Caterina Calgaro
email: caterina.calgaro@math.univ-lille1.fr

Olivier Goubet
email: olivier.goubet@u-picardie.fr

Ezzeddine Zahrouni
email: ezzeddine.zahrouni@fsm.rnu.tn

Finite dimensional global

Keywords: AMS: 35Q55, 35B41, 37L30. keywords: Nonlinear Schrödinger equations, Crank-Nicolson scheme, global attractor, fractal dimension

Introduction

The fractional nonlinear Schrödinger in the whole line R reads

u t -i(-∆) α u + i|u| 2 u = 0, (1)
where the unknown u = u(t, x) is a complex valued function defined on R×R. It was derived by Laskin [START_REF] Laskin | Fractional quantum mechanics and Lévy path integrals[END_REF][START_REF] Laskin | Fractional Schrödinger equation[END_REF] as a result of expanding the Feynman path integral, substituting to the usual Brownian path the so-called Levy quantum mechanical paths. This is a fundamental equation of fractional quantum mechanics, where Levy path provide fractional Laplace operators instead of the usual Laplace operator. For the mathematical point of vue, the initial value problem was addressed recently by numerous authors. The well-posedness of (1) for α ∈ ( 1 2 , 1) in L 2 (R) was established by B. Guo and Z. Huo [START_REF] Guo | Global Well-Posedness for the Fractional Nonlinear Schrödinger Equation[END_REF]. A general theory for well-posedness in Sobolev spaces for fractional NLS equations in R N is given by Y. Hong and Y. Sire in [START_REF] Hong | On Fractional Schrödinger Equations in Sobolev Spaces[END_REF]. The equation ( 1) is a conservative equation. In some physical contexts, an external forcing term and some damping effects have to be taken into account. This leads to the following weakly damped fractional nonlinear Schrödinger equation

u t -i(-∆) α u + i|u| 2 u + γu = f, (2)
where γ > 0 denotes the damping parameter and f is the external forcing term. We assume that the source term f is independent of time and that f belongs to L 2 (R).

We supplement [START_REF] Akroune | Regularity of the attractor for a weakly damped nonlinear Schrödinger equation on R[END_REF] with the initial condition u(0, .) = u 0 ∈ H α (R),

where H α (R) is the fractional Sobolev space associated to the equation (see the very definition of H α (R) below). This equation provides an infinite-dimensional dynamical system, in the framework described in [START_REF] Temam | Infinite Dimensional Dynamical Systems in Mechanics and Physics[END_REF], [START_REF] Hale | Asymptotic behavior of Dissipative Systems[END_REF], [START_REF] Raugel | Global attractors in partial differential equations[END_REF] and [START_REF] Miranville | Attractors for dissipative partial differential equations in bounded and unbounded domains[END_REF]. Results concerning the existence of a global attractor in H α for this fractional equation appear in [START_REF] Cheng | The attractor of the dissipative coupled fractional Schrödinger equations[END_REF]. Equation [START_REF] Akroune | Regularity of the attractor for a weakly damped nonlinear Schrödinger equation on R[END_REF] is the fractional analog of the usual weakly damped cubic nonlinear Schrödinger equation that was first studied in [START_REF] Laurençot | Long time Behaviour for weakly damped driven nonlinear Schrödinger equation in R N , N ≤ 3[END_REF] and [START_REF] Wang | An Energy Equation for Weakly Damped Driven Nonlinear Schrödinger Equations and Its Application to Their Attractors[END_REF]. Further issues as the regularity of global attractors for these NLS equations were addressed in [START_REF] Goubet | Regularity of the attractor for the weakly damped nonlinear Schrödinger equations[END_REF], [START_REF] Akroune | Regularity of the attractor for a weakly damped nonlinear Schrödinger equation on R[END_REF] in the one dimensional case, and in [START_REF] Goubet | Asymptotical smoothing effect for nonlinear Schrödinger equation on the two dimensional torus[END_REF], [START_REF] Goubet | Regularity of the attractor for a weakly damped nonlinear Schrödinger equation in R 2[END_REF] for the 2D case.

In this article, we address the discrete counterpart of these continuous dynamical systems. Actually, we consider a suitable semi-discrete fractional nonlinear Schrödinger equation introducing a suitable Crank-Nicolson scheme in time, keeping the space variable continuous. These numerical schemes first introduced in [START_REF] Delfour | Finite-difference solutions of a nonlinear Schrödinger equation[END_REF] (see also [START_REF] Sanz-Serna | Conservative and nonconservative schemes for the solution of the nonlinear Schrödinger equation[END_REF]) for numerical purposes, provide also discrete infinite-dimensional dynamical systems. This point of view for classical NLS equations appear in [START_REF] Goubet | On a time discretization of a weakly damped forced nonlinear Schrödinger equation[END_REF]. We do not address here alternate discretizations for NLS as respectively splitting methods (see [START_REF] Besse | Order estimates in time of splitting methods for the Nonlinear Schrödinger Equations[END_REF] and the references therein) or respectively relaxation methods introduced in [START_REF] Besse | A relaxation scheme for Nonlinear Schrödinger Equations[END_REF].

Fruitful results about Crank-Nicolson scheme appear in the infinite dimensional dynamical system literature: for the case when the the space variable x belongs to a finite interval, with periodic boundary conditions, it was proved in [START_REF] Goubet | On a time discretization of a weakly damped forced nonlinear Schrödinger equation[END_REF] the existence of a global attractor. It turns out that this result was also obtained in the case where the variable space x belongs to T 2 in [START_REF] Ezzoug | Semi-discrete weakly damped nonlinear 2-D Schrödinger equation[END_REF]. It is worth recalling that the discrete Crank-Nicolson scheme in both space and time variable for (2) was studied in [START_REF] Yan | Attractors and dimensions for discretizations of a weakly damped Schrödinger equations and a sine-Gordon equation[END_REF] and [START_REF] Abounouh | Discrete Schrödinger Equations and dissipative dynamical systems[END_REF] for weakly damped Nonlinear Schrödinger Equation. The discrete in time relaxation scheme for a non local nonlinear Schrödinger equation was studied in [START_REF] Goubet | Dynamical properties for a relaxation scheme applied to a weakly damped non local nonlinear Schrodinger equation Anal[END_REF]. The discrete dynamical system provided by the Crank-Nicolson scheme to (2) is new to our knowledge.

The rest of the article is organized as follows. Section 2 is devoted to the derivation of the scheme and the main results. Section 3 deals the mathematical framework, where we state and prove a nonstandard commutator estimate. Section 4, we focus on the existence and uniqueness of the discrete solution. Finally, in section 5 we prove the existence of a compact global attractor in the phase space H α . We then prove, assuming moreover that the external force has some decay at the infinity, that this global attractor has a finite fractal dimension.

2 Derivation of the scheme and main results

The Crank-Nicolson Scheme

In this section, we introduce a Crank-Nicolson scheme associated to [START_REF] Akroune | Regularity of the attractor for a weakly damped nonlinear Schrödinger equation on R[END_REF], following an idea of M. Abounouh. Equation (2) has a unique solution u [START_REF] Hong | On Fractional Schrödinger Equations in Sobolev Spaces[END_REF]). Alternatively, the equation reads

∈ C([0, ∞), H α ) ∩ C 1 ([0, ∞), H -α ) (see
∂ ∂t (e γt u(t)) -i(-∆) α (e γt u(t)) + i|u(t)| 2 (e γt u(t)) = e γt f, (4)
where ∆u = ∂ xx u.

For a given time step τ > 0, we consider the uniform time sequence (t n ) n defined by t n = nτ and we integrate (4) in time on the interval [t n , t n+1 ] to get

e γt n+1 u(t n+1 ) -e γt n u(t n )+ t n+1 t n -i(-∆) α e γt u(t) + i|u(t)| 2 e γt u(t) dt = e γt n+1 -e γt n γ f. (5)
Set (u n ) n to be such that u n ∼ u(t n ), where u is a solution to (2). Using the the trapezoidal rule to approximate the time integral

t n+1 t n g(s)ds ∼ τ 2 (g(t n+1 ) + g(t n )), (6)
and the approximation of the nonlinearity in [START_REF] Delfour | Finite-difference solutions of a nonlinear Schrödinger equation[END_REF] (that goes back to [START_REF] Strauss | Numerical solution of a nonlinear Klein-Gordon equation[END_REF], according to the reference [START_REF] Sanz-Serna | Conservative and nonconservative schemes for the solution of the nonlinear Schrödinger equation[END_REF]), we are led to

1 τ (u n+1 -δu n ) - i 2 (-∆) α (u n+1 + δu n ) + i 4 (|u n+1 | 2 + |u n | 2 )(u n+1 + δu n ) = f 1 -δ γτ f, (7)
where δ = e -γτ , and γ, τ > 0, f ∈ L 2 (R), u 0 ∈ H α . We have used in (5) that f is time independent function and the approximation 1-δ γτ 1 that is valid if τ is small enough.

Main results

Our first main result is the following. The result is concerned with the wellposedness of the scheme.

Theorem 1 For any u n in H α , there exists at least one solution u n+1 of [START_REF] Chueshov | Attractors for Second-Order Evolution Equations with a Nonlinear Damping[END_REF]. Let S : H α -→ H α , u n → u n+1 the multivalued function defined by [START_REF] Chueshov | Attractors for Second-Order Evolution Equations with a Nonlinear Damping[END_REF]. There exists a bounded set E ⊂ H α such that S(E) ⊂ E, and such that for τ > 0 small enough, the map S is continuous and one-to-one on E.

Our second result is concerned with the existence of the global attractor for the map S : E → E.

Theorem 2 The discrete dynamical system S n n∈N defined on the set E by Su n = u n+1 possesses a compact global attractor A τ in H α .

Our last result is concerned with the fractal dimension of the attractor, assuming that the external force f has some decay at the infinity.

Theorem 3 Assume moreover that the external force f belongs to

L 2 (R, (1 + x 2 )dx), i.e |f (x)| 2 (1 + x 2 )dx < +∞. Then A τ is included in L 2 (R, (1 + x 2 )dx) and has finite fractal dimension in H α ∩ L 2 (R, (1 + x 2 )dx).

Remark 1

The assumption that the datum f belongs to some weighted space in order to ensure that the global attractor has finite dimension is natural, since it is still an open question to remove this assumption for classical NLS equation.

Mathematical framework and notations

Notations

The Fourier transform of u is denoted by either û or F(u) and defined by

F(u)(ξ) = R u(x)e -ixξ dx. (8)
We consider the homogeneous fractional pseudo-differential operator Λ s defined by

Λ s (u)(x) = F -1 (|ξ| s û(ξ)) (x). (9)
We recall some functional spaces that we will use in the sequel. For 1 ≤ r ≤ ∞, s ∈ R, the spaces L r and H s will denote the usual Lebesgue L r (R) and Sobolev spaces H s (R) of complex valued functions. The space S denotes the Schwartz class. The spaces L 2 and H s are Hilbert spaces endowed respectively with the following inner product,

(u, v) L 2 = Re R uvdx, and 2π((u, v)) H s = Re R (1 + |ξ| 2 ) s û(ξ)v(ξ)dξ.
The norms on these space will be denoted simply by . H s for instance. Throughout this article C and K denote numerical constants that may vary from one line to one another, K depending on the data γ, f L 2 .

Useful results

We recall here results from functional analysis that will be used in the sequel. The first one is related to the Brouwer fixed point theorem.

Lemma 1 ([5] and [START_REF] Temam | Navier Stokes Equations[END_REF]) Let X a finite dimensional space endowed with a scalar product [., .] and consider a continuous mapping F : X → X. Suppose that there exists R 0 > 0, such that for all w satisfying [w, w]

1 2 = R 0 we have [F (w), w] > 0. Then there exists w * , [w * , w * ] 1 2 < R 0 such that F (w * ) = 0.
For α > 1 2 , we recall that H α is an algebra and that it is included in L ∞ . Moreover, we have the interpolation inequality

u L ∞ ≤ C α u 1-1 2α L 2 Λ α u 1 2α L 2 . ( 10
)
To check [START_REF] Goubet | Regularity of the attractor for the weakly damped nonlinear Schrödinger equations[END_REF], we proceed as follows. Thanks to the inverse Fourier transform we just have to provide an upper bound to the L 1 norm of û. Consider R > 0. We have, thanks to Cauchy-Schwarz inequality

û L 1 ≤ |ξ|≤R |û| + |ξ|≥R |ξ| α |ξ| α |û| ≤ c α ( √ R u L 2 + R 1 2 -α ||Λ α u|| L 2 ); (11)
a suitable choice of R yields [START_REF] Goubet | Regularity of the attractor for the weakly damped nonlinear Schrödinger equations[END_REF]. It is worth to point out that we may substitute u H α to Λ α u L 2 in the right hand side of [START_REF] Goubet | Regularity of the attractor for the weakly damped nonlinear Schrödinger equations[END_REF].

We finally state and prove the following commutator estimate.

Proposition 1 Consider ϕ ∈ S and u ∈ H α . Then ||[(-∆) α , ϕ]u|| 2 ≤ C α ||ξ φ|| L 1 ||u|| H 2α-1 + || |ξ| 2α φ|| L 1 ||u|| 2 (12)
Remark 2 Let us recall that 2α -1 < α. We then for instance can bound by above ||u|| H 2α-1 by ||u|| H α or any norm

H β for β ∈ (2α -1, α].
Proof. We note that

F(uv)(ξ) = R û(η)v(ξ -η)dη, (13)
so

F(-∆) α (ϕu)(ξ) = |ξ| 2α R φ(η)û(ξ -η)dη, (14)
and similarly,

F(ϕ(-∆) α u)(ξ) = R φ(η)|ξ -η| 2α û(ξ -η)dη. (15)
Hence for the commutator [(-∆) α , ϕ] we have,

F([(-∆) α , ϕ]u)(ξ) = R |ξ| 2α -|ξ -η| 2α φ(η)û(ξ -η)dη. (16)
We now use, recalling that 2α -1 ∈ (0, 1),

||ξ| 2α -|ξ -η| 2α | ≤ cα max(|ξ| 2α-1 , |ξ -η| 2α-1 )|η| ≤ c α (|ξ| 2α-1 |η| + |η| 2α ), (17)
using that |ξ| 2α-1 ≤ c α (|ξ-η| 2α-1 +|η| 2α-1 ). Hence we bound by above the integral in ( 16) by two integrals, accordingly to the upper bound [START_REF] Hong | On Fractional Schrödinger Equations in Sobolev Spaces[END_REF]. The square of the first integral is bounded as follows

R |η|| φ(η)||ξ -η| 2α-1 |û(ξ -η)|dη 2 ≤ R |η|| φ(η)|dη R |η|| φ(η)||ξ -η| 4α-2 |û(ξ -η)| 2 dη . (18)
Integrating in ξ leads to an upper bound in c||u|| H 2α-1 ||ξ φ|| L 1 due to Plancherel formula. We then have the first term in the right hand side of [START_REF] Goubet | Regularity of the attractor for a weakly damped nonlinear Schrödinger equation in R 2[END_REF]. The second integral reads | R | φ(η)||η| 2a |û(ξ -η)|dη|, and we proceed exactly as above. This completes the proof of the lemma.

We complete this section by the discrete Gronwall lemma that will be of constant use throughout this article.

Lemma 2 Set δ < 1. Consider a sequence a n such that a n+1 ≤ δa n + c. Then a n ≤ δ n a 0 + c 1-δ .
4 Proof of Theorem 1

In this section, we prove that the Crank-Nicolson scheme ( 7) is well-posed and provides the existence of a discrete dynamical system S : u n → u n+1 . We follow here the method in [START_REF] Ezzoug | Semi-discrete weakly damped nonlinear 2-D Schrödinger equation[END_REF]. We first prove that for a given u n there exists at least a u n+1 that solves the equation [START_REF] Chueshov | Attractors for Second-Order Evolution Equations with a Nonlinear Damping[END_REF]. The existence results relies on some compactness argument. The drawback of the method is that the uniqueness is not ensured. We then prove that there exists some absorbing set that is invariant by S. We eventually prove that restricted to this absorbing set, and that for τ small enough, S is a continuous one-to-one map.

A Galerkin Method

Consider here (ϕ k ) k∈N * a basis of H α . We assume that this family is orthogonal in

L 2 . Set V N = Span{ϕ 1 , • • • , ϕ N }
the finite dimensional subspace, and introduce

P N : H α → V N the L 2 -orthogonal projector.
We seek w N in V N that is solution to a Galerkin approximation of ( 7) that reads as, for any ϕ N ∈ V N ,

w N -δP N u n τ -iΛ 2α (w N + δP N u n ) 2 + i(|w N | 2 + |u n | 2 ) (w N + δP N u n ) 4 , ϕ N L 2 = P N f, ϕ N L 2 . ( 19
)
We now state and prove

Proposition 2

The equation [START_REF] Laskin | Fractional Schrödinger equation[END_REF] has at least a solution w N that satisfies moreover

w N L 2 ≤ 4δ u n L 2 + τ f L 2 . ( 20
)
Proof :

Consider P * N : H -α → V N defined as (P * N µ, φ) L 2 =< µ, P N φ > H -α ,H α , for any φ ∈ L 2 . Introduce, setting w = w N + δP N u n , F (w) = P * N ( w -2δP N u n τ -iΛ 2α w 2 + i(|w -δP N u n | 2 + |u n | 2 ) w 4 -P N f ). ( 21
)
This defines a continuous map F :

V N → V N such that w N solves (19) if and only if F (w N + δP N u n ) = 0. Let us observe that (F (w), w) L 2 = 1 τ w N 2 L 2 -δ 2 P N u n 2 L 2 -P N f, w N + δP N u n . (22)
Therefore

(F (w), w) L 2 ≥ w N L 2 + δ P N u n L 2 w N L 2 -δ P N u n L 2 τ -f L 2 . ( 23
) Set R 0 = 3δ||u n || L 2 + τ f L 2 . For w ∈ V N such that ||w|| L 2 = R 0 we have (F (w), w) L 2 > 0.
Then by the Brouwer's type Lemma 1 there exists

w * ∈ V N , such that ||w * || L 2 ≤ R 0 and F (w * ) = 0. Hence w N = w * -δP N u n ∈ V N is solution of (19) and satisfies w N L 2 ≤ 4δ u n L 2 + τ f L 2 . ( 24
)
This completes the proof of the Lemma.

Passing to the limit

In order to let N goes to +∞ in [START_REF] Laskin | Fractional Schrödinger equation[END_REF], we need some compactness argument. To this end, we first state and prove

Lemma 3

The sequence (w N ) N is bounded in H α .

Proof : On the one hand, taking

ϕ N = w N + δP N u n in (19) leads to w N 2 L 2 ≤ δ 2 P N u n 2 L 2 + τ (f, w N + δP N u n ) L 2 , (25)
that implies

w N L 2 ≤ δ u n L 2 + τ f L 2 . ( 26
)
Then, since u n belongs to H α then the sequence w N is bounded in L 2 . On the other hand, taking ϕ N = i(w N -δP N u n ) in [START_REF] Laskin | Fractional Schrödinger equation[END_REF], we get

Λ α w N 2 L 2 = δ 2 Λ α P N u n 2 L 2 + 1 2 R (|w N | 2 + |u n | 2 )(|w N | 2 -δ 2 |P N u n | 2 ) dx+ -2Im R f (w N -δP N u n ) dx. (27)
To get an upper bound for the right hand side of ( 27), we control the main term as follows

R (|w N | 2 + |u n | 2 )|w N | 2 dx ≤ w N 2 L ∞ + u n 2 L ∞ w N 2 L 2 ≤ K + 1 8 ||Λ α w N || 2 L 2 . ( 28
)
using moreover the interpolation inequality [START_REF] Goubet | Regularity of the attractor for the weakly damped nonlinear Schrödinger equations[END_REF], Young inequality and the L 2 upper bound. This completes the proof of the Lemma.

Proposition 3 There exists at least a solution to [START_REF] Chueshov | Attractors for Second-Order Evolution Equations with a Nonlinear Damping[END_REF].

Proof : We now pass to the limit in [START_REF] Laskin | Fractional Schrödinger equation[END_REF], for N ≥ N 0 and for a given ϕ N 0 in V N 0 . We know that there exist a subsequence still denoted by w N that converges weakly in H α towards u n+1 . Passing to the limit in the linear terms of ( 19) is standard and then omitted. We focus on the nonlinear term, and we aim to prove that

R (|w N | 2 w N -|u n+1 | 2 u n+1 ) φdx → 0. (29)
We proceed as follows. we consider a smooth cut off function

ρ ∈ C ∞ c (R) such that ρ(s) = 1 if |s| ≤ 1, ρ(s) = 0 if |s| ≥ 2.
For each r > 0, let ρ r (s) = ρ( s r ). On the one hand, up to another subsequence extraction, since H α (-2r, 2r) is compactly embedded into C(-2r, 2r), we have that

R (|w N | 2 w N -|u n+1 | 2 u n+1 )ρ r φdx → 0. ( 30
)
On the other hand, using the previous L 2 and L ∞ estimates

R (|w N | 2 w N -|u n+1 | 2 u n+1 )(1 -ρ r ) φdx ≤ K( |x|≥r |ϕ| 2 dx) 1 2 . ( 31
)
Then letting first N → +∞ and then r → +∞ leads to (29). Hence we can pass to the limit for a given ϕ N 0 in V N 0 . To complete the proof of the Proposition 3 we use that N 0 ≥0 V N 0 is dense in H α and we conclude by a density argument.

Absorbing sets that are positively invariant

We state and prove

Proposition 4 There exists a bounded set E ⊂ H α that is an absorbing set for S, and that is positively invariant by S.

Proof : We first construct a L 2 absorbing ball. Set M 0 = 2 f L 2 γ
. Consider the L 2 scalar product of ( 7) with u n+1 + δu n . We thus obtain

||u n+1 || L 2 ≤ δ||u n || L 2 + (1 -δ) ||f || L 2 γ . ( 32
)
Therefore the set E 0 = {u ∈ H α , u L 2 ≤ M 0 } is positively invariant, and, thanks to the discrete Gronwall Lemma, absorbing for the L 2 topology; actually

||u n || L 2 ≤ δ n ||u 0 || L 2 + (1 -δ n ) ||f || L 2 γ , (33)
and u n belongs to E 0 for n large enough. We now seek for an absorbing set E in H α as a susbset of E 0 . Without loss of generality assume then that u 0 (and then u n ) belongs to the L 2 bounded absorbing set. We now follow the same lines as in the continuous case (see [START_REF] Cheng | The attractor of the dissipative coupled fractional Schrödinger equations[END_REF]). That is, we consider the functional

J(u) = Λ α u 2 L 2 - 1 2 R |u| 4 dx + 2Im R f u dx. ( 34
)
Using the Cauchy-Schwarz inequality and bounding by above the nonlinear part of J(u), we get ∀u ∈ H α

J(u) ≥ Λ α u 2 L 2 - 1 2 u 2 L ∞ u 2 L 2 -2 f L 2 u L 2 . ( 35
)
Then by interpolation inequality [START_REF] Goubet | Regularity of the attractor for the weakly damped nonlinear Schrödinger equations[END_REF] and Young inequality, we obtain ∀u ∈ E 0

J(u) ≥ 1 2 Λ α u 2 L 2 -K 0 ( 36
)
for a fixed K 0 = K 0 (γ, f ).

We seek an equality satisfied by this functional; we then take the scalar product of [START_REF] Chueshov | Attractors for Second-Order Evolution Equations with a Nonlinear Damping[END_REF] with i(u n+1 -δu n ), to get

Λ α u n+1 2 L 2 -δ 2 Λ α u n 2 L 2 - 1 2 R (|u n+1 | 2 + |u n | 2 )(|u n+1 | 2 -δ 2 |u n | 2 ) dx = -2Im R f u n+1 -δu n dx. ( 37
)
This leads to

J(u n+1 ) = δ 2 J(u n ) + 1 2 (1 -δ 2 ) R |u n | 2 |u n+1 | 2 dx + 2δ(1 -δ)Im R f u n dx. ( 38
)
On the one hand, we infer from Cauchy-Schwarz inequality and (33) that

Im R f u n dx ≤ f L 2 (R) u n L 2 (R) ≤ K, ( 39
)
since u n remains trapped into the L 2 absorbing ball. On the other hand, using once more the L 2 upper bound, the interpolation inequality [START_REF] Goubet | Regularity of the attractor for the weakly damped nonlinear Schrödinger equations[END_REF] and Young inequality, we have

R |u n | 2 |u n+1 | 2 dx ≤ u n 2 L 2 u n+1 2 L ∞ ≤ K + 1 2 Λ α u n+1 2 L 2 . ( 40
)
Gathering ( 35) to (40), we thus obtain that

δJ(u n+1 ) ≤ (1 - 1 2 (1 -δ 2 ))J(u n+1 ) ≤ δ 2 J(u n ) + (1 -δ)K. ( 41
)
Using the discrete Gronwall lemma we then infer that there exists K 1 that depends on the data γ, f such that the set E = {u ∈ E 0 , J(u) ≤ K 1 } is a bounded absorbing set in H α that is positively invariant by S. In the remaining of the paper, we denote by M 1 the radius of the smallest ball that contains E.

Continuity of the map S on E.

We prove here that if τ is small enough depending on M 1 , then the map S : E → E is one to one and continuous for the H α topology. This completes the proof of Theorem 1.

Lemma 4 There exists a constant C such that if Cτ M 2 1 < 1. Then the map S : E → E is a Lipschitz map for the H α topology.

Proof : Set u n+1 = Su n , v n+1 = Sv n with u n , v n ∈ E. We denote w n = u n -v n the difference between the two discrete trajectories. Then, w n verifies

1 τ (w n+1 -δw n ) - i 2 (-∆) α (w n+1 + δw n ) + i 4 ((|u n+1 | 2 + |u n | 2 )(w n+1 + δw n )+ i 4 (|u n+1 | 2 -|v n+1 | 2 + |u n | 2 -|v n | 2 )(v n+1 + δv n ) = 0. (42)
We first take the scalar product of (42) with i(w n+1 -δw n ) to obtain,

||Λ α w n+1 || 2 L 2 -δ 2 ||Λ α w n || 2 L 2 = 1 2 R (|u n+1 | 2 + |u n | 2 )(|w n+1 | 2 -δ 2 |w n | 2 ) + 1 2 Re R (|u n+1 | 2 -|v n+1 | 2 + |u n | 2 -|v n | 2 )(v n+1 + δv n )(w n+1 -δw n ). ( 43
)
Using the fact that u n , v n remains trapped in E that is a bounded set of L ∞ , we then obtain straightforwardly

||Λ α w n+1 || 2 L 2 -δ 2 ||Λ α w n || 2 L 2 ≤ CM 2 1 (||w n || 2 L 2 + ||w n+1 || 2 L 2 ) (44)
We now take the scalar product of (42) with (w n+1 + δw n ) to get

||w n+1 || 2 L 2 -δ 2 ||w n || 2 L 2 = τ 4 Im R (|u n+1 | 2 -|v n+1 | 2 +|u n | 2 -|v n | 2 )(v n+1 +δv n )(w n+1 + δw n )
(45) and proceeding as for (43), we get

||w n+1 || 2 L 2 -δ 2 ||w n || 2 L 2 ≤ Cτ M 2 1 (||w n || 2 L 2 + ||w n+1 || 2 L 2 ). (46)
We choose τ > 0 small enough such that Cτ M 2 1 < 1 and setting

C 2 = δ 2 +C 1 τ M 2 1 1-Cτ M 2 1
then we infer from (46)

||w n+1 || 2 L 2 ≤ C 2 ||w n || 2 L 2 . (47)
Using (47) in (44), we deduce that

||w n+1 || H α ≤ C 3 ||w n || H α (48)
that completes the proof of the Lemma.

Proof of Theorems 2 and 3

In this section, we prove the existence of a compact global attractor in H α . The proof relies on a splitting argument first used in [START_REF] Laurençot | Long time Behaviour for weakly damped driven nonlinear Schrödinger equation in R N , N ≤ 3[END_REF] in the continuous case, that allows us to apply the asymptotic compactness Theorem (see Theorem I.1.1 [START_REF] Temam | Infinite Dimensional Dynamical Systems in Mechanics and Physics[END_REF]).

The key proposition reads as follows Proposition 5 For any η > 0, a trajectory u n splits as u n = v n + w n where, for n large enough, w n H α ≤ η, and, for any n, v n is trapped into a bounded set

B η of H 2α ∩ L 2 (R, (1 + x 2 )dx).
The proof of this Proposition uses the commutator estimate. We then prove that thanks to the famous J. Ball argument, it turns out that this global attractor is also a compact subset in H α . This is standard and omitted for the sake of conciseness.

The splitting method

For any small η in ]0, 1], we consider a function f η ∈ S, such that,

||f -f η || L 2 ≤ η. (49)
We split u n solution of (7) as u n = v n + w n , where v n is the solution to

1 τ (v n+1 -δv n ) - i 2 (-∆) α (v n+1 + δv n ) + i 4 (|u n+1 | 2 + |u n | 2 )(v n+1 + δv n ) = f η , (50)
v 0 = 0. (51)
Then

w n = u n -v n is the solution to 1 τ (w n+1 -δw n ) - i 2 (-∆) α (w n+1 + δw n ) + i 4 (|u n+1 | 2 + |u n | 2 )(w n+1 + δw n ) = f -f η , (52)
Using estimate (58) and the fact that u n remains in a bounded set of L ∞ we have that

Λ α w n+1 2 L 2 ≤ δ 2 Λ α w n 2 L 2 + Kη 2 . ( 60
)
This completes the proof of Lemma 5, thanks to discrete Gronwall Lemma.

We then prove an estimate on the sequence v n Proposition 7 The sequence (v n ) is uniformly bounded in H 2α ∩ L 2 (R, (1 +

x 2 )dx), that is there exists a constant K such that for all u 0 ∈ E we have, for any n ≥ 0

||xv n || L 2 + τ ||v n || H 2α ≤ K. (61)
Proof : Since v n = u n -w n we already know that v n remains bounded in H α . We begin with the bound for xv n in L 2 . Introduce a cutoff function ϕ in S such that ϕ is compactly supported, ϕ = 1 on a neighborhood of 0, and

0 ≤ ϕ ≤ 1. Set V n (x) = xϕ( x N )v n , and ϕ N (x) = ϕ( x N
). Let us observe that equation (50) reads also

1 τ (V n+1 -δV n ) - i 2 (-∆) α (V n+1 + δV n ) + i 4 (|u n+1 | 2 + |u n | 2 )(V n+1 + δV n ) = xϕ N f η + i 2 [xϕ N , (-∆) α ](v n+1 + δv n ), (62)
Now, taking the L 2 inner product of (62) with V n+1 + δV n we obtain

1 τ V n+1 2 L 2 -δ 2 V n 2 L 2 = Re R xϕ N f η V n+1 + δV n dx + Im 1 2 R [xϕ N , (-∆) α )](v n+1 + δv n )V n+1 + δV n dx. (63)
We are going to estimate the right hand side of (63). First, the first term is bounded by above as follows

| R xϕ N f η V n+1 + δV n dx| ≤ ||xϕ N f η || 2 ||V n+1 || 2 + δ||V n || 2 . (64)
Then, the second term is bounded by above by

1 2 ||[xϕ N , (-∆) α ](v n+1 + δv n )|| L 2 ||V n+1 || 2 + δ||V n || 2 . ( 65
)
We now use Proposition 1 to get, using also that v n remains bounded in

H α ||[xϕ N , (-∆) α ](v n+1 + δv n )|| L 2 ≤ K(||ξ xϕ N || L 1 + ||ξ 2α xϕ N || L 1 ). (66)
Actually

||ξ xϕ N || L 1 = N 2 R |ξ||∂ ξ φ(N ξ)dξ| ≤ C. (67)
The same computation leads to

||ξ 2α xϕ N || L 1 converges to 0, (68)
when N → +∞. Therefore we have, gathering these in (63)

V n+1 L 2 -δ V n L 2 ≤ τ K. (69)
Again thanks to the discrete Gronwall lemma, we deduce that

xϕ( x N )v n+1 L 2 ≤ K. (70)
Hence, letting N goes to +∞, thanks to the Fatou Lemma we deduce that,

xv n+1 L 2 ≤ K. (71)
Thus, we deduce that v n is bounded in H α ∩ L 2 (R, (1 + x 2 )dx).

We now proceed to the H 2α estimate. Taking the inner product of (50) with i(-∆) α (v n+1 -δv n ), we get

1 2 (-∆) α v n+1 2 L 2 -δ 2 (-∆) α v n 2 L 2 = 1 4 Re R |u n+1 | 2 + |u n | 2 v n+1 + δv n (-∆) α v n+1 -δv n dx -Im R f η (-∆) α v n+1 -δv n dx. (72)
Applying Cauchy-Schwarz inequality and using the fact that the sequences (u n ) n and (v n ) n are bounded in H α we deduce that,

v n+1 H 2α ≤δ v n H 2α + K. (73)
Thus by the discrete Gronwall Lemma and thanks to v 0 = 0 we get,

v n+1 H 2α ≤ K τ .

Existence of the global Attractor

We have constructed a splitting u n = v n + w n where w n is small enough in H α and v n belongs to a bounded set of

H 2α ∩ L 2 (R, (1 + x 2 )dx) ⊂ H α .
To establish the proof of compactness of the trajectories in H α , we use the following classical compact embedding.

Lemma 6

The embedding

H 2α ∩ L 2 (R, (1 + x 2 )dx) → H α is compact.
For any η, any trajectory u n is trapped into the sum of a compact set B η in H α (since B η is bounded in H 2α ∩ L 2 (R, (1 + x 2 )dx)) and a ball of radius η centered at 0. Then the trajectory is covered by a finite sum of balls or radii 2η that ensures the asymptotic compactness of the trajectory. Now, the assumptions of Theorem I.1.1 [START_REF] Temam | Infinite Dimensional Dynamical Systems in Mechanics and Physics[END_REF] are satisfied and we have a compact global attractor A τ in H α for S.

Dimension of the attractor

We prove that the global attractor has finite fractal dimension. For this purpose, we rely on a result given in [START_REF] Chueshov | Attractors for Second-Order Evolution Equations with a Nonlinear Damping[END_REF].

Proposition 8 Let X be a separable Hilbert space and M is a bounded closed set in X. Assume that there exists a mapping S : M → X such that M ⊆ SM and 1. S is Lipschitz on M, i.e., there exists L 0 > 0 such that,

||Sv 1 -Sv 2 || X ≤ L 0 ||v 1 -v 2 || X , v 1 , v 2 ∈ M. 2. For . Y that is compact seminorm with respect to . X (i.e X ⊂ Y is compact) such that ||Sv 1 -Sv 2 || X ≤ µ||v 1 -v 2 || X + K( v 1 -v 2 Y + Sv 1 -Sv 2 Y ), v 1 , v 2 ∈ M ;
where 0 < µ < 1 and K > 0 are constants. Then M is a compact set in X with finite fractal dimension.

To begin with, we prove that if the external force is in L 2 (R, (1 + x 2 )dx), then the global attractor is also included in this space.

Proposition 9 Let f ∈ L 2 (R, (1 + x 2 )dx). Then the global attractor A τ is a bounded set in H α ∩ L 2 (R, (1 + x 2 )dx).
Proof: Consider (u n ) n be a global trajectory that belongs the global attractor A τ . Set U n (x) = xϕ( x N )u n and ϕ N (x) = ϕ( x N ) as above. Let us observe that equation ( 7) reads also

1 τ (U n+1 -δU n ) - i 2 (-∆) α (U n+1 + δU n ) + i 4 (|u n+1 | 2 + |u n | 2 )(U n+1 + δU n ) = xϕ N f + i 2 [xϕ N , (-∆) α ](u n+1 + δu n ). (74)
Now, taking the L 2 inner product of (74) with U n+1 + δU n we obtain

1 τ U n+1 2 L 2 -δ 2 U n 2 L 2 = Re R xϕ N f U n+1 + δU n dx + Im 1 2 R [xϕ N , (-∆) α ](u n+1 + δu n )U n+1 + δU n dx. (75)
We are going to estimate the right hand side of (75). The first term is bounded by above as follows

| R xϕ N f U n+1 + δU n dx| ≤ ||xϕ N f || 2 ||U n+1 || 2 + δ||U n || 2 . ( 76
)
and the second term is bounded by above by

1 2 ||[xϕ N , (-∆) α ](u n+1 + δu n )|| L 2 ||U n+1 || 2 + δ||U n || 2 . ( 77
)
Using that the global attractor is bounded in H α ∩ L 2 (R, (1 + x 2 )dx) we then obtain, for some constant K

1 τ W n+1 L 2 -δ W n L 2 ≤ K w n+1 L ∞ + w n L ∞ + [xϕ N , (-∆) α ](w n+1 + δw n ) L 2 .
(84)

A straightforward application of Proposition 1, using also (67) and (68), leads to

1 τ W n+1 L 2 -δ W n L 2 ≤ K w n+1 L ∞ + w n L ∞ + w n+1 H 2α-1 + w n H 2α-1 . (85)
We now let N → +∞ in (85). Using that H α ⊂ L ∞ ∩ H 2α-1 and the fact that S : w n → w n+1 is Lipschitz on H α we then infer that the map S is Lipschitz on H α ∩ L 2 (R, (1 + x 2 )dx).

We now proceed to the second assumption in Proposition 8. On the one hand, considering the equation satisfied by the difference w n = u n+1 -v n+1 of two solutions u n+1 = Su n and v n+1 = Sv n , we have, adding (44) to (46)

||w n+1 || 2 H α ≤ δ 2 ||w n || 2 H α + K ||w n+1 || 2 L 2 + ||w n || 2 L 2 . ( 86
)
On the other hand, it remains to prove the same assumption for the weighted norm. Letting N → +∞ in (85) lead to

xw n+1 L 2 ≤ δ xw n L 2 + K w n+1 L ∞ + w n L ∞ + w n+1 H 2α-1 + w n H 2α-1 . (87)
Thanks to (86) and (87), we deduce :

w n+1 X ≤ δ w n X + K w n+1 L ∞ + w n L ∞ + w n+1 H 2α-1 + w n H 2α-1 + ||w n+1 || L 2 + ||w n || L 2 . ( 88
)
Since the embedding X ⊂ L ∞ ∩ H 2α-1 is compact, this completes the proof of Theorem 3.

Remark 3 It is worth to point out that our upper bound for the dimension of the attractor depends on τ and blows up as τ converges to 0. This is not satisfactory since the dimension of the continuous limit equation is finite. To our knowledge, to prove an upper bound that is independent of τ is a difficult open issue.

ACKNOWLEDGEMENTS

The first author was supported in part by the Labex CEMPI (ANR-11-LABX-0007-01). The two last authors acknowledge the financial support of PHC Utique ASEO.

(53) Proposition 6 The sequence (v n ) n is well defined and belongs to H 2α .

Proof : the equation (50) reads also

The operator

) is invertible and its inverse maps L 2 into H 2α . We then prove recursively on n that v n belongs to H 2α . We now prove that the sequence w n remains asymptotically small. Lemma 5 There exists κ such that for n large enough

Proof : we consider the scalar product of (52) with w n+1 + δw n . This yields

We use the Cauchy-Schwarz inequality and (49) to obtain

The discrete Gronwall lemma yields that the sequence w n remains small in L 2 , i.e.

We now proceed to the H α estimate. We consider the scalar product of (52) with i(w n+1 -δw n ) to obtain

We now use Proposition 1 to get, using also that u n remains bounded in

Proceeding as in ( 67) and (68), we know that the right hand side of (78) is bounded independently of N . Therefore

Using the discrete Gronwall lemma we then have

Letting n → +∞ leads to

Letting N → +∞ completes the proof of the proposition.

We now complete the proof of Theorem 3. We know that the attractor is included in H α ∩ L 2 (R, (1 + x 2 )dx). We plan to apply Proposition 8 with

Due to (48), we know that S is a Lipschitz map for the H α norm; to check the first assumption it remains to prove the same property for the weighted L 2 norm. We proceed as follows; consider u n , v n ∈ A τ and set u n+1 = Su n , v n+1 = Sv n . Consider also

Now, taking the L 2 inner product of (82) with W n+1 + δW n we obtain


Caitlin M Gillis

Priscila P Zenatti

David A Mancardi

Héloïse Beutier

Laurence Fiette

Lynn E Macdonald

Andrew J Murphy

Susanna Celli

Philippe Bousso

Friederike Jönsson

Pierre Bruhns
email: bruhns@pasteur.fr

In vivo effector functions of high-affinity mouse IgG receptor FcγRI in disease and therapy models

INTRODUCTION

IgG receptors (FcγR) in both humans and mice are divided into high-affinity IgG receptors that are able to retain monomeric IgG, and low-affinity IgG receptors that do not. Both high-and low-affinity FcγRs are, however, able to bind to IgG-immune complexes or IgGopsonised cells and surfaces. In humans only hFcγRI is a high-affinity IgG receptor, for human IgG1, IgG3 and IgG4; and in mice both mFcγRI (for mouse IgG2a) and mFcγRIV (for mouse IgG2a and IgG2b) are high-affinity receptors [1]. Although it was proposed that high-affinity FcγRs are occupied by circulating IgG in vivo (discussed in [2]), multiple effector roles for hFcγRI, mFcγRI and mFcγRIV have been reported using mouse models of disease and therapy [3][START_REF] Van Der Poel | Functional characteristics of the high affinity IgG receptor, FcgammaRI[END_REF][START_REF] Gillis | Contribution of Human FcgammaRs to Disease with Evidence from Human Polymorphisms and Transgenic Animal Studies[END_REF][START_REF] Bogdanovich | Human IgG1 antibodies suppress angiogenesis in a target-independent manner[END_REF]. hFcγRI has been studied by exogenous expression in hFcγRI tg mice, demonstrating its role on dendritic cells in the enhancement of antigen presentation and cross-presentation [START_REF] Bevaart | CpG oligodeoxynucleotides enhance FcgammaRI-mediated cross presentation by dendritic cells[END_REF], and on neutrophils and monocyte/macrophages in inflammation, autoimmunity and systemic anaphylaxis [START_REF] Mancardi | The high-affinity human IgG receptor FcgammaRI (CD64) promotes IgG-mediated inflammation, anaphylaxis, and antitumor immunotherapy[END_REF]. These studies indicate that hFcγRI can, by itself, induce clinical signs of autoimmune diseases, by triggering local inflammation (e.g. autoimmune rheumatoid arthritis) or phagocytosis (e.g. autoimmune thrombocytopenia and anaemia [START_REF] Van Vugt | FcR gamma-chain is essential for both surface expression and function of human Fc gamma RI (CD64) in vivo[END_REF]). Additionally, hFcγRI was reported to induce allergic shock (anaphylaxis) triggered by IgG-immune complexes [START_REF] Mancardi | The high-affinity human IgG receptor FcgammaRI (CD64) promotes IgG-mediated inflammation, anaphylaxis, and antitumor immunotherapy[END_REF]. Finally, hFcγRI may also be a therapeutic target as it can mediate antibody-based therapies such as antimalaria [START_REF] Mcintosh | The importance of human FcgammaRI in mediating protection to malaria[END_REF], anti-metastatic melanoma [START_REF] Mancardi | The high-affinity human IgG receptor FcgammaRI (CD64) promotes IgG-mediated inflammation, anaphylaxis, and antitumor immunotherapy[END_REF] and angiogenesis prevention [START_REF] Bogdanovich | Human IgG1 antibodies suppress angiogenesis in a target-independent manner[END_REF]. The mouse counterpart of hFcγRI, mFcγRI, has been so far studied only by the effect of its absence.

Compared to wild-type, mFcγRI -/-mice demonstrate reduced reaction severity in models of autoimmune diseases such as experimental haemolytic anaemia and arthritis [START_REF] Ioan-Facsinay | Fc gamma RI (CD64) contributes substantially to severity of arthritis, hypersensitivity responses, and protection from bacterial infection[END_REF][START_REF] Van Lent | Role of activatory Fc gamma RI and Fc gamma RIII and inhibitory Fc gamma RII in inflammation and cartilage destruction during experimental antigen-induced arthritis[END_REF]. In addition mFcγRI -/-mice are less susceptible to IgG-mediated systemic anaphylaxis, Arthus reactions [START_REF] Barnes | FcgammaRIdeficient mice show multiple alterations to inflammatory and immune responses[END_REF][START_REF] Baumann | A codominant role of Fc gamma RI/III and C5aR in the reverse Arthus reaction[END_REF], and show reduced efficacy of anti-melanoma [START_REF] Bevaart | The high-affinity IgG receptor, FcgammaRI, plays a central role in antibody therapy of experimental melanoma[END_REF][START_REF] Otten | Experimental antibody therapy of liver metastases reveals functional redundancy between Fc gammaRI and Fc gammaRIV[END_REF][START_REF] Albanesi | Cutting Edge: FcgammaRIII (CD16) and FcgammaRI (CD64) Are Responsible for Anti-Glycoprotein 75 Monoclonal Antibody TA99 Therapy for Experimental Metastatic B16 Melanoma[END_REF], anti-lymphoma [START_REF] Minard-Colin | Lymphoma depletion during CD20 immunotherapy in mice is mediated by macrophage FcgammaRI, FcgammaRIII, and FcgammaRIV[END_REF] and antiangiogenic therapies [START_REF] Bogdanovich | Human IgG1 antibodies suppress angiogenesis in a target-independent manner[END_REF]. mFcγRIV was also initially studied by its absence in mFcγRIV -/-mice or by using blocking anti-mFcγRIV mAbs. These studies reported reduced IgG-mediated autoimmune anaemia, thrombocytopenia, rheumatoid arthritis and experimental nephrotoxic nephritis, but also reduced anaphylaxis and less efficient subcutaneous melanoma therapy in the absence or after blockade of mFcγRIV [START_REF] Nimmerjahn | Fc gamma RIV: a novel FcR with distinct IgG subclass specificity[END_REF][START_REF] Nimmerjahn | FcgammaRIV deletion reveals its central role for IgG2a and IgG2b activity in vivo[END_REF]; some of the latter results should be taken with caution since mFcγRIV-blocking antibody 9E9 may also block mFcγRIII in vivo [START_REF] Tipton | Anti-mouse FcgammaRIV antibody 9E9 also blocks FcgammaRIII in vivo[END_REF]. Effector functions could nevertheless be definitively attributed to mFcγRIV through the generation of mice expressing mFcγRIV without other FcγR. Indeed, using mFcγRIV only mice we could demonstrate that mFcγRIV can individually induce autoimmune thrombocytopenia and rheumatoid arthritis, as well as IgG-mediated airway inflammation and anaphylaxis, but not antimetastatic melanoma therapy [START_REF] Mancardi | The high-affinity human IgG receptor FcgammaRI (CD64) promotes IgG-mediated inflammation, anaphylaxis, and antitumor immunotherapy[END_REF][START_REF] Mancardi | FcgammaRIV is a mouse IgE receptor that resembles macrophage FcepsilonRI in humans and promotes IgE-induced lung inflammation[END_REF][START_REF] Jönsson | Mouse and human neutrophils induce anaphylaxis[END_REF][START_REF] Mancardi | The murine high-affinity IgG receptor Fc(gamma)RIV is sufficient for autoantibody-induced arthritis[END_REF]. Altogether these data propose multiple effector functions for high-affinity receptors hFcγRI, mFcγRI and mFcγRIV in autoimmune and inflammatory disease models, and therapy, with direct evidence provided by studies using hFcγRI tg mice and mFcγRIV only mice, but only indirect evidence provided by mFcγRI -/-mice.

The in vivo effector functions proposed for mFcγRI in IgG-mediated autoimmune disease and therapy models are surprising, considering its expression is restricted to monocytes, monocyte-derived dendritic cells and some tissue-resident macrophages, and is absent on neutrophils. As several reports suggest redundant functions among mFcγRs (reviewed in [1,2]), it is uncertain if mFcγRI can induce IgG-mediated autoimmune diseases and therapeutic efficacy by itself, or if this receptor is indirectly involved: either for optimal activation via other mFcγRs or priming of effector cells. Therefore we analysed the in vivo effector functions of mFcγRI in mFcγRI only mice, i.e. in the absence of mFcγRIIB, mFcγRIII and mFcγRIV, in comparison with mFcγR null mice that express no mFcγR. Our results identify the effector functions of mFcγRI as more restricted than initially reported, but confirm that mFcγRI does function independently in vivo, in particular for depletion of IgG-opsonised cells.

MATERIALS & METHODS

Mice

C57BL/6J mice (WT) were purchased from Charles River. VG1505 (FcγRI only ) mice were reported previously [START_REF] Albanesi | Cutting Edge: FcgammaRIII (CD16) and FcgammaRI (CD64) Are Responsible for Anti-Glycoprotein 75 Monoclonal Antibody TA99 Therapy for Experimental Metastatic B16 Melanoma[END_REF] and generated by Regeneron Pharmaceuticals, Inc. FcγR null mice were generated by crossing FcγRI only mice to FcγRI -/-mice. FcγRI only and FcγR null mice were bred at Institut Pasteur, used for experiments at 8-11 weeks of age and all protocols were approved by the Animal Ethics committee CETEA (Institut Pasteur, Paris, France) registered under #C2EA-89.

K/BxN serum-induced passive arthritis

K/BxN serum was generated from a pooled collection of >40 animals. Arthritis was induced by i.v. transfer of indicated volumes of K/BxN serum, and scored as described [START_REF] Bruhns | Colony-stimulating factor-1dependent macrophages are responsible for IVIG protection in antibody-induced autoimmune disease[END_REF]. In some experiments mice were sacrificed on day 8 for blinded histological assessment.

Airway Inflammation

As previously described [START_REF] Jönsson | Human FcgammaRIIA induces anaphylactic and allergic reactions[END_REF], mice were injected with 50 µL of rabbit anti-OVA serum i.n. and 500 µg of OVA i.v. 16-18h later 4 bronchoalveolar lavages (BALs) were performed with cold PBS (1x 0.5ml, then 3x 1mL) under lethal anaesthetic. Cells were pooled and stained for flow cytometry after RBC lysis; and haemorrhage was determined by OD570nm in the supernatant.

We confirmed that mFcγRI, like all mouse FcγR, can indeed bind rabbit IgG immune complexes (Supplementary Fig. 2).

Passive Systemic Anaphylaxis (PSA)

Mice were sensitised by i.v. injection of 500µg anti-DNP IgG2a (clone Hy1.2) and challenged 16h later with 200µg TNP(21-31)-BSA i.v. Alternatively, mice were injected with 1mg of heataggregated (1 hour at 63°C in BBS pH8) human IVIG; considering that mouse FcγRI cross-binds human IgG subclasses 1, 3 and 4 [START_REF] Overdijk | Crosstalk between Human IgG Isotypes and Murine Effector Cells[END_REF]. Central temperature was monitored using a digital thermometer with rectal probe (YSI).

Experimental Thrombocytopenia

Blood samples were taken in EDTA before and at indicated time points after i.v. injection of 3 or 10µg anti-platelet mAb 6A6. Some mice were treated 32h before 6A6 injection with 300µL PBSor clodronate-liposomes i.v. Platelet counts were determined using an ABC Vet automatic blood analyser (Horiba ABX).

Tumour Immunotherapy

Mice were depilated and received 5x10 5 B16-Luc2+ cells s.c. on d0. Where indicated, mice were injected i.v. with 200µg mAb TA99 on d1, d2 and d3 (Figure 4A, closed symbols), and control groups were untreated (Figure 4A, open symbols). Bioluminescence was acquired from anaesthetised mice on d1, d7 and d13, 10 min after injection of 75µg luciferin s.c. (IVIS Spectrum CT, Caliper Life Sciences), and images were analysed with Living Image software.

Anti-mouse CD20 treatment

Mice received a single i.v. injection of 50µg anti-mouse CD20 (clone 5D2, IgG2a, Genentech) to deplete endogenous B cells, or saline control, and CD19 + B220 + B cells in the blood, spleen and inguinal lymph nodes were assessed 16 hours later by flow cytometry. Remaining B cells were calculated as a percentage of the average of vehicle-treated controls (Fig. 4B).

Statistics

Data was analysed using one-way ANOVA with Bonferroni post-test (Fig. 2C) or a Tukey's multiple comparisons test (Fig. 1 B) to compare individual timepoints (Fig. 2 B, D &E, and Fig. 3 A, B & E), or a Kruskal-Wallis test with Dunn's multiple comparisons (Fig. 2A & 2C, bottom panel); a Student's t-test (Fig. 3 C&D, Fig. 4B) or a Mann-Whitney test (Fig. 4A). Statistical significance is indicated (ns: p>0.05; *p<0.05; **p<0.01, *** p<0.001, **** p<0.0001).

Please refer to supplemental Methods for information on reagents, flow cytometry, histology, surgical procedures and Active Systemic Anaphylaxis (ASA).

RESULTS & DISCUSSION

To evaluate the in vivo effector functions of mouse FcγRI, we investigated mice expressing this receptor in the absence of other endogenous classical FcγR (FcγRIIB, FcγRIII and FcγRIV-deficient), termed FcγRI only mice [START_REF] Albanesi | Cutting Edge: FcgammaRIII (CD16) and FcgammaRI (CD64) Are Responsible for Anti-Glycoprotein 75 Monoclonal Antibody TA99 Therapy for Experimental Metastatic B16 Melanoma[END_REF] or VG1505 mice [START_REF] Gillis | Mechanisms of anaphylaxis in human low-affinity IgG receptor locus knock-in mice[END_REF], in comparison with mice deficient for all four endogenous classical FcγRs (FcγRI, FcγRIIB, FcγRIII and FcγRIVdeficient), termed FcγR null mice (Fig. 1A). Both strains retain FcRn and non-classical IgG receptor expression. FcγRI only and FcγR null mice display normal breeding patterns and development, and no overt pathological signs up until 1 year of age. We assessed the circulating immune cell populations in these novel strains, compared to WT mice, using an automated blood cell analyser (Fig. 1B). Across a large sample size, WT, FcγRI only and FcγR null mice display comparable total leukocyte counts in the blood, and similar frequencies of leukocytes, granulocytes and eosinophils (Fig. 1B). Notably, FcγRI only and FcγR null mice have a slightly higher frequency of circulating monocytes. The pattern of expression of FcγRI in FcγRI only mice was comparable to wt mice: FcγRI was detected on circulating Ly6C hi and Ly6C low monocytes (Fig. 1D), with greater expression on the latter, and on CD11b + CD11c + spleen cells, liver kupffer cells and liver macrophages, alveolar macrophages and bone marrow monocytes (not shown) and prominently on CD11b + skin resident cells (Fig. 1E), but barely detected on F4/80 + peritoneal macrophages, in agreement with previous reports [START_REF] Mancardi | The murine high-affinity IgG receptor Fc(gamma)RIV is sufficient for autoantibody-induced arthritis[END_REF][START_REF] Tan | Unique monoclonal antibodies define expression of Fc gamma RI on macrophages and mast cell lines and demonstrate heterogeneity among subcutaneous and other dendritic cells[END_REF][START_REF] Langlet | CD64 Expression Distinguishes Monocyte-Derived and Conventional Dendritic Cells and Reveals Their Distinct Role during Intramuscular Immunization[END_REF]].

An understanding of the participation of FcγRI in autoimmune and inflammatory pathologies remains elusive [START_REF] Van Der Poel | Functional characteristics of the high affinity IgG receptor, FcgammaRI[END_REF]. FcγRI -/-mice seem to experience reduced inflammation associated with immune complex tissue deposition [START_REF] Barnes | FcgammaRIdeficient mice show multiple alterations to inflammatory and immune responses[END_REF] and severity of antigen-induced arthritis [START_REF] Ioan-Facsinay | Fc gamma RI (CD64) contributes substantially to severity of arthritis, hypersensitivity responses, and protection from bacterial infection[END_REF][START_REF] Van Lent | Role of activatory Fc gamma RI and Fc gamma RIII and inhibitory Fc gamma RII in inflammation and cartilage destruction during experimental antigen-induced arthritis[END_REF]. Yet redundancy between FcγRI and FcγRIII certainly exists [START_REF] Van Lent | Role of activatory Fc gamma RI and Fc gamma RIII and inhibitory Fc gamma RII in inflammation and cartilage destruction during experimental antigen-induced arthritis[END_REF] and many of these earlier studies did not consider the potential contribution of FcγRIV [START_REF] Baumann | A codominant role of Fc gamma RI/III and C5aR in the reverse Arthus reaction[END_REF][START_REF] Mancardi | The murine high-affinity IgG receptor Fc(gamma)RIV is sufficient for autoantibody-induced arthritis[END_REF]. To address whether FcγRI can play an effector role in inflammatory autoimmunity, we subjected FcγRI only and FcγR null mice to the K/BxN passive serum transfer model of autoimmune rheumatoid arthritis (K/BxN PA) [START_REF] Monach | The K/BxN arthritis model[END_REF]. Transfer of K/BxN serum into wt mice induced inflammatory signs of arthritis, and an arthritic score which peaked at day 4-6 and remained elevated until day 10 following serum transfer (Fig. 2A&B), yet the same volume transferred into FcγRI only or FcγR null mice did not induce arthritis. Histological assessment of ankle joints confirmed marked to severe arthritis in wt mice, and no microscopic signs of arthritis observed in either FcγRI only or FcγR null mice (Supplementary Fig. 1A-F). Moreover, K/BxN serum transfer at a dose higher that which was sufficient to induce arthritis in FcγRIV only mice [START_REF] Mancardi | The murine high-affinity IgG receptor Fc(gamma)RIV is sufficient for autoantibody-induced arthritis[END_REF] did not result in arthritis induction in FcγRI only mice (Supplementary Fig. 1G), indicating that FcγRI alone cannot induce significant cellular infiltration and inflammation associated with this K/BxN model of autoimmune rheumatoid arthritis.

During experimental antigen-induced arthritis (EAIA), FcγRI -/-mice were reported to experience comparable joint swelling but a reduction in severe cartilage destruction compared to wt mice [START_REF] Ioan-Facsinay | Fc gamma RI (CD64) contributes substantially to severity of arthritis, hypersensitivity responses, and protection from bacterial infection[END_REF][START_REF] Van Lent | Role of activatory Fc gamma RI and Fc gamma RIII and inhibitory Fc gamma RII in inflammation and cartilage destruction during experimental antigen-induced arthritis[END_REF]. Importantly, EAIA relies first on antigen uptake and presentation, a known functionality of FcγRI [START_REF] Barnes | FcgammaRIdeficient mice show multiple alterations to inflammatory and immune responses[END_REF] on monocyte-derived dendritic cells, leading to the activation of antigen-specific T and B cells and to pathogenic antibody production, and thereafter to FcγRmediated effector functions. Antigen uptake and presentation is not required in the K/BxN passive arthritis model, as pathogenic antibodies are directly injected into the recipient mouse.

Because the severity of K/BxN passive arthritis is not affected in FcγRI -/-mice [START_REF] Bruhns | Colony-stimulating factor-1dependent macrophages are responsible for IVIG protection in antibody-induced autoimmune disease[END_REF], and because FcγRI only mice are resistant to this arthritis model (this report), FcγRI evidently does not induce the required effector functions to engender arthritic symptoms after K/BxN serum transfer. If this inability extends also to the EAIA model, one may propose that the reduced severity of EAIA observed in FcγRI -/-mice is attributable rather to reduced antigen uptake and presentation. Furthermore, the cardinal marker of cartilage destruction used in EAIA studies relies on the release of matrix metalloproteinases (MMPs) and the creation of neo-epitopes [START_REF] Ioan-Facsinay | Fc gamma RI (CD64) contributes substantially to severity of arthritis, hypersensitivity responses, and protection from bacterial infection[END_REF][START_REF] Singer | VDIPEN, a metalloproteinase-generated neoepitope, is induced and immunolocalized in articular cartilage during inflammatory arthritis[END_REF]. Therefore although FcγRI does not mediate cell recruitment and joint inflammation in either of these arthritis models (Fig. 2A and [START_REF] Ioan-Facsinay | Fc gamma RI (CD64) contributes substantially to severity of arthritis, hypersensitivity responses, and protection from bacterial infection[END_REF][START_REF] Van Lent | Role of activatory Fc gamma RI and Fc gamma RIII and inhibitory Fc gamma RII in inflammation and cartilage destruction during experimental antigen-induced arthritis[END_REF]), it may contribute to reaction severity via local events in the tissue, i.e. MMP release and cartilage destruction.

The K/BxN model of autoimmune rheumatoid arthritis relies on local generation of IgGimmune complexes on the cartilage surface, that trigger the activation of FcγR-expressing cells.

To determine if FcγRI can induce local inflammation in another context, we induced IgGimmune complexes at another anatomical site, the airways, and examined the resulting alveolitis.

The inflammatory response in wt mice is characterised in the bronchoalveolar lavage (BAL) by massive neutrophil infiltration, significant infiltration of Ly-6C + monocytes, and damage to the airways with haemorrhage into the BAL 16h after challenge (Fig. 2C) [START_REF] Jönsson | Human FcgammaRIIA induces anaphylactic and allergic reactions[END_REF]. Neither FcγR null mice nor FcγRI only mice experienced significant local inflammation: only a mild neutrophil infiltration was observed in the absence of inflammatory monocytes and haemorrhage. Together, our data support the notion that FcγRI does not mediate significant immune cell recruitment and inflammation associated with IgG-immune complex deposition, but may contribute to reaction severity via local events in the tissue, as suggested by studies using FcγRI -/-mice [START_REF] Barnes | FcgammaRIdeficient mice show multiple alterations to inflammatory and immune responses[END_REF][START_REF] Baumann | A codominant role of Fc gamma RI/III and C5aR in the reverse Arthus reaction[END_REF].

Local inflammatory reactions, such as that which drive autoimmune arthritis, require the activation of several cell populations including monocytes/macrophages and neutrophils. Systemic inflammatory reactions like severe hypersensitivity reactions, or anaphylaxis, can also proceed through pathways dependent on IgG and IgG receptors, yet symptoms may arise following the activation of only one cell population, among monocyte/macrophages, neutrophils and basophils (reviewed in [START_REF] Gillis | Contribution of Human FcgammaRs to Disease with Evidence from Human Polymorphisms and Transgenic Animal Studies[END_REF][START_REF] Jonsson | Neutrophils in local and systemic antibody-dependent inflammatory and anaphylactic reactions[END_REF]). FcγRI may participate in such reactions by its expression on monocyte/macrophages. Although FcγRI -/-mice were reported to experience mouse IgG2ainduced passive systemic anaphylaxis (PSA) with reduced severity [START_REF] Ioan-Facsinay | Fc gamma RI (CD64) contributes substantially to severity of arthritis, hypersensitivity responses, and protection from bacterial infection[END_REF], we could not reproduce these findings [START_REF] Beutier | IgG subclasses determine pathways of anaphylaxis in mice[END_REF]. Here we demonstrate that FcγRI only , like FcγR null mice, were resistant to mouse IgG2a-mediated anaphylaxis (Fig. 2D), heat-aggregated human IVIG-mediated anaphylaxis (Fig. 2E) and even to an active model of anaphylaxis induced by BSA immunization and challenge (Supplementary Fig. 3). Two possibilities emerge from these findings: firstly, that FcγRI alone is incapable of triggering sufficient inflammatory mediator release to cause systemic symptoms; secondly, that FcγRI expression is insufficient when expressed only on monocyte/macrophages to mediate anaphylaxis induction.

Since we found that FcγRI is not sufficient to mediate IgG-induced local or systemic inflammation that require cell recruitment and activation, and release of mediators, we wondered if FcγRI was able to induce IgG-mediated cell depletion through phagocytosis and/or ADCC mechanisms that contribute to several autoimmune diseases (e.g. autoimmune thrombocytopenia and anaemia) and to immunotherapies. Earlier studies using FcγRI -/-mice indicated that FcγRI contributes to experimental autoimmune haemolytic anaemia induced by RBC-targeting mIgG2a antibodies [START_REF] Ioan-Facsinay | Fc gamma RI (CD64) contributes substantially to severity of arthritis, hypersensitivity responses, and protection from bacterial infection[END_REF], particularly to more severe manifestations at high Ab doses [START_REF] Baudino | Differential contribution of three activating IgG Fc receptors (FcgammaRI, FcgammaRIII, and FcgammaRIV) to IgG2a-and IgG2b-induced autoimmune hemolytic anemia in mice[END_REF]. To determine if FcγRI has autoimmune destructive properties in FcγRI only mice, we examined another model of autoimmunity characterised by circulating immune complexes, immune thrombocytopenic purpura (ITP) induced by injecting anti-platelet antibodies intravenously. ITP could be induced in wt and FcγRI only mice, but not FcγR null mice (Fig. 3A). Within 4 hours of mAb injection, circulating platelet levels were reduced to <20% of their initial concentration in both wt and FcγRI only mice (Fig. 3A&B), and platelet counts remained low even 24 hours later. Administration of platelet-targeting mAb at a threefold-reduced dose was also sufficient to induce platelet clearance in FcγRI only mice, comparable to that of WT mice (Supplementary Fig. 4A&B). Highdose toxic clodronate-containing liposomes, administered i.v. to deplete monocyte/macrophages mainly in the blood, spleen, and liver, protected against ITP induction in both WT and FcγRI only mice (Fig. 3C), indicating that monocytes and macrophages are responsible for platelet clearance.

To investigate the organ-specific macrophage population responsible for FcγRI-dependent autoimmune platelet clearance, we subjected FcγRI only mice to either removal of the spleen (splenectomy), or partial removal of the liver (hemi-hepatectomy), prior to ITP induction (Figure 3D&E). ITP induction in FcγRI only mice was mildly inhibited by splenectomy (Figure 3D): comprising an average reduction in platelet clearance from 88% (±4.2%) to 77% (±9%) 3.5 hours after mAb injection. Splenectomised FcγRI only and WT mice had somewhat elevated platelet counts at baseline, whether sampled 1 week or 3 weeks post-surgery, yet notably spleen removal did not inhibit ITP induction in WT mice (Supplementary Fig 4C, and data not shown). These findings suggest that splenic macrophages have a minor contribution to FcγRI-dependent ITP.

Conversely, partial removal of the liver, which amounted to about 50% reduction in liver mass [START_REF] Montalvao | The mechanism of anti-CD20-mediated B cell depletion revealed by intravital imaging[END_REF], did not affect ITP induction in FcγRI only mice (Figure 3E and Supplementary Fig 4F ), suggesting that liver macrophages may not be mandatory for FcγRI-dependent ITP. We performed ITP experiments starting 4-5 days after hepatectomies or sham surgeries, due to the rapid regenerative potential of the liver. Importantly, despite the inflammatory effect of the surgical procedures, we did not see differences in FcγRI expression on circulating cells between sham operated and hemi-hepatectomised mice (data not shown). It is difficult to completely exclude a role for the liver in this model, as the part of the organ remaining after hemi-hepatectomy may be sufficient to efficiently mediate ITP. Moreover, considerable platelet clearance was still observed in splenectomised mice, which implies the involvement of another physiological site; potentially blood monocytes [START_REF] Biburger | Monocyte subsets responsible for immunoglobulin G-dependent effector functions in vivo[END_REF]. Collectively, our data indicate a minor contribution of splenic macrophages to FcγRI-dependent ITP, but do not provide evidence of a role for liver macrophages.

Antibody-mediated therapies are now a frontline treatment for many malignancies, including a number of autoimmune diseases. Reports using FcγRI -/-mice suggest that FcγRI contributes to IgG-induced tumour cell depletion in the lung [START_REF] Bevaart | The high-affinity IgG receptor, FcgammaRI, plays a central role in antibody therapy of experimental melanoma[END_REF] and liver [START_REF] Otten | Experimental antibody therapy of liver metastases reveals functional redundancy between Fc gammaRI and Fc gammaRIV[END_REF] but not the skin [START_REF] Nimmerjahn | FcgammaRIV deletion reveals its central role for IgG2a and IgG2b activity in vivo[END_REF]. As the latter finding may be due to redundant functions among mouse FcγRs in the skin, we followed TYRP-1 + Luc2 + B16 melanoma cells tumour growth by bioluminescent imaging in vivo [START_REF] Albanesi | Cutting Edge: FcgammaRIII (CD16) and FcgammaRI (CD64) Are Responsible for Anti-Glycoprotein 75 Monoclonal Antibody TA99 Therapy for Experimental Metastatic B16 Melanoma[END_REF] in mice treated or not with anti-TYRP-1 mouse IgG2a TA99 mAb. Identical growth kinetics were detected in FcγRI only and FcγR null mice (Fig. 2F; WT mice in Supplementary Fig. 5A), and repeated TA99 injections dramatically reduced tumour load in FcγRI only mice, but not in FcγR null mice, to that of background levels (Fig. 4A and Supplementary Fig. 5). Thus FcγRI only mice reveal FcγRI-mediated functions that can remain masked in FcγRI -/-mice. Furthermore, these data reinforce the previously reported anti-tumour effector function of FcγRI in lung and liver tissues [START_REF] Bevaart | The high-affinity IgG receptor, FcgammaRI, plays a central role in antibody therapy of experimental melanoma[END_REF][START_REF] Otten | Experimental antibody therapy of liver metastases reveals functional redundancy between Fc gammaRI and Fc gammaRIV[END_REF][START_REF] Albanesi | Cutting Edge: FcgammaRIII (CD16) and FcgammaRI (CD64) Are Responsible for Anti-Glycoprotein 75 Monoclonal Antibody TA99 Therapy for Experimental Metastatic B16 Melanoma[END_REF][START_REF] Minard-Colin | Lymphoma depletion during CD20 immunotherapy in mice is mediated by macrophage FcgammaRI, FcgammaRIII, and FcgammaRIV[END_REF] and extends it to the skin tissue.

Anti-CD20 therapy to deplete B cells has been highly successful in the treatment of B cell malignancies and autoimmune disorders. B cell depletion is known to depend on FcR-dependent mechanisms [START_REF] Uchida | The innate mononuclear phagocyte network depletes B lymphocytes through Fc receptor-dependent mechanisms during anti-CD20 antibody immunotherapy[END_REF], primarily phagocytosis by Kupffer cells in the liver [START_REF] Montalvao | The mechanism of anti-CD20-mediated B cell depletion revealed by intravital imaging[END_REF]. A role for FcγRI in the clearance of both malignant and endogenous B cells, in cooperation with FcγRIII and FcγRIV, has been suggested by several studies [START_REF] Minard-Colin | Lymphoma depletion during CD20 immunotherapy in mice is mediated by macrophage FcgammaRI, FcgammaRIII, and FcgammaRIV[END_REF][START_REF] Hamaguchi | Antibody isotype-specific engagement of Fcgamma receptors regulates B lymphocyte depletion during CD20 immunotherapy[END_REF], but has not been formally demonstrated.

Since FcγRI was sufficient to mediate destructive platelet clearance (Fig. 3), and is expressed on liver Kupffer cells, we tested the capacity of FcγRI to deplete endogenous B cells in a model of anti-CD20 therapy. Administration of mouse CD20-targeting mAb 5D2 (IgG2a) induced B cell depletion in the blood, spleen and lymph nodes of FcγRI only , but not FcγR null mice. 16 hours after treatment 85% of B cells were cleared from the blood and 25-30% of B cells from the secondary lymphoid organs of anti-CD20 treated FcγRI only mice. These data demonstrate that FcγRI, in the absence of FcγRIII and FcγRIV, is sufficient to mediate endogenous B cell clearance, and support a contribution for FcγRI to the efficacy of anti-CD20 therapy in models of lymphoma and autoimmunity.

In conclusion, genetically modified FcγRI only mice enabled us to demonstrate that the mouse high-affinity IgG receptor FcγRI is sufficient to mediate IgG-induced autoimmune thrombocytopenia and IgG-based immunotherapy targeting either B cells (anti-CD20) or subcutaneous melanoma, in the absence of FcγRIIB, FcγRIII and FcγRIV. FcγRI alone is, however, insufficient to induce IgG-induced autoimmune rheumatoid arthritis, airway inflammation and systemic anaphylaxis, probably due to its inability to efficiently mediate Abinduced cell recruitment or release of inflammatory mediators. Rather we identify that FcγRI mediates Ab-induced cell depletion/destruction, in both pathogenic autoimmune and therapeutic anti-tumour contexts; which agrees with the important contributions of this receptor to pathogen elimination [START_REF] Ioan-Facsinay | Fc gamma RI (CD64) contributes substantially to severity of arthritis, hypersensitivity responses, and protection from bacterial infection[END_REF][START_REF] Mittal | Fcgamma receptor I alpha chain (CD64) expression in macrophages is critical for the onset of meningitis by Escherichia coli K1[END_REF][START_REF] Esser-Von Bieren | Antibody-mediated trapping of helminth larvae requires CD11b and Fcgamma receptor I[END_REF] and antigen uptake and presentation [START_REF] Barnes | FcgammaRIdeficient mice show multiple alterations to inflammatory and immune responses[END_REF]. Furthermore, our data attribute FcγRI-dependent phagocytic function to macrophages in the skin and the spleen. Finally, the effector capabilities of mouse FcγRI appear aligned with its restricted expression profile: low to moderate expression on monocytes, tissue macrophages and monocyte-derived DCs, who are indeed responsible for clearance of foreign bodies and antigen uptake, whereas that of its human homolog hFcγRI extend to pro-inflammatory and pro-anaphylactic functions attributable to more promiscuous expression, particularly high on circulating monocytes and neutrophils. In conclusion, mice models expressing only one particular IgG receptor, e.g. FcγRI only or FcγRIV only mice [START_REF] Albanesi | Cutting Edge: FcgammaRIII (CD16) and FcgammaRI (CD64) Are Responsible for Anti-Glycoprotein 75 Monoclonal Antibody TA99 Therapy for Experimental Metastatic B16 Melanoma[END_REF][START_REF] Mancardi | FcgammaRIV is a mouse IgE receptor that resembles macrophage FcepsilonRI in humans and promotes IgE-induced lung inflammation[END_REF][START_REF] Jönsson | Mouse and human neutrophils induce anaphylaxis[END_REF][START_REF] Mancardi | The murine high-affinity IgG receptor Fc(gamma)RIV is sufficient for autoantibody-induced arthritis[END_REF], are particularly useful to ascribe independent functions to FcγRs, as distinct from potential cooperative roles with other FcγR, the latter of which may be implied from studies using specific FcγR -/-mice.

ACKNOWLEDGMENTS

We

CONFLICT OF INTEREST STATEMENT

L.E.M and A.M. are employees of Regeneron Pharmaceuticals, Inc. and hold stock in the company. C.G., H.B., F.J., L.F., S.C., D.A.M., P.P.Z., P.Bousso and P.Bruhns declare no competing financial interests.

FIGURE LEGENDS

Supplemental Materials and Methods

Reagents

Human IVIG (Gamunex®) was from Grifols, containing 63% hIgG1, 29% hIgG2, 5% hIgG3 and 3% hIgG4. B16-Luc2+ cells were from Caliper-Life Sciences. IgG were purified by Protein G-affinity purification from supernatants of hybridomas producing anti-gp75 mAb (TA99) from American Type Culture Collection, mIgG2a antiplatelet mAb (clone 6A6) provided by Dr R. Good (USFCM, Tampa, FL, USA), and mIgG2a anti-TNP mAb (Hy1.2) provided by Shozo Izui (University of Geneva, Geneva, Switzerland). Luciferin was from Invitrogen, rabbit anti-OVA antiserum, OVA, BSA, and Freund's adjuvant (CFA/IFA) were from Sigma-Aldrich, TNP [START_REF] Tipton | Anti-mouse FcgammaRIV antibody 9E9 also blocks FcgammaRIII in vivo[END_REF][START_REF] Mancardi | FcgammaRIV is a mouse IgE receptor that resembles macrophage FcepsilonRI in humans and promotes IgE-induced lung inflammation[END_REF][START_REF] Jönsson | Mouse and human neutrophils induce anaphylaxis[END_REF][START_REF] Mancardi | The murine high-affinity IgG receptor Fc(gamma)RIV is sufficient for autoantibody-induced arthritis[END_REF][START_REF] Bruhns | Colony-stimulating factor-1dependent macrophages are responsible for IVIG protection in antibody-induced autoimmune disease[END_REF][START_REF] Jönsson | Human FcgammaRIIA induces anaphylactic and allergic reactions[END_REF][START_REF] Overdijk | Crosstalk between Human IgG Isotypes and Murine Effector Cells[END_REF][START_REF] Gillis | Mechanisms of anaphylaxis in human low-affinity IgG receptor locus knock-in mice[END_REF][START_REF] Tan | Unique monoclonal antibodies define expression of Fc gamma RI on macrophages and mast cell lines and demonstrate heterogeneity among subcutaneous and other dendritic cells[END_REF][START_REF] Langlet | CD64 Expression Distinguishes Monocyte-Derived and Conventional Dendritic Cells and Reveals Their Distinct Role during Intramuscular Immunization[END_REF][START_REF] Monach | The K/BxN arthritis model[END_REF] -BSA was from Santa Cruz and PBS-and clodronate-liposomes were prepared as previously described [1]. Vivotag-680 was from Perkin Elmer and OVA-vivotag-680 was prepared as recommended by the manufacturer.

Tissue processing and flow cytometry

Spleens were dissociated through a 70µm cell strainer into MACS buffer (PBS /0.5%BSA /2mM EDTA) and RBC lysis was performed using an ammonium chloride-based buffer. For isolation of skin cells, ears were split into dorsal and ventral halves and roughly chopped before digestion with 0.25mg/mL Liberase TL ResearchGrade (Roche) + 0.1mg/mL DNase (Sigma) for 1h at 37°C (800rpm; Eppendorf Thermomixer), washed with 10x volume of PBS/ 10%FBS /2mM EDTA and processed through a 100µm cell strainer. Livers were perfused with cold PBS before dissection, and liver leukocytes were isolated using the Liver Dissociation Kit and gentleMACS Octo Dissociator from Miltenyi, according to the manufacturer's instructions. Cells were isolated from the peritoneum by lavage with 6mL cold PBS; BALs were performed 3x with 1mL PBS. For blood leukocyte analysis, heparinised blood was subjected to RBC lysis with either Red Blood Cell Lysis Solution (Miltenyi) or BD Pharm Lyse Lysing Buffer (BD Biosciences) and washed with MACS buffer. Single cell suspensions were washed with MACS buffer, incubated with 2.4G2 (Fab') 2 fragments (anti-CD16/32, 40µg/mL; 15min on ice) and stained with fluorochrome-conjugated antibodies in MACS buffer for 30min on ice. Data was collected on a MACSQuant flow cytometer (Miltenyi), and analysed using FlowJo Software (TreeStar, Inc.).

Cell populations were defined by FSC/SSC properties and surface markers as indicated (Fig. 1B), or in the BAL (Fig. 2B): alveolar macrophages (CD11c + / SiglecF + ), eosinophils (CD11c neg / SiglecF + ), neutrophils (CD11c neg / SiglecF neg / CD11b + / Ly-6G + ), Ly-6C + macrophages (CD11c neg / SiglecF neg / CD11b + / Ly-6G neg / Ly-6C + ).

Active Systemic Anaphylaxis (ASA)

Mice were immunised i.p. on d0 with 200µg BSA in Complete Freund's Adjuvant, and boosted on d14 with 200µg BSA in Incomplete Freund's Adjuvant. BSA-specific IgG1, IgG2a/b/c and IgE serum antibodies were titered by ELISA on d21 as described [2]. Mice with comparable antibody titers were challenged 13-14 days after the last immunisation i.v. with 500µg BSA. Central temperature and mortality was monitored.

Partial hepatectomy and splenectomy

Partial hepatectomy was perfomed as described [3]. Mice were anaesthetised and a transverse abdominal incision was made. The superior lobes of the liver were laid on the diaphragm and the ligaments of the caudate lobe dissected. The caudate lobe was then pulled in front of the stomach and resected after in-bloc ligature of its hilum (6/0 silk). The lateral left lobe was resected using the same technique. The abdomen was closed using 4/0 silk running sutures. The procedure removed approximately half of the initial liver mass. Mice were rested for 2 days before experimental procedure.

For splenectomy, a small vertical incision was made on the left flank, the spleen was gently pulled outside the abdomen, and the splenic ligaments and vessels were cut.

The abdomen was closed with a 4/0 silk suture, and the skin with a surgical staple.

Mice were rested for 1-3 weeks before experimental procedure.

Histology

Ankle joints and surrounding tissues (from the extremity of the femur/tibia to the digits) were sampled on day 8 after K/BxN serum transfer (batch #1, 5mL/kg), then fixed and simultaneously decalcified using Formical-4 ® (StatLab Medical Products) for 2 weeks. Samples were routinely embedded in paraffin, and 4 mm sections were stained with haematoxylin and eosin (H&E). Sections were evaluated microscopically and histological changes (i.e. inflammation, pannus formation, bone erosion and cartilage damage) were scored from 0 (no change) to 5 (severe).

Supplemental references

  are thankful to O. Godon, B. Iannascoli, B. Todorova and O. Richard-LeGoff for technical help; the members of the Unit of Dynamics of Immune Responses for experimental advice and discussion; the Service Communication Institutionnelle et Image, Institut Pasteur, Paris, for photography work (Supplementary Figure 3D); A-M. Nicola (Plate-Forme d'Imagerie Dynamique, Institut Pasteur, Paris) for help with bioluminescence experiments; and D. Sinnaya for administrative help (Institut Pasteur, Paris). We are thankful to our colleagues for their generous gifts: D. Mathis and C. Benoist (HMS, Boston, MA, USA) and IGBMC (Illkirch, France) for K/BxN mice, R. Good (USFCM, Tampa, FL, USA) for IgG2a mAb 6A6, S. Izui (University of Geneva, Geneva, Switzerland) for IgG2a mAb Hy1.2, Genentech for anti-mouse CD20 mAb 5D2, and N. Van Rooijen (VU Medical Center, The Netherlands) and Roche Diagnostics GmbH for liposomes and Cl2MDP, respectively. This work was mainly supported by the Institut Pasteur and the Institut National de la Santé et de la Recherche Médicale (INSERM); anaphylaxis studies were specifically supported by funding from the European Research Council (ERC)-Seventh Frame-work Program (ERC-2013-CoG 616050). C.G. was supported partly by a stipend from the Pasteur -Paris University (PPU) International PhD program and by the Institut Carnot Pasteur Maladies Infectieuses, and partly by the Balsan company. P.P.Z. was supported by FAPESP, process number 2014/233533-9. F.J. is an employee of the Centre National de La Recherche Scientifique (CNRS). H.B. is supported by a fellowship from the University Pierre et Marie Curie.

Figure 1 :

 1 Figure 1: FcγRI only mice have normal blood leukocyte composition and show comparable FcγRI expression to that of WT mice. (A) Schematic representation of WT, FcγRI only and FcγR null mice. (B) Leukocyte counts and relative percentages of immune cell populations in the blood of WT (n=20), FcγRI only (n=35) and FcγR null (n=24) mice were enumerated using an

Figure 2 :

 2 Figure 2: FcγRI alone is insufficient to mediate IgG-induced arthritis, airway inflammation

Figure 3 :

 3 Figure 3: FcγRI-mediated thrombocytopenia is dependent on monocyte/macrophages, and

Figure 4 :

 4 Figure 4: FcγRI is sufficient for anti-melanoma and B cell depletion therapies. (A) FcγRI only (circles) or FcγR null (squares) mice were injected with B16-Luc2+ cells s.c. on d0

  Figure 2

2 G

 2 Figure 4

Supplementary Figure 1 (A-F) H&E stained paraffinembedded sections of decalcified ankle joints on day 8 after K/BxN serum transfer (batch #1, 5mL/kg). Histological signs of marked to severe arthritis were noted in WT mice (A, D), and were characterised by neutrophil and mononuclear cell infiltrates (arrow), pannus formation (asterisk), cartilage damage and bone resorption (arrowhead). No microscopic signs of arthritis were observed