Submitted / under review
Jonathan Bauermann, Sudarshana Laha, Patrick M. McCall, Frank Jülicher, and Christoph A. Weber
The kinetics of chemical reactions are determined by the law of mass action, which has been successfully applied to homogeneous, dilute mixtures. At non-dilute conditions, interactions among the components can give rise to coexisting phases, which can significantly alter the kinetics of chemical reactions. Here, we derive a theory for chemical reactions in coexisting phases at phase equilibrium. We show that phase equilibrium couples the rates of chemical reactions of components with their diffusive exchanges between the phases. Strikingly, the chemical relaxation kinetics can be represented as a flow along the phase equilibrium line in the phase diagram. A key finding of our theory is that differences in reaction rates between coexisting phases stem solely from phase- dependent reaction rate coefficients. Our theory is key to interpret how concentration levels of reactive components in condensed phases control chemical reaction rates in synthetic and biological systems.
Wolfram Pönisch, Thomas C.T. Michaels, and Christoph A. Weber
Condensates in living cells can exhibit a complex rheology including viscoelastic and glassy behaviour. This rheological behavior of condensates was suggested to regulate polymerisation of cytoskeletal filaments and aggregation of amyloid fibrils. Here, we theoretically investigate how the rheological properties of condensates can control the formation of linear aggregates. To this end, we propose a kinetic theory for the aggregate size distribution and the exchange of aggregates between inside and outside of condensates. The rheology of condensates is accounted for by aggregate mobilities that depend on aggregate size. We show that condensate rheology can control whether either aggregates of all sizes or dominantly small aggregates are exchanged between condensate inside and outside on the time-scale of aggregation. As a result, the ratio of aggregate numbers inside to outside of condensates differs significantly. Strikingly, we also find that weak variations in the rheological properties of condensates can lead to a switch-like change of the number of aggregates. These results suggest a possible physical mechanism for how living cells could control linear aggregation in a switch-like fashion through variations in condensate rheology.
Thomas C.T. Michaels, L. Mahadevan and Christoph A. Weber
In living cells, liquid condensates form in the cytoplasm and nucleoplasm via phase separation and regulate physiological processes. They also regulate aberrant aggregation of amyloid fibrils, a process linked to Alzheimer's and Parkinson's diseases. In the absence of condensates it has been shown that amyloid aggregation can be inhibited by molecular chaperones and rationally designed drugs. However it remains unknown how this drug- or chaperone-mediated inhibition of amyloid fibril aggregation is affected by phase-separated condensates. Here we study the interplay between protein aggregation, its inhibition and liquid-liquid phase separation. Our key finding is that the potency of inhibitors of amyloid formation can be strongly enhanced. We show that the corresponding mechanism relies on the colocalization of inhibitors and aggregates inside the liquid condensate. We provide experimentally testable physicochemical conditions under which the increase of inhibitor potency is most pronounced. Our work highlights the role of spatio-temporal heterogeneity in curtailing aberrant protein aggregation and suggests design principles for amyloid inhibitors accounting for partitioning of drugs into liquid condensates.
Jonathan Bauermann, Christoph A. Weber, Frank Jülicher
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.
Joël Mabillard, Christoph A. Weber, Frank Jülicher
Chemically active systems such as living cells are maintained out of thermal equilibrium due to chemical events which generate heat and lead to active fluctuations. A key question is to understand on which time and length scales active fluctuations dominate thermal fluctuations. Here, we formulate a stochastic field theory with Poisson white noise to describe the heat fluctuations which are generated by stochastic chemical events and lead to active temperature fluctuations. We find that on large length and time scales, active fluctuations always dominate thermal fluctuations. However, at intermediate length and time scales, multiple crossovers exist which highlight the different characteristics of active and thermal fluctuations. Our work provides a framework to characterize fluctuations in active systems and reveals that local equilibrium holds at certain length and time scales.
Xueping Zhao, Giacomo Bartolucci, Alf Honigmann, Frank Jülicher, Christoph A. Weber
In living cells, protein-rich condensates can wet the cell membrane and surfaces of membrane-bound organelles. Interestingly, many phase-separating proteins also bind to membranes leading to a molecular layer of bound molecules. Here we investigate how binding to membranes affects wetting, prewetting and surface phase transitions. We derive a thermodynamic theory for a three-dimensional bulk in the presence of a two-dimensional, flat membrane. At phase coexistence, we find that membrane binding facilitates complete wetting and thus lowers the wetting angle. Moreover, below the saturation concentration, binding facilitates the formation of a thick layer at the membrane and thereby shifts the prewetting phase transition far below the saturation concentration. The distinction between bound and unbound molecules near the surface leads to a large variety of surface states and complex surface phase diagrams with a rich topology of phase transitions. Our work suggests that surface phase transitions combined with molecular binding represent a versatile mechanism to control the formation of protein-rich domains at intra-cellular surfaces.
Stefano Bo, Lars Hubatsch, Jonathan Bauermann, Christoph A. Weber, and Frank Jülicher
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.
Giacomo Bartolucci, Omar Adame-Arana, Xueping Zhao, and Christoph A. Weber
Phase separation and transitions among different molecular states are ubiquitous in living cells. Such transitions can be governed by local equilibrium thermodynamics or by active processes controlled by biological fuel. It remains largely unexplored how the behavior of phase-separating systems with molecular transitions differs between thermodynamic equilibrium and cases in which the detailed balance of the molecular transition rates is broken because of the presence of fuel. Here, we present a model of a phase-separating ternary mixture in which two components can convert into each other. At thermodynamic equilibrium, we find that molecular transitions can give rise to a lower dissolution temperature and thus reentrant phase behavior. Moreover, we find a discontinuous thermodynamic phase transition in the composition of the droplet phase if both converting molecules attract themselves with similar interaction strength. Breaking the detailed balance of the molecular transition leads to quasi-discontinuous changes in droplet composition by varying the fuel amount for a larger range of intermolecular interactions. Our findings showcase that phase separation with molecular transitions provides a versatile mechanism to control properties of intracellular and synthetic condensates via discontinuous switches in droplet composition.
Lars Hubatsch, Louise M. Jawerth, Celina Love, Jonathan Bauermann, T-Y Dora Tang, Stefano Bo, Anthony A. Hyman, and Christoph A. Weber
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.
Local thermodynamics govern formation and dissolution of Caenorhabditis elegans P granule condensates
Anatol W. Fritsch, Andrés F. Diaz-Delgadillo, Omar Adame-Arana, Carsten Hoege, Matthäus Mittasch, Moritz Kreysing, Mark Leaver, Anthony A. Hyman, Frank Jülicher, Christoph A. Weber
Membraneless compartments, also known as condensates, provide chemically distinct environments and thus spatially organize the cell. A well-studied example of condensates is P granules in the roundworm Caenorhabditis elegans that play an important role in the development of the germline. P granules are RNA-rich protein condensates that share the key properties of liquid droplets such as a spherical shape, the ability to fuse, and fast diffusion of their molecular components. An outstanding question is to what extent phase separation at thermodynamic equilibrium is appropriate to describe the formation of condensates in an active cellular environment. To address this question, we investigate the response of P granule condensates in living cells to temperature changes. We observe that P granules dissolve upon increasing the temperature and recondense upon lowering the temperature in a reversible manner. Strikingly, this temperature response can be captured by in vivo phase diagrams that are well described by a Flory–Huggins model at thermodynamic equilibrium. This finding is surprising due to active processes in a living cell. To address the impact of such active processes on intracellular phase separation, we discuss temperature heterogeneities. We show that, for typical estimates of the density of active processes, temperature represents a well-defined variable and that mesoscopic volume elements are at local thermodynamic equilibrium. Our findings provide strong evidence that P granule assembly and disassembly are governed by phase separation based on local thermal equilibria where the nonequilibrium nature of the cytoplasm is manifested on larger scales.
Patrick Schwarz, Sudarshana Laha, Jacqueline Janssen, Tabea Huss, Job Boekhoven, Christoph A. Weber
Non-equilibrium, fuel-driven reaction cycles serve as model systems of the intricate reaction networks of life. Rich and dynamic behavior is observed when reaction cycles regulate assembly processes, such as phase separation. However, it remains unclear how the interplay between multiple reaction cycles affects the success of emergent assemblies. To tackle this question, we created a library of molecules that compete for a common fuel that transiently activates products. Often, the competition for fuel implies that a competitor decreases the lifetime of these products. However, in cases where the transient competitor product can phase-separate, such a competitor can increase the survival time of one product. Moreover, in the presence of oscillatory fueling, the same mechanism reduces variations in the product concentration while the concentration variations of the competitor product are enhanced. Like a parasite, the product benefits from the protection of the host against deactivation and increases its robustness against fuel variations at the expense of the robustness of the host. Such a parasitic behavior in multiple fuel-driven reaction cycles represents a lifelike trait, paving the way for the bottom-up design of synthetic life.
Drops in cells: Liquid droplets act as microreactors. Can they also serve as a control mechanism for cellular biochemistry?
Christoph A. Weber and Christoph Zechner
A major challenge in cell biology remains unraveling how cells control their biochemical reaction cycles. For instance, how do they regulate gene expression in response to stress? How does their metabolism change when resources are scarce? Control theory has proven useful in understanding how networks of chemical reactions can robustly tackle those and other tasks.1 The essential ingredients in such approaches are chemical feedback loops that create control mechanisms similar to the circuits that regulate, for example, the temperature of a heating system, the humidity of an archive, or the pH of a fermentation tank.
Christoph A. Weber
The physics of phase separation and the formation of protein-rich droplets play an important role for biochemical processes in living cells. The shape, size and composition of such droplets can change with time and thereby affect biochemical reactions. Such reactions also affect the dynamics of droplets. Obtaining insights into this interplay is key to better understand the mechanisms underlying the spatio-temporal organization of biological cells.
Christoph A. Weber, Lars Hubatsch, Frank Jülicher
Zellen führen biochemische Prozesse aus, um zentrale Abläufe wie die Zellteilung zu realisieren. Bei der dafür erforderlichen raumzeitlichen Organisation der zellulären Prozesse kommt den Organellen eine wichtige Rolle zu. Organellen wie die Mitochondrien oder der Zellkern sind durch Membranen von ihrer Umgebung getrennt. Diese ermöglichen es ihnen, in ihrem Inneren geeignete biochemische Bedingungen für biologische Prozesse zu erzeugen. Doch es gibt auch membranlose Organellen. Wie bewahren diese ihre chemische Identität? Hierbei kommt die Koexistenz proteinreicher, flüssiger Phasen ins Spiel. Ausgehend von der Physikder Phasenseparation ist ein tieferes Verständnis der Dynamik und raumzeitlichen Organisation biochemischer Prozesse möglich.
Omar Adame-Arana, Christoph A. Weber, Vasily Zaburdaev, Jacques Prost, Frank Jülicher
We present a minimal model to study liquid phase separation in a fixed pH ensemble. The model describes a mixture composed of macromolecules that exist in three different charge states and have a tendency to phase separate. We introduce the pH dependence of phase separation by means of a set of reactions describing the protonation and deprotonation of macromolecules, as well as the self-ionisation of water. We use conservation laws to identify the conjugate thermodynamic variables at chemical equilibrium. Using this thermodynamic conjugate variables we perform a Legendre transform which defines the corresponding free energy at fixed pH. We first study the possible phase diagram topologies at the isoelectric point of the macromolecules. We then show how the phase behavior depends on pH by moving away from the isoelectric point. We find that phase diagrams as a function of pH strongly depend on whether oppositely charged macromolecules or neutral macromolecules have a stronger tendency to phase separate. We predict the existence of reentrant behavior as a function of pH. In addition, our model also predicts that the region of phase separation is typically broader at the isoelectric point. This model could account for both, the protein separation observed in yeast cells for pH values close to the isoelectric point of many cytosolic proteins and also for the in vitro experiments of single proteins exhibiting phase separation as a function of pH.
Marta Tena-Solsona, Jacqueline Janssen, Caren Wanzke, Fabian Schnitter, Hansol Park, Benedikt Rieß, Julianne M. Gibbs, Christoph A. Weber, Job Boekhoven
Chemically fueled emulsions are solutions with droplets made of phase-separated molecules that are activated and deactivated by a chemical reaction cycle. These emulsions play a crucial role in biology as a class of membrane-less organelles.Moreover, theoretical studies show that droplets in these emulsions can evolve to the same size or spontaneously self-divide when fuel is abundant. All of these exciting properties,i. e., emergence, decay, collective behavior, and self-division, are pivotal to the functioning of life. However, these theoretical predictions lack experimental systems to test them quantitively.Here, we describe the synthesis of synthetic emulsions formed by a fuel-driven chemical cycle, and we find a surprising new behavior, i. e., the dynamics of droplet growth is regulated by the kinetics of the fuel-driven reaction cycle. Consequently, the average volume of these droplets grows orders of magnitude faster compared to Ostwald ripening. Combining experiments and theory, we elucidate the underlying mechanism.
Nicholas J. Derr, David C. Fronk, Christoph A. Weber, Amala Mahadevan, Chris H. Rycroft, L. Mahadevan
Channel formation and branching is widely seen in physical systems where movement of fluid through a porous structure causes the spatiotemporal evolution of the medium in response to the flow, in turn causing flow pathways to evolve. We provide a simple theoretical framework that embodies this feedback mechanism in a multi-phase model for flow through a fragile porous medium with a dynamic permeability. Numerical simulations of the model show the emergence of branched networks whose topology is determined by the geometry of external flow forcing. This allows us to delineate the conditions under which splitting and/or coalescing branched network formation is favored, with potential implications for both understanding and controlling branching in soft frangible media.
Thomas CT. Michaels, Christoph A. Weber, L Mahadevan
A range of medical conditions, such as Alzheimer’s disease, Parkinson’s disease, and type II diabetes, are linked to protein aggregation. Thus, it is imperative to develop effective therapeutic strategies to combat protein aggregation. Here, we lay out a general approach for optimizing inhibition strategies based on small molecules that suppress nucleation or growth of aggregates. Our model reveals that the optimal timing of drug administration crucially depends on whether the compound inhibits primary nucleation, secondary nucleation, or the growth of aggregates. This approach could guide the rational design of therapeutic strategies to target protein aggregation diseases.
Christoph A. Weber, David Zwicker, Frank Jülicher and Chiu Fan Lee
Phase separating systems that are maintained away from thermodynamic equilibrium via molecular processes represent a class of active systems, which we call active emulsions. These systems are driven by external energy input, for example provided by an external fuel reservoir. The external energy input gives rise to novel phenomena that are not present in passive systems. For instance, concentration gradients can spatially organise emulsions and cause novel droplet size distributions. Another example are active droplets that are subject to chemical reactions such that their nucleation and size can be controlled, and they can divide spontaneously. In this review, we discuss the physics of phase separation and emulsions and show how the concepts that govern such phenomena can be extended to capture the physics of active emulsions. This physics is relevant to the spatial organisation of the biochemistry in living cells, for the development of novel applications in chemical engineering and models for the origin of life.
Christoph A. Weber, Thomas C. T. Michaels, L. Mahadevan
Liquid cellular compartments spatially segregate from the cytoplasm and can regulate aberrant protein aggregation, a process linked to several medical conditions, including Alzheimer's and Parkinson's diseases. Yet the mechanisms by which these droplet-like compartments affect protein aggregation remain unknown. Here, we combine kinetic theory of protein aggregation and liquid-liquid phase separation to study the spatial control of irreversible protein aggregation in the presence of liquid compartments. We find that, even for weak interactions between the compartment constituents and the aggregating monomers, aggregates are strongly enriched inside the liquid compartment relative to the surrounding cytoplasm. We show that this enrichment is caused by a positive feedback mechanism of aggregate nucleation and growth which is mediated by a flux maintaining the phase equilibrium between the compartment and the cytoplasm. Our model predicts that the compartment volume that maximizes aggregate enrichment in the compartment is determined by the reaction orders of aggregate nucleation. The underlying mechanism of aggregate enrichment could be used to confine cytotoxic protein aggregates inside droplet-like compartments suggesting potential new avenues against aberrant protein aggregation. Our findings could also represent a common mechanism for the spatial control of irreversible chemical reactions in general.
Wolfram Pönisch, Christoph A. Weber, Vasily Zaburdaev
Many bacteria rely on active cell appendages, such as type IV pili, to move over substrates and interact with neighboring cells. Here, we study the motion of individual cells and bacterial colonies, mediated by the collective interactions of multiple pili. It was shown experimentally that the substrate motility of Neisseria gonorrhoeae cells can be described as a persistent random walk with a persistence length that exceeds the mean pili length. Moreover, the persistence length increases for a higher number of pili per cell. With the help of a simple, tractable stochastic model, we test whether a tug-of-war without directional memory can explain the persistent motion of single Neisseria gonorrhoeae cells. While the persistent motion of single cells indeed emerges naturally in the model, a tug-of-war alone is not capable of explaining the motility of microcolonies, which becomes weaker with increasing colony size. We suggest sliding friction between the microcolonies and the substrate as the missing ingredient. While such friction almost does not affect the general mechanism of single cell motility, it has a strong effect on colony motility. We validate the theoretical predictions by using a three-dimensional computational model that includes explicit details of the pili dynamics, force generation and geometry of cells.