Figure 6. Results for the polarity sorting simulation in a spherical shell.

Initially, 100,000 0.25 µm-long filaments modeling microtubules and 200,000 motors modeling crosslinking kinesin-like proteins are placed between two concentric spherical shells with radii $r_{\mathrm{in}}=5\mathrm{\mu }\mathrm{m}$ and $r_{\mathrm{out}}=5.102\mathrm{\mu }\mathrm{m}$, to maintain the volume fraction of filaments between these two shells at 40%. Initially, all filaments are evenly distributed on the spherical shell, with their orientation randomly chosen to be either ${\displaystyle \pm {\mathit{e}}_{\theta }}$ at each point, where ${\displaystyle {\mathit{e}}_{\theta }}$ is the polar basis norm vector of spherical coordinate system. The pure filament system is relaxed for $1\mathrm{s}$ to resolve the overlaps in the initial configuration. Afterwards at $t=0$, $200,000$ motors are added to the system homogeneously distributed between the two shells. Sample points are evenly placed to measure the statistics by averaging the volume within $0.25\mathrm{\mu }\mathrm{m}$ from each sample point. (A) The configuration at $t=0$. Filaments are colored by their polarity, while motors are colored as black dots. Only randomly selected 10% of all motors (same after) are shown in the image to illustrate the distribution. (B) Randomly selected trajectories of filaments from $t=0\mathrm{s}$ to $1\mathrm{s}$. Trajectories are colored by time. It is clear that filaments move along the meridians. (C1-C5) Configuration and statistics at $t=4\mathrm{s}$. (C1) The filaments and motors. Motors clearly concentrate in some areas. (C2) The motor number density, that is, number of motors per $1{\mathrm{\mu }\mathrm{m}}^3$. (C3) The nematic director field (shown as black bars) and the nematic order parameter $S$. (C4) The filament volume fraction. (C5) The divergence of polarity field ${\displaystyle \mathrm{\nabla }\cdot \mathit{p}}$ non-dimensionalized by filament length, that is, change of mean polarity per filament length. (D) The development of the correlation between motor number density $n/n_{\mathrm{ave}}$ and the polarity divergence field, at different times of the simulation. Clearly high $n/n_{\mathrm{ave}}$ are correlated with positive polarity divergence. (E1-E5) Configuration and statistics at $t=0.27\mathrm{s}$ for a comparative simulation where motors have end-pausing, arranges in the same style as C1-C5. This case shows significant contraction instead of polarity sorting as filaments are pulled away from the north and south poles and the overall volume fraction significantly increases to approximately 60%. The structure becomes densely packed and does not significantly evolve further. (F) The correlation between motor number density $n/n_{\mathrm{ave}}$ and the local filament volume fraction. For the polarity sorting case at $t=4\mathrm{s}$ the motor number density correlates with low filament volume fraction. This is not seen in the end pausing (EP) case. (G) A schematic for the correlations shown in D and F.