Taking directions: the role of microtubule-bound nucleation in the self-organization of the plant cortical array

Microtubule simulations in plant biology: A field coming to maturity

The plant cortical microtubule array is an important determinant of cell wall structure and, therefore, plant morphology and physiology. The array consists of dynamic microtubules interacting through frequent collisions. Since the discovery by Dixit and Cyr (2004) that the outcome of such collisions depends on the collision angle, computer simulations have been indispensable in studying array behaviour. Over the last decade, the available simulation tools have drastically improved: multiple high-quality simulation platforms exist with specific strengths and applications. Here, we review how these platforms differ on the critical aspects of microtubule nucleation, flexibility, and local orienting cues; and how such differences affect array behaviour. Building upon concepts and control parameters from theoretical models of collective microtubule behaviour, we conclude that all these factors matter in the debate about what is most important for orienting the array: local cues like mechanical stresses or global cues deriving from the cell geometry.

Mean-field theory approach to three-dimensional nematic phase transitions in microtubules

Microtubules are dynamic intracellular fibers that have been observed experimentally to undergo spontaneous self-alignment. We formulate a three-dimensional (3D) mean-field theory model to analyze the nematic phase transition of microtubules growing and interacting within a 3D space, then make a comparison with computational simulations. We identify a control parameter G_{eff} and predict a unique critical value G_{eff}=1.56 for which a phase transition can occur. Furthermore, we show both analytically and using simulations that this predicted critical value does not depend on the presence of zippering. The mean-field theory developed here provides an analytical estimate of microtubule patterning characteristics without running time-consuming simulations and is a step towards bridging scales from microtubule behavior to multicellular simulations.

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

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.

Mechanical stress effects on transcriptional regulation of genes encoding microtubule- and actin-associated proteins

Plant cytoskeleton regulation has been studied using a new approach based on both (1) pharmacological analysis of tubulin and actin inhibitors and (2) mechanical stimulation achieved by using a slow-rotating (2 rpm) clinostat in combination with transcriptional analysis of genes encoding TUA6, ACT2, MAP65-1, CLASP, PLDδ, FH4 and FH1 proteins in Arabidopsis thaliana seedling roots. The obtained data suggest feedback between the organization of microtubule (MT) and actin filament (AF) networks and the expression of the ACT2, TUA6, MAP65-1, CLASP and FH1/FH4 genes. Different regulation of feedback between MT/AF organization and TUA6, ACT2, MAP65-1, CLASP, FH4 and FH1 gene expression was noted during slow clinorotation, possibly due to altered mechanical impact on the cortical cytoskeleton. For the first time, the expression of the tubulin-associated gene MAP65-1 was shown to be dependent upon the organization of AFs. TUA6, MAP65-1, CLASP, FH1 and FH4 likely participate in mechanical signal transduction. Our work demonstrated that slow clinorotation is able to cause mechanical stress.

Long-term single-cell imaging and simulations of microtubules reveal principles behind wall patterning during proto-xylem development

Plants are the tallest organisms on Earth; a feature sustained by solute-transporting xylem vessels in the plant vasculature. The xylem vessels are supported by strong cell walls that are assembled in intricate patterns. Cortical microtubules direct wall deposition and need to rapidly re-organize during xylem cell development. Here, we establish long-term live-cell imaging of single Arabidopsis cells undergoing proto-xylem trans-differentiation, resulting in spiral wall patterns, to understand microtubule re-organization. We find that the re-organization requires local microtubule de-stabilization in band-interspersing gaps. Using microtubule simulations, we recapitulate the process in silico and predict that spatio-temporal control of microtubule nucleation is critical for pattern formation, which we confirm in vivo. By combining simulations and live-cell imaging we further explain how the xylem wall-deficient and microtubule-severing KATANIN contributes to microtubule and wall patterning. Hence, by combining quantitative microscopy and modelling we devise a framework to understand how microtubule re-organization supports wall patterning.

Plant cell wall formation is directed by cortical microtubules, which produce complex patterns needed to support xylem vessels. Here, the authors perform live-cell imaging and simulations of Arabidopsis cells during proto-xylem differentiation to show how local microtubule dynamics control pattern formation.

A plausible mechanism for longitudinal lock-in of the plant cortical microtubule array after light-induced reorientation

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 show that the assumption of a small orientation-dependent shift in microtubule dynamics is sufficient to explain the long-term lock-in 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.

Robust banded protoxylem pattern formation through microtubule-based directional ROP diffusion restriction

In plant vascular tissue development, different cell wall patterns are formed, offering different mechanical properties optimised for different growth stages. Critical in these patterning processes are Rho of Plants (ROP) proteins, a class of evolutionarily conserved small GTPase proteins responsible for local membrane domain formation in many organisms. While te spotted metaxylem pattern can easily be understood as a result of a Turing-style reaction-diffusion mechanism, it remains an open question how the consistent orientation of evenly spaced bands and spirals as found in protoxylem is achieved. We hypothesise that this orientation results from an interaction between ROPs and an array of transversely oriented cortical microtubules that acts as a directional diffusion barrier. Here, we explore this hypothesis using partial differential equation models with anisotropic ROP diffusion and show that a horizontal microtubule array acting as a vertical diffusion barrier to active ROP can yield a horizontally banded ROP pattern. We then study the underlying mechanism in more detail, finding that it can only orient curved pattern features but not straight lines. This implies that, once formed, banded and spiral patterns cannot be reoriented by this mechanism. Finally, we observe that ROPs and microtubules together only form ultimately static patterns if the interaction is implemented with sufficient biological realism.

Multi-scale regulation of cell branching: Modeling morphogenesis

Plant growth and development are driven by extended phases of irreversible cell expansion generating cells that increase in volume from 10- to 100-fold. Some specialized cell types define cortical sites that reinitiate polarized growth and generate branched cell morphology. This structural specialization of individual cells has a major importance for plant adaptation to diverse environments and practical importance in agricultural contexts. The patterns of cell shape are defined by highly integrated cytoskeletal and cell wall systems. Microtubules and actin filaments locally define the material properties of a tough outer cell wall to generate complex shapes. Forward genetics, powerful live cell imaging experiments, and computational modeling have provided insights into understanding of mechanisms of cell shape control. In particular, finite element modeling of the cell wall provides a new way to discover which cell wall heterogeneities generate complex cell shapes, and how cell shape and cell wall stress can feedback on the cytoskeleton to maintain growth patterns. This review focuses on cytoskeleton-dependent cell wall patterning during cell branching, and how combinations of multi-scale imaging experiments and computational modeling are being used to unravel systems-level control of morphogenesis.

A Plausible Microtubule-Based Mechanism for Cell Division Orientation in Plant Embryogenesis

Oriented cell divisions are significant in plant morphogenesis because plant cells are embedded in cell walls and cannot relocate. Cell divisions follow various regular orientations, but the underlying mechanisms have not been clarified. We propose that cell-shape-dependent self-organization of cortical microtubule arrays is able to provide a mechanism for determining planes of early tissue-generating divisions and may form the basis for robust control of cell division orientation in the embryo. To show this, we simulate microtubules on actual cell surface shapes, from which we derive a minimal set of three rules for proper array orientation. The first rule captures the effects of cell shape alone on microtubule organization, the second rule describes the regulation of microtubule stability at cell edges, and the third rule includes the differential effect of auxin on local microtubule stability. These rules generate early embryonic division plane orientations and potentially offer a framework for understanding patterned cell divisions in plant morphogenesis.

The self-organization of plant microtubules inside the cell volume yields their cortical localization, stable alignment, and sensitivity to external cues

Many cell functions rely on the ability of microtubules to self-organize as complex networks. In plants, cortical microtubules are essential to determine cell shape as they guide the deposition of cellulose microfibrils, and thus control mechanical anisotropy of the cell wall. Here we analyze how, in turn, cell shape may influence microtubule behavior. Building upon previous models that confined microtubules to the cell surface, we introduce an agent model of microtubules enclosed in a three-dimensional volume. We show that the microtubule network has spontaneous aligned configurations that could explain many experimental observations without resorting to specific regulation. In particular, we find that the preferred cortical localization of microtubules emerges from directional persistence of the microtubules, and their interactions with each other and with the stiff wall. We also identify microtubule parameters that seem relatively insensitive to cell shape, such as length or number. In contrast, microtubule array anisotropy depends on local curvature of the cell surface and global orientation follows robustly the longest axis of the cell. Lastly, we find that geometric cues may be overcome, as the network is capable of reorienting toward weak external directional cues. Altogether our simulations show that the microtubule network is a good transducer of weak external polarity, while at the same time, easily reaching stable global configurations.

Plants exhibit an astonishing diversity in architecture and morphology. A key to such diversity is the ability of their cells to coordinate and grow to reach a broad spectrum of sizes and shapes. Cell growth in plants is guided by the microtubule cytoskeleton. Here, we seek to understand how microtubules self-organize close to the cell surface. We build upon previous two-dimensional models and we consider microtubules as lines growing in three dimensions, accounting for interactions between microtubules or between microtubules and the cell surface. We show that microtubule arrays are able to adapt to various cell shapes and to reorient in response to external signals. Altogether, our results help to understand how the microtubule cytoskeleton contributes to the diversity of plant shapes and to how these shapes adapt to environment.

A computational framework for cortical microtubule dynamics in realistically shaped plant cells

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.

How selective severing by katanin promotes order in the plant cortical microtubule array

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.

Microtubule encounter-based catastrophe in Arabidopsis cortical microtubule arrays

BACKGROUND

The cortical microtubules (CMTs) that line the plasma membrane of interphase plant cells are extensively studied owing to their importance in forming cell walls, and their usefulness as a model system for the study of MT dynamic instability and acentrosomal MT organization. CMTs influence the orientation and structure of cellulose microfibrils in the cell wall by cooperatively forming arrays of varied patterns from parallel to netted. These CMT patterns are controlled by the combined activities of MT dynamic instability and MT-MT interactions. However, it is an open question as to how CMT patterns may feedback to influence CMT dynamics and interactions.

RESULTS

To address this question, we investigated the effects of CMT array patterning on encounter-based CMT catastrophe, which occurs when one CMT grows into another and is unable to cross over. We hypothesized that the varied CMT angles present in disordered (mixed CMTs) arrays will create more opportunities for MT-MT interactions, and thus increase encounter-based catastrophe rates and distribution. Using live-cell imaging of Arabidopsis cotyledon and leaf epidermal cells, we found that roughly 87% of catastrophes occur via the encounter-based mechanism, with the remainder occurring without encounter (free). When comparing ordered (parallel) and disordered (mixed orientation) CMT arrays, we found that disordered configurations show higher proportions of encounter-based catastrophe relative to free. Similarly, disordered CMT arrays have more catastrophes in general than ordered arrays. Encounter-based catastrophes were associated with frequent and sustained periods of pause prior to depolymerization, and CMTs with tight anchoring to the plasma membrane were more prone to undergo encounter-based catastrophe than weakly-attached ones. This suggests that encounter-based catastrophe has a mechanical basis, wherein MTs form physical barriers to one another. Lastly, we show that the commonly used measure of catastrophe frequencies (Fcat) can also be influenced by CMT ordering and plasma membrane anchoring.

CONCLUSIONS

Our observations add a new layer of complexity to our current understanding of MT organization in plants, showing that not only do individual CMT dynamics influence CMT array organization, but that CMT organization itself has a strong effect on the behavior of individual MTs.

The augmin connection in the geometry of microtubule networks

Microtubules mediate important cellular processes by forming highly ordered arrays. Organization of these networks is achieved by nucleating and anchoring microtubules at centrosomes and other structures collectively known as microtubule-organizing centers (MTOCs). However, the diverse microtubule configurations found in different cell types may not be generated and maintained by MTOCs alone. Work over the last few years has revealed a mechanism that has the capacity to generate cell-type-specific microtubule arrays independently of a specific organizer: nucleation of microtubules from the lateral surface of pre-existing microtubules. This type of nucleation requires cooperation between two different multi-subunit protein complexes, augmin and the γ-tubulin ring complex (γTuRC). Here we review recent molecular insight into microtubule-dependent nucleation and discuss the possibility that the augmin-γTuRC module, initially described in mitosis, may broadly contribute to microtubule organization also in non-mitotic cells.

Stochastic models for plant microtubule self-organization and structure

One of the key enablers of shape and growth in plant cells is the cortical microtubule (CMT) system, which is a polymer array that forms an appropriately-structured scaffolding in each cell. Plant biologists have shown that stochastic dynamics and simple rules of interactions between CMTs can lead to a coaligned CMT array structure. However, the mechanisms and conditions that cause CMT arrays to become organized are not well understood. It is prohibitively time-consuming to use actual plants to study the effect of various genetic mutations and environmental conditions on CMT self-organization. In fact, even computer simulations with multiple replications are not fast enough due to the spatio-temporal complexity of the system. To redress this shortcoming, we develop analytical models and methods for expeditiously computing CMT system metrics that are related to self-organization and array structure. In particular, we formulate a mean-field model to derive sufficient conditions for the organization to occur. We show that growth-prone dynamics itself is sufficient to lead to organization in presence of interactions in the system. In addition, for such systems, we develop predictive methods for estimation of system metrics such as expected average length and number of CMTs over time, using a stochastic fluid-flow model, transient analysis, and approximation algorithms tailored to our problem. We illustrate the effectiveness of our approach through numerical test instances and discuss biological insights.

Cortical microtubule rearrangements and cell wall patterning

Plant cortical microtubules, which form a highly ordered array beneath the plasma membrane, play essential roles in determining cell shape and function by directing the arrangement of cellulosic and non-cellulosic compounds on the cell surface. Interphase transverse arrays of cortical microtubules self-organize through their dynamic instability and inter-microtubule interactions, and by branch-form microtubule nucleation and severing. Recent studies revealed that distinct spatial signals including ROP GTPase, cellular geometry, and mechanical stress regulate the behavior of cortical microtubules at the subcellular and supercellular levels, giving rise to dramatic rearrangements in the cortical microtubule array in response to internal and external cues. Increasing evidence indicates that negative regulators of microtubules also contribute to the rearrangement of the cortical microtubule array. In this review, I summarize recent insights into how the rearrangement of the cortical microtubule array leads to proper, flexible cell wall patterning.

Modelling the role of microtubules in plant cell morphology

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.

Role of nucleation in cortical microtubule array organization: variations on a theme

The interphase cortical microtubules (CMTs) of plant cells form strikingly ordered arrays in the absence of a dedicated microtubule-organizing center. Considerable research effort has focused on activities such as bundling and severing that occur after CMT nucleation and are thought to be important for generating and maintaining ordered arrays. In this review, we focus on how nucleation affects CMT array organization. The bulk of CMTs are initiated from γ-tubulin-containing nucleation complexes localized to the lateral walls of pre-existing CMTs. These CMTs grow either at an acute angle or parallel to the pre-existing CMT. Although the impact of microtubule-dependent nucleation is not fully understood, recent genetic, live-cell imaging and computer simulation studies have demonstrated that the location, timing and geometry of CMT nucleation have a considerable impact on the organization and orientation of the CMT array. These nucleation properties are defined by the composition, position and dynamics of γ-tubulin-containing nucleation complexes, which represent control points for the cell to regulate CMT array organization.

Cortical microtubule arrays are initiated from a nonrandom prepattern driven by atypical microtubule initiation

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.

Progressive transverse microtubule array organization in hormone-induced Arabidopsis hypocotyl cells

The acentriolar cortical microtubule arrays in dark-grown hypocotyl cells organize into a transverse coaligned pattern that is critical for axial plant growth. In light-grown Arabidopsis thaliana seedlings, the cortical array on the outer (periclinal) cell face creates a variety of array patterns with a significant bias (>3:1) for microtubules polymerizing edge-ward and into the side (anticlinal) faces of the cell. To study the mechanisms required for creating the transverse coalignment, we developed a dual-hormone protocol that synchronously induces ∼80% of the light-grown hypocotyl cells to form transverse arrays over a 2-h period. Repatterning occurred in two phases, beginning with an initial 30 to 40% decrease in polymerizing plus ends prior to visible changes in the array pattern. Transverse organization initiated at the cell's midzone by 45 min after induction and progressed bidirectionally toward the apical and basal ends of the cell. Reorganization corrected the edge-ward bias in polymerization and proceeded without transiting through an obligate intermediate pattern. Quantitative comparisons of uninduced and induced microtubule arrays showed a limited deconstruction of the initial periclinal array followed by a progressive array reorganization to transverse coordinated between the anticlinal and periclinal cell faces.

Early, nonciliary role for microtubule proteins in left-right patterning is conserved across kingdoms

Many types of embryos' bodyplans exhibit consistently oriented laterality of the heart, viscera, and brain. Errors of left-right patterning present an important class of human birth defects, and considerable controversy exists about the nature and evolutionary conservation of the molecular mechanisms that allow embryos to reliably orient the left-right axis. Here we show that the same mutations in the cytoskeletal protein tubulin that alter asymmetry in plants also affect very early steps of left-right patterning in nematode and frog embryos, as well as chirality of human cells in culture. In the frog embryo, tubulin α and tubulin γ-associated proteins are required for the differential distribution of maternal proteins to the left or right blastomere at the first cell division. Our data reveal a remarkable molecular conservation of mechanisms initiating left-right asymmetry. The origin of laterality is cytoplasmic, ancient, and highly conserved across kingdoms, a fundamental feature of the cytoskeleton that underlies chirality in cells and multicellular organisms.

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

There is rising interest in modeling the noncentrosomal cortical microtubule cytoskeleton of plant cells, particularly its organization into ordered arrays and the mechanisms that facilitate this organization. In this review, we discuss quantitative models of this highly complex and dynamic structure both at a cellular and molecular level. We report differences in methodologies and assumptions of different models as well as their controversial results. Our review provides insights for future studies to resolve these controversies, in addition to underlining the common results between various models. We also highlight the need to compare the results from simulation and mathematical models with quantitative data from biological experiments in order to test the validity of the models and to further improve them. It is our hope that this review will serve to provide guidelines for how to combine quantitative and experimental techniques to develop higher-level models of the plant cytoskeleton in the future.