Novel simulation methods for Coulomb and hydrodynamic interactions

This thesis presents new methods to simulate systems with hydrodynamic and electrostatic interactions. Part 1 is devoted to computer simulations of Brownian particles with hydrodynamic interactions. The main influence of the solvent on the dynamics of Brownian particles is that it mediates hydrodynamic interactions. In the method, this is simulated by numerical solution of the Navier--Stokes equation on a lattice. To this end, the Lattice--Boltzmann method is used, namely its D3Q19 version. This model is capable to simulate compressible flow. It gives us the advantage to treat dense systems, in particular away from thermal equilibrium. The Lattice--Boltzmann equation is coupled to the particles via a friction force. In addition to this force, acting on {it point} particles, we construct another coupling force, which comes from the pressure tensor. The coupling is purely local, i.~e. the algorithm scales linearly with the total number of particles. In order to be able to map the physical properties of the Lattice--Boltzmann fluid onto a Molecular Dynamics (MD) fluid, the case of an almost incompressible flow is considered. The Fluctuation--Dissipation theorem for the hybrid coupling is analyzed, and a geometric interpretation of the friction coefficient in terms of a Stokes radius is given. Part 2 is devoted to the simulation of charged particles. We present a novel method for obtaining Coulomb interactions as the potential of mean force between charges which are dynamically coupled to a local electromagnetic field. This algorithm scales linearly, too. We focus on the Molecular Dynamics version of the method and show that it is intimately related to the Car--Parrinello approach, while being equivalent to solving Maxwell's equations with freely adjustable speed of light. The Lagrangian formulation of the coupled particles--fields system is derived. The quasi--Hamiltonian dynamics of the system is studied in great detail. For implementation on the computer, the equations of motion are discretized with respect to both space and time. The discretization of the electromagnetic fields on a lattice, as well as the interpolation of the particle charges on the lattice is given. The algorithm is as local as possible: Only nearest neighbors sites of the lattice are interacting with a charged particle. Unphysical self--energies arise as a result of the lattice interpolation of charges, and are corrected by a subtraction scheme based on the exact lattice Green's function. The method allows easy parallelization using standard domain decomposition. Some benchmarking results of the algorithm are presented and discussed.

Mathematical aspects of Feynman integrals

In the present dissertation we consider Feynman integrals in the framework of dimensional regularization. As all such integrals can be expressed in terms of scalar integrals, we focus on this latter kind of integrals in their Feynman parametric representation and study their mathematical properties, partially applying graph theory, algebraic geometry and number theory. The three main topics are the graph theoretic properties of the Symanzik polynomials, the termination of the sector decomposition algorithm of Binoth and Heinrich and the arithmetic nature of the Laurent coefficients of Feynman integrals.rnrnThe integrand of an arbitrary dimensionally regularised, scalar Feynman integral can be expressed in terms of the two well-known Symanzik polynomials. We give a detailed review on the graph theoretic properties of these polynomials. Due to the matrix-tree-theorem the first of these polynomials can be constructed from the determinant of a minor of the generic Laplacian matrix of a graph. By use of a generalization of this theorem, the all-minors-matrix-tree theorem, we derive a new relation which furthermore relates the second Symanzik polynomial to the Laplacian matrix of a graph.rnrnStarting from the Feynman parametric parameterization, the sector decomposition algorithm of Binoth and Heinrich serves for the numerical evaluation of the Laurent coefficients of an arbitrary Feynman integral in the Euclidean momentum region. This widely used algorithm contains an iterated step, consisting of an appropriate decomposition of the domain of integration and the deformation of the resulting pieces. This procedure leads to a disentanglement of the overlapping singularities of the integral. By giving a counter-example we exhibit the problem, that this iterative step of the algorithm does not terminate for every possible case. We solve this problem by presenting an appropriate extension of the algorithm, which is guaranteed to terminate. This is achieved by mapping the iterative step to an abstract combinatorial problem, known as Hironaka's polyhedra game. We present a publicly available implementation of the improved algorithm. Furthermore we explain the relationship of the sector decomposition method with the resolution of singularities of a variety, given by a sequence of blow-ups, in algebraic geometry.rnrnMotivated by the connection between Feynman integrals and topics of algebraic geometry we consider the set of periods as defined by Kontsevich and Zagier. This special set of numbers contains the set of multiple zeta values and certain values of polylogarithms, which in turn are known to be present in results for Laurent coefficients of certain dimensionally regularized Feynman integrals. By use of the extended sector decomposition algorithm we prove a theorem which implies, that the Laurent coefficients of an arbitrary Feynman integral are periods if the masses and kinematical invariants take values in the Euclidean momentum region. The statement is formulated for an even more general class of integrals, allowing for an arbitrary number of polynomials in the integrand.

Die Synthese von Glycopeptiden und Glycopeptid-Protein-Konjugaten mit einer Partialstruktur des tumorassoziierten Mucins MUC1 zur Entwicklung von Tumorvakzinen

„Synthese von Glycopeptiden und Glycopeptid-Protein-Konjugaten mit einer Partialstruktur des tumorassoziierten Mucins MUC1 zur Entwicklung von Tumorvakzinen“ Das Glycoprotein MUC1 ist in Tumorepithelzellen sonderlich stark überexprimiert und wegen der vorzeitig einsetzenden Sialylierung sind die Saccharid-Epitope der O-Glycanketten stark verkürzt (sog. tumorassoziierte Antigene). Dadurch werden auch bisher verborgene Peptidepitope des Glycoprotein-Rückgrates auf der Zelloberfläche der Epithelzellen zugänglich, die als fremd von den Zellen des Immunsystems erkannt werden können. Dies macht das MUC1-Zelloberfächenmolekül zu einem Zielmolekül in der Entwicklung von Tumorvakzinen. Diese beiden strukturellen Besonderheiten wurden in der Synthese von Glycohexadecapeptiden verbunden, indem die veränderten tumorassoziierten Saccharidstrukturen TN-, STN- und T-Antigen als Glycosylaminosäure-Festphasenbausteine synthetisiert wurden und in das Peptidepitop der Wiederholungseinheit des MUC1 durch Glycopeptid-Festphasensynthese eingebaut wurden. Wegen der inhärenten schwachen Immunogenität der kurzen Glycopeptide müssen die synthetisierten Glycopeptidstrukturen an ein Trägerprotein, welches das Immunsystem stimuliert, gebunden werden. Zur Anbindung der Glycopeptide ist ein selektives Kupplungsverfahren nötig, um definierte und strukturell einheitliche Glycopeptid-Protein-Konjugate zu erhalten. Es konnte eine neue Methode entwickelt werden, bei der die Konjugation durch eine radikalische Additionsreaktion von als Allylamide funktionalisierten Glycopeptiden an ein Thiol-modifiziertes Trägerprotein erfolgte. Dazu wurde anhand von synthetisierten, als Allylamide modifizierten Modellaminosäuren untersucht, ob diese Reaktion generell für eine Biokonjugation geeignet ist und etwaige Nebenreaktionen auftreten können. Mit dieser Methode konnten verschiedene MUC1-Glycopeptid-Trägerprotein-Konjugate hergestellt werden, deren immunologische Untersuchung noch bevorsteht. Das tumorassoziierte MUC1 nimmt in der immundominanten Region seiner Wiederholungseinheit eine knaufartige Struktur ein. Für die Entwicklung von selektiven Tumorvakzinen ist es von großer Bedeutung möglichst genau die Struktur der veränderten Zelloberflächenmoleküle nachzubilden. Durch die Synthese von cyclischen (Glyco)Peptiden wurde dieses Strukturelement fixiert. Dazu wurden olefinische Aminosäure Festphasenbausteine hergestellt, die zusammen mit den oben genannten Glycosylaminosäuren mittels einer Glycopeptid-Festphasensynthese in acyclische Glycopeptide eingebaut wurden. Diese wurden dann durch Ringschlussmetathese zyklisiert und im Anschluss reduziert und vollständig deblockiert. In einem dritten Projekt wurde der Syntheseweg zur Herstellung einer C-Glycosylaminosäure mit einer N-Acetylgalactosamin-Einheit entwickelt. Wichtige Schritte bei der von Glucosamin ausgehenden Synthese sind die Keck-Allylierung, eine Epimerisierung, die Herstellung eines Brom-Dehydroalanin-Derivates und eine B-Alkyl-Suzuki-Miyaura-Kreuzkupplung sowie Schutzgruppenoperationen. Der racemische Baustein konnte dann in der Peptid-Festphasensynthese eines komplexen MUC1-Tetanustoxin-Konjugates eingesetzt werden.

A mesoscopic simulation method for electrolyte solutions

This thesis deals with the development of a novel simulation technique for macromolecules in electrolyte solutions, with the aim of a performance improvement over current molecular-dynamics based simulation methods. In solutions containing charged macromolecules and salt ions, it is the complex interplay of electrostatic interactions and hydrodynamics that determines the equilibrium and non-equilibrium behavior. However, the treatment of the solvent and dissolved ions makes up the major part of the computational effort. Thus an efficient modeling of both components is essential for the performance of a method. With the novel method we approach the solvent in a coarse-grained fashion and replace the explicit-ion description by a dynamic mean-field treatment. Hence we combine particle- and field-based descriptions in a hybrid method and thereby effectively solve the electrokinetic equations. The developed algorithm is tested extensively in terms of accuracy and performance, and suitable parameter sets are determined. As a first application we study charged polymer solutions (polyelectrolytes) in shear flow with focus on their viscoelastic properties. Here we also include semidilute solutions, which are computationally demanding. Secondly we study the electro-osmotic flow on superhydrophobic surfaces, where we perform a detailed comparison to theoretical predictions.