Anomalous diffusion and directed coalescence of condensates out of equilibrium

Abstract

Phase separation into domains with distinct composition and properties has widespread implications, ranging from alloys and emulsions to biomolecular condensates in cells. In living and nonliving matter, the organization of these domains can be controlled by nonequilibrium chemical reactions, external fields, or mechanical stresses. In this context, stationary states can emerge from long-range monopolar interactions analogous to electrostatics. More generally, as discussed here, because fluxes induce dipolar force fields, externally controlled boundary motion effectively polarizes the domain even for microscopically nonpolarizable matter. The dipole-dipole interactions resulting from this translation-induced polarization cause directed coalescence of domains. This coarsening mechanism complements Ostwald ripening and coalescence due to Brownian motion or Marangoni flows, and has implications for controlling domains by electric fields or concentration gradients. Interestingly, the chemical potential gradients around a domain that nucleates material are exactly opposite to the hydrodynamic pressure gradients around an impermeable colloid that pushes the fluid, suggesting a competition between phase separation and hydrodynamics. In addition to chemical control, the motion of domains can also be driven by mechanical stresses. An example is the cell interior, where mechanical stresses are actively generated by molecular motors and opposed by passive viscoelastic stresses in the cytoplasm and nucleoplasm. The resulting fluid flows lead to Brownian motion with a suppressed or enhanced size scaling which modifies collision-coalescence. For active stresses with a long correlation time, the domains show superdiffusion on intermediate time scales. Together, these findings shed new light on the dynamics of domains in viscoelastic media and conserved order parameters in general.

Figures (6)

• Figure 1: Emergent phenomena induced by translational motion (velocity $\boldsymbol{v}$, black arrow) of biomolecular condensates and in domain-forming systems in general. A) Translational motion of domains of conserved order parameters (outlined by black line) induces long-ranged forces that are reminiscent of electrostatic dipoles (stream lines). Streamlines indicate the gradient of the chemical potential (color code; red corresponds to high values and blue to low values). B) Impermeable boundaries push fluid when they move. Therefore, the pressure gradients around moving colloidal particles in a Stokes fluid landau_lifshitz_1987 are opposite to the chemical potential gradients around moving droplets. This suggests a competition between phase separation and fluid flow. Streamlines indicate the gradient of the pressure (color code; red corresponds to high values and blue to low values). C) Translational motion leads to directed coalescence, where smaller condensates circle around and flow (black stream lines) towards the leading edge of larger condensates (black circle with label "1"). If hydrodynamic interactions dominate, smaller condensates may circle towards the trailing edge of a larger condensates.
• Figure 2: Effective diffusion coefficient $D_\text{eff}(t) \coloneqq \partial_t \mathop{\mathrm{MSD}}\nolimits(t)$ of a condensate in a viscoelastic medium compared to Brownian motion $D_\text{eq} \coloneqq 6 k_B T/(5\pi \eta R)$ in thermal equilibrium. A) For $\tau_\alpha > \tau_f > 0$, condensate motion is suppressed. Smaller values of $\tau_f / \tau_\alpha$ further reduce the growth rate of $\mathop{\mathrm{MSD}}\nolimits(t)$ and hence the decay of $\exp[-q^2 \mathop{\mathrm{MSD}}\nolimits(t) / 6]$ in Eq. \ref{['eq:ode_transformed_final']}. This affects the fidelity of the asymptotic limits [Eq. \ref{['eq:ode_transformed_boundary_condition']} and Eq. \ref{['eq:diffusion_coefficient_long']}, red lines] compared to the numerical solution of Eq. \ref{['eq:ode_transformed_final']} [solid lines]. B) For $\tau_\alpha < \tau_f$, condensate motion is enhanced relative to thermal equilibrium. For the parameters chosen here, the diffusivity depends only weakly on the time scale. In both panels, the Fourier number, which relates the stirring time to the characteristic time of diffusive mass transport, is $\text{Fo} = D_\text{eq} \tau_\alpha/R^2 = 100$ and hydrodynamic screening is assumed to be negligible $\gamma \approx 0$.
• Figure 3: Diffusion coefficient of condensates on long time scales, relative to thermal equilibrium ($\tau_f = \tau_\alpha$), as a function of A) the Fourier number $\text{Fo} \coloneqq 6 k_B T \tau_\alpha/(5\pi \eta R^3)$ and B) the ratio between the fluid relaxation time $\tau_f$ and the stress correlation time $\tau_\alpha$. Solid lines represent analytical approximations, dots represent numerical calculations. For simplicity, hydrodynamic screening is assumed negligible, $\gamma \approx 0$.
• Figure 4: Effective diffusion coefficient of a condensate with radius $R$, compared to a reference condensate with radius $R_\text{ref}$. Solid lines indicate analytical approximations, while dots represent numerical calculations, where $\text{Fo} = 1000$ for the reference radius $R_\text{ref}$. A) For $\tau_f < \tau_\alpha$, there is a crossover regime where the diffusion coefficient is largely independent of condensate radius. Note that the analytical approximation fails when $\tau_f \ll \tau_\alpha$. B) For $\tau_f > \tau_\alpha$, there is a crossover regime where the diffusion coefficient departs from the Stokes-Einstein scaling. The width of this crossover regime increases with $\tau_f / \tau_\alpha$. For simplicity, hydrodynamic screening is assumed negligible, $\gamma \approx 0$.
• Figure 5: Scaling of the effective diffusion coefficient with condensate radius, $D_\infty \sim 1/R^n$, compared to a reference condensate with radius $R_\text{ref}$ and corresponding Fourier number $\text{Fo} = 1000$. In the blue region of the phase diagram, condensate motion is almost independent of condensate size. In the red region of the phase diagram, the dependence of Brownian motion on condensate size is amplified. In all other regions (bright yellow), the Stokes-Einstein relation $D_\infty \sim 1/R$ is recovered. Dashed line corresponds to a passive system where the fluctuation-dissipation theorem (FDT) holds.