nonlinearity f : C → C is phase covariant, i.e. there exists g : R + → R such that f (z) = g(|z| 2 )z, appears in various physical contexts, for example in nonlinear optics or in the modelling of Bose-Einstein condensates. Mathematically speaking, it is one of the model nonlinear dispersive PDE, along with the Korteweg-De Vries equation and the nonlinear wave equation. The local Cauchy theory in the energy space H 1 (R d ) for (nls) is well understood (see e.g. [START_REF] Cazenave | Semilinear Schrödinger equations[END_REF] and the references cited therein). In this survey we are interested in the long time dynamics of global solutions. Essentially, two effects may be at play at large time. First of all, if the nonlinearity is not too strong, the linear part of the equation can dominate and solutions may behave as if they were solutions to the free linear Schrödinger equation. This is the scattering effect. On the other hand, in some cases the nonlinear term dominates and the solution tends to concentrate, with possible blow-up in finite time. This is the focusing effect. At the equilibrium between these two effects, one may encounter many different types of structures that neither scatter nor focus. The most common of these non-scattering global structures are the solitons, but there exist also dark solitons, kinks, etc. A generic conjecture for nonlinear dispersive PDE is the Soliton Resolution Conjecture. Roughly speaking, it says that, as can be observed in physical settings, any global solution will eventually decompose at large time into a scattering part and well separated non-scattering structures, usually a sum of solitons. Apart from integrable cases (see e.g. [START_REF] Zakharov | Exact theory of two-dimensional self-focusing and onedimensional self-modulation of waves in nonlinear media[END_REF]), such conjecture is usually out of reach (see nevertheless the recent breakthrough [START_REF] Duyckaerts | Classification of radial solutions of the focusing, energy-critical wave equation[END_REF] on energy critical wave equation). Intermediate steps toward this conjecture are existence and stability results of configurations with well separated non-scattering structure, like multi-solitons, multi-kinks, infinite soliton and kink-soliton trains, etc. Our purpose in this paper is to review some of the existing results on this topic.

Solitons, dark solitons and kinks

We review in this section some of the known elementary non-scattering structure solutions of (nls).

Let us first consider special solitons, the standing waves. A standing wave is a solution to (nls) of the form u(t, x) = e iωt φ(x), where ω ∈ R and 0 = φ ∈ H 1 (R d ) is a localized solution to the elliptic stationary equation ( 1)

-∆φ + ωφ -f (φ) = 0.
Since the ground work of Berestycki and Lions [START_REF] Berestycki | Nonlinear scalar field equations I[END_REF][START_REF] Berestycki | Nonlinear scalar field equations II[END_REF], it is well known that (1) admits solutions in

H 1 (R d ) if ω > 0 and f (z) = g(|z| 2
)z verify the following hypotheses.

Assumption 1. (energy-subcritical) Let d ≥ 1. Suppose f (z) = g(|z| 2 )z with g ∈ C 0 ([0, ∞), R) ∩ C 2 ((0, ∞), R), g(0) = 0, |sg (s)| + |s 2 g (s)| ≤ C • (s α1/2 + s α2/2 ), ∀ s > 0,
where

C > 0, 0 < α 1 ≤ α 2 < α max , α max = +∞ if d = 1, 2, α max = 4 d-2 if d 3. (focusing) There exists s 0 > 0, such that G(s 0 ) := s0 0 g(s)ds > ωs 0 .
The profile solutions of (1) in H 1 (R d ) are in general called bound states. Among bound states, it is common to distinguish between the ground states and the excited states. Recall first that three quantities are conserved along the H 1 -flow of (nls): the energy, the mass and the momentum, defined as follows.

E(u) := 1 2 ∇u 2 2 - R d F (u)dx, F (z) := |z| 0 f (s)ds, M (u) := 1 2 u 2 2 , P (u) := R d ū∇udx.
The ground states are minimizers of the action related to (1) (defined by S = E + ωM ) and are in general positive, radial and unique (see [START_REF] Gidas | Symmetry of positive solutions of nonlinear elliptic equations in R n[END_REF][START_REF] Kwong | Uniqueness of positive solutions of ∆u -u + u p = 0 in R n[END_REF] and the recent progresses [START_REF] Byeon | Symmetry and monotonicity of least energy solutions[END_REF][START_REF] Mariş | On the symmetry of minimizers[END_REF]). For d = 1, there exist only ground states, whereas for d 2 there are infinitely many excited states.

A soliton is a standing wave of (nls) that has been given a speed thanks to a Galilean transform. Since (nls) is Galilean invariant, a soliton is still a solution of (nls). Explicitly, a soliton with frequency ω > 0, speed v ∈ R d , initial phase γ ∈ R and position x 0 ∈ R d has the form

R φ,ω,γ,x0,v := φ(x -vt -x 0 ) exp i 1 2 v • x - 1 4 |v| 2 t + ωt + γ .
The dynamical properties of solitons are mostly known when the nonlinearity f is of power-type f (z) = |z| α z and the profile is a ground state (we will refer to such solitons as ground state solitons, opposed to excited state soliton or in general bound state soliton). For L 2 -subcritical α (α < 4 d ), the ground state soliton is orbitally stable [START_REF] Cazenave | Orbital stability of standing waves for some nonlinear Schrödinger equations[END_REF] (i.e. stable up to phase shifts and translations), whereas for L 2 -critical and supercritical α (α = 4 d , α > 4 d ) it is unstable by blow-up [3, 37] (i.e. there exists an initial data in any neighborhood of the soliton such that the corresponding solution to (nls) blows up in finite time). In general, excited state solitons are expected to be unstable [START_REF] Grillakis | Linearized instability for nonlinear Schrödinger and Klein-Gordon equations[END_REF][START_REF] Jones | An instability mechanism for radially symmetric standing waves of a nonlinear Schrödinger equation[END_REF][START_REF] Mizumachi | Instability of bound states for 2D nonlinear Schrödinger equations[END_REF][START_REF] Mizumachi | Vortex solitons for 2D focusing nonlinear Schrödinger equation[END_REF][START_REF] Mizumachi | Instability of vortex solitons for 2D focusing NLS[END_REF].

For the focusing power-type nonlinearities, solitons are the only non-scattering solutions with a fixed profile, which is always localized. However, for other nonlinearities, there exist also other types of non-scattering solutions with a fixed profile, for example the so-called dark and grey solitons or the kinks.

A dark soliton is a travelling wave solution of (nls) of the form φ(x -ct), where c ∈ R d is the velocity and φ is a profile which has the particularity to be nonlocalized but with a constant modulus at infinity. The analysis of such type of solitons is much less developped than for localized (bright) solitons. Most of the works deal with the case where (nls) is the Gross-Pitaevskii equation, i.e. when f (z) = (1 -|z| 2 )z. In this case, existence of dark solitons in dimension 1 follows from direct computations. One may refer to [START_REF] Chiron | Travelling waves for the nonlinear Schrödinger equation with general nonlinearity in dimension one[END_REF][START_REF] Chiron | Stability and instability for subsonic traveling waves of the nonlinear Schrödinger equation in dimension one[END_REF] and the reference cited therein for a study of the existence and stability of dark solitons with generic nonlinearities in dimension 1. In higher dimension, existence of dark solitons has been a long time open problem. In the recent breakthrough [START_REF] Mariş | Traveling waves for nonlinear Schrödinger equations with nonzero conditions at infinity[END_REF], it has been proved that, for generic nonlinearities and in dimension d 3, dark solitons exist for any speed between 0 and the speed of sound ( √ 2 in the case of the Gross-Pitaevskii nonlinearity). A kink is a soliton-type solution to (nls) when d = 1,

K φ K ,ω,c,γ,x0,v := φ K (x -(c + v)t -x 0 ) exp i 1 2 v • x - 1 4 |v| 2 t + ωt + γ ,
but with a profile φ K which has different limits at -∞ and +∞. Here c is the intrinsic velocity associated to φ K . When f is the Gross-Pitaevskii nonlinearity, i.e. f (z) = (1 -|z| 2 )z, there exists an explicit family of kink solutions (which are particular cases of dark solitons) given by

K(t, x) = φ K (x -ct), (2)
φ K (x) = 2 -c 2 2 tanh x √ 2 -c 2 2 + i c √ 2 , |c| < √ 2.
In this paper, we will be particularly interested in cases where the kink profile φ has different limits with different modulus at -∞ and +∞. This will be the case if the nonlinearity f verifies the following assumption.

Assumption 2. For some ω 0 > 0, there is a first b > 0 such that for h

(s) = ω 0 s -f (s), h(b) = 0, h (b) > 0, b 0 h(s)ds = 0.
-φ K + ω 0 φ K -f (φ K ) = 0, lim x→-∞ φ(x) = b, lim x→+∞ φ(x) = 0.
To our knowledge, except in our works [START_REF] Le Coz | Fast-moving finite and infinite trains of solitons for nonlinear Schrödinger equations[END_REF][START_REF] Coz | Infinite soliton and kink-soliton trains for nonlinear Schrödinger equations[END_REF], such kinks have never been investigated in the analysis of the long time behavior of solutions to (nls). One reason for that is that they are genuinely infinite energy solutions and that no proper renormalization exists to make them energy finite.

Multi-solitons, infinite soliton trains and soliton-kink solutions

As mentioned in Section 1, it is expected that global solutions to (nls) will eventually decompose into a scattering part and well separated non-scattering structures. In this section, we review the existence and stability results for solutions composed of several of the elementary non-scattering structures described in Section 2. The basic example is the so-called multi-soliton, a solution of (nls) build upon a finite number of solitons, Figure 1 contains a schematic representation of a multi-soliton. To be a little more precise, let

(3) R(t, x) = N j=1 R φj ,ωj ,γj ,xj ,vj (t, x) =: N j=1 R j (t, x),
where each R j is a soliton made from some parameters (ω j , γ j , x j , v j ) and bound state φ j . As (nls) is a nonlinear equation, R is not a solution. What we call a multi-soliton is a solution u of (nls) such that

(4) lim t→+∞ u -R X([t,+∞)×R d ) = 0,
where X is some space-time function space, e.g.

L ∞ ([0, +∞), L 2 (R d )).
3.1. The integrable cases. The first result of existence of multi-solitons was obtained in Zakharov and Shabat [START_REF] Zakharov | Exact theory of two-dimensional self-focusing and onedimensional self-modulation of waves in nonlinear media[END_REF] in the case of the 1-d focusing cubic (i.e. d = 1, f (z) = |z| 2 z) nonlinear Schrödinger equation via the inverse scattering method. Indeed, in this particular case the equation is completely integrable and one can obtain multi-solitons in a rather explicit manner. The companion situation to the 1-d cubic case is the Gross-Pitaevskii case, i.e. d = 1 and f (z) = (1 -|z| 2 )z. In that case (nls) is again completely integrable, however no localized solitons exist. The counter parts of the solitons in this case are the dark solitons (2), which have modulus 1 at infinity. In this situation, it is also possible to show via the inverse scattering transform the existence of a solution of (nls) behaving at large time as decoupled well-separated dark solitons. Note that in that case, due to the non-zero condition at infinity, the profile cannot be given simply as the sum of the dark solitons.

For the 1-d cubic (nls), Kamvissis [START_REF] Kamvissis | Focusing nonlinear Schrödinger equation with infinitely many solitons[END_REF] showed that it is possible to push the inverse scattering analysis forward and obtain the existence of an infinite soliton train, i.e. a solution u of (nls) defined as in (4) but with N = +∞ in the definition of R. In fact, the result given in [START_REF] Kamvissis | Focusing nonlinear Schrödinger equation with infinitely many solitons[END_REF] is sharper: it is shown that, if d = 1, f (z) = |z| 2 z and under some technical hypotheses, any solution to (nls) with initial data in the Schwartz class will eventually decompose at large time as an infinite soliton train and a "background radiation component". The existence of an infinite train of dark solitons has not been proved for the Gross-Pitaevskii case.

Other exotic solutions are available for 1-d cubic NLS, for example breather solutions which are periodic in space and whose magnitudes approach uniform values as |t| → ∞, see e.g. [START_REF] Akhmediev | First-order exact solutions of the nonlinear Schrödinger equation[END_REF].

3.2.

Existence of multi-solitons, the energy method. Apart from the case previously mentioned, (nls) is not completely integrable and it is generically not possible to apply the inverse scattering method to obtain the existence of multisolitons. The first existence result of multi-solitons in a non-integrable setting was obtained by Merle in [START_REF] Merle | Construction of solutions with exactly k blow-up points for the Schrödinger equation with critical nonlinearity[END_REF] as a by-product of the proof of existence of multiple blowup points solutions for L 2 -critical (nls), i.e. f (z) = |z| 4 d z. The techniques initiated in [START_REF] Merle | Construction of solutions with exactly k blow-up points for the Schrödinger equation with critical nonlinearity[END_REF] were then developed in [START_REF] Côte | High-speed excited multi-solitons in nonlinear Schrödinger equations[END_REF][START_REF] Côte | Construction of multi-soliton solutions for the L 2supercritical gKdV and NLS equations[END_REF][START_REF] Martel | Multi solitary waves for nonlinear Schrödinger equations[END_REF] to obtain the following result.

Theorem 1 (Existence of multi-soliton by energy method, [START_REF] Côte | High-speed excited multi-solitons in nonlinear Schrödinger equations[END_REF][START_REF] Côte | Construction of multi-soliton solutions for the L 2supercritical gKdV and NLS equations[END_REF][START_REF] Martel | Multi solitary waves for nonlinear Schrödinger equations[END_REF]). Assume f (z) = |z| α z with 0 < α < α max . Let R be the profile given in (3). Let ω and v be given by

ω = 1 2 min {ω j , j = 1, ..., N } , v = inf j,k=1,...,N,j =k |v j -v k |.
There exist µ = µ(d, N ) and v := v (φ 1 , ..., φ N ) 0 such that if v > v then there exist

T 0 ∈ R and a solution u ∈ C([T 0 , +∞), H 1 (R d )) of (nls) satisfying u(t) -R(t) H 1 e -µ √ ω v t , ∀t ≥ T 0 .
If in addition all φ j are ground states, then the result holds with v = 0, i.e., any v * > 0.

For the sake of simplicity, we have stated the result only for power-type nonlinearities, but its first part is in fact valid for any C 1 nonlinearity verifying hypotheses a little stronger than Assumption 1. As mentioned before, the case α = 4 d was treated by Merle [START_REF] Merle | Construction of solutions with exactly k blow-up points for the Schrödinger equation with critical nonlinearity[END_REF], the ground state case by Martel and Merle [START_REF] Martel | Multi solitary waves for nonlinear Schrödinger equations[END_REF] for α < 4 d and Côte, Martel and Merle [START_REF] Côte | Construction of multi-soliton solutions for the L 2supercritical gKdV and NLS equations[END_REF] for α > 4 d , and the excited state case by Côte and Le Coz [START_REF] Côte | High-speed excited multi-solitons in nonlinear Schrödinger equations[END_REF].

The proofs in [START_REF] Côte | High-speed excited multi-solitons in nonlinear Schrödinger equations[END_REF][START_REF] Côte | Construction of multi-soliton solutions for the L 2supercritical gKdV and NLS equations[END_REF][START_REF] Martel | Multi solitary waves for nonlinear Schrödinger equations[END_REF] follow a similar scheme. The idea is to choose an increasing sequence of time (T n ) with T n ↑ +∞ and consider the solutions (u n ) to (nls) which solve the equation backward in time with final data u n (T n ) = R(T n ). The sequence (u n ) is an approximate sequence for a multi-soliton. To show its convergence, two arguments are at play. First, one shows that there exists a time T 0 independent of n such that u n satisfies on [T 0 , T n ] the uniform estimates

(u n -R)(t) H 1 e -µ √ ω v t .
Second, we have compactness of the sequence of initial data u n (T 0 ), i.e. there exists

u 0 ∈ H 1 (R d ) such u n → u 0 strongly in H s (R d ), 0 s < 1.
To prove the uniform estimates, one first recall that the second derivative of the action around a soliton is coercive up to some L 2 scalar products. Precisely, given a soliton R 0 with parameters (ω 0 , γ 0 , x 0 , v 0 ), the action is given by

S = E + ω 0 + |v 0 | 2 4 M + v 0 • P
and there exist K ∈ N and (ξ 0 k ) k=1,...,K such that we have for any

ε ∈ H 1 (R d ) S (R 0 )ε, ε ε 2 H 1 - K k=1 | ε, ξ 0 k 2 | 2 .
Hence to control the difference ε = u n -R, it is enough to construct a functional S similar to S but adapted to the multisoliton profile R and to get rid of the bad L 2 scalar products. The construction of the functional is done by gluing together each functional S j suitably localized. The localization is possible since each bound state, henceforth each soliton, is exponentially decaying at infinity. Getting rid of the bad L 2 scalar products is the trickiest part. In [START_REF] Martel | Multi solitary waves for nonlinear Schrödinger equations[END_REF], the authors modulated the solitons in scaling, translation and phase to cancel the scalar products. This was possible only because they were in the L 2 -subcritical case with ground states. In [START_REF] Côte | Construction of multi-soliton solutions for the L 2supercritical gKdV and NLS equations[END_REF], the authors used a topological argument to select a final data which was not R(T n ) but close enough to it, and such that for this final data the bad scalar products were vanishing. In [START_REF] Côte | High-speed excited multi-solitons in nonlinear Schrödinger equations[END_REF], a bootstrap argument on the L 2norm of ε was used to get an a priori control of the type | ε, ξ 0 k 2 | 1 v ε 2 , hence allowing to control the scalar products for v large enough.

The compactness argument is based on the virial identity and does not present major difficulty.

The energy method is very flexible and can be adapted to many other situations, e.g. for multi-solitons of Klein-Gordon equations [START_REF] Bellazzini | Multi-solitary waves for the nonlinear Klein-Gordon equation[END_REF][START_REF] Côte | Multi-solitons for nonlinear Klein-Gordon equations[END_REF] or multi-speeds solitary waves of Schrödinger systems [START_REF] Ianni | Multi-speed solitary wave solutions for nonlinear Schrödinger systems[END_REF]. It suits very well situations with a finite number of well localized solitons with finite energy. However, its implementation is far from being trivial when the number of solitons is infinite or when one soliton is replaced by a kink. In [START_REF] Le Coz | Fast-moving finite and infinite trains of solitons for nonlinear Schrödinger equations[END_REF][START_REF] Coz | Infinite soliton and kink-soliton trains for nonlinear Schrödinger equations[END_REF], the authors have developped an approach suitable to situations where the energy technique fails to be directly applicable.

3.3.

Existence of infinite trains and kink-solitons solutions, the fixed point argument. To prove the existence of solutions which have a priori infinite energy such as multi-solitons with kinks attached at both ends or infinite trains of solitons, an approach based on a fixed point argument around the desired profile has been followed in [START_REF] Le Coz | Fast-moving finite and infinite trains of solitons for nonlinear Schrödinger equations[END_REF][START_REF] Coz | Infinite soliton and kink-soliton trains for nonlinear Schrödinger equations[END_REF].

We briefly recall the definition of the Strichartz space. For dimension d ≥ 1, a pair of exponents (q, r) is said (Schrödinger) admissible if

(5) 2 q + d r = d 2 ,
2 ≤ q, r ≤ ∞, and (d, q, r) = (2, 2, ∞).

On I × R d where I ⊂ R, we define the Strichartz norm (6) u S(I) := sup

(q, r) admissible u L q t L r x (I×R d ) .
For d = 2, we also need to impose q > q 1 in the above norm for some q 1 slightly larger than 2 to stay away from the forbidden endpoint. The Strichartz space S(I) is the closure of all test functions in R × R d under this norm. We denote by N (I) the dual space of S(I).

Let us first state a result on the existence of infinite soliton trains. Its conditions are far from optimal for simplicity of presentation. The same method can be applied to construct finite and infinite soliton trains for more general nonlinearities.

Theorem 2 (Existence of infinite soliton trains [START_REF] Le Coz | Fast-moving finite and infinite trains of solitons for nonlinear Schrödinger equations[END_REF][START_REF] Coz | Infinite soliton and kink-soliton trains for nonlinear Schrödinger equations[END_REF]

). Let d ≥ 1. Assume f (z) = |z| α z + g 2 (|z| 2 )z where g 2 ∈ C 0 ([0, ∞), R) ∩ C 2 ((0, ∞), R), g(0) = 0, |sg 2 (s)| + |s 2 g 2 (s)| ≤ C(s α1.5/2 + s α2/2 ), ∀ s > 0, where C > 0, 0 < α < α 1.5 ≤ α 2 < α max . There exist r 0 > max(1, dα 2 ), c 1 > 0 and v 1 such that, if an infinite soliton train profile R ∞ is given by R ∞ = ∞ j=1 R j ,
with parameters ω j > 0, γ j ∈ R, x j = 0, v j ∈ R d satisfying • (uniform bound for bound states) for some 0 < a < 1 and C > 0,

|φ j (x)| + ω -1/2 j |∇φ j (x)| ≤ Cω 1/α j e -aω 1/2 j |x| , ∀x ∈ R d , ∀j ∈ N, • (Integrability)
∞ j=1 ω 1 α -d 2r 0 j < ∞,
• (High relative speeds)

v * = inf j,k∈N,j =k √ ω j |v k -v j | ≥ v ,
• (Gradient bound)

V * = j∈N v j ω 1 α -d 4 j < ∞ if α < α 2 2 + α 2 .
Then there exists a solution u of (nls) satisfying, for some

T 0 ≥ 0, u -R ∞ S([t,∞)) ≤ e -c1v t , ∀t ≥ T 0 .
Moreover, such profiles R ∞ do exist for every such nonlinearity f . Note that, in the integrable case, it was proved that an integrability condition on the parameters of the infinite trains is necessary for their existence, see [START_REF] Kamvissis | Focusing nonlinear Schrödinger equation with infinitely many solitons[END_REF]Remark 2].

We now switch to dimension d = 1 to investigate the existence of solutions of (nls) composed of kinks and solitons as represented in Figure 2.

x 0 + v 0 t x 1 + v 1 t x 3 + v 3 t x 4 + v 4 t x 5 + v 5 t x 2 + v 2 t K 0 R 1 R 3 R 4 K 5 R 2 v 0 v 1 v 3 v 4 v 5 v 2 Figure 2
. Schematic representation of a kink-soliton train Theorem 3 (Existence of finite kink-soliton trains [START_REF] Le Coz | Fast-moving finite and infinite trains of solitons for nonlinear Schrödinger equations[END_REF]). Let d = 1. Assume that f is an energy subcritical nonlinearity such that there exists two kink profiles φ K0 , φ K N +1 with φ K0 and φ K N +1 exponentially localized and the boundary conditions

lim x→-∞ φ K0 (x) = 0, lim x→+∞ φ K0 (x) = 0, lim x→-∞ φ K N +1 (x) = 0, lim x→+∞ φ K N +1 (x) = 0.
Assume also the existence of N soliton profiles φ 1 , . . . , φ N . Define the profile

W (t, x) := K 0 (t, x) + N j=1 R j (t, x) + K N +1 (t, x).
with parameters (v j , x j , ω j , γ j ) j=0,...,N +1 ⊂ R 4 such that v 0 < • • • < v N +1 . Define v by v := inf{|v j -v k |; j, k = 0, . . . , N + 1, j = k}. Then there exist v > 0 (independent of (v j )) large enough, T 0 1 and constants µ 1 , µ 2 > 0 such that if v > v , then there exists a (unique) multi-kink solution u ∈ C([T 0 , +∞), H 1 loc (R)) to (nls) satisfying on [T 0 , +∞) the estimate e µ1v t u -W S([t,+∞)) + e µ2v t ∇(u -W ) S([t,+∞)) 1.

In the above theorem, one kink can be dropped. We may also have a kink attached to an infinite soliton train. For simplicity we choose a special nonlinearity. See [START_REF] Coz | Infinite soliton and kink-soliton trains for nonlinear Schrödinger equations[END_REF] for general assumptions on f . Theorem 4 (Existence of infinite kink-soliton trains [START_REF] Coz | Infinite soliton and kink-soliton trains for nonlinear Schrödinger equations[END_REF]). Let d = 1 and

f (u) = |u| α u -|u| β u, 0 < α < β < ∞.
It satisfies Assumption 2 and there is a kink profile φ K . If either 0 < α < 4/3, or 4/3 ≤ α < √ 2 < β = 2/α, then there exist r 0 > 1, c 1 > 0 and v 1 such that, if an infinite kink-soliton train profile

W = K + R ∞ = K + ∞ j=1 R j
has its parameters satisfying the same assumptions as in Theorem 2 (with index j starting from 0), then there exists a solution u of (nls) satisfying, for some T 0 ≥ 0,

u -W S([t,∞)) ≤ e -c1v t , ∀t ≥ T 0 .
Moreover, such profiles W do exist.

The strategy for the proofs of Theorems 2, 3 and 4 is the following. Let W be a profile around which we want to build a solution. W can be an infinite train, a kink-soliton train, etc. Since W may be badly localized, we look for a solution of (nls) in the form u = W + η, where η satisfies the perturbation equation ( 7)

i∂ t η + ∆η + f (W + η) -f (W ) + H = 0,
where H is a source term coming from the fact that W is not an exact solution, e.g.

H = f (R ∞ ) - +∞ j=1 f (R j )
in the case of an infinite soliton train. In Duhamel formulation, the perturbation equation for η becomes

η(t) = -i ∞ t e i(t-τ )∆ f (W + η) -f (W ) + H dτ,
and the core of the proof is to perform a fixed point argument for this formulation. Here, two approaches are possible: One is based on a combination of the dispersive estimate

e it∆ u p |t| -d( 1 2 -1 p ) u p p-1 , ∀ t = 0, ∀p ∈ [2, +∞],
and Strichartz estimates

u S((t0,∞)) u 0 2 + F N ((t0,∞)) for i∂ t u + ∆u = F, u(t 0 ) = u 0 .
The other uses only Strichartz estimates, but for both u and ∇u.

3.4. Open problems. We conclude this paper by reviewing some open problems related to the exotic solutions presented here.

3.4.1. Uniqueness. In Theorem 1, no uniqueness is proved for the multi-soliton. In Theorems 2, 3 and 4, the solution presented is unique in the class of solutions satisfying a strong decay estimate towards the desired profile; it does not preclude the possibility of existence of the same type of solution, but with a weaker decay towards the desired profile. In fact, it was proved in [START_REF] Côte | High-speed excited multi-solitons in nonlinear Schrödinger equations[END_REF] that as soon as one of the composing soliton is linearly unstable, then there exists a one parameter family of multi-solitons converging toward the same profile. Hence in such cases uniqueness is not to be expected. However, one may hope that classification results hold, e.g. one may expect that for L 2 -supercritical power-type nonlinearities, the multi-solitons converging toward a fixed N -sum of solitons form a N -parameter family. Such results were obtained for the Korteweg-de Vries equation by Combet [START_REF] Combet | Multi-soliton solutions for the supercritical gKdV equations[END_REF]. See also [START_REF] Combet | Multi-existence of multi-solitons for the supercritical nonlinear Schrödinger equation in one dimension[END_REF] for partial results in that direction for (nls). For the kink-solitons solutions or the infinite trains, no result is available yet.

3.4.2. Stability. Another natural question coming to mind when investigating the exotic solutions of nonlinear Schrödinger equation is their stability. Again, the only available results concern finite multisolitons and the problem is completely open for infinite trains or kink-soliton solutions. For power-type nonlinearities, the only case where orbital stability of a multi-soliton holds is the Gross-Pitaevskii case, as has been proved by Béthuel, Gravejat and Smets for multi-dark-solitons in [START_REF] Béthuel | Stability in the energy space for chains of solitons of the one-dimensional Gross-Pitaevskii equation[END_REF]. Stability of multi-solitons has been proved under restrictive hypotheses in [START_REF] Martel | Stability in H 1 of the sum of K solitary waves for some nonlinear Schrödinger equations[END_REF] for orbital stability and in [START_REF] Perelman | Some results on the scattering of weakly interacting solitons for nonlinear Schrödinger equations[END_REF][START_REF] Perelman | Asymptotic stability of multi-soliton solutions for nonlinear Schrödinger equations[END_REF][START_REF] Rodnianski | Asymptotic stability of N -soliton states of nonlinear Schrödinger equations[END_REF] for asymptotic stability. The hypotheses on the nonlinearity (e.g. high regularity or flatness at the origin) exclude in particular the power-type nonlinearities.

3.4.3. Multikinks. One particular feature of the kinks considered in Theorem 3 is that they converge to 0 on one side. There exist however nonlinearities such that the kinks are connecting non-zero constants on both sides, for example -1 and 1 for the black solitons of Gross-Pitaevskii. There may also exist situations with kinks connecting e.g. 0 to 1 and kinks connecting 1 to 2, etc. One would expect that it is possible to construct solutions by gluing together those kinks to get a solution with modulus having increasing terraces shape. Such solutions have never been constructed for nonlinear Schrödinger equation. One reason for that is the current lack of appropriate ansatz for such terraces shape solutions.

Many other related open problems exist, e.g. constructing solutions when one of the composing element is a line-soliton in 2-d.

Figure 1 .

 1 Figure 1. Schematic representation of a multi-soliton

The research of S. Le Coz is supported in part by the french ANR through project ESONSE. The research of Tsai is supported in part by NSERC grant 261356-13 (Canada).


MS a , Sébastien Imbert

MS b , Priscillia Bresler

PhD Alexandre Boissonnas

PharmD Lauraine Gauthier

MD c , Laëtitia Souchet

MD Madalina Uzunov

Véronique Leblond

MD, PhD Dominique Mazier

MD, PhD Stéphanie Nguyen

PharmD Arnaud Fekkar
email: arnaud.fekkar@aphp.fr

MD, PhD Véronique Le- Blond

Calcineurin inhibitors impair neutrophil activity against Aspergillus fumigatus in allogenic hematopoietic stem cell transplant recipients

Keywords: Innate immunity, Calcineurin inhibitor, Invasive fungal infection, Immune reconstitution, Transplantation ABBREVIATIONS HSCT: hematopoietic stem cell transplant GvHD: graft versus host disease NETs: neutrophil extracellular traps ROS: reactive oxygen species

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

INTRODUCTION

Aspergillus fumigatus is the main causative agent of invasive aspergillosis, which is a major threat to immunocompromised patients, including hematopoietic stem cell transplant (HSCT)

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recipients. Polymorphonuclear neutrophil cells (called neutrophils hereafter) are key effectors against fungal infection. In contrast to monocytes and macrophages, which phagocyte resting conidia, neutrophils are also able to act against germinating conidia and hyphae through a trapping mechanism, giving them particular importance against Aspergillus 1, 2 . However, only a few studies have provided data on the behavior of neutrophils following HSCT, and additionally, how these cells regain their basic functions after the transplant is unknown.

The choice of immunosuppressors for the prevention of graft versus host disease (GvHD) can vary depending on the conditioning regimen, the type of donor and the type of graft. The most common prophylaxis strategy associates a calcineurin inhibitor (mainly cyclosporine A) with methotrexate (MTX), anti-thymoglobulin antibodies (ATG) or mycophenolate mofetil (MMF) [START_REF] Choi | Current and emerging strategies for the prevention of graftversus-host disease[END_REF] . Despite those immunosuppressive drugs, acute GvHD occurs in approximately 40-60% of HSCT patients [START_REF] Choi | Current and emerging strategies for the prevention of graftversus-host disease[END_REF] and requires the administration of corticosteroids [START_REF] Choi | Current and emerging strategies for the prevention of graftversus-host disease[END_REF] . The action of cyclosporine A, which is essentially known for its effect on adaptive immunity, is mediated by inhibition of calcineurin, a calcium-dependent phosphatase. Particularly, calcineurin activates the NFAT transcription factors, leading to the transcription and production of many T-cell effector cytokines such as interleukin 2 [START_REF] Samaniego | Drug insight: maintenance immunosuppression in kidney transplant recipients[END_REF][START_REF] Greenblatt | Calcineurin regulates innate antifungal immunity in neutrophils[END_REF][START_REF] Northrop | NF-AT components define a family of transcription factors targeted in T-cell activation[END_REF] . Very few data exist concerning its potential role on innate immunity. Patients receiving cyclosporine run an increased risk of viral, bacterial or fungal infection. Among these pathogens, the mold Aspergillus is particularly dangerous, causing high morbidity and mortality. The administration of cyclosporine or other recognized T-cell immunosuppressors is included in the host criteria for probable invasive fungal disease as established by the European Organisation for Research and Treatment of Cancer (EORTC) and the Mycoses Study Group (MSG) [START_REF] De Pauw | Revised definitions of invasive fungal disease from the European Organization for Research and Treatment of Cancer/Invasive Fungal Infections Cooperative Group and the National Institute of Allergy and Infectious Diseases Mycoses Study Group (EORTC/MSG) Consensus Group[END_REF] .

Thus, for the present study, we aimed at evaluating neutrophil function and activity toward Aspergillus in HSCT recipients over time. We assessed neutrophil activity, including fungal

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growth inhibition, oxidative burst, surface molecule expression, and neutrophil extracellular traps (NETs) production, following different stimuli.

MATERIALS AND METHODS.

Patients and donors. The present longitudinal study involved 37 patients who received a related (36/37) or unrelated (1/37) allogeneic HSCT for a malignant hemopathy in La Pitié-Salpêtrière Hospital, Paris, France. Blood samples were collected from the recipients at engraftment, then at two, six and ten months post-transplant, and were compared to samples taken from their respective donors. This non-interventional study was approved by the local ethics committee (CPP Ile de France IV). Signed informed consent was obtained from all donors and recipients.

Neutrophil isolation. Neutrophils were isolated using the dextran-Ficoll method. Briefly, whole fresh blood was mixed with an equivalent volume of 2.0% dextran solution (Sigma Aldricht) in normal saline and the red blood cells were allowed to settle for 40 minutes at 4°C. Then the leucocyte-rich supernatant was submitted to Ficoll (Eurobio) centrifuge separation for 30 minutes at 700 g; 4°C. After elimination of the remaining red blood cells, neutrophils in the pellet were recovered in RPMI medium and tested immediately.

Aspergillus fumigatus strain. An A. fumigatus sensu stricto strain isolated from clinical samples in La Pitié-Salpêtrière Hospital was used. The strain was maintained on Sabouraud with chloramphenicol and gentamicin agar tubes at 37° for 5-7 days. Conidia were harvested with phosphate-buffered-saline (PBS) containing 0.05% Tween 20, washed three times and suspended in PBS and counted.

Aspergillus growth inhibition. Black, 96-well clear-bottom plates (Greiner) were seeded with 1,500 conidia per well in RPMI medium containing 1% fetal calf serum (FCS) and allowed to germinate for 7 hours at 37°C. After this growth period, Aspergillus measuring ap-

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proximately 15-20 µm can be considered either as germinating conidia or as small hyphae.

The medium was then changed to RPMI without FCS and the isolated neutrophils were added to the wells at different effector:target ratios in triplicate. In others experiments, neutrophils were added directly to resting conidia. The plates were incubated overnight at 37°C with 5% CO 2 then washed with purified water. Uvitex (1% w/v), a fluorescent marker of chitin (similar to calcofluor, which has already been used to assess fungal biomass [START_REF] Leal | Fungal antioxidant pathways promote survival against neutrophils during infection[END_REF] ) was added to the final dilution. Finally, plates were read using a Flexstation analyzer with excitation at 350 nm and emission at 435 nm. For a given effector:target ratio, the fungal growth was determined as the ratio of the fluorescence intensity of the well containing neutrophils mixed with Aspergillus to that of the well containing Aspergillus only. The percentage of inhibition was defined as 100 minus the percent of fungal growth.

Surface molecule expression of neutrophils.

Five-hundred-microliter whole-blood samples were stimulated with either 10 6 Aspergillus conidia (resting or germinating), 5 ng/mL bacterial lipopolysaccharide (LPS) (Sigma Aldrich) or PBS as control for 45 minutes at 37°C. Neutrophils were stained with an anti-human CD11b (or integrin alpha M) antibody (Dako), an anti-human CD62L (or L-selectin) antibody (Becton Dickinson) and/or an anti-human CD66 (or carcinoembryonic antigen) antibody (Becton Dickinson) before cytometry analysis. TLR and dectin expression were assessed using anti-TLR2, anti-TLR4 and anti-dectin-1 antibodies (RD Systems). Cytometry was performed on a Gallios flow cytometer and results were analyzed using Kaluza software (Beckman Coulter).

Measurement of neutrophil oxidative burst.

Neutrophils contained in 500 µL heparinized whole-blood samples were incubated with hydroethidine (Sigma Aldrich) (final concentration 1.5 µg/mL) for 15 minutes at 37°C, then stimulated with either 10 6 Aspergillus conidia (resting or germinating), 5 ng/mL LPS or PBS as control for 45 minutes at 37°C. Then PMA (final Video microscopy and NETs formation assessment. Interactions between Aspergillus and neutrophils were visualized using a Zeiss Axio Microscope (Carl Zeiss, Germany). After 3 hours of co-culture, Sytox green (Life Technologies) was added to each well at a final dilution of 1/5000. Images were processed and hyphal length was measured using ImageJ software.

Quantification of NETs formation was evaluated as previously described [START_REF] Papayannopoulos | Neutrophil elastase and myeloperoxidase regulate the formation of neutrophil extracellular traps[END_REF] with ImageJ.

Statistical analysis. GraphPad Prism 5 was used for statistical analyses (GraphPad software, La Jolla, Calif).

RESULTS.

Patient characteristics.

Patient data are presented in Table 1. Mean donor and recipient ages were 47 (range 19-67) and 44 (20-69) years respectively. There were 23 male and 13 female donors and 19 male and 18 female recipients. No significant differences were observed concerning age (p=0.25 by Student test) or sex ratio (p=0.39 by Chi-square test). The main single indication for HSCT was acute myeloid leukemia (40.5 % of patients) but lymphoproliferative disorders (i.e. lymphoma, lymphoid leukemia and myeloma) collectively accounted for 43% of cases. Other diseases included primary myelofibrosis and myelodysplastic syndrome.

The majority of patients (86.5%) were transplanted with a matched related donor. Conditioning regimens were mainly busulfan-based with reduced intensity (54.1%) or myeloablative (29.7%). All patients received GvHD prophylaxis involving a calcineurin inhibitor-based regimen plus other drugs depending on the type of graft and conditioning.

Neutrophils were collected from patients in the first month (recovery from neutropenia) and at two, six and ten months post-HSCT. Recovery from neutropenia ("recovery" hereafter) was

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defined as the day where the neutrophil count became >500/mm3 (although for one patient we found a posteriori that this value was not reached). At recovery (median 20 days), all patients were receiving a calcineurin inhibitor, usually cyclosporine (88.6%). No patients had corticosteroid therapy. Due to patient death or loss to follow-up, 10-month samples were available for only 10 patients. The characteristics of blood samples and immunosuppressive drugs regimens at one, two, six and ten months post-HSCT are presented in Table 2. Only donor-derived cells were detectable for all included recipients after approximatively two months post-transplant (data not shown).

Neutrophil surface molecule expression after HSCT

The surface expression of CD11b (also known as integrin alpha M), CD62L (or L-selectin) and CD66 was evaluated at the basal level and following stimulation by resting/germinating conidia or LPS. In association with the β 2 integrin (or CD18), CD11b forms the heterodimeric integrin macrophage-1 antigen involved in the adhesion and migration of leukocytes. CD11b is expressed at the surface of neutrophils after degranulation as it is contained in secondary and tertiary neutrophil granules. CD62L is involved in transient tethering of the neutrophils to the endothelial surface. The shedding of CD62L marks an activation of the neutrophils. CD66 is specific to secondary granules [START_REF] Ducker | Subcellular localization of CD66, CD67, and NCA in human neutrophils[END_REF] . Surface expression of the major pattern recognition receptors TLR-2, TLR-4 and dectin-1 were also assessed by flow cytometry. Neutrophils were found to be activated in only a limited manner by resting conidia but they were strongly activated by germinating conidia, as evidenced by an increase in CD11b and CD66 expression and a decrease in CD62L expression (Figure 1a and data not shown). As tested, LPS induced high activation. There were however no observed differences over time. It is also noteworthy that no differences were observed between donors and patients during recovery except as concerns the expression of CD66, which was higher after the graft (Figure 1 a-b). In compari-

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son with the donors, the expression of TLR4 and dectin did not change but TLR2 diminished slightly in recipients during recovery (Figure 1c).

Neutrophil reactive oxygen species production after HSCT.

As it is a major anti-pathogen mechanism of neutrophils, the production of reactive oxygen species (ROS) was analyzed at a basal level and following exposition to resting/germinating conidia or LPS. Resting conidia caused a moderate production of ROS while germinating conidia led to an important oxidative burst (Figure 1 d-e). The level of ROS production was highly dependent on the stimulus (p<0.0001 by 2-way ANOVA), but it remained similar between recipients and their donors, and stable over time (p=0.96).

Neutrophil inhibition of Aspergillus hyphae growth.

Samples were available from 23 donors and 33 recipients at recovery to test Aspergillus hyphae growth inhibition by neutrophils. This test set also permitted 19 paired sample comparisons. Unpaired (figure 2a) and paired (figure 2b) analysis showed a highly statistically significant decrease of the ability of neutrophils to hamper fungal growth during the recovery, compared with healthy donors.

Interestingly, there were no differences concerning the ability of neutrophils to inhibit the development of resting conidia (data not shown). Of note, no correlations were observed between, on one hand, the period between engraftment and sampling, and on the other, the percentage of inhibition. There was also no link between the absolute neutrophil count at the time of sampling and the ability of neutrophils to inhibit Aspergillus growth (data not shown).

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Effect of calcineurin inhibitors on neutrophil impairment of Aspergillus hyphal growth.

During the study, some patients in recovery had neutrophils that were as effective as controls at inhibiting Aspergillus growth. This was not related to time to engraftment, type of graft, type of conditioning, or absolute neutrophil count in blood. But interestingly, we found that these patients had low plasma calcineurin inhibitor trough concentrations. Indeed, these treatments are difficult to balance and therefore regularly monitored. Considering this, we compared the neutrophils of patients with low plasma calcineurin inhibitor trough concentrations (i.e. <120 ng/mL for cyclosporine and <6 ng/mL for tacrolimus) to those of patients with normal trough concentrations (i.e. >150 ng/mL and >10 ng/mL) for Aspergillus inhibition and found that this latter was significantly greater in patients with low plasma concentrations (Figure 3a). Moreover, Pearson correlation analysis showed a certain link between the percentage of inhibition and the plasma calcineurin inhibitor trough level (r = -0.39; p<0.05) (Figure 3b). To strengthen the hypothesis that the observed effect was due to a pharmacological effect and not a generic effect of recovery, which is a very complex phenomenon, we performed growth inhibition tests using neutrophils from patients recovering from autologous stem cell transplantation, where calcineurin inhibitors are not administered. In this setting, we found no defects in the ability of neutrophils to impair Aspergillus growth (Figure 2a). We also performed in vitro experiments to further assess the hypothesis that calcineurin inhibitors were correlated with reduced neutrophil inhibition. When blood sampled from healthy donors was incubated with cyclosporine at 37°C for 2 hours, the subsequently isolated neutrophils showed diminished activity in terms of Aspergillus growth inhibition, in comparison to untreated controls (Figure 4). Cyclosporine thus appears to impair neutrophil activity against A. fumigatus hyphae.

Alterations to NETs production in early post-HSCT period.

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Neutrophils can exert their function through different mechanisms including phagocytosis, ROS production, degranulation and NETs production. As discussed above, neither ROS production nor degranulation (investigated by CD11b and CD66 surface expression) were altered in the present study. Neutrophils cannot phagocytize hyphae due to their size and support adhesion but it has been shown that they produce NETs in contact with Aspergillus consequently inhibiting its growth [START_REF] Mccormick | NETs formed by human neutrophils inhibit growth of the pathogenic mold Aspergillus fumigatus[END_REF][START_REF] Bianchi | Restoration of NET formation by gene therapy in CGD controls aspergillosis[END_REF] . We thus investigated the production of NETs by neutrophils and their effects on fungal growth in both HSCT patients and healthy donors.

Neutrophils were co-cultured with Aspergillus for three hours and NETs production was visualized using Sytox green. In accordance with the results presented above, Aspergillus growth as assessed by hyphal length measurement during the three-hour culture was greater in patients than in controls; inhibition of Aspergillus growth requires contact between the fungus and neutrophils (Figure 5a). The DNA area of the Sytox-positive cells was measured as previously described [START_REF] Papayannopoulos | Neutrophil elastase and myeloperoxidase regulate the formation of neutrophil extracellular traps[END_REF] for NETs quantification. Controls had a greater number of Sytox-positive cells than patients did (Figure 5b). The results thus indicated that neutrophils of patients produce less NETs than those of controls (Figure 5c-e).

Evolution of neutrophil inhibition of Aspergillus growth and restoration of inhibition with cessation of immunosuppressive therapy.

The evaluation of neutrophil action against Aspergillus showed significant variations over time (Figure 6). Due to their general and previously reported effect on immunity against Aspergillus 13, 14 , the administration of corticoids increased the impairment of neutrophil activity against hyphae. Importantly, in the 10 patients studied 10 months after the HSCT and for whom immunosuppressive therapies were stopped or considerably reduced, the percentage of Aspergillus growth inhibition was restored to the control (donor) level. These results indicate first that the effect of calcineurin inhibitors may add to other immunosuppressive effects, and

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second, that these effects are not permanent, at least when treatment is stopped or decreased within some number of months.

DISCUSSION.

The few studies that have focused on the behavior of neutrophils following HSCT provide a range of results suggesting moderate to no alterations in oxidative burst or antimicrobial activity [START_REF] Ramaprasad | Neutrophil function after bone marrow and hematopoietic stem cell transplant[END_REF] . For the present study, we aimed at evaluating neutrophil function in HSCT recipients compared to that in healthy donors as concerns the major human pathogen mold Aspergillus.

We found that oxidative burst and surface molecule expression at basal levels and following stimulation did not vary significantly over time, with the exceptions of CD66, which was more highly expressed during recovery, and TLR2, which was slightly less expressed. However, our results show that the recovery period is associated with a dramatic decrease in the ability of neutrophils to inhibit Aspergillus hyphae growth, and that the use of calcineurin inhibitors may play a large role in this impairment. It is well known that HSCT recipients are particularly at risk for invasive aspergillosis, not only during the neutropenia period but also after engraftment. It should be acknowledged however that neutrophil recovery is not the most at-risk period since antifungal prophylaxis is now widely used, and furthermore that the use of a calcineurin inhibitor is probable not singly sufficient to trigger invasive aspergillosis although cases of aspergillosis in patients receiving cyclosporine have been reported [START_REF] Caroli | Aspergillus fumigatus pneumonia during cyclosporine treatment for ulcerative colitis[END_REF] . Nevertheless, aspergillosis breakthrough in patients receiving antifungals is not rare [START_REF] Winston | Efficacy, safety, and breakthrough infections associated with standard long-term posaconazole antifungal prophylaxis in allogeneic stem cell transplantation recipients[END_REF][START_REF] Pang | Breakthrough invasive mould infections in patients treated with caspofungin[END_REF] and the assessment of the ability of neutrophils to correctly impair (or not) Aspergillus growth could be used to evaluate the risk of invasive aspergillosis in at-risk patients and thus contribute to reducing the fungal risk. Nonetheless, calcineurin inhibitors do appear to add to other acquired or potential innate immune deficiencies that, together, favor the appearance of fungal disease, which can occur any time after HSCT. Indeed, approximately 20% of invasive mold

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infections are diagnosed early (<40 days after HSCT), 40% late (between 40 and 100 days after HSCT) and 40% very late (>100 days after HSCT) [START_REF] Garcia-Vidal | Epidemiology of invasive mold infections in allogeneic stem cell transplant recipients: biological risk factors for infection according to time after transplantation[END_REF] . Whether this defect favors the occurrence of invasive aspergillosis and adds to genetic disorders (e.g. TLR-4 [START_REF] Bochud | Toll-like receptor 4 polymorphisms and aspergillosis in stem-cell transplantation[END_REF] or IL-1β 21 ) will need to be assessed in further studies.

For years, calcineurin inhibitors have been thought to exert their activity almost exclusively by targeting lymphocytes. However, over the past few years, a growing body of evidence suggests that they also have important effects on innate immunity. In mouse models of invasive aspergillosis, the administration of cyclosporine has been shown to shorten survival [START_REF] High | Invasive aspergillosis in mice immunosuppressed with cyclosporin A, tacrolimus (FK506), or sirolimus (rapamycin)[END_REF][START_REF] Roilides | In vitro and ex vivo effects of cyclosporin A on phagocytic host defenses against Aspergillus fumigatus[END_REF] .

However, non-concordant results have been reported in other animal studies and it remains unclear if cyclosporine alone is sufficient to favor the development of invasive aspergillosis 23, 24 . More recently, Greenblatt et al used in vitro and murine models to show that calcineurin regulates neutrophil immunity against the yeast Candida albicans [START_REF] Greenblatt | Calcineurin regulates innate antifungal immunity in neutrophils[END_REF] . They reported that mice treated with cyclosporine were more highly susceptible to disseminated Candida infection than were controls and that both calcineurin deficient neutrophils and cyclosporine treated neutrophils showed impaired response toward Candida. More recently, Tourneur et al showed that cyclosporine impaired human neutrophil function, but their patients were kidney transplant recipients who were also receiving corticoid therapy, a known modifier of neutrophil function [START_REF] Tourneur | Cyclosporine A impairs nucleotide binding oligomerization domain (Nod1)-mediated innate antibacterial renal defenses in mice and human transplant recipients[END_REF] .

Neutrophils can act against extracellular pathogens by releasing neutrophil extracellular traps (NETs), which are composed of a DNA web containing histones and proteins with antimicrobial activity, such as lactoferrin or elastase [START_REF] Kolaczkowska | Neutrophil recruitment and function in health and inflammation[END_REF] . McCormick et al reported that NETs were able to reduce the polar growth of Aspergillus hyphae [START_REF] Mccormick | NETs formed by human neutrophils inhibit growth of the pathogenic mold Aspergillus fumigatus[END_REF] . Interestingly, a recent study reported that cyclosporine reduced interleukin-8-induced NETs formation [START_REF] Gupta | Efficient neutrophil extracellular trap induction requires mobilization of both intracellular and extracellular calcium pools and is modulated by cyclosporine A[END_REF] .

Calcineurin inhibitors modulate several pathways, including NFATc, NF-κB or AP-1 [START_REF] Aramburu | Affinitydriven peptide selection of an NFAT inhibitor more selective than cyclosporin A[END_REF]29 . Thus, they not only inhibit the phosphatase activity of calcineurin but also the peptidyl-prolyl

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cis-trans isomerase activity of their respective receptor called immunophilin. It would be of interest to test the range of inhibitors that target the complex calcineurin pathway 29 to uncover the mechanism by which NETs are processed. It would also be of great interest to know whether targeting one or the other of these pathways would inhibit only alloreactive memory/effector T cells, therefore preventing rejection or GvHD, without impairing the innate immunity necessary for infection prevention. For now, there is not enough data to accurately describe the mechanism by which calcineurin inhibitors limit NETs formation. Additionally, published results often appear contradictory. Gupta et al found that NETosis was reduced dramatically by cyclosporine and ascomycin (an analogue of tacrolimus,) while rapamycin, which targets the mammalian target of rapamycin (mTOR), had only a small effect [START_REF] Gupta | Efficient neutrophil extracellular trap induction requires mobilization of both intracellular and extracellular calcium pools and is modulated by cyclosporine A[END_REF] . Inversely, McInturff et al found that NETosis was inhibited by rapamycin but not by tacrolimus 30 . We underline however that those two studies used different agents to induce NETosis: interleukin-8 for the former and LPS for the latter. In our work, we found that neutrophils sampled from patients in recovery were less efficient at inhibiting Aspergillus growth than those collected from healthy donors. ROS production was not impaired during recovery and no correlation was observed between the percentage of inhibition and the level of oxidative burst following stimulation by germinating conidia (data not shown). In contrast, Stuehler et al reported recently that the ROS production of neutrophils collected from HSCT patients and stimulated by Aspergillus was altered 30 days after the graft 31 . Interestingly however, their results also indicate that the percentage of fungal damage mediated by neutrophils was decreased in HSCT patients compared to controls, even in patients with normal ROS production 31 . In accordance with that, it has been shown that cyclosporine does not impair ROS production [START_REF] Roilides | In vitro and ex vivo effects of cyclosporin A on phagocytic host defenses against Aspergillus fumigatus[END_REF] . Finally, in our microscopy experiments, we observed that in co-cultures with Aspergillus, neutrophils sampled from patients in recovery produced less NETs than those sampled from healthy donors, with a higher rate of fungal growth. However, rather than a specific ef-

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fect on NETosis, it seems that this diminished NET production is more generally a reflection of a lower neutrophil death rate. The implication of calcineurin in cell death was first reported long ago 32 and the enhancement of cell survival with the use of calcineurin inhibitors has been observed with different cellular types 33,34 .

CONCLUSION.

This study exposes a previously unknown deficiency in the antifungal response of innate immunity of grafted patients. In hematopoietic stem cell transplant recipients, neutrophil-driven immunity against Aspergillus fumigatus is altered during the first month post-transplant; administration of calcineurin inhibitors plays an important role in this impairment, in stark contrast to the classical view of these inhibitors as affecting only adaptive immunity. The specific pathway by which these drugs alter neutrophil antifungal response and NETs formation must now be investigated in further studies, with the particular goal of enabling the development of more specific therapeutic alternatives capable of inhibiting alloreactive effector T cells without impairing the basic functions of innate immunity cells. Attaining this goal will be challenging but it is also vital for improving care in both HSCT and solid organ transplants.

FUNDING.

This study was supported by the Agence de Biomédecine and Aviesan

POTENTIAL CONFLICTS OF INTEREST.

None for all authors.

CONTRIBUTIONS.

SI performed experiments

  µM) or PBS was added for 5 minutes. Samples were then analyzed by flow cytometry.

  , PB performed experiments, AB designed and performed microscopic experiments and participated in writing the paper, LG performed experiments; LS, MU M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT and VL participated in the design of the study and the inclusion of patients, DM participated in scientific discussions and writing of the paper, SNG participated in the design of the study, scientific discussions, inclusion of patients and writing of the paper, AF designed the study, performed experiments and wrote the paper. Baumgrass R. Novel inhibitors of the calcineurin/NFATc hub -alternatives to CsA and FK506? Cell Commun Signal 2009; 7:25.30.McInturff AM, Cody MJ, Elliott EA, Glenn JW, Rowley JW, Rondina MT, et al. Mammalian target of rapamycin regulates neutrophil extracellular trap formation via induction of hypoxia-inducible factor 1 alpha. Blood 2012; 120:3118-25. 31. Stuehler C, Kuenzli E, Jaeger VK, Baettig V, Ferracin F, Rajacic Z, et al. Immune Reconstitution After Allogeneic Hematopoietic Stem Cell Transplantation and Association With Occurrence and Outcome of Invasive Aspergillosis. J Infect Dis 2015. 32. Shibasaki F, McKeon F. Calcineurin functions in Ca(2+)-activated cell death in mammalian cells. J Cell Biol 1995; 131:735-43. 33. Sachewsky N, Hunt J, Cooke MJ, Azimi A, Zarin T, Miu C, et al. Cyclosporin A enhances neural precursor cell survival in mice through a calcineurin-independent pathway. Dis Model Mech 2014; 7:953-61. 34. He H, Liu X, Lv L, Liang H, Leng B, Zhao D, et al. Calcineurin suppresses AMPKdependent cytoprotective autophagy in cardiomyocytes under oxidative stress. Cell Death Dis 2014; 5:e997.

Table 1 :

 1 Characteristics of patients included in the study and their conditioning regimens RIC: reduced-intensity conditioning; MAC: myeloablative conditioning M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT a Myeloablative conditioning involved either total body irradiation and cyclophosphamide (n=6) or busulfan and cyclophosphamide (n=5) Mean age in years (minimum-maximum) 44

Figure 1 :Figure 2 :Figure 3 :

 123 Figure 1: Post-engraftment evolution of surface molecule expression and reactive oxy-

  Figure 3b: Correlation (Pearson test) between the percentage of inhibition of Aspergillus by neutrophils (y-axis) and the percentage of target plasma calcineurin inhibitor trough level (x-axis). Target concentration was defined as 150 ng/mL for cyclosporine and 10 ng/mL for tacrolimus. Analysis included 26 patients receiving cyclosporine and 3 patients receiving tacrolimus. Concentrations were determined the day or the day before neutrophil sampling, except for two patients for whom concentration values had been determined two days before sampling and one patient five days before.

Figure 4 :Figure 5 :

 45 Figure 4: A calcineurin inhibitor diminishes the ability of neutrophils to inhibit Aspergil-

Figure 6 :

 6 Figure 6: evolution of Aspergillus hyphae growth inhibition by neutrophils sampled

  Figure 5d

  Figure 5e

Table 2 :

 2 Characteristics of blood samples and immunosuppressive drugs regimens 1, 2, 6, and

	M A N U S C R I P T

ACKNOWLEDGEMENTS.

We thank all patients and healthcare providers and express our gratitude to Coralie Pannelier, Nathalie Miranda and Audrey Casabona for their assistance in the inclusion of patients, Christophe Parizot for his technical assistance with the flow cytometer, and Dr Noël Zahr for calcineurin inhibitor assaying. Parts of the data were presented at the Trends in Medical Mycology Congress held October 2015 in Lisbon.