Structure–property relation and relevance of beam theories for microtubules: a coupled molecular and continuum mechanics study

Quasi-one-dimensional microtubules (MTs) in cells enjoy high axial rigidity but large transverse flexibility due to the inter-protofilament (PF) sliding. This study aims to explore the structure–property relation for MTs and examine the relevance of the beam theories to their unique features. A molecular structural mechanics (MSM) model was used to identify the origin of the inter-PF sliding and its role in bending and vibration of MTs. The beam models were then fitted to the MSM to reveal how they cope with the distinct mechanical responses induced by the inter-PF sliding. Clear evidence showed that the inter-PF sliding is due to the soft inter-PF bonds and leads to the length-dependent bending stiffness. The Euler beam theory is found to adequately describe MT deformation when the inter-PF sliding is largely prohibited. Nevertheless, neither shear deformation nor the nonlocal effect considered in the ‘more accurate’ beam theories can fully capture the effect of the inter-PF sliding. This reflects the distinct deformation mechanisms between an MT and its equivalent continuous body.     

Anterograde Microtubule Transport Drives Microtubule Bending in LLC-PK1 Epithelial Cells

Microtubules (MTs) have been proposed to act mechanically as compressive struts that resist both actomyosin contractile forces and their own polymerization forces to mechanically stabilize cell shape. To identify the origin of MT bending, we directly observed MT bending and F-actin transport dynamics in the periphery of LLC-PK1 epithelial cells. We found that F-actin is nearly stationary in these cells even as MTs are deformed, demonstrating that MT bending is not driven by actomyosin contractility. Furthermore, the inhibition of myosin II activity through the use of blebbistatin results in microtubules that are still dynamically bending. In addition, as determined by fluorescent speckle microscopy, MT polymerization rarely results, if ever, in bending. We suppressed dynamic instability using nocodazole, and we observed no qualitative change in the MT bending dynamics. Bending most often results from anterograde transport of proximal portions of the MT toward a nearly stationary distal tip. Interestingly, we found that in an in vitro kinesin-MT gliding assay, MTs buckle in a similar manner. To make quantitative comparisons, we measured curvature distributions of observed MTs and found that the in vivo and in vitro curvature distributions agree quantitatively. In addition, the measured MT curvature distribution is not Gaussian, as expected for a thermally driven semiflexible polymer, indicating that thermal forces play a minor role in MT bending. We conclude that many of the known mechanisms of MT deformation, such as polymerization and acto-myosin contractility, play an inconsequential role in mediating MT bending in LLC-PK1 cells and that MT-based molecular motors likely generate most of the strain energy stored in the MT lattice. The results argue against models in which MTs play a major mechanical role in LLC-PK1 cells and instead favor a model in which mechanical forces control the spatial distribution of the MT array.     

Circumferential vibration of microtubules with long axial wavelength

This paper uses an orthotropic shell model to investigate in detail the long axial wavelength circumferential vibration of microtubules (MTs). The deformation patterns in the vibrations were explored and their phonon dispersion relations were presented for MTs with increasing radius. It was shown that with the growth of the axial wavelength, the associated frequency of MTs would finally approach a nonzero asymptotic value, rising considerably with the increase of circumferential wave number but dropping linearly with the growing radius. This study corrects the previous misunderstanding drawn by an oversimplified model, and points out that a parabolic dispersion law does not apply to the circumferential modes when the MT bending stiffness is properly considered.     

Self-organization of anastral spindles by synergy of dynamic instability, autocatalytic microtubule production, and a spatial signaling gradient

Assembly of the mitotic spindle is a classic example of macromolecular self-organization. During spindle assembly, microtubules (MTs) accumulate around chromatin. In centrosomal spindles, centrosomes at the spindle poles are the dominating source of MT production. However, many systems assemble anastral spindles, i.e., spindles without centrosomes at the poles. How anastral spindles produce and maintain a high concentration of MTs in the absence of centrosome-catalyzed MT production is unknown. With a combined biochemistry-computer simulation approach, we show that the concerted activity of three components can efficiently concentrate microtubules (MTs) at chromatin: (1) an external stimulus in form of a RanGTP gradient centered on chromatin, (2) a feed-back loop where MTs induce production of new MTs, and (3) continuous re-organization of MT structures by dynamic instability. The mechanism proposed here can generate and maintain a dissipative MT super-structure within a RanGTP gradient.     

Microtubules, MAPs and plant directional cell expansion

Plant microtubules (MTs) polymerize and depolymerize in a process termed dynamic instability. This allows the assembly, reorganization, and disassembly of at least four MT arrays throughout the cell cycle. The cortical MT array lines the plasma membrane during interphase and plays a central role in directional cell expansion. Microtubule-associated proteins (MAPs) decorate cortical MTs with distinct patterns, regulating MT dynamic instability, MT severing, and other array-ordering processes. The Arabidopsis root has emerged as a highly useful system for identifying and studying cell-expansion-related MAPs. Here, we review how cortical MTs are thought to behave and become ordered in expanding root cells, and we discuss the emerging picture of how MAPs fundamentally govern MT ordering and directional growth processes.     

Microtubule Simulations Provide Insight into the Molecular Mechanism Underlying Dynamic Instability

The dynamic instability of microtubules (MTs), which refers to their ability to switch between polymerization and depolymerization states, is crucial for their function. It has been proposed that the growing MT ends are protected by a “GTP cap” that consists of GTP-bound tubulin dimers. When the speed of GTP hydrolysis is faster than dimer recruitment, the loss of this GTP cap will lead the MT to undergo rapid disassembly. However, the underlying atomistic mechanistic details of the dynamic instability remains unclear. In this study, we have performed long-time atomistic molecular dynamics simulations (1 μs for each system) for MT patches as well as a short segment of a closed MT in both GTP- and GDP-bound states. Our results confirmed that MTs in the GDP state generally have weaker lateral interactions between neighboring protofilaments (PFs) and less cooperative outward bending conformational change, where the difference between bending angles of neighboring PFs tends to be larger compared with GTP ones. As a result, when the GDP state tubulin dimer is exposed at the growing MT end, these factors will be more likely to cause the MT to undergo rapid disassembly. We also compared simulation results between the special MT seam region and the remaining material and found that the lateral interactions between MT PFs at the seam region were comparatively much weaker. This finding is consistent with the experimental suggestion that the seam region tends to separate during the disassembly process of an MT.     

Buckling Behavior of Individual and Bundled Microtubules

As the major structural constituent of the cytoskeleton, microtubules (MTs) serve a variety of biological functions that range from facilitating organelle transport to maintaining the mechanical integrity of the cell. Neuronal MTs exhibit a distinct configuration, hexagonally packed bundles of MT filaments, interconnected by MT-associated protein (MAP) tau. Building on our previous work on mechanical response of axonal MT bundles under uniaxial tension, this study is focused on exploring the compression scenarios. Intracellular MTs carry a large fraction of the compressive loads sensed by the cell and therefore, like any other column-like structure, are prone to substantial bending and buckling. Various biological activities, e.g., actomyosin contractility and many pathological conditions are driven or followed by bending, looping, and buckling of MT filaments. The coarse-grained model previously developed in our lab has been used to study the mechanical behavior of individual and bundled in vivo MT filaments under uniaxial compression. Both configurations show tip-localized, decaying, and short-wavelength buckling. This behavior highlights the role of the surrounding cytoplasm and MAP tau on MT buckling behavior, which allows MT filaments to bear much larger compressive forces. It is observed that MAP tau interconnections improve this effect by a factor of two. The enhanced ability of MT bundles to damp buckling waves relative to individual MT filaments, may be interpreted as a self-defense mechanism because it helps axonal MTs to endure harsher environments while maintaining their function. The results indicate that MT filaments in a bundle do not buckle simultaneously implying that the applied stress is not equally shared among the MT filaments, that is a consequence of the nonuniform distribution of MAP tau proteins along the bundle length. Furthermore, from a pathological perspective, it is observed that axonal MT bundles are more vulnerable to failure in compression than tension.     

How the transition frequencies of microtubule dynamic instability (nucleation, catastrophe, and rescue) regulate microtubule dynamics in interphase and mitosis: analysis using a Monte Carlo computer simulation.

Microtubules (MTs) in newt mitotic spindles grow faster than MTs in the interphase cytoplasmic microtubule complex (CMTC), yet spindle MTs do not have the long lengths or lifetimes of the CMTC microtubules. Because MTs undergo dynamic instability, it is likely that changes in the durations of growth or shortening are responsible for this anomaly. We have used a Monte Carlo computer simulation to examine how changes in the number of MTs and changes in the catastrophe and rescue frequencies of dynamic instability may be responsible for the cell cycle dependent changes in MT characteristics. We used the computer simulations to model interphase-like or mitotic-like MT populations on the basis of the dynamic instability parameters available from newt lung epithelial cells in vivo. We started with parameters that produced MT populations similar to the interphase newt lung cell CMTC. In the simulation, increasing the number of MTs and either increasing the frequency of catastrophe or decreasing the frequency of rescue reproduced the changes in MT dynamics measured in vivo between interphase and mitosis.     

Nonlocal Mechanism of Self-Organization and Centering of Microtubule Asters

Fragments of fish melanophore cells can form and center aggregates of pigment granules by dynein-motor-driven transport along a self-organized radial array of microtubules (MTs). We present a quantitative model that describes pigment aggregation and MT-aster self-organization and the subsequent centering of both structures. The model is based on the observations that MTs are immobile and treadmill, while dynein-motor-covered granules have the ability to nucleate MTs. From assumptions based on experimental observations, we derive partial integro-differential equations describing the coupled granule–MT interaction. We use scaling arguments and perturbation theory to study the model in two limiting cases. The model analysis explains the mechanism of aster self-organization as a positive feedback loop between motor aggregation at the MT minus ends and MT nucleation by motors. Furthermore, the centering mechanism is explained by the spontaneous nucleation of MTs throughout the cytosol which acts as a volume sensing tool. Numerical simulations lend additional support to the analysis. The model sheds light on role of polymer dynamics and polymer–motor interactions in cytoskeletal organization.     

Microtubule dynamics in the spindle. II. A thermodynamic and kinetic description

We have previously presented a model for the assembly and disassembly of mitotic spindle microtubules (MTs) (Pickett-Heaps et al., 1986). In this paper, we describe the thermodynamics of such spindle MT assembly and present equations to describe the polymerization kinetics of different classes of spindle MTs. These equations are used to predict, in terms of kinetics parameters, the magnitude of forces extant on spindle MTs and to define the critical force needed to halt MT assembly. We calculate several of these forces for a hypothetical model cell; our predicted value for the force generated along kinetochore fibers is in close agreement with measured values taken from living cells. The model and its implications are discussed with reference to other recent models of spindle and MT dynamics.     

Morphologically distinct microtubule ends in the mitotic centrosome of Caenorhabditis elegans

During mitosis, the connections of microtubules (MTs) to centrosomes and kinetochores are dynamic. From in vitro studies, it is known that the dynamic behavior of MTs is related to the structure of their ends, but we know little about the structure of MT ends in spindles. Here, we use high-voltage electron tomography to study the centrosome- and kinetochore-associated ends of spindle MTs in embryonic cells of the nematode, Caenorhabditis elegans. Centrosome-associated MT ends are either closed or open. Closed MT ends are more numerous and are uniformly distributed around the centrosome, but open ends are found preferentially on kinetochore-attached MTs. These results have structural implications for models of MT interactions with centrosomes.     

Actomyosin-based Retrograde Flow of Microtubules in the Lamella of Migrating Epithelial Cells Influences Microtubule Dynamic Instability and Turnover and Is Associated with Microtubule Breakage and Treadmilling

We have discovered several novel features exhibited by microtubules (MTs) in migrating newt lung epithelial cells by time-lapse imaging of fluorescently labeled, microinjected tubulin. These cells exhibit leading edge ruffling and retrograde flow in the lamella and lamellipodia. The plus ends of lamella MTs persist in growth perpendicular to the leading edge until they reach the base of the lamellipodium, where they oscillate between short phases of growth and shortening. Occasionally “pioneering” MTs grow into the lamellipodium, where microtubule bending and reorientation parallel to the leading edge is associated with retrograde flow. MTs parallel to the leading edge exhibit significantly different dynamics from MTs perpendicular to the cell edge. Both parallel MTs and photoactivated fluorescent marks on perpendicular MTs move rearward at the 0.4 μm/min rate of retrograde flow in the lamella. MT rearward transport persists when MT dynamic instability is inhibited by 100-nM nocodazole but is blocked by inhibition of actomyosin by cytochalasin D or 2,3-butanedione–2-monoxime. Rearward flow appears to cause MT buckling and breaking in the lamella. 80% of free minus ends produced by breakage are stable; the others shorten and pause, leading to MT treadmilling. Free minus ends of unknown origin also depolymerize into the field of view at the lamella. Analysis of MT dynamics at the centrosome shows that these minus ends do not arise by centrosomal ejection and that ~80% of the MTs in the lamella are not centrosome bound. We propose that actomyosin-based retrograde flow of MTs causes MT breakage, forming quasi-stable noncentrosomal MTs whose turnover is regulated primarily at their minus ends.     

A computational model predicts Xenopus meiotic spindle organization

The metaphase spindle is a dynamic bipolar structure crucial for proper chromosome segregation, but how microtubules (MTs) are organized within the bipolar architecture remains controversial. To explore MT organization along the pole-to-pole axis, we simulated meiotic spindle assembly in two dimensions using dynamic MTs, a MT cross-linking force, and a kinesin-5-like motor. The bipolar structures that form consist of antiparallel fluxing MTs, but spindle pole formation requires the addition of a NuMA-like minus-end cross-linker and directed transport of MT depolymerization activity toward minus ends. Dynamic instability and minus-end depolymerization generate realistic MT lifetimes and a truncated exponential MT length distribution. Keeping the number of MTs in the simulation constant, we explored the influence of two different MT nucleation pathways on spindle organization. When nucleation occurs throughout the spindle, the simulation quantitatively reproduces features of meiotic spindles assembled in Xenopus egg extracts.     

Reconstitution of dynamic microtubules with Drosophila XMAP215, EB1, and Sentin

Dynamic microtubules (MTs) are essential for various intracellular events, such as mitosis. In Drosophila melanogaster S2 cells, three MT tip-localizing proteins, Msps/XMAP215, EB1, and Sentin (an EB1 cargo protein), have been identified as being critical for accelerating MT growth and promoting catastrophe events, thus resulting in the formation of dynamic MTs. However, the molecular activity of each protein and the basis of the modulation of MT dynamics by these three factors are unknown. In this paper, we showed in vitro that XMAP215^msps had a potent growth-promoting activity at a wide range of tubulin concentrations, whereas Sentin, when recruited by EB1 to the growing MT tip, accelerated growth and also increased catastrophe frequency. When all three factors were combined, the growth rate was synergistically enhanced, and rescue events were observed most frequently, but frequent catastrophes restrained the lengthening of the MTs. We propose that MT dynamics are promoted by the independent as well as the cooperative action of XMAP215^msps polymerase and the EB1–Sentin duo.     

Slide-and-cluster models for spindle assembly

BACKGROUND: Mitotic and meiotic spindles are assemblies of microtubules (MTs) that form during cell division to physically separate sister chromosomes. How the various components of spindles act together to establish and maintain the dynamic bipolar structure of spindles is not understood. Interactions between MTs and motors have been studied both experimentally and theoretically in many contexts, including the self-organization of arrays of MTs by motors and the competition between different classes of motors to move a single load. This work demonstrates how the interplay between two types of motors together with continual nucleation of MTs by chromosomes could organize the MTs into spindles. RESULTS: We propose a slide-and-cluster model based on four known molecular activities: MT nucleation near chromosomes, the sliding of MTs by a plus-end-directed motor, the clustering of their minus ends by a minus-end-directed motor, and the loss of MTs by dynamic instability. Our model applies to overlapping, nonkinetochore MTs in anastral spindles, and perhaps also to interpolar MTs in astral spindles. We show mathematically that the slide-and-cluster mechanism robustly forms bipolar spindles with sharp poles and a stable steady-state length. This model accounts for several experimental observations that were difficult to explain with existing models. Three new predictions of the model were tested and verified in Xenopus egg extracts. CONCLUSIONS: We show that a simple two-motor model could create stable, bipolar spindles under a wide range of physical parameters. Our model is the first self-contained model for anastral spindle assembly and MT sliding (known as poleward flux). Our experimental results support the slide-and-cluster scenario; most significantly, we find that MT sliding slows near spindle poles, confirming the model's primary prediction.     

Microtubule Dynamic Instability Controls Podosome Patterning in Osteoclasts through EB1, Cortactin,and Src

In osteoclasts (OCs) podosomes are organized in a belt, a feature critical for bone resorption. Although microtubules (MTs) promote the formation and stability of the belt, the MT and/or podosome molecules that mediate the interaction of the two systems are not identified. Because the growing “plus” ends of MTs point toward the podosome belt, plus-end tracking proteins (+TIPs) might regulate podosome patterning. Among the +TIPs, EB1 increased as OCs matured and was enriched in the podosome belt, and EB1-positive MTs targeted podosomes. Suppression of MT dynamic instability, displacement of EB1 from MT ends, or EB1 depletion resulted in the loss of the podosome belt. We identified cortactin as an Src-dependent interacting partner of EB1. Cortactin-deficient OCs presented a defective MT targeting to, and patterning of, podosomes and reduced bone resorption. Suppression of MT dynamic instability or EB1 depletion increased cortactin phosphorylation, decreasing its acetylation and affecting its interaction with EB1. Thus, dynamic MTs and podosomes interact to control bone resorption.     

Free vibration analysis of microtubules based on the molecular mechanics and continuum beam theory

A molecular structural mechanics (MSM) method has been implemented to investigate the free vibration of microtubules (MTs). The emphasis is placed on the effects of the configuration and the imperfect boundaries of MTs. It is shown that the influence of protofilament number on the fundamental frequency is strong, while the effect of helix-start number is almost negligible. The fundamental frequency is also found to decrease as the number of the blocked filaments at boundaries decreases. Subsequently, the Euler–Bernoulli beam theory is employed to reveal the physics behind the simulation results. Fitting the Euler–Bernoulli beam into the MSM data leads to an explicit formula for the fundamental frequency of MTs with various configurations and identifies a possible correlation between the imperfect boundary conditions and the length-dependent bending stiffness of MTs reported in experiments.     

Regulation of microtubule dynamics by TOG-domain proteins XMAP215/Dis1 and CLASP

The molecular mechanisms by which microtubule-associated proteins (MAPs) regulate the dynamic properties of microtubules (MTs) are still poorly understood. We review recent advances in our understanding of two conserved families of MAPs, the XMAP215/Dis1 and CLASP family of proteins. In vivo and in vitro studies show that XMAP215 proteins act as microtubule polymerases at MT plus ends to accelerate MT assembly, and CLASP proteins promote MT rescue and suppress MT catastrophe events. These are structurally related proteins that use conserved TOG domains to recruit tubulin dimers to MTs. We discuss models for how these proteins might use these individual tubulin dimers to regulate dynamic behavior of MT plus ends.     

Induction of cytokinesis is independent of precisely regulated microtubule dynamics

Astral microtubules (MTs) emanating from the mitotic apparatus (MA) during anaphase are required for stimulation of cytokinesis in eggs. We have used green fluorescent protein-labeled EB1 to observe MT dynamics during mitosis and cytokinesis in normal sea urchin eggs. Analysis of astral MT growth rates during anaphase shows that MTs contact the polar cortex earlier than the equatorial cortex after anaphase onset but that a normal cleavage furrow is not induced until contact with MTs has been achieved throughout the cortex. To assess the role of MT dynamics in initiation of cytokinesis, we used a collection of small molecule drugs to affect dynamics. Hexylene glycol resulted in rapid astral elongation due to decreased MT catastrophe and precocious furrowing. Taxol suppressed MT dynamics but did not inhibit furrow induction when the MA was manipulated toward the cortex. Urethane resulted in short, highly dynamic astral MTs with increased catastrophe that also stimulated furrowing upon being brought into proximity to the cortex. Our findings indicate that astral MT contact with the cortex is necessary for furrow initiation but that the dynamic state of astral MTs does not affect their competency to stimulate furrowing.     

Properties of the kinetochore in vitro. II. Microtubule capture and ATP- dependent translocation

We have studied the interaction of preformed microtubules (MTs) with the kinetochores of isolated chromosomes. This reaction, which we call MT capture, results in MTs becoming tightly bound to the kinetochore, with their ends capped against depolymerization. These observations, combined with MT dynamic instability, suggest a model for spindle morphogenesis. In addition, ATP appears to mobilize dynamic processes at captured MT ends. We used biotin-labeled MT seeds to follow assembly dynamics at the kinetochore. In the presence of ATP and unlabeled tubulin, labeled MT segments translocate away from the kinetochore by polymerization of subunits at the attached end. We have termed this reaction proximal assembly. Further studies demonstrated that translocation could be uncoupled from MT assembly. We suggest that the kinetochore contains an ATPase activity that walks along the MT lattice toward the plus end. This activity may be responsible for the movement of chromosomes away from the pole in prometaphase.