Justin Dzuche

Christian Deffo Tassak
email: dtassak@yahoo.com

Jules Sadefo Kamdem

Louis Aimé Fono

On Two Dominances of Fuzzy Variables based on a Parametric Fuzzy Measure and Application to Portfolio Selection with Fuzzy Return

Keywords: Fuzzy variable, Parametric fuzzy measure, Generalized First order dominance, Optiminism Dominance, Set of best portfolios

 and the second one, based on optimism's level and called optimisnism dominance, is stronger than the first one. We study properties of these dominances on trapezoidal fuzzy numbers and we characterize them. We implement the optiminism dominance in a numerical example to display that its set of efficient portfolios enlarges the set of efficient portfolios obtained by Tassak et al. [17] through their first order dominance.

Introduction

Portfolio selection theory consists on the choice of "best portfolios" among several other ones in the context that assets' future returns are uncertain ( [START_REF] Markowitz | Portofolio selection[END_REF], [START_REF] Carlsson | A possibilistic approach to selecting portfolios with highest utility score[END_REF], [START_REF] Sharpe | A linear programming approximation for the general portfolio analysis problem[END_REF], [START_REF] Ogryczak | From stochastic dominance to mean-risk models: Semideviations as risk measures[END_REF], [START_REF] Stone | A linear programming formulation of the general portfolio selection problem[END_REF], [START_REF] Huang | Mean-semivariance models for fuzzy portfolio selection[END_REF], [START_REF] Li | Mean-variance-skewness model for portfolio selection with fuzzy returns[END_REF], [START_REF] Sadefo | Moments and semi-moments for fuzzy portfolio selection[END_REF]). Therefore, the literature presents a huge variety of methods for the determination of efficient portfolios. More precisely, one can subdivide those methods following two approaches: the first one deals with investors' preferences through proposed target values for benefit or investment risk and the second one doesn't take into consideration any target value and thus, proposes various efficient portfolios on which each investor can make a choice based on his preference. However, two mathematical tools are used as supports to analyze those approaches. Moments and semi-moments of a (random or fuzzy) variable, defined by means of crisp or fuzzy measure, are used in optimization models for the first approach ( [START_REF] Markowitz | Portofolio selection[END_REF], [START_REF] Ogryczak | From stochastic dominance to mean-risk models: Semideviations as risk measures[END_REF], [START_REF] Huang | Mean-semivariance models for fuzzy portfolio selection[END_REF], [START_REF] Li | Mean-variance-skewness model for portfolio selection with fuzzy returns[END_REF], [START_REF] Sadefo | Moments and semi-moments for fuzzy portfolio selection[END_REF], [START_REF] Dzuche | On the first moments and semi-moments of a fuzzy variable based on a new measure and application for portfolio selection with fuzzy returns[END_REF]) whereas the core of portfolios or the set of best portfolios based on the dominance relation on (random or fuzzy) variables defined by means of the fuzzy measure are main tools of the second approach ( [START_REF] Peng | Fuzzy dominance: a new approach for ranking fuzzy variables via credibility measure[END_REF], [START_REF] Tassak | Characterization of order dominances on fuzzy variables for portfolio selection with fuzzy returns[END_REF]). In this paper, we focus on the second approach, namely Game theoretical approach, by introducing, characterizing and studying properties of the two new dominance relations comparing variables. However, the literature also presents more than one way to describe uncertainty in various phenomena and some usual concepts related to human being perceptions. As a consequence, in portfolio selection, future asset's return can be described by a random or fuzzy variable according to the investment context. It means that, in some context, only randomness can influence future assets' returns and leads to random variable whereas in other context, uncertainty is due to imprecise or vague concepts used to describe future assets' returns (around 3UM, between 4UM and 5UM, very expensive...) and leads to a fuzzy variable. This work deals with fuzzy variable which describes uncertainty using the generalized fuzzy measure. Notice that this recent measure, defined by Iwamura [START_REF] Yang | Fuzzy chance-constrained programming with linear combination of possibility measure and necessity measure[END_REF] as a convex combination of possibility and necessity measures defined by Zadeh [START_REF] Zadeh | Fuzzy Set as a basis of theory of possibility[END_REF], takes into consideration an investor optimism's level towards risk ( [START_REF] Yang | Fuzzy chance-constrained programming with linear combination of possibility measure and necessity measure[END_REF], [START_REF] Dzuche | On the first moments and semi-moments of a fuzzy variable based on a new measure and application for portfolio selection with fuzzy returns[END_REF]). This paper is a modest contribution for the study of dominance relations on fuzzy variables using the new generalized measure and the application of obtained results to determine best portfolios made up of assets with fuzzy return. More precisely, we define a distribution function of a fuzzy variable by means of that measure and we study its properties. This distribution is used to define two new dominance relations that allow to compare fuzzy variables according to two main assumptions: investors have the same optimism's level towards investment risk or not. Then, these dominance relations are characterized on trapezoidal fuzzy variables and other properties are established. However, since it seems more natural to compare two entities before selecting them, binary comparisons of fuzzy variables (through distribution functions) can lead to portfolio selection. We prove in particular that the second dominance relation is a pre-order and it is complete in a particular subset of trapezoidal fuzzy variables. This "local completeness" is a specificity of the second order dominance in that sense it can allow in some situations to classify fuzzy variables and in this particular context, it allows to an investor to classify different assets of his portfolio, which naturally makes easier the selection. Furthermore, we describe the sets of non dominated portfolios according to the two dominances and we establish a link between the two sets. We use the seven assets proposed by Huang [START_REF] Huang | Mean-semivariance models for fuzzy portfolio selection[END_REF] to implement the second dominance in order to obtain non dominated portfolios. Those obtained portfolios complement the set of best portfolios obtained by first authors ( [START_REF] Huang | Mean-semivariance models for fuzzy portfolio selection[END_REF], [START_REF] Li | Mean-variance-skewness model for portfolio selection with fuzzy returns[END_REF], [START_REF] Sadefo | Moments and semi-moments for fuzzy portfolio selection[END_REF], [START_REF] Tassak | Characterization of order dominances on fuzzy variables for portfolio selection with fuzzy returns[END_REF]). The paper is organized as follows. Section 2 gives some preliminaries on fuzzy subsets and on fuzzy measures. It presents the new distribution functions of a fuzzy variable and links with known distribution functions. Section 3 presents the new dominance relations, their characterizations and their properties. In Section 4, the new dominance relation is implemented on a set of seven fuzzy variables and some non dominated portfolios are proposed. Section 5 gives concluding remarks and the Appendix made up of some proofs.

Preliminaries

Throughout this paper, X is a nonempty set namely the universal set.

Fuzzy subsets and fuzzy measures

A fuzzy subset A of X is defined by its membership function: µ A : X → [0, 1] such that, to each x ∈ X, is associated µ A (x). For x ∈ X, µ A (x) represents the membership grade of x to A. If ∀x ∈ A, µ A (x) ∈ {0, 1}, then A becomes a crisp subset of X. The support of A is the crisp subset denoted by Supp(A) and defined by:

Supp(A) = {x ∈ X, µ A (x) > 0}.
Let us recall the parametric measure M λ defined by Yang and Iwamura [START_REF] Yang | Fuzzy chance-constrained programming with linear combination of possibility measure and necessity measure[END_REF] that we will use in this paper. For A ⊂ X and ξ : X → R a fuzzy variable with membership function µ, that means, each element x of X is associated to a real number ξ(x) by means of ξ with the membership grade µ(ξ(x)), M λ (A) = λP os(A) + (1 -λ) Nec(A) with λ ∈ [0, 1]. We recall that if λ = 1 (respectively λ = 0), then M λ becomes possibility measure defined as P os(A) = sup x∈A µ(ξ(x)) (respectively necessity measure defined as N ec(A) = 1-P os(A c ) = inf x∈A µ(ξ(x))) by Zadeh [START_REF] Zadeh | Fuzzy Set as a basis of theory of possibility[END_REF]. If λ = 1 2 , then M λ becomes credibility measure defined as Cred(A) = 1 2 [P os(A) + N ec(A)] by Liu [START_REF] Liu | Expected value of fuzzy variable and fuzzy expected value models[END_REF].

Distribution function of a fuzzy variable with respect to M λ -measure

Denote that λ is the optimistic level of an investor or a decision maker. Thus, M λ 1 and M λ 2 can be considered as two different ways to measure fuzzy event A, depending of two investors with optimistic levels λ 1 and λ 2 , by the combination of possibility and necessity measures.

In the next Subsection, we define and study the distribution function of a fuzzy variable with respect to M λ -measure.

Distribution function of a fuzzy variable with respect to

M λ -measure Definition 1. Let ξ be a fuzzy variable and λ ∈ [0, 1]. The distribution function φ λ of ξ with respect to M λ -measure is given by: ∀x ∈ R,

φ λ,ξ (x) = M λ {ξ ≤ x}.
If there is no confusion, φ λ,ξ will be simply denoted φ λ . In addition, if λ = 1 2 , that is, M λ is the credibility measure and φ λ,ξ is simply denoted φ ξ or simply φ defined by Liu [START_REF] Liu | Uncertainty Theory: An Introduction to its Axiomatics Foundations[END_REF].

Let us define the distribution function of a trapezoidal fuzzy variable.

Example 1. Let ξ = (a, b, c, d) be a trapezoidal fuzzy variable, λ ∈ [0, 1] and x ∈ R. We have:

φ λ (x) =              1 if d ≤ x λ(d-x)+x-c d-c if c ≤ x ≤ d λ if b ≤ x ≤ c λ(x-a) b-a if a ≤ x ≤ b 0 if x ≤ a . ( 1
)
Let us end this Subsection by establishing the two next results on properties of φ λ .

The first result shows that the cumulative function of a fuzzy variable is not decreasing w.r.t. the optimistic level.

Proposition 1. Let ξ be a fuzzy variable. The distribution function φ λ of ξ is nondecreasing with respect to λ.

Proof : This result is justified by the fact that the M λ -measure is nondecreasing with respect to λ. In fact, by setting f (λ) = λP os(A) + (1 -λ)N ec(A) with λ ∈ [0, 1] and for a fixed subset A of R, we have:

f (λ) = P os(A) -N ec(A) ≥ 0 as P os(A) ≥ N ec(A).
The second result establishes that in the set of trapezoidal fuzzy variables, if two cumulative functions coincide for two optimistic levels, then they will coincide as credibilistic cumulative functions. The proof of this result is given in Appendix. with respect to M λ -measure.

If there exists

λ 1 , λ 2 ∈ [0; 1] such that: φ 1 λ 1 (x) = φ 2 λ 2 (x), ∀x ∈ R, then φ 1 (x) = φ 2 (x), ∀x ∈ R.
In the next Section, we introduce and study two new order dominances of fuzzy variables which take into account investors's optimism level λ. Therefore, the first one compares fuzzy variables with the same optimism level from investors whereas the second one compares them with different optimism levels from investors.

3 Dominance relations of fuzzy variables with respect to M λ -measure and defined by:

ξ 1 d λ 1 ξ 2 if ∀x ∈ R, φ 1 λ (x) ≤ φ 2 λ (x)
. We establish four first properties of the dominance d λ 1 for trapezoidal fuzzy variables. The three first ones shows that d λ 1 is a pre-order on the set A of trapezoidal fuzzy variables whereas the last one establishes sufficient condition for comparison on A.

Proposition 3. Let λ ∈ [0; 1]. 1. Reflexivity: ∀ξ ∈ A, ξ d λ 1 ξ; 2. Symmetry: ∀(ξ, η) ∈ A 2 , if ξ d λ 1 η and η d λ 1 ξ then ξ ∼ η; 3. Transitivity: ∀(ξ, η, θ) ∈ A 3 , if ξ d λ 1 η and η d λ 1 θ then ξ d λ 1 θ; 4. ∀(ξ, η) ∈ A 2 , inf Supp(ξ) > sup Supp(η) ⇒ ξ d λ 1 η.
Proof : Proofs are similar to ones established by Tassak et al. [START_REF] Tassak | Characterization of order dominances on fuzzy variables for portfolio selection with fuzzy returns[END_REF] for the first order dominance with respect to credibility measure.

The following result characterizes

ξ 1 d λ 1 ξ 2 ⇔        a 1 ≥ a 2 b 1 ≥ b 2 c 1 ≥ c 2 d 1 ≥ d 2 .
Proof : This proof is similar to the proof of the characterization of first order dominance with respect to credibility measure proposed by Tassak et al. [START_REF] Tassak | Characterization of order dominances on fuzzy variables for portfolio selection with fuzzy returns[END_REF].

Let us notice that according to Eq. 1, all dominances d λ 1 with λ ∈ [0, 1] are equivalent on A. But, it is not longer the case for the comparison of two fuzzy variables with different membership functions' sharps.

The following result establishes the compatibility of d λ 1 with respect to the sum.

ξ 1 d λ 1 ξ 2 ⇒ ξ 1 + ξ 3 d λ 1 ξ 2 + ξ 3 .
Proof : The proof is easily obtained through the characterization of d λ 1 for the case of trapezoidal fuzzy variables and the Extension Principle of Zadeh [START_REF] Zadeh | Fuzzy Set as a basis of theory of possibility[END_REF].

Optiminisn Dominance: Definition and Properties

Let us illustrate our idea with the particular case of two investors with different optimism levels α, β ∈ [0; 1]. Let us set Π = {α, β} and propose the following dominance for two fuzzy variables ξ 1 , ξ 2 whose respective distribution functions φ 1 λ and φ 2 λ with respect to M λ -measure:

ξ 1 ≥ D 1 ξ 2 ⇒ ∃λ ∈ Π, ∀ε ∈ Π, φ 1 λ ≤ φ 2 ε . (2)
Intuitively, (2) means that fuzzy return ξ 1 is preferred to fuzzy return ξ 2 in that sense: there is at least an optimism level or a particular distribution of ξ 1 under which ξ 1 gives less chance to obtain lower return than ξ 2 whatever investor's optimism level considered for ξ 2 (that is whatever the way ξ 2 is distributed).

Let us generalize (2) by the following definition.

Definition 3. Let ξ 1 and ξ 2 be two fuzzy variables whose respective distribution functions are φ 1 λ and φ 2 λ with respect to M λ -measure. The optiminisn dominance is the binary relation on fuzzy variables denoted D 1 and defined by:

ξ 1 D 1 ξ 2 ⇒ ∃λ ∈ [0; 1], ∀β ∈ [0; 1], φ 1 λ,ξ 1 ≤ φ 2 β,ξ 2 .
The next result establishes that D 1 is a pre-order on A.

Proposition 5.

1. Reflexivity: ∀ξ ∈ A, ξ D 1 ξ;

2. Symmetry: ∀(ξ, η) ∈ A 2 , if ξ D 1 η and η D 1 ξ then ξ ∼ η; 3. Transitivity: ∀(ξ, η, θ) ∈ A 3 , if ξ D 1 η and η D 1 θ then ξ D 1 θ.
Proof : Let us prove the three precede properties. i) Reflexivity: ξ D 1 ξ. We assume that φ λ is the distribution function of ξ with respect to M λmeasure. Let us find

λ 0 ∈ [0; 1] such that ∀β ∈ [0; 1], ∀x ∈ R, φ λ 0 (x) ≤ φ β (x).
According to Proposition 1, it suffices to take λ 0 = 0 and we get:

φ 0 (x) ≤ φ β (x), ∀x ∈ R, ∀β ∈ [0; 1]. ii) Symmetry: ξ D 1 η and η D 1 ξ ⇒ ξ ∼ η, that is φ 1 (x) = φ 2 (x), ∀x ∈ R,
where φ 1 and φ 2 are respectively the distribution functions of ξ and η, defined by Liu [START_REF] Liu | Uncertainty Theory: An Introduction to its Axiomatics Foundations[END_REF] and φ 1 λ , φ 2 λ are respectively the distribution functions of ξ, η with respect to M λ -measure. Let us assume that ξ D 1 η and η

D 1 ξ. ξ D 1 η ⇒ ∃λ 1 ∈ [0; 1], ∀β ∈ [0; 1], ∀x ∈ R, φ 1 λ 1 (x) ≤ φ 2 β (x) : (R 1 ), η D 1 ξ ⇒ ∃λ 2 ∈ [0; 1], ∀β ∈ [0; 1], ∀x ∈ R, φ 1 λ 2 (x) ≤ φ 2 β (x) : (R 2
). According to (R 1 ),by taking β = λ 2 , we have: φ 1 λ 1 (x) ≤ φ 2 λ 2 (x) : (R 3 ); according to (R 2 ),by taking β = λ 1 , we have: φ 2 λ 2 (x) ≤ φ 1 λ 1 (x) : (R 4 ). Finally, according to (R 3 ), (R 4 ) and Proposition 2, we have:

φ 1 λ 1 (x) = φ 2 λ 2 (x) ⇒ φ 1 (x) = φ 2 (x), ∀x ∈ R. iii) Transitivity: ξ D 1 η and η D 1 θ ⇒ ξ D 1 θ.
Let us assume that ξ D 1 η and η D 1 θ where φ 1 λ , φ 2 λ , φ 3 λ are respectively the distribution functions of ξ, η, θ with respect to M λ -measure.

ξ D 1 η ⇒ ∃λ 3 ∈ [0; 1], ∀β ∈ [0; 1], ∀x ∈ R, φ 1 λ 3 (x) ≤ φ 2 β (x) : (R 5 ), η ≥ D 1 θ ⇒ ∃λ 4 ∈ [0; 1], ∀β ∈ [0; 1], ∀x ∈ R, φ 2 λ 4 (x) ≤ φ 3 β (x)
. According to (R 5 ), by taking β = λ 4 , we have:

φ 1 λ 3 (x) ≤ φ 2 λ 4 (x). Thus, φ 1 λ 3 (x) ≤ φ 2 λ 4 (x) ≤ φ 3 β (x), ∀β ∈ [0; 1], ∀x ∈ R, that is, φ 1 λ 3 (x) ≤ φ 3 β (x), ∀β ∈ [0; 1], ∀x ∈ R. Therefore, ξ D 1 θ.
Let us characterize the optiminisn dominance D 1 by showing that a trapezoidal fuzzy variable dominates a second one if the support of the first is on right and the one of the second is on left.

ξ 1 D 1 ξ 2 ⇐⇒ a 1 ≥ d 2 .
Proof : Let us assume that φ λ 1 and φ λ 2 are respective distribution functions of ξ 1 and ξ 2 with respect to M λ -measure. ⇐) Let us assume that a 1 ≥ d 2 .

According to [START_REF] Asady | Ranking of fuzzy numbers by minimize distance[END_REF], it is obvious that:

∀λ ∈ [0; 1], ∀β ∈ [0; 1], ∀x ∈ R, φ λ 1 (x) ≤ φ β 2 (x). Thus, ξ 1 ≥ D 1 ξ 2 . ⇒) Let us assume that a 1 < d 2 . Let be λ ∈ [0; 1] and let us find γ ∈ [0; 1], x 0 ∈ R such that: φ λ 1 (x 0 ) > φ γ 2 (x 0 ). ( 3
) Let x 0 ∈ [a 1 ; d 2 ] such that x 0 ≤ b 1 and x 0 ≥ c 2 .
According to ( 1) and ( 3), we have:

λ x 0 -a 1 b 1 -a 1 > 1 + (γ -1) x 0 -d 2 c 2 -d 2 . ( 4
)
Let us set θ =

d 2 -a 1 b 1 -a 1 +d 2 -c 2
where θ is the solution of the equation:

x-a 1 b 1 -a 1 = x-d 2 c 2 -d 2 .
We have: (4) ⇒ λθ > 1 + (γ -1)θ, that is, γ < λθ-1 θ + 1 and we set:

ε = λθ-1 θ + 1. i) As    d 2 > a 1 b 1 > a 1 d 2 > c 2
, it is obvious that θ > 0 and thus ε exits.

ii) We also have:

θ -1 = c 2 -b 1 b 1 -a 1 +d 2 -c 2 , that leads to, θ < 1 ⇐⇒ c 2 < b 1 . Therefore, two cases have to be considered: -If c 2 < b 1 Then, θ < 1 ⇒ ε < 1 and it suffices to take γ ∈ [0; ε[. -Else We have: θ ≥ 1 ⇒ ε ≥ 0. It suffices to take γ ∈ [0; ε ∧ 1[.
The following result establishes the compatibility of D 1 with respect to the sum on A.

Proposition 6. Let ξ 1 = (a 1 , b 1 , c 1 , d 1 ), ξ 2 = (a 2 , b 2 , c 2 ,
ξ 1 D 1 ξ 2 ⇒ ξ 1 + ξ 3 D 1 ξ 2 + ξ 3 .
Proof : The proof is easily obtained through the characterization of D 1 for the case of trapezoidal fuzzy variables and the Extension Principle 8 of Zadeh [START_REF] Zadeh | Fuzzy Set as a basis of theory of possibility[END_REF].

The next result establishes that the optiminisn dominance D 1 is strongest than the first order dominance d λ 1 in the set of trapezoidal fuzzy variables. Proposition 7. Let λ ∈ [0; 1] and ξ 1 , ξ 2 be two trapezoidal fuzzy variables whose respective distribution functions are φ 1 λ and φ 2 λ with respect to M λmeasure. Then

ξ 1 D 1 ξ 2 ⇒ ξ 1 d λ 1 ξ 2 .
Proof : The proof is easily obtained through characterization of dominances D 1 and d λ 1 for the case of trapezoidal fuzzy variables.

Let us end this section by establishing that dominance D 1 is a complete relation on the subset of trapezoidal fuzzy variables with disjoint supports. Proposition 8. Let A be a subset of A which is the set of trapezoidal fuzzy variables with disjoint supports. The optiminisn dominance D 1 is complete on A .

Proof : Let ξ = (a, b, c, d) and ξ = (a , b , c , d ) be two elements of A . As Supp(ξ) ∩ Supp(ξ ) = ∅, we necessarily have: a > d or a > d. According to Theorem 2, we have:

ξ D 1 ξ or ξ D 1 ξ.
In the next Section, we use the two proposed dominances to determine the set of best portfolios of assets with fuzzy returns and we implement the obtained result on a given set of triangular fuzzy variables proposed by Huang [START_REF] Huang | Mean-semivariance models for fuzzy portfolio selection[END_REF].

Application of dominances for Portfolio Selection with fuzzy retun

Let us consider the family A = (ξ) 1≤i≤n of n assets where returns are described by trapezoidal fuzzy numbers. A portfolio return ξ associated with A is a convex linear combination of the n assets returns defined by ξ = n i=1 x i ξ i where x i represents the proportion of capital invested in asset i. The set of portfolios associated with

A is P = {ξ = n i=1 x i ξ i , x i ∈ [0, 1], n i=1 x i = 1 and ξ i ∈ A}.
A main question is to determine non dominated portfolios by means of a dominance relation. Based on Game Theory terminology, we will determine the core C R (P ) of (P, R) where R is a dominance on P . In particular, we introduce the core of the dominances D 1 and d λ 1 as follows:

C d λ 1 (P ) = {ξ ∈ P, ∀η ∈ P \ {ξ}, η d λ 1 ξ} (5)
and

C D 1 (P ) = {ξ ∈ P, ∀η ∈ P \ {ξ}, η D 1 ξ}. (6)
The following result gives a link between the two core.

Proposition 9. Let A = (ξ) 1≤i≤n be a family of n assets where returns are described by trapezoidal fuzzy numbers and P the set of portfolios associated with A. We have: C Notice that if λ = 1 2 , then Proposition 9 implies:

C d 1 2 1 (P ) ⊂ C D 1 (P ),
that is, the new set C D 1 (P ) of non dominated portfolios contains non dominated portfolios obtained with the first order dominance associated with credibility measure obtained by Tassak et al. [START_REF] Tassak | Characterization of order dominances on fuzzy variables for portfolio selection with fuzzy returns[END_REF].

The following result characterizes some non dominated portfolios in C D 1 (P ). 1. Γ 1 ⊂ C D 1 (P ).

Γ 2 ⊂ C D 1 (P ).

Proof : The proof is easily obtained by the characterization of D 1 according to Theorem 2.

By considering the family A = (ξ = (a i , b i , c i , d i )) 1≤i≤n of n assets where returns are described by trapezoidal fuzzy numbers and P the set of portfolios associated with A, Proposition 10 leads us to propose two models for the determination in Γ 1 and Γ 2 of non dominated portfolios of with respect to the dominance D 1 as follows:

   max a 1 x 1 + ... + a n x n n i=1 x i = 1 x i ∈ [0, 1] , ∀i ∈ {1, . . . , n} . (7)
and

   max d 1 x 1 + ... + d n x n n i=1 x i = 1 x i ∈ [0, 1] , ∀i ∈ {1, . . . , n} . (8)
In what follows, we implement the two previous optimization models to determine best portfolios in Γ 1 and Γ 2 of the set of seven assets described by seven triangular fuzzy variables in the following table proposed by Huang [START_REF] Huang | Mean-semivariance models for fuzzy portfolio selection[END_REF].

Security i Fuzzy return Security i

Fuzzy return 1 ξ 1 = (-0. Implementation of models ( 7) and ( 8) made with Matlab provides respectively two non dominated portfolios with respect to dominance D 1 defined by: ξ = ξ 1 and η = 0.4214ξ 6 + 0.5786ξ 7 .

We observe that: ξ is obtained by investing all the capital in asset 1 which expresses the "best loss" (a = -0.3) whereas η is obtained by investing the capital only in asset 6 and asset 7 which express the "highest gain" (c = 3.0). So, best portfolios ξ and η are respectively choice of extreme risk averse and extreme risk taker investors.

The following table presents characteristics such as mean, variance and semi-variance, kurtosis and skewness of our optimal portfolios and those obtained by previous authors (Huang [START_REF] Huang | Mean-semivariance models for fuzzy portfolio selection[END_REF] 2, optimal portfolios' characteristics present two different investors' behaviors: averse risk investors and risky investors respectively from and let us prove that:

φ 1 (x) = φ 2 (x), ∀x ∈ R. 1) First step Let us prove that        a 1 = a 2 , b 1 = b 2 , c 1 = c 2 , d 1 = d 2 i)If a 1 = a 2 (
without loss of generality, we assume that a 1 < a 2 ). For x ∈]a 1 ; a 2 [, according to (1) we have:

φ 1 λ 1 (x) = 0 and φ 2 λ 2 (x) = 0, that contradicts (9).
ii) If b 1 = b 2 (without loss of generality, we assume that b 2 < b 1 ). For x 1 , x 2 ∈]b 2 ; b 1 [ such that x 1 = x 2 and x 1 < c 1 , x 2 < c 2 , according to (1) we have:

φ 1 λ 1 (x 1 ) = 0 or φ 2 λ 2 (x 2 ) = 0 or φ 1 λ 1 (x 1 ) = φ 2 λ 2 (x 2 ) = λ 2 .
The first two cases are obvious and we focus only on the third case:

φ 1 λ 1 (x 1 ) = φ 2 λ 2 (x 2 ) = λ 2 =⇒ λ 1 (x 1 -a 1 ) b 1 -a 1 = λ 1 (x 2 -a 1 )
b 1 -a 1 = λ 2 , according to (1) and ( 9). We have: ( λ 1 (x 1 -a 1 ) b 1 -a 1 = λ 1 (x 2 -a 1 ) b 1 -a 1 = λ 2 ) ⇒ x 1 = x 2 , which contradicts x 1 = x 2 .

iii) If c 1 = c 2 (without loss of generality, we assume that c 2 < c 1 ). For x 1 , x 2 ∈]c 2 ; c 1 [ such that x 1 = x 2 and x 1 > b 1 , x 2 > b 2 , according to (1) we have: φ 1 λ 1 (x 1 ) = 1 or φ 2 λ 2 (x 2 ) = 1 or φ 1 λ 1 (x 1 ) = φ 2 λ 2 (x 2 ) = λ 1 . The first two cases are obvious and we focus only on the third case:

φ 1 λ 1 (x 1 ) = φ 2 λ 2 (x 2 ) = λ 1 =⇒ λ 2 (d 2 -x 1 )+x 1 -c 2 d 2 -c 2 = λ 2 (d 2 -x 2 )+x 2 -c 2 d 2 -c 2
= λ 1 according to (1) and [START_REF] Liu | Uncertainty Theory: An Introduction to its Axiomatics Foundations[END_REF]. We have:

( λ 2 (d 2 -x 1 )+x 1 -c 2 d 2 -c 2 = λ 2 (d 2 -x 2 )+x 2 -c 2 d 2 -c 2 = λ 1 ) =⇒ x 1 = x 2 , that contradicts x 1 = x 2 .
iv) If d 1 = d 2 (without loss of generality, we assume that d 1 < d 2 ). For x ∈]d 1 ; d 2 [, according to (1) we have: φ 1 λ 1 (x 1 ) = 1 and φ 2 λ 2 (x) < 1, that contradicts [START_REF] Liu | Uncertainty Theory: An Introduction to its Axiomatics Foundations[END_REF]. , that means in the particular case where λ = 1 2 (Liu [START_REF] Liu | Uncertainty Theory: An Introduction to its Axiomatics Foundations[END_REF]), φ 1 (x) = φ 2 (x), ∀x ∈ R.

Proposition 2 .

 2 Let ξ 1 = (a 1 , b 1 , c 1 , d 1 ) and ξ 2 = (a 2 , b 2 , c 2 , d 2 ) be two trapezoidal fuzzy variables whose respective distribution functions are φ 1 λ and φ 2 λ

3. 1

 1 First order dominance: Definition and Properties Definition 2. Let λ ∈ [0; 1] and ξ 1 , ξ 2 be two fuzzy variables whose respective distribution functions are φ 1 λ and φ 2 λ with respect to M λ -measure. The first order dominance is the binary relation on fuzzy variables denoted by d λ 1

d λ 1

 1 for trapezoidal fuzzy variables. Theorem 1. Let ξ 1 = (a 1 , b 1 , c 1 , d 1 ) and ξ 2 = (a 2 , b 2 , c 2 , d 2 ) be two trapezoidal fuzzy variables and λ ∈ [0, 1]. Then,

Proposition 4 .

 4 Let ξ 1 = (a 1 , b 1 , c 1 , d 1 ), ξ 2 = (a 2 , b 2 , c 2 ,d 2 ) and ξ 3 = (a 3 , b 3 , c 3 , d 3 ) be three trapezoidal fuzzy variables and λ ∈ [0, 1].

Theorem 2 .

 2 Let ξ 1 = (a 1 , b 1 , c 1 , d 1 ) and ξ 2 = (a 2 , b 2 , c 2 , d 2 ) be two trapezoidal fuzzy variables.

d 2 )

 2 and ξ 3 = (a 3 , b 3 , c 3 , d 3 ) be three trapezoidal fuzzy variables and λ ∈ [0, 1].

d λ 1 ( 1 ( 1 (

 111 P ) ⊆ C D 1 (P ). Proof : Let ξ = (a, b, c, d) be a trapezoidal fuzzy variable such that ξ ∈ C d λ P ). Let us prove that ξ ∈ C D 1 (P ). Let η = (a , b , c , d ) ∈ P \{ξ}. P ) ⇒ a > a or b > b or c > c or d > d . Without loss of generality, let us assume that b > b . If a ≥ d then we obtain: b ≥ a ≥ d ≥ b, that is b ≥ b, which contradicts the assumption. So, we conclude that a < d. More precisely, η D 1 ξ and ξ ∈ C D 1 (P ).

Proposition 10 .

 10 Let A = (ξ = (a i , b i , c i , d i )) 1≤i≤n be a family of n assets where returns are described by trapezoidal fuzzy numbers and P the set of portfolios associated with A. Let us set Γ 1 = {ξ = (a, b, c, d)/a = max 1≤i≤n n i=1 x i a i } and Γ 2 = {ξ = (a, b, c, d)/d = max 1≤i≤n n i=1 x i d i }.

Finally, we have a 1 =a 1 =

 11 a 2 , b 1 = b 2 , c 1 = c 2 , d 1 = d 2 . 2) Second step Let us prove that λ 1 = λ 2 . According to the first step, we have [b 2 , c 2 ] = [b 1 , c 1 ]. Moreover, x ∈ [b 1 , c 1 ] =⇒ φ 1 λ 1 (x 1 ) = λ 1 and x ∈ [b 2 , c 2 ] =⇒ φ 2 λ 2 (x) = λ 2 . Thus: (x ∈ [b 1 , c 1 ] ⇐⇒ x ∈ [b 2 , c 2 ]) and φ 1 λ 1 (x 1 ) = φ 2 λ 2 (x), that is, λ 1 = λ 2 .133) Third step Let us prove that φ 1 (x) = φ 2 (x), ∀x ∈ R. By the fact that,φ 1 λ 1 (x) = φ 2 λ 2 (x), ∀x ∈ R, λ 1 = λ 2 and  a 2 , b 1 = b 2 , c 1 = c 2 , d 1 = d 2 we can conclude according to (1) that: φ 1 λ 1 (x 1 ) = φ 2 λ (x), ∀x ∈ R, ∀λ ∈ [0, 1]

Table 1 :

 1 Fuzzy returns of 7 securities (units per stock).

		3, 1.8, 2.3)	5	ξ 5 = (-0.7, 2.4, 2.7)
	2	ξ 2 = (-0.4, 2.0, 2.2)	6	ξ 6 = (-0.8, 2.5, 3.0)
	3	ξ 3 = (-0.5, 1.9, 2.7)	7	ξ 7 = (-0.6, 1.8, 3.0)
	4	ξ 4 = (-0.6, 2.2, 2.8)

Table 2 :

 2 , Li et al.[START_REF] Li | Mean-variance-skewness model for portfolio selection with fuzzy returns[END_REF], Sadefo et al.[START_REF] Sadefo | Moments and semi-moments for fuzzy portfolio selection[END_REF] and Tassak et al.[START_REF] Tassak | Characterization of order dominances on fuzzy variables for portfolio selection with fuzzy returns[END_REF]). Characteristics of optimal portfolios obtained from our two models and previous models From Table

		Mean Variance Kurtosis Skewness
	Huang [5]	1.6	0.7235	1.7972	-0.7543
	Li et al. [6]	1.6	0.7019	1.7291	-0.6823
	Sadefo et al. [14]	1.6	0.7018	1.7290	-0.6823
	Tassak et al. [17] 1.5605	0.6973	1.7033	-0.6666
	Model (7)	1.4	0.4434	0.6922	-0.3380
	Model (8)	1.5975	0.9863	2.4179	-0.7950

model [START_REF] Liu | Expected value of fuzzy variable and fuzzy expected value models[END_REF] and model [START_REF] Liu | Expected value operator of random fuzzy variable and random fuzzy expected models[END_REF]. The first category prefers best risk values described by smallest variance, smallest kurtosis and highest skewness, even though the obtained benefit is lower than the one obtained by previous authors ( [START_REF] Huang | Mean-semivariance models for fuzzy portfolio selection[END_REF], [START_REF] Li | Mean-variance-skewness model for portfolio selection with fuzzy returns[END_REF], [START_REF] Sadefo | Moments and semi-moments for fuzzy portfolio selection[END_REF], [START_REF] Tassak | Characterization of order dominances on fuzzy variables for portfolio selection with fuzzy returns[END_REF]). Thus, since risk values obtained by model [START_REF] Liu | Expected value of fuzzy variable and fuzzy expected value models[END_REF] are better than those obtained by the afore-mentioned authors, this portfolio is the most convenient for risk averse investors. The second category prefers high benefit (1.5975) even though risk values are very uncomfortable. Except the fact that model [START_REF] Liu | Expected value of fuzzy variable and fuzzy expected value models[END_REF] and model [START_REF] Liu | Expected value operator of random fuzzy variable and random fuzzy expected models[END_REF] proposed optimal portfolios for two different types of investors as the recent ones obtained respectively by Sadefo et al. [START_REF] Sadefo | Moments and semi-moments for fuzzy portfolio selection[END_REF] and Tassak et al. [START_REF] Tassak | Characterization of order dominances on fuzzy variables for portfolio selection with fuzzy returns[END_REF], they focus in particular on extreme cases: investors extremely risk averse (model ( 7)) and investors extremely risky (model ( 8)). Moreover, those results justified that, as future portfolio's return is determined through capital's proportions investment on assets, future assets' returns comparisons (through dominance) brings more efficiency on the way of sharing a capital on different portfolio's assets.

Concluding remarks

In this paper, two new dominance relations based on the parametric fuzzy measure were introduced, characterized following two main assumptions and their properties were studied. The obtained results on these new dominances complement previous contributions on various studies made on ranking fuzzy variables, in particular on trapezoidal fuzzy variables, by many scholars ( [START_REF] Asady | Ranking of fuzzy numbers by minimize distance[END_REF], [START_REF] Ezzati | An approach for ranking of fuzzy numbers[END_REF], [START_REF] Wang | Reasonable properties for the ordering of fuzzy quantities[END_REF], [START_REF] Peng | Fuzzy dominance: a new approach for ranking fuzzy variables via credibility measure[END_REF]). Furthermore, we implement the optiminisn dominance (the stronger one) on the set of seven assets proposed by Huang [START_REF] Markowitz | Portofolio selection[END_REF] to enlarge the set of best portfolios obtained by Tassak et al. [START_REF] Tassak | Characterization of order dominances on fuzzy variables for portfolio selection with fuzzy returns[END_REF] through the first order dominance. Those obtained portfolios complement previous and recent studies ([5], [START_REF] Li | Mean-variance-skewness model for portfolio selection with fuzzy returns[END_REF], [START_REF] Sadefo | Moments and semi-moments for fuzzy portfolio selection[END_REF], [START_REF] Tassak | Characterization of order dominances on fuzzy variables for portfolio selection with fuzzy returns[END_REF]) on portfolio selection with fuzzy return by means of the credibility fuzzy measure (when parameter is 1 2 ). In the next research, we intend to introduce and study a new dominance relations which take into account investors optimism's level to compare two fuzzy variables and based on partial moment of those fuzzy variables defined through the generalized fuzzy measure.

Appendix

Proof of Proposition 2:

Let ξ 1 = (a 1 , b 1 , c 1 , d 1 ) and ξ 2 = (a 2 , b 2 , c 2 , d 2 ) be two trapezoidal fuzzy variables whose respective distribution functions φ 1 λ and φ 2 λ with respect to M λ -measure. Let us assume that there exist λ 1 , λ 2 ∈ [0; 1] such that


Audrey De Jong

Samir Jaber

est destinée au dépôt


est destinée au dépôt

  Impact of the driving pressure on mortality in obese and non-obese ARDS patients: a retrospective study of 362 cases Audrey de Jong, Jeanne Cossic, Daniel Verzilli, Clément Monet, Julie Carr, Mathieu Conseil, Marion Monnin, Moussa Fafa Cisse, Fouad Belafia, Nicolas Molinari, et al. To cite this version: Audrey de Jong, Jeanne Cossic, Daniel Verzilli, Clément Monet, Julie Carr, et al.. Impact of the driving pressure on mortality in obese and non-obese ARDS patients: a retrospective study of 362 cases. Intensive Care Medicine, 2018, pp.1106-1114. 10.1007/s00134-018-5241-6. hal-01819466


PharmD Delphine Sterlin

MD Jehane Fadlallah

Emma Slack Phd

MD Guy Gorochov
email: guy.gorochov@sorbonne--universite.fr

Affiliations

The antibody / microbiota interface in health and disease

Keywords: Gut microbiota, secretory Ig, humoral response, autoimmunity

The human intestine is densely colonized with commensal microbes that stimulate the immune system. While secretory Immunoglobulin (Ig) A is known to play a crucial role in gut microbiota compartmentalization, secretory IgM and systemic IgG have recently been highlighted in host--microbiota interactions as well. In this review, we discuss important aspects of secretory IgA biology, but rather than focusing on mechanistic aspects of IgA impact on microbiota, we stress the current knowledge of systemic antibody responses to whole gut microbiota, in particular their generation, specificities and function. We also provide a comprehensive picture of secretory IgM biology. Finally, therapeutic and diagnostic implications of these novel findings for the treatment of various diseases are outlined.

Introduction

Humoral immunity, in particular antibody--mediated responses, plays a key role in the host defense against extracellular pathogens. Notably, the function of secretory IgA, given its mucosal localization, has been considered for a long time to be restricted to immune exclusion, i.e. neutralization of luminal pathogens prevention of host invasion.

In the past decade, with the advent of microbial metagenomics and the improvement of bacterial flow cytometry, murine and human studies have shed new light on IgA/microbiota commensalism. IgA is now viewed as playing a pivotal role in supporting the establishment, selection, diversification and long--term maintenance of a healthy commensal gut microbiota.

Novel functions have also been ascribed, albeit putatively, to IgM and IgG at the gut-microbial interface. Whereas early studies have shown the presence of IgM plasmocytes in gut biopsies of IgA deficient patients (IgAd) [START_REF] Brandtzaeg | The clinical condition of IgA--deficient patients is related to the proportion of IgD--and IgM--producing cells in their nasal mucosa[END_REF][START_REF] Mellander | Secretory antibodies in IgA--deficient and immunosuppressed individuals[END_REF] , more recent studies provide evidence that secretory IgM (sIgM) binds a fraction of commensal microbiota in IgA deficiency thereby enhancing bacterial diversity [START_REF] Fadlallah | Microbial ecology perturbation in human IgA deficiency[END_REF] . Moreover, sIgM in healthy humans seem to bind a very restricted fraction of gut microbiota along with secretory IgA [START_REF] Magri | Human Secretory IgM Emerges from Plasma Cells Clonally Related to Gut Memory B Cells and Targets Highly Diverse Commensals[END_REF] . The functional consequences of these observations are unclear, but could be clinically relevant as it is well established that patients selectively lacking IgA rarely develop inflammatory bowel diseases (IBD), which is instead more common and severe in those lacking both IgA and IgM [START_REF] Agarwal | Pathogenesis and treatment of gastrointestinal disease in antibody deficiency syndromes[END_REF][START_REF] Agarwal | Diagnosis and Treatment of Gastrointestinal Disorders in Patients With Primary Immunodeficiency[END_REF] . Finally, systemic IgG responses targeting gut microbial communities were initially described in IBD and considered merely secondary to bacterial translocations in relation with gut barrier defects. However, it has been reported recently that IgG targeting either bacterial antigens, whole bacterial cells, or whole microbial consortia, can also be retrieved from healthy subjects.

Here, we review recent murine and human studies in an attempt to provide a comprehensive picture of our current knowledge of the interplay between gut antibodies and microbiota.

Secretory IgA bind a wide range of commensals

Early studies of gut microbiota revealed that secretory IgA coat only a fraction of human and murine microbiota [START_REF] Tsuruta | The amount of secreted IgA may not determine the secretory IgA coating ratio of gastrointestinal bacteria[END_REF][START_REF] Waaij | van der In vivo IgA coating of anaerobic bacteria in human faeces[END_REF] . Recently, a combination of bacterial flow cytometry with high--throughput 16S gene sequencing offered an opportunity to identify IgA--coated bacteria ex vivo. Several such IgAseq studies have independently described IgA + communities in mice [START_REF] Bunker | Innate and Adaptive Humoral Responses Coat Distinct Commensal Bacteria with Immunoglobulin A[END_REF][START_REF] Bunker | Natural polyreactive IgA antibodies coat the intestinal microbiota[END_REF][START_REF] Kawamoto | Foxp3(+) T cells regulate immunoglobulin a selection and facilitate diversification of bacterial species responsible for immune homeostasis[END_REF][START_REF] Koch | Maternal IgG and IgA Antibodies Dampen Mucosal T Helper Cell Responses in Early Life[END_REF][START_REF] Kubinak | MHC variation sculpts individualized microbial communities that control susceptibility to enteric infection[END_REF][START_REF] Wilmore | Commensal Microbes Induce Serum IgA Responses that Protect against Polymicrobial Sepsis[END_REF] and in humans [START_REF] Fadlallah | Microbial ecology perturbation in human IgA deficiency[END_REF][START_REF] Magri | Human Secretory IgM Emerges from Plasma Cells Clonally Related to Gut Memory B Cells and Targets Highly Diverse Commensals[END_REF][START_REF] Kau | Functional characterization of IgA--targeted bacterial taxa from undernourished Malawian children that produce diet--dependent enteropathy[END_REF][START_REF] Planer | Development of the gut microbiota and mucosal IgA responses in twins and gnotobiotic mice[END_REF][START_REF] Dzidic | Aberrant IgA responses to the gut microbiota during infancy precede asthma and allergy development[END_REF][START_REF] D'auria | Active and secreted IgA--coated bacterial fractions from the human gut reveal an under--represented microbiota core[END_REF] . Table 1 gathers bacterial taxa that are significantly targeted by IgA in healthy individuals. Despite some variations, core features of the IgA--coated subset emerge from these data. First, IgA bound a highly diverse fraction of microbiota, including numerous rare genera (generally less than 1% in frequency). Second, the four major phyla, namely Actinobacteria, Bacteroidetes, Firmicutes and Proteobacteria were detected in IgA--coated microbiota. Consistently, most microbes, irrespective of their microbial phylogeny, can induce specific IgA in monocolonized gnotobiotic mice [START_REF] Geva--Zatorsky | Mining the Human Gut Microbiota for Immunomodulatory Organisms[END_REF][START_REF] Gaboriau--Routhiau | The Key Role of Segmented Filamentous Bacteria in the Coordinated Maturation of Gut Helper T Cell Responses[END_REF] . Of note, multiple taxa of the phylum Firmicutes, in particular those belonging to the family Lachnospiraceae, are significantly enriched within the IgA + fraction whatever the human cohort (Table 1). The relatively high frequency of these organisms in the human colon might account for their prevalence in IgA + fractions. This observation supports earlier studies showing that Lachnospiraceae members are able to elicit innate immune responses and modulate adaptive immune responses in the gut [START_REF] Wu | Commensal A4 bacteria inhibit intestinal Th2--cell responses through induction of dendritic cell TGF--β production[END_REF][START_REF] Liu | Role of the Gut Microbiome in Modulating Arthritis Progression in Mice[END_REF][START_REF] Lodes | Bacterial flagellin is a dominant antigen in Crohn disease[END_REF] .

While it was concluded from each of the above mentioned studies that IgA preferentially binds a taxonomically distinct subset of the whole microbiota, it remains difficult to define a core sIgA--coated community in healthy individuals. A similar observation has been made in uniform patient groups. One likely reason is that inter-study variability might not only result from technical factors, but also from different strategies to define the sIgA--coated microbiota. Key variations in immuno--staining protocols, such as the use of either monoclonal [START_REF] Magri | Human Secretory IgM Emerges from Plasma Cells Clonally Related to Gut Memory B Cells and Targets Highly Diverse Commensals[END_REF][START_REF] Palm | Immunoglobulin A coating identifies colitogenic bacteria in inflammatory bowel disease[END_REF] or polyclonal anti--IgA antibody [START_REF] Fadlallah | Microbial ecology perturbation in human IgA deficiency[END_REF][START_REF] Bunker | Innate and Adaptive Humoral Responses Coat Distinct Commensal Bacteria with Immunoglobulin A[END_REF][START_REF] Kau | Functional characterization of IgA--targeted bacterial taxa from undernourished Malawian children that produce diet--dependent enteropathy[END_REF][START_REF] Planer | Development of the gut microbiota and mucosal IgA responses in twins and gnotobiotic mice[END_REF][START_REF] D'auria | Active and secreted IgA--coated bacterial fractions from the human gut reveal an under--represented microbiota core[END_REF] , have a substantial impact on background staining. Differences in sequencing methods may also explain the variations observed between reports. IgA + bacterial identification usually relies on 16S gene sequencing [START_REF] Magri | Human Secretory IgM Emerges from Plasma Cells Clonally Related to Gut Memory B Cells and Targets Highly Diverse Commensals[END_REF][START_REF] Kau | Functional characterization of IgA--targeted bacterial taxa from undernourished Malawian children that produce diet--dependent enteropathy[END_REF][START_REF] Planer | Development of the gut microbiota and mucosal IgA responses in twins and gnotobiotic mice[END_REF][START_REF] Dzidic | Aberrant IgA responses to the gut microbiota during infancy precede asthma and allergy development[END_REF][START_REF] D'auria | Active and secreted IgA--coated bacterial fractions from the human gut reveal an under--represented microbiota core[END_REF][START_REF] Palm | Immunoglobulin A coating identifies colitogenic bacteria in inflammatory bowel disease[END_REF] , yet 16S--based classifications cannot discriminate among strains, or even species, within the same bacterial genera, which may substantially differ in terms of immunogenicity [START_REF] Group | Evaluation of 16S rDNA--Based Community Profiling for Human Microbiome Research[END_REF][START_REF] Greenblum | Extensive Strain--Level Copy--Number Variation across Human Gut Microbiome Species[END_REF] . Indeed, Geva--Zatorsky et al. found that two related species of Bacteroides (B. thetaiotaomicron and B. vulgatus) displayed differential IgA induction potential in monocolonized gnotobiotic mice [START_REF] Geva--Zatorsky | Mining the Human Gut Microbiota for Immunomodulatory Organisms[END_REF] . By contrast, whole genome shotgun sequencing has extended taxonomic classification at both species and strain level [START_REF] Jovel | Characterization of the Gut Microbiome Using 16S or Shotgun Metagenomics[END_REF] , thereby revealing IgA predilections at lower taxonomic levels. In addition, whereas some authors focused their analysis on the IgA high population [START_REF] Fadlallah | Microbial ecology perturbation in human IgA deficiency[END_REF][START_REF] Magri | Human Secretory IgM Emerges from Plasma Cells Clonally Related to Gut Memory B Cells and Targets Highly Diverse Commensals[END_REF][START_REF] Palm | Immunoglobulin A coating identifies colitogenic bacteria in inflammatory bowel disease[END_REF] , other studies included both IgA high and IgA dim populations [START_REF] Bunker | Innate and Adaptive Humoral Responses Coat Distinct Commensal Bacteria with Immunoglobulin A[END_REF][START_REF] Dzidic | Aberrant IgA responses to the gut microbiota during infancy precede asthma and allergy development[END_REF] in the definition of IgA + microbiota. Beyond these practical issues, inter-study variations also reflect the considerable diversity of human gut microbiota [START_REF] Eckburg | Diversity of the Human Intestinal Microbial Flora[END_REF][START_REF] Lozupone | Diversity, stability and resilience of the human gut microbiota[END_REF] .

Numerous factors such as age, diet, antibiotics, and genetic background impact gut microbiota composition [START_REF] Zhernakova | Population--based metagenomics analysis reveals markers for gut microbiome composition and diversity[END_REF][START_REF] Bonder | The effect of host genetics on the gut microbiome[END_REF][START_REF] Wu | Linking long--term dietary patterns with gut microbial enterotypes[END_REF][START_REF] Ruiz | A single early--in--life macrolide course has lasting effects on murine microbial network topology and immunity[END_REF] . Interestingly, environmental factors have also been shown to influence IgA responses to bacteria. For instance, antibiotic exposure is associated with increased IgA coating of Enterococcus sp and Lactobacillales [START_REF] Džunková | Active and Secretory IgA--Coated Bacterial Fractions Elucidate Dysbiosis in Clostridium difficile Infection[END_REF] and it is therefore expected that each individual harbors a distinct IgA--coated bacterial repertoire. Another potential challenge for IgAseq studies is that enchained growth, driven by high--affinity IgA, can exert a negative selective pressure [START_REF] Moor | High--avidity IgA protects the intestine by enchaining growing bacteria[END_REF] . Enchained growth can also contribute to an environment with lower levels of temperate phage reactivation [START_REF] Diard | Inflammation boosts bacteriophage transfer between Salmonella spp[END_REF] . It is therefore expected that bacteria persisting in the gut in the presence of IgA have evolved traits to avoid, evade or benefit from enchained growth (such as large capsules preventing cross-linking and protect from phage predation, or enchained growth--induced microcolony deep mucus settling [START_REF] Diard | Inflammation boosts bacteriophage transfer between Salmonella spp[END_REF][START_REF] Donaldson | Gut microbiota utilize immunoglobulin A for mucosal colonization[END_REF] ). Strains that are susceptible to IgA--mediated enchained growth are expected to be absent from IgAseq data, and are logically extremely difficult to identify by IgAseq alone. We also need to take care in interpreting species presence/absence data in terms of function: the microbiota is a complex ecosystem and minor shifts in the abundance or function of one species can have chaotic effects on the rest of the consortium.

Analysis of IgA--coated bacterial communities to unravel IgA function

The identification of IgA--coated bacteria has provided new insights in our understanding of the role of secretory IgA in mucosal homeostasis. The contribution of human studies to describe functions of IgA will be discussed below while other important roles of secretory IgA are reviewed elsewhere in this issue.

The work published by Palm et al. suggests that IgA could coat a group of disease-prone intestinal bacteria in humans [START_REF] Palm | Immunoglobulin A coating identifies colitogenic bacteria in inflammatory bowel disease[END_REF] . This conclusion is based on human IgA+ microbiota transfer to murine disease models. The latter view proposing that IgA would not induce bacterial clearance, but instead might minimize inflammatory effects of targeted bacteria, has been confirmed in other studies in mice. In B. thetaiotaomicron monocolonized mice, Peterson et al. showed that specific IgA selects against a specific phase--variable capsular polysaccharide associated with reduced intestinal innate immune activation [START_REF] Peterson | IgA response to symbiotic bacteria as a mediator of gut homeostasis[END_REF] . Affinity--maturated IgA appear also necessary for SFB (Segmented Filamentous Bacteria) pathobiont control, since AID-- and IgA--deficient mice suffer from SFB overgrowth and intestinal inflammation [START_REF] Fagarasan | Critical roles of activation--induced cytidine deaminase in the homeostasis of gut flora[END_REF][START_REF] Suzuki | Aberrant expansion of segmented filamentous bacteria in IgA-deficient gut[END_REF] . Thus, the selective and/or neutralizing effects of IgA could modulate bacterial fitness toward a non--inflammatory profile.

Another interpretation is that bacteria with a better grafting potential in mice have a more pronounced impact on the immune system and therefore are more likely to induce a strong IgA response. These two effects likely coexist in vivo.

These observations notwithstanding, it is also possible that IgA might be beneficial for commensal growth and therefore could play a protective role in commensal ecology. In line with this idea, we observed that bacterial species known to be IgA--coated in human samples are more likely to be underrepresented than overrepresented in fecal microbiota from sIgAd patients [START_REF] Fadlallah | Microbial ecology perturbation in human IgA deficiency[END_REF] . Therefore, IgA do not necessarily inhibit bacterial growth, since bacteria such as Coprococcus comes, Dorea sp. and Clostridium sp. thrive in the presence of IgA [START_REF] Fadlallah | Microbial ecology perturbation in human IgA deficiency[END_REF][START_REF] Moor | High--avidity IgA protects the intestine by enchaining growing bacteria[END_REF] . This pro--microbial effect of IgA has also been shown using an in vitro system in which human IgA facilitate biofilm formation and enteric bacteria proliferation [START_REF] Bollinger | Human secretory immunoglobulin A may contribute to biofilm formation in the gut[END_REF] . Interestingly, Donaldson et al. demonstrated that IgA binding to Bacteroides fragilis facilitated adherence to intestinal epithelial cells by modulating surface epitopes [START_REF] Donaldson | Gut microbiota utilize immunoglobulin A for mucosal colonization[END_REF] promoting long--term maintenance of this species in the gut microbiota. Furthermore, Nakajima et al showed that IgA interacts with Bacteroides thetaiotaomicron via the O--and N--glycans that extensively decorate its secretory component and hinge regions [START_REF] Nakajima | IgA regulates the composition and metabolic function of gut microbiota by promoting symbiosis between bacteria[END_REF] . The latter results also raise the important issue of the relative importance of Fab--(i.e canonical binding) vs Fc--or glycan--bound (i.e non--canonical bonding) consortia which are not currently distinguished in most IgAseq studies.

IgA as a regulator of bacterial networks

An attractive hypothesis is that IgA might support bacterial symbiosis by facilitating or modifying bacterial networks. Bacterial symbiosis implies that some bacteria depend on other bacteria for their persistence. Within such interdependence, a bacterium, called "satellite", never inhabits independently of another, named "host", in a given microbiota. In sIgAd patients, substantial links were found to be lost between satellites and hosts meaning that bacterial networks are altered in the absence of IgA 3 .

One could consider the possibility that IgA, bound non--canonically or displaying cross-species reactivity, might favor incorporation into symbiotic networks as an ecological extension of their ability to agglutinate pathogens. Despite these exciting insights into the role of IgA in microbiota homeostasis, studies in human patients who lack IgA share a main weak spot: such patients were recruited from hospital and therefore may not be representative of the majority of sIgAd individuals who are asymptomatic. To better understand the physiological role of IgA in humans, a major challenge will be to explore a large number of asymptomatic sIgAd individuals and to design interventional studies in symptomatic patients. Longitudinal studies would also be relevant to look at the effect of IgA on strain stability over time.

Towards the definition of IgA deficiencies at functional level

The impressive development of knowledge surrounding IgA--coated bacterial repertoires may lead to the definition of novel immunopathologic conditions characterized by aberrant patterns of IgA coating. Although such perturbations have already been described in humans, their clinical consequences need to be investigated.

Initially, Dzidic et al. reported that low proportions of IgA--coated gut--derived bacteria in infants at 12 months of age preceded the development of allergic asthma. Furthermore, IgA + repertoires remarkably differed between healthy and allergic children [START_REF] Dzidic | Aberrant IgA responses to the gut microbiota during infancy precede asthma and allergy development[END_REF] . Although these results remain correlative, one could speculate that overall reduction of IgA-coating might reflect weaker stimulation of the mucosal immune system in allergic children. It will be interesting to follow up whether this is due to the presence of less invasive species, or an altered handling by the immune system of very young children at high risk of atopy. Furthermore, fluctuations of IgA+ fraction were measured in multiple sclerosis fecal microbiota depending on disease status (e.g remission or relapse). The IgA--coated subset tended to increase during remission whereas significant decrease of IgA--coated bacteria have been found in active patients. Beyond IgA on their own, the authors discovered that IgA--secreting cells travel from the gut to the brain to attenuate neuroinflammation in an interleukin--10 dependent manner. Translating this result from mice to humans, the authors speculated that a reduction in IgA--bound bacteria might be due to egress of IgA--secreting cells out of the gut [START_REF] Rojas | Recirculating Intestinal IgA--Producing Cells Regulate Neuroinflammation via IL--10[END_REF] . Aberrant patterns of IgA coating have also been analyzed at the bacterial family level in undernourished children. Gordon et al. reported that Enterobacteriaceae IgA coating discriminated healthy from undernourished children, suggesting an interrelationship between IgA responses and nutritional status [START_REF] Kau | Functional characterization of IgA--targeted bacterial taxa from undernourished Malawian children that produce diet--dependent enteropathy[END_REF] . Further work should investigate whether these changes in IgA coating emerge as a consequence of the disease, or play a causative role.

Secretory IgM reinforces the role of secretory IgA

In addition to IgA--secreting cells, the gut mucosa also contains IgM--secreting cells that coexist with clonally related IgM--memory B cells in humans, but not in mice [START_REF] Magri | Human Secretory IgM Emerges from Plasma Cells Clonally Related to Gut Memory B Cells and Targets Highly Diverse Commensals[END_REF] . These newly described IgM--memory B cells harbor a gene signature distinct from splenic marginal zone B cells, yet reflecting antigen experience. Both IgM--secreting--and IgM-memory B cells arise as early as 1.5 months after birth. Numbers of IgM--memory B cells remain stable over time while IgM--secreting cells accumulate over the first 10 years of life [START_REF] Magri | Human Secretory IgM Emerges from Plasma Cells Clonally Related to Gut Memory B Cells and Targets Highly Diverse Commensals[END_REF] . IgM--secreting cells release sIgM into the gut lumen through the polymeric Ig Receptor (pIgR). While pentameric IgM shows higher affinity for pIgR, export of dimeric IgA is favored by a factor 5. This is likely due to the large size of IgM limiting its diffusion through the stroma and its access to the pIgR [START_REF] Natvig | Mechanism for enhanced external transfer of dimeric IgA over pentameric IgM: studies of diffusion, binding to the human polymeric Ig receptor, and epithelial transcytosis[END_REF] . In contrast to sIgA, the extracellular receptor part (secretory component, SC) is non--covalently incorporated to pentameric IgM which explains why sIgM is not as resistant as sIgA to proteolytic degradation in the gut. Importantly, free SC that is released from excess of unoccupied pIgR in most secretions, cooperates with bound SC to stabilize the quaternary structure of sIgM [START_REF] Brandtzaeg | Secretory IgA: Designed for Anti--Microbial Defense[END_REF][START_REF] Brandtzaeg | Human secretory immunoglobulins. 3. Immunochemical and physicochemical studies of secretory IgA and free secretory piece[END_REF] .

Several studies established that sIgM play a protective role in gastrointestinal infections. While immune exclusion is mainly performed by sIgA, sIgM can efficiently bind certain pathogens such as Salmonella enterica Typhimurium and limit their entry into Peyer's patches [START_REF] Bioley | Plasma--Derived Polyreactive Secretory--Like IgA and IgM Opsonizing Salmonella enterica Typhimurium Reduces Invasion and Gut Tissue Inflammation through Agglutination[END_REF] . In an experimental Shigella flexneri mouse model, sIgM also displayed the ability to modulate the expression of bacterial virulence factors, thereby preserving epithelial surface homeostasis [START_REF] Longet | Reconstituted Human Polyclonal Plasma--derived Secretory--like IgM and IgA Maintain the Barrier Function of Epithelial Cells Infected with an Enteropathogen[END_REF] . Unlike sIgA, sIgM can activate the complement system, facilitating pathogen killing through the classical pathway and leading to membrane attack complex formation or opsonisation for phagocytosis.

Kirkland et al. demonstrated that sIgM-- and complement--deficiency are associated with a worse disease course of DSS--induced colitis, supporting a role for IgM in intestinal homeostasis. However, variations of microbiota in WT vs mutant mice were not studied in these immunodeficient animals [START_REF] Kirkland | B cell--intrinsic MyD88 signaling prevents the lethal dissemination of commensal bacteria during colonic damage[END_REF] . The protective role of sIgM is further illustrated in patients with primary selective IgM deficiency that present increased susceptibility to mucosal infections [START_REF] Chovancova | Selective IgM Deficiency: Clinical and Laboratory Features of 17 Patients and a Review of the Literature[END_REF] .

For years, the production of sIgM has been considered to function as a compensatory mechanism in patients lacking IgA, explaining the mild phenotype in the latter. Numerous IgM--producing cells were indeed detected in mucosal samples from sIgAd patients compared to healthy donors [START_REF] Brandtzaeg | The clinical condition of IgA--deficient patients is related to the proportion of IgD--and IgM--producing cells in their nasal mucosa[END_REF][START_REF] Mellander | Secretory antibodies in IgA--deficient and immunosuppressed individuals[END_REF] . Consistently, we found substantial IgM binding to commensals in sIgAd fecal samples and higher free fecal IgM levels compared to healthy donors. However, unlike sIgA, sIgM was found to poorly bind bacteria belonging to Enterobacteriaceae family, which indeed overgrow in sIgAd microbiota.

These observations could explain the susceptibility to enteropathogens in sIgAd patients [START_REF] Fadlallah | Microbial ecology perturbation in human IgA deficiency[END_REF][START_REF] Elinav | NLRP6 inflammasome is a regulator of colonic microbial ecology and risk for colitis[END_REF] . We also observed that sIgM binding preserves Actinobacteria diversity, which, strikingly decreased dramatically in patients with common variable immunodeficiency (CVID) patients, lacking both IgA and IgM [START_REF] Fadlallah | Microbial ecology perturbation in human IgA deficiency[END_REF][START_REF] Jørgensen | Altered gut microbiota profile in common variable immunodeficiency associates with levels of lipopolysaccharide and markers of systemic immune activation[END_REF] , thus highlighting a new role of sIgM in supporting gut microbiota diversity. IgM function is unlikely to be restricted to a particular disease state in humans. Magri et al. indeed profiled mucus-embedded bacteria from ileum and colon of healthy individuals and detected significant IgM--bound commensals [START_REF] Magri | Human Secretory IgM Emerges from Plasma Cells Clonally Related to Gut Memory B Cells and Targets Highly Diverse Commensals[END_REF] . A greater bacterial diversity has been found in IgM--coated communities, corroborating the notion that sIgM optimize microbiota diversity and likely maximize mucus retention of specific bacterial communities with putative beneficial effects. Accordingly, the microbiota of CVID patients is characterized by a drastic reduction in bacterial diversity and a high dysbiosis index, as compared to sIgAd microbiota [START_REF] Jørgensen | Altered gut microbiota profile in common variable immunodeficiency associates with levels of lipopolysaccharide and markers of systemic immune activation[END_REF][START_REF] Jørgensen | Selective IgA deficiency in humans is associated with reduced gut microbial diversity[END_REF] . Moreover, the prevalence of IBD is higher in CVID patients than in sIgAd patients [START_REF] Agarwal | Pathogenesis and treatment of gastrointestinal disease in antibody deficiency syndromes[END_REF] . Altogether, sIgM appear to play a dual role in the intestine, on one hand providing protection from pathogens and on the other hand preserving gut microbiota diversity and intestinal homeostasis, particularly in the case of barrier--or IgA insufficiency.

Spatial heterogeneity of secretory Ig

Profiling of Ig--bound commensals enabled by bacterial flow cytometry provides a valuable picture of the spatial heterogeneity of bacterial communities and secretory antibodies abundance. For instance, Ig--bound bacteria are more frequent in ileum than in colon and in mucus compared to central gut lumen [START_REF] Fadlallah | Microbial ecology perturbation in human IgA deficiency[END_REF][START_REF] Magri | Human Secretory IgM Emerges from Plasma Cells Clonally Related to Gut Memory B Cells and Targets Highly Diverse Commensals[END_REF][START_REF] Bunker | Innate and Adaptive Humoral Responses Coat Distinct Commensal Bacteria with Immunoglobulin A[END_REF][START_REF] Kau | Functional characterization of IgA--targeted bacterial taxa from undernourished Malawian children that produce diet--dependent enteropathy[END_REF][START_REF] Palm | Immunoglobulin A coating identifies colitogenic bacteria in inflammatory bowel disease[END_REF] . The mucus layer is thin in the small intestine but divided in two distinct layers in colon with an inner, attached layer and an outer, loose mucus layer. Swab samples or colon biopsy were used to profile mucus--associated bacteria while fecal content served as central gut lumen representative [START_REF] Bajaj | Linkage of gut microbiome with cognition in hepatic encephalopathy[END_REF][START_REF] Richard | Mucosa--associated microbiota dysbiosis in colitis associated cancer[END_REF][START_REF] Johansson | The gastrointestinal mucus system in health and disease[END_REF] . In the mucus layer, healthy microbiota can be divided into three fractions: sIgA + sIgM + , sIgA + sIgM --and sIgA --sIgM --commensals although in central gut lumen sIgM coated bacteria have not been detected in healthy donors. Thus, only two fractions can be discriminated: sIgA + sIgM --and sIgA --sIgM --bacteria (Figure 1) [START_REF] Fadlallah | Microbial ecology perturbation in human IgA deficiency[END_REF][START_REF] Magri | Human Secretory IgM Emerges from Plasma Cells Clonally Related to Gut Memory B Cells and Targets Highly Diverse Commensals[END_REF] . One could speculate that sIgM cannot be detected in the central gut lumen due to its slow, size--dependent diffusion through mucus and rapid proteolysis, although this remains to be demonstrated in vivo. The existence of an antibody gradient from epithelium to central gut lumen and from proximal to distal compartments could complicate our understanding of secretory Ig function. Secretory Ig may operate differently depending on bacterial density, antibody concentration, and an enriched glycoprotein mucus environment. One study revealed that bacterial density determines how sIgA clump bacteria [START_REF] Moor | High--avidity IgA protects the intestine by enchaining growing bacteria[END_REF] . At high pathogen densities, IgA efficiently agglutinates pathogens as they randomly collide in intestinal compartment. However, at lower densities, pathogen collisions are too rare and IgA--mediated clumping can only occur by enchaining growing bacteria (which are necessarily in close proximity [START_REF] Moor | High--avidity IgA protects the intestine by enchaining growing bacteria[END_REF] ). Because location will also influence bacterial growth rates, future work should determine how biogeography might have implications for secretory Ig functions.

Systemic IgG responses to microbiota

The initial observation that patients with IBD presented high titers of serum IgG against intestinal commensal bacteria was thought to reflect a leaky epithelial barrier in these patients [START_REF] Landers | Selected loss of tolerance evidenced by Crohn's disease-associated immune responses to auto--and microbial antigens[END_REF][START_REF] Macpherson | Mucosal antibodies in inflammatory bowel disease are directed against intestinal bacteria[END_REF] . However, several recent studies have demonstrated that robust levels of serum IgG capable of binding symbiotic bacteria are also detectable in healthy humans [START_REF] Haas | Systemic antibody responses to gut commensal bacteria during chronic HIV--1 infection[END_REF][START_REF] Harmsen | Crohn's Disease Patients Have More IgG--Binding Fecal Bacteria than Controls[END_REF][START_REF] Christmann | Human seroreactivity to gut microbiota antigens[END_REF][START_REF] Fadlallah | Synergistic convergence of microbiota--specific systemic IgG and secretory IgA[END_REF] . Similarly, peripheral CD4 + T cells specific for intestinal are present in healthy donors [START_REF] Bradley | Segmented Filamentous Bacteria Provoke Lung Autoimmunity by Inducing Gut--Lung Axis Th17 Cells Expressing Dual TCRs[END_REF][START_REF] Hegazy | Circulating and Tissue--Resident CD4+ T Cells With Reactivity to Intestinal Microbiota Are Abundant in Healthy Individuals and Function Is Altered During Inflammation[END_REF] . Using novel sensitive assays such as microbial protein arrays and bacterial flow cytometry, it has been shown that serum IgG levels with measurable affinity for commensals arise by 6 months of age, increase from 12 to 24 months of age and remain stable in adulthood [START_REF] Christmann | Human seroreactivity to gut microbiota antigens[END_REF][START_REF] Fadlallah | Synergistic convergence of microbiota--specific systemic IgG and secretory IgA[END_REF] . This timing correlates well with the pattern of gut colonization at birth, diversification of the microbiota after weaning and microbiota stabilization during adulthood [START_REF] Lim | Early life dynamics of the human gut virome and bacterial microbiome in infants[END_REF][START_REF] Faith | The long--term stability of the human gut microbiota[END_REF] . Longitudinal analysis within a time frame of 5--6 years showed that commensal--binding antibody responses do not significantly change over time in humans [START_REF] Haas | Systemic antibody responses to gut commensal bacteria during chronic HIV--1 infection[END_REF] . In line with these data, anti--commensal IgG antibodies have been found in naïve mice housed under steady state conditions [START_REF] Koch | Maternal IgG and IgA Antibodies Dampen Mucosal T Helper Cell Responses in Early Life[END_REF][START_REF] Zeng | Gut Microbiota--Induced Immunoglobulin G Controls Systemic Infection by Symbiotic Bacteria and Pathogens[END_REF] . In addition, an elegant model of reversible E. coli colonization unequivocally demonstrated that intestinally restrained bacteria are able to induce persistent levels of specific IgG in periphery [START_REF] Gomez De Agüero | The maternal microbiota drives early postnatal innate immune development[END_REF][START_REF] Hapfelmeier | Reversible microbial colonization of germ--free mice reveals the dynamics of IgA immune responses[END_REF] .

Early studies using bacterial flow cytometry showed that systemic IgG bound a wide range of commensal bacteria (E. coli, E. faecalis, B. fragilis), as well as certain commensal fungi such as C. albicans [START_REF] Haas | Systemic antibody responses to gut commensal bacteria during chronic HIV--1 infection[END_REF] . We recently reported that about 1% of the gut microbiota of healthy individuals can be bound by serum IgG. Strikingly, serum IgG converge with secretory IgA toward the same bacterial targets with each subject exhibiting a private set of preferred IgG--targeted commensals. However, a "core" IgG repertoire to particular commensals emerged, genera belonging to Proteobacteria (Serratia, Achromobacter and Delftia for instance) being preferentially enriched in the IgG--targeted fraction of the microbiota [START_REF] Fadlallah | Synergistic convergence of microbiota--specific systemic IgG and secretory IgA[END_REF] . Of note, multiple taxa of the phylum Proteobacteria have been previously described as secretory IgA targets in several human, as well as murine, studies [START_REF] Bunker | Innate and Adaptive Humoral Responses Coat Distinct Commensal Bacteria with Immunoglobulin A[END_REF][START_REF] Wilmore | Commensal Microbes Induce Serum IgA Responses that Protect against Polymicrobial Sepsis[END_REF][START_REF] Planer | Development of the gut microbiota and mucosal IgA responses in twins and gnotobiotic mice[END_REF] . Together, these observations suggest that serum IgG coat a particular subset of microbiota, including a broad range of bacteria, enriched in Proteobacteria. Because 16S sequencing analysis cannot resolve species and substrains variations that directly modulate IgG responses, Christmann et al. developed an alternative approach to decipher bacterial determinants of IgG responses using printing slides with a panel of known antigens from the gut microbiota [START_REF] Christmann | Human seroreactivity to gut microbiota antigens[END_REF] . Using this technique, IgG reactivity to different bacterial components, both extracellular and intracellular, from the three most prominent phyla of the human gut microbiota has been uncovered. Four antigens were found to be universally recognized among the cohort. It remains to be shown whether this is due to cross--reactivity with yet unidentified common pathogen antigens, or whether the intriguing hypothesis that postulates the existence of a core microbiota--binding IgG reactivity can be validated. The results of a recent study supported the idea of cross--reactivity of microbiota--binding IgG by showing that monoclonal antibodies specific for the Klebsiella O3 O--antigen, a mannose polymer, also bind yeast and mammalian--derived mannose--containing glycans.

Such antibodies were shown to bind an array of commensals, likely sharing mannose-containing surface structures [START_REF] Rollenske | Cross--specificity of protective human antibodies against Klebsiella pneumoniae LPS O--antigen[END_REF]

The origin of systemic anti--microbiota IgG

Several studies explored commensal--binding IgG in mice and humans yet it remains unclear how this systemic response to gut microbiota develops. It likely reflects episodic exposure to gut commensals that breach the intestinal barrier. Increased gut permeability has been observed in human after high--fat meals, alcohol binging or the use of proton pump inhibitors and non--steroidal anti--inflammatory drugs absorption.

Various intestinal pathogens, including bacteria and viruses, are also able to disrupt the intestinal epithelial barrier [START_REF] Bala | Acute Binge Drinking Increases Serum Endotoxin and Bacterial DNA Levels in Healthy Individuals[END_REF][START_REF] König | Human Intestinal Barrier Function in Health and Disease[END_REF] . In such conditions, commensal translocation take place in an inflammatory context, likely in the presence of cytokines promoting the production of IgG, instead of IgA, in the lamina propria. It is of note that Ig gene sequence analysis has revealed that gut IgG-- and IgA--secreting cells are not clonally related and therefore originate from independent precursors [START_REF] Benckert | The majority of intestinal IgA+ and IgG+ plasmablasts in the human gut are antigen--specific[END_REF] . Consistently, direct switching from IgM to IgA1 or IgA2 predominates in colonic mucosa, in contrast to the rare sequential event during which IgM switches to IgA2 via IgG2 [START_REF] Lin | IgA subclass switch recombination in human mucosal and systemic immune compartments[END_REF] , reinforcing the idea that microbiota-specific IgA and IgG B cells arise via independent mechanisms. Experimental data in mouse models have detailed the ontogeny of anti--commensal IgG B cells showing that many commensal--binding antibodies are generated in a T--independent manner dependent on TLR signaling in B cells. These B cells might derive from B1 cells in mice, although the relevance of this finding for human biology remains unclear [START_REF] Koch | Maternal IgG and IgA Antibodies Dampen Mucosal T Helper Cell Responses in Early Life[END_REF][START_REF] Descatoire | A human equivalent of mouse B--1 cells?[END_REF] . The microbiota also non--specifically supports systemic antibody production through the production of short--chain fatty acids (SCFAs), the major microbial metabolites of dietary fiber fermentation. SCFAs promote B cell differentiation into antibody--secreting cells and boost B cell metabolism resulting in antibody production [START_REF] Kim | Gut Microbial Metabolites Fuel Host Antibody Responses[END_REF] .

Gut--associated lymphoid tissue (GALT) is the primary site of IgA induction [START_REF] Lécuyer | Segmented Filamentous Bacterium Uses Secondary and Tertiary Lymphoid Tissues to Induce Gut IgA and Specific T Helper 17 Cell Responses[END_REF] . By contrast, the extent to which GALT contributes to the IgG repertoire remains still to be determined. Although the intestinal B cell pool is highly dominated by IgA--producing cells, 3--4% of intestinal antibody--secreting cells express IgG at steady state in humans. About two thirds of monoclonal antibodies derived from intestinal IgG + plasmablasts react with common gut bacteria 73--71 . Moreover, the majority of anti--commensal IgG antibodies were of the IgG2 subclass and IgG2--secreting cells have been detected in the lamina propria throughout the intestine [START_REF] Fadlallah | Synergistic convergence of microbiota--specific systemic IgG and secretory IgA[END_REF] . Additionally, higher serum anti--E. coli IgG levels in patients with IBD were associated with a substantial increase of gut IgG + B cells [START_REF] Benckert | The majority of intestinal IgA+ and IgG+ plasmablasts in the human gut are antigen--specific[END_REF] . In mice, mesenteric lymph nodes have also been implicated in generating systemic IgG responses to invasive bacteria [START_REF] Martinoli | Entry Route of Salmonella typhimurium Directs the Type of Induced Immune Response[END_REF] .

Together, these data suggest that intestinal IgG + B cells might contribute to the serum IgG repertoire in humans (Figure 1). Recently, Wilmore et al. demonstrated that members of gut microbiota enhance serum IgA levels and induce large numbers of IgA-secreting cells in the bone marrow [START_REF] Wilmore | Commensal Microbes Induce Serum IgA Responses that Protect against Polymicrobial Sepsis[END_REF] . However, the existence of such a gut/bone marrow axis supporting the production of anti--commensal IgG remains to be proved both in humans and in mice.

A raison d'être for anti--IgG microbiota

The functional relevance of systemic anti--commensal IgG has been first revealed by producing evidence for their protective role in gastrointestinal and systemic infections. Mice lacking the innate immune signaling components MyD88 and TRIF spontaneously produce very high levels of microbiota--targeting IgG. Gnotobiotic colonization demonstrated that these responses were T cell--dependent and, at least partly, capable of providing protection from sepsis [START_REF] Zeng | Gut Microbiota--Induced Immunoglobulin G Controls Systemic Infection by Symbiotic Bacteria and Pathogens[END_REF][START_REF] Slack | Innate and adaptive immunity cooperate flexibly to maintain host-microbiota mutualism[END_REF] . The clinical relevance of anti-microbiota IgG is consistent with observations that patients with CVID, lacking circulating IgG, but not intestinal IgA, nevertheless suffer from gastrointestinal infections and lymphoproliferative disorders [START_REF] Agarwal | Diagnosis and Treatment of Gastrointestinal Disorders in Patients With Primary Immunodeficiency[END_REF][START_REF] Oksenhendler | Infections in 252 patients with common variable immunodeficiency[END_REF] .

Anti--commensal IgG also play a role in early life in modulating the development of the neonatal immune system, with both stimulatory and regulatory effects. Maternal IgG is transported to the neonate across the placenta via the neonatal Fc receptor (FcRn) [START_REF] Leach | Isolation from human placenta of the IgG transporter, FcRn, and localization to the syncytiotrophoblast: implications for maternal--fetal antibody transport[END_REF] that also mediates neonatal uptake of maternal milk--ingested IgG into the small intestine [START_REF] Roopenian | FcRn: the neonatal Fc receptor comes of age[END_REF][START_REF] Pyzik | The Architect Behind the Immune and Nonimmune Functions of IgG and Albumin[END_REF] . Transfer of IgG during gestation drives the development of intestinal NKp46 + innate lymphoid cells (ILC3) and causes changes in the intestinal gene expression profile, in particular genes involved in mucus secretion and metabolism of sugar or dietary xenobiotics whose expression is upregulated in an antibody--dependent manner. Moreover, the study of Gomez de Agüero et al. indicated that microbial products from the maternal microbiota, likely bound by IgG, were transferred to the neonate before birth. Thus, IgG appear to prepare the neonate in utero to the massive bacterial colonization occurring after birth [START_REF] Gomez De Agüero | The maternal microbiota drives early postnatal innate immune development[END_REF] . The specificity of these IgG transferred to the neonate remains to be defined, but it is tempting to speculate that anti--commensal IgG is an important component.

Anti--commensal IgG ingested from maternal milk, have also been found to have regulatory effects on the neonatal immune system. Maternal milk contains both secretory IgA and IgG that cooperate to discourage bacterial translocation and thereby limiting mucosal T cell responses. In mice lacking maternal antibodies, an increased frequency of T follicular helper cells and germinal center B cells has been described, resulting in an increased antibody production aimed to restore homeostasis. The absence of either IgA or IgG during suckling is sufficient to fuel aberrant immune responses in gut mucosa [START_REF] Koch | Maternal IgG and IgA Antibodies Dampen Mucosal T Helper Cell Responses in Early Life[END_REF] .

In summary, systemic anti--commensal IgG (i) provide a pre--existing set of antibodies against a variety of pathogens and (ii) might play a critical role in modulating immune system development in early life. Although these findings raise exciting possibilities in terms of novel therapeutic strategies, human data are still lacking. At present, multiple clinical trials have failed to demonstrate consistent beneficial effects of pooled human IgG (IVIGs) in sepsis, suggesting that IVIGs do not contain the appropriate specificities [START_REF] Almansa | The original sins of clinical trials with intravenous immunoglobulins in sepsis[END_REF] .

Anti--commensal IgG and cancer

Current evidence supports a model in which some (potentially invasive) commensals are able to induce and IgG responses in the context of a healthy microbiota and an intact intestinal barrier. Other microbes however, might only induce systemic responses in the context of dysbiosis or intestinal barrier disorders. This hypothesis has been extensively studied in IBD patients who presented elevated and biased serum anti-commensal IgG responses [START_REF] Landers | Selected loss of tolerance evidenced by Crohn's disease-associated immune responses to auto--and microbial antigens[END_REF][START_REF] Macpherson | Mucosal antibodies in inflammatory bowel disease are directed against intestinal bacteria[END_REF][START_REF] Beaugerie | Clinical, serological and genetic predictors of inflammatory bowel disease course[END_REF][START_REF] Targan | Antibodies to CBir1 Flagellin Define a Unique Response That Is Associated Independently With Complicated Crohn's Disease[END_REF] . More recently, we reported that sIgAd patients have a higher level of anti--commensal IgG, suggesting that bacterial translocation may occur more frequently in the absence of secretory IgA. Moreover, sIgAd patients (SIgAd) harbor a different set of anti--commensal IgG, as compared to healthy donors which is consistent with the presence of a mild dysbiosis [START_REF] Fadlallah | Microbial ecology perturbation in human IgA deficiency[END_REF] . Greater alterations of gut microbiota composition and diversity have been described in fecal samples from CVID patients, lacking both IgA and IgG [START_REF] Jørgensen | Altered gut microbiota profile in common variable immunodeficiency associates with levels of lipopolysaccharide and markers of systemic immune activation[END_REF] . In line with these findings, pooled IgG from healthy donors (IVIGs) poorly bound to fecal bacteria from CVID patients, meaning that these antibodies do not contain appropriate specificities of anti--commensal IgG for the treatment of dysbiotic patients [START_REF] Fadlallah | Synergistic convergence of microbiota--specific systemic IgG and secretory IgA[END_REF] .

In colorectal cancer (CRC), changes in the intestinal tissue result in a leaky gut, with potential aberrant bacterial excursions [START_REF] Brenner | Colorectal cancer[END_REF] . It has also been reported that colorectal CRC is associated with increased prevalence of Fusobacterium nucleatum, a bacteria species with oncogenic potential [START_REF] Rubinstein | Fusobacterium nucleatum promotes colorectal carcinogenesis by modulating E--cadherin/β--catenin signaling via its FadA adhesin[END_REF][START_REF] Kostic | Fusobacterium nucleatum Potentiates Intestinal Tumorigenesis and Modulates the Tumor--Immune Microenvironment[END_REF][START_REF] Brennan | Fusobacterium nucleatum -symbiont, opportunist and oncobacterium[END_REF] . A prospective study recently associated a commensal candidate, Streptococcus gallolyticus subspecies gallolyticus (SGG) with the risk to develop CRC, since anti--SGG IgG were more frequently present in individuals who developed cancer later in life (Table 2) [START_REF] Butt | Prospective evaluation of antibody response to Streptococcus gallolyticus and risk of colorectal cancer[END_REF] .

Although a causative link between the presence of such species and CRC has not been demonstrated, certain anti--commensal IgA [START_REF] Wang | Evaluation of antibody level against Fusobacterium nucleatum in the serological diagnosis of colorectal cancer[END_REF] or IgG [START_REF] Butt | Prospective evaluation of antibody response to Streptococcus gallolyticus and risk of colorectal cancer[END_REF] reactivities might constitute potential diagnostic biomarkers for CRC. The detection of anti--commensal IgG might also offer novel opportunities for predicting immunotherapy efficacy in the treatment of cancers. The dominance of distinct commensal species in gut microbiota such as A.

muciniphila and E. hirae, has been shown to drive immunological changes that influence the anti--cancer efficacy of blockade of the PD1/PDL1 pathway in epithelial tumors [START_REF] Routy | Gut microbiome influences efficacy of PD--1--based immunotherapy against epithelial tumors[END_REF] .

Screening IgG against these specific microbes before treatment could be more convenient than an extensive analysis of stool composition in an attempt to predict resistance to these and other immune checkpoint inhibitors.

Anti--commensal IgG and autoimmunity

For years, the molecular mimicry theory has supported the view that homologies between microbial components and human proteins are likely to take part in the development of autoimmune diseases. This dogma was first supported by studies showing associations between pathogens and the risk to develop autoimmunity, such as Campylobacter jejuni and Guillain--Barré syndrome or Streptococcus pyogenes and juvenile dermatomyositis [START_REF] Cusick | Molecular Mimicry as a Mechanism of Autoimmune Disease[END_REF][START_REF] Massa | Self epitopes shared between human skeletal myosin and Streptococcus pyogenes M5 protein are targets of immune responses in active juvenile dermatomyositis[END_REF][START_REF] Jacobs | Campylobacter jejuni infections and anti--GM1 antibodies in Guillain--Barré syndrome[END_REF] . A few recently published studies also aimed at linking the presence of commensals and anti--commensal IgG with the development of certain autoimmune diseases. Recently, Azzouz et al. found that active SLE patients displayed a fecal expansion of Ruminococcus gnavus strain associated with high and specific systemic IgG responses towards this species, suggesting that some specific components of R. gnavus are nephritogenic by molecular mimicry (Table 2) [START_REF] Azzouz | Lupus nephritis is linked to disease--activity associated expansions and immunity to a gut commensal[END_REF] . The magnitude of serum antibodies titers directed against Enterococcus gallinarum was correlated with that of anti--RNA antibodies in serum from patients with systemic lupus erythematosus (SLE) [START_REF] Vieira | Translocation of a gut pathobiont drives autoimmunity in mice and humans[END_REF] . Greiling et al. also proposed that commensal bacteria expressing Ro60 orthologs might induce hRo60 cross--reactive antibodies and therefore trigger auto--immunity in lupus [START_REF] Greiling | Commensal orthologs of the human autoantigen Ro60 as triggers of autoimmunity in lupus[END_REF] . To test this hypothesis, the authors monocolonized lupus--prone mice with Ro60--expressing bacteria i.e Bacteroides thetaiotaomicron and observed increased autoantibody production and lupus--like nephritis. However, most microbes can induce strong immune responses in monocolonized gnotobiotic mice [START_REF] Geva--Zatorsky | Mining the Human Gut Microbiota for Immunomodulatory Organisms[END_REF] , likely representing an effect of monocolonization due to overabundance of a single species, or immaturity of the germ--free mucosal system on initial encounter with the microbe. Moreover, autoimmune prone nonobese--diabetic (NOD) mice that were monocolonized with B. thetaiotaomicron actually carried hRo60 cross--reactive antibodies but did not develop autoimmune symptoms, indicating that triggering autoimmunity might be the consequence of a much more complex interplay between commensals, host genetics and environmental factors.

Another paradigm, that links anti--commensal IgG responses and immune-mediated diseases is represented by the so--called hygiene hypothesis 100-102 . Christmann et al. associated lower anti--commensal IgG levels during infancy with the risk to later--on eventually develop allergy, suggesting that strong early life anti--microbiota immune responses might be protective against the subsequent development of immune-mediated diseases (Table 2) [START_REF] Christmann | Human seroreactivity to gut microbiota antigens[END_REF] . Likewise, early colonization by microbes, like Bacteroides sp., that potently activate immune pathways plays an important role in immune education and decreased incidence of autoimmune diseases 103 . Of note, such induced beneficial immune deviation effects would not be mediated predominantly by pathogens, but rather result from exposure to diverse commensal consortia 104,105 .

Outlook

The sequencing of immunoglobulin--bound microbiota fractions from both healthy humans and patient populations has provided insight in the nature of bacterial interactions with the host immune system. We and others have noted interesting overlaps and distinctions between IgA, IgM and IgG reactivities which may hint at both differential induction pathways and differential functions. For example, bacterial strains showing strong reactivity with all three isotypes in healthy donors may be particularly strong inducers of humoral immunity. However, in patients with sIgAd there is a counterintuitive (paradoxical) absence of compensatory secretory IgM directed against Gammaproteobacteria, despite evident overgrowth of this gram--negative class of bacteria, suggesting an, as yet, unknown requirement for the induction of sIgM. Another challenge remains in interpreting these results with respect to microbiota control. The mechanisms by which IgA can selectively disadvantage a bacterial species in the gut lumen are well--defined, but while a range of hypotheses exist, there is still limited evidence for pro--colonization mechanisms. Suggestions currently include the generation of mucus--embedded microcolonies or altered bacterial metabolism. Highly relevant questions also remain regarding the function of anti--commensal Ig in health and disease.

Currently, the relatively mild clinical phenotype associated with IgA deficiency is difficult to reconcile with the critical role played by IgA in disease protection in many animal models [START_REF] Bioley | Plasma--Derived Polyreactive Secretory--Like IgA and IgM Opsonizing Salmonella enterica Typhimurium Reduces Invasion and Gut Tissue Inflammation through Agglutination[END_REF][106][107][108] . Further, microbiota robustness is emerging as an important parameter for health in which secretory antibodies could play both beneficial and deleterious roles. Perturbation/recovery studies, as well as hypothesis--driven research in experimental model systems, will be required to address these major questions. We would propose these as priorities, necessary to open up the major untapped potential of secretory antibodies in modulating microbiota for diagnostic and therapeutic purposes. -sIgA and sIgMs are present in decreasing concentrations from mucus to lumen [START_REF] Fadlallah | Microbial ecology perturbation in human IgA deficiency[END_REF][START_REF] Magri | Human Secretory IgM Emerges from Plasma Cells Clonally Related to Gut Memory B Cells and Targets Highly Diverse Commensals[END_REF][START_REF] Bunker | Innate and Adaptive Humoral Responses Coat Distinct Commensal Bacteria with Immunoglobulin A[END_REF][START_REF] Kau | Functional characterization of IgA--targeted bacterial taxa from undernourished Malawian children that produce diet--dependent enteropathy[END_REF][START_REF] Palm | Immunoglobulin A coating identifies colitogenic bacteria in inflammatory bowel disease[END_REF] -Although IgG are absent from the healthy human gut, IgG plasmocytes are found in the in the lamina propria [START_REF] Rollenske | Cross--specificity of protective human antibodies against Klebsiella pneumoniae LPS O--antigen[END_REF] and likely contribute to systemic anti--commensal IgG pools, whose main origin remains unclear.

  . The potential role of such specificities in anti-commensal and cross--reactive autoimmunity is extensively reviewed in this issue (REF O. Pabst and E. Slack).

Figure 1 :

 1 Figure 1 : Secretory and systemic Ig work in concert to control gut microbiota -sIgA, sIgM, and serum IgG converge toward the same microbial targets 4,62 .

Acknowledgments

We thank members of the Gorochov and Slack labs for discussions and H. Yssel for critical reading of the manuscript. DS was supported for this work by a Pasteur/APHP interface fellowship.

Conflict of Interest

The authors declare that they have no conflicts of interest.