Model for the orientational ordering of the plant microtubule cortical array

Abstract

The plant microtubule cortical array is a striking feature of all growing plant cells. It consists of a more or less homogeneously distributed array of highly aligned microtubules connected to the inner side of the plasma membrane and oriented transversely to the cell growth axis. Here we formulate a continuum model to describe the origin of orientational order in such confined arrays of dynamical microtubules. The model is based on recent experimental observations that show that a growing cortical microtubule can interact through angle dependent collisions with pre-existing microtubules that can lead either to co-alignment of the growth, retraction through catastrophe induction or crossing over the encountered microtubule. We identify a single control parameter, which is fully determined by the nucleation rate and intrinsic dynamics of individual microtubules. We solve the model analytically in the stationary isotropic phase, discuss the limits of stability of this isotropic phase, and explicitly solve for the ordered stationary states in a simplified version of the model.

Computer simulation and mathematical models of the noncentrosomal plant cortical microtubule cytoskeleton

“…However, among the studies that consider both bundling and collision-induced depolymerization interactions, there are controversial results regarding the relative contribution of these two mechanisms. Tindemans et al [2010] conclude that collisioninduced depolymerization is sufficient to induce organization even in the absence of bundling, in line with their mathematical model in Hawkins et al [2010]. In their model, bundling has only a minor contribution on organization.…”

“…Tindemans et al [2010] consider a two-state dynamic instability simulation together with bundling and collision-induced depolymerization interactions, complementing their mathematical model in Hawkins et al [2010]. According to the Tindemans et al simulation model, a CMT plus end is either growing or shortening at anytime, whereas the minus end is always static.…”

“…In the mathematical model of Baulin et al [2007], the impact of interactions (that result in CMT stalling) are approximated inspired by the kinetic theory of gases based on the average length, velocity, and density of CMTs in the system, and accounting for the fact that CMTs with similar angles run into each other less frequently. Hawkins et al [2010] developed a stronger mathematical model that considers CMT bundling and collision-induced depolymerization. They used this model to investigate parameter regions where CMT array organization is possible.…”

“…The necessity of CMT interactions for parallel array organization is commonly agreed upon among different modeling studies [Dixit and Cyr, 2004b;Baulin et al, 2007;Allard et al, 2010b;Eren et al, 2010;Hawkins et al, 2010;Tindemans et al, 2010]. However, among the studies that consider both bundling and collision-induced depolymerization interactions, there are controversial results regarding the relative contribution of these two mechanisms.…”

“…Shi and Ma [2010] employ a method that is based on computation of eigenvalues of a standard nematic order matrix [Chaikin and Lubensky, 1995]. Hawkins et al [2010] and Tindemans et al [2010] employ another nematic liquid crystal order parameter based on the orientation and length densities of CMTs. Allard et al [2010b] uses a modified version of the cost function in Baulin et al [2007], that represents the difference between projected polymer length in the dominant direction and its perpendicular direction.…”