Enclosure and non-existence theorems in geometric measure theory

Ein stetiges, lineares Funktional auf der Menge der n-dimensionalen Differentialformen mit kompaktem Träger in einer offenen Menge des R^{n+k} heißt n-dimensionaler Strom mit Kodimension k. Wir betrachten n-dimensionale rektifizierbare Ströme mit kompaktem Träger, welche durch Integration über eine rektifizierbare Menge M entstehen. Diese Objekte erweitern die klassische Differentialgeometrie auf Flächen, welche nicht notwendigerweise glatt sind, durch Verallgemeinerung der Konzepte in einem maßtheoretischen Weg. In dem ersten Teil dieser Arbeit beweisen wir Einschließungssätze für Area-stationäre Ströme, Ströme mit mittlerem Krümmungsvektor sowie für Area-stationäre Ströme in einem Gravitationsfeld und in Untermannigfaltigkeiten. Falls alle Randkomponenten in einer verallgemeinerten nichtkonvexen Menge liegen, dann ist bereits der gesamte Träger des Stroms in dieser Menge enthalten. Dazu setzen wir geeignete Barrierevektorfelder in die jeweiligen ersten Variationsformeln ein. Weiterhin zeigen wir, dass die Träger der Ströme in diesen verschiedenen Situationen nicht durch die Spitze eines Doppelkegels verlaufen können. Das führt zu Nichtexistenzsätzen für zusammenhängende, verallgemeinerte Flächen unter gewissen Bedingungen. In dem Fall der Area-stationären Ströme mit Kodimension Eins vergrößern wir den Öffnungswinkel des Kegels und geben den größtmöglichen Kegel mit der Nichtexistenz-Eigenschaft an. In dem zweiten Abschnitt betrachten wir Krümmungsflüsse für Flächen ohne Ränder, sodass wir mit rektifizierbaren Varifaltigkeiten arbeiten. Wir beweisen ein Einschließungsresultat für den Brakke-Fluss in eine zeitabhängige nichtkonvexe Menge und analysieren die Entwicklung von Singularitäten. Wir enden mit der Definition eines schwachen mittleren Krümmungsflusses in einem Gravitationsfeld. Zusätzlich zu den Beweisen der grundlegenden Eigenschaften konzentrieren wir uns auf Vergleichs- und Einschließungssätze.A continuous linear functional on the set of n-dimensional differential forms with compact support in an open set of R^{n+k} is called an n-dimensional current with codimension k. We consider n-dimensional rectifiable currents with compact support which are obtained by integration over a rectifiable set M. These objects extend the classical differential geometry to surfaces that are not necessarily smooth by generalizing the concepts in a measure theoretic way. In the first part of this thesis we prove enclosure theorems for area stationary currents, currents with mean curvature vector and for area stationary currents in a gravity field as well as in submanifolds. If all boundary components are in a generalized nonconvex set then the whole support of the current is contained in this set. To see this we plug appropriate barrier vector fields into the first variation formulas. Furthermore we show that the supports of currents in all these different situations cannot pass through the vertex of a doublecone. This leads to non-existence theorems for connected generalized surfaces under certain conditions. In case of area stationary currents of codimension one we are able to enlarge the angle of the cone and state the largest cone with the non-existence property. In the second part we consider curvature flows of surfaces without boundaries, thus we deal with rectifiable varifolds. We prove an enclosure result for the Brakke-flow in a time dependent nonconvex set and we analyze the development of singularities. We end with the definition of a weak mean curvature flow in a gravity field. In addition to the proofs of basic properties we focus on compare and enclosure theorems

Rüdiger Klein: Reading the guideposts for axon guidance

Klein uses genetically modified mice to study how axons find the right targets to innervate

Multiple EphB receptor tyrosine kinases shape dendritic spines in the hippocampus

Here, using a genetic approach, we dissect the roles of EphB receptor tyrosine kinases in dendritic spine development. Analysis of EphB1, EphB2, and EphB3 double and triple mutant mice lacking these receptors in different combinations indicates that all three, although to varying degrees, are involved in dendritic spine morphogenesis and synapse formation in the hippocampus. Hippocampal neurons lacking EphB expression fail to form dendritic spines in vitro and they develop abnormal spines in vivo. Defective spine formation in the mutants is associated with a drastic reduction in excitatory glutamatergic synapses and the clustering of NMDA and AMPA receptors. We show further that a kinase-defective, truncating mutation in EphB2 also results in abnormal spine development and that ephrin-B2–mediated activation of the EphB receptors accelerates dendritic spine development. These results indicate EphB receptor cell autonomous forward signaling is responsible for dendritic spine formation and synaptic maturation in hippocampal neurons

Topographically specific effects of ELF-1 on retinal axon guidance in vitro and retinal axon mapping in vivo

Peer reviewedPublisher PD

Ephrin-B2 reverse signaling is required for axon pathfinding and cardiac valve formation but not early vascular development

AbstractVascular development begins with the formation of a primary vascular plexus that is rapidly remodeled by angiogenesis into the interconnected branched patterns characteristic of mature vasculature. Several receptor tyrosine kinases and their ligands have been implicated to control early development of the vascular system. These include the vascular endothelial growth factor receptors (VEGFR-1 and VEGFR-2) that bind VEGF, the Tie-1 and Tie-2 receptors that bind the angiopoietins, and the EphB4 receptor that binds the membrane-anchored ligand ephrin-B2. Targeted mutations in the mouse germline have revealed essential functions for these molecules in vascular development. In particular, protein-null mutations that delete either EphB4 or ephrin-B2 from the mouse have been shown to result in early embryonic lethality due to failed angiogenic remodeling. The venous expression of EphB4 and arterial expression of ephrin-B2 has lead to the speculation that the interaction of these two molecules leads to bidirectional signaling into both the receptor-expressing cell and the ligand-expressing cell, and that both forward and reverse signals are required for proper development of blood vessels in the embryo. Indeed, targeted removal of the ephrin-B2 carboxy-terminal cytoplasmic tail by another group was shown to perturb vascular development and result in the same early embryonic lethality as the null mutation, leading the authors to propose that ephrin-B2 reverse signaling directs early angiogenic remodeling of the primary vascular plexus [Cell 104 (2001) 57]. However, we show here that the carboxy-terminal cytoplasmic domain of ephrin-B2, and hence reverse signaling, is not required during early vascular development, but it is necessary for neonatal survival and functions later in cardiovascular development in the maturation of cardiac valve leaflets. We further show that ephrin-B2 reverse signaling is required for the pathfinding of axons that form the posterior tract of the anterior commissure. Our results thus indicate that ephrin-B2 functions in the early embryo as a typical instructive ligand to stimulate EphB4 receptor forward signaling during angiogenic remodeling and that later in embryonic development ephrin-B2 functions as a receptor to transduce reverse signals involved in cardiac valve maturation and axon pathfinding

An In Vivo Mouse Model of Long-Term Potentiation at Synapses between Primary Afferent C-Fibers and Spinal Dorsal Horn Neurons: Essential Role of Ephb1 Receptor

Abstract Background Long-term potentiation (LTP), a much studied cellular model of synaptic plasticity, has not been demonstrated at synapses between primary afferent C-fibers and spinal dorsal horn (DH) neurons in mice in vivo. EphrinB-EphB receptor signaling plays important roles in synaptic connection and plasticity in the nervous system, but its role in spinal synaptic plasticity remains unclear. Results This study characterizes properties of LTP at synapses of C-fibers onto neurons in the superficial DH following high-frequency stimulation (HFS) of a peripheral nerve at an intensity that activates C-fibers and examines associated activation of Ca2+/calmodulin-activated protein kinase II (p-CaMKII), extracellular signal-regulated kinase (p-ERK) and the cyclic AMP response element binding protein (p-CREB) and expression of c-Fos, and it investigates further roles for the EphB1 receptor in LTP. HFS induced LTP within 5 min and lasts for 3–8 h during the period of recording and resulted in upregulation of p-CaMKII, p-ERK and p-CREB protein levels in the spinal cord and expression of c-Fos in DH. Intrathecal pretreatment of MK-801 or EphB2-Fc prevented LTP and significantly reduced upregulation of p-CaMKII, p-ERK, p-CREB and c-Fos. Further, targeted mutation of EphB1 receptor prevented induction of LTP and associated increases in phosphorylation of CaMKII, ERK, and CREB. Conclusion This study provides an in vivo mouse model of LTP at synapses of C-fibers onto the superficial DH neurons that will be valuable for studying the DH neuron excitability and their synaptic plasticity and hyperalgesia. It further takes advantage of examining functional implications of a specific gene targeted mice and demonstrates that the EphB1 receptor is essential for development of LTP.</p

Mutational analyses of UPIIIA, SHH, EFNB2, and HNF1β in persistent cloaca and associated kidney malformations

OBJECTIVES: ‘Persistent cloaca’ is a severe malformation affecting females in which the urinary, genital and alimentary tracts share a single conduit. Previously, a Uroplakin IIIA (UPIIIA) mutation was reported in one individual with persistent cloaca, and UPIIIA, Sonic Hedgehog (SHH), Ephrin B2 (EFNB2) and Hepatocyte Nuclear Factor 1β (HNF1β) are expressed during the normal development of organs that are affected in this condition. HNF1β mutations have been associated with uterine malformations in humans, and mutations of genes homologous to human SHH or EFNB2 cause persistent cloaca in mice. PATIENTS AND METHODS: We sought mutations of coding regions of UPIIIA, SHH, EFNB2 and HNF1β genes by direct sequencing in a group of 20 patients with persistent cloaca. Most had associated malformations of the upper renal tract and over half had impaired renal excretory function. The majority of patients had congenital anomalies outside the renal/genital tracts and two had the VACTERL association. RESULTS: Apart from a previously described index case, we failed to find UPIIIA mutations, and no patient had a SHH, EFNB2 or HNF1β mutation. CONCLUSION: Persistent cloaca is only rarely associated with UPIIIA mutation. Despite the fact that SHH and EFNB2 are appealing candidate genes, based on their expression patterns and mutant mice phenotypes, they were not mutated in these humans with persistent cloaca. Although HNF1β mutations can perturb paramesonephric duct fusion in humans, HNF1β was not mutated in persistent cloaca

CHMP2B mutants linked to frontotemporal dementia impair maturation of dendritic spines.: CHMP2B and dendritic spines

International audienceThe highly conserved ESCRT-III complex is responsible for deformation and cleavage of membranes during endosomal trafficking and other cellular activities. In humans, dominant mutations in the ESCRT-III subunit CHMP2B cause frontotemporal dementia (FTD). The decade-long process leading to this cortical degeneration is not well understood. One possibility is that, akin to other neurodegenerative diseases, the pathogenic protein affects the integrity of dendritic spines and synapses before any neuronal death. Using confocal microscopy and 3D reconstruction, we examined whether expressing the FTD-linked mutants CHMP2B(intron5) and CHMP2B(Delta10) in cultured hippocampal neurons modified the number or structure of spines. Both mutants induced a significant decrease in the proportion of large spines with mushroom morphology, without overt degeneration. Furthermore, CHMP2B(Delta10) induced a drop in frequency and amplitude of spontaneous excitatory postsynaptic currents, suggesting that the more potent synapses were lost. These effects seemed unrelated to changes in autophagy. Depletion of endogenous CHMP2B by RNAi resulted in morphological changes similar to those induced by mutant CHMP2B, consistent with dominant-negative activity of pathogenic mutants. Thus, CHMP2B is required for spine growth. Taken together, these results demonstrate that a mutant ESCRT-III subunit linked to a human neurodegenerative disease can disrupt the normal pattern of spine development

Dissociation of EphB2 Signaling Pathways Mediating Progenitor Cell Proliferation and Tumor Suppression

SummarySignaling proteins driving the proliferation of stem and progenitor cells are often encoded by proto-oncogenes. EphB receptors represent a rare exception; they promote cell proliferation in the intestinal epithelium and function as tumor suppressors by controlling cell migration and inhibiting invasive growth. We show that cell migration and proliferation are controlled independently by the receptor EphB2. EphB2 regulated cell positioning is kinase-independent and mediated via phosphatidylinositol 3-kinase, whereas EphB2 tyrosine kinase activity regulates cell proliferation through an Abl-cyclin D1 pathway. Cyclin D1 regulation becomes uncoupled from EphB signaling during the progression from adenoma to colon carcinoma in humans, allowing continued proliferation with invasive growth. The dissociation of EphB2 signaling pathways enables the selective inhibition of the mitogenic effect without affecting the tumor suppressor function and identifies a pharmacological strategy to suppress adenoma growth