This work is concerned with the spatio-temporal structures that emerge when non-identical, diffusively coupled oscillators synchronize. It contains analytical results and their confirmation through extensive computer simulations. We use the Kuramoto model which reduces general oscillatory systems to phase dynamics. The symmetry of the coupling plays an important role for the formation of patterns. We have studied the ordering influence of an asymmetry (non-isochronicity) in the phase coupling function on the phase profile in synchronization and the intricate interplay between this asymmetry and the frequency heterogeneity in the system. The thesis is divided into three main parts. Chapter 2 and 3 introduce the basic model of Kuramoto and conditions for stable synchronization. In Chapter 4 we characterize the phase profiles in synchronization for various special cases and in an exponential approximation of the phase coupling function, which allows for an analytical treatment. Finally, in the third part (Chapter 5) we study the influence of non-isochronicity on the synchronization frequency in continuous, reaction diffusion systems and discrete networks of oscillators. / Die vorliegende Arbeit beschäftigt sich in Theorie und Simulation mit den raum-zeitlichen Strukturen, die entstehen, wenn nicht-identische, diffusiv gekoppelte Oszillatoren synchronisieren. Wir greifen dabei auf die von Kuramoto hergeleiteten Phasengleichungen zurück. Eine entscheidene Rolle für die Musterbildung spielt die Symmetrie der Kopplung. Wir untersuchen den ordnenden Einfluss von Asymmetrie (Nichtisochronizität) in der Phasenkopplungsfunktion auf das Phasenprofil in Synchronisation und das Zusammenspiel zwischen dieser Asymmetrie und der Frequenzheterogenität im System. Die Arbeit gliedert sich in drei Hauptteile. Kapitel 2 und 3 beschäftigen sich mit den grundlegenden Gleichungen und den Bedingungen für stabile Synchronisation. Im Kapitel 4 charakterisieren wir die Phasenprofile in Synchronisation für verschiedene Spezialfälle sowie in der von uns eingeführten exponentiellen Approximation der Phasenkopplungsfunktion. Schliesslich untersuchen wir im dritten Teil (Kap.5) den Einfluss von Nichtisochronizität auf die Synchronisationsfrequenz in kontinuierlichen, oszillatorischen Reaktions-Diffusionssystemen und diskreten Netzwerken von Oszillatoren.

Several mechanisms have been proposed to explain the process of vein pattern formation in plant tissues. The most widely accepted amongst biologists is the canalization hypothesis, derived from pea root and stem experiments. According to this hypothesis, a signal, thought to be the phytohormone auxin, is transported polarly from cell to cell from the shoot to the root and is canalized progressively into narrow channels of high auxin fluxes that later differentiate to become vascular tissue. In this project, we set out to test whether auxin canalization drives vein pattern formation, using Arabidopsis thaliana mutants with increased auxin transport (max4-1, max3-9, max2-1 and max1-1). We predicted that the mutants would have distinct vein patterns and especially different angles between the primary and secondary veins, compared to the wild type. First rosette leaves of 15 plants per genotype were harvested for analysis each day from 7 to 17 days after sowing, giving a total of eight hundred twenty-five leaf samples to analyze. Venation patterns were extracted and analyzed using custom-made software written with Matlab. Overall, compared with the wild type, mutants with the highest auxin transport (max4-1 and max3-9) had different vein patterns at early developmental stages, confirming a role for auxin transport in vein patterning. However, veins of mutants and wild type connected at similar angles, which is not consistent with the auxin canalization hypothesis, as originally formulated.

Several mechanisms have been proposed to explain the process of vein pattern formation in plant tissues. The most widely accepted amongst biologists is the canalization hypothesis, derived from pea root and stem experiments. According to this hypothesis, a signal, thought to be the phytohormone auxin, is transported polarly from cell to cell from the shoot to the root and is canalized progressively into narrow channels of high auxin fluxes that later differentiate to become vascular tissue. In this project, we set out to test whether auxin canalization drives vein pattern formation, using Arabidopsis thaliana mutants with increased auxin transport (max4-1, max3-9, max2-1 and max1-1). We predicted that the mutants would have distinct vein patterns and especially different angles between the primary and secondary veins, compared to the wild type. First rosette leaves of 15 plants per genotype were harvested for analysis each day from 7 to 17 days after sowing, giving a total of eight hundred twenty-five leaf samples to analyze. Venation patterns were extracted and analyzed using custom-made software written with Matlab. Overall, compared with the wild type, mutants with the highest auxin transport (max4-1 and max3-9) had different vein patterns at early developmental stages, confirming a role for auxin transport in vein patterning. However, veins of mutants and wild type connected at similar angles, which is not consistent with the auxin canalization hypothesis, as originally formulated.

In many systems, instabilities can lead to time-dependent behavior, and instabilities can act as mechanisms for sustained chaos; an understanding of the dynamical modes governing instability is thus essential for prediction and/or control in such systems. In this thesis work, we have developed an approach toward characterizing instabilities quantitatively, from experiments on the prototypical Rayleigh-Bénard convection system. We developed an experimental technique for preparing a given convection pattern using rapid optical actuation of pressurized SF6, a greenhouse gas. Real-time analysis of convection patterns was developed as part of the implementation of closed-loop control of straight roll patterns. Feedback control of the patterns via actuation was used to guide patterns to various system instabilities. Controlled, spatially localized perturbations were applied to the prepared states, which were observed to excite the dominant system modes. We extracted the spatial structure and growth rates of these modes from analysis of the pattern evolutions. The lifetimes of excitations were also measured, near a particular instability; a critical wavenumber was found from the observed dynamical slowing near the bifurcation. We will also describe preliminary results of using a state estimation algorithm (LETKF) on experimentally prepared non-periodic patterns in a cylindrical convection cell.

Nature uses fibrous surfaces for a wide range of functions such as sensing, adhesion, structural color, and self-cleaning. However, little is known about how fiber properties enable them to self-organize into diverse and complex functional forms. Using polymeric micro/nanofiber arrays with tunable properties as model systems, we demonstrate how the combination of mechanical and surface properties can be harnessed to transform an array of anchored nanofibers into a variety of complex, hierarchically organized dynamic functional surfaces. We show that the delicate balance between fiber elasticity and surface adhesion plays a critical role in determining the shape, chirality, and hierarchy of the assembled structures. We further report a strategy for controlling the long-range order of fiber assemblies by manipulating the shape and movement of the liquid-vapor interface. Our study provides fundamental understanding of the pattern formation by self-organization of bioinspired fibrous surfaces. Moreover, our new strategies offer a foundation for designing a vast assortment of functional surfaces with adhesive, optical, water-repellent, capture and release, and many more capabilities with the structural and dynamic sophistication of their biological counterparts. / Engineering and Applied Sciences

No description available.

No description available.

Several mechanisms have been proposed to explain the process of vein pattern formation in plant tissues. The most widely accepted amongst biologists is the canalization hypothesis, derived from pea root and stem experiments. According to this hypothesis, a signal, thought to be the phytohormone auxin, is transported polarly from cell to cell from the shoot to the root and is canalized progressively into narrow channels of high auxin fluxes that later differentiate to become vascular tissue. In this project, we set out to test whether auxin canalization drives vein pattern formation, using Arabidopsis thaliana mutants with increased auxin transport (max4-1, max3-9, max2-1 and max1-1). We predicted that the mutants would have distinct vein patterns and especially different angles between the primary and secondary veins, compared to the wild type. First rosette leaves of 15 plants per genotype were harvested for analysis each day from 7 to 17 days after sowing, giving a total of eight hundred twenty-five leaf samples to analyze. Venation patterns were extracted and analyzed using custom-made software written with Matlab. Overall, compared with the wild type, mutants with the highest auxin transport (max4-1 and max3-9) had different vein patterns at early developmental stages, confirming a role for auxin transport in vein patterning. However, veins of mutants and wild type connected at similar angles, which is not consistent with the auxin canalization hypothesis, as originally formulated.

Recurrent processes are a general feature of living systems, from the cell cycle to circadian day-night rhythms to hibernation and flowering cycles. During development and life, numerous recurrent processes are controlled by genetic oscillators, a specific class of genetic regulatory networks that generates oscillations in the level of gene products. A vital mechanism controlled by genetic oscillators is the rhythmic and sequential segmentation of the elongating body axis of vertebrate embryos. During this process, a large collection of coupled genetic oscillators gives rise to spatio-temporal wave patterns of oscillating gene expression at tissue level, forming a dynamic prepattern for the precursors of the vertebrae. While such systems of genetic oscillators have been studied extensively over the past years, many fundamental questions about their collective behavior remain unanswered. In this thesis, we study the behavior and the properties of genetic oscillators from the single oscillator scale to the complex pattern forming system involved in vertebrate segmentation. Genetic oscillators are subject to fluctuations because of the stochastic nature of gene expression. To study the effects of noisy biochemical coupling on genetic oscillators, we propose a theory in which both the internal dynamics of the oscillators as well as the coupling process are inherently stochastic. We find that stochastic coupling of oscillators profoundly affects their precision and synchronization properties, key features for their viability as biological pacemakers. Moreover, stochasticity introduces phenomena not known from deterministic systems, such as stochastic switching between different modes of synchrony. During vertebrate segmentation, genetic oscillators play a key role in establishing a segmental prepattern on tissue scale. We study the spatio-temporal patterns of oscillating gene expression using a continuum theory of coupled phase oscillators. We investigate the effects of different biologically relevant factors such as delayed coupling due to complex signaling processes, local tissue growth, and tissue shortening on pattern formation and segmentation. We find that the decreasing tissue length induces a Doppler effect that contributes to the rate of segment formation in a hitherto unanticipated way. Comparison of our theoretical findings with experimental data reveals the occurrence of such a Doppler effect in vivo. To this end, we develop quantification methods for the spatio-temporal patterns of gene expression in developing zebrafish embryos. On a cellular level, tissues have a discrete structure. To study the interplay of cellular processes like cell division and random cell movement with pattern formation, we go beyond the coarse-grained continuum theories and develop a three-dimensional cell-based model of vertebrate segmentation, in which the dynamics of the segmenting tissue emerges from the collective behavior of individual cells. We show that this model is able to describe tissue formation and segmentation in a self-organized way. It provides the first step of theoretically describing pattern formation and tissue dynamics during vertebrate segmentation in a unified framework involving a three-dimensional tissue with cells as distinct mechanical entities. Finally, we study the synchronization dynamics of generic oscillator systems whose coupling is subject to phase shifts and time delays. Such phase shifts and time delays are induced by complex signaling processes as found, e.g., between genetic oscillators. We show how phase shifts and coupling delays can alter the synchronization dynamics while leaving the collective frequency of the synchronized oscillators invariant. We find that in globally coupled systems, fastest synchronization occurs for non-vanishing coupling delays while in spatially extended systems, fastest synchronization can occur on length scales larger than the coupling range, giving rise to novel synchronization scenarios. Beyond their potential relevance for biological systems, these results have implications for general oscillator systems, e.g., in physics and engineering. In summary, we use discrete and continuous theories of genetic oscillators to study their dynamic behavior, comparing our theoretical results to experimental data where available. We cover a wide range of different topics, contributing to the general understanding of genetic oscillators and synchronization and revealing a hitherto unknown mechanism regulating the timing of embryonic pattern formation.

Das Phänomen der Rissmusterentstehung stellt ein anschauliches und numerisch nachvollziehbares Beispiel für Strukturbildung dar. Im Rahmen der vorliegenden Arbeit gelang es, einfache dreidimensionale Rissmuster zu berechnen, die durch Zugspannungen infolge inhomogener Schrumpfung durch Wärme- oder Stofftransport wachsen. Aufgrund wechselseitiger Entlastung bleiben Risse oder Teile einer Rissfront zurück. Für idealisierte periodische Rissanordnungen wurden durch bruchmechanische Verzweigungsanalyse kritische Last- und Geometrieparameter berechnet, für die sich neben der Grundlösung eine weitere Lösung finden lässt. Dadurch konnte die Frage beantwortet werden, warum verschiedene Typen von Rissmustern entstehen. Untersucht wurden Rissmuster in trocknender Stärke und Basalt sowie Tunnelrisse zwischen Glasplatten. Dafür war es nötig, die Risskontur numerisch zu bestimmen, wofür ein neues iteratives Verfahren entwickelt wurde. Für Säulen bildende Rissmuster in Basalt und trocknender Stärke mit konstanten Säulendurchmessern, welche stationär getrieben wachsen, wurde gezeigt, dass entgegen der Annahme in der Literatur eine Verzweigungslösung für zusammenhängende dreidimensionale Rissmuster existiert. Die berechneten kritischen Rissabstände unterscheiden sich nur unwesentlich von denjenigen zweidimensionaler Rissmodelle. Zum Verständnis der Herausbildung der Basaltsäulen wurde ein neuer Mechanismus gefunden, für den die Wechselwirkung zwischen Riss- und Erstarrungsfront der flüssigen Lava eine wesentliche Rolle spielt. Mittels eines iterativen Verfahrens wurde dafür die Erstarrungsfront berechnet. Die Ergebnisse zeigen gute Übereinstimmung mit den experimentellen Werten. Die Durchmesser von Stärkesäulen und Basaltsäulen nehmen nach dem gleichen Potenzgesetz mit zunehmender Wachstumsgeschwindigkeit ab. Für Tunnelrisse sind die Rissabstände nahezu konstant, obwohl entgegen ursprünglicher Annahmen gezeigt werden konnte, dass diese instationär getrieben wachsen. Als Ursache dafür wurde die durch die Rissgeometrie stark reduzierte Wechselwirkung zwischen den Tunnelrissen erkannt. Der Rissabstand nimmt potenzartig mit Dicke der trocknenden Schicht zu. Mittels einer Skalenanalyse konnte gute Übereinstimmung mit den Experimenten gefunden werden. / Crack patterns are a remarkable example of pattern formation. In this work simple three-dimensional crack patterns where analysed that are driven by inhomogeneous shrinkage caused from heat or mass transport. Caused by mutual unloading, parts of the crack front stop to grow. For idealised periodically repeatable crack configurations through fracture mechanics bifurcation analysis critical loading or geometry parameters where calculated for which beside the basic solution (growth of all cracks) another solution is possible. Analysed were crack patterns in drying starch and basalt and tunnelling cracks between glass plates. For this it was necessary to calculate the crack contour numerically, for this a new iteration method was developed. For the connected crack patterns in starch and basalt that form columns, driven stationary, it could be shown that there exists a bifurcation solution. Agreement with experimental data was found by including the solidification front ahead the crack tips. For the tunnelling cracks the crack spacing is nearly constant also they are unsteadily driven. The crack spacing scales with a power law of the thickness of the drying layer. With a scaling analysis good agreement with experimental data was found.