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 (eg. 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 this does not phase separate and a $P$-state that does. However, mean field theory breaks down in the case of two-dimensional (2D) systems, since the concentration around a droplet varies as $lnR$ 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 multi-droplet 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.

中文翻译：

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主动抑制奥斯特瓦尔德熟化：超越平均场理论

主动过程在无膜细胞结构（生物冷凝物）的形成中起主要作用。经典的粗化理论预测，奥斯特瓦尔德熟化后仅残留一个液滴。然而，在细胞核和细胞质中共存有几个基本组成相同的无膜细胞器，这表明存在抑制奥斯特瓦尔德成熟的机制。一种潜在的候选物是通过酶促反应对液体-液相分离进行主动调节，该酶促反应在不同构象状态（例如，不同水平的磷酸化）之间切换蛋白质。近期的理论研究已使用均值场方法来分析三维（3D）系统中奥斯特瓦尔德（Ostwald）成熟的抑制作用，该系统由在两种不同构象状态之间切换的溶质组成，$\mathrm{小号}$-说明这不会相分离，并且 $P$-声明确实如此。然而，在二维（2D）系统中，平均场理论被打破了，因为液滴周围的浓度随$ln\mathrm{[R}$ 而不是 ${\mathrm{[R}}^{-\mathrm{1个}}$，在哪里 $\mathrm{[R}$是距液滴中心的距离。它也无法捕获有限大小的效果。在本文中，我们展示了如何通过使用具有小孔或排除（强烈局部扰动）的区域中的扩散理论超越平均场理论。特别地，我们使用渐近方法来研究在进行主动液-液相分离的2D或3D溶液中对Ostwald成熟的抑制作用。我们通过将液滴外部的区域划分为围绕每个液滴的一组内部区域以及发生均值场相互作用的外部区域。渐近匹配内部和外部解，我们导出了多液滴稳态的存在性和稳定性的先导条件。我们还展示了如何通过在渐近展开中包括高阶项来将有限大小的影响纳入理论，这取决于液滴的位置和2D或3D域的边界。本文开发的理论框架提供了一种通用方法，用于分析有界域（例如在活细胞中发现的域）中的稀液滴的活性相分离。