Advanced techniques for the computer simulation and analysis of biomolecular systems

The Helmholtz free energy is one of the central quantities of classical thermodynamics, as it governs important chemical properties such as equilibria or reaction kinetics. It is, therefore, a desirable quantity to measure, predict, and understand. Unsurprisingly, many methods exist to compute free energy differences between two states of a system. In this thesis, the density of states integration method (DSI) is developed; it detects which subsystems mainly contribute to the free energy difference. The method utilizes the velocity density of states function (VDoS) of each atom to calculate its contribution to the vibrational free energy. It is possible without any approximation to assign fractions of the vibrational free energy to meaningful subsystems, where the local free energy difference is the sum over all atoms comprising that subsystem. In this way, large local changes can be identified (free energy hot-spots), which is crucial for the understanding of free energy differences. The validity and usefulness of DSI is shown via several examples and comparison with state of the art free energy methods. In addition to the development of DSI, this thesis also focuses on free energy barriers in the context of investigating the reaction mechanism of Sirtuin~5, a lysine deacylase class~III. The relationship between the configuration of the enzyme's active site and the height of the reaction barrier is studied by computing minimal energy paths for the catalyzed reaction starting from many different (educt) configurations. Using the power of machine learning, atom-atom distances influencing the activation barrier are identified, allowing for a comprehensive understanding of the interplay of the substrate and residues within the active site of Sirtuin~5. Subsequently, we set out to compute the free energy as a function of the reaction coordinate instead of a minimum energy path. Another theme of this thesis is the computation of spectroscopic observables in a cost effective manner while simultaneously including important features of the experimental setup. The inclusion of solvent molecules and finite temperature effects has a decisive effect on the accuracy of the computed observables. In this context, we highlight the importance of sampling atomic configurations (with and without explicit solvent) and the non-negligible influence of electron correlation on the accuracy of computed observables. Simulation protocols are developed that enable sampling, the inclusion of correlation methods, and large quantum mechanical subsystems at a low computational cost

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described

Protein Structure

Since the dawn of recorded history, and probably even before, men and women have been grasping at the mechanisms by which they themselves exist. Only relatively recently, did this grasp yield anything of substance, and only within the last several decades did the proteins play a pivotal role in this existence. In this expose on the topic of protein structure some of the current issues in this scientific field are discussed. The aim is that a non-expert can gain some appreciation for the intricacies involved, and in the current state of affairs. The expert meanwhile, we hope, can gain a deeper understanding of the topic

SIMULATING SEISMIC WAVE PROPAGATION IN TWO-DIMENSIONAL MEDIA USING DISCONTINUOUS SPECTRAL ELEMENT METHODS

We introduce a discontinuous spectral element method for simulating seismic wave in 2- dimensional elastic media. The methods combine the flexibility of a discontinuous finite element method with the accuracy of a spectral method. The elastodynamic equations are discretized using high-degree of Lagrange interpolants and integration over an element is accomplished based upon the Gauss-Lobatto-Legendre integration rule. This combination of discretization and integration results in a diagonal mass matrix and the use of discontinuous finite element method makes the calculation can be done locally in each element. Thus, the algorithm is simplified drastically. We validated the results of one-dimensional problem by comparing them with finite-difference time-domain method and exact solution. The comparisons show excellent agreement