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Induced charge computation
(2009)
One of the main aspects of statistical mechanics is that the properties of a thermodynamics state point do not depend on the choice of the statistical ensemble. It breaks down for small systems e.g. single molecules. Hence, the choice of the statistical ensemble is crucial for the interpretation of single molecule experiments, where the outcome of measurements depends on which variables or control parameters, are held fixed and which ones are allowed to fluctuate. Following this principle, this thesis investigates the thermodynamics of a single polymer pulling experiments within two different statistical ensembles. The scaling of the conjugate chain ensembles, the fixed end-to-end vector (Helmholtz) and the fixed applied force (Gibbs), are studied in depth. This thesis further investigates the ensemble equivalence for different force regimes and polymer-chain contour lengths. Using coarse-grained molecular dynamic simulations, i.e. Langevin dynamics, the simulations were found to complement the theoretical predictions for the scaling of ensemble difference of Gaussian chains in different force-regimes, giving special attention to the zero force regime. After constructing Helmholtz and Gibbs conjugate ensembles for a Gaussian chain, two different data sets of thermodynamic states on the force-extension plane, i.e. force-extension curves, were generated. The ensemble difference is computed for different polymer-chain lengths by using force-extension curves. The scaling of the ensemble difference versus relative polymer-chain length under different force regimes has been derived from the simulation data and compared to theoretical predictions. The results demonstrate that the Gaussian chain in the zero force limit generates nonequivalent ensembles, regardless of its equilibrium bond length and polymer-chain contour length. Moreover, if polymers are charged in confinement, coarse-graining is problematic, owing to dielectric interfaces. Hence, the effect of dielectric interfaces must be taken into account when describing physical systems such as ionic channels or biopolymers inside nanopores. It is shown that the effect of dielectrics is crucial for the dynamics of a biopolymer or an ion inside a nanopore. In the simulations, the feasibility of an efficient and accurate computation of electrostatic interactions in the presence of an arbitrarily shaped dielectric domain is challenging. Several solutions for this problem have been previously proposed in the literature such as a density functional approach, or transforming problem at hand into an algebraic problem ( Induced Charge Computation (ICC) ) and boundary element methods. Even though the essential concept is the same, which is to replace the dielectric interface with a polarization charge density, these approaches have been analyzed and the ICC algorithm has been implemented. A new superior boundary element method has been devised utilizing the force computation via the Particle-Particle Particle-Mesh (P3M) method for periodic geometries (ICCP3M). This method has been compared to the ICC algorithm, the algebraic solutions, and to density functional approaches. Extensive numerical tests against analytically tractable geometries have confirmed the correctness and applicability of developed and implemented algorithms, demonstrating that the ICCP3M is the fastest and the most versatile algorithm. Further optimization issues are also discussed in obtaining accurate induced charge densities. The potential of mean force (PMF) of DNA modelled on a coarsed-grain level inside a nanopore is investigated with and without the inclusion of dielectric effects. Despite the simplicity of the model, the dramatic effect of dielectric inclusions is clearly seen in the observed force profile.