Organization of chemical reactions by phase separation

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 Bibliograph

Energy and matter supply for active droplets

Chemically active droplets provide simple models for cell-like systems that can grow and divide. Such active droplet systems are driven away from thermodynamic equilibrium and turn over chemically, which corresponds to a simple metabolism. We consider two scenarios of non-equilibrium driving. First, droplets are driven via the system boundaries by external reservoirs that supply nutrient and remove waste (boundary-driven). Second, droplets are driven by a chemical energy provided by a fuel in the bulk (bulk-driven). For both scenarios, we discuss the conservation of energy and matter as well as the balance of entropy. We use conserved and non-conserved fields to analyze the non-equilibrium steady states of active droplets. Using an effective droplet model, we explore droplet stability and instabilities leading to droplet division. Our work reveals that droplet division occurs quite generally in active droplet systems. Our results suggest that life-like processes such as metabolism and division can emerge in simple non-equilibrium systems that combine the physics of phase separation and chemical reactions

Power-Law Population Heterogeneity Governs Epidemic Waves

We generalize the Susceptible-Infected-Removed model for epidemics to take into account generic effects of heterogeneity in the degree of susceptibility to infection in the population. We introduce a single new parameter corresponding to a power-law exponent of the susceptibility distribution that characterizes the population heterogeneity. We show that our generalized model is as simple as the original model which is contained as a limiting case. Because of this simplicity, numerical solutions can be generated easily and key properties of the epidemic wave can still be obtained exactly. In particular, we present exact expressions for the herd immunity level, the final size of the epidemic, as well as for the shape of the wave and for observables that can be quantified during an epidemic. We find that in strongly heterogeneous populations the epidemic reaches only a small fraction of the population. This implies that the herd immunity level can be much lower than in commonly used models with homogeneous populations. Using our model to analyze data for the SARS-CoV-2 epidemic in Germany shows that the reported time course is consistent with several scenarios characterized by different levels of immunity. These scenarios differ in population heterogeneity and in the time course of the infection rate, for example due to mitigation efforts or seasonality. Our analysis reveals that quantifying the effects of mitigation requires knowledge on the degree of heterogeneity in the population. Our work shows that key effects of population heterogeneity can be captured without increasing the complexity of the model. We show that information about population heterogeneity will be key to understand how far an epidemic has progressed and what can be expected for its future course.Comment: 34 pages, 8 figure

Stochastic dynamics of single molecules across phase boundaries

We discuss the stochastic trajectories of single molecules in a phase-separated liquid, when a dense and a dilute phase coexist. Starting from a continuum theory of macroscopic phase separation we derive a stochastic Langevin equation for molecular trajectories that takes into account thermal fluctuations. We find that molecular trajectories can be described as diffusion with drift in an effective potential, which has a steep gradient at phase boundaries. We discuss how the physics of phase coexistence affects the statistics of molecular trajectories and in particular the statistics of displacements of molecules crossing a phase boundary. At thermodynamic equilibrium detailed balance imposes that the distributions of displacements crossing the phase boundary from the dense or from the dilute phase are the same. Our theory can be used to infer key phase separation parameters from the statistics of single-molecule trajectories. For simple Brownian motion, there is no drift in the presence of a concentration gradient. We show that interactions in the fluid give rise to an average drift velocity in concentration gradients. Interestingly, under non-equilibrium conditions, single molecules tend to drift uphill the concentration gradient. Thus, our work bridges between single-molecule dynamics and collective dynamics at macroscopic scales and provides a framework to study single-molecule dynamics in phase-separating systems

Quantitative theory for the diffusive dynamics of liquid condensates

Key processes of biological condensates are diffusion and material exchange with their environment. Experimentally, diffusive dynamics are typically probed via fluorescent labels. However, to date, a physics-based, quantitative framework for the dynamics of labeled condensate components is lacking. Here, we derive the corresponding dynamic equations, building on the physics of phase separation, and quantitatively validate the related framework via experiments. We show that by using our framework, we can precisely determine diffusion coefficients inside liquid condensates via a spatio-temporal analysis of fluorescence recovery after photobleaching (FRAP) experiments. We showcase the accuracy and precision of our approach by considering space- and time-resolved data of protein condensates and two different polyelectrolyte-coacervate systems. Interestingly, our theory can also be used to determine a relationship between the diffusion coefficient in the dilute phase and the partition coefficient, without relying on fluorescence measurements in the dilute phase. This enables us to investigate the effect of salt addition on partitioning and bypasses recently described quenching artifacts in the dense phase. Our approach opens new avenues for theoretically describing molecule dynamics in condensates, measuring concentrations based on the dynamics of fluorescence intensities, and quantifying rates of biochemical reactions in liquid condensates

Liquid spherical shells are a non-equilibrium steady state of active droplets

Liquid-liquid phase separation yields spherical droplets that eventually coarsen to one large, stable droplet governed by the principle of minimal free energy. In chemically fueled phase separation, the formation of phase-separating molecules is coupled to a fuel-driven, non-equilibrium reaction cycle. It thus yields dissipative structures sustained by a continuous fuel conversion. Such dissipative structures are ubiquitous in biology but are poorly understood as they are governed by non-equilibrium thermodynamics. Here, we bridge the gap between passive, close-to-equilibrium, and active, dissipative structures with chemically fueled phase separation. We observe that spherical, active droplets can undergo a morphological transition into a liquid, spherical shell. We demonstrate that the mechanism is related to gradients of short-lived droplet material. We characterize how far out of equilibrium the spherical shell state is and the chemical power necessary to sustain it. Our work suggests alternative avenues for assembling complex stable morphologies, which might already be exploited to form membraneless organelles by cells

Organization of chemical reactions by phase separation

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 Bibliograph

Organization of chemical reactions by phase separation

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 Bibliograph