Buckling Of Microtubules Research Articles

Buckling of microtubules: An insight by molecular and continuum mechanics

The molecular structural mechanics method has been extended to investigate the buckling of microtubules (MTs) with various configurations. The results indicate that for relative short MTs the shear deformation effect, rather than the nonlocal effect, is mainly responsible for the limitation of their widely used Euler beam description and the observed length-dependence of their bending stiffness. In addition, the configuration effect of MTs is also studied and considered as an explanation for the large scattering of the critical buckling force and bending stiffness observed in existing experiments. This configuration effect is also found to mainly originate from the geometry of the MTs and is mainly determined by the protofilament number.

Variational principles for buckling of microtubules modeled as nonlocal orthotropic shells.

A variational principle for microtubules subject to a buckling load is derived by semi-inverse method. The microtubule is modeled as an orthotropic shell with the constitutive equations based on nonlocal elastic theory and the effect of filament network taken into account as an elastic surrounding. Microtubules can carry large compressive forces by virtue of the mechanical coupling between the microtubules and the surrounding elastic filament network. The equations governing the buckling of the microtubule are given by a system of three partial differential equations. The problem studied in the present work involves the derivation of the variational formulation for microtubule buckling. The Rayleigh quotient for the buckling load as well as the natural and geometric boundary conditions of the problem is obtained from this variational formulation. It is observed that the boundary conditions are coupled as a result of nonlocal formulation. It is noted that the analytic solution of the buckling problem for microtubules is usually a difficult task. The variational formulation of the problem provides the basis for a number of approximate and numerical methods of solutions and furthermore variational principles can provide physical insight into the problem.

Nonlocal shear deformable shell model for torsional buckling and postbuckling of microtubules in thermal environments

Buckling and postbuckling analysis is presented for microtubules subjected to torsion in thermal environments. The microtubule is modeled as a nonlocal shear deformable cylindrical shell which contains small scale effects. The governing equations are based on a higher order shear deformation theory. The thermal effects are included and the material properties are assumed to be temperature-dependent. The small scale parameter e0a is estimated by matching the buckling twist angle of microtubules obtained from the nonlocal shear deformable shell model with the existing result. The results show that the small scale effect plays an important role in the postbuckling of microtubules.

Localized buckling of a microtubule surrounded by randomly distributed cross linkers

Microtubules supported by surrounding cross linkers in eukaryotic cells can bear a much higher compressive force than free-standing microtubules. Different from some previous studies, which treated the surroundings as a continuum elastic foundation or elastic medium, the present paper develops a micromechanics numerical model to examine the role of randomly distributed discrete cross linkers in the buckling of compressed microtubules. First, the proposed numerical approach is validated by reproducing the uniform multiwave buckling mode predicted by the existing elastic-foundation model. For more realistic buckling of microtubules surrounded by randomly distributed cross linkers, the present numerical model predicts that the buckling mode is localized at one end in agreement with some known experimental observations. In particular, the critical force for localized buckling, predicted by the present model, is insensitive to microtubule length and can be about 1 order of magnitude lower than those given by the elastic-foundation model, which suggests that the elastic-foundation model may have overestimated the critical force for buckling of microtubules in vivo. In addition, unlike the elastic-foundation model, the present model can capture the effect of end conditions on the critical force and wavelength of localized buckling. Based on the known data of spacing and elastic constants of cross linkers available in literature, the critical force and wavelength of the localized buckling mode, predicted by the present model, are compared to some experimental data with reasonable agreement. Finally, two empirical formulas are proposed for the critical force and wavelength of the localized buckling of microtubules surrounded by cross linkers.

Forces from the rear: deformed microtubules in neuronal growth cones influence retrograde flow and advancement

The directed motility of growth cones at the tip of neuronal processes is a key function in neuronal path-finding and relies on a complex system of interacting cytoskeletal components. Despite intensive research in this field, many aspects of the mechanical roles of actin structures and, in particular, of microtubules throughout this process remain unclear. Mostly, force generation is ascribed to actin–myosin-based structures such as filopodia bundles and the dynamic polymer gel within the lamellipodium. Our analysis of microtubule buckling and deformation in motile growth cones reveals that extending microtubule filaments contribute significantly to the overall protrusion force. In this study, we establish a relationship of the local variations in stored bending energy and deformation characteristics to growth cone morphology and retrograde actin flow. This implies the relevance of microtubule pushing and deformation for general neurite advancement as well as steering processes.

Anomalous Flexural Behaviors of Microtubules

Apparent controversies exist on whether the persistence length of microtubules depends on its contour length. This issue is particularly challenging from a theoretical point of view due to the tubular structure and strongly anisotropic material property of microtubules. Here we adopt a higher order continuum orthotropic thin shell model to study the flexural behavior of microtubules. Our model overcomes some key limitations of a recent study based on a simplified anisotropic shell model and results in a closed-form solution for the contour-length-dependent persistence length of microtubules, with predictions in excellent agreement with experimental measurements. By studying the ratio between their contour and persistence lengths, we find that microtubules with length at ∼1.5 μm show the lowest flexural rigidity, whereas those with length at ∼15 μm show the highest flexural rigidity. This finding may provide an important theoretical basis for understanding the mechanical structure of mitotic spindles during cell division. Further analysis on the buckling of microtubules indicates that the critical buckling load becomes insensitive to the tube length for relatively short microtubules, in drastic contrast to the classical Euler buckling. These rich flexural behaviors of microtubules are of profound implication for many biological functions and biomimetic molecular devices.

Endocytic membrane fusion and buckling-induced microtubule severing mediate cell abscission

Cytokinesis and abscission are complicated events that involve changes in membrane transport and cytoskeleton organization. We have used the combination of time-lapse microscopy and correlative high-resolution 3D tomography to analyze the regulation and spatio-temporal remodeling of endosomes and microtubules during abscission. We show that abscission is driven by the formation of a secondary ingression within the intracellular bridge connecting two daughter cells. The initiation and expansion of this secondary ingression requires recycling endosome fusion with the furrow plasma membrane and nested central spindle microtubule severing. These changes in endosome fusion and microtubule reorganization result in increased intracellular bridge plasma membrane dynamics and abscission. Finally, we show that central spindle microtubule reorganization is driven by localized microtubule buckling and breaking, rather than by spastin-dependent severing. Our results provide a new mechanism for mediation and regulation of the abscission step of cytokinesis.

Predicting buckling behavior of microtubules based on an atomistic-continuum model

An atomistic-continuum model is proposed for microtubules. A higher-order gradient continuum constitutive relationship is established, and elasticity and global buckling of microtubules are studied intensively. As a typical macromolecular bio-system, atomic components and structures are much more complicated. Traditional atomistic simulation methods and classical continuum approaches have their own fundamental drawbacks in dealing with this large atomic system. Adopting a homogenization technique, this paper proposes a concept of fictitious bonds for microtubules to link the large atomic structure with continuum description. After selecting a representative unit, the fictitious bond energy equals to the energy stored in the continuum model. The higher-order Cauchy–Born rule is used to approximate the deformation of fictitious bonds under arbitrary loading conditions. A mesh-free numerical scheme is specifically developed for modeling computation. The elastic modulus and critical compressive force are predicted. Representative case studies are presented, and some results are obtained and discussed.

Buckling of embedded microtubules in elastic medium

Motivated by the application of Winkler-like models for the buckling analysis of embedded carbon nanotubes, an orthotropic Winkler-like model is developed to study the buckling behavior of embedded cytoskeletal microtubules within the cytoplasm. Experimental observations of the buckling of embedded cytoskeletal microtubules reveal that embedded microtubules bear a large compressive force as compared with free microtubules. The present theoretical model predicts that embedded microtubules in an elastic medium bear large compressive forces than free microtubules. The estimated critical pressure is in good agreement with the experimental values of the pressure-induced buckling of microtubules. Moreover, due to the mechanical coupling of microtubules with the surrounding elastic medium, the critical buckling force is increased considerably, which well explains the theory that the mechanical coupling of microtubules with an elastic medium increases compressive forces that microtubules can sustain. The model presented in the paper is a good approximation for the buckling analysis of embedded microtubules.

Dimensional Analysis as a Tool to Introduce Biology Students to Physical Reasoning

As students of biology begin their studies in biological physics, an effective first lesson strives to develop a “feeling for the numbers” of biology [Phillips and Milo, PNAS, 106, 21465-21471 (2009)], in which students become acquainted with the length, time, and energy scales associated with systems of interest and begin thinking quantitatively. The next crucial step is to use these numbers to make more general inferences. To this end, it is both instructive and useful to perform estimates on biological systems in the spirit of “Fermi problems.” Though detailed calculations are unnecessary, estimation often requires extensive physical insight. While the student will ultimately learn the requisite physical principles, the task of performing estimates can initially prove daunting. An important too in nearly all physical contexts, dimensional analysis gives the comfort of being formulaic, yet still requires thought about what parameters might be important, without necessarily requiring deep physical understanding. This provides a gentler, though effective, introduction to physical reasoning for the uninitiated biologist. Furthermore, the process of performing dimensional analysis can expose interesting questions about what physical information is necessary, providing student-discovered segues into further physical topics. We successfully used this technique in a course for biology graduate students at the Max Planck Institute of Molecular Cell Biology and Genetics in Dresden, Germany. In this talk, I will discuss our approach to teaching dimensional analysis to biologists and discuss case studies in Brownian motion, cytoplasmic mixing, interpretation of single-molecule motor experiments, and microtubule buckling [Brangwynne, et al., J. Cell Biol., 173, 733-741 (2006)].

Nonlocal shear deformable shell model for bending buckling of microtubules embedded in an elastic medium

A nonlocal shear deformable shell model is developed for buckling of microtubules embedded in an elastic matrix of cytoplasm under bending in thermal environments. The results reveal that the lateral constraint has a significant effect on the buckling moments of a microtubule when the foundation stiffness is sufficiently large.

Structural basis of stress concentration in the cytoskeleton

Professor Y.C. Fung has shown that living tissues remodel extensively in response to mechanical forces such as blood pressure variations. At the cellular level, those mechanical perturbations must be perceived by individual cells. However, mechanisms of mechanochemical transduction in living cells remain a central challenge to cell biologists. Contrary to predictions by existing models of living cells, we reported previously that a local stress, applied via integrin receptors, is propagated to remote sites in the cytoplasm and is concentrated at discrete foci. Here we report that these foci of strains and stresses in the cytoplasm correspond to local peak deformation or local buckling of microtubules and are near the actin bundles of the cell. Multiple images at different z heights demonstrated more foci of concentrated displacements in the middle of the cell than at the apex or at the cell base. Together with previously published work, these findings underscore the importance of tensed bundled filamentous actin in intracellular mechanical stress distribution and signaling.

Buckling and postbuckling of radially loaded microtubules by nonlocal shear deformable shell model

This paper presents an investigation on the buckling and postbuckling of microtubules (MTs) subjected to a uniform external radial pressure in thermal environments. The microtubule is modeled as a nonlocal shear deformable cylindrical shell which contains small scale effects. The governing equations are based on higher order shear deformation shell theory with a von Kármán–Donnell-type of kinematic nonlinearity and include the extension-twist and flexural-twist couplings. The thermal effects are also included and the material properties are assumed to be temperature-dependent. A singular perturbation technique is employed to determine the buckling pressure and postbuckling equilibrium paths. The small scale parameter e 0 a is estimated by matching the buckling pressure of MTs measured from the experiments with the numerical results obtained from the nonlocal shear deformable shell model. The numerical results show that buckling pressure and postbuckling behavior of MTs are very sensitive to the small scale parameter e 0 a. The results reveal that the 13_3 microtubule has a stable postbuckling path, whereas the 13_2 microtubule has an unstable postbuckling behavior due to the presence of skew angles.

Mechanical breaking of microtubules in axons during dynamic stretch injury underlies delayed elasticity, microtubule disassembly, and axon degeneration

Little is known about which components of the axonal cytoskeleton might break during rapid mechanical deformation, such as occurs in traumatic brain injury. Here, we micropatterned neuronal cell cultures on silicone membranes to induce dynamic stretch exclusively of axon fascicles. After stretch, undulating distortions formed along the axons that gradually relaxed back to a straight orientation, demonstrating a delayed elastic response. Subsequently, swellings developed, leading to degeneration of almost all axons by 24 h. Stabilizing the microtubules with taxol maintained the undulating geometry after injury but greatly reduced axon degeneration. Conversely, destabilizing microtubules with nocodazole prevented undulations but greatly increased the rate of axon loss. Ultrastructural analyses of axons postinjury revealed immediate breakage and buckling of microtubules in axon undulations and progressive loss of microtubules. Collectively, these data suggest that dynamic stretch of axons induces direct mechanical failure at specific points along microtubules. This microtubule disorganization impedes normal relaxation of the axons, resulting in undulations. However, this physical damage also triggers progressive disassembly of the microtubules around the breakage points. While the disintegration of microtubules allows delayed recovery of the "normal" straight axon morphology, it comes at a great cost by interrupting axonal transport, leading to axonal swelling and degeneration.

Both actin and myosin inhibitors affect spindle architecture in PtK1 cells: does an actomyosin system contribute to mitotic spindle forces by regulating attachment and movements of chromosomes in mammalian cells?

Immunocytochemical techniques are used to analyze the effects of both an actin and myosin inhibitor on spindle architecture in PtK(1) cells to understand why both these inhibitors slow or block chromosome motion and detach chromosomes. Cytochalasin J, an actin inhibitor and a myosin inhibitor, 2, 3 butanedione 2-monoxime, have similar effects on changes in spindle organization. Using primary antibodies and stains, changes are studied in microtubule (MT), actin, myosin, and chromatin localization. Treatment of mitotic cells with both inhibitors results in detachment or misalignment of chromosomes from the spindle and a prominent buckling of MTs within the spindle, particularly evident in kinetochore fibers. Evidence is presented to suggest that an actomyosin system may help to regulate the initial and continued attachment of chromosomes to the mammalian spindle and could also influence spindle checkpoint(s).

Small scale effects on the mechanical behaviors of protein microtubules based on the nonlocal elasticity theory

Based on the nonlocal elastic theory, small scale effects are considered in the investigation of the mechanical properties of protein microtubules. A new prediction formula for the persistence lengths of microtubules with the consideration of the small scale effect is presented. Subsequently, the buckling of microtubules is studied based on a nonlocal elastic beam model. The predicted results of our model indicate that the length-dependence of persistence length is related not only to the shear terms, but also to the small scale effect. The Eular beam model, which is always considered unable to explain the length-dependence of microtubules, can capture the length-dependence of the persistence length of microtubules with the consideration of the small scale effect. The elastic buckling behaviors of microtubules in viscoelastic surrounding cytoplasm are also considered using the nonlocal Timoshenko beam model in this paper, and the results indicate that the small scale effect of microtubules also plays an important role in the buckling of microtubules.

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.

Cooperative Movement Of Wild-type Kinesin And Velocity-deficient Mutants

In the classic sliding filament assay, the molecular motor kinesin exhibits processive movement on microtubules with a velocity that is invariant over a large range of motor concentrations. This indicates that kinesin motors move synchronously at high density, but studies examining the motility of 2–3 kinesin motors have shown a surprising lack of synchronization. These results together led us to believe that kinesin motors under high density conditions can pull one another off the microtubule track, accelerating dissociation. Using a computational model, we can demonstrate that this would enable synchronization of MT movement without complete motor to motor synchronization. To test this experimentally, we combined kinesin dimers containing a mutation in the neck-linker (termed VKN) that elicits a 3-fold reduction in velocity when compared to wild-type motor (5.4μm/min vs. 16.2 μm/min) with wild type motors in the sliding filament assay. No significant amount of microtubule buckling was observed for any mixture of wild-type and mutant motors; even at limiting dilutions, and speckled microtubules moved at the same velocities throughout their length, indicating that the motors behave cooperatively, coordinating their movement through a shared interaction with the microtubule. We plan to examine whether this cooperativity is positive (WT motors accelerating VKN mutant movement) or negative (VKN slowing down WT).

Buckling of microtubules under bending and torsion

Microtubules (MTs) in living cells are frequently bend, e.g., with a mean curvature of about 0.4 rad/μm in fibroblast cells [Odde et al., J. Cell Sci. 112, 3283 (1999)]. This raises a natural question whether bending buckling can occur in a MT. In this paper, an orthotropic model is developed to investigate buckling of MTs upon bending and torsion. A critical buckling curvature for a bent MT is predicted to be about 0.03 rad/μm (to which the corresponding bending moment is 0.85 nN nm), indicating that MTs in living cells are likely buckled. Buckling behavior of torsional MTs is also studied, and a critical buckling torque of 0.077 nN nm is obtained. Comparison to the results from an isotropic model shows that anisotropic properties of the MT wall have severe effect on the mechanical behavior of MTs.

A mechanics model of microtubule buckling in living cells

As the most rigid cytoskeletal filaments, microtubules bear compressive forces in living cells, balancing the tensile forces within the cytoskeleton to maintain the cell shape. It is often observed that, in living cells, microtubules under compression severely buckle into short wavelengths. By contrast, when compressed, isolated microtubules in vitro buckle into single long-wavelength arcs. The critical buckling force of the microtubules in vitro is two orders of magnitude lower than that of the microtubules in living cells. To explain this discrepancy, we describe a mechanics model of microtubule buckling in living cells. The model investigates the effect of the surrounding filament network and the cytosol on the microtubule buckling. The results show that, while the buckling wavelength is set by the interplay between the microtubules and the elastic surrounding filament network, the buckling growth rate is set by the viscous cytosol. By considering the nonlinear deformation of the buckled microtubule, the buckling amplitude can be determined at the kinetically constrained equilibrium. The model quantitatively correlates the microtubule bending rigidity, the surrounding filament network elasticity, and the cytosol viscosity with the buckling wavelength, the buckling growth rate, and the buckling amplitude of the microtubules. Such results shed light on designing a unified experimental protocol to measure various critical mechanical properties of subcellular structures in living cells.