Survival of the aligned: ordering of the plant cortical microtubule 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.

Self-organization of mortal filaments and its role in bacterial division ring formation

Filaments in the cell commonly treadmill. Driven by energy consumption, they grow on one end while shrinking on the other, causing filaments to appear motile even though individual proteins remain static. This process is characteristic of cytoskeletal filaments and leads to collective filament self-organization. Here we show that treadmilling drives filament nematic ordering by dissolving misaligned filaments. Taking the bacterial FtsZ protein involved in cell division as an example, we show that this mechanism aligns FtsZ filaments in vitro and drives the organization of the division ring in living Bacillus subtilis cells. We find that ordering via local dissolution also allows the system to quickly respond to chemical and geometrical biases in the cell, enabling us to quantitatively explain the ring formation dynamics in vivo. Beyond FtsZ and other cytoskeletal filaments, our study identifies a mechanism for self-organization via constant birth and death of energy-consuming filaments.

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.

Approximate simulation of cortical microtubule models using dynamical graph grammars

Dynamical graph grammars (DGGs) are capable of modeling and simulating the dynamics of the cortical microtubule array (CMA) in plant cells by using an exact simulation algorithm derived from a master equation; however, the exact method is slow for large systems. We present preliminary work on an approximate simulation algorithm that is compatible with the DGG formalism. The approximate simulation algorithm uses a spatial decomposition of the domain at the level of the system's time-evolution operator, to gain efficiency at the cost of some reactions firing out of order, which may introduce errors. The decomposition is more coarsely partitioned by effective dimension (d= 0 to 2 or 0 to 3), to promote exact parallelism between different subdomains within a dimension, where most computing will happen, and to confine errors to the interactions between adjacent subdomains of different effective dimensions. To demonstrate these principles we implement a prototype simulator, and run three simple experiments using a DGG for testing the viability of simulating the CMA. We find evidence indicating the initial formulation of the approximate algorithm is substantially faster than the exact algorithm, and one experiment leads to network formation in the long-time behavior, whereas another leads to a long-time behavior of local alignment.

The mechanics of plant morphogenesis

Understanding the mechanism by which patterned gene activity leads to mechanical deformation of cells and tissues to create complex forms is a major challenge for developmental biology. Plants offer advantages for addressing this problem because their cells do not migrate or rearrange during morphogenesis, which simplifies analysis. We synthesize results from experimental analysis and computational modeling to show how mechanical interactions between cellulose fibers translate through wall, cell, and tissue levels to generate complex plant tissue shapes. Genes can modify mechanical properties and stresses at each level, though the values and pattern of stresses differ from one level to the next. The dynamic cellulose network provides elastic resistance to deformation while allowing growth through fiber sliding, which enables morphogenesis while maintaining mechanical strength.

Robustness of the microtubule network self-organization in epithelia

Robustness of biological systems is crucial for their survival, however, for many systems its origin is an open question. Here, we analyze one subcellular level system, the microtubule cytoskeleton. Microtubules self-organize into a network, along which cellular components are delivered to their biologically relevant locations. While the dynamics of individual microtubules is sensitive to the organism’s environment and genetics, a similar sensitivity of the overall network would result in pathologies. Our large-scale stochastic simulations show that the self-organization of microtubule networks is robust in a wide parameter range in individual cells. We confirm this robustness in vivo on the tissue-scale using genetic manipulations of Drosophila epithelial cells. Finally, our minimal mathematical model shows that the origin of robustness is the separation of time-scales in microtubule dynamics rates. Altogether, we demonstrate that the tissue-scale self-organization of a microtubule network depends only on cell geometry and the distribution of the microtubule minus-ends.

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.

Quantification of Cytoskeletal Dynamics in Time-Lapse Recordings

The cytoskeleton is key to many essential processes in a plant cell, e.g., growth, division, and defense. Contrary to what "skeleton" implies, the cytoskeleton is highly dynamic, and is able to re-organize itself continuously. The advent of live-cell microscopy and the development of genetically encoded fluorophores enabled detailed observation of the organization and dynamics of the cytoskeleton. Despite the biological importance of the cytoskeletal dynamics, quantitative analyses remain laborious endeavors that only a handful of research teams regularly conduct. With this protocol, we provide a standardized step-by-step guide to analyze the dynamics of microtubules. We provide example data and code for post-processing in Fiji that enables researchers to modify and adapt the routine to their needs. More such tools are needed to quantitatively assess the cytoskeleton and thus to better understand cell biology. © 2019 by John Wiley & Sons, Inc.

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.

Regulation of plant cell wall stiffness by mechanical stress: a mesoscale physical model

A crucial question in developmental biology is how cell growth is coordinated in living tissue to generate complex and reproducible shapes. We address this issue here in plants, where stiff extracellular walls prevent cell migration and morphogenesis mostly results from growth driven by turgor pressure. How cells grow in response to pressure partly depends on the mechanical properties of their walls, which are generally heterogeneous, anisotropic and dynamic. The active control of these properties is therefore a cornerstone of plant morphogenesis. Here, we focus on wall stiffness, which is under the control of both molecular and mechanical signaling. Indeed, in plant tissues, the balance between turgor and cell wall elasticity generates a tissue-wide stress field. Within cells, mechano-sensitive structures, such as cortical microtubules, adapt their behavior accordingly and locally influence cell wall remodeling dynamics. To fully apprehend the properties of this feedback loop, modeling approaches are indispensable. To that end, several modeling tools in the form of virtual tissues have been developed. However, these models often relate mechanical stress and cell wall stiffness in relatively abstract manners, where the molecular specificities of the various actors are not fully captured. In this paper, we propose to refine this approach by including parsimonious biochemical and biomechanical properties of the main molecular actors involved. Through a coarse-grained approach and through finite element simulations, we study the role of stress-sensing microtubules on organ-scale mechanics.

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.

On genes and form

The mechanisms by which organisms acquire their sizes and shapes through growth was a major focus of D'Arcy Thompson's book On Growth and Form. By applying mathematical and physical principles to a range of biological forms, Thompson achieved fresh insights, such as the notion that diverse biological shapes could be related through simple deformations of a coordinate system. However, Thompson considered genetics to lie outside the scope of his work, even though genetics was a growing discipline at the time the book was published. Here, we review how recent advances in cell, developmental, evolutionary and computational biology allow Thompson's ideas to be integrated with genes and the processes they influence to provide a deeper understanding of growth and morphogenesis. We consider how genes interact with subcellular-, cellular- and tissue-level processes in plants to yield patterns of growth that underlie the developmental and evolutionary shape transformations Thompson so eloquently described.

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.

Mechanism of dynamic reorientation of cortical microtubules due to mechanical stress

Directional growth caused by gravitropism and corresponding bending of plant cells has been explored since 19th century, however, many aspects of mechanisms underlying the perception of gravity at the molecular level are still not well known. Perception of gravity in root and shoot gravitropisms is usually attributed to gravisensitive cells, called statocytes, which exploit sedimentation of macroscopic and heavy organelles, amyloplasts, to sense the direction of gravity. Gravity stimulus is then transduced into distal elongation zone, which is several mm far from statocytes, where it causes stretching. It is suggested that gravity stimulus is conveyed by gradients in auxin flux. We propose a theoretical model that may explain how concentration gradients and/or stretching may indirectly affect the global orientation of cortical microtubules, attached to the cell membrane and induce their dynamic reorientation perpendicular to the gradients. In turn, oriented microtubule arrays direct the growth and orientation of cellulose microfibrils, forming part of the cell external skeleton and determine the shape of the cell. Reorientation of microtubules is also observed in reaction to light in phototropism and mechanical bending, thus suggesting universality of the proposed mechanism.

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.

Quantitative analyses of the plant cytoskeleton reveal underlying organizational principles

The actin and microtubule (MT) cytoskeletons are vital structures for cell growth and development across all species. While individual molecular mechanisms underpinning actin and MT dynamics have been intensively studied, principles that govern the cytoskeleton organization remain largely unexplored. Here, we captured biologically relevant characteristics of the plant cytoskeleton through a network-driven imaging-based approach allowing us to quantitatively assess dynamic features of the cytoskeleton. By introducing suitable null models, we demonstrate that the plant cytoskeletal networks exhibit properties required for efficient transport, namely, short average path lengths and high robustness. We further show that these advantageous features are maintained during temporal cytoskeletal rearrangements. Interestingly, man-made transportation networks exhibit similar properties, suggesting general laws of network organization supporting diverse transport processes. The proposed network-driven analysis can be readily used to identify organizational principles of cytoskeletons in other organisms.

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.

Microtubule initiation from the nuclear surface controls cortical microtubule growth polarity and orientation in Arabidopsis thaliana

The nuclear envelope in plant cells has long been known to be a microtubule organizing center (MTOC), but its influence on microtubule organization in the cell cortex has been unclear. Here we show that nuclear MTOC activity favors the formation of longitudinal cortical microtubule (CMT) arrays. We used green fluorescent protein (GFP)-tagged gamma tubulin-complex protein 2 (GCP2) to identify nuclear MTOC activity and GFP-tagged End-Binding Protein 1b (EB1b) to track microtubule growth directions. We found that microtubules initiate from nuclei and enter the cortex in two directions along the long axis of the cell, creating bipolar longitudinal CMT arrays. Such arrays were observed in all cell types showing nuclear MTOC activity, including root hairs, recently divided cells in root tips, and the leaf epidermis. In order to confirm the causal nature of nuclei in bipolar array formation, we displaced nuclei by centrifugation, which generated a corresponding shift in the bipolarity split point. We also found that bipolar CMT arrays were associated with bidirectional trafficking of vesicular components to cell ends. Together, these findings reveal a conserved function of plant nuclear MTOCs and centrosomes/spindle pole bodies in animals and fungi, wherein all structures serve to establish polarities in microtubule growth.

Morphology and growth of polarized tissues

We study and classify the time-dependent morphologies of polarized tissues subject to anisotropic but spatially homogeneous growth. Extending previous studies, we model the tissue as a fluid, and discuss the interplay of the active stresses generated by the anisotropic cell division and three types of passive mechanical forces: viscous stresses, friction with the environment and tension at the tissue boundary. The morphology dynamics is formulated as a free-boundary problem, and conformal mapping techniques are used to solve the evolution numerically. We combine analytical and numerical results to elucidate how the different passive forces compete with the active stresses to shape the tissue in different temporal regimes and derive the corresponding scaling laws. We show that in general the aspect ratio of elongated tissues is non-monotonic in time, eventually recovering isotropic shapes in the presence of friction forces, which are asymptotically dominant.

Reorganization of the plant cortical microtubule array

The interphase microtubule arrays in flowering plant cells assemble at the cell cortex into patterns that affect cellular morphogenesis. A decade of live cell imaging studies has provided significant information about the in vivo properties of the microtubule polymers. Efforts to extrapolate individual properties to larger roles in organizing or patterning the microtubule array have produced models focused on self-organization and local levels of biological control. Recent studies looking at cortical microtubule arrays as they transition from an existing pattern to a new pattern have re-emerged as a testbed for examining these models and the molecular hypotheses underpinning them. The evidence suggests that microtubule patterning is locally controlled on the scale of a cell face, using or circumventing self-organizating properties as necessary.

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.

How mechanical stress controls microtubule behavior and morphogenesis in plants: history, experiments and revisited theories

Microtubules have a key role in plant morphogenesis, as they control the oriented deposition of cellulose in the cell wall, and thus growth anisotropy. The idea that mechanical stress could be one of the main determinants behind the orientation of microtubules in plant cells emerged very soon after their discovery. The cause of mechanical stress in plant cells is turgor pressure, which can build up to 1 MPa and is restrained by cell wall stiffness. On the tissue scale, this can lead to regional patterns of tension, in particular in the epidermis of aerial organs, which resist the stress generated by cells in internal tissues. Here we summarize more than 50 years of work on the contribution of mechanical stress in guiding microtubule behavior, and the resulting impact on growth anisotropy and growth heterogeneity. We propose a conceptual model on microtubule dynamics and their ability to self-organize in bundles parallel to the direction of maximal stress, as well as a synthetic representation of the putative mechanotransducers at play.

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.

Large-scale vortex lattice emerging from collectively moving microtubules

Spontaneous collective motion, as in some flocks of bird and schools of fish, is an example of an emergent phenomenon. Such phenomena are at present of great interest and physicists have put forward a number of theoretical results that so far lack experimental verification. In animal behaviour studies, large-scale data collection is now technologically possible, but data are still scarce and arise from observations rather than controlled experiments. Multicellular biological systems, such as bacterial colonies or tissues, allow more control, but may have many hidden variables and interactions, hindering proper tests of theoretical ideas. However, in systems on the subcellular scale such tests may be possible, particularly in in vitro experiments with only few purified components. Motility assays, in which protein filaments are driven by molecular motors grafted to a substrate in the presence of ATP, can show collective motion for high densities of motors and attached filaments. This was demonstrated recently for the actomyosin system, but a complete understanding of the mechanisms at work is still lacking. Here we report experiments in which microtubules are propelled by surface-bound dyneins. In this system it is possible to study the local interaction: we find that colliding microtubules align with each other with high probability. At high densities, this alignment results in self-organization of the microtubules, which are on average 15 µm long, into vortices with diameters of around 400 µm. Inside the vortices, the microtubules circulate both clockwise and anticlockwise. On longer timescales, the vortices form a lattice structure. The emergence of these structures, as verified by a mathematical model, is the result of the smooth, reptation-like motion of single microtubules in combination with local interactions (the nematic alignment due to collisions)--there is no need for long-range interactions. Apart from its potential relevance to cortical arrays in plant cells and other biological situations, our study provides evidence for the existence of previously unsuspected universality classes of collective motion phenomena.

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.

Nanoscale and geometric influences on the microtubule cytoskeleton in plants: thinking inside and outside the box

The dynamic microtubule (MT) cytoskeleton found in the cell cortex of plants drives cell expansion via cell wall modifications. In the last decade, live cell imaging studies employing green fluorescent protein have helped unravel the mechanisms behind how cells arrange cortical MTs into complex arrays and shape cell expansion. In this review, we explore the reverse scenario: how cell geometry and organelles influence and constrain the organization and behavior of cortical MTs. This newly emerging principle explains how cells perceive local nanoscale structural input from MT-organizing centers, such as the nucleus, endomembranes, and cell edges, and translate this into global cell-wide order via MT self-organization. Studies primarily using the model plant Arabidopsis thaliana and tobacco BY-2 suspension cultures have broadened our understanding of how cells form not only elegant parallel arrays but also more complex MT configurations, including the prominent MT bundles found in preprophase bands, leaf epidermal cells, and developing xylem.

A CLASP-modulated cell edge barrier mechanism drives cell-wide cortical microtubule organization in Arabidopsis

It is well known that the parallel order of microtubules in the plant cell cortex defines the direction of cell expansion, yet it remains unclear how microtubule orientation is controlled, especially on a cell-wide basis. Here we show through 4D imaging and computational modelling that plant cell polyhedral geometry provides spatial input that determines array orientation and heterogeneity. Microtubules depolymerize when encountering sharp cell edges head-on, whereas those oriented parallel to those sharp edges remain. Edge-induced microtubule depolymerization, however, is overcome by the microtubule-associated protein CLASP, which accumulates at specific cell edges, enables microtubule growth around sharp edges and promotes formation of microtubule bundles that span adjacent cell faces. By computationally modelling dynamic 'microtubules on a cube' with edges differentially permissive to microtubule passage, we show that the CLASP-edge complex is a 'tuneable' microtubule organizer, with the inherent flexibility to generate the numerous cortical array patterns observed in nature.

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

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.

VirtualLeaf: an open-source framework for cell-based modeling of plant tissue growth and development

Plant organs, including leaves and roots, develop by means of a multilevel cross talk between gene regulation, patterned cell division and cell expansion, and tissue mechanics. The multilevel regulatory mechanisms complicate classic molecular genetics or functional genomics approaches to biological development, because these methodologies implicitly assume a direct relation between genes and traits at the level of the whole plant or organ. Instead, understanding gene function requires insight into the roles of gene products in regulatory networks, the conditions of gene expression, etc. This interplay is impossible to understand intuitively. Mathematical and computer modeling allows researchers to design new hypotheses and produce experimentally testable insights. However, the required mathematics and programming experience makes modeling poorly accessible to experimental biologists. Problem-solving environments provide biologically intuitive in silico objects ("cells", "regulation networks") required for setting up a simulation and present those to the user in terms of familiar, biological terminology. Here, we introduce the cell-based computer modeling framework VirtualLeaf for plant tissue morphogenesis. The current version defines a set of biologically intuitive C++ objects, including cells, cell walls, and diffusing and reacting chemicals, that provide useful abstractions for building biological simulations of developmental processes. We present a step-by-step introduction to building models with VirtualLeaf, providing basic example models of leaf venation and meristem development. VirtualLeaf-based models provide a means for plant researchers to analyze the function of developmental genes in the context of the biophysics of growth and patterning. VirtualLeaf is an ongoing open-source software project (http://virtualleaf.googlecode.com) that runs on Windows, Mac, and Linux.

A mechanochemical model explains interactions between cortical microtubules in plants

Microtubules anchored to the two-dimensional cortex of plant cells collide through plus-end polymerization. Collisions can result in rapid depolymerization, directional plus-end entrainment, or crossover. These interactions are believed to give rise to cellwide self-organization of plant cortical microtubules arrays, which is required for proper cell wall growth. Although the cell-wide self-organization has been well studied, less emphasis has been placed on explaining the interactions mechanistically from the molecular scale. Here we present a model for microtubule-cortex anchoring and collision-based interactions between microtubules, based on a competition between cross-linker bonding, microtubule bending, and microtubule polymerization. Our model predicts a higher probability of entrainment at smaller collision angles and at longer unanchored lengths of plus-ends. This model addresses observed differences between collision resolutions in various cell types, including Arabidopsis cells and Tobacco cells.

Model for the orientational ordering of the plant microtubule cortical array

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.

Computational cell biology at the home of the helix

The Computational Cell Biology Conference, held jointly by the Cold Spring Harbor Laboratory and the Wellcome Trust, was convened in the grand surroundings of Hinxton Hall near Cambridge, UK. The high quality of the research presented at the meeting confirmed that the field of computational cell biology is maturing rapidly, which mirrors the progression of cell biology from being mostly descriptive to a more quantitative discipline.