A presente pesquisa de cunho qualitativo consiste em um estudo de caso que visa experimentar a Modelagem Matemática como Ambiente de Aprendizagem na introdução de conteúdos programáticos de Estatística. A questão que norteou nossa pesquisa foi: “Um Ambiente de Modelagem Matemática favorece a aprendizagem de Estatística na Educação Básica?” Na busca de resposta a essa pergunta, as atividades foram pensadas baseadas no contexto no qual a turma está inserida. Elaboramos uma sequência didática baseada em questionamentos direcionados à reflexão e à investigação. Nesse cenário, o professor tem papel de incentivador da autonomia e capacidade dos alunos produzirem estratégias para resolverem problemas. Trata-se de um plano de natureza aberta, no qual os conhecimentos prévios dos alunos e suas dúvidas têm maior responsabilidade no processo de aprendizagem. Escolhemos a Modelagem Matemática como metodologia, pois ela atende aos objetivos de nosso trabalho, de dar significado à Matemática à medida que a aproximamos da realidade do estudante, desenvolver a autonomia dos alunos, estimulá-los à reflexão e a crítica de fatos oriundos da sociedade. Queremos que os conteúdos sejam introduzidos dentro de um contexto com referência ao dia a dia do educando. Nossa expectativa é que por meio da compreensão da Estatística e de seu papel na sociedade os alunos consigam utilizá-la como ferramenta de análise da realidade vivida. Essa sequência didática foi aplicada em uma turma de 7º ano de Ensino Fundamental de uma escola pública de Sapucaia do Sul – RS. Baseado nesse trabalho, julgamos que utilizar a Modelagem Matemática como Ambiente de Aprendizagem favorece a aprendizagem de Estatística. Acreditamos que os alunos tiveram maior envolvimento nas atividades à medida que a Matemática se tornava mais próxima à realidade deles. Ao longo do trabalho desenvolvido junto aos alunos, percebemos uma evolução na compreensão dos conteúdos abordados. Atribuímos essa evolução ao maior envolvimento dos alunos nos Ambientes de Aprendizagem proporcionados pela Modelagem Matemática. / This research consists in a case study which experiments Mathematical Modelling as a Learning Environment to introduce statistical contents. This work seeks to answer the following question: “Does a Mathematical Modelling Environment favors statistical learning on lower secondary education?” In order to answer that, activities were created based on questions that consider the context of the class. In this scenario, the teacher has the role of encouraging autonomy and the students the ability of to producing strategies to solve problems. It is an open plan in which the students' previous knowledge and their doubts have greater responsibility in the learning process. We chose Mathematical Modelling as methodology because it meets the objectives of our work, to give meaning to Mathematics as we approach the reality of the student, to develop students' autonomy, to stimulate them to reflect and critique facts from society. We want the contents to be introduced within a context with reference to the student's day-to-day life. Our expectation is that through the understanding of Statistics and its role in society, students will be able to use it as a tool for analyzing their reality. This didactical sequence was applied on a 7th grade elementary public school class of Sapucaia do Sul – RS. Based on this work, we believe that using Mathematical Modeling as a Learning Environment favors the learning of Statistics. We also believe that students were more involved in activities as Mathematics became closer to their reality. Throughout the work developed with the students, we perceived an evolution in the comprehension of the covered contents. We attribute this evolution to the greater involvement of students in the Learning Environments provided by Mathematical Modeling.

L'athérosclérose est une maladie inflammatoire qui commence quand les lipoprotéines de faible densité (LDL) entrent dans l'intima du vaisseau sanguin où elles sont oxydées (ox-LDL). Le ox-LDL est considéré comme un agent dangereux par le système immunitaire provoquant ainsi une réponse immunitaire. Cette réponse immunitaire déclenche le recrutement des monocytes dans l'intima où elles se transforment en macrophages et ensuite en cellules spumeuses. Ce dernier amplifie la production des cytokines inflammatoires et davantage de recrutement des monocytes. Ce processus auto-amplifié est compensé par la sécrétion de cytokines anti-inflammatoires (anti-inflammation biochimique) et la migration des cellules musculaires lisses pour former une chape fibreuse qui couvre le noyau lipidique. Cette chape fibreuse avec le noyau lipidique s'appellent la plaque d'athérosclérose. Celle-ci change la géométrie du vaisseau sanguin en le rétrécissant et interagit avec du flux sanguin. Cette interaction peut avoir des conséquences dangereuses liées à la rupture de plaque ou à la formation du caillot de sang. La thèse est consacrée à la modélisation mathématique de ces phénomènes. Elle est composée de deux parties : Nous développons des modèles mathématiques basés sur des équations de réaction diffusion afin de décrire le processus inflammatoire. Le premier modèle est unidimensionnel. Il nous permet d'expliquer comment le développement de l'athérosclérose dépend de la concentration en cholestérol (ox-LDL). Si cette concentration dans l'intima est basse, alors la maladie ne se développera pas. Les concentrations intermédiaires de ox-LDL peuvent mener au développement de la maladie dans certaines conditions. Nous montrons que l'inflammation se propage en front d'ondes de réaction-diffusion. Les concentrations élevées de ox-LDL engendre le développement de la maladie. Même une petite perturbation du cas non inflammatoire mène à une propagation d'ondes qui correspond à l'inflammation. Ensuite nous étudions un modèle bidimensionnel qui représente un système d'équations type réaction-diffusion sur une bande. La deuxième dimension correspond à la section transversale de l'intima et une condition aux limites non-linéaire décrit le recrutement des monocytes. Cette condition aux limites est une fonction des concentrations des cytokines. Nous démontrons l'existence des fronts de propagation d'onde et confirmons les résultats précédents qui montrent que l'athérosclérose se développe en tant qu'onde de réaction-diffusion. Les résultats théoriques des deux modèles sont confirmés par des simulations numériques qui montrent que le cas bidimensionnel converge vers le cas unidimensionnel quand l'épaisseur de l'intima tend vers zéro. Une fois la plaque se forme, elle interagit avec le flux sanguin engendrant de différentes conséquences mécaniques et biochimiques. Nous développons un modèle d'interaction fluide-structure. La plaque d'athérome composée d'un dépôt lipidique couvert par une chape fibreuse, les deux étant modélisés en tant que matériaux hyper-élastiques. Le sang est considéré comme un fluide non-Newtonien avec une viscosité variable modélisée selon la loi de Carreau. Les paramètres utilisés dans nos simulations sont tirés de données expérimentales mentionnées dans la littérature. Nous étudions les effets non-Newtoniens sur les recirculations du sang en aval de la plaque d'athérome et aussi sur les contraintes sur celle-ci. Les simulations montrent que le modèle Newtonien surestime les recirculations de manière significative par rapport au modèle non-Newtonien. Elles montrent aussi que le modèle Newtonien sous-estime légèrement les contraintes sur la plaque pour des taux de cisaillement usuels, mais cette sous-estimation devient importante pour des taux de cisaillement bas. / Atherosclerosis is an inflammatory disease which starts when low density lipoproteins (LDL) enter the intima of blood vessel where they are oxidized (ox-LDL). The ox-LDL is considered as a dangerous agent by the immune system provoking an anti-inflammatory response. This immune response triggers the recruitment of monocytes into the intima where they differentiate into macrophages and foam cells. The latter amplifies the production of inflammatory cytokines and further recruitment of monocytes. This auto-amplified process is compensated by the secretion of anti-inflammatory cytokines (biochemical anti-inflammation) and triggers the migration of smooth muscle cells to form a fibrous cap that covers the lipid core. These fibrous caps with the lipid core are called atherosclerosis plaque. It changes the geometry of the blood vessel by narrowing it and interacts with the blood flow. This interaction may have dangerous consequences related to the plaque rupture or to the formation of blood clot. The PhD thesis is devoted to mathematical modelling of these phenomena. It consists of two major parts : We develop mathematical models based on reaction-diffusion equations in order to describe the inflammatory process. The first model is one-dimensional. It allows us to explain how the development of atherosclerosis depends on the cholesterol (ox-LDL) concentration. If its concentration in the intima is low, then the disease will not develop. Intermediate ox-LDL concentrations can lead to the disease development under certain conditions. We show that the inflammation propagates as a reaction-diffusion wave. High ox-LDL concentrations will necessary result in the disease development. Even a small perturbation of the non inflammatory case leads to a travelling wave propagation which corresponds to a chronic inflammatory response. We then study a two-dimensional model which represents a reaction-diffusion system in a strip. The second dimension corresponds to the cross-section of the intima, nonlinear boundary conditions describe the recruitment of monocytes as a function of the cytokines concentration. We prove the existence of travelling waves and confirm our previous results which show that atherosclerosis develops as a reaction-diffusion wave. The theoretical results of the two models are confirmed by numerical simulations that show that the two-dimensional model converge to the one-dimensional one if the thickness of the intima tends to zero. When the plaque is formed, it interacts with blood flow resulting in various mechanical and bio-chemical effects. We develop a fluid-structure interaction model. The atheroma plaque is composed of a lipid pool and a fibrous cap and both are modeled as hyper elastic materials. The blood is supposed to be a non-Newtonian fluid with a variable viscosity modeled by the Carreau law. The parameters used in our simulations are taken from experimental data found in literature. We investigate the non-Newtonian effects on the re circulations downstream of the atheroma plaque and on the stress over the plaque. The simulations show that the Newtonian model significantly overestimates the re circulations in comparison with the non-Newtonian model. They also show that the Newtonian model slightly underestimates the stress over the plaque for usual shear rates, but this underestimation can become significant for low shear rates.

Orientador: Tiago de Carvalho / Resumo: O objetivo desta dissertação é abordar aspectos qualitativos de sistemas de equações diferenciais ordinárias e sistemas contínuos suaves por partes aplicados à dinâmica do Vírus da Imunodeficiência Humana (HIV). Neste trabalho, apresentamos um modelo matemático que descreve a dinâmica do HIV no corpo humano e o analisamos através da matriz da próxima geração e teoria de estabilidade, com a finalidade de prever se a doença fica ou não controlada. Posteriormente, estudamos um sistema de equações diferenciais ordinárias usado para modelar a dinâmica do vírus para diferentes tipos de tratamentos. Tal modelo foi explorado qualitativamente de duas maneiras: por um sistema contínuo (pelo método de Korobeinikov) e por um descontínuo (pelas convenções de Filippov). Analisamos o comportamento dinâmico de terapias antirretrovirais, visando a diminuição das concentrações virais no sangue, de acordo com a análise da estabilidade realizada. / Abstract: The goal of this dissertation is to study qualitative aspects about systems of ordinary differential equations and piecewise smooth systems applied to the dynamic of Human Immunodeficiency Virus (HIV). In this work, we present a mathematical model that describes the dynamic of HIV in the human body and we analyze this model by next-generation matrix and stability theory in order to predict if the disease becomes stable, and thus stop virus transmission. In addition, we studied another system of ordinary differential equations that were proposed to model the HIV dynamics assuming different therapies. We have explored qualitatively the model by two distinct approaches: a continuous system (by Korobeinikov method) and a discontinuous system (by Filippov theory). Due to the stability analysis, it was possible to understand the dynamics of anti-retroviral therapies, which are responsible for decreasing the concentration of detectable HIV in blood. / Mestre

Angiogenesis is the process by which new blood vessels develop from existing vessels. Angiogenesis is important in a number of conditions such as embryogenesis, wound healing and cancer. It has been modelled phenomenologically at the macroscale, using the well-known 'snail-trail' approach in which trailing endothelial cells follow the paths of other, leading endothelial cells. In this thesis, we systematically determine the collective behaviour of endothelial cells from their behaviour at the cell-level during corneal angiogenesis. We formulate an agent-based model, based on the snail-trail process, to describe the behaviour of individual cells. We incorporate cell motility through biased random walks, and include processes which produce (branching) and annihilate (anastomosis) cells to represent sprout and loop formation. We use the transition probabilities associated with the discrete model and a mean-field approximation to systematically derive a system of non-linear partial differential equations (PDEs) of population behaviour that impose physically realistic density restrictions, and are structurally different from existing snail-trail models. We use this framework to evaluate the validity of a classical snail-trail model and elucidate implicit assumptions. We then extend our framework to explicitly account for cell volume. This generates non-linear PDE models which vary in complexity depending on the extent of volume exclusion incorporated on the microscale. By comparing discrete and continuum models, we assess the extent to which continuum models, including the classical snail-trail model, account for single and multi-species exclusion processes. We also distinguish macroscale exclusion effects introduced by each cell species. Finally, we compare the predictive power of different continuum models. In summary, we develop a microscale to macroscale framework for angiogenesis based on the snail-trail process, which provides a systematic way of deriving population behaviour from individual cell behaviour and can be extended to account for more realistic and/or detailed cell interactions.

Esta dissertação trata da hidráulica da queda livre em canal de seção retangular. A análise bibliográfica do tema está calcada nos trabalhos pioneiros, nos clássicos e nos recentes. Com base nos princípios da Física: Conservação de Massa, Quantidade de Movimento, e Primeira Lei da Termodinâmica, foi desenvolvido um modelo matemático para a queda livre. O modelo proposto, após ser analisado do ponto de vista de sua consistência, foi validado em comparações com resultados fornecidos por outros pesquisadores, geralmente com modelos empíricos (ou semi-empíricos), ajustados a dados experimentais. Os resultados obtidos pelo modelo matemático proposto nesta dissertação, correspondem a boas estimativas das grandezas envolvidas nos escoamentos em queda livre, o que credencia o modelo proposto como uma ferramenta apropriada para projetos em engenharia hidráulica, principalmente quando se tem em conta que a queda livre é o mecanismo de dissipação mais presente na natureza. / This study addresses free fall hydraulics in rectangular channel section. References were based on not only earlier studies, but also on classical and most recent ones. Based on principles of Physics, such as mass conservation, momentum and the First Law of Thermodynamics, a mathematical model has been developed as an example of free fall hydraulic. After extensive consistency analyses the proposed model has been validated by comparing different results furnished by other researchers, generally based on empirical or semiempirical treatment adjusted to experimental data. Results obtained from the mathematical model proposed here correspond to good estimates of greatnesses involved in the free overfall and that turns the proposed model into an adequate tool for Hydraulic Engineering projects, especially when we all know free overfall is the most common dissipation mechanism in Nature.

O crescimento de tumores vem chamando a atenção de físicos e matemáticos há mais de sessenta anos. Entretanto, a conversa com biólogos e a interação teoria-experimento têm aparecido apenas recentemente. Equações fenomenológicas e simulações computacionais continuam sendo uma ferramenta comum entre todos os modelos que conhecemos. Assim, nesse trabalho nós estudamos o problema do crescimento de tumores monocamada através das abordagens experimental, teórica e computacional, fortalecendo assim a interação teoria-experimento. Cultivamos células das linhagens HeLa (carcinoma cervical humano), HCT-15 (adenocarcinoma coloretal humano), NIH-HN-13 (carcinoma de células escamosas humanas) e U-251 (glioblastoma neuronal humano), obtendo a dimensão fractal e o comportamento do raio médio com o número de células, além de analisarmos os dados da literatura para a linhagem HT-29 (adenocarcinoma coloretal humano). A seguir nós modelamos a taxa de crescimento do raio médio através de uma curva sigmoidal. A solução analítica dessa equação nos permitiu ajustar bem os dados obtidos experimentalmente, e os parâmetros obtidos serviram para a simulação Monte Carlo dinâmico. Para essa, transformamos a taxa de crescimento do raio em taxa de crescimento do número de células, cujos resultados novamente concordaram muito bem com os dados experimentais. A dimensão fractal dos agregados esteve entre 1; 12 df 1; 21, e concordou com os dados da literatura. Novos resultados foram produzidos: i) O raio médio como uma função do número de células nos permitiu um ajuste do tipo Rc(t) = a[Nc(t) ? N~0]1=2 + R~0, mais geral que a comumente aceita relação Rc(t) = cNc(t)1=2; e ii) os tempos de espera no procedimento MCD se distribuem log-normalmente (ou Gaussianamente em alguns casos), diferentemente da distribuição Poissoniana frequêntemente assumida. A distribuição log-normal nos permitiu também conjecturar que um parâmetro , da relação ht(nT)i / n? T , possa caracterizar o crescimento monocamada de tumores devido à sua estreita abrangência 0; 69 0; 81. Nossos resultados nos permitiram concluir que diferentes condições de cultivo podem gerar diferentes respostas dos parâmetros, além disso, dois fenômenos podem caracterizar esse crescimento no âmbito mesoscópico: A competição por espaços livres e a cooperação entre as células. / Tumor growth has been calling attention of physicists and mathematicians for more than sixty years. However, cross-talking with biologists and the interplay between theory and experiment have emerged just recently. Phenomenological equations and computational simulations are still the common toolbox among all the models we know. Thus, in this work, we have studied the problem of monolayer tumor growth through the experimental, theoretical and computational approaches, enhancing the interaction between theory and experiment. We cultivate HeLa (human cervical carcinoma), HCT-15 (human colorectal adenocarcinoma), NIH-HN-13 (human squamous cell carcinoma) and U-251 (human neuronal glioblastoma) cells, calculating the fractal dimension and the behavior of the mean radius with cell number, and analyzing the literature data from HT-29 (human colorectal adenocarcinoma) lineage. Then we modeled the growth rate of mean radius through a sigmoidal curve. The analytical solution of this equation allowed us to fit well the experimental data and the obtained parameters were used into dynamical Monte Carlo simulation. To do this, we transform the radius growth rate in number of cells growth rate, which again agreed with the experimental data. The fractal dimensions of the aggregates ranged from 1; 12 df 1; 21, and agree with the literature. New findings were produced: i) the mean radius as a function of the number of cells enabled us to adjust the function Rc(t) = a[Nc(t) ? N~0]1=2 + R~0, differently from widely accepted relation Rc(t) = cNc(t)1=2; and ii) the waiting times in the MCD procedure are log-normally distributed (sometimes Gaussian), unlike the Poisson distribution often used. The lognormal distribution also allowed us to conjecture that a parameter , from the power law relation ht(nT)i / n? T , might caracterize the tumor monolayer growth due to its narrow range 0; 69 0; 81. Our findings led us to conclude that different culture conditions may produce different parameter responses, furthermore, two phenomenona can describe the growth in mesoscopic level: the competition for free space and the cooperation between cells.

Filamentous fungi are important in an enormous variety of ways to our life, with examples ranging from bioremediation, through the food and drinks industry to human health. These organisms can form huge networks stretching metres and even kilometres. However, their mode of growth is by the extension of individual hyphal tips only a few microns in diameter. Tip growth is mediated by the incorporation of new wall building materials at the soft apex. Just how this process is controlled (in fungi and in cell elongation in other organisms) has been the subject of intense study over many years and has attracted considerable attention from mathematical modellers. In this thesis, we consider mathematical models of fungal tip growth that can be classified as either geometrical or biomechanical. In every model we examine, a 2-D axisymmetric semihemisphere-like curve represents half the medial section of fungal tip geometry. A geometrical model for the role of the Spitzenkorper in the tip growth was proposed by Bartnicki-Garcia et al (1989), where a number of problems with the mathematical derivation were pointed out by Koch (2001). A suggestion is given as an attempt to revise the derivation by introducing a relationship between arc length of a growing tip, deposition of wall-building materials and tip curvature. We also consider two types of geometrical models as proposed by Goriely et al (2005). The first type considers a relationship between the longitudinal curvature and the function used to model deposition of wall-building materials. For these types of models, a generalized formulae for the tip shape is introduced, which allows localization of deposition of wall-building materials to be examined. The second type considers a relationship between longitudinal and latitudinal curvatures and the function used to model deposition of wall-building materials. For these types of models, a new formulation of the function used to model deposition of wall-building materials is introduced. Finally, a biomechanical model as proposed by Goriely et al (2010). Varying arc length of the stretchable region on the tip suggests differences in geometry of tip shape and the effective pressure profile. The hypothesis of orthogonal growth is done by focusing only on the apex of a "germ tube". Following that, it suggests that material points on the tip appear to move in a direction perpendicular to the tip either when surface friction is increased or decreased.

abstract: Carbon dioxide (CO2) levels in the atmosphere have reached unprecedented levels due to increasing anthropogenic emissions and increasing energy demand. CO2 capture and utilization can aid in stabilizing atmospheric CO2 levels and producing carbon-neutral fuels. Utilizing hollow fiber membranes (HFMs) for microalgal cultivation accomplishes that via bubbleless gas-transfer, preventing CO2 loss to the atmosphere. Various lengths and geometries of HFMs were used to deliver CO2 to a sodium carbonate solution. A model was developed to calculate CO2 flux, mass-transfer coefficient (KL), and volumetric mass-transfer coefficient (KLa) based on carbonate equilibrium and the alkalinity of the solution. The model was also applied to a sparging system, whose performance was compared with that of the HFMs. Typically, HFMs are operated in closed-end mode or open-end mode. The former is characterized by a high transfer efficiency, while the latter provides the advantage of a high transfer rate. HFMs were evaluated for both modes of operation and a varying inlet CO2 concentration to determine the effect of inert gas and water vapor accumulation on transfer rates. For pure CO2, a closed-end module operated as efficiently as an open-end module. Closed-end modules perform significantly worse when CO2-enriched air was supplied. This was shown by the KLa values calculated using the model. Finally, a mass-balance model was constructed for the lumen of the membranes in order to provide insight into the gas-concentration profiles inside the fiber lumen. For dilute CO2 inlet streams, accumulation of inert gases -- nitrogen (N2), oxygen (O2), and water vapor (H2O) -- significantly affected module performance by reducing the average CO2 partial pressure in the membrane and diminishing the amount of interfacial mass-transfer area available for CO2 transfer. / Dissertation/Thesis / Masters Thesis Chemical Engineering 2018

O metabolismo das células S2 (Schneider 2) de Drosophila melanogaster ainda não é totalmente conhecido. Existem poucos estudos específicos sobre o metabolismo de células S2, sejam elas selvagens ou recombinantes (rS2), como por exemplo aquelas transfectadas para a expressão da glicoproteína do vírus da raiva (GPV). Como o genoma da Drosophila melanogaster já foi mapeado, as principais enzimas que atuam nos processos metabólicos em geral já foram identificadas e estão à disposição no KEGG (Kyoto Encyclopedia of Genes and Genomes). Assim, o KEGG apresenta todas as possíveis vias metabólicas com as enzimas que podem ser codificadas. Diante deste quadro, foi proposto um modelo metabólico baseado em um conjunto de vias de assimilação de glicose e glutamina e foram encontrados os modos elementares característicos do sistema através do programa Metatool. Em seguida, foi definido o modelo matemático mediante o equacionamento desses modos elementares. Esse processo se repetiu até se encontrar um conjunto de vias metabólicas que, através da modelagem matemática, respondesse coerentemente a um conjunto de dez ensaios em diferentes condições de concentrações iniciais de glicose, glutamina e oxigênio dissolvido. Chegou-se então, a um metabolismo básico para a rS2 contendo 33 vias metabólicas englobando a glicólise, a via das pentoses, o ciclo de Krebs e a fosforilação oxidativa. Dados anteriores indicavam elevada flexibilidade metabólica dessa célula, o que foi prevista através de algumas reações propostas como reversíveis nas vias de degradação e síntese de glutamina. Essa proposta de metabolismo resultou em 37 modos elementares. Outra característica interessante da modelagem foi a utilização da produção de purinas e pirimidinas para a estimativa do crescimento celular. Depois de realizada a modelagem, as mesmas condições iniciais dos ensaios foram simuladas através de um programa de simulação do comportamento cinético das células rS2 desenvolvido em MATLAB. Esse simulador foi utilizado também para simulação com diferentes meios e condições iniciais de cultivo. Chegando-se a um ajuste geral entre valores experimentais e simulados com coeficiente de correlação de 0,88. / The metabolism of the S2 cells (Schneider 2) Drosophila melanogaster is not yet fully known. There have been few specific studies on the metabolism of S2 cells, whether recombinant or wild (rS2), such as those transfected for expressing the rabies virus glycoprotein (RVGP). As the genome of Drosophila melanogaster have been mapped, the key enzymes that act on the metabolic processes in general have been identified and are available in the KEGG (Kyoto Encyclopedia of Genes and Genomes). Thus, KEGG presents all possible pathways with the enzymes that can be encoded. Given this context, it was proposed a metabolic model based on a set metabolic glucose and glutamine assimilation pathways and were found characteristic elementary modes of the system through the Metatool program. Then the mathematical model was defined by addressing these elementary modes. This process was repeated until a set of metabolic pathways, by mathematical modelling, consistently responded to a set of ten experiments (in various conditions). We came to a basic metabolism for rS2 containing 33 pathways comprising glycolysis, pentose, Krebs cycle and oxidative phosphorylation. Previous data indicate that rS2 is a cell with high metabolic flexibility, which was confirmed by some reactions in the process proposed as reverse breakdown and synthesis of glutamine. The proposed metabolism resulted in 37 elementary modes. Another interesting model characteristic was the use of the production of purines and pyrimidines for the estimation of cell growth. After the modelling performed, the same initial runs conditions were simulated using a software of Simulation of the Kinetic behaviour of rS2 cells, developed in MATLAB. This simulator was also used for simulation of other experiments with different initial conditions and methods of cultivation. Coming to a general adjustment of experimental and simulated values with correlation coefficient of 0.88.

L’ATP est synthétisée par les mitochondries à partir de réactions d’oxydoréduction catalysées les complexes de la chaîne respiratoire. Ces réactions impliquent des transferts d’électrons intra-protéine. Une capacité de production de l’anion superoxyde, formé par la réaction de l’oxygène avec un électron, a été identifiée pour les complexes I et III. Les espèces actives de l’oxygène (EAOs) sont des molécules dérivées de l’anion superoxyde. Si elles ne sont pas correctement régulées par les défenses antioxydantes de la cellule, ces EAOs peuvent réagir avec les composants de la cellule et nuire à son fonctionnement : ce déséquilibre est appelé stress oxydatif. L’altération d’un ou plusieurs complexes respiratoires associée à un stress oxydatif cellulaire est un mécanisme commun à de nombreuses maladies neurodégénératives. Dans ce travail nous nous intéressons plus particulièrement à l’atrophie optique autosomique dominante de type 1 (ADOA-1). L’ADOA-1 est une maladie neurodégénérative principalement causée par des mutations du gène codant la protéine mitochondriale OPA1 impliquée dans la dynamique mitochondriale. Les tableaux cliniques et l’âge de début de la maladie sont variables. Il n’existe pas de corrélation claire entre génotypes et phénotypes permettant d’expliquer cette variabilité ni de traitement à cette pathologie. L’hypothèse d’un stress oxydatif a été proposée pour expliquer la variabilité de ces symptômes. C’est pourquoi notre objectif est d’améliorer la compréhension des mécanismes physiopathologiques impliqués dans cette maladie en développant des modèles mathématiques de la production des EAOs par la chaîne respiratoire. Nous avons utilisé deux méthodes de modélisation. Dans le premier cas, nous modélisons l’activité des complexes respiratoires et la production d’anion superoxyde par les complexes I et III par des équations de vitesse que nous construisons en trois étapes. Nous analysons d’abord les données biochimiques disponibles dans la littérature. Nous proposons ensuite des interprétations physiques à ces comportements et les traduisons sous forme de règles floues. Nous modélisons enfin ces règles en utilisant des fonctions données par le formalisme de Michaelis-Menten. Les équations de vitesse sont fonction de variables chimiques telles que la concentration des espèces chimiques impliquées dans les réactions des complexes respiratoires et ne prennent pas en compte le détail des réactions intra-protéine impliquées dans le fonctionnement des complexes. Cette méthode permet de construire un modèle simple, permettant de simuler l’activité des complexes I et III et leur production de superoxyde dans différentes conditions, et qui est facilement modifiable ou intégrable dans un modèle plus complet de la mitochondrie. Le modèle du complexe I que nous avons créé, est capable de simuler l’activité catalytique et la production des EAOs en mode direct par le complexe I pour différentes configurations et concentrations de substrats et produits. / Mitochondria are cellular organelles involved in ATP (adenosine triphosphate) supply to cells. Mitochondrial ATP is produced by the oxidative phosphorylation which involves redox reactions catalysed by the four protein complexes of the mitochondrial respiratory chain. These redox reactions require intra-protein electron transfers. The complex I and complex III of the respiratory chain are able to generate superoxide anion, which is formed by the reaction of oxygen with one electron. Reactive oxygen species (ROS) are molecules derived from the superoxide anion. ROS which are not regulated by cellular antioxidant defences can react with the components of the cells and disturb its functioning: this imbalance between ROS and antioxidant defences has been termed “oxidative stress”. Dysfunctions of one or several respiratory complexes associated to an oxidative stress is a mechanism common to numerous neurodegenerative diseases. In this work, we focus on autosomal dominant optic atrophy 1 (ADOA-1 or DOA-1). DOA-1 is a neurodegenerative pathology mainly caused by mutations in the gene OPA1 which codes for a mitochondrial protein involved in mitochondrial dynamics. The symptoms and ages of onset of the disease are variable. There is no clear correlation between genotypes and phenotypes which can explain this variability and to date, there is no established medical treatment for the disease. The hypothesis of an oxidative stress has been proposed to explain the variability of symptoms observed in patients. Indeed, the mitochondrial energetic metabolism is altered in biological models (cell cultures and animal models) of DOA-1 and low levels of antioxidant defences have been measured in cells from patients suffering from severe forms of the pathology. Hence, our objective is to improve the understanding of the physio-pathological mechanisms involved in this disease by developing mathematical models of ROS production by the respiratory chain. We use two modelling methods. The first method consists in modelling the activities of respiratory complexes and the superoxide production by complexes I and III with rate equations that we build in three steps. We first analyse the biochemical data available in the literature. We subsequently interpret this data physically and translate them in the form of fuzzy rules. We then model these rules with mathematical functions provided by the formalism of Michaelis-Menten. The rate equations depend on chemical variables such as the concentrations of chemical species involved in the reactions catalysed by the respiratory complexes. They do not include the details of intra-protein electron transfers, occurring during the catalysis performed by the complexes. This method enables us to build a simple model simulating the activities and superoxyde productions of complexes I and III in different conditions and that can easily be modified or integrated in a more comprehensive model of the mitochondrion. Our model of complex I can simulate the forward and reverse activities and ROS productions of the enzyme for different concentrations of substrates and products.