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Mapping of triangles from the 3D surface mesh to the 2D plane. ΔABC and ΔADC are connected through a shared edge AC in the 3D x − y − z space, but upon quaternion rotation operation end up separated in the 2D x − y plane. https://doi.org/10.1371/journal.pcbi.1005959.g004
Source publication
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 dy...
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Citations
... The state of cortical microtubule arrays is significantly influenced by a combination of global and local cues. For example, global cues such as cell geometry can favour array orientation along the shortest closed circumferences of the cell [15,16]. Conversely, local cues, including the alignment of microtubules with mechanical stresses [17], have the potential to override global cue and select for a specific orientation for the alignment of the array [18,19]. ...
... Despite this importance, most computational research on microtubule arrays has traditionally employed isotropic nucleation, where microtubules are initiated at random positions with random orientations. While helpful for understanding the effects of microtubule collisions [8,10,23] and geometrical constraints on alignment [15,18,[24][25][26], this approach simplifies the actual complexity, overlooking that microtubules often nucleate from existing ones [20,27]. Efforts to include microtubule-based nucleation in models have faced challenges, highlighting the necessity for more realistic simulation methods. ...
... Previous implementation of microtubule-based nucleation [21,28], as in the GDD case in this work, posses lower informative power as the final alignment of an array, by strongly depending on any initial emerging orientation due the positive feedback that attracts new nucleations in the same direction of extant microtubules, minimises the impact of biases. Conversely, we showed that isotropic nucleation was more susceptible to local and global biases, in agreement with previous research [15,18], with the risk to exaggerate small imbalances that, in a real situation, would be irrelevant for the final array orientation. ...
The self-organisation of cortical microtubules into aligned arrays plays a key role in plant cell morphogenesis and organismal growth. Computational studies have highlighted the impact of simulation domain topology and directional biases in microtubule dynamics on this alignment. Despite this, simulating microtubule nucleation has been challenging for years, as modelling nucleation of new microtubules from extant microtubules lead to inhomogeneity within cortical arrays, characterized by large empty spaces and few densely populated areas. We introduce a novel, efficient algorithm that realistically simulates microtubule nucleation by approximating the diffusion of nucleation complexes, which yields uniform arrays by maintaining realism in the nucleation process without the computational burden of explicit diffusion simulation. With our model, we show that strong biases towards specific orientations of alignment in self-organised arrays can emerge naturally when nucleation is modelled with enhanced biological realism. More specifically, we observe that cell geometry has a strong influence in driving the array towards a preferred transverse orientation. Our approach opens up new avenues for quantitative comparisons of different factors influencing array orientation and, as such, can be a powerful tool for re-assessing conclusions about the drivers of microtubule alignment in plant cells.
... The constricted regions of lobed pavement cells are emphasized as locations for microtubule alignment [22,23], and quantitative analyses have detected subtle enrichments of transfacial microtubules at subsets of cell indentations [24][25][26]. Stresses in the anticlinal wall that are orthogonal to the epidermal surface can strongly orient microtubules in this cell face [15,27]. ...
... Both types of the boundary condition induced transverse alignment even if collision-induced catastrophe between microtubules was disabled. A recent 3D model by Chakrabortty et al. showed that a catastrophe-inducing boundary and cell-face-specific microtubule-destabilizing surfaces can potently generate oriented microtubule arrays [23]. ...
Background:
Morphological properties of tissues and organs rely on cell growth. The growth of plant cells is determined by properties of a tough outer cell wall that deforms anisotropically in response to high turgor pressure. Cortical microtubules bias the mechanical anisotropy of a cell wall by affecting the trajectories of cellulose synthases in the wall that polymerize cellulose microfibrils. The microtubule cytoskeleton is often oriented in one direction at cellular length-scales to regulate growth direction, but the means by which cellular-scale microtubule patterns emerge has not been well understood. Correlations between the microtubule orientation and tensile forces in the cell wall have often been observed. However, the plausibility of stress as a determining factor for microtubule patterning has not been directly evaluated to date.
Results:
Here, we simulated how different attributes of tensile forces in the cell wall can orient and pattern the microtubule array in the cortex. We implemented a discrete model with transient microtubule behaviors influenced by local mechanical stress in order to probe the mechanisms of stress-dependent patterning. Specifically, we varied the sensitivity of four types of dynamic behaviors observed on the plus end of microtubules - growth, shrinkage, catastrophe, and rescue - to local stress. Then, we evaluated the extent and rate of microtubule alignments in a two-dimensional computational domain that reflects the structural organization of the cortical array in plant cells.
Conclusion:
Our modeling approaches reproduced microtubule patterns observed in simple cell types and demonstrated that a spatial variation in the magnitude and anisotropy of stress can mediate mechanical feedback between the wall and of the cortical microtubule array.
... When the growing end of a microtubule collides with another microtubule, it may turn to follow the microtubule (zippering) or undergo depolymerization (collision-induced catastrophe) (47). Computer simulations show that such interactions in a population of microtubules can generate alignments (i.e., near-parallel arrangements) that maximize microtubule survival probability (48,49). In a spherical cell without cues, such alignments are randomly oriented. ...
... A cell polarity protein in protoplasts gives a polarity axis that aligns with subsequent growth orientation (63). Computer simulations show that microtubules tend to adopt orientations parallel to faces or edges where they are preferentially destabilized, because such orientations increase microtubule survival probability (48,64). If polarity proteins at opposite endfaces or edges of a cylindrical cell destabilize microtubules, microtubule orientations parallel to the edges (i.e., circumferential) would therefore be favored. ...
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.
... A key structure in this process is the cortical microtubule array, which determines where cell wall materials are inserted (7)(8)(9)(10) and guides the deposition and, hence, orientation of cellulose microfibrils, the main load-bearing component of cell walls and determinant of their anisotropic mechanical properties (7,(11)(12)(13). The cortical array responds to various mechanical (14,15), geometrical (16)(17)(18), developmental (19)(20)(21), and environmental (19) cues, integrating this information for future plant growth and function. This ability to respond to local wall stresses and other cues introduces a morphomechanical feedback loop that is considered the central ingredient of current plant growth models (22). ...
... The second option for balancing the positive feedback and thus limiting the local increase of microtubule density may come naturally with revisiting the nucleation process. From pioneering work (27,28,30,31) to current studies (17,18,33,45), great progress has been made using isotropic nucleation, i.e., with uniform random location and orientation of new microtubules. In reality, however, most nucleations occur from nucleation complexes bound to existing microtubules, with new microtubules either parallel to their parent microtubules or branching at angles around 35 • (38,39,53). ...
... Our observations of nucleation complex behavior and the solution they provide to the inhomogeneity problem pave the way for the next generation of microtubule simulation models. Some pressing biological questions that require detailed simulations including realistic nucleation are the following: 1) How do cells integrate all the different cues affecting array orientation (14)(15)(16)(17)(18)(19)(20)(21) and resolve conflicts between them? 2) How can changes in the distribution of parent-offspring nucleation angles lead to substantial changes in cell morphology as, for example, in the tonneau2/fass (ton2) mutant (78)? 3) How can a continuous interaction between ROPs and their downstream effectors on the one hand (61,62) and the microtubule array on the other hand lead to various complex wall patterns like in protoxylem and metaxylem? In summary, our work enables various lines of quantitative, mechanistic research that will improve our understanding of how cell wall properties are dynamically controlled. ...
Plant cell walls are versatile materials that can adopt a wide range of mechanical properties through controlled deposition of cellulose fibrils. Wall integrity requires a sufficiently homogeneous fibril distribution to cope effectively with wall stresses. Additionally, specific conditions, such as the negative pressure in water transporting xylem vessels, may require more complex wall patterns, e.g., bands in protoxylem. The orientation and patterning of cellulose fibrils are guided by dynamic cortical microtubules. New microtubules are predominantly nucleated from parent microtubules causing positive feedback on local microtubule density with the potential to yield highly inhomogeneous patterns. Inhomogeneity indeed appears in all current cortical array simulations that include microtubule-based nucleation, suggesting that plant cells must possess an as-yet unknown balancing mechanism to prevent it. Here, in a combined simulation and experimental approach, we show that a limited local recruitment of nucleation complexes to microtubules can counter the positive feedback, whereas local tubulin depletion cannot. We observe that nucleation complexes preferentially appear at the plasma membrane near microtubules. By incorporating our experimental findings in stochastic simulations, we find that the spatial behavior of nucleation complexes delicately balances the positive feedback, such that differences in local microtubule dynamics—as in developing protoxylem—can quickly turn a homogeneous array into a banded one. Our results provide insight into how the plant cytoskeleton has evolved to meet diverse mechanical requirements and greatly increase the predictive power of computational cell biology studies.
... These functions rely on spatiotemporally regulated microtubule interactions with distinct microtubuleassociated proteins (Akhmanova and Kapitein, 2022;Brouhard and Rice, 2018;Gudimchuk and McIntosh, 2021). In plant and animal cells, microtubules enable cell shape maintenance, and microtubule organization is influenced by cell shape (Bidhendi et al., 2019;Chakrabortty et al., 2018;Gomez et al., 2016;Mirabet et al., 2018). Microtubule network integrity is necessary for morphogenesis that depends on apical constriction (Booth et al., 2014;Fernandes et al., 2014;Kasioulis et al., 2017;Lee et al., 2007) but whether and how the organization and dynamics of the microtubule cytoskeleton aids constriction remains poorly understood. ...
Apical constriction powers amnioserosa contraction during Drosophila dorsal closure. The nucleation, movement and dispersal of apicomedial actomyosin complexes generates pulsed apical constrictions during early closure. Persistent apicomedial and circumapical actomyosin complexes drive unpulsed constrictions that follow. Here, we show that the microtubule end-binding proteins EB1 and Patronin pattern constriction dynamics and contraction kinetics by coordinating the balance of actomyosin forces in the apical plane. We find that microtubule growth from moving Patronin platforms governs the spatiotemporal dynamics of apicomedial myosin through the regulation of RhoGTPase signaling by transient EB1-RhoGEF2 interactions. We uncover the dynamic reorganization of a subset of short non-centrosomally nucleated apical microtubules that surround the coalescing apicomedial myosin complex, trail behind it as it moves and disperse as the complex dissolves. We demonstrate that apical microtubule reorganization is sensitive to Patronin levels. Microtubule depolymerization compromised apical myosin enrichment and altered constriction dynamics. Together, our findings uncover the importance of reorganization of an intact apical microtubule meshwork, by moving Patronin platforms and growing microtubule ends, in enabling the spatiotemporal modulation of actomyosin contractility and, through it, apical constriction.
... In tandem with experiments, this process has been studied in mathematical and computational models and analyzed in simulations to understand the importance of each factor. Simulations have been run on geometries with varying levels of complexity: connected planes representing a rectangular prism [4], cylindrical prisms [5,6], triangular meshes approximating cells [7], and within volumes of smooth solids [8]. In many of these works, the assumption is made that MTs travel along geodesics; this is justified if MT segments lengths between anchors are very short relative to the radii of curvature of the cell surface. ...
... The different models are shown in Figure 1. spanning length scales comparable to that of the cell circumference prior to anchoring in the case on inaccessible end caps [9], b) MTs with shorter segment lengths and uniform anchoring [10], c) MTs with a stochastic anchoring process examined in the present model, d) MTs with infinite anchoring density following geodesics [5,7]. Cases b)-d) are assumed on an infinite cylinder. ...
... Previous modelling works exploring plant cell cortex MT array organization have reproduced the in vivo observation of transverse coiling of MTs during the elongation phase, while assuming MTs grow along geodesics. In the case of planes representing faces of polyhedra that approximate cell geometries [7,4], these are straight lines. In the case of cylinders [5,6], these are helices. ...
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 was 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. 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.
... This lends support to the theory that cell edges can act as a site for microtubule nucleation. Computational modeling of plant cell shapes has been an informative tool that can recapitulate the observed microtubule patterns in vivo by inputting parameters such as catastrophe events induced by cell edge curvature Chakrabortty et al., 2018). Recent experimental evidence suggests that microtubules also reorient by mechanisms that do not rely on cell edges. ...
The transition from cell division to differentiation in primary roots is dependent on precise gradients of phytohormones, including auxin, cytokinins and brassinosteroids. The reorganization of microtubules also plays a key role in determining whether a cell will enter another round of mitosis or begin to rapidly elongate as the first step in terminal differentiation. In the last few years, progress has been made to establish connections between signaling pathways at distinct locations within the root. This review focuses on the different factors that influence whether a root cell remains in the division zone or transitions to elongation and differentiation using Arabidopsis thaliana as a model system. We highlight the role of the microtubule-associated protein CLASP as an intermediary between sustaining hormone signaling and controlling microtubule organization. We discuss new, innovative tools and methods, such as hormone sensors and computer modeling, that are allowing researchers to more accurately visualize the belowground growth dynamics of plants.
... A key structure in this process is the cortical microtubule array, which determines where cell wall materials are inserted [7,8,9,10] and guides the deposition, and hence, orientation of cellulose microfibrils, the main load-bearing component of cell walls and determinant of their anisotropic mechanical properties [7,11,12,13]. The cortical array responds to various mechanical [14,15], geometrical [16,17,18], developmental [19,20,21], and environmental [19] cues, integrating this information for future plant growth and function. This ability to respond to local wall stresses and other cues introduces a morpho-mechanical feedback loop that is considered the central ingredient of current plant growth models [22]. ...
... The second option for balancing the positive feedback and thus limiting the local increase of microtubule density may come naturally with revisiting the nucleation process. From pioneering work [31,30,28,27] to current studies [33,45,17,18], great progress has been made using isotropic nucleation, i.e., with uniform random location and orientation of new microtubules. In reality, however, most nucleations occur from nucleation complexes bound to existing microtubules, with new microtubules either parallel to their parent microtubules or branching at angles around 35° [49,38,39]. ...
... Our novel observations of nucleation complex behaviour and the solution they provide to the inhomogeneity problem pave the way for the next generation of microtubule simulation models. Some pressing biological questions that require detailed simulations including realistic nucleation are: 1) How do cells integrate all the different cues affecting array orientation [14,15,16,17,18,19,20,21] and resolve conflicts between them? 2) How can changes in the distribution of parent-offspring nucleation angles lead to substantial changes in cell morphology as, e.g., in the tonneau2/fass (ton2) mutant [72]? 3) How can a continuous interaction between ROPs and their downstream effectors on the one hand [55,56] and the microtubule array on the other hand lead to various complex wall patterns like in proto-and metaxylem? ...
Plant cell walls are versatile materials that can adopt a wide range of mechanical properties through controlled deposition of cellulose fibrils. Wall integrity requires a sufficiently homogeneous fibril distribution to cope effectively with wall stresses. Additionally, specific conditions, such as the negative pressure in water transporting xylem vessels, may require more complex wall patterns, e.g., bands in protoxylem. The orientation and patterning of cellulose fibrils is guided by dynamic cortical microtubules. New microtubules are predominantly nucleated from parent microtubules causing positive feedback on local microtubule density with the potential to yield highly inhomogeneous patterns. Inhomogeneity indeed appears in all current cortical array simulations that include microtubule-based nucleation, suggesting that plant cells must possess an as-yet unknown balancing mechanism to prevent it. Here, in a combined simulation and experimental approach, we show that the naturally limited local recruitment of nucleation complexes to microtubules can counter the positive feedback, whereas local tubulin depletion cannot. We observe that nucleation complexes are preferentially inserted at microtubules. By incorporating our experimental findings in stochastic simulations, we find that the spatial behaviour of nucleation complexes delicately balances the positive feedback, such that differences in local microtubule dynamics – as in developing protoxylem – can quickly turn a homogeneous array into a patterned one. Our results provide insight into how the plant cytoskeleton is wired to meet diverse mechanical requirements and greatly increase the predictive power of computational cell biology studies.
Significance statement
The plant cortical microtubule array is an established model system for self-organisation, with a rich history of complementary experiments, computer simulations, and analytical theory. Understanding how array homogeneity is maintained given that new microtubules nucleate from existing microtubules has been a major hurdle for using mechanistic (simulation) models to predict future wall structures. We overcome this hurdle with detailed observations of the nucleation process from which we derive a more “natural” nucleation algorithm. With this algorithm, we enable various new lines of quantitative, mechanistic research into how cells dynamically control their cell wall properties. At a mechanistic level, moreover, this work relates to the theory on cluster coexistence in Turing-like continuum models and demonstrates its relevance for discrete stochastic entities.
... Epidermal cells with similar shapes adopt multiple types of microtubule configurations on the outer cell cortex over time (16), and different faces of the cell can simultaneously display distinct orientations (8,17). The constricted regions of lobed pavement cells are emphasized as a location for microtubule alignment (18,19). In reality, there is only subtle enrichment of transfacial microtubules at subsets of cell indentations (20), and tensile force in the wall is likely to be a patterning element (8). ...
... Both types of the boundary condition induced transverse alignment even if collision-induced catastrophe between microtubules was disabled. A recent 3D model by Chakrabortty et al. showed that a catastrophe-inducing boundary and cell-face-specific microtubule-destabilizing surfaces can potently generate oriented microtubule arrays (19). ...
Functional properties of cells, tissues, and organs rely on predictable growth outputs. A shape change in plant cells is determined by properties of a tough cell wall that deforms anisotropically in response to high turgor pressure. This morphogenesis requires tight coordination and feedback controls among cytoskeleton-dependent wall patterning, its material properties, and stresses in the wall. Cortical microtubules bias the mechanical anisotropy of cell wall by defining the trajectories of cellulose synthase motility as they polymerize bundled microfibrils in their wake. Cortical microtubules could locally align and orient relative to cell geometry; however, the means by this orientation occurs is not known. Correlations between the microtubule orientation, cell geometry, and predicted tensile forces are strongly established, and here we simulate how different attributes of tensile force can orient and pattern the microtubule array in the cortex. We implemented a discrete model with three-state transient microtubule behaviors influenced by local mechanical stress in order to probe the mechanisms of stress-dependent patterning. We varied the sensitivity of four types of dynamic behaviors observed on the plus ends of microtubules – growth, shrinkage, catastrophe, and rescue – to local stress and then evaluated the extent and rate of microtubule alignments in a square computational domain. We optimized constitutive relationships between local stress and the plus-end dynamics and employed a biomechanically well-characterized cell wall to analyze how stress can influence the density and orientation of microtubule arrays. Our multiscale modeling approaches predict that spatial variability in stress magnitude and anisotropy mediate mechanical feedback between the wall and organization of the cortical microtubule array.
Author Summary
Plant cell growth involve multiple steps and processes. During growth, cell shape changes continuously while responding to external cues from the surroundings. Since growth is mainly driven by pressure, mechanical properties of cell wall are crucial in regulating multiple biological processes that underlie cell expansion and growth. Cell wall assembly is dynamically coupled to the remodeling of subcellular proteins. Experimental evidence has confirmed there exists potential mechanical feedback between wall assembly and protein-protein interactions. However, the actual mechanism remains unknown. In this study, we develop a computational model to study how mechanical stress could affect subcellular protein dynamics or interactions and lead to their reorganization, reminiscent of continuous changes in global pattern and cell morphology. Our results identify key parameters that can respond to external mechanical stimuli at the cellular scale. We also show that a biological stress pattern could induce protein filament organization and bundles that mimic real subcellular structure from experimental images. These computational results could benefit design of experiments for studying and discovering the potential protein candidates that underlie the mechanical feedback between multiple cellular components. In this way, a more systematic understanding about plant cell growth could be achieved, with an integrated theory that combine biology, chemistry, mechanics, and genetics.
... In that case, katanin severing can become a way of amplifying discordant microtubules, kickstarting the reorientation of the microtubule array [31,[141][142][143][144]. Models have also provided insight into other aspects of microtubule self-organisation, including selection of the cell division plane [42], formation of the preprophase band [145], and feedback from cell shape [125,127]. ...
... A key structure in this process is the cortical microtubule array, which determines where cell wall materials are inserted [117][118][119]122] and guides the deposition, and hence, orientation of cellulose microfibrils, the main load-bearing component of cell walls and determinant of their anisotropic mechanical properties [115,117,120,121]. The cortical array responds to various mechanical [128,129], geometrical [124,125,127], developmental [31,33,130], and environmental [31] cues, integrating this information for future plant growth and function. This ability to respond to local wall stresses and other cues introduces a morpho-mechanical feedback loop that is considered the central ingredient of current plant growth models [131]. ...
... The second option for limiting the local increase of microtubule density may come naturally with a sufficiently detailed description of nucleation complexes, which so far have been considered only implicitly. From pioneering work [134][135][136][137] to current studies [125][126][127]138], great progress has been made using isotropic nucleation, i.e., with uniform random location and orientation. In reality, however, most nucleations occur from nucleation complexes bound to existing microtubules, with new microtubules either parallel to their parent microtubules or branching at angles around 35° [107][108][109]. ...
