Yannick Privat
email: yannick.privat@unistra.fr

Rémi Robin
email: remi.robin@inria.fr

Mario Sigalotti
email: mario.sigalotti@inria.fr

Existence of surfaces optimizing geometric and PDE shape functionals under reach constraint

, can be easily extended to costs involving different mathematical objects associated with the domain, such as solutions of elliptic equations on the hypersurface.

1 Framework and main results

Introduction

In this paper, we are interested in the question of the existence of optimal sets for shape optimization problems involving surfaces. More precisely, we are interested in shape functionals written as JpΩq " ż BΩ jpx, ν BΩ pxq, B BΩ pxqq dµ BΩ pxq

where Ω denotes a smooth subset of R d , the wording 'smooth' being understood at this stage such that all the involved quantities make sense, ν denotes the outward pointing normal vector to BΩ and B BΩ is either a purely geometric quantity such as the mean curvature, or the solution of a PDE on BΩ or on Ω.

We are then interested in the existence of solutions for the optimization problem inf

Ω admissible
JpΩq .

This kind of problem is very generic. What matters here is that the standard techniques, exposed and developed for example in [START_REF] Delfour | Shapes and geometries[END_REF][START_REF] Henrot | Shape variation and optimization[END_REF], do not apply to d´1 objects and it is necessary to adopt a particular approach. The first question to ask is the choice of the set O ad of all admissible domains. Since the shape functionals we consider involve geometric quantities of the type "outward normal vector to the boundary" or "mean curvature", it is necessary that the manipulated surfaces are not too irregular. For this reason, we choose to impose a constraint that guarantees a uniform regularity, say C 1,1 , of the manipulated sets. This uniform regularity constraint is imposed by using the notion of "reach". Thus, the set O ad represents the set of surfaces having a reach uniformly bounded by below. The precise definition of this notion will be given in Section 1.3. This kind of problem has been the subject of recent studies and results [START_REF] Guo | On convergence of boundary Hausdorff measure and application to a boundary shape optimization problem[END_REF][START_REF] Dalphin | Uniform ball property and existence of optimal shapes for a wide class of geometric functionals[END_REF][START_REF] Dalphin | Existence of optimal shapes under a uniform ball condition for geometric functionals involving boundary value problems[END_REF], which have provided positive answers to the existence issues. In their approach, the authors used an efficient, but nevertheless laborious, approach based on the parametrization of the manipulated surfaces, seen as regular manifolds, using local charts.

The objective of this paper is to promote a different approach, based on the extension of the functions defined on the manipulated surfaces to volume neighborhoods, the introduction of an extruded surface and the rewriting of the surface integrals as volume integrals using ad-hoc variable changes. This is a methodological paper, in which a proof method is presented that may work in many cases. The results contained in the article illustrate this point. We discuss possible generalizations of these results in a concluding section.

This method allows to gain conciseness and provides much shorter and direct existence proofs than in the above references. The method also allows to extend the field of investigation to new families of problems, involving the solution of a PDE defined on a hypersurface. Nevertheless, some arguments used by the authors of [START_REF] Guo | On convergence of boundary Hausdorff measure and application to a boundary shape optimization problem[END_REF][START_REF] Dalphin | Uniform ball property and existence of optimal shapes for a wide class of geometric functionals[END_REF][START_REF] Dalphin | Existence of optimal shapes under a uniform ball condition for geometric functionals involving boundary value problems[END_REF] cannot be shortened by using our approach. We have therefore chosen to expose our method in a short article, in which we detail all the parts of the proof that can be condensed and we make the necessary reminders concerning the results that cannot be condensed.

The article is organized as follows: we introduce the definition of the reach of a surface as well as the class of admissible sets we will deal with in Section 1.3. The main results of this article, regarding several existence results for shape optimization problems involving surfaces, are provided in Section 1.4. The whole section 2 is devoted to the proofs of the main results. In these proofs, we detail the arguments based on our approach and leading to simplified proofs of the results in [START_REF] Guo | On convergence of boundary Hausdorff measure and application to a boundary shape optimization problem[END_REF][START_REF] Dalphin | Uniform ball property and existence of optimal shapes for a wide class of geometric functionals[END_REF][START_REF] Dalphin | Existence of optimal shapes under a uniform ball condition for geometric functionals involving boundary value problems[END_REF]. In order to illustrate the potential of our approach, we also provide an existence result involving a general functional depending on the solution of a PDE on the sought manifold.

Notations

Let us recall some classical notations used throughout this paper:

• For the sake of notational simplicity, we will sometime use the notation Γ (resp. Γ n ) to denote the hypersurfaces BΩ (resp. BΩ n ).

• The Euclidean inner product (resp. norm) will be denoted x¨, ¨y (resp. } ¨} or sometimes | ¨| when no confusion with other notations is possible).

• Given two positive integers k ď d and Ω Ă R d , H k pΩq denotes the k-dimensional Hausdorff measure of Ω.

• Given Ω Ă R d , the distance (resp., signed distance) to Ω is defined for all x P R d by

d Ω pxq " inf yPΩ }x ´y} (resp., b Ω pxq " d Ω pxq ´dR d zΩ pxqq.
• Given Ω Ă R d and h ą 0, the tubular neighborhood U h pΩq is defined as

U h pΩq " tx P R d | d Ω pxq ď hu. • Given Ω Ă R d , the reach of Ω is defined as ReachpΩq " supth ą 0 | d Ω is differentiable in U h pΩqzΩu.
Recall that, if BΩ is a nonempty compact C 1,1 -hypersurface of R d , then there exists h ą 0 such that Ω satisfies a uniform ball condition, namely

@x P BΩ, Dd x P R n | }d x } R d " 1, B h px ´hd x q Ă Ω and B h px `hd x q Ă R n zΩ, (B h )
where B h pxq stands for the open ball of radius h centered in x. Furthermore, assuming H d pBΩq " 0, we have the simpler characterization

ReachpBΩq " supth | Ω satisfies (B h )u.
Conversely, if BΩ is nonempty and satisfies Condition (B h ), then its reach is larger than h and the Lebesgue measure of BΩ in R n is equal to 0. Furthermore, BΩ is a C 1,1 hypersurface of R n . We refer for instance to Theorems 2.6 and 2.7 in [START_REF] Dalphin | Uniform ball property and existence of optimal shapes for a wide class of geometric functionals[END_REF].

• For a given oriented C 1,1 -hypersurface BΩ, we denote by ∇ BΩ or ∇ Γ the tangential gradient and by ∇ the full gradient in R d . When needed, each gradient will be assimilated to a line vector in R d .

• N denotes N Y t`8u.

• S d´1 denotes the unit sphere of R d .

• M d pRq denotes the linear space of d ˆd matrices with real entries, endowed with the Euclidean operator norm } ¨}. Id denotes the identity matrix in R d .

• For a given C 1,1 hypersurface BΩ, we denote by H BΩ : BΩ Ñ R, its mean curvature. We refer to Appendix A for proper definitions.

Preliminaries on sets of uniformly positive reach

Given r 0 ą 0 and a nonempty compact set D Ă R d , let us introduce the set O r0 of admissible shapes whose reach is bounded by r 0 , namely

O r0 " tΩ Ă D | Ω is closed, ReachpBΩq ě r 0 , Ω " H, and
H d pBΩq " 0u. ( 1
)
The elements of O r0 are known to satisfy the following properties.

Lemma 1. Let Ω P O r0 . Then 1. BΩ is a C 1,1 pd ´1q-submanifold. Conversely, O r0 " tΩ Ă D | Ω is closed, ReachpBΩq ě r 0 , BΩ is a pd ´1q-submanifoldu.
2. For x P BΩ, ∇b Ω pxq is the unit outward normal vector.

3. For h ă r 0 , ∇b Ω is 2 r0´h -Lipschitz continuous on the tubular neighborhood U h pBΩq. We will endow the set O r0 with a 'sequential' topology, by introducing a notion of convergence in this set.

The restriction of ∇b

Ω to BΩ is 1 r0 -Lipschitz continuous.

Definition 1 (R-convergence in O r0

). Given pΩ n q nPN P O N r0 , we say that pΩ n q nPN R-converges to Ω 8 P O r0 and we write

Ω n R Ý Ñ Ω 8 if b Ωn Ñ b Ω8 $ ' & ' % in C pDq, in C 1,
α pU r pBΩ 8 qq, @r ă r 0 , @α P r0, 1q, weakly-star in W 2,8 pU r pBΩ 8 qq, @r ă r 0 .

(2)

The next result justifies the interest of the class O r0 endowed with the R-convergence for existence issues.

Proposition 1. O r0 is sequentially compact for the R-convergence.

The proof of this proposition can be found in Section B. Let us end this section by providing several additional properties of the R-convergence.

Lemma 2. If Ω n R Ý Ñ Ω 8 then
1. H d´1 pBΩ n q converges toward H d´1 pBΩ 8 q as n Ñ `8.

2. H d pΩ n q converges toward H d pΩ 8 q as n Ñ `8.

3. If all the BΩ n belong to the same isotopic class, then BΩ 8 also belongs such a class.

The proof of this lemma is given in Section 2.2.

Remark 1.

According to Lemma 2, we obtain for example that for a given Ω 0 P O r0 , and a ď b,

tΩ P O r0 | a ď H d´1 pBΩq ď b, BΩ is isotopic to BΩ 0 u is a sequentially compact set.

Main results

Let us introduce the general shape functional

F 1 pΩq " ż BΩ j 1 px, νpxq, H BΩ pxqq dµ BΩ pxq, (3)
where j 1 is continuous from R d ˆSd´1 ˆR to R and convex with respect to its last variable. We recall that ν and H BΩ denote respectively the outward pointing normal vector and the mean curvature.

According to Proposition 1, the set O r0 is sequentially compact for the R-convergence. Therefore, in order to infer the existence of an optimal surface minimizing F 1 over O r0 it is enough to prove the lower semicontinuity of functional F 1 (under suitable assumptions on the function j 1 ). This is the main purpose of the following result.

Theorem 1 ([4], Theorem 1.3). Let us assume that j 1 is continuous with respect to all variables and convex with respect to its last one. Then, F 1 is a lower semi-continuous shape functional for the R-convergence, i.e., for every sequence pΩ n q nPN P O N r0 that R-converges toward Ω 8 , one has

lim inf nÑ`8 F 1 pΩ n q ě F 1 pΩ 8 q. ( 4
)
As a consequence, the shape optimization problem

inf ΩPOr 0 F 1 pΩq has a solution.
It is notable that, by applying Theorem 1 both to j 1 and ´j1 , we get the following corollary.

Corollary 1. If j 1 is continuous and linear in the last variable, then F 1 is a continuous shape functionals for the R-convergence.

Remark 2. In the case where d " 3, it is proved in [START_REF] Dalphin | Uniform ball property and existence of optimal shapes for a wide class of geometric functionals[END_REF]Theorem 1.3] that Theorem 1 holds if we replace the mean curvature by the Gaussian one in the definition of F 2 . We do not provide a proof here since most of the difficulties are related to the convergence of a product of weak-star converging sequences and our approach does not change the proof in a significant way.

Let us now consider two classes of shape optimization problems involving either an elliptic PDE inside Ω or an elliptic PDE on the C 1,1 hypersurface BΩ.

Problems involving an elliptic PDE on a C 1,1 -hypersurface of R d . Given f P C 0 pDq, we consider the problem of minimizing a shape functional depending on the solution v BΩ of the equation

∆ Γ v BΩ pxq " f pxq in BΩ, (5)
where ∆ BΩ denotes the Laplace-Beltrami operator on BΩ. Since we are not considering C 8 manifolds but rather C 1,1 ones, we need to explain how the PDE must be understood. We use here an energy formulation, defining, for a closed and nonempty hypersurface BΩ, the functional

E BΩ : H 1 ˚pBΩq Q u Þ Ñ 1 2 ż BΩ |∇ Γ upxq| 2 dµ BΩ ´żBΩ f pxqupxqdµ BΩ (6)
where H 1 ˚pBΩq denotes the Sobolev space of functions in H 1 pBΩq with zero mean on BΩ. We hence define v BΩ as the unique solution of the minimization problem min uPH The proof of this result is postponed to Section C. Let us introduce the shape functional

F 2 pΩq " ż BΩ j 2 px, νpxq, v BΩ pxq, ∇ Γ v BΩ pxqq dµ BΩ pxq,
where j 2 : R d ˆSd´1 ˆR ˆRd Ñ R is assumed to be continuous.

Theorem 2. The shape functional F 2 is lower semi-continuous for the R-convergence, i.e., for every sequence

pΩ n q nPN P O N r0 that R-converges toward Ω 8 , one has lim inf nÑ`8 F 2 pΩ n q ě F 2 pΩ 8 q. ( 8
)
As a consequence, the shape optimization problem

inf ΩPOr 0 F 2 pΩq has a solution.
Problems involving an elliptic PDE in a domain of R d . Finally, let us investigate the case of a shape criterion involving the solution of a PDE on a domain of R d . We consider hereafter a Poisson equation with non-homogeneous boundary condition, but we claim that all conclusions can be easily extended to a larger class of elliptic PDEs. Let h P L 2 pDq, g P H 2 pDq, and define u Ω as the solution of

" ∆u Ω " h in Ω, u Ω " g in BΩ. (9)
Let us introduce the shape functional F 3 given by

F 3 pΩq " ż BΩ j 3 px, νpxq, u Ω pxq, ∇u Ω pxqq dµ BΩ pxq,
where j 3 : R d ˆSd´1 ˆR ˆRd Ñ R is continuous.

Theorem 3 ([5], Theorem 2.1). The shape functional F 3 is lower semi-continuous for the R-convergence.

We mention this theorem demonstrated in [START_REF] Dalphin | Existence of optimal shapes under a uniform ball condition for geometric functionals involving boundary value problems[END_REF]. Nevertheless, it is notable that by adapting the proof of Theorem 2, it is possible to obtain a much shorter proof of this theorem. In order not to make this article unnecessarily heavy, we only give the main steps of the proof in Section 2.5. This example is mentioned both for the sake of completeness, in order to review the existing literature, and also to underline the potential of the approach introduced here, which allows to find more direct proofs of all the known results and to extend them.

In addition, it is interesting to notice that our approach allows to deal with problems involving PDEs both using weak formulations as in [START_REF] Guo | On convergence of boundary Hausdorff measure and application to a boundary shape optimization problem[END_REF] and also whose solutions are obtained using a minimization principle, as is the case in [START_REF] Dalphin | Existence of optimal shapes under a uniform ball condition for geometric functionals involving boundary value problems[END_REF]. The approach thus seems robust and we believe that it can be easily adapted to general families of problems (for example to a general non-degenerate elliptic PDE).

Proofs

The extruded surface approach

One of the key ideas to prove sequential continuity of functionals involving an integral on the boundary is to approximate such an integral by an integral on a small tubular neighborhood (as done e.g. in [START_REF] Delfour | Tangential differential calculus and functional analysis on a C 1,1 submanifold[END_REF]).

Let us first illustrate the method by proving Point 5 of Lemma 1.

Proof of Lemma 1, Point 5

For 0 ă h ă r 0 , consider

T : p´h, hq ˆBΩ Ñ U h pBΩq pt, xq Þ Ñ x `t∇b Ω pxq. ( 10
)
Since T is Lipschitz continuous, it is differentiable at almost every pt 0 , x 0 q, with d pt0,x0q T ps, yq " y `s∇b Ω px 0 q `t0 d x0 ∇b Ω pyq, @ps, yq P R ˆTx0 BΩ.

Remark 3. Note that as ∇b Ω px 0 q is a normal unit vector to BΩ at x 0 , we can identify the tangent hyperplane T x0 BΩ with R d´1 endowed with an Euclidean structure inherited from that of R d . We will use this identification several times in this paper.

As a result, we can identify R ˆTx0 BΩ Q ps, yq Þ Ñ y `s∇b Ω px 0 q with an orthogonal matrix. Moreover, up to the choice of a different orientation on T x0 BΩ, such a matrix belongs to the special orthogonal group SOpnq. We use the same coordinate representation to identify R ˆTx0 BΩ Q ps, yq Þ Ñ d x0 ∇b Ω pyq with a n ˆn matrix. By uniform continuity of the determinant around SOpdq, there exists C 0 ą 0 such that, for every M P SOpdq and every l P M d pRq such that }l} ď C 0 ,

1 2 ď detpM `lq ď 3 2 .
As ∇b Ω is 2 r0 -Lipschitz continuous on BΩ, we have that for almost every x 0 P BΩ and every

t 0 P R, }t 0 d x0 ∇b Ω } ď 2|t0| r0 .
Let us fix h ă minpr 0 , r 0 C 0 {2q (independent of Ω), so that }t 0 d x0 ∇b Ω } ď C 0 for almost every x 0 P BΩ and every t 0 P p´h, hq. By the change of variable formula we then have

H d´1 pBΩq " ż BΩ dµ BΩ " 1 2h ż U h pBΩq detpd T ´1pyq T qdy ď 3 4h H d pU h pDqq,
whence the conclusion.

Extruded surface and R-convergence

Let us now illustrate the power of this approach in the case of a R-converging sequence.

Let Ω n R Ý Ñ Ω 8 . From now on, we will use the notation Γ n :" BΩ n for the hypersurfaces. For h ă r 0 and n P N, let us define a parametrization of a neighborhood of Γ n by

T n : p´h, hq ˆΓn Ñ U h pΓ n q pt, xq Þ Ñ x `t∇b Ωn pxq. (12)
Lemma 4. For every ε ą 0, there exists h ą 0 such that for all n P N, 1 ´ε ď detpd pt0,x0q T n q ď 1 `ε, for a.e. pt 0 , x 0 q P p´h, hq ˆΓn .

Proof. We follow the same argument as in Section 2.1.1. Namely, for a given ε ą 0, there exists C 0 ą 0 such that for every M P SOpdq and every l P M d pRq such that }l} ď C 0 ,

1 ´ε ď detpM `lq ď 1 `ε.
Let us fix h ă minpr 0 , r 0 C 0 {2q (independent of n). As ∇b Ωn is 2 r0 -Lipschitz continuous on Γ n , we get }t 0 d x0 ∇b Ωn } ď C 0 for almost every x 0 P Γ and every t 0 P p´h, hq. Whence, using Equation [START_REF] Jost | Riemannian geometry and geometric analysis[END_REF] with the previous estimate, we conclude the proof.

Remark 4.

In what follows, we will use the Bachmann-Landau notation o hÑ0 p1qfor a function converging to 0 in L 8 as h goes to 0 and for a given n, large enough. For example, Lemma 4 implies that detpdT n q " 1 `ohÑ0 p1q, on p´h, hq ˆΓn , which means @ε ą 0, DN 0 P N, Dh ą 0, @n P N, n ě N 0 implies ˇˇdetpd pt0,x0q T n q ´1ˇˇď ε, for a.e. pt, xq P p´h, hq ˆΓn .

Let us now introduce the orthogonal projection p n onto Γ n , defined on U h pΓ n q for every h P p0, r 0 q.

Lemma 5. The following properties hold:

1. p n coincides with the second component of T ´1 n : U h pΓ n q Ñ p´h, hq ˆΓn .

2. For all x P U h pΓ n q, p n pxq " x ´bΩn pxq∇b Ωn pxq. We can now state the key equality to relate surface and volume integrals. Apply Lemma 4 with ε P p0, 1q to select h ą 0 such that T n : p´h, hq ˆΓn Ñ U h pΓ n q is invertible for every n P N1 . Lemma 6. For all n P N, f P L 1 pΓ n q, and t P p0, hq we have

ż Γn f pxq dµ Γn pxq " 1 2t ż UtpΓnq f ˝pn pyq detpd T ´1 n pyq T n q dy.
Proof. Using the change of variable formula (also known as area formula for Lipschitz continuous functions), one gets

ż J ´t ż Γn f pxq dµ Γn pxqdt " ż UtpΓnq f ˝pn pyq detpd T ´1 n pyq T n q dy.
From now on, we will the T ´1 n pyq inside the determinant to improve the readability.

Lemma 7.

For every h ď r 0 {2 and 0 ă t ă h, there exists N 0 such that @n ě N 0 , U h´t pΓ 8 q Ă U h pΓ n q Ă U h`t pΓ 8 q.

Proof. By uniform convergence of b Ωn toward b Ω8 , we have that for n large enough

b ´1 Ω8 ppt ´h, h ´tqq Ă b ´1 Ωn pp´h, hqq Ă b ´1 Ω8 pp´h ´t, h `tqq.
In order to perform changes of variable in surface integrals, it is convenient to use directly p n as a way to map Γ 8 onto Γ n . To this aim, we define

τ n : Γ 8 Ñ Γ n x Þ Ñ p n pxq. (13)
Note that for n large enough, Lemma 7 ensure that τ n is well-defined. We also introduce Jacpτ n q to denote the Jacobian of τ n . Then we have the following lemma. This is a contradiction, hence τ n is injective which implies that it is a diffeomorphism from Γ 8 to Γ n .

Proof of Lemma 2

Suppose that pΩ n q nPN P O N r0 R-converges toward Ω 8 P O r0 .

Proof of Point 1

For h ă r 0 and using Lemma 6, we have

H d´1 pΓ n q " ż Γn dµ Γn pxq " 1 2h ż U h pΓnq
detpdT n q dy. By Lemma 7, moreover,

H d´1 pΓ n q " 1 2h ż U h´t pΓ8q detpdT n q dy `1 2h ż U h pΓnqzU h´t pΓ8q
detpdT n q dy for t P p0, hq and n large enough. Let us compare the first term in the right-hand side with

H d´1 pΓ 8 q " 1 2ph ´tq ż U h´t pΓ8q detpdT 8 q dy.
Using Lemma 4, detpdT 8 q " detpdT n q `ohÑ0 p1q on p´h, hq ˆΓ8 . Besides,

1 2h ´1 2ph´tq " O `t h ˘. Hence, 1 2h ż U h´t pΓ8q
detpdT n q dy " H d´1 pΓ 8 q `ohÑ0 p1q `O ˆt h ˙.

On the other hand, using again the relation detpdT 8 q " detpdT n q `ohÑ0 p1q,

1 2h ż U h pΓnqzU h´t pΓ8q
detpdT n q dy ď 1 2h

ż U h`t pΓ8qzU h´t pΓ8q
detpdT n q dy " 1 2h `żU h`t pΓ8q detpdT n q dy ´żU h´t pΓ8q detpdT n q dy " 1 2h p2ph `tqH d´1 pΓ 8 q ´2ph ´tqH d´1 pΓ 8 q `ohÑ0 phqq

" 2t h H d´1 pΓ 8 q `ohÑ0 p1q " O ˆt h ˙`o hÑ0 p1q.
By taking h arbitrary small while t " h 2 , we prove that H d´1 pΓ n q Ñ H d´1 pΓ 8 q.

Proof of Point 2

Using the uniform convergence of b Ωn to b Ω8 , we deduce that for every ε ą 0 there exists

N 0 P N such that b ´1 Ω8 pp´8, ´εsq Ă b ´1 Ωn pp´8, 0qq Ă b ´1 Ω8 pp´8, εqq, @n ě N 0 .
Hence, we get

H d pb Ω8 ď ´εq ď H d pΩ n q ď H d pb Ω8 ă εq.
By inner regularity of

H d , H d pb Ω8 ď ´εq εÑ0 Ý ÝÝ Ñ H d pb Ω8 ă 0q " H d pΩ 8 q.
Similarly, by outer regularity

H d pb Ω8 ă εq εÑ0 Ý ÝÝ Ñ H d pb Ω8 ď 0q " H d pΩ 8 q
, where we used that Ω 8 belongs to O r0 .

Proof of Point 3

We want to prove that Γ n is isotopic to Γ 8 for n large enough. We consider

ϕ n pt, xq : r0, 1s ˆΓ8 Ñ R 3 pt, xq Þ Ñ x `tpp n pxq ´xq.
According to Lemma 8, ϕ n p1, ¨q " τ n is a diffeomorphism from Γ 8 onto Γ n . Besides, following the proof of Lemma 8, we easily get that for t P p0, 1q, ϕ n pt, ¨q is a diffeomorphism onto its image.

Proof of Theorem 1

Suppose that pΩ n q nPN P O N r0 R-converges toward Ω 8 P O r0 . Let 0 ă t ă h small enough (to be fixed later) and n large enough.

We recall that the unit normal vector to Γ n is given by ∇b Ωn (see Lemma 5). Then, according to Lemma 6, The key idea is to prove that all arguments of j 1 in the first term convergence toward their analogues for n " 8 and to ensure that the second term is small for t small. Let us start with comparing the first term in the right-hand side with F 1 pΩ 8 q. Notice that detpdT n q detpdT 8 q j 1 pp n pyq, ∇b Ωn pp n pyqq, H Γn pp n pyqqq detpdT 8 q dy.

F 1 pΩ n q " ż Γn j 1 px
By Lemma 4, we have

› › › › 2ph ´tq 2h detpdT n q detpdT 8 q ´1› › › › L 8 pU h pΓ8qq " o hÑ0 p1q `O ˆt h ˙. (18)
Let us now investigate the mean curvature term. Note that this term is slightly technical to handle for two reasons:

• the mean curvature H Γn is defined as the trace of the shape operator, which is itself defined as the differential of the restriction to the hypersurface of ∇b Ωn (see A);

• the Hessian of b Ωn converges only in a weak sense.

We will use the following lemma, which is obtained thanks to the chain rule. Besides, one has that ∇ 2 b Ωn pτ ´1 n pp n pxqqq exists as well. Notice that the last part of the statement is not explicitly contained in [START_REF] Delfour | Tangential differential calculus and functional analysis on a C 1,1 submanifold[END_REF] but can be obtained by straightforwardly adapting the proof of its Theorem 4.4.

As ∇ 2 b Ωn is uniformly bounded on a neighboorhood of Γ 8 and that b Ωn pxq ď h for x P U h pΓ n q, there exists C ą 0 such that ess sup xPU h pΓnq }rId ´bΩn pxq∇ 2 b Ωn pxqs ´1 ´Id } ď Ch, for h small. As a consequence, using Lemma A.1 (given in the appendix), one has

H Γn pp n pxqq " Tr ∇ 2 b Ωn pp n pxqq " Tr ∇ 2 b Ωn pxq `Ophq.
Note also that H Γn ď 1 r0 on Γ n . We can use the uniform continuity of j 1 on a compact set to ensure that for n large enough and n " 8, weak star in L 8 pU r 0 2 pΓ 8 qq. Thus, using for example [START_REF] Berkovitz | Lower semicontinuity of integral functionals[END_REF], we have lim inf

1
nÑ8 1 2ph ´tq ż U h´t pΓ8q j 1 pp n pyq, ∇b Ωn pp n pyqq, Tr ∇ 2 b Ωn pyqq detpdT 8 q dy ě 1 2ph ´tq ż U h´t pΓ8q j 1 pp 8 pyq, ∇b Ω8 pp 8 pyqq, Tr ∇ 2 b Ω8 pyqq detpdT 8 q dy " 1 2ph ´tq ż U h´t pΓ8q j 1 pp 8 pyq, ∇b Ω8 pp 8 pyqq, Tr ∇ 2 b Ω8 pp 8 pyqqq detpdT 8 q dy `Ophq " F 1 pΩ 8 q `Ophq. (20)
In order to conclude, let us check that the term in line (17) is small. Since j 1 is continuous on a compact set, it admits a minimum m 0 P R. Let m 1 " minp0, m 0 q ď 0. Then,

1 2h ż U h pΓnqzU h´t pΓ8q j 1 pp n pyq, ∇b Ωn pp n pyqq, H Γn pp n pyqqq detpdT n q dy ě 1 2h ż U h pΓnqzU h´t pΓ8q m 1 detpdT n q dy ě 1 2h ż U h`t pΓ8qzU h´t pΓ8q m 1 detpdT n q detpdT 8 q detpdT 8 q dy. Using › › › › detpdT n q detpdT 8 q ´1› › › › L 8 pU 2h pΓ8qq
" o hÑ0 p1q and ż

U h˘t pΓ8q m 1 detpdT 8 q dy " 2ph ˘tqm 1 H d´1 pΓ 8 q, we get

1 2h ż U h pΓnqzU h´t pΓ8q j 1 pp n pyq, ∇b Ωn pp n pyqq, H Γn pp n pyqqq detpdT n q dy ě m 1 H d´1 pΓ 8 q ˆohÑ0 p1q `O ˆt h ˙˙. (21)
Finally, combining Equations ( 19)-( 21), we obtain lim inf nÑ`8

F 1 pΩ n q ě pF 1 pΩ 8 q `Ophqq ˆ1 `O ˆt h ˙˙`m 1 H d´1 pΓ 8 q ˆohÑ0 p1q `O ˆt h ˙˙.
Hence, taking h Ñ 0 while ensuring t " ophq gives lim inf nÑ`8

F 1 pΩ n q ě F 1 pΩ 8 q,
and finishes the proof.

Proof of Theorem 2

Let pΩ n q nPN denote a sequence that R-converges to Ω 8 , and let v n denote the unique solution v Γn to Problem [START_REF] Delfour | Shapes and geometries[END_REF] for Ω " Ω n . The difficult part here is that v Γn is not defined on Γ 8 . Our main tool will be τ n , the restriction to Γ 8 of the orthogonal projection p n on Γ n . Those objects were introduced in Section 2.1.2 and we proved that τ n is a diffeomorphism between Γ 8 and Γ n in Lemma 8.

We also have to be careful when we transport the tangential gradient of a function. In order to relate the tangential gradient and the ambient gradient, we establish the following pointwise estimate. Lemma 10. Let n P N and f n P H 1 pΓ n q. Then f n ˝τn P H 1 pΓ 8 q and, for almost every x P Γ 8 ,

∇ Γ8 pf n ˝τn qpxq " ∇ Γn f n pτ n pxqqpId `Cn pxqq, ( 22
)
where tangential gradients are understood as d-dimensional line vectors and C n pxq " p∇b Ωn pxq J ∇b Ωn pxq ´Idq∇b Ω8 pxq J ∇b Ω8 pxq `bΩn pxq∇ 2 b Ωn pxqp∇b Ω8 pxq J ∇b Ω8 pxq ´Idq.

Besides, C n converges toward zero in the L 8 norm:

ess sup xPΓ8 }C n pxq} ÝÝÝÑ nÑ8 0. ( 23
)
Proof. First notice that ∇ Γ8 pf n ˝pn qpxq " ∇pf n ˝pn ˝p8 qpxq

for almost every x P BΩ 8 , since the directional derivative of f n ˝pn ˝p8 at the point x in the direction ∇b Ω8 pxq is zero. By Lemma 9, ∇ 2 b Ωn pxq is well-defined for almost every x in Γ 8 . By Lemma 5 and the chain rule we obtain, almost everywhere on Γ 8 , ∇pf n ˝pn ˝p8 qpxq " pp∇f n q ˝pn qpId ´∇b J Ωn ∇b Ωn ´bΩn ∇ 2 b Ωn qpId ´∇b J Ω8 ∇b Ω8 ´bΩ8 ∇ 2 b Ω8 q " pp∇ Γn f n q ˝τn qpId ´pId ´∇b J Ωn ∇b Ωn q∇b J Ω8 ∇b Ω8 ´bΩn ∇ 2 b Ωn pId ´∇b J Ω8 ∇b Ω8 qq, where we used that ∇f n " ∇ Γn f n , p n " τ n , and b Ω8 " 0 on Γ 8 . This shows Eq. ( 22).

Let us now bound the L 8 norm of C n . There exists C ą 0 such that, for every n satisfying Γ 8 Ă U r 0 2 pΓ n q, ess sup xPΓ8 }∇ 2 b Ωn pxqp∇b J Ω8 pxq∇b Ω8 pxq ´Idq} ď C. Besides, }b Ωn } L 8 pΓ8q converges to zero. Finally, using the uniform convergence of ∇b Ωn toward ∇b Ω8 , we get

∇b J Ωn ∇b Ωn ∇b J Ω8 ∇b Ω8 L 8 pΓ8q Ý ÝÝÝÝ Ñ nÑ8 ∇b J Ω8 p∇b Ω8 ∇b J Ω8 q∇b Ω8 " ∇b J Ω8 ∇b Ω8 .
This concludes the proof of Eq. ( 23).

From the solution v n in H 1 ˚pΓ n q, we introduce the function w n defined on Γ 8 by

w n " v n ˝τn ´1 H d´1 pΓ 8 q ż Γ8 v n ˝τn dµ Γ8 . ( 24
)
Note that, defined as such, w n belongs to H 1 ˚pΓ 8 q.

Step 1: convergence of pw n q nPN . Let us start by considering the sequence of energies pE Γn pv n qq nPN . This sequence is upper bounded by 0, since E Γn pv n q ď E Γn p0q " 0 for every n. By using the uniform Poincaré inequality stated in Proposition C.1 combined with the Cauchy-Schwarz inequality, we get that p ş Γn |v n | 2 dµ Γn pxqq nPN is bounded. We now compute 1

H d´1 pΓ 8 q ż Γ8 v n ˝τn dµ Γ8 " 1 H d´1 pΓ 8 q ż Γn v n Jacpτ n qdµ Γn " ˜aH d´1 pΓ n q H d´1 pΓ 8 q }v n } L 2 pΓnq ¸onÑ8 p1q,
where we used Lemma 8, the Cauchy-Schwarz inequality, and the fact that v n has zero average on Γ n . Hence, we infer that w n " v n ˝τn `onÑ8 p1q. Besides, by performing a change of variable and by using Lemmas 8 and 10, we get

ż Γn ˆ1 2 |∇ Γn v n pyq| 2 ´f pyqv n pyq ˙dµ Γn pyq " ż Γ8 ˆ1 2 |∇ Γn v n pτ n pyqq| 2 ´f pτ n pyqqpv n ˝τn qpyq ˙Jacpτ n q ´1dµ Γ8 pyq " ż Γ8 ˆ1 2 |∇ Γ8 w n pyq| 2 ´f pτ n pyqqw n pyq ˙dµ Γ8 pyq `onÑ8 p1q,
where we used that ∇ Γ8 w n " ∇ Γ8 pv n ˝τn q by definition of w n . By using Proposition C.1 and again the Cauchy-Schwarz inequality, we successively infer that the sequences p ş Γ8 |w n | 2 dµ Γ8 pxqq nPN and p ş Γ8 |∇ Γ8 w n | 2 dµ Γ8 pxqq nPN are bounded. By using Theorem C.1, the sequence pw n q nPN converges up to a subsequence toward w 8 P H 1 ˚pΓ 8 q, weakly in H 1 pΓ 8 q and strongly in L 2 pΓ 8 q. Up to extracting a subsequence, we get ż

Γ8 |∇ Γ8 w 8 pxq| 2 dµ Γ8 ď lim inf nÑ`8 ż Γ8 |∇ Γ8 w n pxq| 2 dµ Γ8 , lim nÑ8 ż Γ8 w n pxqf pτ n pxqq dµ Γ8 " ż Γ8 w 8 pxqf pxq dµ Γ8 .
As a consequence, E Γ8 pw 8 q ď lim inf nÑ`8

E Γn pv n q.
Step 2: Minimality of w 8 . Let u P H 1 ˚pΓ 8 q be given and define z n in H 1 ˚pΓ n q by

z n " u ˝τ ´1 n ´1 H d´1 pΓ n q ż Γn u ˝τ ´1 n dµ Γn . ( 25
)
Let n P N. By minimality, one has E Γn pv n q ď E Γn pz n q.

By mimicking the arguments and computations of the first step, we easily get that

E Γn pz n q " E Γ8 puq `onÑ8 p1q, (26)
yielding at the end E Γ8 pw 8 q ď E Γ8 puq. We infer that w 8 is the unique solution to the variational problem [START_REF] Delfour | Shapes and geometries[END_REF]. Since the reasoning above holds for any closure point of pw n q nPN , it follows that the whole sequence pw n q nPN converges toward w 8 , weakly in H 1 pΓ 8 q and strongly in L 2 pΓ 8 q. Finally, using u " w 8 in (26), we obtain that E Γ8 pw 8 q " lim inf nÑ`8 E Γn pv n q.

In particular p}w n } 2 H 1 pΓ8q q nPN converges toward }w 8 } 2 H 1 pΓ8q which implies the strong convergence of w n in H 1 pΓ 8 q.

Step 3: lower semi-continuity of F 2 . Let us use the same notations as previously. Using a change of variable, we get

F 2 pΩ n q " ż Γn j 2 px, ∇b Ωn pxq, v n pxq, ∇ Γn v n pxqq dµ Γn pxq " ż Γ8 j 2 pτ n pxq, ∇b Ωn pτ n pxqq, v n pτ n pxqq, ∇ Γn v n ˝τn pxqq Jacpτ n q ´1dµ Γ8 pxq.
Besides, according to the results above and Lemma 5, the following convergences hold

$ ' ' ' ' ' ' ' & ' ' ' ' ' ' ' % Jacpτ n q ´1 ÝÝÝÑ nÑ8 1 strongly in L 8 pΓ 8 q τ n ÝÝÝÑ nÑ8 Id | Γ8 strongly in L 8 pΓ 8 q ∇b Ωn ˝τn ÝÝÝÑ nÑ8 ∇b Ω8 strongly in L 8 pΓ 8 q v n ˝τn ÝÝÝÑ nÑ8 w 8 strongly in L 2 pΓ 8 q ∇ Γn v n ˝τn ÝÝÝÑ nÑ8 ∇ Γ8 w 8 strongly in L 2 pΓ 8 q,
where w 8 is the unique solution to the variational problem [START_REF] Delfour | Shapes and geometries[END_REF].

By applying [1, Theorem 1], one has lim inf nÑ`8

F 2 pΩ n q ě F 2 pΩ 8 q.
This is the desired conclusion.

Main steps in the proof of Theorem 3

First note that u Ω ´g solves Eq. ( 9) with source term h ´∆g and Dirichlet boundary condition. As a consequence, we can reduce our study to the case of homonegeous Dirichlet condition (i.e., u Ω " 0 on Γ).

The method relies on a uniform extension property proved by Chenais in [START_REF] Chenais | On the existence of a solution in a domain identification problem[END_REF] for surfaces satisfying an ε-cone condition, which is weaker than the uniform ball condition.

Lemma 11 ([3, Theorem II.1]

). There exists a positive constant C (depending only on r 0 and D) such that for every Ω P O r0 there exists an extension operator E Ω P LpH 2 pΩq, H 2 pDqq satisfying

E Ω puq| Ω " u, }E Ω } LpH 2 pΩq,H 2 pDqq ď C. ( 27
)
We will use this lemma to extend the solution of the PDEs to the whole box D. The next step is to find a uniform H 2 estimate of the solutions. In our case such an estimate was proved by Dalphin who extended a result for domains with C 2 boundary obtained by Grisvard in [START_REF] Grisvard | Elliptic Problems in Nonsmooth Domains[END_REF].

Lemma 12 ([5, Proposition 3.1]

). There exists C ą 0 (depending only on r 0 and D) such that for every Ω P O r0 and f P H 2 pΩq X H

The next step is to prove that the restriction to Ω 8 of u ˚is u Ω8 .

To this aim, let us consider an arbitrary compact set K contained in the interior of Ω 8 and a C 8 function ϕ with compact support included in K. For n large enough, K is contained in the interior of Ω n (see Lemma 7), and, therefore, one has ϕ P H 1 0 pΩ n q for such integers n. Using the variational formulation of the PDE (9), we get ż D x∇E Ωn pu Ωn q, ∇ϕy ´f ϕ " 0.

(31)

Using the density of C 8 functions with compact support in H 1 0 pΩ 8 q and passing to the limit yields that u ˚|Ω8 " u Ω8 . The last step is to relate F 3 pΩ n q and F 3 pΩ 8 q. Since the involved functions belong to Sobolev spaces and since one aims at comparing surface integrals with tubular ones, we need a suitable uniform trace result.

Lemma 13. There exists C such that for every h ă r0 2 , every n P N and every f P H 1 pU r 0 2 pΓ n qq,

}f ´f ˝pn } L 2 pU h pΓnqq ď Ch}f } H 1 pU r 0 2 pΓnqq , ( 32
)
where f denotes the trace of f on Γ n .

Proof. Let f be a smooth function. According to Lemma 5, every point y P U h pΓ n q can be written in a unique way as y " x `t∇b

q dxdt ď ż h ´h ż Γn C 2 }B ∇b Ωn f px `y∇b Ωn pxqq} 2 L 2 y p´r 0 2 , r 0 2 q |t| detpdT n q dxdt ď C 2 h 2 ż Γn }B ∇b Ωn f px `y∇b Ωn pxqq} 2 L 2 y p´r 0 2 , r 0 2 q p1 `ohÑ0 p1qq dx ď C 2 h 2 }f } 2 H 1 pU r 0 2 pΓnqq p1 `ohÑ0 p1qq.
We conclude thanks to the density of the smooth functions in H 1 .

Using that u Ωn is uniformly bounded in H 2 pDq, let us apply Lemma 13 to u Ωn and ∇u Ωn . We obtain }u Ωn ´uΩn ˝pn } 2 L 2 pU h pΓnqq `}∇u Ωn ´p∇u Ωn q ˝pn } 2 L 2 pU h pΓnqq " o hÑ0 phq. The end of the proof is similar to the one of Theorem 1 and consists in using the extruded surface approach to prove lim inf nÑ`8 F 3 pΩ n q ě p1 `ohÑ0 p1qq lim inf nÑ`8

Conclusion

In this paper, we have introduced a new method to tackle the existence issue for shape optimization problems under uniform reach constraints on the considered shapes, of the type inf ΩPOr 0 ż BΩ jpx, ν BΩ pxq, B BΩ pxqq dµ BΩ pxq.

While several references such as [START_REF] Guo | On convergence of boundary Hausdorff measure and application to a boundary shape optimization problem[END_REF][START_REF] Dalphin | Uniform ball property and existence of optimal shapes for a wide class of geometric functionals[END_REF][START_REF] Dalphin | Existence of optimal shapes under a uniform ball condition for geometric functionals involving boundary value problems[END_REF] have already addressed similar questions on the same type of problems, we believe that the approaches developed in this paper are on the one hand simpler, but also sufficiently robust to allow easy extension of the results to more general settings. For example, we believe that minor adaptations of the developed proof techniques allow one to extend our results to the following cases without much effort:

-Under weaker regularity hypotheses, one could think of replacing the continuity assumption by lower semicontinuity on the integrand j t1,2,3u . Another example would be to assume that f in equation ( 5) belongs to H 1{2 pDq instead of CpDq.

-More general PDEs could be considered Theorems 2 and 3. Extension to general elliptic equations associated with differential operators of the kind ∇ Γ ¨pσ∇ Γ q satisfying a coercivity property should be straightforward. We also believe that our framework allows extensions to nonlinear elliptic PDEs under reasonable assumptions.

-One could consider costs involving the solution of a minimization problem depending on Ω but not necessarily related to a PDE. Indeed, in the proof of Theorem 2, our study of the variational problem does not rely on the underlying PDE. We treated a case involving a convex minimization problem over the set of divergence-free vectors fields on BΩ in [START_REF] Privat | Optimal shape of stellarators for magnetic confinement fusion[END_REF].

All those generalizations do not seem obvious when using other methods.

B R-convergence: proof of Proposition 1

The compactness property follows from two facts. First, the Arzelà-Ascoli theorem, combined with the fact that every function b Ω , for Ω P O r0 , is 1-Lipschitz continuous. Second, the reach constraint which imposes a uniform bound on the second derivative of b Ω . These two facts are used in [START_REF] Delfour | Shapes and geometries[END_REF] and [START_REF] Dalphin | Uniform ball property and existence of optimal shapes for a wide class of geometric functionals[END_REF] to get the sequential compactness results used below. Let pΩ n q nPN denote a sequence in O r0 . By the compactness property of sets of uniformly positive reach proved in [START_REF] Delfour | Shapes and geometries[END_REF]Chapter 6], it follows that, up to a subsequence, b Ωn converges to b Ω8 for the C 0 topology on D. In [START_REF] Dalphin | Uniform ball property and existence of optimal shapes for a wide class of geometric functionals[END_REF] the convergence is shown to hold also for the strong C 1,α topology (for α ă 1) and for the weak W 2,8 topology in a r-tubular neighborhood of BΩ 8 , with r ă r 0 .

As a consequence, 8 q ě r 0 . In particular, according to Lemma 1, b Ω8 is C 1,1 on U r pΓ 8 q.

C The Laplace-Beltrami equation on a manifold: proof of Lemma 3

Let pBΩ, gq denote a closed compact manifold. We explain hereafter how to understand the equation ∆ BΩ v " h in BΩ in a weak sense, whenever Ω P O r0 . Indeed, under this assumption, BΩ is a C [START_REF] Delfour | Tangential differential calculus and functional analysis on a C 1,1 submanifold[END_REF]Th 4.5.ii], since BΩ is C 1,1 , the L 2 norm } ¨}L 2 pBΩq on the surface BΩ and the L 2 norm L 2 pBΩq Q u Þ Ñ }u ˝pΩ } L 2 pU h pBΩqq on the thickened surface U h pBΩq are equivalent whenever h ą 0 is small enough, where p Ω pxq denotes the orthogonal projection of x onto BΩ, that is, p Ω pxq " x ´bΩ pxq∇b Ω pxq, and U h pBΩq " tx P R d | |b Ω pxq| ă h and p Ω pxq P BΩu.

Similarly, according to [6, Th 4.7.v], since BΩ is C 1,1 , the norm } ¨}H w n pxq dµ Γ8 " 0.

By using the first equality, we get that w ˚is constant on Γ and we obtain a contradiction with the two last equalities above.

Let us now prove Lemma 3. Let pu n q nPN denote a minimizing sequence for Problem [START_REF] Delfour | Shapes and geometries[END_REF]. Since pE Γ pu n qq nPN is bounded, and since E Γ pu n q ě Cpd, r 0 q}u n } 2 L 2 pΓq ´}h} L 2 pΓq }u n } L 2 pΓq according to Proposition C.1, we infer that p}u n } L 2 pΓq q nPN is bounded. Since ż Γ |∇ Γ u n pxq| 2 dµ Γ " E Γ pu n q `żΓ u n pxqhpxq dµ Γ ď }h} L 2 pΓq }u n } L 2 pΓq , we infer the existence of u ˚P H 1 ˚pΓq such that, up to a subsequence, pu n q nPN converges weakly in H 1 ˚pΓq and strongly in L 2 pΓq. Up to extracting a subsequence, we get inf

5 .

 5 There exists a constant C depending only on d, r 0 , and D such that H d´1 pBΩq ď C. Points 1 and 2 are proved in [7, Theorem 8.2, Chapter 7]. Points 3 and 4 are proved in [4, Theorems 2.7 and 2.8]. The proof of Point 5 is given in Section 2.1.1.

3 .

 3 p n converges toward p 8 in L 8 pU h pΓ 8 qq. Proof. Items 1 and 2 are obviously equivalent and are proved in [7, Theorem 7.2, Chapter 7]. Item 3 follows from the C 1 convergence of b Ωn toward b Ω8 .

Lemma 9 (∇ 2 b

 92 Theorem 4.4 of[START_REF] Delfour | Tangential differential calculus and functional analysis on a C 1,1 submanifold[END_REF]). Let h ă r 0 and n P N. If ∇ 2 b Ωn pxq exists for x P U h pΓ 8 q, then ∇ 2 b Ωn pp n pxqq exist and Ωn pp n pxqq " ∇ 2 b Ωn pxqrId ´bΩn pxq∇ 2 b Ωn pxqs ´1.

Remark 5 .

 5 In order to replace Dirichlet boundary conditions by Neumann's ones, one can follow similar steps as those leading to Equation (30). Then, by considering the variational formulation with ϕ P C 8 pDq and passing to the limit in ż Γn gB ν ϕ Ñ ż Γ8 gB ν ϕ, (consequence of Corollary 1 if g P C 0 pDq) one gets that u ˚|Ω8 " u Ω8 .

uPH 1 ˚pΓqE 8 E

 18 Γ puq " lim inf nÑ`Γ pu n q ě ż Γ |∇ Γ u ˚pxq| 2 dµ Γ ´żΓ u ˚pxqhpxq dµ Γ " E Γ pu ˚qand the existence follows. The uniqueness is standard and follows from the strong convexity of the functional E Γ .

  Let Ω P O r0 . Problem[START_REF] Delfour | Shapes and geometries[END_REF] has a unique solution v BΩ . Furthermore, if BΩ is C 2 and if f P C 0 pDq, then v BΩ satisfies[START_REF] Dalphin | Existence of optimal shapes under a uniform ball condition for geometric functionals involving boundary value problems[END_REF] almost everywhere in BΩ.

	1	E BΩ puq.	(7)
	˚pBΩq
	Lemma 3.

  Proof. Let x P Γ 8 . We take v P T x Γ 8 , and identify it with an element of the tangent hyperplane (see Remark 3). As v is tangent to Γ 8 at x, we get xv, ∇b Ω8 pxqy " 0.Let us now fix h ă r03 . For n large enough, thanks to Lemma 7, we have Γ n Ă U h pΓ 8 q. Thus,}dx p n pvq ´v} ď }∇b Ωn pxq ´∇b Ω8 pxq}}v} `}b Ωn } L 8 pΓ8q }∇ 2 b Ωn pxq}}v} ď }v} ˆ}∇b Ωn ´∇b Ω8 } L 8 pU r 0 3 pΓ8qq `}b Ωn } L 8 pΓ8q }∇ 2 b Ωn pxq} L 8 pU r 0 We recall that both }∇b Ωn ´∇b Ω8 } L 8 pU r 0 3 pΓ8qq and }b Ωn } L 8 pΓ8q converge toward zero. Besides, the quantity }∇ 2 b Ωn pxq} L 8 pU r 0Using a similar argument to the one used in Lemma 4, we take the determinant and obtain Eq. (14).As a result we know that τ n is a local diffeomorphism. It remains to prove that τ n is injective. To this aim, we suppose that n is large enough to ensure that }∇b Ωn ´∇b Ω8 } L 8 pU r 0

						˙.
						pΓ8qq
						3
	pΓ8qq is uniformly bounded. As a consequence,
	3
	sup xPΓ8	sup vPTxΓn	}d x p n pvq ´v} ÝÝÝÑ nÑ8	0.	(15)
		}v}"1
				3	pΓ8qq ă	1 2	.
	sup xPΓ8	|Jacpτ n qpxq ´1| ÝÝÝÑ nÑ8	0.	(14)

Lemma 8. For n large enough, τ n : Γ 8 Ñ Γ n is a diffeomorphism. Besides Using Lemma 5, Item 2, we get, d x p n pvq " v ´x∇b Ωn pxq, vy∇b Ωn pxq ´bΩn pxq∇ 2 b Ωn pxqv. Let x, y P Γ 8 such that p n pxq " p n pyq. If x " y, it implies that there exists t P p´2 r0 3 , 2r0 3 qzt0u such that x " y `t∇b Ωn pp n pyqq " y `t∇b Ωn pyq As Ω 8 P O r0 , it satisfies the r 0 uniform ball property (see (B h )). Thus, one has B r0 py `r0 sign t∇b Ω8 pyqq X Γ 8 " H. But we have |x ´y ´r0 sign t∇b Ω8 pyq| " |t∇b Ωn pyq ´r0 sign t∇b Ω8 pyq| ď t 2 `|t ´r0 sign t| ă r 0 .

  , ∇b Ωn pxq, H Γn pp n pyqqqdµ Γn pxq 1 pp n pyq, ∇b Ωn pp n pyqq, H Γn pp n pyqqq detpd T ´1 n pyq T n q dy. 1 pp n pyq, ∇b Ωn pp n pyqq, H Γn pp n pyqqq detpdT n q dy.

		"	1 2h	ż U h pΓnq
	Using Lemma 7, moreover,
	F 1 pΩ n q "	1 2h	ż U h´t pΓ8q	(16)
				ż
		`1 2h	U h pΓnqzU h´t pΓ8q	(17)

j j 1 pp n pyq, ∇b Ωn pp n pyqq, H Γn pp n pyqqq detpdT n q dy j

  [START_REF] Berkovitz | Lower semicontinuity of integral functionals[END_REF] pp n pyq, ∇b Ωn pp n pyqq, H Γn pp n pyqqq detpdT 8 q dy 1 pp n pyq, ∇b Ωn pp n pyqq, Tr ∇ 2 b Ωn pyqq detpdT 8 q dy `Ophq.

		ż
	2ph ´tq	U h´t pΓ8q
	"	1 2ph ´tq	ż U h´t pΓ8q	(19)
	The next step is to pass to the limit within the integral. Note that, by definition of R-convergence,
		$ ' & ' %	p n ÝÝÝÑ nÑ8 ∇b Ωn ˝pn ÝÝÝÑ nÑ8 Tr ∇ 2 b Ωn ÝÝÝÑ	p 8 ∇b Ω8 ˝p8 strongly in L 8 pU r 0 strongly in L 8 pU r 0 2 pΓ 8 qq, 2 pΓ 8 qq,

j j nÑ8 Tr ∇ 2 b Ω8

  1 0 pΩq, we have }f } H 2 pΩq ď C}∆f } L 2 pΩq . (28) As a consequence, we have a uniform H 2 pDq estimate on the extension of the solution u Ω , namely, }E Ω pu Ω q} H 2 pDq ď C}h} L 2 pDq , @Ω P O r0 . Ω 8 . Using Eq. (29), we get that pE Ωn pu Ωn qq nPN is uniformly bounded in H 2 pDq. Up to extracting a subsequence, we can assume that E Ωn pu Ωn q ÝÝÝÑ

			(29)
	Let us now consider Ω n	R Ý Ñ nÑ8	u ˚" weakly in H 2 pDq strongly in H 1 pDq.

  Ωn pxq with x " p n pyq P Γ n and t P p´h, hq. Moreover, one has |f px `t∇b Ωn pxqq ´f pxq| 2 ď C 2 }B ∇b Ωn pxq f px `y∇b Ωn pxqq} 2 Ωn stands for the derivative in the direction ∇b Ωn pxq and C is the norm of the continuous embedding of H 1 pr´r 0 2 , r0 2 sq into the space C Γn |f px `t∇b Ωn pxqq ´f pxq| 2 detpdT n

		L 2 y p´r 0 2 ,	r 0 2 q |t|,
	where B ∇b 1 2 of 1 2 -Hölder continuous functions. Hence, using Lemma 4, we get
	}f ´f ˝pn } 2 L 2 pU h pΓnqq "	ż h ´h ż

  3 px, ∇b Ωn pp n pxqq, E Ωn pu Ωn qpxq, ∇E Ωn pu Ωn qpxqq dx 3 px, ∇b Ω8 pp 8 pxqq, u ˚pxq, ∇u ˚pxqq dx

					1	ż
					2h	U h pΓnq
	ě p1 `ohÑ0 p1qqp1	`O ˆt h	˙q lim inf nÑ`8	1 2ph ´tq	ż U h´t pΓ8q
	`ohÑ0 p1q	`O ˆt h	ě
	F 3 pΩ 8 q `ohÑ0 p1q	`O ˆt h	˙,
	which concludes the proof.

j j

  1,1 submanifold according to Lemma 1, not necessarily C 2 , which justifies why such an equation cannot be understood in a strong sense. The key ingredient in what follows is the Rellich-Kondrachov lemma, stating the compactness of the embedding H 1 ˚pBΩq ãÑ L 2 pBΩq. Theorem C.1 (Rellich-Kondrachov theorem on surfaces). Let Ω P O r0 . Let pu n q nPN denote a sequence in H 1 ˚pBΩq such that p ş BΩ |∇u n pxq| 2 dµ BΩ q nPN is bounded. There exists u ˚P H 1 ˚pBΩq such that, up to a subsequence, pu n q nPN converges to u ˚weakly in H 1 ˚pBΩq and strongly in L 2 pBΩq. Proof. According to

  |∇ Γ u ˝pΩ | 2 dµ BΩ are equivalent whenever h ą 0 is small enough. We conclude by using the standard Rellich-Kondrakov theorem (see e.g. [2, Section 9.3]) on the thickened surface U h pBΩq.The following result is a Poincaré type lemma, uniform with respect to the chosen surface in the set O r0 .Proposition C.1 (Poincaré lemma on a surface). Let r 0 ą 0 and Ω P O r0 . There exists Cpr 0 , Dq ą 0 such that Proof. Let pΩ n , v n q nPN , with v n P H 1 ˚pΓ n q, be a minimizing sequence for the problem inf Γn |v n pxq| 2 dµ Γn " 1 by homogeneity of the Rayleigh quotient. According to Proposition 1, we can assume without loss of generality that pΩ n q nPN R-converges toward Ω 8 P O r0 .Let p n denote the orthogonal projection on Γ n and let us introduce the function w n defined in U h pΓ n q for h as in Lemma 4 and n large enough by w n " v n ˝pn . We follow exactly the same lines as in the first step of the proof of Theorem 2. A direct adaptation of the first step of the proof of Theorem 2 yieldsż Γ8 |∇ Γ8 w n pyq| 2 dµ Γ8 pyq " Γn |∇ Γn v n |2 dµ Γn pxq `op1q.By using Theorem C.1, we get that the sequence pw n q nPN converges up to a subsequence toward w ˚P H 1 ˚pΓ 8 q weakly in H 1 pΓ 8 q and strongly in L 2 pΓ 8 q. Up to extracting a subsequence, we get ż Γ8 |∇ Γ8 w ˚pxq| 2 dµ Γ8 ď lim inf

					ż
					(C.1)
	We infer that
		ż
		Γ8	|∇ Γ8 w n | 2 dµ Γ8 pxq ď 1 n `op1q.
					ż
					nÑ`8	Γ8	|∇ Γ8 w n pxq| 2 dµ Γ8 " 0,
		ż
		|w ˚pxq| 2 dµ Γ8 " 1,
		Γ8
		ż
		Γ8
					1 ˚pBΩq defined as
					ż
			}u} 2 H 1 ˚pBΩq "	BΩ	|∇ Γ u| 2 dµ BΩ ,
	and the norm } ¨}H 1 U h pBΩq	given by
		}u} H 1 U h pBΩq	"	1 2h	ż U h pBΩq
			ż
		@u P H 1 ˚pΓq,
			ΩPOr 0	inf uPH 1 ˚pΓq	ş Γ |∇ Γ upxq| 2 dµ Γ ş Γ |upxq| 2 dµ Γ	.
	Let us argue by contradiction, assuming that
		ż

Γ |∇ Γ upxq| 2 dµ Γ ě Cpr 0 , Dq ż Γ |upxq| 2 dµ Γ . Γn |∇ Γn v n pxq| 2 dµ Γn ď 1 n and ż

It is actually well-known that the domain of invertibility of Tn contains Ur 0 pΓnq.

Note that in differential geometry it is common to define the mean curvature as the trace of the shape operator divided by pd ´1q.

Acknowledgements

This work has been supported by the Inria AEX StellaCage. The first author were partially supported by the ANR Project "SHAPe Optimization -SHAPO".

Appendix

A Curvatures of a submanifold

Let us quickly review the definition of the mean curvature for an oriented pd ´1q-submanifold of R d with C 1,1 regularity. To stick with our notation, we consider the submanifold to be the boundary of some Ω P O r0 .

Definition A.1. The Gauss map is the application which assigns to each x P Γ " BΩ the direct unit normal vector to Γ at x. In our setting it can be defined as

We can now define the following objects:

• the shape operator (or Weingarten map) is the differential of the Gauss map. For every x P Γ, the tangent spaces T x Γ and T N pxq S d´1 are equal as linear subspaces of R d and, the shape operator at x is self-adjoint where it is defined. See e.g. [START_REF] Jost | Riemannian geometry and geometric analysis[END_REF]Chapter 5] for a general introduction.

• The trace of the shape operator is called the mean curvature and is denoted H 2 .

• The determinant of the shape operator is called the Gauss curvature.

Remark A.1. The Gauss map is 1 r0 -Lipschitz continuous (see Lemma 1), where r 0 is the reach of Γ. Thus, the shape operator is in L 8 and at almost every x P Γ, all the eigenvalues κ 1 pxq, . . . , κ d´1 pxq of the shape operator are bounded in modulus by 1 r0 .

We insist on the fact that N is defined only on Γ and thus the shape operator is not defined on R d or any tubular neighborhood of Γ. Nevertheless, we have the following property.

Lemma A.1. The mean curvature coincides with the trace of ∇ 2 b Ω on Γ.

Proof. Let x P Γ and let B be an orthonormal basis of T x Γ. Using the identification between T x Γ and the tangent hyperplane (see Remark 3), we obtain that t∇b Ω pxqu Y B is an orthonormal basis of R d . ∇b Ω is constant along the direction ∇b Ω pxq (see e.g. [START_REF] Delfour | Shapes and geometries[END_REF]Theorem 7.8.5.ii]). As a consequence, the trace of ∇ 2 b Ω and the mean curvature coincide.


Will increasing demand of cobalt impact environment? Olivier Pourret

Consumption of most raw materials has increased steadily since World War II, and demand is expected to continue to grow in response to the burgeoning global population and economic growth, especially in Brazil, Russia, India and China and other emerging economies.

We are also using a greater variety of metals than ever before.

Figure 1 Cobalt crust sample

New technologies such as those required for modern communication and computing and to produce clean renewable, low-carbon energy require considerable quantities of many metals (Figure 1). In the light of these trends there is increasing global concern over the long-term availability of secure and adequate supplies of the minerals and metals needed by society. Of particular concern are 'critical' raw materials (Figure 2), so called because of their growing economic importance and essential contribution to emerging 'green' technologies, yet which have a high risk of supply shortage. Moreover, over a century, mining has disseminated large quantities of wastes in the environment and become a major environmental issue [START_REF] Pourret | Assessment of soil metal distribution and environmental impact of mining in Katanga (Democratic Republic of Congo)[END_REF]). As a consequence, these activities have created secondary metalliferous substrates with elevated concentrations of trace elements including Co (more than 1000 times greater than the concentration in normal soil). Such exploitation results in huge occupation of soil (over 30 000 km 2 ) and implies soil degradation. In southeastern Central Africa, trace elements contamination constitutes a major environmental issue related to low pH of Ferralsol increasing trace elements mobility, intense erosion by rainfall in the rainy season, and aerial dispersal of metal particles by wind in the dry season [START_REF] Pourret | Assessment of soil metal distribution and environmental impact of mining in Katanga (Democratic Republic of Congo)[END_REF]. Besides, Co is highly toxic and directly impacts human heath, given as there is an elevated Co exposure in the general population of southern D.R. Congo.

Figure 2

 2 Figure 2 Criticity of metals

Figure 3

 3 Figure 3 Incremental commodity demand in a 100% electric vehicle world (proportion of

Figure 4

 4 Figure 4 Global cobalt production in kt (data collected from mineral commodity summaries


Lucie Sancey

Elisabeth Garanger

Stéphanie Foillard

Guy Schoehn

Amandine Hurbin

Corinne Albiges-Rizo

Didier Boturyn

Catherine Souchier

Alexeï Grichine

Pascal Dumy

Sancey Lucie

Garanger Elisabeth

Foillard Stéphanie

Schoehn Guy

Hurbin Amandine

Albiges-Rizo Corinne

Boturyn Didier

Souchier Catherine

Grichine Alexeï

Dumy Pascal

Coll Jean-Luc
email: jean-luc.coll@ujf-grenoble.fr

Dr Jean - Coll

Luc

Clustering and internalization of integrin alphavbeta3 with a tetrameric RGD

Keywords: integrin internalization, RGD-peptide, angiogenesis, cancer, carrier

teaching and research institutions in France or abroad, or from public or private research centers.

Introduction

Targeting tumor angiogenesis for specific drug transfer into tumor masses and metastasis has been identified as a promising approach for three main reasons: i) angiogenesis is a common and genetically stable characteristic of most solid tumors, ii) it is readily accessible from the blood stream and iii) it can be targeted by specific RGD-containing peptides binding integrin α v β 3 . This integrin is indeed poorly expressed on quiescent vessels and is selectively overexpressed on activated endothelial cells of growing vessels. In addition, integrin α v β 3 is also frequently overexpressed on tumor cells, as observed in lung cancers [START_REF] Chen | Integrin alpha v beta 3-targeted imaging of lung cancer[END_REF][START_REF] Sato | Vascular integrin beta 3 and its relation to pulmonary metastasis of colorectal carcinoma[END_REF], melanomas [START_REF] Seftor | Role of the alpha v beta 3 integrin in human melanoma cell invasion[END_REF], brain tumors [START_REF] Gladson | Glioblastoma expression of vitronectin and the alpha v beta 3 integrin. Adhesion mechanism for transformed glial cells[END_REF] or breast cancers [START_REF] Rolli | Activated integrin alphavbeta3 cooperates with metalloproteinase MMP-9 in regulating migration of metastatic breast cancer cells[END_REF].

Integrins are membrane-spanning heterodimers of  and  subunits, both of which comprise a short cytoplasmic tail, a single transmembrane helix and a large extracellular domain [START_REF] Hynes | Integrins: bidirectional, allosteric signaling machines[END_REF]. Most integrins are expressed in a default low-affinity ligand-binding state but their conformation and affinity can vary in response to cellular and microenvironment stimulations [START_REF] Calderwood | Integrin activation[END_REF][START_REF] Ginsberg | Integrin regulation[END_REF]. This will also affect their lateral assembly and clustering on the surface of the cell [START_REF] Cluzel | The mechanisms and dynamics of (alpha)v(beta)3 integrin clustering in living cells[END_REF]. Several groups have developed multimeric RGD-presenting molecules, with the aim not only to increase integrin affinity and clustering but also to induce an active integrin-mediated internalization [START_REF] Garanger | Tumor targeting with RGD peptide ligands-design of new molecular conjugates for imaging and therapy of cancers[END_REF][START_REF] Gestwicki | Influencing receptor-ligand binding mechanisms with multivalent ligand architecture[END_REF][START_REF] Temming | RGD-based strategies for selective delivery of therapeutics and imaging agents to the tumour vasculature[END_REF].

We have developed a Regioselectively Addressable Functionalized Template (RAFT) cyclodecapeptide scaffold, able to present 4 cyclic RGD pentapeptide motifs. We have shown using nuclear or optical imaging methods, that RAFT-RGD allows an improved and  v  3 -specific targeting and drug delivery [START_REF] Foillard | Synthesis and biological characterisation of targeted pro-apoptotic peptide[END_REF] as well as in vivo imaging of tumors as compared to the monomeric cyclic RGD (cRGD) [START_REF] Jin | Noninvasive optical imaging of ovarian metastases using Cy5-labeled RAFT-c(-RGDfK-)4[END_REF][START_REF] Jin | In vivo optical imaging of integrin alphaV-beta3 in mice using multivalent or monovalent cRGD targeting vectors[END_REF][START_REF] Sancey | In vivo imaging of tumour angiogenesis in mice with the alpha(v)beta (3) integrin-targeted tracer (99m)Tc-RAFT-RGD[END_REF][START_REF] Garanger | New multifunctional molecular conjugate vector for targeting, imaging, and therapy of tumors[END_REF].

Surprisingly, while the interaction between RGD-ligands and integrins has been known for a long time [START_REF] Cheresh | Arg-Gly-Asp recognition by a cell adhesion receptor requires its 130-kDa alpha subunit[END_REF][START_REF] Dedhar | A cell surface receptor complex for collagen type I recognizes the Arg-Gly-Asp sequence[END_REF] and RGD-containing molecules been widely used to deliver various kinds of

* Manuscript

Tetrameric RGD vector targeting integrin v3 2 cargos including nanoparticles (liposomes or polymers), cytotoxic peptides, low molecular weight drugs and contrast-enhancing agents (fluorochromes, radiotracers) [START_REF] Temming | RGD-based strategies for selective delivery of therapeutics and imaging agents to the tumour vasculature[END_REF][START_REF] Schiffelers | Cancer siRNA therapy by tumor selective delivery with ligand-targeted sterically stabilized nanoparticle[END_REF][START_REF] Haubner | Noninvasive visualization of the activated alphavbeta3 integrin in cancer patients by positron emission tomography and [18F]Galacto-RGD[END_REF][START_REF] Ellerby | Anti-cancer activity of targeted pro-apoptotic peptides[END_REF], very little is known about the internalization mechanism of RGD peptides binding to integrin  v  3 .

Two studies have described how an antibody directed against integrin α v β 3 (mAb 17E6) and monomeric or multimeric RGD peptides are internalized. Both concluded that the internalization of monomeric RGD ligands is independent of its α v β 3 receptor and occurs via a fluid-phase endocytic pathway. In contrast, multimeric RGD molecules are co-internalized with their receptor [START_REF] Castel | RGD peptides and monoclonal antibodies, antagonists of alpha(v)-integrin, enter the cells by independent endocytic pathways[END_REF][START_REF] Schraa | Endothelial cells internalize and degrade RGD-modified proteins developed for tumor vasculature targeting[END_REF], evidence in favor of integrin aggregation and clustering.

The integrin endo/exocytic cycle [START_REF] Pellinen | Integrin traffic[END_REF][START_REF] Pizarro-Cerda | Bacterial adhesion and entry into host cells[END_REF][START_REF] Ramsay | Integrin trafficking and its role in cancer metastasis[END_REF] suggests that there are, at least, three types of pathways associated with integrin internalization: 1/ clathrin-mediated endocytosis was described for α v β 5 integrins [START_REF] Nemerow | Role of alpha(v) integrins in adenovirus cell entry and gene delivery[END_REF]; 2/ caveolae-mediated endocytosis for α 2 integrins [START_REF] Ning | Activated epidermal growth factor receptor induces integrin alpha2 internalization via caveolae/raft-dependent endocytic pathway[END_REF] and 3/ clathrin-caveolaeindependent endocytosis. To our knowledge, endocytosis of  v  3 integrins was described to occur through clathrin-dependent endocytosis [START_REF] De Deyne | The vitronectin receptor associates with clathrin-coated membrane domains via the cytoplasmic domain of its beta5 subunit[END_REF] or uncoated vesicles [START_REF] Alam | Intracellular delivery of an anionic antisense oligonucleotide via receptor-mediated endocytosis[END_REF]. Clathrin-dependent endocytosis has also been described for viruses such as adenoviruses that enter the cells after binding to the  v  3 integrin secondary receptor [START_REF] Wickham | Integrins alpha v beta 3 and alpha v beta 5 promote adenovirus internalization but not virus attachment[END_REF].

Here, we studied the biological properties of the tetrameric RAFT-RGD peptide (coupled or not to fluorescent probes) as compared to those of its monomeric counterpart cRGD so as to better understand and improve the potential of RGD-based vectors to specifically deliver therapeutic drugs directly inside the target cells. We measured their affinity (K D ) for the purified integrin α v β 3 in solution.

Results

Tetrameric RGD vector targeting integrin v3 3 Affinities of RAFT-RGD versus cRGD for the purified, soluble integrin α v β 3 .

Fluorescence correlation spectroscopy (FCS) is commonly used to characterize the dynamics of fluorescent molecules in solution. This technique allows users to measure fluorescence intensity fluctuations due to diffusion phenomenon, chemical reactions, aggregation... We first established the diffusion properties of each fluorescently labeled RGD-containing molecules in solution and then measured the variation of this parameter in the presence of a large excess of purified integrins. This provided quantitative information allowing the determination of a constant of association (K D ).

FCS analysis indicated that the tetrameric RAFT-RGD-Cy5 had a 10-fold higher affinity for its soluble receptor integrin  v  3 in HBSS (containing Ca 2+ /Mg 2+ ) than the monomeric cRGD-Cy5 (Fig. 1b). Its dissociation constant (K D ), obtained by curve fitting with a two-component model, was 3.87 nM while the K D of cRGD-Cy5 reached 41.70 nM. The non-specific RAFT-RAD-Cy5 did not interact with integrin  v  3 : the data fitted neither a two-nor even a three-component model but only fitted a one-component model, corresponding to free RAFT-RAD-Cy5. The K D of RAFT-RGD-Cy5 was also determined for a non-specific receptor integrin  3  1 : in this case, the measured K D was 1147 nM, i.e. 300 fold higher than the one obtained with  v  3 .

RAFT-RGD slows down integrin α v β 3 mobility in the cell membrane

We measured the mobility of integrin  v  3 in the membrane of adherent HEK293( 3 ) cells in the presence of the different peptides by FRAP analysis using a confocal microscope. These cells expressing natural amount of the human  v chain were stably transfected with a plasmid encoding for the human  3 chain. We focused the laser beam on the apical membrane for two reasons: 1/ integrins from this region are mobile because they are not engaged in cell-matrix adhesions and 2/ this area is less affected by cell shrinkage usually observed in the presence of RGD peptides. Adherent cells were co-incubated for 8 min with the different peptides and the R-PhycoErythrin-labeled LM609 antibody. After the RPE photobleaching in a defined area (ROI), the time for fluorescence recovery due to the lateral movement of RPE-labeled integrins on the membrane was measured (Fig. 2). Importantly, we initially verified that LM609's binding was not affected by the presence of the peptides (data not shown). Also, no significant cellular movement or changes in membrane curvature occurred during fluorescence sampling. Results presented in Fig. 2 indicate that the presence of RAFT-RGD-FITC peptide dramatically slowed down the recovery of the integrin signal into the bleached area as compared to untreated cells. In contrast, no decrease in the time of recovery was observed either by using the negative control peptide RAFT-RAD-FITC or with the monovalent cRGD-FITC. The apparent diffusion time,

RAFT-RGD can bind two  v  3 integrins simultaneously

Tetrameric RGD vector targeting integrin v3 5

In order to confirm the data obtained by FRAP, we aimed at visualizing the possible formation of integrin clusters induced by RAFT-RGD. We used negative staining electron microscopy to observe  v  3 integrin and  v  3 /RGD-peptides mixed in 1 mM Mg 2+ /Ca 2+ as illustrated in Fig. 3.

Peptides were used in excess as compared to the integrin concentration. These conditions are not supposed to maximize the number of dimers. Integrins alone or mixed with cRGD or RAFT-RAD displayed compact particles representing single heterodimers of  v  3 ± RGD as expected.

Indeed, the monomeric cRGD does not have the possibility to interact with several integrins at the same time, while the RAFT-RAD is not able to recognize them at all. Conversely  v  3 /RAFT-RGD micrographs were frequently showing larger particles corresponding to complexes of two  v  3 integrins probably linked by the multimeric RAFT-RGD. Visually, we estimated that about 10% of  v  3 /RAFT-RGD were forming integrin dimers. But, this percentage was most probably underestimated because we counted only the aggregates laying on the grid in a proper angle of examination and providing particles with this typical dimer-shape.

For example, clusters viewed down the long axis would appear more compact [START_REF] Adair | Three-dimensional EM structure of the ectodomain of integrin {alpha}V{beta}3 in a complex with fibronectin[END_REF] and were not included (the staining agent outlines only those parts of the objects that are in contact with the carbon film). Nevertheless, electron microscopy is a qualitative technique and not a quantitative technique: dimers of integrins may bind with less affinity to the carbon than monomeric integrins, thus leading to underestimate the number of dimers.

RAFT-RGD-mediated integrin α v β 3 internalization

RAFT-RGD-Cy5 and cRGD-Cy5 internalizations were observed by confocal microscopy on live HEK293( 3 ) cells (Fig. 4). RAFT-RGD-Cy5 was rapidly internalized in small vesicles after 10 min (Fig. 4a) but was also found in the cytoplasm and at cell-cell contacts. Monomeric cRGD-Tetrameric RGD vector targeting integrin v3 6 Cy5 internalization is less extensive than that of RAFT-RGD-Cy5 and the laser intensity had to be increased three times in order to obtain comparable signal intensities (Fig. 4E).

We then developed a special ELISA assay to demonstrate that RGD-peptides were inducing integrin  v  3 internalization. Briefly, integrins exposed on the surface of HEK293(β 3 ) cells were biotinylated and the cells were incubated in the presence of 0 to 1 M RAFT-RGD or 0 to 4 M cRGD for 10 min in order to keep the number of RGD motifs constant. The cells were then lysed, fractionated and the concentration of biotinylated- v  3 -integrins present into each fraction measured using ELISA. The absence of peptide (control condition) established the normal endocytosis of integrin  v  3 ; we found that 12  1% of the labeled integrins are internalized "naturally" in 10 min (Table 1). In the presence of RAFT-RGD, internalization increased in a dose-dependent manner and reached 21  2% at 1 M, corresponding to an increase of 79% vs. control. In contrast, increasing doses of cRGD (from 0.1 to 4 M) did not affect integrin internalization at all, which remained similar to that of the control (i.e. 12  1% at 1 M).

Altogether this indicated that RAFT-RGD internalization was correlated with integrin  v  3 endocytosis, while the monomeric cRGD did not affect  v  3 natural endocytosis.

RAFT-RGD internalization occurs via clathrin-mediated endocytosis

RAFT-RGD-Cy5 and cRGD-Cy5 internalization pathways were analyzed using confocal microscopy in the presence of specific inhibitors (Fig. 4). In the presence of the clathrin-inhibitor amantadine at 1 mM (Fig. 4b), RAFT-RGD-Cy5 internalization was extensively inhibited. The fluorescence was found at the cell surface mainly and especially at the cell-cell contacts. In contrast, amantadine did not affect cRGD-Cy5 internalization (compare Fig. 4e and4f). One µM of nystatin, an inhibitor of caveolae-dependent internalization had no effect on either peptide (Fig. 4c and4g). In the presence of 1 mM amiloride, internalization of the peptides remained unchanged (Fig. 4d and4h) although, we proved by using 70 kDa-dextran-FITC, that macropinocytose was correctly inhibited in these cells (data not shown).

These results were confirmed by the ELISA measurements of the integrin  v  3 amount internalized after binding to RAFT-RGD in the presence of amantadine, nystatin or amiloride (Table 1). In the presence of amantadine, 1 M of RAFT-RGD was not able any more to induce integrin internalization and the % of internalized integrins was exactly similar to the control values (12  2%). In contrast, nystatin or amiloride did not prevent RAFT-RGD-induced integrin  v  3 internalization. These data suggested that RAFT-RGD was internalized with integrin  v  3 via the clathrin-dependent pathway. Furthermore, peptides internalization was quantified from confocal microscopy analysis. The related peptide internalization indexes are reported Figure 4 (mean of Cy5-intensity/pixels). Those indexes confirmed that RAFT-RGD internalization occurred in a clathrin-dependent pathway, but also that RAFT-RGD gets into the cells more efficiently than cRGD.

Discussion

Drug vectorization mediated by specific tumor-targeting molecules could allow specific delivery of cytotoxic agents to tumors therefore limiting their systemic toxicity. Based on this concept, RGD-containing peptides have been largely used for the targeting of  v  3 -integrin expressing tumors and/or of their microvasculature. Our group contributed to the development of a synthetic multimeric RGD-based vector, called RAFT-RGD. This peptide proved to be particularly efficient for the delivery of drugs, imaging agents or both [14-16, 17 , 34 ]. However, although RGD-integrin interaction has been discovered a long time ago [START_REF] Cheresh | Arg-Gly-Asp recognition by a cell adhesion receptor requires its 130-kDa alpha subunit[END_REF][START_REF] Dedhar | A cell surface receptor complex for collagen type I recognizes the Arg-Gly-Asp sequence[END_REF], the mechanism of internalization of monomeric or multimeric RGD peptides is a poorly documented process. In this study, we focused our attention to the study of the mechanism by which the well-known cRGD, similar to the original cyclic peptide developed by Kessler et al. [START_REF] Dechantsreiter | N-Methylated cyclic RGD peptides as highly active and selective alpha(V)beta(3) integrin antagonists[END_REF][START_REF] Heckmann | Probing integrin selectivity: rational design of highly active and selective ligands for the alpha5beta1 and alphavbeta3 integrin receptor[END_REF] and its RAFTsupported tetrameric version RAFT-RGD are internalized, with a particular emphasis on the internalization pathways involved after recognition and binding to the  v  3 receptor.

Multimeric RGD-peptides are expected to present an increased affinity for the  v  3 integrin as compared to their monomeric counterpart. This has been demonstrated when comparing cyclic versus linear RGD-based peptides [START_REF] Kato | Using model substrates to study the dependence of focal adhesion formation on the affinity of integrin-ligand complexes[END_REF][START_REF] Kumar | Biochemical characterization of the binding of echistatin to integrin alphavbeta3 receptor[END_REF]. We confirmed this characteristic for the RAFT-RGD using an FCS assay. RAFT-RGD-Cy5 bound specifically to integrin  v  3 with a 10 fold higher affinity than cRGD (Fig. 1b). Surprisingly, although integrins are known to present at least two affinity states, we measured only one K D value for both RAFT-RGD-Cy5 and cRGD-Cy5. In addition, the K D value measured for cRGD was about 25 times higher than previously reported values [START_REF] Kumar | Biochemical characterization of the binding of echistatin to integrin alphavbeta3 receptor[END_REF][START_REF] Pfaff | Selective recognition of cyclic RGD peptides of NMR defined conformation by alpha IIb beta 3, alpha V beta 3, and alpha 5 beta 1 integrins[END_REF]. This suggested that, in our FCS assay, integrins were exclusively in their activated form as a result of the combined presence of octyl-β-D-glucopyranoside (unpublished observations), Mg 2+ ions [START_REF] Takagi | Global conformational rearrangements in integrin extracellular domains in outside-in and inside-out signaling[END_REF][START_REF] Dormond | Manganese-induced integrin affinity maturation promotes recruitment of alpha V beta 3 integrin to focal adhesions in endothelial cells: evidence for a role of phosphatidylinositol 3-kinase and Src[END_REF] and of cRGD peptides [START_REF] Cluzel | The mechanisms and dynamics of (alpha)v(beta)3 integrin clustering in living cells[END_REF][START_REF] Legler | Superactivation of integrin alphavbeta3 by low antagonist concentrations[END_REF] in the media, each of these factors being known to switch integrins in their high affinity state. In addition, it must be noticed that previous measurements of the K D described in the literature were based on solid-phase receptor binding assays. In our case, integrins were in solution and this certainly modified their constant of affinity.

Using FRAP, we also demonstrated that the multimeric RAFT-RGD decreased the lateral mobility of  v  3 receptors on the surface of HEK293( 3 ) cells. This suggested that the presence of four cRGD motifs onto the RAFT scaffold allowed the clustering of integrin  v  3 . This result is important because it suggested at least two cRGD motifs presented by a single RAFT molecule can bind two integrins. This was an open challenge for the RAFT scaffold, which is no more than 10 Å large. Indeed, the three-dimensional structure of purified  v  3 integrin showed that the diameter of this integrin is close to 100 Å [START_REF] Pfaff | Selective recognition of cyclic RGD peptides of NMR defined conformation by alpha IIb beta 3, alpha V beta 3, and alpha 5 beta 1 integrins[END_REF], but that the RGD binding site is on the periphery of the molecule. The 2 RGD motifs presented by a single RAFT could thus bridge 2 integrins positioned back to back. This was confirmed by EM results, which indicated that the tetrameric, but not the monomeric cRGD, could form clusters of 2 integrins. Formation of these  v  3 clusters was then immediately followed by an active internalization of the tetrameric RAFT-RGD-Cy5 via, and concomitantly with integrin  v  3 . Indeed, the natural endocytosis of this integrin almost doubled in less than 10 min in the presence of RAFT-RGD and its internalization was mainly involving clathrin-mediated endocytosis. Accordingly, this process was abolished in the presence of amantadine, a specific inhibitor of the clathrin-mediated endocytosis. Macropinocytosis and caveolae-mediated endocytosis may not be implicated since their inhibitors like amiloride or nystatin had no effect on RAFT-RGD internalization.

Interestingly, the monomeric cRGD peptide interacted in a completely different manner. Its internalization did not rely on clathrin-or caveolae-mediated endocytosis and was most probably independent of  v  3 because of the internalization of the integrin which was not affected. These results are in agreement with a previous report and indicated that cRGD can probably cross cell membranes via a fluid-phase pathway [START_REF] Schraa | Endothelial cells internalize and degrade RGD-modified proteins developed for tumor vasculature targeting[END_REF]. The corresponding efficiency of internalization is however much less efficient than that of the RAFT-RGD, which explains the lower intensity of staining of the inside of cells labeled with cRGD-Cy5 and as demonstrated by the indexes in Figure 4.

Tetrameric RGD vector targeting integrin v3 10 Viruses such as foot-and-mouth disease virus [START_REF] Verdaguer | Flexibility of the major antigenic loop of foot-and-mouth disease virus bound to a Fab fragment of a neutralising antibody: structure and neutralisation[END_REF] or adenovirus present several RGD motifs allowing their interaction with the  v  3 integrin. This interaction is a prerequisite to their internalization [START_REF] Wickham | Integrins alpha v beta 3 and alpha v beta 5 promote adenovirus internalization but not virus attachment[END_REF][START_REF] Fender | Adenovirus dodecahedron cell attachment and entry are mediated by heparan sulfate and integrins and vary along the cell cycle[END_REF], which also occurs via clathrin-coated vesicles [START_REF] Bartlett | Infectious entry pathway of adeno-associated virus and adeno-associated virus vectors[END_REF]. RAFT-RGD may thus mimic some properties of these viruses.

In summary, the tetrameric RAFT-RGD binds 10 times more strongly to its  v  3 receptor than cRGD. RAFT-RGD is actively and efficiently internalized with integrin  v  3 via clathrin-coated pits as previously described for the  v  3 integrin [START_REF] Memmo | The alphavbeta5 integrin functions as an endocytic receptor for vitronectin[END_REF]. This contrasts with the trafficking route followed by the β 1 integrin, which was shown to use preferentially a caveolae-dependent pathway [START_REF] Del Pozo | Phospho-caveolin-1 mediates integrin-regulated membrane domain internalization[END_REF]. Efficient internalization is of course an important issue for drug delivery and we proved that this RAFT-RGD molecule is indeed capable of inducing a specific and efficient targeted intracellular delivery of a toxic peptide able to destabilize mitochondria [START_REF] Foillard | Synthesis and biological characterisation of targeted pro-apoptotic peptide[END_REF].

From this study, multimeric presentation of cRGD motifs appears to be a prerequisite for the development of efficient integrin targeting and cell internalizing vectors for drug delivery to tumors.

Materials and Methods

Material

Integrin  v  3 was purchased from Chemicon International (CC1021, St Quentin en Yvelines, France). Monoclonal antibody anti-human integrin  v  3 LM609 conjugated to R-PhycoErythrin (RPE-LM609) and anti-human CD61 were purchased from Chemicon and Beckman Coulter respectively (IM0540, Villepinte, France). Cycloheximide was from Sigma Aldrich (Lyon, France). NHS-SS-biotin was from Pierce (21441, Brebières, France).

RGD-Peptides Synthesis and Fluorescent Labeling

Compounds were synthetized according to previously reported procedures [START_REF] Boturyn | Template assembled cyclopeptides as multimeric system for integrin targeting and endocytosis[END_REF]

FCS analysis

FCS study was performed on the ConfoCor 2 system (Carl Zeiss, Jena, Germany) using a 40x The goodness-of-fit (Xhi 2 ) was the mean end point for the quality of the fit (in our condition, 5E - 4 <Xhi 2 <1E -6 for a good fit). Furthermore, the residual curves had no wavy shape (see example in supplementary data).

Cell Lines and Culture Conditions

HEK293(β 3 ), stable transfectants of human β 3 from the human embryonic kidney cell line (kindly provided by J-F. Gourvest, Aventis, France), were cultured as described in Jin et al. [START_REF] Jin | In vivo optical imaging of integrin alphaV-beta3 in mice using multivalent or monovalent cRGD targeting vectors[END_REF].

The cell line was cultured at 37°C in a humidified 95% air / 5% CO 2 atmosphere.

Confocal Laser Scanning Microscopy and FRAP Experiments

HEK293( 3 ) cells were grown for 24 h on 18 mm round cover glasses placed in the wells of a

                                  t d Ι t d Ι t d F F F F(t) 1 0 0 0    2 2 2 exp (1)
Tetrameric RGD vector targeting integrin v3 14 where t is time, F 0 and F  are initial and final mean fluorescence intensities after photobleaching respectively, I 0 and I 1 are modified Bessel functions. The diffusion time values obtained for each peptide conditions are the mean of 20 individual cells.

Electron microscopy

Soluble human integrin  v  3 (Chemicon, #CC1021) was diluted to 0.095 mg/ml (≈ 3.65 fmol) in PBS containing MgCl 2 and CaCl 2 at 1 mM and mixed with RAFT-RGD 0.45 mg/ml (≈ 1 nmol), cRGD 0.1 mg/ml (≈ 1 nmol) or RAFT-RAD 0.45 mg/ml (≈ 1 nmol) for 2 min before addition on top of a carbon-coated electron microscope grid. Thirty sec later, the excess of liquid was removed by blotting with a filter paper. Four ml of a 2 % Uranyl Acetate solution were placed on the grid and incubated for 30 sec -1 min at room temperature. The staining solution was subsequently removed by filter paper adsorption, and the grid was dried on a paper filter for 2 min and then examined using an electron microscope. Micrographs were taken under low-dose conditions with a Jeol 1200-EX II microscope at 100 kV or a FEI CM12 microscope at 120 kV and a respectively calibrated magnification of 40000 and 45000 times. Selected negatives films were digitalized on a Zeiss scanner (Photoscan TD) with a pixel size of 14 mm, corresponding to 3.5 Å or 3.1 Å at the sample scale.

Confocal Microscopy of Peptide Internalization

HEK293(β 3 ) cells were grown as described in 4-wells Lab-Tek I chambered coverglass. Cells were starved 30 min and incubated with DMEM w/o red phenol alone or containing amantadine 1 mM, nystatin 1 M or amiloride 1 mM for another 30 min at 37°C, 5% CO 2 . Then, 1 mM RAFT-RGD-Cy5 or 1 mM cRGD-Cy5 were added to the culture medium, together with 5 M of Hoechst, for 10 min. Confocal microscopy was performed on the Axiovert 200 LSM510 LNO Meta microscope (Carl Zeiss, Jena, Germany) using a 40x oil immersion objective of 1.2 N.A., after addition of fresh medium. The 633 nm laser intensity was set up on request at 10 or 30% of its maximum intensity depending on the peptide. The following inhibitors were used in order to block caveolae, clathrin-coated pits or macropinocytosis: nystatin 1 M, amantadine 1 mM, or amiloride 1 mM (Sigma Aldrich).

Peptides internalization was quantified from confocal microscopy analysis. The mean Cy5 intensity was related to the cell area (in pixel); the related index is reported on the pictures.

Integrin internalization assay

Surface biotin labeling and internalization HEK293( 3 ) cells were cultured at ~85% confluence in 90 mm dishes and starved at 37°C for 30 min in DMEM containing 15 M of cycloheximide (Sigma Aldrich, St Quentin Fallavier, France). Membrane labeling was adapted from Roberts et al. [START_REF] Roberts | PDGF-regulated rab4-dependent recycling of alphavbeta3 integrin from early endosomes is necessary for cell adhesion and spreading[END_REF]. Depending of the condition, cells were kept at 4°C or placed 10 min at 37°C in DMEM alone (control) or containing RAFT-RGD or cRGD from 0.1 to 4 M, in order to allow receptor internalization.

In order to measure endocytosis inhibition, amantadine 1 mM, nystatin 1 M or amiloride 1mM were added to the medium 30 min before biotin labeling. Those inhibitors were kept in the medium during biotin labeling and peptides internalization.

ELISA

Integrin internalization was quantified using Enzyme Linked ImmunoSorbent Assay in 96-wells plate, through gentle agitation. The previous day, 0.2 g of mAb anti-human CD61 (IM0540, Beckman Coulter, France) were used to coat wells (n=3 wells/condition) by incubating overnight at 4°C under gentle agitation. Antibodies were removed and unspecific sites were blocked with 300 l of PBS/BSA 3%/0.05% tween for 1 h et RT. Wells were washed 3 times with 300 l PBS/0.05% tween for 5 min before addition of 50 g of protein lysates adjusted to 200 l with lysis buffer, for 1 h at RT. Wells were washed again 5 times with 300 l PBS/0.05% tween for 5 min. Then, 200 l of streptavidin-POD/PBS/0.05% tween (1:10000) (11 089 153 001, Roche Diagnostic, Meylan, France) were added on anti-CD61 / biotinylated integrin complex for 1 h at RT. Samples were washed 3 times with 300 l PBS/0.05% tween and 2 times with 300 l PBS for 5 min. At last, integrin internalization was revealed using ABTS kit (00-2011, Zymed, Cergy Pontoise, France) and quantified as described by the manufacturer.

Results were expressed as mean of OD +/-S.E.M. and each experiment was performed in quadruplet at least. Representative examples of negatively stained electron micrographs of the soluble  v  3 integrin alone or mixed with RGD-peptides. The  v  3 integrins remain in monomeric state except when using RAFT-RGD; in this condition, integrins can be found as dimers on the grid. As expected, RGD-peptides (<6 kDa) were not distinguishable. Upper panels: Original images. Lower panels:

Photoshop enhanced visualization of the complexes.

  calculated from the fluorescence recovery curves obtained on twenty individual cells (three separated experiments), increased from 46  14 seconds (RAFT-RAD-FITC, cRGD-FITC and PBS) to 144  22 seconds (RAFT-RGD-FITC). No concomitant change in peptide distribution was induced during FRAP experiments as monitored in the FITC detection channel. The fluorescence recovery being directly correlated to the mobility of the receptor, these results suggested that the presence of the tetrameric RAFT-RGD-FITC slowed down  v  3 integrin diffusion within the cell membrane by linking several integrins together, i.e. by clustering integrins.

  and chemical structures are presented in Fig. 1a. Briefly, RAFT is a cyclic decapeptide (c [-Lys(Boc)-Lys(Alloc)-Lys(Boc)-Pro-Gly-Lys(Boc)-Lys(Alloc)-Lys(Boc)-Pro-Gly-]) having two orthogonally addressable domains pointing on either side of the cyclopeptide backbone. On the upper face, four copies of the c[-RGDfK-] peptide were grafted via an oxime bond (R1-O-N = C-R2) for recognition of the integrin α v β 3 . On the other side of the RAFT, either Cy5 mono NHS (N-hydroxysuccinimide) ester (Amersham Biosciences, Uppsala, Sweden) or FITC (Sigma-Aldrich, St Quentin Fallavier, France) was added on the lysine chain (c [-KKKPGKAKPG-]) [17]. As a negative control probe, Cy5-labeled RAFT(c[-RβADfK-]) 4 (RAFT-RAD) was also synthesized in a similar way.

  water immersion C-Apochromat objective lens (numerical aperture (N.A.) = 1.2). The measurements were carried out at room temperature in 8-well Lab-Tek I chambered coverglass (Nalge Nunc Int., Illkirch, France). The 633 nm He-Ne laser beam was focused into 50 μl solutions at 150 μm over the cover glass. The fluorescence emission was collected through a pinhole and a 650 nm-long pass filter. Photon counts were detected by an Avalanche PhotoDiode (APD) at 20 MHz for 30 sec. For each sample, FCS measurements were repeated 15 times. The data evaluation was performed using the Zeiss FCS Fit software. Most of the intensity autocorrelation curves were fitted using a free diffusion model with two components: the peptide coupled to the fluorochrome alone and the fluorescent peptide-integrin complex. Preliminary studies enabled us to determine the diffusion time value of the first component and structural parameter. Moreover, a calibration step with 4 nM Cy5 allowed the evaluation of the size of the confocal volume (≈ 1 fl). Interaction assays were performed at RT in HBSS containing Mg 2+ and Ca 2+ . One to 40 nM of soluble integrin  v  3 (CC1021, Chemicon Int., France) were mixed with 0.6 nM of RAFT-RGD-Cy5 and RAFT-RAD-Cy5 or 2.4 nM of cRGD-Cy5. FCS measurements were performed 2 min after mixing. Theoretical calculation was made using the Origin software.

12 -

 12 well plate (seeding density of 7 x 10 4 cells per well). Immediately before running the experiment, cells were incubated for 8 min at RT (22°C) in a mixture of R-PhycoErythrinconjugated LM609 monoclonal antibody (RPE-LM609, Chemicon Int.) and 0.5 M FITC-labeled RGD peptides. The incubation with monovalent cRGD was performed at either 0.5 M or 2 M. The antibody and peptide solutions were extemporaneously prepared in HBS buffer enriched with 1 mM MgCl 2 . For microscopic observations, coverslips were rinsed once in HBS buffer and disposed on a custom-made incubation chamber containing 200 µL of the FITClabeled RGD peptide solutions (0.5 M or 2 M with cRGD). The confocal imaging and FRAP measurements were carried out on an inverted confocal microscope (LSM510, Carl Zeiss, Jena, Germany) using a 40x water immersion objective of 1.2 N.A. A pinhole adjustment resulted in a 2.5 µm optical slice used for the visualization of a 25 µm circular region of the cell apex membrane at scan zoom 4. For FRAP experiments, a 3 µm circular ROI was uniformly bleached for 2 sec with 100% intensity of the 543 nm line (fluorescence bleaching ratio > 90%). The fluorescence recovery was then sampled on the whole region for 170 sec every 5 sec with 0.1% laser intensity set with AOTF. Thanks to the extremely small excitation power and short acquisition times, no photobleaching was induced during sampling as observed on control cells or on the membrane out of the bleached ROI. Neither lateral nor axial displacement of ROI was observed during FRAP measurements and no recovery of fluorescence was observed on the entirely bleached control cells.Data analysis was performed in assumption that the recovery of fluorescence in the ROI was solely due to the two-dimensional cytoplasmic diffusion of fluorescent species. The diffusion time d was determined by fitting the normalized fluorescence recovery curves F(t) to the recovery kinetics equation:[START_REF] Soumpasis | Theoretical analysis of fluorescence photobleaching recovery experiments[END_REF][START_REF] Klein | Determination of plasma membrane fluidity with a fluorescent analogue of sphingomyelin by FRAP measurement using a standard confocal microscope[END_REF]

4 9 .

 9 24 ± 1.23 -23% NS * P < 0.05, ** P < 0.01, *** P < 0.001 vs. control (a) Variations of internalization were compared to the control condition (absence of peptide). Results were expressed as mean +/-S.E.M. and each experiment was performed in quadruplet at least. (b) P values calculated vs. RAFT-RGD 1 M.

Figure 1 :

 1 Figure 1: RGD-peptides affinities.

Figure 2 :

 2 Figure 2: RAFT-RGD-FITC reduces  v  3 integrin lateral mobility.

Figure 3 :

 3 Figure 3: RGD-peptides/integrin  v  3 complexes.

Figure 4 :

 4 Figure 4: Confocal imaging on HEK293( 3 ) living cells.

Figure

  Figure-2 Click here to download high resolution image

Table 1 :

 1 Integrin  v  3 internalization assay.

	Peptide	Concentrations (M)	% Internalized integrins	Variation (%) (a)
		none	11.99 ± 1.13	-
		0.1	17.18 ± 1.08	+ 43% *
		0.5	20.05 ± 2.08	+ 67% **
		1	21.41 ± 1.85	+ 79% ***
	RAFT-RGD	1 + Amantadine	12.03 ± 1.58	+ 0% ** (b)
		1 + Nystatin	19.67 ± 1.42	+ 64% NS (b)
		1 + Amiloride	19.91 ± 2.80	+ 66% NS (b)
		0.1	11.67 ± 1.31	-3% NS
		0.5	10.76 ± 1.08	-10% NS
	cRGD	1	11.77 ± 1.00	-2% NS

Table 1 :

 1 Integrin α v β 3 internalization assay. HEK293(β 3 ) membrane proteins were biotinylated and the cells were incubated in presence of 0 to 1 M RAFT-RGD or 0 to 4 M cRGD for 10 min. The cells were then lysed and the concentration of biotinylated- v  3integrins present into each fraction measured using ELISA. Variations of internalization were compared to the control condition (absence of peptide). P < 0.05, ** P < 0.01, *** P < 0.001 vs. control;(1) P values calculated vs. RAFT-RGD 1 M.

	Peptide	Concentrations (M)	% Internalized integrins	Variation (%)
		none	11.99 ± 1.13	-
		0.1	17.18 ± 1.08	+ 43% *
		0.5	20.05 ± 2.08	+ 67% **
		1	21.41 ± 1.85	+ 79% ***
	RAFT-RGD	1 + Amantadine	12.03 ± 1.58	+ 0% ** (1)
		1 + Nystatin	19.67 ± 1.42	+ 64% NS (1)
		1 + Amiloride	19.91 ± 2.8	+ 66% NS (1)
		0.1	11.67 ± 1.31	-3% NS
		0.5	10.76 ± 1.08	-10% NS
	cRGD	1	11.77 ± 1.00	-2% NS
		4	9.24 ± 1.23	-23% NS

*

Table Figure -1 Click here to download high resolution image

 Figure

Acknowledgments

This work was supported by the Institut National de la Santé et de la Recherche Médicale (INSERM), the INCA (Institut National for Cancer), the Association for Research on Cancer (ARC, France), the Agence Nationale pour la Recherche (ANR (V2IP and Pepvec programs)) and the EMIL and N2L NoE of the 6 th FWP. We also acknowledged Marc Block (ERL CNRS 3148, Institut Albert Bonniot BP 170, 38 042 Grenoble cedex 9, France) for helpful discussion during the redaction of this manuscript.

Supplemental Figure Click here to download high resolution image

Supplementary data

Fig. 1Sup: Curve fitting of FCS analysis. Curve fitting of RAFT-RGD-Cy5 mixed with integrin  v  3 and the corresponding residuals curves using a one-compartment (A) and a twocompartments model (B). Xhi 2 values are 5.3E -2 and 1.9E -4 , respectively. Using the threecompartments model, the same kinds of curves as in (B) was found, with two similar diffusion times of the complex RAFT-RGD-Cy5/integrin ( 3 s).

Supplemental Information (legend supp figure)