Heide Narnhofer
email: heide.narnhofer@univie.ac.at

Entropy for quantum fields in curved space time

Keywords: numbers: 04.62+v, 05.90+m quantum fields, black hole entropy

published or not. The documents may come

Introduction

In [1] it is argued that a black hole has to have some entropy. In [START_REF] Hartle | [END_REF] it is shown that a quantum field on a Schwarzschild space time is in a temperature state where the temperature is determined by the size of the black hole. This inspired [START_REF] Verlinde | On the origin of Gravity and the Laws of Newton[END_REF] to look for natural connections between gravitational forces and temperature as consequence of quantum theory. On the other hand it was argued [START_REF] Sewell | [END_REF] that the Hawking temperature is just an analogue to the Bisognano-Wichmann theorem [START_REF] Bisognano | [END_REF] arising from mathematical equivalences and therefore is an effect related to an observer who in the Bisognano-Wichmann situation is an accelerated observer moving in the Rindler wedge and his acceleration determines the temperature. For the de Sitter space the observer moves in the wedge and observes a temperature depending on the curvature of the metric as well as on his acceleration [6] [7]. In the case of a black hole it is an observer following the time evolution of the Schwarzschild time. Altogether the temperature interpretation depends on the observer and so does the notion of an entropy, if we rely on its definition from thermodynamics.

In this note we take the conservative view point that entropy is just a property of a state and does not refer to any dynamics or any observer. We consider the state of a quantum field over a curved space, so that the quantum field itself cannot influence the geometry. We are interested in the entropy of this state restricted to some local domains and its dependence on the choice of the local domain. Already here problems arise and we have to choose the local domain carefully to ensure that the entropy is not necessarily infinite. To be more precise we are unable to fully characterize permitted local domains and as a consequence are also unable to give explicit numbers. However we are able to give upper limits on the entropy, where we need the assumption of nuclearity [5], and also lower limits, where we demand that the system admits a scaling law [8]. Both assumptions are satisfied for free fields and are taken to be necessary properties of physically meaningful theories. These upper and lower limits are strong enough to find qualitative differences between the entropy of local domains in flat space, in de Sitter space and in the Schwarzschild-Kruskal space time. In all these three situations our analysis is based on the existence of a local Killing vector field and the corresponding analyticity properties of the Wightman functions.. However it will also become clear that its existence is only a useful tool in the mathematical analysis that enables explicit calculations but suggests that the relevant behaviour will also hold in a more general context.

The main observations are the following: Given two local domains one included in the other with some distance between the boundaries, then it is possible to find a subalgebra containing all the observables corresponding to the smaller domain but still belonging to the observables of the larger domain in such a way, that the entropy of this algebra is finite. Its value increases with the size of the smaller domain but also with its inverse distance to the larger domain tending to infinity if the distance tends to 0. However it is possible to let the size of the smaller domain tend to zero together with the distance, where the distance has to be scaled appropriately. If we are in flat space we can arrange the scaling in such a way, that in the limit the entropy tends to 0. The same holds if we are in de Sitter space. It also holds if we are in Schwarzschild space outside of the horizon. If however our local domain contains the black hole then we have to become careful in applying the scaling law. With an appropriate definition for the algebra corresponding to an arbitrary small neighborhood of the black hole its entropy becomes infinite.

Another question concerns how the entropy depends on the size of the smaller domain, when we keep the distance of the larger domain fixed. Here we can find a qualitative behaviour that distinguishes between flat space and curved space. The entropy of a finite domain depends on one hand on the size of the domain, on the other hand on the size of the boundary. In flat space the dominating term is the surface effect. In de Sitter space the lack of a global time evolution that corresponds to a Killing vector field only admits weaker estimates on the upper and lower limit of the entropy, and this is reflected on a dependence on the size of the domain.The entropy is far from being monotonically increasing with the size of the domain due to the interplay of its dependence on the size of the boundary but it becomes maximal when the domain approaches halfspace. A similar behaviour can be found in Schwarzschild space, when we consider domains outside of the horizon.

The paper is organized as follows: first we collect the relevant results obtained in flat space. Here we present also the necessary concepts as nuclearity and split property. These properties are known to be satisfied for quasifree quantum fields on flat space, and it is a reasonable assumption that they hold also for interacting quantum fields and also on curved space. We will argue that the essential features guaranteeing the finiteness of the entropy do not change or can find a replacement that is as powerful if we consider quantum fields in curved space away from singularities. We will consider first the similarities and changes in de Sitter space where we have a Killing horizon but no singularity. The main tool in our analysis will be the application of the modular theory offered by the existence of a Killing vector field. Finally we will apply the methods to Schwarzschild space time and its extension to Kruskal space time. Here the existence of a singularity is the cause that we can assign an entropy to the black hole.

Facts in flat space

The algebra of quantum fields [START_REF] Haag | Local Quantum Physics[END_REF] can be considered to be built by Wightman fields W (f ) where f (x) are functions over R 4 . Local subalgebras A Λ with Λ some compact domain in R 4 are built by W (f ) with supp(f) ⊂ Λ. A special example is the algebra on the Rindler wedge Λ R where

Λ R = {x ∈ R 4 ; x 1 ≥ 0, x 1 < |x 0 |} (1)
The vacuum state over the quantum field reduced to the Rindler wedge is a temperature state with respect to the Lorenz boost

τ (s) =      cosh s sinh s 0 0 sinh s cosh s 0 0 0 0 1 0 0 0 0 1      (2)
at temperature 2π [START_REF] Bisognano | [END_REF]. Under the assumption of a scaling law where the limit of the Wightman functions lim

λ→0 λ 2 ω(λx 1 , ...λx n ) = ω 0 (x 1 , ..x n ) ( 3 )
exists and corresponds again to a quantum field theory that we may even expect to be massless then [8] has shown that the vacuum state over the causal completion of a compact domain in R 3 , e.g. a double cone, is again a state constructed over a cyclic and separating vector and therefore a temperature state with respect to its modular evolution, and that further this modular evolution has the same spectral properties as the generator of the boost. But if the modular evolution has continuous spectrum then necessarily the algebra is of type III 1 and the entropy of any state over such an algebra is ∞.

In order to find a qualitative expression to the fact that in physically reasonable systems the phase space over a finite domain should again be essentially finite with its size limited by its energy [5] formulated the nuclearity condition: We assume that we can express e -βH A|Ω >=

n φ n (A)|Ψ n > ∀A ∈ A Λ (4)
with ||Ψ n || normalized vectors and φ n (A) linear functionals over A Λ , depending on β.

Nuclearity holds with respect to some nuclearity norm that e.g is defined as

ν p (Λ, β) = inf[ n |φ n | p ] 1/p (5)
if this norm is finite. The infimum in (5) is taken over all possible {φ n , Ψ n }. Variations in the norm are possible and can be adjusted to the special demands, especially we can also consider ν ln defined with respect to xlnx.

For free massless theories this nuclearity index ν (for various choices of the index p) was estimated in [12] by ν p (Λ r , β) ≤ c p e r/β , r < β

where Λ r is the causal completion of a ball of radius r in R 3 . Of course we have to keep in mind that the fact that a quantum field allows finite nuclearity indices is an additional assumption and should not be taken for granted.

Based on the assumption of nuclearity [11] showed that for

A ∈ A Λ 1 and B ∈ A Λ C 2 with a distance d(Λ 1 , Λ C 2 ) = inf x∈Λ 1 ,y∈Λ C 2 ||x -y|| the vacuum state satisfies ω(AB) = φ 1,k (A)φ 2,k (B). (7)
Here φ 1,k are linear functionals on A Λ 1 and φ 2,k linear functionals on A Λ C 2 implemented by Ψ n as they entered in the estimation of the nuclearity index. If we fix |φ 2k | = 1 then they satisfy

k |φ 1,k | p < c p ν p (Λ 1 , d(Λ 1 , Λ 2 )) (8)
for all choices of the norm, including the one given by xlnx. Notice that the parameter β that in the definition was related to the Hamiltonian has been shown to be essentially linearly related to the distance [11].

Based on this fact it can be shown, that it is possible to find algebras that are local but also have finite entropy. We start with an inclusion

A Λ 1 ⊂ A Λ 2 where Λ C 2 is the causal complement to Λ 2 and d(Λ 1 , Λ C 2 ) > 0. Then it is possible to construct some algebra N such that A Λ 1 ⊂ N ⊂ A Λ 2 (9)
and N is a type I algebra in such a way that the vacuum state restricted to N can be expressed by a density matrix belonging to N . The algebra N is not unique, however relating the construction of the algebra to the decomposition (7) the nuclearity index gives a bound on the entropy [13], [START_REF] Narnhofer | Local type I algebras and their entropy in quantum field theory[END_REF]. To assign a well defined entropy to the inclusion Λ 1 ⊂ Λ 2 there are several possibilities. One strategy was followed in [13]: Though S(A Λ 1 ) = ∞ ( we will surpress that the entropy depends on the state since all our considerations concentrate on the vacuum state or on a well defined state for curved space time )and also the entanglement of formation E(A Λ 1 ) = ∞ (considered with respect to A Λ 2 and expressing the quantum correlations between the domain Λ 1 and its causal complement in Λ 2 ) the difference [14]

H A Λ 2 (A Λ 1 ) = sup λ i ω i =ω λ i S(ω|ω i ) A Λ 1 (10)
(the supremum is taken over all possible decompositions of ω into states ω i in A Λ 2 ) stays finite under the appropriate definition of the nuclearity index ν (5) with respect to the norm given byn |Φ n | ln |Φ n | as it enters in all entropy expressions. An alternative is to consider the usual expression for the entropy S(ρ) = -T rρ ln ρ and define

S(Λ 1 , Λ 2 ) = inf A Λ 1 ⊂N ⊂A Λ 2 S(N ) ≥ H A Λ 2 (A Λ 1 ) (11)
However it is not clear that this optimization already defines a unique algebra N . The results in [START_REF] Narnhofer | Local type I algebras and their entropy in quantum field theory[END_REF] give an upper bound for some explicitly constructed algebra N that is sufficient for our purpose. The available estimates permit to consider lim

Λ 1 →R 3 lim Λ 2 →R 3 H A Λ 2 (A Λ 1 ) = lim Λ 1 →R 3 lim Λ 2 →R 3 S(Λ 1 , Λ 2 ) = 0 (12)
in agreement with the fact that the vacuum is the ground state for the quantum field and therefore pure [13]. But the same estimates also allow to consider lim

Λ 1 ⊂Λ 2 →0 H A Λ 2 (A Λ 1 ) = lim Λ 1 ⊂Λ 2 →0 S(Λ 1 , Λ 2 ) = 0 ( 13
)
where we have to specify how the larger domain tends to 0 in comparison to the smaller domain. This specification can be done on the basis of the nuclearity condition that provides an upper limit. However estimates to obtain bounds on the nuclearity index are rather rough and we expect they can be improved. This belief is supported by the behaviour of the lower limit.

Here we use the relative entropy

S(ω ⊗ ω|ω) A Λ 1 ⊗A Λ 2 . ( 14
)
That this expression makes sense is based on the fact that A Λ 1 ∪Λ C 2 = A Λ 1 ⊗A Λ 2 due to the results in [11]. The relative entropy is monotonically increasing in A Λ 1 and in A Λ 2 , therefore monotonically decreasing in A Λ 2 . Especially the relative entropy is monotonically increasing with the algebra so that

S(ω ⊗ ω|ω) A Λ 1 ⊗A Λ 2 ≤ S(ω ⊗ ω|ω) N ⊗N = S(N ) + S(N ) -S(N ⊗ N ) = 2S(N ) ( 15
)
provided N is a factor. Otherwise there is the inequality for type I algebras

S(N ) ≤ S(ω ⊗ ω|ω) N ∨N ≤ 2S(N ) ( 1 6 )
where the left side becomes an equality if N is abelian. From this view point, combined with the results in [START_REF] Narnhofer | Local type I algebras and their entropy in quantum field theory[END_REF] (15) can be read that the nuclearity index gives an upper bound on the relative entropy or that the relative entropy gives a lower bound for the nuclearity index.

The algebras A Λ 1 and A Λ 2 are type III algebras. Therefore the relative entropy cannot be expressed in terms of density matrices but instead we can use the Kosaki-formula [START_REF] Kosaki | [END_REF]

S(ω|φ)

A = sup [ ω(1) 1 + t -ω(y * (t)y(t)) - 1 t φ(x(t)x * (t))] dt t ( 17
)
where the supremum is taken over x(t) ∈ A and y(t) = 1 -x(t) as operator valued step functions. Especially for states ω, φ we have the general inequality

S(ω|φ) ≥ 1 2 ||ω -φ|| 2 .
Therefore the entropy can only tend to 0 if we choose Λ 2 sufficiently large with respect to Λ 1 so that ω nearly factorizes. We concentrate on shrinking balls. The nuclearity index (6) tends to 1 as long as β >> r, but still β → 0 is possible so that

S(Λ r , Λ β ) → 0. ( 18
)
Next we are interested how S(Λ r , Λ r+β ) depends on r, β, i.e. how far the entropy is a volume effect and how far it is a surface effect. The upper limit was discussed in [13], but the estimates are rather rough. We rely more on the lower limit that is based on an analysis of [START_REF] Wald | General Relativity[END_REF]. The motivation comes from the additivity of the relative entropy for tensor products. This additivity is however a bit hidden in [START_REF] Wald | General Relativity[END_REF].

Having chosen some x(t) for [START_REF] Wald | General Relativity[END_REF] we can optimize its size in the Kosaki formula with inf

f (t) [ω((1 -f (t)x(1 -f (t)x * ) + φ(x * xf 2 (t)]/t which is attained for f (t) = t(ω(x) + ω(x * )) 2(φ(x * x) + tω(xx * )) (19)
This gives a contribution

[ ω(1) t + 1 -ω(1) + tω(x + x * ) 2 2(φ(x + x) + tω(xx * )) 2 - tω(x + x * )) 2 4(φ(x + x) + tω(xx * )) ] dt t = (20) = [ ω(x + x * ) 2 4(φ(x * x) + tω(xx * )) - 1 1 + t ]dt
For large t the main contribution comes from

ω(x + x * ) 2 4tω(xx * ) - 1 t
which can only be positive if x = 1. Large contributions to the relative entropy can only be obtained for small t where we can obtain

ω(x + x * ) 2 φ(x * x) >> c (21)
Scaling cannot increase this value. But we can recover the additivity behaviour of the relative entropy for tensor products ω = ⊗ω i and φ = ⊗φ i if we choose x = n i x i with

ω i (x i + x * i ) 2 -φ i (x * i x i ) = c
and also φ i (x i ) = 0 for the individual contributions so that

ω( n i (x i + x * i )) 2 4φ( n i x * i n i x i ) = c 2 n (22)
If we do not work with tensor products the same scaling behaviour remains as long as φ( i,j x i x j ) does not contribute as n 2 , as it happens if the state nearly factorizes with increasing |i -j|.

If we therefore consider a ball or a similar domain and divide it into cells then the cells inside the ball will nearly factorize with the cells outside the ball and the main contribution will come from the cells at the boundary coupled to the corresponding cell outside but close. Therefore the entropy apart from a negligible contribution from the volume will essentially scale like its surface with a parameter depending on β -2 .

Estimates in de Sitter space

We consider the de Sitter space as subspace of a five dimensional space satisfying

x 2 1 +x 2 2 + x 2 3 +x 2 4 -x 2 0 = R 2 .
On this space we have Killing vector fields L 0,i = x 0 ∂ i -x i ∂ 0 very similar to the boost in the Rindler wedge of flat space. Like the boosts they do not represent global time evolution, and as the boost they give rise to a horizon:

{x 1 = |x 0 |, x 2 2 + x 2 3 + x 2 4 = R 2 }.
We consider a quantum field on de Sitter space with the condition that its state is invariant under the full de Sitter group and that in the scaling limit it reduces to a free massless theory [6]. Then the state is a temperature state with respect to the Killing vector field L 0,i with temperature 1/2πR. The Killing flow can be most easily expressed if we change to the coordinates

x 0 = (R 2 -x 2 2 -x 2 3 -x 2 4 ) 1/2 sinh t, x 1 = (R 2 -x 2 2 -x 2 3 -x 2 4 ) 1/2 cosh t, x 2 , x 3 , x 4 (23)
where it corresponds to the shift in t. We observe that the domain

W = {t, x 2 , x 3 , x 4 ; x 1 ≥ 0} ( 24
)
is invariant under the Killing flow and corresponds to the Rindler wedge. Reduced to A W the state is a temperature state with respect to L 0,1 . Extending the state to the whole algebra we can use the modular conjugation:

JA|Ω >= e -H/2 A * |Ω >= JAJ|Ω > (25)
where A belongs to A W and JAJ belongs to A W , H implements the boost (i.e. the modular evolution ) on A W and J is the modular conjugation that maps

A W into A W . If we consider operators A = Π k W (f k (x))
we can express the effect of the boost as

e iHt Ae -iHt = Π k W (f k (L t x)).
We know that the Wightman functions defining the state are analytic with respect to the Killing parameter. We apply this analyticity to L t x and consider sinh t → sinh(t + iπ), y cosh t → y cosh(t + iπ)

For t = 0 we obtain W = {x > 0, t} → {x < 0, t} so that the wedge is mapped into its causal complement. The algebras on the two wedges generate the whole algebra. Therefore our state is pure and the algebra satisfies Haag duality. We are interested in the entropy of algebras over local domains. For these local domains we choose local domains in R 4 with t = 0 and their causal completion. Again the scaling law implies that the state on these algebras is a KMS state with respect to a modular automorphism whose generator has the same spectrum as the Killing generator and therefore the algebra is type III and has infinite entropy. Again we consider the replacement A Λ 1 ⊂ N ⊂ A Λ 2 . In order to obtain an upper bound the nuclearity condition of [5] cannot be used because a global time like vector field is never a Killing vector field and therefore does not define a unitary group resp. a hamiltonian. However for Minkowski space [8] has shown that the nuclearity condition guarantees that for an inclusion Λ 1 ⊂ Λ 2 we can consider the modular evolution implemented by e iH 2 t with respect to A Λ 2 and can obtain a nuclear bound for the set

e -λH 2 A|Ω >, A ∈ A Λ 1 , ||A|| = 1, 0 < λ < 1/2. ( 26
)
The restriction on λ stems from the fact that differently to the Hamiltonian in full flat space H 2 is not bounded from below and we can only be sure that A|Ω > belongs to the domain of e -λH 2 for 0 ≤ λ < 1/2. Otherwise the definition of the nuclearity index follows the definition (5). The optimal value for the nuclearity bound is obtained for λ = 1/4. Further from the order relation between the modular operators (the forms of the corresponding modular operators are defined on increasing domains) it follows that ν(Λ 2 ) is decreasing with increasing Λ 2 . If Λ 2 approaches Λ 1 then the nuclearity bound diverges.

The nuclearity bound in Minkowski space was introduced as a measure on something that replaces the size of the phase space in classical mechanics. Giving such an interpretation to a nuclearity bound for inclusions might be done on the basis of a scaling at a point far away from the horizon, where modular automorphism, boost and time evolution nearly coincide in first order in the appropriate scaling. However this is not very convincing. Its relation to entanglement might give some insight: since the nuclearity index decreases with increasing Λ 2 this implies that also the entanglement with Λ 1 and the causal complement of Λ 2 though existing gets smaller. Thus the entanglement between A Λ and A Λ is concentrated on the boundary as a consequence of the nuclearity condition.

For the de Sitter space we can consider the inclusion of a small local domain into a wedge that we choose such that the local domain is sufficiently separated from the boundary. Then we can use the Killing vector field to examine whether nuclearity holds. We assume that this is satisfied. For free fields modifications of the estimates in [12] are possible. Now take two domains that are causally independent. They will stay causally independent when one domain is moved by the Killing flow for a sufficiently small interval. Therefore as in [13] the methods of [11] can be generalized to guarantee that also the nuclearity condition with respect to a general inclusion Λ 1 ⊂ Λ 2 holds. As in Minkowski space also in de Sitter space we can find a type I algebra

A Λ 1 ⊂ N ⊂ A Λ 2 with S(N ) < ∞.
The construction of the quantum field is based on the scaling law. Therefore again lim

Λ 1 ⊂Λ 2 →0 inf A Λ 1 ⊂N ⊂A Λ 2 S(N ) = 0 (27)
with the same conditions on the distance as in Minkowski space. Similarly lim

Λ 2 →deSitterspace H A Λ 2 (A Λ 1 ) = 0 ∀Λ 1 Since H A Λ 2
is monotonically decreasing and obtained as a supremum, we get this result as a consequence that the state is pure. It remains to look for a difference between the behaviour in Minkowski space and in de Sitter space. In Minkowski space we have argued that the entropy of a local domain depends essentially on its surface. The argument was based on the lower estimate where we assumed that only operators at the boundary contribute to [START_REF] Wald | General Relativity[END_REF] whereas for operators concentrated inside of the domain the nuclearity (6) index can be regarded with respect to the boundary, i.e. with large β in (4). Now we can only estimate the nuclearity with respect to the boost and for limited values of β. We examine again which contributions we have to expect in [START_REF] Wald | General Relativity[END_REF]. We divide our local domain into cells. The cells at the boundary will contribute similarly as in Minkowski space being sensitive to the distance of the outer domain. For the cell inside of the local domain we can only use the nuclearity bound up to λ < 1/2. For an operator A in such a cell we will find some JAJ in the other wedge such that

< Ω|JAJA|Ω >=< Ω|A * e -H W /2 A|Ω > = | < Ω|A|Ω > | 2 . ( 28
)
How much it differs determines how much it can contribute to the relative entropy. Of course the definition of the local entropy is done without referring to a wedge. Obviously we have to choose the wedge such that ||(e -H W /2 A -A)|Ω > || is minimal. This means, that the horizon of W should be far away. This can be achieved in flat space, but not in de Sitter space.

As a consequence the state does not cluster approximately and therefore we can use such A for a contribution in [START_REF] Wald | General Relativity[END_REF] as indicated in (22). Therefore the entropy of a local domain will also depend on its size and not only on its surface. Notice however, that tacitly we assumed that N belongs to the wedge which gives a limit on the including algebra. If the including algebra approaches the total algebra again the entropy of the smaller algebra will vanish. Notice however keeping the algebras small the amount of the entropy will depend on R and will decrease with R, in correspondence to the observation in flat space, that the boost far away from the horizon has to be scaled to estimate the local domain.

Schwarzschild-Kruskal space time

We consider quantum fields on the Schwarzschild space time

g = 1 1 -r/r 0 dr 2 + r 2 (dθ 2 + sin 2 θdφ 2 ) -(1 -r/r 0 )dt 2 (29)
together with its extension to the Kruskal space time with the coordinates

u = ( r r 0 -1) 1/2 e r/2r 0 Ch t 2r 0 v = ( r r 0 -1) 1/2 e r/2r 0 Sh t 2r 0 (30) with g = 4r 3 0 r e -r/r 0 (du 2 -dv 2 ) + r 2 (dθ 2 + sin 2 θdφ 2 ) ( 3 1 )
where now u, v ∈ R, u 2 -v 2 > -1 and the domain u 2 -v 2 > 0 corresponds to the Schwarzschild domain.

We concentrate on a free quantum field over the Schwarzschild -Kruskal space time. According to our description with respect to the local structure we consider the algebra to be built by Weyl operators W (f ) (now not Wightman fields but their exponent so that W (f ) are bounded operators) with supp(f ) ⊂ Λ satisfying the commutation relations

W (f 1 )W (f 2 ) = e -σ(f 1 ,f 2 ) W (f 1 + f 2 ) ( 3 2 )
where

σ(f 1 , f 2 ) = -σ(f 2 , f 1 ) is a symplectic form σ(f, g) = f 1 (x)G(x, x )f 2 (x )dΩ x dΩ x (33) where (-∂ µ g µν √ -g∂ ν + m 2 √ -g)G(x, x ) = (34) √ -g(-2 x + m 2 )G(x, x ) = √ -g(-2 x + m 2 )G(x, x ) = 0 with G(x, x ) = -G(x , x
) and G(x, x ) = 0 if x and x are relatively spacelike and normalized such that they reflect the commutation relations. With

h 1 (x) = f (x )G(x, x )dΩ x , h 2 = f (x )∂ µ G(x, x )dΩ x (35)
h 1 , h 2 can serve as the initial data determining uniquely W (f ) in accordance to the equivalence relation

W (f 1 ) = W (f 2 ) if σ(f 1 , f 3 ) = σ(f 2 , f 3 ) ∀f 3 .
According to [START_REF] Hartle | [END_REF] we choose the state so that for the free quantum field it is a quasifree state satisfying for the two point function the conditions formulated in [17]

w (2) (x 1 , x 2 ) = α s + β ln |s| + γ (36)
where α, β, γ are smooth functions of x, x and s is the square of the geodesic distance. Notice that this state has been constructed as a global state on the Kruskal space time and therefore we have left the Schwarzschild domain.

Based on (30) in the Schwarzschild domain we have again the same analyticity properties as in the Rindler wedge in Minkowski space or in the wedge in de Sitter space. We have the time evolution, that is now a Killing flow and therefore implemented by a unitary one parameter group. With its extension into the complex plane we can map the Schwarzschild domain u > 0, u 2 -v 2 > 0 into the causal complement u < 0, u 2 > v 2 . Therefore again the state is a temperature state with temperature T = 1/4πr 0 in the Schwarzschild domain with respect to the time evolution. Considered as a state over the whole Kruskal space time the state is pure and satisfies Haag duality

A Λ = A Λ C with Λ C the causal complement to Λ. Notice that the causal complement to {u > 0, u 2 -v 2 > 0} ∪ {v > 0, 0 < v 2 -u 2 <
1} is empty. Therefore the quantum field considered as acting on the physical space with r ≥ 0 is again in a pure state.

In the Schwarzschild domain we can apply the same considerations as in de Sitter space. Due to the scaling law that now is a consequence of (36) local algebras with compact Λ have modular operators with continuous spectrum, are type III and have infinite entropy. Due to the decrease properties of the two point function we can find algebras containing A Λ , remaining localized but with finite entropy. Again as in de Sitter space the entropy will not only scale with the surface but due to the fact that the state is a temperature state we have again to apply (28) telling us that also an operator far away from the horizon is quantum mechanically correlated to an operator in the commutant. Therefore the entropy of a local domain will contain a term depending on the size of the domain, not only on its boundary. From this view point we can observe the effect of the Hawking temperature even if we are far away from the horizon and therefore the Hawking temperature is not only an analogue to the temperature with respect to the boost in Rindler space, since now the temperature is given by the time evolution and not by a boost and therefore no scaling is necessary.

Our main interest however concerns the behaviour on the horizon and inside of the horizon. First we ask whether again the entropy of a point vanishes. In flat space we argued with the scaling law. This scaling law is evident for our free field described by (32) and (33). We consider functions f with finite support. With scaling f (x 0 + x) → f (x 0 + λx) their support shrinks with λ → ∞. This scaling can be absorbed in the integration dΩ x and in the Green function G(x, x ). In this sense we obtain some f λ (x 0 + x) that is close to λ 3 cf (x 0 + λx) where c depends on the point x 0 around which we scale. The weight of f is irrelevant for our description, only its support characterizes whether the Weyloperator belongs to the local domain. Therefore as in flat space we can conclude that it is possible to arrange the scaling in such a way that the local domain shrinks and so does its surrounding, though with a slightly different scaling that demands a relatively increasing distance if we approach the singularity. The entropy with respect to this inclusion tends to zero, also inside the horizon. Of course we have to stay away from the singularity

u 2 -v 2 = -1.
Concentrating on the horizon we can argue that every local region that is sufficiently smeared out and stays away from the singularity r = 0 has an entropy determined by its size respectively the size of its surface. This holds also for a covering of the horizon, so that for such a covering the entropy is proportional to r 2 0 . Our main interest however is the behaviour of the local entropy inside of the Schwarzschild radius or inside of an arbitrary small neighborhood of the black hole. We consider as physically relevant domains those that lie in the future of the Schwarzschild space time, i.e. the domain {u 2 -v 2 > -1, v > 0}. If we consider the local algebra A Λ with Λ the causal completion of the domain {δ < r < , θ, φ} that we interpret as included into the total space as {u = 0, (1 -δ) 2 > v > (1 -) 2 , θ, φ} this domain shrinks and if we rely on the scaling law the entropy tends to zero if we scale δ together with in such a way that the boundary remains sufficiently large. However the choice of this domain is rather artificial and of no relevance for any observation. We can go back to flat space to recall, why the choice of double cones was taken to be the natural basic description. The starting point is the assumption that the observables are essentially located in 3-dimensional space and that these observables form a von Neumann algebra. Observables located in domains that are space like separated commute with one another. Therefore the von Neumann algebra is obtained as A Λ , the double commutant of the original algebra, and this justifies the choice of causally closed domains. If however we choose domains in the Kruskal domain including 0 < r < r 1 < r 0 and compare with a domain r 1 < r < r 0 the corresponding observables will not commute, since the points are not space like any more.

The other observation in flat space uses Haag duality: If we evaluate the entropy for some inclusion Λ 1 ⊂ Λ 2 i.e. via

S(Λ 1 , Λ 2 ) = inf A Λ 1 ⊂N ⊂A Λ 2 S(N ) ( 3 7 )
then the fact that we are in a pure state implies S(N ) = S(N ) so that

S(Λ 1 , Λ 2 ) = S(Λ C 2 , Λ C 1 ). ( 38
)
As a consequence the entropy of the causal complement of a finite domain is finite. However the entropy of the wedge is infinite, even when we consider not the wedge itself but any algebra containing the wedge and belonging to a shifted wedge, again based on the estimates (28). According to our construction Haag duality is also satisfied for the Kruskal-Schwarzschild metric, i.e. the quantum field is in a pure state. Instead of evaluating S(A 0≤r<r 1 ) we can evaluate S(A r 1 <r<∞ ), i.e. we consider as relevant what can be observed in the total domain that just stays away from the singularity. Next we recall that in flat space a diamond can be considered to be the causal completion of small diamonds covering a spatial cut of the large diamond. In this spirit we construct the complement of a neighborhood of the black hole. We specify which algebra we take for A r 1 <r<∞ = A >r 1 : A possible choice is A r 1 <r<∞ = (A r 1 <r<r 0 ,u=0 ∨ A r 0 <r<∞ ) where A r 1 <r<r 0 ,u=0 is a diamond in the region {v > 0, v 2 -u 2 > 0}.Its commutant is not A 0<r<r 1 ,u=0 but the von Neumann algebra is located over an infinitely extended domain in {u 2 -v 2 > 0, u < 0}. Therefore, based on the Kosaki formula, most of the local subalgebras of the Schwarzschild algebra (A r 0 <r<∞ ) = A S will contribute to the relative entropy and will make it infinite. Therefore

S(A >r 2 , A >r 1 ) = ∞, 0 < r 1 < r 2 (39)
With our identificationA r<r 1 = A >r 1 this gives an infinite entropy to the black hole. Notice however, that in these considerations the horizon r = r 0 is not of special significance.

We have to admit that these arguments do not offer a way to assign an entropy to the black hole that scales like r 2 0 with a scaling factor that is justified by the smearing of the surface that should be chosen of the size of the Planck length. However such a behaviour cannot be extracted from the model, because on one hand an infinite region is in a temperature state and has therefore infinite entropy, whereas we assume that the total system is in a pure state, so that also the complement of this region has infinite entropy. This discrepancy with the usual arguments might be the result that our quantum field was constructed on a fixed background without the possibility that the gravitational field can react on the quantum field whereas in the usual approach the entropy is obtained by the possibility of an energy exchange and entropy exchange of the black hole with the quantum field. Imagine several quantum fields corresponding to different non interacting particle species, built over the gravitational field. For such a model the local entropies are additive, and therefore necessarily our entropies are model dependent. Therefore any scaling parameter cannot be model independent. Notice that the gravitational constant disappeared in the calculations and only r 0 appears. If we express r 0 given by the mass of the black hole and the gravitational constant we can scale these parameters keeping r 0 fixed. That the quantum field cannot effect the gravitational background corresponds that we take the gravitational constant to be 0 together with the mass of the black hole M being ∞ whereas r 0 is kept constant. This would be in agreement with the fact that the entropy does not scale with r 0 but with M. Our considerations were determined by the analyticity properties of the quantum field together with Haag duality. Of course the assumption to be in a pure state was essential. Another question is whether the precise form of the state in the Schwarzschild region was essential or if it is possible to reduce the entropy in the Schwarzschild region by appropriate correlations between the horizon and infinity without violating the regularity conditions at the horizon.

Conclusion

The fact that we can assign a temperature to the state of quantum fields over a curved background has to refer to an automorphism group on the algebra. This can be done only in special cases in such a way, that we can give a geometrical interpretation to the automorphism group. These cases include the boost in the Rindler wedge, the boosts in the wedges of de Sitter space and time evolution in Schwarzschild space time.

In contrast a local entropy, now without interpretation with respect to thermodynamics but considered as property of a state, can be defined under some mild restrictions for inclusions of local algebras. Also this local entropy shows different behaviour on a qualitative level, reflecting the properties of the underlying quantum field. In all cases the entropy vanishes for shrinking regular domains. In flat space the entropy is mainly determined by the surface of the domain, reflecting the entanglement of domains that is concentrated on the surface. This contribution remains unchanged also in gravitational fields. In the Sitter space the temperature that we can assign to the wedge induces that the entropy of local domains scales roughly also with their volume. The same is true for Schwarzschild space time. From this view point the temperature is more than an analogue to the Rindler wedge, where the temperature is observed by an accelerated observer and the temperature is only determined by the acceleration. In addition we have to extend the quantum field over the Schwarzschild space time if we want to keep Haag duality, i.e. to consider a pure state. The obvious extension is the extension to Kruskal space. Here the origin corresponds to a singularity. We give an interpretation to the quantum field around this singularity as the complement, i.e. commutant to the algebra outside the singularity. With this interpretation the algebra around the origin has infinite entropy without the possibility to regularize it by smearing.

Acknowledgement

This work was inspired by a discussion with Bert Schroer. I am grateful to Helmuth Rumpf and Helmuth Urbantke for their interest and many critical remarks.


Camilla Pilati

Mohamed Amessou

Michel P Bihl

Charles Balabaud

Jeanne Tran

Van Nhieu

Valérie Paradis

Jean Charles Nault

Tina Izard

Jeanne Tran Van Nhieu

Paulette Bioulac-Sage

Gabrielle Couchy

Karine Poussin

Jessica Zucman- Rossi

Jessica Zucman-Rossi

Somatic mutations activating STAT3 in

Keywords: STAT3, gp130, IL6ST, IL-6, -catenin, CTNNB1, HNF1A, IFN, SAA, CRP, SOCS3, JAK1, JAK2, Src

Introduction

Inflammatory hepatocellular adenomas (IHCA) are benign liver tumors that predominantly develop in women and that are frequently associated with obesity and alcohol abuse [START_REF] Bioulac-Sage | Hepatocellular adenoma subtype classification using molecular markers and immunohistochemistry[END_REF][START_REF] Zucman-Rossi | Genotype-phenotype correlation in hepatocellular adenoma: new classification and relationship with HCC[END_REF]. These tumors are characterized by overexpression of the acute-phase inflammatory response in tumor hepatocytes and highly polymorphous inflammatory infiltrates. We recently showed that 60% of IHCA activate STAT3 (Signal Transducers and Activators of Transcription type

3) via somatic heterozygous activating mutations in IL6ST that encodes the gp130 coreceptor and signal transducer of IL-6 family cytokines [START_REF] Rebouissou | Frequent in-frame somatic deletions activate gp130 in inflammatory hepatocellular tumours[END_REF].

Inflammatory adenoma is a homogenous disease, where gp130 mutated and non-mutated IHCAs are similar in their expression profiles, in particular regarding the expression of acute phase inflammatory response target genes [START_REF] Rebouissou | Frequent in-frame somatic deletions activate gp130 in inflammatory hepatocellular tumours[END_REF]. To identify genetic alterations causing the inflammatory phenotype, defined as an acute-phase inflammatory response with elevated levels of Serum Amyloid A (SAA) and C-Reactive protein (CRP) observed in non-mutated gp130 IHCA, we used a candidate gene strategy. We focused on STAT3 as it encodes a key nuclear transcription factor that induces the acute-phase inflammatory response in hepatocytes. Moreover, STAT3 has oncogenic roles in epithelial cell tumorigenesis [START_REF] Grivennikov | IL-6 and Stat3 are required for survival of intestinal epithelial cells and development of colitis-associated cancer[END_REF]Grivennikov et al., 2010;[START_REF] Yu | STATs in cancer inflammation and immunity: a leading role for STAT3[END_REF]. STAT proteins are latent transcription factors activated by phosphorylation, typically following the binding of extracellular ligands to cytokine receptors that activate associated JAK kinases, via growth factor receptors having intrinsic tyrosine kinase activity (e.g., EGFR, PDGFR and CSF1R), as well as by non-receptor tyrosine kinases such as Src family kinases [START_REF] Levy | Stats: transcriptional control and biological impact[END_REF].

STAT3 mutations in hepatocellular adenoma 5 STAT3, one of seven STAT family members, is activated by gp130 in response to interleukin-6 (IL-6) [START_REF] Inoue | The amino acid residues immediately carboxyl-terminal to the tyrosine phosphorylation site contribute to interleukin 6-specific activation of signal transducer and activator of transcription 3[END_REF][START_REF] Zhong | Stat3: a STAT family member activated by tyrosine phosphorylation in response to epidermal growth factor and interleukin-6[END_REF]. STAT3 is persistently phosphorylated in many human cancer cell lines and in epithelial cells and primary tumors, including hepatocellular carcinoma, breast, prostate, and head and neck cancers, and also in several hematological malignancies. STAT3 mutations have never been described in human tumors, yet we reasoned that alterations activating its function would explain the inflammatory phenotype of IHCA lacking mutations in gp130.

Results and Discussion

Identification of STAT3 mutations in IHCA

Among 114 screened hepatocellular adenomas, we identified seven STAT3 mutations in 6 tumors (Table 1 and Supplementary Fig. S1a-b). All of these adenomas were inflammatory (6 out of 75 IHCA) and no additional somatic mutations were identified in HNF1A, CTNNB1 or IL6ST, genes recurrently altered in IHCA and HCA [START_REF] Bluteau | Bi-allelic inactivation of TCF1 in hepatic adenomas[END_REF][START_REF] Rebouissou | Frequent in-frame somatic deletions activate gp130 in inflammatory hepatocellular tumours[END_REF][START_REF] Zucman-Rossi | Genotype-phenotype correlation in hepatocellular adenoma: new classification and relationship with HCC[END_REF]. Thus, STAT3 IHCA mutations are exclusive from all other known genetic alterations in hepatocellular tumors. STAT3 mutated IHCA were associated with obesity (n = 4) and/or alcohol abuse (n = 3) and multiple (≥4) nodules were present in half of these cases. All STAT3 mutations were somatic, as they were not observed in adjacent non-tumor liver tissues that were steatotic in 5/6 cases. All mutations were also monoallelic and led to amino acid substitutions in 5 cases or to in-frame insertions of 1 to 4 amino acids in the remaining two cases. In one case (#379) we identified three nucleotide mutations that led to amino acid substitutions at codons 502 and 658. Sequencing cloned STAT3 cDNA from this tumor showed that the three different nucleotide mutations were carried by the same allele. We also showed in three different cases (#379, #966 and #1351) that the mutations were harbored by all of the isoforms of STAT3, including splicing of nucleotides 2099-2101 and 2145-2194 (STAT3β) that lead to p.Ser701del and p.Thr716PhefsX8, respectively (Supplementary Fig. S1c) [START_REF] Schaefer | Cooperative transcriptional activity of Jun and Stat3 beta, a short form of Stat3[END_REF].

Sequencing STAT3 RT-PCR products in mutated tumors showed that all cases expressed the normal and mutated STAT3 alleles at comparable levels (Supplementary Fig. S1a-b). Thus, the expression of one mutated STAT3 allele is a rare but recurrent genetic event in IHCA that is selected for during benign hepatocellular tumorigenesis.

Four of the seven identified mutations were located in the domain of STAT3

(residues 585-688) that shares homology with Src-homology-2 (SH2) domains (Fig. 1a).

The STAT3 SH2 domain mediates STAT3 dimerization via binding of phosphotyrosine residue Y705 [START_REF] Shuai | Interferon activation of the transcription factor Stat91 involves dimerization through SH2-phosphotyrosyl peptide interactions[END_REF][START_REF] Wen | Maximal activation of transcription by Stat1 and Stat3 requires both tyrosine and serine phosphorylation[END_REF]. Moreover, three of these mutations clustered in a hotspot in the SH2 domain at codons 657 and 658. Interestingly, introduction of cysteine at residues 662 and 664 promotes STAT3 dimerization and creates a constitutively active transcription factor (Stat3-C) [START_REF] Bromberg | Stat3 as an oncogene[END_REF]. The fourth mutation, Y640F, was in the PYTK motif conserved in STAT1, STAT2 and STAT3. Interestingly, substitution of the corresponding Y631 to phenylalanine in STAT2 promotes type I IFN signaling [START_REF] Constantinescu | Mining for JAK-STAT mutations in cancer[END_REF][START_REF] Scarzello | A Mutation in the SH2 domain of STAT2 prolongs tyrosine phosphorylation of STAT1 and promotes type I IFN-induced apoptosis[END_REF]. The three other mutations found in IHCA were distributed throughout the protein; the altered leucine-78 that contributes to dimerization [START_REF] Chen | A reinterpretation of the dimerization interface of the N-terminal domains of STATs[END_REF], glutamate-166 that is part of helix 1 involved in the interaction with gp130 (Zhang et al., 2000a) and aspartate-mutations identified in IHCA were similar to the 150 germline inactivating STAT3 heterozygous mutations described in patients with Job's syndrome that causes the hyper-IgE syndrome [START_REF] Holland | STAT3 mutations in the hyper-IgE syndrome[END_REF][START_REF] Minegishi | Dominant-negative mutations in the DNA-binding domain of STAT3 cause hyper-IgE syndrome[END_REF].

IHCA mutations activate STAT3 in absence of IL-6

To investigate possible functional consequences of these STAT3 IHCA mutations, we introduced the seven different IHCA mutations into a STAT3α (full length) cDNA and analyzed their effects in Hep3B cells, a hepatocellular carcinoma line that activates the acute inflammatory phase following treatment with IL-6 [START_REF] Coulouarn | Genome-wide response of the human Hep3B hepatoma cell to proinflammatory cytokines, from transcription to translation[END_REF]. The results were also validated in two additional hepatocellular cell lines (HepG2 and Huh7).

As described [START_REF] Bromberg | Stat3 as an oncogene[END_REF], in the absence of IL-6 ligand and serum, overexpression of wild-type STAT3 (WT) alone was not sufficient to activate a STAT3 reporter (Fig. 1b) nor the downstream acute-phase inflammatory genes (Fig. 1c). In contrast, the six IHCA mutants and the mouse Stat3-C mutant [START_REF] Bromberg | Stat3 as an oncogene[END_REF] activated the STAT3 reporter in the absence of IL-6 and induced targets typical of the acute inflammatory response, including CRP and SOCS3 (Fig. 1b and1c). Moreover, activation of STAT3 was proportional to the amount of the transfected mutant STAT3 (Fig. 1d). The last IHCA mutant D502Y/K658Y, identified in tumor #379, has two mutations on the same allele. The D502Y mutation did not impart constitutive activity, whereas K658Y activated the inflammatory response. Further, the dual mutant (D502Y + K658Y) produced a hyperactive STAT3 mutant compared to K658Y alone (Fig. 1e).

Thus, mutation at D502 alone is not sufficient to activate STAT3 but cooperates with the mutation at K658 present in the SH2 domain. Tyr705-STAT3 in the nucleus of these tumor hepatocytes (Fig. 2a). Similarly, in Hep3B transfected cells, we showed that STAT3 mutants L78R, E166Q, Y640F and G656_Y657insF were constitutively phosphorylated on Tyr705 and translocated to the nucleus, whereas overexpressed wild-type STAT3 was unphosphorylated and cytoplasmic in the absence of cytokine (Fig. 2b-d). STAT3 IHCA mutants did not exhibit increased phosphorylation of serine 727, a second residue also involved in STAT3 activation [START_REF] Wen | Maximal activation of transcription by Stat1 and Stat3 requires both tyrosine and serine phosphorylation[END_REF], when compared to IL-6 activated wild-type STAT3 (Supplementary Fig. S2a). Moreover, in STAT3 Y640 and G656 mutants substitution of tyrosine 705 with phenylalanine totally abrogated activation of the inflammatory response. Thus, tyrosine phosphorylation is essential for activation of IHCA mutant STAT3 (Fig. 2e).

IHCA STAT3 mutants dimerize in absence of cytokine

A critical step in STAT3 activation is the formation of homodimers through reciprocal phospho-Tyr705-SH2 domain interactions. To assess dimerization by IHCA STAT3 mutants, we performed co-immunoprecipitation assays with epitope-tagged STAT3

proteins. The Y640F mutant homodimerized or heterodimerized with wild-type STAT3 independent of IL-6 (Fig. 3a). Moreover, mutant homodimers were highly phosphorylated at tyrosine 705. In contrast, homodimers of wild-type STAT3 were only faintly detected and were not phosphorylated on tyrosine 705. As expected, IL-6 treatment does not enhance dimer formation, but slightly increases tyrosine 705

phosphorylation [START_REF] Haan | Cytoplasmic STAT proteins associate prior to activation[END_REF]. In addition, overexpression of wild-type STAT3 impaired the activity of the STAT3 mutant Y640F in a dose dependent manner (Fig. 3b).

Thus, wild-type STAT3 can compete with the mutant STAT3 and impair the formation of active IHCA mutant homodimers. Interestingly, analysis of the STAT3 homodimer structure [START_REF] Becker | Three-dimensional structure of the Stat3beta homodimer bound to DNA[END_REF] showed that 4 out of 6 STAT3 mutants targeted amino acids (Y640, K658, Y657 and Y657-M660) that reside near the transcription activation domain (TAD, residues 701-722) (Fig. 3c and Supplementary Fig. S2b). For example, Tyr-640 engages in hydrophobic interactions with two-fold related Phe-710 and residues 656-659 pack against TAD residues 711-716 (Fig. 3c and Supplementary Fig. S2b).

These modifications could thus enhance mutant STAT3 function, and they must also somehow facilitate phosphorylation of tyrosine 705 by kinases in hepatocytes, which would then drive dimerization and nuclear localization of mutant STAT3. Following cytokine exposure, phosphorylated STAT3 homodimers translocate to the nucleus where they bind to specific IL-6 responsive elements to activate the transcription of STAT3 mutations in hepatocellular adenoma 10 target genes [START_REF] Hattori | Acute-phase reaction induces a specific complex between hepatic nuclear proteins and the interleukin 6 response element of the rat alpha 2-macroglobulin gene[END_REF]. In Hep3B cells, nuclear Y640 and G656 STAT3 mutants bind to recognition elements found in endogenous STAT3 target genes in the absence of IL-6 whereas, as expected, wild-type STAT3 was defective in binding in the absence of IL-6 (Fig. 3d).

IHCA STAT3 mutants are hyper-responsive to IL-6

To investigate the role of the IL-6-JAK pathway in the activation status of IHCA STAT3 mutants, we tested the effects of increasing doses of IL-6 on their activity. Indeed, even low doses of IL-6 (1 ng/ml) dramatically augmented the constitutive activity of the Y640 STAT3 mutant when compared to the wild-type STAT3 (Fig. 4a). This hypersensitive response to IL-6 signaling was confirmed for L78, E166, G656 and Y657 STAT3 mutants (Fig. 4b and Supplementary Fig. S2c). Together with gp130, JAK1 is known as the major kinase that phosphorylates and activates STAT3 in IL-6 signaling.

Accordingly, silencing JAK1 by siRNA abolished the hyper-stimulation of the STAT3 mutants by IL-6 (Fig. 4b).

Activity of IHCA STAT3 mutants requires Src

In the absence of IL-6 signaling, silencing of gp130 or JAK2 did not affect the constitutive activity of IHCA STAT3 mutants (Fig. 4c-d). In contrast, JAK1 knockdown by siRNA partially impaired constitutive IHCA STAT3 mutant activity (Fig. 4e). Similarly, AG490, an inhibitor that disables the kinase activity of both JAK1 and JAK2, Ruxolitinib, a selective inhibitor of Jak1, and curcumin, a STAT3 inhibitor, partially decreased the activity of IHCA STAT3 mutants (Supplementary Fig. S3a-b). In contrast, treatment with SrcI-1 kinase inhibitor dramatically impaired both the constitutive and IL-6 induced activation of IHCA STAT3 mutants (Fig. 4f and Supplementary Fig. S3c). Two other Src kinase inhibitors also impaired the activity of IHCA STAT3 mutants (Supplementary Fig. S3d-e). Thus, phosphorylation and activation of mutant STAT3 does not require gp130 or JAK2 but requires JAK1 and Src signaling. Depending on cell context, STAT3 can be activated directly by Src family kinases or via Jak1 (in NIH 3T3 cells) as an intermediate kinase between Src and STAT3 (Zhang et al., 2000). Our findings suggest that in hepatocellular cell lines, the requirement of Jak1 is necessary for maximal activation of IHCA STAT3 mutants, whereas Src is essential for their constitutive activation.

STAT3-mutated IHCA are similar to gp130-mutated IHCA

In vivo, expression analyses of 16 IHCA revealed no significant differences in 28 target genes of IL-6 and interferon signalling pathways in STAT3 mutated adenomas (n=5) versus gp130 mutated IHCA (n = 11; Fig. 4g). Thus, the functional effects of mutating STAT3 and gp130 in human hepatocellular adenomas appear redundant.

In the liver, activation of the IL-6-STAT3 signaling is necessary for the acute-phase response during inflammation [START_REF] Lutticken | Association of transcription factor APRF and protein kinase Jak1 with the interleukin-6 signal transducer gp130[END_REF] and liver regeneration, particularly following transplantation [START_REF] Cressman | Liver failure and defective hepatocyte regeneration in interleukin-6-deficient mice[END_REF]. The connection between inflammation and tumorigenesis has been extensively investigated over the past decade (see ref. During these processes and in numerous tumor types, STAT3 activation has been recognized as a major oncogenic event. However, in our cohort, STAT3 activating mutations are found in 8% of the inflammatory hepatocellular adenoma, but were not present in a series of 218 malignant liver tumors, including hepatocellular carcinomas and cholangiocarcinomas (n = 11). Thus, STAT3 activating mutations appear to be largely or quasi-exclusively associated with benign proliferation of hepatocytes in human.

Collectively our findings show that most of the genetic alterations of HCA subtypes, (HNF1A inactivation, ß-catenin activation or activation of STAT3 via mutation of STAT3 or gp130), are associated with benign hepatocyte tumorigenesis, where there remains a largely controlled rate of tumor cell proliferation. At present, only ß-catenin activating mutations are associated with malignant transformation to hepatocellular carcinoma [START_REF] Zucman-Rossi | Genotype-phenotype correlation in hepatocellular adenoma: new classification and relationship with HCC[END_REF], and here there are likely additional cooperating mutations that lead to frank malignancy. Regardless, to our knowledge this is the first identification of somatic mutation of STAT3 in human tumors, and these findings underscore the important role of STAT3 activation in hepatocellular benign tumorigenesis and reveal a new mechanism of recurrent STAT3 activation.

Materials and Methods

Tumors and Patients

A series of 364 hepatocellular tumors including 114 hepatocellular adenoma (HCA), 218 Tumor and non-tumor liver samples were frozen immediately after surgery or biopsy and stored at -80°C. Tissues samples were also included in 10% formalin, paraffin-embedded and stained with haematin-eosin and masson trichrome. The diagnosis of HCA, HCC, cholangiocarcinoma and hepatocholangiocarcinoma was performed on established histological criteria (1995).

Quantitative RT-PCR

Quantitative RT-PCR was performed as described [START_REF] Rebouissou | Frequent in-frame somatic deletions activate gp130 in inflammatory hepatocellular tumours[END_REF] using predesigned primers and probe sets from Applied Biosystems (listed in Supplementary Table S1). Ribosomal 18S RNA (R18S) was used to normalize expression data and the 2-∆∆CT method was apllied.

DNA sequencing

DNA sequencing was performed as described [START_REF] Rebouissou | Frequent in-frame somatic deletions activate gp130 in inflammatory hepatocellular tumours[END_REF] using primers described in Supplementary Table S2. All HCA samples were previously sequenced for CTNNB1 (exons 2-4), HNF1A (exons 1-10), and IL6ST (exon 6). All HCC samples were previously sequenced for CTNNB1 (exons 2-4) and P53 (exons 2-10) as described [START_REF] Rebouissou | Frequent in-frame somatic deletions activate gp130 in inflammatory hepatocellular tumours[END_REF][START_REF] Zucman-Rossi | Genotype-phenotype correlation in hepatocellular adenoma: new classification and relationship with HCC[END_REF]. All mutations were validated by sequencing a second independent PCR product on both strands. In all cases the somatic origin of the mutation found in tumor was verified by sequencing the corresponding adjacent, normal liver sample.

Generation of STAT3 mutants

A full-length STAT3 open reading frame cloned in pCMV6-XL4 vector was purchased from OriGene (SC124165, NM_139276). Mutagenesis reactions were performed using the QuickChange XL site-directed mutagenesis kit (Stratagene) using primers described in Supplementary Table S3. All constructs were verified by sequencing. The pCMV6-XL4 expression vector plasmid was used as a control. Stat3-C plasmid was purchased from Addgene, Inc. (plasmid 8722).

Cell culture

Hep3B, Huh7 and HepG2 cells (ATCC) were grown in DMEM containing 10% FCS supplemented with penicillin (100 units/mL) and streptomycin (0.1 mg/mL). For transfections, cells were plated 16 hr prior to transfection to produce monolayers that were 60% confluent, and these were transfected by using either Lipofectamine™ LTX (plasmid alone) or Lipofectamine™ 2000 (plasmid and siRNA) according to the manufacturer's instructions (Invitrogen). Transfection efficiency was monitored by measuring the level of either wild-type and mutated STAT3 mRNA using quantitative RT-PCR or western blot analysis. The same STAT3 expression level was observed in wild-type STAT3 and IHCA mutant STAT3 transfected cells. For luciferase assays, Hep3B cells were co-transfected with a STAT3 luciferase reporter expressing a firefly luciferase reporter gene that contains three copies of a STAT3 consensus binding site linked to a minimal thymidine kinase promoter (kindly provided by Dr. H. Gascan, Institut National de la Santé et de la Recherche Médicale UMR564, Angers, France) (pSIEM-Luc 1μg, [START_REF] Coqueret | Functional interaction of STAT3 transcription factor with the cell cycle inhibitor p21WAF1/CIP1/SDI1[END_REF]) and the expression plasmid for wild-type or IHCA mutant STAT3 (1μg). One day after transfection, cells were starved in a freeserum medium for 3 hr, then left untreated or treated with IL-6 (100 ng/ml) for 3 hr, and they were then lysed and the luciferase activity determined according to the manufacturer's recommendations (Promega). The activities were normalized to the protein amounts in each cell lysate. All analyses were performed in triplicate.

Knockdown with siRNA

The efficiency of knockdown using siRNAs was monitored by measuring the level of either JAK1, JAK2 or IL6ST mRNA using quantitative RT-PCR (Supplementary Fig. S3f)

or by western blot analysis. Knockdown efficiency was greater than 80%. JAK1 siRNA (S7646), JAK2 siRNA (S7651) and IL6ST (gp130) siRNA (S229006) were purchased from Applied Biosystems; siRNA control (Block-IT Alexa Fluor Red) was purchased from Invitrogen.

Pharmacological treatments

Cells were exposed, in serum-free medium, for 6 hr to Tyrphostin AG490 (Sigma France), 9 hr to Src Inhibitor-1 (Sigma France), Src Inhibitor-5 (JS Res Chemical Trading, Germany) or PP2 (Sigma), 18 hr to Ruxolitinib (INCB-018424, JS Res Chemical Trading, Germany) and 24 hr to Curcumin (Sigma). In the last 3 hr of treatment, cells were stimulated with IL-6 (100ng/ml), or not, as indicated.

Western blot analysis

Western blot analyses were performed as described [START_REF] Rebouissou | Frequent in-frame somatic deletions activate gp130 in inflammatory hepatocellular tumours[END_REF]

Immunohistochemistry

Immunohistochemistry was performed using a Dako autostainer, on paraffin sections of 10% fixed tumor tissue using a monoclonal anti-phospho-STAT3 Tyr705 (Cell Signaling Technology, 1:50). For each immunohistochemical procedure, antigen retrieval was performed in citrate buffer, detection was amplified by the Dako Envision system.

STAT3 DNA-binding assays

One day after transfection, cells were starved in a serum-free medium overnight, and the TransFactor Extraction Kit (Clontech Inc.) was used to prepare nuclear extracts following the manufacturer's protocol. Nuclear extracts were analyzed for STAT3 DNA binding activity using the TransFactor Universal STAT3-specific kits (Clontech Inc.), which is an ELISA-based method.

Immunofluorescence 24 hr after transfection cells were fixed in 4% formaldehyde for 15 min, washed with phosphate-buffered saline (PBS), then permeabilized with 0.1% triton for 15 min. Cells were saturated with 2% BSA for 30 min, washed with PBS, then incubated with primary antibody overnight at 4°C. After three washes with PBS, cells were incubated with secondary antibodies for 1 hr. The slides were washed, then mounted with VECTASHIELD® Mounting Medium with DAPI (Vector labs). Immunofluorescence images were obtained using a Carl Zeiss HBO 100 optic microscope. All data from one experiment were collected at the same intensity and background level. The following monoclonal antibodies were used: rabbit anti-phospho-STAT3 Tyr705 (Cell Signaling Technology, 1:200), mouse anti-total STAT3 (Cell Signaling Technology, 1:100) and the secondary antibodies were anti-mouse (Invitrogen, 1:200) and anti-rabbit (Invitrogen, 1:200).

  phosphorylation of tyrosine 705, dimerization and translocation to the nucleus. Accordingly, in STAT3 mutated IHCA, immunohistochemical analyses demonstrated marked overexpression of phospho-

(

  Grivennikov and Karin, 2010) for review). Depending of the model, inflammation can trigger tumor initiation or function to enhance tumor progression and/or metastasis.

  using antibodies specific for STAT3 (Cell Signaling Technology, diluted 1:500) and phospho-STAT3 Tyr705, (Cell Signaling Technology, diluted 1:200). For dimerization assays, cell lysates were incubated with Immobilized Protein G agarose (Pierce) and anti-flag (Cell signaling technology, 1:50) or anti-myc (Cell signaling technology, 1:200) antibody, at 4°C overnight, prior to western blot analysis.

Figure 1 .

 1 Figure 1. Gain-of-function mutations in STAT3 in IHCA. a, Distribution of STAT3 mutations identified in IHCA are represented according to the different protein domains (SH2: Src-homology-2 domain; TAD: Transactivation domain). * indicates the two mutations that are found on the same allele in tumor #379. Stat3-C location is indicated in grey. b, STAT3 mutants L78R (L 78 ), E166Q (E 166 ), Y640F (Y 640 ), D502Y K658Y (D 502 /K 658 ),G656_Y657insF (G 656 ), Y657_M660dup (Y 657 ) and Stat3-C (S C) or control STAT3 wild-type (WT) and empty plasmid (EP) were transfected in Hep3B, HepG2 and Huh7 cells expressing a STAT3-driven luciferase (Luc) reporter construct. STAT3 activation was measured after 6 hr of serum starvation and when indicated were treated for 3 hr with IL-6 (100 ng/ml). Shown is the Luc activity (mean) determined from triplicate co-transfections (± s.d.) relative to pSIEM-luc alone (EP) without IL-6. c, Graphs are qRT-PCR results showing the induction of endogenous CRP, STAT3 and SOCS3 mRNA after overexpression of mutant STAT3 relative to unstimulated empty-plasmidtransfected Hep3B cells (EP) (mean ± s.d.). d, Endogenous CRP mRNA expression in Hep3B cells transfected with increasing amounts of expression plasmids encoding WT or E 166 STAT3 mutant (dotted line: mock-transfected cells). e, Activity of the STAT3 mutants, including D502Y (D 502 ), K658Y (K 658 ) and the double mutant D502Y K658Y (D 502 /K 658 ) were evaluated using STAT3-driven Luc in Hep3B cell line. Data are from triplicate transfections (± s.d.) relative to pSIEM-luc alone (EP). ns, not significant. *, ** and *** indicate differences between groups at P<0.05, P<0.01 and P<0.001, respectively (two-tailed t-test).

Figure 2 .

 2 Figure 2. Subcellular localization and role of tyrosine 705 phosphorylation in STAT3 IHCA mutants. a, Immunochemistry of phospho-STAT3-Y705 in STAT3mutated IHCA and adjacent non-tumor liver (NTL). b-d, Vectors driving the expression of STAT3 mutants, including L78R (L 78 ), E166Q (E 166 ), Y640F (Y 640 ), G656_Y657insF (G 656 ) or wild-type (WT) STAT3 were transfected into Hep3B cells. b, Expression of total and phosphorylated (pY705) STAT3 proteins were assessed by western blot and bands were quantified with ChemiDoc (Bio-Rad) using the Quantity One software; bars show the expression of pY705 STAT3 relative to total STAT3. c, d, Subcellular localization of total and phosphorylated (pY705) STAT3 proteins were analyzed by western blot (c) and by immunofluorescence (d). Results are representative of three independent experiments. Scale bars = 20 µm. e, CRP mRNA expression after transfection of wildtype (WT), Y705F (Y 705 ), Y640F (Y 640 ), Y640F/Y705F (Y 640 /Y 705 ), G656_Y657insF (G 656 ) or G656_Y657insF/Y705F (G 656 /Y 705 ) STAT3 in unstimulated Hep3B cells. Data are from triplicate transfections (± s.d.) relative to empty plasmid (EP). ns, not significant. *, ** and *** indicate differences between groups at P<0.05, P<0.01 and P<0.001, respectively (two-tailed t-test).

F

Role of IL-6-JAK pathway and Src kinases on STAT3 IHCA mutant activation.

  Flag and pY705-STAT3 specific antibodies. Results are representative of three independent experiments. b, Effects of increasing WT STAT3 expression (wedge) on the constitutive activity of Y 640 mutant STAT3 without IL-6. The graph plots the mean level of expression of SOCS3 and CRP relative to cells transfected with EP(± s.d.). Units are plasmid µg. c, Several STAT3 IHCA mutations cluster near the transcription activation domain (TAD) of STAT3 (crystal structure PDB: 1BG1)[START_REF] Becker | Three-dimensional structure of the Stat3beta homodimer bound to DNA[END_REF]. Residues 640, 656-660 (shown in green) that are mutated in IHCA tumors, the phosphorylated tyrosine (Tyr-705, black), as well as TAD residues 711-716 (blue) are shown in stick representation. One polypeptide chain and both TADs are shown as a cartoon. The two-fold related molecule, except the TAD, is shown as a surface representation. For residues of the TAD, Tyr-710 of the two-fold related molecule is labeled while residues 711-716 are not for clarity. d, DNA-binding activity of 10 µg of nuclear extracts of Hep3B cells engineered to express wild-type (WT), Y 640 or G 656 STAT3 mutants. Bars indicate the DNA-binding activity to consensus STAT3 oligonucleotides relative to activity measured in cells transfected with empty plasmid (EP). ns, not significant. *, ** and *** indicate differences between groups at P<0.05, P<0.01 and P<0.001, respectively (two-tailed t-test). Data are from triplicate transfections(± s.d.). STAT3 mutants Y 640 and E 166 were over-expressed in Hep3B cells.

	), G656_Y657insF (G 656 ) or G656_Y657insF/Y705F (G 656 /Y 705 ) STAT3 in unstimulated Hep3B cells. Data are from triplicate transfections (± s.d.) relative to empty plasmid (EP). ns, not significant. *, ** and *** indicate differences between groups at P<0.05, P<0.01 and P<0.001, respectively (two-tailed t-test). Figure 3. Dimerization of STAT3 IHCA mutants. a, Flag-and Myc-tagged constructs encoding either WT or mutant Y640 STAT3 were co-transfected (1:1) into Hep3B cells treated, or not, with IL-6 (100 ng/ml). STAT3 dimer formation was detected after immunoprecipitation using the anti-Flag antibody, followed by immunoblot (IB) analysis Y 640 ) or empty vector (EP) were co-transfected with gp130 siRNA (+) or control siRNA (-) into Hep3B cells. d. STAT3 mutants (E 166 and Y 640 ) or empty vector (EP) were co-transfected with JAK2 siRNA (+) or control siRNA (-) into Hep3B cells. e. STAT3 mutants (E 166 and Y 640 ) or empty vector (EP) were co-transfected with JAK1 siRNA (+) or control siRNA (-) into Hep3B cells. f. STAT3 mutants (E 166 and Y 640 ) or empty vector (EP) were transfected into Hep3B cells exposed to Src inhibitor 1 (SrcI-1) (10 µM, 9h). with Myc, Figure 4. 166 and ns, not significant.

Activation of STAT3 is shown by luciferase (Luc) activity (mean) determined from triplicate transfections (± s.d.) relative to pSIEM-luc alone (EP) with serum starvation. a, Hep3B cells were exposed to increasing concentrations of IL-6. b. STAT3 mutants (E 166 and Y 640 ) or empty vector (EP) were co-transfected with JAK1 siRNA (+) or control siRNA (-) into Hep3B cells exposed to IL-6 (100 ng/ml). c, STAT3 mutants (E *, ** and *** indicate differences between groups at P<0.05, P<0.01 and P<0.001, respectively (two-tailed t-test). Data are from triplicate transfections

(± s.d.)

. g

, Expression profiles of STAT3 mutated IHCA and gp130 mutated IHCA.

	Comparison was carried out for the 28 genes validated by quantitative RT-PCR that
	were significantly differentially expressed between IHCA and non-tumorous livers. 5
	STAT3 mutated IHCA (in grey) were compared to 11 gp130 mutated IHCA (in white).
	Results are expressed as mean ± s.d.; differences between groups are not significant
	(two-tailed Mann-Whitney test).
	Supplementary Figure S1.

Somatic DNA alterations identified on STAT3 gene and distribution of mutations according to the different isoforms. a-b,

  Selection of forwards and reverse sequences in three mutated IHCA (T) and their matching nontumor DNA (N) showing the somatic origin of all mutations identified (arrows). Amino acids and nucleotides are numbered according to STAT3 cDNA ORF. Sequences of cDNA show the expression of both normal and mutated STAT3 alleles in tumors. c, Schematic representation of STAT3 isoforms obtained by sequencing full-length cloned STAT3 cDNA from tumor #379 and according to the different protein domains (SH2: mutants E 166 or Y 640 were over-expressed in Hep3B and exposed to Src inhibitor 5 (SrcI-5) or to PP2, 10 µM during 9 hr. The graph plots the luciferase (Luc) activity relative to the transfected cells not treated with the inhibitor. All results are from triplicate transfections (± s.d.) relative to pSIEM-luc alone (EP) with serum starvation. ns, not significant. *, ** and *** indicate differences between groups at P<0.05, P<0.01 and P<0.001, respectively (two-tailed t-test). e, Vectors driving the expression of Y640F (Y 640 ) STAT3 mutant or empty vector (EP) were transfected into Hep3B cells and exposed to Src inhibitor 1 (SrcI-1) (10 µM, 9 hr) and IL-6 (100 ng/ml, during the last 3 hr of treatment). Expression of total and phosphorylated (pY705) STAT3 proteins were analyzed by western blot. f, Knockdown efficiency by JAK1 siRNA. Hep3B cells were co-transfected with Y 640 STAT3 mutant or empty vector (EP) and JAK1 siRNA (+) or control siRNA (-). Shown is the endogenous JAK1 mRNA expression determined from triplicate transfections (± s.d.) relative to EP with serum starvation. Remaining JAK1 expression in JAK1 siRNA transfected cells was less than 20% of control.

Table 1 . STAT3 somatic mutations identified in inflammatory hepatocellular tumours

 1

	Tumours	Nucleotide change	Amino acid change	Mutant ID
	CHC341T	T233>G	L78>R	L	78
	CHC379T	G1504>T A1972>T; G1974>T	D502>Y K658>Y	502 D K 658
	CHC574T	G496>C	E166>Q	E 166
	CHC966T	A1919>T	Y640>F	Y 640
	CHC1021T T1969_G1980dup	Y657_M660dup	Y 657
	CHC1351T C1968>T; 1969_1970insTTT	G656_Y657insF	G 656
	Codons and mutated nucleotides are numbered according to STAT3 cDNA ORF

Table S2 . List of primers used in PCR and sequencing

 S2

Acknowledgments:

We warmly thank Gaelle Cubel for her participation to this work and Hugues Gascan who kindly provided the pSIEM-Luc plasmid. We also thank Jean Saric, Christophe Laurent, Brigitte Le Bail, Anne Rullier, Antonio Sa Cunha for contributing to the tissue collection (CHU Bordeaux) and all the participant to GENTHEP network (Génétique des tumeurs hépatiques). This work was supported by the association pour la recherche Contre le Cancer (ARC, grant n°3194), Inserm (Réseaux de recherche clinique et réseaux de recherche en santé des populations), Collection Nationale des Carcinomes Hépatocellulaires, the Ligue Nationale Contre le Cancer ("Cartes d'identité des tumeurs" program) and the BioIntelligence collaborative program. C.P., M.A. and J-C.N. are supported by a fellowship from the MENRT, Inca and ARC, respectively. T.I. is supported by the National Institutes of Health grants GM071596, AI055894, and AI067949. The authors have no conflicting financial interests

Src-homology-2 domain; TAD: Transactivation domain). Mutations were harbored by all the different STAT3 isoforms, including or not splicing of nucleotides 2099-2101 leading to in-frame Ser701 deletion (S701del); and splicing of nucleotides 2145-2194 leading to a frame shift and a premature termination stop, eight codons downstream (p.Thr716PhefsX8) (STAT3β) [START_REF] Schaefer | Cooperative transcriptional activity of Jun and Stat3 beta, a short form of Stat3[END_REF]. The two mutations (D502Y/K658Y) found are carried by the same allele.

Supplementary Figure S2. a, Serine 727 phosphorylation of STAT3 mutants. Vectors driving the expression of STAT3 mutants, including L78R (L 78 ), E166Q (E 166 ), Y640F (Y 640 ), or wild-type (WT) STAT3 were transfected into Hep3B cells. Expression of total and phosphorylated (pS727) STAT3 proteins were analyzed by western blot and the bands were quantified with ChemiDoc (Bio-Rad) using the Quantity One software. Bars show the expression of pS727 STAT3 relative to total STAT3. Results are representative of three independent experiments. b, Several STAT3 mutations in IHCA tumors reside near the TAD and Tyr-705. The crystal structure (PDB entry 1bg1) reveals that several of residues mutated or duplicated in IHCA tumors (Tyr-640 and 656-660) reside near the TAD. The DNA, one entire polypeptide chain, and both TADs are shown as a cartoon (bottom) and are colored as shown in the primary structure (top): the N-terminal domain that mediates cooperative DNA binding, grey, residues 1-130 and not part of the crystal structure; the coiled-coil domain, red, residues 130-320; the DNA binding domain, orange, residues 321-465; a linker region, yellow, residues 466-585; the SH2 domain, green, residues 586-689; and the transcriptional activation domain (TAD), blue, residues 689-722. The crystallographically related residues 136-688 are shown as a surface representation; carbon atoms are in white, nitrogen in blue, oxygen in red, and sulfur in yellow. Residues mutated or duplicated in inflammatory tumors (166,502,640,(656)(657)(658)(659)(660) are shown as spheres (carbon in white, nitrogen in blue, and oxygen in red) and some are labeled. The phosphorylated tyrosine (Tyr-705) is also shown as a sphere in both two-fold related molecules with the carbon drawn in black. For clarity, residues labeled on the molecule shown in surface representation are identified by their residue number. c, Effect of IL-6 on STAT3 mutant activity. Hep3B transfected with EP, WT or STAT3 mutants (L 78 , G 656 or Y 657 ) with or without IL-6 exposure (100 ng/ml). Graph plots the luciferase activity (Luc) relative to cells transfected with pSIEM-Luc alone (EP) ± s.d. (transfections in triplicate). ns, not significant. *, ** and *** indicate differences between groups at P<0.05, P<0.01 and P<0.001, respectively (two-tailed t-test). S3. Effect of inhibitors on STAT3 IHCA mutants activation. a, Hep3B cells transfected with STAT3 mutants (E 166 and Y 640 ) or empty vector (EP) treated with 100 ng/ml of IL-6 were exposed 3 hr to increasing concentrations of AG490 or 18 hr to increasing concentrations of Ruxolitinib. The graph plots the luciferase (Luc) activity relative to the transfected cells not treated with the inhibitor. b, Y 640 STAT3 mutant was over-expressed in Hep3B and exposed to increasing concentration of STAT3 inhibitor Curcumin during 24 hr. The graph plots the luciferase (Luc) activity relative to the transfected cells not treated with the inhibitor. c, STAT3 mutants E 166 or Y 640 were over-expressed in Hep3B and exposed to Src inhibitor 1 (SrcI-1) (10 µM, 9 hr) and IL-6 (100 ng/ml, during the last 3 hr of treatment). The graph plots the luciferase (Luc) activity relative to the transfected cells not treated with the inhibitor. d, STAT3

Supplementary Figure

Supplementary Table S1. TaqMan® pre-designed gene expression assays used for quantitative RT-PCR analyses