Benoît Pier

Nigel Peake

Global nonlinear dynamics of thin aerofoil wakes

In the present investigation of thin aerofoil wakes we compare the global nonlinear dynamics, obtained by direct numerical simulations, to the associated local instability features, derived from linear stability analyses. A given configuration depends on two control parameters: the Reynolds number Re and the adverse pressure gradient m (with m < 0) prevailing at the aerofoil trailing edge. Global instability is found to occur for large enough Re and |m|; the naturally selected frequency is determined by the local absolute frequency prevailing at the trailing edge.

Introduction

The global dynamics of spatially developing shear flows is known to closely depend on its local stability features. Previous linear stability analyses [START_REF] Woodley | Global linear stability analysis of thin aerofoil wakes[END_REF][START_REF] Taylor | A note on the absolute instability of wakes[END_REF] have established that thin aerofoil wakes display local absolute instability near the trailing edge. In the present investigation we revisit this flow and examine its fully nonlinear régime within the framework of nonlinear global mode theory [START_REF] Pier | Bifurcation to fully nonlinear synchronized structures in slowly varying media[END_REF][START_REF] Chomaz | Fully nonlinear dynamics of parallel wakes[END_REF].

With x and y denoting streamwise and cross-stream coordinates respectively, we consider the semi-infinite flow domain (x > 0, -∞ < y < +∞) corresponding to the region downstream of the aerofoil trailing edge, where a symmetric velocity profile is imposed, represented by the double Falkner-Skan boundary layer solutions with negative pressure gradient m. The pa- 2 Global nonlinear dynamics obtained by direct numerical simulations

Global flow dynamics are obtained by direct numerical simulation of the governing equations, implementing a method similar to that used in [START_REF] Pier | Local and global instabilities in the wake of a sphere[END_REF]. Timeintegration is performed via a fractional step method of second-order accuracy in time. Spatial discretization combines fifth-order finite differences in the x-direction and Chebyshev collocation points in the y-direction. The streamwise grid is uniformly stretched in the downstream direction and the collocation points are mapped onto the entire cross-stream domain via an algebraic transformation. Non-reflecting boundary conditions [START_REF] Jin | A nonreflecting outlet boundary condition for incompressible unsteady Navier-Stokes calculations[END_REF] are used at the downstream boundary of the computational domain.

At low values of Re the wake is globally stable and the system converges towards a steady y-symmetric solution of the Navier-Stokes equations. Above a critical Reynolds number, which depends on the value of the parameter m, global instability occurs, and time-dependent fluctuations develop in the wake. After a transient régime, a fully developed self-sustained downstreampropagating vortex-street is observed. Figure 1 shows snapshots of streamwise and cross-stream velocity fields of such a saturated régime obtained with Re = 5000 and m = -0.085. The naturally selected vortex-shedding frequencies are obtained by recording time-series at different locations and computing the associated frequency spectra. These spectra display a wide range of harmonics but obtaining them at different spatial locations demonstrates that the entire flow is tuned to a single fundamental frequency. The dependence of the selected global frequency on both control parameters is given in figure 2. From these curves it is concluded that, for given m, the global frequency is nearly independent of Re, except near onset. However, the global frequency significantly depends on m: the vortex shedding slows down when m is decreased and separation of the inlet velocity profile is approached.

Basic flow and local absolute frequencies

This section investigates the unperturbed basic flow and its local stability features, covering the same parameter ranges as the previous section. The basic flow is, by definition, a steady solution of the Navier-Stokes equations. Basic flow fields may be computed by imposing a symmetry condition along the x-axis and considering only the domain y ≥ 0. At very large Reynolds numbers, however, this symmetry condition is not enough to stabilize the basic flow, and the selective frequency damping method [START_REF] Åkervik | Steady solutions of the Navier-Stokes equations by selective frequency damping[END_REF] has therefore also been implemented. Thus, the basic flows are readily obtained by numerical integration of the governing equations, using the same code as in the previous section with only minor modifications.

For given values of Re and m, local stability characteristics of the associated basic flow are then derived at a given streamwise station by freezing the x-coordinate and studying the equivalent parallel shear flow of corresponding velocity profile. Such an approach can be justified by rigorous asymptotic analyses based on the assumption of slow streamwise development of the basic flow. In the case of thin aerofoil wakes, this is certainly a reasonable assumption, except very close to the trailing edge; and even in the very near wake region, the assumption of weak spatial inhomogeneity only breaks down for small values of both Re and |m|.

Local linear instability waves developing on these velocity profiles are then governed by the Orr-Sommerfeld equation, which yields the local linear dispersion relation ω = Ω(k; x; Re, m)

between the complex frequency ω and complex wavenumber k at streamwise station x for the basic flow characterized by Re and m.

In the context of open shear flows, a crucial feature is the complex absolute frequency defined for parallel flows as the frequency observed at a fixed spatial location in the long-time linear response to an initial impulse. The local absolute frequency ω 0 (x; Re, m) is derived in classical fashion by applying a pinch-point criterion [START_REF] Briggs | Electron-stream interaction with plasmas[END_REF][START_REF] Bers | Space-time evolution of plasma instabilities -absolute and convective[END_REF], equivalent to a vanishing group velocity condition

ω 0 (x; Re, m) = Ω(k 0 ; x; Re, m) with ∂Ω ∂k (k 0 ; x; Re, m) = 0. ( 2
)
Figure 3 illustrates the streamwise evolution of the local absolute growth rate ω 0,i and real absolute frequency ω 0,r . It is seen that the largest absolute growth rate ω 0,i occurs near the trailing edge at x = 0. The growth rate increases with (adverse) pressure gradient |m|, and the size of the absolutely unstable domain (where ω 0,i > 0) increases with Reynolds number. All basic wake flows of this investigation are at least marginally absolutely unstable at x = 0. However, for moderate values of m, the extent of the absolutely unstable domain is rather limited. It is only for stronger adverse pressure gradients, close to detachment, that relatively strong absolute instability is obtained.

Discussion

The spatial structure of the global mode is the following. The upstream region, close to the trailing edge, is covered by small-amplitude perturbations that propagate downstream. While propagating downstream, their amplitude grows exponentially until nonlinear saturation occurs. Further downstream, a finite-amplitude wavetrain prevails, the amplitude and wavenumber of which slowly vary with downstream distance. In the far wake, the wave amplitude slowly decays while small-scale features may develop for high-Reynoldsnumber configurations.

A comparison of the vortex shedding frequency measured by DNS and the real part of the absolute frequency ω 0,r (x = 0), prevailing at the trailing edge, is shown in figure 4. This plot shows that the naturally selected frequency is in excellent agreement with the absolute frequency prevailing at the trailingedge, thereby demonstrating that the global dynamics of thin aerofoil wakes is closely linked to the local absolute stability characteristics of the associated base flow. 0.16 0.17 A nonlinear saturation station x sat may be defined as the streamwise position where the envelope of the global mode reaches 90% of its maximum. Figure 5 shows how the thus computed x sat depends on Re for selected values of m. The saturation length appears to converge towards a constant value for Re → ∞ while it would diverge at the critical Reynolds number for onset of global instability. A somewhat intriguing result is also derived from this plot: the nonlinear saturation may occur downstream of the absolutely unstable domain. It has been verified that this observation does not depend on the length of the computational domain. Nevertheless, we are currently implementing a completely different numerical method to verify the robustness of this counterintuitive result.

We are also studying the controllability of this flow by applying smallamplitude harmonic forcing near the trailing edge. Preliminary results in- dicate that the open-loop control technique developed in a different context [START_REF] Pier | Open-loop control of absolutely unstable domains[END_REF][START_REF] Pier | Primary crossflow vortices, secondary absolute instabilities and their control in the rotating-disk boundary layer[END_REF] is capable of modifying the present flow.

Computing resources from Fédération lyonnaise de calcul hautes performances (FLCHP) and Institut national de physique nucléaire et de physique des particules (IN2P3) are gratefully acknowledged.

  Benoît PIER Laboratoire de mécanique des fluides et d'acoustique (CNRS -Université de Lyon) École centrale de Lyon, 36 avenue Guy-de-Collongue, 69134 Écully cedex, France. Nigel PEAKE Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Centre for Mathematical Sciences, Wilberforce Road, Cambridge, CB3 OWA, UK. 2 Benoît PIER and Nigel PEAKE rameter m is varied from m = 0 (double-Blasius wake) to m = -0.09 (near flow separation); Reynolds numbers from Re = 50 up to Re = 5000 are considered. The entire study is carried out for two-dimensional incompressible flows governed by the Navier-Stokes equations.

Fig. 1

 1 Fig. 1 Snapshot of fully developed nonlinear régime at Re = 5000 and m = -0.085. Equispaced isocontours of (a) streamwise and (b) cross-stream velocity fields.

Fig. 2

 2 Fig. 2 Reynolds number dependence of the naturally selected global frequencies for different values of the parameter m.

Fig. 3

 3 Fig. 3 Streamwise variation of (a) imaginary and (b) real parts of local absolute frequencies obtained with m = -0.085 and Re = 100, 200, 300, 400, 500, 700, 1000, 1500, . . . , 5000.

Fig. 4

 4 Fig. 4 Comparison of measured vortex shedding frequency ωg and trailing-edge absolute frequency ω 0,r (x = 0).

Fig. 5

 5 Fig. 5 Variation of nonlinear saturation station with control parameters Re and m.


Marc Ciligot-Travain

Sado Traoré

On a robustness property in single-facility location in continuous space

Keywords: Single-Facility Location, Robustness, Fermat-Weber problem, 1-center problem Mathematics Subject Classification (2000) MSC 90B85

We consider a single-facility location problem in continuous space, here the problem of minimizing a sum or the maximum of the possibly weighted distances from a facility to a set of points of demand. The main result of this paper shows that every solution (optimal facility location) of this problem has an interesting robustness property. Any optimal facility location is the most robust in the following sense: given a suitable highest admissible cost, it allows the greatest perturbation of the locations of the demand without exceeding this highest admissible chosen cost.

Introduction

Our work is concerned with the classical single-facility location problem on continuous space, with discrete demand and mixed norms. This problem comes from a particular approach of the more general following decision problem: one has to choose the location of a facility serving the demand of a discrete set of points. The problem is parameterized by the data (description of the demand) and this decision has a cost. For generalities and more about facility location problem, one can consult [START_REF] Drezner | Facility location[END_REF] and Drezner and Hamacher (2002) and references therein.

Perhaps the most immediate concern about this problem was/is to minimize the cost which leads to an enormous literature devoted to various aspects of this question (see above for references). We will call it the initial problem.

Particular instances of this problem are the Fermat-Weber problem (1-median problem) and the 1-center problem (see below). Recall also that the mathematical problem behind appears in Descriptive Data Summarization where the optimal facility location can be interpret as a measure of the central tendency of our data (the demand), it has been used, for example, in Cluster Analysis.

Minimizing the cost is an important purpose but other considerations may lead to different concerns. In this paper, we propose to take into account uncertainty in the location of the demand. In facility location problem, points of demand can be derived from aggregation, as gravity centers of some districts. Having in mind that facility location decisions are very long term decisions, it seems reasonnable to be concerned with demographic changes which may cause displacement of these gravity centers. Secondly, in descriptive data summarization, some noise in the data can be assimilated as changing in the position.

All this considerations can make interesting looking for a facility location that tolerate the greatest perturbations in the locations of the demand in the sense that the cost doesn't exceed a given highest admissible cost. This highest admissible cost can be chosen with the help of the value of the initial single-facility location problem evocated above. This is our robust counterpart of the initial problem.

Our main result is that, a bit surprisingly, in the single-facility location problem considered, both the initial and the robust counterpart problems have the same solution. In other words, the solutions of our initial single facility location problem exhibit a kind robustness property.

We situate our work in the continuation of [START_REF] Carrizosa | Robust facility location[END_REF] where Carrizosa and S. Nickel introduced in Locational Analysis this very natural and interesting idea of robustness, appearing yet in robust control and which has been also developed in infogap theory (see below for references). In their work, they consider a similar problem but paying attention to uncertainty on the demand.

Let us add one last motivation: as said above, the classical, old and intensively studied Fermat-Weber problem is one instance of the problem considered in our work so, our main result gives a new (essentially qualitative) property of this problem.

The plan of the paper is as follows: in the next section, the single-facility location problem in a continuous space with mixed norms is described and we give some elements about robustness which leads to consider the maximization of the radius of robustness. In the section 2, we give the main Theorem which asserts that maximizing the radius of robustness in the single-facility location problem with mixed norms is equivalent to the initial problem which means that the solutions of this problem present a certain robustness property.

Preliminaries

Consider n ∈ N ⋆ points of demand, numeroted from 1 to n. Let x = (x 1 , ..., x n ) represents the location of the demand (where x i ∈ R d is the location of the i-th point of demand), y ∈ R d be the location of the facility. Satisfaction of the demand by the facility has a cost that takes the general form

C(x, y) = N(γ 1 (x 1 -y), ..., γ n (x n -y)) (1)
where N is an absolute norm on R n (one can consult [START_REF] Bauer | Absolute and monotonic norms[END_REF] for more details about absolute norms.), i.e. a norm which satisfies:

∀(v 1 , ..., v n ) ∈ R n , N(v 1 , ..., v n ) = N(|v 1 |, ..., |v n |),
and, for i = 1, ..., n, γ i is a norm on R d . In all the sequel, if γ is a norm on a vectorial space E, the unit ball associated with γ is

B γ = {x ∈ E | γ(x) ≤ 1} and the distance associated with γ on E, is d γ : (x, y) → γ(x -y).
This setting is convenient to include some classical cases we are going to describe briefly now.

When N = N 1 : (ν 1 , ..., ν n ) → ∑ i=1,...,n |ν i |, the ℓ 1 -norm on R n , and, for i = 1, ..., n, γ i = ω i |.| i , where ω i > 0 is the demand at the ith-point and the distance from the ith point is measured by the distance associated with a norm |.| i or will be approximated by the distance associated with a norm |.| i (see [START_REF] Brimberg | Estimating distances[END_REF] and references therein), the cost

C(x, y) = ∑ i=1,...,n γ i (x i -y) = ∑ i=1,...,n ω i |x i -y| i
can be seen as a total transportation cost. It is a cost associated with an idea of efficacy/efficiency.

When

N = N ∞ : (ν 1 , ..., ν n ) → max i=1,...,n |ν i |, the ℓ ∞ -norm on R n , the cost C(x, y) = max i=1,...,n γ i (x i -y)
is the maximal distance of a point of demand from the facility. It is a cost associated with an idea of equity, In all the sequel, the norms γ i , i = 1, ..., n, are fixed and the norm N in the cost (1) equals N 1 or N ∞ .

If we set

W not. = N • γ 1 × • • • × γ n : (v 1 , ..., v n ) → N(γ 1 (v 1 ), ..., γ n (v n )),
using the fact that N is absolute, W is a norm on (R d ) n and the cost given by ( 1) is, for

x = (x 1 , ..., x n ) ∈ (R d ) n , y ∈ R d and setting ŷ = (y, ..., y) ∈ (R d ) n , C(x, y) = W (x -ŷ).
As said in the introduction, our initial problem is

Minimize C(x, y) = W (x -ŷ), y ∈ R d . (P x )
As noted in [START_REF] Carrizosa | Generalized goal programming: polynomial methods and applications[END_REF], it is interesting to note that solving (P x ) is equivalent to solve the following problem

Minimize W (x -y), y ∈ ∆ , (P ′ x )
where

∆ = {(z, ..., z) | z ∈ R d } ⊂ (R d ) n .
So the problem (P x ) is equivalent to a projection problem of the data x on a subspace ∆ of (R d ) n using the distance associated with the norm

W = N • γ 1 × • • • × γ n .
One can refer to [START_REF] Plastria | Continuous Location Problems[END_REF], [START_REF] Drezner | The Weber problem[END_REF], [START_REF] Hansen | Location theory, dominance, and convexity: Some further results[END_REF], [START_REF] Durier | A general framework for the one center location problem[END_REF], [START_REF] Durier | The general one center location problem[END_REF] and references therein for more details about the singlefacility location problem considered here.

Now suppose that we want to take into account some (deterministic) uncertainty about the locations of the demand described by x. More precisely, suppose that, once the facility location y chosen on the basis of locations of the demand x, there may be some variations on the locations of the demand (from the reference x). A priori, just choosing y minimizing the criterion y → C( x, y) seems unsatisfying.

Without going into too much detail, to better understand the approach considered in [START_REF] Carrizosa | Robust facility location[END_REF], we find instructive to develop some general considerations around robustness.

The matter discussed until the end of this section being largely independent of our facility location decision problem and because it implies only very light modifications, we adopt an abstract framework in order to both show the real scope of the concepts and ideas introduced, and better situate the different approaches of robustness. The decision problem is the following:

(D x ) Given data x in a set X, choose an element y in a set Y knowing that it generates a cost C(x, y) ∈ R + .
Of course, we will seek a decision satisfying some properties that we consider desirable. Probably one of the most classical procedure to solve (D x ) and choose y consists in minimizing the cost:

Minimize C(x, y), y ∈ Y.

(P x )
But in the decision problem (D x ), you can choose just to control the cost. Given a maximal admissible cost α ∈ R + , in presence of uncertainty, supposing that (X, d) is a metric space and that the true data x ′ belongs to the open ball B(x, r) = {z ∈ X | d(x, z) < r} for some r > 0, this naturally leads to the robust parameterized feasibility problem

Find y ∈ Y such that B(x, r) ⊂ [C(., y) ≤ α] (2)
where [C(., y) ≤ α] = {x ′ ∈ X |C(x ′ , y) ≤ α}. Now you can introduce optimization in the problem (2) by two very natural ways.

A first approach consists in optimizing the maximal cost. This leads to the following problem, fixing x ∈ X, r > 0:

Minimize α, (y, α) ∈ Y × R + satisfying B(x, r) ⊂ [C(., y) ≤ α] (3)
It is easily seen that ( ȳ, ᾱ) is a solution of (3) if and only if ȳ is solution of Minimize sup

x ′ ∈ B(x,r) C(x ′ , y), y ∈ Y, ( 4
) and ᾱ = sup x ′ ∈ B(x,r) C(x ′ , ȳ).
This is probably the most classical procedure to solve (D x ) robustly and choose y in this context consists in minimizing the cost in the worst case. Allowing a more general uncertainty set then an open ball leads to a problem sometimes called the robust counterpart of the initial problem (P x ).

This method, often named robust optimization, has been intensely studied for many continuous optimization problems, see [START_REF] Ben-Tal | Robust optimization[END_REF], [START_REF] Kouvelis | Robust discrete optimization and its applications, volume 14 of Nonconvex Optimization and its Applications[END_REF] and references therein.

Another way to introduce optimization in the problem (2) is, for data x ∈ X, to fix a maximal cost α ∈ R + and to maximize the radius of the perturbation.

This leads to the following optimization problem:

Maximize r, (y, r)

∈ Y × R + satisfying B(x, r) ⊂ [C(., y) ≤ α] ( 5
)
and it is easy to check that ( ȳ, r) is a solution of (5) if and only if ȳ is a solution of Maximize ρ(y;

x, α) = d(x, [C(., y) > α]), y ∈ Y, (R x )
and r = ρ( ȳ; x, α). This is the approach considered in [START_REF] Carrizosa | Robust facility location[END_REF]. As far as we know, it is a particular case of a general method known on one hand as infogap theory and developped essentially by Y. Ben-Haim since the end of the 90's (see [START_REF] Ben-Haim | Information-gap decision theory[END_REF] and references therein), and on the other hand, some similar considerations have been made in robust control, beginning with the problem of maximizing the radius of stability of linear systems (see Hinrichsen et al. (1990), Hinrichsen and Pritchard (1990)). So, this way to consider robustness leads to the introduction of what we call the radius of robustness ρ(y; x, α) = d(x, [C(., y) > α]) called itself robustness in [START_REF] Carrizosa | Robust facility location[END_REF], robustness function or stability radius in previous works around this method.

In the sequel, the problem (R x ) will be our robust counterpart of the initial problem (P x ).

In [START_REF] Carrizosa | Robust facility location[END_REF], they consider uncertainty on the weights ω where, s + = max(0, s) for s ∈ R, for i = 1, ..., n, d i (y) is the distance from y to the ith point of demand, TC(ω, y) = ∑ i=1,...,n ω i d i (y) and .

= (ω 1 , ..., ω n ) ∈ R n + (Y = S ⊂ R 2 , X = R n + ).
• is the dual norm of an absolute norm . chosen to measure the distances between the weights on R n . Moreover, they consider the problem (R x ) for some classical norms on R 2 and S convex.

To conclude this section, one can consult the surveys Snyder ( 2004), [START_REF] Drezner | Application of decision analysis techniques to the weber facility location problem[END_REF] for some considerations about robustness in facility location problems, and [START_REF] Beyer | Robust optimization -a comprehensive survey[END_REF], [START_REF] Roy | Robustness in operational research and decision aiding: A multi-faceted issue[END_REF] for more general considerations about robustness in Optimization, Operation Research and Aid Decision.

Maximizing the radius of robustness for the single-facility location problem in a continuous space

As said before, the purpose of this paper is to show that for the facility location problem considered, optimizing the minisum or minimax criterion is equivalent to optimizing the radius of robustness.

So let us clarify our robust counterpart of the problem (P x ). On one hand, the cost in the problem (P x ) is based on the choice of a certain norm W on (R d ) n . As we have seen, for our problem, the norms of most interest are

W = N 1 • γ 1 × • • • × γ n or W = N ∞ • γ 1 × • • • × γ n .
On the other hand, in order to consider the radius of robustness, we have to choose a distance on the space of data (R d ) n . Due to the interpretation of the problem, it seems natural to consider the distance d W associated with the norm W

= N ∞ • γ 1 × • • • × γ n , i.e. for x = (x 1 , ..., x n ), x ′ = (x ′ 1 , ..., x ′ n ) ∈ (R d ) n , d W (x, x ′ ) = N ∞ (γ 1 (x 1 -x ′ 1 ), ..., γ n (x n -x ′ n )).
Nevertheless, as in [START_REF] Carrizosa | Robust facility location[END_REF], one can use

W = Ñ • γ 1 × • • • × γ n where Ñ is any absolute norm on R n . If y ∈ R d , α > 0, [C(., y) > α] = [W (. -ŷ) > α] = B W ( ŷ, α) c ,
where B W ( ŷ, α) is the closed ball centered at ŷ of radius α associated with W and

B W ( ŷ, α) c = {z ∈ (R d ) n | z / ∈ B W ( ŷ, α)} is the complementary set of B W ( ŷ, α). So, for x = (x 1 , ..., x n ) ∈ (R d ) n , the radius of robustness is ρ(y; x, α) = d W (x, B W ( ŷ, α) c ) = d W (x -ŷ, αB c W ) = αd W x - ŷ α , B c W . (6)
It follows that our robust counterpart of the problem (P x ) is:

Maximize d W (x -ŷ, αB c W ), y ∈ R d . (R x )
To begin with, we consider the trivial case

W = W . Since for z ∈ (R d ) n , d W (z, B c W ) = (1 -W (z)) + , so for y ∈ R d , x ∈ (R d ) n : d W (x -ŷ, αB c W ) = (α -W (x -ŷ)) + . (7)
This leads easily to the following result concerning the minimax criterion, where

v(P x ) = inf y∈R d W (x -ŷ)
is the value of the initial problem (P x ):

Theorem 1 Let x = (x 1 , ..., x n ) ∈ (R d ) n . For any α > v(P x ), a point y ∈ R d is a solution of the 1-center problem (P x ) (where N = N ∞ ) if and only if it is a solution of (R x ) (where N = N ∞ too)
and measuring the distance between the data using

N ∞ • γ 1 × • • • × γ n . Moreover, the radius of robustness is ρ(y; x, α) = α -N ∞ (γ 1 (x 1 -y), ..., γ n (x n -y)) = α -v(P x ).
Proof One always has sup

y∈R d ρ(y; x, α) = sup y∈R d (α -W (x -ŷ)) + = (α -inf y∈R d W (x -ŷ)) + = (α -v(P x )) + = α -v(P x ). (8) Now if y is a 1-center then for all y ′ ∈ R d , W (x -ŷ′ ) ≥ W (x -ŷ) so one obtains ρ(y; x, α) = (α -W (x -ŷ)) + ≥ (α -W (x -ŷ′ )) + = ρ(y ′ ; x, α).
If y maximizes the radius of robustness then, due to ( 8),

(α -W (x -ŷ)) + = α -v(P x ) > 0, so α -W (x -ŷ) ≥ 0 and one obtains, for all y ′ ∈ R d : α -W (x -ŷ) = α -v(P x ) ≥ α -W (x -ŷ′ )
which implies that y is a 1-center.

Note that this theorem still holds if we replace N ∞ by any absolute norm on R n , in particular N 1 .

Let us now consider the case when

W = N 1 • γ 1 × • • • × γ n and W = N ∞ • γ 1 × • • • × γ n .
Due to the special relation between N 1 and N ∞ and the polyhedrality of N ∞ , we shall establish a formula similar to (7) leading to a theorem analogous to the Theorem 1. This is the main result of our paper. Its proof is based on several technical lemmas. We begin by describing the setting and some notations. One can consult Hiriart-Urruty and [START_REF] Hiriart-Urruty | Fundamentals of convex analysis[END_REF] for the well known results of Convex Analysis we used in the sequel.

Let E be a finite dimensional real vector space, provided with a duality ., . . We provide E = R n with its usual duality ((x 1 , ..., x n ), (y 1 , ..., y n )) -→ x 1 y 1 + • • • + x n y n and E = (R d ) n with its usual duality too, ((x 1 , ..., x n ), (y 1 , ...,

y n )) -→ x 1 , y 1 + • • • + x n , y n .
If C is a convex set of E, we denote by ext(C) the set of extreme points of C. Now if W is a norm on E, we denote by W • , the dual norm of W , defined by W • (x) = sup x ′ ∈B W x, x ′ , for x ∈ E.

We will need the following direct consequence of a immediate generalization of the Ascoli formula, suppose that E is provided with a norm W (see e.g., in the case of a norm, [START_REF] Singer | Duality for nonconvex approximation and optimization[END_REF], Corollary 1.4 and references therein, or, in the case of a gauge, [START_REF] Plastria | Gauge distances and median hyperplanes[END_REF] Theorem 1.1). If z, x ∈ E, z = 0, then the distance from x to the half space [ .,

z > α] = {z ′ ∈ E | z ′ , z > α} is given by d W (x, [ ., z > α]) = (α -z, x ) + W • (z) , (9)
and using (9), if W , W are two norms on E, the for all z ∈ E,

d W (z, B c W ) = inf z ′ ∈ext(B W • ) (1 -z ′ , z ) + W • (z ′ ) . ( 10
)
Lemma 1 [START_REF] Carrizosa | Generalized goal programming: polynomial methods and applications[END_REF])

Let W = N • γ 1 × • • • × γ n where N is an absolute norm on R n . Then W • = N • • γ • 1 × • • • × γ • n .
Lemma 2 (Dowling and Saejung (2008

)) Let W = N •γ 1 ו • •×γ n where N is an ab- solute norm on R n , if x = (x 1 , ..., x n ) is an extreme point of B W then (γ 1 (x 1 ), ..., γ n (x n ))
is an extreme point of B N .

Lemma 3 Let W = N 1 • γ 1 × • • • × γ n , W = Ñ • γ 1 × • • • × γ n , where Ñ is an absolute norm on R n . Then, for all z ∈ (R d ) n , d W (z, B c W ) = (1 -W (z)) + Ñ• (1, ..., 1
) .

Proof We know from (10) that:

d W (z, B c W ) = inf z ′ ∈ext(B W • ) (1 -z ′ , z ) + W • (z ′ ) . Due to lemma 1, W • = (N 1 • γ 1 × • • • × γ n ) • = N ∞ • γ • 1 × • • • × γ • n . If z ′ = (z ′ 1 , ..., z ′ n ) ∈ ext(B W • ) then, due to lemma 2, (γ • 1 (z 1 ), ..., γ • n (z ′ n )) ∈ ext(B N ∞ ). But ext(B N ∞ ) = (s 1 , ...., s n ) , s 1 , ..., s n = ±1 , so γ • 1 (z ′ 1 ) = ... = γ • n (z n ) = 1. Now, using lemma 1 again, W • = Ñ• • γ • 1 × • • • × γ • n so for all z ′ ∈ ext(B W • ), knowing that γ • 1 (z ′ 1 ) = ... = γ • n (z ′ n ) = 1, one obtains W • (z ′ ) = Ñ• (1, ..., 1
) and we conclude, using that inf

z ′ ∈ext(B W • ) (1 -z ′ , z ) + = 1 -sup z ′ ∈B W • z ′ , z + = (1 -W •• (z)) + . ⊓ ⊔ Theorem 2 Let x = (x 1 , ..., x n ) ∈ (R d ) n .
For any α > v(P x ), a point y ∈ R d is a solution of the generalized Fermat-Weber problem (P x ) (where N = N 1 ) if and only if it is a solution of (R x ) (with N = N 1 too) and measuring the distance between the data using Ñ

• γ 1 × • • • × γ n , where Ñ is an absolute norm on R d . Moreover, the radius of robustness is ρ(y; x, α) = α -N 1 (γ 1 (x 1 -y), ..., γ n (x n -y)) Ñ• (1, ..., 1) = α -v(P x ) Ñ• (1, ..., 1) .
Proof The Theorem is a direct consequence of the Lemma 3. Using this Lemma 3 and ( 6), one has for every z

∈ R d , α > 0, x = (x 1 , ..., x n ) ∈ (R d ) n ,
ρ(z; x, α) = (α -N 1 (γ 1 (x 1z), ..., γ n (x nz))) + Ñ• (1, ..., 1) .

One concludes as in Theorem 1.

⊓ ⊔

The following result may be proved in much the same way as Theorem 2, supposing that Ñ is symmetric, which means that for all (s 1 , ..., s n ) ∈ R n and permutation σ of {1, ..., n}, Ñ(s 1 , ..., s n ) = Ñ(s σ (1) , ..., s σ (n) ), using the facts that Ñ• is also symmetric and ext(B N 1 ) = (s 1 , ...., s n ) , ∃i ∈ {1, ..., n}, s i = ±1 and s j = 0, ∀ j = i .

For example, the ℓ p -norm on R n , p ∈ [1, +∞], are symmetric.

Theorem 3 Let x = (x 1 , ..., x n ) ∈ (R d ) n . For any α > v(P x ), a point y ∈ R d is a solution of the 1-center problem (P x ) (where N = N ∞ ) if and only if it is a solution of (R x ) (with N = N ∞ too) and measuring the distance between the data using Ñ • γ 1 × • • • × γ n , where Ñ is an absolute norm on R d . Moreover, the radius of robustness is ρ(y; x, α) = α -N ∞ (γ 1 (x 1y), ..., γ n (x n -y))

Ñ• (1, 0, ..., 0) = αv(P x ) Ñ• (1, 0, ..., 0) .

Conclusion

In this work, we have established a robustness property of solutions of two classical single-facility location problems with mixed norms on continuous space. We have proved that, in a sense, the solutions of the classical Weber problem (Minisum problem) and Minimax problem (1-center problem) are the most robust relatively to perturbations of the locations of the demand. Moreover, we give some quantitative information about this robustness by an explicit calculus of the radius of robustness of the classical solutions, calculation which requires only the easy computation of the value of the convex initial problem.

Acknowledgements

The authors gratefully ackowledge the many helpful suggestions of the referees.

This work was supported by the French ANR project ROLSES (Robust and Optimal Locations for Sustainable Environment and Systems)


PhD Margarita Rivera
email: margarita.rivera_sanchez@kcl.ac.uk

PhD Sarah Cohen-Woods

PhD Karen Kapur

PhD Gerome Breen

PhD Mandy Y Ng

PhD Amy W Butler

PhD Nick Craddock

PhD Michael Gill

MD Ania Korszun

PhD Wolfgang Maier

MD Ole Mors

PhD Michael J Owen

PhD Martin Preisig

MD Sven Bergmann

PhD Federica Tozzi

MD John Rice

PhD Marcella Rietschel

MD James Rucker

MD Alexandra Schosser

MD PhD Katherine J Aitchison

PhD Rudolf Uher

PhD Ian W Craig

PhD Cathryn M Lewis

PhD Anne E Farmer

MD Peter Mcguffin

DEPRESSIVE DISORDER MODERATES THE EFFECT OF THE FTO GENE ON BODY MASS INDEX

Keywords: Depression, FTO gene, obesity, BMI, Body Mass Index

There is evidence that obesity related disorders are increased among people with depression. Variation in the FTO (fat mass and obesity associated) gene has been shown to contribute to common forms of human obesity. This study aimed to investigate the genetic influence of polymorphisms in FTO in relation to Body Mass Index (BMI) in two independent samples of major depressive disorder (MDD) cases and controls.

We analysed 88 polymorphisms in the FTO gene in a clinically ascertained sample of 2442 MDD cases and 809 controls (Radiant Study). Eight of the top 10 SNPs showing the strongest associations with BMI were followed-up in a population-based cohort (PsyCoLaus Study) consisting of 1292 depression cases and 1690 controls.

Linear regression analyses of the FTO variants and BMI yielded 10 SNPs significantly associated with increased BMI in the depressive group but not the control group in the Radiant sample. The same pattern was found in the PsyCoLaus sample. We found a significant interaction between genotype and affected status in relation to BMI for 7 SNPs in Radiant (p<0.0057), with PsyCoLaus giving supportive evidence for 5 SNPs (pvalues between 0.03-0.06) which increased in significance when the data were combined in a meta-analysis. This is the first study investigating FTO and BMI within the context of MDD, and the results indicate that having a history of depression moderates the effect of FTO on BMI. This finding suggests that FTO is involved in the mechanism underlying the association between mood disorders and obesity.

Introduction

In addition to unipolar major depressive disorder (MDD) being a major public health problem in its own right [START_REF] Murray | Global mortality, disability, and the contribution of risk factors: Global Burden of Disease Study[END_REF] there is growing evidence suggesting that rates of physical disease, such as obesity and diabetes, are elevated amongst people with depression. [START_REF] Farmer | Medical disorders in people with recurrent depression[END_REF][START_REF] Moussavi | Depression, chronic diseases, and decrements in health: results from the World Health Surveys[END_REF] The study by Farmer et al. demonstrated a striking relationship between recurrent clinically sampled depression and increased BMI, type II diabetes, coronary heart disease and hypertension. Regression analyses showed that the association of depression with this cluster of disorders (sometimes referred to a 'metabolic syndrome') was largely accounted for BMI. [START_REF] Farmer | Medical disorders in people with recurrent depression[END_REF] A recent World Health Survey (WHS) study involving 245404 participants from 60 countries in regions all over the world found that the number of participants with one or more chronic physical disease, and comorbid depression, ranges from 9.3 to 23%. [START_REF] Moussavi | Depression, chronic diseases, and decrements in health: results from the World Health Surveys[END_REF] Furthermore, recent studies support the hypothesis that there may be shared aetiological factors, including genetic factors, between recurrent unipolar depression, obesity and physical disorders. [START_REF] Farmer | Medical disorders in people with recurrent depression[END_REF][START_REF] Patten | Long-term medical conditions and major depression: strength of association for specific conditions in the general population[END_REF] The fat mass and obesity associated gene, FTO, on chromosome 16q has been reported multiple times to contribute to common forms of human obesity. [START_REF] Loos | FTO: the first gene contributing to common forms of human obesity[END_REF] In 2007, two groups independently identified a common single nucleotide polymorphism (SNP) in FTO (rs9939609) that was associated with BMI and an increased risk for adult obesity. [START_REF] Dina | Variation in FTO contributes to childhood obesity and severe adult obesity[END_REF][START_REF] Frayling | A common variant in the FTO gene is associated with body mass index and predisposes to childhood and adult obesity[END_REF] In the same year, a third group performing a genome-wide association (GWA) study of BMI also found variants in FTO strongly associated with BMI and obesity-related traits. 8 Recently the FTO association with BMI has also been replicated in two large GWA studies for obesity. [START_REF] Thorleifsson | Genome-wide association yields new sequence variants at seven loci that associate with measures of obesity[END_REF][START_REF] Willer | Six new loci associated with body mass index highlight a neuronal influence on body weight regulation[END_REF] These two studies strongly support the association between FTO variants and obesity.

Animal studies reported that FTO is widely expressed in the brain, with high expression in hypothalamic nuclei regulating energy balance. [START_REF] Gerken | The obesity-associated FTO gene encodes a 2-oxoglutarate-dependent nucleic acid demethylase[END_REF] Human studies have also found that the FTO variant (rs9939609) risk allele is associated with an increased energy intake [START_REF] Cecil | An obesity-associated FTO gene variant and increased energy intake in children[END_REF][START_REF] Wardle | The FTO gene and measured food intake in children[END_REF] and diminished satiety, [START_REF] Wardle | 3643. acetylcholine receptor M2 (CHRM2) gene in recurrent major depressive disorder[END_REF] implicating FTO in the regulation of appetite.

This study explores the hypothesis that shared genetic aetiological factors exist between depression and obesity. Specifically, we aimed to investigate the influence of polymorphisms in FTO in relationship with BMI on a sample of depressed patients and psychiatrically healthy controls and follow-up the SNPs showing the strongest associations in an independent population-based cohort.

Materials and methods

Clinical Sample

Radiant Study

The depression case sample included 2442 individuals (740 men, 1702 women; mean age ± s.d.: 45.25±12.15) and was sourced from several studies: the Depression Case-Control (DeCC) study 15 , Depression Network (DeNT) study [START_REF] Farmer | The Depression Network (DeNT) Study: methodology and sociodemographic characteristics of the first 470 affected sibling pairs from a large multi-site linkage genetic study[END_REF][START_REF] Mcguffin | Whole genome linkage scan of recurrent depressive disorder from the depression network study[END_REF] and the Genome Based Therapeutic Drugs for Depression (GENDEP) study. [START_REF] Uher | Genetic predictors of response to antidepressants in the GENDEP project[END_REF] The DeCC sample comprises subjects with recurrent unipolar depression (minimum two episodes) of at least moderate severity as defined by DSM-IV or ICD-10 criteria, recruited from three UK sites (London, Cardiff and Birmingham). 15 Probands from the DeNT affected sibling pair linkage study consisting of cases with recurrent unipolar depression of at least moderate severity collected at seven European sites and one US site. [START_REF] Farmer | The Depression Network (DeNT) Study: methodology and sociodemographic characteristics of the first 470 affected sibling pairs from a large multi-site linkage genetic study[END_REF][START_REF] Mcguffin | Whole genome linkage scan of recurrent depressive disorder from the depression network study[END_REF] The GENDEP study includes individuals with an episode of depression of at least moderate severity recruited from nine European centres, [START_REF] Uher | Genetic predictors of response to antidepressants in the GENDEP project[END_REF] not necessarily recurrent. Either DSM-IV or ICD-10 diagnosis of depression were ascertained using the Schedules for Clinical Assessment in Neuropsychiatry (SCAN) interview in all three studies. [START_REF] Wing | Schedules for Clinical Assessment in Neuropsychiatry[END_REF] Individuals were excluded if they, or a first-degree relative, had a history of mania, hypomania, schizophrenia or bipolar disorder, or if the depression occurred in relation to alcohol, substance misuse or a medical illness or medication.

Eight hundred and nine control individuals from the UK (313 men, 496 women; mean age ± s.d.: 39.9±13.71) were screened for lifetime absence of any psychiatric disorder using a modified version of the Past History Schedule. [START_REF] Mcguffin | Past and present state examination: the assessment of 'lifetime ever' psychopathology[END_REF] Participants were excluded if they, or a first-degree relative, ever fulfilled the criteria for depression, bipolar disorder or schizophrenia or if they scored 10 or above on the Beck Depression Inventory. [START_REF] Beck | study: a population-based study to investigate the epidemiology and genetic determinants of cardiovascular risk factors and metabolic syndrome[END_REF] All cases and controls were of white European ancestry.

PsyCoLaus Study

The sample is part of the PsyCoLaus study which focused on psychiatric symptoms in a population-based cohort originally assessed for cardiovascular risk factors (CoLaus).

CoLaus is a cross-sectional study including 6188 extensively phenotyped Caucasian individuals randomly selected from the list of residents of the city of Lausanne (Switzerland). All 35 to 66 year-old subjects of the CoLaus sample were invited to participate in the psychiatric evaluation for the PsyCoLaus sub-study (see 22,[START_REF] Preisig | The PsyCoLaus study: methodology and characteristics of the sample of a population-based survey on psychiatric disorders and their association with genetic and cardiovascular risk factors[END_REF] for a detailed description). The PsyCoLaus sample included 1296 cases (431 men, 862 women; mean age ± s.d.: 49.69±8.68) who fulfilled lifetime ever criteria for MDD according to DSM-IV based on assessment using the Diagnostic Interview for Genetics Studies (DIGS [START_REF] Nurnberger | Diagnostic interview for genetic studies. Rationale, unique features, and training. NIMH Genetics Initiative[END_REF] ). The 1690 PsyCoLaus subjects (974 men, 724 women; mean age ± s.d.: 50.59±8.94) who have never fulfilled criteria for MDD were included as controls.

All studies were approved by the local ethical committees and informed written consent was obtained from all participants. Table 1 summarizes the characteristics of the subjects in both studies.

Phenotypic data

Body Mass Index (BMI) was defined as weight (kilograms) / height (metres) [START_REF] Farmer | Medical disorders in people with recurrent depression[END_REF] . In the Radiant study self-reported height and weight were obtained during the SCAN interview for cases and during telephone interview for controls. In the PsyCoLaus sample weight and height were measured with participants standing without shoes in light indoor clothes. 22 In both samples the distribution of BMI was positively skewed. We therefore transformed the data to log 10 (BMI) to achieve a closer approximation to normal distribution.

Genotyping

Radiant Study

Depression cases and controls were genotyped using the Illumina HumanHap610-Quad BeadChips by the Centre National de Génotypage (CNG) as previously described. [START_REF] Lewis | Genome-Wide Association Study of Major Recurrent Depression in the U.K. Population[END_REF] After applying stringent quality control criteria for missing genotypes, departure from Hardy-Weinberg equilibrium and low minor allele frequency, a total of 88 SNPs spanning FTO (+/-20 Kb either side, chr16: 52,295,376-52,705,882. UCSC Mar.2006) were included in the analyses.

PsyCoLaus Study

Genotyping for depressive cases and controls was performed using the Affymetrix 500 K SNP chip. The genotyping are described in detail in the CoLaus 22 and PsyCoLaus [START_REF] Preisig | The PsyCoLaus study: methodology and characteristics of the sample of a population-based survey on psychiatric disorders and their association with genetic and cardiovascular risk factors[END_REF] studies. IMPUTE v0.2 was used to impute the SNPs used in the Radiant study based on the HapMap CEU (Phase II, release 21) population.

Statistical analyses

T-Test analyses were carried out to test the association between BMI and affected status in both studies.

In Radiant study linear regression models for quantitative traits assuming an additive genetic model were performed for each SNP to test for association between LogBMI and FTO variants. The analyses were first carried out in the whole sample and then separately in the depressive cases and controls. Following the same procedures the most significant SNPs associated with LogBMI in the Radiant sample were analysed in PsyCoLaus.

Affected status, gender and age were included as covariates in the regression analyses in both studies. Principal components were used to control for possible population stratification. In the Radiant study we included centre of ascertainment as a covariate. We also tested for the interaction between FTO polymorphisms and affected status for an effect on LogBMI.

The number of effective/independent tests in the Radiant data was calculated using the method of Galwey et al. [START_REF] Galwey | A new measure of the effective number of tests, a practical tool for comparing families of non-independent significance tests[END_REF] The significance was assessed by dividing 0.05 by the number of effective tests (calculated to be 44.8115), giving threshold of p< 0.0011.

The statistical analyses were performed using the statistical package PLINK v1.06 [START_REF] Purcell | PLINK: a tool set for whole-genome association and population-based linkage analyses[END_REF] and the software Quicktest 28 in the Radiant and PsyCoLaus studies, respectively.

Since the most strongly associated FTO polymorphism reported in literature so far is rs9939609, we used Beagle [START_REF] Browning | A unified approach to genotype imputation and haplotype-phase inference for large data sets of trios and unrelated individuals[END_REF] to impute the genotypes for this SNP in the UK subsample of the Radiant study (1361 cases; 813 controls) using HapMap CEU samples as a reference panel. Two genotyped SNPs (rs8050136, rs3751812) are in absolute linkage disequilibrium (LD) (r 2 =1) with rs9939609, allowing imputation with high certainty. (For LD structure of the variants most significantly associated see supplementary Figure 1). Analysis of imputed data was performed using gene dosage values. The regression analyses with the imputed variant were performed using R software.

Finally, we performed an interaction analysis between the most significant SNPs associated with LogBMI and affected status in the combined Radiant and PsyCoLaus samples.

Results

BMI and Depressive status

In Radiant, as previously reported in a subset of the same data [START_REF] Farmer | Medical disorders in people with recurrent depression[END_REF] , there was a significant association between BMI and affected status. Depressed patients had significantly higher BMI values in comparison with controls in both men and women (Men: t=-5.049, p=0.007; Women: t=-8.755, p<0.0001). These results differ in the PsyCoLaus sample where there were no significant differences in BMI values between depressed cases and controls (Men: t=-1.57, p=0.117; Women: t=-0.11, p=0.913) (Table 1).

FTO and LogBMI

All SNPs tested were in Hardy Weinberg Equilibrium (Radiant: p> 0.01; PsyCoLaus: p> 10 -4 ) in depressive cases and controls.

Radiant Study

Following linear regression between the 88 FTO polymorphisms and LogBMI, 9 of these SNPs located in the first intron of the gene were the most significantly associated with BMI using affected status, gender, age and principal components as covariates. Although these SNPs are mainly in high LD with each other (Figure 1), conditional analyses conducted on the most significant SNP (rs3751812) suggested that they represent two independent signals. One signal includes SNPs rs7205986 and rs6499640, and the second signal includes the rest of the SNPs. Linear regression carried out with the imputed rs9939609 SNP in the UK subsample, also correcting for affected status, gender, age and principal components yielded a significant association with LogBMI (ß=-0.006, p=0.0149).

Linear regression in the depressive group with gender, age, principal components and study as covariates presented evidence of association for the same 9 SNPs that were associated with LogBMI in the combined sample. Moreover, the association between these SNPs and BMI was stronger when analysing the depressive group alone versus the combined sample. Seven remained statistically significant following multiple testing correction (see Table 2).

None of the SNPs tested for association between FTO and LogBMI reached significant pvalues in the control group alone. Results for the combined, case and control analyses are shown in Figure 1.

Controlling for gender, age and principal components, rs9939609 was associated with LogBMI in the depressive group (ß=-0.011, p=0.0007) but not in the control group (ß=0.003, p=0.39).

PsyCoLaus Study

Eight of the top 10 SNPs associated with BMI in the Radiant study had very high imputation accuracy (rSqHat > 0.9) and were included in association analyses. Linear regression analyses in combined depressive cases and controls yielded 6 SNPs significantly associated with LogBMI when affected status, gender, age and principal components were used as covariates. Results carried out in the depressive group alone with gender, age and principal components co-varied in the regression model yielded the same 6 SNPs significantly associated with LogBMI (Table 3). As in the Radiant study, the analyses in the control group alone showed no significant associations with logBMI (Table 3).

Interaction between FTO and affected status

Radiant Study

We found a significant interaction between genotype and affected status in relationship to LogBMI, and found no significant interaction with gender (data not shown). An interaction between genotype and affected status was observed in 7 SNPs (p<0.0057) (Table 2).

We also found an interaction between the previously reported rs9939609 (the polymorphism most commonly associated with BMI in previous studies) and affected status in the UK Radiant subsample (ß=-0.01, p=0.0047) taking gender, age and principal components as covariates in the model.

Radiant and PsyCoLaus combined analysis

Five of the seven SNPs with high imputation quality genotypes in PsyCoLaus, showed evidence (p-values between 0.03-0.06) of interaction between genotype and MDD affected status in their effect on LogBMI (Table 3). Additionally, these SNPs were statistically significant in a meta-analysis of both cohorts (Table 4). The Cochran Q pvalues from the test of heterogeneity of effects in each cohort were non significant indicating that estimates were similar across the two cohorts and justifying the effects meta-analysis (data not shown).

Discussion

We have investigated the influence of variants in FTO on BMI in a large clinical sample of depressed patients and controls (Radiant) to explore the genetic mechanism underlying the reported association between BMI or obesity and psychiatric disorders. [START_REF] Barry | Gender differences in associations between body mass index and DSM-IV mood and anxiety disorders: results from the National Epidemiologic Survey on Alcohol and Related Conditions[END_REF][START_REF] Desai | Gender differences in the association between body mass index and psychopathology[END_REF][START_REF] Scott | Obesity and mental disorders in the adult general population[END_REF] We have then followed up our findings in a population-based cohort (PsyCoLaus).We found associations between several SNPs in FTO and BMI that are in keeping with previous reports. However an intriguing novel finding in the Radiant sample is that the association observed in the whole sample could be attributed to the depressive group, with none of the SNPs being significantly associated with BMI in the control sample alone, and the association strengthening when analysing the patients alone. A similar pattern was found in the PsyCoLaus sample, strongly arguing against this being a chance finding.

There are two potential limitations of our study. The first is in the Radiant sample selfreported data were used to calculate BMI. Against this, height and weight were actually measured in the replication sample (PsyCoLaus) and the same pattern of results was found. The second limitation is that FTO polymorphism most frequently reported to be associated with BMI (rs9939609) in previous studies was here imputed rather than genotyped directly. Against this, the imputation was based on two genotyped SNPs (rs8050136, rs3751812) that are in absolute LD (r 2 =1) with rs9939609.

In the Radiant study evidence that the observed FTO-BMI association is specific to the depressive sample is further supported by the interaction between FTO variants and affected status. This interaction was found in 7 of the SNPs associated with BMI and indicates that the failure to observe an association in the control sample is unlikely to be a consequence of a lack of power. This suggests that the effect of the FTO gene on BMI is modified by having a depressive disorder. Furthermore, the interaction between FTO variants and affected status found in the Radiant data was corroborated by the combined results obtained from the Radiant and PsyCoLaus samples.

As this was the first study of its kind concerning the relationship between BMI and depression we decided to interrogate the entire FTO gene. However the evidence for the association with BMI was, as in previous studies not involved with depression, concentrated in a group of SNPs located in the first intron of the gene. Previous studies report association with SNPs located in the same region. [START_REF] Dina | Variation in FTO contributes to childhood obesity and severe adult obesity[END_REF][START_REF] Frayling | A common variant in the FTO gene is associated with body mass index and predisposes to childhood and adult obesity[END_REF][8][START_REF] Thorleifsson | Genome-wide association yields new sequence variants at seven loci that associate with measures of obesity[END_REF][START_REF] Willer | Six new loci associated with body mass index highlight a neuronal influence on body weight regulation[END_REF] Interestingly, 5 associated SNPs (rs6499640, rs8050136, rs3751812, rs7190492, rs8044769) in our sample have also been found to be genome-wide significantly associated with variation in BMI in the study by Thorleifsson et al. [START_REF] Thorleifsson | Genome-wide association yields new sequence variants at seven loci that associate with measures of obesity[END_REF] Two of the most significant SNPs associated with BMI in Radiant and PsyCoLaus studies (rs3751812, rs8050136) are in absolute LD (r 2 =1) with the FTO variant rs9939609. The results obtained with this imputed variant in both studies are in the same direction and confirm previous findings. [START_REF] Dina | Variation in FTO contributes to childhood obesity and severe adult obesity[END_REF][START_REF] Frayling | A common variant in the FTO gene is associated with body mass index and predisposes to childhood and adult obesity[END_REF][8][START_REF] Andreasen | Low physical activity accentuates the effect of the FTO rs9939609 polymorphism on body fat accumulation[END_REF][START_REF] Hinney | Genome wide association (GWA) study for early onset extreme obesity supports the role of fat mass and obesity associated gene (FTO) variants[END_REF][35][36][37] In the Radiant study depressed patients higher BMI than controls and this could be attributable to a side effect of antidepressant treatment, or because of people who are depressed carry out less physical activity and/or have an increased food intake. These effects are impossible to tease apart in the Radiant sample because all of the cases had at some point received antidepressants. Also, our control sample has been screened so as to have no history of psychiatric disorder in themselves or a first-degree relative. This is something previous studies investigating BMI, obesity and FTO have not done and could further explain why no association is observed in our control group.

In contrast, in the PsyCoLaus sample there were no statistically significant differences in BMI between depressed cases and controls. These differences possibly reflect the fact that Radiant is a clinically ascertained sample of mainly recurrent depression and all of the cases recruited have received antidepressant treatment, whereas the PsyCoLaus cases were recruited from the community and were an arguably less severe and more broadly defined group, where only one episode was required for inclusion and in which only 37.5% of cases had ever received treatment with antidepressants. However, it is noteworthy that the regression of BMI on antidepressant status showed no hint of association (p = 0.573) nor was there any effect when antidepressant status was included as a covariate in the regression of BMI on FTO SNPs. These findings indicate that moderating effect of depressive disorder on the BMI/FTO association is unlikely to be simply a result of taking antidepressants.

An alternative but not necessarily exclusive explanation that depression and obesity involve pathophysiologically overlapping mechanisms. The hypothalamic-pituitaryadrenocortical (HPA) system governs the stress response and has been implicated in the aetiology of depression. 38 FTO is highly expressed in the hypothalamus, pituitary and adrenal glands indicating a possible role in the HPA axis which is involved in body weight regulation. 39 Although several studies have investigated the influence of FTO variants on BMI, and the association between obesity and psychiatric disorders separately, to the authors' knowledge this is the first study investigating the relationship between FTO, BMI and psychiatric disorders (depression) concurrently. Our research suggests that although an association is observed in the whole sample, having a history of depression does in fact moderate the effect of FTO on BMI. The results found initially in the large clinical sample of depressed cases and controls (Radiant) have been replicated in a populationbased cohort (PsyCoLaus). In addition, since milder forms of depression and depressive symptoms are very common in the general population 40 it is possible that experiencing such symptoms moderates the effect of FTO in the population as a whole and partly determines whom of those carrying ' at risk' FTO variants go on to become overweight or obese. Supplementary Figure 1. LD plot structure for the most significant SNPs in FTO gene.

Conflict of interest

Red represents very high LD (r 2 ) between SNPs.

Figure 1 .

 1 Figure 1. Plot showing -log p-values for the SNPs analysed in FTO gene in the Radiant sample.

Table 1 .

 1 Aitchison, Farmer and McGuffin have received consultancy fees and honoraria for participating in expert panels for pharmaceutical companies including GlaxoSmithKline. variants near TCF7L2, SLC30A8, HHEX, CDKAL1, CDKN2A/B, IGF2BP2, and FTO in type 2 diabetes and obesity in 6,719 Asians. Diabetes 2008; 57: 2226-2233. 36 Renstrom F, Payne F, Nordstrom A, Brito EC, Rolandsson O, Hallmans G, et al. Replication and extension of genome-wide association study results for obesity in 4923 adults from northern Sweden. Hum Mol Genet 2009; 18: 1489-1496. 37 Tan JT, Dorajoo R, Seielstad M, Sim XL, Ong RT, Chia KS, et al. FTO variants are associated with obesity in the Chinese and Malay populations in Singapore. Diabetes Demographic characteristics of the Radiant and PsyCoLaus samples.

	2008; 57: 2851-2857.
	38 Holsboer	F.	The	corticosteroid	receptor	hypothesis	of	depression.
	Neuropsychopharmacology 2000; 23: 477-501.
	39 Su AI, Wiltshire T, Batalov S, Lapp H, Ching KA, Block D, et al. A gene atlas of the
	mouse and human protein-encoding transcriptomes. Proc Natl Acad Sci U S A 2004;
	101: 6062-6067.
	40 Kessler RC, Berglund P, Demler O, Jin R, Koretz D, Merikangas KR, et al. The
	epidemiology of major depressive disorder: results from the National Comorbidity
	Survey Replication (NCS-R). JAMA 2003; 289: 3095-3105.

Table 2 .

 2 Association analyses between FTO and LogBMI and the interaction with affected status in the Radiant study.

	p-values

a Position (chr 16) according to the HapMap database. b Risk Alleles are underlined. c Interaction between FTO SNPs and affected status. MAF: minor allele frequency. Minor allele is in brackets.

Table 3 .

 3 Association analyses between FTO and LogBMI and the interaction with affected status in the PsyCoLaus study.

	p-values

a Position (chr 16) according to the HapMap database. b Coded Alleles are underlined. c Interaction between FTO SNPs and affected status. MAF: minor allele frequency. Minor allele is in brackets.

Table 4 .

 4 Interaction analyses between FTO and affected status in the combined sample.

	SNP	Position a	Coded Allele	ß	SE	p-value
	rs9930333	52357478	G	0.01	0.0029	7x10 -4
	rs10852521	52362466	T	-0.007	0.0029	0.0199
	rs8050136	52373776	A	0.01	0.03	5x10 -4
	rs3751812	52375961	T	0.01	0.03	6x10 -4
	rs9939609	52378028	T	-0.01	0.03	0.001
	rs9941349	52382989	T	0.01	0.029	0.001

a Position (chr 16) according to the HapMap database. SE is the standard error.

Aitchison's declares interests through Advisory Boards for & Johnson, Lundbeck, Roche Diagnostics, and Bristol-Myers Squibb; membership of Bristol-Myers Squibb UK Steering group 2003 to present; consultancy work for Roche Diagnostics, Johnson & Johnson Pharmaceutical Research and Development, Lundbeck, and Bristol-Myers Squibb Pharmaceuticals Limited; grants awarded by Johnson & Johnson Pharmaceutical Research and Development, Bristol-Myers Squibb Pharmaceuticals Limited, and E Merck Pharmaceuticals. Maier is member of the Advisory Boards/ has received fees for speaking from: Lilly, Lundbeck. Tozzi was full time employee of GlaxoSmithKline at the time when the work was performed. All other authors declare no conflicts of interest.

Acknowledgements

The authors would like to thank those who agreed to participate in the studies and the many colleagues who contributed to collection and phenotypic characterization of the clinical samples, to genotyping and to statistical analyses. This study was funded by the Medical Research Council, UK and GlaxoSmithKline (G0701420). The GENDEP study was funded by a European Commission Framework 6 grant, EC Contract Ref.: LSHB-CT-2003-503428. This research was supported by a Marie Curie Intra European Fellowship within the 7 th European Community Framework Programme. Dr Cohen-Woods and Dr Schosser were both funded by Postdoctoral Fellowships from the National Institute for Health Research (NIHR) specialist Biomedical Research Centre for Mental Health. Dr Rucker is funded by a Research Training Fellowship from the Wellcome Trust. The CoLaus/PsyCoLaus study was supported by grants from the Swiss National Science Foundation (#3200B0-105993, #3200B0-118'308, 33CSC0-122661) and from Sven Bergmann is grateful for financial support from the Giorgi-Cavaglieri Foundation, the Swiss National Science Foundation (Grant #3100AO-116323/1), the Swiss Institute of Bioinformatics and the European Framework Project 6 (through the AnEuploidy and EuroDia projects).

Supplementary information is available at Molecular Psychiatry's website.


Anna ; Passoja

Matti Knuuttila

Liisa Hiltunen

Riitta Karttunen

Onni Niemelä

Taina Raunio

Olli Vainio

Pirjo Hedberg

Tellervo Tervonen
email: tellervo.tervonen@oulu.fi

Hiltunen L Karttunen

Serum IL-6 may modulate periodontal inflammation in type 1 diabetic subjects

Keywords: Periodontal disease, Gingivitis, periodontal pocket, periodontitis bleeding on probing, inflammation, interleukin (IL)-6, periodontitis, diabetes mellitus, serum

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

Introduction

Hyperglycemia is known to evoke a pro-inflammatory state characterised by increased concentrations of systemic inflammatory biomarkers such as cell adhesion molecules, C-reactive protein (CRP) and pro-inflammatory cytokines, including interleukin (IL)-6 and tumour necrosis factor (TNF)-α [START_REF] Devaraj | Evidence of increased inflammation and microcirculatory abnormalities in patients with type 1 diabetes and their role in microvascular complications[END_REF][START_REF] Kaul | Is inflammation a common retinal-renal-nerve pathogenic link in diabetes?[END_REF]. While it is generally accepted that the major causative factor behind diabetic complications is hyperglycemia, growing evidence exists that the development and progression of diabetic micro-and macro-vascular complications are also linked with systemic inflammation [START_REF] Saraheimo | Diabetic nephropathy is associated with low-grade inflammation in Type 1 diabetics[END_REF][START_REF] Devaraj | Evidence of increased inflammation and microcirculatory abnormalities in patients with type 1 diabetes and their role in microvascular complications[END_REF]). Both gingivitis and periodontitis have been considered as hyperglycemia-related diabetic complications [START_REF] Taylor | Bidirectional interrelationships between diabetes and periodontal diseases: an epidemiologic perspective[END_REF], Lalla et al 2007a). Since the very first reports by Gislen and co-workers (1980) and by our own group [START_REF] Ervasti | Relation between control of diabetes and gingival bleeding[END_REF][START_REF] Tervonen | Relation of diabetes control to periodontal pocketing and alveoral bone level[END_REF], the role of the metabolic control of diabetes mellitus as the main determinant of increased periodontal inflammation has been confirmed in a large number of studies [START_REF] Taylor | Bidirectional interrelationships between diabetes and periodontal diseases: an epidemiologic perspective[END_REF], Lalla et al 2007a). Importantly, both micro-vascular complications and periodontal inflammation seem to share at least some potential hyperglycemiarelated pathways in their development, including hyperglycemia-enhanced formation of advanced glycation end products (AGEs), protein kinase C activation, increased oxidative stress and apoptosis [START_REF] Graves | Diabetes-enhanced inflammation and apoptosis -impact on periodontal pathology[END_REF][START_REF] Devaraj | Evidence of increased inflammation and microcirculatory abnormalities in patients with type 1 diabetes and their role in microvascular complications[END_REF][START_REF] Nassar | Diabetic periodontitis: a model for activated innate immunity and impaired resolution of inflammation[END_REF][START_REF] Van Den Oever | Endothelial dysfunction, inflammation, and apoptosis in diabetes mellitus[END_REF].

It is nowadays generally agreed that chronic periodontitis is associated with low-grade systemic inflammation both in non-diabetic [START_REF] Loos | Systemic markers of inflammation in periodontitis[END_REF][START_REF] Nibali | Severe periodontitis is associated with systemic inflammation and a dysmetabolic status: a case-control study[END_REF][START_REF] Saxlin | Role of serum cytokines tumour necrosis factor-alpha and interleukin-6 in the association between body weight and periodontal infection[END_REF]) and in diabetic [START_REF] Engebretson | Plasma levels of tumour necrosis factor-α in patients with chronic periodontitis and type 2 diabetes[END_REF][START_REF] Lim | Relationship between markers of metabolic control and inflammation on severity of periodontal disease in patients with diabetes mellitus[END_REF][START_REF] Chen | Association of periodontal parameters with metabolic level and systemic inflammatory markers in patients with type 2 diabetes[END_REF] individuals. On the contrary, less is known about the contribution of elevated levels of serum inflammatory mediators to periodontal inflammation. This biological pathway was recently studied by Andriankaja and co-workers (2009), who observed a significant association between serum IL-6 and gingivitis in type 2 diabetic subjects with no deepened periodontal pockets (≥ 3 mm) and bleeding on probing ≥ 10%. One of the We determined the periodontal health status and systemic inflammation in a group of T1DM subjects before and after periodontal therapy to explore the possible bidirectional nature of the association between periodontal health and systemic inflammatory status. Moreover, our aim was to study whether the possible link between periodontal and systemic inflammation is analogous to the link between systemic inflammation and the development and progression of micro-vascular complications of diabetes mellitus.

Material and Methods

The study protocol was approved by the Ethical Committee of Oulu University Hospital, Oulu, Finland, and an informed consent form was signed by the subjects.

Subjects

Originally, a total of 80 patients with T1DM were recruited on a volunteer basis and examined clinically by a periodontal specialist (T.R.) at the Specialist Dental Health Care Unit, City of Oulu, Finland [START_REF] Tervonen | Resolution of periodontal inflammation does not guarantee improved glycemic control in type 1 diabetic subjects[END_REF]. Subjects needing prophylactic antibiotic medication in association with periodontal probing and those with immunosuppressive medication or antibiotics during the past four months were excluded. After anti-infective periodontal therapy a total of 65 subjects were re-examined. Data on glycosylated haemoglobin (HbA1c), high-density HDL cholesterol, lowdensity LDL cholesterol, IL-6 and usCRP were available for all subjects at the baseline and for 59 subjects after therapy. One subject with a severe and acute wound infection and maxillary sinusitis presented a pronounced increase in serum IL-6 (from 3.6 pg/ml to 8.2 pg/ml) during the study and was excluded, leaving a total of 58 treated subjects for the analyses.

Patient records were used to retrieve data regarding the general health status of the subjects. Four subjects had either chronic maxillary sinusitis, skin infection or symptoms of a urinary tract infection during the study (indicated in Fig. 1A). Smoking data were obtained by an interview and the subjects were categorised as smokers and non-smokers. A total of 56 subjects of those examined at the baseline and 44 out of the treated subjects reported to be non-smokers. The subjects' body mass index (BMI kg m -2 ) and age were also recorded. A total of 58 subjects out of those studied at the baseline and 44 of the treated subjects had BMI ≤ 26 kg m -2 (Table 1) (according to the WHO, the lower limit for overweight is 25 kg m -2 ).

Clinical periodontal examination and periodontal therapy

Periodontal variables were recorded on mesiobuccal, midbuccal, distobuccal and midlingual surfaces of each tooth as described earlier [START_REF] Tervonen | Resolution of periodontal inflammation does not guarantee improved glycemic control in type 1 diabetic subjects[END_REF]. Shortly, the presence of visible plaque according to the criteria of the scores 2 and 3 of the plaque index [START_REF] Silness | Periodontal disease in pregnancy (II). Correlation between oral hygiene and periodontal condition[END_REF], probing depth (PD) and bleeding on probing (BOP) were registered. Anti-infective periodontal therapy including oral hygiene education, scaling and root planing were performed. As the goal was to minimise periodontal inflammation, these treatments were repeated and periodontal surgery was performed whenever indicated. In the follow-up examination approximately 8 weeks after the completion of the treatment 1-9 bleeding sites were found in 14 subjects, 10-19 sites in 19 and > 20 sites in 16 subjects. Moreover, bleeding and PD ≥ 4 mm were detected at 1-2 sites in 14 subjects and at 3-10 sites in 7 subjects.

Laboratory assays

Venous blood samples were drawn at the baseline and in the follow-up examination. Glycosylated haemoglobin (HbA1c, %) was analysed using a latex immunoturbidimetric method (Siemens Medical Solutions Diagnostics, Tarrytown, NY 10591-5097 USA). Serum samples were stored at -70 о C until assayed. Serum IL-6 level (pg/ml) was measured with a sensitive ELISA Quantikine HS Immunoassay kit (R&D Systems GmbH, Wiesbaden-Nordenstadt, Germany). The usCRP (ultrasensitive CRP, mg/l) analyses were made using an Innotrac Aio! analyser (Innotrac Diagnostics Oy, Turku, Finland), that utilises a unique all-in-one (Aio!) dry-reagent concept and [START_REF] Hedberg | Highly sensitive determination of C-reactive protein on the Innotrac Aio! immunoanalyzer[END_REF]). Low-density LDL cholesterol (mmol/l) and high-density HDL cholesterol (mmol/l) were measured using direct enzymatic methods implemented in the Advia 2400 Chemistry systems (Siemens Medical Solutions Diagnostics, Tarrytown, NY 10591-5097 USA).

An 'increase' in IL-6 level between the baseline and the follow-up examination was recorded if the level was elevated by ≥ 0.2 pg/ml, a 'decrease' if the level dropped by ≥ 0.2 pg/ml and a lack of either of the previous was considered a 'stable' level (Fig. 1A).

Statistical methods

Statistical analyses were performed using statistical software (SPSS Statistical Package for Social Sciences, version 16.0 for Windows; SPSS Inc., Chicago, IL, USA) and the level of statistical significance was set at < 5% (p < 0.05). All analyses were based on subject-level data. The extent of periodontal inflammation was measured as the number of affected sites per subject. Mean serum IL-6 level before and after therapy were compared using the Wilcoxon signed ranks test. Spearman´s rank correlation analysis was used to study unadjusted associations between periodontal parameters and serum inflammatory markers at the baseline (Table 2). Depending on the distribution of the outcome, variable-adjusted associations between periodontal inflammation and serum inflammatory markers were studied using either linear regression or binomial regression analyses. In the binomial regression models the ln of the number of measured sites was used as an offset variable. In order to control the confounding/modifying effect of smoking and BMI on the association between serum IL-6 and periodontal inflammation, we used the interaction terms BMI*IL-6 and smoking*IL-6 in the models. We also stratified the sample by BMI (≤ 26 kg m -2 / > 26 kg m -2 ) and smoking (nonsmokers / smokers) to study the associations in the stratified samples. The association between periodontal healing and serum IL-6 level was studied using linear regression analysis with the reduction in the number of sites with bleeding and PD ≥ 4 mm (number of sites after therapy deducted from the number of sites at the baseline, range -2 -90) as the outcome.

Results

The subject characteristics are presented in Table 1. The group of treated subjects was comparable with the whole study group in terms of age, gender, smoking habits, BMI, duration of DM and number of teeth. Regardless of a favorable response to periodontal therapy, no essential changes in the mean levels of serum inflammatory markers from the baseline to the follow-up examination could be observed.

Based on the unadjusted associations between serum inflammatory markers and the periodontal parameters at the baseline (Table 2), we studied the adjusted associations between the number of sites with BOP and bleeding and PD ≥ 4 mm (the outcome variables) and serum IL-6 and HbA1c in further analyses using multiple regression models. The regression analyses of all subjects both at the baseline and after periodontal therapy (non-stratified samples) revealed statistically significant associations between serum IL-6 and periodontal inflammation. We then controlled for possible interaction between serum IL-6 and smoking and serum IL-6 and BMI by adding the interaction term smoking*IL-6 and BMI*IL-6 into the model and found that the interaction terms were not statistically significant in the models. Neither was there any statistically significant association between serum IL-6 level and periodontal inflammation in the new model (data not shown).

Therefore, further analyses were done in samples stratified separately by smoking and BMI. While serum HbA1c level was associated significantly with BOP and bleeding and PD ≥ 4 mm both in subjects with BMI ≤ 26 kg m -2 and in non-smokers at the baseline, serum IL-6 level turned out to be a statistically significant determinant of the number of sites with bleeding and PD ≥ 4 mm in the same subjects (Table 3, Models I and II). At the baseline, the adjusted associations between serum IL-6 level and the number of sites with BOP and bleeding and PD ≥ 4 mm were not statistically significant in subjects with BMI > 26 kg m -2 (n = 22) and in smokers (n = 24) (data not shown). After periodontal therapy, a statistically significant association was found between the level of serum IL-6 and the number of sites with bleeding and PD ≥ 4 mm in subjects with BMI ≤ 26 kg m -2 (p = 0.043) (Table 4, Model III). In non-smokers this association was of borderline significance (p = 0.064) (Table 4, Model IV). The number of sites with bleeding and PD ≥ 4 mm after therapy was not associated with smoking. The actual serum levels of IL-6 in subjects with BMI ≤ 26 kg m -2 and in non-smokers according to the number of residual sites with bleeding and PD ≥ 4 mm after periodontal therapy are presented in Table 5. These results also indicate that the level of serum IL-6 is higher in subjects with higher numbers of residual sites with inflammation. The adjusted associations between serum IL-6 level and the number of sites with BOP and bleeding and PD ≥ 4 mm were not statistically significant in subjects with BMI > 26 kg m -2 (n = 14) and in smokers (n = 14) after therapy (data not shown).

A decrease (≥ 0.2 pg/ml) in serum IL-6 level was observed in 23 (39.7%) subjects between the baseline and the follow-up examination. The level remained stable in 18 (31.0%) subjects and an increase (≥ 0.2 pg/ml) was observed in 17 (29.3%) subjects (Fig. 1A). Subjects with chronic infections were evenly distributed into the previous groups (Fig. 1A). The reduction in the number of sites with bleeding and PD ≥ 4 mm, was similar in subjects with a decreased, stable or increased serum IL-6 level (Fig. 1 B). After controlling for confounding factors a statistically significant association (p = 0.003) was found between periodontal healing (the outcome variable) and the aftertherapy serum IL-6 level, indicating poorer periodontal healing in subjects with a high IL-6 level (Table 6). In a second model (data not shown) including the interaction terms BMI*IL-6 and smoking*IL-6 the above association turned out to be weaker (p=0.022). The interaction terms were, however, not statistically significant.

Discussion

In line with earlier studies [START_REF] Loos | Elevation of systemic markers related to cardiovascular diseases in the peripheral blood of periodontitis patients[END_REF][START_REF] Buhlin | Risk factors for cardiovascular disease in patients with periodontitis[END_REF]) reporting higher serum IL-6 levels in patients with chronic periodontitis than in healthy control subjects, we found a statistically significant association between the level of serum IL-6 and the number of sites with bleeding and PD ≥ 4 mm at the baseline. A new finding was that this association was evident also after periodontal therapy. We also observed that periodontal therapy was not consistently followed by a decrease in serum IL-6 level and that periodontal healing turned out to be poorer in subjects with a high after-therapy IL-6 level than in those with a low level.

When interpreting the above observations, we started from the premise that the causal pathway between serum IL-6 and periodontal inflammation may be bidirectional. Increased serum levels of inflammatory markers, such as IL-6, in subjects with chronic periodontitis have been explained by dumping of locally concentrated IL-6 from the inflamed areas into the circulation, its production by distant immune cells as a response to bacterial irritation from the site of infection or both. In light of the above, a high serum IL-6 level observed at the baseline of this study may be interpreted as reflecting a high extent of periodontal inflammation.

The association between serum IL-6 level and the number of sites with bleeding and PD ≥ 4mm after therapy on one hand, and the reduction in the number of sites with bleeding and PD ≥ 4mm on the other hand, was interpreted to indicate poorer periodontal healing in subjects with higher serum IL-6 levels and to be suggestive of a causal pathway, where a high serum IL-6 level acts as a susceptibility factor for periodontal inflammation. IL-6 is known for its pro-inflammatory properties [START_REF] Kaplanski | IL-6: a regulator of the transition from neutrophil to monocyte recruitment during inflammation[END_REF][START_REF] Gabay | Interleukin-6 and chronic inflammation[END_REF][START_REF] Kishimoto | Diabetes mellitus promotes periodontal destruction in children[END_REF]) and it also accelerates bone resorption [START_REF] Tamura | Soluble interleukin-6 reseptor triggers osteoclast formation by interleukin-6[END_REF]. Therefore, a high serum IL-6 level may boost inflammation and modulate tissue responses locally in the periodontal area. Although we consider the above direction of the causal pathway more likely, one could also speculate that the resolution of inflammation contributed to the decreased serum IL-6 levels; the higher the number of healed sites, the less IL-6 could be detected in serum after therapy. Supporting this, slightly decreased levels of serum inflammatory markers after anti-infective therapies have been reported (D´Aiuto et al. 2004[START_REF] Loos | Systemic markers of inflammation in periodontitis[END_REF][START_REF] Kardeşler | Adipokines and inflammatory mediators after initial periodontal treatment in patients with type 2 diabetes and chronic periodontitis[END_REF]. However, in our subjects, the decrease in the mean serum IL-6 level from the baseline to the follow-up examination (0.1 pg/ml) was not statistically significant. In addition, similar periodontal healing was followed by a decrease in IL-6 level in 40%, a stable level in 31% and an increase in 29% of the subjects (Figs. 1A,B). These results further support our interpretation that serum IL-6 acts as a susceptibility factor for periodontal inflammation. We can't fully exclude that there were also other mechanisms/conditions, such as sub-clinical or unreported infections, which may have contributed to the serum IL-6 levels. Also, including a non-diabetic group with a comparable periodontal health status would have made it possible to assess the influence of diabetes mellitus itself on the systemic IL-6.

One of the methodological challenges in studies of the periodontal-systemic relationship is to adequately control the confounding/modifying effects of smoking and BMI. Smoking is known to deteriorate periodontal health [START_REF] Bergström | Periodontitis and smoking[END_REF]) and evidently increase the serum IL-6 level [START_REF] Tappia | Cigarette smoking influences cytokine production and antioxidant defences[END_REF]. On the other hand, adipose tissue is considered an important source of IL-6 (Mohamed-Ali et al. 1997[START_REF] Fried | Omental and subcutaneous adipose tissues of obese subjects release interleukin-6: depot difference and regulation by glucocorticoid[END_REF], and obesity/BMI has been associated with chronic periodontitis [START_REF] Ylöstalo | Association between body weight and periodontal infection[END_REF]. We took into consideration the above both by using respective interaction terms in the multivariate models and by stratifying the samples according to smoking and BMI. Due to the small sample size, stratification according to smoking and BMI at the same time was not possible. As regards a more complete control of the confounding/modifying effect of smoking and BMI, a need for verification of our result in larger samples remains. Moreover, a more definite measure of smoking, such as pack-years, should be used.

As the goal of periodontal therapy was to minimise periodontal inflammation, anti-infective treatments were continued until this goal was reached. The lack of a statistically significant association between smoking and periodontal inflammation after therapy can be explained by the setting of the study where additional treatments were delivered to poorly responding subjects such as smokers. Considering the treatment protocol used and the suggested poorer healing response of subjects with higher IL-6 levels, we speculate that the strength of the association between serum IL-6 and the number of inflamed sites after therapy was underestimated.

The shortcomings of this study include the fairly small sample size and lack of power analysis for the sample size. Moreover, this study allows us to suggest a bidirectional relationship between serum IL-6 level and periodontal inflammation exclusively in T1DM subjects. Regarding the observed association between serum IL-6 level and periodontal inflammation, the chance of a spurious association must also be considered; in that case there are one or more common background factors contributing to both the increased serum level and the higher extent of periodontal inflammation.

In summary, while serum IL-6 level was associated significantly with periodontal inflammation at the baseline, resolution of inflammation was not followed by a decrease in the serum IL-6 level in all subjects. The response to anti-infective periodontal therapy was poorer in subjects with a high after-therapy level of serum IL-6 than in those with a low level. Thus, a high serum IL-6 level may modulate inflammatory responses in the periodontal area as it modulates the development and progression of diabetic micro-and macro-vascular complications. The evident bidirectional causal pathway between serum IL-6 and periodontal inflammation is essential in elucidating both the pathogenesis of periodontitis and periodontal-systemic relationships. Therefore future studies should be focused on confirming the observed associations also in non-diabetic individuals.

  the authors was that serum IL-6 might have a specific role in modulating local inflammatory reaction in individuals with type 2 diabetes.

  Fig. 1A. Subject-level changes in serum IL-6 level. A decrease in serum IL-6 level (≥ 0.2 pg/ml) between the baseline and the follow-up examination was observed in 23 subjects (lines on the left)and an increase (≥ 0.2 pg/ml) in 17 subjects (lines on the right). The lines in the middle present subjects with a stable IL-6 level (change < 0.2 pg/ml). Subjects with chronic infections, other than gingivitis or periodontitis, are indicated using dashed lines. Fig.1B. Healing of periodontal inflammation indicated as the reduction in the number of sites with bleeding and PD ≥ 4 mm in subjects presenting increased, stable or decreased serum IL-6 level between the baseline and the follow-up examination (see Fig.1A).289x171mm (150 x 150 DPI)

Table 1 .

 1 Subject characteristics presented as mean values/subject (± SD) and numbers of subjects. BOP, bleeding on probing; bleeding and PD ≥ 4mm, bleeding pocket ≥ 4mm in depth ; HbA1c, glycosylated haemoglobin; HDL, high-density lipoprotein; LDL, low-density lipoprotein; usCRP, ultrasensitive C-reactive protein; BMI, body mass index

	All subjects (n = 80)	Treated subjects (n = 58)	(n = 58)

Table 2 .

 2 Unadjusted associations, based on Spearman's rank correlation analysis, between the numbers of sites presenting bleeding on probing (BOP) and bleeding and PD ≥ 4 mm, serum inflammatory markers and body mass index (BMI) at the baseline.

	Correlation coefficients, p-values; statistically significant values are presented in boldface.

Table 3 .

 3 Adjusted associations between the number of bleeding sites (BOP) and sites with bleeding and PD ≥ 4 mm (the outcome variables) and serum IL-6 level at the baseline in subjects with BMI ≤ 26 kg m -2 and in non-smokers.

	Boldface denotes statistical significance.

*Linear regression and †negative binomial regression models:

Model I; adjusted for age, gender, smoking, HbA1c, serum HDL, plaque and number of sites. Model II; adjusted for age, gender, HbA1c, BMI, serum HDL, plaque and number of sites. ‡ Smoking as a categorized variable; non-smokers vs. smokers (reference).

Table 4 .

 4 Adjusted associations between the number of bleeding sites (BOP) and sites with bleeding and PD ≥ 4 mm (the outcome variables) and serum IL-6 level after therapy in subjects with BMI ≤ 26 kg m -2 and in non-smokers.

	Boldface denotes statistical significance.

Negative binomial regression models:

Model III, adjusted for age, gender, smoking, HbA1c, serum HDL, plaque and number of sites. Model IV, adjusted for age, gender, HbA1c, BMI, serum HDL, plaque and number of sites. *Smoking as a categorized variable; non-smokers vs. smokers (reference).

Table 5 .

 5 Mean serum IL-6 levels (pg/ml) according to the number of sites with bleeding and PD ≥ 4 mm after periodontal therapy.

		BMI ≤ 26 kg m -2		Non-smokers
	N	IL-6 pg/ml	N	IL-6 pg/ml

Journal of Clinical Periodontology -PROOF

Journal of Clinical Periodontology -PROOFJournal of Clinical Periodontology -PROOF

Conflict of interest and source of funding statement:

The authors declare that they have no conflicts of interest.

The study was self-funded by the authors and their institutions.

Clinical Relevance

Scientific rationale for the study

While some studies indicate a causative role of periodontal infection on systemic inflammatory responses, less is known of the influence of high serum levels of pro-inflammatory mediators on local host responses.

Principal findings

The level of serum IL-6 was associated with the extent of bleeding and PD ≥ 4 mm at the baseline in non-smokers and in subjects with BMI ≤ 26 kg m -2 . These associations were also evident after periodontal therapy. Subjects with a high IL-6 level presented poorer periodontal healing than those with a low level.

Practical implications

Serum IL-6 may modulate inflammatory responses in the periodontal area and contribute to both the progression and healing of periodontal inflammation as it contributes to the initiation and progression of other inflammation-associated diabetic complications. The outcome variable: the number of sites with bleeding and PD ≥ 4mm after therapy deducted from the number of the sites at the baseline (range -2 -90).

Journal of Clinical Periodontology -PROOF

Journal of Clinical Periodontology -PROOF F o r P e e r R e v i e w

Smoking as a categorized variable; non-smokers vs. smokers (reference)