Piotr Graczyk

Tomasz Jakubowski

On exit time of stable processes

Keywords:

We study the exit time τ = τ (0,∞) for 1-dimensional strictly stable processes and express its Laplace transform at t α as the Laplace transform of a positive random variable with explicit density. Consequently, τ satisfies some multiplicative convolution relations. For some stable processes, e.g. for the symmetric 2 3 -stable process, explicit formulas for the Laplace transform and the density of τ are obtained as an application.

Introduction

Let α ∈ (0, 2) and (X t , P x ) be a strictly α-stable process in R with characteristic function

E 0 e iXtz = exp -t|z| α 1 -iβ tan πα 2 sgn z ,
where β ∈ [-1, 1] and β = 0 for α = 1. For any D ⊂ R let τ D = inf{t ≥ 0 : X t ∈ D} be the first exit time from D of the process X t . Throughout this article we shall consider the starting point x > 0 and τ = τ (0,∞) , the exit time of X t from the positive half-line.

The question of the first exit time from domains are basic for all stochastic processes. Surprisingly few exact formulas are known for stable processes. The only exceptions are Brownian motion, completely asymmetric stable processes with α > 1 (see [START_REF] Bingham | Maxima of sums of random variables and suprema of stable processes[END_REF], [START_REF] Kyprianou | Introductory lectures on fluctuations of Lévy processes with applications[END_REF], [START_REF] Simon | Fonctions de Mittag-Leffler et processus de Lévy stables sans sauts négatifs[END_REF]) and symmetric Cauchy process ( [START_REF] Darling | The maximum of sums of stable random variables[END_REF], see also [START_REF] Kulczycki | Spectral properties of the Cauchy process on half-line and interval[END_REF]). The quotient τ /τ was studied for independent τ and dual τ in [START_REF] Doney | On the maxima of random walks and stable processes and the arc-sine law[END_REF].

Some recent results on this problem in the completely asymmetric case were obtained by T. Simon in [START_REF] Simon | Fonctions de Mittag-Leffler et processus de Lévy stables sans sauts négatifs[END_REF] and next were applied in [START_REF] Patie | Intertwining certain fractional derivatives[END_REF] and [START_REF] Simon | Hitting densities for spectrally positive stable processes[END_REF]. On the other hand, some formulas were found by A. Kuznetsov in [START_REF] Kuznetsov | On extrema of stable processes[END_REF], however the final expressions are complicated. M. Kwaśnicki in [START_REF] Kwaśnicki | Spectral analysis of subordinated Brownian motions in half-line[END_REF] gives an integral representation of the density of τ in the case of symmetric stable processes (β = 0).

In this article we study the exit time τ = τ (0,∞) for 1-dimensional stable processes and give in Theorem 3 a new formula for its Laplace transform. It follows that τ satisfies some multiplicative convolution relations(Corollary 6); in particular for α > 1 the exit time τ is the multiplicative convolution of a 1/α-stable subordinator with an explicitly given random variable M α,ρ . We generalize in this way the result of [START_REF] Simon | Fonctions de Mittag-Leffler et processus de Lévy stables sans sauts négatifs[END_REF] for all stable processes. Applications of Theorem 3 are next given in the final part of the article. New explicit formulas for the Laplace transform and the density of τ are proven for the processes dual to those of the Doney's class C 1,1 , in particular for the symmetric 2 3 -stable process (Proposition 7 and Corollary 8). Further applications of Theorem 3 will be presented in a forthcoming paper.

The main tool to prove the results of this article is a series representation that we obtained in [START_REF] Graczyk | On Wiener Hopf factors of stable processes[END_REF] for the logarithm of the bivariate Laplace exponent κ(η, θ) of the ascending ladder process built from the process X t . This application of the series representation of ln κ was announced in [START_REF] Graczyk | On Wiener Hopf factors of stable processes[END_REF]. It allows to determine explicitly in Proposition 1 the inverse Stieltjes transform of the function 1/κ(1, θ).

Stieltjes transform and Wiener-Hopf factors

In this part of the article we will exploit our series representation of κ(1, θ) from [START_REF] Graczyk | On Wiener Hopf factors of stable processes[END_REF] by inverting a Stieltjes transform.

Recall that if µ is a positive Borel measure on [0, ∞) then for any x ∈ (0, ∞) the Stieltjes transform of µ is defined by

Sµ(θ) = ∞ 0 1 θ + x dµ(x) (1)
whenever the integral converges. According to [START_REF] Berg | Quelques remarques sur le cône de Stieltjes[END_REF], a function G on (0, ∞) is of the form G(θ) = a + Sµ(θ) for a positive measure µ and a ≥ 0 if and only if (S1) G extends to a holomorphic function in the cut plane

C \ R - (S2) G(θ) ≥ 0 for θ > 0 (S3) ImG(z) ≤ 0 for Imz > 0. Then the inverse Stieltjes transform is S -1 (G)(x) = -(1/π) lim y→0 + ImG(-x + iy) , x > 0 ,
where the limit, in general, is in the vague sense and equals µ. If µ is absolutely continuous with a continuous density, the limit is equal to the density of µ for all x > 0 ( [START_REF] Widder | The Laplace transform[END_REF]).

Let α ∈ (0, 2) and (X t , P x ) be a strictly α-stable process in R. By κ α,ρ (η, θ) we denote the bivariate Laplace exponent of the ascending ladder process built from X t . We normalize it requiring that κ α,ρ (1, 0) = 1. To simplify the notation we will write κ(η, θ) for a fixed pair α, ρ (or equivalently a fixed process X t ). By κ we denote the Laplace exponent for the dual process Xt = -X t . As usually we write the positivity coefficient

ρ = P 0 (X t ≥ 0) = 1 2 + 1 πα arctan β tan πα 2 .
Proposition 1. For ρ ∈ (0, 1] \ {1/α} we have

sin(ραπ) π ∞ 0 1 x + θ x α κ(1, x) x 2α + 2x α cos(ραπ) + 1 dx = 1 κ(1, θ) . (2)
Proof. Denote G(θ) = 1/κ(1, θ). The function G(θ) extends to a holomorphic function h 1 (z) on C \ R -(see [START_REF] Fourati | Inversion de l'espace et du temps des processus de Lévy stables[END_REF],(i) p.205). Let L be the set of Liouville numbers. For θ ∈ (0, 1) and α ∈ L ∪ Q we have by [START_REF] Graczyk | On Wiener Hopf factors of stable processes[END_REF]

G(θ) = exp - ∞ m=1 (-1) m+1 θ m sin(ρmπ) m sin( mπ α ) - ∞ k=1 (-1) k+1 θ αk sin(ραkπ) k sin(αkπ) .
The right hand side of the last formula may be extended to a holomorphic function

h 2 (z) on {z ∈ C : |z| < 1} \ R -defining w α = exp(αLogw)
where Logw = ln |w| + iArgw, Argw ∈ (-π, π], is the principal value of the complex logarithm. We note that h 1 = h 2 on (0, 1), hence

h 1 = h 2 on {z ∈ C : |z| < 1} \ R -and h 2 extends to a holomorphic function on C \ R -, equal for |z| > 1 to the holomorphic extension of 1 κ(1,θ) for θ > 1.
In the first part of the proof we will compute

l(x) = - 1 π lim y→0 + ImG(-x + iy)
for positive x. Denote by h(z) the expression under exponential of h 2 . Let us compute for 0 < x < 1

l(x) = - 1 π lim y→0 + Im exp(h(-x + iy)) = - 1 π e Re(w) sin(Im(w)),
where

w = - ∞ m=1 (-1) m+1 (-x) m sin(ρmπ) m sin( mπ α ) - ∞ k=1
(-1) k+1 e iαkπ x αk sin(ραkπ) k sin(αkπ) .

The last limit is justified by a standard estimation argument, that implies that in a converging power series one can enter the limit under the series. Moreover, the same argument shows that when 0 < x < 1, we have

l(x) = - 1 π lim w→-x,Imw>0
ImG(w) = -1 π e Re(w) sin(Im(w)).

(

) 3
Now we evaluate

Re(w) = ∞ m=1 x m sin(ρmπ) m sin( mπ α ) - ∞ k=1 (-1) k+1 cos(αkπ)x αk sin(ραkπ) k sin(αkπ) , Im(w) = - ∞ k=1 (-1) k+1 sin(αkπ)x αk sin(ραkπ) k sin(αkπ) = ∞ k=1 (-1) k x αk sin(ραkπ) k .
We will need the following formulas from [10]

∞ k=1 p k sin(kϕ) k = arctan p sin ϕ 1 -p cos ϕ , ϕ ∈ (0, 2π), p 2 ≤ 1. ( 4
) ∞ k=1 p k cos(kϕ) k = - 1 2 log(1 -2p cos ϕ + p 2 ) , ϕ ∈ (0, 2π), p 2 ≤ 1. (5)
Therefore applying a formula sin(arctan u) = u √ 1+u 2 we get sin(Im(w)) = sin(arctan

-x α sin(ραπ) 1 + x α cos(ραπ) ) = -x α sin(ραπ) 1+x α cos(ραπ) 1 + x α sin(ραπ) 1+x α cos(ραπ) 2 = -x α sin(ραπ) x 2α + 2x α cos(ραπ) + 1 .
Now we compute Re(w). By (5) we get

Re(w) = ∞ m=1 (-1) m+1 x m sin((1 -ρ)mπ) m sin( mπ α ) + ∞ k=1 (-1) k+1 x αk sin((1 -ρ)αkπ) k sin(αkπ) - ∞ k=1 (-1) k+1 x αk cos(ραkπ) k = log κ(1, x) - 1 2 log(1 + 2x α cos(ραπ) + x 2α ).
Hence

- 1 π e Re(w) sin(Im(w)) = sin(ραπ) π x α κ(1, x) x 2α + 2x α cos(ραπ) + 1 = l(x) > 0.
By [9, Lemma 5] we have for θ > 1

κ(1, θ) = θ αρ κ(1, 1/θ)
and we use the same method and (3) to obtain

l(x) = - 1 π lim w→-x,Imw>0 ImG(w) = sin(ραπ) π x α κ(1, x) x 2α + 2x α cos(ραπ) + 1 (6) for x > 0, x = 1.
As the function l(x) is continuous at x = 1 and by [8, p.205] the limit lim w→-1,Imw>0 ImG(w) exists, it follows that the convergence in (6) holds also for x = 1.

Let us now justify the fact that the function G(θ) = 1/κ(1, θ) is a Stieltjes transform of a positive measure µ on R + . We will check the conditions (S1-3) given in the beginning of this section.

The function G(θ) is strictly positive for θ ∈ (0, ∞) and it extends to a holomorphic function h 1 (z) on C \ R -. Thus the conditions (S1) and (S2) are verified. In order to justify (S3), we use the following property that we proved above: the harmonic function -(1/π)ImG extends continuously to the closed upper half-space {Imz ≥ 0} and its boundary values on R are l(-x) > 0 when x < 0 and 0 for x ≥ 0. Taking into account the fact that lim |z|→∞ G(z) = 0 ([8, p.205]), the maximum principle([2, 1.10]) implies that ImG(z) ≤ 0 on {Imz > 0} and (S3) also holds.

It follows that for a certain a ≥ 0 we have G(θ) = a+S(l)(θ). Considering θ → ∞ we determine a = 0.

Finally consider any α ∈ (0, 2]. Since the Lebesgue measure of the set L ∪ Q is 0 we can take a sequence α n tending to α. Passing to the limit we obtain (2) for all α ∈ (0, 2].

Remark. Other proofs of the fact that 1/κ(1, θ) is the Stieltjes transform of a positive measure µ seem possible, using properties of Bernstein functions ( [START_REF] Schilling | Bernstein functions, Theory and Applications[END_REF]).

We deduce immediately from Proposition 1 the following corollary.

Corollary 2. For ρ ∈ [0, 1) \ {1 -1/α} we have

sin((1 -ρ)απ) π ∞ 0 1 x + θ x α κ(1, x) x 2α + 2x α cos((1 -ρ)απ) + 1 dx = 1 κ(1, θ) . (7)
3 Laplace transform of τ and applications

The following theorem is the main result of the article.

Theorem 3. Let X t be a non-spectrally positive strictly α-stable process on R. For any t > 0 we have

E 1 e -tτ = sin((1 -ρ)απ) π ∞ 0 e -t 1/α x x α-1 κ(1, x) x 2α + 2x α cos((1 -ρ)απ) + 1 dx. ( 8
)
Remark. Observe that the only case excluded from the Theorem 3 is well known: when X t is a spectrally positive α-stable process starting from X 0 = x, 1 < α < 2, then (τ x (0,∞) ) x>0 is a 1/α-stable subordinator and E 1 e -tτ = e -t 1/α ([4] p.281). When X t is spectrally negative, the formula (8) was obtained recently by T. Simon([17]).

Proof. We note that if g(0) = 1 then

∞ 0 1 x + t f (x)dx = g(t) =⇒ ∞ 0 1 x + t f (x) x dx = 1 t - g(t) t . (9)
Indeed

1 = ∞ 0 x + t x + t f (x) x dx = g(t) + ∞ 0 t x + t f (x)
x dx

From [START_REF] Kyprianou | Introductory lectures on fluctuations of Lévy processes with applications[END_REF] we know that

∞ 0 e -θy E y e -ητ dy = 1 θ - κ(η, 0) θ κ(η, θ) , (10)
where τ = τ (0,∞) and η, θ > 0. Putting η = 1 and applying ( 9) to ( 7) we get

∞ 0 e -θy E y e -τ dy = sin((1 -ρ)απ) π ∞ 0 1 x + θ x α-1 κ(1, x) x 2α + 2x α cos((1 -ρ)απ) + 1 dx = sin((1 -ρ)απ) π ∞ 0 ∞ 0 e -y(x+θ) x α-1 κ(1, x) x 2α + 2x α cos((1 -ρ)απ) + 1 dy dx = ∞ 0 e -θy sin((1 -ρ)απ) π ∞ 0 e -yx x α-1 κ(1, x) x 2α + 2x α cos((1 -ρ)απ) + 1 dx dy. Therefore E y e -τ = sin((1 -ρ)απ) π ∞ 0 e -yx x α-1 κ(1, x) x 2α + 2x α cos((1 -ρ)απ) + 1 dx
and assertion of the theorem follows from scaling property E y e -τ = E 1 e -y α τ of stable processes.

For α ≥ 1 we immediately obtain from Theorem 3 a formula for the density h = h α,ρ of τ under P 1 . Corollary 4. Let α ∈ (1, 2). The density of τ = τ (0,∞) under P 1 is given by

h(s) = sin((1 -ρ)απ) π ∞ 0 η 1/α (x, s) x α-1 κ(1, x) x 2α + 2x α cos((1 -ρ)απ) + 1 dx, ( 11
)
where η γ (t, x) is the transition density of γ-stable subordinator.

Proof. Since

∞ 0 e -xs η γ (t, x)dx = e -ts γ we obtain by Theorem 3

∞ 0 e -st h α (s) ds = sin((1 -ρ)απ) π ∞ 0 ∞ 0 e -st η 1/α (x, s) x α-1 κ(1, x) x 2α + 2x α cos((1 -ρ)απ) + 1 ds dx = ∞ 0 e -st sin((1 -ρ)απ) π ∞ 0 η 1/α (x, s) x α-1 κ(1, x) x 2α + 2x α cos((1 -ρ)απ) + 1 dx ds
and the assertion follows.

Corollary 5. If X t is a Cauchy process on R (α = 1, ρ = 1/2) then the density of the exit time τ = τ (0,∞) under P 1 is given by

h(x) = 1 π κ(1, x) x 2 + 1 , x ≥ 0.
Remark. The above formula for h 1,1/2 was obtained previously by Darling in [START_REF] Darling | The maximum of sums of stable random variables[END_REF] (see also [11, (7.13)]).

For α = 1, Theorem 3 gives interesting multiplicative convolution relations verified by τ . We present them in the following subsection.

Interpretation in terms of multiplicative convolutions

For a given strictly α-stable process X t with ρ ∈ [0, 1) \ {1 -1/α} we define the following function on

R + m α,ρ (x) = sin((1 -ρ)απ) πα κ(1, x 1/α ) x 2 + 2x cos((1 -ρ)απ) + 1 , x ≥ 0.
Observe that m α,ρ (x) is a probability density on R + . This follows from the formula

sin((1 -ρ)απ) π ∞ 0 x α-1 κ(1, x) x 2α + 2x α cos((1 -ρ)απ) + 1 dx = 1
obtained from (8) when t → 0 and by a change of variables x = y 1/α . Denote by M α,ρ a positive random variable with density m α,ρ . The variable M α,1/α , 1 < α < 2 appeared for the first time in [START_REF] Simon | Fonctions de Mittag-Leffler et processus de Lévy stables sans sauts négatifs[END_REF] for the special case of a completely asymmetric α-stable process, 1 < α < 2 (in our context a spectrally negative process), when κ(1, x 1/α ) = 1 + x 1/α .

Let γ ∈ (0, 1) and η γ (t, x) be the transition density of γ-stable subordinator. Denote by N(γ) a random variable with the density η γ (1, x), i.e.

E exp(-xN(γ)) = e -x γ , x ≥ 0.
Recall that if Y and Z are independent random variables on (0, ∞) with densities f and g respectively, then the multiplicative convolution Y × Z p is a random variable with the density

P[Y × Z p ∈ dt] = ∞ 0 f ( t u p )g(u) du u p . (12)
Corollary 6. (i) Let 1 < α < 2. Suppose that random variables M α,ρ and N(1/α) are independent and that X 0 = 1. We have

τ d = M α,ρ × N(1/α),
(ii) Let 0 < α < 1. Suppose that random variables τ and N(α) are independent. We have

τ × N(α) α d = M α,ρ .
Proof. Part (i) follows immediately from Corollary 4. In [START_REF] Kulczycki | Spectral properties of the Cauchy process on half-line and interval[END_REF] we substitute x α = u and use the scaling property η 1/α (u 1/α , s) = u -1 η 1/α (1, su -1 ).

In order to prove (ii), we use e -ty = E[-(ty) 1/α N(α)] in the left-hand side of (8), we apply Fubini and change variables x = y 1/α u. By unicity of the Laplace transform we get

∞ 0 h α ( x α u α )η α (u) du u α = sin((1 -ρ)απ) πα κ(1, x) x 2α + 2x α cos((1 -ρ)απ) + 1
Replacing x α by x in the last formula and using [START_REF] Kuznetsov | On extrema of stable processes[END_REF] ends the proof of (ii).

Application for the Doney's class Ĉ1,1

Let α ∈ [ 1 2 , 1) and 1 -ρ = 1/α -1, i.e. we consider a process dual to a process from the class C 1,1 from R. Doney's article [START_REF] Doney | On Wiener-Hopf factorisation and the distribution of extrema for certain stable processes[END_REF]. We denote the class of such processes by Ĉ1,1 . In this case we have by [START_REF] Doney | On Wiener-Hopf factorisation and the distribution of extrema for certain stable processes[END_REF] or [START_REF] Graczyk | On Wiener Hopf factors of stable processes[END_REF] κ(1, x) =

x 2α -2x α cos απ + 1 1 + x .

We recall the definition of the confluent hypergeometric functions 1 F 1 and U (see [START_REF] Abramowitz | Handbook of mathematical functions with formulas, graphs, and mathematical tables[END_REF])

1 F 1 (a, b, z) = ∞ k=0 (a) k z k (b) k k! , z ∈ C, U(a, b, z) = π sin πb 1 F 1 (a, b, z) Γ(1 + a -b)Γ(b) -z 1-b 1 F 1 (1 + a -b, 2 -b, z) Γ(a)Γ(2 -b) , z ∈ C \ R -,
where (a) k = a(a + 1) . . . (a + k -1), (a) 0 = 1 is Pochhammer symbol. Using Theorem 3 we get the following formulas for Laplace and Stieltjes transforms of τ Proposition 7. If α ∈ [ 1 2 , 1) and X t ∈ Ĉ1,1 then

(i) E 1 e -τ t = sin(απ) π Γ(α)Γ(1 -α, t 1/α )e t 1/α , (13)
(ii)

E 1 1 x + τ = ∞ 0 e -u e u 1/α x -1/α x - u 1/α-1 x 1/α 1 F 1 (1, 2 -α, u 1/α x -1/α ) Γ(2 -b) du, (14)
where Γ(a, z) = ∞ z t a-1 e -t dt is the incomplete Gamma function.

Proof. To prove (i) we use simple transformations of integrals

E 1 e -τ t = sin(α(1/α -1)π) π ∞ 0 e -t 1/α x x α-1 (x 2α -2x α cos απ + 1) (1 + x)(x 2α + 2x α cos(α(1/α -1)π) + 1) dx = sin(απ) π ∞ 0 e -t 1/α x x α-1 1 + x dx (15) = sin(απ) π ∞ 0 ∞ 0 e -t 1/α x e -s(x+1) x α-1 ds dx = sin(απ) π ∞ 0 e -s Γ(α) (t 1/α + s) α ds = sin(απ)Γ(α) π ∞ t 1/α e t 1/α -r 1 r α dr = sin(απ) π Γ(α)Γ(1 -α, t 1/α )e t 1/α .
To prove (ii) we use the following integral representation of U (see [START_REF] Abramowitz | Handbook of mathematical functions with formulas, graphs, and mathematical tables[END_REF])

U(a, b, z) = 1 Γ(a) ∞ 0 e -zt t a-1 (1 + t) b-a-1 dt, Re z > 0.
Applying this to [START_REF] Patie | Intertwining certain fractional derivatives[END_REF] we get

E 1 1 x + τ = sin(απ) π ∞ 0 ∞ 0 e -xt e -t 1/α s s α-1 1 + s ds dt = sin(απ) π ∞ 0 ∞ 0 e -u e -u 1/α x -1/α s s α-1 x(1 + s) ds du = sin(απ) π ∞ 0 e -u Γ(α) U(α, α, u 1/α x -1/α ) x du.
Hence

E 1 1 x + τ = ∞ 0 e -u x 1 F 1 (α, α, u 1/α x -1/α ) - u 1/α-1 x 1/α-1 1 F 1 (1, 2 -α, u 1/α x -1/α ) Γ(2 -b) du = ∞ 0 e -u e u 1/α x -1/α x - u 1/α-1 x 1/α 1 F 1 (1, 2 -α, u 1/α x -1/α ) Γ(2 -b) du.
It is possible to invert Stieltjes transform in [START_REF] Kyprianou | Introductory lectures on fluctuations of Lévy processes with applications[END_REF] and the resulting density of τ is given by a series either in x or in 1/x (cf. [START_REF] Kuznetsov | On extrema of stable processes[END_REF]). For α = 2/3 the process X t is symmetric and the density of τ has a nice integral representation involving hypergeometric function

1 F 2 (a; b, c; z) = ∞ k=0 (a) k z k (b) k (c) k k! , z ∈ C.
Corollary 8. Let X t be a symmetric 2 3 -stable process on R with X 0 = 1. Then the density of τ = τ (0,∞) is given by the formula

P 1 (τ ∈ dx) = 1 π
∞ 0 e -tx sin(t 3/2 ) + t 1/2 1 F 2 (1; 2/3, 7/6; -t 3 /4) Γ(4/3) dt [START_REF] Schilling | Bernstein functions, Theory and Applications[END_REF] and its Laplace transform is

E 1 e -tτ = √ 3 2π
Γ(2/3)Γ(1/3, t 3/2 )e t 3/2 , t > 0.

Proof. The second part is a direct consequence of [START_REF] Kwaśnicki | Spectral analysis of subordinated Brownian motions in half-line[END_REF]. By ( 14)

E 1 1 x + τ = ∞ 0
e -u e u 3/2 x -3/2 x -u 1/2 x 3/2 1 F 1 (1, 4/3, u 3/2 x -3/2 ) Γ(4/3) du.

Inverting Stieltjes transform we get

P 1 (τ ∈ dx) = - 1 π Im ∞ 0
e -u e u 3/2 x -3/2 e -3iπ/2 -x -u 1/2 x 3/2 e 3iπ/2 1 F 1 (1, 4/3, u 3/2 x -3/2 e -3iπ/2 ) Γ(4/3) du

= - 1 π Im ∞ 0
e -u e u 3/2 x -3/2 i -x -i u 1/2 x 3/2 1 F 1 (1, 4/3, u 3/2 x -3/2 i) Γ(4/3) du.

Since

Im i 1 F 1 (1, 4/3, yi) = Re Remark. It is possible to obtain the formula ( 16) from the results of M. Kwaśnicki [START_REF] Kwaśnicki | Spectral analysis of subordinated Brownian motions in half-line[END_REF]. However, in order to do this, one has to make several non-elementary transformations of integrals and our approach seems simpler than using [START_REF] Kwaśnicki | Spectral analysis of subordinated Brownian motions in half-line[END_REF].

( 1 )

 1 k (-y 2 ) k k!(4/6) k (7/6) k 2 2k = 1 F 2 (1; 2/3, 7/6; -y 2 /4)we finally get P 1 (τ ∈ dx)

Acknowledgement. We thank Christian Berg for discussions on the Stieltjes transform and Thomas Simon for helpful comments and bibliographical indications.

This research was partially supported by grants MNiSW and ANR-09-BLAN-0084-01.


Céline Cudejko

Kristiaan Wouters

Lucía Fuentes

Sarah Anissa Hannou

Charlotte Paquet

Kadiombo Bantubungi

Emmanuel Bouchaert

Jonathan Vanhoutte

Sébastien Fleury

Patrick Remy

Anne Tailleux

Giulia Chinetti

David Dombrowicz

Bart Staels
email: bart.staels@pasteur-lille.fr

Réjane Paumelle

p16INK4a deficiency promotes IL-4-induced polarization and inhibits proinflammatory signaling in macrophages

INTRODUCTION

The tumor suppressor p16 INK4a is encoded by the CDKN2A locus on the human chromosome 9p21 and on the murine chromosome 4. p16 INK4a belongs to the INK4 family of cyclindependent kinase (CDK) inhibitors, also including p15 INK4b , p18 INK4c , and p19 . p16 INK4a inhibits cell cycle progression by preventing cyclin D-CDK 4/6 complex formation. As a consequence, pRb hyperphosphorylation and its association with E2F, which induces transcription of S phase genes, are inhibited. p16 INK4a inactivation by deletion, point mutation or promoter methylation, occurs frequently in most tumors [START_REF] Sharpless | INK4a/ARF: a multifunctional tumor suppressor locus[END_REF] .

Besides its role in cancer as an inhibitor of cell cycle progression, p16 INK4a plays a crucial role in senescence and aging [START_REF] Alcorta | Involvement of the cyclin-dependent kinase inhibitor p16 (INK4a) in replicative senescence of normal human fibroblasts[END_REF][START_REF] Hara | Regulation of p16CDKN2 expression and its implications for cell immortalization and senescence[END_REF] . Indeed, expression of p16 INK4a increases with age in various tissues from several species 9, -11 . A genome-wide association study has shown association of the CDKN2A locus with an increased risk of the age-related frailty syndrome [START_REF] Melzer | A common variant of the p16(INK4a) genetic region is associated with physical function in older people[END_REF] .

Additionally, increased p16 INK4a expression causes the age-dependent decline in proliferation of self-renewing cellular compartments such as haematopoietic stem cells [START_REF] Janzen | Stem-cell ageing modified by the cyclindependent kinase inhibitor p16INK4a[END_REF] , which give rise to immune cells.

Although the role of p16 INK4a in mature immune cells has not yet been investigated, a number of studies has shown that the CDKN2A locus is associated with an increased risk for coronary heart disease [START_REF] Mcpherson | A common allele on chromosome 9 associated with coronary heart disease[END_REF] , atherosclerosis [START_REF] Liu | INK4/ARF transcript expression is associated with chromosome 9p21 variants linked to atherosclerosis[END_REF] , and type 2 diabetes (T2D) [START_REF] Saxena | Genome-wide association analysis identifies loci for type 2 diabetes and triglyceride levels[END_REF] . In these pathologies, immune cells, such as macrophages, play a crucial role. Besides their pleiotropic immune functions, macrophages also play a role in the development and homeostasis of several tissues, such as adipose tissue [START_REF] Wei | Modulation of CSF-1-regulated post-natal development with anti-CSF-1 antibody[END_REF] and liver [START_REF] Baffy | Kupffer cells in non-alcoholic fatty liver disease: the emerging view[END_REF] . Depending upon the cytokine environment, macrophages differentiate into distinct subclasses with specific characteristics. Classically activated macrophages (CAM) differentiate in presence of Th1 cytokines, such as interferon gamma (IFN), or in presence of bacterial products such as lipopolysaccharide (LPS). CAM trigger pro-inflammatory responses required to kill intracellular pathogens [START_REF] Gordon | Alternative activation of macrophages[END_REF] . Alternatively activated macrophages (AAM), induced by Th2 cytokines such as interleukin (IL)-4 and IL-13, are associated with Th2-type immune responses as seen in helminth parasite infections [1]. During inflammation, AAM play a key role in protecting the organism against tissue damage [2]. However, little is known about the mechanisms underlying the acquisition of the AAM phenotype.

In the present study, we investigated whether p16 INK4a -deficiency influences macrophage activation in vitro, using bone marrow-derived macrophages (BMDM), and in vivo by infection with the parasite Schistosoma mansoni. BMDM from p16 INK4a -deficient (p16 -/-) mice exhibit a phenotype resembling IL-4-induced macrophage polarization and an enhanced response to the Th2 cytokine IL-4 compared to BMDM from wild type (p16 +/+ ) mice. By contrast, their response to Th1 stimuli is diminished. Moreover, Schistosoma mansoni infection of mice transplanted with p16 -/-bone marrow resulted in an increased hepatic AAM signature in vivo. Finally, we show that p16 INK4a -deficiency does not influence the IL-4-induced STAT6 pathway. By contrast, the lower response to classical activation stimuli likely occurs through an inhibition of the phosphorylation of both STAT1 and IKK,, which correlated with decreased JAK2 phosphorylation and increased acetylation of STAT1 and IKK,,

METHODS

Mice. C57BL/6J p16 INK4a -deficient and littermate control mice on a C57Bl6 background (>97%) were kindly provided by P. Krimpenfort. p16 -/-mice showed neither abnormalities in biochemical parameters nor spontaneous tumor development at the studied age. Flow cytometry analysis of splenic T-and B-cells or macrophages showed no differences between p16 -/-and p16 +/+ mice (data not shown). All protocols were conducted with the approval of the Pasteur Institute ethical review board, Lille, France.

Bone marrow chimeras. 8 week-old C57BL/6J mice were lethally irradiated (8 Gy) and injected in the tail vein the next day with 10x10 6 bone marrow cells isolated from p16 -/-or p16 +/+ donor mice. Mice were supplied with autoclaved acidified water (pH=2) supplemented with neomycin 100 mg/L (Cat.N1142, Sigma-Aldrich), polymyxin B sulphate 60000 U/L (Cat.21850029, Invitrogen) 1 week before and 4 weeks after transplantation. Mice were studied 6 weeks post-transplantation to allow repopulation by the donor bone marrow. To ensure that donor bone marrow replaced the resident blood cell population, DNA was extracted from whole blood with an Illustra blood kit (GE Healthcare). PCR was performed with forward 5'-GCA-GTG-TTG-CAG-TTT-GAA-CCC-3' and reverse 5'-TGT-GGC-AAC-TGA-TTC-AGT-TTG-3' primer sets yielding products of different lengths, which were separated on a 1.5% agarose gel and quantified with the Gel Doc XR system (Bio-Rad). Over 95% of host blood cells were from donor origin.

Primary cell isolation and culture. LPS-elicited neutrophils from air pouches were isolated as described [START_REF] Bellosta | HMG-CoA reductase inhibitors reduce MMP-9 secretion by macrophages[END_REF] . Splenic CD4 + T-cells and CD43 -B-cells were isolated from female mice respectively using Dynal (Invitrogen), biotin-conjugated anti-CD43 (Becton Dickinson) and streptavidin conjugated to magnetic beads (Miltenyi Biotech) according to manufacturer's instructions. Bone marrow-derived dendritic cells were prepared as described [START_REF] Dubois | Regulation of Th2 Responses and Allergic Inflammation through Bystander Activation of CD8+ T Lymphocytes in Early Life[END_REF] . BMDM were obtained from femoral and tibial bone marrow suspensions, plated at 10x10 6 cells in 10 cm plates and differentiated in BMDM medium (RPMI 1640 containing Hepes 25 mM supplemented with 10% low endotoxin fetal bovine serum, 15% L929-conditioned medium, 2 mM L-glutamine, 1 mM gentamycine). Stimulations and microarray analysis were carried out at day 8. IL-4 (15 ng/mL) and IFN (2.5 ng/mL and 10 ng/mL) were obtained from Peprotech. LPS (100 ng/mL) from E. coli serotype 0111B4 was from Sigma-Aldrich. CINK4 (#219492) was obtained from Merck. For indirect co-culture experiments, conditioned medium of alternatively differentiated p16 +/+ and p16 -/-macrophages was added to cultures of BMDM 24h before LPS (100 ng/mL) stimulation. After 4h, CAM were washed and cultured for 24h in fresh medium. The resulting media were assayed for cytokine secretion by ELISA.

Flow cytometry.

During in vitro differentiation, asynchronously growing cultures of BMDM were gently scraped every two days with PBS 1X/EDTA 10 mM/2% trypsine, washed with PBS and collected by centrifugation (5 min, 900 rpm). For protein analysis, non-specific staining was avoided by incubating cells with purified rat anti-mouse CD16/CD32 (Cat.553141, BD Pharmingen) for 20 min at 4°C and labeled with anti-mouse F4/80-FITC (Cat.RM2901, Caltag Laboratories) for 30 min at 4°C. Cells were analyzed using a Coulter EPICS XL4-MCL ® Flow cytometer (BeckmanCoulter) and Expo32 ® software (applied cytometry systems). For cell sorting, cells were incubated with anti-mouse F4/80-FITC antibodies (11-4801, eBioscience) and Rat IgG2a K isotype (11-4321, eBioscience) and sorted on an EPICS-ALTRA cell sorter (Beckman Coulter). For FACS-based cell cycle analysis, cells were fixed with 70% ethanol for 1h at -20°C and labeled with 50 µg/mL propidium iodide containing 50 µg/mL RNase A for 30 min at room temperature. The percentage of cell cycle distribution was obtained using the Multicycle ® software (Phoenix Flow systems).

Microarray studies. Mouse genome MOE 430 2.0 gene chips (Affymetrix ® ) were used in triplicate for each condition. Sample preparation, hybridization and scanning were performed according the One-cycle protocol (Affymetrix ® ). Only genes significantly regulated (p<0.05) were analyzed further with Genomatix Bibliosphere ® . Microarray data have been deposited to http://www.ebi.ac.uk/arrayexpress.

RNA extraction and analysis. Total RNA was obtained from BMDM using EXTRACT-ALL ® , DNase-I treated, and cDNA was generated using cDNA reverse transcription kit (AB Applied Biosystems). Real-time QPCR detection was conducted on a Stratagene Mx3005P ® (Agilent technologies) using Brilliant II SYBR Green reagent ® (Agilent Technologies) and specific primers (Table S1). mRNA levels were normalized to 36B4 mRNA and fold induction was calculated using the 2 -ΔΔCt method.

Cytokine production measurement. IL-6, TNF, IL-1Rn, IL-4, IL-10 and IFN levels in culture supernatants were measured with ELISA kits according to the manufacturer's instructions (R&D systems). Plasma IgE from infected animals was determined by ELISA using purified rat anti-mouse IgE (Cat.553416, BD Pharmingen) as a capture antibody, and biotin rat anti-mouse IgE as a detection antibody (Cat.553414, BD Pharmingen). Standards were prepared using purified mouse IgE  isotype control (Cat.557079, BD Pharmingen).

Protein extraction and Western blotting. BMDM lysates were prepared using Cell Lysis Buffer (#9803, Cell Signaling Technology) supplemented with 1 mM PMSF. Lysates were resolved in 16% or 10% SDS-PAGE and analyzed by western-blot. Anti-p16 (sc-1207), -actin (sc-1616) and -p38 (sc-535) antibodies were purchased from Santa Cruz Biotechnology.

Anti-phospho-STAT6 and -STAT6 (#9361 and 9362), anti-phospho-STAT1 and -STAT1 (#9171 and 9172), anti-phospho-p38 (#9211), anti-phospho-IKK, (#2681), anti-IBα (#9242) and anti-phospho-JAK2 and -JAK2 (#3771 and 3230) antibodies were purchased from Cell Signaling Technology. Hybridizations were done 1:1000 in 5% BSA, 1X TBS, 0.1% Tween-20 overnight at 4°C. Anti-rabbit IgG Peroxidase conjugate secondary antibody 1:10000 (Cat.A-0545, Sigma-Aldrich) was added for 1h at room temperature. Signals were visualized using the enhanced chemiluminescence Western blot detection kit (Amersham ECL plus Western blotting Detection System; GE Healthcare). Membranes were stripped with Reblot Plus Strong Antibody Stripping Solution ® (Cat.2504, Millipore) and re-blotted with the indicated antibodies.

Immunoprecipitation (IP) assay. 500 µg of BMDM lysates were prepared using lysis buffer supplemented with trichostatin A 1 µM (Cat.T8552, Sigma-Aldrich). Lysates were mixed with 100 µL of Protein G/Protein A Agarose (Millipore) and rotated for 1h at 4°C. After centrifugation for 2 min at 3000 rpm, anti-acetyl-Lysine antibody 1:100 (ab21623, Abcam) was added. Samples were rotated overnight at 4°C. The Protein G/Protein A Agarose mix (100 µL) was added to the samples and incubated for 1h at 4°C. The beads were washed 3x and the pellets dissolved in 40 µL of Laemmli 2X buffer. Proteins were analyzed by western blotting as previously described.

S. mansoni infection. Chimeric mice were anesthetized and percutaneously infected with 40 S. mansoni cercariae and sacrificed 9 weeks later. The liver was perfused and adult worms were recovered and quantified. A piece of liver was weighed and dissolved in a KOH solution (5%, 37°C during 16h) and eggs were counted. Organs were snap frozen or fixed in paraformaldehyde for further analysis. Resin-embedded livers were cut at 5 µm and Mallory's Trichrome staining was performed to determine granuloma size and hepatic collagen.

Immunostaining was performed using antibodies directed to Moma-2 (sc59332, Santa Cruz), Arginase1 (Arg-I) (ab60176-100), Fizz1 (ab39626-50) and Ym1 (ab93034) from Abcam. May Grunwald Giemsa staining was performed and eosinophils were counted per granuloma area. Granulomas were isolated using an Arcturus Laser capture microdissection system from 14 µm sections of snap frozen liver samples. For ex vivo restimulation, spleens were homogenized, filtered (70 µm) and erythrocytes were lysed. Cells were cultured in RPMI 1640 (Gibco) supplemented with 10% low endotoxin fetal bovine serum and 1 mM gentamycin. 5x10 5 cells were stimulated with 10 µg/ml soluble egg antigen (SEA) or anti-CD3 (#555274; BD Pharmingen) antibodies during 4 days after which medium was collected for cytokine determination by ELISA.

Statistical analysis.

Groups were compared using 2-tailed non-paired t-tests using GraphPad Prism ® software. In vitro data are expressed as means ± SD and in vivo data are expressed as means ± SEM.

10

RESULTS p16 INK4a expression increases during macrophage maturation, but p16 INK4a -deficiency does not modulate macrophage cell cycle progression.

In order to assess whether and in which mature immune cell types p16 INK4a is expressed, p16 INK4a mRNA was measured in dendritic cells (DC), bone marrow-derived macrophages (BMDM) and neutrophils, all of myeloid origin, and in B-and T-lymphocytes. p16 INK4a was highly expressed in BMDM and DC, while its expression was markedly lower in neutrophils and virtually absent in lymphocytes (Figure 1A). Since the CDKN2A locus has been associated to diseases with inflammatory components in which macrophages play a central role, we focused on the role of p16 INK4a in these cells. BMDM displayed a gradual increase in p16 INK4a mRNA expression during in vitro differentiation (Figure 1B). Cell sorting analysis showed that p16 INK4a mRNA expression increased specifically in F4/80-positive cells (macrophages) (Figure S1). Together, these data show that increased expression occurs in a higher number of macrophages which also express higher p16 INK4a levels. In line, p16 INK4a protein was expressed in differentiated macrophages (Figure 1C). Absence of p16 INK4a (Figure 1C) did not influence macrophage maturation, since it affected neither mRNA (Figure 1D) nor protein (Figure 1E) expression of the macrophage surface marker F4/80 during differentiation. Interestingly, absence of p16 INK4a did not affect cell cycle progression at any time point of in vitro macrophage differentiation (Figure 1F). Expression analysis of the INK4 family members p15 INK4b , p18 INK4c , p19 INK4d and p19 ARF , another product of the INK4A/ARF locus which inhibits the p53 pathway, as well as the KIP/CIP family members p21 WAF1 and p27 KIP1 revealed increased expression of p15 INK4b in p16 INK4a -deficient (p16 -/-) BMDM (Fig. S2). Together with the observation that expression of Cyclin D, a direct E2F target gene, is similar in both genotypes (Figure S3), these data suggest that the induction of p15 INK4b may compensate for p16 INK4a -deficiency in the control of the cell cycle.

These data indicate that p16 INK4a expression increases during macrophage differentiation and that p16 INK4a -deficiency influences neither maturation nor the cell cycle in BMDM.

Macrophage p16 INK4a -deficiency leads to a phenotype resembling IL-4-induced macrophage polarization.

Microarray analysis of p16 -/-BMDM revealed that the expression of a large number of inflammatory genes was significantly lower compared to wild type (p16 +/+ ) BMDM under basal conditions (Figure 2A). Several of these genes are typically associated with a CAM phenotype [START_REF] Van Ginderachter | Classical and alternative activation of mononuclear phagocytes: picking the best of both worlds for tumor promotion[END_REF][START_REF] Vats | Oxidative metabolism and PGC-1beta attenuate macrophage-mediated inflammation[END_REF][START_REF] Martinez | Transcriptional profiling of the human monocyte-to-macrophage differentiation and polarization: new molecules and patterns of gene expression[END_REF][START_REF] Noël | Alternatively activated macrophages during parasite infections[END_REF][START_REF] Mantovani | The chemokine system in diverse forms of macrophage activation and polarization[END_REF] . By contrast, a number of AAM-associated genes, linked to IL-4-induced STAT6 activation [START_REF] Van Ginderachter | Classical and alternative activation of mononuclear phagocytes: picking the best of both worlds for tumor promotion[END_REF][START_REF] Vats | Oxidative metabolism and PGC-1beta attenuate macrophage-mediated inflammation[END_REF][START_REF] Martinez | Transcriptional profiling of the human monocyte-to-macrophage differentiation and polarization: new molecules and patterns of gene expression[END_REF][START_REF] Noël | Alternatively activated macrophages during parasite infections[END_REF][START_REF] Mantovani | The chemokine system in diverse forms of macrophage activation and polarization[END_REF] , were significantly higher expressed in p16 -/-BMDM (Figure 2B). However, the AAM markers Ym1/2 and Fizz1 were not spontaneously induced by p16 INK4adeficiency. The differential expression of a number of these genes was confirmed by QPCR (Table S2).

To further characterize this IL-4-like phenotype of p16 -/-BMDM, the gene signature of these cells was compared with the gene signature of p16 +/+ BMDM polarized in vitro by addition of IL-4 from day 0 of differentiation, generating p16 +/+ AAM [START_REF] Bouhlel | PPARgamma activation primes human monocytes into alternative M2 macrophages with anti-inflammatory properties[END_REF] . Compared to p16 +/+ BMDM, 14311 probesets were differentially expressed in p16 +/+ AAM (Figure 2C, green dots), whereas 7605 probesets were regulated in p16 -/-BMDM (Figure 2C, red dots), of which the majority (78%) was also differentially expressed in p16 +/+ AAM (Figure 2C, blue dots). In addition, a strong correlation (p<0.0001; Pearson R=0.7; R 2 =0.5) was found between the two experimental conditions (Figure 2C), as also illustrated by a heat map representation of a selection of AAM and CAM marker genes (Figure 2D and Table S3). Since p16 -/-BMDM displayed lower expression levels of inflammatory genes (Figure 2A,D), the secretion of inflammatory cytokines produced in p16 -/-BMDM compared to p16 +/+ BMDM was measured. Under basal conditions, secretion of the pro-inflammatory cytokines IL-6 and TNF was lower (Figure 3A,B) in p16 -/-BMDM. By contrast, secretion of anti-inflammatory IL-1Rn was, albeit modestly, higher (Figure 3C). These observations are in line with the microarray data and suggest that p16 -/-BMDM are functionally different from their wild type counterparts.

Next, we compared the response of p16 -/-vs. p16 +/+ BMDM to polarization into AAM induced by IL-4 added at the start (day 0) of differentiation. Induction of the AAM marker genes Arg-I (Figure 3D) and Ym1/2 (Figure 3E) was higher in p16 -/-AAM. However, induction of Mgl2 (Figure 3F) was not different in these cells, suggesting that the IL-4responsiveness of certain, but not all, genes is dependent on p16 INK4a . This heterogeneity in regulation of AAM marker genes in IL-4-polarized p16 -/-AAM confirms the microarray results (Figure 2D). Since p16 -/-macrophages display increased AAM marker expression after IL-4-induced polarization and since AAM exert paracrine anti-inflammatory effects [START_REF] Katakura | CCL17 and IL-10 as effectors that enable alternatively activated macrophages to inhibit the generation of classically activated macrophages[END_REF] , the functional effects of p16 INK4a -deficiency on IL-4-induced macrophage polarization were analyzed by indirect co-culture experiments. Hereto, conditioned medium from IL-4-polarized p16 -/-or p16 +/+ AAM was added to p16 +/+ BMDM and their response to LPS was measured (Figure 3G). Conditioned medium from p16 -/-AAM resulted in a more pronounced inhibition of LPSinduced secretion of IL-6 and TNF (Figure 3H,I). Thus, medium from p16 -/-AAM more potently inhibits pro-inflammatory responses.

Collectively, these results evidence that p16 INK4a -deficiency results in a phenotype partially resembling IL-4-induced macrophage polarization. Moreover, IL-4-polarized p16 -/- macrophages display a more pronounced AAM phenotype. Since p16 -/-BMDM resemble IL-4-polarized macrophages and since p16 -/-AAM show more potent anti-inflammatory effects than wild type macrophages, the response of p16 -/-BMDM to acute (24h) IL-4 activation was assessed. Interestingly, IL-4 treatment resulted in a larger increase of Arg-I, Ym1/2 and Mgl2 mRNA in p16 -/-compared to p16 +/+ BMDM (Figure 4A-C).

AAM are known to be less sensitive to inflammatory stimuli [START_REF] Gordon | Alternative activation of macrophages[END_REF] . Therefore, the effects of IFN and LPS, two pro-inflammatory stimuli, were tested on these cells. Interestingly, p16 -/-BMDM were less responsive to IFN, shown by the lower induction of the IFN target genes iNOS, Cxcl10 and MHCII (Figure 4D-F), as well as to LPS, shown by the decreased response of the LPS-regulated genes TNF, IL-6 and MCP-1 (Figure 4G-I). Expression levels of the cognate receptors for IL-4, IFN and LPS was similar in both genotypes (Figure S4A-C), suggesting that the altered responses were not due to differences in receptor expression.

Collectively, these data show that p16 -/-BMDM are more responsive to alternative activation by IL-4 and less responsive to the classical macrophage activators IFN and LPS.

Haematopoietic p16 INK4a -deficiency increases hepatic expression of alternatively activated macrophage markers upon S. mansoni infection.

Since helminth parasite infections induce a strong Th2 immune response resulting in AAM differentiation [START_REF] Herbert | Alternative macrophage activation is essential for survival during schistosomiasis and downmodulates T helper 1 responses and immunopathology[END_REF][START_REF] Pearce | Th2 response polarization during infection with the helminth parasite Schistosoma mansoni[END_REF] , the influence of p16 INK4a -deficiency on macrophage polarization in vivo was investigated during S. mansoni infection. To assess the role of p16 INK4a specifically in haematopoietic cells, p16 +/+ and p16 -/-bone marrow was transplanted in lethally irradiated wild type recipients. Mice were sacrificed 9 weeks after infection, corresponding to the peak of the Th2 immune response, and the hepatic pathology was subsequently characterized.

Expression of Th2 and Th1 cytokines in livers (Figure S5A-D) and spleens (Figure S6A-C), hepatic infection parameters (Table 1), plasma IgE levels (Figure S7), eosinophil infiltration (Table 1), as well as mortality and weight loss (data not shown) did not differ between p16 -/- and p16 +/+ transplanted animals. These results indicate that there are no differences in the lymphocyte-dependent Th2 response, in line with the very low expression of p16 INK4a in Tcells (Figure 1A).

Since macrophages express significant levels of p16 INK4a (Figure 1A) and since these cells are abundant in S. mansoni-induced granulomas [START_REF] Herbert | Alternative macrophage activation is essential for survival during schistosomiasis and downmodulates T helper 1 responses and immunopathology[END_REF] , macrophage markers were measured in the livers of infected mice. Hepatic mRNA levels of F4/80, CD68 and CD14 were equally induced upon infection (Figure 5A-C), indicating that macrophage numbers were not modified. However, the induction of the AAM markers Ym1/2 (Figure 5D), Fizz1 (Figure 5E) and Mgl2 (Figure 5F) was higher in mice transplanted with p16 -/-vs. p16 +/+ bone marrow. Moreover, expression of the tissue remodeling markers SMA and Timp1 (Fig. S5E,F) was higher in mice transplanted with p16 -/-bone marrow, consistent with the tissue remodeling function of AAM [START_REF] Wynn | Fibrotic disease and the T(H)1/T(H)2 paradigm[END_REF] .

Immunohistochemical analysis showed that the parasite-induced granulomas are highly macrophage-enriched (Figure S8). Moreover, Arg-I, Fizz1 and Ym1 co-localized with the general macrophage marker Moma-2. Surprisingly, the macrophage population within the granulomas appeared heterogeneous, since these markers did not entirely co-localize (Figure S8). Nevertheless, these data suggest that the increase in hepatic AAM marker expression occurs mainly in macrophages. To unequivocally demonstrate this, the macrophage-positive areas of the granulomas were isolated by laser-capture microdissection from p16 -/-and p16 +/+ bone marrow transplanted mice and mRNA levels were measured. p16 -/-granulomas displayed increased Mgl2 and Fizz1 mRNA levels, whereas Ym1/2 mRNA tended to be increased (Figure 5G-I). F4/80, CD68 and CD14 expression were identical, indicating that macrophage numbers did not differ between genotypes (data not shown). [START_REF] Herbert | Alternative macrophage activation is essential for survival during schistosomiasis and downmodulates T helper 1 responses and immunopathology[END_REF][START_REF] Elliott | Schistosome egg antigen(s) presentation and regulatory activity by macrophages isolated from vigorous or immunomodulated liver granulomas of Schistosoma mansoni-infected mice[END_REF] , the ex vivo response of splenocytes from mice transplanted with p16 +/+ or p16 -/-bone marrow to soluble S. mansoni egg antigen (SEA) was measured. Antigen-independent stimulation of naïve splenocytes by anti-CD3 showed similar IL-4 production (data not shown) suggesting an equal ability of Tcells to synthesize Th2 cytokines, regardless of macrophages within the splenocyte preparation. By contrast, antigen-dependent restimulation with SEA showed a lower production of both IL-4 and IFN in splenocytes from p16 -/-transplanted mice (Figure S6D, E), while IL-10 production was similar (Figure S6F), hence resulting in an unchanged IL-4/IFN balance [START_REF] Mosmann | TH1 and TH2 cells: different patterns of lymphokine secretion lead to different functional properties[END_REF] .

Since macrophage polarization influences T-cell activation

Collectively, these results provide evidence for a role of p16 INK4a in AAM skewing in vivo.

p16 INK4 -deficiency in macrophages decreases STAT1 and IKK, signaling without altering STAT6 phosphorylation.

To unravel the mechanisms underlying the phenotype induced by p16 INK4a -deficiency, the microarray data were analyzed to identify which pathways are affected. Interestingly, many genes with a decreased expression in p16 -/-BMDM respond to activation of the IFN and NF-B inflammatory signaling pathways (Figure S9), of which STAT1 and IKK, respectively are crucial components. Detailed examination of these gene expression patterns revealed that expression levels were comparable (Cxcl10, ifi27, ifi203, ifi47, ifit1, Ccl5, Irak3, Birc3 and Ly6e) or even lower (ifi44, CD55 and NFBia) in p16 -/-BMDM under basal conditions compared to IL-4-polarized p16 +/+ AAM (Figure 6). Moreover, most inflammatory genes were more strongly inhibited in IL-4-induced p16 -/-AAM than in p16 +/+ AAM (Figure 6 and Figure 2D), suggesting an additive effect of p16 INK4a -deficiency on IL-4 treatment. By contrast, such a synergism could not be observed with respect to the AAM marker genes since many of these AAM marker genes responded similarly in IL-4-induced p16 -/-and p16 +/+ AAM (Table S4 and Figure 2D). Collectively, these data show that the pathways controlling IFN, NF-B, and IL-4 signaling are affected by p16 INK4a -deficiency in macrophages.

To determine whether the observed differences were related to indirect effects through pRb, BMDM were treated with CINK4, a potent inhibitor of pRb hyperphosphorylation [START_REF] Soni | Selective in vivo and in vitro effects of a small molecule inhibitor of cyclin-dependent kinase 4[END_REF] . CINK4 treatment resulted in a dose-dependent decrease of cyclin D mRNA levels both in p16 +/+ and p16 -/-BMDM (Figure S3). Furthermore, treatment with CINK4 in combination with IFN did not influence the different phenotype between the two genotypes (Figure S10), thus excluding a role for pRb therein. STAT6 is the major transcription factor activated by IL-4. Surprisingly, IL-4 treatment induced STAT6 phosphorylation to a similar extent in p16 -/-and p16 +/+ BMDM (Figure 7A). By contrast, IFN-induced STAT1 phosphorylation (Figure 7B) and LPS-induced IKK, phosphorylation were markedly diminished (Figure 7C) in p16 -/-vs. p16 +/+ BMDM. Interestingly, IB protein levels were increased in p16 -/-BMDM under basal conditions (Figure 7C). However, phosphorylation of p38, another downstream target of TLR-4, was not different between p16 -/-and p16 +/+ BMDM (Figure 7C), suggesting that p16 INK4a -deficiency specifically alters the NF-B response.

Several genes coding for proteins with acetyl transferase activity were induced in p16 -/- BMDM (Table 2). Acetylation of STAT1 and IKK, inhibits their phosphorylation and thereby their transcriptional activity [START_REF] Mittal | Acetylation of MEK2 and I kappa B kinase (IKK) activation loop residues by YopJ inhibits signaling[END_REF][START_REF] Krämer | Acetylation of Stat1 modulates NF-kappaB activity[END_REF] . Immunoprecipitation of acetylated proteins revealed that STAT1 and IKK, were acetylated to a higher extent in p16 -/-than in p16 +/+ BMDM (Figure 7D). To identify whether STAT1 acetylation is the cause or the consequence of decreased STAT1 phosphorylation, JAK2 phosphorylation, which is upstream in this pathway, was assessed. JAK2 phosphorylation was less pronounced under basal conditions and after activation with IFN in p16 -/-BMDM (Figure 7E). Since acetylation of STAT1 can interfere with NF-B signaling [START_REF] Krämer | Acetylation of Stat1 modulates NF-kappaB activity[END_REF] and both pathways act synergistically [START_REF] Hiroi | The transcriptional coactivator CREB-binding protein cooperates with STAT1 and NF-kappa B for synergistic transcriptional activation of the CXC ligand 9/monokine induced by interferon-gamma gene[END_REF] , the decreased STAT1 and NF-B signaling and their increased acetylation probably results from decreased JAK2 phosphorylation.

DISCUSSION

We identify a novel role for the tumor suppressor protein p16 INK4a in the regulation of the macrophage phenotype in vitro and in vivo. p16 INK4a -deficiency promotes a STAT6independent IL4-induced macrophage polarization, whereas LPS and IFN responses are lower. These actions of p16 INK4a -deficiency likely occur through modification of the JAK2-STAT1 and NF-B pathways, Although p16 INK4a is mainly known as a cell cycle inhibitor, cell cycle and macrophage maturation were unaffected in the absence of p16 INK4a , and an E2F target gene signature [START_REF] Vernell | Identification of target genes of the p16INK4A-pRB-E2F pathway[END_REF][START_REF] Stanelle | Gene expression changes in response to E2F1 activation[END_REF] , which would be expected to be increased by p16 INK4a -deficiency, could not be identified in p16 -/-BMDM. Since p15 INK4b is a critical tumor suppressor in the absence of p16 INK4a 40 and since an increase of p15 INK4b expression was observed in p16 -/-BMDM, a compensatory mechanism between these CDKI might have taken place with respect to cell cycle control. Interestingly, p16 INK4a has been previously shown to modulate transcriptional activities of the pro-inflammatory transcription factors NF-B and AP-1. Via its ankyrin motifs, p16 INK4a binds c-jun [START_REF] Choi | The tumor suppressor p16(INK4a) prevents cell transformation through inhibition of c-Jun phosphorylation and AP-1 activity[END_REF] and NF-B [START_REF] Wolff | INK4 cell cycle inhibitors direct transcriptional inactivation of NF-kappaB[END_REF] , inhibiting their activity in several immortalized cell lines. However, this mechanism does not appear to play a role in primary murine macrophages, where the loss of p16 INK4a has anti-inflammatory effects. Another CDKI, p21 WAF1/CIP1 , was also reported to modulate inflammation without altering cell cycle or differentiation. However, in contrast with p16 INK4a -deficiency, p21 WAF1/CIP1 -deficiency in macrophages resulted in a higher sensitivity to inflammatory stimuli [START_REF] Scatizzi | The CDK domain of p21 is a suppressor of IL-1beta-mediated inflammation in activated macrophages[END_REF][START_REF] Trakala | Regulation of macrophage activation and septic shock susceptibility via p21(WAF1/CIP1)[END_REF] . Additionally, p21 WAF1/CIP1 expression was unchanged in p16 -/- BMDM.

In a model of AAM differentiation in vivo, infection with the helminth S. mansoni resulted in equal Th2-associated responses in p16 -/-and p16 +/+ bone marrow transplanted mice. These observations indicate unaltered T-lymphocyte function, in line with the low expression levels of p16 INK4a in these cells. By contrast, AAM markers were significantly increased in the livers and in hepatic granulomas of p16 -/-bone marrow transplanted animals, probably due to a cell-autonomous activity of p16 INK4a in macrophages. Since total bone marrow was transplanted, a contribution of other haematopoietic cell types to this cell-autonomous macrophage response cannot be excluded. However, since expression of the measured markers was also up-regulated in the macrophage-rich granulomas and since AAM markers co-localized with macrophages in the granuloma, the effects are, at least in part, due to p16 INK4a -deficiency in the macrophages. Surprisingly, different AAM markers did not strictly co-localize, suggesting the existence of different macrophage populations in defined regions of the granuloma, which could be due to different gradients of stimuli within the granuloma.

Microarray analysis showed that p16 -/-BMDM under basal conditions display a gene signature resembling that of IL-4-induced macrophage polarization [START_REF] Mantovani | The chemokine system in diverse forms of macrophage activation and polarization[END_REF] , with the exception of the AAM marker genes Ym1/2 and Fizz1. Moreover, p16 -/-BMDM were more responsive to IL-4 activation and displayed weaker pro-inflammatory responses to IFNγ and LPS. Importantly, our data show that p16 INK4a -deficiency does not influence pRb-mediated activation of the E2F transcription factor. The decreased IFN-induced phosphorylation of STAT1 and the LPS-induced IKK, phosphorylation in p16 -/-BMDM probably account for the attenuated inflammatory responses. Decreased IKK, phosphorylation combined with increased IBα protein levels correlated with the profound decrease of NF-B target gene expression in p16 -/-BMDM. Since a cluster of genes coding for proteins with acetyl transferase activity was induced in p16 -/-BMDM and since STAT1 [START_REF] Krämer | Acetylation of Stat1 modulates NF-kappaB activity[END_REF] and IKK, [START_REF] Mittal | Acetylation of MEK2 and I kappa B kinase (IKK) activation loop residues by YopJ inhibits signaling[END_REF] acetylation reduces their ability to be phosphorylated, the decrease in STAT1 and IKK, phosphorylation may result from increased acetylation of these proteins in p16 -/-BMDM. The acetylation-phosphorylation balance may be influenced by alterations upstream in the signaling pathway. Since phosphorylation of JAK2, the first component to be phosphorylated upon receptor dimerization [START_REF] Shuai | Regulation of JAK-STAT signalling in the immune system[END_REF] , was decreased, it appears that the p16 INK4a -deficiency-induced defect in JAK2 phosphorylation may be implicated in the reduced Stat1 phosphorylation. The increased acetylation status may be a consequence of this mechanism, contributing to the phenotype of p16 -/-BMDM. Moreover, JAK2 phosphorylation was less pronounced already under basal culture conditions, in line with the observation that p16 INK4a -deficiency confers a lower basal inflammatory phenotype, which is further exacerbated upon stimulation. Since p16 INK4a -deficiency alters the IFN response early in the pathway and TLR4 signaling only at the level of NF-B, but not, for example, at the level of p38, p16 INK4a -deficiency thus likely acts directly on the IFN and indirectly on the NF-B signaling pathway. Surprisingly, STAT6 phosphorylation, the major transcription factor for IL-4-mediated signaling response, was comparable in BMDM of both genotypes, although IL-4-responsive genes were increased in p16 -/-BMDM, suggesting that other pathways are involved. It has been shown that STAT1 activation antagonizes expression of STAT6-induced AAM marker genes [START_REF] Dickensheets | Interferons inhibit activation of STAT6 by interleukin 4 in human monocytes by inducing SOCS-1 gene expression[END_REF][START_REF] Ohmori | IL-4-induced STAT6 suppresses IFN-gamma-stimulated STAT1-dependent transcription in mouse macrophages[END_REF] and that decreased STAT1 phosphorylation can result in an increase of AAM marker genes [START_REF] Porta | Tolerance and M2 (alternative) macrophage polarization are related processes orchestrated by p50 nuclear factor kappaB[END_REF] by a number of molecular mechanisms. First, IL-4 action is inhibited by STAT1 activation through inhibition of STAT6 phosphorylation and its subsequent nuclear translocation in human monocytes [START_REF] Dickensheets | Inhibition of IL-4-inducible gene expression in human monocytes by type I and type II interferons[END_REF] . However, p16 INK4a -deficiency did not modify STAT6 phosphorylation. Second, IFN inhibits STAT6 activation by inducing the expression of the suppressor of cytokine signaling 1 (SOCS1) in macrophages [START_REF] Dickensheets | Interferons inhibit activation of STAT6 by interleukin 4 in human monocytes by inducing SOCS-1 gene expression[END_REF] . Although SOCS1 mRNA levels were not different between p16 -/-and p16 +/+ BMDM (data not shown), we cannot exclude its involvement via post-transcriptional mechanisms. Third, STAT1 and STAT6 have been shown to compete for binding the same promoters [START_REF] Ohmori | IL-4-induced STAT6 suppresses IFN-gamma-stimulated STAT1-dependent transcription in mouse macrophages[END_REF] , since decreased STAT1 activity increases binding of STAT6 to the promoters of several of its target genes. To our knowledge, an effect of NF-B signaling on STAT6-dependent gene expression has not yet been demonstrated. Therefore, it is possible that the AAM-like phenotype of p16 -/-BMDM is secondary to the decreased CAM-associated response via JAK2-STAT1.

In conclusion, our results identify p16 INK4a as a novel modulator of macrophage polarization and show that p16 INK4a -deficiency skews macrophages towards an IL-4-like phenotype. Since the CDKN2A locus is predictive for the risk of T2D and CVD [START_REF] Mcpherson | A common allele on chromosome 9 associated with coronary heart disease[END_REF][START_REF] Liu | INK4/ARF transcript expression is associated with chromosome 9p21 variants linked to atherosclerosis[END_REF] , in which macrophage polarization plays a role [START_REF] Bouhlel | PPARgamma activation primes human monocytes into alternative M2 macrophages with anti-inflammatory properties[END_REF][START_REF] Odegaard | Macrophage-specific PPARgamma controls alternative activation and improves insulin resistance[END_REF] , the role of p16 INK4a in the control of macrophage function could be one mechanism contributing to the association of this locus with these diseases. Microarray analysis using mRNA from p16 -/-BMDM compared to p16 +/+ BMDM showed (A) decreased mRNA expression of classically activated macrophages-associated genes and (B) increased mRNA expression of alternatively activated macrophages-associated genes.

ABBREVIATIONS

Data is expressed as fold change relative to p16 +/+ BMDM. (C) Differential gene expression in p16 -/-BMDM relative to p16 +/+ BMDM was correlated with the changes induced in IL-4induced p16 +/+ AAM. The figure shows 2log values of the probesets significantly (p<0.05) regulated only in p16 -/-BMDM (red dots), only in IL-4-polarized p16 +/+ AAM (green dots) and by both conditions (blue dots), compared to p16 +/+ BMDM. The X-axis represents differences in gene expression induced by IL-4, whereas the Y-axis represents the effect of p16 INKadeficiency. These comparisons are depicted in the schematic representation of the protocol in the corresponding colors. Pearson Correlation analysis was done for probesets differentially expressed by both conditions (blue). (D) Heat map of p16 +/+ BMDM, p16 -/- BMDM, IL-4-polarized p16 +/+ and p16 -/-AAM gene expression profiles. Colors fluctuate from blue (poorly expressed) to green (intermediate expression) and yellow (high expression).

Additional information regarding gene description, fold induction, and p-value can be found in Table S3. Total protein extracts of p16 +/+ and p16 -/-BMDM treated with 15 ng/mL IL-4 (A), 2.5 ng/mL IFN (B) and (E), and 100 ng/mL LPS (C) for the times indicated. Western blots were performed with the indicated antibodies. (D) Total protein extracts from p16 +/+ and p16 -/- BMDM treated with 2.5 ng/mL IFN for 5h were then immunoprecipated (IP) with an anti-Acetyl-Lys antibody and then immunoblotted with an anti-STAT1 antibody or with an anti-IKK, antibody, as indicated. Input corresponds to 5% of total protein extract used for IP.

p16

  INK4a -deficient macrophages are more responsive to IL-4, while they respond less to IFN or LPS.

Figure 1 .

 1 FIGURE LEGENDSFigure 1. p16 INK4a is expressed, but does not influence maturation or the cell cycle in macrophages.(A) p16 INK4a mRNA expression was measured in immune cells isolated from mice: dendritic

Figure 2 .

 2 Figure 2. p16 INK4a -deficiency induces a gene expression profile resembling IL-4-

Figure 3 .

 3 Figure 3. p16 INK4a -deficient macrophages phenotypically and functionally resemble IL-4-polarized alternatively activated macrophages.

Figure 4 .

 4 Figure 4. p16 INK4a -deficiency modulates macrophage responses to alternatively and

Figure 5 .

 5 Figure 5. Haematopoietic deficiency of p16 INK4a exacerbates hepatic alternative

Figure 6 .Figure 7 .

 67 Figure 6. p16 INK4a -deficiency results in an alteration of STAT1 and NF-B signaling

ACKNOWLEDGMENTS P.Krimpenfort provided p16 INK4a -deficient mice. Discussions with G.Raes, J.Vanginderachter and M.de Winther are greatly acknowledged. We thank E.Vallez for mouse breeding, J.Brozek (Genfit SA, Loos, France) for microarray raw data analysis, T.Coevoet , N.Jouy and A.Lucas for technical assistance. This work was supported by the Fondation pour la Recherche Médicale (DCV20070409276 to B.S.), the EFSD/GSK Program 2009, the Cost Action (BM0602) and the Conseil régional Nord Pas-de-Calais and FEDER. C.Cudejko was supported by a doctoral fellowship from the Nouvelle Société Française d'Athérosclérose/Schering-Plough/MSD. K.Wouters was supported by a European FP7 Marie Curie grant (PIEF-GA-2009-235221) and a European Atherosclerosis Society grant.

The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.