A área conhecida como Análise Não Standard consiste na aplicação dos métodos da Teoria dos Modelos e da Teoria dos Ultrafiltros para a obtenção de extensões peculiares de sistemas matemáticos infinitos. As novas estruturas construídas segundo esse procedimento satisfazem ao Princípio da Transferência, uma propriedade de suma importância e influência a qual afirma que as mesmas sentenças de primeira ordem com quantificadores limitados são verdadeiras para o sistema original e a sua extensão. Concebida em 1961 por Abraham Robinson e aprimorada por vários matemáticos nos anos subsequentes, tal área de pesquisa provou ser bastante proveitosa e esclarecedora para diversas outras partes da Matemática, como a Topologia, a Teoria das Probabilidades, a Análise Funcional e a Análise Complexa. Manifesta-se uma reavaliação da Teoria dos Domínios Ordenados seguida de um tratamento completo e gradual das fundações da Análise Não Standard assumindo a perspectiva dos Monomorfismos Não Standard, onde adota-se como metateoria a teoria dos conjuntos de Neumann-Bernays-Gödel com o Axioma da Escolha. A fim de impulsionar a assimilação da metodologia abordada, o estudo explora as propriedades do corpo não arquimediano dos números hiper-reais de maneira intuitiva e informal, utilizando-se destas para revelar demonstrações alternativas e relativamente diretas de alguns dos principais resultados do Cálculo Diferencial e Integral, como o Teorema do Valor Intermediário, o Teorema de Bolzano-Weierstrass, o Teorema do Ponto Crítico, o Teorema da Função Inversa e o Teorema Fundamental do Cálculo. / The field known as Non-standard Analysis consists in the application of the methods of Model Theory and Ultrafilter Theory to the attainment of peculiar extensions of infinite mathematical systems. The new structures produced under that procedure satisfy the Transfer Principle, a property of the utmost importance and influence which states that the same first-order sentences with bounded quantifiers are true for the original system and its extension. Conceived in 1961 by Abraham Robinson and improved by a number of mathematicians in the following years, such area of research has proved to be very fruitful and illuminating to many other parts of Mathematics, such as Topology, Probability Theory, Functional Analysis and Complex Analysis. The work presents a reexamination of the Theory of Ordered Domains followed by a thorough and gradual treatment of the foundations of Non-standard Analysis under the perspective of Non-standard Monomorphisms, where Neumann-Bernays-Gödels set theory with the Axiom of Choice is adopted as metatheory. In order to boost the assimilation of the methodology put forward, the study explores the properties of the non-archimedean field of hyperreal numbers in an intuitive and informal fashion, employing them to reveal alternative and relatively direct proofs of some of the main results of Differential and Integral Calculus, such as the Intermediate Value Theorem, the Bolzano-Weierstrass Theorem, the Extreme Value Theorem, the Inverse Function Theorem and the Fundamental Theorem of Calculus.

A área conhecida como Análise Não Standard consiste na aplicação dos métodos da Teoria dos Modelos e da Teoria dos Ultrafiltros para a obtenção de extensões peculiares de sistemas matemáticos infinitos. As novas estruturas construídas segundo esse procedimento satisfazem ao Princípio da Transferência, uma propriedade de suma importância e influência a qual afirma que as mesmas sentenças de primeira ordem com quantificadores limitados são verdadeiras para o sistema original e a sua extensão. Concebida em 1961 por Abraham Robinson e aprimorada por vários matemáticos nos anos subsequentes, tal área de pesquisa provou ser bastante proveitosa e esclarecedora para diversas outras partes da Matemática, como a Topologia, a Teoria das Probabilidades, a Análise Funcional e a Análise Complexa. Manifesta-se uma reavaliação da Teoria dos Domínios Ordenados seguida de um tratamento completo e gradual das fundações da Análise Não Standard assumindo a perspectiva dos Monomorfismos Não Standard, onde adota-se como metateoria a teoria dos conjuntos de Neumann-Bernays-Gödel com o Axioma da Escolha. A fim de impulsionar a assimilação da metodologia abordada, o estudo explora as propriedades do corpo não arquimediano dos números hiper-reais de maneira intuitiva e informal, utilizando-se destas para revelar demonstrações alternativas e relativamente diretas de alguns dos principais resultados do Cálculo Diferencial e Integral, como o Teorema do Valor Intermediário, o Teorema de Bolzano-Weierstrass, o Teorema do Ponto Crítico, o Teorema da Função Inversa e o Teorema Fundamental do Cálculo. / The field known as Non-standard Analysis consists in the application of the methods of Model Theory and Ultrafilter Theory to the attainment of peculiar extensions of infinite mathematical systems. The new structures produced under that procedure satisfy the Transfer Principle, a property of the utmost importance and influence which states that the same first-order sentences with bounded quantifiers are true for the original system and its extension. Conceived in 1961 by Abraham Robinson and improved by a number of mathematicians in the following years, such area of research has proved to be very fruitful and illuminating to many other parts of Mathematics, such as Topology, Probability Theory, Functional Analysis and Complex Analysis. The work presents a reexamination of the Theory of Ordered Domains followed by a thorough and gradual treatment of the foundations of Non-standard Analysis under the perspective of Non-standard Monomorphisms, where Neumann-Bernays-Gödels set theory with the Axiom of Choice is adopted as metatheory. In order to boost the assimilation of the methodology put forward, the study explores the properties of the non-archimedean field of hyperreal numbers in an intuitive and informal fashion, employing them to reveal alternative and relatively direct proofs of some of the main results of Differential and Integral Calculus, such as the Intermediate Value Theorem, the Bolzano-Weierstrass Theorem, the Extreme Value Theorem, the Inverse Function Theorem and the Fundamental Theorem of Calculus.

Au cours de ces dernières années, des études ont montré l’existence d’une compartimentation latéraledes composants de la membrane plasmique végétale, de manière analogue à ce qui avait été montréchez les animaux et les levures. L’objectif de cette thèse était d’apporter de nouveaux éléments decaractérisation de cette compartimentation (propriétés physiques de domaines particuliers, mécanismesde mise en place de ces domaines, de contrôle de leur taille, etc…) et d’étudier son rôle dans laphysiologie de la cellule végétale.Le développement d’une méthodologie de microscopie confocale spectrale couplée à l’utilisationd’une sonde environnementale a permis d’apporter la première description à l'échellesubmicrométrique de l’organisation du plasmalemme en territoires aux propriétés physiquesdifférentiées. Ces domaines coexistent au sein de la membrane plasmique de cellules en suspension,comme à celle de membranes artificielles composées de lipides modèles ou de lipides de membranescellulaires, de vésicules géantes constituées de membrane plasmique purifiée, ou de protoplastes.Cependant, les différences de l’organisation latérale observées chez ces différentes membranes ontpermis de montrer l’importance des phytostérols qui seraient, par le biais d'interactions spécifiquesavec d’autres lipides végétaux tels que les GIPCs, des composés essentiels pour la formation locale dedomaines lipidiques ordonnés. La grande diversité des lipides végétaux organiserait ainsi lacompartimentation de la membrane plasmique permettant la ségrégation dynamique des composantsmembranaires. Si les stérols augmentent de manière importante le degré de compaction de la bicouche,les protéines le diminuent. Le cytosquelette et la paroi ne semblent, quant à eux, modifier ni laprésence, ni l’organisation des domaines ordonnés de la membrane plasmique. Nous avons égalementmontré que l’organisation de ces domaines évolue transitoirement lors des étapes précoces de lacascade de signalisation induite par des réactions de défense. De fait, nous avons identifié desmodifications des propriétés physiques globales et de l’organisation fine de la membrane provoquéespar différents éliciteurs de réactions de défense, dont la cryptogéine, une protéine sécrétée parl’oomycète Phytophthora cryptogea. Nous avons montré que ces modifications sont un élémentgénérique de la signalisation de défense, sous la dépendance de phénomènes de phosphorylation, leburst oxydatif étant également une étape clé de l’augmentation du degré d’ordre observé dans lesphases précoces de cette signalisation. La cryptogéine, qui présente une aptitude singulière pour piégerles stérols, a également montré une capacité spécifique à augmenter la fluidité membranaire, ceparamètre pouvant contrôler l’intensité de la cascade de signalisation, mesurée par la production deformes actives d’oxygène.Ces résultats ouvrent de nouvelles perspectives dans la compréhension des interactions cellule-élicitineet apportent un nouvel éclairage sur le rôle des lipides végétaux dans l’organisation latérale de lamembrane plasmique végétale et positionne la dynamique membranaire comme un élément designalisation de défense des plantes. / Recent studies have shown the existence of lateral sub-compartmentalization of plant plasmamembrane similar to that of animal cells and yeasts. The aim of this thesis was to provide newelements to characterize this compartmentalization (physical properties of specific domains,mechanisms of their formation, determination of their size, etc...) and to study its role in thephysiology of plant cells.The development spectral confocal microscopy coupled with the use of an environment-sensitiveprobe enabled to obtain the first description at the submicron scale of plasma membrane organizationinto domains exhibiting various physical properties. These domains coexist at the plasma membranesurface of tobacco suspension cells as well as the membrane of vesicles composed of models lipids orcell plasma membrane lipids, purified plasma membrane vesicles, and protoplasts. However,differences in the lateral organization observed in these membranes have shown the importance ofphytosterols which are, through specific interactions with neighboring plant lipids such as GIPCs,essential for local formation of ordered domains. The huge diversity of plant lipids drives thecompartmentalization of the plasma membrane allowing the dynamic segregation of membranecomponents. Sterols greatly increase membrane order, whereas proteins tends to decrease it.Cytoskeleton and cell wall do alter neither presence nor organization of ordered domains of the plasmamembrane. We have also shown that the organization of these domains is transiently modified duringthe early stages of defense signaling cascade. In fact, we have identified changes in overall physicalproperties and fine lateral organization of the membrane caused by various elicitors of defensereactions, including cryptogein, a protein secreted by the oomycete Phytophthora cryptogea. We haveshown that these changes are a generic element of defense signaling cascade and depend onphosphorylation processes; oxidative burst being also a major actor of the control of the increase ofmembrane order observed during the very early stages of the signalling process. Cryptogein, whichexhibits the particular ability to trap sterols, also showed a specific capacity to increase membranefluidity and amplify the intensity of the signalling cascade, as measured by the production of reactiveoxygen species.These results open new perspectives in the understanding of cell-elicitin interactions and provide anew view on the central role of sterol composition in the lateral organization of plant plasmamembrane. They also identify membrane dynamics as a new player in the signalling cascade occurringduring plant defense.