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

In unserem täglichen Leben begegnen wir Tropfen oft in physikalischen Systems, beispielsweise als Öltropfen in Salatsoße. Diese Tropfen sind meist chemisch inaktiv. In biologischen Zellen bilden Proteine und RNA zusammen Tropfen. Zellen sind chemisch aktiv, so dass die Tropfenkomponenten neu gebildet, abgebaut und modifiziert werden können. In dieser Doktorarbeit wird das dynamische Verhalten von chemisch aktiven Tropfen mit analytischen und numerischen Methoden untersucht. Um das dynamische Verhalten von solchen aktiven Tropfen zu untersuchen, benutzen wir ein Minimalmodell mit zwei Komponenten, die zwei Phasen bilden und durch chemische Reaktionen ineinander umgewandelt werden. Die chemischen Reaktionen werden durch das Brechen von Detailed Balance aus dem Gleichgewicht gehalten, so dass die Tropfen chemisch aktiv sind. Wir konzentrieren uns auf den Fall, in dem Tropfenmaterial im Tropfen in die äußere Komponente umgewandelt wird, und in der äußeren Phase erzeugt wird. Wir finden ein vielfältiges dynamisches Phasendiagramm mit Regionen, in denen Tropfen schrumpfen und verschwinden, Regionen, in denen Tropfen eine stabile stationäre Größe besitzen, und Regionen, in denen eine Forminstabilität zu komplexer Tropfen-Dynamik führt. In der letzten Region deformieren sich Tropfen typischenweise prolat, verformen sich zu einer Hantel, und teilen sich in zwei Tochtertropfen, die wieder anwachsen. Dies kann zu Zyklen von Wachstum und Teilung von Tropfen führen, bis die Tropfen das gesamte Volumen füllen. Während spherische Tropfen durch die chemischen Reaktionen entgegen ihrer Oberflächenspannung deformiert werden, können Tropfen- Zylinder und Platten durch chemische Reaktionen stabilisiert werden. Generell ist die Dynamik von Tropfen ein hydrodynamisches Problem, da die Oberflächenspannung von deformierten Tropfen hydrodynamische Flüsse erzeugt. Wir finden, dass chemische Reaktionen entgegen die Oberflächenspannung Arbeit verrichten können, so dass die Tropfenteilung auch unter Berücksichtigung hydrodynamischer Flüsse möglich ist. Diese Doktorarbeit zeigt, dass die Kombination von chemische Reaktionen und Phasenseparation unter Nichtgleichgewichtsbedingungen zu neuem dynamischen Verhalten führen kann. Die Ergebnisse zeigen die Relevanz von chemischen Reaktionen zum Verständnis von Phasenseparation in biologischen Systemen auf, und können bei der Umsetzung der diskutierten Phänomene in experimentellen Systemen helfen. Die Tropfenteilung, die in dieser Doktorarbeit diskutiert wird, erinnert an die Teilung von biologischen Zellen. Davon motiviert schlagen wir vor, dass die Teilung von chemisch aktiven Tropfen ein Mechanismus für die Replikation von Tropfen-artigen Protozellen am Ursprung des Lebens gewesen sein könnte.:1. Introduction 2. Theory of multi-component phase-separating systems with chemical reactions 3. Minimal model for chemically active droplets in two formulations 4. Shape instability of spherical droplets with chemical reactions 5. Dynamical behavior of chemically active droplets 6. Shape instability of droplets with various geometries 7. Role of hydrodynamic flows in chemically driven droplet division 8. Chemically active droplets as a model for protocells at the origin of life 9. Conclusion Appendices / In our everyday environment, we regularly encounter liquid-liquid phase separation in physical systems such as oil droplets in vinegar. These droplets tend to be chemically inert. In biological cells, protein and RNA may together form liquid droplets. Cells are chemically active, so that droplet components can be created, degraded and modified. In this thesis we study the influence of nonequilibrium chemical reactions on the shape dynamics of a droplet theoretically, using analytical and numerical methods. To discuss the dynamical behavior that results from combining phase separation and chemical reactions in sustained nonequilibrium conditions, we introduce a minimal model with only two components that separate into distinct phases. These two components are converted into each other by chemical reactions. The reactions are kept out of equilibrium by breaking of detailed balance, so that the droplet becomes active. We concentrate on the case where the reaction inside the droplet degrades droplet material into the outer component, and where the reaction outside creates new droplet material. We find that chemically active droplets have a rich dynamic phase space, with regions where droplets shrink and vanish, regions where droplets have a stable stationary size, and regions where the flux-driven instability leads to complex dynamic behavior of droplets. In the latter, droplets typically elongate into a dumbbell shape and then split into two symmetrical daughter droplets. These droplets then grow until they have the same size as the initial droplet. This can lead to cycles of growth and division, so that an initial droplet divides until droplets fill the simulation volume. We analyze the stationary spherical state of the droplet, which is created by a balance of the fluxes driven by the chemical reactions. We find that stationary droplets may have a shape instability, which is driven by the continuous fluxes across the droplet interface and which may trigger the division. We also find that while reactions may destabilize spherical droplet shapes despite the surface tension of the droplet, they can have stabilizing effects on cylindrical droplets and droplet plates. Generally, the shape dynamics of droplets is a hydrodynamic problem because surface tension in non-spherical droplets drives hydrodynamic flows that redistribute material and deform the droplet shape. We therefore study the influence of hydrodynamic flows on the shape changes of chemically active droplets. We find that chemical reactions in active droplets can perform work against surface tension and flows, so that the droplet division is possible even in the presence of hydrodynamic flows. The present thesis highlights how the combination of basic physical behaviors – phase separation and chemical reactions – may create novel dynamic behavior under sustained nonequilibrium conditions. The results demonstrate the importance of considering chemical reactions for understanding the dynamics of droplets in biological systems, as well as proposes a minimalist model for experimentalists that are interested in creating a system of dividing droplets. Finally, the division of chemically active droplets is reminiscent of the division of biological cells, and it motivates us to propose that chemically active droplets could have provided a simple mechanism for the self-replication of droplet-like protocells at the origin of life.:1. Introduction 2. Theory of multi-component phase-separating systems with chemical reactions 3. Minimal model for chemically active droplets in two formulations 4. Shape instability of spherical droplets with chemical reactions 5. Dynamical behavior of chemically active droplets 6. Shape instability of droplets with various geometries 7. Role of hydrodynamic flows in chemically driven droplet division 8. Chemically active droplets as a model for protocells at the origin of life 9. Conclusion Appendices

Biological cells consist of many subunits that form distinct compartments and work together to allow for life. These compartments are clearly separated from each other and their sizes are often strongly correlated with cell size. Examples for those structures are centrosomes, which we consider in this thesis. Centrosomes are essential for many processes inside cells, most importantly for organizing cell division, and they provide an interesting example of cellular compartments without a membrane. Experiments suggest that such compartments can be described as liquid-like droplets. In this thesis, we suggest a theoretical description of the growth phase of centrosomes. We identify a possible mechanism based on phase separation by which the centrosome may be organized. Specifically, we propose that the centrosome material exists in a soluble and in a phase separating form. Chemical reactions controlling the transitions between these forms then determine the temporal evolution of the system. We investigate various possible reaction schemes and generally find that droplet sizes and nucleation properties deviate from the known equilibrium results. Additionally, the non-equilibrium effects of the chemical reactions can stabilize multiple droplets and thus counteract the destabilizing effect of surface tension. Interestingly, only a reaction scheme with autocatalytic growth can account for the experimental data of centrosomes. Here, it is important that the centrioles found at the center of all centrosomes also catalyze the production of droplet material. This catalytic activity allows the centrioles to control the onset of centrosome growth, to stabilize multiple centrosomes, and to center themselves inside the centrosome. We also investigate a stochastic version of the model, where we find that the autocatalytic growth amplifies noise. Our theory explains the growth dynamics of the centrosomes of the round worm Caenorhabditis elegans for all embryonic cells down to the eight-cell stage. It also accounts for data acquired in experiments with aberrant numbers of centrosomes and altered cell volumes. Furthermore, the model can describe unequal centrosome sizes observed in cells with disturbed centrioles. Our example thus suggests a general picture of the organization of membrane-less organelles. / Biologische Zellen bestehen aus vielen Unterstrukturen, die zusammen arbeiten um Leben zu ermöglichen. Die Größe dieser meist klar voneinander abgegrenzten Strukturen korreliert oft mit der Zellgröße. In der vorliegenden Arbeit werden als Beispiel für solche Strukturen Zentrosomen untersucht. Zentrosomen sind für viele Prozesse innerhalb der Zelle, insbesondere für die Zellteilung, unverzichtbar und sie besitzen keine Membran, welche ihnen eine feste Struktur verleihen könnte. Experimentelle Untersuchungen legen nahe, dass solche membranlose Strukturen als Flüssigkeitstropfen beschrieben werden können. In dieser Arbeit wird eine theoretische Beschreibung der Wachstumsphase von Zentrosomen hergeleitet, welche auf Phasenseparation beruht. Im Modell wird angenommen, dass das Zentrosomenmaterial in einer löslichen und einer phasenseparierenden Form existiert, wobei der Übergang zwischen diesen Formen durch chemische Reaktionen gesteuert wird. Die drei verschiedenen in dieser Arbeit untersuchten Reaktionen führen unter anderem zu Tropfengrößen und Nukleationseigenschaften, welche von den bekannten Ergebnissen im thermodynamischen Gleichgewicht abweichen. Insbesondere verursachen die chemischen Reaktionen ein thermisches Nichtgleichgewicht, in dem mehrere Tropfen stabil sein können und der destabilisierende Effekt der Oberflächenspannung unterdrückt wird. Konkret kann die Wachstumsdynamik der Zentrosomen nur durch eine selbstverstärkende Produktion der phasenseparierenden Form des Zentrosomenmaterials erklärt werden. Hierbei ist zusätzlich wichtig, dass die Zentriolen, die im Inneren jedes Zentrosoms vorhanden sind, ebenfalls diese Produktion katalysieren. Dadurch können die Zentriolen den Beginn des Zentrosomwachstums kontrollieren, mehrere Zentrosomen stabilisieren und sich selbst im Zentrosom zentrieren. Des Weiteren führt das selbstverstärkende Wachstum zu einer Verstärkung von Fluktuationen der Zentrosomgröße. Unsere Theorie erklärt die Wachstumsdynamik der Zentrosomen des Fadenwurms Caenorhabditis elegans für alle Embryonalzellen bis zum Achtzellstadium und deckt dabei auch Fälle mit anormaler Zentrosomenanzahl und veränderter Zellgröße ab. Das Modell kann auch Situationen mit unterschiedlich großen Zentrosomen erklären, welche auftreten, wenn die Struktur der Zentriolen verändert wird. Unser Beispiel beschreibt damit eine generelle Möglichkeit, wie membranlose Zellstrukturen organisiert sein können.

In this thesis extensions to the formalism of endoreversible thermodynamics for multi-extensity fluxes and chemical reactions are introduced. These extensions make it possible to model a great variety of systems which could not be investigated with standard endoreversible thermodynamics. Multi-extensity fluxes are important when studying processes with matter fluxes or processes in which volume and entropy are exchanged between subsystems. For including reversible as well as irreversible chemical reaction processes a new type of subsystems is introduced - the so called reactor. It is similar to endoreversible engines, because the fluxes connected to it are balanced. The difference appears in the balance equations for particle numbers, which contain production or destruction terms, and in the possible entropy production in the reactor. Both extensions are then applied to an endoreversible fuel cell model. The chemical reactions in the anode and cathode of the fuel cell are included with the newly introduced subsystem -- the reactor. For the transport of the reactants and products as well as the proton transport through the electrolyte membrane, the multi-extensity fluxes are used. This fuel cell model is then used to calculate power output, efficiency and cell voltage of a fuel cell with irreversibilities in the proton and electron transport. It directly connects the pressure and temperature dependencies of the cell voltage with the dissipation due to membrane resistance. Additionally, beside the listed performance measures it is possible to quantify and localize the entropy production and dissipated heat with only this one model. / In dieser Arbeit erweitere ich den Formalismus der endoreversiblen Thermodynamik, um Flüsse mit mehr als einer extensiven Größe sowie chemische Reaktionsprozesse modellieren zu können. Mit Hilfe dieser Erweiterungen eröffnen sich zahlreiche neue Anwendungsmöglichkeiten für endoreversible Modelle. Flüsse mit mehreren extensiven Größen sind für die Betrachtung von Masseströmen ebenso nötig wie für Prozesse, bei denen sowohl Volumen als auch Entropie zwischen zwei Teilsystem ausgetauscht werden. Für sowohl reversibel wie auch irreversibel geführte chemische Reaktionsprozesse wird ein neues Teilsystem - der "Reaktor" - vorgestellt, welches sich ähnlich wie endoreversible Maschinen durch Bilanzgleichungen auszeichnet. Der Unterschied zu den Maschinen besteht in den Produktions- bzw. Vernichtungstermen in den Teilchenzahlbilanzen sowie der möglichen Entropieproduktion innerhalb des Reaktors. Beide Erweiterungen finden dann in einem endoreversiblen Modell einer Brennstoffzelle Anwendung. Dabei werden Flüsse mehrerer gekoppelter Extensitäten für den Zustrom von Wasserstoff und Sauerstoff sowie für den Protonentransport durch die Elektrolytmembran benötigt. Chemische Reaktionen treten in der Anode und Kathode der Brennstoffzelle auf. Diese werden mit dem neu eingeführten Teilsystem, dem Reaktor, eingebunden. Mit Hilfe des Modells werden dann Wirkungsgrad, Zellspannung und Leistung einer Brennstoffzelle unter Berücksichtigung der Partialdrücke der Substanzen, der Temperatur sowie der Dissipation beim Protonentransport berechnet. Dabei zeigt sich, dass experimentelle Daten für die Zellspannung sowohl qualitativ als auch näherungsweise quantitativ durch das Modell abgebildet werden können. Der Vorteil des endoreversiblen Modells liegt dabei in der Möglichkeit, mit nur einem Modell neben den genannten Kenngrößen auch die abgegebene Wärme sowie die Entropieproduktion zu quantifizieren und den einzelnen Teilprozessen zuzuordnen.

Biological cells consist of many subunits that form distinct compartments and work together to allow for life. These compartments are clearly separated from each other and their sizes are often strongly correlated with cell size. Examples for those structures are centrosomes, which we consider in this thesis. Centrosomes are essential for many processes inside cells, most importantly for organizing cell division, and they provide an interesting example of cellular compartments without a membrane. Experiments suggest that such compartments can be described as liquid-like droplets. In this thesis, we suggest a theoretical description of the growth phase of centrosomes. We identify a possible mechanism based on phase separation by which the centrosome may be organized. Specifically, we propose that the centrosome material exists in a soluble and in a phase separating form. Chemical reactions controlling the transitions between these forms then determine the temporal evolution of the system. We investigate various possible reaction schemes and generally find that droplet sizes and nucleation properties deviate from the known equilibrium results. Additionally, the non-equilibrium effects of the chemical reactions can stabilize multiple droplets and thus counteract the destabilizing effect of surface tension. Interestingly, only a reaction scheme with autocatalytic growth can account for the experimental data of centrosomes. Here, it is important that the centrioles found at the center of all centrosomes also catalyze the production of droplet material. This catalytic activity allows the centrioles to control the onset of centrosome growth, to stabilize multiple centrosomes, and to center themselves inside the centrosome. We also investigate a stochastic version of the model, where we find that the autocatalytic growth amplifies noise. Our theory explains the growth dynamics of the centrosomes of the round worm Caenorhabditis elegans for all embryonic cells down to the eight-cell stage. It also accounts for data acquired in experiments with aberrant numbers of centrosomes and altered cell volumes. Furthermore, the model can describe unequal centrosome sizes observed in cells with disturbed centrioles. Our example thus suggests a general picture of the organization of membrane-less organelles.:1 Introduction 1.1 Organization of the cell interior 1.2 Biology of centrosomes 1.2.1 The model organism Caenorhabditis elegans 1.2.2 Cellular functions of centrosomes 1.2.3 The centriole pair is the core structure of a centrosome 1.2.4 Pericentriolar material accumulates around the centrioles 1.3 Other membrane-less organelles and their organization 1.4 Phase separation as an organization principle 1.5 Equilibrium physics of liquid-liquid phase separation 1.5.1 Spinodal decomposition and droplet formation 1.5.2 Formation of a single droplet 1.5.3 Ostwald ripening destabilizes multiple droplets 1.6 Non-equilibrium phase separation caused by chemical reactions 1.7 Overview of this thesis 2 Physical Description of Centrosomes as Active Droplets 2.1 Physical description of centrosomes as liquid-like droplets 2.1.1 Pericentriolar material as a complex fluid 2.1.2 Reaction-diffusion kinetics of the components 2.1.3 Centrioles described as catalytic active cores 2.1.4 Droplet formation and growth kinetics 2.1.5 Complete set of the dynamical equations 2.2 Three simple growth scenarios 2.2.1 Scenario A: First-order kinetics 2.2.2 Scenario B: Autocatalytic growth 2.2.3 Scenario C: Incorporation at the centrioles 2.3 Diffusion-limited droplet growth 2.4 Discussion 3 Isolated Active Droplets 3.1 Compositional fluxes in the stationary state 3.2 Critical droplet size: Instability of small droplets 3.3 Droplet nucleation facilitated by the active core 3.4 Interplay of critical droplet size and nucleation 3.5 Perturbations of the spherical droplet shape 3.5.1 Linear stability analysis of the spherical droplet shape 3.5.2 Active cores can center themselves in droplets 3.5.3 Surface tension stabilizes the spherical shape 3.5.4 First-order kinetics destabilize large droplets 3.6 Discussion 4 Multiple Interacting Active Droplets 4.1 Approximate description of multiple droplets 4.2 Linear stability analysis of the symmetric state 4.3 Late stage droplet dynamics and Ostwald ripening 4.4 Active droplets can suppress Ostwald ripening 4.4.1 Perturbation growth rate in the simple growth scenarios 4.4.2 Parameter dependence of the stability of multiple droplets 4.4.3 Stability of more than two droplets 4.5 Discussion 5 Active Droplets with Fluctuations 5.1 Stochastic version of the active droplet model 5.1.1 Comparison with the deterministic model 5.1.2 Ensemble statistics and ergodicity 5.1.3 Quantification of fluctuations by the standard deviation 5.2 Noise amplification by the autocatalytic reaction 5.3 Transient growth regime of multiple droplets 5.4 Influence of the system geometry on the droplet growth 5.5 Discussion 6 Comparison Between Theory and Experiment 6.1 Summary of the experimental observations 6.2 Estimation of key model parameters 6.3 Fits to experimental data 6.4 Dependence of centrosome size on cell volume and centrosome count 6.5 Nucleation and stability of centrosomes 6.6 Multiple centrosomes with unequal sizes 6.7 Disintegration phase of centrosomes 7 Summary and Outlook Appendix A Coexistence conditions in a ternary fluid B Instability of multiple equilibrium droplets C Numerical solution of the droplet growth D Diffusion-limited growth of a single droplet E Approximate efflux of droplet material F Determining stationary states of single droplets G Droplet size including surface tension effects H Distortions of the spherical droplet shape H.1 Harmonic distortions of a sphere H.2 Physical description of the perturbed droplet H.3 Volume fraction profiles in the perturbed droplet H.4 Perturbation growth rates I Multiple droplets with gradients inside droplets J Numerical stability analysis of multiple droplets K Numerical implementation of the stochastic model / Biologische Zellen bestehen aus vielen Unterstrukturen, die zusammen arbeiten um Leben zu ermöglichen. Die Größe dieser meist klar voneinander abgegrenzten Strukturen korreliert oft mit der Zellgröße. In der vorliegenden Arbeit werden als Beispiel für solche Strukturen Zentrosomen untersucht. Zentrosomen sind für viele Prozesse innerhalb der Zelle, insbesondere für die Zellteilung, unverzichtbar und sie besitzen keine Membran, welche ihnen eine feste Struktur verleihen könnte. Experimentelle Untersuchungen legen nahe, dass solche membranlose Strukturen als Flüssigkeitstropfen beschrieben werden können. In dieser Arbeit wird eine theoretische Beschreibung der Wachstumsphase von Zentrosomen hergeleitet, welche auf Phasenseparation beruht. Im Modell wird angenommen, dass das Zentrosomenmaterial in einer löslichen und einer phasenseparierenden Form existiert, wobei der Übergang zwischen diesen Formen durch chemische Reaktionen gesteuert wird. Die drei verschiedenen in dieser Arbeit untersuchten Reaktionen führen unter anderem zu Tropfengrößen und Nukleationseigenschaften, welche von den bekannten Ergebnissen im thermodynamischen Gleichgewicht abweichen. Insbesondere verursachen die chemischen Reaktionen ein thermisches Nichtgleichgewicht, in dem mehrere Tropfen stabil sein können und der destabilisierende Effekt der Oberflächenspannung unterdrückt wird. Konkret kann die Wachstumsdynamik der Zentrosomen nur durch eine selbstverstärkende Produktion der phasenseparierenden Form des Zentrosomenmaterials erklärt werden. Hierbei ist zusätzlich wichtig, dass die Zentriolen, die im Inneren jedes Zentrosoms vorhanden sind, ebenfalls diese Produktion katalysieren. Dadurch können die Zentriolen den Beginn des Zentrosomwachstums kontrollieren, mehrere Zentrosomen stabilisieren und sich selbst im Zentrosom zentrieren. Des Weiteren führt das selbstverstärkende Wachstum zu einer Verstärkung von Fluktuationen der Zentrosomgröße. Unsere Theorie erklärt die Wachstumsdynamik der Zentrosomen des Fadenwurms Caenorhabditis elegans für alle Embryonalzellen bis zum Achtzellstadium und deckt dabei auch Fälle mit anormaler Zentrosomenanzahl und veränderter Zellgröße ab. Das Modell kann auch Situationen mit unterschiedlich großen Zentrosomen erklären, welche auftreten, wenn die Struktur der Zentriolen verändert wird. Unser Beispiel beschreibt damit eine generelle Möglichkeit, wie membranlose Zellstrukturen organisiert sein können.:1 Introduction 1.1 Organization of the cell interior 1.2 Biology of centrosomes 1.2.1 The model organism Caenorhabditis elegans 1.2.2 Cellular functions of centrosomes 1.2.3 The centriole pair is the core structure of a centrosome 1.2.4 Pericentriolar material accumulates around the centrioles 1.3 Other membrane-less organelles and their organization 1.4 Phase separation as an organization principle 1.5 Equilibrium physics of liquid-liquid phase separation 1.5.1 Spinodal decomposition and droplet formation 1.5.2 Formation of a single droplet 1.5.3 Ostwald ripening destabilizes multiple droplets 1.6 Non-equilibrium phase separation caused by chemical reactions 1.7 Overview of this thesis 2 Physical Description of Centrosomes as Active Droplets 2.1 Physical description of centrosomes as liquid-like droplets 2.1.1 Pericentriolar material as a complex fluid 2.1.2 Reaction-diffusion kinetics of the components 2.1.3 Centrioles described as catalytic active cores 2.1.4 Droplet formation and growth kinetics 2.1.5 Complete set of the dynamical equations 2.2 Three simple growth scenarios 2.2.1 Scenario A: First-order kinetics 2.2.2 Scenario B: Autocatalytic growth 2.2.3 Scenario C: Incorporation at the centrioles 2.3 Diffusion-limited droplet growth 2.4 Discussion 3 Isolated Active Droplets 3.1 Compositional fluxes in the stationary state 3.2 Critical droplet size: Instability of small droplets 3.3 Droplet nucleation facilitated by the active core 3.4 Interplay of critical droplet size and nucleation 3.5 Perturbations of the spherical droplet shape 3.5.1 Linear stability analysis of the spherical droplet shape 3.5.2 Active cores can center themselves in droplets 3.5.3 Surface tension stabilizes the spherical shape 3.5.4 First-order kinetics destabilize large droplets 3.6 Discussion 4 Multiple Interacting Active Droplets 4.1 Approximate description of multiple droplets 4.2 Linear stability analysis of the symmetric state 4.3 Late stage droplet dynamics and Ostwald ripening 4.4 Active droplets can suppress Ostwald ripening 4.4.1 Perturbation growth rate in the simple growth scenarios 4.4.2 Parameter dependence of the stability of multiple droplets 4.4.3 Stability of more than two droplets 4.5 Discussion 5 Active Droplets with Fluctuations 5.1 Stochastic version of the active droplet model 5.1.1 Comparison with the deterministic model 5.1.2 Ensemble statistics and ergodicity 5.1.3 Quantification of fluctuations by the standard deviation 5.2 Noise amplification by the autocatalytic reaction 5.3 Transient growth regime of multiple droplets 5.4 Influence of the system geometry on the droplet growth 5.5 Discussion 6 Comparison Between Theory and Experiment 6.1 Summary of the experimental observations 6.2 Estimation of key model parameters 6.3 Fits to experimental data 6.4 Dependence of centrosome size on cell volume and centrosome count 6.5 Nucleation and stability of centrosomes 6.6 Multiple centrosomes with unequal sizes 6.7 Disintegration phase of centrosomes 7 Summary and Outlook Appendix A Coexistence conditions in a ternary fluid B Instability of multiple equilibrium droplets C Numerical solution of the droplet growth D Diffusion-limited growth of a single droplet E Approximate efflux of droplet material F Determining stationary states of single droplets G Droplet size including surface tension effects H Distortions of the spherical droplet shape H.1 Harmonic distortions of a sphere H.2 Physical description of the perturbed droplet H.3 Volume fraction profiles in the perturbed droplet H.4 Perturbation growth rates I Multiple droplets with gradients inside droplets J Numerical stability analysis of multiple droplets K Numerical implementation of the stochastic model

The docotoral thesis titled 'Modeling the effects of Transient Stream Flow on Solute Dynamics in Stream Banks and Intra-meander Zones' investigates flow and solute dynamcis across surface water-groundwater interface under dynamic flow conditons through numerical simulations. The abstract of the thesis is as follows: Waters from various sources meet at the interface between streams and groundwater. Due to their different origins, these waters often have contrasting chemical signatures and therefore mixing of water at the interface may lead to significant changes in both surface and subsurface water quality. The riparian zone adjacent to the stream serves as transition region between groundwater and stream water, where complex water and solute mixing and transport processes occur. Predicting the direction and the magnitude of solute exchanges and the extent of transformations within the riparian zone is challenging due to the varying hydrologic and chemical conditions as well as heterogeneous morphological features which result in complex, three-dimensional flow patterns. The direction of water flow and solute transport in the riparian zone typically varies over time as a result of fluctuating stream water and groundwater levels. Particularly, increasing groundwater levels can mobilize solutes from the unsaturated zone which can be subsequently transported into the stream. Such complex, spatially and temporally varying processes are hard to capture with field observations alone and therefore modeling approaches are required to predict the system behavior as well as to understand the role of individual factors. In this thesis, we investigate the inter-connectivity of streamthe s and adjacent riparia zones in the context of water and solute exchanges both laterally for bank storage and longitudinally for hyporheic flow through meander bends. Using numerical modeling, the transient effect of stream flow events on solute transport and transformation within the initially unsaturated part of stream banks and meander bends have been simulated using a systematic set of hydrological, chemical and morphological scenarios. A two dimensional variably saturated media groundwater modeling set up was used to explore solute dynamics during bank flows. We simulated exchanges between stream and adjacent riparian zone driven by stream stage fluctuations during stream discharge events. To elucidate the effect of magnitude and duration of discharge events, we developed a number of single discharge event scenarios with systematically varying peak heights and event duration. The dominant solute layer was represented by applying high solute concentration in upper unsaturated riparian zone profile. Simulated results show that bank flows generated by high stream flow events can trigger solute mobilization in near stream riparian soils and subsequently export significant amounts of solutes into the stream. The timing and amount of solute export is linked to the shape of the discharge event. Higher peaks and increased duration significantly enhance solute export, however, peak height is found to be the dominant control for overall lateral mass export. The mobilized solutes are transported towards the stream in two stages (1) by return flow of stream water that was stored in the riparian zone during the event and (2) by vertical movement to the groundwater under gravity drainage from the unsaturated parts of the riparian zone, which lasts for significantly longer time (> 400 days) resulting in a theoretically long tailing of bank outflows and solute mass outfluxes. Our bank flow simulations demonstrate that strong stream discharge events are likely to mobilize and export significant quantity of solutes from near stream riparian zones into the stream. Furthermore, the impact of short-term stream discharge variations on solute exchange may sustain for long times after the flow event. Meanders are prominent morphological features of stream systems which exhibit unique hydrodynamics. The water surface elevation difference across the inner bank of a meander induces lateral hyporheic exchange flow through the intrameander region, leading to solute transport and reactions within intra-meander region. We examine the impact of different meander geometries on the intra-meander hyporheic flow field and solute mobilization under both steady-state and transient flow conditions. In order to explore the impact of meander morphology on intrameander flow, a number of theoretical meander shape scenarios, representing various meander evolution stages, ranging from a typical initial to advanced stage (near cut off ) meander were developed. Three dimensional steady-state numerical groundwater flow simulations including the unsaturated zone were performed for the intra-meander region for all meander scenarios. The meandering stream was implemented in the model by adjusting the top layers of the modeling domain to the streambed elevation. Residence times for the intra-meander region were computed by advective particle tracking across the inner bank of meander. Selected steady state cases were extended to transient flow simulations to evaluate the impact of stream discharge events on the temporal behavior of the water exchange and solute transport in the intra-meander region. Transient hydraulic heads obtained from the surface water model were applied as transient head boundary conditions to the streambed cells of the groundwater model. Similar to the bank storage case, a high concentration of solute (carbon source) representing the dominant solute layer in the riparian profile was added in the unsaturated zone to evaluate the effect of stream flow event on mobilization and transport from the unsaturated part of intrameander region. Additionally, potential chemical reactions of aerobic respiration by the entry of oxygen rich surface water into subsurface as well denitrification due to stream and groundwater borne nitrates were also simulated. The results indicate that intra-meander mean residence times ranging from 18 to 61 days are influenced by meander geometry, as well as the size of the intra-meander area. We found that, intra-meander hydraulic gradient is the major control of RTs. In general, larger intra-meander areas lead to longer flow paths and higher mean intra-meander residence times (MRTs), whereas increased meander sinuosity results in shorter MRTs. The vertical extent of hyporheic flow paths generally decreases with increasing sinuosity. Transient modeling of hyporheic flow through meanders reveals that large stream flow events mobilize solutes from the unsaturated portion of intra-meander region leading to consequent transport into the stream via hyporheic flow. Advective solute transport dominates during the flow event; however significant amount of carbon is also consumed by aerobic respiration and denitrification. These reactions continue after the flow events depending upon the availability of carbon source. The thesis demonstrates that bank flows and intra-meander hyporheic exchange flows trigger solute mobilization from the dominant solute source layers in the RZ. Stream flow events driven water table fluctuations in the stream bank and in the intra-meander region transport substantial amount of solutes from the unsaturated RZ into the stream and therefore have significant potential to alter stream water quality.:Declaration Abstract Zusammenfassung 1 General Introduction 1.1 Background and Motivation 1.2 Hydrology and Riparian zones 1.2.1 Transport processes driven by fluctuation in riparian water table depth 1.2.1.1 Upland control 1.2.1.2 Stream control 1.2.2 Biochemical Transformations within the Riparian Zone 1.3 Types and scales of stream-riparian exchange 1.3.1 Hyporheic Exchange 1.3.1.1 Small Scale Vertical HEF 1.3.1.2 Large Scale lateral HEF 1.3.2 Bank Storage 1.4 Methods for estimation of GW-SW exchanges 1.4.1 Field Methods 1.4.1.1 Direct measurement of water flux 1.4.1.2 Tracer based Methods 1.4.2 Modeling Methods 1.4.2.1 Transient storage models 1.4.2.2 Physically based models 1.5 Research gaps and need 1.6 Objectives of the research 1.7 Thesis Outline 2 Flow and Transport Dynamics during Bank Flows 2.1 Introduction 2.2 Methods 2.2.1 Concept and modeling setup 2.2.2 Numerical Model 2.2.3 Stream discharge events 2.2.4 Model results evaluation 2.3 Results and discussion 2.3.1 Response of water and solute exchange to stream discharge events 2.3.1.1 Water exchange time scales 2.3.1.2 Stream water solute concentration 2.3.2 Solute mobilization within the riparian zone 2.3.3 Influence of peak height and event duration on solute mass export towards the stream 2.3.4 Effects of event hydrograph shape on stream water solute concentration 2.3.5 Model limitations and future studies 2.4 Summary and Conclusions Appendix 2 3 Flow and Transport Dynamics within Intra-Meander Zone 3.1 Introduction 3.2 Methods 3.2.1 Meander Shape Scenarios 3.2.2 Surface Water Simulations 3.2.3 3D Groundwater Flow Simulations with Modeling code MIN3P 3.2.3.1 Steady Flow Simulations 3.2.3.2 Stream flow event and Solute Mobilization Set-up 3.2.4 Reactive Transport 3.3 Results and Discussion 3.3.1 Groundwater heads and flow paths in the saturated intrameander zone 3.3.1.1 Groundwater heads 3.3.1.2 Flow paths and isochrones 3.3.1.3 Vertical extent of flow paths 3.3.2 Intra-Meander Residence Time Distribution 3.3.3 Factors affecting intra-meander flow and residence times 3.3.3.1 intra-meander hydraulic gradient 3.3.3.2 Maximum penetration depth 3.3.3.3 Meander sinuosity 3.3.3.4 intra-meander area (A) 3.3.4 Influence of Discharge Event on intra-meander Flow and Solute Transport 3.3.4.1 Spatial distribution of groundwater head and solute concentration 3.3.4.2 Time scales of intra-meander groundwater heads and solute transport 3.3.4.3 Solute export during stream discharge event 3.3.5 Intra-meander reactive transport during stream discharge event 3.3.5.1 Impact of stream discharge on aerobic respiration and denitrification 3.3.5.2 DOC mass removal during stream discharge event 3.4 Summary and Conclusions Appendix 3 4 General Summary and Conclusions 4.1 Summary 4.2 Conclusions 4.2.1 Flow and Transport Dynamics in Near Stream Riparian Zone (Bank Flows) 4.2.2 Flow and Transport Dynamics within Intra-Meander Zone 4.3 Model Limitations and Future Studies Bibliography Acknowledgement

In this thesis extensions to the formalism of endoreversible thermodynamics for multi-extensity fluxes and chemical reactions are introduced. These extensions make it possible to model a great variety of systems which could not be investigated with standard endoreversible thermodynamics. Multi-extensity fluxes are important when studying processes with matter fluxes or processes in which volume and entropy are exchanged between subsystems. For including reversible as well as irreversible chemical reaction processes a new type of subsystems is introduced - the so called reactor. It is similar to endoreversible engines, because the fluxes connected to it are balanced. The difference appears in the balance equations for particle numbers, which contain production or destruction terms, and in the possible entropy production in the reactor. Both extensions are then applied to an endoreversible fuel cell model. The chemical reactions in the anode and cathode of the fuel cell are included with the newly introduced subsystem -- the reactor. For the transport of the reactants and products as well as the proton transport through the electrolyte membrane, the multi-extensity fluxes are used. This fuel cell model is then used to calculate power output, efficiency and cell voltage of a fuel cell with irreversibilities in the proton and electron transport. It directly connects the pressure and temperature dependencies of the cell voltage with the dissipation due to membrane resistance. Additionally, beside the listed performance measures it is possible to quantify and localize the entropy production and dissipated heat with only this one model. / In dieser Arbeit erweitere ich den Formalismus der endoreversiblen Thermodynamik, um Flüsse mit mehr als einer extensiven Größe sowie chemische Reaktionsprozesse modellieren zu können. Mit Hilfe dieser Erweiterungen eröffnen sich zahlreiche neue Anwendungsmöglichkeiten für endoreversible Modelle. Flüsse mit mehreren extensiven Größen sind für die Betrachtung von Masseströmen ebenso nötig wie für Prozesse, bei denen sowohl Volumen als auch Entropie zwischen zwei Teilsystem ausgetauscht werden. Für sowohl reversibel wie auch irreversibel geführte chemische Reaktionsprozesse wird ein neues Teilsystem - der "Reaktor" - vorgestellt, welches sich ähnlich wie endoreversible Maschinen durch Bilanzgleichungen auszeichnet. Der Unterschied zu den Maschinen besteht in den Produktions- bzw. Vernichtungstermen in den Teilchenzahlbilanzen sowie der möglichen Entropieproduktion innerhalb des Reaktors. Beide Erweiterungen finden dann in einem endoreversiblen Modell einer Brennstoffzelle Anwendung. Dabei werden Flüsse mehrerer gekoppelter Extensitäten für den Zustrom von Wasserstoff und Sauerstoff sowie für den Protonentransport durch die Elektrolytmembran benötigt. Chemische Reaktionen treten in der Anode und Kathode der Brennstoffzelle auf. Diese werden mit dem neu eingeführten Teilsystem, dem Reaktor, eingebunden. Mit Hilfe des Modells werden dann Wirkungsgrad, Zellspannung und Leistung einer Brennstoffzelle unter Berücksichtigung der Partialdrücke der Substanzen, der Temperatur sowie der Dissipation beim Protonentransport berechnet. Dabei zeigt sich, dass experimentelle Daten für die Zellspannung sowohl qualitativ als auch näherungsweise quantitativ durch das Modell abgebildet werden können. Der Vorteil des endoreversiblen Modells liegt dabei in der Möglichkeit, mit nur einem Modell neben den genannten Kenngrößen auch die abgegebene Wärme sowie die Entropieproduktion zu quantifizieren und den einzelnen Teilprozessen zuzuordnen.