Ce travail se situe dans le domaine de l'optimisation combinatoire. Nous étudions plus particulièrement des caractérisations d'objets pour lesquels des problèmes, qui dans le cas général sont NP-complets, deviennent polynomiaux. Nous traitons d'abord le problème de la faisabilité d'un multiflot, qui possède des applications trés importantes en recherche opérationnelle. C'est à dire, étant donnée la spécification du problème, avec le réseau, les capacités et les demandes, on veut démontrer l'existence ou la non-existence d'une solution. Une façon d'aborder ce problème est de donner des conditions nécessaires et suffisantes pour l'existence d'un multiflot, comme celle connue par condition de coupe. Nous présentons la condition (CC, K_5, F_7), qui généralise la condition de coupe et "raffine" une autre condition existante, la (CC3). La structure du problème de multiflot nous permet aussi de regarder un problème étroitement associé, celui du "packing" de métriques. Nous traitons le cas des packing entiers et demi-entiers, quand la famille de métriques comprend les métriques CC3 et les métriques K_5 et F_7. Nous caractérisons la classe de graphes, et plus généralement de matroïdes, ou l'on peut trouver des packings entiers et demi-entiers, sous quelques hypothèses additionnelles. Puis nous nous intéressons aux propriétés générales des graphes h- et t-parfaits, et au problème de coloration associé. Les résultats que nous présentons donnent des bornes pour leur nombres chromatiques, et des classes qui satisfont une conjecture de Shepherd. Enfin nous présentons la hiérarchie des graphes étudiés, qui est obtenu grâce à des outils comme les graphes faiblement bipartis, les clutters binaires et les matrices à composantes 0,1. Nous clôturons ce mémoire en précisant quelques directions de recherche qui pourront donner suite à ce travail, aussi bien sur le sujet de la faisabilité des problèmes de multiflot, que sur la coloration des graphes h- et t-parfaits.

Les calculs numériques ont été effectués à l'aide du logiciel SAGE. / En 1993, Conway et Schneeberger fournirent un critère simple permettant de déterminer si une forme quadratique donnée représente tous les entiers positifs ; le théorème des 15. Dans ce mémoire, nous nous intéressons à un problème analogue, soit la recherche d’un critère similaire permettant de détecter si une forme quadratique en trois variables représente tous les entiers impairs. On débute donc par une introduction générale à la théorie des formes quadratiques, notamment en deux variables, puis on expose différents points de vue sous lesquels on peut les considérer. On décrit ensuite le théorème des 15 et ses généralisations, en soulignant les techniques utilisées dans la preuve de Bhargava. Enfin, on démontre deux théorèmes qui fournissent des critères permettant de déterminer si une forme quadratique ternaire représente tous les entiers impairs. / In 1993, Conway and Schneeberger gave a simple criterion allowing one to determine whether a given quadratic form represents all positive integers ; the 15-theorem. In this thesis, we investigate an analogous problem, that is the search for a similar criterion allowing one to detect if a quadratic form in three variables represents all odd integers. We start with a general introduction to the theory of quadratic forms, namely in two variables, then, we expose different points of view under which quadratic forms can be considered. We then describe the 15-theorem and its generalizations, with a particular emphasis on the techniques used in Bhargava’s proof of the theorem. Finally, we give a proof of two theorems which provide a criteria to determine whether a ternary quadratic form represents all odd integers.

En 1993, Conway et Schneeberger fournirent un critère simple permettant de déterminer si une forme quadratique donnée représente tous les entiers positifs ; le théorème des 15. Dans ce mémoire, nous nous intéressons à un problème analogue, soit la recherche d’un critère similaire permettant de détecter si une forme quadratique en trois variables représente tous les entiers impairs. On débute donc par une introduction générale à la théorie des formes quadratiques, notamment en deux variables, puis on expose différents points de vue sous lesquels on peut les considérer. On décrit ensuite le théorème des 15 et ses généralisations, en soulignant les techniques utilisées dans la preuve de Bhargava. Enfin, on démontre deux théorèmes qui fournissent des critères permettant de déterminer si une forme quadratique ternaire représente tous les entiers impairs. / In 1993, Conway and Schneeberger gave a simple criterion allowing one to determine whether a given quadratic form represents all positive integers ; the 15-theorem. In this thesis, we investigate an analogous problem, that is the search for a similar criterion allowing one to detect if a quadratic form in three variables represents all odd integers. We start with a general introduction to the theory of quadratic forms, namely in two variables, then, we expose different points of view under which quadratic forms can be considered. We then describe the 15-theorem and its generalizations, with a particular emphasis on the techniques used in Bhargava’s proof of the theorem. Finally, we give a proof of two theorems which provide a criteria to determine whether a ternary quadratic form represents all odd integers. / Les calculs numériques ont été effectués à l'aide du logiciel SAGE.

Dans ce manuscrit de thèse, nous nous intéressons à la description des noyaux atomiques composés d'un nombre impair de nucléons dans la méthode dite de la fonctionnelle énergie de la densité (EDF). Plus précisément, nous présentons et appliquons dans le cas de ces noyaux, des extensions à cette méthode : (i) la projection sur les bons nombres quantiques, (ii) le mélange de configurations à travers la méthode des coordonnées génératrices (GCM), qui permettent de prendre en compte dans nos calculs des corrélations de type " au-delà du champ moyen " entre les nucléons constituant le noyau. De telles extensions n'avaient jusqu'alors été employées, dans leurs versions les plus générales, qu'aux noyaux ayant à la fois un nombre pair de neutrons et de protons, et nous nous proposons donc de démontrer leurs applicabilités également dans le cas des noyaux impairs. Dans la première partie de ce travail, nous présentons le formalisme mathématique de la méthode EDF, en mettant tout particulièrement l'accent sur le traitement des symétries dans cette approche. Dans la seconde partie du manuscrit, nous appliquons notre modèle au cas du noyau de Mg25 et analysons les résultats sous différents angles (ex : précision numérique des résultats, convergence du mélange de configurations, comparaison avec les données expérimentales connues). Les premiers résultats obtenus dans ce travail de thèse sont encourageants et démontrent l'intérêt de notre approche pour les calculs théoriques de structure nucléaire.

Dans ce manuscrit de thèse, nous nous intéressons à la description des noyaux atomiques composés d’un nombre impair de nucléons dans la méthode dite de la fonctionnelle énergiede la densité (EDF). Plus précisément, nous présentons et appliquons dans le cas de ces noyaux,des extensions à cette méthode : (i) la projection sur les bons nombres quantiques, (ii) le mélange de configurations à travers la méthode des coordonnées génératrices (GCM), qui permettent deprendre en compte dans nos calculs des corrélations de type « au-delà du champ moyen » entre les nucléons constituant le noyau. De telles extensions n’avaient jusqu’alors été employées, dansleurs versions les plus générales, qu’aux noyaux ayant à la fois un nombre pair de neutrons etde protons, et nous nous proposons donc de démontrer leurs applicabilités également dans le cas des noyaux impairs. Dans la première partie de ce travail, nous présentons le formalisme mathématique de la méthode EDF, en mettant tout particulièrement l’accent sur le traitement des symétries dans cette approche. Dans la seconde partie du manuscrit, nous appliquons notre modèle au cas du noyau de 25Mg et analysons les résultats sous différents angles (ex : précision numérique des résultats, convergence du mélange de configurations, comparaison avec les données expérimentales connues). Les premiers résultats obtenus dans ce travail de thèse sontencourageants et démontrent l’intérêt de notre approche pour les calculs théoriques de structure nucléaire. / In this work, we are interested in the treatment of odd-mass atomic nuclei in energydensity functional (EDF) models. More precisely, the goal of this thesis is to develop and to applyto odd-mass nuclei, the theoretical extensions of the EDF method that are: (i) the projectiontehnique, and (ii) the configuration mixing by the generator coordinate method (GCM). Thesetwo extensions are part of the so-called multi-reference energy density functional (MR-EDF)formalism and allow for taking into account, within an EDF context, the "beyond-mean-field"correlations between the nucleons forming the nucleus. Until now, the MR-EDF formalism hasbeen applied, in its full-fledged version, only to the calculation of even-even nuclei. In this thesis,we want to demonstrate the applicability of such a model also for the description of odd-massnuclei. In the first part of this thesis, we describe the theoretical formalism of the EDF models,giving particular attention to the treatment of symmetries within our approach. In the secondpart of the manuscript, we apply our model to the nucleus 25Mg and investigate different aspectsof the method (e.g. numerical accuracy, convergence of the configuration mixing, comparison toknown experimental data). The results obtained in this work are encouraging and demonstratethe potential of our approach for theoretical nuclear structure calculations.

L’approche au coeur de cette étude est une recherche-action qui a pour but de mettre à l’essai un réseau littéraire, à caractère interdisciplinaire, pour le réinvestissement du vocabulaire disciplinaire et des concepts en arithmétique auprès des élèves de deuxième année du primaire. Cette recherche comporte trois objectifs spécifiques : 1) mettre à l’essai un réseau littéraire pour décrire et analyser les pratiques enseignantes 2) décrire et analyser le réinvestissement du vocabulaire disciplinaire en arithmétique des élèves du premier cycle du primaire 3) mettre en relation les pratiques enseignantes et le réinvestissement du vocabulaire disciplinaire en arithmétique des élèves. Pour y arriver, une enseignante de 2e année du primaire du centre de services scolaire Marguerite-Bourgeoys a participé à notre recherche en expérimentant le réseau littéraire proposé dans cette étude. Nous avons conduit trois séances d’entretien en tout avec l’enseignante avant, pendant et après l’expérimentation. Une analyse de contenu a été effectuée à partir des propos et des impressions de l’enseignante. Durant l’expérimentation, nous avons participé à trois séances d’observation dont l’analyse a été effectuée en nous basant sur le modèle multi-agenda. Finalement, quatre élèves ont participé à un entretien qui visait notamment à analyser et à décrire le réinvestissement du vocabulaire disciplinaire en arithmétique et leurs résultats ont été analysés à l’aide d’une grille d’évaluation bâtie par nous. À la lumière des données, nous constatons l’importance de l’engagement professionnel et des pratiques expertes de l’enseignante pour favoriser le réinvestissement du vocabulaire disciplinaire associé aux concepts en arithmétique chez les élèves. Concernant les apports du réseau littéraire, les albums jeunesse ont constitué une source de motivation, de participation et d’engagement pour les élèves. Ils ont aussi favorisé le réinvestissement du vocabulaire disciplinaire. Les limites de cette recherche se situent davantage au niveau de la longueur et du sens de certaines questions étant donné que certains élèves n’ont pas le français comme langue maternelle, ce qui pouvait avoir un impact sur le niveau de compréhension des questions en lien avec les oeuvres du réseau. / This research project fits according to an action research approach and aims at putting into practice a literacy network, being of interdisciplinary nature, for the reinvestment of disciplinary vocabulary in arithmetic with second grade students. This present research has the following three specific objectives: 1) to put into practice a literacy network to analyze and describe the teacher’s practices 2) to analyze and describe the second-grade student’s reinvestment of disciplinary vocabulary in arithmetic 3) to put into relation the impact of teacher practices in the reinvestment of the student’s disciplinary vocabulary in arithmetic. In order to succeed, a second-grade elementary teacher from the Centre de services scolaire Marguerite-Bourgeoys participated in our research project by experimenting the literacy network proposed in this study. During the experimentation, we participated in three sessions of observation in which the analysis was performed according to the multi-agenda model. In addition, we conducted three semi-structured interview sessions before, during and after the experimentation. A content analysis was carried out based on the teacher’s comments and impressions. Lastly, four students participated in our interview questionnaire and their results were analyzed by an evaluation grid that we have built. At the light of the results and of our analysis, we notice the importance of professional engagement and the second-grade teacher’s expert practices pertaining to the disciplinary vocabulary of arithmetic of the students. Concerning the contributions, children’s literature has constituted an important source of student motivation and participation. It also favored the reinvestment of disciplinary vocabulary. The limits of this research are more pertaining to the length and the meaning of certain questions considering the fact that certain students do not have French as their native language which could’ve had an impact on their level of comprehension relating to the questions on the literacy network.

O Vírus da Imunodeficiência Humana (HIV) é o agente etiológico da Síndrome da Imunodeficiência Adquirida (AIDS). A AIDS é uma doença de distribuição mundial, e estima-se que existam atualmente pelo menos 36,9 milhões de pessoas infectadas com o vírus. Durante o seu ciclo replicativo, o HIV promove diversas alterações na fisiologia da célula hospedeira a fim de promover sua sobrevivência e potencializar a replicação. A rápida progressão da infecção pelo HIV-1 em humanos e em modelos animais está intimamente ligada à função da proteína acessória Nef. Dentre as diversas ações de Nef está a regulação negativa de proteínas importantes na resposta imunológica, como o receptor CD4. Sabe-se que esta ação resulta da indução da degradação de CD4 em lisossomos, mas os mecanismos moleculares envolvidos ainda são totalmente elucidados. Nef forma um complexo tripartite com a cauda citosólica de CD4 e a proteína adaptadora 2 (AP-2), em vesículas revestidas por clatrina nascentes, induzindo a internalização e degradação lisossomal de CD4. Pesquisas anteriores demonstraram que o direcionamento de CD4 aos lisossomos por Nef envolve a entrada do receptor na via dos corpos multivesiculares (MVBs), por um mecanismo atípico, pois, embora não necessite da ubiquitinação de carga, depende da ação de proteínas que compõem os ESCRTs (Endosomal Sorting Complexes Required for Transport) e da ação de Alix, uma proteína acessória da maquinaria ESCRT. Já foi reportado que Nef interage com subunidades dos complexos AP-1, AP-2, AP-3 e Nef não parece interagir com subunidades de AP-4 e AP-5. Entretanto, o papel da interação de Nef com AP-1 e AP-3 na regulação negativa de CD4 ainda não está totalmente elucidado. Ademais, AP-1, AP-2 e AP-3 são potencialmente heterogêneos devido à existência de isoformas múltiplas das subunidades codificadas por diferentes genes. Todavia, existem poucos estudos para demonstrar se as diferentes combinações de isoformas dos APs são formadas e se possuem propriedades funcionais distintas. O presente trabalho procurou identificar e caracterizar fatores celulares envolvidos na regulação do tráfego intracelular de proteínas no processo de regulação negativa de CD4 induzido por Nef. Mais especificamente, este estudo buscou caracterizar a participação do complexo AP-1 na modulação negativa de CD4 por Nef de HIV-1, através do estudo funcional das duas isoformas de ?-adaptina, subunidades de AP-1. Utilizando a técnica de Pull-down demonstramos que Nef é capaz de interagir com ?2. Além disso, nossos dados de Imunoblot indicaram que a proteína ?2-adaptina, e não ?1-adaptina, é necessária no processo de degradação lisossomal de CD4 por Nef e que esta participação é conservada para degradação de CD4 por Nef de diferentes cepas virais. Ademais, por citometria de fluxo, o silenciamento de ?2, e não de ?1, compromete a diminuição dos níveis de CD4 por Nef da membrana plasmática. A análise por imunofluorêsncia indireta também revelou que a diminuição dos níveis de ?2 impede a redistribuição de CD4 por Nef para regiões perinucleares, acarretando no acúmulo de CD4, retirados por Nef da membrana plasmática, em endossomos primários. A depleção de ?1A, outra subunidade de AP-1, acarretou na diminuição dos níveis celulares de ?2 e ?1, bem como, no comprometimento da eficiente degradação de CD4 por Nef. Além disso, foi possível observar que, ao perturbar a maquinaria ESCRT via super-expressão de HRS (uma subunidade do complexo ESCRT-0), ocorreu um acumulo de ?2 em endossomos dilatados contendo HRS-GFP, nos quais também detectou-se CD4 que foi internalizado por Nef. Em conjunto, os resultados indicam que ?2-adaptina é uma importante molécula para o direcionamento de CD4 por Nef para a via ESCRT/MVB, mostrando ser uma proteína relevante no sistema endo-lisossomal. Ademais, os resultados indicaram que as isoformas ?-adaptinas não só possuem funções distintas, mas também parecem compor complexos AP-1 com diferentes funções celulares, já que apenas a variante AP-1 contendo ?2, mas não ?1, participa da regulação negativa de CD4 por Nef. Estes estudos contribuem para o melhor entendimento dos mecanismos moleculares envolvidos na atividade de Nef, que poderão também ajudar na melhor compreensão da patogênese do HIV e da síndrome relacionada. Em adição, este trabalho contribui para o entendimento de processos fundamentais da regulação do tráfego de proteínas transmembrana no sistema endo-lisossomal. / The Human Immunodeficiency Virus (HIV) is the etiologic agent of Acquired Immunodeficiency Syndrome (AIDS). AIDS is a disease which has a global distribution, and it is estimated that there are currently at least 36.9 million people infected with the virus. During the replication cycle, HIV promotes several changes in the physiology of the host cell to promote their survival and enhance replication. The fast progression of HIV-1 in humans and animal models is closely linked to the function of an accessory protein Nef. Among several actions of Nef, one is the most important is the down-regulation of proteins from the immune response, such as the CD4 receptor. It is known that this action causes CD4 degradation in lysosome, but the molecular mechanisms are still incompletely understood. Nef forms a tripartite complex with the cytosolic tail of the CD4 and adapter protein 2 (AP-2) in clathrin-coated vesicles, inducing CD4 internalization and lysosome degradation. Previous research has demonstrated that CD4 target to lysosomes by Nef involves targeting of this receptor to multivesicular bodies (MVBs) pathway by an atypical mechanism because, although not need charging ubiquitination, depends on the proteins from ESCRTs (Endosomal Sorting Complexes Required for Transport) machinery and the action of Alix, an accessory protein ESCRT machinery. It has been reported that Nef interacts with subunits of AP- 1, AP-2, AP-3 complexes and Nef does not appear to interact with AP-4 and AP-5 subunits. However, the role of Nef interaction with AP-1 or AP-3 in CD4 down-regulation is poorly understood. Furthermore, AP-1, AP-2 and AP-3 are potentially heterogeneous due to the existence of multiple subunits isoforms encoded by different genes. However, there are few studies to demonstrate if the different combinations of APs isoforms are form and if they have distinct functional properties. This study aim to identify and characterize cellular factors involved on CD4 down-modulation induced by Nef from HIV-1. More specifically, this study aimed to characterize the involvement of AP-1 complex in the down-regulation of CD4 by Nef HIV-1 through the functional study of the two isoforms of ?-adaptins, AP-1 subunits. By pull-down technique, we showed that Nef is able to interact with ?2. In addition, our data from immunoblots indicated that ?2- adaptin, not ?1-adaptin, is required in Nef-mediated targeting of CD4 to lysosomes and the ?2 participation in this process is conserved by Nef from different viral strains. Furthermore, by flow cytometry assay, ?2 depletion, but not ?1 depletion, compromises the reduction of surface CD4 levels induced by Nef. Immunofluorescence microscopy analysis also revealed that ?2 depletion impairs the redistribution of CD4 by Nef to juxtanuclear region, resulting in CD4 accumulation in primary endosomes. Knockdown of ?1A, another subunit of AP-1, resulted in decreased cellular levels of ?1 and ?2 and, compromising the efficient CD4 degradation by Nef. Moreover, upon artificially stabilizing ESCRT-I in early endosomes, via overexpression of HRS, internalized CD4 accumulates in enlarged HRS-GFP positive endosomes, where co-localize with ?2. Together, the results indicate that ?2-adaptin is a molecule that is essential for CD4 targeting by Nef to ESCRT/MVB pathway, being an important protein in the endo-lysosomal system. Furthermore, the results indicate that ?-adaptins isoforms not only have different functions, but also seem to compose AP-1 complex with distinct cell functions, and only the AP-1 variant comprising ?2, but not ?1, acts in the CD4 down-regulation induced by Nef. These studies contribute to a better understanding on the molecular mechanisms involved in Nef activities, which may also help to improve the understanding of the HIV pathogenesis and the related syndrome. In addition, this work contributes with the understanding of primordial process regulation on intracellular trafficking of transmembrane proteins.

O Vírus da Imunodeficiência Humana (HIV) é o agente etiológico da Síndrome da Imunodeficiência Adquirida (AIDS). A AIDS é uma doença de distribuição mundial, e estima-se que existam atualmente pelo menos 36,9 milhões de pessoas infectadas com o vírus. Durante o seu ciclo replicativo, o HIV promove diversas alterações na fisiologia da célula hospedeira a fim de promover sua sobrevivência e potencializar a replicação. A rápida progressão da infecção pelo HIV-1 em humanos e em modelos animais está intimamente ligada à função da proteína acessória Nef. Dentre as diversas ações de Nef está a regulação negativa de proteínas importantes na resposta imunológica, como o receptor CD4. Sabe-se que esta ação resulta da indução da degradação de CD4 em lisossomos, mas os mecanismos moleculares envolvidos ainda são totalmente elucidados. Nef forma um complexo tripartite com a cauda citosólica de CD4 e a proteína adaptadora 2 (AP-2), em vesículas revestidas por clatrina nascentes, induzindo a internalização e degradação lisossomal de CD4. Pesquisas anteriores demonstraram que o direcionamento de CD4 aos lisossomos por Nef envolve a entrada do receptor na via dos corpos multivesiculares (MVBs), por um mecanismo atípico, pois, embora não necessite da ubiquitinação de carga, depende da ação de proteínas que compõem os ESCRTs (Endosomal Sorting Complexes Required for Transport) e da ação de Alix, uma proteína acessória da maquinaria ESCRT. Já foi reportado que Nef interage com subunidades dos complexos AP-1, AP-2, AP-3 e Nef não parece interagir com subunidades de AP-4 e AP-5. Entretanto, o papel da interação de Nef com AP-1 e AP-3 na regulação negativa de CD4 ainda não está totalmente elucidado. Ademais, AP-1, AP-2 e AP-3 são potencialmente heterogêneos devido à existência de isoformas múltiplas das subunidades codificadas por diferentes genes. Todavia, existem poucos estudos para demonstrar se as diferentes combinações de isoformas dos APs são formadas e se possuem propriedades funcionais distintas. O presente trabalho procurou identificar e caracterizar fatores celulares envolvidos na regulação do tráfego intracelular de proteínas no processo de regulação negativa de CD4 induzido por Nef. Mais especificamente, este estudo buscou caracterizar a participação do complexo AP-1 na modulação negativa de CD4 por Nef de HIV-1, através do estudo funcional das duas isoformas de ?-adaptina, subunidades de AP-1. Utilizando a técnica de Pull-down demonstramos que Nef é capaz de interagir com ?2. Além disso, nossos dados de Imunoblot indicaram que a proteína ?2-adaptina, e não ?1-adaptina, é necessária no processo de degradação lisossomal de CD4 por Nef e que esta participação é conservada para degradação de CD4 por Nef de diferentes cepas virais. Ademais, por citometria de fluxo, o silenciamento de ?2, e não de ?1, compromete a diminuição dos níveis de CD4 por Nef da membrana plasmática. A análise por imunofluorêsncia indireta também revelou que a diminuição dos níveis de ?2 impede a redistribuição de CD4 por Nef para regiões perinucleares, acarretando no acúmulo de CD4, retirados por Nef da membrana plasmática, em endossomos primários. A depleção de ?1A, outra subunidade de AP-1, acarretou na diminuição dos níveis celulares de ?2 e ?1, bem como, no comprometimento da eficiente degradação de CD4 por Nef. Além disso, foi possível observar que, ao perturbar a maquinaria ESCRT via super-expressão de HRS (uma subunidade do complexo ESCRT-0), ocorreu um acumulo de ?2 em endossomos dilatados contendo HRS-GFP, nos quais também detectou-se CD4 que foi internalizado por Nef. Em conjunto, os resultados indicam que ?2-adaptina é uma importante molécula para o direcionamento de CD4 por Nef para a via ESCRT/MVB, mostrando ser uma proteína relevante no sistema endo-lisossomal. Ademais, os resultados indicaram que as isoformas ?-adaptinas não só possuem funções distintas, mas também parecem compor complexos AP-1 com diferentes funções celulares, já que apenas a variante AP-1 contendo ?2, mas não ?1, participa da regulação negativa de CD4 por Nef. Estes estudos contribuem para o melhor entendimento dos mecanismos moleculares envolvidos na atividade de Nef, que poderão também ajudar na melhor compreensão da patogênese do HIV e da síndrome relacionada. Em adição, este trabalho contribui para o entendimento de processos fundamentais da regulação do tráfego de proteínas transmembrana no sistema endo-lisossomal. / The Human Immunodeficiency Virus (HIV) is the etiologic agent of Acquired Immunodeficiency Syndrome (AIDS). AIDS is a disease which has a global distribution, and it is estimated that there are currently at least 36.9 million people infected with the virus. During the replication cycle, HIV promotes several changes in the physiology of the host cell to promote their survival and enhance replication. The fast progression of HIV-1 in humans and animal models is closely linked to the function of an accessory protein Nef. Among several actions of Nef, one is the most important is the down-regulation of proteins from the immune response, such as the CD4 receptor. It is known that this action causes CD4 degradation in lysosome, but the molecular mechanisms are still incompletely understood. Nef forms a tripartite complex with the cytosolic tail of the CD4 and adapter protein 2 (AP-2) in clathrin-coated vesicles, inducing CD4 internalization and lysosome degradation. Previous research has demonstrated that CD4 target to lysosomes by Nef involves targeting of this receptor to multivesicular bodies (MVBs) pathway by an atypical mechanism because, although not need charging ubiquitination, depends on the proteins from ESCRTs (Endosomal Sorting Complexes Required for Transport) machinery and the action of Alix, an accessory protein ESCRT machinery. It has been reported that Nef interacts with subunits of AP- 1, AP-2, AP-3 complexes and Nef does not appear to interact with AP-4 and AP-5 subunits. However, the role of Nef interaction with AP-1 or AP-3 in CD4 down-regulation is poorly understood. Furthermore, AP-1, AP-2 and AP-3 are potentially heterogeneous due to the existence of multiple subunits isoforms encoded by different genes. However, there are few studies to demonstrate if the different combinations of APs isoforms are form and if they have distinct functional properties. This study aim to identify and characterize cellular factors involved on CD4 down-modulation induced by Nef from HIV-1. More specifically, this study aimed to characterize the involvement of AP-1 complex in the down-regulation of CD4 by Nef HIV-1 through the functional study of the two isoforms of ?-adaptins, AP-1 subunits. By pull-down technique, we showed that Nef is able to interact with ?2. In addition, our data from immunoblots indicated that ?2- adaptin, not ?1-adaptin, is required in Nef-mediated targeting of CD4 to lysosomes and the ?2 participation in this process is conserved by Nef from different viral strains. Furthermore, by flow cytometry assay, ?2 depletion, but not ?1 depletion, compromises the reduction of surface CD4 levels induced by Nef. Immunofluorescence microscopy analysis also revealed that ?2 depletion impairs the redistribution of CD4 by Nef to juxtanuclear region, resulting in CD4 accumulation in primary endosomes. Knockdown of ?1A, another subunit of AP-1, resulted in decreased cellular levels of ?1 and ?2 and, compromising the efficient CD4 degradation by Nef. Moreover, upon artificially stabilizing ESCRT-I in early endosomes, via overexpression of HRS, internalized CD4 accumulates in enlarged HRS-GFP positive endosomes, where co-localize with ?2. Together, the results indicate that ?2-adaptin is a molecule that is essential for CD4 targeting by Nef to ESCRT/MVB pathway, being an important protein in the endo-lysosomal system. Furthermore, the results indicate that ?-adaptins isoforms not only have different functions, but also seem to compose AP-1 complex with distinct cell functions, and only the AP-1 variant comprising ?2, but not ?1, acts in the CD4 down-regulation induced by Nef. These studies contribute to a better understanding on the molecular mechanisms involved in Nef activities, which may also help to improve the understanding of the HIV pathogenesis and the related syndrome. In addition, this work contributes with the understanding of primordial process regulation on intracellular trafficking of transmembrane proteins.