Exchange controls coarsening of surface condensates

TL;DR

The paper addresses how material exchange between a membrane surface and a bulk modulates pattern formation of surface condensates, introducing a thermodynamically consistent model with surface field $\phi(\boldsymbol r,t)$ and bulk variable $\psi(t)$. It combines passive and active exchange via fluxes that couple surface and bulk dynamics, and uses linear stability analysis, thin-interface approximations, and flux-based timescale estimates to reveal when coarsening is enhanced, slowed, or arrested, yielding distinct scaling laws such as $R \sim t^{1/3}$, $R \sim t^{1/2}$, or ballistic $R \sim t$ regimes. Active exchange can accelerate coarsening by biasing unbinding toward dilute regions (and can arrest coarsening when biased toward dense regions), with multi-stability and pattern selection emerging from non-local bulk transport. The findings illuminate how cells could exploit active, non-local exchange to form a single polarity spot quickly and robustly, with broader implications for PAR protein patterns and other membrane-associated condensates.

Abstract

Biological membranes often exhibit heterogeneous protein patterns, which cells control. Strong patterns, like the polarity spot in budding yeast, can be described as surface condensates, formed by physical interactions between constituents. However, it is unclear how these interactions affect the material exchange with the bulk. To study this, we analyze a thermodynamically consistent model, which reveals that passive exchange generally accelerates the coarsening of surface condensates. Active exchange can further accelerate coarsening, although it can also fully arrest it and induce complex patterns involving various length scales. We reveal how these behaviors are related to non-local transport via diffusion through the bulk, rationalizing the various scaling laws we observe and allowing us to interpret biologically relevant scenarios.

Figures (5)

• Figure 1: Passive exchange with bulk accelerates coarsening in a surface. (A) Schematic of our model describing interacting solutes that phase separate in a 2D surface and exchanges with a 3D bulk. (B) Snapshots of numerical simulations with and without passive exchange at the same time $t=10^3\,t_0$ for $\Phi_{\mathrm{tot}}=0.5$, $\chi =3.5$, $\kappa =2 \, k_\mathrm{B}T w^2 / \nu_\mathrm{s}$, $\omega_\mathrm{b} -\omega_\mathrm{s}= 6.5$, $\gamma A = 100$, $k_\mathrm{a} =0$, $w = (\nu_\mathrm{s} \kappa/ k_\mathrm{B}T)^{1/2}$, and $t_0 = w^2/(k_\mathrm{B}T \Lambda)$. (C) Mean droplet radius $R$ as a function of $t$ for $k_\mathrm{p}=0$ (red line) and $k_\mathrm{p} t_0 = 10^{-3}, 10^{-2}, 10^{-1}$ (lighter to darker blue lines) for the same parameters as in (A). (D) Fluxes from a droplet toward the dilute phase ($J_\mathrm{diff}$) and the bulk ($J_\mathrm{int}$) as a function of radius $R$ for reaction-diffusion lengths $\ell_\mathrm{in} = \ell_\mathrm{out}=\ell=40 \, l_\mathrm{c}$ (solid lines) and $\ell=2 \, l_\mathrm{c}$ (dashed lines) for $\phi_\mathrm{out}^{(0)} = 0.1$, $\phi_\mathrm{in}^{(0)} = 0.9$, $k_0=D/l_\mathrm{c}^2$ with $D=D_\mathrm{in} = D_\mathrm{out}$. (E) Time scale of growing a single droplet ($T_\mathrm{single}$, blue line) and coarsening of two droplets ($T_\mathrm{coars}$, orange line) as a function of $\ell$. Limiting cases ($T_\mathrm{coars}^{\mathrm{diff}}$, $T_\mathrm{coars}^{\mathrm{int}}$, and $T_\mathrm{load}$) are discussed in main text. Gray disk indicates estimates for polarity spots in budding yeast (SI. \ref{['app:timescales_estimation']}). Parameters are $R_* = 10^2 \, l_\mathrm{c}$, $L= 10^3 \, l_\mathrm{c}$, $\Phi_{\mathrm{tot}}=0.5$, $\eta = 100$, $\phi_\mathrm{dil} =0.101$, and given in (D).
• Figure 2: Active unbinding from dilute regions accelerates coarsening. (A) Mean droplet radius $R$ as a function of time $t$ for $\alpha=-0.5$ (active system, orange line) and $\alpha=0$ (passive system, gray line) as well as $k=10^{-1} /t_0$ (light colors) and $k=10^{-2}/t_0$ (dark colors). Parameters are $\Delta \mu=2\,k_\mathrm{B}T$ and given in Fig. \ref{['fig:passive_coarsening']}(B--C). (B) Diffusive efflux $J_\mathrm{diff}$ and internal unbinding flux $J_\mathrm{int}$ as a function of $R$ for $\Delta\phi_\mathrm{in} = 0.01$ (light colors) and $\Delta\phi_\mathrm{in} = 0.05$ (dark colors). Grey dashed line corresponds to $J_\mathrm{tot} = J_\mathrm{diff} + J_\mathrm{int}$. Vertical line indicates estimated transition between area- and interface-limited flux $J_\mathrm{int}$ (SI. \ref{['app:diff_int']}). Model parameters are $\Delta\phi_\mathrm{out} = -10^{-3}$, $\ell = 40 \,l_\mathrm{c}$, and given in Fig. \ref{['fig:passive_coarsening']}(D--E).
• Figure 3: Active unbinding from dense regions can arrest coarsening. (A) Mean droplet radius $R$ as a function of time $t$ for $\alpha=0.5$, $\Delta \mu=2 \,k_\mathrm{B}T$, and $k t_0= 5 \times 10^{-1}, 10^{-1}, 10^{-2},10^{-3}, 10^{-4}$ (fast to slow). Dashed curves corresponds to passive limits ($\alpha = 0$). Remaining parameters are given in Fig. \ref{['fig:passive_coarsening']}(B--C). (B) Diffusive efflux and internal unbinding fluxes for $\Delta\phi_\mathrm{in} = -0.1$, $\Delta\phi_\mathrm{out} = 0.05$, $\ell=10\, l_\mathrm{c}$. Remaining parameters and units are given in Fig. \ref{['fig:passive_coarsening']}(D--E). The grey dashed line corresponds to $J_\mathrm{tot} = J_\mathrm{diff} + J_\mathrm{int}$. (C--E) Top panels: Stationary concentration field $c(\boldsymbol r, t)$ (top panels) and corresponding exchange fluxes (bottom panels) for various parameters: (C) $k = 10^{-2} /t_0$ (corresponding to pink line in panel A), (D) $k = 5 \times 10^{-1} /t_0$ (yellow line in panel A), and (E) $\alpha = 1$ with $\chi=3$, $\Phi_{\mathrm{tot}}=0.6$, and $k=0.1$. Remaining parameters are the same as in panel A.
• Figure S1: (A) Dispersion relation from linear stability analysis for different choices of activity inhomogeneity coefficient $\alpha$. Parameters are $\chi = 2.7$, $\Phi_{\mathrm{tot}} = 0.6$, $k=0.1$. Remaining parameters are the same as Fig. \ref{['fig:enhanced_coarsening']}A. (B) Effective passive simulations showing scalings associated with different types of fluxes. (C) Mean radius as a function of time for $\alpha =-0.5$, $\Delta \mu = 5 \, k_\mathrm{B}T$, $\Phi_{\mathrm{tot}} = 1$. From lighter to darker orange the curves correspond to $k t_0= 5 \times 10^{-3}, 10^{-2}, 3 \times 10^{-2}$. Remaining parameters are the same as Fig. \ref{['fig:enhanced_coarsening']}A in the main text.
• Figure S2: Active unbinding from dense regions exhibits multistability (A) Initial condition given by perturbing homogeneous state that is a zero of the homogeneous binding flux. (B) Inital condition with multiple droplets. Left panels: concentration profiles at early times. Central panel: concentration profiles at stationarity. Right panel: corresponding exchange fluxes at stationarity. Remaining parameters are the same as Fig. \ref{['fig:active_binding']}D in the main text.