Mechanisms of self-organization of cortical microtubules in plants revealed by computational simulations

Abstract

Microtubules confined to the two-dimensional cortex of elongating plant cells must form a parallel yet dispersed array transverse to the elongation axis for proper cell wall expansion. Some of these microtubules exhibit free minus-ends, leading to migration at the cortex by hybrid treadmilling. Collisions between microtubules can result in plus-end entrainment ("zippering") or rapid depolymerization. Here, we present a computational model of cortical microtubule organization. We find that plus-end entrainment leads to self-organization of microtubules into parallel arrays, whereas catastrophe-inducing collisions do not. Catastrophe-inducing boundaries (e.g., upper and lower cross-walls) can tune the orientation of an ordered array to a direction transverse to elongation. We also find that changes in dynamic instability parameters, such as in mor1-1 mutants, can impede self-organization, in agreement with experimental data. Increased entrainment, as seen in clasp-1 mutants, conserves self-organization, but delays its onset and fails to demonstrate increased ordering. We find that branched nucleation at acute angles off existing microtubules results in distinctive sparse arrays and infer either that microtubule-independent or coparallel nucleation must dominate. Our simulations lead to several testable predictions, including the effects of reduced microtubule severing in katanin mutants.

Figure 1.

Simulation snapshots at t = 60 min with collision-induced catastrophe only, using parameters from wild type at 31°C (Kawamura and Wasteneys 2008) (A) and collision-induced pauses, using the single-state model of Baulin et al. (2007) (B).

Figure 2.

Collision-induced catastrophe at steep angles (>40°) and entrainment at shallow angles (<40°) for four sets of kinetic parameters from Kawamura and Wasteneys (2008) and continuously depolymerizing minus-end (Shaw et al., 2003). The top (A and B) and bottom (C and D) rows are wild-type and mor1-1 kinetic parameters, respectively, and the left (A and C) and right (B and D) are at 21 and 31°C, respectively. New MTs are inserted randomly at a rate of k₀ = 10 μm⁻² min⁻¹. The boundaries are periodic in both directions. After 60 min, order emerges in local domains in all cases except mor1-1 at 31°C. The direction of the red arrow indicates the dominant direction of the MT array, whereas their lengths are proportional to the order parameter.

Figure 3.

Data from wild-type (WT) simulation runs. (A) Average length of an MT. (B) Order parameter S, given by the equation in Materials and Methods [a modified version of the Baulin et al. (2007) order parameter]. All simulations use kinetics from wild type at 31°C in Table 1. (C–E) Histograms from selected runs depicted in Figure 2. Blue curves in A and B and histogram in C have a depolymerizing minus end and a critical entrainment angle of θ_Z = 40°. Green curves in A and B and histogram in D have a static minus-end (v^p_s = 0) with θ_Z = 40°, and red curves in A and B and histogram in E have depolymerizing minus end with θ_Z = 60°. Time series from 10 independent simulations are shown in each case.

Figure 4.

Data from simulations of the mor1-1 mutant. (A) Average length of an MT. (B) Order parameter S, given by the equation in Materials and Methods [a modified version of the Baulin et al. (2007) order parameter]. All simulations use kinetics from mor1-1 at 31°C in Table 1. Blue curves in A and B and histogram in C have a depolymerizing minus end, whereas green curves and histogram in D have a static minus-end (v^p_s = 0).

Figure 5.

Simulation snapshots at t = 60 min using kinetic parameters from wild type at 31°C. (A) Entrainment at shallow angles (<40°) and no collision-induced catastrophe (p_cat = 0). (B) A biased, transverse dominant angle that arises if two edges (here, the top and bottom) induce catastrophe. This provides a possible mechanism for selecting a direction transverse to the cell elongation axis. (C) Sparse array that results if all nucleation is branched MT-dependent nucleation. (D) Moderately sparse array arising from a combination of MT-dependent and MT-independent nucleation.