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Liquid-liquid phase separation is the process in which two immiscible liquids demix. This spontaneous phenomenon 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, nonequilibrium reaction cycle. Chemically fueled phase separation yields dissipative structures sustained by a continuous fuel conversion. Such dissipative structures are ubiquitous in biology but 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 transition into a new morphology—a liquid, spherical shell of droplet material. A spherical shell would be highly unstable at equilibrium. Only by continuously converting chemical energy, this dissipative structure can be sustained. We demonstrate the transition mechanism, which is related to the activation of a product outside of the droplet, and the deactivation within the droplets leading to gradients of 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 new avenues for assembling complex stable morphologies, which might already be exploited to form membraneless organelles by cells.

The kinetics of chemical reactions are determined by the law of mass action, which has been successfully applied to homogeneous, dilute mixtures. At nondilute 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 interpreting how concentration levels of reactive components in condensed phases control chemical reaction rates in synthetic and biological systems.

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. Two scenarios of nonequilibrium driving are considered. 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, the conservation of energy and matter as well as the balance of entropy are discussed. Conserved and nonconserved fields are used to analyse the nonequilibrium steady states of active droplets. Using an effective droplet model, droplet stability and instabilities leading to droplet division are explored. This work reveals that droplet division occurs quite generally in active droplet systems. The results suggest that life-like processes such as metabolism and division can emerge in simple nonequilibrium systems that combine the physics of phase separation and chemical reactions.

To unravel the biological functions of membraneless liquid condensates it is crucial to develop a quantitative understanding of the physics underlying their dynamics. Key processes within such condensates are diffusion and material exchange with their environment. Experimentally, diffusive dynamics are typically probed via fluorescent labels. However, to date we lack a physics-based quantitative framework for the dynamics of labeled condensate components. Here, we derive the corresponding theory, building on the physics of phase separation, and quantitatively validate this framework via experiments. We show that using our theory 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. Strikingly, our theory can also be used to determine the diffusion coefficient in the dilute phase and the partition coefficient, without relying on fluorescence measurements in the dilute phase. This bypasses recently described quenching artefacts in the dense phase, which can underestimate partition coefficients by orders of magnitude. Our experimentally verified theory 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.

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.

We study simple integrate-and-fire type models with multiplicative noise and consider the transmission of a weak and slow signal, i.e. a signal that evokes a small modulation of the instantaneous firing rate on time scales that are much larger than the membrane time scale and the mean interspike interval. The specific question of interest is whether and how the state-dependence of the noise can be optimized with respect to information transmission. First, in a simple model in which the noise intensity varies linearly with the state variable, we show analytically that multiplicative fluctuations may benefit the signal transfer and we elucidate the mechanism for this improvement. In a conductance-based integrate-and-fire model with synaptically filtered shot-noise input, we show by means of extended numerical simulations that also in a biophysically more relevant situation, multiplicative noise can enhance the signal-to-noise ratio. Our results shed light on a so far unexplored aspect of stochastic signal transmission in neural systems.