Alberto Medina
email: alberto.medina@umontpellier.fr

O Saldarriaga
email: omar.saldarriaga@udea.edu.co

A Villabon

FLAT AFFINE MANIFOLDS AND THEIR TRANSFORMATIONS

Keywords: Flat affine manifolds, Infinitesimal affine transformations, Associative Envelope 2020 Mathematics Subject Classification. Primary: 57S20, 54H15; Secondary: 53C07, 17D25 Partially

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Flat affine transformations and their transformations

Introduction

The objects of study of this paper are flat affine paracompact smooth manifolds with no boundary and their affine transformations. A well understanding of the category of Lagrangian manifolds assumes a good knowledge of the category of flat affine manifolds (Theorem 7.8 in [Wei], see also [Fuk]). Recall that flat affine manifolds with holonomy reduced to GL n (Z) appear naturally in integrable systems and Mirror Symmetry (see [KoSo]).

For simplicity, in what follows M is a connected real n-dimensional manifold, P = L(M ) its bundle of linear frames, θ its fundamental 1-form, Γ a linear connection on P of connection form ω and ∇ the covariant derivative on M associated to Γ. The pair (M, ∇) is called a flat affine manifold if the curvature and torsion tensors of ∇ are both null. In the case where M is a Lie group and the connection is left invariant, we call it a flat affine Lie group. It is well known that a Lie group is flat affine if and only if its Lie algebra is endowed with a left symmetric product compatible with the bracket. For a rather complete overview of flat affine manifolds, the reader can refer to [Gol].

An affine transformation of (M, ∇) is a diffeomorphism f of M whose derivative map f * : T M -→ T M sends parallel vector fields into parallel vector fields, and therefore geodesics into geodesics (together with its affine parameter). An infinitesimal affine transformation of (M, ∇) is a smooth vector field X on M whose local 1-parameter groups φ t are local affine transformations. We will denote by a(M, ∇) the real vector space of infinitesimal affine transformations of (M, ∇) and for X(M ) the Lie algebra of smooth vector fields on M . An element X of X(M ) belongs to a(M, ∇) if and only if it satisfies

L X • ∇ Y -∇ Y • L X = ∇ [X,Y ] , for all Y ∈ X(M ), (1)
where L X is the Lie derivative. For a vector field X ∈ X(M ), we will denote by L(X) its natural lift on P . If φ t is the flow of X, the flow of L(X) is the natural lift L(φ t ) of φ t given by L(φ t )(u) = (φ t (m), (φ t ) * ,m X ij ), where u = (m, X ij ). It is also well known that X ∈ a(M, ∇) if and only if L L(X) ω = 0.

(2)

This is also equivalent to saying that L(X) commutes with every standard horizontal vector field B(ξ) (i.e., vector fields satisfying that θ(B(ξ)) := ξ for all ξ ∈ R n ). We will call a(P, ω) the set of vector fields Z on P satisfying Z is invariant by R a for every a ∈ GL n (R)

L Z θ = 0 (3) L Z ω = 0,
where R a is the right action of GL n (R) on P . The map X → L(X) is an isomorphism of Lie algebras from a(M, ∇) to a(P, ω). The vector subspace aff(M, ∇) of a(M, ∇) whose elements are complete, with the usual bracket of vector fields, is the Lie algebra of the group Aff(M, ∇) of affine transformations of (M, ∇) (see [KoNo] chapter VI, page 229). The image of aff(M, ∇) under the isomorphism X → L(X) will be denoted by a c (P ). For the purposes of this work, it is useful to recall that (P, ω) admits an absolute parallelism, that is, its tangent bundle admits n 2 + n sections independent at every point. In fact, {B(e 1 ), . . . , B(e n ), E * 11 , . . . , E * nn } is a paralelism of P , where (e 1 , . . . , e n ) and {E 11 , . . . , E nn } are respectively the natural basis of R n and gl n (R) (see [KoNo] p 122).

Remark 1.1. If X ∈ a(M, ∇) has flow φ t , the flow L(φ t ) of Z = L(X) determines a local isomorphism of (P, ω). This implies the existence of an open covering (U α ) α∈A trivializing the bundle (L(M ), M, π) such that the maps L(φ t ) : (π -1 (U α ), ω) -→ (π -1 (φ t (U α )), ω) are isomorphisms. When Z is complete, the maps L(φ t ) are global automorphisms of (P, ω).

Notice that, for every t, the map L(φ t ) preserves the fibration and the horizontal distribution associated to ω.

This paper is organized as follows. In Section 2 we state a characterization of flat affine manifolds and provide some of its consequences. In the theory of flat affine manifolds appears, in a natural way, a finite dimensional associative algebra (see Lemma 3.1). This will allow to show the existence of an associative envelope of the Lie algebra of the Lie group relative to a subgroup H of the group of affine transformations of the manifold. In Section 3 we specialize our study to the case of flat affine Lie groups and exhibit some examples. Section 4 deals with the study of the following question posed by A. Medina.

Question 1.1. Does the group of affine transformations, Aff(M, ∇), of a flat affine manifold (M, ∇) admit a left invariant flat affine or, by default, projective structure determined by ∇?

This question was answered positively in some particular cases in the special case of some flat affine Lie groups (see in [M-S-G]). In Section 4. we exhibit more cases where the answer to this question is positive.

Characterization of flat affine manifolds

Before stating the main result of the section, let us denote by A * the fundamental vector field on P = L(M ) associated to A ∈ g = gl n (R) and by B(ξ) the basic or standard horizontal vector fields (see [KoNo] page 63). Recall that the fundamental form

θ of P is a tensorial 1-form of type (GL(R n ), R n ), that is, it verifies (R * a θ)(Z) = a -1 (θ(Z))
, for all a ∈ GL n (R) and Z ∈ X(P ).

If Γ is a linear connection over M , its connection form ω is a gl n (R)-valued 1-form satisfying

ω(A * ) = A, for all A ∈ gl n (R), (4)
R * a ω = Ad a -1 ω, for all a ∈ GL n (R). ( 5
)
It is well known that if Ω and Θ are respectively the curvature and torsion forms of the connection ω, they satisfy Cartan's structure equations

dω(X, Y ) = - 1 2 [ω(X), ω(Y )] + Ω(X, Y ), dθ(X, Y ) = - 1 2 (ω(X) • θ(Y ) -ω(Y ) • θ(X)) + Θ(X, Y ),
for all X, Y ∈ T u (P ), u ∈ P (see [KoNo] page 75).

Given the presheaf U → aff(R n ), where U is an open set in P , consider the sheaf F (called simple faisceau by the French school, see [God], see also [Chr]) of base P and fibre the Lie algebra aff(R n ) generated by the presheaf so that the restriction operations are reduced to the identity.

Lemma 2.1. The simple sheaf F acts on R n by

(u, s(u)) • w = (u, (v u , f u )) • w := v u + f u (w) (6)
Moreover, if η : Sec(T P ) -→ Sec(P × aff(R n )) is a homomorphism of sheaves of Lie algebras, then Sec(T P ) acts on R n via η.

Proof. Recall the construction of the étale space corresponding to F. As F(u) is a direct limit we have F(u) = aff(R n ) for any u ∈ P . Moreover the sections over U considered as maps from U to aff(R n ) are locally constant. Hence the topology of F = P × aff(R n ) is the product topology of that of P by the discrete topology of aff(R n ).

As a consequence, Equation ( 6) can be written locally as

s • w = (v, f ) • w = v + f (w).
This gives an infinitesimal action of F on R n . The last assertion easily follows.

As a consequence the Lie algebras Sec(T P ), a(P ) and a c (P ) also act infinitesimally on R n .

In these terms we have the following result.

Theorem 2.2. Let M be a smooth connected manifold, P = L(M ) be the principal bundle of linear frames of the manifold, θ the fundamental form of P and Γ a linear connection on P of connection form ω. The following assertions are equivalent (i) The linear connection Γ on P is flat affine.

(ii) The map η :

Sec(T P ) -→ Sec(P × aff(R n )) Z → (θ(Z), ω(Z)) ,
is a homomorphisms of sheaves of real Lie algebras.

Proof. As aff(R n ) = R n id gl(R n ) is the semidirect product of the abelian Lie algebra R n and the Lie algebra of commutators of linear endomorphisms of R n , the map

η : Sec(T P ) -→ Sec(P × aff(R n )) Z → (θ(Z), ω(Z))
is well defined and R-linear.

That η is a homomorphism of sheaves of Lie algebras means that

η([Z 1 , Z 2 ]) = (ω(Z 1 ) • θ(Z 2 ) -ω(Z 2 ) • θ(Z 1 ), [ω(Z 1 ), ω(Z 2 )]
), for all Z 1 , Z 2 ∈ Sec(T P ). ( 7)

In other words, ω is a linear representation of the Lie algebra Sec(T P ) and θ is a 1-cocycle relative to this representation. Suppose that Γ is flat and torsion free, then the structure equations for ω reduce to

dθ(Z 1 , Z 2 ) = - 1 2 (ω(Z 1 ) • θ(Z 2 ) -ω(Z 2 ) • θ(Z 1 )) (8) dω(Z 1 , Z 2 ) = - 1 2 [ω(Z 1 ), ω(Z 2 )]. (9)
It is clear that (7) holds when both vector fields are vertical or if one is vertical and the other is horizontal. Now let us suppose that both Z 1 and Z 2 are horizontal vector fields. As the distribution determined by ω is integrable, it follows that [Z 1 , Z 2 ] is horizontal and Equation (9) implies that ω([Z 1 , Z 2 ]) = [ω(Z 1 ), ω(Z 2 )]. Also notice that ω(Z 1 ) • θ(Z 2 ) -ω(Z 2 ) • θ(Z 2 ) = 0. Now, we can suppose that Z 1 and Z 2 are basic vector fields, that is, Z 1 = B(ξ 1 ) and Z 2 = B(ξ 2 ). Hence from (8) we get that θ([Z 1 , Z 2 ]) = 0. Therefore assertion (ii) follows from (i).

To prove that (ii) implies (i), we will show that Equation (7) implies that the curvature and the torsion vanish, i.e., Equations ( 8) and (9) hold.

As η is a homomorphism of sheaves of Lie algebras it follows that for Z 1 , Z 2 ∈ Sec(T P )

ω([Z 1 , Z 2 ]) = [ω(Z 1 ), ω(Z 2 )] and θ([Z 1 , Z 2 ]) = ω(Z 1 ) • θ(Z 2 ) -ω(Z 2 ) • θ(Z 1 )
The first equality means that the local horizontal distribution on P determined by ω is completely integrable, hence Ω vanishes.

On the other hand, as the torsion Θ is a tensorial 2-form on

P of type (GL n (R), R n ), we have Θ(Z 1 , Z 2 ) = 0 if Z 1 or Z 2 is vertical. Now, if Z 1 = B(ξ 1 ) and Z 2 = B(ξ 2 ) for some ξ 1 , ξ 2 ∈ R n , we obtain Θ(Z 1 , Z 2 ) = dθ(Z 1 , Z 2 ) = 1 2 (Z 1 (θ(Z 2 )) -Z 2 (θ(Z 1 )) -θ([Z 1 , Z 2 ])) = 1 2 (Z 1 (ξ 2 ) -Z 2 (ξ 1 ) -ω(Z 1 ) • θ(Z 2 ) + ω(Z 2 ) • θ(Z 1 )) = 0.
Consequently the torsion form Θ vanishes.

Recall that there are topological obstructions to the existence of a flat affine connection. In particular, a closed manifold with finite fundamental group does not admit flat affine structures (see [AuMa], [Mil] and [Smi]).

As P = L(M ) has a natural parallelism, the group K of transformations of this parallelism is a Lie group of dimension at most dim P = n 2 + n. More precisely, given σ ∈ K and any u ∈ P , the map σ → σ(u) is injective and its image {σ(u) | σ ∈ K} is a closed submanifold of P . The submanifold structure of {σ(u) | σ ∈ K} turns K into a Lie transformation group of P (see [Kob]). The group Aut(P, ω), of diffeomorphisms of P preserving ω (therefore θ), is a closed Lie subgroup of K, so it is a Lie group of transformations of P .

Remark 2.1. The group Aut(L(R n ), ω 0 ) of diffeomorphisms of L(R n ) preserving the usual connection ω 0 on R n is isomorphic to the group Aff(R n , ∇ 0 ), called the classic affine group, that is, the group generated by translations and linear automorphisms. Recall also that having a flat affine structure ∇ on M is equivalent to have a smooth atlas whose change of coordinates are elements of Aut(L(R n ), ω 0 ).

The following technical result is a direct consequence of Theorem 2.6, it concerns certain G-structures to which a connection can be attached.

Corollary 2.3. Let Γ be flat affine and η the homomorphism of Theorem 2.2, then we have a. The connection Γ is a metric connection, if and only if η takes values in the Lie algebra e (p,q) (n) = R n o (p,q) . b. The connection Γ preserves a volume form if and only if η takes values in the Lie algebra R n sl n (R). In particular, if n = 2m, Γ preserves a symplectic form σ if and only if η takes values in the Lie algebra R n sp m (R).

Proof. Consider the G-structure on M , whenever exists, given respectively by O (p,q) (with p+q = n), SL n (R) and Sp m (R). In each respective case, the homomorphism η takes values in the sheaf of Lie algebras Sec(P × (R n o (p,q) )), Sec(P × (R n sl n (R)) and Sec(P × (R n sp m (R)), respectively.

Remark 2.2. If Γ is flat affine and X ∈ a(M, ∇) with flow φ t , then its natural lift L(φ t ) preserves the foliation defined by ω. Moreover, if L(X) is complete then L(φ t ) is an automorpism of the bi-foliated manifold L(M ).

The groups Aut(P, ω) and Aff(M, ∇) are isomorphic. We denote by Aut(P, ω) 0 the unit component of Aut(P, ω) and by Aut(P, ω) 0 its universal covering Lie group.

Example 2.1. Identify C with the plane R 2 and C * with the punctured plane R 2 \{(0, 0)} and let p : R 2 -→ R 2 \{(0, 0)} be the complex exponential map. It is clear that p is a covering map with automorphism group given by H

= {F k : R 2 -→ R 2 | F k (x, y) = (x, y + 2πk) with k ∈ Z}.
As H is a subgroup of affine transformations of R 2 relative to ∇ 0 , there exists a unique linear connection ∇ on M := R 2 \ {(0, 0)} so that p is an affine map. Consequently the Aff(R 2 )-principal fiber bundle of affine frames A(M ) has a connection Γ , of curvature zero, determined by ∇. The geodesic curves in M determined by ∇ are logarithmic spirals. The elements of the group Aff(M, ∇), of affine transformations of M relative to ∇, are diffeomorphisms of M locally given by F (r, θ) = (ar b , c ln(r) + θ + d) where a, b, c, d ∈ R with a > 0 and (r, θ) are polar coordinates of M .

Let u 0 be in A(M ) and H(u 0 ) be the affine holonomy bundle through u 0 . Let us denote by h affine holonomy representation of (M, ∇). It is well known that H(u 0 ) is a covering manifold of M with structure group h(π 1 (M )) and that A(M ) is the disjoint union of holonomy bundles. Let p : H(u 0 ) -→ M be the covering map and F the natural affine lift of F ∈ Aff(M, ∇) to A(M ), i.e., F ((q, X 1 , X 2 ) (x,y) ) = ((F * ,(x,y) (q), F * ,(x,y) (X 1 ), F * ,(x,y) (X 2 )) F (x,y) ), with F * ,(x,y) (q) the end point of the vector F * ,(x,y) ( -→ q ) and -→ q the vector on T (x,y) from the origin to q. It can be verified that F sends the affine holonomy bundle H(u 0 ) through u 0 to the holonomy bundle

H( F (u 0 )) through F (u 0 ).
Recall that ω is flat affine if and only if its affine holonomy group is discrete (see [KoNo] p 210). Nevertheless, there are flat affine manifolds whose linear holonomy group is not discrete.

The Lie algebra homomorphism η of Theorem 2.2 could be integrated, in some particular cases, into a Lie group homomorphism. For instance, applying Lie's third Theorem (see [Ch] for details on the proof of this theorem) and Cartan's Theorem to the Lie algebra homomorphism η : a c (P ) -→ aff(R n ), obtained from η restricted to a c (P ), we have.

Corollary 2.4. Let (M, ∇) be a flat affine connected manifold of connection form ω. Then there exists a Lie group homomorphism ρ :

Aut(P, ω) 0 -→ Aut(L(R n ), ω 0 ) determined by (θ, ω).
Lemma 2.5. Let M be a smooth connected manifold and Γ a flat affine connection on P = L(M ) with dim(a c (P )) ≥ dim(M ). Then there exists u ∈ P so that the map φ u :

a c (P ) -→ R n defined by φ u (Z) = η(0)(Z) = θ u (Z u ) is onto.
Proof. Let β = (Z 1 , . . . , Z m ) be a linear basis of a c (P ). We claim that there are linear frames u where the set (η u (0

)(Z 1 ) = θ u (Z 1,u ), . . . , (η u (0)(Z m ) = θ u (Z m,u )) spans R n .
Otherwise, for every u and every subset (X 1 = Z i 1 , . . . , X n = Z in ) of β with n = dim(M ) and 1 ≤ i 1 < • • • < i n ≤ m, the set {θ u (X 1,u ), . . . , θ u (X n,u )} is linearly dependent. Thus, there exists real constants α 1 , . . . , α n so that 0 = α i θ u (X i,u ). It follows that

0 = α i θ u (X i,u ) = θ u α i X i,u = u -1 (π * ,u (X u ))
where u is seen as a linear transformation from R n to T π(u) M and X = α i X i . It follows that π * ,u (X u ) = 0 for every u, that is, X ≡ 0. But this contradicts the fact that the set {X 1 , . . . , X n } is linearly independent. As the map φ u is linear, we conclude that it is onto.

The following example exhibits a flat affine manifold M whose bundle P = L(M ) has frames u where the map φ u of the previous lemma is not onto.

Example 2.2. Consider the manifold M = R 2 \ {(0, 0), (0, 1)} with the connection ∇ induced by ∇ 0 . A simple calculation shows that the real space aff(M, ∇) is two dimensional with linear basis given by X 1 = x ∂ ∂x , X 2 = x ∂ ∂y . One can also verify that, for a linear frame u over a point (x, y) ∈ M , one has that θ u ( X 1 ) = u -1 x 0 and θ u ( X 2 ) = u -1 0 x , where X 1 and X 2 are the natural lifts of X 1 and X 2 , respectively. Notice that the set {θ u ( X 1 ), θ u ( X 2 )} is linearly independent if and only if x = 0.

Remark 2.3. If M = G is a Lie group endowed with a flat affine left invariant connection ∇ + , then the map φ u of the previous lemma is onto for every u ∈ P . This follows from the fact that right invariant vector fields are complete infinitesimal affine transformations relative to ∇ + forming a global parallelism of G.

Theorem 2.6. Let M be a smooth connected n-dimensional manifold and Γ a flat affine connection on P = L(M ) of connection form ω. If dim(Aut(P, ω)) ≥ n, there exists a homomorphism ρ : Aut(P, ω) 0 -→ Aut(R, ω 0 ) having a point of open orbit. Moreover, if dim(Aut(P, ω)) = n, the point also has discrete isotropy, that is, ρ is an étale affine representation of Aut(P, ω) 0 in R n . Proof. Let u ∈ P be a linear frame, φ u as in the previous lemma, η : a c (P ) -→ aff(R n ) the map defined by η (Z) = (θ u (Z), ω u (Z)) and ρ : Aut(P, ω) -→ Aut(L(R n ), ω 0 ) the homomorphism satisfying that ρ * ,Id = η . If π : Aut(P, ω) -→ Orb 0 is the orbital map, since π * ,Id (Z) = η(0)(Z), it follows that π * ,Id = φ u . As φ u is onto, we get that 0 is a point of open orbit. The second assertion in immediate.

The following result is a direct consequence of the preceding theorem.

Corollary 2.7. Under the hypothesis of Theorem 2.6, manifolds M satisfying the condition dim(Aff(M, ∇)) = dim(M ) give a positive answer to Question 1.1. That is, the group Aff(M, ∇) is endowed with a flat affine left invariant connection determined by ∇.

Recall that there are flat affine manifolds (M, ∇) with dimension equal to dim(Aff(M, ∇)). For instance, flat affine tori other than the Hopf torus listed in [NaYa] (see also [Ben]) and one of the manifolds of Example 2.3 below.

Notice that the proof of Lemma 2.5, implies that if dim(a c (P )) < n, the map φ u is injective, so we have Corollary 2.8. Under the conditions of Theorem 2.6, if dim(a c (P )) < n, there exists an injective homomorphism ρ : Aut(P, ω) 0 -→ Aut(L(R n , ω 0 )), that is, a faithful representation of Aut(P, ω) 0 by classical affine transformations of R n .

Proof. Let η be as in the proof of Theorem 2.6 and suppose that η (Z) = η (Y ), for some Y, Z ∈ a c (P ), hence θ u (Z u ) = θ u (Y u ). As θ u is inyective for some u ∈ P , we have that Y u = Z u . Since Y and Z are natural lifts of infinitesimal affine transformations, it follows that Y = Z.

Remark 2.4. There are flat affine manifolds (M, ∇) of dimension greater than dim(Aff(M, ∇)). This condition is satisfied by some of the flat affine Klein bottles listed in [FuAr]. More precisely, flat affine Klein bottles that are not double covered by a Hopf torus.

Example 2.3. Let M 1 , M 2 and M 3 be the plane without one, two and three non colineal points, respectively, and let ∇ i , i = 1, 2, 3, be the connection ∇ 0 restricted to M i . Let us suppose that the points removed are p 1 = (0, 0), p 2 = (0, 1) and p 3 = (1, 0). An easy calculation shows that the corresponding groups of affine transformations of M i relative to ∇ i are given by

Aff(M 1 , ∇ 1 ) = {F : M 1 -→ M 1 | F (x, y) = (ax + by, cx + dy) such that ad -bc = 0}. Aff(M 2 , ∇ 2 ) = {F ∈ Diff(M 2 ) | F (x, y) = (ax, bx + y) or F (x, y) = (ax, bx -y + 1), a = 0}
notice that this is a 2-dimensional non-commutative group with four connected components.

The group Aff(M 3 , ∇ 3 ) is discrete, its elements are the set of affine transformations of (R 2 , ∇ 0 ) permuting the points p 1 , p 2 and p 3 .

The group of diffeomorphisms Aut(P, ω 1 ) of P = L(M 1 ) preserving the connection form ω 1 is given by where (x, y, X 11 , X 12 , X 21 , X 22 ) are local coordinates of L(M 1 ).

F ∈ Diff(P) F (x,
The connected component of the unit of the group Aut(L(M 2 ), ω 2 ), of automorphisms of P = L(M 2 ) preserving the connection form ω 2 associated to ∇ 2 , is formed by diffeomorphisms F a,b , with a > 0, given by F a,b (x, y, X 11 , X 12 , X 21 , X 22 ) = (ax, bx + y, aX 11 , aX 12 , bX 11 + X 21 , bX 12 + X 22 ) , where (x, y, X 11 , X 12 , X 21 , X 22 ) are local coordinates of L(M 2 ).

The homomorphism of Corollary 2.4 corresponding to the manifold M 1 is given by

ρ 1 : Aff(M 1 , ∇ 1 ) → Aff(R 2 ) F = F (Y 1 ,Y 2 ,Y 3 ,Y 4 ) → ρ 1 (F )
where the linear part of ρ 1 (F ) is given by 

  X 11 X 22 Y 1 -X 12 X 21 Y 4 -X 11 X 12 Y 3 +X 21 X 22 Y 2 X 11 X 22 -X 12 X 21 X 2 22 Y 2 -x 2 12 Y 3 +(Y 1 -Y 4 )X 12 X 22 X 11 X 22 -X 12 X 21 X 2 11 Y 3 -X 2 21 Y 2 +(Y 4 -Y 1 )X 11 X 21 X 11 X 22 -X 12 X 21 X 11 X 22 Y 4 -X 12 X 21 Y 1 +X 11 X 12 Y 3 -X 21 X 22 Y 2 X 11 X 22 -X 12 X 21    ,
and the cocycle part of

ρ 1 (F ) is   u 2 X 12 -u 1 X 22 +u 1 X 22 Y 1 -u 2 X 12 Y 4 -u 1 X 12 Y 3 +u 2 X 22 Y 2 X 11 X 22 -X 12 X 21 u 1 X 21 -u 2 X 11 -u 1 X 21 Y 1 +u 2 X 11 Y 4 +u 1 X 11 Y 3 -u 2 Xx 21 Y 2 X 11 X 22 -X 12 X 21   with (u 1 , u 2 ) are local coordinates on M 1 , (X 11 , X 12 , X 21 , X 22 ) are local coordinates for GL 2 (R) and F = F (Y 1 ,Y 2 ,Y 3 ,Y 4 ) defined by F (u 1 , u 2 ) = (Y 1 u 1 + Y 3 u 2 , Y 2 u 1 + Y 4 u 2 ) with Y 1 Y 4 -Y 2 Y 3 = 0.
Moreover, Theorem 2.6 guarantees that the action of Aff(M 1 , ∇ 1 ) on R 2 determined by ρ 1 leaves and open orbit. In fact, the open orbit is the whole plane. Now, a direct calculation shows that the homomorphism of Corollary 2.4 corresponding to the manifold M 2 is given by ρ

: G(a c (P )) -→ Aff(R 2 ) F a,b → ρ(F a,b
) where G(a c (P )) is the connected and simply connected Lie group of Lie algebra a c (P ), F a,b the affine map of (M 2 , ∇ 2 ) defined by F a,b (x, y) = (ax, bx + y) with a > 0 and ρ(F a,b ) is given by

1 D   aX 11 X 22 -bX 11 X 12 -X 12 X 21 (a -1)X 12 X 22 -bX 2 12 (a -1)xX 22 -bxX 12 (1 -a)X 11 X 21 + bX 2 11 -aX 12 X 21 + bX 11 X 12 + X 11 X 22 (1 -a)xX 21 + bxX 11 0 0 D   where (X 11 , X 12 , X 21 , X 22 ) is a system of local coordinates of GL 2 (R), D = X 11 X 22 -X 12 X 21
and (x, y) local coordinates of M 2 . According to Theorem 2.6, the representation ρ is étale whenever x = 0, hence, Aff(M 2 , ∇ 2 ) admits a flat affine (left invariant) connection.

If M = G is a Lie group, we will identify its Lie algebra Lie(G) = g with the real vector space of left invariant vector fields on G and also with the tangent space at the unit ε ∈ G. For any x ∈ T ε G, we denote by x + (respectively by x -) the left (right) invariant vector field determined by x. The group G acts on itself on the left (respectively right), and we will denote by L σ (respectively R σ ) the left (respectively right) action of σ ∈ G. These actions naturally lift to actions of G on P = L(G) given by ψ

1,σ (u) = σ • u = (στ, (L σ ) * ,τ (X ij )) (respectively ψ 2,σ (u) = u • σ = (τ σ, (R σ ) * ,τ (X ij ))
where u = (τ, (X ij )). A linear connection Γ on G is said left invariant (respectively right invariant) if the action ψ 1 (respectively ψ 2 ) preserves the horizontal distribution. The connection is bi-invariant if the horizontal distribution is preserved by ψ 1 and ψ 2 .

The existence of a left invariant flat affine connection on a Lie group G, is equivalent to the existence of an étale affine representation of G, i.e, there exists a homomorphism of Lie groups τ : G -→ Aff(R n ) having at least an open orbit with discrete isotropy.

In these terms we get the following consequence of Theorem 2.2 and Corollary 2.4. In the next result the Lie group G can be supposed simply connected Theorem 2.9. Let G be a connected n-dimensional Lie group, of Lie algebra g, endowed with a left invariant connection ∇ + of connection form ω + . The following assertions are equivalent 1 The connection ω + is flat affine. 2 There exists a unique Lie group homomorphism ρ :

Aut(L(G), ω + ) 0 -→ Aut(L(R n ), ω 0 ) with derivative ρ * ,Id = (θ, ω +
) leaving an open orbit. 3 There exists an affine étale representation ρ : G op -→ Aff(R n , ∇ 0 ) determined by (θ, -ω + ).

Proof. That 1. implies 2. follows from Corollary 2.4, Theorem 2.6 and Remark 2.3. Now, suppose that there is a Lie group homomorphism ρ : Aut(L(G), ω + ) 0 -→ Aut(L(R n ), ω 0 ) with a point of open orbit so that ρ * ,Id = (θ, ω + ). That is, ρ * ,Id = η /ac(P ) where η is the homomorphism of Theorem 2.2. As right invariant vector fields on G are infinitesimal affine transformations, we have that g op , the opposite Lie algebra of g, can be considered as a Lie subalgebra of a c (L(G)). Therefore the restriction η = η/ g op : g op -→ aff(R n ) is also a Lie algebra homomorphism. Using Cartan's and Lie's third Theorems we get a Lie group homomorphism ρ : G op -→ Aff(R n ). Let π : G op -→ Orb(0) be the orbital map. It is easy to verify that π * , (Z) = θ u (Z), hence by mimicking the proof of Lemma 2.5, we get that π * , (Z) is a linear isomorphism, therefore ρ is étale. Hence 2. implies 3. is true.

Finally suppose 3., i.e., there exists an affine étale representation ρ : G op -→ Aff(R n , ∇ 0 ) with ρ * , = (θ, -ω + ). Then it is easy to verify that the map λ : g -→ aff(R n , ∇ 0 ) defined by λ = (θ, ω + ) is a representation of g. By Lie's third theorem, the map ρ : G -→ Aff(R n , ∇ 0 ) with ρ * , = λ is an affine étale representation of G. Therefore ω + is a flat affine left invariant connection on G.

The associative envelope of the Lie algebra of a Lie group of flat affine transformations

Given a flat affine manifold (M, ∇) and a Lie subgroup H of the group Aff(M, ∇), we show the existence of a simply connected Lie group endowed with a flat affine bi-invariant connection whose Lie algebra contains Lie(H) as a Lie subalgebra (see Theorem 3.3).

For this purpose we recall some known facts and introduce some notation. Let M be an n dimensional manifold with a linear connection Γ and corresponding covariant derivative ∇, consider the product XY := ∇ X Y, for X, Y ∈ X(M ).

(10) If the curvature tensor relative to ∇ is identically zero, this product verifies the condition

[X, Y ]Z = X(Y Z) -Y (XZ), for all X, Y, Z ∈ X(M ).
If both torsion and curvature vanish identically we have

(XY )Z -X(Y Z) = (Y X)Z -Y (XZ), for all X, Y, Z ∈ X(M ). ( 11
)
A vector space endowed with a bilinear product satisfying Equation ( 11) is called a left symmetric algebra (see [Vin], see also [Kos]). These algebras are also called Kozsul-Vinberg algebras or simply KV-algebras. The product given by [X, Y ] 1 := XY -Y X defines a Lie bracket on the space. Moreover, if ∇ is torsion free, this Lie bracket agrees with the usual Lie bracket of

X(M ), that is [X, Y ] = ∇ X Y -∇ Y X (12)
Recall that product (10) satisfies (f X)Y = f (XY ) and X(gY ) = X(g)Y + g(XY ), for all f, g ∈ C ∞ (M, R), i.e., the product defined above is R-bilinear and C ∞ (M, R)-linear in the first component.

In what follows, given an associative or a left symmetric algebra (A, •), we will denote by A - the Lie algebra of commutators of A, i.e., the Lie algebra with bracket given by

[a, b] = a • b -b • a.
To our knowledge, the next result seems to appear for the first time in J. Vey's thesis (see [Vey], see also [Yag]).

Lemma 3.1. Let (M, ∇) be an n-dimensional flat affine manifold. Then Product (10) turns the vector space a(M, ∇) into an associative algebra, of dimension at most n 2 + n, whose commutator is the Lie bracket of vector fields on M . In particular, if (G, ∇ + ) is a flat affine Lie group of Lie algebra g, the space a(G, ∇ + ) -contains g op as a Lie subalgebra, where g op is the opposite Lie algebra of g.

Proof. Since ∇ is flat affine, it follows from (1) that a smooth vector field X is an infinitesimal affine transformation if and only if

∇ ∇ Y Z X = ∇ Y ∇ Z X, (13)
for all Y, Z ∈ X(M ). This equality implies that ∇ X Y ∈ a(M, ∇), whenever X, Y ∈ a(M, ∇) and also gives that the product XY = ∇ X Y is associative. On the other hand, it follows from Equation ( 12) that the commutator of this product agrees with the Lie bracket of a(M, ∇).

In particular if M = G is a Lie group of Lie algebra g and ∇ + is left invariant and flat affine, the real vector space g of right invariant vector fields on G is a subspace of aff(G, ∇ + ). Hence from (12) we get that g op is a Lie subalgebra of a(G, ∇ + ) -.

The following well known result will be used in what follows (see for instance [Med] or [BoMe]). Proposition 3.2. Let G be a Lie group of Lie algebra g. The group G admits a flat affine bi-invariant structure if and only if g is the underlying Lie algebra of an associative algebra so that [a, b] = ab -ba, for all a, b ∈ g.

We have the following consequence of Lemma 3.1.

Theorem 3.3. Given a flat affine manifold (M, ∇) and a Lie subgroup H of Aff(M, ∇), there exists a connected finite dimensional Lie group G endowed with a flat affine bi-invariant structure containing a connected Lie subgroup locally isomorphic to H.

Proof. From Lemma 3.1, the real vector space a(M, ∇) is an associative algebra under the product determined by ∇. Let E be the subalgebra of the associative algebra a(M, ∇) generated by the vector subspace h =Lie(H) and A = E ⊕ R1 the associative algebra obtained from E by adjoining a unit element 1. Consider U ( A) the group of units of A, this is open and dense

in A. Let G := u ∈ U ( A) | u = 1 + a, with a ∈ E
and G = G 0 the connected component of the unit of G. Then the Lie group G verifies the conditions of the theorem (for more details see [BoMe]).

It is clear that h is a Lie subalgebra of E -. Consequently there exists a connected Lie subgroup H of G of Lie algebra h, hence the groups H and H are locally isomorphic.

Remark 3.1. In general aff(M, ∇) is not a subalgebra of the associative algebra a(M, ∇). This fact is contrary to what some authors state. This can be observed in Examples 3.2 and 3.4.

The previous remark and Theorem 3.3 motivates the following. Definition 3.1. If (M, ∇) is a flat affine manifold and H is a Lie subgroup of Aff(M, ∇), we set (a) The smallest subalgebra of the associative algebra a(M, ∇) containing the vector space h =Lie(H), denoted by env ∇ (h), will be called the associative envelope of h relative to ∇. (b) Any Lie group of Lie algebra env ∇ (h) will be said an enveloping Lie group of H.

That any enveloping Lie group is endowed with a flat affine bi-invariant connection determined by ∇ follows from Proposition 3.2.

Corollary 3.4. Given a flat affine Lie group (G, ∇ + ) of Lie algebra g, there exists a finite dimensional associative algebra A containing g so that g is a Lie subalgebra of A -.

Proof. As g op is a subalgebra of the associative algebra a(G, ∇ + ), there exists a Lie subgroup H of Aff(G, ∇ + ) of Lie algebra g op . By taking A =env ∇ (h) op , the opposite algebra of env ∇ + (h), we get an an associative algebra satisfying the statement.

Definition 3.2. The algebra of the previous corollary, denoted by env ∇ + (g), will be called the associative envelope algebra of the left symmetric algebra g. Remark 3.2. Although the elements of the associative envelope env ∇ + (g) of the left symmetric algebra g =Lie(G) are differential operators of order less than or equal to 1 on G, the associative envelope is not a subalgebra of the universal enveloping algebra of g.

Example 3.1. Let M = R 2 \ {(0, 0)} and ∇ be the connection on M induced by the connection ∇ 0 . That G = Aff(M, ∇) 0 is an enveloping Lie group of itself results from the following observations. It can be verified that a linear basis for the real space aff(M, ∇) is given by

x ∂ ∂x , y ∂ ∂x , x ∂ ∂y , y ∂ ∂y
. Also, this space is a subalgebra of the associative algebra a(M, ∇).

As a consequence G = Aff(M, ∇) is endowed with a flat affine bi-invariant connection determined by ∇.

A direct calculation shows that a linear basis of a(G, ∇ + ) is given by the following vector fields

e - 1 = x ∂ ∂x + y ∂ ∂y , e - 2 = ∂ ∂y , C 3 = 1 x ∂ ∂x , C 4 = y x ∂ ∂x , C 5 = x + y 2 x ∂ ∂x and C 6 = -xy - y 3 x ∂ ∂x + (x 2 + y 2 ) ∂ ∂y ,
where e - 1 and e - 2 denote the right invariant vector fields. As the connection is left invariant, the vector fields e - 1 and e - 2 are complete. Moreover, it can be checked that no real linear combination of the fields C 3 , C 4 , C 5 and C 6 is complete. Consequently (e - 1 , e - 2 ) is a linear basis of aff(G, ∇ + ) = g op .

On the other hand, the multiplication table of the product defined by ∇ + , i.e., the product XY = ∇ + X Y , on the basis of a(G, ∇ + ) displayed above is given by

e - 1 e - 2 C 3 C 4 C 5 C 6 e - 1 e - 1 + C 5 C 4 0 C 4 2C 5 2C 6 e - 2 e - 2 + C 4 C 3 0 C 3 2C 4 2e - 1 -2C 5 C 3 2C 3 0 0 0 2C 3 2e - 2 -2C 4 C 4 2C 4 0 0 0 2C 4 2e - 1 -2C 5 C 5 2C 5 0 0 0 2C 5 2C 6 C 6 C 6 C 5 0 C 5 0 0 (15)
It follows from Table 15 that the real associative subalgebra of a(G, ∇ + ) generated by {e - 1 , e - 2 } has linear basis β = (e - 1 , e - 2 , C 3 , C 4 , C 5 ). That is, env ∇ + (g) is the real 5-dimensional associative algebra with linear basis β and the opposite product of Table (15).

The animation below shows the lines of flow of each of the vector fields e - 1 , e - 2 , C 3 , C 4 , C 5 and C 6 with the initial condition (1.5, -1). The fact that the flows of the vector fields C 3 , C 4 , C 5 and C 6 cross the boundary of the orbit of (0, 0) determined by the affine étale representation relative to ∇, correspond to the fact that the vector fields are not complete in G 0 . (To play the animation click on the image on the pdf version, only accessible from the online version).

Remark 3.3. The reader can verify that the simply connected Lie group of Lie algebra env ∇ + (g) - of the previous example is isomorphic to the group E := (R 2 θ 1 R) θ 2 G, where G = Aff(R) 0 is the connected component of the unit of Aff(R) and the actions θ 1 and θ 2 are respectively given by θ 1 (t)(x, y) = (e 2t x, e 2t y) and θ

2 (x, y)(z 1 , z 2 , z 3 ) = x 2 z 1 -xyz 2 + y 2 z 3 , xz 2 -2yz 3 , z 3
The Lie group E is also isomorphic to the semidirect product H 3 ρ R 2 of the additive group R 2 acting on the 3-dimensional Heisenberg group where ρ is given by ρ(a, b)(x, y, z) = e a x, e 2a+2b y, e a (e 2b -1)x + e a+2b z .

(16)

At this point we have some questions for which we do not have an answer.

Questions. 1. To determine a transformation group E of diffeomorphisms of P = L(M ) so that Lie(E) = env ∇ (aff(M, ∇)) in the case when dim(env ∇ (aff(M, ∇))) < n 2 + n. 2. Is it possible to realize an enveloping Lie group of a flat affine Lie group (G, ∇ + ) as a group of transformations of L(G) (eventually as a subgroup of the Lie group K)? If a c (P ) = a(P, ω) the answer to the first question is positive. When dim(Env(a c (P ))) < n 2 +n, the method described on Theorem 3.3 could give an answer to these questions.

The following example describes the generic case of flat affine connections on Aff(R).

Example 3.3. Consider the family of left symmetric products on g = aff(R) given by

• e 1 e 2 e 1 αe 1 e 2 e 2 0 0

where α = 0 and denote by ∇ + α the corresponding left invariant flat affine connection on Aff(R) 0 =: G. A calculation shows that aff(G, ∇ + α ) is generated by the vector fields

C 1 = x ∂ ∂x + y ∂ ∂y , C 2 = ∂ ∂y , C 3 = y ∂ ∂y , and
C 4 = 1 α (x α -1) ∂ ∂y .
The product defined as in Equation 10gives

C 1 C 2 C 3 C 4 C 1 αC 1 -(α -1)C 3 0 C 3 αC 4 + C 2 C 2 C 2 0 C 2 0 C 3 C 3 0 C 3 0 C 4 C 4 0 C 4 0 Hence env ∇ + α (aff(G, ∇ + α )) = aff(G, ∇ + α )
. Now, for α = 1, the associative envelope of the left symmetric algebra (g, •) is 3-dimensional with basis (C 1 , C 2 , C 3 ). Whereas, for α = 1, the associative envelope of (g, •) is two dimensional with linear basis (C 1 , C 2 ).

Furthermore, using Theorem 3.3, an enveloping Lie group of Aff(G, ∇ + α ) is given by

G =      1 + β 1 β 2 0 0 1 + β 3 0 β 1 /α β 2 /α + β 4 1   β 1 = -1 and β 3 = -1    .
We finish the section with a more general example.

Example 3.4. Let G = GL n (R) endowed with the flat affine bi-invariant connection D determined by composition of linear endomorphisms. Given the local coordinates [x ij ] with i, j = 1, . . . , n, it is easy to check that linear bases of left and right invariant vector fields are given by

E + rs = n i=1 x ir ∂ ∂x is and E - rs = n i=1 x si ∂ ∂x ri , with r, s = 1, . . . , n. The group Aff(G, D) is of dimension 2n 2 -1 ([BoMe]
) and the Lie bracket of its Lie algebra is the bracket of vector fields on G. Using the product determined by D on left and right invariant vector fields one gets

D E + pq E - rs = x sp ∂ ∂x rq
, for all p, q, r, s = 1, . . . , n.

It follows that an enveloping Lie group of Aff(G, D) is locally isomorphic to GL n 2 (R).

The following remark describes an algorithm to compute the associative envelope of a finite dimensional left symmetric algebra.

Remark 3.4. Given a finite dimensional real or complex left symmetric algebra A, do as follows.

Find, applying Lie's third theorem, the connected and simply connected Lie group G(A) of Lie algebra A -of commutators of A. This group is endowed with a flat affine left invariant connection ∇ + determined by the product on A.

Compute the associative subalgebra B of a(G(A), ∇ + ) generated by the subspace of right invariant vector fields.

The associative envelope env ∇ + (A) of A is B op .

To finish the section, let us pose the following interesting problem.

Open Problem. To find an algebraic method to determine the associative envelope, in the sense of the Definition 3.2, of a real or complex finite dimensional left symmetric algebra.

Affine Transformation Groups endowed with a flat Affine bi-invariant structure

To finish, let us present some general cases of flat affine manifolds giving a positive answer to Question 1.1. Proof. It follows from the fact that the completeness of ∇ implies that a(M, ∇) = aff(M, ∇) (see [KoNo] page 234).

Corollary 4.2. Let (M, ∇) be a connected and simply connected flat affine manifold. If dim Aff x (M, ∇) ≥ n 2 for some x ∈ M, then the group Aff(M, ∇) is endowed with a flat affine bi-invariant structure determined by ∇.

Proof. The result follows by recalling that the only obstruction to the completeness of ∇ is the existence of a point x ∈ M so that dim Aff x (M ) < n 2 (see [BoMe] and [Tse]).

Corollary 4.3. Let (G, ∇ + ) be a flat afffine Lie group, then the group Aff(G, ∇ + ) admits a flat affine bi-invariant connection if any of the following conditions hold

(1) (G, ω + ) is a unimodular symplectic Lie group and ∇ + is the flat affine connection naturally determined by ω + . (2) G is unimodular and ∇ + is the Levi-Civita connection of a flat pseudo-Riemannian metric.

Proof. Cases (1) and (2) imply that the connection is complete (see [LiMe] and [AuMe], respectively).

The following result is due to [Vey], we present a different proof .

Proposition 4.4. Let (M, ∇) be a compact flat affine manifold, then the Lie group Aff(M, ∇) is endowed with a flat affine bi-invariant connection determined by ∇.

Proof. As M is a compact manifold every vector field on M is complete, in particular every infinitesimal affine transformation of (M, ∇) is complete, hence aff(M, ∇) = a(M, ∇). The conclusion follows from Lemma 3.1. Notice that the converse of Proposition 4.1 is not true. A counterexample is given by the Hopf torus that admits a flat affine connection determined by the usual connection on the punctured plane (see [NaYa] p 200, see also [Ben]), and therefore non-complete, but its group of affine transformations is locally isomorphic to GL 2 (R) (see [NaYa]).

Corollary 4.6. The group of isometries I(M, g) of a compact flat Riemannian manifold (M, g), admits a flat affine bi-invariant structure determined by the Levi-Civita connection.

Proof. The hypothesis imply that I(M, g) 0 = Aff(M, ∇) 0 (see [Yan] and [KoNo] page 244).

Corollary 4.7. If M is compact and ∇ is the Levi-Civita connection of a flat Lorentzian metric, the group of Aff(M, ∇) admits a flat affine bi-invariant connection.

Proof. Under these assumptions, the flat manifold is geodesically complete (see [Car]).

  Example 3.2. Let ∇ + be the flat affine left invariant connection on G = Aff(R) defined by

  Proposition 4.1. If (M, ∇) is a complete flat affine manifold, then Aff(M, ∇) has a flat affine bi-invariant connection determined by ∇.

Corollary 4. 5 .

 5 Let (G, ∇ + ) be a flat affine Lie group, D a discrete cocompact subgroup of G and ∇ the connection naturally induced by ∇ + on M = D\ G. Then the group Aff(M, ∇) has a flat affine bi-invariant structure determined by ∇.

Acknowledgment. We are very grateful to Martin Bordemann and to the referee for their many comments and suggestions that allowed to improve the redaction of this work. We also thank to the referee for communicating non published Jack Vey's results.


Redouane Beniazza

Natalia Bayo

Damien Jardel

Ruben Rust

Bosi Mao

Thibaut Divoux

Marc Schmutz

Frédéric Castet

Guillaume Raffy

André Del

André Del Guerzo

Nathan D Mcclenaghan

Thierry Buffeteau

Jean-Marc Vincent

A fluorous sodium l -prolinate derivative as low molecular weight gelator for perfluorocarbons

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.

exhibit key attractive features: they are biocompatible, inherently accessible as enantiomerically pure compounds, while they can be easily modified with chemical groups to tune their solubility/aggregation properties, and/or bring new functionalities. To date, α-amino acid-based LMWGs have been developed only for aqueous and organic media. Perfluorocarbons (PFCs) are the least polar and least polarisable known fluids. On the π* scale of solvent polarity/polarizability, perfluoorooctane has an extremely low value of -0.41 compared to cyclohexane which defines 0, while water is at 1.09. 4 As a consequence PFCs form biphasic systems at room temperature with water and organic solvents, including alkanes. PFCs are non-toxic biocompatible liquids, nonflammable, chemically inert and dissolve gases in large amounts including O2 and CO2. [START_REF] Gladysz | Fluorous solvents and related media[END_REF] Therefore, PFC gels/emulsions 6 have strong potential for bioapplications requiring oxygen delivery such as blood substitutes, [START_REF] Riess | [END_REF] wound-healing treatments, 8 photodynamic therapy (PDT), 9 or cell culture. 10 Also, PFCs are the liquids which exhibit the lowest intermolecular cohesiveness, a key property of high performance lubricants. On the other hand, such low cohesiveness makes PFCs very challenging to gelate. To date, only a few LMWGs of PFCs have been described (Fig. 1). 11 These gelators were shown to gelate PFCs including perfluorodecalin (PFD), n-perfluorooctane, perfluorotributylamine (FC-43), perfluorotripentylamine (FC-70), bromoperfluorooctane, or the 1,1,1,2,2,4,4,5,5,5decafluoropentan-3-one. From previous results it thus emerged that PFC gels of an α-amino acid-derived LMWG might be of great value for biomedical applications such as PDT. Here we report the first example of an α-amino acid-based LMWG for PFCs. In this preliminary account, the synthesis of the fluorous sodium L-prolinate 1 (Fig. 1), its gelating properties and the characterization of the PFC gels using rheology, microscopy, vibrational circular dichroism and modelling studies, are described. Preliminary data on photosensitized singlet oxygen generation in a gel of perfluorodecalin are also reported.

1 Molecular structures of LMWGs for perfluorocarbons.

Fluorous sodium L-prolinate 1 was prepared in 3 steps from 3,5diiodobenzoic acid and L-proline methyl ester (see ESI, Fig. S1). Solubility tests showed that 1 is insoluble at room temperature in water, organic solvents and PFCs. Moreover, gelation tests revealed that 1 was unable to gelate water or any of the organic solvents tested (see ESI). Conversely, 1 is an excellent gelator for PFCs. The gels were obtained by dispersing 1 by sonication (0.5 mL of solvent in 2 mL vials), followed by rapid heating to the solvent boiling point to obtain a transparent solution, before cooling down to 20 °C. Remarkably, fully transparent gels (Fig. 2) for perfluoromethylcyclohexane (PFMC, bp 76 °C, d 1.79 g/cm 3 ), perfluorodecalin (PFD, bp 142 °C, d 1.92 g/cm 3 ) and perfluorotripentylamine (FC-70, bp 215 °C, d 1.93 g/cm 3 ) containing as little as 0.5-0.6 wt% of 1 for PFD and PFMC, and 1 wt% for FC-70, were obtained. These weight fractions correspond to the lowest gelation concentrations of 9.5 mM and 17 mM, respectively. Gels of n-perfluoorooctane (bp 103-104 °C, d 1.77 g/cm 3 ) were successfully prepared as well, but syneresis occurred within minutes after gelation. Interestingly, the hydrofluoroether HFE-7500 could also be gelated by 1 with a minimum gelling concentration of 9.5 mM (0.56 wt%) yielding a fully transparent gel (Fig. 2). The linear viscoelastic properties of perfluorodecalin gels were characterized using a parallel-plate geometry connected to a stress-controlled rheometer. The sample temperature is controlled by a Peltier module placed under the bottom plate. The liquid PFD solutions were introduced at a temperature of about 120 °C in the geometry pre-heated at 80 °C, which is then sealed with a layer of Sunflower seed oil to prevent evaporation over the duration of the experiment. The gelation was induced by a decreasing ramp of temperature from 80 °C down to 20 °C (Fig. 3a), while the viscoelastic properties were monitored by small amplitude oscillations. As illustrated in Fig. 3b for a sample containing 3 wt% of 1, the elastic modulus ( ′) and the viscous modulus ( ′′) increase during the temperature ramp before reaching steady-state values after 20 min verifying ′ ≫ ′′ , which means that the sample, initially liquid, turned into a soft solid. Beyond the gelation point associated with the crossing point of ′ and ′′ at a temperature higher than 70 °C, the sample experiences a vertical contraction, as evidenced by the decrease of the gap width separating the two plates (see grey symbols in Fig. 3b). This contraction is strongly reminiscent of that observed during the gelation of solutions of polysaccharides such as agarose and carrageenan, 12 although here the contraction takes place after the gelation point. Repeating these rheological measurements on samples prepared with different gelling contents c shows that the terminal elastic modulus ′ increases as a power law of c (Fig. 3c), which is a common feature of polymer gels. [START_REF] Djabourov | Physical gels from biological and synthetic polymers[END_REF] Finally, the sample contraction seems to increase for increasing amount of gelling agent (Fig. 3d). As gelator 1 is chiral, we used vibrational circular dichroism (VCD) to characterize the PFC gels. The infrared (IR) and VCD spectra measured for 1 wt% of 1 in PFD are reported in Fig. 4 in the 1700 -1550 cm -1 spectral range, since no VCD contribution appears at lower wavenumbers (Fig. S2). The rotation of the cell around the light beam axis gave no change in the VCD spectra (Fig. S3), revealing no linear dichroism contribution. A strong positive couplet is observed around 1660 cm -1 with negative and positive components at 1664 and 1658 cm -1 , respectively, assigned to the stretching vibration (νC=O) of the amide carbonyl group located at 1661 cm -1 in the IR spectrum (See the calculated IR spectra Fig. S10). This couplet arises from the coupling of adjacent carbonyl groups (transition dipole coupling model) in the helical supramolecular structure. [START_REF] Tinoco | [END_REF] Furthermore, the intensity of this couplet is related to the number of molecules in the supramolecular structure. 15 The anisotropic ratio (also called g factor) 16 is equal to 0.0064 ± 0.0001, which is in the range of values obtained for amyloid fibrils, 17,18 and ten times lower than values obtained for fluorinated gelators based on 1,2-diaminocyclohexane. 19 Three other VCD bands of weak and negative intensities are observed at 1631, 1623 and 1614 cm -1 , related to the antisymmetric stretching vibration (νaCOO -) of the carboxylate groups and coupled modes with the amide νC=O vibration (Fig. S10). The g factor of the νaCOO -band (-2.4 ×10 -4 ) is significantly lower than the one of the amide carbonyl band. Information about the chirality of perfluoroalkyl chains between 1300 and 1100 cm -1 is not possible in our study due to the interference associated with the strong absorptions of the PFD solvent in this region. Finally, replacing the sodium counterion with potassium results in a small decrease of the g

(∆h/h 0 ) max (%)
This journal is © The Royal Society of Chemistry 20xx J. Name., 2013, 00, 1-3 | 3

Please do not adjust margins

Please do not adjust margins factor for the couplet related to the carbonyl band, with a value equal to 0.0058 ± 0.0002 (Fig. 4). To get more insights on the native 3D fibrillar network of PFC gels, Cryo-Scanning Electron Microscopy (cryo-SEM) and Atomic Force Microscopy (AFM) were used (Fig. 5). First, we emphasize that we could not observe fibers deposited on a surface that are devoid of solvent. This is because the fluorous fibers do not adhere on the carbon grid or the glass coverslip on which they are deposited, but migrate with the solvent front upon evaporation, to reach a point where their density becomes too high to allow their observation. Consequently, attempts to exploit classical high-resolution TEM or SEM techniques failed.

On the contrary, the microstructure of a perfluorodecalin gel was obtained by cryo-SEM. The sample was prepared by fracturing the upper surface of a quickly frozen piece of gel followed by short sublimation of the solvent (2 min at -90 °C) before metallization with a thin layer of platinum. Fig. 5a and Fig. S5 reveal that in the perfluorodecalin solvent the gel is composed of a dense network of nanofibrils, the average diameter of which is about 10 nm. Higher-resolution images were obtained by AFM, in particular using alternative contact mode phase image, which provides images of the fibrils that are still surrounded by the solvent (Fig. 5b, Fig. S6-S8). These images unambiguously show the presence of nano-helices that are all left-handed (M), whose average width is 10.6 ± 2.4 nm, in agreement with the width observed by cryo-SEM, while the average pitch is 13.1 ± 4.1 nm (Fig. S9). Taken together, the IR, VCD and microscopy studies reveal a gelation mechanism in which the fluorous proline 1 self-assembles in PFCs into a dense network of thin M nano-helices leading to very effective formation of gels.

When considering the possible mechanism of the molecular self-assembly leading to the fiber formation, previous studies on LMWGs have revealed that the carboxylate-sodium interaction was a key driving force in the gelation processes. 3a,11d,20 The carboxylic acid of 1, which exhibits improved solubility compared to 1, fails to gelate the PFCs but leads to the formation of crystalline materials instead.

Comparison of the experimental IR spectrum of the PFD gel of 1 with the calculated spectra for four structures of 1 having various carboxylate/Na + ratios and binding modes (structures A-D in Fig. S10), led us to propose structure D as the most likely.

In this structure, each carboxylate is coordinated to three sodium ions by both non-bridging bidentate and bridging bidentate coordination modes, with the oxygen atom of the amide group also binding one sodium. Upon self-assembly, such arrangement should favour the formation of coordination polymers that could drive the elongation process necessary to form fibers. Notably, such carboxylate/sodium binding modes are found in the X-ray structures of compounds structurally related to 1, namely, the antihypertensive drug Fosinopril Sodium 2, 21 and the sodium salt of the (R)-Phenylalanine functionalized norbornene 3, 3a a LMWG for organic solvents (Fig. S11a). In the crystal, these compounds are assembled into coordination polymers forming 1D infinite double-strands for 2 (Fig. S11b), 21 or helical columns for 3. 3a Finally, having in mind the potential of PFCs gels for application in topical PDT we conducted, for the first time, studies on the photogeneration of singlet oxygen in a PFC gel. Gels of perfluorodecalin containing the highly fluorophilic porphyrin 4 9a,22 as photosensitizer (6.8 x 10 -5 M) were prepared (Fig. S12), and the amount and lifetime of 1 O2 generated by photoirradiation by irradiation at 532 nm were evaluated. We find that for a 0.6 wt% PFC gel, the amount of 1 O2 luminescence detected (λmax = 1270 nm) is about 4.5 lower than that of the PFD porphyrin solution, as shown in Fig. 6. This result is mirrored by the 1 O2 lifetime, which decreases from 17.3 ms in solution to 3.7 ms in the gel, i.e. by a factor of about 4.7. A similar radiative rate constant can, therefore, be inferred in solution and inside the gel, while non-radiative deexcitation processes are enhanced which is consistent with modest dynamic quenching of 1 O2 by interaction with 1. Also, a Stern-Volmer type analysis (Table S1, Fig. S13) shows a linear dependence of singlet oxygen luminescence quenching in gels with increasing concentrations of gelator molecule (quencher) consistent with dynamic quenching, rather than coconfinement-enhanced quenching. Despite that, the long 1 O2 lifetime (which determines its effective diffusion distance) of 3.7 ms measured in the 0.6 wt% PFD gel is still much higher than those measured in acetonitrile (54.4 µs), benzene (26.7 µs) or water (4 µs). 23,24 To conclude, we have shown that the fluorous sodium Lprolinate 1 self-assembles in perfluorocarbons to form a threedimensional network of left-handed nano-helices resulting in the formation of transparent PFC gels. Combined IR and modelling studies suggest that the fiber elongation process This journal is © The Royal Society of Chemistry 20xx

Please do not adjust margins Please do not adjust margins originates from the formation of a coordination polymer. Data showing the long singlet oxygen lifetime in PFC gels are also reported. These results bode well for application of such transparent biocompatible PFC gels for PDT, gels which should facilitate topical application on skin, while favouring woundhealing and effective 1 O2 delivery.

Fig. 2

 2 Fig. 2 Photos of PFD, PFMC and HFE-7500 gels (inverted test tubes) at 9.5 mM, and a FC-70 gel at 17 mM.

Fig. 3

 3 Fig. 3 (a) Temperature ramp imposed on the sample to induce gelation. The cooling rate is 5°C/min. (b) Evolution of the elastic (G,' in red) and viscous modulus (G," in blue) measured by oscillations of small amplitude (strain γ = 0.01%) during the formation of a perfluorodecalin gel composed of 3 wt% of 1. The continuous lines correspond to a box average of the data. The experiment is performed at constant normal force ( 0N), while the gap width (initial value ℎ ) is free to adapt to maintain such a constrain. The relative change in the gap width is plotted as grey open squares (right axis). The sample contracts by about 3% over the gelation transition. (c) Evolution of the terminal values of the elastic modulus ′ with the amount of 1 reported in logarithmic scales. The red line corresponds to the best power-law fit of the data: ′ 0.26 0.01 . .. The error bar was determined by repeating three times the experiment on the 3% sample.(d) Maximal relative gap decrease ∆ℎ/ℎ during gelation of samples with different contents in 1.

Fig. 4

 4 Fig. 4 VCD (top) and IR (bottom) spectra measured for 1 wt% perfluorodecalin gels of 1 (solid line), and fluorous potassium L-prolinate derivative (dotted line) in PFD solvent.

Fig. 5

 5 Fig. 5 (a) Cryo-SEM image (scale bar 1 µm) of a perfluorodecalin gel of 1 (0.5 wt%, see the text and ESI for details on sample preparation). (b) AFM phase image (scale bar 100 nm) of a perfluorodecalin gel of 1 (1 wt%) recorded during solvent evaporation (See ESI for experimental details). (c) Zoom on a helix (scale bar 25 nm).

Fig. 6

 6 Fig.6Singlet oxygen phosphorescence from a perfluorodecalin solution (red line) of porphyrin photosensitizer 4 (6.8 x 10 -5 M, λex = 532 nm) and corresponding opticallytransparent gel (black line) comprising gelator 1 (0.6 wt%).

Conflicts of interest

There are no conflicts to declare.


Julie Pham

Alvaro Hernandez

Anna Cioce

Silvia Achilli

Giulio Goti

Corinne Vivès

Michel Thepaut

Anna Bernardi

Franck Fieschi

Niels-Christian Reichardt
email: nreichardt@cicbiomagune.es

Chemo-Enzymatic Synthesis of S. mansoni O-Glycans and Their Evaluation as Ligands for C-Type Lectin Receptors MGL, DC-SIGN, and DC-SIGNR

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Introduction

S. mansoni glycans play an important role in host infection by helminths and mounting evidence suggests that they contribute to the parasite evasion of the host immune system by hijacking C-type lectin receptors (CLRs). CLRs are crucial components in the immune system acting as pattern recognition receptors (PRRs). [1,2] Their interaction with pathogen associated molecular patterns (PAMPs) at the pathogen surface can lead to a series of signalling events ultimately resulting in a skewed immune response. [3,4] The glycans on the surface of S. mansoni are thought to be responsible for the characteristic TH2/Treg response observed during chronic schistosomiasis. [5] Concomitantly, reduced disease related inflammation was observed in autoimmune pathologies such as multiple sclerosis, rheumatoid arthritis, type I diabetes and inflammatory bowel diseases for patients infected with helminths or treated with parasite extracts. [6] Although yet to be fully assigned, the S. mansoni glycome displays a rich array of immunogenic glycan epitopes such as Le X and LDNF which are known to interact with host CLRs. Moreover, interesting similarities can be observed between helminth and host glycans. Indeed, the S. mansoni specific O-glycan core, which is predominantly found in the cercariae, closely resembles an O-glycan core found in the intermediate snail host Biomphalaria glabrata. [7,8] On the other hand, the mucin core 2, which is abundantly found in mammalian host mucosal tissues, is predominantly found in the helminth eggs. [9] Thus, S. mansoni is strongly suspected to employ glycan mimicry as a strategy to subvert the host immune system to allow parasitic survival. [10] The immunomodulatory properties of S. mansoni O-glycans place them as potentially interesting lead compounds for the development of carbohydrate-based drugs to treat immuno-compromised patients with autoimmune diseases, allergies or cancer. However, most studies have focused on parasite N-glycans or the antigenic motifs alone and only few describe the relevance of parasite O-glycans and their interactions with CLRs. [11,12] This is mainly due to the challenge of isolating pure compounds in sufficient amounts for functional and diagnostic tests. [13][14][15] The aim of our study was to investigate the interaction of S. mansoni O-glycans with key CLRs and identify lead structures for improved targeting of immune cells via specific CLRglycan interactions. To this end, we report, here for the first time, the chemoenzymatic synthesis of several S. mansoni O-glycans and evaluate their interactions with fluorescently labelled human CLRs DC-SIGN, DC-SIGNR and MGL.

Results and Discussion

The retrosynthetic analysis of the O-glycan cores O1 and O2 (Scheme 1) suggested a convergent strategy involving three building blocks. After conjugation of the GalNAc donor 1 with a protected amine linker 2, glycosylation in O-3 with galactose donor 7 would provide the Galβ1-3GalNAc disaccharide 5 common to both cores. A second glycosylation in O-6 with galactose (Gal) 7 or N-acetylglucosamine (GlcNAc) donor 8 would furnish the S. mansoni core O1 or mucin 2 core O2, respectively. Scheme 1. Retrosynthetic analysis for the S. mansoni core and core 2 trisaccharides O1 and O2 To facilitate the immobilisation of final compounds as ligands on glycan arrays or to carrier proteins, the known galactosamine trichloroacetimidate donor 1 [16] (Scheme 1) was conjugated with N-benzyl-benzylcarbamate protected aminopentyl linker 2 [17] to obtain glycoconjugate 3 in 73%. Treatment of 3 with a guanidine/guanidium solution quantitatively hydrolyzed all acetates without affecting the base-sensitive N-Troc group. [18] Subsequent acetal protection of O-4 and O-6 with benzaldehyde dimethyl acetal under acid catalysis provided 4 in 70%. Glycosylation of the remaining free O-3 with the galactose imidate 7 under TMSOTf catalysis then afforded disaccharide 5 in 62% yield, as a common intermediate in the synthesis of both core structures. Reductive ring opening of the benzylidene acetal in 5 with BH3•THF and catalytic TMSOTf furnished the 4-benzyl ether 6 in a moderate 61% yield. Finally, glycosylation at O-6 either with galactose donor 7 or GlcNAc donor 8 afforded trisaccharides 9 and 10 in 40% and 73% yield, respectively. For the enzymatic diversification of the core structures with recombinant glycosyltransferases both O-glycan cores were partially deprotected. The benzyl protection was left untouched at this stage of the synthesis to aid in the separation of products by HPLC after enzymatic modifications. Deprotection of the N-Troc functionality under standard conditions with LiOH or zinc/acetic acid amalgam failed to provide clean products but refluxing the compounds 9 and 10 in 1M TBAF in THF afforded the clean S. mansoni core 11 in 95% and the mucin core 2 trisaccharide 12 in 78% yield after three steps. [19][20][21][22] Both core structures were then diversified by enzymatic glycosylations to access the Gal1-4GlcNAc (LN) and GalNAc1-4GlcNAc (LDN) motifs typically observed in the cercarial and egg glycomes of S. mansoni. Towards this end, we employed three recombinant glycosyltransferases, a bovine beta-1,4-galactosyltransferase (β4Gal-T1), a double mutant (Y289LC342T) conferring beta-1,4-N-galactosaminyltransferase (DMGalT1) activity and a Neisseira meningitidis beta-1,3-N-acetylglucosaminyltransferase (LgtA_X) construct prepared exclusively for this purpose (scheme 3). [23][24][25] The new LgtA_X construct contains two polyhistidine tags and a thioredoxin domain to increase LgtA purity and solubility, respectively. Scheme 3.Enzymatic diversification of cores structures 11 and 12 yielding glycans 13 (57%), 14 (28%), 15 (49%), 16 (61%) and 17(59%).

The general substrate and donor specificity of LgtA, including a comparison with the human homologue B3GnT2, has previously been described by Blixt et al. Employing N-glycan G5 (Figure 1) we confirmed the enzymatic activity of LgtA_X to be similar to LgtA in solution, i.e. for the elongation of terminal galactose with GlcNAc. To evaluate the usefulness of the enzyme for diversification of the S. mansoni core type trisaccharide we assessed the enzyme selectivity and activity on trisaccharide 11. Incubation of the O-glycan core with 2 potential acceptor sites afforded 57% of a β1-6 monosubstituted product 13, next to only 7% of the bis-substituted product. [26]Errore. Il segnalibro non è definito.

Before enzymatic introduction of an acid-labile fucose residue we decided to remove remaining benzyl and Cbz protecting groups by hydrogenation. Protected glycans 11-17 were dissolved in a mixture of H2O/MeOH/AcOH, and hydrogenated under Pd/C(10%) catalysis under atmospheric pressure of H2(g), which afforded the deprotected final glycans O1-O7 in 40-100% yield (Figure 1). [27] For the synthesis of the proposed Galβ1-4[Fucα1-3]GlcNAc (Le X )-type and GalNAcβ1-4[Fucα1-3]GlcNAc (LDNF) structures we employed a recombinant α1,3-fucosyltransferase from Caenorhabditis elegans (CeFUT6) and a commercial Heliobacter pylori α-1,3 fucosyltransferase (HP-FucT). CeFUT6 has a known Lewis-X fucosylation activity which was employed in this study to yield O8 from O7 in 73% yield. [28] HP-FucT has a broad substrate and donor specificity including the C-6 azido-fucose surrogate (FucZ). [29,30] Introducing an azide function as a biorthogonal handle in the molecule opened the possibility of generating glycomimetics by copper-catalyzed cycloaddition with alkynes to afford a library of novel glycomimetics with potentially improved affinity towards CLRs. [31][32][33][34] We decided to employ a microarray platform for the rapid preparation of fucosylated and azido-fucosylated compounds. Performing the reactions in parallel and on a microscale decreases the need of expensive azido-donor by several orders of magnitude compared to the solution phase synthesis of individual compounds. The synthetic parasite O-glycans O1-O8 were printed alongside a set of N-glycans which were chosen as additional fucose acceptors from a glycan library available in our laboratory onto NHS activated indium tin oxide (ITO) slides (Figure 1) . [35,36] The bio-functionality of the printed glycans was confirmed by incubating the array with the plant lectin peanut agglutinin (PNA) ECD recognizing Galβ1-3GalNAc residues (see SI, Figure S4) [37,38] Figure 1. a) Selection of ligands immobilized on the microarray; b) surface functionalization of the ITO slide Initially, we screened donor and HP-FucT enzyme concentrations to optimize on-chip fucosylation for printed glycans. Yields for the enzymatic on-chip glycosylation were determined by MALDI-TOF MS as ratios of the sum of product signal intensities (glycan#-F for fucosylation and glycan#-Z for azido-fucosylation) to remaining starting material signal intensities (Figure 2,A). Plotting the product ratios for the different glycans showed no increase in product formation beyond an enzyme concentration of 189 mU ml -1 (Figure 2,B) possibly due to enzyme precipitation at higher concentration.

Repeated glycosylation with freshly prepared enzyme solution and 1mM of GDP-fucose produced higher product conversion than a single exposure with 2mM donor concentration (Figure 2,C). While for glycosylation with the natural GDP-fucose donor, two incubation cycles were sufficient to reach a maximum conversion between 65-90%, similar conversion yields employing the GDP-6-azido-fucose required 3 cycles. Product conversion yields showed a certain degree of structure-dependent variability but were altogether above 80% for the fucose addition and above 75% for the azido-fucose addition (see SI, Figure S5).

As a general note to the reader unfamiliar with glycan array binding experiments, the binding data between different arrays were only compared in relative terms i.e. by comparing ligand binding profiles. Apart from the underlying binding affinity between carbohydrate ligand and lectin, the absolute fluorescence intensities given as RFU values depend on the degree of protein labelling, the incubation time, the washing conditions and image acquisition parameters among others. Some of these parameters are more easily controlled than others and efforts are under way in many labs to improve reproducibility of glycan arrays to allow for a comparison of data between arrays and different immobilisation platforms. Unless an internal reference is used to normalise fluorescence values between different slides or arrays, or extreme measures are put in to place to reproduce incubation and analysis conditions for all slides, absolute fluorescence values between different arrays should not be compared. The effect of the spatial organization of carbohydrate binding domains on glycan interaction was first assessed by comparing the fluorescent binding profiles of the monovalent CRD and the tetrameric ECD of DC-SIGNR on the unmodified array (Figure 3). DC-SIGNR is an endothelial tetrameric C-type lectin receptor that binds to high mannose and complex glycans found on surface glycoproteins of ebola, HIV or hepatis virus. [39] It has also been suggested as an entry path for S. mansoni larvae that begin their migration towards the intestine and liver. [START_REF] Alberts | Molecular Biology of the Cell[END_REF] The glycan binding profile of DC-SIGNR has been previously studied by glycan array and frontal affinity chromatography. Strong binders include high mannose and hybrid structures and to a lesser extent galactosylated complex glycans with terminal GlcNAc residues. [START_REF] Yabe | [END_REF][42][43] Lacking high mannose and hybrid structures on our array, we found strongest binding to the ECD domain for glycans with terminal GlcNAc (G4, O2 and O3). The CRD showed preferential binding towards glycans with a S. mansoni core and galactose residues G10, G14 and O4, while mucin core derivatives O2 and O3 which strongly bound the ECD construct did not show interaction with the isolated carbohydrate recognition domain CRD. In addition, and contrary to the ECD, fucosylated glycans (O8, G11, G13 and G7) displayed enhanced lectin binding compared to their nonfucosylated counterparts for the CRD (O7 and G5). The observed differences in binding profiles towards monovalent CRD and tetravalent ECD receptor organization are striking and have been observed previously for other systems. [44] Based on our array binding data we can only speculate on the underlying molecular basis determining the observed change in specificity. Rapid rebinding to multiple equivalent binding sites in close proximity could improve binding to some structures, while others could face additional binding penalties due to steric clashes in the tetrameric lectin organization. In addition, chelation could favor binding to compounds which are capable of interacting with more than one binding site. This observation could have practical implications for lectin binding studies as for simplicity and practical reasons these are often

G1 G2 G3 G4 G5 G6 G7 G8 G9 G10 G11 G12 G13 G14 G15 G16 O1 O2 O3 O4 O5 O6 O7 O8
Tetravalent ECD Relative glycan recognition (%)

G10 G10
carried out on the monomeric CRD, which is easier to express and purify. The natural presentation of DC-SIGNR, however, is determined by the organization via the stem region into defined tetramers. Any development of multivalent inhibitors should therefore take the natural receptor organization into account. DC-SIGNR, unlike DC-SIGN, binds to neoglycoconjugates presenting all Lewis antigens except Lewis X epitopes. [42] We therefore turned our attention to DC-SIGN. Recently, we reported complementary binding selectivity towards one of the isomeric N glycans G3/G8 and G4/G9 by DC-SIGN and DC-SIGNR. [43] DC-SIGN is found on dendritic cells and macrophages as a tetrameric lectin and displays specificity for mannose, fucose and GlcNAc residues. N-glycans bearing Le X and LDNF epitopes in the soluble egg antigens have been shown to bind to DC-SIGN and inhibit DC activation. [45,46] Incubation of the unmodified initial array with fluorescently labeled DC-SIGN ECD showed strongest binding to glycans presenting Le X (G10, O8) or LDNF (G11-G13) glycan determinants but not to the bis-core fucosylated N-glycan G7 (Figure 4) possibly due to the impact of core fucosylation on glycan conformation and hence lectin recognition. [47] Glycans G3, G4 and O3 presenting a free GlcNAc residue on the 6-arm barely showed interaction with the lectin. However, in a previous screen on a glycan array with different surface architecture they have been shown to bind to DC-SIGN with similar strength as LDNF structures. [43] Among the strongest binders in our previously reported screen were biantennary glycans with terminal GlcNAc residues and high mannose structures, not included in the present array. After on-chip fucosylation with HP-FucT, DC-SIGN bound to all enzymatically extended structures, though the spot-to-spot variability increased considerably, possibly due to partial removal of the non-covalently attached glycoconjugates during prolonged washing procedures (Figure 5). Nevertheless, the population of fucosylated O-glycans in our array bound with intensity similar to the fucosylated N-glycans, making them equally as valuable in the analyses of CLR-glycan interaction, as previously stated by Van Diepen et al. [48] Lewis X binding to DC-SIGN CRD is thought to be favoured by Van der Waals interactions between the fucose C-2 and the neighbouring V351. [49] However, given the limitations in glycan-lectin interaction predictions, it was difficult to predict whether the azide modification at C-6 would affect glycan binding. [50,51] Despite also showing an increased degree of variability, DC-SIGN bound to all azido-fucosylated structures of our array, indicating that the C-6 modification does not significantly contribute to glycan-CLR interaction (Figure 6). Additionally, it was interesting to note that glycans G11Z and G13Z containing an LDNF and azido-LDNF epitope were observed to be the highest binders.

G1 G2 G3 G4 G5 G6 G7 G8 G9 G10 G11 G12 G13 G14 G15 G16 O1 O3 O4 O5 O2 O6 O7 O8 RFU/1000
G1F G2F G3F G4F G5F G6F G7F G8F G9F G10 G11F G12 G13F G14F G15F G16F O1 O3 O4F O5F O2 O6F O7F O8 RFU/1000 0 5 10 15 20 G1Z G2Z G3Z G4Z G5Z G6Z G7Z G8Z G9Z G10 G11Z G12Z G13Z G14Z G15Z G16Z O1 O3 O4Z O5Z O2 O6Z O7Z O8
RFU/1000

N 3 N 3
Finally, we looked at macrophage galactose lectin (MGL) binding to our collection of immobilised O-and N-glycans. MGL is a PRR expressed at the cell surface of macrophages and dendritic cells that recognizes predominantly terminal GalNAc residues found on Tn antigen glycosylated mucins overexpressed on cancer cells or the GalNAcβ1-4GlcNAc (LDN) disaccharide motif of the parasite S.mansoni. [52] Targeting MGL with high affinity glycans or glycomimetics has been suggested as a strategy for macrophage targeting in cancer immune therapy and drug delivery. [53,54] In line with its reported binding specificity, we found strong interactions of MGL with all glycans presenting terminal GalNAc residues of the unmodified array (Figure 7). We observed a low degree of oligosaccharide context dependent alteration of the interaction of GalNac residues with MGL. A similar binding strength was seen for the isolated LDN epitope G16 and glycans presenting the LDN motif on one or both antennae. [43] Glycan O5 with an asymmetrically extended S. mansoni core, presenting a single GalNAc residue, was the strongest binding ligand on the array. Molecular modelling studies suggest that the presence of a α1,3 fucose residue would disrupt the chelation of the GalNAc 3-OH with calcium at the binding site but our MGL binding profile results does not support this generalized rule. [55] Although lower binding was observed for the LDNF glycan G12 than for the isolated LDN epitope G16, both monofucosylated bi-antennary glycans G11 and G13 showed higher fluorescence intensities than the non-fucosylated compound G6. In addition, we were surprised to observe binding of the Le X epitope G10 with a similar binding strength as for the LDNF epitope G12, neither of which was anticipated to bind MGL as fucosylation should inhibit binding. These results suggest that fucosylation may indeed, induce alternate binding modes for non-GalNAc ligands in MGL CRD. On-chip enzymatic fucosylation led to a generally reduced binding to MGL with the exception of O5F and G2F which maintained high affinities (Figure 8, a). Despite the spot-to-spot variability increasing considerably here too, the comparison of relative binding profiles showed that the elongation with 6-azido-fucose decrease binding to most of the immobilized glycans with the exception of G2Z, G4Z and G6Z which remained relatively unchanged (Figure 8,b).

Conclusions

We have chemo-enzymatically synthesized eight novel O-glycans based on the S. mansoni specific core and the mammalian mucin core 2. The unexpected regioselectivity for the 6arm of the S. mansoni core, observed for the recombinant glycosyltransferase LgtA_X, provided a simple entry into products with enzymatic elongations on the 6-arm. The eight synthesized O-glycans were printed alongside a number of N-glycans and were enzymatically modified in good to excellent yields with fucose and non-natural 6-azidofucose. Glycan array based preliminary binding studies with the human C-type lectin receptors MGL, DC-SIGNR and DC-SIGN showed interesting novel insights into the fine specificity of CLR-glycan binding. This study highlights the potential of employing glycan arrays to identify leads for the further development of glycomimetics with improved CLR targeting properties. In addition, our comparison of the binding profiles of CRD and ECD domains of DC-SIGNR underscores the importance of multivalent receptor presentation for inducing binding with an additional level of glycan selectivity. To a solution of 1 [16] (11.8 g, 18.9 mmol) in dry DCM (30ml) on activated molecular sieves and under argon was added a solution of N-benzyl-N-(5-hydroxypentyl)carbamate (7.42 g, 22.7 mmol) 2 in dry DCM (200 ml). The solution was placed at 0°C before TMSOTf (0.690 ml, 3.8 mmol) was added dropwise. The reaction was allowed to warm to RT and stirred for 2hours after which TLC showed full consumption of starting material. The reaction mixture was quenched with Et3N and filtered through celite. The crude sample was purified by flash column chromatography (1540% EtOAc:Tol) to yield 3 as a white gum, 11 g, 73%. 1 [α]D 20 = -11.8° (c=1, CHCl3)

Experimental

5-(benzyl(benzyloxycarbonyl)amino)pentyl4,6-O-benzylidene-2-deoxy-2-((2,2,2-trichloroethoxycarbonylamino) β-D-galactopyranoside 4

Compound 3 (10.2 g, 12.9 mmol) was dissolved in G/GHNO3 [18] (700 ml) and the solution was stirred at RT for 15 mins until full consumption of starting material was observed by TLC. The reaction mixture was concentrated in vacuo then washed thoroughly with DCM and filtered through celite. The filtrate was concentrated to yield a pale pink residue. This was dissolved in dry acetonitrile (100 ml) under argon and benzaldehyde dimethyl acetal (3.4 ml, 2.25 mmol) and camphor sulfonic acid (600 mg, 2.6 mmol) were added. The reaction was stirred at RT overnight after which successful conversion was observed by TLC.

Et3N was added and the reaction mixture was concentrated in vacuo. The resulting residue was purified by flash column chromatography (310% MeOH:DCM) to yield 4 as a white solid, 6.78 g, 70%. To a solution of 4 (6.26 g, 8.3mmol) and 7 [56] (5.0 g, 10.1 mmol) in dry DCM (120 ml) on activated molecular sieves at -40°C and under argon was added TMSOTf (0.301 ml, 1.7 mmol) and the reaction was stirred at -20 for 1hr as monitored by TLC. Triethylamine was added and the reaction was filtered through celite then concentrated to be purified by FCC (2070% EtOAc:Hex) yielding 5 as a white foam, 5.6 g, 62%.

5-(benzyl (benzyloxycarbonyl)amino) pentyl 2,3,4,6-tetra-O-acetyl-β-D-galactopyranosyl-(1→3)-4-O-benzyl-2-deoxy-2-((2,2,2-trichloroethoxycarbonylamino)β-D-galactopyranoside 6

To a solution of 5 (5.6 g, 5.2 mmol) in dry DCM (35ml) under Ar at 0°C was added 1M BH3.THF (20.7 ml, 20.7 mmol) dropwise. The solution was allowed to cool before adding TMSOTf (0.467 ml, 2.59 mmol) dropwise and the reaction was stirred at 0°C under Ar for 1.5 hours after which full conversion was observed by TLC. The ice bath was removed and the solution was quenched with a solution of MeOH:Et3N (10:1) until effervescence ceased. The reaction mixture was concentrated in vacuo and purified by FCC (50100% EtOAc:Hex) to yield 6 as a white solid, 3.4 g, 61%. 1 H NMR (500 MHz, CDCl3) δ 7.43 (d, J = 6.8 Hz, 2H, Ar), 7.39 -7.13 (m, 13H, Ar), 5.43 -5.38 (S, 1H, H-4'), 5.27 (dd, J = 10.5, 8.0 Hz, 1H, H-2'), 5.17 (d, J = 10.9 Hz, 2H, OCH2Bn), 4.99 (dd, J = 10.5, 3. To a solution of 6 (470 mg, 0.434 mmol) and 7 (256 mg, 0.520 mmol) in dry DCM (5ml) on activated molecular sieves, under argon and at -60°C was added TMSOTf (12 μl, 65.1 μmol) dropwise. The reaction was stirred between -40°C and -20°C for 1hr as monitored by TLC.

Triethylamine was added and the reaction was filtered through celite then concentrated in vacuo before purification by FCC (40100% EtOAc:Hex) to yield 9 as a white foam, 252 mg, 41%. To a solution of 6 (420 mg, 0.387 mmol) and with the imidate 8 (291 mg, 0.465 mmol) in dry DCM (5ml) on activated molecular sieves, under argon and at -60°C was added TMSOTf (10.5 µL, 58.1 µmol). The reaction was stirred between -40°C and -20°C for 1hr until full reaction was observed as monitored by TLC. Triethylamine was added and the reaction was filtered through celite. The reaction crude was then concentrated in vacuo and purified by FCC (4060% EtOAc:Hex) yielding 10 as a white solid, 438 mg, 73%.

5-(benzyl(benzyloxycarbonyl)amino)pentylβ-D-galactopyranosyl-(1→3)-[2-deoxy-2acetamido-β-D-glucopyranosyl-(1→6)]-4-O-benzyl-2-deoxy-2-acetamido-β-Dgalactopyranoside 12

To a solution of 10 (390 mg, 0.253 mmol) in THF (6ml) was added TBAF (1M in THF, 0.607 ml, 0.607 mmol) dropwise. The yellow reaction mixture was refluxed for 1.5 hours after which it was placed on ice and quenched with MeOH. The reaction mixture was concentrated to dryness in vacuo before being redissolved in anhydrous pyridine (3ml). Acetic anhydride (0.6 ml, 6.32 mmol) was added and the reaction was left to stir overnight. MeOH was added and the reaction mixture was concentrated in vacuo.

Scheme 2 .

 2 Scheme 2. Convergent synthesis of the partially protected S. mansoni and mucin 2 cores 11 and 12. Reagents and conditions: a)Linker 2, TMSOTf, DCM, -40°C, 73%; b)i) G/GHNO3, MeOH/DCM; ii) PhCH(OMe)2, cat. CSA, DCM, 70%; c) Donor, TMSOTf, DCM, -40°C, 62%; d) 1M BH3.THF, TMSOTf, DCM, 61%; e) Donor 7, TMSOTf, DCM, -41°C, 40%; f) Donor 8, TMSOTf, DCM, -40°C, 73%; g)i)1M TBAF, THF, reflux; ii) Ac2O, pyridine; iii) 0.5M NaOMe, MeOH, 70-90%

Figure 2 .

 2 Figure 2. Optimisation of reaction conditions for on-chip fucosylation; a) MALDI-TOF MS of fucosylation of O7; b) Effect of enzyme concentration; c) Fucose donor concentration; D) Azido fucose donor concentration on product conversion

Figure 3 .

 3 Figure 3. Relative fluorescence intensities and fluorescence image for the unmodified array after incubation with DC-SIGNR CRD (10μg/ml, orange bars) and DC-SIGNR ECD (50µg/ml, blue bars). Each bar in the histogram represents the average of fluorescence from 3 spots which were normalized to G10 and the propagation of error is shown as an error bar.

Figure 4 .

 4 Figure 4. Binding profile of the array immobilized glycans with DC-SIGN ECD (10μg/ml). Histogram bars represent the average fluorescence from 3 spots and error bars represent the standard deviation.

Figure 5 .

 5 Figure 5. Glycan binding of glycan array with tetravalent DC-SIGN ECD (10µg/ml) after onchip fucosylation with HP-FucT. Each bar in the histogram represents the average of fluorescence from 3 spots and the standard deviation is given as an error bar.

Figure 6 .

 6 Figure 6. Glycan binding profile for DC-SIGN ECD (10μg/ml) after modification of the glycan array with the C6-azido-fucose. Each bar in the histogram represents the average fluorescence for 3 replicate spots and the standard deviations are given as an error bar.

Figure 7 .

 7 Figure 7. Glycan binding profile with MGL ECD (10μg/ml). Histogram bars represent the average fluorescence from 3 spots and the standard deviation is given as an error bar.

Figure 8 .

 8 Figure 8. Fluorescence intensities for a) fucosylated y and b) the azido-fucosylated glycan array after incubation with MGL ECD (10μg/ml). Histogram bars represent the average fluorescence from 3 spots and the standard deviation is given as an error bar.

  The crude was purified by FCC (05% MeOH:DCM) to yield 5-(benzyl (benzyloxycarbonyl)amino) pentyl 2,3,4,6-tetra-O-acetyl-β-D-galactopyranosyl-(1→3)-[3,4,6-tri-O-acetyl-2-deoxy-2acetamido-β-D-glucopyranosyl-(1→6)]-4-O-benzyl-2-deoxy-2-acetamido-β-Dgalactopyranoside as a brown solid, 226 mg, 70%.

  .25, 95.67, 74.30, 70.50, 69.86, 69.73, 67.21, 66.76, 61.48, 52.73, 50.48, 50.37, 47.28, 46.10, 29.04, 28.66, 27.76, 26.99, 22.96, 20.69, 20.65.

	H NMR
	(500 MHz, CDCl3) δ 7.41 -7.23 (m, 9H, Ar), 7.17 (d, J = 8.1 Hz, 1H, Ar), 5.36 (d, J = 3.3 Hz, 1H,
	H-4), 5.25 -5.09 (m, 3H, H-3, CH2Bn), 4.79 -4.59 (m, 2H, CH2Troc), 4.59 -4.40 (m, 3H, H-
	1,CH2Bn), 4.14 (qd, J = 11.2, 6.7 Hz, 2H, 2H-6), 3.94 -3.70 (m, 3H, H-5, H-2, CHlinker), 3.50 -

3

.25 (m, 1H, CHlinker), 3.23 -3.12 (m, 2H, CH2), 2.14 (s, 3H, OCH3), 2.05 (s, 3H, OCH3), 1.99 (s, 3H, OCH3), 1.59 -1.43 (m, 4H, 2CH2linker), 1.39 -1.18 (m, 2H, CH2linker).

13

C NMR (126 MHz, CDCl3) δ 170.45, 170.37, 156.77, 156.30, 154.38, 137.79, 136.74, 128.57, 128.44, 127.92, 127.81, 127.73, 127.35, 127.20, 101HRMS (MALDI-Tof): m/z calcd. for C35H43Cl3N2O12 [M+Na] + : 811.1778, found: 811.1728.

  .43, 125.16, 125.10, 124.98, 124.88, 124.57, 124.54, 124.39, 123.71, 99.78, 99.53, 99.33, 94.60, 75.14, 75.07, 74.49, 70.66, 70.37, 69.68, 69.62, 69.52, 67.75, 67.72, 66.93, 56.43, 51.84, 51.71, 48.87, 47.78, 31.72, 31.33, 30.52, 29.75, 26.02, 25.90. HRMS (MALDI-Tof) m/z calcd. for C36H41Cl3N2O9 [M+Na] + : 773.1774, found: 773. 1772. [α]D 20 = -3.5° (c=1, CHCl3)

	1 H NMR (500 MHz, CDCl3) δ 7.51 (d, J = 5.2 Hz, 2H, Ar), 7.42 -7.22 (m,
	11H, Ar), 7.17 (d, J = 7.0 Hz, 2H, Ar), 5.57 (s, 1H, CHPh), 5.23 -5.13 (m, 2H, CH2Ph), 4.72 (d, J
	= 11.7 Hz, 2H, CH2Troc), 4.50 (m, 3H, H-1, CH2Bn), 4.32 (d, J = 12.4 Hz, 1H, H-6a), 4.20 (d, J = 3.6

Hz, 1H, H-4), 4.08 (dd, J = 12.4, 1.9 Hz, 1H, H-6b), 4.03 -3.82 (m, 2H, H-3, CH2), 3.65(m, 1H, H-2), 3.5 (s, 1H, H-5), 3.43-3.25(m, 1H, CHlinker), 3.25-3.15(m, 2H, CH2), 1.62 -1.41 (m, 2H, 2CH2linker), 1.39 -1.16 (m, 2H, CH2linker).

13

C NMR (126 MHz, CDCl3) δ 152.

34, 151.88, 150.76, 150.49, 134.39, 134.36, 134.34, 134.15, 133.47, 133.32, 126.34, 125.72, 125.69, 125.60, 125

  .55 , 128.46 , 128.35 , 128.03 , 127.94 , 127.79 , 127.30 , 127.19 , 102.45 , 99.95 , 95.60 , 78.73 , 74.53 , 74.42 , 74.22 , 73.97 , 73.85 , 70.74 , 70.68 , 69.88 , 69.53 , 69.04 , 67.20 , 67.09 , 61.79 , 61.27 , 54.89 , 50.37 , 47.24 , 45.97 , 29.12 , 28.75 , 27.78 , 27.19 , 23.28 , 23.07 , 20.72 , 20.63 , 20.56 . HRMS (MALDI-Tof) m/z calcd. for C50H61Cl3N2O18 [M+Na] + : 1105.2881, found: 1105.2839. [α]D 20 = -18.4° (c=1, CHCl3)

	4 Hz, 1H, H-3'), 4.92 (d, J = 11.8 Hz, 1H, 4-
	OCH2Bn), 4.84 -4.64 (m, 5H, CH2Troc, 4-OCH2Bn, H-1, H-1'), 4.47 (d, J = 8.8 Hz, 3H, H-3, NCH2Bn

), 4.17 (ddq, J = 15.9, 11.1, 6.1, 4.7 Hz, 2H, H-6'), 3.98 -3.65 (m, 4H, H5', H-4, CHlinker, H-6a), 3.53 -3.14 (m, 6H, H-6b, H-2, H-5, CHlinker, CH2linker), 2.14 (s, 3H, CH3), 2.10 (s, 3H, CH3), 2.02 (s, 3H, CH3), 1.99 (s, 3H, CH3), 1.53 (d, J = 18.6 Hz, 4H, 2CH2linker), 1.31 (s, 2H, CH2linker).

13

C NMR (126 MHz, CDCl3) δ 170.41 , 170.15 , 170.07 , 169.51 , 138.31 , 137.82 , 129.09 , 128

  .12, 169.52, 156.32, 154.11, 138.38, 137.83, 136.67, 128.90, 128.58, 128.49, 128.18, 127.97, 127.79, 127.34, 127.23, 102.31, 101.03, 100.07, 95.63, 78.22, 75.09, 74.40, 74.23, 72.18, 71.82, 70.79, 70.70, 69.00, 68.50, 67.24, 67.09, 62.03, 61.18, 56.38, 54.80, 50.39, 47.35, 28.83, 27.81, 27.20, 23.37, 20.75, 20.68, 20.65, 20.59. HRMS (MALDI-Tof) m/z calcd. for C65H79Cl6N3O27 [M+Na] + : 1566.2928, found: 1566.2886.[α]D 20 = -6.2° (c=1, CHCl3) The yellow reaction mixture was refluxed for 1.5 hours after which it was placed on ice and quenched with MeOH. The reaction mixture was concentrated to dryness in vacuo before being redissolved in anhydrous pyridine (1 ml). Acetic anhydride (0.3 ml, 3.27 mmol) was added and the reaction was left to stir overnight. MeOH was added and the reaction mixture was concentrated in vacuo.

	170.22, 1705-(benzyl(benzyloxycarbonyl)amino)pentylβ-D-galactopyranosyl-(1→3)-[β-D-galacto-
	pyranosyl-(1→6)]-4-O-benzyl-2-deoxy-2-acetamido-β-D-galactopyranoside 11
	To a solution of 9 (154 mg, 0.109 mmol) in THF (3ml) was added TBAF (1M in THF, 0.163
	ml, 0.163 mmol) dropwise. The crude was
	purified by FCC (50100% EtOAc:Hex) to yield 5-(benzyl (benzyloxycarbonyl)amino) pentyl
	2,3,4,6-tetra-O-acetyl-β-D-galactopyranosyl-(1→3)-[2,3,4,6-tetra-O-acetyl-β-D-
	galactopyranosyl-(1→6)]-4-O-benzyl-2-deoxy-2-acetamido-β-D-galactopyranoside a pale
	yellow solid, 125 mg, 90%. 1 H NMR (500 MHz, Chloroform-d) δ 7.43 -7.13 (m, 15H,Ar), 5.38
	(dd, J = 12.7, 3.5, 1.1 Hz, 2H, H-4',H-4"), 5.24 (t, J = 9.2 Hz, 1H,H-2'), 5.19 -5.11 (m, 3H,
	OCH2Bn, H-2"), 5.02 (dd, J = 10.4, 3.4 Hz, 1H, H-3'), 4.95 (dd, J = 10.5, 3.4 Hz, 1H, H-3"), 4.88
	(dd, 2H, H-1, 4-OCHBn), 4.75 -4.63 (m, 3H, H-1', H-3, 4-OCHBn), 4.60 -4.38 (m, 3H, H-
	1",NCH2Bn), 4.22 -4.09 (m, 4H, 2H-6', 2H-6"), 3.96 -3.55 (m, 7H, H-4, H-5, H-5', H-5", 2H-6,
	CHlinker), 3.43 -3.28 (m, 2H, H-2, CHlinker), 3.16 (s, 2H, CH2linker), 2.15 -1.93 (m, 27H, 9CH3),
	1.54 (s, 4H, 2CH2linker), 1.37 -1.21 (m, 2H, CH2linker). 13 C NMR (126 MHz, CDCl3) δ 170.37,
	170.31, 170.25, 170.17, 170.10, 170.07, 169.18, 138.26, 137.74, 101.95, 101.18, 99.26,
	77.88, 75.12, 74.13, 70.83, 70.58, 69.88, 69.21, 68.91, 67.18, 67.06, 67.02, 61.26, 60.98,
	55.21, 50.30, 47.29, 45.91, 29.63, 29.10, 28.71, 27.41, 23.59, 23.12, 20.81, 20.64, 20.61,
	20.54. HRMS (MALDI-Tof) m/z calcd. for C63H80N2O26 [M+Na] + : 1303.4893, found:
	1303.5106, [α]D 20 =-20,5(c=1 CHCl3). The product (120 mg, 94 mmol) was redissolved in dry
	MeOH (4 ml) and 0.5M NaOMe (1.5 ml, 750 µmol) was added. The RM was stirred at RT for
	1 hour after which it was quenched with Amberlite®IR 120 (H). The filtrate was
	concentrated, purified by Sephadex LH-20 (D=3.5 cm, H=45 cm, MeOH) and lyophilized to
	yield 11 as a white powder, 87.5 mg, 99%. 1 H NMR (500 MHz, Methanol-d4) δ 7.46 -7.16
	(m, 15H,Ar), 5.15 (d, J = 16.3 Hz, 2H, OCH2Bn), 4.99 (d, J = 11.6 Hz, 1H, 4-OCHBn), 4.70 (d, J =
	11.6 Hz, 1H, 4-OCHBn), 4.50 (s, 2H, NCH2Bn), 4.47 (s, 1H, H-1), 4.30 (d, J = 7.6 Hz, 1H, H-1'),
	4.24 (d, J = 7.5 Hz, 1H, H-1"), 4.08 (dd, J = 15.4, 6.2 Hz, 2H, H-2, H-4), 3.90 (dd, J = 10.9, 2.9
	Hz, 1H, H-3), 3.87 -3.69 (m, 10H, H-4', H-4", 2H-6, 2H-6', 2H-6", H-5, CHlinker), 3.58 (dd, J =
	1 H NMR (500 MHz, 9.7, 7.6 Hz, 1H, H-2'), 3.48 (tdd, J = 20.9, 10.2, 7.3 Hz, 7H, H-2", H-3', H-3", H-5',H-5",H-2",
	CDCl3) δ 7.42 -7.13 (m, 15H, Ar), 5.98 (s, 0.5H, N-HTroc), 5.45 (s, 0.5H, N-HTroc), 5.39 (s, 1H, CHlinker), 3.22 (dd, J = 14.3, 6.9 Hz, 2H, CH2linker), 1.97 -1.89 (m, 3H, CH3), 1.57 -1.42 (m, 4H,
	H-4'), 5.30 -5.13 (m, 4H, H-2', H-3", OCH2Bn), 5.06 (t, J = 9.6 Hz, 1H, H-4"), 4.98 (dd, J = 10.5, 2CH2linker), 1.28 -1.21 (m, 2H, CH2linker). 13 C NMR (126 MHz, MeOD) δ 174.30, 140.51,
	3.5 Hz, 1H, H-3'), 4.91 (d, J = 11.3 Hz, 1H, CHTroc), 4.84 -4.59 (m, 6H, CHTroc, 4-OCH2Bn,H-1,H-129.59, 129.55, 129.05, 128.36, 107.18, 105.20, 102.59, 81.67, 77.15, 76.96, 76.54, 75.63,
	1',H-1") 4.46 (dt, J = 22.7, 12.8 Hz, 3H, NCH2Bn, H-3), 4.27 (dd, J = 12.3, 4.7 Hz, 2H, H-6'a,H-75.13, 74.97, 74.56, 72.63, 72.54, 70.28, 70.09, 69.83, 68.47, 62.85, 62.26, 53.74, 49.85,
	6"a), 4.12 (ddd, J = 26.1, 11.8, 4.5 Hz, 2H, H-6'b,H6"b), 3.96 -3.25 (m, 10H, H-4,H-5,H-5', H-30.24, 24.
	5", 2H-6, CH2linker, H-2, H-2"), 3.19 (s, 2H, CH2linker), 2.16 -1.95 (m, 21H, 7CH3), 1.65 -1.44
	(m, 4H, 2CH2linker), 1.38 -1.28 (m, 2H, CH2linker). 13 C NMR (126 MHz, CDCl3) δ 170.73, 170.58,

24, 23.27. HRMS (MALDI-Tof) m/z calcd. for C47H64N2O23 [M+Na] + : 967.4052, found: 967.408, [α]D 20 =-5.7(c= 1, MeOH)

  1 H NMR (500 MHz, CDCl3) d 7.43 -7.13 (m, 15H, Ar), 5.39 (d, J = 3.5 Hz, 1H, H-4'), 5.26 -5.10 (m, 4H, H-2', H-3", OCH2Bn), 5.08 -4.98 (m, 2H, H-4", H-3'),4.87 (d, J = 11.4 Hz, 1H, 4-OCHBn), 4.82 (d, J = 8.3 Hz, 1H, H-1), 4.64 (dt, J = 27.6, 9.0 Hz, 4H, 4-OCHBn, H-1', H-1", H-3), 4.49 (m, J = 17.7, 16.6 Hz, 2H,NCH2Bn), 4.30 -4.03 (m, 4H, 2H-6', 2H-6"), 3.97 -3.71 (m, 5H, H-2", H6a, H-4, H-5', CHlinker), 3.70 -3.57 (m, 3H, H6b, H-5", H-5), 3.52 -3.25 (m, 3H, H-2, CH2linker), 3.24 -3.11 (m, 2H, CH2linker), 2.15 -1.84 (m, 24H), 1.64 -1.42 (m, 4H, CH2linker), 1.36 -1.26 (m, 2H, CH2Linker). 13 C NMR (126 MHz, CDCl3) δ 170.88, 170.75, 170.22, 170.12, 169.44, 169.23, 138.52, 137.76, 128.86, 128.60, 128.51, 128.12, 127.72, 127.68, 127.38, 127.27, 101.87, 100.95, 99.53, 75.34, 74.33, 73.63, 72.62, 71.76, 70.88, 70.77, 69.66, 69.26, 68.96, 68.53, 67.23, 67.18, 62.08, 61.17, 54.88, 50.35, 47.32, 28.82, 27.34, 23.66, 23.30, 20.85, 20.79, 20.75, 20.71, 20.68, 20.65, 20.59. HRMS (MALDI-Tof) m/z calcd. for C63H81N3O25 [M+Na] + : 1302.5057, found: 1302.5106, [α]D20 =-18.2° (c= 1, CHCl3). The product (218 mg, 0.170 mmol) was redissolved in dry MeOH(9 ml) and 0.5M NaOMe (2.5 ml, 1.24 mmol) was added. The RM was stirred at RT for 1 hour after which it was quenched with Amberlite®IR 120 (H). The filtrate was concentrated, purified by Sephadex LH-20 (D=3.5cm, H=45cm, MeOH) and lyophilized to yield 12 as a white powder, 153 mg, 100%.1 H NMR (500 MHz, Methanol-d4) δ 7.45 -7.40 (m, 2H, Ar), 7.40 -7.20 (m, 12H, Ar), 7.18 (s, 1H, Ar), 5.15 (d, J = 15.8 Hz, 2H, OCH2Bn), 4.98 (d, J = 11.6 Hz, 1H, 4-OCHBn), 4.67 (d, J = 11.5 Hz, 1H, 4-OCHBn), 4.50 (s, 2H, NCH2Bn), 4.41 (brs, 1H, H-1), 4.36 (d, J = 8.4 Hz, 1H, H-1"), 4.30 (d, J = 7.6 Hz, 1H, H-1'), 4.06 (brs, 1H, H-2), 4.02 (d, J = 3.1 Hz,1H, H-4), 3.91 -3.73 (m, 6H,H6'a, H6"b,H6c, H-3, H-4'), 3.71 -3.54 (m, 5H,H6'a, H6"b,H6c, H-5, H-2', H-2", CHLinker), 3.54 -3.50 (m, 1H, H-5'), 3.49 -3.33 (m, 3H, H-3', H-3", CHLinker), 3.24 (m, 3H, H-5', CH2Linker), 1.92 (d, 6H, 2CH), 1.52 (s, 4H, 2CH2linker), 1.29 (s, 2H, CH2linker).13 C NMR (126 MHz, MeOD) δ 174.24, 173.63, 158.36, 157.84, 140.36, 139.09, 137.95, 129.55, 129.48, 129.03, 128.88, 128.63, 128.36, 128.28, 107.00, 102.61, 102.42, 81.45, 77.79, 77.30, 76.93, 75.98, 75.65, 74.75, 74.47, 72.57, 71.99, 70.24, 70.12, 69.74, 68.29, 62.82, 62.67, 57.18, 53.64, 51.42, 51.22, 49.85, 47.44, 30.14, 28.85, 28.35, 24.19, 23.28, 23.20. HRMS (MALDI-Tof) m/z calcd. for C49H67N3O18 [M+Na] + : 1008.4317, found: 1008.437, [α]D 20 =-6.3 (c=0,1, MeOH)

Acknowledgements J.P., S.A., A.C., N.C.R, and F.F. were supported by the EU Horizon 2020 Research and Innovation Program (Marie Sklodowska-Curie Grant 642870, ETN-Immunoshape). N.C.R additionally acknowledges funding from the Ministry of Science and Education (MINECO) Grant No. CTQ2017-90039-R and RTC-2017-6126-1 and the Maria de Maeztu Units of Excellence Program from the Spanish State Research Agency-Grant No.MDM-2017-0720.F.F.acknowledge also the French Agence Nationale de la Recherche (ANR) PIA for Glyco@Alps (ANR-15-IDEX-02).The Multistep Protein Purification Platform (MP3) was exploited for human DC-SIGN/R and MGL ECD production with support from FRISBI (ANR-10-INSB-05-02) and GRAL (ANR-10-LABX-49-01) within the Grenoble Partnership for Structural Biology.