Active suppression of Ostwald ripening: Beyond mean-field theory

Active processes play a major role in the formation of membraneless cellular structures (biological condensates). Classical coarsening theory predicts that only a single droplet remains following Ostwald ripening. However, in both the cell nucleus and cytoplasm there coexist several membraneless organelles of the same basic composition, suggesting that there is some mechanism for suppressing Ostwald ripening. One potential candidate is the active regulation of liquid-liquid phase separation by enzymatic reactions that switch proteins between different conformational states (e.g., different levels of phosphorylation). Recent theoretical studies have used mean-field methods to analyze the suppression of Ostwald ripening in three-dimensional (3D) systems consisting of a solute that switches between two different conformational states, an S state that does not phase separate and a P state that does. However, mean-field theory breaks down in the case of 2D systems, since the concentration around a droplet varies as ln R rather than R-1, where R is the distance from the center of the droplet. It also fails to capture finite-size effects. In this paper we show how to go beyond mean-field theory by using the theory of diffusion in domains with small holes or exclusions (strongly localized perturbations). In particular, we use asymptotic methods to study the suppression of Ostwald ripening in a 2D or 3D solution undergoing active liquid-liquid phase separation. We proceed by partitioning the region outside the droplets into a set of inner regions around each droplet together with an outer region where mean-field interactions occur. Asymptotically matching the inner and outer solutions, we derive leading-order conditions for the existence and stability of a multidroplet steady state. We also show how finite-size effects can be incorporated into the theory by including higher-order terms in the asymptotic expansion, which depend on the positions of the droplets and the boundary of the 2D or 3D domain. The theoretical framework developed in this paper provides a general method for analyzing active phase separation for dilute droplets in bounded domains such as those found in living cells.