Paolo Bellingeri

Luis

Luis Paris

Virtual braids and permutations

Introduction

The study of the homomorphisms from the braid group B n on n strands to the symmetric group S n goes back to Artin [START_REF] Artin | Braids and permutations[END_REF] himself. He was right thinking that this would be an important step toward the determination of the automorphism group of B n . As pointed out by Lin [START_REF] Lin | Artin braids and related groups and spaces (Russian)[END_REF][START_REF] Lin | Braids and permutations[END_REF], all the homomorphisms from B n to S m for n ≥ m are easily deduced from the ideas of Artin [START_REF] Artin | Braids and permutations[END_REF]. The automorphism group of B n was then determined by Dyer-Grossman [START_REF] Dyer | The automorphism groups of the braid groups[END_REF] using in particular Artin's results in [START_REF] Artin | Braids and permutations[END_REF]. The homomorphisms from B n to B m were determined by Lin [START_REF] Lin | Artin braids and related groups and spaces (Russian)[END_REF][START_REF] Lin | Braids and permutations[END_REF] for n > m and by Castel [START_REF] Castel | Geometric representations of the braid groups[END_REF] for n = m and n ≥ 6.

Virtual braids were introduced by Kauffman [START_REF] Kauffman | Virtual knot theory[END_REF] together with virtual knots and links. They have interpretations in terms of diagrams (see Kauffman [START_REF] Kauffman | Virtual knot theory[END_REF], Kamada [START_REF] Kamada | Braid presentation of virtual knots and welded knots[END_REF] and Vershinin [START_REF] V V Vershinin | On homology of virtual braids and Burau representation[END_REF]) and also in terms of braids in thickened surfaces (see Cisneros de la Cruz [START_REF] De La | Virtual braids from a topological viewpoint[END_REF]) and now there is a quite extensive literature on them. Despite their interest in low-dimensional topology, the virtual braid groups are poorly understood from a combinatorial point of view. Among the known results for these groups there are solutions to the word problem in Bellingeri-Cisneros de la Cruz-Paris [START_REF] Bellingeri | A simple solution to the word problem for virtual braid groups[END_REF] and in Chterental [START_REF]Virtual braids and virtual curve diagrams[END_REF] and the calculation of some terms of its lower central series in Bardakov-Bellingeri [START_REF] V G Bardakov | Combinatorial properties of virtual braids[END_REF], and some other results but not many more. For example, it is not known whether these groups have a solution to the conjugacy problem or whether they are linear.

In the present paper we prove results on virtual braid groups in the same style as those previously mentioned for the braid groups. More precisely, we take n, m ∈ N such that n ≥ 5, m ≥ 2 and n ≥ m and

(1) we determine all the homomorphisms from VB n to S m , [START_REF] Artin | Braids and permutations[END_REF] we determine all the homomorphisms from S n to VB m , [START_REF] Bardakov | The virtual and universal braids[END_REF] we determine all the homomorphisms from VB n to VB m .

From these classifications it will follow that Out(VB n ) is isomorphic to Z/2Z × Z/2Z, and that VB n is both, Hopfian and co-Hopfian.

Our study is totally independent from the works of Dyer-Grossman [START_REF] Dyer | The automorphism groups of the braid groups[END_REF], Lin [START_REF] Lin | Artin braids and related groups and spaces (Russian)[END_REF][START_REF] Lin | Braids and permutations[END_REF] and Castel [START_REF] Castel | Geometric representations of the braid groups[END_REF] cited above and all similar works on braid groups. Our viewpoint/strategy is also new for virtual braid groups, although some aspects are already present in Bellingeri-Cisneros de la Cruz-Paris [START_REF] Bellingeri | A simple solution to the word problem for virtual braid groups[END_REF]. Our idea/main-contribution consists on first observing that the virtual braid group VB n decomposes as a semi-direct product VB n = KB n S n of an Artin group KB n by the symmetric group S n , and secondly and mainly on making a deep study of the action of S n on KB n . From this study we deduce that there exists a unique embedding of the symmetric group S n into the virtual braid group VB n up to conjugation (see Lemma 5.1). From there the classifications are deduced with some extra work.

The group VB 2 is isomorphic to Z * Z/2Z. On the other hand, fairly precise studies of the combinatorial structure of VB 3 can be found in Bardakov-Bellingeri [START_REF] V G Bardakov | Combinatorial properties of virtual braids[END_REF], Bardakov-Mikhailov-Vershinin-Wu [START_REF] V G Bardakov | On the pure virtual braid group PV 3[END_REF] and Bellingeri-Cisneros de la Cruz-Paris [START_REF] Bellingeri | A simple solution to the word problem for virtual braid groups[END_REF]. From this one can probably determine (with some difficulties) all the homomorphisms from VB n to S m , from S n to VB m , and from VB n to VB m , for n ≥ m and n = 2, 3. Our guess is that the case n = 4 will be much more tricky.

Our paper is organized as follows. In Section 2 we give the definitions of the homomorphisms involved in the paper, we state our three main theorems, and we prove their corollaries. Section 3 is devoted to the study of the action of the symmetric group S n on the above mentioned group KB n . The results of this section are technical but they provide key informations on the structure of VB n that are essential in the proofs of our main theorems. We also think that they are interesting by themselves and may be used for other purposes. Our main theorems are proved in Section 4, Section 5 and Section 6, respectively.

Definitions and statements

In the paper we are interested in the algebraic and combinatorial aspects of virtual braid groups, hence we will adopt their standard definition in terms of generators and relations. So, the virtual braid group VB n on n strands is the group defined by the presentation with generators σ 1 , . . . , σ n-1 , τ 1 , . . . , τ n-1 and relations

τ 2 i = 1 for 1 ≤ i ≤ n -1 , σ i σ j = σ j σ i , τ i τ j = τ j τ i , τ i σ j = σ j τ i for |i -j| ≥ 2 , σ i σ j σ i = σ j σ i σ j , τ i τ j τ i = τ j τ i τ j , τ i τ j σ i = σ j τ i τ j for |i -j| = 1 .
For each i ∈ {1, . . . , n -1} we set s i = (i, i + 1). It is well-known that the symmetric group S n on n letters has a presentation with generators s 1 , . . . , s n-1 and relations

s 2 i = 1 for 1 ≤ i ≤ n -1 , s i s j = s j s i for |i -j| ≥ 2 , s i s j s i = s j s i s j for |i -j| = 1 .
Now, we define the homomorphisms that are involved in the paper. Let G, H be two groups. We say that a homomorphism ψ : G → H is Abelian if the image of ψ is an Abelian subgroup of H . Note that the abelianization of S n is isomorphic to Z/2Z, hence the image of any Abelian homomorphism ϕ : S n → H is either trivial or a cyclic group of order 2. On the other hand, the abelianization of VB n is isomorphic to Z × Z/2Z, where the copy of Z is generated by the class of σ 1 and the copy of Z/2Z is generated by the class of τ 1 . Thus, the image of any Abelian homomorphism

ψ : VB n → H is a quotient of Z × Z/2Z.
From the presentations of VB n and S n given above we see that there are epimorphisms

π P : VB n → S n and π K : VB n → S n defined by π P (σ i ) = π P (τ j ) = s i for all 1 ≤ i ≤ n -1 and by π K (σ i ) = 1 and π K (τ j ) = s i for all 1 ≤ i ≤ n -1, respectively.
The kernel of π P is called the virtual pure braid group and is denoted by VP n . A presentation of this group can be found in Bardakov [START_REF] Bardakov | The virtual and universal braids[END_REF]. It is isomorphic to the group of the Yang-Baxter equation studied in Bartholdi-Enriquez-Etingof-Rains [START_REF] Bartholdi | Groups and Lie algebras corresponding to the Yang-Baxter equations[END_REF]. The kernel of π K does not have any particular name. It is denoted by KB n . It is an Artin group, hence we can use tools from the theory of Artin groups to study it and get results on VB n itself (see for example Godelle-Paris [START_REF] Godelle | K(π, 1) and word problems for infinite type Artin-Tits groups, and applications to virtual braid groups[END_REF] or Bellingeri-Cisneros de la Cruz-Paris [START_REF] Bellingeri | A simple solution to the word problem for virtual braid groups[END_REF]). This group will play a prominent role in our study. It will be described and studied in Section 3.

Using again the presentations of S n and VB n we see that there is a homomorphism ι : S n → VB n that sends s i to τ i for all 1 ≤ i ≤ n -1. Observe that ι is a section of both, π P and π K , hence ι is injective and we have the decompositions VB n = VP n S n and VB n = KB n S n .

The two main automorphisms of VB n that are involved in the paper are the automorphisms

ζ 1 , ζ 2 : VB n → VB n defined by ζ 1 (σ i ) = τ i σ i τ i and ζ 1 (τ i ) = τ i for all 1 ≤ i ≤ n -1, and by ζ 2 (σ i ) = σ -1 i and ζ 2 (τ i ) = τ i for all 1 ≤ i ≤ n -1,
) isomorphic to Z/2Z × Z/2Z.
The last homomorphism (automorphism) concerned by the paper appears only for n = 6. This is the automorphism ν 6 : S 6 → S 6 defined by

ν 6 (s 1 ) = (1, 2)(3, 4)(5, 6) , ν 6 (s 2 ) = (2, 3)(1, 5)(4, 6) , ν 6 (s 3 ) = (1, 3)(2, 4)(5, 6) , ν 6 (s 4 ) = (1, 2)(3, 5)(4, 6) , ν 6 (s 5 ) = (2, 3)(1, 4)(5, 6) .
It is well-known that Out(S n ) is trivial for n = 6 and that Out(S 6 ) is a cyclic group of order 2 generated by the class of ν 6 . Note also that ν 2 6 is the conjugation by w 0 = (1, 6, 2, 5, 3).

We give a last definition before passing to the statements. Let G, H be two groups. For each α ∈ H we denote by c α : H → H , β → αβα -1 , the conjugation by α. We say that two homomorphisms ψ 1 , ψ 2 : G → H are conjugate and we write ψ 1 ∼ c ψ 2 if there exists α ∈ H such that ψ 2 = c α • ψ 1 .

In Section 4 we determine the homomorphisms from VB n to S m up to conjugation. More precisely we prove the following. Theorem 2.1 Let n, m ∈ N such that n ≥ 5, m ≥ 2 and n ≥ m, and let ψ : VB n → S m be a homomorphism. Then, up to conjugation, one of the following possibilities holds.

(1) ψ is Abelian, (2) n = m and ψ ∈ {π K , π P },

(3) n = m = 6 and ψ ∈ {ν 6 • π K , ν 6 • π P }.
In Section 5 we determine the homomorphisms from S n to VB m up to conjugation. More precisely we prove the following. Theorem 2.2 Let n, m ∈ N such that n ≥ 5, m ≥ 2 and n ≥ m, and let ϕ : S n → VB m be a homomorphism. Then, up to conjugation, one of the following possibilities holds.

(1) ϕ is Abelian,

(2) n = m and ϕ = ι,

(3) n = m = 6 and ϕ = ι • ν 6 .
Finally, in Section 6 we determine the homomorphisms from VB n to VB m up to conjugation. More precisely we prove the following.

Theorem 2.3 Let n, m ∈ N such that n ≥ 5, m ≥ 2 and n ≥ m, and let ψ : VB n → VB m be a homomorphism. Then, up to conjugation, one of the following possibilities holds.

(1) ψ is Abelian,

(2) n = m and ψ ∈ {ι • π K , ι • π P }, (3) n = m = 6 and ψ ∈ {ι • ν 6 • π K , ι • ν 6 • π P }, (4) n = m and ψ ∈ {id, ζ 1 , ζ 2 , ζ 1 • ζ 2 } = ζ 1 , ζ 2 .
A group G is called Hopfian if every surjective homomorphism ψ : G → G is also injective. On the other hand, it is called co-Hopfian if every injective homomorphism ψ : G → G is also surjective. It is known that the braid group B n is Hopfian but not co-Hopfian. However, it is quasi-co-Hopfian by Bell-Margalit [START_REF] Bell | Braid groups and the co-Hopfian property[END_REF]. The property of hopfianity for the braid group B n follows from the fact that it can be embedded in the automorphism group of the free group F n (see Artin [START_REF] Artin | Theory of braids[END_REF]). We cannot apply such an argument for the virtual braid group VB n since we do not know if it can be embedded into the automorphism group of a finitely generated free group. A first consequence of Theorem 2.3 is the following. Proof We see in Theorem 2.3 that, up to conjugation, the only surjective homomorphisms from VB n to VB n are the elements of {id,

ζ 1 , ζ 2 , ζ 1 • ζ 2 } = ζ 1 , ζ 2 ,
and they are all automorphisms, hence VB n is Hopfian. We show in the same way that VB n is co-Hopfian.

Recall that, by Dyer-Grossman [START_REF] Dyer | The automorphism groups of the braid groups[END_REF], the group Out(B n ) is isomorphic to Z/2Z. Here we show that Out(VB n ) is a little bigger, that is: The proof of Corollary 2.5 relies on the following lemma whose proof is given in Section 3.

Corollary 2.5 Let n ∈ N, n ≥ 5. Then Out(VB n ) is isomorphic to Z/2Z × Z/2Z
Lemma 2.6 Let n ∈ N, n ≥ 5. Then ζ 1 is not an inner automorphism of VB n .
Proof of Corollary 2.5 It follows from Theorem 2.3 that Out(VB n ) is generated by the classes of ζ 1 and ζ 2 . We also know that these two automorphisms are of order two and commute. So, it suffices to show that none of the elements

ζ 1 , ζ 2 , ζ 1 • ζ 2 is an inner automorphism.
It is easily seen from its presentation that the abelianization of VB n is isomorphic to Z × Z/2Z, where the copy of Z is generated by the class of σ 1 and the copy of Z/2Z is generated by the class of τ 1 . Since ζ 2 (σ 1 ) = σ -1 1 , ζ 2 acts non-trivially on the abelianization of VB n , hence ζ 2 is not an inner automorphism. The transformation ζ 1 • ζ 2 is not an inner automorphism for the same reason, and ζ 1 is not an inner automorphism by Lemma 2.6.

In order to determine Out(B n ), Dyer and Grossman [START_REF] Dyer | The automorphism groups of the braid groups[END_REF] use also another result of Artin [START_REF] Artin | Braids and permutations[END_REF] which says that the pure braid group on n strands is a characteristic subgroup of B n . From Theorem 2.3 it immediately follows that the equivalent statement for virtual braid groups holds. More precisely, we have the following.

Corollary 2.7 Let n ∈ N, n ≥ 5. Then the groups VP n and KB n are both characteristic subgroups of VB n .

Preliminaries

Let n ≥ 3. Recall that we have an epimorphism π K : VB n → S n which sends σ i to 1 and τ i to s i for all 1 ≤ i ≤ n -1. Recall also that KB n denotes the kernel of π K and that we have the decomposition VB n = KB n S n . The aim of the present section is to prove three technical results on the action of S n on KB n (Lemma 3.7, Lemma 3.10 and Lemma 3.12). These three lemmas are key points in the proofs of Theorem 2.2 and Theorem 2.3. We think also that they are interesting by themselves and may be use in the future for other purposes. The two main tools in the proofs of these lemmas are the Artin groups and the amalgamated products.

The following is proved in Rabenda's master thesis at the Université de Bourgogne in Dijon in 2003. This thesis is actually unavailable but the proof of the proposition can also be found in Bardakov-Bellingeri [START_REF] V G Bardakov | Combinatorial properties of virtual braids[END_REF].

Proposition 3.1 For 1 ≤ i < j ≤ n we set δ i,j = τ i τ i+1 • • • τ j-2 σ j-1 τ j-2 • • • τ i+1 τ i , δ j,i = τ i τ i+1 • • • τ j-2 τ j-1 σ j-1 τ j-1 τ j-2 • • • τ i+1 τ i .
Then KB n has a presentation with generating set

S = {δ i,j | 1 ≤ i = j ≤ n} ,
and relations

δ i,j δ k, = δ k, δ i,j for i, j, k, pairwise distinct , δ i,j δ j,k δ i,j = δ j,k δ i,j δ j,k for i, j, k pairwise distinct .
Moreover, the action of each w ∈ S n on a generator δ i,j is by permutation of the indices, that is, w(δ i,j ) = δ w(i),w(j) .

Let S be a finite set. A Coxeter matrix over S is a square matrix M = (m s,t ) s,t∈S indexed by the elements of S such that m s,s = 1 for all s ∈ S and m s,t = m t,s ∈ {2, 3, 4, . . . } ∪ {∞} for all s, t ∈ S , s = t. For s, t ∈ S , s = t, and m ≥ 2, we denote by Π(s, t, m) the word sts • • • of length m. In other words, we have Π(s, t, m) = (st)

m 2 if m is even and Π(s, t, m) = (st) m-1
2 s if m is odd. The Artin group associated with the Coxeter matrix M = (m s,t ) s,t∈S is the group A = A M defined by the presentation

A = S | Π(s, t, m s,t ) = Π(t, s, m s,t ) for s, t ∈ S, s = t, m s,t = ∞ . Example Let S = {δ i,j | 1 ≤ i = j ≤ n}. Define a Coxeter matrix M = (m s,t ) s,t∈S
as follows. Set m s,s = 1 for all s ∈ S . Let s, t ∈ S , s = t. If s = δ i,j and t = δ k, , where i, j, k, are pairwise distinct, then set m s,t = 2. If s = δ i,j and t = δ j,k , where i, j, k are pairwise distinct, then set m s,t = m t,s = 3. Set m s,t = ∞ for all other cases. Then, by Proposition 3.1, KB n is the Artin group associated with M = (m s,t ) s,t∈S .

If X is a subset of S , then we set M[X ] = (m s,t ) s,t∈X and we denote by A[X ] the subgroup of A generated by X . Theorem 3.2 (Van der Lek [START_REF] Van Der Lek | The homotopy type of complex hyperplane complements[END_REF]) Let A be the Artin group associated with a Coxeter matrix M = (m s,t ) s,t∈S , and let X be a subset of S . Then A[X ] is the Artin group associated with M

[X ]. Moreover, if X and Y are two subsets of S , then A[X ]∩A[Y] = A[X ∩ Y].
In our example, for X ⊂ S we denote by KB n [X ] the subgroup of KB n generated by X . So, by Theorem 3.2, KB n [X ] is still an Artin group and it has a presentation with generating set X and only two types of relations:

• st = ts if m s,t = 2 in M[X ], • sts = tst if m s,t = 3 in M[X ].
The following result can be easily proved from Theorem 3.2 using presentations, but it is important to highlight it since it will be often used throughout the paper. Lemma 3.3 Let A be an Artin group associated with a Coxeter matrix M = (m s,t ) s,t∈S . Let X and Y be two subsets of S such that X ∪ Y = S and m

s,t = ∞ for all s ∈ X \ (X ∩ Y) and t ∈ Y \ (X ∩ Y). Then A = A[X ] * A[X ∩Y] A[Y].
Let G be a group and let H be a subgroup of G. A transversal of H in G is a subset T of G such that for each α ∈ G there exists a unique θ ∈ T such that αH = θH . For convenience we will always suppose that a transversal T contains 1 and we set T * = T \ {1}. The following is classical in the theory and is proved in Serre [20, Section 1.1, Theorem 1]. Theorem 3.4 (Serre [START_REF] Serre | amalgames[END_REF]) Let G 1 , . . . , G p , H be a collection of groups. We suppose that H is a subgroup of G j for all j ∈ {1, . . . , p} and we consider the amalgamated product

G = G 1 * H G 2 * H • • • * H G p .
For each j ∈ {1, . . . , p} we choose a transversal T j of H in G j . Then each element α ∈ G can be written in a unique way in the form α = θ 1 θ 2 • • • θ β such that:

(a) β ∈ H and, for each i ∈ {1, . . . , }, there exists j = j(i) ∈ {1, . . . , p} such that

θ i ∈ T * j = T j \ {1}, (b) j(i) = j(i + 1) for all i ∈ {1, . . . , -1}.
In particular, we have α ∈ H if and only if = 0 and α = β .

The expression of α given in the above theorem is called the normal form of α. It depends on the amalgamated product and on the choice of the transversal of H in G j for all j.

Consider the notation introduced in

Theorem 3.4. So, G = G 1 * H • • • * H G p and T j is a transversal of H in G j for all j ∈ {1, . . . , p}. Let α ∈ G. We suppose that α is written in the form α = α 1 • • • α such that ≥ 1, (a) for each i ∈ {1, . . . , } there exists j = j(i) ∈ {1, . . . , p} such that α i ∈ G j \ H , (b) j(i) = j(i + 1) for all i ∈ {1, . . . , -1}.
We define β i ∈ H and θ i ∈ T * j(i) for i ∈ {1, . . . , } by induction on i as follows. First, θ 1 is the element of T * j(1) such that α 1 H = θ 1 H and β 1 is the element of H such that α 1 = θ 1 β 1 . We suppose that i ≥ 2 and that

β i-1 ∈ H is defined. Then θ i is the element of T * j(i) such that β i-1 α i H = θ i H and β i is the element of H such that β i-1 α i = θ i β i .
The following result is quite obvious and its proof is left to the reader. Proposition 3.5 Under the above notations and hypothesis α = θ 1 • • • θ β is the normal form of α. In particular, α ∈ H (since ≥ 1).

We turn now to the proofs of our technical lemmas. Lemma 3.6 Let G 1 , G 2 , H be three groups. We suppose that H is a common subgroup of G 1 and G 2 and we set

G = G 1 * H G 2 . Let τ : G → G be an automorphism of order 2 such that τ (G 1 ) = G 2 and τ (G 2 ) = G 1 . Let G τ = {α ∈ G | τ (α) = α}. Then G τ ⊂ H . Proof Since τ (G 1 ) = G 2 , τ (G 2 ) = G 1 and H = G 1 ∩ G 2 , we have τ (H) = H . Let T 1 be a transversal of H in G 1 . Set T 2 = τ (T 1 ). Then T 2 is a transversal of H in G 2 . Let α ∈ G and let α = θ 1 θ 2 • • • θ β be the normal form of α. Notice that the normal form of τ (α) is τ (θ 1 ) τ (θ 2 ) • • • τ (θ ) τ (β). Suppose that ≥ 1.
Without loss of generality we can assume that θ 1 ∈ T * 1 . Then τ (θ 1 ) ∈ T * 2 , hence θ 1 = τ (θ 1 ), thus α = τ (α), and therefore α ∈ G τ . So, if α ∈ G τ , then = 0 and α = β ∈ H .

Recall from Proposition 3.1 that VB n = KB n S n and that the action of S n on KB n is defined by w(δ i,j ) = δ w(i),w(j) for all w ∈ S n and all δ i,j ∈ S . Recall also that for each generator s i of S n and each α ∈ KB n we have s i (α) = τ i ατ i . Throughout the paper we will use both interpretations, as action of s i and as conjugation by

τ i . For each 1 ≤ k ≤ n we set S k = {δ i,j | k ≤ i = j ≤ n}. Lemma 3.7 Let X be a subset of S invariant under the action of s 1 . Then KB n [X ] s 1 = KB n [X ∩ S 3 ]. Proof Set U = {δ i,j ∈ X | (i, j) ∈ {(1, 2), (2, 1)}}. We first prove that KB n [X ] s 1 ⊂ KB n [U]. If X ⊂ U
there is nothing to prove. We can therefore suppose that X ⊂ U . Since X is invariant under the action of s 1 , we have

X = U ∪ {δ 1,2 , δ 2,1 }. Set U = U ∪{δ 1,2 } and U = U ∪{δ 2,1 }. By Lemma 3.3, KB n [X ] = KB n [U ] * KBn[U ] KB n [U ]. Moreover, s 1 (KB n [U ]) = KB n [U ] and s 1 (KB n [U ]) = KB n [U ]. So, by Lemma 3.6, KB n [X ] s 1 ⊂ KB n [U]. For 2 ≤ k ≤ n we set V k = {δ i,j ∈ X | (i, j) ∈ ({1, 2} × {1, 2, . . . , k})}. We show by induction on k that KB n [X ] s 1 ⊂ KB n [V k ]. Since V 2 = U the case k = 2 holds.
Suppose that k ≥ 3 and that the inductive hypothesis holds, that is, KB n

[X ] s 1 ⊂ KB n [V k-1 ]. If V k = V k-1 there is nothing to prove. We can therefore suppose that V k = V k-1 . Since V k-1 is invariant under the action of s 1 we have V k-1 = V k ∪ {δ 1,k , δ 2,k }. Set V k = V k ∪ {δ 1,k } and V k = V k ∪ {δ 2,k }. By Lemma 3.3, KB n [V k-1 ] = KB n [V k ] * KBn[V k ] KB n [V k ]. Moreover, s 1 (KB n [V k ]) = KB n [V k ] and s 1 (KB n [V k ]) = KB n [V k ]. So, by Lemma 3.6, KB n [X ] s 1 ⊂ KB n [V k ]. For 2 ≤ k ≤ n we set W k = {δ i,j ∈ X | (i, j) ∈ ({1, 2} × {1, 2, . . . , n}) and (i, j) ∈ ({1, 2, . . . , k} × {1, 2})}. We show by induction on k that KB n [X ] s 1 ⊂ KB n [W k ]. Since W 2 = V n the case k = 2 holds. Suppose that k ≥ 3 and that the inductive hypothesis holds, that is, KB n [X ] s 1 ⊂ KB n [W k-1 ]. If W k = W k-1 there is nothing to prove. We can therefore suppose that W k = W k-1 . Since W k-1 is invariant under the action of s 1 we have W k-1 = W k ∪ {δ k,1 , δ k,2 }. Set W k = W k ∪ {δ k,1 } and W k = W k ∪ {δ k,2 }. By Lemma 3.3, KB n [W k-1 ] = KB n [W k ] * KBn[W k ] KB n [W k ]. Moreover, s 1 (KB n [W k ]) = KB n [W k ] and s 1 (KB n [W k ]) = KB n [W k ]. So, by Lemma 3.6, KB n [X ] s 1 ⊂ KB n [W k ]. The inclusion KB n [X ] s 1 ⊂ KB n [X ∩ S 3 ] therefore follows from the fact that W n = X ∩ S 3 . The reverse inclusion KB n [X ∩ S 3 ] ⊂ KB n [X ] s 1 is obvious.
By applying the action of the symmetric group, from Lemma 3.7 it immediately follows: Corollary 3.8 Let k ∈ {1, . . . , n-1}, let X be a subset of S invariant under the action of s k , and let

U k = {δ i,j ∈ S | i, j ∈ {k, k + 1}}. Then KB n [X ] s k = KB n [X ∩ U k ].
Lemma 3.9 Let G 1 , G 2 , H be three groups. We suppose that H is a common subgroup of G 1 and G 2 and we set

G = G 1 * H G 2 . Let τ : G → G be an automorphism of order 2 such that τ (G 1 ) = G 2 and τ (G 2 ) = G 1 . Let α ∈ G such that τ (α) = α -1 . Then there exist α ∈ G and β ∈ H such that τ (β ) = β -1 and α = α β τ (α -1 ). Proof Since τ (G 1 ) = G 2 and τ (G 2 ) = G 1 , we have τ (H) = H . Let T 1 be a transversal of H in G 1 . Then T 2 = τ (T 1 ) is a transversal of H in G 2 . Let α ∈ G such that τ (α) = α -1 . Let α = θ 1 θ 2 • • • θ β
be the normal form of α. We prove by induction on that there exist α ∈ G and β ∈ H such that τ (β ) = β -1 and α = α β τ (α -1 ). The case = 0 being trivial we can suppose that ≥ 1 and that the inductive hypothesis holds. We have

1 = α τ (α) = θ 1 • • • θ β τ (θ 1 ) • • • τ (θ ) τ (β).
By Proposition 3.5 we must have θ β τ (θ 1 ) ∈ H , namely, there exists β 1 ∈ H such that θ β = β 1 τ (θ 1 ) -1 . Note that this inclusion implies that is even (and therefore ≥ 2), since we should have either θ , τ (θ 1 )

∈ G 1 \ H or θ , τ (θ 1 ) ∈ G 2 \ H . Thus, α = θ 1 θ 2 • • • θ -1 β 1 τ (θ -1 1 ) = θ 1 α 1 τ (θ -1 1 )
, where

α 1 = θ 2 • • • θ -1 β 1 . We have τ (α 1 ) = α -1 1 since τ (α) = α -1
, so, by induction, there exist α 1 ∈ G and β ∈ H such that τ (β ) = β -1 and α 1 = α 1 β τ (α -1 1 ). We set α = θ 1 α 1 . Then τ (β ) = β -1 and α = α β τ (α -1 ). Lemma 3.10 Let X be a subset of S invariant under the action of

s 1 . Let α ∈ KB n [X ] such that s 1 (α) = α -1 . Then there exists α ∈ KB n [X ] such that α = α s 1 (α -1 ). Proof Let α ∈ KB n [X ] such that s 1 (α) = α -1 . Let U = {δ i,j ∈ X | (i, j) ∈ {(1, 2
), (2, 1)}}. We show that there exist α ∈ KB n [X ] and β ∈ KB n [U] such that s 1 (β ) = β -1 and α = α β s 1 (α -1 ). If X = U there is nothing to prove. We can therefore suppose that X = U . Since X is invariant under the action of s 1 we have

X = U ∪ {δ 1,2 , δ 2,1 }. Set U = U ∪ {δ 1,2 } and U = U ∪ {δ 2,1 }. By Lemma 3.3, KB n [X ] = KB n [U ] * KBn[U ] KB n [U ]. Moreover, s 1 (KB n [U ]) = KB n [U ] and s 1 (KB n [U ]) = KB n [U ]
. So, by Lemma 3.9, there exist α ∈ KB n [X ] and β ∈ KB n [U] such that s 1 (β ) = β -1 and α = α β s 1 (α -1 ).

For 2 ≤ k ≤ n we set V k = {δ i,j ∈ X | (i, j) ∈ ({1, 2} × {1, 2, . . . , k})}. We show by induction on k that there exist α ∈ KB n [X ] and β ∈ KB n [V k ] such that s 1 (β ) = β -1 and α = α β s 1 (α -1 ). Since V 2 = U the case k = 2 follows from the previous paragraph. Suppose that k ≥ 3 and that the inductive hypothesis holds. So, there exist α 1 ∈ KB n [X ] and

β 1 ∈ KB n [V k-1 ] such that s 1 (β 1 ) = β -1 1 and α = α 1 β 1 s 1 (α -1 1 ). If V k = V k-1
there is nothing to prove. Thus, we can suppose that

V k = V k-1 . Since V k-1 is invariant under the action of s 1 we have V k-1 = V k ∪ {δ 1,k , δ 2,k }. Set V k = V k ∪ {δ 1,k } and V k = V k ∪ {δ 2,k }. By Lemma 3.3, KB n [V k-1 ] = KB n [V k ] * KBn[V k ] KB n [V k ]. Moreover, s 1 (KB n [V k ]) = KB n [V k ] and s 1 (KB n [V k ]) = KB n [V k ]. By Lemma 3.9 it follows that there exist α 2 ∈ KB n [V k-1 ] and β ∈ KB n [V k ] such that s 1 (β ) = β -1 and β 1 = α 2 β s 1 (α -1 2 ). Set α = α 1 α 2 . Then β ∈ KB n [V k ], s 1 (β ) = β -1 and α = α β s 1 (α -1 ). For 2 ≤ k ≤ n we set W k = {δ i,j ∈ X | (i, j) ∈ ({1, 2} × {1, 2
, . . . , n}) and (i, j) ∈ ({1, 2, . . . , k} × {1, 2})}. We show by induction on k that there exist α ∈ KB n [X ] and β ∈ KB n [W k ] such that s 1 (β ) = β -1 and α = α β s 1 (α -1 ). Since W 2 = V n the case k = 2 follows from the previous paragraph. Suppose that k ≥ 3 and that the inductive hypothesis holds. So, there exist α 1 ∈ KB n [X ] and

β 1 ∈ KB n [W k-1 ] such that s 1 (β 1 ) = β -1 1 and α = α 1 β 1 s 1 (α -1 1 ). If W k = W k-1
there is nothing to prove. We can therefore suppose that

W k = W k-1 . Since W k-1 is invariant under the action of s 1 we have W k-1 = W k ∪ {δ k,1 , δ k,2 }. Set W k = W k ∪ {δ k,1 } and W k = W k ∪ {δ k,2 }. By Lemma 3.3, KB n [W k-1 ] = KB n [W k ] * KBn[W k ] KB n [W k ]. Moreover, s 1 (KB n [W k ]) = KB n [W k ] and s 1 (KB n [W k ]) = KB n [W k ]
. By Lemma 3.9 it follows that there exist Lemma 3.7). By the above there exist α ∈ KB n [X ] and β ∈ KB n [W n ] such that s 1 (β ) = β -1 and α = α β s 1 (α -1 ). So, β = s 1 (β ) = β -1 , hence β 2 = 1, and therefore β = 1, since, by Godelle-Paris [START_REF] Godelle | K(π, 1) and word problems for infinite type Artin-Tits groups, and applications to virtual braid groups[END_REF], KB n is torsion free. So, α = α s 1 (α -1 ). Lemma 3.11 Let G 1 , G 2 , . . . , G p , H be a collection of groups. We suppose that H is a subgroup of G j for all j ∈ {1, . . . , p} and we consider the amalgamated product

α 2 ∈ KB n [W k-1 ] and β ∈ KB n [W k ] such that s 1 (β ) = β -1 and β 1 = α 2 β s 1 (α -1 2 ). Set α = α 1 α 2 . Then β ∈ KB n [W k ], s 1 (β ) = β -1 and α = α β s 1 (α -1 ). Notice that W n = X ∩ S 3 . Recall that s 1 (β ) = β for all β ∈ KB n [X ∩ S 3 ] = KB n [W n ] (see
G = G 1 * H G 2 * H • • • * H G p . Let τ 1 , τ 2 : G → G be two automorphisms satisfying the following properties: (a) τ 2 1 = τ 2 2 = 1 and τ 1 τ 2 τ 1 = τ 2 τ 1 τ 2 . (b)
For all i ∈ {1, 2} and j ∈ {1, . . . , p} there exists k ∈ {1, . . . , p} such that τ i (G j ) = G k .

(c) For all j ∈ {1, . . . , p} there exists i ∈ {1, 2} such that τ i (G j ) = G j .

(d) For all i ∈ {1, 2} we have τ i (H) = H .

(e) For all i ∈ {1, 2} and γ ∈ H such that τ i (γ) = γ -1 there exists δ ∈ H such that γ = δ τ i (δ -1 ).

Let α ∈ G satisfying the following equation:

(3-1) α τ 2 (α -1 ) (τ 2 τ 1 )(α) (τ 2 τ 1 τ 2 )(α -1 ) (τ 1 τ 2 )(α) τ 1 (α -1 ) = 1 .
Then there exist α , α ∈ G and β ∈ H such that α = α βα , τ 1 (α ) = α , τ 2 (α ) = α and β satisfies Equation (3-1).

Proof Let α ∈ G satisfying Equation (3-1). It is easily checked that, if α is written α = α βα , where τ 1 (α ) = α and τ 2 (α ) = α , then β also satisfies Equation (3-1). So, it suffices to show that there exist α , α ∈ G and β ∈ H such that α = α βα , τ 1 (α ) = α and τ 2 (α ) = α . If α ∈ H there is nothing to prove. We can therefore suppose that α ∈ H . We write α in the form α = α 1 α 2 • • • α such that (a) for all i ∈ {1, . . . , } there exists j = j(i) ∈ {1, . . . , p} such that α i ∈ G j \ H , (b) j(i) = j(i + 1) for all i ∈ {1, . . . , -1}.

We argue by induction on .

Suppose first that = 1. We can assume without loss of generality that α ∈ G 1 \ H . We set

β 1 = α, β 2 = τ 2 (α -1 ), β 3 = (τ 2 τ 1 )(α), β 4 = (τ 2 τ 1 τ 2 )(α -1 ), β 5 = (τ 1 τ 2 )(α),
and

β 6 = τ 1 (α -1 )
. By hypothesis we have β 1 β 2 β 3 β 4 β 5 β 6 = 1 and, for each i ∈ {1, . . . , 6}, there exists j = j(i) ∈ {1, . . . , p} such that β i ∈ G j \H . If we had τ 1 (G 1 ) = G 1 and τ 2 (G 1 ) = G 1 , then we would have j(i) = j(i + 1) for all i ∈ {1, . . . , 5}, thus, by Proposition 3.5, we would have

β 1 β 2 β 3 β 4 β 5 β 6 = 1: contradiction. So, either τ 1 (G 1 ) = G 1 or τ 2 (G 1 ) = G 1 .
On the other hand, by Condition (c) in the statement of the lemma, either

τ 1 (G 1 ) = G 1 or τ 2 (G 1 ) = G 1 . Thus, either τ 1 (G 1 ) = G 1 and τ 2 (G 1 ) = G 1 , or τ 1 (G 1 ) = G 1 and τ 2 (G 1 ) = G 1 . We assume that τ 1 (G 1 ) = G 1 and τ 2 (G 1 ) = G 1 . The case τ 1 (G 1 ) = G 1 and τ 2 (G 1 ) = G 1 is proved in a similar way. So, j (1) = j(2), j(2) = j(3), j(3)
= j(4), j(4) = j(5), and j(5) = j(6). Since

β 1 β 2 β 3 β 4 β 5 β 6 = 1, by Proposition 3.5, either β 2 β 3 ∈ H , or β 4 β 5 ∈ H , that is, α -1 τ 1 (α) = γ ∈ H . Now, we have τ 1 (γ) = τ 1 (α -1 )α = γ -1
, hence, by Condition (e) in the statement of the lemma, there exists δ ∈ H such that γ = δ τ 1 (δ -1 ). We set α = αδ , α = 1 and

β = δ -1 . Then α = α βα , τ 1 (α ) = α , τ 2 (α ) = α and β ∈ H .
Suppose that ≥ 2 and that the inductive hypothesis holds. It follows from Proposition 3.5 and Equation (3-1) that either α τ 2 (α -1 ) ∈ H or τ 1 (α -1 1 ) α 1 ∈ H . Suppose that α τ 2 (α -1 ) ∈ H . Then there exists γ ∈ H such that α = γ τ 2 (α ). By applying τ 2 to this equality we obtain τ 2 (α ) = τ 2 (γ) α , and therefore τ 2 (γ) = γ -1 . By hypothesis there exists δ ∈ H such that γ = δ -1 τ 2 (δ). We set

α = 1, α = δα , α -1 = α -1 δ -1 and β 1 = α 1 • • • α -2 α -1 . We have α = α β 1 α , τ 1 (α ) = α and τ 2 (α ) = α , hence β 1 satisfies Equation (3-1). By induction, there exists γ , γ ∈ G and β ∈ H such that β 1 = γ βγ , τ 1 (γ ) = γ and τ 2 (γ ) = γ . Set α = α γ and α = γ α . Then α = α βα , τ 1 (α ) = α and τ 2 (α ) = α . The case τ 1 (α -1
1 ) α 1 ∈ H can be proved in a similar way.

Lemma 3.12 Let X be a subset of S invariant under the action of s 1 and under the action of

s 2 . Let α ∈ KB n [X ] such that (3-2) α s 2 (α -1 ) (s 2 s 1 )(α) (s 2 s 1 s 2 )(α -1 ) (s 1 s 2 )(α) s 1 (α -1 ) = 1 .
Then there exist α , α

∈ KB n [X ] such that s 1 (α ) = α , s 2 (α ) = α and α = α α . Proof For 4 ≤ k ≤ n we set U k = {δ i,j ∈ X | (i, j) ∈ ({1, 2, 3} × {4, . . . , k})}.
We set also U 3 = X . We show by induction on k that there exist α , α ∈ KB n [X ] and

β ∈ KB n [U k ] such that α = α βα , s 1 (α ) = α , s 2 (α ) = α and β satisfies Equation (3-2)
. The case k = 3 is true by hypothesis. Suppose that 4 ≤ k ≤ n and that the inductive hypothesis holds. So, there exist α 1 , α 1 ∈ KB n [X ] and

β 1 ∈ KB n [U k-1 ] such that α = α 1 β 1 α 1 , s 1 (α 1 ) = α 1 , s 2 (α 1 ) = α 1 and β 1 satisfies Equation (3-2). If U k-1 = U k there is nothing to prove. Suppose that U k-1 = U k . Since U k-1 is invariant
under the action of s 1 and under the action of s 2 , we have

U k-1 = U k ∪{δ 1,k , δ 2,k , δ 3,k }. Set G j = KB n [U k ∪ {δ j,k }] for j ∈ {1, 2, 3} and H = KB n [U k ]. By Lemma 3.3, KB n [U k-1 ] = G 1 * H G 2 * H G 3 .
Moreover, we have the following properties.

• For each i ∈ {1, 2} and each j ∈ {1, 2, 3} there exists k ∈ {1, 2, 3} such that s i (G j ) = G k .

• For each j ∈ {1, 2, 3} there exists i ∈ {1, 2} such that s i (G j ) = G j .

• For each i ∈ {1, 2} we have s i (H) = H .

• By Lemma 3.10, for each i ∈ {1, 2} and each γ ∈ H such that s i (γ) = γ -1 , there exists δ ∈ H such that γ = δ s i (δ -1 ).

By Lemma 3.11 it follows that there exist

α 2 , α 2 ∈ KB n [U k-1 ] and β ∈ KB n [U k ] such that β 1 = α 2 βα 2 , s 1 (α 2 ) = α 2 , s 2 (α 2 )
= α 2 and β satisfies Equation (3-2). We set α = α 1 α 2 and α = α 2 α 1 . Then α = α βα , s 1 (α ) = α and s 2 (α ) = α .

For 4 ≤ k ≤ n we set V k = {δ i,j ∈ U n | (i, j) ∈ ({4, . . . , k} × {1, 2, 3})}. We set also V 3 = U n . We show by induction on k that there exist α , α ∈ KB n [X ] and β ∈ KB n [V k ] such that α = α βα , s 1 (α ) = α , s 2 (α ) = α and β satisfies Equation . The case k = 3 is true by the above. Suppose that 4 ≤ k ≤ n and that the inductive hypothesis holds. So, there exist α 1 , α 1 ∈ KB n [X ] and

β 1 ∈ KB n [V k-1 ] such that α = α 1 β 1 α 1 , s 1 (α 1 ) = α 1 , s 2 (α 1 ) = α 1 and β 1 satisfies Equation (3-2). If V k-1 = V k there is nothing to prove. Suppose that V k-1 = V k . Since V k-1 is invariant
under the action of s 1 and under the action of s 2 , we have

V k-1 = V k ∪{δ k,1 , δ k,2 , δ k,3 }. Set G j = KB n [V k ∪ {δ k,j }] for j ∈ {1, 2, 3} and H = KB n [V k ]. By Lemma 3.3, KB n [V k-1 ] = G 1 * H G 2 * H G 3 .
Moreover, we have the following properties.

• For each i ∈ {1, 2} and each j ∈ {1, 2, 3} there exists k ∈ {1, 2, 3} such that s i (G j ) = G k .

• For each j ∈ {1, 2, 3} there exists i ∈ {1, 2} such that s i (G j ) = G j .

• For each i ∈ {1, 2} we have s i (H) = H .

• By Lemma 3.10, for each i ∈ {1, 2} and each γ ∈ H such that s i (γ) = γ -1 , there exists δ ∈ H such that γ = δ s i (δ -1 ).

By Lemma 3.11 it follows that there exist

α 2 , α 2 ∈ KB n [V k-1 ] and β ∈ KB n [V k ] such that β 1 = α 2 βα 2 , s 1 (α 2 ) = α 2 , s 2 (α 2 )
= α 2 and β satisfies Equation (3-2). We set α = α 1 α 2 and α = α 2 α 1 . Then α = α βα , s 1 (α ) = α and s 2 (α ) = α .

Set W 1 = {δ i,j ∈ X | (i, j) ∈ ({1, 2, 3} × {1, 2, 3})} and W 2 = {δ i,j ∈ X | (i, j) ∈ ({4, . . . , n} × {4, . . . , n})}. Notice that

V n = W 1 W 2 , KB n [V n ] = KB n [W 1 ] × KB n [W 2 ]
, and s 1 (γ) = s 2 (γ) = γ for all γ ∈ KB n [W 2 ]. By the above there exist α 1 , α 1 ∈ KB n [X ] and

β 1 ∈ KB n [V n ] such that α = α 1 β 1 α 1 , s 1 (α 1 ) = α 1 , s 2 (α 1 ) = α 1 and β 1 satisfies Equation (3-2). Let β ∈ KB n [W 1 ] and α 2 ∈ KB n [W 2 ] such that β 1 = βα 2 . Set α = α 1 and α = α 2 α 1 . Then α = α βα , s 1 (α ) = α , s 2 (α ) = α and β satisfies Equation (3-2). Let W 1,1 = {δ i,j ∈ X | (i, j) ∈ {(1, 2), (2, 3), (3, 1)}} and W 1,2 = {δ i,j ∈ X | (i, j) ∈ {(2, 1), (3, 2), (1, 3)}}. Set G 1 = KB n [W 1,1 ] and G 2 = KB n [W 1,2 ]. By Lemma 3.3, KB n [W 1 ] = G 1 * G 2 . Moreover, s 1 (G 1 ) = s 2 (G 1 ) = G 2 and s 1 (G 2 ) = s 2 (G 2 ) = G 1 .
From Lemma 3.11 applied with H = {1} it follows that there exist β , β ∈ KB n [W 1 ] such that β = β β , s 1 (β ) = β and s 2 (β ) = β . Actually, by Lemma 3.7,

β = β = 1, hence β = 1. So, α = α α , s 1 (α ) = α and s 2 (α ) = α .
As announced in Section 2, we take advantage of the results of the present section to prove Lemma 2.6.

Proof of Lemma 2.6 Suppose instead that ζ 1 is an inner automorphism, that is,

ζ 1 = c γ : VB n → VB n , δ → γδγ -1
, for some γ ∈ VB n . We have γ = 1 since ζ 1 = id. We write γ in the form γ = α ι(w) with α ∈ KB n and w ∈ S n . For each i ∈ {1, . . . , n-1} we have

s i = π K (τ i ) = π K (ζ 1 (τ i )) = π K (γτ i γ -1 ) = ws i w -1
, hence w lies in the center of S n which is trivial, and therefore w = 1 and γ = α ∈ KB n .

Note that ζ 1 (δ i,j ) = δ j,i for all i, j ∈ {1, . . . , n}, i = j. Take i, j ∈ {1, . . . , n}, i < j, and set U i,j = S \ {δ i,j , δ j,i }, U i,j = U i,j ∪ {δ i,j } and U i,j = U i,j ∪ {δ j,i }. By Lemma

3.3, KB n = KB n [U i,j ] * KBn[U i,j ] KB n [U i,j ]. Moreover, ζ 1 (KB n [U i,j ]) = KB n [U i,j ] and ζ 1 (KB n [U i,j ]) = KB n [U i,j ]. From Lemma 3.6 it follows that KB ζ 1 n ⊂ KB n [U i,j ]. Since 1≤i<j≤n U i,j = ∅, by Theorem 3.2, 1≤i<j≤n KB n [U i,j ] = KB n [∅] = {1}, thus KB ζ 1 n = {1}. But α ∈ KB ζ 1
n and α = γ = 1, which is a contradiction. So, ζ 1 is not an inner automorphism.

From virtual braid groups to symmetric groups

The following is well-known and can be easily deduced from Artin [START_REF] Artin | Braids and permutations[END_REF] and Lin [START_REF] Lin | Artin braids and related groups and spaces (Russian)[END_REF][START_REF] Lin | Braids and permutations[END_REF]. It is a preliminary to the proof of Theorem 2.1. Lemma 5.1 Let n ∈ N, n ≥ 3, and let ϕ : S n → VB n be a homomorphism such that π K • ϕ = id. Then ϕ is conjugate to ι.

Proof Since π K • ϕ = id, for each i ∈ {1, . . . , n -1} there exists α i ∈ KB n such that ϕ(s i ) = α i τ i . We prove by induction on k that there exists a homomorphism ϕ : S n → VB n conjugate to ϕ such that π K • ϕ = id and ϕ (s i ) = τ i for all i ∈ {1, . . . , k}. The case k = n -1 ends the proof of the lemma.

Suppose that k = 1. We have 1 = ϕ(s 1 ) 2 = α 1 s 1 (α 1 ) τ 2 1 = α 1 s 1 (α 1 ), hence s 1 (α 1 ) = α -1 1 . By Lemma 3.10 there exists

β 1 ∈ KB n such that α 1 = β 1 s 1 (β -1 1 ). Thus, ϕ(s 1 ) = α 1 τ 1 = β 1 s 1 (β -1 1 ) τ 1 = β 1 τ 1 β -1 1 . Set ϕ = c β -1 1 • ϕ. Then ϕ is conjugate to ϕ, π K • ϕ = id, since β 1 ∈ KB n , and ϕ (s 1 ) = τ 1 .
We assume that k = 2 and ϕ(s

1 ) = τ 1 . Since ϕ(s 1 s 2 s 1 ) = ϕ(s 2 s 1 s 2 ), we have τ 1 α 2 τ 2 τ 1 = α 2 τ 2 τ 1 α 2 τ 2 , hence s 1 (α 2 ) = α 2 (s 2 s 1 )(α 2 )
. On the other hand, as in the previous paragraph, from the equality ϕ(s 2 ) 2 = 1 it follows that there exists

β 2 ∈ KB n such that α 2 = β 2 s 2 (β -1 2 ). Thus, s 1 β 2 s 2 (β -1 2 ) = β 2 s 2 (β -1 2 ) (s 2 s 1 ) β 2 s 2 (β -1 2 ) ⇒ β 2 s 2 (β -1 2 ) (s 2 s 1 )(β 2 ) (s 2 s 1 s 2 )(β -1 2 ) (s 1 s 2 )(β 2 ) s 1 (β -1 2 ) = 1 . By Lemma 3.12 there exist β 2 , β 2 ∈ KB n such that β 2 = β 2 β 2 , s 1 (β 2 ) = β 2 and s 2 (β 2 ) = β 2 . Thus, ϕ(s 2 ) = α 2 τ 2 = β 2 τ 2 β -1 2 = β 2 τ 2 β -1 2 . Set ϕ = c β -1 2 • ϕ. Then ϕ is conjugate to ϕ, π K • ϕ = id, since β 2 ∈ KB n , ϕ (s 1 ) = τ 1 , since s 1 (β 2 ) = β 2 ,
and ϕ (s 2 ) = τ 2 by construction.

We assume that k ≥ 3 and ϕ(s

i ) = τ i for all i ∈ {1, . . . , k-1}. Let ∈ {1, . . . , k-2}. Since ϕ(s k s ) = ϕ(s s k ), we have α k τ k τ = τ α k τ k , hence s (α k ) = α k . By Corollary 3.8 it follows that α k ∈ KB n [U ], where U = {δ i,j ∈ S | i, j ∈ { , + 1}}. Recall that S k = {δ i,j ∈ S | k ≤ i = j ≤ n}. We have 1≤ ≤k-2 U = S k hence, by Theorem 3.2, α k ∈ KB n [S k ]. Since ϕ(s k-1 s k s k-1 ) = ϕ(s k s k-1 s k ), we have τ k-1 α k τ k τ k-1 = α k τ k τ k-1 α k τ k , hence s k-1 (α k ) = α k (s k s k-1 )(α k ).
On the other hand, as in the two previous paragraphs, from the equality ϕ(s k ) 2 = 1 it follows that there exists

β k ∈ KB n [S k ] such that α k = β k s k (β -1 k ). So, s k-1 β k s k (β -1 k ) = β k s k (β -1 k ) (s k s k-1 ) β k s k (β -1 k ) ⇒ β k s k (β -1 k ) (s k s k-1 )(β k ) (s k s k-1 s k )(β -1 k ) (s k-1 s k )(β k ) s k-1 (β -1 k ) = 1 .
Note that S k is not invariant under the action of s k-1 , but S k-1 is and S k ⊂ S k-1 . By Lemma 3.12 there exist β k , β k ∈ KB n [S k-1 ] such that

β k = β k β k , s k-1 (β k ) = β k and s k (β k ) = β k . So, ϕ(s k ) = α k τ k = β k τ k β -1 k = β k τ k β -1 k . Since s k-1 (β k ) = β k ,
Assume that n = m and ψ • ι = ι, that is, ψ(τ i ) = τ i for all i ∈ {1, . . . , n -1}. For each i ∈ {1, . . . , n -1} we set ψ(σ i ) = α i ι(w i ), where α i ∈ KB n and w i ∈ S n . For each i ∈ {3, . . . , n-1} we have s i w 1 = (π K •ψ)(τ i σ 1 ) = (π K •ψ)(σ 1 τ i ) = w 1 s i , hence w 1 lies in the centralizer of s 3 , . . . , s n-1 in S n , which is equal to s 1 = {1, s 1 }, hence either w 1 = s 1 or w 1 = 1.

Suppose that w 1 = s 1 , that is, ψ(σ 1 ) = α 1 τ 1 . For each k ∈ {3, . . . , n -1} we have α 1 τ 1 τ k = ψ(σ Let ω ∈ F 2 such that α 1 = ω(δ 1,2 , δ 2,1 ). Note that, by Theorem 3.2, KB n [{δ 1,2 , δ 2,1 }] is a free group freely generated by {δ 1,2 , δ 2,1 }, hence ω is unique. We have σ 2 = τ 1 τ 2 σ 1 τ 2 τ 1 , hence ψ(σ 2 ) = ω(δ 2,3 , δ 3,2 )τ 2 . On the other hand, since σ 1 σ 2 σ 1 = σ 2 σ 1 σ 2 , we have ω(δ 1,2 , δ 2,1 )τ 1 ω(δ 2,3 , δ 3,2 )τ 2 ω(δ 1,2 , δ 2,1 )τ 1 = ω(δ 2,3 , δ 3,2 )τ 2 ω(δ 1,2 , δ 2,1 )τ 1 ω(δ 2,3 , δ 3,2 )τ 2 , hence (6-1) ω(δ 1,2 , δ 2,1 )ω(δ 1,3 , δ 3,1 )ω(δ 2,3 , δ 3,2 ) = ω(δ 2,3 , δ 3,2 )ω(δ 1,3 , δ 3,1 )ω(δ 1,2 , δ 2,1 ) .

1 τ k ) = ψ(τ k σ 1 ) = τ k α 1 τ 1 ,
Let W = {δ i,j | 1 ≤ i = j ≤ 3}, W 1 = {δ 1,2 , δ 2,3 , δ 3,1 } and W 2 = {δ 2,1 , δ 3,2 , δ 1,3 }. By Lemma 3.3, KB n [W] = KB n [W 1 ] * KB n [W 2 ]. From this decomposition and Lemma 6.1 it is easily deduced that the only element of F 2 which satisfies Equation (6-1) is ω = 1, thus ψ(σ 1 ) = τ 1 . Then from the equalities σ i = τ i-1 τ i σ i-1 τ i τ i-1 , 2 ≤ i ≤ n -1, it follows that ψ(σ i ) = τ i for all i ∈ {2, . . . , n -1}, hence ψ = ι • π P .

Suppose that w 1 = 1, that is, ψ(σ 1 ) = α 1 . For each k ∈ {3, . . . , n -1} we have α 1 τ k = ψ(σ 1 τ k ) = ψ(τ k σ 1 ) = τ k α 1 , hence s k (α 1 ) = α 1 . By Corollary 3.8 this implies that α 1 ∈ KB n [U k ], where U k = {δ i,j | i, j ∈ {k, k + 1}}. As above, 3≤k≤n-1 U k = {δ 1,2 , δ 2,1 }, hence α 1 ∈ KB n [{δ 1,2 , δ 2,1 }]. Let ω ∈ F 2 such that α 1 = ω(δ 1,2 , δ 2,1 ). Again, since KB n [{δ 1,2 , δ 2,1 }] is a free group freely generated by {δ 1,2 , δ 2,1 }, the element ω is unique. We have σ 2 = τ 1 τ 2 σ 1 τ 2 τ 1 , hence ψ(σ 2 ) = (s 1 s 2 )(α 1 ) = ω(δ 2,3 , δ 3,2 ). On the other hand, since σ 1 σ 2 σ 1 = σ 2 σ 1 σ 2 , we have (6-2) ω(δ 1,2 , δ 2,1 )ω(δ 2,3 , δ 3,2 )ω(δ 1,2 , δ 2,1 ) = ω(δ 2,3 , δ 3,2 )ω(δ 1,2 , δ 2,1 )ω(δ 2,3 , δ 3,2 ) .

Recall that KB n

[W] = KB n [W 1 ] * KB n [W 2 ]
, where W = {δ i,j | 1 ≤ i = j ≤ 3}, W 1 = {δ 1,2 , δ 2,3 , δ 3,1 } and W 2 = {δ 2,1 , δ 3,2 , δ 1,3 }. From this decomposition and Lemma 6.1 it is easily seen that, if ω satisfies Equation (6-2), then ω is of the form ω = z k with z ∈ {x, y} and k ∈ Z. By Crisp-Paris [START_REF] Crisp | The solution to a conjecture of Tits on the subgroup generated by the squares of the generators of an Artin group[END_REF], if k ∈ {1, 0, -1}, then the subgroup of KB n [W 1 ] generated by {δ k 1,2 , δ k 2,3 , δ k 3,1 } is a free group freely

Corollary 2 . 4

 24 Let n ∈ N, n ≥ 5. Then VB n is Hopfian and co-Hopfian.

  , and is generated by the classes of ζ 1 and ζ 2 .

  respectively. It is easily checked that ζ 1 and ζ 2 are of order two and commute, hence they generate a subgroup of Aut(VB n

  hence s k (α 1 ) = α 1 . By Corollar 3.8 this implies thatα 1 ∈ KB n [U k ], where U k = {δ i,j | i, j ∈ {k, k + 1}}. We have 3≤k≤n-1 U k = {δ 1,2 , δ 2,1 } hence, by Theorem 3.2, α 1 ∈ KB n [{δ 1,2 , δ 2,1 }].

Proof of Theorem 2.1 Let n, m ∈ N such that n ≥ 5, m ≥ 2 and n ≥ m, and let ψ : VB n → S m be a homomorphism. By Proposition 4.1 one of the following possibilities holds up to conjugation.

• ψ • ι is Abelian,

• n = m and ψ • ι = id,

• n = m = 6 and ψ • ι = ν 6 .

Suppose that ψ • ι is Abelian. Then there exists w 1 ∈ S m such that w 1 = (ψ • ι)(s i ) = ψ(τ i ) for all i ∈ {1, . . . , n-1}. Since s 2 i = 1, we have

) for all i ∈ {1, . . . , n -2}, hence ψ(σ i ) = w 2 for all i ∈ {1, . . . , n -1}. Finally, from the relation τ 1 σ 3 = σ 3 τ 1 it follows that w 1 w 2 = w 2 w 1 . So, ψ is Abelian.

Suppose that n = m and ψ • ι = id. Then ψ(τ i ) = s i for all i ∈ {1, . . . , n -1}. From the relations σ 1 τ i = τ i σ 1 , 3 ≤ i ≤ n -1, it follows that ψ(σ 1 ) lies in the centralizer of s 3 , . . . , s n-1 in S n , which is equal to

for all i ∈ {1, . . . , n -1}, hence ψ = π K . Assume that ψ(σ 1 ) = s 1 . We show by induction on i that ψ(σ i ) = s i for all i ∈ {1, . . . , n -1}. The case i = 1 is true by hypothesis. Suppose that i ≥ 2 and ψ(σ i-1 ) = s i-1 . Then, since τ i-

Suppose that n = m = 6 and ψ • ι = ν 6 . Then ν -1 6 • ψ • ι = id, hence, by the above, either ν -1 6 • ψ = π K or ν -1 6 • ψ = π P , and therefore either ψ = ν 6 • π K or ψ = ν 6 • π P .

From symmetric groups to virtual braid groups

The core of the proof of Theorem 2.2 lies in the following lemma. by Corollary 3.8,

Proof of Theorem 2.2 Let n, m ∈ N such that n ≥ 5, m ≥ 2, and n ≥ m, and let ϕ : S n → VB m be a homomorphism. By Proposition 4.1 one of the following possibilities holds up to conjugation.

this element is of finite order, since s 1 s i is of finite order, and, by Godelle-Paris [START_REF] Godelle | K(π, 1) and word problems for infinite type Artin-Tits groups, and applications to virtual braid groups[END_REF], KB m is torsion free, hence

Suppose that n = m and π K • ϕ = id. Then, by Lemma 5.1, ϕ is conjugate to ι.

Suppose that n = m = 6 and π K • ϕ = ν 6 . We have

6 From virtual braid groups to virtual braid groups Lemma 6.1 Let n ≥ 3, let i, j, k ∈ {1, . . . , n} pairwise distinct, and let 1 , 2 ∈ Z.

Then

By Crisp-Paris [START_REF] Crisp | The solution to a conjecture of Tits on the subgroup generated by the squares of the generators of an Artin group[END_REF] the subgroup of KB n [{δ i,j , δ j,k }] generated by {δ 2 i,j , δ 2 j,k } is a free group freely generated by {δ 2 i,j , δ 2 j,k }, hence t 1 = t 2 = 0. We show in the same way that, if

Lemma 6.2 Let n = 6. Set u i = ν 6 (s i ) for all i ∈ {1, 2, 3, 4, 5}. Let H be the subgroup of S 6 generated by {u 3 , u 4 , u 5 }. Then KB H 6 = {1}, where KB H 6 = {α ∈ KB 6 | w(α) = α for all w ∈ H}.

Proof Let U = {u 3 , u 4 , u 5 , u 3 u 4 u 3 , u 4 u 5 u 4 , u 3 u 4 u 5 u 4 u 3 }. We have

] by Lemma 3.3. On the other hand, it is easily observed that there exists v ∈ U such that (i, j) is a cycle in the decomposition of v as a product of disjoint cycles. For that v we have v(KB 6 [U i,j ]) = KB 6 [U i,j ] and v(KB 6 [U i,j ]) = KB 6 [U i,j ]. By Lemma 3.6 we deduce that KB H 6 ⊂ KB v 6 ⊂ KB 6 [U i,j ]. We have 1≤i<j≤6 U i,j = ∅, hence, by Theorem 3.2, KB H 6 ⊂ KB 6 [∅] = {1}.

We will use the following notation in the proof of Theorem 2.3. Let F 2 = F 2 (x, y) be the free group of rank 2 freely generated by {x, y}. If G is a group, α, β ∈ G, and ω ∈ F 2 , then ω(α, β) denotes the image of ω under the homomorphism F 2 → G which sends x to α and y to β .

Proof of Theorem 2.3 Let n, m ∈ N such that n ≥ 5, m ≥ 2 and n ≥ m, and let ψ : VB n → VB m be a homomorphism. By Theorem 2.2 one of the following possibilities holds up to conjugation.

• ψ • ι is Abelian,

Assume that ψ • ι is Abelian. We argue in the same way as in the Abelian case in the proof of Theorem 2.1. There exists

), for all i ∈ {1, . . . , n -2}, hence ψ(σ i ) = β 2 for all i ∈ {1, . . . , n -1}. Finally, from the relation

Now, we assume that n = m = 6 and ψ • ι = ι • ν 6 . For each i ∈ {1, 2, 3, 4, 5} we set u i = ν 6 (s i ) and β i = ι(u i ). Thus, ψ(τ i ) = β i for all i ∈ {1, 2, 3, 4, 5}. For each i ∈ {1, 2, 3, 4, 5} we set ψ(σ i ) = α i ι(w i ), where α i ∈ KB 6 and w i ∈ S 6 . For each i ∈ {3, 4, 5} we have

in the centralizer of u 3 , u 4 , u 5 in S 6 , which is equal to u 1 = {1, u 1 }, thus either w 1 = u 1 or w 1 = 1. On the other hand, for each i ∈ {3, 4, 5}, we have α 1 ι(w 1 ) β i = ψ(σ 1 τ i ) = ψ(τ i σ 1 ) = β i α 1 ι(w 1 ) and ι(w 1 ) β i = ι(w 1 u i ) = ι(u i w 1 ) = β i ι(w 1 ), hence u i (α 1 ) = β i α 1 β -1 i = α 1 . By Lemma 6.2 it follows that α 1 = 1. So, either ψ(σ 1 ) = β 1 or ψ(σ 1 ) = 1. If ψ(σ 1 ) = β 1 , then, since σ i = τ i-1 τ i σ i-1 τ i τ i-1 for all i ∈ {2, 3, 4, 5}, we have ψ(σ i ) = β i for all i ∈ {1, 2, 3, 4, 5}, and therefore ψ = ι • ν 6 • π P . If ψ(σ 1 ) = 1, then, since σ i = τ i-1 τ i σ i-1 τ i τ i-1 for all i ∈ {2, 3, 4, 5}, we have ψ(σ i ) = 1 for all i ∈ {1, 2, 3, 4, 5}, and therefore ψ = ι • ν 6 • π K .


F Kong

Yonghua Li
email: yonghua.li@cea.fr

Lipid catabolism in microalgae

Lipid degradation processes are important in microalgae because survival and growth of microalgal cells under fluctuating environmental conditions require permanent remodeling or turnover of membrane lipids as well as rapid mobilization of storage lipids. Lipid catabolism comprises two major spatially and temporarily separated steps, namely lipolysis, which releases fatty acids and head groups and is catalyzed by lipases at membranes or lipid droplets, and degradation of fatty acids to acetyl-CoA, which occurs in peroxisomes through the β-oxidation pathway in green microalgae, and can sometimes occur in mitochondria in some other algal species. Here we review the current knowledge on the enzymes and regulatory proteins involved in lipolysis and peroxisomal β-oxidation and highlight gaps in our understanding of lipid degradation pathways in microalgae. Metabolic use of acetyl-CoA products via glyoxylate cycle and gluconeogenesis is also reviewed. We then present the implication of various cellular processes such as vesicle trafficking, cell cycle and autophagy on lipid turnover. Finally, physiological roles and the manipulation of lipid catabolism for biotechnological applications in microalgae are discussed.

Introduction

Microalgae are a group of unicellular photosynthetic eukaryotic organisms that are found in various aquatic habitats ranging from cold seas to desert microbiotic crusts, freshwaters and soils. They represent a polyphyletic group because they are derived from several heterotrophic eukaryotic lineages [START_REF] Keeling | The endosymbiotic origin, diversification and fate of plastids[END_REF]. However, microalgae have in common the presence of one or several plastids, which host the photosynthetic apparatus and account for a major part of the microalgal membranes. Depending on the microalgal lineage, plastids are derived from engulfment of a cyanobacterium or another alga and they are thus bounded by two membranes (glaucophytes, green algae and red algae), or three (euglenophytes and many dinoflagellates) or four membranes (diatoms and all other algae). From this complex evolutionary history, it is expected that a wide variety of lipid compositions, lipid metabolism and lipid trafficking pathways might have evolved within algal diversity.

In the past 10 yr, interest in lipid-based biofuels and ω-3 fatty acids (FAs) has triggered intensive research on algal lipid metabolism, which has been mostly performed in a few algal models, namely the soil-living green microalga Chlamydomonas reinhardtii, the marine diatom Phaeodactylum tricornutum and two species of the brown algae-related genus Nannochloropsis. This research has resulted in a wealth of new data in microalgae on FA and membrane lipid synthesis, as well as storage lipid accumulation [START_REF] Harwood | The versatility of algae and their lipid metabolism[END_REF][START_REF] Li-Beisson | Metabolism of acyl-lipids in Chlamydomonas reinhardtii[END_REF][START_REF] Du | Triacylglycerol accumulation in photosynthetic cells in plants and algae[END_REF]. It has also led to the realization that lipid catabolism plays an important role in microalgae. Indeed, faced with rapid changes in temperature, light and/or nutrient conditions, microalgal cells must rapidly remodel or degrade membrane and storage lipids from plastids, lipid droplets and other organelles to ensure survival and growth. Here, we focus on recent advances in the characterization of microalgal enzymes and regulatory proteins involved in the lipolysis of FA-based lipids (i.e. acyl-lipids) and the βoxidation of FAs, with an emphasis on Chlamydomonas reinhardtii(hereafter referred to as Chlamydomonas).

Lipolysis

A schematic overview of lipolysis in a subcellular context and the subsequent FA degradation processes that occur in algal peroxisomes are shown in Fig. 1. The function of lipolysis is multifold, including providing carbon and energy for growth, remodeling membrane lipid composition, or generating signaling molecules. Here we do not review membrane lipid remodeling or lipid signaling per se, but focus on the breakdown of bulk membrane or storage lipids and the degradation of FAs to two-carbon compounds. In the laboratory, lipid degradation is mostly studied via manipulation of nitrogen (N) content in the culture medium: N depletion triggers accumulation of triacylglycerols (TAGs) in the lipid droplets (LDs), which is partly reliant on de novo FA synthesis and partly on recycling of membrane lipids [START_REF] Siaut | Oil accumulation in the model green alga Chlamydomonas reinhardtii: characterization, variability between common laboratory strains and relationship with starch reserves[END_REF]. By contrast, N resupply causes massive degradation of the

TAG breakdown

Owing to its economic importance, TAG lipases have been intensively studied, which resulted in the discovery of major TAG lipase in Arabidopsis thaliana (SDP1, the sugar-dependent 1)

and Saccharomyces cerevisae (TGL3P and TGL4P). These TAG lipases are located to LDs, the major site of neutral lipid storage [START_REF] Athenstaedt | Tgl4p and Tgl5p, two triacylglycerol lipases of the yeast Saccharomyces cerevisiae are localized to lipid particles[END_REF][START_REF] Eastmond | SUGAR-DEPENDENT1 encodes a patatin domain triacylglycerol lipase that initiates storage oil breakdown in germinating Arabidopsis seeds[END_REF]. A TAG lipase (TGL1) has also been identified in the diatom Phaeodactylum tricornutum [START_REF] Barka | Identification of a triacylglycerol lipase in the diatom Phaeodactylum tricornutum[END_REF].

TGL1 showed high sequence similarity to Arabidopsis SDP1. In vitro, the recombinant TGL1 possessed TAG lipase activity. Artificial microRNA knockdown lines had almost twice the amount of TAGs compared with the wild-type (WT), without compromising growth. A SDP1 homolog from the oleaginous alga Lobosphaera incisa (LiSDP1) was also studied [START_REF] Siegler | Analysis of the lipid body proteome of the oleaginous alga Lobosphaera incisa[END_REF]. LiSDP1 shares 44% identity to Arabidopsis SDP1, and contains a patatin domain harboring a putative catalytic dyad, which includes a serine located within a GXSXG motif. Interestingly the LiSDP1, when expressed heterologously in the tobacco pollen tube system, was not located to the LDs. Moreover, when expressed in the Arabidopsismutant sdp1, only partial restoration of the seedling growth phenotype of sdp1 was observed. Collectively, this study highlighted potential differences in the functions of SDP1 in an evolutionary context.

Eight putative TAG lipases from Chlamydomonas showing reduction in transcription during N starvation compared with the N-sufficient condition were tested using a function complementation approach (Li et al., 2012b). Among the eight candidate lipases (CrLIP1-8), five were cloned as full-length cDNAs and expressed in the yeast TAG lipase mutant (tgl3Δtgl4Δ), respectively. Only CrLIP1 was able to complement tgl3Δtgl4Δ. This result highlights the difficulty in assigning functions to putative lipases. Artificial microRNA silencing of CrLIP1 resulted in slower TAG degradation during N recovery. However, the in vitro lipase assay on recombinant CrLIP1 showed that the protein was not able to degrade TAG but rather DAG and polar lipids. Overall, CrLIP1 is proposed to be a DAG lipase with broad substrate specificity, therefore acting downstream of the TAG lipase in oil breakdown.

In addition, CrLIP1 is highly expressed during optimal growth where TAG synthesis is low; therefore it might also play a key role in degrading DAGs resulting from removal of head groups in some membrane lipids.

Thaps3_264297, a homolog of the human Comparative Gene Identification-58 (CGI-58), was recently identified as involved in lipid catabolism in the diatom Thalassiosira pseudonana [START_REF] Trentacoste | Metabolic engineering of lipid catabolism increases microalgal lipid accumulation without compromising growth[END_REF]. CGI-58 belongs to the family 5 of α/β-hydrolase fold proteins. It is a soluble enzyme that associates with LDs in certain conditions, and its absence results in overaccumulation of TAGs in plants and mammals [START_REF] James | Disruption of the Arabidopsis CGI-58 homologue produces Chanarin-Dorfman-like lipid droplet accumulation in plants[END_REF]. [START_REF] Trentacoste | Metabolic engineering of lipid catabolism increases microalgal lipid accumulation without compromising growth[END_REF] reported that Thaps3_264297 exhibits TAG lipase, phospholipase, and acyltransferase activities in in vitro assays, but in a later study, CGI-58 was concluded as only acting as a co-activator of TAG lipase [START_REF] Mcmahon | Comparative gene identification 58/α/β hydrolase domain 5 lacks lysophosphatidic acid acyltransferase activity[END_REF]. Targeted knockdown of this enzyme increased lipid yields by two-to fourfold during exponential growth and under silicon starvation in Thalassiosira. These suggested a role for Thaps3_264297 in lipid homeostasis.

A putative ortholog of CGI-58 is encoded in many other sequenced algal genomes, including the one of Chlamydomonas [START_REF] Merchant | The Chlamydomonas genome reveals the evolution of key animal and plant functions[END_REF], but whether and how it is involved in lipid metabolism in these algae remain to be elucidated.

Recently, the function of a highly abundant putative TAG lipase (LiLBP36) of the LDs isolated from the green alga Lobosphaera incisa was investigated [START_REF] Siegler | Analysis of the lipid body proteome of the oleaginous alga Lobosphaera incisa[END_REF]). An LD location was further confirmed via in vivo expression in tobacco pollen tube. LiLBP36 was identified as a class 3 lipase, and contains a putative catalytic triad of serine, aspartate and histidine that is found in many lipases, with the serine residue being part of a conserved GXSXG motif. The structure of LiLBP36 matches very closely a group of known fungal-and yeast-secreted lipases that act on MAG, DAG and TAG [START_REF] Siegler | Analysis of the lipid body proteome of the oleaginous alga Lobosphaera incisa[END_REF]. Moreover, the transcription of LiLBP36 is up-regulated by N starvation, and down-regulated upon N resupply. However, only moderate functional restoration of the Arabidopsis sdp1 mutant phenotype was observed, and evidence for lipase activity of this protein remains to be determined.

Membrane lipid breakdown

Plastid Galactoglycerolipid Degradation 1 (PGD1) is the only lytic enzyme that has been demonstrated to be involved in membrane lipid turnover in Chlamydomonas (Li et al., 2012a). of Chlamydomonas has been shown, in addition to acyltransferase activity, to exhibit lipase activity toward a wide range of lipids (TAG, phospholipids, galactolipids and cholesteryl esters) [START_REF] Yoon | Phospholipid: diacylglycerol acyltransferase is a multifunctional enzyme involved in membrane lipid turnover and degradation while synthesizing triacylglycerol in the unicellular green microalga Chlamydomonas reinhardtii[END_REF]. PDAT contains an α/β-hydrolase fold domain, where the conserved lipase motif (GXSXG) is present. PDAT is known to synthesize TAG by transferring acyl moiety from phospholipids to DAG in yeasts and higher plant cells [START_REF] Dahlqvist | Phospholipid: diacylglycerol acyltransferase: an enzyme that catalyzes the acyl-CoAindependent formation of triacylglycerol in yeast and plants[END_REF]. This transacylation pathway mediated by PDAT has been also shown to contribute to TAG synthesis in Chlamydomonas, because the pdat1 mutant accumulates 25% less TAG compared with the WT [START_REF] Boyle | Three acyltransferases and nitrogen-responsive regulator are implicated in nitrogen starvation-induced triacylglycerol accumulation in Chlamydomonas[END_REF]. It is therefore believed that PDAT is involved in both membrane lipid turnover and TAG synthesis in microalgae, similar to its role in higher plants [START_REF] Zhang | DGAT1 and PDAT1 acyltransferases have overlapping functions in Arabidopsis triacylglycerol biosynthesis and are essential for normal pollen and seed development[END_REF].

Fatty acid degradation

Fatty acid degradation is an oxidative process that results in production of C2 acetyl units that are linked to coenzyme A (acetyl-CoA). FA degradation is known to occur in peroxisomes of land plants and yeast, and in both peroxisomes and mitochondria in mammalian cells. In microalgae both mitochondrial and/or peroxisomal FA breakdown have been reported depending on algal lineage [START_REF] Stabenau | Microbodies in different algae[END_REF]Stabenau et al., 1984aStabenau et al., ,b, 1988;;[START_REF] Winkler | Beta-oxidation of fatty acids in algaelocalization of thiolase and acyl-CoA oxidizing enzymes in 3 different organisms[END_REF]. Differences in the compartmentation of FA β-oxidation enzymes have been suggested as a consequence of different phylogenetic development. Owing to the limited literature on microalgal β-oxidation, we focus our discussion here only on green microalgae, where peroxisomal FA β-oxidation has recently been demonstrated through study of aChlamydomonas mutant knocked-out for an acyl-CoA oxidase [START_REF] Kong | Chlamydomonas carries out fatty acid beta-oxidation in ancestral peroxisomes using a bona fide acyl-CoA oxidase[END_REF].

Microalgal peroxisomes

Peroxisomes (also called microbodies) are globular organelles (0.1-1 μm) surrounded by a single membrane and are present in all eukaryotic cells. Peroxisomes are defined as organelles that carry out oxidative reactions producing hydrogen peroxide (H2O2), which is subsequently converted to water by catalase. Therefore, catalase is often considered a marker for peroxisomes. Peroxisomes have been reported to occur in various algal species, such as Mougeotia, Spirogyra, Dunaliella, Etemosphaera, Chlorogonium elongatum [START_REF] Silverberg | Cytochemical localization of oxidase activities with diaminobenizidine in the green alga Chlamydomonas dysosmos[END_REF][START_REF] Silverberg | Cytochemical localization of oxidase activities with diaminobenizidine in the green alga Chlamydomonas dysosmos[END_REF]Stabenau et al., 1984b[START_REF] Stabenau | Localization of glycolate dehydrogenase in two species of Dunaliella[END_REF]. In Chlamydomonas, the first trackable report on peroxisomes was published in 1974 [START_REF] Stabenau | Verteilung von microbody-enzymen aus Chlamydomonas in dichtegradienten[END_REF]. This paper studied the distribution of microbody enzymes on sucrose gradient. Not until 2009 has another paper focusing on peroxisomes from Chlamydomonas appeared [START_REF] Shinozaki | Peroxisomal targeting signals in green algae[END_REF], where the authors reported the functional presence of a peroxisomal targeting sequence (PTS)

in Chlamydomonas [START_REF] Shinozaki | Peroxisomal targeting signals in green algae[END_REF]. They further showed that the Chlamydomonas PTS can target GFP to peroxisomes in the green alga Closterium ehrenbergii. This study provided additional evidence that the protein import machinery between the two algal species is conserved. Peroxisomes of Chlamydomonas were visualized only as recently as 5 yr ago under both confocal microscopy and transmission electron microscopy [START_REF] Hayashi | Visualization of microbodies in Chlamydomonas reinhardtii[END_REF]. The same group further showed that the number and size of peroxisomes in a cell can be manipulated via changing carbon source; for example, the number of peroxisomes in Chlamydomonas can be boosted up to three times when cultured with acetate [START_REF] Hayashi | Increase in peroxisome number and the gene expression of putative glyoxysomal enzymes in Chlamydomonas cells supplemented with acetate[END_REF]. This suggests dynamic turnover of the algal peroxisomes and their active participation to metabolism.

Acyl delivery to peroxisomes

Degradation of lipids by lipases involves trafficking of FA released from all organelles to peroxisomes.Whether this occurs by transport processes used for lipid synthesis or by specific means is unknown. Studies in mammals and plants have suggested that TAG stored in LDs serves as an intermediate in the transport of otherwise toxic free FAs released from membrane lipid breakdown [START_REF] Listenberger | Triglyceride accumulation protects against fatty acid-induced lipotoxicity[END_REF][START_REF] Fan | Arabidopsis Lipins, PDAT1 Acyltransferase, and SDP1 triacylglycerol lipase synergistically direct fatty acids toward β-oxidation, thereby maintaining membrane lipid homeostasis[END_REF]. Physical interactions between LDs and peroxisomes have been observed in plants and also in Chlamydomonasand other algal species [START_REF] Hayashi | Increase in peroxisome number and the gene expression of putative glyoxysomal enzymes in Chlamydomonas cells supplemented with acetate[END_REF][START_REF] Schwarz | Carbon starvation induces lipid degradation via autophagy in the model alga Micrasterias[END_REF] (Fig. 3). It remains to be determined how

LDs move in the cell to approach peroxisomes, or vice versa.

Figure 3 Transmission electron micrograph (TEM) of a section through an algal cell showing

the close proximity of lipid droplets and peroxisomes. The freshwater alga Micrasterias denticulata(Streptophyta) was kept in the dark for 9 wk before being fixed and imaged under TEM [START_REF] Schwarz | Carbon starvation induces lipid degradation via autophagy in the model alga Micrasterias[END_REF]. LB, lipid body; P, peroxisome; D, dictyosome.

Once FAs get close to the peroxisome, studies in Arabidopsis show that they are transported across the single membrane by the ATP-binding cassette transporter (ABCD1, also known as PXA1/CTS/PED3 [START_REF] Zolman | The Arabidopsis pxa1 mutant is defective in an ATPbinding cassette transporter-like protein required for peroxisomal fatty acid β-oxidation[END_REF]. ABCD1 has been shown to possess, in addition to the transporter activity, a thioesterase activity [START_REF] De | Intrinsic acyl-CoA thioesterase activity of a peroxisomal ATP binding cassette transporter is required for transport and metabolism of fatty acids[END_REF]. This means that FA has to be activated to its CoA ester before being transported by ABCD1 across the peroxisomal membrane. After entering peroxisome, FAs are activated to their CoA esters via the action of two peroxisomal long-chain acyl-CoA synthetases (LACS6/7) [START_REF] Fulda | Peroxisomal acyl-CoA synthetase activity is essential for seedling development in Arabidopsis thaliana[END_REF]. A physical interaction between the ABCD1 and LACS6/7 has been demonstrated [START_REF] De | Intrinsic acyl-CoA thioesterase activity of a peroxisomal ATP binding cassette transporter is required for transport and metabolism of fatty acids[END_REF]. Chlamydomonas encodes one putative ABCD1 and three LCS (=LACS) proteins [START_REF] Li-Beisson | Metabolism of acyl-lipids in Chlamydomonas reinhardtii[END_REF], among which two are associated with LDs in Chlamydomonas [START_REF] Nguyen | Proteomic profiling of oil bodies isolated from the unicellular green microalga Chlamydomonas reinhardtii: with focus on proteins involved in lipid metabolism[END_REF]. Only the LD-associated LCS2 has been shown to participate in TAG synthesis during LD formation [START_REF] Li | An indexed, mapped mutant library enables reverse genetics studies of biological processes in Chlamydomonas reinhardtii[END_REF]. The function of the putative peroxisomal isoform remains to be validated.

Reactions and enzymes of fatty acid β-oxidation

Fatty acid β-oxidation is the principal pathway implicated in degradation of FAs in the peroxisomes of land plants [START_REF] Poirier | Peroxisomal beta-oxidationa metabolic pathway with multiple functions[END_REF][START_REF] Graham | Seed storage oil mobilization[END_REF]. The core reactions of FA βoxidation require a cyclic reaction of four enzymatic steps: oxidation, hydration, dehydrogenation and thiolytic cleavage of an acyl-CoA, and are catalyzed by three enzymes:

acyl-CoA oxidase (ACX), multifunctional protein and 3-ketoacyl-CoA thiolase. The end product of these reactions is acetyl-CoA (C2). Genes homologous to known Arabidopsis FA βoxidation enzymes are encoded in the genome of Chlamydomonas [START_REF] Merchant | The Chlamydomonas genome reveals the evolution of key animal and plant functions[END_REF], and their IDs and number of homologs are described in [START_REF] Li-Beisson | Metabolism of acyl-lipids in Chlamydomonas reinhardtii[END_REF]. Except for ACX2, none of the other enzymes of FA β-oxidation has been characterized.

We have recently isolated and characterized a knockout mutant for ACX2. The acx2 mutant is impaired in oil remobilization upon N resupply [START_REF] Kong | Chlamydomonas carries out fatty acid beta-oxidation in ancestral peroxisomes using a bona fide acyl-CoA oxidase[END_REF]. Oil remobilization is severely (60-80%) but not completely blocked in acx2, suggesting functional redundancy with the other four ACXs [START_REF] Li-Beisson | Metabolism of acyl-lipids in Chlamydomonas reinhardtii[END_REF]. Indeed, overlapping substrate specificities between different ACX isoenzymes have been reported for Arabidopsis where no phenotype can be observed with any of the single acx mutants [START_REF] Eastmond | The Arabidopsis acyl-CoA oxidase gene family[END_REF]. Furthermore, except for ACX2, all four other Chlamydomonas ACXs contain a putative PTS (either PTS1 or PTS2) sequence [START_REF] Kong | Chlamydomonas carries out fatty acid beta-oxidation in ancestral peroxisomes using a bona fide acyl-CoA oxidase[END_REF], suggesting potential functions in peroxisome metabolism.

In in vitro assays, the recombinant Chlamydomonas ACX2 catalyzes the conversion of acyl-CoA to trans-2-enoyl-CoA, requiring FAD as a co-factor and producing H2O2. Thus, it possesses all the characteristics of a classical plant ACX [START_REF] Eastmond | The Arabidopsis acyl-CoA oxidase gene family[END_REF]. Using protein tagging to mCherry, ACX2 was localized to peroxisomes in Chlamydomonas. Therefore, this study demonstrated for the first time that Chlamydomonasuses a peroxisomal pathway for FA degradation, and that H2O2-producing activities had already evolved in green microalgae.

In addition to acetyl-CoAs, FA β-oxidation produces H2O2 and NADH. In higher plants, the highly oxidative H2O2 is usually decomposed to water by the peroxisome-resident catalase, whereas there is still debate about the subcellular localization of catalases in Chlamydomonas [START_REF] Kato | Characterization of catalase from green algae Chlamydomonas reinhardtii[END_REF]. Nevertheless, homologs of ascorbate peroxidases are encoded in the genome of Chlamydomonas [START_REF] Merchant | The Chlamydomonas genome reveals the evolution of key animal and plant functions[END_REF], and these enzymes could be used by cells to quench H2O2. Continued operation of FA β-oxidation requires a steady supply of oxidized NAD + . This can occur either through oxidation of NADH to NAD + by a peroxisomal malate dehydrogenase (MDH; [START_REF] Pracharoenwattana | Arabidopsis peroxisomal malate dehydrogenase functions in β-oxidation but not in the glyoxylate cycle[END_REF] or through import of NAD + from the cytosol to the peroxisome with the aid of a homolog of the NAD + carrier protein (PXN) [START_REF] Bernhardt | A peroxisomal carrier delivers NAD+ and contributes to optimal fatty acid degradation during storage oil mobilization[END_REF]. Both proteins have been demonstrated to play a role in FA respiration in land plants, and their homologs occur in microalgae but experimental demonstration for their function is lacking.

In parallel to the core activities of the β-oxidation spiral which are mainly responsible for oxidation of saturated straight chain fatty acids, microalgae are rich in unsaturated fatty acids, and which can make up to 70% of their membranes in Chlamydomonas [START_REF] Nguyen | The green microalga Chlamydomonas reinhardtii has a single ω-3 fatty acid desaturase that localizes to the chloroplast and impacts both plastidic and extraplastidic membrane lipids[END_REF].

The β-oxidation of unsaturated fatty acids are proposed to go through two alternative pathways in land plants [START_REF] Li-Beisson | Acyl-lipid metabolism[END_REF]. The best characterized protein of these auxiliary pathways is the enoyl-CoA isomerase (ECI) [START_REF] Goepfert | Peroxisomal Δ3, Δ2-enoyl CoA isomerases and evolution of cytosolic paralogues in embryophytes[END_REF], and a protein encoded by the locus Cre03.g190850 showing weak homology to the plant ECI can be identified in Chlamydomonas but its function remains to be validated.

Metabolic fate of acetyl-CoA produced from FA β-oxidation

Acetyl units (C2), generated by the β-oxidation of FAs, can usually be utilized by cells after being synthesized to 4-carbon compounds (i.e. succinate) through the glyoxylate cycle.

Subsequently, succinate enters the tricarboxylic acid cycle in mitochondria, releasing malate which can be converted to hexose and sucrose in the cytosol via gluconeogenesis. Coordinate regulation of transcription and enzyme activities of the three pathways (i.e. FA β-oxidation, glyoxylate cycle, and gluconeogenesis) at the onset of oilseed germination have been demonstrated, suggesting global regulatory mechanisms [START_REF] Rylott | Co-ordinate regulation of genes involved in storage lipid mobilization in Arabidopsis thaliana[END_REF]. However, the identity of the signal(s) or regulatory proteins that induces all these genes is not known. Chlamydomonas encodes all known enzymes of the glyoxylate cycle, that is, citrate synthase (CYS), isocitrate lyase (ICL), malate synthase (MS), aconitase (AOS) and MDH [START_REF] Merchant | The Chlamydomonas genome reveals the evolution of key animal and plant functions[END_REF], which have shown up-regulation in the dark, similar to core enzymes of FA β-oxidation [START_REF] Zones | High-resolution profiling of a synchronized diurnal transcriptome from Chlamydomonas reinhardtii reveals continuous cell and metabolic differentiation[END_REF]. Either N-terminal or C-terminal fusion to fluorescence proteins revealed that all of these (CYS, MS, AOS, MDH), except for ICL which is cytosolic, are localized in the peroxisomes [START_REF] Lauersen | Peroxisomal microbodies are at the crossroads of acetate assimilation in the green microalga Chlamydomonas reinhardtii[END_REF]. The localization of the glyoxylate enzymes in Chlamydomonas differs from that of plants, where ICL is peroxisomal.

The icl mutant cannot grow in the dark and grows at a slower rate when cultivated mixotrophically in the light [START_REF] Plancke | Lack of isocitrate lyase in Chlamydomonas leads to changes in carbon metabolism and in the response to oxidative stress under mixotrophic growth[END_REF], demonstrating that the glyoxylate cycle is important for acetate metabolism in Chlamydomonas. Moreover, the icl mutant overaccumulated both total FAs and neutral lipids when cultivated mixotrophically. Based on quantitative proteomic data, the authors consider that the observed changes in lipid could be the result of a slowdown in the β-oxidation of FAs and is less likely to be a result of increased de novo FA synthesis. The availability of the icl mutant provides an exciting opportunity to address the question of whether, in addition to the glyoxylate cycle, there are alternative routes to transport acetyl units from the peroxisome to mitochondria in microalgae.

Lipid degradation in relation to other cellular processes

Cellular lipid content and composition vary according to developmental stage, phase of cell cycle, and in response to nutrient availability, as well as changes in environmental conditions (e.g. light, temperature). This suggests tight and complex control of lipid synthesis and degradation. We review here the processes shown to be implicated in lipid catabolism in microalgae.

Vesicle trafficking

Brefeldin A (BFA), is a known inhibitor of ADP-ribosylation factor 1 that promotes the formation of COPI-coated vesicles (important for peroxisome vesiculation) and is thought to inhibit the delivery of TAG lipase to LDs in mammalian systems [START_REF] Farese | Lipid droplets finally get a little R-E-S-P-E-C-T[END_REF].

Studies in Arabidopsis seeds have recently shown that mutants defective in the retromer function are impaired in the delivery of SDP1 lipase from peroxisome to LDs through peroxisome extensions, thereby perturbing FA degradation and seed germination [START_REF] Thazar-Poulot | Peroxisome extensions deliver the Arabidopsis SDP1 lipase to oil bodies[END_REF]. Two independent studies have observed an increase in cellular TAG content in Chlamydomonas upon addition of BFA [START_REF] Kato | Triacylglycerol mobilization is suppressed by brefeldin A in Chlamydomonas reinhardtii[END_REF][START_REF] Kim | Rapid induction of lipid droplets in Chlamydomonas reinhardtii and Chlorella vulgaris by brefeldin A[END_REF]. [START_REF] Kato | Triacylglycerol mobilization is suppressed by brefeldin A in Chlamydomonas reinhardtii[END_REF] showed that the increased TAG accumulation is a result of inhibition of lipid degradation. Progression of FA β-oxidation in pre-mature peroxisomes was shown to be delayed. This was further supported by FA feeding assays, as, in the presence of BFA, fluorescent FAs are not degraded. Indeed, proteins implicated in vesicle trafficking are present in the LD proteome of Chlamydomonas [START_REF] Moellering | RNA interference silencing of a major lipid droplet protein affects lipid droplet size in Chlamydomonas reinhardtii[END_REF][START_REF] Nguyen | Proteomic profiling of oil bodies isolated from the unicellular green microalga Chlamydomonas reinhardtii: with focus on proteins involved in lipid metabolism[END_REF].

Taken together, these studies highlight the participation of vesicle trafficking in lipid catabolism in microalgae.

Cell cycle

Lipids of Chlamydomonas are observed to show specific alterations in relation to the phase in the cell cycle; and three unique patterns can be seen [START_REF] Juppner | Dynamics of lipids and metabolites during the cell cycle of Chlamydomonas reinhardtii[END_REF]. TAGs have shown a bimodal accumulation, with degradation occurring in the middle of the day, as well as at the beginning of the night. This suggests fine-tuning of lipolysis to cell cycle progression. Indeed, a direct link between cell cycle and TAG degradation has been demonstrated in S. cerevisiae,

where the major TAG lipase (TGL4) is phosphorylated and activated by cyclin-dependent kinase 1 (Cdk1/Cdc28) [START_REF] Kurat | Cdk1/Cdc28-dependent activation of the major triacylglycerol lipase Tgl4 in yeast links lipolysis to cell-cycle progression[END_REF]. Whether such a system occurs in microalgae remains to be tested. Upon N starvation, algal cells enter a quiescent state and accumulate TAGs. Upon N resupply, cells degrade TAGs and exit quiescence thereby restarting growth. [START_REF] Tsai | The protein Compromised Hydrolysis of Triacylglycerols 7 (CHT7) acts as a repressor of cellular quiescence in Chlamydomonas[END_REF] showed that exiting quiescence is linked to TAG remobilization, because the cht7 (compromised in hydrolysis of TAG) mutant, which is defective in a gene required for exiting quiescence, is severely impaired in TAG degradation. CHT7 encodes a CXC-domain containing DNA-binding protein and is proposed as a transcription factor regulating quiescence and proliferation. This study provides evidence that TAG remobilization is tuned to cell division and growth in algae.

Autophagy-mediated lipid turnover

In addition to lipase-mediated lipid turnover, autophagy was also reported to be involved in lipid degradation in mammals, plant kingdoms and fungi [START_REF] Singh | Autophagy regulates lipid metabolism[END_REF][START_REF] Kurusu | OsATG7 is required for autophagy-dependent lipid metabolism in rice postmeiotic anther development[END_REF][START_REF] Van Zutphen | Lipid droplet autophagy in the yeast Saccharomyces cerevisiae[END_REF]. In yeast cells, LDs were observed to be taken up by vacuoles/lysosomes and subsequently degraded [START_REF] Van Zutphen | Lipid droplet autophagy in the yeast Saccharomyces cerevisiae[END_REF]. Inhibition of autophagy in a mouse model increased TAG content, thereby providing evidence that autophagy plays a role in lipid turnover [START_REF] Singh | Autophagy regulates lipid metabolism[END_REF]. Formation of autophagosomes has been observed in N-starved Chlamydomonas cells, and could be averted upon acetate boost, thereby explaining the additional obesity in acetate-boosted cells [START_REF] Goodenough | The path to triacylglyceride obesity in the sta6 strain of Chlamydomonas reinhardtii[END_REF].

This provides the first evidence that averting autophagy can increase oil content in microalgae.

It is also reported that in the green alga Auxenochlorella protothecoides, LDs were observed to be directly sequestered by the vacuole during the transition from heterotrophy to autotrophy when a large amount of TAGs were degraded [START_REF] Zhao | Autophagy-like processes are involved in lipid droplet degradation in Auxenochlorella protothecoides during the heterotrophy-autotrophy transition[END_REF]. The authors suggest that LD autophagy observed in A. protothecoides resembles more a microautophagy-like mechanism rather than a classical autophagy, as during this process no autophagosome is observed. By contrast, LD degradation during carbon starvation in the green alga Micrasterias denticulata (Streptophyta) involves a classical autophagy pathway [START_REF] Schwarz | Carbon starvation induces lipid degradation via autophagy in the model alga Micrasterias[END_REF].

Finally, inhibition of autophagy in Chlorella zofingiensis by the autophagy inhibitor 3methyladenine during N starvation resulted in cells accumulating 15-20% less TAG than control cells [START_REF] Zhang | Inhibition of autophagy modulates astaxanthin and total fatty acid biosynthesis in Chlorella zofingiensis under nitrogen starvation[END_REF]. The authors considered that the reduced TAG accumulation during N starvation upon inhibition of autophagy could be the result of inhibition of lipolysis of membrane lipids, which contributes to TAG amount in addition to de novo synthesis.

Detailed studies of the type of autophagy (lipophagy, pexophagy, nonselective autophagy…) associated with each condition should help to clarify the complex interrelationship between autophagy and lipid metabolism.

Physiological functions and biotechnological applications of lipid catabolism

Microalgae are exposed to constant changes in their habitat, light, temperature, and nutrient availability; therefore the need to remodel their membranes to adapt to these changes is crucial for survival. This explains the high expression of many genes encoding lipolytic enzymes and enzymes of fatty acid β-oxidation under stress [START_REF] Miller | Changes in transcript abundance in Chlamydomonas reinhardtii following nitrogen deprivation predict diversion of metabolism[END_REF]. In rapidly growing cells, lipid degradation plays also a 'housekeeping' role in removing harmful free FAs which can originate from acylated-protein and lipid degradation. During the N recovery, lipid catabolism is important to degrade TAG storage to supply cells with carbon precursors for synthesis of structural membranes. Moreover, FA β-oxidation is found to be important in regulating the amount of FAs during day/night cycles, and is essential for FA utilization upon carbon starvation [START_REF] Kong | Chlamydomonas carries out fatty acid beta-oxidation in ancestral peroxisomes using a bona fide acyl-CoA oxidase[END_REF]. In addition, lipid turnover has also been found to contribute to resource allocation during aging or senescence in land plants [START_REF] Troncoso-Ponce | Lipid turnover during senescence[END_REF].

Whether such a function holds true in algal cells remains to be determined.

In addition to their importance in cell metabolism and physiology, lipid catabolisms are also very useful targets for biotechnological applications: discovery of new lipases can provide the industry with novel enzymes; shutting down lipid catabolism has been shown to increase cellular oil content [START_REF] Trentacoste | Metabolic engineering of lipid catabolism increases microalgal lipid accumulation without compromising growth[END_REF][START_REF] Tsai | The protein Compromised Hydrolysis of Triacylglycerols 7 (CHT7) acts as a repressor of cellular quiescence in Chlamydomonas[END_REF][START_REF] Kong | Chlamydomonas carries out fatty acid beta-oxidation in ancestral peroxisomes using a bona fide acyl-CoA oxidase[END_REF]; and mutants defective in FA β-oxidation, which prevents futile cycling, can serve as background strains for genetic engineering strategies aiming to produce exotic or unusual FAs which are often otherwise degraded.

Remaining questions

Although considerable progress has been made towards our understanding of lipid catabolism in microalgae, many unknowns remain. For example, the identity of a true TAG lipase remains to be identified in Chlamydomonas. The function of the major lipid droplet protein is still not clear. Does it possess enzymatic activities, similar to that of oleosin [START_REF] Parthibane | Oleosin is bifunctional enzyme that has both monoacylglycerol acyltransferase and phospholipase activities[END_REF]?

As lipases are water-soluble, while their substrates are mostly insoluble in water, lipolysis is an interfacial process. What are the processes or enzymes involved in opening up the protein coat of LDs, therefore allowing the access to lipase to the TAG core? Are lipolysis and FA βoxidation regulated by a common signal or transcription factor? In addition to acetyl-CoAs, FA β-oxidation produces H2O2 and NADH. How these products are metabolized or into or out of the algal peroxisomes remains unknown. Furthermore, in addition to FA catabolism, do algal peroxisomes also play roles in the metabolism of substrates other than lipids, as is the case in higher plants? Answering these and many other questions should help to expand our basic knowledge on lipid catabolism and aid in the design of engineering strategies in the creation of high TAG strains for biofuel, nutrition and green chemistry applications.

Figure 1

 1 Figure 1 Pathways and processes involved in lipid catabolism in microalgae. Fatty acids (FAs) are made in the chloroplast; one part of them is being used to make plastidial membranes, and the other part is exported to the endoplasmic reticulum (ER) where they are assembled to glycerol to make the extraplastidial membranes. In response to developmental signals or adverse conditions, cells accumulate some of these acyl-chains as triacylglycerols (TAGs) in lipid droplets. Lipid turnover starts from lipolysis of structural or storage lipids through action of various lipases. The released free FAs enter peroxisomes for complete degradation. FA βoxidation operates as a spiral reaction consisting of four repeating enzymatic steps. Acetyl-Coenzyme A (CoA), the final product of FA degradation, is then used to make a four-carbon compound by the glyoxylate cycle partly operating in the peroxisomes. Succinates produced are used to fuel the tricarboxylic acid cycle in the mitochondria and subsequently to produce sucrose for growth through the gluconeogenesis pathway in the cytoplasm. It is worth mentioning here that this pathway is drawn mostly based on what we know in the model alga Chlamydomonas, and in some other algal species, FA β-oxidation is known to occur in mitochondria. Known enzymes are written in red and bold. Biosynthetic steps are shown as gray lines, whereas degradative steps are in black. ABC, ATP binding cassette; ACP, acyl-

  PGD1 catalyzes the hydrolysis of acyl chains at the sn-1 position of the glycerol backbone of the plastidial galactolipid monogalacosyldiacylglycerol (MGDG). The null mutant contained 50% fewer TAGs with a reduced amount of oleic acid (18 : 1 Δ9 ) during N starvation. FA fluxes from plastidial lipids to TAGs are also reduced in pgd1. Although the steady-state concentrations of membrane lipids of pgd1 were not significantly different from those of WT, the pulse-chase labeling assay and in vitro lipase assay with recombinant protein suggested that MGDG is a substrate of PGD1. In in vitro assay, PGD1 hydrolyzed only the newly synthesized MGDG containing 18 : 1 Δ9 . Taken together, these results indicate that PGD1 acts as a lipase removing low desaturated acyl chains from MGDG, which are then used for TAG synthesis under N deprivation. Such a route has not yet been reported for land plants, but as putative orthologs of PGD1 are present in plant and other algal genomes, the transfer of acyl-chains from MGDG to TAG might be operating in plants and a wider group of algae.No enzyme responsible for degradation of other major membrane lipids (i.e. diacylglycerylN,N,N-trimethylhomoserine, sulfoquinovosyldiacylglycerol, phosphatidylglycerol, phosphatidylethanolamine, and digalactosyldiacylglycerol) has been identified in microalgae. Nevertheless, the phospholipid:diacylglycerol acyltransferase (PDAT)

Acknowledgements

The research in the authors' laboratory is funded by French ANR funding. I.T.R. acknowledges the CEA for an international PhD studentship, and J.W. thanks the Junior Research Fellowship Program from the French Embassy in Bangkok (Thailand) for funding a 6-month scholarship.