Melvin G Renault

Jordi Zamarreño Beas

Badreddine Douzi

Maïalène Chabalier

Abdelrahim Zoued

Yannick R Brunet

Christian Cambillau

Laure Journet

Eric Cascales
email: cascales@imm.cnrs.fr

The gp27-like hub of VgrG serves as adaptor to promote Hcp tube assembly

Contractile injection systems are multiprotein complexes that use a spring-like mechanism to deliver effectors into target cells. In addition to using a conserved mechanism, these complexes share a common core known as the tail. The tail comprises an inner tube tipped by a spike, wrapped by a contractile sheath, and assembled onto a baseplate. Here, using the Type VI secretion system (T6SS) as a model of contractile injection systems, we provide molecular details on the interaction between the inner tube and the spike. Reconstitution into the E. coli heterologous host in absence of other T6SS components and in vitro experiments demonstrated that the Hcp tube component and the VgrG spike interact directly. VgrG deletion studies coupled to functional assays showed that the N-terminal domain of VgrG is sufficient to interact with Hcp, to initiate proper Hcp tube polymerization, and to promote sheath dynamics and Hcp release. The interaction interface between Hcp and VgrG was then mapped using docking simulations, mutagenesis and cysteine-mediated cross-links. Based on these results, we propose a model in which the VgrG base serves as adaptor to recruit the first Hcp hexamer and initiates inner tube polymerization.

INTRODUCTION

The Type VI secretion system (T6SS) is a multiprotein machine, widespread in Proteobacteria and responsible for the delivery of effectors into prokaryotic as well as eukaryotic target cells, thus participating to interbacterial competition and pathogenesis [START_REF] Cascales | The type VI secretion toolkit[END_REF][START_REF] Silverman | Structure and regulation of the type VI secretion system[END_REF][START_REF] Coulthurst | The Type VI secretion system -a widespread and versatile cell targeting system[END_REF][START_REF] Mogk | Deadly syringes: type VI secretion system activities in pathogenicity and interbacterial competition[END_REF][START_REF] Russell | Type VI secretion system effectors: poisons with a purpose[END_REF][START_REF] Durand | and beyond: the versatile arsenal of type VI secretion effectors[END_REF][START_REF] Diniz | Molecular weaponry: diverse effectors delivered by the type VI secretion system[END_REF][START_REF] Cianfanelli | Aim, load, fire: the type VI secretion system, a bacterial nanoweapon[END_REF]. The T6SS comprises a trans-envelope complex constituted of the TssJ, TssL and TssM proteins [START_REF] Ma | An IcmF family protein, ImpLM, is an integral inner membrane protein interacting with ImpKL, and its walker a motif is required for type VI secretion system-mediated Hcp secretion in Agrobacterium tumefaciens[END_REF][START_REF] Aschtgen | The SciZ protein anchors the enteroaggregative Escherichia coli type VI secretion system to the cell wall[END_REF][START_REF] Durand | Biogenesis and structure of a type VI secretion membrane core complex[END_REF]. This TssJLM membrane complex serves as a docking station for a tail-like structure [START_REF] Durand | Biogenesis and structure of a type VI secretion membrane core complex[END_REF][START_REF] Brunet | The type VI secretion TssEFGK-VgrG phage-like baseplate is recruited to the TssJLM membrane complex via multiple contacts and serves as assembly platform for tail tube/sheath polymerization[END_REF][START_REF] Zoued | Structure-function analysis of the TssL cytoplasmic domain reveals a new interaction between the Type VI secretion baseplate and membrane complexes[END_REF][START_REF] Logger | Molecular dissection of the interface between the Type VI secretion TssM cytoplasmic domain and the TssG baseplate component[END_REF][START_REF] Nguyen | Type VI secretion TssK baseplate protein exhibits structural similarity with phage receptor-binding proteins and evolved to bind the membrane complex[END_REF]. The T6SS taillike structure is structurally, evolutionarily and functionally related to the tail of contractile injection systems (CIS), a broad family of machines that use a spring-like mechanism to deliver macromolecules into target cells [START_REF] Zoued | Architecture and assembly of the type VI secretion system[END_REF][START_REF] Ho | A view to a kill: the bacterial Type VI secretion system[END_REF][START_REF] Sarris | A phage tail-derived element with wide distribution among both prokaryotic domains: a comparative genomic and phylogenetic study[END_REF][START_REF] Basler | Type VI secretion system: secretion by a contractile nanomachine[END_REF][START_REF] Böck | In situ architecture, function, and evolution of a contractile injection system[END_REF][START_REF] Cascales | Microbiology: and Amoebophilus invented the machine gun![END_REF]. A number of CIS act as autonomous machineries and are released in the external medium where they bind to specific receptors located at the cell surface of target cells. These include the prototypical contractile bacteriophages of the Myoviridae family that injects their genome into bacterial host cells, and particles such as R-pyocins, anti-feeding prophages and Photorhabdus virulence cassettes [START_REF] Leiman | Contractile tail machines of bacteriophages[END_REF][START_REF] Bönemann | Tubules and donuts: a type VI secretion story[END_REF][START_REF] Heymann | Three-dimensional structure of the toxin-delivery particle antifeeding prophage of Serratia entomophila[END_REF][START_REF] Michel-Briand | The pyocins of Pseudomonas aeruginosa[END_REF]. Other CIS act intracellularly and are oriented toward the cell exterior. These include the Pseudoalteromonas luteoviolacea metamorphosis-associated and Amoebophilus asiaticus contractile machines, as well as the T6SS [START_REF] Zoued | Architecture and assembly of the type VI secretion system[END_REF][START_REF] Basler | Type VI secretion system: secretion by a contractile nanomachine[END_REF][START_REF] Böck | In situ architecture, function, and evolution of a contractile injection system[END_REF][START_REF] Shikuma | Marine tubeworm metamorphosis induced by arrays of bacterial phage tail-like structures[END_REF][START_REF] Penz | The genome of the amoeba symbiont "Candidatus Amoebophilus asiaticus" encodes an afp-like prophage possibly used for protein secretion[END_REF][START_REF] Brackmann | Using force to punch holes: mechanics of contractile nanomachines[END_REF]. These CIS are responsible for the delivery of effector proteins that stimulate the genetic program of eukaryotic cells to induce morphogenesis, interfere with the cytoskeleton or the signalling pathway in eukaryotic cells, or cause cell damages in bacterial cells [START_REF] Russell | Type VI secretion system effectors: poisons with a purpose[END_REF][START_REF] Durand | and beyond: the versatile arsenal of type VI secretion effectors[END_REF][START_REF] Böck | In situ architecture, function, and evolution of a contractile injection system[END_REF][START_REF] Shikuma | Marine tubeworm metamorphosis induced by arrays of bacterial phage tail-like structures[END_REF][START_REF] Jank | Tyrosine glycosylation of Rho by Yersinia toxin impairs blastomere cell behaviour in zebrafish embryos[END_REF][START_REF] Hachani | Type VI secretion and antihost effectors[END_REF]. Hence, CIS participate to establishment of symbiosis, pathogenesis, modification of the host or bacterial competition.

The mechanism of action is conserved between all these machines: a needle is propelled by the contraction of a sheath and penetrates into the target cell to deliver macromolecules [START_REF] Brackmann | Using force to punch holes: mechanics of contractile nanomachines[END_REF].

However, these systems have evolved additional modules to fulfil specific functions [START_REF] Cascales | Microbiology: and Amoebophilus invented the machine gun![END_REF].

Extracellular CIS have fibers that mediate contact with the target cell by binding to specific receptors. Intracellular CIS are anchored to the cell envelope by a membrane anchor or membrane complex that orients the CIS toward the exterior and serves as channel for the passage of the needle during sheath contraction [START_REF] Cascales | Microbiology: and Amoebophilus invented the machine gun![END_REF]. At the molecular level, the CIS is constituted of a needle surrounded by a contractile sheath and built on an assembly platform, the baseplate [START_REF] Brunet | The type VI secretion TssEFGK-VgrG phage-like baseplate is recruited to the TssJLM membrane complex via multiple contacts and serves as assembly platform for tail tube/sheath polymerization[END_REF][START_REF] Leiman | Contractile tail machines of bacteriophages[END_REF][START_REF] Büttner | Baseplate assembly of phage Mu: Defining the conserved core components of contractile-tailed phages and related bacterial systems[END_REF][START_REF] Arisaka | Molecular assembly and structure of the bacteriophage T4 tail[END_REF]. The needle is composed of an inner tube tipped by a puncturing device [START_REF] Basler | Type VI secretion system: secretion by a contractile nanomachine[END_REF][START_REF] Leiman | Contractile tail machines of bacteriophages[END_REF][START_REF] Brackmann | Using force to punch holes: mechanics of contractile nanomachines[END_REF]. The baseplate comprises conserved components that form wedges that assemble around the central hub [START_REF] Brunet | The type VI secretion TssEFGK-VgrG phage-like baseplate is recruited to the TssJLM membrane complex via multiple contacts and serves as assembly platform for tail tube/sheath polymerization[END_REF][START_REF] Büttner | Baseplate assembly of phage Mu: Defining the conserved core components of contractile-tailed phages and related bacterial systems[END_REF][START_REF] Taylor | Structure of the T4 baseplate and its function in triggering sheath contraction[END_REF]. In most cases, the central hub connects the inner tube with the puncturing device [START_REF] Watts | Structure of the bacteriophage T4 baseplate as determined by chemical cross-linking[END_REF][START_REF] Kanamaru | Structure of the cell-puncturing device of bacteriophage T4[END_REF][START_REF] Arisaka | Protein interactions in the assembly of the tail of bacteriophage T4[END_REF]. However, in T6SSs, the central hub and the puncturing device are fused as a single polypeptide, a chimera protein called VgrG [START_REF] Cascales | The type VI secretion toolkit[END_REF][START_REF] Pukatzki | Type VI secretion system translocates a phage tail spike-like protein into target cells where it cross-links actin[END_REF][START_REF] Leiman | Type VI secretion apparatus and phage tail associated protein complexes share a common evolutionary origin[END_REF].

Hence it is predicted that VgrG represents both the central part of the baseplate and the tail tube-puncturing device connector [START_REF] Büttner | Baseplate assembly of phage Mu: Defining the conserved core components of contractile-tailed phages and related bacterial systems[END_REF][START_REF] Leiman | Morphogenesis of the T4 tail and tail fibers[END_REF][START_REF] Cascales | Structural biology of type VI secretion systems[END_REF]. Indeed, interactions of VgrG with the inner tube Hcp protein has been evidenced in Agrobacterium tumefaciens and with several baseplate components including TssK and the TssFG complex in enteroaggregative Escherichia coli (EAEC) [START_REF] Brunet | The type VI secretion TssEFGK-VgrG phage-like baseplate is recruited to the TssJLM membrane complex via multiple contacts and serves as assembly platform for tail tube/sheath polymerization[END_REF][START_REF] Lin | Systematic dissection of the agrobacterium type VI secretion system reveals machinery and secreted components for subcomplex formation[END_REF][START_REF] Zoued | TssK is a trimeric cytoplasmic protein interacting with components of both phagelike and membrane anchoring complexes of the Type VI secretion system[END_REF]. The structure of the uropathogenic E. coli VgrG Nterminal fragment corresponding to the hub has been solved [START_REF] Leiman | Type VI secretion apparatus and phage tail associated protein complexes share a common evolutionary origin[END_REF], as well as the full-length VgrG1 protein from Pseudomonas aeruginosa [START_REF] Spínola-Amilibia | The structure of VgrG1 from Pseudomonas aeruginosa, the needle tip of the bacterial type VI secretion system[END_REF]. VgrG form trimers and is constituted of several domains: (i) an N-terminal domain that resembles the bacteriophage T4 gp27 and other phages hub proteins and that has, in the trimer, the overall structure of two superimposed disks, (ii) an oligonucleotide/oligosaccharide-binding (OB)-fold domain which is structurally similar to the bacteriophage T4 gp5 OB-fold domain, and (iii) the puncturing device which is comparable to the membrane-penetrating gp5 triple-stranded β-helix [START_REF] Kanamaru | Structure of the cell-puncturing device of bacteriophage T4[END_REF][START_REF] Pukatzki | Type VI secretion system translocates a phage tail spike-like protein into target cells where it cross-links actin[END_REF][START_REF] Leiman | Type VI secretion apparatus and phage tail associated protein complexes share a common evolutionary origin[END_REF][START_REF] Spínola-Amilibia | The structure of VgrG1 from Pseudomonas aeruginosa, the needle tip of the bacterial type VI secretion system[END_REF][START_REF] Veesler | A common evolutionary origin for tailed-bacteriophage functional modules and bacterial machineries[END_REF][START_REF] Uchida | Structure and properties of the C-terminal β-helical domain of VgrG protein from Escherichia coli O157[END_REF]. In addition to the core gp27-gp5 structure, additional domains could be found at the VgrG C-terminal extremity, such as effector domains bearing an enzymatic activity or adaptor domains that binds specific effectors [START_REF] Durand | and beyond: the versatile arsenal of type VI secretion effectors[END_REF][START_REF] Diniz | Molecular weaponry: diverse effectors delivered by the type VI secretion system[END_REF][START_REF] Pukatzki | Type VI secretion system translocates a phage tail spike-like protein into target cells where it cross-links actin[END_REF][START_REF] Unterweger | Adaptor proteins of Type VI secretion system effectors[END_REF]. Therefore, T6SS-associated VgrG proteins have a modular architecture and serve as hub for baseplate assembly, syringe for target cell membrane penetration, docking site for effector translocation, and connector for tail tube. In T6SS, the tail polymerizes once the baseplate docks to the membrane complex [START_REF] Durand | Biogenesis and structure of a type VI secretion membrane core complex[END_REF][START_REF] Brunet | The type VI secretion TssEFGK-VgrG phage-like baseplate is recruited to the TssJLM membrane complex via multiple contacts and serves as assembly platform for tail tube/sheath polymerization[END_REF]. Assembly of the inner tube proceeds by the addition of Hcp hexameric rings that stack on each other in a head-to-tail organization [START_REF] Brunet | Type VI secretion and bacteriophage tail tubes share a common assembly pathway[END_REF]. Hcp shares the same fold as tailed phages major tail proteins that are structurally related to the upper ring domain of the phage T4 gp27 hub [START_REF] Leiman | Type VI secretion apparatus and phage tail associated protein complexes share a common evolutionary origin[END_REF][START_REF] Veesler | A common evolutionary origin for tailed-bacteriophage functional modules and bacterial machineries[END_REF][START_REF] Pell | The phage lambda major tail protein structure reveals a common evolution for long-tailed phages and the type VI bacterial secretion system[END_REF][START_REF] Mougous | A virulence locus of Pseudomonas aeruginosa encodes a protein secretion apparatus[END_REF]. Polymerization of the sheath proceeds by the addition of TssBC complexes at the distal end of the growing tail structure [START_REF] Basler | Type VI secretion system: secretion by a contractile nanomachine[END_REF][START_REF] Brackmann | Using force to punch holes: mechanics of contractile nanomachines[END_REF][START_REF] Kudryashev | Structure of the type VI secretion system contractile sheath[END_REF][START_REF] Vettiger | The type VI secretion system sheath assembles at the end distal from the membrane anchor[END_REF][START_REF] Wang | Cryo-EM structure of the extended type VI secretion system sheath-tube complex[END_REF]. The assembly of the tube is coordinated with that of the sheath: in absence of sheath, the Hcp hexameric rings do not stack properly whereas no sheath extension occurs in absence of Hcp [START_REF] Brunet | Type VI secretion and bacteriophage tail tubes share a common assembly pathway[END_REF]. The coordinated polymerization of the tail is mediated by TssA, a protein that associates to the baseplate and then recruits and incorporates new tube and sheath blocks at the distal extremity [START_REF] Zoued | Priming and polymerization of a bacterial contractile tail structure[END_REF][START_REF] Zoued | TssA: The cap protein of the Type VI secretion system tail[END_REF]. However, it is not yet known how the first ring of Hcp is recruited to the baseplate and how Hcp interacts with VgrG in the final needle structure. Interestingly, the N-terminal disk of VgrG has a structure comparable to that of the hexameric Hcp, and therefore has been proposed to connect the axial 6-fold symmetry of Hcp with the 3-fold axial symmetry of the β-helix needle [START_REF] Leiman | Type VI secretion apparatus and phage tail associated protein complexes share a common evolutionary origin[END_REF][START_REF] Veesler | A common evolutionary origin for tailed-bacteriophage functional modules and bacterial machineries[END_REF]. VgrG therefore constitutes a good candidate to serve as nucleation factor for the inner tube. Here, bacterial two-hybrid (BACTH) and co-immunoprecipitation (co-IP) experiments show that VgrG1 interacts with Hcp1 in EAEC. Using shorter variants of VgrG1, we provide evidence that the gp27-like hub domain of VgrG1 is sufficient to interact with Hcp1, and that the strength of this interaction is comparable to that between Hcp1 hexamers. We then demonstrate that the VgrG hub domain is sufficient for the assembly and the mechanism of action of the T6SS, as it promotes assembly of the inner tube and of the contractile sheath, and permits Hcp1 release in the supernatant. Finally, targeted mutagenesis and cysteine cross-linking studies reveal that the base of the VgrG1 gp27-like hub interacts with the head surface of Hcp1. Taken together, these results allow to propose a model for the assembly and the architecture of the hub/tube interface.

RESULTS

VgrG1 interacts with Hcp1.

To test whether VgrG and Hcp interact, the vgrG and hcp genes of the EAEC sci-1 gene cluster (vgrG1, EC042_4533, accession number (GI): 284924254; hcp1, EC042_4529, GI: 284924250) were cloned into BACTH vectors, in frame with the T18 and T25 domains of the Bordetella adenylate cyclase, respectively. The results shown on reporter plates demonstrate that Hcp1 and VgrG1 interact (Fig. 1A). The interaction between the two proteins was confirmed by co-immunoprecipitation (co-IP) in the heterologous host E. coli K-12: VSV-Gtagged VgrG1 specifically co-immunoprecipitated with FLAG-tagged Hcp1 (Fig. 1B and1C).

These results show that the two proteins interact, and that this interaction is likely to be direct because it occurs in the absence of the other T6SS genes in E. coli K-12.

The VgrG1 N-terminal gp27-like hub is sufficient to mediate interaction with Hcp1.

VgrG proteins form trimers with a modular architecture, as a result of fusion events between gp27-like, gp5-like and adaptor/effector domains. In EAEC, the Sci-1 VgrG protein, VgrG1, is predicted to adopt a classic VgrG fold with the gp27-like hub and gp5 OB-fold and β-helix domains. In addition, EAEC VgrG1 carries a C-terminal extension, comprising DUF2345 and TTR domains, which binds to the Tle1 phospholipase effector [START_REF] Flaugnatti | A phospholipase A1 antibacterial Type VI secretion effector interacts directly with the C-terminal domain of the VgrG spike protein for delivery[END_REF]. To determine the region of VgrG1 that interacts with Hcp1, we conducted a deletion mutagenesis coupled to co-IP assays. First, VgrG1 boundaries were defined by modelling its structure based on the structure of the P. aeruginosa VgrG1 protein (PDB: 4UHV; [START_REF] Spínola-Amilibia | The structure of VgrG1 from Pseudomonas aeruginosa, the needle tip of the bacterial type VI secretion system[END_REF]): gp27-like hub (residues 1-386), gp5 OB-fold (residues 387-490), gp5 β-helix (residues 491-573), and C-terminal extension (residues 574-841) (Fig. 2A and2B). Fragments corresponding to the gp27-OB-gp5-C (∆DUF), gp27-OB (∆gp5) and gp27 (∆OB) regions (Fig. 2B) were fused to the VSV-G epitope, and the interaction of the VgrG truncated variants with Hcp1 was assayed using co-IP. Fig. 2C shows that none of the deletions affects the interaction with Hcp1, and hence that the gp27-like hub domain is sufficient to promote VgrG1-Hcp1 complex formation. To quantify the strength of this association, VgrG1-∆OB and Hcp1 were purified. However, while the isolated gp27 domain of VgrG1 was not stable, a variant comprising the OB fold (i.e., VgrG1-∆gp5) was purified to homogeneity (Fig. 2D, left upper panel). The VgrG1-∆gp5 interaction with Hcp1 was recorded by microscale thermophoresis (MST). VgrG1-∆gp5 was labelled with the amine-reactive blue fluorescent NHS dye and titrated against increasing concentrations of Hcp1 (Fig. 2D, right upper panel). The fitted traces confirmed that VgrG1-∆gp5 interacts with Hcp1 with a K D value of ∼ 2.8 µM (Fig. 2D, lower panel). Interestingly, the VgrG-Hcp association is not restricted to the EAEC Sci-1 T6SS, as MST assays demonstrated that (i) Hcp2, the Hcp protein encoded by the second T6SS gene cluster in the EAEC 17-2 strain, interacts with its cognate VgrG, VgrG2-∆gp5, with a K D of ∼ 9.4 µM (Fig. S1A), and (ii) the V. cholerae Hcp (Hcp Vc ) and VgrG (VgrG Vc ) interact with a K D of ∼ 4.1 µM (Fig. S1B). Hence, VgrG proteins interact with cognate Hcp proteins with an affinity in the 1-10 µM range. To gain further information about the specificity of Hcp protein towards VgrG, we performed cross-experiments where we tested the interaction of Hcp1 and Hcp2 with VgrG2-∆gp5 and VgrG1-∆gp5, respectively. MST results showed that VgrG1-∆gp5 and Hcp2 (K D ∼ 35.5 µM; Fig. S1C), and VgrG2-∆gp5 and Hcp1 (K D ∼ 22.4 µM; Fig. S1D) interact, although with lower affinity compared to the VgrG1-∆gp5-Hcp1 and VgrG2-∆gp5-Hcp2 complexes. These results suggest that VgrG and Hcp proteins interact but that the affinity is significantly higher between cognate pairs. However, in vivo interbacterial competition assays showed that VgrG2 and Hcp2 are unable to compensate for the absence of vgrG1 and hcp1, respectively (Fig. S2). It is therefore unlikely that non-cognate VgrG and Hcp proteins work together in EAEC cells.

The VgrG1 gp27-like hub is sufficient to promote Hcp1 tube assembly, sheath dynamics and Hcp1 release in the medium.

T6SS biogenesis starts with the formation of the membrane complex and the recruitment of the baseplate prior to coordinated assembly of the tail tube and sheath [START_REF] Durand | Biogenesis and structure of a type VI secretion membrane core complex[END_REF][START_REF] Brunet | The type VI secretion TssEFGK-VgrG phage-like baseplate is recruited to the TssJLM membrane complex via multiple contacts and serves as assembly platform for tail tube/sheath polymerization[END_REF][START_REF] Zoued | Priming and polymerization of a bacterial contractile tail structure[END_REF][START_REF] Gerc | Visualization of the Serratia Type VI secretion system reveals unprovoked attacks and dynamic assembly[END_REF]. Once assembled, the T6SS sheath contracts and propels the Hcp tube toward the target cell to deliver effectors [START_REF] Vettiger | Type VI secretion system substrates are transferred and reused among sister cells[END_REF]. A recent study has demonstrated that EAEC T6SS-mediated killing requires the full-length VgrG protein, which comprises the C-terminal extension that binds the Tle1 phospholipase effector [START_REF] Flaugnatti | A phospholipase A1 antibacterial Type VI secretion effector interacts directly with the C-terminal domain of the VgrG spike protein for delivery[END_REF]. However no information is available regarding the contribution of VgrG1 domains for T6SS biogenesis. We thus tested whether VgrG1-∆OB is sufficient to promote tail tube and tail sheath assembly, and Hcp1 release in the medium (Fig. 3).

It has been proposed that assembly of the tail tube and sheath is concomitant, the addition of a new Hcp ring immediately preceding the polymerization of a TssBC sheath strand [START_REF] Brunet | Type VI secretion and bacteriophage tail tubes share a common assembly pathway[END_REF]. The arrangements of Hcp rings could be probed by disulfide bond formation between cysteine side-chains located on each side of the Hcp ring [START_REF] Brunet | Type VI secretion and bacteriophage tail tubes share a common assembly pathway[END_REF][START_REF] Ballister | In vitro self-assembly of tailorable nanotubes from a simple protein building block[END_REF]. In vivo, Hcp1 rings organize in a head-to-tail manner with an estimated K D of 7 µM to form the tail tube [START_REF] Douzi | Crystal structure and self-interaction of the Type VI Secretion tail-tube orotein from Enteroaggregative Escherichia coli[END_REF]. This organization is controlled by components of the baseplate, including VgrG1: in their absence, Hcp1 rings associate randomly in head-to-tail, head-to-head and tail-to-tail arrangements [START_REF] Brunet | The type VI secretion TssEFGK-VgrG phage-like baseplate is recruited to the TssJLM membrane complex via multiple contacts and serves as assembly platform for tail tube/sheath polymerization[END_REF][START_REF] Brunet | Type VI secretion and bacteriophage tail tubes share a common assembly pathway[END_REF]. We thus tested the minimal VgrG1 requirements for proper Hcp1 tube assembly. Formation of disulfide bonds between pairs of cysteines that specifically report head-to-head, tail-to-tail and head-to-tail Hcp1 assembly was tested in cells producing the full-length VgrG1 or VgrG1 gp27-like hub (VgrG1-∆OB). Cysteine-mediated cross-linking in oxidative conditions showed that VgrG1-∆OB is sufficient to control proper assembly of Hcp1 tube exclusively in head-to-tail arrangements (Fig. 3A). Furthermore, dynamics of the tail sheath could be observed by fluorescence microscopy using fluorescently-labelled TssB.

Time-lapse recordings show that the VgrG1-∆OB construct is sufficient to mediate extension and contraction of the T6SS tail sheath (Fig. 3B and Fig. S3). These results suggest that the VgrG1 gp27-like hub is sufficient for proper assembly and action of the EAEC Sci-1 T6SS. This hypothesis was confirmed by the observation that VgrG1-∆OB is sufficient to enable Hcp1 release in the milieu, a reporter of sheath contraction (Fig. 3C). The ability of VgrG1-∆OB to promote T6SS assembly is not due to the formation of functional heterologous VgrG2/VgrG1-∆OB trimer assemblies as production of VgrG1-∆OB in vgrG1-vgrG2 mutant cells is sufficient to support Hcp release in the culture supernatant (Fig. S4).

The head side of Hcp1 interacts with VgrG1.

To define the side of Hcp1 that contacts VgrG1, we used tryptophan substitution variants of Hcp1 previously described to prevent assembly of Hcp tubes by sterically perturbing Hcp hexameric rings association [START_REF] Brunet | Type VI secretion and bacteriophage tail tubes share a common assembly pathway[END_REF][START_REF] Douzi | Crystal structure and self-interaction of the Type VI Secretion tail-tube orotein from Enteroaggregative Escherichia coli[END_REF]. These tryptophan substitutions were positioned at the head (S158W) or tail (N93W) side of Hcp1. Bacterial two-hybrid assays showed that introduction of a tryptophan residue at position Ser-158 disrupts the interaction with VgrG1 whereas a tryptophan at position Asn-93 does not impact Hcp1-VgrG1 complex formation (Fig. 4A). The strong impact of the S158W mutation on VgrG1-Hcp1 complex formation was further confirmed by co-immunoprecipitation (Fig. 4B). This result suggests that the head side of Hcp1 contacts VgrG1 (Fig. 4C). The current model of EAEC T6SS assembly proposes that TssA controls T6SS tail polymerization and binds to the distal end of the tube/sheath during extension [START_REF] Zoued | Priming and polymerization of a bacterial contractile tail structure[END_REF][START_REF] Zoued | TssA: The cap protein of the Type VI secretion system tail[END_REF]. This model implies that TssA should contact Hcp1 at the opposite side relative to VgrG1 (i.e., the tail side). Bacterial two-hybrid experiments using TssA and the Hcp1 tryptophan variants mirrored the VgrG1-Hcp1 interaction: the Hcp1 N93W variant does not interact with TssA whereas a tryptophan at position 158 does not affect Hcp1-TssA interaction (Fig. 4A and4B). Taken together, these results indicate that TssA and VgrG1 bind to the two opposite sides of Hcp1: VgrG1 contacts the head side whereas TssA interacts with the tail side (Fig. 4C).

The VgrG1 loop 360-362 is required for VgrG1-Hcp1 complex formation.

Because Hcp1 likely interacts with the basal side of the VgrG1 gp27-like domain, we engineered a VgrG1 variant in which one of the protruding loops (Fig. 4C), comprising residues Ser-360, Arg-361 and Lys-362, was exchanged by three glycine residues (S360G-R361G-K362G; hereafter called VgrG1 loop ). These mutations had no impact on VgrG1 multimer formation, as monitored by bacterial two-hybrid (Fig. 4D). However, proteinprotein interaction assays demonstrated that the VgrG1 360-362 region is important for VgrG1-Hcp1 complex formation, as no interaction was detected between Hcp1 and VgrG1 loop by bacterial two-hybrid and co-immunoprecipitation (Fig. 4B and4D). Finally, production of the VgrG1 loop variant does not compensate the absence of VgrG1 for in vivo Hcp1 tube formation (Fig. 4E) and inter-bacterial activity (Fig. 4F). These results demonstrate that the base of the VgrG1 hub-like domain, or at least the 360-362 loop, is critical for mediating functional contacts with Hcp1.

Disulfide bond formation identifies the T6SS hub-tube interface.

To gain detailed insights onto the VgrG1-Hcp1 interface, we introduced cysteine residues at the base of the VgrG1 gp27 hub domain and at each side of the Hcp1 hexamer. We reasoned that the thiol groups of cysteine side-chains would form disulfide bonds if located at appropriate distances (Cα-Cα distance < 7 Å). We first performed molecular docking simulations using the structure of Hcp1 [START_REF] Douzi | Crystal structure and self-interaction of the Type VI Secretion tail-tube orotein from Enteroaggregative Escherichia coli[END_REF] and the generated structural model of VgrG1.

Based on the best matches, cysteine residues were rationally positioned in loops, substituting residues sharing a 3 to 7-Å Cα-Cα distance. Six positions were substituted on the cysteineless (C38S) Hcp1: Gln-24, Asn-46, Gly-48, Ala-95, Gly-96 and Ser-158. Gln-24, Ala-95 and Gly-96 locate at the Hcp1 tail side whereas Asn-46, Gly-48 and Ser-158 locate at the Hcp1 head side. The Hcp1 Q24C, G48C, A95C, G96C, and S158C mutants were previously shown to be functional [START_REF] Brunet | Type VI secretion and bacteriophage tail tubes share a common assembly pathway[END_REF]. The VgrG1 protein possesses five native cysteine residues, including two in the gp27-like hub domain (Cys-27 and Cys-379). Because the bacterial cytoplasm is considered as reducing, formation of disulfide bonds between the VgrG1 and Hcp1 cysteine variants was catalyzed in presence of the oxidative agent copper(II) orthophenanthroline. Fig. 5A shows that none of the VgrG1 native cysteines engages in disulfide bond formation with the Hcp1 cysteine variants. Thus, two residues located in the VgrG1 360-362 loop that is important for VgrG1-Hcp1 complex formation were substituted by cysteines in the wild-type VgrG1 protein: Arg-361 and Lys-362. Interbacterial competition assays demonstrated that the Hcp1 N46C and VgrG1 R361C and K362C mutants are functional (Fig. S5). Fig. 5B shows that higher molecular weight species (denoted with * and ** in the anti-VSV-G and anti-FLAG immunoblots, respectively) are observable for four combinations involving VgrG1 R361C and K362C and Hcp1 N46C and G48C. The apparent molecular weight of these species (~ 120 kDa) is compatible with a complex between VgrG1 (theoretical weight: 93 kDa) and Hcp1 (theoretical weight: 18 kDa). These complexes are likely to be covalently bound by disulfide bridges as they dissociate upon addition of a reducing agent (Fig. 5C), and likely comprise both VgrG1 and Hcp1 as they are recognized by both the VSV-G and FLAG antibodies (Fig. 5D).

DISCUSSION

In this study, BACTH, co-IP, MST and disulfide bond cross-linking assays demonstrated that the component of the T6SS inner tube, Hcp1, interacts with the needle spike VgrG1 in EAEC. A complex comprising the Agrobacterium tumefaciens VgrG and Hcp proteins has been previously reported using co-immunoprecipitation [START_REF] Lin | Systematic dissection of the agrobacterium type VI secretion system reveals machinery and secreted components for subcomplex formation[END_REF]. Interaction between the VgrG spike and the Hcp tube component is likely to be conserved in all T6SS. Indeed, additional results demonstrated that the Hcp and VgrG proteins associated with the EAEC Sci-2 and V. cholerae T6SS gene clusters also interact. Interestingly, we also showed that cross-interactions between non-cognate Hcp and VgrG proteins from the two EAEC T6SS gene clusters exist, albeit with a significant decrease in affinity compared to cognate pairs. However, in vivo studies showed that VgrG2 and Hcp2 cannot functionally replace VgrG1 and Hcp1, respectively. Our VgrG1 deletion studies demonstrated that its gp27-like domain is sufficient to interact with Hcp1. Interestingly, we showed that neither the gp5 OB-fold nor the β-helix or the C-domains are essential for the assembly for the T6SS apparatus since the gp27 domain is sufficient to promote Hcp1 assembly, sheath extension and contraction, and Hcp1 release in the milieu. In addition, a previous report from our laboratory had shown that the VgrG1 C-terminal extension is necessary to cause target cell lysis by binding and delivering the Tle1 phospholipase effector [START_REF] Flaugnatti | A phospholipase A1 antibacterial Type VI secretion effector interacts directly with the C-terminal domain of the VgrG spike protein for delivery[END_REF]. Taken together, these information suggest that the VgrG1 gp27like hub is sufficient to promote assembly of the T6SS, and that the β-helix and C-terminal extension are required to penetrate the cell envelope of the target cell and to deliver effectors.

The observation that Hcp1 is released in the medium of cells producing the VgrG1 gp27-like domain further suggests that the VgrG1 β-helix is not required for piercing the outer membrane of the producing cell, and therefore that the needle should cross the outer membrane through a dedicated channel such as that proposed by a conformational change of the outermost portion of the TssJLM complex [START_REF] Durand | Biogenesis and structure of a type VI secretion membrane core complex[END_REF]. Our data also showed that the VgrG1 OBfold domain is not necessary for promoting Hcp1 tube formation, which is in agreement with the suggestion that the OB-fold acts as a symmetry connector between the six-fold gp27-like hub domain and the three-fold β-helix needle domain [START_REF] Leiman | Type VI secretion apparatus and phage tail associated protein complexes share a common evolutionary origin[END_REF].

Mutational studies and docking simulations coupled to disulfide bond assays detailed the contacts between Hcp1 and VgrG1 (Fig. 6). Using substitutions with tryptophan, a bulky side-chain aminoacid, we showed that the head side of the Hcp1 hexameric ring mediates contacts with the gp27-like base of VgrG1. In agreement with this organization, the opposite side of Hcp engages contacts with TssA, a subunit that coordinates the assembly of the inner tube with the polymerization of the sheath and that remains associated with the distal end of the tail during its extension [START_REF] Zoued | Priming and polymerization of a bacterial contractile tail structure[END_REF]. Formation of disulfide bonds between cysteine residues located at positions 46 and 48 of Hcp1 and 361 and 362 of VgrG1 allows to propose a model for Hcp1 binding to VgrG1 (Fig. 6).

Interestingly, Leiman et al. noted that the gp27-like domain of the VgrG trimer has a fold similar to that of an Hcp hexamer [START_REF] Leiman | Type VI secretion apparatus and phage tail associated protein complexes share a common evolutionary origin[END_REF]. When superimposed, the VgrG basal side corresponds to the tail side of Hcp. By interacting with the Hcp1 head side, VgrG1 engages into a head-to-tail organization with Hcp1, and hence in an association comparable to that between two Hcp1 hexamers. We thus propose that the hub formed by the VgrG gp27-like domain trimer serves as an adaptor to recruit the first Hcp hexameric ring and thus initiates the assembly of the inner tube. In agreement with this proposal, in vivo tube formation assays have shown that the Hcp1 tube does not assemble properly in absence of vgrG1 [START_REF] Brunet | Type VI secretion and bacteriophage tail tubes share a common assembly pathway[END_REF].

Interestingly, affinity measurements demonstrate that Hcp proteins interact with their cognate VgrG with a K D in the 1-10 micromolar range. With a K D of ~ 3 µM, the affinity of Hcp1 for VgrG1 is comparable -albeit better -to Hcp1 hexamers association (7 µM) [START_REF] Douzi | Crystal structure and self-interaction of the Type VI Secretion tail-tube orotein from Enteroaggregative Escherichia coli[END_REF]. Based on the slightly better affinity for VgrG1, we propose that an Hcp1 hexamer binds to the free base of VgrG1, which then becomes unavailable. Hcp1 tube polymerization can then proceed by the addition of new Hcp1 hexamers on each other at the distal end of the growing structure (Fig. 7). This model agrees with recent data showing that the tail extends by the distal end [START_REF] Vettiger | The type VI secretion system sheath assembles at the end distal from the membrane anchor[END_REF][START_REF] Zoued | Priming and polymerization of a bacterial contractile tail structure[END_REF]

MATERIAL AND METHODS

Bacterial strains, medium, and growth conditions. Strains used in this study are listed in Table S1.

Escherichia coli DH5α (New England Biolabs) was used for cloning procedures, W3110 for coimmunoprecipitations, BTH101 [START_REF] Karimova | A bacterial two-hybrid system based on a reconstituted signal transduction pathway[END_REF] for bacterial two-hybrid assays and BL21(DE3)pLys (Invitrogen) for protein purification. The enteroaggregative E. coli (EAEC) 17-2 strain and its ∆vgrG1, ∆hcp1, ∆vgrG1∆hcp1, and ∆vgrG1-tssB-mCh derivatives [START_REF] Brunet | The type VI secretion TssEFGK-VgrG phage-like baseplate is recruited to the TssJLM membrane complex via multiple contacts and serves as assembly platform for tail tube/sheath polymerization[END_REF] have been used for in vivo studies. Cells were routinely grown in Lysogeny broth (LB) at 30°C or 37°C. sci-1 gene expression was induced in SIM (Sci-1 inducing medium; M9 minimal medium supplemented with LB 10%, glycerol 0.4%, casaminoacids 40 µg.mL -1 , MgCl 2 2 mM, CaCl 2 0.1 mM, Vitamin B1 200 µg.mL -1 ) [START_REF] Brunet | An epigenetic switch involving overlapping fur and DNA methylation optimizes expression of a type VI secretion gene cluster[END_REF]. Terrific broth (TB; 1.2% peptone, 2.4% yeast extract, 72 mM K 2 HPO 4 , 17 mM KH 2 PO 4 , and 0.4% glycerol) was used for protein overproduction. Plasmids were maintained by addition of ampicillin (100 µg.mL -1 ), kanamycin (50 µg.mL -1 for E. coli K-12; 100 µg.mL -1 for EAEC) or chloramphenicol (40 µg.mL -1 ).

Gene expression was induced with 0.1-0.5 mM isopropyl-β-thio-galactoside (IPTG) or L-arabinose (0.02%).

Plasmid construction.

Plasmids used in this study are listed in Table S1. Polymerase Chain Reactions (PCR) were performed using a Biometra thermocycler using Pfu Turbo DNA polymerase (Agilent Technology). Custom oligonucleotides, listed in Table S1, were synthesized by Sigma Aldrich. EAEC 17-2 chromosomal DNA was used as a template for all PCRs. E. coli strain DH5α was used for cloning procedures. Bacterial two-hybrid plasmids encoding T18 and T25 fusions to Hcp1 and VgrG1, plasmid pBAD-VgrG1 V , encoding C-terminally VSV-G-tagged VgrG1 from the pBAD33 vector, plasmids pUC-Hcp1 F and pUC-Hcp2 F , encoding N-terminally FLAG-tagged Hcp1 and Hcp2 from the pUC12 vector, and plasmids pDest14-Hcp1 and pDest14-Hcp2, encoding Hcp1 and Hcp2 fused to a C-terminal 6×His tag have been previously described [START_REF] Aschtgen | The SciZ protein anchors the enteroaggregative Escherichia coli type VI secretion system to the cell wall[END_REF][START_REF] Zoued | TssK is a trimeric cytoplasmic protein interacting with components of both phagelike and membrane anchoring complexes of the Type VI secretion system[END_REF][START_REF] Douzi | Crystal structure and self-interaction of the Type VI Secretion tail-tube orotein from Enteroaggregative Escherichia coli[END_REF][START_REF] Aschtgen | SciN is an outer membrane lipoprotein required for type VI secretion in enteroaggregative Escherichia coli[END_REF]. Plasmids encoding truncated fragments of VgrG have been constructed by restriction/ligation-free cloning, as previously described [START_REF] Aschtgen | SciN is an outer membrane lipoprotein required for type VI secretion in enteroaggregative Escherichia coli[END_REF][START_REF] Van Den Ent | RF cloning: a restriction-free method for inserting target genes into plasmids[END_REF]. Briefly, fragments of interest were amplified with oligonucleotides introducing extensions annealing to the target vector. The double-stranded product of the first PCR has then been used as oligonucleotides for a second PCR using the target vector as template. Except the N46C variant, cysteine and tryptophan variants of Hcp1 have been previously described [START_REF] Brunet | Type VI secretion and bacteriophage tail tubes share a common assembly pathway[END_REF][START_REF] Douzi | Crystal structure and self-interaction of the Type VI Secretion tail-tube orotein from Enteroaggregative Escherichia coli[END_REF]. Hcp1 N46C and VgrG1 R361C, K362C and S360G-R361G-K362G (VgrG1 loop ) substitutions have been introduced by site-directed mutagenesis using complementary pairs of oligonucleotides and the Pfu Turbo high fidelity polymerase (Agilent technologies). For protein purification, the DNA sequences encoding the VgrG1 (VgrG1-∆gp5, Met-1 to Asn-490), VgrG2 (VgrG2-∆gp5, Met-1 to Asn-465) and V. cholerae strain O395 VgrG (VgrG Vc -∆gp5, Met-1 to Gln-471) gp27/OB regions were amplified from EAEC 17-2 or V. cholerae O395 and cloned into the pETG20A expression vector using standard Gateway procedures to yield pETG20A-VgrG1-∆gp5, pETG20A-VgrG2-∆gp5, and pETG20A-VgrG Vc -∆gp5, respectively. The resulting constructions allow the production of VgrG1-∆gp5, VgrG2-∆gp5, and VgrG Vc -∆gp5 fused to an N-terminal thioredoxin followed by a 6×His tag and a Tobacco Etch Virus (TEV) cleavage site. The DNA sequence encoding V. cholerae Hcp was amplified from strain O395 and cloned into the pDEST14 vector to yield pDest14-Hcp Vc . All constructs have been verified by DNA sequencing (Eurofins).

In vivo T6SS activity reporter assays. The interbacterial competition assay has been performed as described [START_REF] Flaugnatti | A phospholipase A1 antibacterial Type VI secretion effector interacts directly with the C-terminal domain of the VgrG spike protein for delivery[END_REF]. The Hcp release assay was performed as described [START_REF] Aschtgen | SciN is an outer membrane lipoprotein required for type VI secretion in enteroaggregative Escherichia coli[END_REF] except that cells were grown in SIM to absorbance at λ=600 nm (A 600 ) of 0.6. The periplasmic TolB marker was used as control for cell integrity.

Fluorescence microscopy. Overnight cultures of EAEC 17-2 derivative strains were diluted 1:100 in SIM medium and grown to an A 600 ~ 0.6 to maximise the expression of the sci-1 T6SS gene cluster [START_REF] Brunet | An epigenetic switch involving overlapping fur and DNA methylation optimizes expression of a type VI secretion gene cluster[END_REF]. Cells were concentrated to an A 600 = 5 in SIM, spotted on a thin pad of 1.5% agarose in SIM and covered with a cover slip. Microscopy recordings and digital image processing have been performed as previously described [START_REF] Zoued | TssK is a trimeric cytoplasmic protein interacting with components of both phagelike and membrane anchoring complexes of the Type VI secretion system[END_REF][START_REF] Brunet | Type VI secretion and bacteriophage tail tubes share a common assembly pathway[END_REF][START_REF] Zoued | Priming and polymerization of a bacterial contractile tail structure[END_REF][START_REF] Brunet | Imaging type VI secretionmediated bacterial killing[END_REF]. The Z project (average intensity) plugin has been used to merge and flatten all Z-planes. Microscopy analyses were performed at least three times, each with technical triplicate, and a representative experiment is shown.

Overproduction and purification of the EAEC Hcp and VgrG proteins. E. coli BL21(DE3) pLysS

(Invitrogen) cells were transformed with the pDEST14 derivatives encoding Hcp proteins or pETG20A derivatives encoding VgrG gp27/OB domains. Overnight cultures grown in TB supplemented with ampicillin (100 µg.mL -1 ) and chloramphenicol (35 µg.mL -1 ) at 37°C were diluted in TB and grown at 37°C to A 600 ~ 0.6. The temperature was then decreased to 25°C and the expression of the hcp or vgrG genes was induced with 500 mM IPTG for 18 h. Cells were harvested, resuspended in buffer A (50 mM Tris-HCl (pH 8.0), 300 mM NaCl) supplemented with 1 mM of EDTA, 0.5 mg.mL -1 of lysozyme, and phenylmethylsulfonyl fluoride (PMSF), submitted to three freeze-thawing cycles and sonicated after the addition of 20 µg.mL -1 of DNase and 20 mM of MgCl 2 .

Insoluble material was discarded by centrifugation for 30 min at 16,000 × g. All the subsequent purification steps were performed using an AKTA FPLC system. For Hcp proteins, the soluble fraction was loaded onto a 5-mL HisTrap FF column (GE Health Sciences). After extensive washing, the Hcp proteins were eluted in one step gradient of Imidazole 250 mM in Buffer A. For VgrG proteins, we followed the same protocol used for Hcp proteins with additional purification steps.

Briefly, the soluble proteins purified after the first HisTrap purification step were desalted on a HiPrep 26/10 column (Sephadex TM G-25, Amersham Biosciences), and VgrG proteins were obtained by cleavage using 2 mg of TEV protease for 18 h at 4°C and collected in the flow-through of a 5-mL HisTrap FF column. The resulting soluble proteins were concentrated using the centricon technology (Millipore, 10-kDa cut-off). After concentration, the soluble Hcp or VgrG proteins were passed through a Sephadex 200 26/60 column pre-equilibrated with 25 mM Tris-HCl (pH7.5), 100 mM NaCl.

Protein-protein interaction assays. Bordetella adenylate cyclase-based bacterial two-hybrid [START_REF] Zoued | TssK is a trimeric cytoplasmic protein interacting with components of both phagelike and membrane anchoring complexes of the Type VI secretion system[END_REF], coimmuno-precipitation [START_REF] Zoued | Structure-function analysis of the TssL cytoplasmic domain reveals a new interaction between the Type VI secretion baseplate and membrane complexes[END_REF] and oxidative cross-linking [START_REF] Brunet | Type VI secretion and bacteriophage tail tubes share a common assembly pathway[END_REF] experiments have been performed as previously described. Microscale thermophoresis experiments were performed using a Monolith NT.115 apparatus (NanoTemper). VgrG1-∆gp5, VgrG2-∆gp5 and VgrG Vc -∆gp5 were labeled with the blue-fluorescent dye NT-495-NHS (NanoTemper) and the buffer was exchanged for the assay buffer (25 mM Tris-HCl (pH 7.5), 200 mM NaCl, glycerol 5%, 0.05% Tween-20) using a Nap5 column (GE Healthcare). Titrations were conducted with a constant 200 nM fluorophore-labeled VgrG protein solution against up to 600 µM of Hcp proteins. Each data point was measured in triplicate. Single-site fitting was performed using the NanoTemper data analysis software.

Protein modeling and molecular docking simulations. The available structure of full-length VgrG1 protein from P. aeruginosa (PDB: 4MTK [Sycheva, Shneider and Leiman, unpublished] and 4UHV [START_REF] Spínola-Amilibia | The structure of VgrG1 from Pseudomonas aeruginosa, the needle tip of the bacterial type VI secretion system[END_REF]; 25.5/58.4% identity/similarity with EAEC VgrG1) was used as template to generate a model of the full-length VgrG1 from EAEC using the Homology modeling server accessible from the WHAT IF interactive interface. The HADDOCK2.2 webserver interface was used for molecular docking simulations [START_REF] Van Zundert | The HADDOCK2.2 web server: userfriendly integrative modeling of biomolecular complexes[END_REF].

Miscellaneous. Sodium Dodecyl Sulfate-Poly Acrylamide Gel Electrophoresis (SDS-PAGE) was performed using standard protocols. For immunostaining, proteins were transferred onto nitrocellulose membranes, and immunoblots were probed with primary antibodies and goat secondary antibodies coupled to alkaline phosphatase and were developed in alkaline buffer in presence of 5-bromo-4-chloro-3-indolylphosphate and nitroblue tetrazolium. The anti-FLAG (clone M2, Sigma-Aldrich), and anti-VSV-G (clone P5D4, Sigma-Aldrich) monoclonal antibodies, and alkaline phosphataseconjugated goat anti-mouse secondary antibodies (Beckman Coulter) have been purchased as indicated and used as recommended by the manufacturer. The anti-TolB polyclonal antibody is from our laboratory collection. tube/sheath starts after docking of the baseplate (BC) onto the membrane complex (MC) (1) [START_REF] Durand | Biogenesis and structure of a type VI secretion membrane core complex[END_REF][START_REF] Brunet | The type VI secretion TssEFGK-VgrG phage-like baseplate is recruited to the TssJLM membrane complex via multiple contacts and serves as assembly platform for tail tube/sheath polymerization[END_REF]. The first ring of Hcp hexamer is recruited and docks to the VgrG base [START_REF] Silverman | Structure and regulation of the type VI secretion system[END_REF]. Then, Hcp hexameric rings and TssBC strands are recruited and the tail tube/sheath extends from the distal end (3) [START_REF] Vettiger | The type VI secretion system sheath assembles at the end distal from the membrane anchor[END_REF][START_REF] Zoued | Priming and polymerization of a bacterial contractile tail structure[END_REF]. The VgrG protein is shown with the same color code as Fig. 2A and Fig. 6. The tail tube/sheath coordinator TssA is shown in red. OM, outer membrane; PG, peptidoglycan; IM, inner membrane. 7 "

Legend to Figures

V! Hcp1 F! !! 64! !! 98! !! 30! !! !! 36! 16! IP! T! IP! T! VgrG1 V! Hcp1 F! VgrG1 V! B VgrG1 V! Hcp1 F! !! 64! !! 98! !! 30! !! 16! IP! T! IP! T! Hcp1 F! VgrG1 V! Hcp1 F! Figure 2! D ! C B ! A gp27 DUF2345 and TTR coiled-coil gp5-C (β-helix) OB 615! 386! 490! 573! Δgp5! gp27! OB! ΔDUF! gp27! gp5-C! OB! VgrG1! gp27! gp5-C! OB! ΔOB! gp27! cc! cc! 615! 386! 490! 573! TTR! DUF2345! Δgp5 V! Hcp1 F ! ΔOB V ! Hcp1 F ! ΔDUF V ! Hcp1 F! !! 36! 16! !! 98! !! 50! !! 30! 64! !! !! *! *! *! T! IP! T! IP! T! IP! Hcp1 F! K D (μM) = 2.8 ± 0.1! Hcp1 concentration (log nM)! 5! 4! 1! 3! 2! 8! 9! 0! ΔF (10 2 )! time (sec)! 50! 40! 10! 30! 20! 0! 0.7! 1! Fluorescence! 0.9! 0.8! !! 45! 15! !! 130! !! 55! 70! !! !! !! 35! !! 25! Figure 3! C !! 16! !! 50! !! 30! !! 64! SN! C! TolB ! Hcp1 F! vgrG1-ΔOB ! SN! C! ΔvgrG1! SN! C! vgrG1 +! !! 36! A B vgrG1 ! vgrG1-ΔOB ! 0 ! 120 ! 90 ! 60 ! 30 ! vgrG1 + ! !! 36! !! 16! !! 50! !! 64! !! 98! 250! !! vgrG1 +! ΔvgrG1 ! Hcp1 F! Hcp1 F! ΔOB ! * ! !! 30! Figure 4! A T25-X ! VgrG1-T18! T18-Pal! T18-TssA! VgrG1-T25! T25-TolB! VgrG1 loop -T25! !! 50! 16! !! 64! !! 30! 98! !! !! VgrG1 V! Hcp1 F! VgrG1 V! Hcp1 S158W ! VgrG1 loop! Hcp1 F ! VgrG1 V ! Hcp1 F! T! IP! T! IP! T! IP! WT! ΔvgrG1 vgrG1 +! 10 ! 8 ! 7 ! 6 ! 9 ! E. coli K-12 survival (log cfu) ! ΔvgrG1 vgrG1 loop ! ΔvgrG1! D E B C N93W! S158W! TssA! tail! head! Hcp1! VgrG1! !! 36! !! 16! !! 50! !! 64! !! 98! 250! !! !! 30! Hcp1 F! Hcp1 F! * ! VgrG1 loop ! F Figure 5! A! C! D! -! +! -! +! -! +! -! +! VgrG1 V! Hcp1 F! ** ! * ! !! 36! !! 16! !! 50! !! 64! !! 98! !! 250! !! 30! !! 64! !! 98! 250! !! G48C! N46C! VgrG1-K362C! G48C! N46C! VgrG1-R361C! β-ME ! -! VgrG1-R361C! -! VgrG1-K362C! !! 36! !! 16! !! 50! !! 64! !! 98! !! 250! !! 30! !! 64! !! 98! 250! !! 36! 16! 50! 30! 64! 98! 250! 64! 98! !! !! !! !! !! !! !! !! !! !! 250! VgrG1 V! Hcp1 F! * ** ! !! 36! !! 16! !! 50! !! 30! !! 64! !! 98! 250! !! , ! Figure 6!
6 "

9 " ΔvgrG1 vgrG2 + "

ΔvgrG1"

E. coli K-12 survival (log cfu) "

WT" Δhcp1 hcp1 +" 10 " 8 "

7 "

6 "

9 " Δhcp1 hcp2 + "

Δhcp1"

E. coli K-12 survival (log cfu) " B A

*"

vgrG1-ΔOB " ΔvgrG1" vgrG1 +" WT " sheath dynamics (% of total sheaths) " 100 " 50 "

"

0 " 75 " sheaths with extension-contraction behaviors " non-dynamic tubular structures " n=122 " n=187 " n=150 " n=138 " WT"

ΔvgrG1 " R361C " K362C " Δhcp1" hcp1 +" N46C " vgrG1 +"

Figure 1 .

 1 Figure 1. Hcp1 interacts with VgrG1. (A) Bacterial two-hybrid analysis. BTH101 reporter

Figure 2 .

 2 Figure 2. The gp27-like base of VgrG1 is sufficient to mediate interaction with Hcp1. (A)

Figure 3 .

 3 Figure 3. The gp27-like base of VgrG1 is sufficient to promote Hcp1 tube formation,

Figure 4 .

 4 Figure 4. The head and tail sides of the Hcp1 hexamer engage contacts with VgrG1 and

Figure 5 .

 5 Figure 5. Disulfide bond formation between VgrG1 and Hcp1. Cytoplasmic extracts from

Figure 6 .

 6 Figure 6. Model of the EAEC VgrG1-Hcp1 complex. The modelled EAEC VgrG1 trimer

Figure 7 .

 7 Figure 7. Model for the early stages of T6SS tail biogenesis. The assembly of the T6SS tail

Figure S3 .

 S3 Figure S3. Statistical analyses of sheath dynamics. Distribution of the dynamic behavior of

Figure S4 .

 S4 Figure S4. The gp27-like base of VgrG1 does not require VgrG2 to promote Hcp1

Figure S5 .

 S5 Figure S5. Hcp1 and VgrG1 cysteine variants are functional for T6SS-dependent interbacterial activity. E. coli K-12 recipient cells (W3110 gfp + , kan R ) were mixed with the

Figure 1

 1 Figure1!

Figure 7 Figure S1 .

 7S1 Figure 7!

  Figure S1"

Figure S2 .

 S2 Figure S2. VgrG2 and Hcp2 do not compensate the absence of VgrG1 and Hcp1 for Sci1 T6SS-mediated inter-bacterial activity. E. coli K-12 recipient cells (W3110 gfp + , kan R ) were mixed with the indicated attacker cells: WT, EAEC 17-2; ∆vgrG1, ∆vgrG1 cells carrying the pBAD33 empty vector; vgrG1 + , ∆vgrG1 cells producing VgrG1; vgrG2 + , ∆vgrG1 cells producing VgrG2 (panel A); WT, EAEC 17-2; ∆hcp1, ∆hcp1 cells carrying the pUC12 empty vector; hcp1 + , ∆hcp1 cells producing Hcp1; hcp2 + , ∆hcp1 cells producing Hcp2 (panel B). The mixtures were spotted onto Sci-1 inducing medium (SIM) agar plates and incubated for 4 hours at 37°C. The image of a representative bacterial spot is shown below the graph reporting the number of surviving E. coli prey cells (counted on selective kanamycin medium; in log10 of colony-forming units (cfu)). The open circles indicate values from three independent assays, and the average is indicated by the bar.

Figure S3 "Figure S3 .

 S3S3 Figure S3"

Figure S4 .

 S4 Figure S4. The gp27-like base of VgrG1 does not require VgrG2 to promote Hcp1 release. FLAG-tagged Hcp1 (Hcp1 F ) release was assessed by separating total cell (C) and cell-free culture supernatant (SN) fractions from 5×10 8 ∆vgrG1-∆vgrG2 cells carrying the empty pBAD33 vector (-), the pBAD33 vector producing VgrG1 (vgrG1 + ), VgrG1-∆OB (vgrG1-∆OB) or VgrG2 (vgrG2 + ). Proteins were separated by 12.5%-acrylamide SDS-PAGE and periplasmic TolB and Hcp1 F were immunodetected using anti-TolB (upper panel) and anti-FLAG (lower panel) antibodies. Molecular weight markers (in kDa) are indicated on the left.

Figure S5 .

 S5 Figure S5. Hcp1 and VgrG1 cysteine variants are functional for T6SS-dependent inter-bacterial activity. E. coli K-12 recipient cells (W3110 gfp + , kan R ) were mixed with the indicated attacker cells (WT, EAEC 17-2; ∆hcp1, ∆hcp1 cells carrying the pUC12 empty vector; hcp1 + , ∆hcp1 cells producing Hcp1; N46C, ∆hcp1 cells producing the Hcp1 N46C variant; ∆vgrG1, ∆vgrG1 cells carrying the pBAD33 empty vector; vgrG1 + , ∆vgrG1 cells producing VgrG1; R361C and K362C, ∆vgrG1 cells producing the VgrG1 R361C and K362C variants, respectively), spotted onto Sci-1 inducing medium (SIM) agar plates and incubated for 4 hours at 37°C. The image of a representative bacterial spot is shown below the graph reporting the number of surviving E. coli prey cells (counted on selective kanamycin medium; in log10 of colony-forming units (cfu)). The open circles indicate values from three independent assays, and the average is indicated by the bar.

ACKNOWLEDGEMENTS

We thank the members of the Cascales, Cambillau, Lloubès, Bouveret and Sturgis research groups for insightful discussions, and Moly Ba, Isabelle Bringer, Annick Brun and Olivier Uderso for technical assistance. This work was supported by the Aix-Marseille Université (AMU), the Centre National de la Recherche Scientifique (CNRS) and grants from the Agence Nationale de la Recherche to E.C. (ANR-10-JCJC-1303-03, ANR-14-CE14-0006-02 and ANR-17-CE11-0039-01). Undergraduate laboratory works of M.G.R. and J.Z.B. were supported by the ANR-14-CE14-0006-02 and ANR-10-JCJC-1303-03 grants, respectively. B.D. was supported by a Fondation pour la Recherche Médicale (FRM) grant to C.C. (DEQ2011-0421282). A.Z. was supported by a doctoral fellowship from the French Ministère de la Recherche and a FRM end-of-thesis fellowship (FDT20140931060). Y.R.B. was supported by a doctoral fellowship from the French Ministère de la Recherche.

1! 3! 2! Hcp! OM! IM! PG! MC! BP! TssBC! Hcp!

Asn93-to-Trp substitution introduced into pT25-Hcp1 Brunet et al., 2014 pT25-Hcp1-S158W Ser158-to-Trp substitution introduced into pT25-Hcp1 Brunet et al., 2014 pVgrG1-T18 sci-1 vgrG cloned upstream T18 into pEB355 Zoued et al., 2013! pT18-TssA sci-1 tssA cloned downstream T18 into pEB355 Zoued et al., 2013 pT25-VgrG1 sci-1 vgrG cloned downstream T25 into pEB354 Zoued et al., 2013 pT25-VgrG1loop Ser360-to-Gly, Arg361-to-Gly and Lys362-to-Gly substitution introduced into pT25-VgrG1 This study pBAD33 Cm R , P15A, pAraBAD, AraC, cloning vector. Guzman et al., 1995 pBAD-VgrG1 sci-1 vgrG fused to a C-terminal VSV-G tag cloned into pBAD33 This study pBAD-VgrG1-∆DUF sci-1 vgrG (1-573) fused to a C-terminal VSV-G tag cloned into pBAD33 This study pBAD-VgrG1-∆gp5 sci-1 vgrG (1-490) fused to a C-terminal VSV-G tag cloned into pBAD33 This study pBAD-VgrG1-∆OB sci-1 vgrG (1-386) fused to a C-terminal VSV-G tag cloned into pBAD33 This study pBAD-VgrG1-R361C Arg361-to-Cys substitution introduced into pBAD-VgrG1 This study pBAD-VgrG1-K362C Lys362-to-Cys substitution introduced into pBAD-VgrG1 This study pBAD-VgrG1loop Ser360-to-Gly, Arg361-to-Gly and Lys362-to-Gly substitution introduced into pBAD-VgrG1 This study pUC12 Amp R , ColE1, pLac, cloning vector This study pUC12-Hcp1 sci-1 hcp fused to a C-terminal FLAG tag cloned into pUC12 Aschtgen et al., pUC12-Hcp2 sci-2 hcp fused to a C-terminal FLAG tag cloned into pUC12 Aschtgen et al., pUC12-Hcp1-Q24C Gln24-to-Cys substitution introduced into pUC12-Hcp1 Brunet et al., 2014 pUC12-Hcp1-N46C Asn46-to-Cys substitution introduced into pUC12-Hcp1 This study pUC12-Hcp1-G48C Gly48-to-Cys substitution introduced into pUC12-Hcp1 Brunet et al., 2014 pUC12-Hcp1-A95C Ala95-to-Cys substitution introduced into pUC12-Hcp1 Brunet et al., 2014 pUC12-Hcp1-G96C Gly96-to-Cys substitution introduced into pUC12-Hcp1 Brunet et al., 2014 pUC12-Hcp1-S158C Ser158-to-Cys substitution introduced into pUC12-Hcp1 Brunet et al., 2014 pUC12-Hcp1-Q24C-A95C Ala95-to-Cys substitution introduced into pUC12-Hcp1-Q24C Brunet et al., 2014 pUC12-Hcp1-G96C-S158C Gly96-to-Cys substitution introduced into pUC12-Hcp1-S158C Brunet et al., 2014

Legend to Supplementary Figures


Abdelmalek Aboussoror

Samir Adly
email: samir.adly@unilim.fr

Fatima Ezzahra Saissi
email: saissi-fz@hotmail.fr

Professor Michel Théra

Abdelmalek

A Duality Approach for a Class of Semivectorial Bilevel Programming Problems

Keywords: Bilevel programming, Vectorial programming, Convex analysis, Duality, Stability Mathematics Subject Classification (2010) 91A65, 90C29, 46N10, 90C46

In this paper, we present a duality approach using conjugacy for a semivectorial bilevel programming problem (S) where the upper and lower levels are vectorial and scalar respectively. This approach uses the Fenchel-Lagrange duality and is given via a regularization of problem (S) and the operation of scalarization. Then, using this approach, we provide necessary optimality conditions for a class of properly efficient solutions of (S). Finally, sufficient optimality conditions are given for (S) regardless of the duality approach. Examples are given for illustration.

Introduction

In this paper, we are concerned with the following semivectorial minimization problem

(S) v-min x∈M F (x),
where M is the solution set of the following scalar non parameterized lower level problem

(P) min x∈X f (x)
F : R p → R k , f : R p → R, are convex functions, and X is a nonempty compact convex subset of R p , and "v-min" stands for vectorial minimization.

The scalar case of problem (S) yields the following bilevel programming problem

( Ŝ) min x∈M F (x),
where F : R p → R is a function, which was first considered by Solodov in [START_REF] Solodov | An explicit descent method for bilevel convex optimization[END_REF]. The author showed that the class of problems having the form of ( Ŝ) includes the following large class of standard optimization problems of the form min x∈H ĥ(x),

where H = {x ∈ X/ ĝ(x) ≤ 0, Ax = a}, ĥ : R p → R, ĝ : R p → R m are differentiable convex functions, A ∈ R q×p , a ∈ R q and X is a closed convex subset of R p . Theoretical and numerical studies of such a class of scalar bilevel programming problems were considered in [1, 4-6, 11, 14]. In [START_REF] Aboussoror | A duality approach and optimality conditions for simple convex bilevel problems[END_REF], a Fenchel-Lagrange duality approach and optimality conditions are given for ( Ŝ). In this paper, we will extend this duality approach via scalarization to the vectorial case where the corresponding upper level is vectorial and the lower level is non parameterized and scalar. The duality that we consider is the so-called Fenchel-Lagrange duality (see [START_REF] Wanka | On the relations between different dual problems in convex mathematical programming[END_REF]). We note that in order to establish strong duality, we will need the Slater constraint qualification. Because of the lack of such a property by the scalarized problem of (S) defined according to the definition given in [START_REF] Geoffrion | Proper efficiency and the theory of vector maximization[END_REF], we will consider first a regularized problem (S ) of (S) ( > 0), whose scalarized problem (S s ) satisfies this condition. The regularization is based on the use of -approximate solutions of the lower level problem (P). As a stability result, we show that any accumulation point of a sequence of regularized properly efficient solutions is a properly efficient solution of the original problem (S). Afterwards, we consider the Fenchel-Lagrange dual (D s ) of the scalarized problem (S s ). Under appropriate assumptions, we establish strong duality and provide optimality conditions for the primal-dual pair (S s )-(D s ). Then, via the duality given for the regularized scalarized case, we deduce necessary optimality conditions for a class of properly efficient solutions of problem (S). These properly efficient solutions are accumulation points of sequences composed of some particular feasible points of the scalarized problems (S s n ), n ∈ N, n 0 + . Finally, sufficient optimality conditions are established for problem (S) without using duality. Illustrative examples are also given.

The outline of the paper is as follows. In Section 2, we recall some definitions and results related to convex analysis and multiobjective optimization. In Section 3, we introduce a regularized problem (S ) of (S). As a stability result, we show that any accumulation point of regularized properly efficient solutions is a properly efficient solution of (S). In Section 4, we associate the Fenchel-Lagrange dual to the scalarized problem (S s ). We show that strong duality holds and provide necessary and sufficient optimality conditions for the primal and dual problems. In Section 5, based on the results given in the previous section, we give necessary optimality conditions for a class of properly efficient solutions of problem (S). Finally, in Section 6, we provide sufficient optimality conditions for problem (S).

Preliminaries

In this section, we recall some fundamental definitions and results relating essentially to convex analysis and multiobjective optimization that we will use in the sequel.

Let A be a nonempty subset of R n . We shall denote by ψ A and σ A the indicator and the support functions of the set A respectively, defined on R n by

ψ A (x) = 0 if x ∈ A, +∞ if x / ∈ A, σ A (x) = sup a∈A x, a .
In the sequel, the following conventions in R = R ∪ {±∞} will be adopted

(+∞) -(+∞) = (-∞) -(-∞) = (+∞) + (-∞) = +∞, 0 × (+∞) = +∞, 0 × (-∞) = 0, α(-∞) = -∞, α(+∞) = +∞ for α ∈ R * + , α(-∞) = +∞, α(+∞) = -∞ for α ∈ R * -.
Definition 1 Let g : R n → R be a function.

i) The conjugate function of g relative to the set A is denoted by g * A and defined on R n by

g * A (p) = sup x∈A { p, x -g(x)},
where •, • denotes the inner product for two vectors in R n , i.e., for x = (x 1 , . . . , x n ) , and y = (y 1 , . . . , y n ) , x, y = n i=1 x i y i . When A = R n , we get the usual Legendre-Fenchel conjugate function of g denoted by g * .

ii) The effective domain of g denoted by dom g is the set defined by

dom g = {x ∈ R n /g(x) < +∞}.
We say that g is proper if g(x) > -∞ for all x ∈ R n , and dom g is nonempty.

Definition 2 Let g : R n → R ∪ {+∞} be a proper convex function. Let x ∈ dom g.

i) The subdifferential in the sense of convex analysis of g at x denoted by ∂g( x) is the set defined by

∂g( x) = x * ∈ R n /g(x) ≥ g( x) + x * , x -x ∀x ∈ R n .
An element x * ∈ ∂g( x) is called a subgradient of g at x. ii) Let ≥ 0. The -subdifferential of g at x denoted by ∂ g( x) is the set defined by

∂ g( x) = x * ∈ R n /g(x) ≥ g( x) + x * , x -x -∀x ∈ R n . An element x * in ∂ g( x) is called an -subgradient of g at x. When = 0, we have ∂ g( x) = ∂g( x). Remark 1 We have the following properties i) x * ∈ ∂g( x) ⇐⇒ x * , x = g( x) + g * (x * ). ii) g(x) + g * (x * ) ≥ x * , x ∀x, x * ∈ R n (the Young-Fenchel inequality).
Definition 3 Let C be a nonempty convex subset of R n and x ∈ C.

i) The normal cone to C at x denoted by N C ( x) is the set defined by

N C ( x) = x * ∈ R n / x * , x -x ≤ 0 ∀x ∈ C . It is not difficult to verify that ∂ψ C ( x) = N C ( x) and if x ∈ int C, then N C ( x) = {0 R n }, where int C denotes the topological interior of C. ii) Let > 0. The set of -normal directions to C at x denoted by N (C, x) is defined as follows N (C, x) = x * ∈ R n / x * , x -x ≤ ∀x ∈ C . We have ∂ ψ C ( x) = N (C, x).
Theorem 1 [9, Theorem 24.7] Let g : R n → R ∪ {+∞} be a proper convex and lower semicontinuous function, and C be a nonempty compact subset of int(dom g). Then, the set

x∈C ∂g(x) is compact. Theorem 2 [2, Theorem IV.23] Let f 1 , f 2 : R n → R ∪ {+∞} be proper convex functions.
Assume that there exists x 0 ∈ dom f 1 such that f 2 is continuous at x 0 . Then, for every

x ∈ R n ∂(f 1 + f 2 )(x) = ∂f 1 (x) + ∂f 2 (x). Theorem 3 [8, Theorem 3.3.1] Let ≥ 0 and f 1 , f 2 : R n → R ∪ {+∞} be proper convex functions such that ri(dom f 1 ) ∩ ri(dom f 2 ) = ∅. Then, for every x ∈ (dom f 1 ) ∩ (dom f 2 ), we have ∂ (f 1 + f 2 )(x) = 1 ≥0, 2 ≥0 1 + 2 ≤ {∂ 1 f 1 (x) + ∂ 2 f 2 (x)},
where for a nonempty subset A of R n , ri A denotes the relative interior of A, i.e., the interior of A relative to the smallest affine set containing A.

Theorem 4 [START_REF] Rockafellar | Convex Analysis[END_REF]Theorem 16.4] Let f 1 , . . . , f m : R n → R be proper convex functions. Assume that m i=1 ri(dom f i ) = ∅. Then, for any x ∈ R n , we have

m i=1 f i * (x) = inf x 1 ,...,x m ∈R n x 1 +•••+xm=x m i=1 f * i (x i )
and the infimum is attained.

Now, let us recall some definitions and fundamental results relating to multiobjective optimization that we are concerned in our study. Consider the following vectorial optimization problem

(Q) v-min x∈ X g(x)
where g = ( g1 , . . . , gr ) : R n → R r , gi : R n → R, i = 1, . . . , r, r ∈ N * , are functions and X is a nonempty subset of R n . Throughout the paper, the following partial order on R r will be adopted. For y = (y 1 , . . . , y r ) and z = (z 1 , . . . , z r ) y ≤ z ⇐⇒ y i ≤ z i ∀ i ∈ {1, . . . , r}. Definition 4 An element x ∈ X is said to be an efficient solution with respect to problem (Q) if g(x) ≤ g( x), for some x ∈ X, then g(x) = g( x). An efficient solution is also called a Pareto-efficient solution.

In the sequel, we will adopt the following definition of properly efficient solution in the sense of Geoffrion [START_REF] Geoffrion | Proper efficiency and the theory of vector maximization[END_REF].

Definition 5 An element x ∈ X is said to be a properly efficient solution to problem (Q), if it is efficient and if there exists a positive real number M such that for each i ∈ {1, . . . , r} and x ∈ X satisfying gi (x) < gi ( x), there exists j ∈ {1, . . . , r} such that

i) gj ( x) < gj (x), ii) gi ( x) -gi (x) gj (x) -gj ( x) ≤ M.
The following well-known result characterizes properly efficient solutions via scalarization.

Theorem 5 [START_REF] Geoffrion | Proper efficiency and the theory of vector maximization[END_REF]Theorem 2] Assume that X and g are convex. Let x ∈ X. Then, x is a properly efficient solution of problem (Q) if and only if there exists λ = (λ 1 , . . . , λ r ) ∈ int(R r + ) such that x solves the following scalar minimization problem

(Q s ) min x∈ X r i=1 λ i gi (x).

Regularization of Problem (S)

In the sequel, for our investigation, we adopt the following equivalent formulation of problem (S) (S) v-min

x∈X f (x)≤v F (x),
where v denotes the infimal value of the lower level problem (P). This formulation will allow us to extend the work given in [START_REF] Aboussoror | A duality approach and optimality conditions for simple convex bilevel problems[END_REF] for the scalar case to the vectorial one. Remark that since X is compact and f is continuous as a finite convex function, then the problem (P) admits at least one solution. So, v is a finite real number. Besides, we note that the problem (S) does not satisfy the Slater constraint qualification. The aim of this paper is to give a duality approach for problem (S). Our investigation is inspired from the work given in [START_REF] Aboussoror | A duality approach and optimality conditions for simple convex bilevel problems[END_REF], and will use some results established in [START_REF] Bot | Duality for multiobjective optimization problems with convex objective functions and D.C. constraints[END_REF] concerning a duality for multiobjective optimization problems with d.c. constraints. The duality considered by the authors in [START_REF] Bot | Duality for multiobjective optimization problems with convex objective functions and D.C. constraints[END_REF], is given via scalarization, and strong duality is established under a constraint qualification fulfilled by the considered scalarized problem. In our case, the scalarized problem of (S) defined according to the definition given in [START_REF] Geoffrion | Proper efficiency and the theory of vector maximization[END_REF] has the following form

(S s ) min x∈X f (x)≤v k i=1 λ i F i (x),
where (λ 1 , . . . , λ k ) ∈ int(R k + ) is fixed. Then, the problem (S s ) has the same set of constraints as (S). It follows that it cannot also satisfy the Slater condition which corresponds to the constraint qualification considered in [START_REF] Bot | Duality for multiobjective optimization problems with convex objective functions and D.C. constraints[END_REF]. Such a property will be used in our investigation to establish strong duality. To overcome this obstacle, in this section, we consider a regularized problem (S ) of (S) that satisfies this condition. As a consequence, the scalarized problem (S s ) of (S ) that we consider in the next section will also satisfy this condition. On the other hand, as a stability result, we show that any accumulation point of a sequence of regularized properly efficient solutions is a properly efficient solution to (S).

For > 0, we consider the following semivectorial regularized problem of (S)

(S ) v-min x∈M F (x),
where M is the set of -approximate solutions of problem (P), i.e.,

M = {x ∈ X/f (x) ≤ v + }.
Define on R p the following functions

g 1, (x) = ψ X (x), g 2, (x) = f (x), h 1, (x) = 0, h 2, (x) = v + .
Then, the regularized semivectorial bilevel programming problem (S ) can be written as

(S ) v- min x∈R p g 1, (x)-h 1, (x)≤0 g 2, (x)-h 2, (x)≤0 F (x).
In the sequel, and in order to apply some results given in [START_REF] Bot | Duality for multiobjective optimization problems with convex objective functions and D.C. constraints[END_REF] for optimization problems having d.c. constraints, we will view the convex constraints

g 1, (x) -h 1, (x) = ψ X (x) -0 and g 2, (x) -h 2, (x) = f (x) -(v + )
as d.c. constraints. In this case, (S ) can be viewed as a multiobjective minimization problem with convex vectorial objective function and two inequality d.c. constraints.

Let n 0 + . For simplicity, we shall denote the problem (S n ) by (S n ). The following theorem establishes that any accumulation point of a sequence of regularized properly efficient solutions is a properly efficient solution of problem (S).

Theorem 6 Let n 0 + . For n ∈ N,

let xn be a properly efficient solution of problem (S n ). Then, any accumulation point of the sequence ( xn ) n is a properly efficient solution of problem (S).

Proof Feasibility: Let x be an accumulation point of the sequence ( xn ) n . Let N be an infinite subset of N such that xn → x as n → +∞, n ∈ N . Since X is compact, it follows that x ∈ X. Moreover, from the feasibility of xn to problem (S n ), we have

f ( xn ) ≤ v + n .
Passing to the limit as n → +∞, n ∈ N , and using the continuity of the function f , we obtain f ( x) ≤ v, i.e., x ∈ M.

Optimality: Since the solution set M and the objective function F are convex, it follows that the semivectorial bilevel programming problem (S) is convex. Accordingly, it is sufficient to show that there exists λ = (λ 1 , . . . , λ k ) ∈ int(R k + ) such that x solves the scalar minimization problem

(S s ) min x∈M k i=1 λ i F i (x)
and the result will follow from Theorem 5. By contradiction, assume that for all λ = (λ 1 , . . . , λ k ) ∈ int(R k + ), there exists

x λ ∈ M such that k i=1 λ i F i (x λ ) < k i=1 λ i F i ( x).
We have xn → x as n → +∞, n ∈ N . Then, using the continuity of the functions F i , i = 1, . . . , k (as finite convex functions), we deduce that there exists n 0 ∈ N such that

k i=1 λ i F i (x λ ) < k i=1 λ i F i ( xn ) ∀n ≥ n 0 , n ∈ N . ( 1
)
Since x λ ∈ M ⊂ M n , then, the strict inequality in (1) yields a contradiction with the fact that xn is a properly efficient solution of problem (S n ), n ∈ N , n ≥ n 0 .

Duality for the Scalarized Problem of (S )

In this section, we consider the so-called Fenchel-Lagrange dual of a scalarized problem (S s ) of (S ). Then, we establish strong duality and provide optimality conditions for the scalar primal-dual pair. Define on R p the functions

g1, = g 1, -h 1, g2, = g 2, -h 2,
, and g = ( g1, , g2, ) T .

Then, we have g1, = ψ X and g2, = f (•) -(v + ). For > 0 and a fixed λ = (λ 1 , . . . , λ k ) ∈ int(R k + ), consider the following well-known scalar optimization problem associated to the regularized problem (S ) (see [START_REF] Geoffrion | Proper efficiency and the theory of vector maximization[END_REF])

(S s ) min x∈R p g (x)≤0 k i=1 λ i F i (x).
Remark 2 Let > 0. Then, from the characterization of the infimal value of problem (P), there exists x ∈ X verifying f (x ) < v + which is the so-called Slater condition relative to the set X, and corresponds to the constraint qualification used in [START_REF] Bot | Duality for multiobjective optimization problems with convex objective functions and D.C. constraints[END_REF] to establish strong duality. So that, such a constraint qualification is always satisfied in our case by the problem (S s ).

To start our duality approach, let us consider the following dual problem of (S s ) (see [START_REF] Wanka | On the relations between different dual problems in convex mathematical programming[END_REF])

( D s ) sup a∈R p q∈R 2 + ⎧ ⎨ ⎩ - k i=1 λ i F i * (
a i ∈R p a= k i=1 a i k i=1 (λ i F i ) * (a i ). Hence - k i=1 λ i F i * (a) -q g * (-a) = -min a i ∈R p a= k i=1 a i k i=1 (λ i F i ) * (a i ) -(q g )(-a) = -min a i ∈R p k i=1 λ i F * i (a i ) -q g * - k i=1 λ i a i = sup a i ∈R p - k i=1 λ i F * i (a i ) -q g * - k i=1 λ i a i .
Set q = (q 1 , q 2 ) . Then

q g * - k i=1 λ i a i = sup x∈R p - k i=1 λ i a i , x -(q g )(x) = sup x∈R p - k i=1 λ i a i , x -(q 1 g1, + q 2 g2, )(x)
Using the conventions introduced in Section 2, we have

q g * - k i=1 λ i a i = sup x∈R p - k i=1 λ i a i , x -ψ X (x) -q 2 g2, (x) = sup x∈X - k i=1 λ i a i , x -q 2 g2, (x) = (q 2 g2, ) * X - k i=1 λ i a i .
Therefore, after changing the notation of the coefficient of g2, , the problem ( D s ) and the following problem

(D s ) sup a i ∈R p i=1,...,k q∈R + - k i=1 λ i F * i (a i ) -(q g2, ) * X - k i=1 λ i a i
have the same optimal value. Therefore, in the sequel for our investigation we will use the problem (D s ) instead of ( D s ) that we will also call it the Fenchel-Lagrange dual of (S s ).

Via simple calculations, we obtain the following formulation of (D s )

(D s ) sup a i ∈R p i=1,...,k q∈R + inf x∈X - k i=1 λ i F * i (a i ) + k i=1 λ i a i , x + qf (x) -q(v + ) .
Then, we have the following result concerning weak duality.

Proposition 1 Let > 0. Then inf(S s ) ≥ sup(D s ).
Proof The result uses the fact that sup( D s ) = sup(D s ) and the well-known result of weak Fenchel-Lagrange duality between (S s ) and ( D s ) ( [START_REF] Wanka | On the relations between different dual problems in convex mathematical programming[END_REF][START_REF] Wanka | On the relations between different duals assigned to composed optimization problems[END_REF]).

The following theorem establishes strong duality between (S s ) and (D s ).

Theorem 7 Let > 0. Then, the dual problem (D s ) has a solution and strong duality holds, i.e., inf(S s ) = max(D s ).

Proof The result follows from [3, Theorem 3.3].

As necessary and sufficient optimality conditions for the primal-dual pair (S s )-(D s ), we have the following.

Theorem 8 Let > 0.

1) Necessary optimality conditions.

Let x be a solution of problem (S s ). Then, there exists a solution (a , α ) of the dual problem (D s ), with a = (a 1 , . . . , a k ), a i ∈ R p , i = 1, . . . , k, α ≥ 0, verifying together with x the following optimality conditions

i) F i (x ) + F * i (a i ) = a i , x , i = 1, . . . , k, ii) α (f (x ) -v -) = 0, iii) inf x∈X k i=1 λ i a i , x + α f (x) = k i=1 λ i a i , x + α (v + ).
2) Sufficient optimality conditions.

Let x and (a , α ) be feasible points of (S s ) and (D s ) respectively. Assume that they satisfy together the above conditions i)-iii). Then, x and (a , α ) solve (S s ) and (D s ) respectively. Moreover, strong duality holds for the primal-dual pair (S s )-(D s ).

Proof The result follows from [START_REF] Bot | Duality for multiobjective optimization problems with convex objective functions and D.C. constraints[END_REF]Theorem 3.4].

Remark 3

In terms of subdifferential and normal cone, the properties i) and iii) in Theorem 8 can be written respectively as follows 1)

a i ∈ ∂F i (x ) ∀i = 1, . . . , k, 3) -k i=1 λ i a i ∈ ∂(α f )(x ) + N X (x ).
If moreover α > 0, then, in terms of -subdifferential and the set of -normal directions to X, the property ii) is equivalent to the following

2) 0 R p ∈ ∂ (f + ψ X )(x ) = 1 ≥0, 2 ≥0 1 + 2 ≤ {∂ 1 f (x ) + N 2 (X, x )}.
The assertion concerning property 1) is obvious. To show property 3), first remark that by using ii), we can write iii) as follows

inf x∈X k i=1 λ i a i , x + α f (x) = k i=1 λ i a i , x + α f (x ). That is x solves the problem min x∈X k i=1 λ i a i , x + α f (x) . It follows that 0 R p ∈ ∂(α f )(x ) + k i=1 λ i a i + N X (x ).
Hence,

- k i=1 λ i a i ∈ ∂(α f )(x ) + N X (x ).
For property 2), it suffices to apply Theorem 3. Let us give the following example in which we apply the results of Theorems 8 and 6.

Example 1 Let F and f be the functions defined on R 2 by

F (x 1 , x 2 ) = (F 1 (x 1 , x 2 ), F 2 (x 1 , x 2 )) = (x 2 1 + x 2 2 , -x 1 + x 2 2 ) , f (x 1 , x 2 ) = - 3 2 x 1 -x 2 and X = (x 1 , x 2 ) ∈ R 2 /x 2 1 -4x 1 -x 2 + 4 ≤ 0, 3 2 x 1 + x 2 -4 ≤ 0 .
Then, the functions F and f are convex and X is a convex compact set. It is not difficult to check that inf P = v = -4 and M = conv (0, 4) , ( 52 , 1 4 ) . In order to solve the problem (S), we will apply the duality considered for the regularized scalarized case associated to problem (S). For > 0 sufficiently small, we first consider the scalarized problem (S s ) of (S ) with (λ 1 , λ 2 ) = (1, 1) ∈ int(R 2 + ) which is written as (S s ) min

(x 1 ,x 2 )∈M (F 1 + F 2 )(x 1 , x 2 ). Let x = (x 1 , x 2 ) , (a , α ), a = (a 1 , a 2 ), a i ∈ R 2 , i ∈ {1, 2}
, α > 0, be feasible points of problems (S s ) and (D s ) respectively. Note that from Theorem 8, we have strong duality. Using again this theorem, we deduce that x and (a , α ) solve (S s ) and (D s ) respectively if and only if the following conditions are satisfied (Remark 3)

i) a i ∈ ∂F i (x ) ∀i ∈ {1, 2}. ii) α (f (x ) -v -) = 0. iii) -2 i=1 a i ∈ ∂(α f )(x ) + N X (x ).
Assume that α > 0. Then, the complementary slackness condition ii) yields 3 2

x 1 + x 2 -4 + = 0. ( 2
)
From i), we have

(a 1 , a 2 ) ∈ 2 i=1 ∂F i (x ). ( 3
)
So that using (3) and iii), we obtain

0 ∈ ∂F 1 (x ) + ∂F 2 (x ) + ∂(α f )(x ) + N X (x ). Assume that x ∈ int X, if it exists. Then, N X (x ) = {0 R 2 } and 0 ∈ ∂F 1 (x ) + ∂F 2 (x ) + α ∂f (x )
which is explicitly written as .

2x 1 -1 -3 2 α 4x 2 -α = 0 0 . Using (
We have x → x = ( 25 11 , 13 11 ) , as → 0 + . Then, according to Theorem 6, the point x = ( 25 11 , 13 11 ) is a properly efficient solution of the original semivectorial bilevel programming problem (S).

Necessary Optimality Conditions for the Semivectorial Bilevel Programming Problem (S)

In this section, from the results obtained in the previous section for the scalarized problem (S s ) of (S ), we will deduce necessary optimality conditions for problem (S). The following additional assumptions will be needed.

(H 1 ) For every > 0 sufficiently small, there exists x ∈ int X such that f (x

) ≤ v + , (H 2 ) inf x∈R p F i (x) < inf x∈X F i (x), i = 1, . . . , k, (H 3 ) inf x∈R p f (x) < inf x∈X f (x).
Let us give the following simple example where assumptions (H 1 )-(H 3 ) and assumptions of convexity and compactness are satisfied.

Example 2 Let X = [START_REF] Aboussoror | A duality approach and optimality conditions for simple convex bilevel problems[END_REF][START_REF] Azé | Elements d'Analyse Convexe et Variationnelle Ellipses[END_REF], F and f be the functions defined on R by

F (x) = (F 1 (x), F 2 (x)) = (x 2 , x + 1), f (x) = |x| + 3 2 .
Then, X is a convex compact set and the functions F and f are convex. Let us verify if assumptions (H 1 )-(H 3 ) are satisfied.

-Assumption (H 1 ): Let > 0 sufficiently small. We have v = 5 2 and M =

[1, 1 + ]. Let x = 1 + 2 . Then, x ∈ int X and assumption (H 1 ) is satisfied. -Assumption (H 2 ): We have inf x∈R F 1 (x) = 0 < inf x∈X F 1 (x) = 1, inf x∈R F 2 (x) = -∞ < inf x∈X F 2 (x) = 2. So that, (H 2 ) is satisfied. -Assumption (H 3 ): We have inf x∈R f (x) = 3 2 < inf x∈X f (x) = 5 2 .
So that, (H 3 ) is satisfied.

Let λ = (λ 1 , . . . , λ k ) ∈ int(R k + ) be fixed. For simplicity, in the sequel, the problem (S s n ) will be denoted by (S s n ). The following theorem gives necessary optimality conditions for (S). The necessary optimality conditions are given for the class of properly efficient solutions of (S) which are accumulation points of a sequence of some particular feasible points of problems (S s n ), n ∈ N.

Theorem 9 Let n 0 + . Assume that assumptions (H 1 )-(H 3 ) are satisfied. Let x n be the feasible point of the scalarized problem (S s n ) given by assumption (H 1 ) corresponding to n . Moreover, assume that x n satisfies together with a certain feasible point (a n , α n ) of the dual problem (D s n ) the conditions i)-iii) in Theorem 8, with a n = (a 1n , . . . , a kn ), a in ∈ R p , i = 1, . . . , k, and α n > 0. Let x be an accumulation point of the sequence (x n ) n . Then,

x is a properly efficient solution of (S) and there exists (a, α), a = (a 1 , . . . , a k ), a i ∈ R p , α ∈ R * + , such that the following optimality conditions are satisfied

a) a ∈ k i=1 ∂F i ( x), b) 0 ∈ ∂f ( x) + N X ( x), c) -k i=1 λ i a i ∈ α∂f ( x). Furthermore, x solves the scalar unconstrained minimization problem min x∈R n k i=1 λ i F i + αf (x).
Proof Let N be an infinite subset of N such that x n → x, n → +∞, n ∈ N . Theorem 8 implies that x n solves the problem

(S s n ) min x∈R p ψ X (x)≤0 f (x)-v-n≤0 k i=1 λ i F i (x).
Then, from Theorem 5, x n is a properly efficient solution of problem (S n ). Therefore, from Theorem 6, x is a properly efficient solution of problem (S) as an accumulation point of the sequence (x n ). On the other hand, for every n ∈ N , we have i)

F i (x n ) + F * i (a in ) = a in , x n ∀i ∈ {1, . . . , k}, ii) α n (f (x n ) -v -n ) = 0, iii) (α n f ) * X -k i=1 λ i a in = -k i=1 λ i a in , x n -α n (v + n ).
Since α n = 0, then, the complementary slackness condition ii) yields

f (x n ) = v + n .
Letting n → +∞, n ∈ N , we get f ( x) = v. That is x solves the lower level problem (P). Hence 0 ∈ ∂f ( x) + N X ( x).

The first condition i) is equivalent to

a in ∈ ∂F i (x n ) ∀i ∈ {1, . . . , k}.
Let i ∈ {1, . . . , k}. We have ∂F i (x n ) ⊂ x∈X ∂F i (x). Since X is compact, it follows that the set ∂F i (X) = x∈X ∂F i (x) is compact (Theorem 1). Then, there exists an infinite subset N 1 ⊂ N such that the sequence (a in ) n∈N 1 converges to a certain point a i . Since, for any n ∈ N 1 , we have

F i (y) ≥ F i (x n ) + a in , y -x n ∀y ∈ R p
then, passing to the limit as n → +∞, n ∈ N 1 , we get

F i (y) ≥ F i (x) + a i , y -x ∀y ∈ R p .
That is

a i ∈ ∂F i ( x) ∀i ∈ {1, . . . , k}. ( 4
)
It follows that

a = (a 1 , . . . , a k ) ∈ k i=1 ∂F i ( x).
On the other hand, for every n ∈ N 1 , from the condition iii), we have

(α n f ) * X - k i=1 λ i a in = - k i=1 λ i a in , x n -α n (v + n ) = - k i=1 λ i a in , x n -α n f (x n ). So - k i=1 λ i a in , x n -α n f (x n ) ≥ - k i=1 λ i a in , x -α n f (x) ∀x ∈ X.
Hence

α n f (x) + k i=1 λ i a in , x ≥ α n f (x n ) + k i=1 λ i a in , x n ∀x ∈ X.
Therefore, for all n ∈ N 1 , x n is a solution of the minimization problem

min x∈X α n f (x) + k i=1 λ i a in , x . It follows that 0 ∈ α n ∂f (x n ) + k i=1 λ i a in + N X (x n ). Since x n ∈ int X, then N X (x n ) = {0}. We deduce that - 1 α n k i=1 λ i a in ∈ ∂f (x n ) ∀n ∈ N 1 .
Using the fact that ∂f (x n ) ⊂ x∈X ∂f (x) for all n ∈ N 1 , and that x∈X ∂f (x) is compact, we deduce that there exist an infinite subset N 2 of N 1 and x * ∈ R p such that -1

α n k i=1 λ i a in → x * , as n → +∞, n ∈ N 2 .
On the other hand, for n ∈ N 2 , we have

f (y) ≥ f (x n ) + - 1 α n k i=1 λ i a in , y -x n ∀y ∈ R p .
Therefore, passing to the limit as n → +∞, n ∈ N 2 , we obtain

f (y) ≥ f ( x) + x * , y -x ∀y ∈ R p .
That is x * ∈ ∂f (x). Assumptions (H 2 ) and (H 3 ) imply respectively that a i = 0 ∀i ∈ {1, . . . , k} and x * = 0. For n ∈ N 2 , set x * n = -1

α n k i=1 λ i a in . Since x * n → x * > 0 as n → +∞, n ∈ N 2
then, there exists n 0 ∈ N 2 such that

x * n > 0 ∀n ≥ n 0 , n ∈ N 2 .
Hence, for n ∈ N 2 , n ≥ n 0 , we have

α n = k i=1 λ i a in x * n which converges to α = k i=1 λ i a i x *
, with α = 0. Besides, the sequence (-

1 α n k i=1 λ i a in ) n∈N 2 converges to - k i=1 λ i a i α = x * . Hence - k i=1 λ i a i ∈ α∂f ( x). (5)
Using ( 4), ( 5) and Theorem 2, we obtain

0 ∈ ∂ k i=1 λ i F i + αf ( x).
Therefore, x solves the scalar minimization problem min

x∈R p k i=1 λ i F i + αf (x).

Sufficient Optimality Conditions for (S)

In this section, we will provide sufficient optimality conditions for the semivectorial bilevel programming problem (S). These optimality conditions are given regardless of the duality approach.

Theorem 10 Let x ∈ R p . Assume that there exists (a, α, λ), a = (a 1 , . . . , a k ),

a i ∈ R p , α ∈ R * + , λ = (λ 1 , . . . , λ k ) ∈ int(R k + ) such that the following optimality conditions are satisfied i) a ∈ k i=1 ∂F i ( x), ii) 0 ∈ ∂f ( x) + N X ( x), iii) -k i=1 λ i a i ∈ α∂f ( x).

Then, x is a properly efficient solution of problem (S).

Proof Feasibility: The property ii) implies that x solves the problem (P). So, it is a feasible point of problem (S).

Optimality: From property i), we have

k i=1 λ i a i ∈ ∂ k i=1 λ i F i ( x). (6)
Hence, from property iii) and ( 6), we obtain

0 ∈ ∂ k i=1 λ i F i + αf ( x).
Therefore, x solves the problem min

x∈R n k i=1 λ i F i + αf ( x).
It follows that

k i=1 λ i F i ( x) + αf ( x) ≤ k i=1 λ i F i (x) + αf (x) ∀x ∈ R p . Then k i=1 λ i F i ( x) ≤ k i=1 λ i F i (x) + α(f (x) -f ( x)) ∀x ∈ R p . Let x ∈ M. Since x ∈ M, then f (x) = f ( x). Hence k i=1 λ i F i ( x) ≤ k i=1 λ i F i (x) ∀x ∈ M. So, x is a solution of problem min x∈M k i=1 λ i F i (x).
Therefore, from Theorem 5, x is a properly efficient solution of (S).

Theorem 11 Let x ∈ R p . Assume that x satisfies the following conditions

i) 0 ∈ ∂ k i=1 ∂F i + ∂f ( x), ii) 0 ∈ ∂f ( x) + N X ( x).

Then, x is a properly efficient solution of problem (S).

Proof Feasibility: The feasibility of x to problem (S) follows from the property ii).

Optimality: Property i) implies that x solves the problem min

x∈R p k i=1 F i + f (x). So k i=1 F i (x) + f (x) ≥ k i=1 F i ( x) + f ( x) ∀x ∈ R p and hence k i=1 F i (x) ≥ k i=1 F i ( x) + (f ( x) -f (x)) ∀x ∈ R p . (7) Let x ∈ M. Then, f (x) = f ( x). It follows from (7) that k i=1 F i (x) ≥ k i=1 F i ( x) ∀x ∈ M.
That is x solves the problem

min x∈M k i=1 F i (x).
Therefore, Theorem 5 implies that x is a properly efficient solution of problem (S).

We give the following illustrative example for Theorem 10 in a nondifferentiable case.

Example 3 Let X = [0, 1] × [1, 2
], F and f be the functions defined on R 2 by

F (x 1 , x 2 ) = (F 1 (x 1 , x 2 ), F 2 (x 1 , x 2 ), F 3 (x 1 , x 2 ), F 4 (x 1 , x 2 )) = |x 1 | + x 2 2 -x 2 , x 2 1 -4x 1 + 1, 2|x 1 | -3x 2 , x 1 + 1 2 x 2 2 , f (x 1 , x 2 ) = x 2 1 + x 1 + 1.
Then, the functions F and f are convex and X is a convex compact set. We have

v = inf (x 1 ,x 2 ) ∈X f (x 1 , x 2 ) = 1 and M = {0} × [1, 2].
Note that the function F is not differentiable on the set {0} × R. The semivectorial bilevel programming problem that we consider is

(S) v-min (x 1 ,x 2 ) ∈M |x 1 | + x 2 2 -x 2 , x 2 1 -4x 1 + 1, 2|x 1 | -3x 2 , x 1 + 1 2 x 2 2 ,
where M is the solution set of problem

(P) min (x 1 ,x 2 ) ∈[0,1]×[1,2] (x 2 1 + x 1 + 1).
Let us determine a point x = ( x1 , x2 ) that satisfies the sufficient optimality conditions in Theorem 10. Then, we are led to verify if there exists (a, α, λ), a = (a 1 , . . . , a 4 ), a i ∈ R 2 , i = 1, . . . , 4, α > 0, and λ = (λ 1 , . . . , λ 4 ) ∈ int(R 4 + ), such that the conditions i)-iii) in Theorem 10 are satisfied. These conditions are written as follows i) (a 1 , . . . , a 4 ) ∈ . We can take for example α = 1, a 11 = 0, and a 31 = 2. Therefore, (0, 4 3 ) is a properly efficient solution of problem (S). The conditions i)-iii) are satisfied with a = (a 1 , . . . , a 4 ) , a 1 = (0, 5 3 ) , a 2 = (-4, 0) , a 3 = (2, -3) , a 4 = (1, 4 3 ) , α = 1 and λ = (1, 1, 1, 1) .

Conclusion

In this paper, we have given a duality approach for problem (S). Our investigation which is based on the operations of regularization and scalarization, is inspired from the work given in [START_REF] Aboussoror | A duality approach and optimality conditions for simple convex bilevel problems[END_REF] and uses some results established in [START_REF] Bot | Duality for multiobjective optimization problems with convex objective functions and D.C. constraints[END_REF]. Since the semivectorial bilevel programming problem (S) and its scalarized problem does not satisfy the Slater condition which we use as a sufficient condition to establish strong duality, we have first proceeded to a regularization of problem (S). The obtained regularized problem (S ) and its scalarized problem (S s ) satisfy this condition. Afterwards, under the data, we have established strong Fenchel-Lagrange duality and provided optimality conditions for the scalarized primal-dual pair (S s )-(D s ). Then, via the results established for the regularized scalarized case, we have provided necessary optimality conditions for a class of properly efficient solutions of problem (S). These properly efficient solutions are accumulation points of sequences of feasible points of problem (S s n ) that satisfy the property considered in (H 1 ), n 0 + . Finally, we have provided sufficient optimality conditions for (S) regardless of the duality approach. We emphasize that the obtained results are new in the literature of multiobjective bilevel optimization, and extend the results given in [START_REF] Aboussoror | A duality approach and optimality conditions for simple convex bilevel problems[END_REF] from the scalar case to the vectorial one. This extension corresponds to the case where the upper and lower levels are respectively vectorial and scalar.

  -Lagrange dual. Let us give an explicit formulation of the objective function of problem ( D s ). Using Theorem 4, we have k i=1 λ i F i * (a) = min

4 4 3 .

 443 i=1 ∂F i ( x1 , x2 ), ii) 0 ∈ ∂f ( x1 , x2 ) + N X ( x1 , x2 ), iii) -4 i=1 λ i a i ∈ α∂f ( x1 , x2 ). Condition ii) implies that x ∈ M. Then, x1 = 0 and x2 ∈ [1, 2]. We have ∂f ( x1 , x2 ) = {(1, 0) } and ∂F 1 ( x1 , x2 ) = [-1, 1] × {2 x2 -1}, ∂F 2 ( x1 , x2 ) = {(-4, 0) } ∂F 3 ( x1 , x2 ) = [-2, 2] × {-3}, ∂F 4 ( x1 , x2 ) = {(1, x2 ) }.Take λ = (1, 1, 1, 1) and set a i = (a i1 , a i2 ) , i ∈ {1, . . . , 4}. From condition i), we obtaina 11 ∈ [-1, 1], a 12 = 2 x2 -1, a 2 = (-4, 0) , a 31 ∈ [-2,2], a 32 = -3, a 4 = (1, x2 ) and condition iii) yields -a 11a 31α + 3 = 0, x2 = So α ∈ [0, 6]