>Magister Scientiae - MSc / HIV/AIDS is nowadays considered as the greatest public health disaster of modern time. Its progression has challenged the global population for decades. Through mathematical modelling, researchers have studied different interventions on the HIV pandemic, such as treatment, education, condom use, etc. Our research focuses on different compartmental models with emphasis on the effect of public health education. From the point of view of statistics, it is well known how the public health educational programs contribute towards the reduction of the spread of HIV/AIDS epidemic. Many models have been studied towards understanding the dynamics of the HIV/AIDS epidemic. The impact of ARV treatment have been observed and analysed by many researchers. Our research studies and investigates a compartmental model of HIV with treatment and education campaign. We study the existence of equilibrium points and their stability. Original contributions of this dissertation are the modifications on the model of Cai et al. [1], which enables us to use optimal control theory to identify optimal roll-out of strategies to control the HIV/AIDS. Furthermore, we introduce randomness into the model and we study the almost sure exponential stability of the disease free equilibrium. The randomness is regarded as environmental perturbations in the system. Another contribution is the global stability analysis on the model of Nyabadza et al. in [3]. The stability thresholds are compared for the HIV/AIDS in the absence of any intervention to assess the possible community benefit of public health educational campaigns. We illustrate the results by way simulation The following papers form the basis of much of the content of this dissertation, [1 ] L. Cai, Xuezhi Li, Mini Ghosh, Boazhu Guo. Stability analysis of an HIV/AIDS epidemic model with treatment, 229 (2009) 313-323. [2 ] C.P. Bhunu, S. Mushayabasa, H. Kojouharov, J.M. Tchuenche. Mathematical Analysis of an HIV/AIDS Model: Impact of Educational Programs and Abstinence in Sub-Saharan Africa. J Math Model Algor 10 (2011),31-55. [3 ] F. Nyabadza, C. Chiyaka, Z. Mukandavire, S.D. Hove-Musekwa. Analysis of an HIV/AIDS model with public-health information campaigns and individual with-drawal. Journal of Biological Systems, 18, 2 (2010) 357-375. Through this dissertation the author has contributed to two manuscripts [4] and [5], which are currently under review towards publication in journals, [4 ] G. Abiodun, S. Maku Vyambwera, N. Marcus, K. Okosun, P. Witbooi. Control and sensitivity of an HIV model with public health education (under submission). [5 ] P.Witbooi, M. Nsuami, S. Maku Vyambwera. Stability of a stochastic model of HIV population dynamics (under submission).

No description available.

The freshwater dinoflagellate species, Ceratium hirundinella (C. hirundinella) possesses unique characteristics, such as a thecal-plate cell covering of cellulose, spines and flagella. Unlike most other algae and cyanobacteria, C. hirundinella cells are relatively large in size (up to 450 μm in length and 50 μm in width). These unique characteristics (e.g. cell covering and flagella) and adaptations (e.g. spines) give the dinoflagellate cells the ability to reduce their sinking rate from the euphotic zone and to migrate easily through the water column. When source water contains high concentrations of C. hirundinella cells, water treatment problems and poor aesthetic water quality can be expected. These water treatment problems may include 1) the disruption of coagulation and flocculation, 2) clogging of sand filters and 3) taste and odour problems when cells penetrate into the final water. In Chapter 9 of this study, a list of operational guidelines (including alert levels) and recommendations to assist managers and operators of plants when C. hirundinella cells are causing water treatment problems. During events of high C. hirundinella concentrations in source water, managers and operators of conventional water treatment plants need strategies to optimize coagulants and unit processes. Thus when source water contains motile nuisance algae, such as C. hirundinella, in moderate or abundant quantities, it is advisable to conduct jar stirring test experiments using both turbidity and total photosynthetic pigment (or chlorophyll-a) analyses as indicators of appropriate coagulant choice and dosages. The aims of this study are summarized as follows:  To optimize coagulants and conventional water treatment processes by implementing relevant algal removal strategies and indicators during jar stirring test experiments,  To investigate the changes in surface charge (known as zeta potential) on C. hirundinella cells before and after adding coagulants as part of the treatment processes,  To investigate the physical and chemical impacts on the morphology of C. hirundinella cells after coagulation, flocculation and sedimentation,  To identify organic compounds that may be responsible for taste and odour problems associated with C. hirundinella,  To investigate the efficiency of pre-chlorination on the removal C. hirundinella cells when dosing various coagulants, and  Give recommendations and operational guidelines relevant for a conventional water treatment plant to improve C. hirundinella removal A combined water treatment system (Phipps and Bird Model), consisting of a six paddle jar test apparatus and six sand filter columns, was used to simulate conventional processes (coagulation, flocculation, sedimentation and rapid sand filtration). Source water samples containing relatively high C. hirundinella concentrations (> 500 cell/mℓ) were collected from Benoni Lake (26º10’50.40’’S; 28º17’50.11’’ E) in plastic containers and stored as a homogenous sample in a 200 litre container under laboratory conditions (± 22 °C). Samples were collected from the source water as well as after sedimentation (from the supernatant or sludge) to determine turbidity, total photosynthetic pigment analyses (chlorophyll) and for phytoplankton analyses. Flocs (containing C. hirundinella cells) were collected from the sludge or sediment for scanning electron microscopy investigations and to perform zeta potential analyses. Concentrated C. hirundinella samples were frozen at -80 °C according to the proposed sampling protocol for organic compound analyses. Results obtained from this study proved that using the relevant indicators to determine the appropriate coagulant dosages during jar stirring tests may generally improve the removal of problem-causing algae, such as C. hirundinella cells. Improved algal removal efficiencies will subsequently ensure final water with good aesthetic quality. The surface charge (zeta potential) on C. hirundinella cells can be used to evaluate the best coagulation conditions within an operating window of -10 mV to +3 mV when dosing various coagulants. Scanning electron microscopy investigations revealed major damaging effects to C. hirundinella cells when dosing high Ca(OH)2 concentrations. However, when dosing lower Ca(OH)2 concentrations, in combination with organic polymer, better C. hirundinella cell removal efficiencies with less damaging effects to cells was observed. This study also indicated that the pre-chlorination, without causing cell lyses, can be applied to render the highly motile cells immobile which will subsequently assist the coagulation unit process. The aesthetic quality (e.g. tastes and odours) of drinking water may be influenced when C. hirundinella cells release organic material into the water as a result of cell lyses. Organic compounds, such as fatty acids and dicarboxylic acids can lead to taste and odour problems which associate with the presence of C. hirundinella. Organic compounds also serve as precursors for the formation of harmful chlorine by-products formed during chlorination. / PhD (Environmental Sciences), North-West University, Potchefstroom Campus, 2015

The freshwater dinoflagellate species, Ceratium hirundinella (C. hirundinella) possesses unique characteristics, such as a thecal-plate cell covering of cellulose, spines and flagella. Unlike most other algae and cyanobacteria, C. hirundinella cells are relatively large in size (up to 450 μm in length and 50 μm in width). These unique characteristics (e.g. cell covering and flagella) and adaptations (e.g. spines) give the dinoflagellate cells the ability to reduce their sinking rate from the euphotic zone and to migrate easily through the water column. When source water contains high concentrations of C. hirundinella cells, water treatment problems and poor aesthetic water quality can be expected. These water treatment problems may include 1) the disruption of coagulation and flocculation, 2) clogging of sand filters and 3) taste and odour problems when cells penetrate into the final water. In Chapter 9 of this study, a list of operational guidelines (including alert levels) and recommendations to assist managers and operators of plants when C. hirundinella cells are causing water treatment problems. During events of high C. hirundinella concentrations in source water, managers and operators of conventional water treatment plants need strategies to optimize coagulants and unit processes. Thus when source water contains motile nuisance algae, such as C. hirundinella, in moderate or abundant quantities, it is advisable to conduct jar stirring test experiments using both turbidity and total photosynthetic pigment (or chlorophyll-a) analyses as indicators of appropriate coagulant choice and dosages. The aims of this study are summarized as follows:  To optimize coagulants and conventional water treatment processes by implementing relevant algal removal strategies and indicators during jar stirring test experiments,  To investigate the changes in surface charge (known as zeta potential) on C. hirundinella cells before and after adding coagulants as part of the treatment processes,  To investigate the physical and chemical impacts on the morphology of C. hirundinella cells after coagulation, flocculation and sedimentation,  To identify organic compounds that may be responsible for taste and odour problems associated with C. hirundinella,  To investigate the efficiency of pre-chlorination on the removal C. hirundinella cells when dosing various coagulants, and  Give recommendations and operational guidelines relevant for a conventional water treatment plant to improve C. hirundinella removal A combined water treatment system (Phipps and Bird Model), consisting of a six paddle jar test apparatus and six sand filter columns, was used to simulate conventional processes (coagulation, flocculation, sedimentation and rapid sand filtration). Source water samples containing relatively high C. hirundinella concentrations (> 500 cell/mℓ) were collected from Benoni Lake (26º10’50.40’’S; 28º17’50.11’’ E) in plastic containers and stored as a homogenous sample in a 200 litre container under laboratory conditions (± 22 °C). Samples were collected from the source water as well as after sedimentation (from the supernatant or sludge) to determine turbidity, total photosynthetic pigment analyses (chlorophyll) and for phytoplankton analyses. Flocs (containing C. hirundinella cells) were collected from the sludge or sediment for scanning electron microscopy investigations and to perform zeta potential analyses. Concentrated C. hirundinella samples were frozen at -80 °C according to the proposed sampling protocol for organic compound analyses. Results obtained from this study proved that using the relevant indicators to determine the appropriate coagulant dosages during jar stirring tests may generally improve the removal of problem-causing algae, such as C. hirundinella cells. Improved algal removal efficiencies will subsequently ensure final water with good aesthetic quality. The surface charge (zeta potential) on C. hirundinella cells can be used to evaluate the best coagulation conditions within an operating window of -10 mV to +3 mV when dosing various coagulants. Scanning electron microscopy investigations revealed major damaging effects to C. hirundinella cells when dosing high Ca(OH)2 concentrations. However, when dosing lower Ca(OH)2 concentrations, in combination with organic polymer, better C. hirundinella cell removal efficiencies with less damaging effects to cells was observed. This study also indicated that the pre-chlorination, without causing cell lyses, can be applied to render the highly motile cells immobile which will subsequently assist the coagulation unit process. The aesthetic quality (e.g. tastes and odours) of drinking water may be influenced when C. hirundinella cells release organic material into the water as a result of cell lyses. Organic compounds, such as fatty acids and dicarboxylic acids can lead to taste and odour problems which associate with the presence of C. hirundinella. Organic compounds also serve as precursors for the formation of harmful chlorine by-products formed during chlorination. / PhD (Environmental Sciences), North-West University, Potchefstroom Campus, 2015

Dans cette thèse nous étudions les théorèmes limites dans l’analyse statistique dessystèmes dynamiques. Le premier chapitre est consacré aux notions des bases des systèmesdynamiques ainsi que la théorie ergodique. Dans le deuxième chapitre nous introduisonsun cadre fonctionnel abstrait pour lequel la version quenched du théorème de la limitecentrale (TLC) en dimension 1 pour les systèmes dynamiques uniformément dilatantsest satisfaite sous une condition de validité nécessaire et suffisante. Le troisième chapitreest consacré au principe d’invariance presque sûr (PIPS) pour les application aléatoiresdilatantes par morceaux. Nous présentons certaines hypothèses sous lesquelles le (PIPS)est vérifié en utilisant la méthode d’approximation des martingales de Cuny et Merlèvede.Nous étudions aussi le théorème de Sprindzuk et ses conséquences. Nous établissons dansle chapitre quatre la décroissance des corrélations pour les systèmes dynamiques aléatoiresuniformément dilatants par la méthode de couplage en dimension 1. Nous terminons cetravail par une présentation des concepts de base de la théorie des mesures et probabilitéset une présentation de l’espace des fonctions à variation bornée. / In this thesis we study the limit theorems in the statistical analysis of dynamicalsystems. The first chapter is devoted to the basic notions in dynamical systems as well asthe ergodic theory. In the second chapter we introduce an abstract functional frameworkunder which the quenched version of the central limit theorem (CLT) in dimension 1for uniformly expanding dynamic systems is satisfied under a necessary and sufficientcondition validity. The third chapter is devoted to the almost sure invariance principle(ASIP) for random piecewise expanding maps. We present some hypotheses under whichthe (ASIP) is verified using the method of approximation of the martingales of Cuny andMerlèvede. We also study the Sprindzuk theorem and its consequences. In chapter four,we define the decay of correlations for the random dynamical systems uniformly expandingby the coupling method in dimension 1. We finish this work with a presentation of thebasic concepts of the theory of measures and probabilities and a presentation of the spaceof functions with bounded variation.

Le travail présenté dans ce mémoire concerne une méthode de conception sûre de systèmes(COTS). Un COTS est un composant matériel ou logiciel générique qui est naturellement conçu pour être réutilisable et cela se traduit par une forme de flexibilité dans la mise en oeuvre de sa fonctionnalité : en clair, une même fonction peut être réalisée par un ensemble (potentiellement infini) de scénarios différents, tous réalisables par le COTS. La complexité grandissante des fonctions implémentées fait que ces situations sont très difficiles à anticiper d’une part, et encore plus difficiles à éviter par un codage correct. Réaliser manuellement une fonction composite correcte sur un système de taille industrielle, s’avère être très coûteuse. Elle nécessite une connaissance approfondie du comportement des COTS assemblés. Or cette connaissance est souvent manquante, vu qu’il s’agit de composants acquis, ou développés par un tiers, et dont la documentation porte sur la description de leur fonction et non sur sa mise en IJuvre. Par ailleurs, il arrive souvent que la correction manuelle d’une faute engendre une ou plusieurs autres fautes, provoquant un cercle vicieux difficile à maîtriser. En plus, le fait de modifier le code d’un composant diminue l’avantage lié à sa réutilisation. C’est dans ce contexte que nous proposons l’utilisation de la technique de synthèse du contrôleur discret (SCD) pour générer automatiquement du code de contrôle commande correct par construction. Cette technique produit des composants, nommés contrôleurs, qui agissent en contraignant le comportement d’un (ou d’un assemblage de) COTS afin de garantir si possible la satisfaction d’une exigence fonctionnelle. La méthode que nous proposons possède plusieurs étapes de conception. La première étape concerne la formalisation des COTS et des propriété de sûreté et de vivacité (P) en modèles automate à états et/ou en logique temporelle. L’étape suivante concerne la vérification formelle du modèle d’un(des) COTS pour l’ensemble des propriétés (P). Cette étape découvrir les états de violation des propriétés (P) appelés états d’erreur. La troisième étape concerne la correction automatique des erreurs détectées en utilisant la technique SCD. Dans cette étape génère on génère un composant correcteur qui sera assemblé au(x) COTS original(aux) pour que leur comportement général respecte les propriétés souhaitées. L’étape suivante concerne la vérification du système contrôlé pour un ensemble de propriétés de vivacité pour assurer la passivité du contrôleur et la vivacité du système. En fin, une étape de simulation est proposée pour observer le comportement du système pour quelque scénarios intéressent par rapport à son implémentation finale. / This PhD dissertation contributes to the safe design of COTS-based control-command embedded systems. Due to design constraints bounding delays, costs and engineering resources, component re-usability has become a key issue in embedded design. Our proposal is a design method which ensures correction of COTS-based designs. This method uses in synergy a number of design techniques and tools. It starts from modeling of the COTS components which are stored in a generic COTS library, and ends with a design of the global control-command system, verified to be free of errors and ready to be implemented over a hardware chip such as an ASIC or an FPGA "Field Programmable Gate Array". The designer starts by modeling the temporal and logical local preconditions and postconditions of each COTS component, then the global pre/post conditions of the assembly which are not necessary a simple combination of local properties. He models also a list of properties that must be satisfied by the assembly. Any violation of these properties is defined as a design error. Then, by using the model checking approach the model of the assembly is verified against the predefined local and global properties. Some design errors can be corrected automatically through the Discrete Controller Synthesis method (DCS), others however must be manually corrected. After the correction step, the controlled control-command system is verified. Finally a global simulation step is proposed in order to perform a system-level verification beyond the capabilities of available formal tools. We apply the method on two different systems, one concerns transferring data from senders to receivers through FIFO unit, the other is controlcommand system of a train passengers’ access.

對樣本資料之機率密度函數f(x)的無母數估計方法，一直是統計推論領域的研究重點之一，而且在通訊理論與圖形辨別上有非常重要的地位。傳統的文獻對密度函數的估計方法大部分著重於獨立樣本的情形。對於時間數列的相關樣本（例如：經濟指標或加權股票指數資料）比較少提到。本文針對具有弱相關性的穩定時間數列樣本，嘗試提出一個核密度估計的方法並探討其性質。 / For a sample data, the nonparametric estimation of a probability density f(x) is always one point of research problem in statistical inference and plays an important role in communication theory and pattern recognition. Traditionally, the literature dealing with density estimation when the observations are independent is extensive. Time series sample with weak dependence, (for example, an economic indicator or a stock market index data), less in this aspect of discussion. Our main purpose is concerned with the estimation of the probability density function f(x) of a stationary time series sample and discusses some properties of this kernel density.

Cette Thèse est composée de deux parties portant respectivement sur la discrétisation des équations différentielles stochastiques et sur le théorème de la limite centrale presque sûre pour les martingales.<br /><br />La première Partie est composée de trois chapitres: Le premier chapitre introduit le cadre de l'étude et présente les résultats obtenus. Le deuxième chapitre est consacré à l'étude d'une nouvelle méthode d'accélération de convergence, appelée méthode de Romberg statistique, pour le calcul d'espérances de fonctions ou de fonctionnelles d'une diffusion.<br />Ce chapitre est la version augmentée d'un article à paraître dans la revue Annals of Applied Probability.<br /><br />Le troisième chapitre traite de l'application de cette méthode à l'approximation de densité par des méthodes de noyaux.<br />Ce chapitre est basé sur un travail en collaboration avec Arturo Kohatsu-Higa.<br /><br />La deuxième partie de la thèse est composée de deux chapitres: le premier chapitre présente la littérature récente concernant le théorème de la limite centrale presque sûre et ses extensions. Le deuxième chapitre, basé sur un travail en collaboration avec Faouzi Chaâbane, étend divers résultats de type TLCPS à des martingales quasi-continues à gauche.

L'avant-pays de Chartreuse est une zone de transition entre les chaînes subalpines (massif de la Grande-Sure ) , la terminaison méridionale du Jura (chaînon du Ratz) et le bassin molassique du Bas-Dauphiné. La grande diversité des dépôts (calcaire karstique , molasse marneuse et conglomératique, alluvions fluviatiles , glaciaires et interglaciaires ) a justifié l'emploi de méthodes variées dans le but de définir le comportement hydrodynamique des différents aquifères et d'en déterminer les potentialités. La description des particularités lithologiques et hydrogéologiques des différentes formations a permis de préciser les conditions aux limites des réservoirs alluviaux. L'étude du remplissage alluvial, réalisée à partir d'une campagne de prospection géophysique et de nombreux forages , a défini la géométrie de ces réservoirs et a permis l'ébauche d'un schéma de la configuration des dépôts en fonction des phases glaciaires et interglaciaires. Les limites de bassins versants ont été déterminées par l'étude des circulations karstiques dans les massifs calcaires . L'analyse de la structure et de la fracturation, l 'observation des débits et des caractères physico- chimiques des eaux des émergences ont donné un schéma cohérent du fonctionnement des systèmes karstiques qui participent pour une grande part à l 'alimentation des aquifères alluviaux. L'évaluation des volumes d'eau infiltrée à partir des données hydroclimatiques a mis en évidence de remarquables potentialités d'exploitation de la nappe dans la vallée de Saint-Laurent-du-Pont . L'étude hydrodynamique de cette nappe a fait apparaître que l 'effet transitoire des apports est rapidement effacé vers l'aval, la nappe acquérant en permanence un régime de quasi-équilibre entre les effets d ' alimentations amont et le drainage des zones argileuses . L' ensemble des connaissances concernant l'aquifère et ses limites a donc pu être contrôlé par un modèle d'écoulement permanent qui permet de simuler les effets d 'un captage.

X-ray crystallography is a powerful method for determining the three-dimensional structures of biological macromolecules at atomic resolution. Crystallography can reliably provide the answer to many structure related questions, from global folds to atomic details of bonding. Crystallographic techniques find wide applications in understanding macromolecular assembly, enzyme mechanism, mode of activation of enzymes, substrate-specificity, ligand-binding properties, domain movement etc. The knowledge of accurate molecular structures is also a prerequisite for rational drug design and for structure based functional studies to aid the development of effective therapeutic agents. The current thesis can be broadly divided into two major parts. The first four chapters deal with assembly studies that have been carried out on Sesbania mosaic virus and the next two chapters describe the structure and function of a stationary phase survival protein, SurE from Salmonella typhimurium. In both studies X-ray crystallographic techniques have been used extensively for the structural studies. Viruses are obligate parasites with a proteinaceous capsid enclosing the genetic material. For genetic economy, several copies of capsid proteins self assemble to form complex virus capsids. Due to their intricate symmetric structures, viruses are considered as minute marvels of molecular architecture and study of virus structures serve as a paradigm for solutions to problems concerning macromolecular assembly and function in general. Crystallography provides a means of visualizing intact virus particles as well as their isolated constituent proteins and enzymes at near-atomic resolution, and is thus an extraordinarily powerful tool for understanding the function of these biological systems. Protein-protein interactions, protein-nucleic acid interactions, metal-ion mediated interactions, interactions between capsid proteins and auxillary or scaffolding proteins and particle maturation or post processing of capsid protein subunits are various elements that play a role in capsid assembly. Many structural and sequential motifs have been proposed as important conformational switches of capsid assembly. A functional analysis of these motifs by way of mutations in the capsid protein and structural studies of these mutants can provide further insight into capsid assembly pathways. Interaction between capsid protein subunits can determine the size and robustness of the capsid. Analysis of protein-protein interactions can help in understanding the principles of self-assembly. Arresting capsid assembly by disrupting intersubunit interactions and trapping the assembly intermediates will be helpful to delineate the changes that happen in capsid protein during the course of assembly and understand assembly pathways. Sesbania mosaic virus (SeMV) is a plant virus with a positive sense single-stranded RNA genome and belongs to the Sobemovirus genus. The protein and nucleic acids of SeMV can be separated and reassembled in vitro. Also, expression of the coat protein (CP) gene of SeMV in E. coli leads to the formation of virus like particles (VLPs). Therefore, SeMV is an excellent model system to study the assembly pathways that lead to the formation of complex virus shells. Earlier structural and functional studies on the native virus and the recombinant capsid protein and its various mutants have revealed the following: SeMV is a T=3 virus with chemically identical A-, B- and C-subunits occupying quasi equivalent positions in the icosahedral asymmetric unit of the virus particle. The A-type subunits form pentamers at the five-fold, and the B- and C- type subunits form hexamers at the icosahedral three-fold axes. The amino terminus of the polypeptide is ordered from residue 72 in the A- and B- subunits whereas it is ordered from residue 44 in the C-subunit. The disordered segment in all the subunits has an arginine rich motif (N-ARM). The segment ordered only in C-subunits has a -annulus structure that promotes intersubunit interactions at the quasi six-fold and a -segment (A). The virus is stabilized by protein-protein, protein–RNA and Ca2+ mediated protein-protein interactions. Virus like particles (VLPs) formed by the expression of full length CP encapsidate 23 S E. coli rRNA and CP mRNA. Expression of a deletion mutant lacking the N-terminal 65 residues (rCP∆N65) which results in the removal of the N-ARM, the -annulus and the A leads to the formation of stable T=1 particles. The -annulus, which was earlier believed to be an important molecular switch controlling the assembly of T=3 VLPs was found to be dispensable. The N-ARM, though important for RNA encapsidation, was not essential for capsid assembly . Depletion of Ca2+ ions led to slight swelling of virus particles and significantly reduced stability. Extensive studies on the VLPs suggested that the assembly is most likely initiated by the dimers of the capsid protein. Following a brief account of the historical highlights in the field of structural virology, a review of current literature on the available crystal structures of viruses and various assembly studies on viruses that have been carried out with emphasis on role of nucleic acid mediated interactions, protein-protein interactions and role of specific residues and ion-mediated interactions in assembly are presented in Chapter I of the thesis. A separate section in this chapter deals with the disassembly experiments that have led to the formation of smaller oligomers of spherical viruses. This chapter also gives an account of the earlier work that has been carried out on SeMV, which is the model system of study for the present thesis. Chapter II describes in detail the structural studies on the β-annulus deletion mutant of SeMV. A unique feature of several T = 3 icosahedral viruses is the presence of a structure called the β-annulus formed by extensive hydrogen bonding between protein subunits related by icosahedral three-fold axis of symmetry. This unique structure has been suggested as a molecular switch that determines the T = 3 capsid assembly. In order to examine the importance of the β-annulus, a deletion mutant of Sesbania mosaic virus coat protein in which residues 48–59 involved in the formation of the β-annulus were deleted retaining the rest of the residues in the amino terminal segment (rCP (Δ48–59)) was constructed. When expressed in Escherichia coli, the mutant protein assembled into virus like particles of size close to that of the wild type virus particles. The purified capsids were crystallized and their three dimensional structure was determined at 3.6Å resolution by X-ray crystallography. The mutant capsid structure closely resembled that of the native virus particles. However, surprisingly, the structure revealed that the assembly of the particles has proceeded without the formation of the β-annulus. Therefore, the β-annulus is not essential for T = 3 capsid assembly as speculated earlier and may be formed as a consequence of the particle assembly. This is the first structural demonstration that the virus particle morphology with and without the β-annulus could be closely similar. Chapter III begins with a detailed description of the interfacial residue mutations that have been carried out in SeMV with the aim of disrupting assembly and trapping an assembly intermediate. These mutations were performed in rCP as well as rCP∆N65 gene. Among these, a single point mutation of a Trp 170 to a charged residue (either Glu or Lys) arrested virus assembly and resulted in stable dimers of the capsid protein. The chapter also gives an account of the biophysical characterization of these mutants. rCP∆N65 dimer mutants showed a characteristic 230 nm peak in CD spectral studies which may be due to the interactions of a stretch of aromatic residues in the capsid protein. The isolated dimers were more susceptible to trypsin cleavage compared to the assembled capsids due to the exposed basic amino terminus. Thermal melting studies showed that the isolated dimer mutants were much less stable when compared to the assembled capsids, probably due to the loss of intersubunit interactions and Ca2+ mediated interactions. The structure of one of the isolated dimer mutant- rCP∆N65W170K was solved to a resolution of 2.65Å. Chapter IV describes the crystal structure analysis of the rCP∆N65W170K mutant dimer and compares its structure with the dimers of native virus, T=3 and T=1 VLPs. A number of structural changes occur especially in the loop and interfacial regions during the course of assembly. The dimer in solution was “more relaxed” than the dimer that initiates assembly. Ca2+ ion is not bound and consequently the C-terminal residues are disordered. The FG loop, which interacts with RNA, was found to be flexible and adopts a different conformation in the unassembled dimer. The present thesis also deals with the structural and functional studies of a phosphatase, SurE, the stationary phase survival protein from Salmonella typhimurium. Chapter V provides a general introduction on Salmonella, which is a mesophilic food borne pathogen, its general features, classification and stress responses. This chapter also gives an account of stationary phase in bacteria and stress responses. A brief description about phosphatases and their classification is also presented in this chapter. Following this, a review of the current literature on the structural, biochemical and functional role of stress related proteins and phylogenetic and enzymatic studies of various homologues of SurE are described in detail. Chapter VI deals with the detailed crystal structure analysis of SurE, the first stationary phase survival protein from a mesophilic organism. SurE, of Salmonella typhimurium forms part of a stress survival operon regulated by the stationary phase RNA polymerase alternative sigma factor. SurE is known to improve bacterial viability during stress conditions. It functions as a phosphatase specific to nucleoside monophosphates. Here we report the X-ray crystal structure of SurE from Salmonella typhimurium (St SurE). The protein crystallized in two forms- orthorhombic F222 and monoclinic C2. The two structures were determined to resolutions of 1.7Å and 2.7Å, respectively. The protein exists as a domain swapped dimer. The residue Asp 230 is involved in several interactions that are probably crucial for domain swapping. A divalent metal ion is found at the active site of the enzyme, which is consistent with the divalent metal-ion dependent activity of the enzyme. Interactions of the conserved DD motif present at the N-terminus with the phosphate and the Mg2+ present in the active site suggest that these residues play an important role in enzyme activity. The divalent metal ion specificity and the kinetic constants of SurE were determined using the generic phosphatase substrate- para- Nitro Phenyl Phosphate. The enzyme was inactive in the absence of divalent cations and was most active in the presence of Mg2+. Thermal denaturation studies showed that St SurE is much less stable compared to its homologues and an attempt was made to understand the molecular basis of the lower thermal stability based on solvation free energy. The thesis concludes with a brief summary of the entire work that have been presented and future prospects. The various crystallographic, biochemical and biophysical techniques employed in the investigations are described under the section experimental techniques in Appendix I and the NCS matrices used in the structure solution of the β-annulus deletion mutant are listed in Appendix II.