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Mathematical and numerical models for studying the electrophoresis of topologically nontrivial molecules in two and three dimensions are presented. The molecules are modeled as polygons residing on a square lattice and a cubic lattice whereas the electrophoretic media of obstacle network are simulated by removing vertices from the lattices at random. The dynamics of the polymeric molecules are modeled by configurational readjustments of segments of the polygons. Configurational readjustments arise from thermal fluctuations and they correspond to piecewise reptation in the simulations. A Metropolis algorithm is introduced to simulate these dynamics, and the algorithms are proven to be reversible and ergodic. Monte Carlo simulations of steady field random obstacle electrophoresis are performed and the results are presented.

We suggested a geometric approach to address the external field influence on the DNA molecules, described by the WLC model via geometric coupling. It consists in the introduction of the effective metrics depending on the potential of the external field, with further re-definition of the arc-length parameter and of the extrinsic curvatures of the DNA molecules. It yields the nontrivial impact of the external field in the internal energy of macromolecules. We give the Hamiltonian formulation of this model and perform its preliminary analysis in the redefinition of the initial energy density.

A hard-sphere model strictly guaranteeing the uncrossability of polymer chains is used to simulate the relaxation process of the linear and ring polymers. The statistical and dynamic properties of both linear and ring chains are obtained by using Monte Carlo algorithm. We find that the relaxation time of a self-avoiding ring polymer is smaller than the time required for the polymer to move a distance of order of its own size. Moreover, the stretching properties of linear and ring polymers are also studied, and the results show that the elongation curve of a ring polymer is almost identical to that of a linear polymer except under the situation of small external force.

Physics-based computational approaches to predicting the structure of macromolecules such as proteins are gaining increased use, but there are remaining challenges. In the current work, it is demonstrated that in energy-based prediction methods, the degree of optimization of the sampled structures can influence the prediction results. In particular, discrepancies in the degree of local sampling can bias the predictions in favor of the oversampled structures by shifting the local probability distributions of the minimum sampled energies. In simple systems, it is shown that the magnitude of the errors can be calculated from the energy surface, and for certain model systems, derived analytically. Further, it is shown that for energy wells whose forms differ only by a randomly assigned energy shift, the optimal accuracy of prediction is achieved when the sampling around each structure is equal. Energy correction terms can be used in cases of unequal sampling to reproduce the total probabilities that would occur under equal sampling, but optimal corrections only partially restore the prediction accuracy lost to unequal sampling. For multiwell systems, the determination of the correction terms is a multibody problem; it is shown that the involved cross-correlation multiple integrals can be reduced to simpler integrals. The possible implications of the current analysis for macromolecular structure prediction are discussed.

Density-function-theory calculations were performed to find the performance of a series of 2,2’-(1,2,4,5-tetrazine-3,6diyl) dihydrazinecarboxamide-based nitrogen-rich macrocyclic compounds as an energetic plasticizer. Reliable methods have been used to predict energetic properties such as gas-phase and solid-phase heat of formation (HOF), density, detonation velocity, detonation pressure, explosive power, heat of combustion, heat of detonation, specific impulse, flame temperature, brisance, and sensitivity. All the designed macrocycles possess a nitrogen content of over 48%. The designed compounds show positive HOFs and high predicted densities ranging from 1.81g/cm³ to 1.86g/cm³. The predicted properties were compared with GAP, polyGLYN and their monomers which establish the designed macrocycles of interest for further investigations concerning their suitability as plasticizers in energetic formulations.

The application of graph theory in structural biology offers an alternative means of studying 3D models of large macromolecules such as proteins. The radius of gyration, which scales with exponent $\sim 0.4$, provides quantitative information about the compactness of the protein structure. In this study, we combine two proven methods, the graph-theoretical and the fundamental scaling laws, to study 3D protein models. This study shows that the mean node degree (MND) of the protein graphs, which scales with exponent 0.038, is scale-invariant. In addition, proteins that differ in size have a highly similar node degree distribution. Linear regression analysis showed that the graph parameters (radius, diameter, and mean eccentricity) can explain up to 90% of the total radius of gyration variance. Thus, the graph parameters of radius, diameter, and mean eccentricity scale match along with the same exponent as the radius of gyration. The main advantage of graph eccentricity compared to the radius of gyration is that it can be used to analyze the distribution of the central and peripheral amino acids/nodes of the macromolecular structure.

The synthesis of a new self-assembled porphyrin macrostructure based on disulfide bonds, is presented. This constitutes a new way to directly connect porphyrins in macromolecular arrays, to complement the usual methods of intermolecular hydrogen bonds and metal coordination bonding.

The dynamics and efficiency of electronic excitation energy transfer (EET), migration and trapping are investigated in a series of novel macromolecular systems. These systems have been engineered specifically with the aim of controlling the efficiency of the energy migration though the structure of the macromolecule. Three systems are discussed. Firstly, time-resolved fluorescence measurements of poly(acenaphthylene) homopolymer, P(ACE), are compared with similar measurements on an acenaphthylene/maleic anhydride copolymer, P(ACE/MA), in which the MA groups act as spacers between the ACE chromophores. The rate of energy migration is shown to be extremely rapid in P(ACE) but restricted in the copolymer. Secondly, we illustrate the structural control now achievable through living free radical polymerization methods, which permit the design and synthesis of polymers with structures optimized for energy migration and trapping. Finally, we discuss the results of time-resolved fluorescence anisotropy measurements in the study of EET in several generations of porphyrin-based dendrimers.