Anders als menschengemachte Maschinen verfügen Zellen über keinen festgeschriebenen Bauplan und die Positionen einzelner Elemente sind häufig nicht genau festgelegt, da die Moleküle diffusiven Zufallsbewegungen unterworfen sind. Darüber hinaus sind einzelne Bauteile auch nicht auf eine einzelne Funktion festgelegt, sondern können parallel in verschiedene Prozesse einbezogen sein. Basierend auf Selbstorganisation und Selbstassemblierung muß die Organisation von Anordnung und Funktion einer lebenden Zelle also bereits in ihren einzelnen Komponenten inhärent enthalten sein. Die intrazelluläre Organisation wird zum großen Teil durch ein internes Biopolymergerüst reguliert, das Zytoskelett. Biopolymer-Netzwerke und –Fasern durchdringen die gesamte Zelle und sind verantworlich für mechanische Integrität und die funktionale Architektur. Unzählige essentielle biologische Prozesse hängen direkt von einem funktionierenden Zytoskelett ab. Die vorliegende Arbeit zielt auf ein besser Verständnis und den Nachbau zweier verschiedener funktionaler Module lebender Zellen anhand stark reduzierter Modellsysteme. Als zentrales Element wurde Aktin gewählt, da dieses Biopolymer eine herausragende Rolle in nahezu allen eukaryotischen Zellen spielt. Mit dem ersten Modellsystem wird der bewegliche Aktin-Polymerfilm an der Vorderkante migrierender Zellen betrachtet. Die wichtigsten Elemente dieser hochdynamischen Netzwerke sind bereits bekannt und wurden in dieser Arbeit benutzt um ein experimentelles Modellsystem zu etablieren. Vor allem aber lieferten detailierte Computersimulationen und ein mathematisches Modell neue Erkenntnisse über grundlegende Organisationsprinzipien dieser Aktinnetzwerke. Damit war es nicht nur möglich, experimentelle Daten erfolgreich zu reproduzieren, sondern das Entstehen von Substrukturen und deren Charakteristika auf proteinunabhängige, generelle Mechanismen zurückzuführen. Das zweite studierte System betrachtet die Selbstassemblierung von Aktinnetzwerken durch entropische Kräfte. Aktinfilamente aggregieren hierbei durch Kondensation multivalenter Ionen oder durch Volumenausschluss hochkonzentrierter inerter Polymere. Ein neu entwickelter Experimentalaufbau bietet die Möglichkeit in gut definierten zellähnlichen Volumina, Konvektionseinflüsse zu umgehen und Aggregationseffekte gezielt einzuschalten. Hierbei wurden neuartige, regelmäßige Netzwerkstrukturen entdeckt, die bislang nur im Zusammenhang mit molekularen Motoren bekannt waren. Es konnte ferner gezeigt werden, dass die Physik der Flüssigkristalle entscheidend zu weiteren Variationen dieser Netzwerke beiträgt. Dabei wird ersichtlich, dass entstehende Netzwerke in ihrer Architektur direkt die zuvor herrschenden Anisotropien der Filamentlösung widerspiegeln.

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.

A highly structured tissue is formed from an unstructured accumulation of cells during morphogenesis. The pioneering works by Thompson, Turing and Meinhardt introduced physical principles allowing the breaking of symmetry, i.e. the emergence of patterns. This started an ongoing effort to understand the physics behind morphogenesis. In this thesis I will analyze spatial and temporal aspects of morphogenesis for different biological systems in separate parts. The planarian is an ideal model animal to understand mechanisms of spatial body axis formation. This is due to the possibility to measure its body orientation field which utilizes the orientation of the cilia of the planarian’s ventral tissue. Moreover, their astonishing regeneration capabilities allow extensive perturbation experiments. I propose a minimal model which demonstrates the emergence of the wild type body orientation as well as the development of dual-head body orientation due to beta-Catenin RNAi treatment. The topological defects of the body orientation field are calculated on a lattice for simulations and lattice-free for experimental data. These topological defects are a robust way to analyze and compare experiments with simulations. My minimal model reveals sufficient components and mechanisms for robust body axis regeneration. The second important aspect of morphogenesis is the growth regulation of tissues which is often driven by cell proliferation. The regulation of growth is not only important during growth, but also to maintain homeostasis. As fast renewing tissues are very dynamic they may have more pathways of morphogenesis active than non-renewing tissues which points to mechanisms of morphogenesis. The in vivo measurement of this proliferation rate is a challenging task. In this thesis the analysis of DNA labelling assays and the carbon 14 dating method are extended. The carbon 14 dating method can be used to determine cell renewal rates on time scales as long as the lifetime of organism last. Moreover, this method can be applied for tissues in any terrestrial organism because it utilizes the change of carbon 14 in the atmosphere due to atmospheric nuclear bomb tests in the 1960s. The method is extended to gain a better understanding of the tissue dynamics of liver, muscles and amygdala. On the other hand, the DNA labelling assays are used to estimate cell cycle parameters for fast cycling cells. The measurements are fatal to the samples and involve plenty of labor resulting in few sampled data for a time series. The deterministic Nowakowski model is extended to a stochastic model accounting for cell-to-cell and sample-to-sample variability to fully exploit the information contained even in the fluctuations of the data points. A comprehensive parameter recovery study with synthetic ground truth data is performed to evaluate the models. The new stochastic model shows no bias, a good accuracy and scales well with the number of measurements in contrast to the deterministic text-book method. I conclude with proposed applications of my new models and methods that can advance our understanding of growth and pattern formation during morphogenesis. All python software developed in this thesis is shared as open source and a website makes the stochastic analysis of DNA labelling assays available to experimentalists in a user-friendly way.

Die vorliegende Arbeit beschäftigt sich mit der Oberflächenglättung und selbstorganisierten Musterbildung auf Si(100) und Ge(100) durch Beschuss mit niederenergetischen Edelgasionen (Ne, Ar, Kr, Xe). Die Untersuchungen wurden für Ionenenergien zwischen 400 eV und 2000 eV für Ioneneinfallswinkel von 0° bis 85° durchgeführt. Zudem wurde die zeitliche Entwicklung spezifischer Erosionsformen durch die Variation der Fluenz über zwei Größenordnungen analysiert. In den Experimenten finden sich deutliche Anzeichen einer Facettierung sowie einer Vergröberung der Strukturen mit zunehmender Erosionszeit. Diese Beobachtungen deuten darauf hin, dass von Beginn an gradientenabhängiges Zerstäuben und die Reflexion von Primärionen einen wesentlichen Einfluss auf die Strukturentwicklung haben. Die Ergebnisse werden im Kontext bestehender Musterbildungsmodelle diskutiert.

All living things are driven by chemical reactions. Reactions provide energy and transform matter. Thus, maintaining the system out of equilibrium. However, these chemical reactions have to be organized in space. One way for this spatial organization is via the process of phase separation. Motivated by the recent discovery of liquid-like droplets in cells, this thesis studies the organization of chemical reactions in phase-separated systems, with and without broken detailed balance. After introducing the underlying thermodynamic principles, we generalize mass-action kinetics to systems with homogeneous compartments formed by phase separation. Here, we discuss the constraints resulting from phase equilibrium on chemical reactions. We study the relaxation kinetics towards thermodynamic equilibrium and investigate non-equilibrium states that arise when detailed balance is broken in the rates of reactions such that phase and chemical equilibria contradict each other. We then turn to spatially continuous systems with spatial gradients within formed compartments. We derive thermodynamic consistent dynamical equations for reactions and diffusion processes in such systems. Again, we study the relaxation kinetics towards equilibrium and discuss non-equilibrium states. We investigate the dynamics of droplets in the presence of reactions with broken detailed balance. Furthermore, we introduce active droplet systems maintained away from equilibrium via coupling to reservoirs at their boundaries and organizing reactions solely within droplets. Here, detailed balance is only broken at the boundaries. Nevertheless, stationary chemically active droplets exist in open systems, and droplets can divide. To quantitatively study chemically active droplet systems in multi-component mixtures, we introduce an effective description. Therefore, we couple linearized reaction-diffusion equations via a moving interface within a sharp interface limit. At the interface, the boundary conditions are set by a local phase equilibrium and the continuity of fluxes. Equipped with these tools, we introduce and study protocell models of chemically active droplets. We explicitly model these protocells’ nutrient and waste dynamics, leading to simple models of their metabolism. Next, we study the energetics of these droplets and identify processes responsible for growth or shrinkage and maintaining the system out of equilibrium. Furthermore, we discuss the energy balance leading to the heating and cooling of droplets. Finally, we show why chemically active droplets do not spontaneously divide in two-dimensional systems with bulk-driven reactions. Here, droplets can elongate but do not pinch off. To have a minimal two-dimensional model with droplet division, we introduce additional reactions. When these reactions are localized at the interface and dependent on its mean curvature, droplets robustly divide in 2D. In summary, this thesis contributes to the theoretical understanding of how the existence of droplets changes the kinetics of reactions and, vice versa, how chemical reactions can alter droplet dynamics.:1 Introduction 1.1 Thermodynamics of phase separation 1.1.1 Phase equilibrium in the thermodynamic limit 1.1.2 Relaxation dynamics towards equilibrium 1.1.3 Local stability of homogeneous phases 1.2 Thermodynamics of chemical reactions in homogenous mixtures 1.2.1 Conserved densities and reaction extents 1.2.2 Equilibrium of chemical reactions 1.2.3 Mass-action kinetics towards equilibrium 1.3 Simultaneous equilibrium of chemical reactions and phase separation 1.4 Chemical reactions maintained away from equilibrium 1.5 Structure of this thesis 2 Chemical reactions in compartmentalized systems 2.1 Mass-action kinetics for compartments built by phase separation 2.1.1 Dynamical equations for densities and phase volumes 2.1.2 Relaxation kinetics in a simple example 2.2 Driven chemical reactions in compartmentalized systems 2.2.1 Non-equilibrium steady states at phase equilibrium 2.2.2 The tie line selecting manifold 2.3 Discussion 3 Dynamics of concentration fields in phase-separating systems with chemical reactions 3.1 Reaction-diffusion equations for phase-separating systems 3.2 Relaxation towards thermodynamic equilibrium in spatial systems 3.2.1 Relaxation kinetics and fast diffusion 3.2.2 Relaxation kinetics with spatial gradients 3.3 Driven chemical reactions in phase-separating systems 3.3.1 Driven chemical reaction and fast diffusion 3.3.2 Non-equilibrium steady states and spatial gradients 3.3.3 Droplets growth and ripening with driven chemical reactions 3.4 Boundary-driven chemically active droplets 3.4.1 Droplets in open systems 3.4.2 Non-equilibrium steady droplets and shape instabilities 3.5 Discussion 4 Chemically active droplets in the sharp interface limit 4.1 Droplet dynamics via reaction-diffusion equations coupled by a moving interface 4.2 Stationary interface positions in spherical symmetry 4.2.1 Interface conditions in closed systems 4.2.2 Interface conditions in open systems 4.3 Shape instabilities of spherical droplets 4.4 Discussion 5 Models of protocells and their metabolism as chemically active droplets 5.1 Breaking detailed balance in protocell models 5.1.1 Boundary-driven protocell models 5.1.2 Bulk-driven protocell models 5.2 Protocell dynamics 5.2.1 Steady states droplets 5.2.2 Shape stability of spherical symmetric droplets 5.3 Energetics of protocells 5.3.1 Mass conservation and droplet growth or shrinkage 5.3.2 Energy conservation and droplet heating or cooling 5.4 Discussion 6 The role of dimensionality on droplet division 6.1 Stability of chemically active droplets in 2D vs. 3D 6.1.1 Stationary droplets in 1D, 2D and 3D 6.1.2 Elongation instability 6.1.3 Pinch-off instability 6.2 Pinch-off in 2D via curvature-dependent chemical reactions 6.2.1 Determining the mean curvature of the droplet interface 6.2.2 Chemical reactions at the interface 6.3 Discussion 7 Conclusion and Outlook A Free energy considerations B Surface tension in multi-component mixtures C Figure details Bibliography

During animal development numerous organs with functions ranging from fluid transport to signal propagation develop into highly branched shapes and forms. To ensure organ function, the formation of their geometrical and topological as well as size-dependent properties is crucial. For example, organ geometry serves to maximize exchange area with its surroundings and organ topology controls the response to fluctuations and damage. Most importantly, organ size and proportion need to scale throughout animal growth to meet the demands of increasing body size. However, how organ geometry and topology are established and scaled in a self-organized manner, remains poorly understood. In this thesis, we present a novel theoretical framework to study the self-organized growth and scaling of branched organs. In this framework, we represent the organ outline by an infinitely thin interface and consider morphogen-controlled interface evolution in growing domains. We demonstrate that an instability in interface motion can lead to the self-organized formation of complex branched morphologies and show how the interplay between interface motion, morphogen dynamics, and domain growth controls the geometrical, topological, and size-dependent properties of the resulting structures. To understand the formation of branched structures from instabilities in morphogen-controlled interface growth, we first consider a range of different interface growth scenarios in non-growing domains. In a first approach, we present a stochastic lattice model with interface growth driven by a morphogen concentration gradient. We find a range of branched morphologies extending from self-similar fractal structures to almost circular structures with only a few branches depending on the morphogen gradient length scale. We present the Euler characteristic as an example of a topological invariant and employ it to introduce topological constraints into interface growth, leading to the formation of tree-like structures. In a second approach, we study a continuum model for morphogen-controlled interface growth. In this model, the interface has a constant tendency to grow and is inhibited by morphogen concentration. Additionally, we take into account a curvature dependency of interface growth, which leads to an effective stabilization of interface motion at small length scales. We identify branch distance and thickness as key morphological properties and discuss their regulation. We relate branch distance regulation to the interplay of destabilization from morphogen inhibition and stabilization from the curvature dependency of interface growth and explain branch thickness regulation in terms of mutual branch inhibition. By considering interface instability in different scenarios, we overall demonstrate the robustness of our approach. Finally, we apply our theoretical framework to study the branching morphogenesis of the planarian gut. The planarian gut is a highly branched organ that spans the entire organism and is responsible for the delivery of nutrients to the planarian body. Planarians undergo massive body size changes of more than one order of magnitude in organism length and thus constitute an ideal model organism to study the growth and scaling of branched organs. We reconsider our continuum model and include novel features needed to account for the organization of the planarian gut. We take into account external guiding cues that alter the orientation of branches and, most importantly, consider branching morphogenesis in a growing domain. We demonstrate that our model can account for the geometrical and topological properties of the gut and show that gut scaling can arise from to the interplay of branch growth and organism growth. Overall, we present a novel theoretical framework to study the growth and scaling of branched organs. In this framework, we demonstrate the self-organized formation of branched morphologies from instabilities in morphogen-controlled interface growth and show how the interplay of interface motion, morphogen dynamics, and system size determine geometry, topology, and size-dependent properties of the resulting structures.

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.

Anders als menschengemachte Maschinen verfügen Zellen über keinen festgeschriebenen Bauplan und die Positionen einzelner Elemente sind häufig nicht genau festgelegt, da die Moleküle diffusiven Zufallsbewegungen unterworfen sind. Darüber hinaus sind einzelne Bauteile auch nicht auf eine einzelne Funktion festgelegt, sondern können parallel in verschiedene Prozesse einbezogen sein. Basierend auf Selbstorganisation und Selbstassemblierung muß die Organisation von Anordnung und Funktion einer lebenden Zelle also bereits in ihren einzelnen Komponenten inhärent enthalten sein. Die intrazelluläre Organisation wird zum großen Teil durch ein internes Biopolymergerüst reguliert, das Zytoskelett. Biopolymer-Netzwerke und –Fasern durchdringen die gesamte Zelle und sind verantworlich für mechanische Integrität und die funktionale Architektur. Unzählige essentielle biologische Prozesse hängen direkt von einem funktionierenden Zytoskelett ab. Die vorliegende Arbeit zielt auf ein besser Verständnis und den Nachbau zweier verschiedener funktionaler Module lebender Zellen anhand stark reduzierter Modellsysteme. Als zentrales Element wurde Aktin gewählt, da dieses Biopolymer eine herausragende Rolle in nahezu allen eukaryotischen Zellen spielt. Mit dem ersten Modellsystem wird der bewegliche Aktin-Polymerfilm an der Vorderkante migrierender Zellen betrachtet. Die wichtigsten Elemente dieser hochdynamischen Netzwerke sind bereits bekannt und wurden in dieser Arbeit benutzt um ein experimentelles Modellsystem zu etablieren. Vor allem aber lieferten detailierte Computersimulationen und ein mathematisches Modell neue Erkenntnisse über grundlegende Organisationsprinzipien dieser Aktinnetzwerke. Damit war es nicht nur möglich, experimentelle Daten erfolgreich zu reproduzieren, sondern das Entstehen von Substrukturen und deren Charakteristika auf proteinunabhängige, generelle Mechanismen zurückzuführen. Das zweite studierte System betrachtet die Selbstassemblierung von Aktinnetzwerken durch entropische Kräfte. Aktinfilamente aggregieren hierbei durch Kondensation multivalenter Ionen oder durch Volumenausschluss hochkonzentrierter inerter Polymere. Ein neu entwickelter Experimentalaufbau bietet die Möglichkeit in gut definierten zellähnlichen Volumina, Konvektionseinflüsse zu umgehen und Aggregationseffekte gezielt einzuschalten. Hierbei wurden neuartige, regelmäßige Netzwerkstrukturen entdeckt, die bislang nur im Zusammenhang mit molekularen Motoren bekannt waren. Es konnte ferner gezeigt werden, dass die Physik der Flüssigkristalle entscheidend zu weiteren Variationen dieser Netzwerke beiträgt. Dabei wird ersichtlich, dass entstehende Netzwerke in ihrer Architektur direkt die zuvor herrschenden Anisotropien der Filamentlösung widerspiegeln.:1 Introduction 1 2 General background 7 2.1 General concepts 7 2.1.1 Coarse-graining as hierarchical reduction 8 2.1.2 Functional modules and redundancies 10 2.1.3 Emergence 11 2.1.4 Self-organization and self-assembly 13 2.1.5 Bottom-up and top-down 13 2.2 The cytoskeleton 15 2.2.1 From actin monomers to filaments 16 2.2.2 Accessory proteins and actin networks 21 2.3 Biopolymer pattern formation 25 2.3.1 Random networks and nematic phases 25 2.3.2 Linker and motor induced networks 28 3 Lamellipodial actin network formation 33 3.1 Background: crawling cell migration 33 3.1.1 Leading edge actin structures 35 3.1.2 Lamellipodial self-organization into oriented branches? 40 3.1.3 Lamellipodial modeling 41 3.1.4 Beyond the lamellipodium: adhesion and network contraction 42 3.2 Methods: lamellar treadmilling model 45 3.2.1 Assumptions 45 3.2.2 Choice of model parameters 51 3.2.3 Computer simulation (implementation) 52 3.2.4 Mathematical modeling 56 3.3 Modeling results 63 3.3.1 Reproduction of motile cell characteristics 64 3.3.2 Self-organization into lamellipodium and lamellum 65 3.3.3 Filament severing and annealing influence network properties 70 3.3.4 Unconfined network growth 74 3.4 Feasible model extensions 76 3.4.1 Alternative nucleation mechanisms 77 3.4.2 Convergence zone through myosin-driven network contraction 80 3.5 Experimental bottom-up approach 82 3.6 Discussion: Arp2/3 induced actin networks 87 4 Actin network patterns in confined systems 91 4.1 Background: counterion condensation and depletion forces 91 4.1.1 Actin, a polyelectrolyte: counterion condensation 92 4.1.2 Actin and molecular crowding: depletion forces 95 4.2 Methods: Experimental design and data analysis 97 4.2.1 Protein purification and handling 98 4.2.2 Droplet formation 98 4.2.3 Volume monitoring and pattern analysis 100 4.3 Actin pattern formation 105 4.3.1 Counterion-induced network formation 105 4.3.2 Depletion force induced network formation 111 4.4 First modeling attempts: bundling simulation 116 4.4.1 Model concept and assumptions 116 4.5 Discussion: Counterion and depletion-based network assembly 119 5 Discussion & Outlook 125 Appendix 129 A. Variation of filament orientation 129 B. Analytical solution of the mathematical model 131 C. Pre-alignment of filaments 132 D. Protocols 134 d1. Acetone Powder Prep 134 d2. Actin prep 135 d3. Actin labling with rhodamine dye 137 Bibliography 141 Acknowledgements 157

The actin cortex of an animal cell is a thin polymeric layer attached to the inner side of the plasma membrane. It plays a key role in shape regulation and pattern formation on the cellular and tissue scale and, in particular, generates the contractile ring during cell division. Experimental studies showed that the cortex is fluid-like but highly viscous on long time scales with a mechanics that is sensitively regulated by active and passive cross-linker molecules that tune active stress and shear viscosity. Here, we use an established minimal model of active surface dynamics of the cell cortex supplemented with the experimentally motivated feature of mechanosensitivity in cross-linker binding dynamics. Performing linear stability analysis and computer simulations, we show that cross-linker mechanosensitivity significantly enhances the versatility of pattern formation and enables self-organized formation of contractile rings. Furthermore, we address the scenario of concentration-dependent shear viscosities as a way to stabilize ring-like patterns and constriction in the mid-plane of the active surface.

In this thesis a new frequency domain approach for the analysis of time delay systems is presented. After linearization of a nonlinear delay differential equation (DDE) with constant distributed delay around a constant or periodic reference solution the so-called Hill-Floquet method can be used for the analysis of the resulting linear DDE. In addition, systems with fast or slowly time-varying delays, systems with variable transport delays originating from a transport with variable velocity, and the corresponding spatially extended systems are presented, which can be also analyzed with the presented method. The newly introduced Hill-Floquet method is based on the Hill’s infinite determinant method and enables the transformation of a system with periodic coefficients to an autonomous system with constant coefficients. This makes the usage of a variety of existing methods for autonomous systems available for the analysis of periodic systems, which implies that the typical calculation of the monodromy matrix for the time evolution of the solution over the principle period is no longer required. In this thesis, the Chebyshev collocation method is used for the analysis of the autonomous systems. Specifically, in this case the periodic part of the solution is expanded in a Fourier series and the exponential behavior of the solution is approximated by the discrete values of the Fourier coefficients at the Chebyshev nodes, whereas in classical spectral or pseudo-spectral methods for the analysis of linear periodic DDEs the complete solution is expanded in terms of basis functions. In the last part of this thesis, new results for three applications with time delay effects are presented, which were analyzed with the presented methods. On the one hand, the occurrence of diffusion-driven instabilities in reaction-diffusion systems with delay is investigated. It is shown that wave instabilities are possible already for single-species reaction diffusion systems with distributed or time-varying delay. On the other hand, the stability of metal cutting vibrations at machine tools is analyzed. In particular, parallel orthogonal turning processes with multiple discrete delays and turning processes with a time-varying delay due to a spindle speed variation are studied. Finally, the stability of the synchronized solution in networks with heterogeneous coupling delays is studied. In particular, the eigenmode expansion for synchronized periodic orbits is derived, which includes an extension of the classical master stability function to networks with heterogeneous coupling delays. Numerical results are shown for a network of Hodgkin-Huxley neurons with two delays in the coupling. / In dieser Dissertation wird ein neues Verfahren zur Analyse von Systemen mit Totzeiten im Frequenzraum vorgestellt. Nach Linearisierung einer nichtlinearen retardierten Differentialgleichung (DDE) mit konstanter verteilter Totzeit um eine konstante oder periodische Referenzlösung kann die sogenannte Hill-Floquet Methode für die Analyse der resultierende linearen DDE angewendet werden. Darüber hinaus werden Systeme mit schnell oder langsam variierender Totzeit, Systeme mit einer variablen Totzeit, resultierend aus einem Transport mit variabler Geschwindigkeit, und entsprechende räumlich ausgedehnte Systeme vorgestellt, welche ebenfalls mit der vorgestellten Methode analysiert werden können. Die neu eingeführte Hill-Floquet Methode basiert auf der Hillschen unendlichen Determinante und ermöglicht die Transformation eines Systems mit periodischen Koeffizienten auf ein autonomes System mit konstanten Koeffizienten. Dadurch können zur Analyse periodischer Systeme auch eine Vielzahl existierender Methoden für autonome Systeme genutzt werden und die Berechnung der Monodromie-Matrix für die Lösung des Systems über eine Periode entfällt. In dieser Arbeit wird zur Analyse des autonomen Systems die Tschebyscheff-Kollokationsmethode verwendet. Im Speziellen wird bei diesem Verfahren der periodische Teil der Lösung in einer Fourierreihe entwickelt und das exponentielle Verhalten durch die Werte der Fourierkoeffizienten an den Tschebyscheff Knoten approximiert, wohingegen bei klassischen spektralen Verfahren die komplette Lösung in bestimmten Basisfunktionen entwickelt wird. Im Anwendungsteil der Arbeit werden neue Ergebnisse für drei Beispielsysteme präsentiert, welche mit den vorgestellten Methoden analysiert wurden. Es wird gezeigt, dass Welleninstabilitäten schon bei Einkomponenten-Reaktionsdiffusionsgleichungen mit verteilter oder variabler Totzeit auftreten können. In einem zweiten Beispiel werden Schwingungen an Werkzeugmaschinen betrachtet, wobei speziell simultane Drehbearbeitungsprozesse und Prozesse mit Drehzahlvariationen genauer untersucht werden. Am Ende wird die Synchronisation in Netzwerken mit heterogenen Totzeiten in den Kopplungstermen untersucht, wobei die Zerlegung in Netzwerk-Eigenmoden für synchrone periodische Orbits hergeleitet wird und konkrete numerische Ergebnisse für ein Netzwerk aus Hodgkin-Huxley Neuronen gezeigt werden.