Two-dimensional droplet ripening in a concentration gradient

The discovery of various membraneless subcellular structures (biological condensates) in the cytoplasm and nucleus of cells has generated considerable interest in the effects of non-equilibrium chemical reactions on liquid-liquid phase separation and droplet ripening. Examples include the suppression of droplet ripening due to ATP-driven protein phosphorylation and the spatial segregation of droplets due to regulation by protein concentration gradients. Most studies of biological phase separation have focused on 3D droplet formation, for which mean field methods can be applied. However, mean field theory breaks down in the case of 2D systems, since the concentration around a droplet varies as lnRrather thanR(-1), whereRis the distance from the center of a droplet. In this paper we use the asymptotic theory of diffusion in domains with small holes or exclusions (strongly localized perturbations) to study the segregation of circular droplets in gradient systems. 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 dynamical equations for the position-dependent growth and drift of droplets. We thus show how a gradient of regulatory proteins leads to the segregation of droplets to one end of the domain, as previously found for 3D droplets.