Enzyme-Enriched Condensates Show Self-Propulsion, Positioning, and Coexistence

Enzyme-enriched condensates can organize the spatial distribution of their substrates by catalyzing nonequilibrium reactions. Conversely, an inhomogeneous substrate distribution induces enzyme fluxes through substrate-enzyme interactions. We find that condensates move toward the center of a confining domain when this feedback is weak. Above a feedback threshold, they exhibit self-propulsion, leading to oscillatory dynamics. Moreover, catalysis-driven enzyme fluxes can lead to interrupted coarsening, resulting in equidistant condensate positioning, and to condensate division.

Figure 1

Steady-state concentration profiles for a 1D droplet with no-flux boundary conditions at $x=\pm L$. Arrows indicate reactive (vertical) and diffusive (horizontal) fluxes. The analytical solutions in the sharp interface approximation (lines) match our simulations (dots). We use $c_{+}$ as reference concentration and define the characteristic time ${\tau }_0≔k_2^{-1}$, diffusion length in the absence of enzymes $l_0≔\sqrt{D/k_2}$, and reference energy ${\epsilon }_0≔r{c}_{+}$. The remaining parameters, $c_{-}=0.1c_{+}$, $w=0.1l_0$, $R=l_0$, $L=5l_0$, $M=100D/{\epsilon }_0$, $k_1=k_2/c_{+}$, ${\chi }_s=-0.05r$, ${\chi }_p=-0.01r$, and $s+p=c_{+}$, are fixed for all figures unless stated otherwise.

Figure 2

Self-propulsion instability. (a) Analytical profiles for a droplet moving with velocity $v=2l_0/{\tau }_0$. Lighter colors indicate earlier times. (b) Graphical analysis of the self-consistency relation, Eq. (3), for $M{\epsilon }_0/D=1000$. The solid curve indicates the substrate imbalance $\mathrm{\Delta }s(v)$, while the slope of the dashed line corresponds to the right-hand side of inequality, Eq. (4). Stationary droplets correspond to unstable solutions (empty circle), while self-propelling droplets are stable (filled circles). (c) Theoretical prediction and simulation results for the self-propulsion velocity (color scale) with $M$ and $k_1$ as free parameters. The solid black lines indicate the critical mobility $M^{*}$ (for the explicit closed expression see [25]).

Figure 3

Self-centering and oscillations. (a) Droplet center trajectories in simulations for different values of $M$; symbols indicate the position in the diagram shown in Fig. 2. The droplet center is initially at $x_d(0)=-l_0$ in a domain of size $L=3l_0$. (b) Decay rate (black dots) and frequency (red dots) as functions of $M$, obtained by fitting the respective droplet trajectories. Red triangle indicates overdamped regime. (c) Relaxation rate $\lambda$ as a function of $k_1$ (bottom axis), for $M{\epsilon }_0/D=10$ and $x_d(0)=-0.3l_0$. Top axis relates the domain size to the length scale of the concentration profiles. The analytical predictions (blue line) in the quasi-steady-state approximation [25] match our simulations (black dots).

Figure 4

Coexistence and division of 3D droplets. We consider $M{\epsilon }_0/D=10$ and $w=0.05l_0$. Scale bars indicate unit length $l_0≔\sqrt{D/k_2}$. (a) Simulated pairs of droplets with different initial radii $R_1=1.1l_0$ and $R_2=0.9l_0$ either stay separated (cyan regime) or coalesce (yellow regime) depending on $k_1$ and on the difference in FH parameters $\mathrm{\Delta }\chi ={\chi }_p-{\chi }_s$. Solid black line corresponds to analytical estimate, Eq. (5), for $\mathrm{\Delta }{\chi }^{*}(k_1)$. (b) Simulation snapshots demonstrating a droplet division in 3D. We observed droplet divisions only for very strong attraction of enzymes toward substrates, here ${\chi }_s/r=-0.5$.