Phase separation by polar active transport

Abstract

We propose an active Cahn-Hilliard theory for the dynamics of a new type of phase transition where the driving force is not the direct interactions between the two separating components, but their active sorting by a third polar species. This third species can transport the other two along its polarity in opposite directions, thus separating them. Inspired by recent experiments where molecular motors that walk in opposite directions along microtubules are sorted into separated domains, our theoretical description of this process introduces a new mechanism for active phase separation and could serve as a model for the organization of biological material in space inside cells. We predict the formation of motor domains, and further show that they can either coarsen to form macroscopic phases or reach a finite micro- or mesoscopic steady state size, these latter due to an arrest of coarsening through activity.

Figures (4)

• Figure 1: a. Schematics of the experimental system. Two kinds of motors, plus-end directed (cyan) and minus end directed (magenta) are embedded in a fluid lipid membrane and diffuse freely. b. While bound to lipids, motors move along the filaments (black arrow) in either direction (solid arrow). At the same time, they exert a force on the filament in the opposite direction (dashed arrow). When both motors walk on a filament, their forces add up. c. From an initially disordered state, motors first walk on filaments separating in space. This creates a local force imbalance on filaments, that glide until the motor concentration at their side is such that no net force is exerted on the microtubules. This results in the creation of motors-enriched domains separated by a filament interface. d. When patterns have formed, filaments accumulate at the interface. Their pumping of motors (active transport) is countered by passive diffusion.
• Figure 2: a Phase diagram for $\bar{\kappa}<1$, in the ($\bar{\zeta}, \;\bar{a}$) plane. Different shaded areas correspond to different phases. We set $\bar{k}=0.56$. b Eigenvalues (eq. S11 in SI) for Type I (solid lines, green for macro and orange for mesophase separation) and Type II (dashed). The eigenvalues are normalized by their maximum value for clarity. Lengthscale rescaled by the microtubule length $L$.
• Figure 3: a Snapshots of simulations ($D_r t=64, 2048, 7000\;$) in the Type I region (top, ${\bar{\zeta}}=2800$, $\bar{a}=1.5$), one showing arrested coarsening (bottom, $\bar{\zeta}=1.87$, $\bar{a}=29.52$). Scale bars are $10L$, notice the difference in scales. b Heatmap of the growth exponent $\xi$, showing the slow down of coarsening also where the instability is of Type I. Obtained by interpolating simulations, marked by stars c-d Length plot $\ell(t)$ at fixed activity ( c), indicating how high $\bar{\zeta}$ favors coarsening, and at varying activity with fixed $\bar{\zeta}$ ( d) showing that activity favors arrested coarsening. Black dashed line indicates a $\sim t^{1/3}$ scaling, solid black line is a constant.
• Figure S1: (a) Snapshots at time $tD_r= 2048$ for different ratios of two opposite polarity motors. (b) Snapshot for micro-phase at time $tD_r= 100$ is shown. The scale bars are $L$, notice the difference in scale bars in two panels. (c) Variation of the average domain length $\ell / L$ with time for different ratio of the motors. We have shown values of $\bar{\zeta}$ in the legends. Dashed and solid lines refer to the slope $1/3$ and $0$, respectively.