Flexural rigidity of microtubules and actin filaments measured from thermal fluctuations in shape

Microtubules are long, proteinaceous filaments that perform structural functions in eukaryotic cells by defining cellular shape and serving as tracks for intracellular motor proteins. We report the first accurate measurements of the flexural rigidity of microtubules. By analyzing the thermally driven fluctuations in their shape, we estimated the mean flexural rigidity of taxol-stabilized microtubules to be 2.2 x 10(-23) Nm2 (with 6.4% uncertainty) for seven unlabeled microtubules and 2.1 x 10(-23) Nm2 (with 4.7% uncertainty) for eight rhodamine-labeled microtubules. These values are similar to earlier, less precise estimates of microtubule bending stiffness obtained by modeling flagellar motion. A similar analysis on seven rhodamine-phalloidin- labeled actin filaments gave a flexural rigidity of 7.3 x 10(-26) Nm2 (with 6% uncertainty), consistent with previously reported results. The flexural rigidity of these microtubules corresponds to a persistence length of 5,200 microns showing that a microtubule is rigid over cellular dimensions. By contrast, the persistence length of an actin filament is only approximately 17.7 microns, perhaps explaining why actin filaments within cells are usually cross-linked into bundles. The greater flexural rigidity of a microtubule compared to an actin filament mainly derives from the former's larger cross-section. If tubulin were homogeneous and isotropic, then the microtubule's Young's modulus would be approximately 1.2 GPa, similar to Plexiglas and rigid plastics. Microtubules are expected to be almost inextensible: the compliance of cells is due primarily to filament bending or sliding between filaments rather than the stretching of the filaments themselves.     

Rigidity of microtubules is increased by stabilizing agents

Microtubules are rigid polymers that contribute to the static mechanical properties of cells. Because microtubules are dynamic structures whose polymerization is regulated during changes in cell shape, we have asked whether the mechanical properties of microtubules might also be modulated. We measured the flexural rigidity, or bending stiffness, of individual microtubules under a number of different conditions that affect the stability of microtubules against depolymerization. The flexural rigidity of microtubules polymerized with the slowly hydrolyzable nucleotide analogue guanylyl-(alpha, beta)- methylene-diphosphonate was 62 +/- 9 x 10(-24) Nm2 (weighted mean +/- SEM); that of microtubules stabilized with tau protein was 34 +/- 3 x 10(-24) Nm2; and that of microtubules stabilized with the antimitotic drug taxol was 32 +/- 2 x 10(-24) Nm2. For comparison, microtubules that were capped to prevent depolymerization, but were not otherwise stabilized, had a flexural rigidity of 26 +/- 2 x 10(-24) Nm2. Decreasing the temperature from 37 degrees C to approximately 25 degrees C, a condition that makes microtubules less stable, decreased the stiffness of taxol-stabilized microtubules by one-third. We thus find that the more stable a microtubule, the higher its flexural rigidity. This raises the possibility that microtubule rigidity may be regulated in vivo. In addition, the high rigidity of an unstabilized, GDP-containing microtubule suggests that a large amount of energy could be stored as mechanical strain energy in the protein lattice for subsequent force generation during microtubule depolymerization.     

Length-dependence of flexural rigidity as a result of anisotropic elastic properties of microtubules

Unexplained length-dependence of flexural rigidity and Young's modulus of microtubules is studied using an orthotropic elastic shell model. It is showed that vibration frequencies and buckling load predicted by the accurate orthotropic shell model are much lower than that given by the approximate isotropic beam model for shorter microtubules, although the two models give almost identical results for sufficiently long microtubules. It is this inaccuracy of the isotropic beam model used by all previous researchers that leads to reported lower flexural rigidity and Young's modulus for shorter microtubules. In particular, much lower shear modulus and circumferential Young's modulus, which only weaken flexural rigidity of shorter microtubules, are responsible for the observed length-dependence of the flexural rigidity. These results confirm that longitudinal Young's modulus of microtubules is length-independent, and the observed length-dependence of the flexural rigidity and Young's modulus is a result of strongly anisotropic elastic properties of microtubules which have a length-dependent weakening effect on flexural rigidity of shorter microtubules.     

The membrane transporter lactose permease increases lipid bilayer bending rigidity

Cellular life relies on membranes, which provide a resilient and adaptive cell boundary. Many essential processes depend upon the ease with which the membrane is able to deform and bend, features that can be characterized by the bending rigidity. Quantitative investigations of such mechanical properties of biological membranes have primarily been undertaken in solely lipid bilayers and frequently in the absence of buffers. In contrast, much less is known about the influence of integral membrane proteins on bending rigidity under physiological conditions. We focus on an exemplar member of the ubiquitous major facilitator superfamily of transporters and assess the influence of lactose permease on the bending rigidity of lipid bilayers. Fluctuation analysis of giant unilamellar vesicles (GUVs) is a useful means to measure bending rigidity. We find that using a hydrogel substrate produces GUVs that are well suited to fluctuation analysis. Moreover, the hydrogel method is amenable to both physiological salt concentrations and anionic lipids, which are important to mimic key aspects of the native lactose permease membrane. Varying the fraction of the anionic lipid in the lipid mixture DOPC/DOPE/DOPG allows us to assess the dependence of membrane bending rigidity on the topology and concentration of an integral membrane protein in the lipid bilayer of GUVs. The bending rigidity gradually increases with the incorporation of lactose permease, but there is no further increase with greater amounts of the protein in the membrane.     

Evaluation of linear finite-element analysis models' assumptions for external fixation devices.

Linear finite-element models (FEMs) have enjoyed an increased use in orthopaedic research, including the use for modeling external fixation devices. These fixator FEMs depend on a number of basic assumptions concerning the overall fixation frame stability and the components' rigidity. Among the more important ones are: (i) rigid fixation at both ends of the pin and sidebar; (ii) that the sidebar can be treated essentially as a rigid entity, with all bending occurring in the bone pins; and (iii) that the system can be treated as linearly elastic. Prior work done by the authors questions some of these assumptions. Thus, this study sought an empirical evaluation of the validity of some of these a priori assumptions. A Hoffmann single half-frame was tested in its standard form and then according to a stepwise protocol wherein the frame was welded to eliminate any possible points of instability. These tests looked at the stability and rigidity in various modes (axial compression, torsion, and medial-lateral and anterior-posterior four-point bending). The basic assumptions concerning the frame stability, frame rigidity and the frame's response to loads were found to be erroneous. Component failure was common under minimal loads and statistically significant differences (p less than 0.05) of up to 75% were noted in frame rigidity among the various frame forms tested. Thus, considerable caution must be exercised when employing the FEM technique for evaluating the fixator properties.     

Nanomechanical model of microtubule translocation in the presence of electric fields

Research efforts in recent years have been directed toward actively controlling the direction of translocation of microtubules on a kinesin-coated glass surface with E-fields (electric fields), opening up the possibility of engineering controllable nanodevices that integrate microtubules and motor proteins into their function. Here, we present a detailed, biophysical model that quantitatively describes our observations on the steering of microtubules by electric fields. A sudden application of an electric field parallel to the surface and normal to the translocation direction of a microtubule bends the leading end toward the anode, because Coulombic (electrophoretic) forces are dominant on negatively charged microtubules. Modeling this bending as a cantilever deflection with uniform loading requires accurate mechanical and electrical properties of microtubules, including their charge density, viscous drag, and flexural rigidity. We determined the charge density of microtubules from measurements of the electrophoretic mobility in a “zero flow” capillary electrophoresis column and estimate it to be 256 e⁻ per micron of length. Viscous drag forces on deflecting microtubules in electroosmotic flows were studied theoretically and experimentally by directly characterizing flows using a caged dye imaging method. The flexural rigidity of microtubules was measured by applying E-fields to microtubules with biotinylated segments that were bound to streptavidin-coated surfaces. From the calculated loading, and the Bernoulli-Euler curvature and moment equation, we find that the flexural rigidity of microtubules depends on their length, suggesting microtubules are anisotropic. Finally, our model accurately predicts the biophysical properties and behavior of microtubules directed by E-fields, which opens new avenues for the design of biomolecular nanotransport systems.     

Dynamic behaviors of microtubules in cytosol

Highly anisotropic microtubules (MTs) immersed in cytosol are a central part of the cytoskeleton in eukaryotic cells. The dynamic behaviors of an MT–cytosol system are of major interest in biomechanics community. Such a solid–fluid system is characterized by a Reynolds number of the order 10^?3 and a slip ionic layer formed at the MT–cytosol interface. In view of these unique features, an orthotropic shell-Stokes flow model with a slip boundary condition has been developed to explore the distinctive dynamic behaviors of MTs in cytosol. Three types of motions have been identified, i.e., (a) undamped and damped torsional vibration, (b) damped longitudinal vibration, and (c) overdamped bending and radial motions. The exponentially decaying bending motion given by the present model is found to be in qualitative agreement with the existing experimental observation [Felgner et al., 1996. Flexural rigidity of microtubules measured with the use of optical tweezers, Journal of Cell Science 109, 509–516 ].     

Bending dynamics of fluctuating biopolymers probed by automated high-resolution filament tracking

Microscope images of fluctuating biopolymers contain a wealth of information about their underlying mechanics and dynamics. However, successful extraction of this information requires precise localization of filament position and shape from thousands of noisy images. Here, we present careful measurements of the bending dynamics of filamentous (F-)actin and microtubules at thermal equilibrium with high spatial and temporal resolution using a new, simple but robust, automated image analysis algorithm with subpixel accuracy. We find that slender actin filaments have a persistence length of approximately 17 microm, and display a q(-4)-dependent relaxation spectrum, as expected from viscous drag. Microtubules have a persistence length of several millimeters; interestingly, there is a small correlation between total microtubule length and rigidity, with shorter filaments appearing softer. However, we show that this correlation can arise, in principle, from intrinsic measurement noise that must be carefully considered. The dynamic behavior of the bending of microtubules also appears more complex than that of F-actin, reflecting their higher-order structure. These results emphasize both the power and limitations of light microscopy techniques for studying the mechanics and dynamics of biopolymers.     

Vibration and length-dependent flexural rigidity of protein microtubules using higher order shear deformation theory

Microtubules are hollow cylindrical filaments of the eukaryotic cytoskeleton characterized by extremely low shear modulus. A remarkable controversy has occurred in the literature, regarding the length dependence of flexural rigidity of microtubules predicted by the classical elastic beam model. In this study, a higher order shear deformable beam model for microtubules is employed to study unexplained length-dependent flexural rigidity and Young’s modulus of microtubules reported in the literature. The formulation allows for warping of the cross-section of the microtubule and eliminates the need for using arbitrary shear correction coefficients as in other theories. It is showed that vibration frequencies predicted by the present parabolic shear deformation theory (PSDT) are much lower than that given by the approximate isotropic beam model for shorter microtubules, although the two models give almost identical results for sufficiently long microtubules. It is confirmed that transverse shearing and the warping of the cross-section of microtubules are mainly responsible for the length-dependent flexural rigidity of an isolated microtubule reported in the literature, which cannot be explained by the widely used Euler-Bernoulli beam model. Indeed, the length-dependent flexural rigidity predicted by the present model is found to be in qualitative agreement with the existing experimental data ( [Kurachi et al., 1995] and [Pampaloni et al., 2006]). These results recommend that the parabolic shear deformation-beam theory offers a unified simple 1D model, which can capture the length dependence of flexural rigidity and be applied to various static and dynamic problems of microtubule mechanics.     

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.     

Microtubular Protofilament Analysis Based on Molecular Level Tubulin Interaction

Nonlinear microstructure of the microtubules (MTs) plays an important role in their mechanical properties. Despite the extensive efforts into the development of continuum models for microtubules, a mesoscale finite element model that can link the molecular level information to the overall performance of microtubules is still missing. The aim of this study is to develop a molecular dynamics model (MDM), finite element model (FEM) and structural mechanics beam model (SMBM) for tubulins of protofilament (PF). In MDM, the backbone atoms of α-tubulin were fixed while the backbone atoms of β-tubulin were attached to a molecular dynamics (MD) atom through a virtual spring. In FEM, both α and β tubulins are modeled as spherical shells and adjacent tubulins are connected by linear springs. The spherical shells were framed as beams in SMBM. Corresponding parameters such as the elasticity of tubulin-tubulin interaction (TTI) and the stiffness of springs and beam are derived from MD simulation. Marginal differences in the force-deflection curve among the FEM, the MDM and SMBM indicate the good accuracy in describing the mechanical properties of microtubules. Simulation results show that the protofilament behaves non-linearly under tension and torsion but linearly under bending. Deformation pattern of a PF from the SMBM frame bending can be well captured by the classical Euler-Bernouli beam theory and the flexural rigidity derived from FEM is in good agreement with SMBM. These findings lend compelling credence in our developed models of PF to deepen our understanding of the underlying mechanism of statics and dynamics of MTs. In perspective our approach provides a tool for the analysis of MTs mechanical behavior under different conditions.     

Computer simulation of bend propagation by axoplasmic microtubules

The generation of bending waves by microtubules in squid nerve axoplasm has been modelled using appropriately modified versions of computer programs developed previously for simulation of flagellar bending waves. The results confirm that a constant longitudinal force directed along the axis of the microtubule is sufficient to cause the generation of regular oscillations and propagated bending waves when the forward gliding movement of the microtubule is obstructed. No control mechanism is required to modulate the active force-generating system. In order to obtain bending waves similar to those observed experimentally, it was necessary to use a model for the force-generating system in which the active force decreases with increasing sliding velocity. If the elastic bending resistance of axoplasmic microtubules is similar to that of microtubules in sperm terminal filaments, the longitudinal force per unit length generated by the axoplasmic microtubules must be of the same order of magnitude as the force generated by dynein arms along the doublet microtubules of eukaryotic flagella.     

The evolution of jumping in frogs: Morphological evidence for the basal anuran locomotor condition and the radiation of locomotor systems in crown group anurans

Our understanding of the evolution of frog locomotion follows from the work of Emerson in which anurans are proposed to possess one of three different iliosacral configurations: 1) a lateral‐bending system found in walking and hopping frogs; 2) a fore‐aft sliding mechanism found in several locomotor modes; and 3) a sagittal‐hinge‐type pelvis posited to be related to long‐distance jumping performance. The most basal living (Ascaphus) and fossil (Prosalirus) frogs are described as sagittal‐hinge pelvic types, and it has been proposed that long‐distance jumping with a sagittal‐hinge pelvis arose early in frog evolution. We revisited osteological traits of the pelvic region to conduct a phylogenetic analysis of the relationships between pelvic systems and locomotor modes in frogs. Using two of Emerson's diagnostic traits from the sacrum and ilium and two new traits from the urostyle, we resampled the taxa originally studied by Emerson and key paleotaxa and conducted an analysis of ancestral‐character state evolution in relation to locomotor mode. We present a new pattern for the evolution of pelvic systems and locomotor modes in frogs. Character analysis shows that the lateral‐bender, walker/hopper condition is both basal and generally conserved across the Anura. Long‐distance jumping frogs do not appear until well within the Neobatrachia. The sagittal‐hinge morphology is correlated with long‐distance jumping in terrestrial frogs; however, it evolved convergently multiple times in crown group anurans with the same four pelvic traits described herein. Arboreal jumping has appeared in multiple crown lineages as well, but with divergent patterns of evolution involving each of the three pelvic types. The fore‐aft slider morph appears independently in three different locomotor modes and, thus, is a more complex system than previously thought. Finally, it appears that the advent of a bicondylar sacro‐urostylic articulation was originally related to providing axial rigidity to lateral‐bending behaviors rather than sagittal bending. J. Morphol., 2011. © 2010 Wiley‐Liss, Inc.     

Anisotropy in sickle hemoglobin fibers from variations in bending and twist

We have studied the variations of twist and bend in sickle hemoglobin fibers. We find that these variations are consistent with an origin in equilibrium thermal fluctuations, which allows us to estimate the bending and torsional rigidities and effective corresponding material moduli. We measure bending by electron microscopy of frozen hydrated fibers and find that the bending persistence length, a measure of the length of fiber required before it starts to be significantly bent due to thermal fluctuations, is 130microm, somewhat shorter than that previously reported using light microscopy. The torsional persistence length, obtained by re-analysis of previously published experiments, is found to be only 2.5microm. Strikingly this means that the corresponding torsional rigidity of the fibers is only 6x10(-27)Jm, much less than their bending rigidity of 5x10(-25)Jm. For (normal) isotropic materials, one would instead expect these to be similar. Thus, we present the first quantitative evidence of a very significant material anisotropy in sickle hemoglobin fibers, as might arise from the difference between axial and lateral contacts within the fiber. We suggest that the relative softness of the fiber with respect to twist deformation contributes to the metastability of HbS fibers: HbS double strands are twisted in the fiber but not in the equilibrium crystalline state. Our measurements inform a theoretical model of the thermodynamic stability of fibers that takes account of both bending and extension/compression of hemoglobin (double) strands within the fiber.     

Developmental studies on the loricate choanoflagellateStephanoeca diplocostata Ellis. V. The cytoskeleton and the effects of microtubule poisons

Summary InStephanoeca diplocostata microtubules are located in four positions namely: within the flagellar axoneme; just beneath the plasmalemma; associated with the silica deposition vesicles (SDVs) during early stages of costal strip deposition; and in the mitotic spindle. At the anterior end of the cell the 50–60 peripheral microtubules, which are organized more or less parallel to the long axis of the cell, converge around the base of the emergent flagellum. A short second flagellar base is positioned between the nucleus and the base of the emergent flagellum. Developing costal strips are located individually within SDVs in the peripheral cytoplasm. During the early stages of silica deposition each SDV is curved and subtended longitudinally on its concave side by two microtubules. When a costal strip has achieved sufficient rigidity to withstand bending the SDV-associated microtubules are depolymerized. Treatment of exponentially growing cells with sublethal concentrations of microtubule poisons, such as colchicine, podophyllotoxin, griseofulvin andVinca alkaloids depresses growth. Treatment with these drugs also affects the length and morphology of developing costal strips perhaps by interfering with the shaping and supporting functions of SDV-associated microtubules. Instead of being long and crescentic with a standard radius of curvature, costal strips of treated cells are usually short and misshapen, with irregular bends. After drug treatment, juveniles produced as a result of cell division do not develop flagella but can still assemble a lorica although it is usually misshapen. The role of microtubules and microfilaments in lorica production is discussed.     

Motion and flexibility in human cytochrome p450 aromatase

The crystal structures of human placental aromatase in complex with the substrate androstenedione and exemestane have revealed an androgen-specific active site and the structural basis for higher order organization. However, X-ray structures do not provide accounts of movements due to short-range fluctuations, ligand binding and protein-protein association. In this work, we conduct normal mode analysis (NMA) revealing the intrinsic fluctuations of aromatase, deduce the internal modes in membrane-free and membrane-integrated monomers as well as the intermolecular modes in oligomers, and propose a quaternary organization for the endoplasmic reticulum (ER) membrane integration. Dynamics of the crystallographic oligomers from NMA is found to be in agreement with the isotropic thermal factors from the X-ray analysis. Calculations of the root mean square fluctuations of the C-alpha atoms from their equilibrium positions confirm that the rigid-core structure of aromatase is intrinsic regardless of the changes in steroid binding interactions, and that aromatase self-association does not deteriorate the rigidity of the catalytic cleft. Furthermore, NMA on membrane-integrated aromatase shows that the internal modes in all likelihood contribute to breathing of the active site access channel. The collective intermolecular hinge bending and twisting modes provide the flexibility in the quaternary association necessary for membrane integration of the aromatase oligomers. Taken together, fluctuations of the active site, the access channel, and the heme-proximal cavity, and a dynamic quaternary organization could all be essential components of the functional aromatase in its role as an ER membrane-embedded steroidogenic enzyme.     

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.     

Vesicle fluctuation analysis of the effects of sterols on membrane bending rigidity

Sterols are regulators of both biological function and structure. The role of cholesterol in promoting the structural and mechanical stability of membranes is widely recognized. Knowledge of how the related sterols, lanosterol and ergosterol, affect membrane mechanical properties is sparse. This paper presents a comprehensive comparison of the effects of cholesterol, lanosterol, and ergosterol upon the bending elastic properties of 1-palmitoyl-2-oleoyl- sn-glycero-3-phosphocholine giant unilamellar vesicles. Measurements are made using vesicle fluctuation analysis, a nonintrusive technique that we have recently improved for determining membrane bending rigidity. Giving a detailed account of the vesicle fluctuation analysis technique, we describe how the gravitational stabilization of the vesicles enhances image contrast, vesicle yield, and the quality of data. Implications of gravity on vesicle behaviour are also discussed. These recent modifications render vesicle fluctuation analysis an efficient and accurate method for determining how cholesterol, lanosterol, and ergosterol increase membrane bending rigidity.