In this study, we used E. coli strain BL21 (DE3) to express the recombinant protein his-tag streptavidin. To find out the optimal production conditions, we studied on the culture conditions, medium composition, induction conditions and the timing of induction. In the purification processes we tried to find out the difference between hydrophobic column and affinity column. We also tested the effect of heat treatment on the crude extract to increase the recombinant protein yield. The results showed that when cultured in LB medium, the optimal culture conditions of recombinant protein expression are 37¢XC, pH 6.0 to 7.0, and the induction temperature is 37¢XC. The best induction time is at late log phase or the early stationary phase when OD600 values reached to the ranged of 1.1 to 1.8. The inducer, IPTG concentration is 0.1 mM, which can also replaced with 2 mM lactose. The best production medium is TB medium. When cultured in 5 liters fermentor with optimal culture and induction condition, the highest recombinant protein yield could be 81.1 mg /L. To improve the purification process, we used a affinity chromatography. The purified high homogeneous recombinant protein had a high biotin binding activity up to 14 U / mg, and the recovery yield could be as high as 97% in comparing with the hydrophobic column was only 51%. When we treated the crude extract with 75 ¢J for 10 min, the biotin binding activity was 14.1 U / mg, but the recovery rate decreased to 64 %.

Cette thèse porte sur deux équations aux dérivées partielles qui modélisent les phénomènes biologiques de l'évolution génétique et mouvement dans l'espace d'une population de cellules. Le premier problème (Partie I, Chapitre 1), il est sur l'évolution phénotypique d'une population de cellules, nous avons réussi à démontrer que la limite asymptotique des solutions de l'équation différentielle partielle proposée est une masse de Dirac. Pour modéliser ce phénomène, nous avons étudié une équation de transport sur le mouvement génétique, y compris des éléments classiques de l'écologie mathématique et ajouter un transport terme dans la variable génétique x pour modéliser le phénomène de sélection naturelle. Nous intégrons un paramètre approprié dans notre modèle, qui a un problème associé normalisée. Ensuite, nous faisons quelques estimations pour donner des propriétés des solutions et obtenir sa limite. Pour ce faire, nous définissons une sous-solution et sur-solution, qui délimitent la solution du problème en appliquant un principe du maximum.Le deuxième problème (Partie II, Chapitre 2), résume les principaux résultats obtenus dans l'étude d'un système d'équations aux dérivées partielles paraboliques inspiré par l'équation Keller-Segel. C'est pourquoi le résultat principal est d'obtenir des conditions optimales sur la masse initiale pour l'existence globale et blow-up des solutions du système étudié, utilisé la méthode des moments et des inégalités de Hardy-Littlewood-Sobolev pour systèmes. / In Chapter 1, we consider a cell population where the individuals live in the same environmental conditions for some fixed period of time where they compete for nutrients among themselves, considering that offspring has the same trait as their parents, we were defining a fitness function that is trait and density dependent, assuming there were a unique trait best adapted at fixed environmental conditions. We modeled this phenomenon using a Transport Equation. The main result have been obtaining a Dirac mass concentration like solutions for the asymptotic behavior, incorporating a parameter, which is biologically sustained. We applied the classical framework to obtain this result. First, we give the apriori estimates and existence result to the simplified problem, next we add terms to have a more realistic model, then we study an approximate problem given some regularity and properties at solutions, finally we obtain this limit. We used tools as BV convergence properties, Anzats, sub and super solutions, maximum principle, etc.Chapter 2 had been publishing in the following papers (see part II):- E. ESPEJO, K. VILCHES, C. CONCA (2012), Sharp conditon for blow-up and global existence in a two species chemotactic Keller-Segel system in R^2, European J. Appl. Math- C. CONCA, E. ESPEJO, K. VILCHES (2011), Remarks on the blow-up and global existence for a two species chemotactic Keller-Segel system in R^2. European J. Appl. Math.In this chapter, we give the main results obtained in these two publications. We have been studying the sharp condition to global existence and Blow-up in time to the parabolic PDE system in R^2, inspired by the studies were done in the one species case. We model the movement for two chemotactic populations produced by one chemical substance. The main result is to extend the result obtained to classical simplified Keller-Segel model in one species case to the multispecies case, using the adequately tools for PDE’s systems. We used the moment method to prove Blow-up and have been bounding the entropy to show global existence.