optomotor-blind and the horizontal and vertical system cells of the drosophila optic lobes

The horizontal and vertical system neurons (HS and VS cells) are part of a conserved set of lobula plate giant neurons (LPGNs) in the optic lobes of the adult brain. Structure and physiology of these cells are well known, predominantly from studies in larger Dipteran flies. Our knowledge about the ontogeny of these cells is limited and stems predominantly from laser ablation studies in larvae of the house fly Musca domestica. These studies suggested that the HS and VS cells stem from a single precursor, which, at least in Musca, has not yet divided in the second larval instar. A regulatory mutation (In(1)omb[H31]) in the Drosophila gene optomotor-blind (omb) leads to the selective loss of the adult HS and VS cells. This mutation causes a transient reduction in omb expression in what appears to be the entire optic lobe anlage (OLA) late in embryogenesis. Here, I have reinitiated the laser approach with the goal of identifying the presumptive embryonic HS/VS precursor cell in Drosophila. The usefulness of the laser ablation approach which has not been applied, so far, to cells lying deep within the Drosophila embryo, was first tested on two well defined embryonic sensory structures, the olfactory antenno-maxillary complex (AMC) and the light-sensitive Bolwing´s organ (BO). In the case of the AMC, the efficiency of the ablation procedure was demonstrated with a behavioral assay. When both AMCs were ablated, the response to an attractive odour (n-butanol) was clearly reduced. Interestingly, the larvae were not completely unresponsive but had a delayed response kinetics, indicating the existence of a second odour system. BO will be a useful test system for the selectivity of laser ablation when used at higher spatial resolution. An omb-Gal4 enhancer trap line was used to visualize the embryonic OLA by GFP fluorescence. This fluorescence allowed to guide the laser beam to the relevant structure within the embryo. The success of the ablations was monitored in the adult brain via the enhancer trap insertion A122 which selectively visualizes the HS and VS cell bodies. Due to their tight clustering, individual cells could not be identified in the embryonic OLA by conventional fluorescence microscopy. Nonetheless, systematic ablation of subdomains of the OLA allowed to localize the presumptive HS/VS precursor to a small area within the OLA, encompassing around 10 cells. Future studies at higher resolution should be able to identify the precursor as (an) individual cell(s). Most known lethal omb alleles do not complement the HS/VS phenotype of the In(1)omb[H31] allele. This is the expected behaviour of null alleles. Two lethal omb alleles that had been isolated previously by non-complementation of the omb hypomorphic allele bifid, have been reported, however, to complement In(1)omb[H31]. This report was based on low resolution paraffin histology of adult heads. Four mutations from this mutagenesis were characterized here in more detail (l(1)omb[11], l(1)omb[12], l(1)omb[13], and l(1)omb[15]). Using A122 as marker for the adult HS and VS cells, I could show, that only l(1)omb[11] can partly complement the HS/VS cell phenotype of In(1)omb[H31]. In order to identify the molecular lesions in these mutants, the exons and exon/intron junctions were sequenced in PCR-amplified material from heterozygous flies. Only in two mutants could the molecular cause for loss of omb function be identified: in l(1)omb[13]), a missense mutation causes the exchange of a highly conserved residue within the DNA-binding T-domain; in l(1)omb[15]), a nonsense mutation causes a C-terminal truncation. In the other two mutants apparently regulatory regions or not yet identified alternative exons are affected. To see whether mutant OMB protein in the missense mutant l(1)omb[13] is affected in DNA binding, electrophoretic shift assays on wildtype and mutant T-domains were performed. They revealed that the mutant no longer is able to bind the consensus palindromic T-box element.

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.

Frobenius polynomials for Calabi-Yau equations

Sei $\pi:X\rightarrow S$ eine \"uber $\Z$ definierte Familie von Calabi-Yau Varietaten der Dimension drei. Es existiere ein unter dem Gauss-Manin Zusammenhang invarianter Untermodul $M\subset H^3_{DR}(X/S)$ von Rang vier, sodass der Picard-Fuchs Operator $P$ auf $M$ ein sogenannter {\em Calabi-Yau } Operator von Ordnung vier ist. Sei $k$ ein endlicher K\"orper der Charaktetristik $p$, und sei $\pi_0:X_0\rightarrow S_0$ die Reduktion von $\pi$ \uber $k$. F\ur die gew\ohnlichen (ordinary) Fasern $X_{t_0}$ der Familie leiten wir eine explizite Formel zur Berechnung des charakteristischen Polynoms des Frobeniusendomorphismus, des {\em Frobeniuspolynoms}, auf dem korrespondierenden Untermodul $M_{cris}\subset H^3_{cris}(X_{t_0})$ her. Sei nun $f_0(z)$ die Potenzreihenl\osung der Differentialgleichung $Pf=0$ in einer Umgebung der Null. Da eine reziproke Nullstelle des Frobeniuspolynoms in einem Teichm\uller-Punkt $t$ durch $f_0(z)/f_0(z^p)|_{z=t}$ gegeben ist, ist ein entscheidender Schritt in der Berechnung des Frobeniuspolynoms die Konstruktion einer $p-$adischen analytischen Fortsetzung des Quotienten $f_0(z)/f_0(z^p)$ auf den Rand des $p-$adischen Einheitskreises. Kann man die Koeffizienten von $f_0$ mithilfe der konstanten Terme in den Potenzen eines Laurent-Polynoms, dessen Newton-Polyeder den Ursprung als einzigen inneren Gitterpunkt enth\alt, ausdr\ucken,so beweisen wir gewisse Kongruenz-Eigenschaften unter den Koeffizienten von $f_0$. Diese sind entscheidend bei der Konstruktion der analytischen Fortsetzung. Enth\alt die Faser $X_{t_0}$ einen gew\ohnlichen Doppelpunkt, so erwarten wir im Grenz\ubergang, dass das Frobeniuspolynom in zwei Faktoren von Grad eins und einen Faktor von Grad zwei zerf\allt. Der Faktor von Grad zwei ist dabei durch einen Koeffizienten $a_p$ eindeutig bestimmt. Durchl\auft nun $p$ die Menge aller Primzahlen, so erwarten wir aufgrund des Modularit\atssatzes, dass es eine Modulform von Gewicht vier gibt, deren Koeffizienten durch die Koeffizienten $a_p$ gegeben sind. Diese Erwartung hat sich durch unsere umfangreichen Rechnungen best\atigt. Dar\uberhinaus leiten wir weitere Formeln zur Bestimmung des Frobeniuspolynoms her, in welchen auch die nicht-holomorphen L\osungen der Gleichung $Pf=0$ in einer Umgebung der Null eine Rolle spielen.