We show that the nonadditive open systems can be studied in a consistent manner by using a generalized version of S-theorem. This new generalized S-theorem can further be considered as an indication of self-organization in nonadditive open systems as prescribed by Haken. The nonadditive S-theorem is then illustrated by using the modified Van der Pol oscillator. Finally, Tsallis entropy as an equilibrium entropy is studied by using Boltzmann's method of orthodes. This part of dissertation shows that Tsallis ensemble is on equal footing with the microcanonical, canonical and grand canonical ensembles. However, the associated entropy turns out to be Renyi entropy.

The thesis concentrates on property of linearity in time series models, its definitions and possibilities of testing. Presented tests focus mainly on the time domain; these are based on various statistical methods such as regression, neural networks and random fields. Their implementation in R software is described. Advantages and disadvantages for tests, which are implemented in more than one package, are discussed. Second topic of the thesis is additivity in nonlinear models. The definition is introduced as well as tests developed for testing its presence. Several test (both linearity and additivity) have been implemented in R for purposes of simulations. The last chapter deals with application of tests to real data. 1

Computational methods such as linear free energy relationships (LFERs) offer a useful high-throughput solution to quickly evaluate drug developability, e.g. membrane permeability, organic solvent/water partition coefficients, and solubility. LFERs typically assume the contribution of structural components/functional groups to the overall properties of a given molecule to be constant and independent. This dissertation describes a series of studies in which linear free energy relationships were developed to predict solvation of small organic molecules in lipid formulations, specifically, triglyceride containing solvents and phospholipid-based liposomes. The formation of intermolecular HBs in triglyceride solvents (homogenous with H-bond accepting ability) and intramolecular HBs within the bilayer barrier domain (hydrocarbon-like) proved to be the major factors to consider in developing LFERs to account for the increased oil/water partition coefficients and enhanced bilayer permeability of small organic molecules. The triglyceride solvent/water partition coefficients of a series of model compounds varying in polarity and H-bond donating/accepting capability were used to establish a correlation between the solvent descriptors and the ester concentration in these solvents using the Abraham LFER approach. The LFER analyses showed that the descriptors representing the polarizability and H-bond basicity of the solvents vary systematically with the ester concentration. A fragment-based LFER to predict membrane permeability or 1,9- decadiene/water partition coefficients of small organic molecules including small peptides was systematically constructed using a total of 47 compounds. Significant nonadditivity was observed in peptides in that the contribution of the peptide backbone amide to the apparent transfer free energy from water into the bilayer barrier domain is considerably smaller than that of a “well-isolated” amide and greatly affected by adjacent polar substituents on the C-termini. In order to explain the phenomenon of nonadditivity, the formation of intramolecular HBs and inductive effects of neighboring polar groups on backbone amide, were investigated using FTIR and MD simulations. Both spectroscopic and computational results provided supportive evidence for the hypothesis that the formation of intramolecular HBs in peptides is the main reason for the observed nonadditivity of Δ(ΔG°)-CONH-. The MD simulation results showed that the inductive effect of neighboring groups is not as important as the effect of intramolecular HBs.