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.

Statistical mechanics of protein complexed and condensed DNA

In this thesis I treat various biophysical questions arising in the context of complexed / ”protein-packed” DNA and DNA in confined geometries (like in viruses or toroidal DNA condensates). Using diverse theoretical methods I consider the statistical mechanics as well as the dynamics of DNA under these conditions. In the first part of the thesis (chapter 2) I derive for the first time the single molecule ”equation of state”, i.e. the force-extension relation of a looped DNA (Eq. 2.94) by using the path integral formalism. Generalizing these results I show that the presence of elastic substructures like loops or deflections caused by anchoring boundary conditions (e.g. at the AFM tip or the mica substrate) gives rise to a significant renormalization of the apparent persistence length as extracted from single molecule experiments (Eqs. 2.39 and 2.98). As I show the experimentally observed apparent persistence length reduction by a factor of 10 or more is naturally explained by this theory. In chapter 3 I theoretically consider the thermal motion of nucleosomes along a DNA template. After an extensive analysis of available experimental data and theoretical modelling of two possible mechanisms I conclude that the ”corkscrew-motion” mechanism most consistently explains this biologically important process. In chapter 4 I demonstrate that DNA-spools (architectures in which DNA circumferentially winds on a cylindrical surface, or onto itself) show a remarkable ”kinetic inertness” that protects them from tension-induced disruption on experimentally and biologically relevant timescales (cf. Fig. 4.1 and Eq. 4.18). I show that the underlying model establishes a connection between the seemingly unrelated and previously unexplained force peaks in single molecule nucleosome and DNA-toroid stretching experiments. Finally in chapter 5 I show that toroidally confined DNA (found in viruses, DNAcondensates or sperm chromatin) undergoes a transition to a twisted, highly entangled state provided that the aspect ratio of the underlying torus crosses a certain critical value (cf. Eq. 5.6 and the phase diagram in Fig. 5.4). The presented mechanism could rationalize several experimental mysteries, ranging from entangled and supercoiled toroids released from virus capsids to the unexpectedly short cholesteric pitch in the (toroidaly wound) sperm chromatin. I propose that the ”topological encapsulation” resulting from our model may have some practical implications for the gene-therapeutic DNA delivery process.

Radiochemical aspects of production and processing of radiometals for preparation of metalloradiopharmaceuticals

Radiometals play an important role in nuclear medicine as involved in diagnostic or therapeutic agents. In the present work the radiochemical aspects of production and processing of very promising radiometals of the third group of the periodic table, namely radiogallium and radiolanthanides are investigated. The 68Ge/68Ga generator (68Ge, T½ = 270.8 d) provides a cyclotron-independent source of positron-emitting 68Ga (T½ = 68 min), which can be used for coordinative labelling. However, for labelling of biomolecules via bifunctional chelators, particularly if legal aspects of production of radiopharmaceuticals are considered, 68Ga(III) as eluted initially needs to be pre-concentrated and purified. The first experimental chapter describes a system for simple and efficient handling of the 68Ge/68Ga generator eluates with a cation-exchange micro-chromatography column as the main component. Chemical purification and volume concentration of 68Ga(III) are carried out in hydrochloric acid – acetone media. Finally, generator produced 68Ga(III) is obtained with an excellent radiochemical and chemical purity in a minimised volume in a form applicable directly for the synthesis of 68Ga-labelled radiopharmaceuticals. For labelling with 68Ga(III), somatostatin analogue DOTA-octreotides (DOTATOC, DOTANOC) are used. 68Ga-DOTATOC and 68Ga-DOTANOC were successfully used to diagnose human somatostatin receptor-expressing tumours with PET/CT. Additionally, the proposed method was adapted for purification and medical utilisation of the cyclotron produced SPECT gallium radionuclide 67Ga(III). Second experimental chapter discusses a diagnostic radiolanthanide 140Nd, produced by irradiation of macro amounts of natural CeO2 and Pr2O3 in natCe(3He,xn)140Nd and 141Pr(p,2n)140Nd nuclear reactions, respectively. With this produced and processed 140Nd an efficient 140Nd/140Pr radionuclide generator system has been developed and evaluated. The principle of radiochemical separation of the mother and daughter radiolanthanides is based on physical-chemical transitions (hot-atom effects) of 140Pr following the electron capture process of 140Nd. The mother radionuclide 140Nd(III) is quantitatively absorbed on a solid phase matrix in the chemical form of 140Nd-DOTA-conjugated complexes, while daughter nuclide 140Pr is generated in an ionic species. With a very high elution yield and satisfactory chemical and radiolytical stability the system could able to provide the short-lived positron-emitting radiolanthanide 140Pr for PET investigations. In the third experimental chapter, analogously to physical-chemical transitions after the radioactive decay of 140Nd in 140Pr-DOTA, the rapture of the chemical bond between a radiolanthanide and the DOTA ligand, after the thermal neutron capture reaction (Szilard-Chalmers effect) was evaluated for production of the relevant radiolanthanides with high specific activity at TRIGA II Mainz nuclear reactor. The physical-chemical model was developed and first quantitative data are presented. As an example, 166Ho could be produced with a specific activity higher than its limiting value for TRIGA II Mainz, namely about 2 GBq/mg versus 0.9 GBq/mg. While free 166Ho(III) is produced in situ, it is not forming a 166Ho-DOTA complex and therefore can be separated from the inactive 165Ho-DOTA material. The analysis of the experimental data shows that radionuclides with half-life T½ < 64 h can be produced on TRIGA II Mainz nuclear reactor, with specific activity higher than any available at irradiation of simple targets e.g. oxides.