Figure 1

Density and orientation of cortical filaments. (a),(b) Fluorescence microscopy images of microtubule patterns in Tobacco BY2 suspension cells. (a) Between cell divisions microtubules form a cortical array and orient preferentially perpendicular to the cell’s long axis. (b) Shortly before cell division microtubules condense into a ringlike structure, the preprophase band. (c) Representation of a stationary solution to the dynamic equations (1, 2, 3, 4, 5) resembling a PPB. The density of filaments is represented by the gray level on the cylinder. The blue bars indicate the nematic order of filaments by their orientation and length. Panels (a) and (b) are courtesy of Vos (Wageningen University). Scale bars are $10\mu \mathrm{m}$.Reuse & Permissions

Figure 2

Schematic state diagrams of an active gel layer in cylindrical geometry with periodic boundary conditions in the $x$ direction are shown for two values of $\tilde{F}$ as a function of two dimensionless parameters ${\tilde{B}}_2$ and $\tilde{D}$. Regions of linear stability of homogenous filament distributions are shaded in dark gray. Outside these regions, the topology of the state diagram as determined numerically is indicated by dotted lines. Asymptotic states include chevron patterns and single or multiple rings that are either stationary or do oscillate. Regions of oscillating solutions are shaded in light gray. The long dashed line indicates a Hopf bifurcation determined by linear stability analysis. Examples for stationary chevron and ring patterns are displayed in Fig. 3. Parameters are defined as $\tilde{D}=-({\tilde{A}}_1+2{\tilde{c}}_0{\tilde{A}}_2)$, ${\tilde{B}}_2=B_2/(B_1{L}^2)$, $\tilde{F}=F/(B_1^2{L}^4)$, and ${\tilde{A}}_2=A_2/(B_1{L}^4)$. Parameters values are ${\tilde{c}}_0=c_{0}L=0.5$, $L(-2B_1/A_1{)}^{1/2}=5$, ${\tilde{A}}_1=A_1/(B_1{L}^2)=-2$, $A_3/(B_1{L}^6)=0.1$, $A_4/(B_1{L}^6)=-1$, $A_5/B_1=1$, $B_3/(B_1{L}^2)=-0.05$, $B_5/B_1=-10$, and $\tilde{F}=-1$ for (a) and $\tilde{F}=1$ for (b). $L$ is the period of the system.Reuse & Permissions

Figure 3

Examples of stationary solutions to the dynamic equations with periodic boundary conditions. (a) Chevron pattern with homogenous filament density for ${\tilde{B}}_2=-1.25$, $\tilde{D}=0.75$, and $\tilde{F}=-1$. (b) Multiple ring pattern for ${\tilde{B}}_2=-1.5$, $\tilde{D}=1.5$, and $\tilde{F}=-1$.Reuse & Permissions

Figure 4

Snapshots at different times of an oscillating filament ring for no-flux boundary conditions and ${\tilde{B}}_2=1$, $\tilde{D}=0.25$, and $\tilde{F}=1$. The pattern consists of a filament ring that forms near the center of the cylinder (a) and moves towards the right pole (b),(c), where the ring disappears. Simultaneously, a new ring is formed near the center (d), which subsequently moves to the opposite pole. The whole process is repeated periodically. The times correspond to phases $\varphi =\omega t$ of the oscillation with (a) $\varphi =0$, (b) $\varphi =(4/36)2\pi$, (c) $\varphi =(7/36)2\pi$, and (d) $\varphi =(15/36)2\pi$.Reuse & Permissions