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