Numerical and combinatorial applications of generalized Appell polynomials

This thesis studies properties and applications of different generalized Appell polynomials in the framework of Clifford analysis. As an example of 3D-quasi-conformal mappings realized by generalized Appell polynomials, an analogue of the complex Joukowski transformation of order two is introduced. The consideration of a Pascal n-simplex with hypercomplex entries allows stressing the combinatorial relevance of hypercomplex Appell polynomials. The concept of totally regular variables and its relation to generalized Appell polynomials leads to the construction of new bases for the space of homogeneous holomorphic polynomials whose elements are all isomorphic to the integer powers of the complex variable. For this reason, such polynomials are called pseudo-complex powers (PCP). Different variants of them are subject of a detailed investigation. Special attention is paid to the numerical aspects of PCP. An efficient algorithm based on complex arithmetic is proposed for their implementation. In this context a brief survey on numerical methods for inverting Vandermonde matrices is presented and a modified algorithm is proposed which illustrates advantages of a special type of PCP. Finally, combinatorial applications of generalized Appell polynomials are emphasized. The explicit expression of the coefficients of a particular type of Appell polynomials and their relation to a Pascal simplex with hypercomplex entries are derived. The comparison of two types of 3D Appell polynomials leads to the detection of new trigonometric summation formulas and combinatorial identities of Riordan-Sofo type characterized by their expression in terms of central binomial coefficients.

Development of polymeric materials to inhibit bacterial quorum sensing

Bacterial infections are an increasing problem for human health. In fact, an increasing number of infections are caused by bacteria that are resistant to most antibiotics and their combinations. Therefore, the scientific community is currently searching for new solutions to fight bacteria and infectious diseases, without promoting antimicrobial resistance. One of the most promising strategies is the disruption or attenuation of bacterial Quorum Sensing (QS), a refined system that bacteria use to communicate. In a QS event, bacteria produce and release specific small chemicals, signal molecules - autoinducers (AIs) - into the environment. At the same time that bacterial population grows, the concentration of AIs in the bacterial environment increases. When a threshold concentration of AIs is reached, bacterial cells respond to it by altering their gene expression profile. AIs regulate gene expression as a function of cell population density. Phenotypes mediated by QS (QSphenotypes) include virulence factors, toxin production, antibiotic resistance and biofilm formation. In this work, two polymeric materials (linear polymers and molecularly imprinted nanoparticles) were developed and their ability to attenuate QS was evaluated. Both types of polymers should to be able to adsorb bacterial signal molecules, limiting their availability in the extracellular environment, with expected disruption of QS. Linear polymers were composed by one of two monomers (itaconic acid and methacrylic acid), which are known to possess strong interactions with the bacterial signal molecules. Molecularly imprinted polymer nanoparticles (MIP NPs) are particles with recognition capabilities for the analyte of interest. This ability is attained by including the target analyte at the synthesis stage. Vibrio fischeri and Aeromonas hydrophila were used as model species for the study. Both the linear polymers and MIP NPs, tested free in solutions and coated to surfaces, showed ability to disrupt QS by decreasing bioluminescence of V. fischeri and biofilm formation of A. hydrophila. No significant effect on bacterial growth was detected. The cytotoxicity of the two types of polymers to a fibroblast-like cell line (Vero cells) was also tested in order to evaluate their safety. The results showed that both the linear polymers and MIP NPs were not cytotoxic in the testing conditions. In conclusion, the results reported in this thesis, show that the polymers developed are a promising strategy to disrupt QS and reduce bacterial infection and resistance. In addition, due to their low toxicity, solubility and easy integration by surface coating, the polymers have potential for applications in scenarios where bacterial infection is a problem: medicine, pharmaceutical, food industry and in agriculture or aquaculture.