Studies in non-Gaussian analysis

This thesis presents general methods in non-Gaussian analysis in infinite dimensional spaces. As main applications we study Poisson and compound Poisson spaces. Given a probability measure μ on a co-nuclear space, we develop an abstract theory based on the generalized Appell systems which are bi-orthogonal. We study its properties as well as the generated Gelfand triples. As an example we consider the important case of Poisson measures. The product and Wick calculus are developed on this context. We provide formulas for the change of the generalized Appell system under a transformation of the measure. The L² structure for the Poisson measure, compound Poisson and Gamma measures are elaborated. We exhibit the chaos decomposition using the Fock isomorphism. We obtain the representation of the creation, annihilation operators. We construct two types of differential geometry on the configuration space over a differentiable manifold. These two geometries are related through the Dirichlet forms for Poisson measures as well as for its perturbations. Finally, we construct the internal geometry on the compound configurations space. In particular, the intrinsic gradient, the divergence and the Laplace-Beltrami operator. As a result, we may define the Dirichlet forms which are associated to a diffusion process. Consequently, we obtain the representation of the Lie algebra of vector fields with compact support. All these results extends directly for the marked Poisson spaces.

PEGylated polyethyleneimine-entrapped gold nanoparticles for enhanced and targeted gene delivery applications

Gene therapy, which involves the transfer of nucleic acid into target cells in patients, has become one of the most important and widely explored strategies to treat a variety of diseases, such as cancer, infectious diseases and genetic disorders. Relative to viral vectors that have high immunogenicity, toxicity and oncogenicity, non-viral vectors have gained a lot of interest in recent years. This is largely due to their ability to mimic viral vector features including the capacity to overcome extra- and intra-cellular barriers and to enhance transfection efficiency. Polyethyleneimine (PEI) has been extensively investigated as a non-viral vector. This cationic polymer, which is able to compact nucleic acid through electrostatic interactions and to transport it across the negatively charged cell membranes, has been shown to effectively transfect nucleic acid into different cell lines. Moreover, entrapment of gold nanoparticles (Au NPs) into such an amine-terminated polymer template has been shown to significantly enhance gene transfection efficiency. In this work, a novel non-viral nucleic acid vector system for enhanced and targeted nucleic acid delivery applications was developed. The system was based on the functionalization of PEI with folic acid (FA; for targeted delivery to cancer cells overexpressing FA receptors on their surface) using polyethylene glycol (PEG) as a linker molecule. This was followed by the preparation of PEI-entrapped Au NPs (Au PENPs; for enhancement of transfection efficiency). In the synthesis process, the primary amines of PEI were first partially modified with fluorescein isothiocyanate (FI) using a molar ratio of 1:7. The formed PEI-FI conjugate was then further modified with either PEG or PEGylated FA using a molar ratio of 1:1. This process was finally followed by entrapment of Au NPs into the modified polymers. The resulting conjugates and Au PENPs were characterized by several techniques, namely Nuclear Magnetic Resonance, Dynamic Light Scattering and Ultraviolet-Visible Spectroscopy, to assess their physicochemical properties. In the cell biology studies, the synthesized conjugates and their respective Au PENPs were shown to be non-toxic towards A2780 human ovarian carcinoma cells. The role of these materials as gene delivery agents was lastly evaluated. In the gene delivery studies, the A2780 cells were successfully transfected with plasmid DNA using the different vector systems. However, FA-modification and Au NPs entrapment were not determinant factors for improved transfection efficiency. In the gene silencing studies, on the other hand, the Au PENPs were shown to effectively deliver small interfering RNA, thereby reducing the expression of the B-cell lymphoma 2 protein. Based on these results, we can say that the systems synthesized in this work show potential for enhanced and targeted gene therapy applications.