Niccoló Di

Andrea Frosini
email: andrea.frosini@unifi.it

William Lawrence Kocay
email: bkocay@cs.umanitoba.ca

Lama Elisa Pergola
email: elisa.pergola@unifi.it

Lama Tarsissi
email: lama.tarsissi@esiee.fr

The null labelling of k-uniform hypergraphs and the complexity of the 2-intersection graph decision problem

Keywords: MSC Classification: 05C65, 05C22, 05C85, 05C99

come

The null labelling of k-uniform hypergraphs and the complexity of the 2-intersection graph decision problem

Introduction

The degree sequence of a graph or hypergraph provides useful information about its geometrical and topological properties. When a graph or hypergraph has been constructed for a given degree sequence, we say that it has been reconstructed. A longstanding problem concerns the use of degree sequences for the reconstruction of the related graphs or hypergraphs. In 1960, in a milestone paper, Erdös and Gallai [START_REF] Erdös | Graphs with prescribed degrees of vertices (Hungarian)[END_REF] characterized the degree sequences of simple graphs and set the basis for the successive definition of fast reconstruction algorithms.

These results can not be easily generalized to hypergraphs, even in the simplest form of the k-uniform ones, with k > 2. In recent years, many different reconstruction strategies have been considered, most of them relying on a general result of Dewdney in [START_REF] Dewdney | Degree sequences in complexes and hypergraphs[END_REF], concerning relevant sub-classes of k-uniform hypergraphs (see [START_REF] Behrens | New Results on Degree Sequences of Uniform Hypergraphs[END_REF][START_REF] Brlek | A Tomographical Interpretation of a Sufficient Condition for h-Graphical Sequences DGCI[END_REF][START_REF] Frosini | On the degree sequences of uniform hypergraphs Lect[END_REF][START_REF] Frosini | New sufficient conditions on the degree sequences of uniform hypergraphs Theoret[END_REF]).

Recently, Deza et al. proved the N P -completeness of the reconstruction problem of 3-uniform hypergraphs from their degree sequences. So, the study of the classes of hypergraphs whose reconstruction can be performed in polynomial time becomes more relevant, as does the characteristics of the solutions related to a single degree sequence.

As an example, in [START_REF] Kocay | On 3-Hypergraphs with Equal Degree Sequences[END_REF], the authors defined a general operator to move among all the 3-hypergraphs sharing the same degree sequence and they introduced the notion of null-labelling obtained as the symmetric difference of the edges of any two 3-hypergraphs. In [START_REF] Frosini | On null 3-hypergraphs[END_REF], this notion, restricted to 3-hypergraphs, is studied through their intersection graph and a sufficient condition for its existence was provided.

Finally, in [START_REF] Marco | A Study on the Existence of Null Labelling for 3-Hypergraphs IWOCA[END_REF], it was shown that as the number of edges of the related 3hypergraph increases, the usefulness of the intersection graph decreases, and the stronger notion of 2-intersection graph was introduced. In particular, the presence of an Hamiltonian cycle in the 2-intersection graph guarantees the existence of, and allows one to compute, the null labelling of the related 3hypergraph.

Our study relies on these results and aims to deepen the knowledge of the structural properties of 2-intersection graphs in order to enhance their use for the null labelling problem. We are concerned with the question of when, given a graph G, it is possible to find a 3-hypergraph H such that I 2 (H) ∼ = G. If it is possible, we say that G is reconstructible or equivalently, it has the 2-intersection property. In the final section of the paper we prove that the reconstruction problem is N P -complete.

Definitions and preliminary results

Let us recall some basic definitions for graph and hypergraphs, borrowing the notation from [START_REF] Berge | Hypergraphs[END_REF]. A graph G is defined as a pair G = (V, E) such that V = {v 1 , . . . , v n } is the set of vertices and E is a subset of pairs of vertices called edges. The notion of graph admits a natural generalization to hypergraph by removing the constraint on the cardinality of the edges: H = (V, E) is a hypergraph with V = {v 1 , . . . , v n } the set of vertices and E a collection of subsets of vertices called hyperedges, or briefly edges, when no misunderstanding may arise. We also choose to abbreviate the vertex notation v i with its only index i, when possible.

In the sequel, we will consider only graphs and hypergraphs that are simple, i.e., they do not allow singleton (hyper)edges or (hyper)edges that are contained in or equal to other edges. The degree of a vertex v ∈ V is the number of (hyper)edges containing v. The degree sequence d = (d 1 , d 2 , . . . , d n ) of a (hyper)graph is the list of its vertex degrees usually arranged in non-increasing order. We focus on a remarkable class of hypergraphs, i.e., k-uniform hypergraphs or simply k-hypergraphs, whose hyperedges have fixed cardinality k.

In particular we are going to consider 3-hypergraphs.

Given a (hyper)graph, we can assign a +1 or -1 label to each (hyper)edge, resulting in positive and negative degrees of the vertices.

An assignment of ±1 to the (hyper)edges of a (hyper)graph is a null labelling if, for each vertex v, the sum d(v) of the labels of its incident edges equals 0. A (hyper)graph that admits a null labelling is said to be a null (hyper)graph.

The study of null (hyper)graphs is strictly related to their reconstruction from the degree sequences. Let H 1 = (V, E 1 ) and H 2 = (V, E 2 ) be two hypergraphs with the same vertex set V , and same degree sequence (d 1 , d 2 , . . . , d n ). Assign +1 to the triples of H 1 and -1 to the triples of H 2 , and construct

H 1 ⊕H 2 = (V, (E 1 ∪E 2 )\(E 1 ∩E 2 )).
It is a hypergraph with a null labelling. This raises the question of whether there is a characterization of null (hyper)graphs. Concerning graphs, the following lemma characterizes those with null labelling. On the other hand, for k-hypergraphs with k ≥ 3, it is shown in [START_REF] Frosini | On null 3-hypergraphs[END_REF] that the problem of finding a null labelling even for the simplest case of 3-hypergraphs is NP-hard.

An obvious necessary condition for a (hyper)graph to have a null labelling is that each vertex must have even degree, i.e., it is an even (hyper)graph.

So, the characterization of subclasses of 3-hypergraphs having a null labelling acquires relevance. In [START_REF] Frosini | On null 3-hypergraphs[END_REF], an exhaustive computation of the simplest cases supported the following The role of 2-intersection graph in the hypergraph null labelling Conjecture 1. All connected even 3-hypergraphs on n vertices with more than 1 2 n 3 edges have a null labelling.

Note that n

3 is the maximum number of edges a (simple) 3-hypergraph on n vertices admits.

A valuable tool to inspect the characteristics of hypergraphs in general and, in our framework of 3-hypergraphs, is their intersection graph. The intersection graph of a 3-hypergraph H is denoted I(H), its vertices are the hyperedges of H and two hyperedges are adjacent, i.e., they are joined by an edge, if their intersection is non-empty. This is an extension of the idea of a line graph to 3-hypergraphs. In [START_REF] Frosini | On null 3-hypergraphs[END_REF], the following result has been proved

Theorem 1 Let H be a connected, even 3-hypergraph, in which every vertex has degree two. Then H has a null labelling if and only if I(H) is bipartite.

However, the intersection graphs dramatically collapse into complete graphs when the number of edges of the related 3-hypergraph increases, so in [START_REF] Marco | A Study on the Existence of Null Labelling for 3-Hypergraphs IWOCA[END_REF] the notion of 2-intersection graph was introduced. The 2-intersection graph of a 3-hypergraph H is denoted I 2 (H) = (V 2H , E 2H ). Its vertices V 2H = {v e1 , . . . , v em } represents the hyperedges E = {e 1 , . . . , e m } of H. Two vertices v ei , v ej ∈ V 2H are adjacent, i.e., {v ei , v ej } ∈ E 2H , if the related hyperedges e i and e j share two vertices in H (see Example 1 and the related Fig. 1). In the sequel, we label the edge {v ei , v ej } ∈ E 2H with the indices of the vertices that are shared by e i and e j , if needed.

The idea of the 2-intersection graph is related to Steiner triple systems, in which any two triplets can intersect in at most one vertex. In the diagrams following, a 3-hypergraph is represented by its Levi graph, that is, a bipartite graph of vertices versus hyperedges, in which each hyperedge is represented by a vertex shaded grey, and of degree three.

Example 1. Consider the 3-hypergraph H = (V, E), with V = (v 1 , . . . , v 5 ) and E = {{v 1 , v 3 , v 5 }, {v 2 , v 3 , v 5 }, {v 2 , v 3 , v 4 }, {v 3 , v 4 , v 5 }} as depicted in Fig. 1, top (a) (vertex v i is indicated by its index i). The related I 2 (H) is in Fig. 1, bottom (a). It has | E |= 4 vertices, labelled by the corresponding triplets of the hyperedges of H, and five edges, labelled by the pair of vertices they contain. Furthermore, the common label {3, 5} is assigned to the gray triangle clique.

The following property is straightforward: Property 1. The edges of a 3-hypergraph H sharing the same pair of vertices determine a clique in the 2-intersection graph I 2 (H).

In the sequel, we extend the common label of the edges of a clique of a 2-intersection graph to label the clique itself.

Since each edge of H has three pairs of vertices to share, the following holds Property 2. Given a 3-hypergraph H, each vertex of I 2 (H) belongs to at most three maximal cliques.

Our main focus is to study the properties of the 2-intersection graph and its connection with the null labelling problem. Moreover, we prove that, given a graph G, deciding if G is the 2-intersection graph of some 3-hypergraph is an NP-complete problem. Therefore, we consider the following questions.

Question 1: is there a subclass of null 3-hypergraphs whose null labelling relies on properties of the 2-intersection graphs (recall from [START_REF] Frosini | On null 3-hypergraphs[END_REF] that the problem is NP-complete in its general form)?

Question 2: given a graph G, is it possible to decide in polynomial time whether there exists a 3-hypergraph H such that G ∼ = I 2 (H)?

The next section presents a brief review of [START_REF] Marco | A Study on the Existence of Null Labelling for 3-Hypergraphs IWOCA[END_REF], i.e. Question 1, while Section 4 will answer Question 2 by providing a proof of its NP-completeness.

A subclass of 3-hypergraphs having null labelling

Recall that in [START_REF] Marco | A Study on the Existence of Null Labelling for 3-Hypergraphs IWOCA[END_REF] the authors proved the following theorem. The proof is based on an iterative algorithm that, moving along the vertices of the 2-intersection graph, reduces step by step the signed degree of the vertices, considered by absolute value. The presence of an Hamiltonian cycle guarantees that all the vertices can be reached by the algorithm.

Define the n-intersection graph of a k-hypergraph H as the graph G in which the vertices are the hyperedges of H and two vertices form an edge if their hyperedges share at least n vertices, and denote it by I n (H). The following obvious generalization of Theorem 2 holds.

Corollary 1 Let H be a k-hypergraph. If I k-1 (H) is Hamiltonian, then H admits a null labelling.
Theorem 2 is a step towards proving Conjecture 1. However, an easy check reveals that not all the 3-hypergraphs with more than half the possible edges and with a null labelling, have a Hamiltonian cycle in I 2 (H). On the other hand, after generating several 3-hypergraphs on 6, 7 and 8 vertices, we noticed that the connectedness of the graph I 2 (H) seems to be related to the presence of a null labelling in H. This property could extend the results in Theorem 2. Unfortunately, successive computations show that I 2 (H) being connected is not sufficient to provide a null labelling of H as shown by the following Example 2. Consider the following even 3-hypergraph

H = {159,
[1, 1, 1, 1, 1, -1, -1, 1, 1, -1, -1, -1, -1, -1]
However it has a disconnected I 2 (H).

Next, we show an extension of Theorem 2. The idea comes from the following observation: consider an even graph G with n connected components such that each component has an even number of edges. We can find a null labelling of G by considering, for each component, an Eulerian cycle, assigning alternatively ±1 to its edges. Note that, if we have an intersection graph I(G) with n components, each having an Hamiltonian cycle, in the same fashion, we can alternately assign ±1 to the vertices of each Hamiltonian cycle to obtain a null labelling. This same property holds for 3-hypergraphs.

Theorem 3 Let H be a 3-hypergraph whose 2-intersection graph G = I 2 (H) has n connected components G 1 , . . . , Gn. If each connected component G i , with 1 ≤ i ≤ n, is Hamiltonian, and the sub-3-hypergraph H i ⊂ H such that G i = I 2 (H i ) is even, then H admits a null labelling.
Proof Since each G i is Hamiltonian and comes from an even sub-3-hypergraph H i , we can apply Theorem 2 to it, obtaining a null labelling of H i . The null labelling of H is obtained by the union of the null labellings of these sub-hypergraphs.

Note that H may be also a connected 3-hypergraph, but the requirement of the previous theorem asks that its 2-intersection graph has to be disconnected. Furthermore, each of its components comes from an even sub-3-hypergraph of H.

The 2-Intersection-Graph Problem

In this section we provide some structural properties of 2-intersection graphs that are related to 3-hypergraphs, with the aim of characterizing them and determining the computational complexity of this problem.

Recall that a graph G has the 2-intersection property if it is the 2intersection graph of a 3-hypergraph H.

So, we approach the problem of the existence of a 3-hypergraph H whose 2-intersection graph is G, i.e., G ∼ = I 2 (H), by first observing that Property 1 does not characterize the cliques of I 2 (H). In fact, there may appear triangles (K 3 cliques) whose edges do not share a common label. In order to distinguish them from the triangles whose edges have a common label, we indicate the first and the second types as T -triangles and K-triangles, respectively.

More precisely, the vertices {v 1 , v 2 , v 3 } of a T -triangle have the general form v 1 = {x, y, z}, v 2 = {x, y, t}, and v 3 = {x, z, t} -only four vertices of H are used; while those of a K triangle (with common label {x, y}) have the form v 1 = {x, y, z}, v 2 = {x, y, t}, and v 3 = {x, y, r} -five vertices of H are used.

Figure 1, (a), shows a configuration of a 2-intersection graph where a Ttriangle and a K-triangle live together and share an edge. The T -triangle is shaded gray. With the previous definitions, a first obvious necessary condition for a graph to be reconstructible is described in the following lemma.

t1t2.pdf

Lemma 2 If G is reconstructible, then every vertex of G must belong to at most three maximal cliques, such that the vertices of each of them have in common the same couple of labels. Furthermore, these maximal cliques can be joined by at most two non-adjacent T -triangles as shown in Fig. 2.

Proof A clique in I 2 (H) can derive from a shared pair or it can be a triangle (in which two pairs are used). In all cases, since a vertex of I 2 (H) has only three pairs, it cannot belong to more than three cliques. However, note that the configuration in which a point is shared by four triangles is reconstructible (see Fig. 2). The role of 2-intersection graph in the hypergraph null labelling The reason is simple: the edges of the T triangles shared with the K triangles count as a unique pair containing the central point. Therefore, it shares exactly three pairs, as desired. Note that the configuration is reconstructible even if the K triangles are replaced by any clique of order at least four.

We encounter a similar situation when dealing with K 4 cliques: Figure 1,(b), shows a clique that may be present in a 2-intersection graph and whose edges do not share a common label. This clique is composed of four intersecting T -triangles. It arises from four vertices of H for which all possible triples occur as hyperedges. So, maintaining the notation introduced for triangles, we denote such a square by S, and we differentiate it from the K-square, i.e., the 4-clique whose edges all have a common label.

Property 3. Let C 1 and C 2 be two cliques of I 2 (H). If | C 1 ∩ C 2 |≥ 3 then C 1 = C 2 . Proof Let C = {v 1 , v 2 , v 3 } ⊆ C 1 ∩ C 2 .
We proceed by contradiction assuming that there exist two non adjacent vertices x ∈ C 1 and y ∈ C 2 such that C 1 = C ∪ x and C 2 = C ∪ y are two different cliques. The following two cases arise (see cases (a) and (b) in Fig. 1).

C is a T -triangle: we suppose w.l.g. v 1 = {1, 2, 3}, v 2 = {1, 2, 4}, and v 3 = {1, 3, 4}. Since C 1 is a clique, then x = {2, 3, 4} is the only possibility for x. The same reasoning holds for y, so it follows x = y, against the hypothesis.

C is a K-triangle: we suppose w.l.g. v 1 = {1, 2, 3}, v 2 = {1, 2, 4}, and v 3 = {1, 2, 5}. Both x and y have to contain the pair {1, 2}, shared by all the elements of C, so there exists an edge joining them, against the hypothesis.

In the proof of Property 3, we note that if C is a K-triangle, then only two of its elements are enough to determine the common pair {1, 2}, so that:

Corollary 2 Let C 1 and C 2 be two cliques of I 2 (H) not containing any T -triangle. If | C 1 ∩ C 2 |≥ 2 then C 1 = C 2 . Corollary 3 Let C 1 and C 2 be two (maximal) cliques of I 2 (H). If | C 1 ∩ C 2 |= 2, then either C 1 or C 2 , but not both, are T -triangles.
We emphasize that, if both C 1 and C 2 are T -triangles such that | C 1 ∩C 2 |= 2, then they are not maximal, i.e., they form an S-square as in Fig. 1,(b).

The following example shows that S is the biggest clique that does not admit a common pair in all of its edges.

Example 4. Consider a clique K 5 of a 2-intersection graph. It is not possible to label its vertices without a common pair. In fact, let us start by labelling a vertex with {1, 2, 3} (see Fig. 3) and one of its neighbours with {1, 2, 4}. Suppose that we label a third vertex with a pair different from {1, 2}, say {1, 3, 4}. Then we have only one label possible for the fourth vertex, namely {2, 3, 4}. We observe that there does not exist any label for the fifth vertex, since all possible pairs have been already used. So a labelling of the vertices of K 5 without a common pair is not possible. The same argument can be used to prove that any clique of order greater than four has the edges labelled with the same pair of indices. Many other conditions can help to restrict the cases when G is a 2-intersection graph.

Theorem 4 Let G be a graph containing two distinct cliques of order n and m and intersecting in 2 ≤ i ≤ min(m, n) -1 vertices. Moreover, assume that the condition of Lemma 2 holds. Then the following statements hold: (i) if n = 3 and m ≥ 3, then G is reconstructible;

(ii) let n = 4 and m ≥ 4, then G is reconstructible if and only if i = 2;

(iii) if n > 4 and m > 5, then G is not reconstructible.

Proof (i): by hypothesis, i = 2, so the two cliques must intersect in two vertices. From Property 3 and the corollaries, a clique Km, with m ≥ 3 shares a common edge, say with label {x, y}, with a T -triangle. We proceed in labelling the The role of 2-intersection graph in the hypergraph null labelling three vertices of T by {x, y, z 1 }, {x, y, z 2 }, {x, z 1 , z 2 }. Finally, we assign the labels {x, y, z 3 }, {x, y, z 4 }, . . . , {x, y, zm} to the remaining vertices of Km. The labels turn out to be the hyperedges of a 3-hypergraph H on {x, y, z 1 , . . . , zn} vertices such that I 2 (H) = G, as desired.

(ii): from Property 3 and corollaries, if i = 2, then the edge shared by the two cliques must intersect a triangle T . Let T be contained in the first clique whose configuration turns out to be S. The other clique is forced to have a common label of the edges, since it can not contain a second T -triangle. So, a labelling similar to that defined in (i) is allowed, providing a 3-hypergraph whose 2-intersection graph is G.

Finally, the case i = 3 causes the collapse of the two cliques into one single clique.

(iii): consider two cliques of order greater than four. Property 3 states that if they share two or more vertices, then they are the same clique, so G is not a 2intersection graph.

The above result can be naturally extended to graphs having more than two cliques, by inspecting the intersections between pairs of them. A labelling of the edges and the determination of a 3-hypergraph compatible with them can be performed.

However, there are several classes of graphs that can be immediately detected as 2-intersection graphs due to their geometric properties. For them, some simple edge labellings lead to the determination of a related 3-hypergraph. One class is the cycle graphs, i.e. a sequence of consecutive 2-cliques.

First, we observe that a cycle of length k admits a labelling involving k integer indices at most. In fact, we start by labelling a randomly chosen vertex v 1 with {1, 2, 3}, then we visit the remaining vertices v 2 , . . . , v k of the cycle according to a chosen direction and we label the i th one, where 1 < i < k -1, with {i, i + 1, i + 2}, so that {i, i + 1} is the pair shared by v i-1 and v i , i.e., the label of the edge joining them. Finally, vertices v k-1 and v k are labelled with {1, k, k + 1} and {1, 2, k}, respectively. An example is shown in Fig. 4.

Note that the labelling of the first k -1 vertices of the cycle defined above can be used to label a path of length k -1 involving k integer indices.

{1,2,3}

{2,3,4} {3,4,5} {4,5,6} {1,5,6} {1,2,6} Fig. 4 The labelling of a cycle of length 6.

Deciding if a graph has the 2-intersection property is N P -hard

In this section it is shown that it is N P -complete to determine whether an arbitrary graph G is the 2-intersection graph of a 3-hypergraph. We refer to this problem as then 2INT problem. We reduce from the following variant of the 3-SAT problem (L02 in [START_REF] Garey | Computers and Intractability -A Guide to the Theory of NP-Completeness[END_REF])

MAX-3-SAT:

Instance: a set U of variables and a set C of clauses over U such that each clause c ∈ C has | c |= 3 literals, where a literal corresponding to a variable x is either x or x. Each variable, as represented by a literal, appears at most three times in the clauses. Furthermore, no three occurrences of a variable are the same literal. Question: Is there a satisfying truth assignment for C? First of all, note that this problem is N P -complete.

Lemma 3 MAX-3-SAT is NP-complete.
Proof We reduce 3-SAT to MAX-3-SAT. Consider an instance of 3-SAT, in which a variable x appears k times, either as x or x. Replace the first instance with x 1 , the second with x 2 and so on up to x k (possibly the complements). Then add (x 1 ∨ x 2 ) ∧ (x 2 ∨ x 3 ) . . . (x k ∨ x 1 ). This ensures that either all x i are true, or all are false. Note that each x i has two literals occurring in this expression, and one more literal occurring in the clauses of 3-SAT, giving three occurrences in total. Doing the same for the others literals we obtain an instance of 3-SAT, but in which each variable appears at most three times, and no three occurrences of a variable are the same literal.

Given an instance C of MAX-3-SAT, we construct a graph G C so that there is a solution of the MAX-3-SAT instance if and only if G C is a 2intersection graph. This will imply that 2INT is NP-complete. We remark that each solution of C will be achieved by one of the possible labellings of G C that determines a 3-hypergraph H such that G = I 2 (H).

To achieve the objective, we need to define some graph configurations that have the 2-intersection property and that are useful to model variables and clauses of 3-SAT. Each vertex of a 2-intersection graph G corresponds to a hyperedge of its related hypergraph H. So each vertex v of G contains three vertices of H. In order to avoid confusion with the use of the word vertex, we will use the term indices for the vertices of H contained in a vertex of G. When a hypergraph H is constructed from a 2-intersection graph G, by assigning indices to the vertices of G, we say that a label is assigned to v, eg., we assign {1, 2, 4} as the label of a vertex of G. In the following properties, we consider subgraphs of G determined by triangles, and show that when G is a The role of 2-intersection graph in the hypergraph null labelling 2-intersection graph, the labels of G have to satisfy certain requirements, i.e., local properties of G extend to properties of H. Property 4. If two triangles intersecting in one vertex have the 2-intersection property, and there are no edges joining the triangles, then they are not both T -triangles.

Proof Observe that the three vertices of a T -triangle use just four different indices from H. If the two triangles are T -triangles, then the common vertex has to share four different pairs of indices with the four vertices with which it is adjacent, and this is not possible if there are no edges joining the triangles (see Fig. 5).

We will call two triangles sharing one vertex, with no edges between them, a ribbon configuration. Figure 5 shows the labels of two one-vertex intersecting triangles when at least one of them is a K-triangle. Property 5. Let T 1 and T 2 be two triangular cliques. Suppose there are just two edges joining a vertex of T 1 to a vertex of T 2 , and that these edges have no common endpoints. Then the obtained configuration has the 2-intersection property. Furthermore, T 1 and T 2 cannot both be T -triangles. It is possible for T 1 and T 2 to be both K-triangles.

Proof Let x 1 and y 1 be the vertices of T 1 adjacent to the vertices x 2 , and y 2 of T 2 by the edges ex and ey, respectively. We first show that that T 1 and T 2 cannot be both T -triangles. Suppose that T 1 is a T -triangle whose vertices have labels x 1 = {1, 2, 3}, y 1 = {2, 3, 4}, and z 1 = {1, 2, 4} (see Fig. 6). Since x 2 is adjacent to only x 1 in T 1 , then the label of ex must be {1, 3}. With the same argument, the label of ey is {3, 4}. Since x 2 and y 2 belong to T 2 , they have a common pair of labels, say {3, 5}. So their labels must be x 2 = {1, 3, 5} and y 2 = {3, 4, 5}. If we assume T 2 to be a T -triangle, then z 2 = {1, 4, 5}. This leads to a contradiction since z 2 is not adjacent to z 1 = {1, 2, 4}. Therefore the only possible label is z 2 = {3, 5, 6}, showing that T 2 is a K-triangle.

To show that T 1 and T 2 can both be K-triangles, we set x 2 = {1, 3, 5} and y 2 = {3, 4, 5} and z 2 = {3, 5, 6} in T 2 (see Fig. 6). We can then take x 1 = {1, 2, 3}, y 1 = {2, 3, 4} and z 1 = {2, 3, 7} in T 1 .

x 1 = {1, 2, 3} x 2 = {1, 3, 5} z 1 = {1, 2, 4} y 1 = {2, 3, 4} y 2 = {3, 4, 5} z 2 = {3, 5, 6} T 1 T 2
Fig. 6 The configuration obtained by two triangles with joined by two edges. One possible labelling is shown.

Representing the variables of U

Define a variable gadget, denoted G x , to represent a variable x in U . The gadget is a 2-intersection graph obtained by the union of different configurations and its definition can be checked in Fig. 7.

The graph G x consists of two gadgets G 1 x and G 2

x that grow around two K 5 cliques C 1 and C 2 that are connected by a ribbon configuration, say R.

The gadget G 1

x includes the clique C 1 whose vertices are in common with five different K 3 triangular cliques, counter-clockwise denoted by T 1 i , with 1 ≤ i ≤ 5, and starting from the triangle facing the ribbon that joins the two gadgets. These triangles are then connected in pairs by their two remaining free vertices, forming a 10-cycle, as shown in Fig. 7.

The triangles T 1 2 and T 1 5 will be associated with the two (at most) occurrences of the variable x, while T 2 3 will be associated with the single occurrence of the variable x.

We show that the graph G x has the 2-intersection property. Furthermore, it allows very few possibilities when setting the triangles as K or T . x and G 2 x are indicated together with their connection through a ribbon configuration. Proof W.l.g. let T 1 , . . . , T 6 be the boundary triangles (labelled counter-clockwise) of a clause gadget. Without loss of generality, let T i have vertices {x i , y i , z i }, such that x i is in common with the central clique K 6 , while y i and z i connect T i with the neighbour triangles T i-1 and T i+1 . Note that the indices i -1 and i + 1 are reduced to the range 1 . . . 6. We first show that there is no labelling when three (non consecutive, according with Property 5) boundary triangles are T -triangles. Let T 1 , T 3 and T 5 be T -triangles that alternate with three K-triangles T 2 , T 4 and T 6 . We show that such a configuration does not have the 2-intersection property, i.e., it does not allow any labelling. Assume that the edges of the central clique K 6 have the common label {1, 2}. Therefore, label every vertex {x 1 , . . . , x 6 } of K 6 with {1, 2, i}, with 3 ≤ i ≤ 8, respectively, and let x 1 belong to a T -triangle. W.l.g, its two remaining vertices have labels y 1 = {1, 3, 9} and z 1 = {2, 3, 9}. Suppose T 1 is connected by the edges (y 1 , y 2 ) and (z 1 , z 6 ) with the triangles T 2 and T 6 , respectively. So, the edge {y 1 , y 2 } has label {1, 9}, and the edge {z 1 , z 6 } has label {2, 9} (check the labelling in Fig. 11, left graph).

So, the labels y 2 = {1, 8, 9} and z 6 = {2, 4, 9} are determined. Since T 2 is a Ktriangle, the label of z 2 requires the introduction of a new index, i.e., z 2 = {1, 8, 10}. The same holds in T 6 , where the label y 6 = {2, 4, 11} can be assumed.

Continuing labelling the boundary triangles, we observe that the edge {z 2 , z 3 } has two possible labels remaining, i.e., either {1, 10} or {8, 10}. Since T 3 is a T -triangle, then z 3 has either label {1, 7, 10} or {2, 7, 10}; the first one only being compatible with one of the possible labels of the edge {z 2 , z 3 }. As a consequence, y 3 = {2, 7, 10}.

Acting similarly on the T -triangle T 5 , we get y 5 = {2, 5, 11} and z 5 = {1, 5, 11}. Finally, a problem occurs in the labelling of the K-triangle T 4 : the edges {z 3 , z 4 } and {y 4 , y 5 } have labels {2, 10} and {1, 11}, respectively, preventing any connection between the vertices y 5 and z 5 .

A similar reasoning reveals that the presence of exactly one boundary T -triangle does not allow a labelling of the clause gadget. The failure of a labelling in this case is shown in Fig. 11, right (T 1 is the T -triangle).

Finally, Fig. 12 shows the possible labellings when either two or no Ttriangles are present as boundary triangles.

The final NP-completeness reduction.

Let us consider the instance C = {c 1 , . . . , c n } of MAX-3-SAT involving the variables in the set U = {x 1 , . . . x m }. Based on the gadgets already defined, we construct a graph G C whose labels determine its 2-intersection property and express the desired valuations of C. The reader can follow an example of the construction of G C in the case C = (x 1 ∨ x 1 ∨ x 2 ) ∧ (x 1 ∨ x 2 ∨ x 3 ) in Fig. 13.

First, from Lemma 3, we can suppose that each variable must appear at most three times: one in a form and two in the opposite form. For each variable x i ∈ U , we define a variable gadget G xi , and associate the triangle T 2 12 Reconstruction in the case of zero or two T -triangles. The bold triangles are the boundary triangles. The role of 2-intersection graph in the hypergraph null labelling the single occurrences of a literal x i . The at-most-two remaining occurrences of the opposite literal are associated with the triangles T 1 2 and T 1 5 . For each clause c j ∈ C, we construct a clause gadget G cj and label its boundary triangles T 1 . . . T 6 as in Lemma 4.

T K K K K K K K K K K K K K K Fig.
Finally, we connect variable gadgets and clause gadgets together as follows: for each clause c j , with 1 ≤ j ≤ n, we use a ribbon to the triangle T 2i-1 of the clause gadget G c , to the corresponding triangle associated with the i th literal of c in the G x gadget of its variable, as in Fig. 13. Now, we prove the main theorem of this section. Proof Let us assume that there exists a valuation for the MAX-3-SAT instance C. Given a variable x ∈ U , for each literal with value true associated with x, we assign the triangles associated with it to be T -triangles, and we assign the triangles associated with its negation to be K-triangles. Corollary 4 assures that the variable gadget Gx has a labelling. Then, for each clause c j ∈ C, in its clause gadget Gc j , we assign the triangles associated with the literals having valuation true to be K-triangles, but T -triangles for the literals having valuation false. Since three or more T -triangles are not allowed in Gc j by Lemma 4, then also the clause gadget has a labelling. The labelling of the connecting ribbons is straightforward. The reader can benefit from an example of such a construction in Fig. 13. On the other hand, suppose G C has the 2-intersection property, i.e., there is a labelling of G C that determines a 3-hypergraph H such that I 2 (H) = G C . For each clause gadget Gc j , with c j ∈ C, there exists at least one triangle among T 1 , T 3 and T 5 , say T of type K, by Lemma 4. Property 4 assures that T having type K leads to a T -triangle in the variable gadget Gx to which a literal, say l, is associated (note that the opposite does not hold, as shown by the red triangle in Fig. 13). We assign such a literal the truth value true. The opposite literal l is then associated with false. Due to Corollary 4 the triangles (one or two) in Gx associated with l are of type K.

We emphasize the following situation: it may happen that there exists a Gx labelling where all three triangles associated with the literals are of type K. In such a case, the value assigned to the variable does not affect the truth value of the clauses of C, so it can be arbitrarily assigned.

So, the valuation defined is a solution of the MAX-3-SAT instance C: each clause gadget Gc j has at most one triangle among T 1 , T 3 and T 5 of type K, so at least one literal in the clause has value true. Furthermore, in each variable gadget, if one literal is set to true, i.e. at least one of the corresponding triangles is of type T , then the the opposite literal has the logical value false. The proof directly follows from Theorem 5, after checking that the construction of the graph G C associated with the instance C can be performed in polynomial time.

Remark 3. In Fig. 13, the two highlighted triangles in different colors indicate the following situation:

• the red triangle in the ribbon connecting T 1 of G c1 to T 1 2 of G x3 shows that the T -triangle related to the value false in a clause may produce the truth value true in the associated literal. This is not a contradiction: it simply means that in the clause, here c 1 , there exists at least a second triangle of type K, i.e. the clause is already satisfied by at least one different literal;

• the blue triangle T 5 1 in G x2 is of type K which is different from the Ttriangle T 1 2 although they are associated with the same literal l. This means that l may have a value false in a second clause in which it is possibly present. As a consequence, a different literal having truth value true is present in that clause. What is relevant in G x2 is that the triangle associated with l, i.e. T 2 3 , has to be of type K, assuring that l has truth value false in all its occurrences. Example 5. Consider the formula F = (x 1 ∨ x 1 ∨ x 3 ) ∧ (x 1 ∨ x 2 ∨ x 3 ). Figure 13 shows the construction of its corresponding gadget. The labelling shown in the same figure corresponds to the valuation x 1 = false, x 3 = false. The value of x 2 doesn't matter since, in any case, the assignment is a solution for F .

Fig. 1

 1 Fig. 1 (a) example of a T and a K sharing 2 vertices. (b) example of S.

Fig. 2

 2 Fig. 2 Reconstruction of the configuration in which a point belongs to 4 triangles.

Fig. 3

 3 Fig.3The labelling of the clique K 5 used in Example 4.

Fig. 5

 5 Fig. 5 Two possible labels of a ribbon configuration. In each of them at most one triangle is a T -triangle.

Fig. 7

 7 Fig. 7 The graph Gx. It is used to model the occurrences of the literals x and x in the clauses of an instance of MAX-3-SAT. The two parts G 1x and G 2 x are indicated together with their connection through a ribbon configuration.

Fig. 9 Fig. 10

 910 Fig. 9 One of the labellings of Gx related to case ii) of Corollary 4. The starting triangle T 2 3 = T is shaded. The type (T or K) of each triangle is also shown

Theorem 5

 5 Given an instance C of MAX-3-SAT, the graph G C has the 2intersection property if and only if the instance C has a solution.

Corollary 5

 5 The 2IN T Problem is NP-complete.

Fig. 13

 13 Fig. 13 Example of the construction of the gadget for C = {c 1 , c 2 }, c 1 = (x 3 , x 1 , x 1 ), c 2 = (x 1 , x 3 , x 2 ). One of the valuations obtained by the labelling of the corresponding G C graph is x 1 =false[Andrea: true], x 2 =true and x 3 =true. The valuation can be obtained by the position of (at least one of) the T configurations in the triangles T 1 2 , T 1 5 and T 2 3 of the variable gadget.

  Lemma 1 A graph G has a null labelling if and only if every connected component is an Eulerian graph with an even number of edges.

  269, 147, 369, 459, 146, 368, 157, 156, 189, 279, 246, 379, 239}. It is easy to check that H does not have a null labelling although its 2-intersection graph I 2 (H) is connected. Remark 1. Connectedness of the 2-intersection graph does not characterize the set of 3-hypergraphs meeting the lower bound in Conjecture 1 as shown by the following example. Example 3. The hypergraph H defined as E = {134, 245, 247, 125, 567, 167, 146, 157, 357, 135, 347, 467, 127, 236, 257, 457, 156, 345} has 7 vertices and 18 hyperedges. It satisfies Conjecture 1 as shown by the following null labelling

  An attempt to define a labelling of a clause gadget with three (on the left) or one (on the right) boundary T -triangles. Both configurations prevent any labelling of the K-triangle T 4 .

		{1,8,9}	{1,3,9}				{1,8,9}	{1,3,9}
	{1,8,10}	T2	{1,2,8}	T1	{2,3,9}		{1,8,10}	T2	{1,2,8}	T1	{2,3,9}
				{1,2,3}						{1,2,3}
	{1,7,10}				{2,4,9}	{1,7,10}				{2,4,9}
	{1,2,7}			{1,2,4}		{1,2,7}			{1,2,4}
	T3				T6		T3				T6
	{2,7,10}				{2,4,11}	{1,7,11}				{2,4,10}
				{1,2,5}						{1,2,5}
		T4	{1,2,6}	T5	{2,5,11}			T4	{1,2,6}	T5	{2,5,10}
	?			{1,5,11}						{2,5,11}
	{1,8,9} {2,4,12} {2,4,9} Fig. 11 {1,2,8} {1,8,10} T {1,2,8} {1,2,3} {1,2,4} {1,2,5} {1,2,6} {1,2,7} {1,7,10} {2,7,10} {2,6,10} {2,6,11} {2,5,11} {2,5,12} {1,3,9} {1,8,9} T {2,3,9} {1,8,10}	{1,3,9} {1,7,10} {1,7,12}	{1,3,14} {1,2,7} {1,6,12} {2,6,12} {1,2,8} {1,2,6} {1,8,9} {1,8,10}	{1,2,3} {1,2,5} {1,2,4} {2,5,12} {2,5,11} {1,3,9} {2,3,9} T	{2,4,11} {2,4,9}
						{1,2,3}
				{1,7,10}				{1,4,14}
				{1,2,7}			{1,2,4}
				{1,7,11}			{1,4,13}
						{1,2,5}
					{1,2,6}
				{1,6,11}			{1,5,13}
					{1,6,12}	{1,5,12}

The role of 2-intersection graph in the hypergraph null labelling

Property 6. The variable gadget G x is a 2-intersection graph.

In Fig. 8 and9 we provide samples of labellings of G x .

Corollary 4 A labelling of the vertices of the graph Gx only produces the following possible types (K or T ) of the triangles T 1 2 , T 1 5 and

Proof i) let us assume T 2 3 = T . Property 5 assures that triangle T 2 1 is a K-triangle. So, by Corollary 3, the triangle adjacent to T 2 1 is a T -triangle, and the triangle adjacent to T 1 1 is a K-triangle by Property 4. Continuing with the configurations, we get T 1 1 = T , and, again by Property 5, we obtain that both T 1 2 and T 1 5 are K-triangles as desired (see a possible labelling in Fig. 8).

ii) let us assume T 1 5 = T (resp. T 1 2 = T ). Property 5 assures that T 1 1 = K. So, by Corollary 3, the triangle adjacent to T 1 1 is a T -triangle, and the triangle adjacent to T 1 1 is a K-triangle by Property 4, and finally T 2 1 = T . Again Property 5 assures that T 2 3 is a K-triangle as desired (see a possible labelling in Fig. 9). Remark 2. We point out that in case ii) of Corollary 4 no assumption involves the type of T 1 2 , that can be either T or K, when T 1 5 = T and vice versa. Figure 9 shows an example of a labelling where T 2 1 = K. An easy check reveals that exchanging the types K and T between T 3 1 and T 2 1 produces a new admissible labelling.


N Guillemot

A Brut

G Wallois

    Quality assessment of the MAJA atmospheric correction processor over a farmland area with a new operational in-situ ROSAS station

 414, 440, 500, 555, 675, 702, 740, 782, 870, 937, 1020 and 1640nmRow effect

  processors validation requires in-situ surface reflectances to go beyond the usual algorithms intercomparison • Pre-existing ROSAS stations in La Crau (France) and Gobabeb (Namibia) have a strong homogeneity of reflectances in space and time, which is good for sensor calibration • The new Fr-Lam ROSAS station is set-up since 2021 in an agricultural field (1) with a 3-crops rotation cycle every 2 years, offering a strong intra-and inter-annual variation of surface reflectances • Opportunities to validate reflectances over vegetation with various BRDF, anisotropy, rows and adjacency effects • The ROSAS protocol designed by the CNES allows to reconstruct in-situ BRDF from a sequence of measurements (2) using a multispectral photometer • A CIMEL CE318 with 12 filters is installed on top of a 10m high mast, and sequentially acquire sun, sky and ground irradiances • Data are quality controlled and postprocessed to reconstruct a BRDF rose (3) • For Level 2 EO products validation, the reflectances are fitted to the sensor's bands spectral functions and sensor view angles. • The Sentinel-2 surface reflectances processed by the Atmospheric Correction processor MAJA are compared with Fr-Lam measurements • To meet quality criteria, only perfectly clear sky acquisitions are used for validation, leading to only 9 samples (5) • RMSEs are below 9.10 -3 for bands 2, 3 and 4, which is better than the specifications ; • Poorer RMSEs were obtained for band 8, reasons of which are still to investigate • Fr-Lam is intended to acquire continuously BRDF in the coming decade, leading to a rich BRDF collection • Started in March 2021, the ROSAS Fr-Lam made its first acquisition over a full growth cycle of silage maize • Resulting surface reflectances reflect changes from bare soil to small sparse plants, reaching up to 2.6m high before harvest (4)The Fr-Lam site aims at becoming a key level-2 validation site for existing and coming missions.

Quality assessment of the MAJA atmospheric correction processor over a farmland area with a new operational in-situ ROSAS station

  . Colin 1 , L. Landier 2 , S. Coustance 2 , O.Hagolle 1,2 , S.Marcq 2 , N. Guillemot 3 , A. Brut 1 , G. Wallois 4 , A. Meygret 2 1 CESBIO, UMR 5126 (CNES/CNRS/INRAE/IRD/Université Toulouse III), Toulouse, France, 2 CNES, Toulouse, France, 3 THALES Services, Toulouse, France, 4 Ecole d'Ingénieur de Purpan, Toulouse, France

J

Objectives ROSAS protocol Measurements Results CIMEL photometer 4G GSM Solar panel GPS & Datalogger 1 2 4 5

	2022 Living Planet Symposium, 23-27 May 2022, World Conference Centre Bonn, Germany Contact: jerome.colin@cesbio.cnes.fr

Lean more and get updates on our blog


Sarah Safieddine

Maya George

Lieven Clarisse

Simon Whitburn

Cathy Clerbaux

Skin temperature from IASI: agriculture and urban heat islands monitoring

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

Introduction Methods

Applications

Skin temperature, (Tskin), also known as Land Surface Temperature is derived from infrared sensors and provides a continuous view of Earth's surface and allows for the monitoring of surface temperature changes relevant for climate trends. Tskin from the Infrared Atmospheric Sounding Interferometer (IASI) has not been properly exploited to date to assess its long-term spatio-temporal variability. In this study, we present a fast retrieval method of Tskin based on an artificial neural network from a set of IASI channels selected using entropy reduction technique.

We show the extent to which Tskin can be used to monitor land use in the past decade. We focus in particular on heatwaves and on agricultural and urban expansion with a focus on the Arabian Peninsula where extreme heat is recorded.

Channels are selected in the shaded region after an analysis of the spectrum. This spectral region [780-980] cm -1 is optimal in terms of SNR, radiometric noise, and emissivity effect and have the best information on skin temperature.

Artificial neural networks (ANN) method is used to approximate the complex radiative transfer function that maps the radiances to skin temperature. The training dataset is constructed out of clear-sky (cloud cover <10%) Level 1C (L1C) IASI radiances over 100 channels selected using Entropy Reduction. We train our ANN with these IASI radiances as input, and we use the T skin from the ERA5 reanalysis as output/target [START_REF] Safieddine | Artificial Neural Networks to Retrieve Land and Sea Skin Temperature from IASI[END_REF].

Clerbaux et al., the conversation, 2020

Conclusions

This study shows the various applications of the IASI-skin temperature product which is a valuable tracer for land use change and could be used to assess heatwaves and fires, urban heat/cool islands, agriculture and irrigation, and climate change.

  over agriculturally irrigated fields during day and nightLarge positive anomalies in IASI Tskin during a heatwave in Russia and Siberia that caused large firesTskin is a climate essential variable (ECV) that can be used to assess and detect the warming of the atmosphere and its effect on land and sea Up: IASI typical spectrum with brightness temperature in red, Tskin jacobians in blue, SNR in orange, and noise in grey. Bottom: surface emissivity over land and sea

ACKNOWLEDGEMENTS

This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 742909) IASI Tskin day and night observations during summer (June, July and August) for Riyadh and Paris. Paris, during the day and the night is warmer than the suburbs. In Riyadh, the city exhibits the opposite effect during the day and acts as an urban cool island


Sophie Desgraupes
email: sophie.desgraupes@pasteur.fr

Patricia Jeannin
email: patricia.jeannin@pasteur.fr

Antoine Gessain
email: antoine.gessain@pasteur.fr

Pierre-Emmanuel Ceccaldi
email: pierre-emmanuel.ceccaldi@pasteur.fr

Aurore Vidy
email: aurore.vidy@pasteur.fr

Excretion of Cell-Free and Cell-Associated Zika Virus into Breast Milk of Infected Dams and Identification of Antiviral Factors

Keywords: mother-to-child transmission, breastfeeding, milk cells, antiviral, fatty acids

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

Introduction

Zika virus (ZIKV) is an arthropod-borne virus (arbovirus) belonging to the Flaviviridae family and to the Flavivirus genus. It was first identified in 1947 in the Zika forest in Uganda and caused little concern for public health as ZIKV infection was reported as being mostly asymptomatic or associated with mild symptoms. During the following 60 years, only 14 human cases were reported in Southeast Asia and Africa [START_REF] Faye | Molecular evolution of Zika virus during its emergence in the 20th century[END_REF]. In 2007, the first Zika fever outbreak ever described occurred in Yap islands, resulting in the infection of approximatively 5000 inhabitants (73% of the population) [START_REF] Duffy | Virus Outbreak on Yap Island, Federated States of Micronesia[END_REF]. An even larger outbreak occurred in 2013-2014 in French Polynesia, causing approximatively 28,000 estimated cases of infection [START_REF] Musso | Rapid spread of emerging Zika virus in the Pacific area[END_REF]. Two small outbreaks occurred that same year in New Caledonia where 1400 cases were described and in the Cook Islands where over 900 cases were reported [START_REF] Roth | Concurrent outbreaks of dengue, chikungunya and Zika virus infections-An unprecedented epidemic wave of mosquito-borne viruses in the Pacific 2012-2014[END_REF]. Following the introduction of ZIKV in Brazil in 2015 [5], a large outbreak happened causing 440,000 to 1,300,000 cases [START_REF] Hennessey | Virus Spreads to New Areas-Region of the Americas, May 2015-January 2016[END_REF], thus compelling the World Health Organization to declare a "Public Health Emergency of International Concern".

As 80% of ZIKV infected individuals remain asymptomatic, the number of cases estimated during these epidemics is undervalued. After a small incubation period, 20% of infected people developed mild symptoms such as fever, skin rashes, and joint pain. However, since 2013, severe neurological symptoms have been associated with ZIKV infection [START_REF] Carod-Artal | Neurological complications of Zika virus infection[END_REF]. In adults, Guillain-Barre syndrome (GBS) is the most common neurological complication, but other pathologies have been reported such as sensory polyneuropathy, sensory neuronopathy [START_REF] Carod-Artal | Neurological complications of Zika virus infection[END_REF], and meningitis [START_REF] Wright | Virus-Associated Aseptic Meningitis and Guillain-Barre Syndrome in a Traveler Returning from Latin America: A Case Report and Mini-Review[END_REF]. During the French Polynesian outbreak, the risk of developing a ZIKV-related GBS was estimated to be 0.24 per 1000 cases of infection [START_REF] Cao-Lormeau | Guillain-Barré Syndrome outbreak associated with Zika virus infection in French Polynesia: A case-control study[END_REF]. Neurological birth defects, called the congenital Zika syndrome, have also been associated to ZIKV infection: children born to mothers infected during pregnancy can develop severe microcephalies, ocular abnormalities, or show decreased brain tissue [START_REF] Rasmussen | Virus and Birth Defects-Reviewing the Evidence for Causality[END_REF].

ZIKV is mostly transmitted to humans by Aedes aegypti mosquitoes but interhuman transmission can also occur during sexual intercourse or from mother-to-child in utero [START_REF] Gregory | Modes of Transmission of Zika Virus[END_REF]. During the past 8 years, ZIKV transmission through breastfeeding has repeatedly been suggested in epidemiological studies [START_REF] Desgraupes | Mother-to-Child Transmission of Arboviruses during Breastfeeding: From Epidemiology to Cellular Mechanisms[END_REF] and has been demonstrated on a mouse model by our team [START_REF] Hubert | Evidence That Zika Virus Is Transmitted by Breastfeeding to Newborn A129 (Ifnar1 Knock-Out) Mice and Is Able to Infect and Cross a Tight Monolayer of Human Intestinal Epithelial Cells[END_REF]. Epidemiological studies have reported the presence of viral RNA (vRNA) and/or infectious particles in the breast milk of infected mothers as well as viral persistence in breast milk after clearance from the blood stream [START_REF] Desgraupes | Mother-to-Child Transmission of Arboviruses during Breastfeeding: From Epidemiology to Cellular Mechanisms[END_REF]. In addition, a case of secondary microcephaly was recently reported in a breastfed infant who was negative for ZIKV at birth and showed a normal head circumference. The newborn was exclusively breastfed and ZIKV was isolated in the mother's breast milk [START_REF] Mello | Development of Secondary Microcephaly After Delivery: Possible Consequence of Mother-Baby Transmission of Zika Virus in Breast Milk[END_REF], raising concern about the transmission of ZIKV during breastfeeding.

Breast milk is mainly composed of water and contains maternal cells, carbohydrates (e.g., lactose), fat, proteins, and minerals [START_REF] Ballard | Human Milk Composition. Nutrients and Bioactive Factors[END_REF]. In human breast milk, both breast-derived and blood-derived cells are found. Blood-derived immune cells represent a small minority of the milk cells (<2% in mature human milk), whereas the vast majority come from the mammary gland (luminal, myoepithelial, progenitor, and stem cells). Among those, luminal and myoepithelial cells represent 98% in human milk [START_REF] Witkowska-Zimny | Cells of human breast milk[END_REF]. The different components of breast milk can be separated by centrifugation into a cell pellet, the lactoserum and the cream fraction. The lactoserum, also called whey, is the liquid fraction rich in proteins, lactose, and minerals, whereas the cream fraction contains the milk fat globules (MFG). The latter are composed of a triple-layered milk fat globule membrane (MFGM) surrounding a triglyceride core [START_REF] Argov | Milk Fat Globule structure & function; Nanosciece comes to milk production[END_REF]. The composition of breast milk varies with different factors such as the species, nutrition, or stage of lactation [START_REF] Ballard | Human Milk Composition. Nutrients and Bioactive Factors[END_REF]. Indeed, during the first hours after birth, the breast milk is rich in immunological components and called colostrum, whereas transitional and mature milk produced in the following weeks become richer in nutritional factors and poorer in leukocytes.

Although protective components are transmitted to the newborn during breastfeeding, deleterious factors such as drugs or pathogens can also be passed on. For humans, three viruses are recognized as efficiently transmitted during breastfeeding: human immunodeficiency virus type 1 (HIV-1), human T-cell leukemia virus type 1 (HTLV-1), and cytomegalovirus (CMV) [START_REF] Desgraupes | Mother-to-Child Transmission of Arboviruses during Breastfeeding: From Epidemiology to Cellular Mechanisms[END_REF]. These milk-borne viruses are excreted in human breast milk in a cell-free (HIV-1 [START_REF] Ndirangu | Cell-Free (RNA) and Cell-Associated (DNA) HIV-1 and Postnatal Transmission through Breastfeeding[END_REF], CMV [START_REF] Asanuma | Role of Milk Whey in the Transmission of Human Cytomegalovirus Infection by Breast Milk[END_REF]) or cell-associated (HIV-1 [START_REF] Ndirangu | Cell-Free (RNA) and Cell-Associated (DNA) HIV-1 and Postnatal Transmission through Breastfeeding[END_REF], CMV [START_REF] Asanuma | Role of Milk Whey in the Transmission of Human Cytomegalovirus Infection by Breast Milk[END_REF], HTLV-1 [START_REF] Ureta-Vidal | Mother-to-child transmission of human T-cell-leukemia/lymphoma virus type I: Implication of high antiviral antibody titer and high proviral load in carrier mothers[END_REF]) form and transmitted to the infant, resulting in a chronic infection.

As breast milk components and viral entities are simultaneously present in the milk, their interaction could lead to an enhanced or decreased viral transmission to the newborn during breastfeeding. Indeed, some breast milk components such as prostaglandins [START_REF] Clarke | Effect of antimicrobial factors in human milk on rhinoviruses and milk-borne cytomegalovirus in vitro[END_REF] have been shown to be proviral, whereas others such as lactoferrin (LF), monolaurin, or free fatty acids [START_REF] Desgraupes | Mother-to-Child Transmission of Arboviruses during Breastfeeding: From Epidemiology to Cellular Mechanisms[END_REF] are antiviral. Breast milk has been suggested to have an antiviral effect on some arboviruses [START_REF] Welsh | Use of Semliki forest virus to identify lipid-mediated antiviral activity and anti-alphavirus immunoglobulin A in human milk[END_REF], but it was further shown that the antiviral effect on ZIKV was caused by the storage of breast milk samples at 4 • C [START_REF] Pfaender | Inactivation of Zika virus in human breast milk by prolonged storage or pasteurization[END_REF]. The authors showed that the storage of the samples leads to an increase in the free fatty acid content [START_REF] Pfaender | Inactivation of Hepatitis C Virus Infectivity by Human Breast Milk[END_REF] and suggested a role of the latter in viral inactivation. Although milk stored at 4 • C exerted an antiviral effect, fresh breast milk from healthy donors was then tested and shown not to inactivate ZIKV [START_REF] Conzelmann | Storage-Dependent Generation of Potent Anti-ZIKV Activity in Human Breast Milk[END_REF], raising alarm concerning its transmission during breastfeeding.

Although cases of probable transmission of ZIKV from infected mothers to breastfed newborns have been reported [START_REF] Desgraupes | Mother-to-Child Transmission of Arboviruses during Breastfeeding: From Epidemiology to Cellular Mechanisms[END_REF], other case reports showed a lack of transmission by breastfeeding [START_REF] Centeno-Tablante | Update on the Transmission of Zika Virus Through Breast Milk and Breastfeeding: A Systematic Review of the Evidence[END_REF]. Therefore, biological or genetic factors controlling the transmission should be investigated (viral load, viral form, immunological status, stage of lactation, etc.).

In this paper, by using an established murine model for ZIKV infection [START_REF] Dowall | A Susceptible Mouse Model for Zika Virus Infection[END_REF], we show the infectious nature of breast milk isolated from ZIKV-infected dams and we identify the nature of viral entities in breast milk as being both cell-free and cell-associated. We also determine that the majority of milk cells in murine breast milk are mammary epithelial cells. Finally, we investigate the effect of several breast milk components on ZIKV infectivity (LF, lactalbumin (LA), and free fatty acids) in order to better understand the biological factors regulating the transmission.

Materials and Methods

Animal Model

A129 mice (129S2/SvPas-Ifnar1tm1Agt) were used to carry out the experiments, a well-established model to study ZIKV infection. They were housed and bred at the animal facilities of the Institut Pasteur which are accredited by the French Ministry of Agriculture for breeding and performing experiments on live rodents.

Ethics Statement

Experiments on animals were performed in compliance with French and European regulations on care and protection of laboratory animals (EC Directive 2010/63, French Law 2013-118, 6 February 2013). All experiments were approved by the Ethics Committee #89 and registered by the French "Ministère de l'Enseignement Supérieur, de la Recherche et de l'Innovation" under the reference "APAFIS#16119-2018071314475930 v1" (date of approval: 19 November 2018). Use of genetically modified mice (A129) was approved by the institutional instances and the French "Ministère de l'Enseignement supérieur, de la Recherche et de l'Innovation" under the reference no. 2194 (date of approval: 6 October 2017).

Virus Strains and Cell Lines

All experiments were carried out using two ZIKV strains belonging to the Asian lineage (Brazil/2016, GenBank: KU991811; H/PF13, GenBank: KX369547), amplified in Vero E6 cells as previously described [START_REF] Hubert | Productive Infection of Mouse Mammary Glands and Human Mammary Epithelial Cells by Zika Virus[END_REF]. ZIKV suspension was added to Vero E6 cell monolayers at 37 • C and 5% CO 2 for 2 h in DMEM supplemented with 2% fetal bovine serum (FBS, Gibco, Thermo Fisher Scientific, Waltham, MA, USA), 100 U/mL penicillin, and 100 µg/mL streptomycin (Gibco, Thermo Fisher Scientific, Waltham, MA, USA). After adsorption, the inoculum was removed and replaced by DMEM containing 2% FBS and 10 mM Hepes (Gibco, Thermo Fisher Scientific, Waltham, MA, USA). After 2 days of culture, supernatants were centrifuged at 500× g for 10 min and aliquoted for storage at -80 • C until titration by plaque forming assay. These supernatants were used as inoculum for animal infections and in vitro experiments.

Vero E6 cells (CRL-1586, ATCC) and Caco-2 cells (Clone TC7, SCC209, Sigma-Aldrich, Saint-Louis, MO, USA) were grown in DMEM medium supplemented with L-glutamine (Gibco, Thermo Fisher Scientific, Waltham, MA, USA), 10% FBS, 100 U/mL penicillin, and 100 µg/mL streptomycin. They were maintained at 37 • C and 5% CO 2 for culture.

Mouse Infection and Sample Collection

For in vivo experiments, 6-to 13-week-old A129 females and males were housed and bred for mating according to animal welfare recommendations. Lactating A129 mice were infected subcutaneously from 1 to 11 days post-partum with 10 6 PFU of the Brazil/2016 strain of ZIKV. Blood samples were collected at 3 days post-infection in tubes containing 15 mM EDTA (Invitrogen, Thermo Fisher Scientific, Waltham, MA, USA) and subjected to plaque forming assays and RNA extractions to monitor the mothers' infection. For milk sample collection at 4 days post-infection, the dams were isolated from their pups for 2 h 30 to 3 h to allow the accumulation of milk in the mammary glands. To stimulate the ejection of milk from the alveoli into the lactiferous ducts, the dams were injected intra-peritoneally with 250 µL of ocytocine (Ocytovem, Ceva Santé Animale, Libourne, France) and put in contact with their pups during a few minutes. The dams were watched closely to let the pups start suckling but prevent them from drinking all the breast milk. After stimulation, the dams were anesthetized by intra-peritoneal injection with 100 µL of an anesthetic mix containing 5 mg/kg of xylazine (Rompun, Bayer, Leverkusen, Germany) and 80 mg/kg of ketamine (Imalgene, Boehringer Ingelheim, Ingelheim am Rhein, Germany). Breast milk was collected with a Pasteur pipette by massaging the mammary glands. To evaluate the transmission, spleens were collected from euthanized pups and subjected to RNA extraction and RT-qPCR.

Milk Fraction Isolation

To separate the different fractions of breast milk, whole milk samples were centrifuged at 5000× g for 15 min. The cream fraction, the whey, and the cell pellet were then separated and washed twice in PBS at 5000× g for 15 min in order to increase their purity.

Viral RNA Extraction

vRNAs were extracted from plasma samples (obtained by blood centrifugation at 2000× g for 15 min), whole milk, cream fractions, whey fractions, and spleens, using the QIAamp viral RNA mini kit (Qiagen, Hilden, Germany). Spleens were stored at -80 • C before vRNA extractions. Spleens were thawed in PBS and dissociated using the Bullet Blender Storm shredder (Dutscher, Bernolsheim, France) in the presence of 3.2 mm steel beads (Next Advance, Troy, MO, USA). Spleen suspensions were centrifuged at 2000× g for 10 min and the supernatants were used for vRNA extractions.

RT-qPCR

Reverse transcription was performed using random hexamers and the Maxima H Minus Reverse-Transcriptase kit (Invitrogen). Quantitative PCR was performed using 5 µL of template cDNA, with 10 µL of MasterMix (iTaq Universal SYBR Green Supermix, Biorad, Hercules, CA, USA), 500 nM of each ZIKV NS5-specific primer (Forward: 5 -AAR TAC ACA TAC CAR AAC AAA GTG GT-3 ; Reverse: 5 -TCC RCT CCC YCT YTG GTC TTG-3 ), and using the following program: 10 min/95 • C, followed by 40 cycles of 15 s/95 • C, 20 s/60 • C, and 30 s/72 • C (Mastercycler Eppendorf Realplex, Hamburg, Germany). Quantification analysis was realized using a standard curve of a ZIKV-encoding plasmid.

Viral Titration by Plaque Forming Assays

Vero cell monolayers were exposed to 10-fold dilutions of plasma, whole milk, cream, and whey samples in DMEM-2% FBS during 2 h at 37 • C and 5% CO 2 . After viral adsorption, the inocula were removed and replaced with DMEM-2% FBS-1.6% carboxymethyl cellulose. The cells were cultured for 5 days at 37 • C and 5% CO 2 . After three washes in PBS, Vero monolayers were fixed in 4% paraformaldehyde (PFA) for 15 min at room temperature (RT) and colored with crystal violet (Sigma-Aldrich, Saint-Louis, MO, USA).

Co-Culture

Infectious center assays were performed by co-culturing milk cells with Vero E6 cells after 3-fold dilutions of 1 million milk cells in DMEM-2% FBS for 2 h at 37 • C and 5% CO 2 . DMEM-2% FBS-1.6% carboxymethyl cellulose was added and the co-cultures were kept for 6 days at 37 • C and 5% CO 2 .

After three washes in PBS, Vero monolayers were fixed in 4% PFA for 15 min at RT and colored with crystal violet.

Immune Staining and Flow Cytometry

After washing the cell pellets from breast milk samples twice in PBS, anti-mouse CD16/CD32 antibodies (Invitrogen) were added for 20 min at RT in order to block mouse antibody recognition. They were diluted in MACS buffer (FBS-2% EDTA-0.5% BSA) at a ratio of 2:50. After a wash in PBS at 400× g for 4 min, the extracellular antigens were stained for 1 h at RT in the dark (ratio 1:50 for antibody:MACS) as follows: anti-mouse CD45 stained with eF450 (Invitrogen), anti-mouse CD24 stained with APC (Invitrogen), and anti-mouse CD49f stained with PE (Invitrogen). The following steps were all performed in the dark. After two washes in PBS, cells were fixed using 4% PFA for 15 min at RT. After two washes in PBS, cells were resuspended in PBS-10% cell fix (BD Biosciences, Franklin Lakes, NJ, USA).

Cytotoxicity Assays

Vero or Caco-2 cells were incubated with DMEM-2% FBS, dimethyl sulfoxide (DMSO), or different concentrations of free fatty acids, LF or LA for 2 h at 37 • C. Cell layers were washed and cultured in DMEM-2% FBS at 37 • C and 5% CO 2 . Cell viability was assessed at 48 h of culture by an MTT colorimetric assay (thiazolyl blue tetrazolium bromide, Sigma-Aldrich). Cytotoxicity results can be found in Supplementary Figures S1 andS2.

2.12. Viral Particle Incubation with Lactoferrin, Lactalbumin, Ethanol, or DMSO ZIKV particles were incubated with DMEM-2% FBS or different concentrations of LF, LA, ethanol, or DMSO for 1 h at 37 • C before infection at a multiplicity of infection (MOI) of 1. ZIKV infectivity was assessed using the Caco-2 cell line as, physiologically, ZIKV and breast milk are in contact with the newborn's intestinal barrier. Viral inocula were added to Caco-2 monolayers in DMEM-2% FBS for 2 h at 37 • C and 5% CO 2 . The inocula were replaced by DMEM-2% FBS and the cells were cultured at 37 • C and 5% CO 2 for 48 h. Cell layers were washed twice in PBS and subjected to vRNA extraction and RT-qPCR as described above.

Viral Particle Incubation with Free Fatty Acids

Viral inocula were incubated at 37 • C for 0 h, 1 h, 4 h, or 6 h and titrated in triplicates by plaque forming assays as described below (Tables 1 and2). Caco-2 cells were incubated with DMEM-2% FBS or different concentrations of free fatty acids, LF, or LA for 1 h at 37 • C before ZIKV infection (MOI 1). Viral inocula were added to Caco-2 monolayers in DMEM-2% FBS for 2 h at 37 • C and 5% CO 2 then replaced by DMEM-2% FBS. The cells were cultured at 37 • C and 5% CO 2 for 48 h. Cell layers were washed twice in PBS and subjected to vRNA extraction and RT-qPCR as described above. The results can be found in Supplementary Figure S4.

Statistical Analysis

All statistical analyses were performed using Prism 9.1.0 software. The number of experiments and the statistical tests used are detailed in the legends of the figures and presented as mean ± SD.

Results

ZIKV Is Efficiently Transmitted to Breastfed Pups

In order to study the transmission of ZIKV to breastfed newborn pups, 6-to 13-weekold female mice were mated with male mice. They were infected once with ZIKV (10 6 PFU) through the subcutaneous route, between 1 and 11 days post-partum depending on the day of parturition. Infection was assessed by RT-qPCR and plaque assays performed on plasma samples collected at 3 days post-infection. Breast milk samples and pup spleens were collected at 4 days post-infection in order to investigate the presence of ZIKV (Figure 1A). Plasma samples were obtained by blood centrifugation and approximatively 10 5 vRNA copies were detected per microliter of plasma and 10 8 infectious particles per milliliter of plasma, confirming the infection of the dams with ZIKV (Figure 1B). Pup infection was investigated by amplifying vRNA from pup spleens by RT-qPCR (Figure 1C). More than 90% of the pups that were breastfed by ZIKV-infected mothers were infected, with an average of 42.3 vRNA copies/mg of spleen.

Infectious Particles Are Excreted in Murine Breast Milk

To evaluate the risk of transmission of ZIKV to the newborn, the presence of ZIKV was investigated in breast milk samples of infected dams. The dam infections and sample collection were performed as previously described and murine breast milk was harvested at 4 days post-infection (Figure 1A). To assess the infectious nature of the breast milk, the presence of infectious particles was investigated by plaque assays. The tests were performed either in whole breast milk samples or in fractions isolated by three spin-wash cycles (Figure 2A). To assess infectivity of breast milk, the latter was subjected to plaque assays and revealed 3.41 × 10 8 PFU/mL (±standard deviation (SD) = 4.8 × 10 8 , 11 samples) of whole milk. The cream and whey fractions contained, respectively, 1.09 × 10 7 (±SD = 10 7 , 5 samples) and 1.12 × 10 9 (±SD = 1.4 × 10 9 , 5 samples) PFU/mL, demonstrating a high concentration of free infectious viral particles in murine breast milk (Figure 2B).

Murine Breast Milk Cells Are in Majority Mammary Epithelial Cells

To determine which cell types are present in murine breast milk, we isolated the cell content from murine breast milk samples collected as previously described (Figure 1A). After separation of the different breast milk fractions by centrifugation, the cell pellet was washed twice in PBS in order to eliminate free viral particles. Immune staining and flow cytometry were performed to determine the cell types present in murine breast milk (Figure 3A). To detect epithelial and immune cells, anti-CD24 and anti-CD45 antibodies were used, respectively. An average of 70.9% (n = 2) and 88.5% (n = 7) of epithelial cells were detected in the breast milk of MOCK-infected and ZIKV-infected dams, respectively. An average of 2.85% and 1.28% of immune cells were detected in MOCK (n = 2) and ZIKV (n = 7) breast milk samples (Figure 3B,D). To further characterize the epithelial cells detected in murine breast milk, CD49f expression was studied as it is differentially expressed by mammary epithelial cells. Indeed, luminal cells are CD24 high /CD49f + while myoepithelial cells are CD24 low /CD49f + , and progenitor cells (immature luminal cells) are CD24 high /CD49f high . In all breast milk samples, the majority of cells were CD24 high /CD49f + with an average of 68% in MOCK samples (n = 2) and 79.1% in ZIKV samples (n = 7). No clear population of CD24 low /CD49f + cells was detected in any of the samples, while some (MOCK: 2/2; ZIKV: 5/7) contained CD24 high /CD49f high cells ranging from 1.24% to 8.6% in MOCK samples and 1.88% to 15.2% in ZIKV samples (Figure 3C,E).

Infectious Cell-Associated ZIKV is Excreted in Murine Breast Milk

In order to determine whether cells present in the milk are capable of transmitting the infection, co-cultures were performed between milk cells (purified as previously) and permissive target cells (Vero) in presence of carboxymethylcellulose (CMC). Vero cell infection was assessed by the observation of plaque formation after crystal violet staining (Figure 4A). In the same murine breast milk samples, the viral titers in whole milk (Figure 4B) and the number of infectious cells per million of total cells (Figure 4C) were deter-

Infectious Cell-Associated ZIKV Is Excreted in Murine Breast Milk

In order to determine whether cells present in the milk are capable of transmitting the infection, co-cultures were performed between milk cells (purified as previously) and permissive target cells (Vero) in presence of carboxymethylcellulose (CMC). Vero cell infection was assessed by the observation of plaque formation after crystal violet staining (Figure 4A). In the same murine breast milk samples, the viral titers in whole milk (Figure 4B) and the number of infectious cells per million of total cells (Figure 4C) were determined. While viral titers in whole breast milk ranged from 6.33 × 10 4 to 1.08 × 10 9 PFU/mL (mean: 3.41 × 10 8 PFU/mL) in samples collected from ZIKV-infected dams, the number of cells capable of transmitting the infection ranged from 1.65 × 10 1 to 2.3 × 10 4 cells/10 6 cells (mean: 5 × 10 3 cells/10 6 cells), with a correlation between whole milk titers and the number of infectious cells (Spearman test, rho = 0.9).

Lactoferrin and Lactalbumin Are Not Antiviral against ZIKV

In order to identify proviral or antiviral breast milk factors, the effect of LF and LA on ZIKV infectivity was studied by pre-incubation of different physiological concentrations (as reported in human breast milk [START_REF] Yang | Concentration of Lactoferrin in Human Milk and Its Variation during Lactation in Different Chinese Populations[END_REF][START_REF] Jackson | A multinational study of α-lactalbumin concentrations in human milk[END_REF]) of these factors with viral particles and subsequent infection of Caco-2 cells. The Caco-2 cell line was chosen to assess ZIKV infectivity because, physiologically, ZIKV and breast milk are in contact with the newborn's intestinal barrier.

The infection of the target cells was quantified by RNA extraction and vRNA amplification by RT-qPCR (Figure 5). Approximatively 10 6 vRNA copies/µg of total RNA were detected in the Caco-2 cells with no significant difference between the non-treated, LF-treated, and LA-treated virus, suggesting the absence of effect of these factors on ZIKV infectivity.

Free Fatty Acids Are Antiviral against ZIKV

In order to identify whether free fatty acids exert antiviral activity against ZIKV, as they have been shown to inactivate other enveloped viruses, viral particles were pre-incubated with six free fatty acids present in human breast milk (C10:0, C14:0, C18:0, C18:1, C18:2, and C18:3) before Vero cell infection and titration by plaque assay (Figure 6A). Viral particles were either pre-incubated with culture medium, PFA 4%, free fatty acids, or the solvent used to resuspend them (either ethanol or DMSO) (Table 1). The cytotoxicity assays of free fatty acids on target cells can be found in Supplementary Figure S1 and only noncytotoxic concentrations were used for the antiviral assays (Table 2). No inactivation of the non-treated virus was observed over time, while pre-incubation with PFA 4% resulted in the inactivation of the virus in less than 1 h (first time point tested) as expected. The saturated fatty acids C10:0 and C14:0 exerted no effect on ZIKV infectivity for up to 6 h of incubation while C18:0 became antiviral after 4 h (Figure 6B). The mono-unsaturated fatty acid C18:1 inactivated ZIKV in less than 1 h of incubation, while the poly-unsaturated fatty acids C18:2 and C18:3 inactivated ZIKV in 4 h (Figure 6B), suggesting a role of the unsaturations in viral inactivation. The pre-incubation of the viral particles with ethanol or DMSO resulted in a slight decrease in viral infectivity for most of the concentrations tested except for the C18:0 control which decreased 5 log at 6 h of incubation. But as C18:0 inactivated ZIKV in less than 4 h of incubation (when the solvent did not), these results suggest an antiviral effect of C18:0.

Discussion

Since the 2007 outbreak, it has repeatedly been suggested that ZIKV can be transmitted to newborns through breastfeeding [START_REF] Desgraupes | Mother-to-Child Transmission of Arboviruses during Breastfeeding: From Epidemiology to Cellular Mechanisms[END_REF]. However, since several case reports described no transmission of ZIKV from infected mothers to their newborn [START_REF] Centeno-Tablante | Update on the Transmission of Zika Virus Through Breast Milk and Breastfeeding: A Systematic Review of the Evidence[END_REF], biological factors regulating the efficiency of this route of transmission might exist (viral load, viral form, immunological status, stage of lactation, breast milk composition, etc.). We investigated the role of breast milk and its components in viral transmission using a well-described animal model for ZIKV infection: A129 mice.

ZIKV has previously been shown, by our team [START_REF] Hubert | Productive Infection of Mouse Mammary Glands and Human Mammary Epithelial Cells by Zika Virus[END_REF] and Regla-Nava et al. [START_REF] Regla-Nava | Detection of Zika virus in mouse mammary gland and breast milk[END_REF], to disseminate to the mammary glands of infected mice. Here, we investigated the ability of ZIKV to be excreted in breast milk. High viral titers (10 9 PFU/mL) were detected in the breast milk of infected breastfeeding dams. Between the 12 milk samples tested, viral titers ranged from 6.33 × 10 4 to 1.08 × 10 9 PFU/mL of milk. Infectious particles had also been detected by Regla-Nava et al. in the breast milk of AG129 mice after amplification on Vero cells [START_REF] Regla-Nava | Detection of Zika virus in mouse mammary gland and breast milk[END_REF]. To better characterize the milk-associated virus, we separated different milk fractions. The cream and whey fractions were separated from the cell pellet by centrifugation and further isolated by several spin-wash cycles. Infectious viral particles were found in both fractions at 10 7 and 10 9 PFU/mL, respectively. These results demonstrate that infectious cell-free viral particles are present in breast milk, thus risking transmission to the offspring. Several mechanisms have been suggested to explain the origin of the viral particles in breast milk [START_REF] Desgraupes | Mother-to-Child Transmission of Arboviruses during Breastfeeding: From Epidemiology to Cellular Mechanisms[END_REF]: Free viral particles in breast milk could have been excreted by mammary luminal epithelial cells as they produce milk and are permissive to ZIKV infection [START_REF] Hubert | Productive Infection of Mouse Mammary Glands and Human Mammary Epithelial Cells by Zika Virus[END_REF]. Although it is less likely, viral particles could also be locally produced in breast milk by infected cells.

The cell pellet was also tested to investigate the ability of breast milk cells to transmit ZIKV. Infectious plaque forming units were detected when co-culturing milk-cells from infected dams with permissive Vero cells (whereas no plaques were observed using milkcells from MOCK-infected mothers), suggesting the existence of cell-associated virus within breast milk.

To describe more accurately the cell content of the breast milk and thus identify the potentially infectious cell types, we performed flow cytometry experiments using epithelial (CD24) and immune (CD45) cell markers. We observed a majority of epithelial cells ranging from 74.4% to 99% versus an average of 1.28% of CD45-positive cells in breast milk from infected dams, comparable to values observed in breast milk from MOCK dams. These results, obtained using murine milk, are comparable to studies performed on human milk [START_REF] Witkowska-Zimny | Cells of human breast milk[END_REF]: Witkowska-Zimny et al. detected from 60% to 98% of epithelial cells in mature human milk and less than 2% of immune cells. As the breast milk composition fluctuates during lactation, the epithelial-immune cell ratio varies over time. As described previously, the colostrum is rich in immunological factors whereas the transitional and mature milks are rich in nutritious elements. Here, in mature milk, we further characterized the epithelial cells to determine whether luminal, myoepithelial or progenitor cells from the mammary gland are found in breast milk. Using the CD49f marker, we showed that murine breast milk of ZIKV-infected mice contains mature luminal cells (74.8%) and immature progenitors (4.28%), but no clear population of myoepithelial cells was detected. These results seem coherent with the structure of the mammary epithelium as luminal and progenitor cells are in contact with the lumen of the lactiferous ducts while myoepithelial cells are found beneath the tight luminal cell monolayer [START_REF] Huebner | Developmental stratification of the mammary epithelium occurs through symmetry-breaking vertical divisions of apically positioned luminal cells[END_REF]. As luminal mammary cells are permissive to ZIKV [START_REF] Hubert | Productive Infection of Mouse Mammary Glands and Human Mammary Epithelial Cells by Zika Virus[END_REF] and predominant in murine breast milk, they could be infected or carry adsorbed ZIKV particles at their surface in the milk. According to a mechanism we suggested [START_REF] Desgraupes | Mother-to-Child Transmission of Arboviruses during Breastfeeding: From Epidemiology to Cellular Mechanisms[END_REF], infected mammary luminal cells could detach from the mammary epithelium, thus introducing cell-associated ZIKV into the breast milk.

The transmission of ZIKV to breastfed pups was evaluated in the spleens and revealed a very high rate of viral transmission (>90%), with splenic viral loads ranging from 0.93 to 4 × 10 2 vRNA copies/mg. The viral loads remain relatively low, most probably due to the very early stage of the pups' infection [START_REF] Hubert | Evidence That Zika Virus Is Transmitted by Breastfeeding to Newborn A129 (Ifnar1 Knock-Out) Mice and Is Able to Infect and Cross a Tight Monolayer of Human Intestinal Epithelial Cells[END_REF].

According to our previous studies, ZIKV is produced in mouse mammary glands with a peak at 6 days post-subcutaneous infection [START_REF] Hubert | Productive Infection of Mouse Mammary Glands and Human Mammary Epithelial Cells by Zika Virus[END_REF]. Because we observed a reduction in milk production as early as day 5, breast milk samples had to be collected at day 4. As the dams were euthanized after harvesting as much breast milk as possible, pup spleens had to be collected at day 4 as well. However, to optimize the study of the transmission during breastfeeding and detect higher viral loads, pup spleens should be harvested a few days after the peak of production and excretion of ZIKV by the mammary gland. Previously, we already demonstrated that ZIKV is efficiently transmitted during breastfeeding to 1-to 3-day-old pups [START_REF] Hubert | Evidence That Zika Virus Is Transmitted by Breastfeeding to Newborn A129 (Ifnar1 Knock-Out) Mice and Is Able to Infect and Cross a Tight Monolayer of Human Intestinal Epithelial Cells[END_REF]. Here, we show that older pups are infected through breastfeeding as well (up to 11 days old), even as the impermeability of the intestinal barrier increases [START_REF] Holmes | Claudin profiling in the mouse during postnatal intestinal development and along the gastrointestinal tract reveals complex expression patterns[END_REF].

To understand which biological factors could regulate the transmission of ZIKV during breastfeeding, we investigated the effect of several antiviral factors of breast milk, such as LF, LA, and free fatty acids, on ZIKV infectivity. No antiviral effect of LF or LA was observed on ZIKV, although they have been shown to decrease viral infectivity of multiple viruses, such as HIV-1 and CMV [START_REF] Desgraupes | Mother-to-Child Transmission of Arboviruses during Breastfeeding: From Epidemiology to Cellular Mechanisms[END_REF][START_REF] Ng | Antiviral activities of whey proteins[END_REF]: LF prevents viral adsorption at the cell membrane, either by interacting with the viral particle (e.g., with gp120 of HIV-1 [START_REF] Swart | Antiviral Effects of Milk Proteins: Acylation Results in Polyanionic Compounds with Potent Activity against Human Immunodeficiency Virus Types 1 and 2in Vitro[END_REF]) or with the target cells (e.g., with heparins [START_REF] Van Berkel | N-terminal stretch Arg2, Arg3, Arg4 and Arg5 of human lactoferrin is essential for binding to heparin, bacterial lipopolysaccharide, human lysozyme and DNA[END_REF]). The effect of LF and LA on ZIKV target cells was also investigated (Supplementary Figure S4) but no antiviral effect was observed at any of the physiological concentrations tested. Free fatty acids were shown to inactivate enveloped viruses by interfering with the viral envelope [START_REF] Thormar | Inactivation of enveloped viruses and killing of cells by fatty acids and monoglycerides[END_REF]. Here, we studied their effect on ZIKV to better understand the antiviral mechanism of breast milk stored at 4 • C. Both the saturated fatty acid C18:0 and unsaturated fatty acids C18:1, C18:2, and C18:3 exerted an antiviral effect against ZIKV at physiological concentrations. These results confirm the hypothetical mechanism of viral inactivation in refrigerated or frozen breast milk suggested by Pfaender et al. [START_REF] Pfaender | Inactivation of Hepatitis C Virus Infectivity by Human Breast Milk[END_REF]. Indeed, they suggested that upon refrigeration, the MFGM becomes altered, thus exposing the triglyceride core to lipases present in breast milk. The free fatty acids produced would then alter the viral envelope integrity. However, as breast milk storage is responsible for the increase in free fatty acids [START_REF] Pfaender | Inactivation of Hepatitis C Virus Infectivity by Human Breast Milk[END_REF], fresh breast milk does not have an antiviral effect, as demonstrated by Conzelmann et al. [START_REF] Conzelmann | Storage-Dependent Generation of Potent Anti-ZIKV Activity in Human Breast Milk[END_REF]. Finally, free fatty acids did not decrease the infection when target cells were treated before viral adsorption, thus showing no effect on the cell receptors or on the early steps of the viral cycle.

Conclusions

ZIKV transmission through breastfeeding has been described in several cases in past years but is not systematic. In previous studies, we described the ability of ZIKV to cross the intestinal barrier of breastfed pups and to be transmitted by breastfeeding to very young pups (1 to 3 days old) [START_REF] Hubert | Evidence That Zika Virus Is Transmitted by Breastfeeding to Newborn A129 (Ifnar1 Knock-Out) Mice and Is Able to Infect and Cross a Tight Monolayer of Human Intestinal Epithelial Cells[END_REF]. Here, we show that even older pups (up to 11 days old) are also very susceptible to infection through breastfeeding. In order to understand the biological factors regulating this transmission, we performed in vivo infections of breastfeeding A129 dams and showed that ZIKV is excreted into breast milk as both cell-free and cell-associated virus. Among the different biological factors we investigated (LF, LA, and free fatty acids), the free fatty acids C18:0, C18:1, C18:2, and C18:3 were shown to exert an antiviral effect. As the free fatty acid content in breast milk varies with nutrition, these results should be considered in risk assessment of ZIKV transmission through breastfeeding in order to better advise infected mothers.

Figure 1 .

 1 Figure 1. ZIKV is efficiently transmitted to breastfed pups. (A): Methodology of in vivo infections and sample collection for all mouse experiments. Female A129 dams were infected at 1 to 11 days post-partum with 10 6 PFU of the Brazil/2016 strain of Zika virus (ZIKV) through the subcutaneous route (SC). Blood samples were collected from the dams at 3 days post-infection to assess ZIKV infection. Milk samples and pups' spleens were collected at 4 days post-infection to investigate the presence of ZIKV. (B): Dam infection was assessed by RT-qPCR (a) (n = 33 mice) and plaque assays (b) (n = 23 mice) in the plasma. (C): Transmission was evaluated by RT-qPCR in the spleens of the pups (n = 83 pups). ****: Significantly different (p < 0.0001) by Mann-Whitney test (p < 0.05).

Figure 2 .

 2 Figure 2. Infectious particles are excreted in murine breast milk. (A): Female dams were MOCK or ZIKV-infected with the Brazil/2016 strain as described in Figure 1. Breast milk samples were collected at 4 days post-infection and centrifuged to separate the cream fraction, the whey, and the cell pellet. The cream and whey fractions were isolated and further centrifuged twice with PBS to increase their purity. (B): The concentration of infectious viral particles in whole breast milk or breast milk fractions was determined by plaque assays (whole milk: n = 11; cream and whey: n = 5).

Viruses 2022 , 16 Figure 3 .

 2022163 Figure 3. Breast milk cells of infected or non-infected mice are in majority mammary epithelial cells. (A): Breast milk samples collected from MOCK or ZIKV-infected dams with the Brazil/2016 strain at 4 days post-infection were centrifuged to separate the cream fraction, the whey, and the cell pellet. The cells were then subjected to two spin-wash cycles in PBS in order to eliminate any free viral particles left from the liquid fractions. Antigen expression was analyzed by immune staining and flow cytometry. (B): Anti-CD45 and anti-CD24 antibodies were used to detect and quantify immune and epithelial cells respectively. (C): CD49f expression was used to identify mammary epithelial cells among the CD45-negative cells determined in (B). The results of one experiment are presented (MOCK: n = 2, ZIKV: n = 7). (D): histogram representation of CD45-positive (immune) or CD24positive (epithelial) cells in breast milk of MOCK or ZIKV-infected dams (MOCK: n = 2, ZIKV: n = 7). (E): histogram representation of mature and immature luminal mammary cells in breast milk, determined among the CD45-negative cells (MOCK: n = 2, ZIKV: n = 7).

Figure 3 .

 3 Figure 3. Breast milk cells of infected or non-infected mice are in majority mammary epithelial cells. (A): Breast milk samples collected from MOCK or ZIKV-infected dams with the Brazil/2016 strain at 4 days post-infection were centrifuged to separate the cream fraction, the whey, and the cell pellet. The cells were then subjected to two spin-wash cycles in PBS in order to eliminate any free viral particles left from the liquid fractions. Antigen expression was analyzed by immune staining and flow cytometry. (B): Anti-CD45 and anti-CD24 antibodies were used to detect and quantify immune and epithelial cells respectively. (C): CD49f expression was used to identify mammary epithelial cells among the CD45-negative cells determined in (B). The results of one experiment are presented (MOCK: n = 2, ZIKV: n = 7). (D): histogram representation of CD45-positive (immune) or CD24-positive (epithelial) cells in breast milk of MOCK or ZIKV-infected dams (MOCK: n = 2, ZIKV: n = 7). (E): histogram representation of mature and immature luminal mammary cells in breast milk, determined among the CD45-negative cells (MOCK: n = 2, ZIKV: n = 7).

Figure 4 :Figure 4 .

 44 Figure 4 : Infectious cell-associated ZIKV is excreted in murine breast milk.

Figure 5 .

 5 Figure 5. Lactoferrin and lactalbumin are not antiviral against ZIKV. ZIKV H/PF13 particles were incubated with culture medium (NT), lactoferrin, or lactalbumin (at physiological concentrations of human milk) for 1 h at 37 • C before infection of Caco-2 cells (MOI 1). At 48 h post-infection, cell layers were washed twice in PBS. After intracellular RNA extraction, viral RNAs were amplified by RT-qPCR (n = 3). ns: Non significantly different by Kruskal-Wallis test (p < 0.05).

Figure 6 .

 6 Figure 6. Free fatty acids are antiviral against ZIKV. (A): The fatty acids tested are either saturated fatty acids (SFA), mono-unsaturated fatty acids (MUFA), or polyunsaturated fatty acids (PUFA). Their nomenclature, structure, and abundance in human milk at 4 weeks after delivery (Cruz-Hernandez, 2013) are listed in the table. (B): To determine whether free fatty acids exert antiviral activity, 10 6 PFU of ZIKV particles (Brazil/2016) were incubated with culture medium (NT), paraformaldehyde 4% (PFA), free fatty acids (FFA), or the solvent used to dilute them (solvent) for 6 h. At several time points (0 h, 1 h, 4 h, and 6 h), the viral titers were determined by plaque assays (n = 3).

Table 1 .

 1 Free fatty acids were purchased from Sigma and resuspended at the following concentrations either in ethanol or DMSO.

	Fatty Acid	Stock Concentration (mg/mL)	Solvent	Reference
	C10:0	30	Ethanol	Sigma C1875
	C14:0	12	DMSO	Sigma M3128
	C18:0	22	Ethanol	Sigma S4751
	C18:1	100	DMSO	Sigma O1008
	C18:2	100	DMSO	Sigma L1376
	C18:3	100	DMSO	Sigma L2376

Table 2 .

 2 ZIKV (10 6 PFU) was incubated either with DMEM-2% FBS, 4% PFA, or with the following non-cytotoxic concentrations of free fatty acids.

	Fatty Acid	Assay Concentration (mg/mL)	Condition
	C10:0	0.6	Physiological [29]
	C14:0	2	Physiological [29]
	C18:0	2	Physiological [29]
	C18:1	4	Maximum non-cytotoxic
	C18:2	3.5	Maximum non-cytotoxic
	C18:3	0.5	Physiological [29]

9

Acknowledgments:

We thank Mathieu Hubert (Institut Pasteur) for his participation in the milk collection during the last experiment, Philippe Afonso (Institut Pasteur) for improving the manuscript, Emeline Perthame (Bioinformatics and Biostatistics HUB, Institut Pasteur) for improving the statistical analysis and Valérie Choumet (Institut Pasteur) for helpful advice.

Supplementary Materials:

The following supporting information can be downloaded at: https: //www.mdpi.com/article/10.3390/v14050851/s1, Figure S1 Informed Consent Statement: Not applicable.

Conflicts of Interest:

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.