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The cortical microtubule array of dark-grown hypocotyl cells of plant seedlings undergoes a striking, and developmentally significant, reorientation on exposure to light. This process is driven by the exponential amplification of a population of longitudinal microtubules, created by severing events localized at crossovers with the microtubules of the pre-existing transverse array. We present a dynamic one-dimensional model for microtubule amplification through this type of templated severing. We focus on the role of the probability of immediate stabilization-after-severing of the newly created lagging microtubule, observed to be a characteristic feature of the reorientation process. Employing stochastic simulations, we show that in the dynamic regime of unbounded microtubule growth, a finite value of this probability is not required for amplification to occur but does strongly influence the degree of amplification and hence the speed of the reorientation process. In contrast, in the regime of bounded microtubule growth, we show that amplification only occurs above a critical threshold. We construct an approximate analytical theory, based on a priori limiting the number of crossover events considered, which allows us to predict the observed critical value of the stabilization-after-severing probability with reasonable accuracy.

Central to the building and reorganizing cytoskeletal arrays is creation of new polymers. Although nucleation has been the major focus of study for microtubule generation, severing has been proposed as an alternative mechanism to create new polymers, a mechanism recently shown to drive the reorientation of cortical arrays of higher plants in response to blue light perception. Severing produces new plus ends behind the stabilizing GTP-cap. An important and unanswered question is how these ends are stabilized in vivo to promote net microtubule generation. Here we identify the conserved protein CLASP as a potent stabilizer of new plus ends created by katanin severing in plant cells. Clasp mutants are defective in cortical array reorientation. In these mutants, both rescue of shrinking plus ends and the stabilization of plus ends immediately after severing are reduced. Computational modeling reveals that it is the specific stabilization of severed ends that best explains CLASP's function in promoting microtubule amplification by severing and array reorientation., (© 2018 Lindeboom et al.)

The cortical microtubule arrays of higher plants are organized without centrosomes and feature treadmilling polymers that are dynamic at both ends. The control of polymer end stability is fundamental for the assembly and organization of cytoskeletal arrays, yet relatively little is understood about how microtubule minus ends are controlled in acentrosomal microtubule arrays, and no factors have been identified that act at the treadmilling minus ends in higher plants. Here, we identify Arabidopsis thaliana SPIRAL2 (SPR2) as a protein that tracks minus ends and protects them against subunit loss. SPR2 function is required to facilitate the rapid reorientation of plant cortical arrays as stimulated by light perception, a process that is driven by microtubule severing to create a new population of microtubules. Quantitative live-cell imaging and computer simulations reveal that minus protection by SPR2 acts by an unexpected mechanism to promote the lifetime of potential SPR2 severing sites, increasing the likelihood of severing and thus the rapid amplification of the new microtubule array., (© 2018 Nakamura et al.)

Plant morphogenesis is strongly dependent on the directional growth and the subsequent oriented division of individual cells. It has been shown that the plant cortical microtubule array plays a key role in controlling both these processes. This ordered structure emerges as the collective result of stochastic interactions between large numbers of dynamic microtubules. To elucidate this complex self-organization process a number of analytical and computational approaches to study the dynamics of cortical microtubules have been proposed. To date, however, these models have been restricted to two dimensional planes or geometrically simple surfaces in three dimensions, which strongly limits their applicability as plant cells display a wide variety of shapes. This limitation is even more acute, as both local as well as global geometrical features of cells are expected to influence the overall organization of the array. Here we describe a framework for efficiently simulating microtubule dynamics on triangulated approximations of arbitrary three dimensional surfaces. This allows the study of microtubule array organization on realistic cell surfaces obtained by segmentation of microscopic images. We validate the framework against expected or known results for the spherical and cubical geometry. We then use it to systematically study the individual contributions of global geometry, cell-edge induced catastrophes and cell-face induced stability to array organization in a cuboidal geometry. Finally, we apply our framework to analyze the highly non-trivial geometry of leaf pavement cells of Arabidopsis thaliana, Nicotiana benthamiana and Hedera helix. We show that our simulations can predict multiple features of the microtubule array structure in these cells, revealing, among others, strong constraints on the orientation of division planes.

We propose a minimal model for the spontaneous and persistent generation of polarity in a spherical cell based on dynamic microtubules and a single mobile molecular component. This component, dubbed the polarity factor, binds to microtubules nucleated from a centrosome located in the center of the cell, is subsequently delivered to the cell membrane, where it diffuses until it unbinds. The only feedback mechanism we impose is that the residence time of the microtubules at the membrane increases with the local density of the polarity factor. We show analytically that this system supports a stable unipolar symmetry-broken state for a wide range of parameters. We validate the predictions of the model by 2D particle-based simulations. Our model provides a route towards the creation of polarity in a minimal cell-like environment using a biochemical reconstitution approach.

Plant morphogenesis requires differential and often asymmetric growth. A key role in controlling anisotropic expansion of individual cells is played by the cortical microtubule array. Although highly organized, the array can nevertheless rapidly change in response to internal and external cues. Experiments have identified the microtubule-severing enzyme katanin as a central player in controlling the organizational state of the array. Katanin action is required both for normal alignment and the adaptation of array orientation to mechanical, environmental, and developmental stimuli. How katanin fulfills its controlling role, however, remains poorly understood. On the one hand, from a theoretical perspective, array ordering depends on the "weeding out" of discordant microtubules through frequent catastrophe-inducing collisions among microtubules. Severing would reduce average microtubule length and lifetime, and consequently weaken the driving force for alignment. On the other hand, it has been suggested that selective severing at microtubule crossovers could facilitate the removal of discordant microtubules. Here we show that this apparent conflict can be resolved by systematically dissecting the role of all of the relevant interactions in silico. This procedure allows the identification of the sufficient and necessary conditions for katanin to promote array alignment, stresses the critical importance of the experimentally observed selective severing of the "crossing" microtubule at crossovers, and reveals a hitherto not appreciated role for microtubule bundling. We show how understanding the underlying mechanism can aid with interpreting experimental results and designing future experiments., Competing Interests: The authors declare no conflict of interest.

The highly orientationally ordered cortical microtubule array in plant cells is a key component for cell growth and development. Recent experimental and computational work has shown that the anisotropic nucleation of new microtubules from pre-existing microtubules has a major effect on the alignment process. We formulate a theoretical model to investigate the role of the microtubule-bound nucleation on the self-organization of the dynamical cortical microtubules. A bifurcation analysis of the stability of the disordered phase of the model reveals that the effective degree of co-aligned nucleation is the main determinant of the location of the transition. Increased co-aligned nucleation creates a positive feedback effect on the ordering process that can significantly widen the ordered region. We validate these predictions by comparing to the results of particle-based simulations.

Environmental and hormonal signals cause reorganization of microtubule arrays in higher plants, but the mechanisms driving these transitions have remained elusive. The organization of these arrays is required to direct morphogenesis. We discovered that microtubule severing by the protein katanin plays a crucial and unexpected role in the reorientation of cortical arrays, as triggered by blue light. Imaging and genetic experiments revealed that phototropin photoreceptors stimulate katanin-mediated severing specifically at microtubule intersections, leading to the generation of new microtubules at these locations. We show how this activity serves as the basis for a mechanism that amplifies microtubules orthogonal to the initial array, thereby driving array reorientation. Our observations show how severing is used constructively to build a new microtubule array.

Normal plant growth requires the anisotropic expansion of cells and the proper orientation of their divisions. Both are controlled by the architecture of the cortical microtubule array. Cortical microtubules interact through frequent collisions. Several modelling studies have shown that these interactions can be sufficient for spontaneous alignment. Further requirements to this self-organization are the homogeneous distribution of microtubule density and reliable control over the array orientation. We review the contribution of computer simulations and mathematical modelling on each of these challenges. These models now provide a good understanding of the basic alignment mechanism and will continue to be very useful tools for investigating more advanced questions, for example how microtubule severing contributes to alignment and array reorientation., (Copyright © 2013 Elsevier Ltd. All rights reserved.)

The ordered arrangement of cortical microtubules in growing plant cells is essential for anisotropic cell expansion and, hence, for plant morphogenesis. These arrays are dismantled when the microtubule cytoskeleton is rearranged during mitosis and reassembled following completion of cytokinesis. The reassembly of the cortical array has often been considered as initiating from a state of randomness, from which order arises at least partly through self-organizing mechanisms. However, some studies have shown evidence for ordering at early stages of array assembly. To investigate how cortical arrays are initiated in higher plant cells, we performed live-cell imaging studies of cortical array assembly in tobacco (Nicotiana tabacum) Bright Yellow-2 cells after cytokinesis and drug-induced disassembly. We found that cortical arrays in both cases did not initiate randomly but with a significant overrepresentation of microtubules at diagonal angles with respect to the cell axis, which coincides with the predominant orientation of the microtubules before their disappearance from the cell cortex in preprophase. In Arabidopsis (Arabidopsis thaliana) root cells, recovery from drug-induced disassembly was also nonrandom and correlated with the organization of the previous array, although no diagonal bias was observed in these cells. Surprisingly, during initiation, only about one-half of the new microtubules were nucleated from locations marked by green fluorescent protein-γ-tubulin complex protein2-tagged γ-nucleation complexes (γ-tubulin ring complex), therefore indicating that a large proportion of early polymers was initiated by a noncanonical mechanism not involving γ-tubulin ring complex. Simulation studies indicate that the high rate of noncanonical initiation of new microtubules has the potential to accelerate the rate of array repopulation.

I formulate a dynamical model for microtubules interacting with a catastrophe-inducing boundary. In this model microtubules are either waiting to be nucleated, actively growing or shrinking, or stalled at the boundary. I first determine the steady-state occupation of these various states and the resultant length distribution. Next, I formulate the problem of the mean first-passage time to reach the boundary in terms of an appropriate set of splitting probabilities and conditional mean first-passage times and solve explicitly for these quantities using a differential equation approach. As an application, I revisit a recently proposed search-and-capture model for the interaction between microtubules and target chromosomes [M. Gopalakrishnan and B. S. Govindan, Bull. Math. Biol. 73, 2483 (2011)]. I show how my approach leads to a direct and compact solution of this problem.

The highly aligned cortical microtubule array of interphase plant cells is a key regulator of anisotropic cell expansion. Recent computational and analytical work has shown that the non-equilibrium self-organization of this structure can be understood on the basis of experimentally observed collisional interactions between dynamic microtubules attached to the plasma membrane. Most of these approaches assumed that new microtubules are homogeneously and isotropically nucleated on the cortical surface. Experimental evidence, however, shows that nucleation mostly occurs from other microtubules and under specific relative angles. Here, we investigate the impact of directed microtubule-bound nucleations on the alignment process using computer simulations. The results show that microtubule-bound nucleations can increase the degree of alignment achieved, decrease the timescale of the ordering process and widen the regime of dynamic parameters for which the system can self-organize. We establish that the major determinant of this effect is the degree of co-alignment of the nucleations with the parent microtubule. The specific role of sideways branching nucleations appears to allow stronger alignment while maintaining a measure of overall spatial homogeneity. Finally, we investigate the suggestion that observed persistent rotation of microtubule domains can be explained through a handedness bias in microtubule-bound nucleations, showing that this is possible only for an extreme bias and over a limited range of parameters.

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.

Microtubules are highly regulated dynamic elements of the cytoskeleton of eukaryotic cells. One of the regulation mechanisms observed in living cells is the severing by the proteins katanin and spastin. We introduce a model for the dynamics of microtubules in the presence of randomly occurring severing events. Under the biologically motivated assumption that the newly created plus end undergoes a catastrophe, we investigate the steady-state length distribution. We show that the presence of severing does not affect the number of microtubules, regardless of the distribution of severing events. In the special case in which the microtubules cannot recover from the depolymerizing state (no rescue events) we derive an analytical expression for the length distribution. In the general case we transform the problem into a single ordinary differential equation that is solved numerically.

The cortical array is a structure consisting of highly aligned microtubules which plays a crucial role in the characteristic uniaxial expansion of all growing plant cells. Recent experiments have shown polymerization-driven collisions between the membrane-bound cortical microtubules, suggesting a possible mechanism for their alignment. We present both a coarse-grained theoretical model and stochastic particle-based simulations of this mechanism, and we compare the results from these complementary approaches. Our results indicate that collisions that induce depolymerization are sufficient to generate the alignment of microtubules in the cortical array.

The recent visualization of the motion of fluorescently labeled cellulose synthase complexes by Alexander Paredez and colleagues heralds the start of a new era in the science of the plant cell wall. Upon drug-induced complete depolymerization, the movement of the complexes does not become disordered but instead establishes an apparently self-organized novel pattern. The ability to label complexes in vivo has provided us with the ideal tool for tackling the intriguing question of the underlying default mechanisms at play.

Microtubules or microtubule bundles in cells often grow longer than the size of the cell, which causes their shape and organization to adapt to constraints imposed by the cell geometry. We test the reciprocal role of elasticity and confinement in the organization of growing microtubules in a confining box-like geometry, in the absence of other (active) microtubule organizing processes. This is inspired, for example, by the cortical microtubule array of elongating plant cells, where microtubules are typically organized in an aligned array transverse to the cell elongation axis. The method we adopt is a combination of analytical calculations, in which the polymers are modeled as inextensible filaments with bending elasticity confined to a two-dimensional surface that defines the limits of a three-dimensional space, and in vitro experiments, in which microtubules are polymerized from nucleation seeds in microfabricated chambers. We show that these features are sufficient to organize the polymers in aligned, coiling configurations as for example observed in plant cells. Though elasticity can account for the regularity of these arrays, it cannot account for a transverse orientation of microtubules to the cell's long axis. We therefore conclude that an additional active, force-generating process is necessary to create a coiling configuration perpendicular to the long axis of the cell.

The light-induced reorientation of the cortical microtubule array in dark-grown Arabidopsis thaliana hypocotyl cells is a striking example of the dynamical plasticity of the microtubule cytoskeleton. A consensus model, based on katanin-mediated severing at microtubule crossovers, has been developed that successfully describes the onset of the observed switch between a transverse and longitudinal array orientation. However, we currently lack an understanding of why the newly populated longitudinal array direction remains stable for longer times and re-equilibration effects would tend to drive the system back to a mixed orientation state. Using both simulations and analytical calculations, we showthat the assumption of a small orientation-dependent shi? inmicrotubule dynamics is sufficient to explain the long-termlockin of the longitudinal array orientation. Furthermore, we show that the natural alternative hypothesis that there is a selective advantage in severing longitudinal microtubules, is neither necessary nor sufficient to achieve cortical array reorientation, but is able to accelerate this process significantly. [ABSTRACT FROM AUTHOR]

Abstract The assembly of microtubules generates forces that play a role in cellular motility processes such as the motion of chromosomes during mitosis. Recently, Mogilner and Oster proposed a model for the growth of microtubules that agrees quantitatively with the force-velocity relation measured for individual microtubules. In addition, the authors predicted that the stall force for any polymer consisting of N independently growing protofilaments should increase as the square root of N. We simulated this model and found that the stall force increases linearly with N, and is in fact consistent with the maximum force predicted by thermodynamic arguments. We show that this discrepancy can be explained by a more careful treatment of the "off-term" in the Mogilner-Oster model. [ABSTRACT FROM AUTHOR]

The cortical microtubule arrays of higher plants are organized without centrosomes and feature treadmilling polymers that are dynamic at both ends. The control of polymer end stability is fundamental for the assembly and organization of cytoskeletal arrays, yet relatively little is understood about how microtubule minus ends are controlled in acentrosomal microtubule arrays, and no factors have been identified that act at the treadmilling minus ends in higher plants. Here, we identify Arabidopsis thaliana SPI RAL2 (SPR2) as a protein that tracks minus ends and protects them against subunit loss. SPR2 function is required to facilitate the rapid reorientation of plant cortical arrays as stimulated by light perception, a process that is driven by microtubule severing to create a new population of microtubules. Quantitative live-cell imaging and computer simulations reveal that minus protection by SPR2 acts by an unexpected mechanism to promote the lifetime of potential SPR2 severing sites, increasing the likelihood of severing and thus the rapid amplification of the new microtubule array. [ABSTRACT FROM AUTHOR]