Arnaud Chéritat
email: arnaud.cheritat@math.univ-toulouse.fr

Relatively compact Siegel disks with non-locally connected boundaries

We construct holomorphic maps f with a Siegel disk whose boundary is not locally connected (and is an indecomposable continuum), yet compactly contained in the domain of definition of the map. Our examples are injective and defined on a subset of C.

As Herman remarked, the construction is very flexible. The reader will find in the conclusion section more properties that these maps can be given. There, we also suggest possible other consequences of the method.

Remark 1. We ignore if the maps can be chosen entire (or entire meromorphic).

Introduction

In [START_REF] Handel | A pathological area preserving C ∞ diffeomorphism of the plane[END_REF], Handel constructed a C ∞ area preserving diffeomorphism of the plane that has a minimal set that is a pseudo circle (see section 3.5 for a definition of pseudo circles). In [START_REF] Michael | Construction of some curious diffeomorphisms of the Riemann sphere[END_REF], Herman adapted the construction to produce a C ∞ diffeomorphism of the sphere that is conjugated to a rotation in the two complementary components of an invariant pseudo circle, and holomorphic in one of them. In the same spirit, in [START_REF] Pérez | Siegel disks with smooth boundary[END_REF] Pérez Marco was able, using tube-log Riemann surfaces, to construct examples of injective holomorphic maps defined in a subset U of C that have a Siegel disk compactly contained in U whose boundary is a C ∞ Jordan curve, which came as a surprise. Again the method is versatile and Kingshook Biswas used Pérez-Marco's construction to produce a set of interesting examples: [START_REF] Biswas | Smooth combs inside hedgehogs[END_REF][START_REF] Biswas | Hedgehogs of Hausdorff dimension one[END_REF]. Here we add an ingredient to this construction and get: Theorem 1. There exists a holomorphic map f defined in a simply connected open subset U of C containing the origin, fixing 0 and having at 0 a Siegel disk ∆ that is compactly contained in U and whose boundary is a pseudo circle.

The new ingredient

Compared to the previous constructions of Siegel disks and hedgehogs, the main new idea is to make use of Runge's theorem. Runge's theorem has been applied in a variety of situations to construct unexpected examples. We will make use of the following version:

Theorem (Runge, simply connected case). For all holomorphic map f defined in a simply connected open set U ⊂ C, there exists a sequence of polynomials that tends to f locally uniformly on U . 1It is easy to see that f can be approximated (in the sense above) by polynomials if and only if it can be approximated by entire maps. In our application, the fact that the approximating sequence consists in polynomials is not so important. We will only need entire maps.

Aknowledgements

The author would like to thank Kingshook Biswas with whom he had very useful discussions, Pérez-Marco for inventing his methods and explaining them to me, Yohann Genzmer for submitting me me a stimulating math problem for which I found a complicated solution using iterated exponentials, Julien Duval for explaining me a simple solution to Genzmer's problem using Runge's theorem, thus bringing back this theorem to my memory and showing me its usefulness. The author would also like to thank the referee for useful comments.

Terminology

Saying that two holomorphic maps commute can mean different things. To be rigorous, we will say that "f and g commute, domain included", if the domain of definition of f • g equals that of g • f and if they take the same values on it. This is what f • g = g • f is supposed to mean. Similarly we will say that "f • g = h • i, domain included" if the domain of definition of f • g equals that of h • i and if they take the same values on it. We will say that "f and g commute on U " if U is included in the domain of f • g and in the domain of g • f and if both compositions take the same value on U . There is another natural notion that we will not use: "f and g commute, wherever defined", if f • g(z) = g • f (z) for all z for which both hand sides are defined.

We will denote the domain of definition of a map f by dom(f ).

The construction

Working in the universal cover of C *

The map f will be the limit of a sequence of compositions

f n = g n • • • • • g 1
of maps g i fixing the origin. However we prefer to work at the level of lifted coordinates: let

E(z) = exp(2iπz).
The map E is a universal cover from C to C * , with deck transformations (Z, +) and induces an isomorphism between the cylinder C/Z and C * . This map can be used to extend the Riemann surface C/Z at its upper end (we will not use the lower end in this article): denote +i∞ the added point, corresponding to

0 ∈ C ⊂ C * .
Let T v denote the translation by v: A holomorphic map F that commutes with T 1 , domain included, and such that

F (z) = z + t + o(1)
for some t ∈ R as Im (z) tends2 to +∞, "projects" to a holomorphic map f with an erasable singularity at the origin, i.e. there exists

f holomorphic defined on {0} ∪ E(domain of f ), with E • F = f • E, f (0) = 0, f (0) = e 2πit = 0, and f (z) = 0 =⇒ z = 0.
Conversely, a holomorphic map f defined in an open subset O of C containing the origin, with f (0) = 0, f (0) = 0, and f (z) = 0 =⇒ z = 0, will have a (non unique) lift

F : E -1 (O) → C that commutes with T 1 , domain included, with E • F = f • E and F (z) -z has a limit when Im (z) → +∞.

Presentation of the actors

Recall that T v denotes the translation by v:

T v (z) = z + v. Let H h = z ∈ C Im (z) > h ,
H = H 0 that we will call upper half planes. Let 1 be the constant vector field in C of expression 1(z) = 1.

Assume that maps R k are given for k = 1, . . . , n, satisfying the following conditions:

-R k is entire -there exists q k ∈ N * such that R k • T 1 = T q k • R k -R k (z) is injective on some upper half-plane. Given the above condition, it is equivalent to "R k (z) = q k z + c + o(1) as Im (z) -→ +∞" -∀z ∈ C, R k (z) = 0
We call these maps renormalizations. They will be inductively defined later in this article.

Let

-X n = R * 1 • • • R * n 1, i.e
. the pull back of the vector field 1 by the composition

R n • • • • • R 1 -G n , the time 1 of the flow of the vector field X n -F n = G n • • • • • G 1
Then X n is a non-vanishing entire vector field. The maps G n and F n are holomorphic but not necessarily defined on all of C.

These objects "project" to C/Z in the following sense:

-T * 1 X n = X n , i.e. X n is T 1 -invariant -G n and F n commute with T 1 , domain included -recall that R n • T 1 = T qn • R n
In particular, they can be thought of objects living on C/Z. Also, they are defined in a neighborhood3 of +i∞, and we have the following expansions as Im (z) -→ +∞:

-R n (z) = q n z + c + o(1) -X n (z) = 1 q 1 . . . q n + o(1) -G n (z) = z + 1 q 1 . . . q n + o(1) -F n (z) = z + n k=1 1 q 1 . . . q k + o(1)
Since each G n , and thus F n commute with T 1 , domains included, the map F n projects by E to a map f n , i.e. there exists a holomorphic map

f n : {0} ∪ E(dom(F n )) → C such that E • F n = f n • E,
domain included, and f n fixes the origin.

Let Dom be the set of points that have a neighborhood on which all F n are defined and converge uniformly. It is an open set. It might be empty. Let F be the map defined on Dom as the limit of the F n . Then F is holomorphic and commutes with T 1 . The following lemma is elementary so we will not explain its proof: it is stated for reference.

Lemma 1. Assume that Dom contains an upper half plane. Define dom = E(Dom) ∪ {0}. Then dom is an open subset of C and f n converges locally uniformly on dom to a holomorphic map f defined on dom and satisfying f

• E = E • F , domain included.
The set of domains containing an upper half plane and on which R n is a bijection to an upper half plane or to C, is non-empty and has a greatest element V n for inclusion. Let U 0 = C and for n ≥ 1 let U n be the set of points

z ∈ V 1 such that for all k < n, R k •• • ••R 1 (z) ∈ V k+1 . Then (U n ) n∈N forms a decreas- ing sequence for inclusion, of T 1 -invariant simply connected open subsets U n of C containing upper half planes, on which R n • • • • • R 1 is an isomorphism to an upper half plane or to C and on which R k • • • • • R 1 is injective for all k ≤ n. Let U n = E(U n ) ∪ {0}.
It is a connected and simply connected open subset of C. Note that each f n is conjugated to a rotation in a neighborhood of the origin. Indeed, let

θ n = n k=1 1/(q 1 • • • q k ). The map z → 1 qn R n • • • • • R 1 (z)
is a conjugacy from F n on U n to T θn on an upper half plane or on C, and commutes with T 1 on U n , domain included. Therefore it projects by E to a conjugacy on U n to the rotation of angle 2πθ n on a disk or on C.

We will use the following lemma, applied to well chosen subsets D n of U n :

Lemma 2. Assume f n is a sequence of holomorphic maps defined on open subsets of C containing the origin. Assume f n fixes 0 and that f n (0) = e i2πθn with θ n ∈ R and θ n -→ θ ∈ R as n → +∞. Assume D n is a sequence of simply connected open subsets of the domain of f n and containing 0, such that f n is analytically conjugated on D n to the rotation of angle 2πθ n on a disk or on C. Assume that there exists a limit D to the sequence D n in the sense of Caratheodory. Then f n tends on D locally uniformly to a map f that is analytically conjugated on D to the rotation of angle θ on a disk or on C.

Proof. Let R α (z) = e i2πα z. The conjugacy φ n on D n can always be chosen so that φ n (0) = 1. Then a theorem of Caratheodory says that φ n converges locally uniformly on D to the unique conformal map φ from D to a disk or C such that φ(0) = 0 and φ (0

) = 1. The reciprocal φ -1 n is locally uniformly convergent to φ -1 . From f n = φ -1 n • R θn • φ n we deduce that f n tends to φ -1 • R θ • φ locally uniformly.
Depending on authors, the definition of Siegel disks mays differ slightly. For us, it is the maximal domain of conjugacy to an irrational rotation. In the lemma above, if θ is irrational, then D is an invariant subset of the Siegel disk of f : could the Siegel disk of f be bigger? It necessarily requires the boundary of D to be an analytic Jordan curve 4 , compactly contained in dom. In fact it is equivalent. So if we get a boundary that is not a Jordan curve or not an analytic one, then we know that D equals the Siegel disk of f . (This implies that U n has the same Caratheodory limit as D n .)

How to build examples with D ⊂⊂ dom? First we will ensure that Dom contains H -h for some h > 0. Second we will ensure that D is contained in H.

Ensuring convergence on big domains

Here we give a slightly informal description, and a rigorous lemma that will be used in the more formal construction described in section 3.6.

It will be handy to use a notation for the set of maps satisfying the conditions we required on R n in the previous section. So let H 1 denote the set of maps R such that:

-R is entire -R • T 1 = T 1 • R -R(z) is injective on some upper half-plane -∀z ∈ C, R (z) = 0 So that R k ∈ q k H 1 , i.e. R k is the product of a member of H 1 with the scalar q k .
For this, we will inductively define the maps R n using the following (elementary) lemma:

Lemma 3. Fix n ∈ N. Assume we are given R k ∈ q k H 1 for k = 1, . . . , n -1 and some B n ∈ H 1 . Let R n = qB n
for some q ∈ N * . The corresponding map f n depends on q but f n-1 does not. Then, as q -→ +∞, f n tends to f n-1 in the following sense: every compact subset K of dom(f n-1 ) is eventually contained in dom(f n ) and f n -→ f n-1 uniformly on K. Note however, that the domain U n defined in section 3.2 is independent of q.

Proof. Indeed, R * n 1 = (qB n ) * 1 = 1 q B * n 1. So X n is the product of the scalar 1 q with an entire vector field that does not depend on q. The map G n is the time 1/q of the flow of the latter vector field. The rest follows easily.

Fix some number h > 0. We will inductively define the sequence R n by first choosing B 1 , then q 1 , then B 2 , then q 2 , etc. . . We will be careful to choose q k big enough so that the domain of definition of f n contains the upper half plane

H -h-1 n and |f n-1 -f n | < 1/2 n on H -h
. This is always possible by the lemma above. We can also choose q n to ensure that the sum θ is convergent5 and has an irrational value: a sufficient condition for this is that q n tends to +∞. Then the half plane H -h is contained in the domain of definition Dom of F .

It shall be stressed out that in the construction, the sequence (B n ) will not be fixed independently of the sequence (q n ): the choice of B n+1 will depend on the choice of all the previous objects B 1 , q 1 , B 2 , q 2 , . . . B n , q n .

Ensuring compactly contained bad boundaries

As in the previous section we give here a slightly informal description, and a rigorous lemma that will be used in the more formal construction described in section 3.6.

The following claim is where we use Runge's theorem:

Lemma 4. Assume φ is a conformal bijection from H/Z to a domain W ⊂ C/Z that extends to a conformal bijection fixing the upper end. Then there exists a sequence of holomorphic maps ψ n : C/Z → C/Z such that:

ψ n does not vanish, ψ n (z) -z has a limit when Im (z) tends to +∞, as n tends to +∞, ψ n tends uniformly to φ -1 on every compact subset of W and in a neighborhood of the upper end of C/Z.

Proof. Conjugate the situation by E: let W = E(W ) ∩ {0} and Ψ : W → D be the conjugate of φ -1 , extended by setting Ψ (0) = 0, where D denotes the unit disk. It is a conformal isomorphism. It can therefore be written as Ψ (z) = ze v(z) for some holomorphic map v : W → C. As a conformal isomorphism its derivative does not vanish, hence 1 + zv (z) = 0. Thus there exists a map u : z) . Since e u(0) = 1, we can choose u such that u(0) = 0. So u(z) = zw(z) for some holomorphic w : W → C. Now apply Runge's theorem: there exists a sequence of entire maps w n (even polynomials) that converge locally uniformly to w on W . There exists an entire map v n (unique) taking the same value as v at the origin and such that 1 + zv n = e zwn . Indeed (e zwn -1)/z has a removable singularity at the origin, so it extends to an entire map, of which v n is just the appropriate primitive. The maps ψ n (z) = z + v n (e z ) satisfy the required conditions. Now we describe informally how we can use this lemma to choose the functions B n to get interesting examples. A more formal, but more specific, description is done in section 3.6.

W → C such that 1 + zv (z) = e u(
Let D 0 = H/Z. Let W 1 be any open simply connected strict subset of H/Z ∪ {+i∞} containing the upper end +i∞. Let φ be any conformal map from H/Z ∪ {+i∞} to W 1 that fixes the upper end. Let ε > 0 and let the Jordan curve J 1 be the image by φ of the Jordan curve of equation "Im (z) = ε" in H/Z. If the boundary of W 1 is a convoluted Jordan curve (or something more complicated than a Jordan curve) and ε is small then J 1 will also be convoluted.

Apply lemma 4 to W = W 1 -{+i∞}. By the uniform convergence of ψ n to φ -1 on compact subsets of W and on a neighborhood of the upper end of C/Z, there is for n big enough a branch φ n of ψ -1 n defined for Im (z) > ε/2 and mapping the upper end to itself. For n big, the Jordan curve J 1 = φ n (R + iε), image by φ n of the Jordan curve of equation "Im (z) = ε" in H/Z, will be close to J 1 so it will also be convoluted. So we will choose B 1 = ψ n (z) -iε for some big value of n. Then we will chose q 1 as in section 3.4. Let D 1 be the domain bounded by J 1 and containing the upper end. Among the integral lines of the vector field X 1 , there are the images of the horizontals by φ n , which loop. This includes J 1 and also a set of curves foliating the domain D 1 . The map F 1 = G 1 , seen as acting on C/Z, is conjugate to a finite order6 rotation on D 1 . The renormalization R 1 induces a conjugacy of the restriction of F 1 to D 1 to the translation T 1 on the half cylinder H/q 1 Z.

The next step in the construction consists in choosing a new simply connected domain W 2 , contained in D 1 , containing the upper end, whose boundary is even more convoluted, yet invariant by F 1 , and very close to J 1 . By the conjugacy R 1 on D 1 it amounts to choosing a domain in H/q 1 Z ∪ {+i∞} that is invariant by T 1 , i.e. to choosing a simply connected domain W 2 in H/Z ∪ {+i∞} (by applying the cover z+q 1 Z ∈ H/q 1 Z → z+Z ∈ H/Z), and whose boundary is very close to R.

Let φ : H/Z ∪ {+i∞} → W 2 be a conformal map fixing the upper end. We choose a new value of ε, which gives a new Jordan curve J 2 = D 1 ∩R -1 1 (φ(R+ iε)) close to the boundary of W 2 . We apply lemma 4 again and get a new sequence of entire maps ψ n tending to φ -1 on W 2 . We set B 2 = ψ n (z) -iε for n big enough so that the curve

J 2 = D 1 ∩ R -1 1 (φ n (R + iε)
) is close to J 2 . We choose q 2 as in section 3.4. The domain D 2 bounded by J 2 is then invariant by G 1 and by the flow of the vector field X 2 . The maps G 1 and G 2 , will be thus conjugated by R 2 • R 1 on D 2 to the translations by respectively q 2 and 1 on H/q 1 q 2 Z.

And so on. . . We get a decreasing sequence of domains

D n ⊂ H/Z on which R n • • • • • R 1 is a conformal bijection to H/(q 1 • • • q n )Z
and conjugating all the maps G k , for k ≤ n, to integer translations, and thus the map F n too. Their boundaries form a sequence of Jordan curves extremely convoluted and close to each other. The construction has enough flexibility to allow for D = n D n to have a non locally connected boundary, as we will prove in the following sections (basically by the same method as Handel in [START_REF] Handel | A pathological area preserving C ∞ diffeomorphism of the plane[END_REF]).

Pseudo circles

Pseudo circles are amazing topological objects. The interested 7 reader may look at [START_REF] Bing | Concerning hereditarily indecomposable continua[END_REF] (Example 2 p.48), [START_REF] Fearnley | The pseudo-circle is unique[END_REF], [START_REF] Ted | Pseudo-circles and universal circularly chainable continua[END_REF], [START_REF] Kuperberg | A short proof of nonhomogeneity of the pseudo-circle[END_REF]. It was chosen by Handel in [START_REF] Handel | A pathological area preserving C ∞ diffeomorphism of the plane[END_REF] as a good example of extreme pathology that a minimal set can have even for an area preserving smooth diffeomorphism of the sphere. Herman did the same choice in [START_REF] Michael | Construction of some curious diffeomorphisms of the Riemann sphere[END_REF] for his construction. So we will continue this tradition.

For our purpose, it will be enough to know the following:

-A pseudo circle is compact, connected, separates the plane into two components, but it is not locally connected thus it is not a Jordan curve. -It is equal to the boundary of both components (in particular it has empty interior). -Any set K obtained by the following procedure is a pseudo circle. PROCEDURE: A circular chain in a topological space X is a sequence e = (e i ) of open subsets of X, called links, indexed by Z/mZ for some m ≥ 4, such that e i ∩ e j = ∅ if and only if i, j are adjacent or equal. 8 For each n ≥ 1, let Q n be a circular chain in the plane 9 such that:

the diameters of the links of Q n are finite for all n and their supremum for a

given n tends to 0 as n → +∞ -Q n+1 is a refinement of Q n : each link of the former is contained in a link of the latter -Q n+1 is crookedly embedded in Q n , which means the following: assume that Q n = (d i ) has m links, and that Q n+1 = (e i ) has m links. It will be convenient to let i ∈ Z and let d i denote dī where ī is the class of i modulo m. Similarly, for i ∈ Z let e i denote eī for the residue class modulo m . To be crookedly embedded means that there exists a map f : Z → Z such that 1. ∀i ∈ Z, the closure of e i is contained in

d f (i) 2. ∀i ∈ Z, f (i + 1) -f (i) ∈ {-1, 0, 1} ( 10 ) 3. ∀i ∈ Z, f (i + m ) = f (i) + m ( 11 )
4. for all i, j with f (i) + 2 < f (j) < f (i) + m, there exists i and j with f (i ) = f (j) -1, f (j ) = f (i) + 1 and either i < i < j < j or i > i > j > j.

7 Let us mention the following facts: There exists a circularly chainable and hereditarily indecomposable planar continuum. For the definition of all these terms, the reader is referred to the introduction of [START_REF] Rogers | Intrinsic rotations of simply connected regions and their boundaries[END_REF]. It was proved to be unique up to homeomorphism (but it is not homogeneous). Such an object is called a pseudo circle.

8 This is probably inspired from Čech cohomology: we want the nerve of the open cover to be a circle. So for chains of length 3 one would also require e 0 ∩ e 1 ∩ e 2 = ∅. 9 it does not matter whether the ei are connected or not 10 By condition 1 it is already the case modulo m because Qn and Qn+1 are circular chains. 11 It translates the idea of Qn+1 having winding number 1 in Qn.

Then define K = n≥1 Q n : it is a pseudo circle.

END OF DESCRIPTION

Let us say that a Jordan curve, parameterized by ι(t), t ∈ R/Z, is stably crooked in a circular chain Q n = (e i ) with i ∈ Z/mZ, whenever:

the curve is contained in the union of the links of the chain, it has winding number 1 in Q n , oriented by ι, ( 12 ) -∀t 1 , t 2 ∈ R with t 1 < t 2 < t 1 + 1 and ∀u, v ∈ Z, if ι(t 1 ) ∈ e u and ι(t 2 ) ∈ e v and if u + 2 < v < u + m then there exists t 1 and t 2 with either t 1 < t 1 < t 2 < t 2 or t 1 > t 1 > t 2 > t 2 such that ι(t 1 ) ∈ e v-1 and ι(t 2 ) ∈ e u+1 .

It is independent of the choice of parameterization of the Jordan curve. The subtlety of taking closures for e u and e v but not e u+1 and e v-1 is to ensure the following: if a Jordan curve is stably crooked in Q n then any nearby Jordan curve is. By nearby we mean a Jordan curve which is parameterized by a map that is close to ι.

The core of the construction

Following Herman (with different conventions), for m ≥ 4 consider the circular chain C m of length m consisting of the rectangular links e i = i -1/4, i + 5/4 × 0, 1 in the cylinder C/mZ. Then -There exists a Jordan curve in C/mZ that is stably crooked in C m and invariant by T 1 . The proof is in [START_REF] Handel | A pathological area preserving C ∞ diffeomorphism of the plane[END_REF] but let us mention that it is quite entertaining to work it out on one's own. Figure 1 gives examples of solutions for m = 5, 6 and 7. -Fix such a Jordan curve J. Consider a conformal map φ from H/mZ ∪ {+i∞} to the component of C/mZ∪{+i∞}-J that contains the upper end, and that fixes the upper end. There exists such maps. 13 There exists ε such that the Jordan curve J ε , image by φ of the curve of equation "Im (z) = ε", is stably crooked in C m . It follows from the definition of stably crooked and 12 The winding number of an oriented closed curve ι in a circular chain Q = (dj) of length m ≥ 4 could be defined in terms of first Čech cohomology groups with coefficients in Z by a diagram looking like this: Z ≈ Ȟ1 (Q, Z) → Ȟ1 (  d, Z) → Ȟ1 (S 1 , Z) ≈ Z, but this is sophisticated. It amounts to doing the following: cut the curve into a finite number of small consecutive pieces eı = ι([tı, t ı+1 ]) for some sequence tı indexed by Z/pZ for some p > 0. If they are small enough, they will all be contained in some d. There is at most two possible values of . Choose anyone and call it g(ı). Since Qn is a chain, g(ı + 1) and g(ı) must differ of at most one in Z/mZ. There exists a lift f : Z → Z of g such that f (i + 1) and f (i) differ of at most one for all i. Then there exists k ∈ Z such that f (i + p) = f (i) + km for all i. This k is the winding number. It is independent of all choices and of the oriented parameterization ι, and invariant by small perturbations of ι. 13 Apply the Riemann mapping theorem to the image of the situation by E.

from the following theorem of Caratheodory: φ extends continuously to the closure of H/mZ and the extension is on R/mZ a parameterization of the Jordan curve J. -Let W be the preimage by the natural projection C → C/mZ of the image of φ. Then W is connected and φ lifts to a conformal bijection, also denoted by φ, from H to W . Since W is T 1 -invariant, φ commutes with T 1 , domain included. So φ induces an isomorphism from H/Z to W/Z that we will also denote by φ. -Now apply lemma 4 to φ and W/Z. We get a sequence of holomorphic maps ψ n : C/Z → C/Z such that: ψ n does not vanish, ψ n (z) -z has a limit when Im (z) tends to +∞, as n tends to +∞, ψ n tends uniformly to φ -1 on every compact subset of W and in a neighborhood of the upper end of C/Z. -By this uniform convergence, there is for n big enough a branch φ n of ψ -1 n defined for Im (z) > ε/2 and mapping the upper end to itself. There exists M ≥ 4 and N ∈ N such that for all m ≥ M and n ≥ N , the image of the chain C m by z → φ n m m z + iε is a circular chain that is crookedly embedded in C m . Proof. Denote (e ı ) this circular chain and C m = (d  ), with ı ∈ Z/m Z and  ∈ Z/mZ. First there is M, N such that for m ≥ M and n ≥ N , for all link e ı , the set of  such that "e ı ⊂ d  " is not empty, for otherwise, taking subsequences, there would be a point of J ε that is a limit of points not contained in any link, but links are open and J ε is contained in their union, which leads to a contradiction. Since C m is a chain, this set of  is at most two consecutive integers modulo m. Let g(ı) be any of them (the "smallest" for instance). Because (e ı ) and (d  ) are circular chains, we have necessarily g(ı + 1)-g(ı) ∈ {-1, 0, 1}. There is thus a unique lift of f from Z to Z that satisfies f (i + 1) -f (i) ∈ {-1, 0, 1} and f (0) ∈ {0, . . . , m -1}. This lift necessarily satisfies f (i+m ) = f (i)+km for some k independent of i. This k is necessarily equal to the winding number in C m of the curve φ n (x + iε) that runs along the lower edges of the "rectangles" defining (e ı ), which is equal to the winding number of J ε in C m , that is 1, so we get condition (3). If condition (4) were not satisfied for all m , n big enough, then taking subsequences would contradict the fact that J ε is stably crooked in C m .

Let us sum up what we will use of the above analysis. Setting ψ(z) = ψ N (z) -iε for the value of N mentioned above, we get: Lemma 5. For all m ≥ 4 there exists M ≥ 4 and a holomorphic map ψ such that ψ is entire, ψ does not vanish, ψ • T 1 = T 1 • ψ, ψ(z) -z has a limit when Im (z) → +∞, ψ has an inverse branch φ defined for Im (z) > -ε/2 for some ε > 0, with φ(z) -z having a limit as Im (z) → +∞, and φ • T 1 = T 1 • φ, for all m ≥ M , the image of the circular chain C m by z → φ m m z is a circular chain crookedly embedded in C m .

Putting it all together

We can now apply this to build our example. Fix h > 0. Let B n ∈ H 1 , q n ∈ N * be defined by induction as follows:

Let B 1 = id on C, and q 1 be any integer ≥ 5.

Assume that B 1 , q 1 , . . . B n , q n have been fixed such that the following holds for all k ≤ n:

1. As in section 3.2, let R k = q k B k , let V k be biggest domain containing an upper half plane and on which R k is a bijection to an upper half plane, let U 0 = C and let U n be the set of points z ∈ V 1 such that for all k < n,

R k • • • • • R 1 (z) ∈ V k+1 . We already saw that R k • • • • • R 1 are bijections from U k to upper half planes.
We require these half-planes to contain H -ε for some ε > 0 which may depend on k. 2. Let Q k be the circular chain defined as the preimage in C/Z by the restriction of

R k • • • • • R 1 to U k /Z of the chain C m ⊂ H/mZ with m = q 1 . . . q k . If k > 1, we require Q k to be crookedly embedded in Q k-1 . 3.
The supremum of the diameters of the links of Q k is ≤ 1/k. 4. Let F k be defined as in section 3.2. We require the domain of definition of

F k to contain H -h-1/k . 5. If k > 1, the supremum on H -h of F k -F k-1 is ≤ 1/2 k . 6. q k ≥ k.
This set of conditions is satisfied for n = 1: for (1) we have R 1 (z) = 5z,

C = V 1 = U 1 = R 1 (U 1 );
(2) is empty; for (3) the diameter of the links C m is √ 13/2 thus the diameter of the links of Q 1 is √ 13/2q 1 and here m = q 1 ≥ 5; (4) F 1 = T 1/q 1 and is defined on C; (5) is empty; (6) q 1 ≥ 5 ≥ 1.

If it is satisfied for some n, then let us explain why it is possible to choose (B n+1 , q n+1 ) such that it is satisfied for n + 1:

-First, we define B n+1 . Let m = q 1 • • • q n . Let ψ, φ, M be given by lemma 5.

Let B n+1 (z) = ψ(z) and for any q n+1 ∈ N * set m = q n+1 m. We obtain at once (1), and (2) follows as soon as q n+1 is big enough so that m ≥ M . -That (3) holds when q n+1 is big enough follows merely from the continuity of φ. -That (4) and (5) hold when q n+1 is big enough follows from section 3.3.

-Point (6) needs no comment.

Then by (5) the sequence F n tends uniformly on H -h to some holomorphic map F , so H -h ⊂ Dom in the notations of section 3.2. Let f the map associated to F in lemma 1. By (6) the number θ =

n 1/(q 1 • • • q n ) is irrational. By (2) and (3) the set K = n Q n is a pseudo circle. Let D n be the preimage of H by the restriction of R k • • • • • R 1 to U k and let D n = E(D n ) ∪ {0}.
By (1), f n is conjugated to a (rational14 ) rotation on D n (see the discussion in section 3.2). The sequence D n is decreasing for inclusion, and tends in the sense of Caratheodory to the connected component D of the complement of the pseudo circle E(K) that contains the origin. The boundary of D is equal to E(K) (it is a property that pseudo circles share with Jordan curves: they are equal to the boundary of both components of their complement). By lemma 2, D is contained in the Siegel disk of f . But since its boundary is not a Jordan curve (it is a pseudo circle) and the rotation number is irrational, D is the Siegel disk. It is contained in the unit disk D, because D 1 = H and D n is decreasing. So the Siegel disk of f is compactly contained in the domain of definition of f , because the latter contains e 2πh D.

Conclusion

With the above proof, we realize that the map f we constructed is injective, as a limit of injective holomorphic maps. The domain of definition can be taken to contain RD for R as big as wanted (while the pseudo circle is contained in D). By injectivity and properties of univalent maps, f will be very close to a rotation on, say, 2D when R gets big. The pseudo circle can be chosen to be very close to the unit circle. Alternatively, with a slight modification of the construction it can be chosen to span between distance ε and 1 -ε .

In [START_REF] Pérez | Siegel disks with smooth boundary[END_REF], it is mentioned that with Pérez Marco's examples, one can get a whole uncountable group of commuting maps sharing the same Siegel disk.

Here the same holds: by taking the u n small enough we can ensure that for any infinite subset J of N, the infinite composition of the G k over k ∈ J will converge on big domains, and they will all leave the Siegel disk of f invariant, so they will have the same Siegel disk since their rotation number remains irrational.

We believe that the new flexibility allowed by Runge's theorem allows to prove the following claims. But this has to be carefully checked.

Claim. Prove that there exists an injective holomorphic map f defined in a simply connected open subset U of C containing the origin, fixing 0 and having at 0 a hedgehog of positive Lebesgue measure compactly contained in U .

Claim. Prove that there exists an injective holomorphic map f defined in a doubly connected open subset U of C and a Jordan curve J with positive Lebesgue measure contained in U that is invariant by f , and carries an invariant line field.

We still believe in the following conjecture:

Conjecture. The boundary of the Siegel disks of all polynomials are Jordan curves.

It seems very likely that the construction requires gigantic values of q k so that we probably always get very Liouvillian rotation numbers (in particular non Brjuno). We can always arrange so that it is the case. So Herman's question in [START_REF] Michael | Construction of some curious diffeomorphisms of the Riemann sphere[END_REF] concerning which rotation numbers allow for non Jordan curve boundaries is still quite topical.

Addendum: Anosov and Katok

After submitting the present article, the author learned by discussion with colleagues that Herman's construction is an instance of a general construction by Anosov and Katok [START_REF] Anosov | New examples in smooth ergodic theory. Ergodic diffeomorphisms[END_REF], [START_REF] Anosov | New examples of ergodic diffeomorphisms of smooth manifolds[END_REF]. The following description of the Anosov and Katok construction is adapted from [START_REF] Fayad | Constructions in elliptic dynamics[END_REF] (with the area preserving assumption removed and h n replaced by its inverse): consider a (real) manifold M together with a smooth action of the group T = R/Z by diffeomorphisms : S α : M → M with S α+β = S α S β . Typically, imagine M = a sphere and S α are rotations of angle 2πα around a common axis. Then one tries to construct f : M → M obtained as a limit:

f = lim n→∞ f n , where f n = H -1 n • S α n+1 • H n with α n ∈ Q and H n = h n • • • • • h 1 ,
where every h n is a diffeomorphism of M that satisfies

h n • S αn = S αn • h n .
The sequence f n converges but usually not the sequence H n . The h n and α n are chosen inductively in the following order:

α 1 , h 1 , α 2 , h 2 , α 3 , . . . The choice of h n is very free (within the condition h n •S αn = S αn •h n ), it does not need to be close to the identity, and usually won't. To ensure convergence of the sequence f n , it is enough to choose each time α n+1 very close to α n , because then f n is close to f n-1 . This can be rewritten as follows:

f n = g n • • • • • g 1 with g n = H -1 n • S α n+1 -αn • H n .

Fig. 1 .

 1 Fig. 1. Examples of T1-invariant stably crooked Jordan curves in circular chains (unrolled) of length 5, 6 and 7. To make the figure more readable, we replaced the rectangles of the chain Cm by smoother regions.

a.k.a. "on every compact subset" of U

The filter understood by "Im (z) -→ +∞" is the one generated by all the half planes H h for h ∈ R. We also imply that the domain of F contains such a half-plane.

i.e. on a set containing a half plane H h for some h that may depend on n

this claim uses the fact that θ is irrational

It already follows from the convergence of fn, though.

Recall that these maps are not entire. Some iterate will be the identity on the component of the domain for the iterate that contains D1.

Some iterate of fn is thus the identity on the component of its domain that contains Un. Recall that fn is not entire: when fn is iterated, its domain decreases and may disconnect.

Arnaud Chéritat: Siegel disks with non-locally connected boundaries Note that g n is the time α n+1 -α n of the flow of a vector field (H n ) * X, the pull back by

For comparison, the construction in the present article also takes the form


Pierre Pontarotti
email: pierre.pontarotti@univ-amu.fr

Laurent Abi-Rached

Jung-Hua Yeh

Julien Paganini

Self-Peptidome Variation Shapes Individual Immune Responses

Keywords: HLA, TCR, Disease, Inherent Modulator Epitopes, Immune responses, Peptide

The relationship between human genetic variation and disease has not been fully elucidated. According to the present view on infectious diseases pathogen resistance is linked to Human Leukocyte Antigen (HLA) class I/II variants and their individual capacity to present pathogen-derived peptides. Yet, T cell education in the thymus occurs though negative and positive selection, and both processes are controlled by a combination of HLA class I/II variants and peptides from the self. Therefore, the capacity of given HLA class I/II variants to bind pathogen-derived peptides is only one part of the selective process to generate effective immune responses. We thus propose that peptidome variation contributes to shaping TCR repertoires and hence individual immune responses, and that this variation represents Inherent Modulator Epitopes.

Introduction: disease resistance/susceptibility is linked to Inherent Modulator

Epitopes (IME)

Despite the rapidly increasing availability of genetic data, the impact of genetic variation on disease resistance/susceptibility is not fully elucidated [START_REF] Antonelli | Host genetics: deciphering the variability in susceptibility to infections[END_REF]. In humans, resistance/susceptibility to pathogens was linked to Human Leukocyte Antigen (HLA) class I and II molecules because of their capacity to present pathogen-derived peptides to induce immune responses. Therefore, a major research focus was to study the genetic diversity of HLA class I/II molecules and the individual capacity of the variants to bind pathogen-derived peptides. This research supported the view that HLA class I and II genetic diversity impacts disease resistance/susceptibility and that this diversity evolved to meet the challenges of the pathogens encountered in different geographical areas [START_REF] Radwan | Advances in the Evolutionary Understanding of MHC Polymorphism[END_REF][START_REF] Spurgin | How pathogens drive genetic diversity: MHC, mechanisms and misunderstandings[END_REF].

We propose here that this conclusion is incomplete and that polymorphism in the peptides derived from the self also impacts disease resistance/susceptibility, playing a role as Inherent Modulator Epitopes (IME). T cell education in the thymus is done through positive and negative selection. First, a positive selection of T cells capable of antigen recognition and then negative selection (see Glossary) to eliminate T cells displaying high affinity for the combination of HLA class I/II plus self-peptide (self-antigen tolerance) [START_REF] Inglesfield | Rethinking Thymic Tolerance: Lessons from Mice[END_REF]. While the shaping of T Cell Receptor (TCR) repertoires is a complex process involving different mechanisms, some of them still not well described or understood, it is accepted that these two selection steps rely, in part, on peptide pools derived from the self (peptidome) [START_REF] Kosmrlj | Effects of thymic selection of the T-cell repertoire on HLA class I-associated control of HIV infection[END_REF]. Peptide polymorphisms can thus impact TCR repertoires and modulate the capacity of these repertoires to recognize pathogen-derived peptides. This would also explain differences in disease resistance/susceptibility for individuals that display the same sets of HLA class I/II variants.

The first part of this opinion article corresponds to a brief state-of-the-art regarding HLA class I/II and TCR structural biology, in particular in the context of thymic selection. We then discuss how thymic selection impacts HLA class I/II polymorphism selection. We then proceed to elaborate on how peptide pools vary between individuals and how such variation can impact immune responses. Finally, we will conclude with the proposal that some of these polymorphic peptides represent IME.

Diversity in the interaction between the HLA class I/II-peptide complex and the TCR

The central interaction in cell-mediated adaptive immunity occurs between the αβ TCR (αβTCR) and the HLA class I/II-peptide complex (Box 1).

HLA class I/II-peptide complex diversity HLA class I/II genes are highly polymorphic with over 15,000 allotypes encoded by the five most polymorphic genes (HLA-A/-B/-C/-DRB1/-DQB1) [START_REF] Robinson | IPD-IMGT/HLA Database[END_REF]. The polymorphism is concentrated around the peptide-binding cleft of the HLA class I/II protein and thus distinct HLA class I/II allotypes typically have distinct peptide-binding preferences that are determined by anchor residues that reside within the PBD [START_REF] Rossjohn | T cell antigen receptor recognition of antigen-presenting molecules[END_REF]. HLA class I/II variants can each present thousands of peptides [START_REF] Sarkizova | A large peptidome dataset improves HLA class I epitope prediction across most of the human population[END_REF], with an overlap between these peptides that is positively correlated with PDB similarity [START_REF] Gfeller | Predicting Antigen Presentation-What Could We Learn From a Million Peptides?[END_REF].

TCR diversity

In contrast to HLA class I/II diversity, TCR diversity is generated during T cell ontogeny in the thymus, via random rearrangements of variable (V), diversity (D) and joining (J) gene segments from TCR β loci, and random re-arrangement of V and J gene segments from the TCR α gene loci (Box 2). Yet, while this diversity is initially randomly generated, selection of αβTCR cells in the thymus involves immature thymocytes being subjected to both positive and negative selection [START_REF] Inglesfield | Rethinking Thymic Tolerance: Lessons from Mice[END_REF] (Figure 1). The first step is the positive selection resulting in the survival of thymocytes that undergo intrathymic migration. This step ensures that the thymus produces T cells capable of antigen recognition. This selection is done in the thymus cortex where the self-peptide involved in positive selection seems to be generated by the thymoproteasome, a specialized type of proteasome [START_REF] Takada | Generation of Peptides That Promote Positive Selection in the Thymus[END_REF][START_REF] Takahama | Thymoproteasome and peptidic self[END_REF].

The second step is the negative selection that ensures that T cell development produces functional thymocytes that are tolerant to self-antigens. This step occurs in the thymus medulla. The selection is carried out by eliminating the high binding T cells and keeping only T cells that can recognize the HLA class I/II-peptide complex with low-to-medium affinity. Because all genes are likely expressed in the thymus tissues, the whole peptidome thus plays a role at both the positive and negative selection steps.

The HLA class I/II-peptide complex impacts TCR repertoires

The role played by HLA class I/II molecules on the resistance/susceptibility to pathogens is twofold: first, through their capacity to bind pathogen-derived peptides and present them to T cells and second, through their effect on T cell education in the thymus with the presentation of self-peptides. While the former was extensively discussed (for example [START_REF] Radwan | Advances in the Evolutionary Understanding of MHC Polymorphism[END_REF][START_REF] Spurgin | How pathogens drive genetic diversity: MHC, mechanisms and misunderstandings[END_REF][START_REF] Pierini | Divergent Allele Advantage at Human MHC Genes: Signatures of Past and Ongoing Selection[END_REF], the latter is not typically considered when discussing disease resistance/susceptibility in light of HLA class I/II variation.

Yet, negative thymic selection prunes T cells that are strongly self-reactive and recognize with high affinity HLA allotypes presenting peptides from the self. While this process does not eliminate all self-specific T cells [START_REF] Yu | Clonal Deletion Prunes but Does Not Eliminate Self-Specific alphabeta CD8(+) T Lymphocytes[END_REF], it does impact TCR repertoires. Similarly, positive thymic selection selects TCR that can recognize self peptide-HLA class I/II complexes with enough affinity. It should be noted that there is uncertainty regarding how the strength of the affinity between HLA class I/II ligands and peptides, and between HLA class I/II-peptide complex and TCR could impact the selection.

Hence with both types of thymic selection, the nature of the peptides from the self (immunopeptidome) contributes to the shaping of TCR repertoires. In the next section, we show that some of these peptides can be polymorphic and such variability thus leads to distinct TCR repertoires and adaptive immune responses.

Impact of the polymorphism of the immunopeptidome on the immune function: the case of the Minor Histocompatibility Antigens

The importance of the polymorphism of the immunopeptidome is best illustrated by the Minor Histocompatibility Antigens (MHA) [START_REF] Mandl | T cell-positive selection uses self-ligand binding strength to optimize repertoire recognition of foreign antigens[END_REF][START_REF] Griffioen | Autosomal Minor Histocompatibility Antigens: How Genetic Variants Create Diversity in Immune Targets[END_REF] that were first discovered when HLA class I/II identical donors and recipients gave immunological responses during bone marrow transplants. HLA class I/II molecules are a major factor associated with graft rejection as distinct HLA class I/II allotypes present distinct sets of peptides to educate T cells in the thymus and such TCR repertoire differences lead to allograft rejections [START_REF] Klein | Natural History of the Major Histocompatibility Complex[END_REF]. Similarly, graft rejection can occur even in autologous situations (same HLA class I/II allotypes) because of the peptides that are polymorphic between these individuals and hence can lead to a differential pruning of TCRs that recognize with high affinity the combination of HLA class I/II allotypes and peptides from the self [START_REF] Wallny | Identification of classical minor histocompatibility antigen as cell-derived peptide[END_REF].

Several polymorphic peptides acting as MHA have long been described (see for example [START_REF] Wallny | Identification of classical minor histocompatibility antigen as cell-derived peptide[END_REF]) but more and more are being discovered thanks to extensive genome sequencing and the development of high throughput approaches to predict or elute HLA class I/II peptides [START_REF] Oostvogels | Identification of minor histocompatibility antigens based on the 1000 Genomes Project[END_REF][START_REF] Granados | Impact of genomic polymorphisms on the repertoire of human MHC class Iassociated peptides[END_REF]. In particular, discovery of new MHA is ensured by either "forward immunology" or "reverse immunology" approaches. The former includes the isolation of alloreactive clones from the patient and subsequent inference of immunogenic missense nucleotide polymorphisms [START_REF] Oostvogels | Identification of minor histocompatibility antigens based on the 1000 Genomes Project[END_REF][START_REF] Torikai | The human cathepsin H gene encodes two novel minor histocompatibility antigen epitopes restricted by HLA-A*3101 and -A*3303[END_REF]. The latter is based on mass spectrometry (MS) analysis of HLA-eluted peptides, or by in silico predictions. The peptides detected by MS are then linked to polymorphic genomic regions [START_REF] Granados | Impact of genomic polymorphisms on the repertoire of human MHC class Iassociated peptides[END_REF][START_REF] Bykova | In Silico Analysis of the Minor Histocompatibility Antigen Landscape Based on the 1000 Genomes Project[END_REF].

The question that arises here is whether immunopeptidome polymorphisms have a role that goes beyond graft transplants and affect more generally resistance/susceptibility to diseases.

Peptidome polymorphism and Inherent Modulator Epitopes

The previous section illustrates how peptidome polymorphisms impact T cell education in the thymus and can lead to histocompatibility. To understand this effect, it is important to re-emphasize how each HLA class I/II allotype presents its own set of peptides and that combinations of HLA class I/II plus self-peptide define T cell repertoires.

However, due to thymic negative selection, mature T cells can display a depleted repertoire that does not fully recognize certain HLA class I/II plus pathogen-derived peptide combinations, even if in theory the HLA class I/II molecules could present such peptides. The same reasoning can be applied to the polymorphic immunopeptidome and such polymorphic peptides that affect T cell repertoires can be referred to as Inherent Modulator Epitopes (IME). These IME can also drive thymus positive and negative selection and select TCR that will recognize pathogen-derived peptides presented by HLA class I/II [START_REF] Takada | Generation of Peptides That Promote Positive Selection in the Thymus[END_REF][START_REF] Takahama | Thymoproteasome and peptidic self[END_REF].

Polymorphic peptides could correspond to non-synonymous mutations on coding genes, but also to the presence/absence of genomic sequences that are potentially coding for proteins. Therefore, a wide variety of variation distributed along the genome can modify the immunopeptidome and hence impact TCR repertoires, and ultimately the ability of T cell to respond to diseases. Peptidome polymorphism can thus explain individual variation in disease resistance/susceptibility through their impact on TCR repertoires (Figure 2). This hypothesis could be tested by analysis of immunopeptidome variation from individuals with the same HLA types but with different disease resistance/susceptibility phenotypes.

Interestingly, while HLA class I/II variation is driven by natural selection, the same could be true for peptide polymorphisms that represent IME.

Genomic plasticity increases peptidome variability: inventory of variation and detail of the common mechanisms of endogenization.

We describe here how the peptidome contributes to shaping TCR repertoires and thus immune responses. any genomic variation modifying the peptidome can potentially impact immune responses. Such variation could occur at different levels, starting with direct and simple non-synonymous mutations in protein-coding regions of the genome but potentially involving more indirect mechanisms like mutations impacting immunoproteome processing [START_REF] Liepe | A large fraction of HLA class I ligands are proteasome-generated spliced peptides[END_REF][START_REF] Lopez De Castro | How ERAP1 and ERAP2 Shape the Peptidomes of Disease-Associated MHC-I Proteins[END_REF].

Other genomic variation, such as presence/absence of genes or indels in coding regions will also change the peptidome repertoire by adding or suppressing specific peptides.

Similarly, variations affecting transcription can play a role: alternative splicing events will produce new peptides for the same coding regions. The transcription step itself could be affected by genomic variation or be under the control of epigenetic factor that can modify peptide expression in the thymus [START_REF] Ucar | Promiscuous Gene Expression in the Thymus: A Matter of Epigenetics, miRNA, and More?[END_REF]. Differences in peptide expression levels could similarly modify T cell maturation, inducing a competition between peptides bound by HLA class I/II. In extreme cases, modulation of gene expression in the thymus could impact the peptidome just like gene absence/presence would. Non-coding regions and their variability could also affect the peptidome as some pseudogenes can be partially translated for example. Other genetic events, like genome rearrangement involving transposons could also lead to the expression of new peptides [START_REF] Ewing | High-throughput sequencing reveals extensive variation in human-specific L1 content in individual human genomes[END_REF][START_REF] Wang | Transposable element activity, genome regulation and human health[END_REF].

Finally, endogenization of exogenous DNA is also potentially a phenomenon that could impact immune responses. Indeed, endogenous viruses and transposon repertoires vary between individuals [START_REF] Mager | Homologous recombination between the LTRs of a human retrovirus-like element causes a 5-kb deletion in two siblings[END_REF][START_REF] Thomas | Variation in proviral content among human genomes mediated by LTR recombination[END_REF][START_REF] Wildschutte | Discovery of unfixed endogenous retrovirus insertions in diverse human populations[END_REF], and such variation could generate IME. This phenomenon has potential to enhance immune responses against exogenous retrovirus. This was for example reported in mouse with endogenous retroviruses shaping T cell repertoires and hence allowing specific responses against a closely related exogenous retrovirus [START_REF] Young | Negative selection by an endogenous retrovirus promotes a higher-avidity CD4+ T cell response to retroviral infection[END_REF] by cross reactivity. Two other adaptive systems are known to use endogenization to fight pathogens: the Clustered regularly interspaced short palindromic repeats (CRISPRs) in archaea and bacteria [START_REF] Wiedenheft | RNA-guided genetic silencing systems in bacteria and archaea[END_REF] and piRNA in eukaryotes [START_REF] Ophinni | piRNA-Guided CRISPR-like Immunity in Eukaryotes[END_REF]. In both cases foreign DNA prevents new infections [START_REF] Ophinni | piRNA-Guided CRISPR-like Immunity in Eukaryotes[END_REF][START_REF] Yu | The piRNA Response to Retroviral Invasion of the Koala Genome[END_REF]. Here, we propose that endogenization could be a widespread phenomenon that would lead to new IME and, through the resulting impact on TCR repertoires, allow immune systems to better fight pathogens.

Conclusion

The genetic basis for disease resistance/susceptibility is a complex mechanism but HLA class I/II variation is commonly involved. Consistent with this, natural selection commonly targets HLA class I/II molecules and they display a high level of polymorphism in human populations. This point is important and necessary to understand the human genetic-based resistance and its variation between individuals and populations, yet it does not explain all the differences in HLA-TCR biology. Indeed, even if HLA polymorphism could be selected to favor binding of pathogen peptides, this selection could not be efficient if T cells do not have adequate TCR to recognize these complexes between pathogen peptides and HLA class I/II. Hence, it is likely that selection also occurs on T cell maturation during the positive and negative selection in the thymus. The whole peptidome is involved in this maturation and thus plays a role in shaping TCR repertoires. Therefore, we suggest that selection could also target the peptidome and allow selection of TCR that can recognize HLA class I/II plus pathogenderived peptides.

Currently, disease association studies target HLA class I/II variants [START_REF] Dendrou | HLA variation and disease[END_REF] or use genomewide association studies to isolate genomic polymorphisms [START_REF] Betz | Genome-wide meta-analysis in alopecia areata resolves HLA associations and reveals two new susceptibility loci[END_REF][START_REF] Okada | Deep whole-genome sequencing reveals recent selection signatures linked to evolution and disease risk of Japanese[END_REF][START_REF] Tian | Genome-wide association and HLA region fine-mapping studies identify susceptibility loci for multiple common infections[END_REF][START_REF] Waage | Genome-wide association and HLA fine-mapping studies identify risk loci and genetic pathways underlying allergic rhinitis[END_REF]. Similarly to what was done to isolate new MHA [START_REF] Fuchs | Optimized Whole Genome Association Scanning for Discovery of HLA Class I-Restricted Minor Histocompatibility Antigens[END_REF], analyses focused on peptidome such as immunopeptidome-wide association studies [START_REF] Vizcaino | The Human Immunopeptidome Project: A Roadmap to Predict and Treat Immune Diseases[END_REF] could help explore the impact of changes in self immuno-peptidome. Such analyses take into account both HLA variation and peptidome variation, in order to identify changes that could impact selfimmunopeptidome due to an increased or a decreased number of self-peptides that can be presented by the HLA class I/II variants of one individual. This type of analysis could hence lead to the discovery of IME and provide ways to test interesting hypotheses, such as the possibility that gender specific differences in the immune response against COVID-19 [START_REF] Takahashi | Sex differences in immune responses that underlie COVID-19 disease outcomes[END_REF] could be explained by the Y chromosome specific immunopeptidome. A self-peptidome variation can impact the binding of a peptide to the HLA class I/II variants of a given individual. This change will affect the affinity of TCRs for the HLA-peptide complex (peptide variant #1 and #2) and hence directly impact TCR repertoires. Moreover, self-peptidome variation could also generate a peptide that cannot be presented so it will not be part of the immune-peptidome, and that will also impact TCR repertoires (peptide variant #3). protected by the peptide-binding groove of HLA class II molecules. This CLIP fragment will be replaced by higher affinity peptides with the help of the dedicated HLA class IIlike chaperone DM, which locally opens the groove to release low-affinity peptides such as CLIP then locks in place the high affinity peptide replacements. The HLA class II-peptide complex will then translocate to the plasma membrane, where it can interact with CD4 + T cells. These cells are "helper cells" that carry out multiple functions in cellular immunity, including cytokine secretion, activation of innate immune cells, and initiation of the adaptive immune responses.

Box 2: How TCR variability is built

The generation of the TCR occurs by random rearrangement of variable (V), diversity (D), and joining (J) gene segments. Moreover, junctional diversity increases the TCR repertoire by addition of non-template encoded nucleotides (N) at V (D) J Junctions.

V(D)J recombination is initiated early during lymphocyte development by a site-specific endonuclease: RAG1 and 2. The RAGs cleave at a conserved recombination signal (RSS) that flanks each V, D, and J segments [START_REF] Teng | Regulation and Evolution of the RAG Recombinase[END_REF]. The random pairing of different TCRα-chain and TCRβ-chains increases the repertoire further. In humans, according to the number 42 V segments and 61 J segments are found for the TCR α-chain locus, while 52 V, 2 D and 13 J genes in the case TCR β-chain locus [START_REF] Mak | Primer to the immune response[END_REF]. Random rearrangements of genes encoding TCRα and TCRβ and random pairing can theoretically generate up to 1 × 10 20 different αβ TCRs [START_REF] Lieber | Site-specific recombination in the immune system[END_REF], although we note that the expressed diversity in any given individual is closer to 5 × 10 6 αβ TCRs [START_REF] Arstila | A direct estimate of the human alphabeta T cell receptor diversity[END_REF][START_REF] Bradley | Using T Cell Receptor Repertoires to Understand the Principles of Adaptive Immune Recognition[END_REF]. Combinations of HLA class I/II plus peptide interact with TCR hypervariable complementarity-determining regions (CDRs). The germ line V genes encode the CDR1 and CDR2 regions and the V(D)J junction encodes the CDR3 region. CDR1 and CDR2 interact with the HLA class I/II molecules, while CDR3 interacts with the peptide presented by HLA class I/II molecules [START_REF] Bradley | Using T Cell Receptor Repertoires to Understand the Principles of Adaptive Immune Recognition[END_REF].

Glossary

Allotype: A distinct variant at the protein level. Several allelic variants can code for the same allotype.

Immunopeptidome:

The complete set of peptides encoded by a particular genome, or present within a particular cell type or organism that can be presented by HLA class I/II molecules.

Inherent Modulator Epitopes (IME): epitopes of the self-proteome whose polymorphism can affect the negative or positive selection of T Cells and can thus contribute to the shaping of immune system responses (resistance, immunotolerance).

HLA:

The Human Leukocyte Antigen (HLA) region is a ~4Mb genomic region on chromosome 6 (6p21.3) that contains a set of highly polymorphic genes named HLA class I and class II. These genes encode cell surface proteins essential for the adaptive immune system because of their capacity to present peptide antigens to immune receptors and hence discriminate 'self' versus 'non-self'.

Minor Histocompatibility Antigens (MHA): peptides of the self that differ between patients and donors, that are presented by HLA molecules at the cell surface, and that give rise to graft rejection in the context of bonne marrow transplantation (stem cells).

Negative selection: (i) evolution: natural selection removing deleterious alleles; (ii) immunology: deletion of T cells whose TCR recognizes with a high affinity HLA class I/IIself-peptide complex complexes of peptides-HLA class I/II molecules.

Polymorphism: A distinct variant of the same element. In the context of peptides, polymorphism means two peptides originating from the exact same part of a protein but with a distinct sequence.

Peptidome:

The complete set of peptides encoded by a particular genome, or present within a particular cell type or organism. Self-peptide variants that affect T cell repertoires represent Inherent Modulator Epitopes.

Figure 1 :

 1 Figure 1: T cell selection depends on the affinity of its TCR to self peptide-HLA class I/II complexes. If TCR affinity is low (orange), T cells are eliminated by positive selection. If TCR affinity is too high, T cells are eliminated by negative selection (blue).

Figure 2 :

 2 Figure 2: Impact of self peptidome variation on the generation of T cell repertoires for the same HLA class I/II variants. A self-peptidome variation can impact the binding

Box 1 :

 1 HLA class I/II proteins present peptide antigens to immune receptors HLA class I molecules present peptides derived from the proteins that are synthesized within the cell. The proteins are degraded by the proteasome into peptides in the cytosol and short peptides are translocated to the Endoplasmic Reticulum (ER) by the transporter associated with antigen presentation (TAP) peptide transporter. Once in the ER, the translocated peptides are loaded into the HLA groove (peptide-binding cleft) with the help of chaperone proteins. These loaded peptides are typically 8-10 amino acid long and have specific anchor residues; longer peptides are trimmed by the aminopeptidase, ERAP1. Exocytosis of newly synthesized HLA class I molecules loaded with peptides on the cell surface initiates the interaction with the TCR of CD8 + T cells. CD8 + T cells are involved in cell-mediated immune responses that can directly induce cytotoxicity in virus-infected cells as well as in tumor cells. HLA class II molecules present peptides that are longer than those presented by HLA class I molecules and that are derived from extracellular proteins degraded in the endosomal pathway. HLA class II molecules are assembled in the ER with the invariant chain Ii. Ii binds the HLA class II peptide-binding groove and directs the transport of HLA class II molecules into the endosomal pathway, where they will encounter exogenous peptides generated by proteases. Once in the endosomal pathway, Ii is degraded by proteases, leaving an invariant chain fragment (CLIP) inaccessible for proteases as it is

  Positive selection: (i) evolution: natural selection increasing frequencies of advantageous mutations; (ii) immunology: retention of T cells whose TCR recognizes complexes of self peptides-HLA class I/II. T cell receptor (TCR): receptor of T cells that recognizes complexes of peptide-HLA class I/II. It is coded by V, D, and J genes: through somatic recombination, these genes produce a vast number of distinct variants in each individual. Highlights T Cell Receptor (TCR) repertoires emerge in the thymus in each individual as T cells undergo positive and negative selection. T cell education is controlled by the combination of Human Leukocyte Antigen (HLA) class I/II molecules and their peptide pools (peptidome). HLA class I/II molecules are highly plastic in human populations but the peptidome is also a source of variation. Hence combined diversity of HLA class I/II molecules and of self-peptides shapes individual immune responses.

Acknowledgements

We thank Jim Kaufman and Pierre Boudinot for helpful discussion This work was supported by the French Government under the «Investissements d'avenir» (Investments for the Future) program managed by the Agence Nationale de la Recherche (ANR, fr:National Agency for Research), (reference: Méditerranée Infection 10-IAHU-03).

Outstanding Questions

 What is the respective contribution of IME vs HLA class I/II for TCR repertoires?  Could some MHA also play a role as IME?

 Could some IME be involved in the development of autoimmune diseases by selecting TCR that can recognize with sufficient affinity self-peptides?

 What is the part of the genome that could be under selection to preserve IME?  How to investigate the impact of IME on immune responses?


Sandra Weller

Claude-Agnès Reynaud
email: reynaud@necker.fr

Jean-Claude Weill
email: weill@necker.fr

Vaccination against encapsulated bacteria in humans: Paradoxes

Keywords: encapsulated bacteria, polysaccharides, conjugated vaccine, T-independent response

Infection with encapsulated bacteria can be prevented by vaccination with capsular polysaccharides, either plain or conjugated to a protein carrier. But results concerning these vaccinations raise several paradoxes. Polysaccharides from encapsulated bacteria are generally considered to be Tindependent antigens unable to trigger a T-dependent germinal center reaction, but strikingly, antipolysaccharide antibodies are often mutated in humans. Polysaccharide-protein conjugate vaccines are able to induce a true T-dependent memory response with a rise in antibody titers and a switch to high affinity-IgG antibodies in children below 2 years of age, but neither the plain nor the conjugate vaccine can induce memory in older infants and adults. We propose some explanations to these paradoxes based on our recent observation that humans display a circulating splenic marginal zone B cell population with a pre-diversified immunoglobulin repertoire in charge of the immune response to T-independent vaccines.

.

Nevertheless, when looking at the abundant literature on immune responses to encapsulated bacteria after vaccination by either plain or conjugated capsular polysaccharide vaccines, some paradoxes emerge which we would like to discuss.

Paradox 1

It is generally accepted that bacterial capsular polysaccharides (CPS) are TI antigens that trigger specific B cells residing in the splenic marginal zone to secrete antibodies with opsonophagocytic properties against these bacteria [START_REF] Harms | In vitro complement-dependent binding and in vivo kinetics of pneumococcal polysaccharide TI-2 antigens in the rat spleen marginal zone and follicule[END_REF]. In rodents in which these responses have been mostly studied, immunization do not induce an extensive proliferation within B cell follicles, neither the formation of germinal centers, and are for most of them taken care by unmutated germline antibodies [START_REF] Dammers | Most marginal zone B cells in rat express germline encoded Ig VH genes and are ligand selected[END_REF][START_REF] Maizels | The T-cell-independent immune response to the hapten NP uses a large repertoire of heavy chain genes[END_REF][START_REF] De Vinuesa | T-independent type 2 antigens induce B cell proliferation in multiple splenic sites, but exponential growth is confined to extrafollicular foci[END_REF]. In humans on the contrary, the splenic marginal zone contains B cells with mutated Ig receptors [START_REF] Dunn-Walters | Analysis of mutations in immunoglobulin heavy chain variable region genes of microdissected marginal zone (MGZ) B cells suggests that the MGZ of human spleen is a reservoir of memory B cells[END_REF][START_REF] Tangye | Identification of functional human splenic memory B cells by expression of CD148 and CD27[END_REF][START_REF] Tierens | Marginal-zone B cells in the human lymph node and spleen show somatic hypermutations and display clonal expansion[END_REF] and mutated antibodies are raised in responses to encapsulated bacteria, these mutations being responsible for the antibodies avidity for the immunizing antigen [START_REF] Lucas | Polysaccharide vaccines as probes of antibody repertoires in man[END_REF][START_REF] Zhou | Somatic hypermutation and diverse immunoglobulin gene usage in the human antibody response to the capsular polysaccharide of Streptococcus pneumoniae type 6B[END_REF]. In all species, these TI responses do not trigger any memory, the B cells engaged being able very rapidly to switch isotype and to secrete large amount of antibodies. These plasma cells can remain in the organism for various lengths of time, after which the response wanes out [START_REF] Zandvoort | The dual function of the splenic marginal zone: essential for initiation of anti-TI-2 responses but also vital in the general first-line defense against blood-borne antigens[END_REF][START_REF] Lopez-Carvalho | Development and selection of marginal zone B cells[END_REF]. To account for the presence in humans of mutated antibodies during a T-independent immune response that cannot normally trigger a cognate T-B dependent germinal center reaction, the classical explanation put forward by authors is that such responses are in fact taken care by bona fide memory B cells that have been primed by a previous encounter with the pathogen, either during an infection or by silent carriage [START_REF] Lucas | Combinatorial library cloning of human antibodies to Streptococcus pneumoniae capsular polysaccharides: variable region primary structures and evidence for somatic mutation of Fab fragments specific for capsular serotypes 6B, 14, and 23[END_REF][START_REF] Barington | The progeny of a single virgin B cell predominates the human recall B cell response to the capsular polysaccharide of Haemophilus influenzae type b[END_REF][START_REF] Baxendale | Immunogenetic analysis of the immune response to pneumococcal polysaccharide[END_REF]. These memory B cells would then eventually reside in the splenic marginal zone where they could acquire marginal zone B cell surface markers [START_REF] Liu | Memory B cells in T-cell-dependent antibody responses colonize the splenic marginal zones[END_REF][START_REF] Vinuesa | Recirculating and germinal center B cells differentiate into cells responsive to polysaccharide antigens[END_REF][START_REF] Shih | Role of BCR affinity in T cell-dependent antibody responses in vivo[END_REF]. This explanation requires that during this natural priming which must have occurred in each case analyzed so far, the polysacharidic capsules from the pathogen are somehow linked to a protein moiety in order to drive the response into the classical germinal center-dependent memory B cell pathway.

Paradox 2

Children below 2 years do not respond to plain CPS vaccines but they do respond when the CPS are conjugated to a protein carrier (Table 1 and Box 1). Moreover these conjugate vaccines will induce a memory response with a rise of antibody titer after several rounds of immunization and a booster response several months later [START_REF] Borrow | Influence of prior meningococcal C polysaccharide vaccination on the response and generation of memory after meningococcal C conjugate vaccination in young children[END_REF][START_REF] Greenberg | Enhanced antibody responses in infants given different sequences of heterogeneous Haemophilus influenzae type b conjugate vaccines[END_REF][START_REF] Richmond | Ability of 3 different meningococcal C conjugate vaccines to induce immunologic memory after a single dose in UK toddlers[END_REF][START_REF] Rennels | Safety and immunogenicity of heptavalent pneumococcal vaccine conjugated to CRM 197 in United States infants[END_REF]. Surprisingly, in older infants and in adults, the same conjugate vaccine does not induce a memory response but a response similar to the one obtained with the plain CPS preparation [START_REF] Goldblatt | Natural and vaccine-induced immunity and immunologic memory to Neisseria menigitidis serogroup C in young adults[END_REF][START_REF] Cryz | Immunization with a Pseudomonas aeruginosa immunotype 5 O polysaccharide-toxin A conjugate vaccine: effect of a booster dose on antibody level in humans[END_REF][START_REF] Kamboj | Significant variation in serotype-specific immunogenicity of the seven-valent Streptococcus pneumoniae Capsular polysaccharide-CRM 197 conjugate vaccine occurs despite vigorous T cell help induced by the carrier protein[END_REF][START_REF] Shelly | Comparison of pneumococcal polysaccharide and CRM 197 conjugated pneumococcal oligosaccharide vaccines in young and elderly adults[END_REF], even though the response to the conjugate appears in some cases more robust [START_REF] Baker | Safety and immunogenicity of capsular polysaccharide-tetanus toxoid conjugate vaccines for group B streptococcal types Ia and Ib[END_REF][START_REF] Guttormsen | Type III group B streptococcal polysaccharide induces antibodies that cross react with streptococcus pneumoniae type 14[END_REF].

A proposed explanation

We have shown that humans display a circulating marginal zone B cell population which is involved in TI response and which diversifies its Ig receptor in the absence of cognate T-B interactions. These conclusions are based on the following experimental facts:

1)

Hyper-IgM patients with either CD40L or CD40 deficiency do not have germinal centers and possess a population of circulating IgM + IgD + CD27 + B cells with a mutated Ig receptor, in the complete absence of switched memory B cells. In both types of patients the frequency of Ig mutations observed in these cells is similar to the one observed in controls, although the number of IgM + IgD + CD27 + blood cells is in general lower in the patients as compared to normal children. The absence of CD40 or CD40L may alter B cell development by hampering the network of cytokines interactions involving dendritic cells, activated T cells and natural killer cells. Nevertheless, the fact that these patients never display any CD27 + switched B cell indicate that the IgM + IgD + CD27 + B cells present are not produced in some cryptic germinal center structures, but rather belong to a separate population.

2) Phenotypic analysis and gene expression profiling in normal individuals indicate that human splenic marginal zone B cells are, as opposed to rodents [START_REF] Maclennan | The lymphocytes of splenic marginal zones: a distinct Bcell lineage[END_REF], recirculating in the blood.

3)

After vaccination with a non-conjugated pneumococcal vaccine, blood and splenic IgM + IgD + CD27 + B cells are specifically mobilized in this T-independent response. This response is taken care of by B cells carrying mutated antibodies prior to immunization. Upon expansion of these cells in the splenic marginal zone during the week following vaccination, the hypermutation process is reactivated.

4)

The blood IgM + IgD + CD27 + B cell population starts to expand after birth. It is well expanded and mutated in children around one year of age, an age at which toddlers do not respond to Tindependent antigens, thus strongly suggesting that diversification results at this stage from a developmental program and not from an immune response.

Our conclusion from these preliminary investigations is, as already proposed in our first report [START_REF] Weller | CD40-CD40L independent Ig gene hypermutation suggests a second B cell diversification pathway in humans[END_REF],

that this so-called memory IgM + IgD + CD27 + B cell population, which represents 10-30% of B cells in blood and spleen, belonged to a different pathway of differentiation and was linked to the response against TI-antigens [START_REF] Weller | Human blood IgM "memory" B cells are circulating splenic marginal zone B cells harboring a pre-diversified immunoglobulin repertoire[END_REF]. A similar conclusion was reached by R. Carsetti and her colleagues in a recent report, based on their study on the responding capacity asplenic or splenectomized patients to T-independent pneumococcal antigens [34].

The model we propose offers obviously another solution to paradox 1 without having to involve an obligatory primary infection or carriage. The B cells involved in the response are already prediversified and can undergo another round of hypermutation after stimulation by the immunizing antigen. These B cells, which are already in a pre-activated state, do not possess any properties attributed to memory B cells. However, being constantly triggered by TI antigens, they could account for most of the so-called « natural » IgM antibodies present in the circulation [START_REF] Bernasconi | Maintenance of serological memory by polyclonal activation of human memory B cells[END_REF].

In order to explain paradox 2 we would like to propose that before two years of age, in the absence of a functional splenic marginal zone, CPS vaccines conjugated to a protein carrier will trigger naive follicular B cells and drive them into a germinal center-dependent reaction including affinity maturation and memory B cell generation. After two years, the now functional splenic marginal zone will sequester polysaccharidic preparations from encapsulated bacteria, whether they are conjugated to a protein carrier or not, thus explaining why a similar marginal zone type of response without any memory characteristics is obtained with both the plain and the conjugated CPS vaccines. Strikingly, bone marrow transplanted patients, which cannot respond to pneumoccocal CPS until approximately 2 years after the transplant, respond to the conjugate vaccine like normal toddlers below 2 years by generating a clear memory response [START_REF] Guinan | Polysaccharide conjugate vaccine responses in bone marrow transplant patients[END_REF][START_REF] Molrine | Donor immunization with pneumococcal conjugate vaccine and early protective antibody responses following allogeneic hematopoietic cell transplantation[END_REF]. Thus in both situations in which the splenic marginal zone is not functional, do we see an uncoupling between the response to the plain and the conjugated CPS preparation.

There may be direct clinical consequences to our propositions that can be tested. Effectively, patients undergoing a splenectomy are usually vaccinated by the plain or the conjugated vaccines prior to the operation. According to our scheme these patients should mount, like toddlers, a memory response if vaccinated by the conjugated vaccine after splenectomy. The same result should be obtained with asplenic patients.

Moreover, having observed that the B cells engaged in these responses are already well developed and mutated in blood at an age at which children cannot respond to the plain CPS vaccine, and taking into account that it will be challenging to conjugate the serotypes of all encapsulated bacteria (e.g. there are 90 serotypes for Streptoccocus pneumoniae and only 11 have been conjugated at this moment), it is tempting to speculate that plain vaccines in conjunction with the appropriate adjuvant specific for these cells may be able to trigger a protective response in toddlers below 2 years. It is clear that such a result if possible would be a life-saving progress in underdeveloped countries.

There remain many unsolved questions concerning these cells and the response against encapsulated bacteria. The splenic marginal zone being obviously a complex lymphoid niche, is there a contribution in humans, even minor, of bona fide memory B cells to TI responses? What are the cells that sequester CPS antigens after 2 years, whether there are in their natural or vaccine form? Does this occur in the splenic marginal zone or also in blood through specific DC which can bring them to the spleen, and what are the specific receptors involved? [START_REF] Gijtenbeek | Marginal zone macrophage express a murine homologue of DC-SIGN that captures blodd-borne antigens in vivo[END_REF][START_REF] Balazs | Blood dendritic cells interact with splenic marginal zone B cells to initiate T-independent immune responses[END_REF] How and where do marginal zone B cells diversify their Ig receptor throughout life? During the first two years of life in infants one can study blood IgM + IgD + CD27 + B cells as they develop and diversify their Ig receptors, prior to any immune response. Thereafter encounter with external TI antigens in the mature environment of the splenic marginal zone will trigger their effector function, i.e. the production of antibodies What pre-activates these cells and drive their final differentiation remains an open question. This pre-diversification step could take place in the splenic marginal zone and in marginal zone-like regions in lymph nodes and Peyer's patches [START_REF] Spencer | Human marginal-zone B cells[END_REF]. It could alternatively occur in isolated B lymphoid follicles along the gut, as has been described for other species [START_REF] Hamada | Identification of multiple isolated lymphoid follicles on the antimesenteric wall of the mouse small intestine[END_REF] (see below). A similar pre-activated stage has been described for NK T cells [START_REF] Brigl | Mechanism of CD1d-restricted natural killer T cell activation during microbial infection[END_REF], and, similarly to them, pre-activation of marginal zone B cells could be driven by self or bacterial commensal antigens. For both these cellular populations, NKT and marginal zone B cells which are intermediate between the innate and adaptive immune system, this pre-activated stage may explain why they react very rapidly upon challenge by external bacterial antigens [START_REF] Lopez-Carvalho | Development and selection of marginal zone B cells[END_REF].

Looking at other B cell immune systems

GALT (gut-associated lymphoid tissues) species, as opposed to bone marrow species, do not use ongoing rearrangement in the bone marrow to generate B cell diversity. They include a large number of mammalian species studied today (sheep, rabbits, cattle, pigs, etc…) and have been named so because they generate their B cell pre-immune repertoire by post-rearrangement diversification processes such as hypermutation and/or gene conversion in gut-associated lymphoid tissues [START_REF] Weill | GALT versus bone marrow models of B cell ontogeny[END_REF][START_REF] Reynaud | Postrearrangement diversification processes in Gutassociated lymphoid tissues[END_REF]. This pre-diversification step is antigen-independent although gut bacterial antigens provide the mitotic stimuli permitting the intense proliferation of these cells in GALT follicles [START_REF] Reynaud | Hypermutation generating the sheep immunoglobulin repertoire is an antigen-independent process[END_REF][START_REF] Lanning | Immunoglobulin genes and generation of antibody repertoires in higher vertebrates: a key role for GALT, in Molecular Biology of B cells[END_REF]. Moreover, in all these species there is only one window of B cell production and diversification that occurs during several months after birth, the peripheral B cell compartment being then installed for the whole life of the animal. The obvious advantage of this mode of generation of the B cell repertoire by gene conversion and hypermutation, which target all three CDRs, is the variety of specificities and affinities produced, allowing these species to also respond to T-independent antigens with a large range of binding capacities.

Despite obvious differences between GALT B cells and human marginal zone B cells, our data imply that humans have most probably conserved for one arm of their B cell system the strategies used by GALT species to diversify their pre-immune repertoire. This strategy which differs from the one used by follicular B cells, allows the generation of a robust immune response against life threatening infections by encapsulated bacteria.

Figure 1 .

 1 Figure 1. A proposed scheme for the response to PS vaccines, plain or coupled to a protein

Acknowledgements

We thank Capucine Picard for critical reading of the manuscript.