A Stochastic Model of Cortical Microtubule Anchoring and Mechanics Provides Regulatory Control of Microtubule Shape

Abstract

The organization of cortical microtubule arrays play an important role in the development of plant cells. Until recently, the direct mechanical influence of cell geometry on the constrained microtubule (MT) trajectories have been largely ignored in computational models. Modelling MTs as thin elastic rods constrained on a surface, a previous study examined the deflection of MTs using a fixed number of segments and uniform segment lengths between MT anchors. It is known that the resulting MT curves converge to geodesics as the anchor spacing approaches zero. In the case of long MTs on a cylinder, buckling has been found for transverse trajectories. There is a clear interplay between two factors in the problem of deflection: curvature of the membrane and the lengths of MT segments. Here, we examine the latter in detail, in the backdrop of a circular cylinder. In reality, the number of segments are not predetermined and their lengths are not uniform. We present a minimal, realistic model treating the anchor spacing as a stochastic process and examine the net effect on deflection. We find that, by tuning the ratio of growth speed to anchoring rate, it is possible to mitigate MT deflection and even prevent buckling for lengths significantly larger than the previously-derived critical buckling length. We suggest that this mediation of deflection by anchoring might provide cells with a means of preventing arrays from deflecting away from the transverse orientation.

Appendix A Steady State

Fig. 9

Numerical confirmation of the steady state tip length approximation (A5), as a function of small values of $l_d$. Starting with $l_0=0$, 1000 trials were simulated with 40 anchoring steps each. The steady state tip length $l^*$ is found by averaging the final tip lengths of the 1000 trials (dotted). The approximate $l^*$ was found by numerically solving (A5) for $l^*$

The two-step process can be reduced to a single step, with the Markov kernel $K(l|l^n)$ describing the probability density of having tip length l given $l^n$ previously. The subscript “$_\text {tip}$” will be dropped for these variables. To find K, we first find the density describing anchoring at point $l_0^{n+1}\in [0,l^n]$, $\psi (l_0^{n+1}|l^n)$. This is uniform. Then we condition the probability of growing to length l given initial $l_0^{n+1}$, $P(l|l_0^{n+1})$, to get $K(l|l_0^{n+1}|l^n)=P(l|l_0^{n+1})\psi (l_0^{n+1}|l^n)$. Integrating over the intermediate variable denoted as $l_0$ rather than $l_0^{n+1}$,

$$\begin{aligned} K(l|l^{n})&= \int _{0}^{l^n}P(l|l_0)\psi (l_0|l^n)\textrm{d}l_0 \nonumber \\&= \frac{\pi }{2 l^n l_d^2}\int _{0}^{l^n} l \, e^{-\frac{\pi }{4 l_d^2}l^2}e^{\frac{\pi }{4 l_d^2}l_0^2}\Theta (l-l_0) \textrm{d}l_0 \nonumber \\&= {\left\{ \begin{array}{ll} \frac{\pi l}{2 l^n l_d^2}e^{-\frac{\pi }{4 l_d^2}l^2} \int _{0}^{l^n} e^{\frac{\pi }{4 l_d^2}l_0^2}\textrm{d}l_0, &{}\quad l^n\le l\\ \frac{\pi l}{2l^n l_d^2}e^{-\frac{\pi }{4 l_d^2}l^2} \int _{0}^{l} e^{\frac{\pi }{4 l_d^2}l_0^2}\textrm{d}l_0, &{}\quad l^n> l \end{array}\right. } \nonumber \\&= {\left\{ \begin{array}{ll} \frac{\pi }{2}\frac{l}{l^n l_d}e^{-\frac{\pi l^2}{4 l_d^2}}\text {erfi}\left( \frac{\sqrt{\pi }l^n}{2 l_d}\right) , &{}\quad l^n\le l\\ \frac{\pi }{2}\frac{l}{l^n l_d}e^{-\frac{\pi l^2}{4 l_d^2}}\text {erfi}\left( \frac{\sqrt{\pi }l}{2 l_d}\right) , &{}\quad l^n> l \end{array}\right. } \end{aligned}$$

(A1)

Suppose there exists a stationary distribution $K_s(l)$. Then, given that this has been reached, the next step will have this same distribution. This is given by the Fredholm integral equation

$$\begin{aligned} K_s(l)&= \int _0^\infty K(l|l') K_s(l') \textrm{d}l' \end{aligned}$$

(A2)

This cannot be solved analytically. Rather, we make a closure approximation that the stationary distribution is approximated by the distribution given by having length $l^*$ in the previous step,

$$\begin{aligned} K_s(l) \approx K(l|l^*). \end{aligned}$$

(A3)

Heuristically, this is true if the stationary distribution is sharply peaked around $l^*$. In such case, the previous step, having reached steady state, will sample from $K_s(l)$ and return values near $l^*$. Indeed, in the limit $K_s(l) = \delta (l-l^*)$,

$$\begin{aligned} K_s(l)&= \int _0^\infty K(l|l') K_s(l') \textrm{d}l' \nonumber \\&= \int _0^\infty K(l|l') \delta (l-l^*) \textrm{d}l' \nonumber \\&=K(l|l^*). \end{aligned}$$

(A4)

Then one is able to solve for $l^*$ with the relation

$$\begin{aligned} l^*&= \int _0^{\infty }l K_s(l)\textrm{d}l = \int _0^{\infty }l K(l|l^*)\textrm{d}l. \end{aligned}$$

(A5)

This is confirmed to be accurate numerically in Fig. 9 for small values of $l_d$. Larger values of $l_d$ requires more a careful implementation of the numerical root-solving for Eq. (A5), not attempted here. One notes that as $l_d$ increases, the approximation worsens. This is expected because, as $l_d$ increases, the tip length distribution widens, diverging from the delta-function distribution limit.