A unique mechanism of protein oligomerization is domain swapping. It is a feature found in some proteins wherein a dimer or a higher oligomer is formed by the exchange of identical structural segments between protomers. Domain swapping is thought to have played a key role in the evolution of stable oligomeric proteins and in oligomerization of amyloid proteins. This thesis deals with studies to understand the significance of hinges involved in domain swapping for protein oligomerization and function. The stationary phase survival protein SurE from Salmonella typhimurium (StSurE) and Sesbania mosaic virus (SeMV) coat protein have been used as models for studies on domain swapping. This thesis has been divided into eight chapters. Chapter 1 provides a brief introduction to domain swapping, while Chapters 2 to 6 describes the studies carried out on StSurE protein, Chapter 7 deals with studies on SeMV coat protein. The final Chapter 8 provides brief descriptions of various experimental techniques employed during these investigations. Chapter 1 deals with a brief introduction to domain swapping in proteins. Examples where different domains are exchanged are cited. Then it describes physiological relevance of domain swapping in proteins and probable factors which promote swapping. Finally it also discusses the uncertainties that are inevitable in protein structure prediction and design. Chapter 2 describes the structure of Salmonella typhimurium SurE (StSurE; Pappachan et al., 2008) determined at a higher resolution. The chapter also deals with the sequence and structure based comparison of StSurE with other known SurE homolog structures. A comparative analysis of the relative conservation of N- and C-terminal halves of SurE protomer and variations observed in the quaternary structures of SurE homologs are presented. Then a brief introduction is provided on function of StSurE. The conserved active site of StSurE that might be important for its phosphatase activity is described. A plausible mechanism for the phosphatase activity as proposed by Pappachan et al. (2008) is presented. Crystal structures of StSurE bound with AMP, pNPP and pNP that was determined with the view of better understanding the mechanism of enzyme function is presented. These structures provide structural evidence for the mechanism proposed by Pappachan et al. (2008). Finally a substrate entry channel inferred from these structures is discussed. SurE from Salmonella typhimurium (StSurE) was selected for studies on domain swapping as there is at least one homologous structure (Pyrobaculum aerophilum - PaSurE) in which swapping of the C-terminal helices appears to have been avoided without leading to the loss of oligomeric structure or function. It was of interest to examine if an unswapped dimer of StSurE resembling PaSurE dimer could be constructed by mutagenesis. To achieve this objective, a crucial hydrogen bond in the hinge involved in C-terminal helix swapping was abolished by mutagenesis. These mutants were constructed with the intention of increasing the flexibility of the hinge which might bring the C-terminal helices closer to the respective protomer as in PaSurE. Chapter 3 presents a comparative analysis of the hinges involved in C-terminal helix swapping in PaSurE and StSurE. Based on the comparison of structure and sequence, crucial residues important for C-terminal helix swapping in StSurE were identified as D230 and H234. The chapter describes the construction of mutants obtained by substituting D230 and H234 by alanine and their biophysical characterization. Finally it describes structural studies carried out on these mutants. The mutation H234A and D230A/H234A resulted in highly distorted dimers, although helix swapping was not avoided. Comparative analysis of the X-ray crystal structures of native StSurE and mutants H234A and D230A/H234A reveal large structural changes in the mutants relative to the native structure. However the crystal structures do not provide information on the changes in dynamics of the protein resulting from these mutations. To gain better insights into the dynamics involved in the native and mutants H234A and D230A/H234A, MD simulations were carried on using GROMACS 4.0.7. Chapter 4 deals with a brief description of the theory of molecular dynamics, followed by results of simulation studies carried out on monomeric and dimeric forms of StSurE and dimeric forms of its mutants H234A and D230A/H234A. The conformational changes and dynamics of different swapped segments are discussed. Crystal structures of H234A and D230A/H234A mutants reveal that they form highly distorted dimers with altered dimeric interfaces. Chapter 5 focuses on comparison of dimeric interfaces of the native StSurE and hinge mutants H234A and D230A/H234A. Based on the analysis, three sets of interactions were selected to investigate the importance of the interface formed by swapped segments in StSurE mutants H234A and D230A/H234A. One of the selected sites corresponds to a novel interaction involving tetramerization loop in the hinge mutants H234A and D230A/H234A resulting in a salt bridge between E112 – R179’ and E112’ – H180 (prime denotes residue from the other chain of the dimeric protein). This salt bridge seems to stabilize the distorted dimer. It is shown by structural studies that the loss of this salt bridge due to targeted mutation restores symmetry and dimeric organization of the mutants. Loss of a crucial hydrogen bond in the hinge region involved in C-terminal helix swapping in SurE not only leads to large structural changes but also alters the conformation of a loop near the active site. It is of interest to understand functional consequences of these structural changes. StSurE is a phosphatase, and its activity could be conveniently monitored using the synthetic substrate para nitrophenyl phosphate (pNPP) at pH 7 and 25 ºC. Chapter 6 deals with the functional studies carried out with various StSurE mutants. The studies suggest that there is a drastic loss in phosphatase activity in hinge mutants D230A, H234A and D230A/H234A, while in the salt bridge mutants the function seems to have been restored. Few of these mutants also exhibit positive cooperativity, which could probably be due to altered dynamics of domains. Sesbania mosaic virus (SeMV) is a plant virus, belonging to genus sobemovirus. SeMV is a T=3 icosahedral virus (532 symmetry) made up of 180 coat protein (CP) subunits enclosing a positive-sense RNA genome. The asymmetric unit of the icosahedral capsid is composed of chemically identical A, B and C subunits occupying quasi-equivalent environments. Residues 48 – 59 of the N-terminal arms of the C subunits interact at the nearby icosahedral three-fold axes through a network of hydrogen bonds to form a structure called the “β-annulus”. Residues 60 – 73 form the “βA-arm” that connects the N-terminal β-annulus to the rest of the protomer. Various studies on SeMV-CP suggest that different lengths of the N-terminal segments affect the assembly of virus. It might be possible to exploit this flexibility of the N-terminus in SeMV-CP to introduce swapping of this segment between two 2-fold related C subunits as is found in Rice yellow mottle virus (RYMV), another sobemovirus, with which SeMV shares significant sequence similarity. Chapter 7 focuses on attempts made to examine the mutational effects planned to introduce domain swapping. The strategy used for introducing swapping in SeMV-CP was based on the sequence of the βA-arm or the hinge involved in swapping of β-annulus in RYMV. TEM images of the mutant virus like particles obtained suggest that they are heterogeneous. These mutants could not be crystallized, probably due to the heterogeneity. However, the assembly of the expressed proteins to virus like particles was profoundly influenced by the mutations. Chapter 8 discusses various crystallographic, biophysical and biochemical techniques used during these investigations. Finally the thesis concludes with Conclusions and Future perspectives of the various studies reported in the thesis. In summary, I have addressed the importance of amino acid residues and interactions of hinges involved in domain swapping for the quaternary structure and function of proteins.

Philosophiae Doctor - PhD / Tuberculosis is a bacterial infection which is a major cause of death worldwide. TB is a curable disease, however the bacterium can become resistant to the first line treatment against the disease. This leads to a disease called drug resistant TB that is difficult and expensive to treat. It is well-known that TB disease thrives in communities in overcrowded environments with poor ventilation, weak nutrition, inadequate or inaccessible medical care, etc, such as in some prisons or some refugee camps. In particular, the World Health Organization discovered that a number of prisoners come from socio-economic disadvantaged population where the burden of TB disease may be already high and access to medical care may be limited. In this dissertation we propose compartmental models of systems of differential equations to describe the population dynamics of TB disease under conditions of crowding. Such models can be used to make quantitative projections of TB prevalence and to measure the effect of interventions. Indeed we apply these models to specific regions and for specific purposes. The models are more widely applicable, however in this dissertation we calibrate and apply the models to prison populations.

Nous nous intéressons dans cette thèse à certains modèles mathématiques permettant une description de systèmes ouverts classiques et quantiques. Dans l'étude de ces systèmes en interaction avec un environnement, nous montrons que la dynamique induite par l'environnement sur le système donne lieu à l'apparition de bruits. Dans une première partie de la thèse, dédiée aux systèmes classiques, le modèle décrit est le schéma d'interactions répétées. Etant à la fois hamiltonien et markovien, ce modèle en temps discret permet d'implémenter facilement la dissipation dans des systèmes physiques. Nous expliquons comment le mettre en place pour des systèmes physiques avant d'en étudier la limite en temps continu. Nous montrons la convergence Lp et presque sûre de l'évolution de certains systèmes vers la solution d'une équation différentielle stochastique, à travers l'étude de la limite de la perturbation d'un schéma d'Euler stochastique. Dans une seconde partie de la thèse sur les systèmes quantiques, nous nous intéressons dans un premier temps aux actions d'environnements quantiques sur des systèmes quantiques aboutissant à des bruits classiques. A cette fin, nous introduisons certains opérateurs unitaires appelés « classiques », que nous caractérisons à l'aide de variables aléatoires dites obtuses. Nous mettons en valeur comment ces variables classiques apparaissent naturellement dans ce cadre quantique à travers des 3-tenseurs possédant des symétries particulières. Nous prouvons notamment que ces 3-tenseurs sont exactement ceux diagonalisables dans une base orthonormée. Dans un second temps, nous étudions la limite en temps continu d'une variante des interactions répétées quantiques dans le cas particulier d'un système biparti, c'est-à-dire composé de deux systèmes isolés sans interaction entre eux. Nous montrons qu'à la limite du temps continu, une interaction entre ces sous-systèmes apparaît explicitement sous forme d'un hamiltonien d'interaction; cette interaction résulte de l'action de l'environnement et de l'intrication qu'il crée / This dissertation is dedicated to some mathematical models describing classical and quantum open systems. In the study of these systems interacting with an environment, we particularly show that the dynamics induced by the environment leads to the appearance of noises. In a first part of this thesis, devoted to classical open systems, the repeated interaction scheme is developed. This discrete-time model, being Hamiltonian and Markovian at the same time, has the advantage to easily implement the dissipation in physical systems. We explain how to set this scheme up in some physical examples. Then, we investigate the continuous-time limit of these repeated interactions. We show the Lp and almost sure convergences of the evolution of the system to the solution of a stochastic differential equation, by studying the limit of a perturbed Stochastic Euler Scheme. In a second part of this dissertation on quantum systems, we characterize in a first work classical actions of a quantum environment on a quantum system. In this study, we introduce some “classical” unitary operators representing these actions and we highlight a strong link between them and some random variables, called obtuse random variables. We explain how these random variables are naturally connected to some 3-tensors having some particular symmetries. We particularly show that these 3 tensors are exactly the ones that are diagonalizable in some orthonormal basis. In a second work of this part, we study the continuous-time limit of a variant of the repeated interaction scheme in a case of a bipartite system, that is, a system made of two isolated systems not interaction together. We prove that an explicit Hamiltonian interaction between them appears at the limit. This interaction is due to the action of the environment and the entanglement between the two systems that it creates

Cette thèse est consacrée à l'étude de quelques problèmes dans le domaine des équations différentielles stochastiques rétrogrades (EDSR), et leurs applications aux équations aux dérivées partielles.Dans le premier chapitre, nous introduisons la notion d'équation différentielle doublement stochastique rétrograde (EDDSR) avec condition terminale singulière. Nous étudions d’abord les EDDSR avec générateur monotone, et obtenons ensuite un résultat d'existence par un schéma d'approximation. Une dernière section établit le lien avec les équations aux dérivées partielles stochastiques, via l'approche solution faible développée par Bally, Matoussi en 2001.Le deuxième chapitre est consacré aux EDSR avec condition terminale singulière et sauts. Comme dans le chapitre précédent la partie délicate sera de prouver la continuité en T. Nous formulons des conditions suffisantes sur les sauts afin d'obtenir cette dernière. Une section établit ensuite le lien entre solution minimale de l'EDSR et équations intégro-différentielles. Enfin le dernier chapitre est dédié aux équations différentielles stochastiques rétrogrades du second ordre (2EDSR) doublement réfléchies. Nous avons établi l'existence et l'unicité de telles équations. Ainsi, il nous a fallu dans un premier temps nous concentrer sur le problème de réflexion par barrière supérieure des 2EDSR. Nous avons ensuite combiné ces résultats à ceux existants afin de donner un cadre correct aux 2EDSRDR. L'unicité est conséquence d'une propriété de représentation et l'existence est obtenue en utilisant les espaces shiftés, et les distributions de probabilité conditionnelles régulières. Enfin une application aux jeux de Dynkin et aux options Israëliennes est traitée dans la dernière section. / This thesis is devoted to the study of some problems in the field of backward stochastic differential equations (BSDE), and their applications to partial differential equations.In the first chapter, we introduce the notion of backward doubly stochastic differential equations (BDSDE) with singular terminal condition. A first work consists to study the case of BDSDE with monotone generator. We then obtain existing result by an approximating scheme built considering a truncation of the terminal condition. The last part of this chapter aim to establish the link with stochastic partial differential equations, using a weak solution approach developed by Bally, Matoussi in 2001.The second chapter is devoted to the BSDEs with singular terminal conditions and jumps. As in the previous chapter the tricky part will be to prove continuity in T. We formulate sufficient conditions on the jumps in order to obtain it. A section is then dedicated to establish a link between a minimal solution of our BSDE and partial integro-differential equations.The last chapter is dedicated to doubly reflected second order backward stochastic differential equations (2DRBSDE). We have been looking to establish existence and uniqueness for such equations. In order to obtain this, we had to focus first on the upper reflection problem for 2BSDEs. We combined then these results to those already existing to give a well-posedness context to 2DRBSDE. Uniqueness is established as a straight consequence of a representation property. Existence is obtained using shifted spaces, and regular conditional probability distributions. A last part is then consecrated to the link with some Dynkin games and Israeli options.

This master thesis deals with design and implementation of betting agent. It covers issues such as theoretical background of an online betting, probability and statistics. In its first part it is focused on data mining and explains the principle of knowledge mining form data warehouses and certain methods suitable for different types of tasks. Second, it is concerned with neural networks and algorithm of back-propagation. All the findings are demonstrated on and supported by graphs and histograms of data analysis, made via SAS Enterprise Miner program. In conclusion, the thesis summarizes all the results and offers specific methods of extension of the agent.