Figure 1

(a) Fluorescently labeled microtubules in a Tobacco BY-2 cell expressing GFP:TUA6. Image courtesy of Jelmer Lindeboom, Wageningen University. (b) Schematic representation of a plant cell cortical array.Reuse & Permissions

Figure 2

Schematic representation of the model interaction. The microtubule of interest is drawn in black and other microtubules that it encounters are in gray. The active segments of the black microtubule have an arrow head indicating growth or shrinkage whereas inactive segments end in the junction with the following segment depicted by a dot.Reuse & Permissions

Figure 3

Probabilities for zippering, crossovers and catastrophes as deduced from the observations of [15] (combined data from MBD-DsRed and YFP-TUA6 labeling). Light gray shaded region: fraction of zippering events. Dark gray shaded region: fraction of induced catastrophes. White region: fraction of crossovers. Every data point is located at the center of the corresponding bin, and the shaded regions have been extended to the boundaries using horizontal lines. The corresponding lowest order Fourier coefficients of the interaction functions are: ${\widehat{c}}_0=0.59$, ${\widehat{c}}_2=-0.36$, and ${\widehat{z}}_0=0.24$ (computed using numerical integration of the product of $\left|\text{sin}\theta \right|$ and a piecewise linear interpolation of the data).Reuse & Permissions

Figure 4

(Color online) Bifurcation diagrams for the simplified interaction functions using three different induced catastrophe parameters. The figures on the left depict the probability $P_{\text{c}}\left(\theta \right)$ to induce a catastrophe upon collision, along with the corresponding values of ${\widehat{c}}_0$. The center and right columns depict the corresponding bifurcation diagrams as a function of $G$, expressed in terms of the total density $K_{\text{total}}$ and the 2D nematic order parameter $S_2$, respectively, where $K_{\text{total}}=\int K\left(\theta \right)d\theta$. The isotropic solutions are by definition disordered, so $S_2=0$, and their density is computed from Eq. (42). The bifurcation point is determined using Eq. (50), with ${\widehat{c}}_2=-1/2$. For each diagram, ordered solutions have been computed for ${\widehat{z}}_0=0$ (black), ${\widehat{z}}_0=1$ (blue/dark gray) and ${\widehat{z}}_0=10$ (red/light gray). The solutions have been computed using the method discussed in appendix A. Solid lines indicate stable solutions and dashed lines indicate unstable solutions (see also Sec. 3E). Note that the case of ${\widehat{c}}_0=1$ in the absence of zippering is a singular case where the stability cannot be determined, because non-isotropic solutions only exist for $G=0$. This has been indicated by a dotted line. The $S_2$-diagrams include the asymptotic limit point at $G=0$ with absolute ordering (at infinite density). The labels a, b and c indicate the parameter values of the results depicted in Fig. 5. The fact that the solutions for $S_2$ in the case ${\widehat{c}}_0=\frac{3}{4}$ do not reach the asymptotic point $\left(G=0,S_2=1\right)$ is a consequence of the slowdown in convergence of the path-following method as $G\uparrow 0$.Reuse & Permissions

Figure 5

Three stable ordered solutions that correspond to the points labeled (a), (b), and (c) in Fig. 4. The (unstable) isotropic solutions for the same parameter values are indicated with a dashed line. The parameter values for (a) and (b) differ only in the value of $G$, whereas the parameter values for (b) and (c) differ only in the value of ${\widehat{z}}_0$. All results have been calculated using the method described in appendix A.Reuse & Permissions