Active suppression of Ostwald ripening: Beyond mean-field theory

Paul C. Bressloff

Phys. Rev. E 101, 042804 – Published 29 April 2020

Abstract

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.

Received 3 January 2020
Accepted 1 April 2020

DOI:https://doi.org/10.1103/PhysRevE.101.042804

Physics Subject Headings (PhySH)

Biomolecular self-assembly Diffusion Nonequilibrium statistical mechanics Nucleation Phase separation

Physics of Living SystemsStatistical Physics & Thermodynamics

Authors & Affiliations

Paul C. Bressloff

Department of Mathematics, University of Utah, Salt Lake City, Utah 84112, USA

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Issue

Vol. 101, Iss. 4 — April 2020

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Images

Figure 1
Ostwald ripening. Schematic diagram showing the concentration profile as a function of $x$ along the axis joining the centers of two well-separated droplets with different radii $R_{1} > R_{2}$ . The solute concentration $ϕ_{a} (R_{1})$ around the larger droplet is lower than the concentration $ϕ_{a} (R_{2})$ around the smaller droplet, resulting in a net diffusive flux from the small droplet to the large droplet. Here $ϕ_{\infty}$ denotes the mean field of LSW theory.
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Figure 2
Construction of the inner solution in terms of stretched coordinates $y = ε^{- 1} (x - x_{i})$ , where $x_{i}$ is the center of the $i th$ droplet. Rescaled radius is $ρ_{i}$ and the region outside the droplet is taken to be $R^{2}$ rather than the bounded domain $Ω$ . The concentration inside the droplet is given by the constant $ϕ_{b}$ , with a discontinuity at the interface so that $Φ_{i} (ρ_{i}^{+}) = ϕ_{a} (ρ_{i}) = ϕ_{a} (1 + 1 / ρ_{i})$ .
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Figure 3
Construction of the outer solution $ϕ$ . Each droplet is shrunk to a single point. The outer solution can be expressed in terms of the corresponding modified Neumann Green's function and then matched with the inner solution $Φ$ around each droplet.
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Figure 4
Interior and exterior droplet regions in stretched coordinates. The $P$ concentration satisfies an inhomogeneous modified Helmholtz equation with length constant $ξ = \sqrt{D / (α + β)}$ and $U$ the total solute concentration (including molecules in the $P$ and $S$ states). The total concentration within the droplet is a constant $Q_{i}$ . The $P$ -state concentration is discontinuous at the interface with ${\hat{Φ}}_{i} (ρ_{i}) = ϕ_{b}$ and $Φ_{i} (ρ_{i}) = ϕ_{a} (ρ_{i}) \equiv ϕ_{a} (1 + 1 / ρ_{i})$ . On the other hand the $S$ -state concentration is continuous at the interface.
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Figure 5
(a) Various $P$ fluxes crossing the $i th$ droplet interface. There is one outward current $J_{i, -} (ρ, Θ)$ and two inward currents $J_{i, +} (ρ, Θ)$ and $j_{i} (ρ, Θ)$ , where $ρ$ is the droplet radius and $Θ$ is the continuous $S$ concentration at the interface. The current $j_{i}$ depends on the local gradient $U_{i}^{'}$ of the total solute concentration outside the droplet. The fluxes balance at the steady-state solution $(ρ^{*}, Θ^{*})$ with $j_{i} (ρ^{*}, Θ^{*}) = 0$ . (b) Corresponding $S$ fluxes. The net $S$ flux is continuous across the interface.
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Figure 6
Active suppression of Ostwald ripening in the small-droplet regime ( $ξ / ρ^{*} ≪ 1$ ). In a multidroplet state $(ρ_{i}, Θ_{i}) = (ρ^{*}, Θ^{*}), i = 1, ..., N$ , the net flux at a droplet interface is a combination of an influx $J_{+}^{*} = J_{+} (ρ^{*})$ and an efflux $J_{-}^{*} = J_{-} (ρ^{*})$ where we have set $Θ^{*} = u_{\infty} - ϕ_{a} (1 + 1 / ρ^{*})$ . A solution is a steady state with respect to droplet growth provided that the fluxes are balanced. For fixed supersaturation $Δ = ϕ_{\infty} - ϕ_{a}$ and interfacial length constant $ξ$ , steady states correspond to points of intersection of the curves $J_{+} (ρ^{*})$ and $J_{-} (ρ^{*})$ . (a) Plots of $J_{+} (ρ^{*})$ as a function of $ρ^{*}$ for various $ξ$ values and fixed supersaturation $Δ = 1.2$ (b) Plots of $J_{+} (ρ^{*})$ as a function of $ρ^{*}$ for various $Δ$ values and fixed $ξ = 100$ . For a range of values of $Δ$ and $ξ$ , there exist two steady-state radii $ρ_{\pm}^{*}$ for which the fluxes cancel. These steady states disappear below a critical supersaturation or above a critical interfacial length constant $ξ$ . Other parameters are $D = 1$ and $ϕ_{a} = 1$ .
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