Partial reconstruction of the trajectories of a discretely observed branching diffusion with immigration and an application to inference

In this treatise we consider finite systems of branching particles where the particles move independently of each other according to d-dimensional diffusions. Particles are killed at a position dependent rate, leaving at their death position a random number of descendants according to a position dependent reproduction law. In addition particles immigrate at constant rate (one immigrant per immigration time). A process with above properties is called a branching diffusion withimmigration (BDI). In the first part we present the model in detail and discuss the properties of the BDI under our basic assumptions. In the second part we consider the problem of reconstruction of the trajectory of a BDI from discrete observations. We observe positions of the particles at discrete times; in particular we assume that we have no information about the pedigree of the particles. A natural question arises if we want to apply statistical procedures on the discrete observations: How can we find couples of particle positions which belong to the same particle? We give an easy to implement 'reconstruction scheme' which allows us to redraw or 'reconstruct' parts of the trajectory of the BDI with high accuracy. Moreover asymptotically the whole path can be reconstructed. Further we present simulations which show that our partial reconstruction rule is tractable in practice. In the third part we study how the partial reconstruction rule fits into statistical applications. As an extensive example we present a nonparametric estimator for the diffusion coefficient of a BDI where the particles move according to one-dimensional diffusions. This estimator is based on the Nadaraya-Watson estimator for the diffusion coefficient of one-dimensional diffusions and it uses the partial reconstruction rule developed in the second part above. We are able to prove a rate of convergence of this estimator and finally we present simulations which show that the estimator works well even if we leave our set of assumptions.

Measurement of branching fractions and CP violation for charmless charged two-body B decays at LHCb

Charmless charged two-body B decays are sensitive probes of the CKM matrix, that parameterize CP violation in the Standard Model (SM), and have the potential to reveal the presence of New Physics. The framework of CP violation within the SM, the role of the CKM matrix, with its basic formalism, and the current experimental status are presented. The theoretical tools commonly used to deal with hadronic B decays and an overview of the phenomenology of charmless two-body B decays are outlined. LHCb is one of the four main experiments operating at the Large Hadron Collider (LHC), devoted to the measurement of CP violation and rare decays of charm and beauty hadrons. The LHCb detector is described, focusing on the technologies adopted for each sub-detector and summarizing their performances. The status-of-the-art of the LHCb measurements with charmless two-body B decays is then presented. Using the 37/pb of integrated luminosity collected at sqrt(s) = 7 TeV by LHCb during 2010, the direct CP asymmetries ACP(B0 -> Kpi) = −0.074 +/- 0.033 +/- 0.008 and ACP(Bs -> piK) = 0.15 +/- 0.19 +/- 0.02 are measured. Using 320/pb of integrated luminosity collected during 2011 these measurements are updated to ACP(B0 -> Kpi) = −0.088 +/- 0.011 +/- 0.008 and ACP(Bs -> piK) = 0.27 +/- 0.08 +/- 0.02. In addition, the branching ratios BR(B0 -> K+K-) = (0.13+0.06-0.05 +/- 0.07) x 10^-6 and BR(Bs -> pi+pi-) = (0.98+0.23-0.19 +/- 0.11) x 10^-6 are measured. Finally, using a sample of 370/pb of integrated luminosity collected during 2011, the relative branching ratios BR(B0 -> pi+pi-)/BR(B0 -> Kpi) = 0.262 +/- 0.009 +/- 0.017, (fs/fd)BR(Bs -> K+K-)/BR(B0 -> Kpi)=0.316 +/- 0.009 +/- 0.019, (fs/fd)BR(Bs -> piK)/BR(B0 -> Kpi) = 0.074 +/- 0.006 +/- 0.006 and BR(Lambda_b -> ppi)/BR(Lambda_b -> pK)=0.86 +/- 0.08 +/- 0.05 are determined.

The central role of Fyn kinase in axon-glial signalling and translation of myelin proteins

During central nervous system myelination, oligodendrocytes extend membrane processes towards an axonal contact site which is followed by ensheathment resulting in a compacted multilamellar myelin sheath. The formation of this axon-glial unit facilitates rapid saltatory propagation of action potentials along the axon and requires the synthesis and transport of copious amounts of lipids and proteins to the axon-glial contact site. Fyn is a member of the Src family of non receptor tyrosine kinases and inserted into the inner leaflet of the oligodendrocyte membrane by acylation. Fyn activity plays a pivotal role in the maturation of oligodendrocytes and the myelination process. It was suggested previously that Fyn kinase can be stimulated by binding of a neuronal ligand to oligodendroglial F3/ contactin, a glycosyl-phosphatidyl-inositol anchored immunoglobulin superfamily (IgSF) member protein. It could be shown here, that neuronal cell adhesion molecule L1 binds to oligodendrocytes in an F3-dependent manner and activates glial Fyn. In the search for downstream participants of this novel axon-glial signalling cascade, heterogeneous nuclear ribonucleoprotein (hnRNP) A2 was identified as a novel Fyn target in oligodendrocytes. HnRNP A2 was known to be involved in the localisation of translationally repressed myelin basic protein (MBP) mRNA by binding to a cis acting A2 response element (A2RE) present in the 3’ untranslated region. Transport of MBP mRNAs occurs in RNA-protein complexes termed RNA granules and translational repression during transport is achieved by hnRNP A2-mediated recruitment of hnRNP E1 to the granules. It could be shown here, that Fyn activity leads to enhanced translation of reporter mRNA containing a part of the 3’ UTR of MBP including the A2RE. Furthermore hnRNP E1 seems to dissociate from RNA granules in response to Fyn activity and L1 binding. These findings suggest a novel form of neuron- glial communication: Axonal L1 binding to oligodendroglial F3 activates Fyn kinase. Activated Fyn phosphorylates hnRNP A2 leading to removal of hnRNP E1 from RNA granules initiating the translation of MBP mRNA. MBP is the second most abundant myelin protein and mice lacking this protein show a severe hypomyelination phenotype. Moreover, the brains of Fyn knock out mice contain reduced MBP levels and are hypomyelinated. Hence, L1-mediated MBP synthesis via Fyn as a central molecule could be part of a regulatory mechanism required for myelinogenesis in the central nervous system.

Mutually catalytic branching at infinite rate

The purpose of this doctoral thesis is to prove existence for a mutually catalytic random walk with infinite branching rate on countably many sites. The process is defined as a weak limit of an approximating family of processes. An approximating process is constructed by adding jumps to a deterministic migration on an equidistant time grid. As law of jumps we need to choose the invariant probability measure of the mutually catalytic random walk with a finite branching rate in the recurrent regime. This model was introduced by Dawson and Perkins (1998) and this thesis relies heavily on their work. Due to the properties of this invariant distribution, which is in fact the exit distribution of planar Brownian motion from the first quadrant, it is possible to establish a martingale problem for the weak limit of any convergent sequence of approximating processes. We can prove a duality relation for the solution to the mentioned martingale problem, which goes back to Mytnik (1996) in the case of finite rate branching, and this duality gives rise to weak uniqueness for the solution to the martingale problem. Using standard arguments we can show that this solution is in fact a Feller process and it has the strong Markov property. For the case of only one site we prove that the model we have constructed is the limit of finite rate mutually catalytic branching processes as the branching rate approaches infinity. Therefore, it seems naturalto refer to the above model as an infinite rate branching process. However, a result for convergence on infinitely many sites remains open.

Comparative analysis of midline guidance and axon pathfinding in the chelicerates Achaearanea tepidariorum and Cupiennius salei

Die Neurogenese und axonale Wegfindung sind in den vergangenen Jahrzehnten Thema einer Vielzahl wissenschaftlicher Untersuchungen in den verschiedensten Organismen gewesen. Die zusammengetragenen Daten in Insekten und Crustaceen geben eine gute Übersicht darüber, wie das Nervensystem in Arthropoden aufgebaut wird. Die entwicklungsbiologischen Prozesse, die daran beteiligt sind, sind in den beiden genannten Gruppen sehr gut verstanden. In den Gruppen der Cheliceraten und Myriapoden jedoch wurden ähnliche Analysen bisher kaum durchgeführt. Das Hauptanliegen dieser Arbeit war es daher, Mechanismen in den Spinnen Achaearanea tepidariorum und Cupiennius salei, zwei Vertretern der Cheliceraten, zu untersuchen, die eine Rolle im Leitsystem der ventralen Mittellinie und bei der axonalen Wegfindung spielen. Eine Vorraussetzung hierfür sind Kenntnisse über die Architektur des Zentralnervensystems. In einem ersten Schritt beschrieb ich daher grundlegend die Morphologie des Nervensystems im Verlauf der gesamten Embryoalentwicklung. Ich konnte zeigen, dass in Spinnen ein für Arthropoden typisches Strickleiternervensystem gebildet wird. Dieses wird von segmental angelegten Neuronen geformt, wobei sowohl Gruppen von Zellen als auch einzelne Neurone daran beteiligt sind, die primären axonalen Trakte zu etablieren. Im Besonderen konnte ich eine Zelle identifizieren, die in Position, Projektionsmuster und der Expression des Markergens even-skipped vergleichbar zum PR2 Neuron in Drosophila ist, welches die posteriore Wurzel des Segmentalnervs anlegt.rnrnIn einem zweiten Ansatz untersuchte ich die ventrale Mittellinie in Spinnen im Bezug auf ihre mögliche Funktion in der axonalen Wegfindung. Es konnte gezeigt werden, dass es sich beim Epithel der Mittellinie, das die Lücke zwischen beiden Keimstreifhälften während des gesamten Prozesses der Inversion überspannt, um eine transiente Struktur handelt, die keine neuralen Zellen hervorbringt. Es ist daher vergleichbar mit der so genannten Floor plate in Vertebraten, die ebenfalls nur vorübergehend existiert. Die Untersuchung von single minded (sim) zeigte, dass es, anders als in Drosophila, wo sim ein wichtiges regulatorisches Gen für die korrekte Spezifizierung von Mittellinienzellen ist, nicht in den Zellen der Mittellinie, sondern in diesen benachbarten Zellen, exprimiert wird. Das ist vergleichbar mit Vertebraten. Zusätzlich konnte ich Expression von sim an den Basen der Gliedmassen und im Kopf nachweisen. Wie in Vertebraten könnte sim an der Musterbildung dieser Gewebe beteiligt sein. Dennoch spielt die Mittellinie in Spinnen eine wichtige Rolle als Organisator für auswachsende, kommissurale Axone. Diese Funktion teilt sie mit anderen Invertebraten und Vertebraten.rnrnDie Signaltransduktionskaskade, die an der axonalen Wegfindung an der Mittellinie beteiligt ist, ist in den verschiedensten Organismen hoch konserviert. In der vorliegenden Arbeit konnte ich sowohl in Achaearanea als auch in Cupiennius ein netrin Homolog identifizieren und eine konservierte Funktion des Wegfindungsmoleküls während der Bildung der Kommissuren aufzeigen. RNAi Experimente belegen, dass, wird die Funktion von netrin herunterreguliert, das Strickleiternervensystem nicht korrekt gebildet wird, ins Besondere die kommissuralen Faszikel. Des Weiteren konnte ich eine neue Funktion von netrin, die bisher in anderen Organsimen noch nicht beschrieben wurde, identifizieren. Neben seiner Rolle in der axonalen Wegfindung, scheint netrin auch an der epithelialen Morphogenese im zentralen Nervensystem beteiligt zu sein. In dieser Funktion scheint netrin in Gliazellen, die die epithelialen Vesikel der Invaginationsgruppen umhüllen, wichtig zu sein, um neurale Vorläuferzellen in einem undifferenzierten Zustand zu halten. Der Abbau von netrin Transkript durch RNA Interferenz führt zu einer verfrühten Segregation neuraler Vorläuferzellen aus dem epithelialen Verband der Invaginationsgruppen und zu einer Zunahme an Zellen, die den frühen Differenzierungsmarker islet exprimieren.

Endocytic trafficking and oligodendroglial exosome secretion in axon-glia communication and myelination

Oligodendrocytes form specialized plasma membrane extensions which spirally enwrap axons, thereby building up the myelin sheath. During myelination, oligodendrocytes produce large amounts of membrane components. Oligodendrocytes can be seen as a complex polarized cell type with two distinct membrane domains, the plasma membrane surrounding the cell body and the myelin membrane. SNARE proteins mediate the fusion of vesicular cargoes with their target membrane. We propose a model in which the major myelin protein PLP is transported by two different pathways. VAMP3 mediates the non-polarized transport of newly synthesized PLP via recycling endosomes to the plasma membrane, while transport of PLP from late endosomes/lysosomes to myelin is controlled by VAMP7. In the second part of the thesis, the role of exosome secretion in glia to axon signaling was studied. Further studies are required to clarify whether VAMP7 also controls exosome secretion. The thesis further focused on putative metabolic effects in the target neurons. Oligodendroglial exosomes showed no obvious influences on neuronal metabolic activity. Analysis of the phosphorylation levels of the neurofilament heavy subunit revealed a decrease in presence of oligodendrocytes, indicating effects of oligodendroglial exosomes on the neuronal cytoskeleton. Finally, candidates for kinases which are possibly activated upon influence of oligodendroglial exosomes and could influence neuronal survival were identified.

Ergodicity and regularity of invariant measure for branching Markov processes with immigration

In this thesis we consider systems of finitely many particles moving on paths given by a strong Markov process and undergoing branching and reproduction at random times. The branching rate of a particle, its number of offspring and their spatial distribution are allowed to depend on the particle's position and possibly on the configuration of coexisting particles. In addition there is immigration of new particles, with the rate of immigration and the distribution of immigrants possibly depending on the configuration of pre-existing particles as well. In the first two chapters of this work, we concentrate on the case that the joint motion of particles is governed by a diffusion with interacting components. The resulting process of particle configurations was studied by E. Löcherbach (2002, 2004) and is known as a branching diffusion with immigration (BDI). Chapter 1 contains a detailed introduction of the basic model assumptions, in particular an assumption of ergodicity which guarantees that the BDI process is positive Harris recurrent with finite invariant measure on the configuration space. This object and a closely related quantity, namely the invariant occupation measure on the single-particle space, are investigated in Chapter 2 where we study the problem of the existence of Lebesgue-densities with nice regularity properties. For example, it turns out that the existence of a continuous density for the invariant measure depends on the mechanism by which newborn particles are distributed in space, namely whether branching particles reproduce at their death position or their offspring are distributed according to an absolutely continuous transition kernel. In Chapter 3, we assume that the quantities defining the model depend only on the spatial position but not on the configuration of coexisting particles. In this framework (which was considered by Höpfner and Löcherbach (2005) in the special case that branching particles reproduce at their death position), the particle motions are independent, and we can allow for more general Markov processes instead of diffusions. The resulting configuration process is a branching Markov process in the sense introduced by Ikeda, Nagasawa and Watanabe (1968), complemented by an immigration mechanism. Generalizing results obtained by Höpfner and Löcherbach (2005), we give sufficient conditions for ergodicity in the sense of positive recurrence of the configuration process and finiteness of the invariant occupation measure in the case of general particle motions and offspring distributions.

Non-parametric estimation of the diffusion coefficient of a branching diffusion with immigration

Wir betrachten Systeme von endlich vielen Partikeln, wobei die Partikel sich unabhängig voneinander gemäß eindimensionaler Diffusionen [dX_t = b(X_t),dt + sigma(X_t),dW_t] bewegen. Die Partikel sterben mit positionsabhängigen Raten und hinterlassen eine zufällige Anzahl an Nachkommen, die sich gemäß eines Übergangskerns im Raum verteilen. Zudem immigrieren neue Partikel mit einer konstanten Rate. Ein Prozess mit diesen Eigenschaften wird Verzweigungsprozess mit Immigration genannt. Beobachten wir einen solchen Prozess zu diskreten Zeitpunkten, so ist zunächst nicht offensichtlich, welche diskret beobachteten Punkte zu welchem Pfad gehören. Daher entwickeln wir einen Algorithmus, um den zugrundeliegenden Pfad zu rekonstruieren. Mit Hilfe dieses Algorithmus konstruieren wir einen nichtparametrischen Schätzer für den quadrierten Diffusionskoeffizienten $sigma^2(cdot),$ wobei die Konstruktion im Wesentlichen auf dem Auffüllen eines klassischen Regressionsschemas beruht. Wir beweisen Konsistenz und einen zentralen Grenzwertsatz.

Estimation of axon counts in a rat model of glaucoma: comparison of fixed-pattern sampling with targeted sampling

Full axon counting of optic nerve cross-sections represents the most accurate method to quantify axonal damage, but such analysis is very labour intensive. Recently, a new method has been developed, termed targeted sampling, which combines the salient features of a grading scheme with axon counting. Preliminary findings revealed the method compared favourably with random sampling. The aim of the current study was to advance our understanding of the effect of sampling patterns on axon counts by comparing estimated axon counts from targeted sampling with those obtained from fixed-pattern sampling in a large collection of optic nerves with different severities of axonal injury.