We propose a hypothesis and a numerical model for the formation of branching plagioclase textures visible at both macroscopic (∼cm to ∼m) and microscopic scale within melagabbro of the Isle of Rum, Scotland, based on macroscopic, microscopic observations and relevant geological history. The plagioclase crystals are typically linked as twins and form meshes of planar stellate structures (m-scale) with a large range in geometrical organization from patchy to radiating. Evidence of macroscopic crystal aggregation and alignment is attributed to interfacial free energy minimization at the microscopic scale during growth. Accordingly, a binary immiscible Lattice Boltzmann model was developed to simulate diffusion of simplified plagioclase in the melt phase. Isothermal phase transitions modelled via first order chemical reactions are subsequently coupled with stochastic dynamics at the crystal growth front to simulate energy minimization processes including twinning during crystallization in an igneous environment. The solid phase and the liquid phase are coupled with a temporal flexibility that sets the overall ratio between the rate of diffusion and chemical enrichment in the liquid state and the rate of crystallization. The parameter space of the model is explored extensively, followed by a reasonable transcription of physical parameters and an estimation of other parameters to construct realistic simulation scenarios yielding synthetic plagioclase stellates. The results are presented, analyzed and discussed. They appear to be in reasonable qualitative agreement with observations, and several aspects of the natural stellates such as the stellate spacing and long continuous stretches of plagioclase with epitaxial junctions seem to be in reasonable quantitative agreement with observations.

Zellmembranen sind hochspezialisierte Mehrkomponentenlegierungen, welche sowohl die Zelle selbst als auch ihre Organellen umgeben. Sie spielen eine entscheidende Rolle bei vielen biologisch relevanten Prozessen wie die Signaltransduktion und die Zellbewegung. Aus diesem Grund ist eine genaue Charakterisierung ihrer Eigenschaften der Schlüssel zum Verständnis der Bausteine des Lebens sowie ihrer Erkrankungen. Besonders Krebs steht im engen Zusammenhang mit Veränderungen der biomechanischen Eigenschaften vom Gewebe, Zellen und ihren Organellen. Während Veränderungen des Zytoskeletts von Krebszellen im Fokus vieler Biophysiker stehen, ist die Bedeutung der Biomechanik von Zellmembran weitgehend unklar. Zellmembranen faszinieren Wissenschaftler jedoch nicht nur wegen ihrer biomechanischen Eigenschaften. Sie sind auch Beispiele für eine selbstorganisierte und heterogene Landschaft, in der Prozesse fernab des Gleichgewichtes, wie z.B. räumliche und zeitliche Musterbildungen, auftreten. Die vorgelegte Dissertation untersucht erstmals umfassend die zentrale Rolle der Zellmembran und ihrer molekularen Architektur für die Signalübertragung, die Biomechanik und die Zellmigration. Hierfür werden einfache Modellmembranen aber auch komplexere Vesikel und ganze Zellen mittels etablierter physikalischer Methoden analysiert. Diese reichen von Fourier- Analysen zur Charakterisierung von thermisch angeregten Membranundulationen über Massenspektrometrie und ‘Optical Stretcher’ Messungen von ganzen Zellen bis hin zur Filmwaagentechnik. Des Weiteren wird ein Modellsystem vorgestellt, welches sowohl einen experimentellen als auch einen mathematischen Zugang zum ‘ME-switch’ ermöglicht. Die vorgelegte Dissertation bietet neue Einblicke in wichtige Funktionen von Zellmembranen und zeigt neue therapeutische Perspektiven in der Membran- und Krebsforschung auf.:1 Introduction 2 Background 2.1 The Cell Membrane 2.1.1 Lipids in Cell Membranes 2.1.2 Membrane Proteins 2.1.3 An Overview on Membrane Models 2.1.4 Lipid Rafts 2.2 Model Membranes – An Experimental Access to Cell Membranes 2.2.1 Surface Tension and Thermodynamic Equilibrium 2.2.2 Langmuir Monolayer 2.2.3 The Polymorphism of Langmuir Monolayers 2.2.4 Membrane Vesicles 2.3 Biological Membranes as Semiflexible Shells 2.3.1 Elasticity of Soft Shells 2.3.2 Helfrichs Theory About Bending Deformations 2.3.3 Membrane Undulation 2.4 Membranes in Cell Signaling 2.4.1 Signal Transduction Fundamentals 2.4.2 Phosphoinositides 2.4.3 Phosphatidylinositol Signaling Pathway 2.4.4 The Myristoyl-Electrostatic Switch 2.5 Reaction-Diffusion Systems 2.5.1 Diffusion 2.5.2 Michaelis-Menten Kinetics 2.5.3 Reaction-Diffusion Systems 3 Methods, Materials and Theory 3.1 Optical Microscopy 3.1.1 Fluorescence Microscopy 3.1.2 Phase Contrast Microscopy 3.2 Cell Culture and GPMV Formation 3.2.1 Tumor Dissociation and Cell Culturing of Primary Cells 3.2.2 Cell Lines and Cell Culturing 3.2.3 Preparation of Giant Plasma Membrane Vesicles 3.3 Optical Stretcher 3.4 Fourier Analysis of Thermally Excited Membrane Fluctuations 3.4.1 The Quasi-Spherical Model – Membrane Fluctuations 3.4.2 Determination of the Bending Rigidity 3.5 Mass Spectrometry 3.5.1 MALDI-TOF Mass Spectrometry 3.5.2 ESI Mass Spectrometry 3.6 Migration, Invasion and Cell Death Assays 3.7 Langmuir-Blodgett Technique 3.7.1 Langmuir Troughs and Film Balances 3.7.2 Experimental Setup and Monolayer Preperation 3.7.3 Phospholipids, Dyes and Buffer Solutions 4 Fluctuations in Cell Membranes 4.1 Cell Membrane Softening in Human Breast and Cervical Cancer Cells 4.1.1 Bending Rigidity of Human Beast and Cervical Cell Membranes 4.1.2 MALDI-TOF Analysis of Lipid Composition 4.1.3 Summary and Discussion 4.2 Targeting of Membrane Rigidity – Implications on Migration 4.2.1 ESI Tandem Analysis of Lipid Composition 4.2.2 Biomechanical Behavior of Whole Cells and Membranes 4.2.3 Migration and Invasion Behavior 4.2.4 Summary and Discussion 5 Oscillations in Cell Membranes 5.1 Mimicking the ME-switch 5.1.1 DPPC/PIP2 monolayers at the presence of MARCKS 5.1.2 Lateral organization of PIP2 in DPPC/PIP2 monolayers 5.1.3 Translocation of MARCKS 5.1.4 Phosphorylation of MARCKS by PKC 5.1.5 Summary and Discussion 5.2 Dynamic Membrane Structure Induces Temporal Pattern Formation 5.2.1 Mechanism of the Oscillation 5.2.2 Modeling the ME-switch 5.2.3 Time Evolution 5.2.4 Phase Diagrams and Open Systems 5.2.5 Summary and Discussion 6 Conclusion and Outlook Appendix Bibliography List of Figures List of Abbreviations Acknowledgement

The correct patterning of embryos requires the exchange of information between cells. This is in part achieved by the proper distribution of signaling molecules, many of which exert their function by establishing gradients of concentration. Because of this property they were named "morphogens", or "form giving" substances. Among these, proteins belonging to the Hedgehog (Hh) family have received much attention, owing to their unusual double lipid modification and their involvement in human disease, causing congenital birth defects and cancer. Great efforts have been made in order to elucidate the mechanisms by which Hh molecules are propagated in the embryo. However, no conclusive evidence exists to date to which structures these molecules localize and how they, despite their membrane association, establish a gradient of concentration. Therefore, I decided to study the distribution of the vertebrate Hh homolog, Sonic Hedgehog (Shh) in developing zebrafish embryos. By fluorescently tagging Shh proteins, I found that these localize to discrete punctate structures at the membranes of expressing cells. These were often regions from which filopodial protrusions emanated from the cells. Puctate deposits of Shh were also located outside of expressing cells. In dividing cells, Shh accumulated at the cleavage plane. Furthermore, by making use of confocal microscopy and time lapse analysis, I visualized Shh proteins moving in filopodial extensions present between cells. This suggests a novel mechanism of Shh distribution, which relies on the direct contact of cells by filopodia for the exchange of signaling proteins. In a second part of my thesis, I characterized genes implicated in regulating Shh protein distribution and signaling function. I cloned three zebrafish genes belonging to the Ext1 (exostosin) family of glycosyltransferases required for the synthesis of Heparan Sulfate Proteoglycans and established a tentative link of these genes to somitic Hh signaling. In addition, I characterized the developmental expression and function of zebrafish Rab23, a small GTPase, which acts as a negative regulator of the Shh signaling pathway. Performing knock-down experiments of zebrafish Rab23, I found that Rab23 functions in left-right axis specification, a process previously shown to depend on proper Shh signaling.

Cellular automata (CA) are fully discrete dynamical systems. Space is represented by a regular lattice while time proceeds in finite steps. Each cell of the lattice is assigned a state, chosen from a finite set of "values". The states of the cells are updated synchronously according to a local interaction rule, whereby each cell obeys the same rule. Formal definitions of deterministic, probabilistic and lattice-gas CA are presented. With the so-called mean-field approximation any CA model can be transformed into a deterministic model with continuous state space. CA rules, which characterise movement, single-component growth and many-component interactions are designed and explored. It is demonstrated that lattice-gas CA offer a suitable tool for modelling such processes and for analysing them by means of the corresponding mean-field approximation. In particular two types of many-component interactions in lattice-gas CA models are introduced and studied. The first CA captures in abstract form the essential ideas of activator-inhibitor interactions of biological systems. Despite of the automaton´s simplicity, self-organised formation of stationary spatial patterns emerging from a randomly perturbed uniform state is observed (Turing pattern). In the second CA, rules are designed to mimick the dynamics of excitable systems. Spatial patterns produced by this automaton are the self-organised formation of spiral waves and target patterns. Properties of both pattern formation processes can be well captured by a linear stability analysis of the corresponding nonlinear mean-field (Boltzmann) equations.

Ausgedehnte dissipative Systeme können fernab vom thermodynamischen Gleichgewicht instabil gegenüber Oszillationen bzw. Wellen oder raumzeitlichem Chaos werden. Die komplexe Ginzburg-Landau Gleichung (CGLE) stellt ein universelles Modell zur Beschreibung dieser raumzeitlichen Strukturen dar. Diese Arbeit ist der theoretischen Analyse komplexer Muster gewidmet. Mittels numerischer Bifurkations- und Stabilitätsanalyse werden Instabilitäten einfacher Muster identifiziert und neuartige Lösungen der CGLE bestimmt. Modulierte Amplitudenwellen (MAW) und Super-Spiralwellen sind Beispiele solcher komplexer Muster. MAWs können in hydrodynamischen Experimenten und Super-Spiralwellen in der Belousov-Zhabotinsky-Reaktion beobachtet werden. Der Grenzübergang von Phasen- zu Defektchaos wird durch den Existenzbereich der MAWs erklärt. Mittels der selben numerischen Methoden wird Bursting vom Fold-Hopf-Typ in einem Modell der Kalziumsignalübertragung in Zellen identifiziert.

In this thesis we investigate the formation of patterns in a simple activator-inhibitor model supplemented with an inhibitory nonlocal coupling term. This model exhibits a wave instability for slow inhibitor diffusion, while, for fast inhibitor diffusion, a Turing instability is found. For moderate values of the inhibitor diffusion these two instabilities occur simultaneously at a codimension-2 wave-Turing instability. We perform a weakly nonlinear analysis of the model in the neighbourhood of this codimension-2 instability. The resulting amplitude equations consist in a set of coupled Ginzburg-Landau equations. These equations predict that the model exhibits bistability between travelling waves and Turing patterns. We present a study of interfaces separating wave and Turing patterns arising from the codimension-2 instability. We study theoretically and numerically the dynamics of such interfaces in the framework of the amplitude equations and compare these results with numerical simulations of the model near and far away from the codimension-2 instability. Near the instability, the dynamics of interfaces separating small amplitude Turing patterns and travelling waves is well described by the amplitude equations, while, far from the codimension-2 instability, we observe a locking of the interface velocities. This locking mechanism is imposed by the absence of defects near the interfaces and is responsible for the formation of drifting pattern domains, i.e. moving localised patches of travelling waves embedded in a Turing pattern background and vice versa.

Networked dynamical systems, i.e., systems of dynamical units coupled via nontrivial interaction topologies, constitute models of broad classes of complex systems, ranging from gene regulatory and metabolic circuits in our cells to pandemics spreading across continents. Most of such systems are driven by irregular and distributed fluctuating input signals from the environment. Yet how networked dynamical systems collectively respond to such fluctuations depends on the location and type of driving signal, the interaction topology and several other factors and remains largely unknown to date. As a key example, modern electric power grids are undergoing a rapid and systematic transformation towards more sustainable systems, signified by high penetrations of renewable energy sources. These in turn introduce significant fluctuations in power input and thereby pose immediate challenges to the stable operation of power grid systems. How power grid systems dynamically respond to fluctuating power feed-in as well as other temporal changes is critical for ensuring a reliable operation of power grids yet not well understood. In this work, we systematically introduce a linear response theory (LRT) for fluctuation-driven networked dynamical systems. The derivations presented not only provide approximate analytical descriptions of the dynamical responses of networks, but more importantly, also allow to extract key qualitative features about spatio-temporally distributed response patterns. Specifically, we provide a general formulation of a LRT for perturbed networked dynamical systems, explicate how dynamic network response patterns arise from the solution of the linearised response dynamics, and emphasise the role of LRT in predicting and comprehending power grid responses on different temporal and spatial scales and to various types of disturbances. Understanding such patterns from a general, mathematical perspective enables to estimate network responses quickly and intuitively, and to develop guiding principles for, e.g., power grid operation, control and design.

[pt] Uma propriedade notável dos sistemas biológicos é a formação de estruturas espaciais. Estas podem surgir por auto-organização, como consequência das próprias interações entre os indivíduos. Para estudar estas estruturas e como elas emergem, têm sido muito úteis modelos simples para a dinâmica da densidade espacial de uma população, que levam em conta apenas certos processos elementares (como reprodução, competição e dispersão). Em particular, a equação de FKPP (Fisher-Kolmogorov- Petrovski-Piskunov), que inclui simplesmente o crescimento logístico mais a difusão normal, é um modelo clássico para a dinâmica de uma população de uma única espécie. Dentro do quadro minimalista da equação de FKPP e suas variantes, a competição à distância (ou, não local) é a principal responsável por produzir oscilações espaciais na densidade da população. Entretanto, a não localidade pode ocorrer também nos demais processos. Assim, um primeiro objetivo desta tese é investigar como as diferentes escalas espaciais presentes podem interferir entre si, afetando a formação de padrões. Para isso, consideramos uma generalização da equação de FKPP em que todos os termos são não locais, em um ambiente homogêneo com condições de contorno periódicas. Enquanto a competição é o principal processo por trás da formação de padrões, mostramos que os outros dois podem agir de forma construtiva ou destrutiva. Por exemplo, a difusão, que comumente homogeniza, pode favorecer a formação de padrões dependendo do formato e alcance das funções de influência de cada processo. Em um segundo estudo, motivado por resultados experimentais, procuramos entender como a variabilidade da difusividade pode impactar a organização espacial da população dentro e fora de um refúgio (região de alta qualidade imersa em um ambiente hostil). Para tanto, consideramos uma outra generalização da equação de FKPP, com não localidade apenas no processo de competição intra-espécie, e modificada para levar em conta a presença do refúgio. Além da dependência espacial da taxa de crescimento, que é a principal característica distintiva de um refúgio em um ambiente hostil, também consideramos o fato de que a mobilidade pode ser heterogênea no espaço ou depender da densidade populacional. Focamos em dois casos em que a difusividade responde à densidade de indivíduos, diminuindo ou aumentando com a densidade populacional. Para comparação, também abordamos a difusividade dependente do espaço, com valores diferentes dentro e fora do refúgio. Observamos que o limiar da formação de padrões, no espaço de parâmetros, é bastante robusto diante destas heterogeneidades. Por outro lado, a dependência com a densidade pode produzir uma realimentação que está ausente em meios homogêneos, e que afeta a forma dos padrões. Em todos os casos, os resultados foram obtidos mediante a integração numérica das equações integro-diferenciais e realizando considerações analíticas. / [en] A remarkable property of biological systems is the formation of spatial structures. These can arise by self-organization, as a consequence of the interactions between individuals. To study these structures and how they emerge, simple models for the dynamics of the spatial density of a population, which take into account only certain elementary processes (such as reproduction, competition and dispersion) have been very useful. In particular, the FKPP (Fisher-Kolmogorov-Petrovski-Piskunov) equation, which simply includes logistic growth plus normal diffusion, is a classic model for the dynamics of a population of a single species. Within the minimalist framework of the FKPP equation and its variants, distance (or, non-local) competition is primarily responsible for producing spatial oscillations in population density. However, non-locality can also be present in other processes. Then, a first objective of this thesis is to investigate how the different spatial scales which are present in each process can interfere between them, affecting the formation of patterns in a homogeneous environment with periodic boundary conditions. For this purpose, we consider a generalization of the FKPP equation in which all terms are nonlocal. While competition is the main process behind pattern formation, we show that the other two can act constructively or destructively. For example, diffusion, which commonly homogenizes, can favor the formation of patterns depending on the format and range of the influence functions of each process. In a second study, motivated by experimental results, we seek to understand how the variability of the diffusivity can impact the spatial organization of the population inside and outside a refuge (a high-quality region immersed in a hostile environment). Therefore, we consider another generalization of the FKPP equation, with non-locality only in the intra-species competition process, modified to take into account the presence of the refuge. In addition to the spatial dependence of the growth rate, which is the main distinguishing feature of a refuge in a hostile environment, we also consider the fact that mobility can be spatially heterogeneous or depend on population density. We focus on two cases in which diffusivity responds to the density of individuals, decreasing or increasing with population density. For comparison, we also address spacedependent diffusivity, with different values inside and outside the refuge. We observed that the threshold of pattern formation in parameter space is quite robust under the presence of these heterogeneities. On the other hand, density dependence can produce a feedback that is absent in homogeneous media, and that affects the shape of the patterns. In all cases, the results were obtained by numerical simulations of the integro-differential equations and by analytical considerations.

Рассмотрены две пространственные модели реакции–диффузии, найдены параметрические зоны диффузионной неустойчивости. Исследованы сценарии детерминированной и стохастической генерации паттернов. Показано, что обе модели мультистабильны с большим количеством сосуществующих аттракторов–паттернов. Под воздействием шума возможны переходы между стационарными неоднородными состояниями, а также генерация паттернов в зоне диффузионной устойчивости. / Two spatial reaction-diffusion models are considered, and parametric zones of diffusion instability are found. Scenarios of deterministic and stochastic pattern formation are investigated. It is shown that both models are multistable with a large number of coexisting attractor-patterns. Under the influence of noise, transitions between stationary inhomogeneous states are possible, as well as the generation of patterns in the zone of diffusion stability.

The main focus of this paper is the evolution of complex behavior in a system of coupled nonlinear memristor circuits depending on the applied coupling conditions. Thereby, the parameter space for the local activity and the edge-of-chaos domain will be determined to enable the emergence of the pattern formation in locally coupled cells according to Chua's principle. Each cell includes a Niobium oxide-based memristor, which may feature a locally active behavior once it is suitably biased on the negative differential resistance region of its DC current-voltage characteristic. It will be shown that there exists a domain of parameters under which each uncoupled cell may become locally active around a stable bias state. More specifically, under these conditions, the coupled cells are on the edge-of-chaos, and can support the static and dynamic pattern formation. The emergence of such complex spatio-temporal behavior in homogeneous structures is a prerequisite for information processing. The theoretical results are confirmed by