Asymmetric positioning of the mitotic spindle in C. elegans embryos is mediated by force-generating complexes that are anchored at the plasma membrane and that pull on microtubules growing out from the spindle poles. Although asymmetric distribution of the force generators is thought to underlie asymmetric positioning of the spindle, the number and location of the force generators has not been well defined. In particular, it has not been possible to visualize individual force generating events at the cortex. We discovered that perturbation of the acto-myosin cortex leads to the formation of long membrane invaginations that are pulled from the plasma membrane toward the spindle poles. Several lines of evidence show that the invaginations, which also occur in unperturbed embryos though at lower frequency, are pulled by the same force generators responsible for spindle positioning. Thus, the invaginations serve as a tool to localize the sites of force generation at the cortex and allow us to estimate a lower limit on the number of cortical force generators within the cell.

Asymmetric positioning of the mitotic spindle in C. elegans embryos is mediated by force-generating complexes that are anchored at the plasma membrane and that pull on microtubules growing out from the spindle poles. Although asymmetric distribution of the force generators is thought to underlie asymmetric positioning of the spindle, the number and location of the force generators has not been well defined. In particular, it has not been possible to visualize individual force generating events at the cortex. We discovered that perturbation of the acto-myosin cortex leads to the formation of long membrane invaginations that are pulled from the plasma membrane toward the spindle poles. Several lines of evidence show that the invaginations, which also occur in unperturbed embryos though at lower frequency, are pulled by the same force generators responsible for spindle positioning. Thus, the invaginations serve as a tool to localize the sites of force generation at the cortex and allow us to estimate a lower limit on the number of cortical force generators within the cell.

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.

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