During Drosophila embryogenesis, an extra-embryonic tissue, known as the amnioserosa (AS), is required for the morphogenetic processes of germ band retraction (GBR) and dorsal closure (DC). Being extra-embryonic, the AS is not part of the embryo proper but is eliminated via programmed cell death (PCD) in the late stages of embryogenesis. Programmed elimination of the AS during normal development can be prevented by directly inhibiting apoptosis, either through the deletion of the pro-apoptotic genes hid, grim and reaper, or through the expression of the pan-caspase inhibitor P-35. PCD in the AS can also be prevented by indirect inhibition of apoptosis via inactivation of autophagy, either through activation of the InR/PI3K pathway, or through activation of the Ras signalling pathway. The timing of AS elimination is critical to development as mutants associated with premature AS loss fail in GBR. To better characterize this premature AS death, a detailed phenotypic analysis of the AS behaviour in the GBR mutant hindsight (hnt) was performed. Direct inactivation of apoptosis failed to rescue the GBR defects in hnt mutant, though the premature AS death was completely rescued. Inactivation of autophagy, however, rescued AS cell behaviour and contacts during GBR, with partial rescue of the GBR defects in the hnt mutant. The nature of premature AS loss is characterized as a possible model for anoikis, a form of cell death that is triggered through reduced cell-cell or cell- matrix contact.

During Drosophila embryogenesis, an extra-embryonic tissue, known as the amnioserosa (AS), is required for the morphogenetic processes of germ band retraction (GBR) and dorsal closure (DC). Being extra-embryonic, the AS is not part of the embryo proper but is eliminated via programmed cell death (PCD) in the late stages of embryogenesis. Programmed elimination of the AS during normal development can be prevented by directly inhibiting apoptosis, either through the deletion of the pro-apoptotic genes hid, grim and reaper, or through the expression of the pan-caspase inhibitor P-35. PCD in the AS can also be prevented by indirect inhibition of apoptosis via inactivation of autophagy, either through activation of the InR/PI3K pathway, or through activation of the Ras signalling pathway. The timing of AS elimination is critical to development as mutants associated with premature AS loss fail in GBR. To better characterize this premature AS death, a detailed phenotypic analysis of the AS behaviour in the GBR mutant hindsight (hnt) was performed. Direct inactivation of apoptosis failed to rescue the GBR defects in hnt mutant, though the premature AS death was completely rescued. Inactivation of autophagy, however, rescued AS cell behaviour and contacts during GBR, with partial rescue of the GBR defects in the hnt mutant. The nature of premature AS loss is characterized as a possible model for anoikis, a form of cell death that is triggered through reduced cell-cell or cell- matrix contact.

The amnioserosa (AS) is an epithelium that plays major roles in two crucial morphogenetic processes during Drosophila embryogenesis: Germ Band Retraction (GBR) and Dorsal Closure (DC). The AS is extraembryonic and as such, it does not contribute to the mature embryo but is eliminated during development by programmed cell death. In this thesis, a comprehensive investigation of the timing and characteristics of the AS death and degeneration is performed. It is demonstrated that AS elimination occurs in two phases: “cell extrusion” during DC, embryonic stages 12 to 14, and “tissue dissociation” following DC, embryonic stages 15 to 16. Ten percent of AS cells are eliminated during phase one while the remaining ninety percent are removed during phase two. It is found that both cell extrusion and tissue dissociation are absent in apoptotic defective backgrounds, as well as in genetic backgrounds associated with increased class I phosphoinositide 3-kinase (PI3K) activity, a key regulator of autophagy. It is also found that extrusion is enhanced two-fold in embryos expressing the pro-apoptotic reaper gene product, and that tissue dissociation also accelerates in this background. Interestingly, our observations suggest that the activation of caspase cascade is not complete until AS cells have lost apical contacts with neighboring cells. Shortly after the loss of apical contact, an apoptotic morphology including membrane blebbing, cell fragmentation, and macrophage engulfment is readily observed. Measurements of the rate of DC demonstrate that this process is protracted in backgrounds lacking extrusion, leading to the conclusion that extrusion contributes towards generating adequate AS tension required for normal DC rates. Overall, our data suggest that phase one extrusion and phase two dissociation are manifestations of the same cellular event and that both are caspase dependent. It is also demonstrated that autophagy is a key component of AS death that acts upstream of apoptosis. Strikingly, our results lead to the suggestion that autophagy may function to trigger apoptosis during the programmed elimination of this extra-embryonic tissue.

The amnioserosa (AS) is an epithelium that plays major roles in two crucial morphogenetic processes during Drosophila embryogenesis: Germ Band Retraction (GBR) and Dorsal Closure (DC). The AS is extraembryonic and as such, it does not contribute to the mature embryo but is eliminated during development by programmed cell death. In this thesis, a comprehensive investigation of the timing and characteristics of the AS death and degeneration is performed. It is demonstrated that AS elimination occurs in two phases: “cell extrusion” during DC, embryonic stages 12 to 14, and “tissue dissociation” following DC, embryonic stages 15 to 16. Ten percent of AS cells are eliminated during phase one while the remaining ninety percent are removed during phase two. It is found that both cell extrusion and tissue dissociation are absent in apoptotic defective backgrounds, as well as in genetic backgrounds associated with increased class I phosphoinositide 3-kinase (PI3K) activity, a key regulator of autophagy. It is also found that extrusion is enhanced two-fold in embryos expressing the pro-apoptotic reaper gene product, and that tissue dissociation also accelerates in this background. Interestingly, our observations suggest that the activation of caspase cascade is not complete until AS cells have lost apical contacts with neighboring cells. Shortly after the loss of apical contact, an apoptotic morphology including membrane blebbing, cell fragmentation, and macrophage engulfment is readily observed. Measurements of the rate of DC demonstrate that this process is protracted in backgrounds lacking extrusion, leading to the conclusion that extrusion contributes towards generating adequate AS tension required for normal DC rates. Overall, our data suggest that phase one extrusion and phase two dissociation are manifestations of the same cellular event and that both are caspase dependent. It is also demonstrated that autophagy is a key component of AS death that acts upstream of apoptosis. Strikingly, our results lead to the suggestion that autophagy may function to trigger apoptosis during the programmed elimination of this extra-embryonic tissue.

<p>I have put my half title as the main thesis title here. I would like to use that as the title displayed online.</p> / <p>The Drosophila heart arises from two bilateral rows of cardioblasts (CB) that migratedorsally towards the midline and contact their contralateral partners to form the dorsal vessel.Generally, migrating cells rely on the extensions at the leading edge domain. Like other migratingcells, we show that the leading edge of the CBs extends finger-like processes which might play arole in sensing guidance cues during guided migration. Expressing an mCherry-Moesin transgenein the CBs enabled us to characterise the dynamic nature and genetic requirements of thesefilopodial processes. While studying the role of filopodial activity during heart assembly weobserved that CBs extended cellular protrusions towards the internalizing amnioserosa cells.Filopodial activity is low during migration, and rises when the CBs are near the amnioserosacells. However, filopodial contacts are stabilized by interaction with contralateral CBs, not theamnioserosa cells. CB cell bodies can contact their contra lateral partners only after theamnioserosa is fully internalized. We propose that filopodia are generated in response to thepresence of sensory guidance molecules excreted by the amnioserosa cells.Robo/Slit signalling has been previously shown to play a role in CB migration, adhesionand lumen formation. Additionally, studies have shown that Robo/Slit signalling plays a role infilopodial extension in the Drosophila nervous system development. We observed that in embryosin which Robo signaling in the CBs was reduced or absent, the CBs were less active at the LE. Inaddition, the migration speed of CBs in mutant embryos was notably decreased. Based on theseresults, we hypothesize that Robo/Slit signaling plays a role in filopodial extensions.</p> / Master of Science (MSc)

Topological defects are distinguished objects in vector fields that occur in a wide range of applications, ranging from material sciences to cosmology to bio-medical imaging and fingerprint recognition. This thesis considers topological point defects, also known as singular points, of two-dimensional vector fields. Besides Euclidean vectors as representation of modulus and direction, this also includes nematic vectors that equally have a modulus but direction is replaced with a head-to-tail symmetric orientation. In both case, a singular point or topological defect is an isolated discontinuity in an otherwise continuous vector field. It is characterized by its index or topological charge, which attains integer values for polar and half-integer values for nematic vector fields. There are different yet equivalent approaches to define the index. They either base on homology groups and the Brouwer degree, or on the first fundamental group and the mapping degree, or relatedly on lifting of a loop path enclosing the singular point. The definition by lift used here translates changes in the vector field along a path into a summed change in orientation angle. This translates to topological defects in discretized vector fields, where topological charge is calculated as sum of finite angle differences along a loop path between discretization points. On closer inspection, this calculation is an estimation, and is guaranteed to yield the correct estimate only with additional assumptions, for example when the underlying continuous-domain vector field is smooth and sampled at sufficiently high spatial resolution. Otherwise, arbitrary locations and charges of topological defects are possible, which yield exactly the same discretized vectors by the periodicity of representative orientation angles. Besides, the estimated topological charge depends discontinuously on each of the discrete input vectors and exhibits discrete jumps. As application data typically is subject to noise and uncertainty, this raises the question how reliable are topological defects identified in it. The present thesis quantifies, how large perturbations of a vector field are admissible without alteration of topological defects and charges. To that end, it introduces a robustness measure for each edge in a discretization grid that are combined along loop paths. Replacing critical edges of minimal robustness within a loop path by other path segments around a minimally larger area allows targeted increase of robustness. This data-dependent method called expansion over the critical edge is iterated until a user-set robustness is satisfied. The final areas of this algorithm are shown to have minimal size and therefore maximal spatial resolution, which also adapts to the local quality of data. The areas are also given as the faces in the graph of sufficiently robust edges after deleting all vertices of degree 1 (leaves) and all their connected edges. The minimal robust areas turn out to be nested by inclusion according to their robustness threshold. This allows to tradeoff detection robustness of topological charges versus their localization accuracy, both within a selection of pre-defined loop path shapes, and for free data-dependent expansion over the critical edge. Differently from defect identification by pattern matching, there is no restriction on the charge detectable. Besides, the robustness is shown to detect the size of unordered cores of defects. Robust defect areas indicate possible defect dynamics comprising motion, defect pair generation and annihilation already from single time point data. The robustness is also applicable to irregular discretization grids thanks to its graph theoretic characterization, and an extension to curved surfaces is foreseeable. The robust data-dependent defect identification is exemplified on microscopy images of the fruit fly Drosophila melanogaster. During Dorsal Closure, a developmental process, a cell sheet called amnioserosa contracts in highly regulated manner, whereby forces are actively generated and propagated along filamentous proteins like actin. Thereby, activity level and visco-elastic properties of the tissue are linked to the topological defects in the actin orientation field. Robust detection of these reveals that the sum over robust charges is clearly positive in the hundreds, whereas the overall sum of charge without robustness consideration fluctuates around zero. Numerous charges are observed, but $\pm 1/2$ dominate and confirm the amnioserosa as nematic material despite polar molecular constituents like actin. The sizes of robust defects span three orders of magnitude, and the largest defects follow the shapes of biological cells. The size distribution decays by a power law with the power for positive defects being more negative. Time courses show slightly higher speed of motion for +1/2 defects than for -1/2 defects, an order of magnitude above material flow velocity. Experiments with a genetic modification in the protein Crumbs had shown excess contraction of the amnioserosa cell layer during development. Comparing defect velocity of these embryos to wildtype suggests that viscosity and rotational viscosity increase stronger than activity level. This hypothesis remains to be tested in a combination of experiments and simulations, yet it would not have been generated in the first place without consideration of robust defects. More generally, the presented robustness measure and optimal data-dependent identification of topological defects could benefit the analysis of defects in discretized vector fields in a variety of disciplines. The optimal data-dependent identification allows for example to calculate error distributions for charge and localization of defects. The size, shape, and nested inclusion of robust defects constitute new observables, that generate numerous follow-up questions already for the fruit fly and enable novel analyses.:1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Application fields of topological defects . . . . . . . . . . . . . . . . . . . . . 1 1.2 Challenges of noisy, discretized vector fields . . . . . . . . . . . . . . . . . . 2 1.3 Thesis contribution and outline . . . . . . . . . . . . . . . . . . . . . . . . . 3 2 Mathematical background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.1 Polar and nematic vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.1.1 Isomorphism between polar and nematic vectors in two dimensions . 9 2.2 Homotopy, and (universal) covering . . . . . . . . . . . . . . . . . . . . . . . 19 2.2.1 Homotopies and the degree of mappings in the sense of homotopies . 19 2.2.2 Lifting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3 Defect identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.1 Index and topological charge in continuous domains . . . . . . . . . . . . . 32 3.1.1 Definition of topological charge by lift . . . . . . . . . . . . . . . . . 39 3.1.2 Differential expressions for topological charge . . . . . . . . . . . . . 46 3.2 Topological charge in discrete domains . . . . . . . . . . . . . . . . . . . . . 48 3.2.1 Correct discretization by sufficiently fine discretization . . . . . . . . 53 3.3 Topological charge estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.4 Comparison: defect identification by loop paths . . . . . . . . . . . . . . . . 61 3.4.1 ... equals fixed-size stencils . . . . . . . . . . . . . . . . . . . . . . . 61 3.4.2 ... equals convolution . . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.4.3 ... resembles “diffusive topological charge” . . . . . . . . . . . . . . . 65 3.4.4 ... does not improve by larger stencil size . . . . . . . . . . . . . . . 66 3.4.5 ... is linked to local maxima of azimuthal change . . . . . . . . . . . 68 3.4.6 ... differs from nematic order parameter thresholds . . . . . . . . . . 68 3.4.7 ... differs from matching with template patterns . . . . . . . . . . . 70 3.4.8 ... extends to irregular and unstructured data . . . . . . . . . . . . . 71 3.5 Discontinuous dependence of defects on discretized vector fields . . . . . . . 72 4 Robustness of defect identification and topological charge estimation in discrete domains . . . 75 4.1 Robustness between two discretization points . . . . . . . . . . . . . . . . . 76 4.2 Robustness of a discrete loop path . . . . . . . . . . . . . . . . . . . . . . . 84 4.3 Magnitude-aware robustness for non-normalized vector fields . . . . . . . . 102 4.4 Robustness for fixed path shapes . . . . . . . . . . . . . . . . . . . . . . . . 116 4.4.1 Robustness of fixed-shape identification for noise-free defects . . . . 117 4.4.2 Robustness of fixed-shape identification for noisy defects . . . . . . . 120 4.5 Data-dependent path shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 4.5.1 Expansion over the critical edge . . . . . . . . . . . . . . . . . . . . 126 4.5.2 Graph-theoretic characterization . . . . . . . . . . . . . . . . . . . . 128 4.5.3 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 4.5.4 Detection of defect core size . . . . . . . . . . . . . . . . . . . . . . . 135 4.5.5 Estimation of defect motion from still images . . . . . . . . . . . . . 143 4.5.6 Estimation of defect pair annihilation and generation, respectively, from still images . . . 145 4.5.7 Application to irregular grids . . . . . . . . . . . . . . . . . . . . . . 146 4.6 Comparison of defect identification methods from a robustness point of view 148 4.7 Extensions of the robustness measure . . . . . . . . . . . . . . . . . . . . . . 154 4.7.1 ... to two-dimensional manifolds . . . . . . . . . . . . . . . . . . . . 155 4.7.2 ... to higher dimensions . . . . . . . . . . . . . . . . . . . . . . . . . 160 5 Application to Dorsal Closure in Drosophila embryos . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 5.1 Dorsal Closure in the fruit fly Drosophila melanogaster . . . . . . . . . . . . 164 5.1.1 Cytoskeleton, motor proteins, and cell junctions . . . . . . . . . . . 164 5.1.2 Active gel models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 5.1.3 Image acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 5.1.4 Orientation estimation . . . . . . . . . . . . . . . . . . . . . . . . . . 172 5.2 Topological charge analysis by robustness . . . . . . . . . . . . . . . . . . . 173 5.2.1 Robustness threshold of edges . . . . . . . . . . . . . . . . . . . . . . 173 5.2.2 Sizes of robust areas . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 5.2.3 Total topological charge in the field of view . . . . . . . . . . . . . . 181 5.2.4 Sum of robust charges . . . . . . . . . . . . . . . . . . . . . . . . . . 186 5.3 Further observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 5.4 Comparison of robust defects . . . . . . . . . . . . . . . . . . . . . . . . . . 194 5.4.1 ... to microscopic defects . . . . . . . . . . . . . . . . . . . . . . . . . 196 5.4.2 ... to image preprocessing . . . . . . . . . . . . . . . . . . . . . . . . 201 5.5 Towards parameter estimation from defect dynamics . . . . . . . . . . . . . 202 5.5.1 The amnioserosa as an active nematic material . . . . . . . . . . . . 202 5.5.2 Defect tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 5.5.3 Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 6 Conclusions and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 / Topologische Defekte sind abgegrenzte Objekte in Vektorfeldern, die in einer großen Bandbreite von Anwendungsfeldern auftreten. Diese reichen von den Materialwissenschaften über Kosmologie und bio-medizinische Bildgebung bis zur Erkennung von Fingerabdrücken. Die vorliegende Dissertation beschäftigt sich mit topologischen Punkt-Defekten, auch bekannt als singuläre Punkte, in zwei-dimensionalen Vektorfeldern. Neben Euklidischen Vektoren als Darstellung von Betrag und Richtung umfasst das auch nematische Vektoren, die genauso einen Betrag haben, aber deren Richtungsinformation symmetrisch zwischen “vorn” und “hinten” ist. In beiden Fällen ist ein singulärer Punkt oder topologischer Defekt als isolierte Unstetigkeit in einem ansonsten stetigen Vektorfeld definiert. Er wird durch seinen Index oder die topologische Ladung charakterisiert, die ganzzahlige Werte für polare Felder annimmt, und halb-zahlige in nematischen Feldern. Es gibt verschiedene, jedoch äquivalente Weisen, den Index zu definieren. Sie basieren entweder auf Homologiegruppen und dem Brouwer'schen Abbildungsgrad, oder auf der Fundamentalgruppe und deren Abbildungsgrad, oder damit verbunden auf der Hochhebung eines geschlossenen Pfades um den singulären Punkt. Hier wird die Definition mittels Hochhebung verwendet, welche die Änderung des Vektorfeldes entlang eines Pfades in eine summierte Änderung des Orientierungswinkels übersetzt. Dies überträgt sich zu diskretisierten Vektorfeldern, wo die topologische Ladung als Summe über endliche Winkeldifferenzen entlang eines Pfades zwischen Diskretisierungspunkten berechnet wird. Diese Berechungsweise ist bei genauer Betrachtung eine Schätzung, und ihre Korrektheit nur unter zusätzlichen Bedingungen garantiert, zum Beispiel wenn ein zugrundeliegendes Vektorfeld mit kontinuierlichem Definitionsbereich glatt ist und mit hinreichender räumlicher Auflösung abgetastet wurde. Aufgrund der periodischen Darstellung jedes Vektors durch Orientierungswinkel sind andernfalls beliebige Positionen und Ladungen von topologischen Defekten möglich, die zu exakt den gleichen diskretisierten Vektoren führen. Außerdem hängt der Schätzwert der topologischen Ladung nicht kontinuierlich von jedem einzelnen der diskreten Vektoren ab, sondern weist diskrete Sprünge auf. Da Anwendungsdaten meist mit Messunsicherheiten behaftet oder verrauscht sind, steht die Frage, wie verlässlich die darin identifizierten Defekte sind. Die vorliegende Dissertation quantifiziert, wie groß die Störungen eines Vektorfeldes sein dürfen, ohne dass sich topologische Defekte und Ladungen ändern. Dafür wird ein Robustheitsmaß eingeführt, zunächst für jede Kante in einem Diskretisierungsgitter, und darauf basierend für Pfade. Das ermöglicht, die Robustheit der Defekt-Identifizierung gezielt zu erhöhen: Kritische Kanten mit der kleinsten Robustheit innerhalb eines Pfades werden durch andere Pfadstücke ersetzt, die eine minimal größere Fläche begrenzen. Diese datenabhängige “Erweiterung über die kritische Kante” (expansion over the critical edge) wird wiederholt, bis eine benutzerdefinierte Robustheit erreicht ist. Es wird gezeigt, dass die finalen Flächen dieses iterativen Algorithmus minimale Größe und damit höchste räumliche Auflösung haben, die sich zudem lokal an die Qualität der Daten anpasst. Die Flächen ergeben sich auch aus dem Graphen aller hinreichend robusten Kanten durch Löschen aller Knoten vom Grad 1 (Blätter) und der damit verbundenen Kanten. Es stellt sich damit heraus, dass die minimalen robusten Flächen je nach Robustheitsgrenze per Inklusion verkettet sind. Das erlaubt, die Robustheit für die Identifizierung topologischer Ladungen gegen die räumliche Genauigkeit abzuwägen, sowohl innerhalb von vorgegebenen Pfadformen, als auch für die freie, datenabhängige Erweiterung über die kritische Kante. Dabei gibt es — anders als bei Methoden der Defekt-Identifizierung mittels Muster-Erkennung — keine Beschränkung für die detektierbare Ladung. Außerdem wird gezeigt, dass man mit dem Robustheitsmaß die Größe von ungeordneten Kernen der Defekte bestimmen kann. Sogar die mögliche Dynamik von Defekten mit Bewegung, Paarbildung und -auslöschung wird aus den robusten Flächen eines einzelnen Zeitpunktes erkennbar. Die graphentheoretische Darstellung erlaubt dabei auch die Anwendung auf unstrukturierten Diskretisierungsgitter, und eine Erweiterung auf gekrümmte Flächen ist absehbar. Die robuste, datenabhängige Identifizierung von Defekten wird exemplarisch auf Mikroskopie-Bilder der Fruchtfliege Drosophila melanogaster angewendet. Während der Dorsal Closure, einem Entwicklungsprozess, zieht sich eine Zellschicht namens Amnioserosa auf genau regulierte Weise zusammen, wobei die wirkenden Kräfte entlang von Filamenten wie Aktin aktiv erzeugt und übertragen werden. Dabei sind der Aktivitätsgrad und viskoelastische Eigenschaften des Gewebes mit den topologischen Defekten im Orientierungsfeld des Aktins verknüpft. Deren robuste Identifizierung zeigt, dass die Summe der robusten Ladungen eindeutig positiv ist mit dreistelligen Werten, während die Gesamtladung ohne Beachtung der Robustheit um Null schwankt. Es werden zahlreiche Ladungen beobachtet; aber $\pm 1/2$ dominieren und bestätigen die Amnioserosa als nematisches Material, obwohl die molekularen Bestandteile wie Aktin polar sind. Die Größen von robusten Defekten umfassen drei Zehnerpotenzen, und die größten Defekte folgen der Form biologischer Zellen. Die Größenverteilung fällt nach einem Potenzgesetz ab, mit stärkerer negativer Potenz für positive Defekte. Zeitreihen zeigen geringfügig höhere Geschwindigkeit von +1/2 Defekten als von -1/2 Defekten, und deutlich über der Geschwindigkeit des Materialflusses. In Experimenten mit Modifikation im Gen des Proteins Crumbs wurde beobachtet, dass sich die Zellschicht der Amnioserosa in der Entwicklung übermäßig zusammenzieht. Ein Vergleich der Defektgeschwindigkeiten zwischen diesen Embryonen und Wildtyp führt zu der Hypothese, dass die Mutation die Viskosität und die Rotationsviskosität stärker steigen lässt als den Aktivitätsgrad. Diese Hypothese muss jedoch noch durch eine Kombination von Experimenten und Simulationen überprüft werden. Sie wäre aber ohne die Betrachtung von robusten Defekten gar nicht erst möglich gewesen. Das vorgestellte Robustheitsmaß könnte allgemein für vielfältige Disziplinen bei der Analyse topologischer Defekte in diskretisierten Vektorfeldern nützen. Auf Basis der optimalen datenabhängige Identifizierung kann zum Beispiele eine Fehlerrechnung für die Ladung und Lage von Defekten durchgeführt werden. Die Größen, Formen und Inklusionsketten von robusten Defekten bilden interessante neue Beobachtungsgrößen, die allein im Fall der Fruchtfliege zahlreiche weiterführende Fragen aufwerfen und bisher unbekannte Untersuchungen ermöglichen.:1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Application fields of topological defects . . . . . . . . . . . . . . . . . . . . . 1 1.2 Challenges of noisy, discretized vector fields . . . . . . . . . . . . . . . . . . 2 1.3 Thesis contribution and outline . . . . . . . . . . . . . . . . . . . . . . . . . 3 2 Mathematical background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.1 Polar and nematic vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.1.1 Isomorphism between polar and nematic vectors in two dimensions . 9 2.2 Homotopy, and (universal) covering . . . . . . . . . . . . . . . . . . . . . . . 19 2.2.1 Homotopies and the degree of mappings in the sense of homotopies . 19 2.2.2 Lifting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3 Defect identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.1 Index and topological charge in continuous domains . . . . . . . . . . . . . 32 3.1.1 Definition of topological charge by lift . . . . . . . . . . . . . . . . . 39 3.1.2 Differential expressions for topological charge . . . . . . . . . . . . . 46 3.2 Topological charge in discrete domains . . . . . . . . . . . . . . . . . . . . . 48 3.2.1 Correct discretization by sufficiently fine discretization . . . . . . . . 53 3.3 Topological charge estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.4 Comparison: defect identification by loop paths . . . . . . . . . . . . . . . . 61 3.4.1 ... equals fixed-size stencils . . . . . . . . . . . . . . . . . . . . . . . 61 3.4.2 ... equals convolution . . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.4.3 ... resembles “diffusive topological charge” . . . . . . . . . . . . . . . 65 3.4.4 ... does not improve by larger stencil size . . . . . . . . . . . . . . . 66 3.4.5 ... is linked to local maxima of azimuthal change . . . . . . . . . . . 68 3.4.6 ... differs from nematic order parameter thresholds . . . . . . . . . . 68 3.4.7 ... differs from matching with template patterns . . . . . . . . . . . 70 3.4.8 ... extends to irregular and unstructured data . . . . . . . . . . . . . 71 3.5 Discontinuous dependence of defects on discretized vector fields . . . . . . . 72 4 Robustness of defect identification and topological charge estimation in discrete domains . . . 75 4.1 Robustness between two discretization points . . . . . . . . . . . . . . . . . 76 4.2 Robustness of a discrete loop path . . . . . . . . . . . . . . . . . . . . . . . 84 4.3 Magnitude-aware robustness for non-normalized vector fields . . . . . . . . 102 4.4 Robustness for fixed path shapes . . . . . . . . . . . . . . . . . . . . . . . . 116 4.4.1 Robustness of fixed-shape identification for noise-free defects . . . . 117 4.4.2 Robustness of fixed-shape identification for noisy defects . . . . . . . 120 4.5 Data-dependent path shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 4.5.1 Expansion over the critical edge . . . . . . . . . . . . . . . . . . . . 126 4.5.2 Graph-theoretic characterization . . . . . . . . . . . . . . . . . . . . 128 4.5.3 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 4.5.4 Detection of defect core size . . . . . . . . . . . . . . . . . . . . . . . 135 4.5.5 Estimation of defect motion from still images . . . . . . . . . . . . . 143 4.5.6 Estimation of defect pair annihilation and generation, respectively, from still images . . . 145 4.5.7 Application to irregular grids . . . . . . . . . . . . . . . . . . . . . . 146 4.6 Comparison of defect identification methods from a robustness point of view 148 4.7 Extensions of the robustness measure . . . . . . . . . . . . . . . . . . . . . . 154 4.7.1 ... to two-dimensional manifolds . . . . . . . . . . . . . . . . . . . . 155 4.7.2 ... to higher dimensions . . . . . . . . . . . . . . . . . . . . . . . . . 160 5 Application to Dorsal Closure in Drosophila embryos . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 5.1 Dorsal Closure in the fruit fly Drosophila melanogaster . . . . . . . . . . . . 164 5.1.1 Cytoskeleton, motor proteins, and cell junctions . . . . . . . . . . . 164 5.1.2 Active gel models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 5.1.3 Image acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 5.1.4 Orientation estimation . . . . . . . . . . . . . . . . . . . . . . . . . . 172 5.2 Topological charge analysis by robustness . . . . . . . . . . . . . . . . . . . 173 5.2.1 Robustness threshold of edges . . . . . . . . . . . . . . . . . . . . . . 173 5.2.2 Sizes of robust areas . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 5.2.3 Total topological charge in the field of view . . . . . . . . . . . . . . 181 5.2.4 Sum of robust charges . . . . . . . . . . . . . . . . . . . . . . . . . . 186 5.3 Further observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 5.4 Comparison of robust defects . . . . . . . . . . . . . . . . . . . . . . . . . . 194 5.4.1 ... to microscopic defects . . . . . . . . . . . . . . . . . . . . . . . . . 196 5.4.2 ... to image preprocessing . . . . . . . . . . . . . . . . . . . . . . . . 201 5.5 Towards parameter estimation from defect dynamics . . . . . . . . . . . . . 202 5.5.1 The amnioserosa as an active nematic material . . . . . . . . . . . . 202 5.5.2 Defect tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 5.5.3 Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 6 Conclusions and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 Bibliography . . . . . . . . . . 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