Kinetic theory of pattern formation in mixtures of microtubules and molecular motors

In this study we formulate a theoretical approach, based on a Boltzmann-like kinetic equation, to describe pattern formation in two-dimensional mixtures of microtubular filaments and molecular motors. Following the previous work by Aranson and Tsimring [Phys. Rev. E 74, 031915 (2006)] we model the motor-induced reorientation of microtubules as collision rules, and devise a semianalytical method to calculate the corresponding interaction integrals. This procedure yields an infinite hierarchy of kinetic equations that we terminate by employing a well-established closure strategy, developed in the pattern-formation community and based on a power-counting argument. We thus arrive at a closed set of coupled equations for slowly varying local density and orientation of the microtubules, and study its behavior by performing a linear stability analysis and direct numerical simulations. By comparing our method with the work of Aranson and Tsimring, we assess the validity of the assumptions required to derive their and our theories. We demonstrate that our approximation-free evaluation of the interaction integrals and our choice of a systematic closure strategy result in a rather different dynamical behavior than was previously reported. Based on our theory, we discuss the ensuing phase diagram and the patterns observed.

Figure 1

Collision rule employed in Eq. (1). Two colliding microtubular bundles are reoriented by the action of the molecular motors to assume a common orientation along the bisector of the original angle between them. Their center of mass does not move in the process.

Figure 2

Model anisotropic distribution of the molecular motors along a microtubular filament.

Figure 3

$B_{1,1}^{\left(1\right)\mathrm{ani}}$ as a function of ${\tau }_0$.

Figure 4

Regions of existence of the density instability for ${\tau }_0=0$ and various values of $\mathrm{\Xi }$. The lines are solutions to Eq. (63), while the shaded regions indicate where the homogeneous and isotropic state is unstable with respect to density fluctuations (the density instability). The lines can therefore be seen as spinodal lines, and the shaded areas as regions of phase separation.

Figure 5

The data from Fig. 4 replotted as $\mathrm{\Xi }$ vs ${\rho }_0$ graph for ${\tau }_0=0$ and various values of $\alpha$. The dotted brown line is the onset of global order, given by Eq. (60). Black circles indicate points for which we perform direct numerical simulations with the ATC and QC models: ${\tau }_0=0$, $\alpha =0.35$, $\mathrm{\Xi }=0.2$, and $\mathrm{\Xi }=0.5$ with ${\rho }_0=2$, 3, $\cdots$, 12.

Figure 6

Instantaneous snapshots from the direct numerical simulations of the ATC model with ${\tau }_0=0$, $\alpha =0.35$, and $\mathrm{\Xi }=0.2$. (a) ${\rho }_0=5$. (b) ${\rho }_0=12$.

Figure 7

Instantaneous snapshots from the direct numerical simulations of the ATC model with ${\tau }_0=0$, $\alpha =0.35$, and $\mathrm{\Xi }=0.5$. (a) ${\rho }_0=5$. (b) ${\rho }_0=7$. (c) ${\rho }_0=12$.

Figure 8

Instantaneous snapshots from the direct numerical simulations of the QC model with ${\tau }_0=0$, $\alpha =0.35$, and $\mathrm{\Xi }=0.2$. (a) ${\rho }_0=5$. (b) ${\rho }_0=11$.

Figure 9

Instantaneous snapshots from the direct numerical simulations of the QC model with ${\tau }_0=0$, $\alpha =0.35$, and $\mathrm{\Xi }=0.5$. (a) ${\rho }_0=4$. (b) ${\rho }_0=8$. (c) ${\rho }_0=12$.

Figure 10

Linear stability diagram of the QC model for ${\tau }_0=0$ and $\alpha =0.35$. As in Fig. 5, the dotted black line is the onset of global order, given by Eq. (60), and the dashed blue line delineates the region of the parameter space where the homogeneous and isotropic state exhibits the density instability (the same as the dashed blue line in Fig. 5). The brown solid line indicates the region where a homogeneous, globally ordered state becomes linearly unstable. Inside this line we also specify the instability mode: magenta-shaded region (inside the magenta dash-dotted line) corresponds to the density and polarization fluctuations modulated along the direction of the global order, while the brown-shaded region corresponds to modulations both perpendicular and parallel to that direction.

Figure 11

Comparison between the long-time dynamics of the ATC and QC models with ${\tau }_0=0$, $\alpha =0.6$, $\mathrm{\Xi }=0.5$, and ${\rho }_0=12$. (a) ATC model. (b) QC model.