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 Possible Contribution of Intracellular Electric Fields to Oriented Assemblage of Microtubules

Mathematical modeling suggests that the intensity of the intracellular electric fields is sufficient to cause directed polymerization of tubulin to microtubules, and thus to determine the direction of the movement of intracellular components.     

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.     

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.     

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.     

Experimental studies of the magnetic structure and plasma dynamics in current sheets (a review)

Based on measurements of magnetic fields in current sheets, spatial distributions of the electric current and electrodynamic forces in successive stages of the sheet evolution are determined. Two new effects manifesting themselves mostly in the late stages of the current sheet evolution have been discovered, namely, expansion of the current flow region at the periphery of the sheet and the appearance of a region with inverse currents, which gradually expands from the periphery toward the center of the sheet. Using spectroscopic methods, generation of superthermal plasma flows accelerated along the sheet width from the center toward the periphery has been revealed and investigated. The measured energies of accelerated plasma ions satisfactorily agree with the Ampère forces determined from magnetic measurements. The excitation of inverse currents additionally confirms the motion of high-speed plasma flows from the center of the current sheet toward its side edges.     

The cytaster,a colchicine-sensitive migration organelle of cleavage nuclei in an insect egg

Time-lapse analyses of nuclear multiplication in the eggs of the gall midge Wachtliella persicariae L., documented in film D 1235 (available from the IWF, Göttingen), give evidence of a special migration organelle of cleavage nuclei. Each of these “migration cytasters” represents one greatly enlarged polar cytaster of the mitotic apparatus, which is connected to one nucleus. From the films it can be concluded that the astral rays temporarily adhere to peripheral egg structures and exert tractive forces toward the cytaster center. These forces combine and pull the accompanying daughter nucleus through the ooplasm after each mitosis. This “active” mode of migration, which is accompanied by extensive polarized transport of yolk particles toward the cytaster center, enables the energids (= cleavage nucleus and its associated island of cytoplasm) to move relative to the surrounding ooplasm. In addition, there is a “passive” mode of nuclear migration: The energids are moved by means of plasmic flows, thereby maintaining their position in relation to the surrounding ooplasm. Electron microscopic studies show solitary microtubules running radially toward the cytaster center. As a result of colchicine injection (1) the microtubules disintegrate, (2) the polarized transport of yolk particles cases, (3) the active nuclear migration stops and the nuclei are only passively moved by rhythmic ooplasmic flows. This inhibition of active nuclear migration gives further evidence that microtubules take an essential part in it. Control experiments with lumicolchicine show no effect on nuclear migration. Conversely, under the influence of cytochalasin B active nuclear migration is continued, while the ooplasmic flows are inhibited. Thus the mechanisms of active and passive nuclear migration can work independently of each other. The generation of tractive forces along the astral rays is discussed with respect to current models of spindle function.     

Plant cell growth responds to external forces and the response requires intact microtubules

Microfibril deposition in most plant cells is influenced by cortical microtubules. Thus, cortical microtubules are templates that provide spatial information to the cell wall. How cortical microtubules acquire their spatial information and are positioned is unknown. There are indications that plant cells respond to mechanical stresses by using microtubules as sensing elements. Regenerating protoplasts from tobacco (Nicotiana tabacum) were used to determine whether cells can be induced to expand in a preferential direction in response to an externally applied unidirectional force. Additionally, an anti-microtubule herbicide was used to investigate the role of microtubules in the response to this force. Protoplasts were embedded in agarose, briefly centrifuged at 28 to 34g, and either cultured or immediately prepared for immunolocalization of their microtubules. The microtubules within many centrifuged protoplasts were found to be oriented parallel to the centrifugal force vector. Most protoplasts elongated with a preferential axis that was oriented 60 to 90 degrees to the applied force vector. Protoplasts treated transiently with the reversible microtubule-disrupting agent amiprophos-methyl (applied before and during centrifugation) elongated but without a preferential growth axis. These results indicate that brief biophysical forces may influence the alignment of cortical microtubules and that microtubules themselves act as biophysical responding elements.     

Dielectrophoretic separation of cells: Continuous separation

Dielectrophoresis is the movement of particles in non-uniform alternating and direct current (AC, DC) electric fields. When nonuniform electric fields are created between microelectrodes, cells will redistribute themselves around the electrodes, the force holding the cells in place dependig on the local electric field and on the electrical properties of the cells themselves and the suspending medium. Steric drag forces produced by a gentle fluid flow in the chamber can be used to separate cells by selectively lifting cells from potential energy wells produced by the electric field. The technique is demonstrated in the batch separation of bacteria, yeast cells, and plant cells. Continuous separation and extraction of two cell types can be achieved by repeated reversing of the fluid flow direction in phase with the switching on and off of the applied voltage, and the efficacy of the technique is demonstrated for viable and nonviable (heat-treated) yeast cells. (c) 1995 John Wiley & Sons, Inc.     

Analysis of cortical flow models in vivo

Cortical flow, the directed movement of cortical F-actin and cortical organelles, is a basic cellular motility process. Microtubules are thought to somehow direct cortical flow, but whether they do so by stimulating or inhibiting contraction of the cortical actin cytoskeleton is the subject of debate. Treatment of Xenopus oocytes with phorbol 12-myristate 13-acetate (PMA) triggers cortical flow toward the animal pole of the oocyte; this flow is suppressed by microtubules. To determine how this suppression occurs and whether it can control the direction of cortical flow, oocytes were subjected to localized manipulation of either the contractile stimulus (PMA) or microtubules. Localized PMA application resulted in redirection of cortical flow toward the site of application, as judged by movement of cortical pigment granules, cortical F-actin, and cortical myosin-2A. Such redirected flow was accelerated by microtubule depolymerization, showing that the suppression of cortical flow by microtubules is independent of the direction of flow. Direct observation of cortical F-actin by time-lapse confocal analysis in combination with photobleaching showed that cortical flow is driven by contraction of the cortical F-actin network and that microtubules suppress this contraction. The oocyte germinal vesicle serves as a microtubule organizing center in Xenopus oocytes; experimental displacement of the germinal vesicle toward the animal pole resulted in localized flow away from the animal pole. The results show that 1) cortical flow is directed toward areas of localized contraction of the cortical F-actin cytoskeleton; 2) microtubules suppress cortical flow by inhibiting contraction of the cortical F-actin cytoskeleton; and 3) localized, microtubule-dependent suppression of actomyosin-based contraction can control the direction of cortical flow. We discuss these findings in light of current models of cortical flow.     

Flexural rigidity of individual microtubules measured by a buckling force with optical traps

We used direct buckling force measurements with optical traps to determine the flexural rigidity of individual microtubules bound to polystyrene beads. To optimize the accuracy of the measurement, we used two optical traps and antibody-coated beads to manipulate each microtubule. We then applied a new analytical model assuming nonaxial buckling. Paclitaxel-stabilized microtubules were polymerized from purified tubulin, and the average microtubule rigidity was calculated as 2.0 x 10(-24) Nm2 using this novel microtubule buckling system. This value was not dependent on microtubule length. We also measured the rigidity of paclitaxel-free microtubules, and obtained the value of 7.9 x 10(-24) Nm2, which is nearly four times that measured for paclitaxel-stabilized microtubules.     

Analysis of the migration behaviour of single microtubules in electric fields

By video contrast microscopy, individual microtubules formed from pure tubulin in the presence of taxol were studied in constant electric fields. At nearly physiological conditions, i.e., in a buffer at pH 6.8 and 120 mM ionic strength, suspended microtubules moved towards the anode with an electrophoretic mobility of approximately 2.6 x 10(-4) cm(2)/V s, corresponding to an unbalanced negative charge of 0.19 electron charges per tubulin dimer. Strikingly, this value is lower by a factor of at least 50 than that calculated from crystallographic data for the non-assembled tubulin dimer. Moreover, the taxol-stabilized microtubules had an isoelectric point of about pH 4.2 which is significantly lower than that known for the tubulin monomers. This indicates that microtubule formation is accompanied by substantial changes of charge distribution within the tubulin subunits. Constant electric fields were shown to affect also the orientation of microtubules gliding across a kinesin-coated surface at pH 6.8.     

Dynamics of the structure of electric currents and electrodynamic forces in current sheets

Specific features of the spatial distributions of the electric current and electrodynamic forces in current sheets are examined by studying the magnetic fields in them. It is shown that the j × B forces should lead to a gradual increase in the kinetic energy of the plasma accelerated along the current sheet surface. Excitation of currents directed oppositely to the main current in the central part of the sheet is observed for the first time, and the time evolution of the forward and reverse currents is investigated. Generation of reversed currents is a manifestation of the dynamic effects caused by the motion of plasma flows in the magnetic field and leading to a change in the magnetic structure of the current sheet.     

小鼠皮肤黑色素瘤细胞在三种不同基质上的趋电性比较

微小直流电场具有指导细胞进行定向迁移的作用。各种细胞外基质的物理、化学性质会影响细胞的迁移。该研究以小鼠皮肤黑色素瘤细胞（B16-F10）为模型,比较微直流电场（250mV／mm）指导下细胞在平滑基底与两种不同市售基质Matrigel及FNC上的趋电性。结果显示,黑色素瘤细胞在三种基底上均有明显的向电场阴极迁移的趋电运动,但在不同基质上细胞趋电的方向性无显著差异,但细胞迁移速度及在细胞沿电场进行定向迁移的持续性有显著差异。     

Molecular organization and in vivo function of the cytoskeleton of amphibian erythrocytes

One prominent cytoskeletal feature of non-mammalian vertebrate erythrocytes is the marginal band (MB), composed of microtubules. However, there have been several reports of MB-associated F-actin. We have further investigated the function of MB-associated F-actin, using newt erythrocytes having large, thick MBs. Confocal microscopy revealed a distinctive band of F-actin colocalizing point- by-point with MB microtubules. Furthermore, the F-actin band was present in isolated elliptical MBs, but absent in membrane skeletons lacking MBs. F-actin depolymerizing agents did not affect F-actin band integrity in isolated MBs, indicating its non-dynamic state. However, exposure to elastase resulted in F-actin removal and MB circularization. These results provide evidence of a strong association of F-actin with MB microtubules in mature ellipsoidal erythrocytes. To assess the true extent of mechanical stress on the cytoskeleton, erythrocytes were observed by video microscopy during flow in vivo. Moving with long axis parallel to flow direction, cells underwent reversible shape distortion as they collided vigorously with other erythrocytes and vessel walls. In addition, cells twisted into figure-8 shapes, a cytoskeletal property that may provide physiological advantages during flow. Our results, together with those of others, yield a consistent picture in which developing erythrocytes undergo transition from spheroids to immature discoids to mature ellipsoids. The causal step in discoid formation is biogenesis of circular MBs with sufficient flexural rigidity to determine cell shape. F-actin binding to MB microtubules then creates a composite system, enhancing flexural rigidity to produce and maintain ellipsoidal shape during the physical challenges of blood flow in vivo.     

Electromagnetic Field of Microtubules: Effects on Transfer of Mass Particles and Electrons

Biological polar molecules and polymer structures with energy supply (such as microtubules in the cytoskeleton) can get excited and generate an endogenous electromagnetic field with strong electrical component in their vicinity. The endogenous electrical fields through action on charges, on dipoles and multipoles, and through polarization (causing dielectrophoretic effect) exert forces and can drive charges and particles in the cell. The transport of mass particles and electrons is analyzed as a Wiener-Lévy process with inclusion of deterministic force (validity of the Bloch theorem is assumed for transport of electrons in molecular chains too). We compare transport driven by deterministic forces (together with an inseparable thermal component) with that driven thermally and evaluate the probability to reach the target. Deterministic forces can transport particles and electrons with higher probability than forces of thermal origin only. The effect of deterministic forces on directed transport is dominant.     

The motor activity of mammalian axonemal dynein studied in situ on doublet microtubules

Flagellar dynein generates forces that produce relative shearing between doublet microtubules in the axoneme; this drives propagated bending of flagella and cilia. To better understand dynein's role in coordinated flagellar and ciliary motion, we have developed an in situ assay in which polymerized single microtubules glide along doublet microtubules extruded from disintegrated bovine sperm flagella at a pH of 7.8. The exposed, active dynein remain attached to their respective doublet microtubules, allowing gliding of individual microtubules to be observed in an environment that allows direct control of chemical conditions. In the presence of ATP, translocation of microtubules by dynein exhibits Michaelis-Menten type kinetics, with V(max) = 4.7 +/- 0.2 microm/s and K(m) = 124 +/- 11 microM. The character of microtubule translocation is variable, including smooth gliding, stuttered motility, oscillations, buckling, complete dissociation from the doublet microtubule, and occasionally movements reversed from the physiologic direction. The gliding velocity is independent of the number of dynein motors present along the doublet microtubule, and shows no indication of increased activity due to ADP regulation. These results reveal fundamental properties underlying cooperative dynein activity in flagella, differences between mammalian and non-mammalian flagellar dynein, and establish the use of natural tracks of dynein arranged in situ on the doublet microtubules of bovine sperm as a system to explore the mechanics of the dynein-microtubule interactions in mammalian flagella.     

Force on spindle microtubule minus ends moves chromosomes

The spindle is a dynamic self-assembling machine that coordinates mitosis. The spindle’s function depends on its ability to organize microtubules into poles and maintain pole structure despite mechanical challenges and component turnover. Although we know that dynein and NuMA mediate pole formation, our understanding of the forces dynamically maintaining poles is limited: we do not know where and how quickly they act or their strength and structural impact. Using laser ablation to cut spindle microtubules, we identify a force that rapidly and robustly pulls severed microtubules and chromosomes poleward, overpowering opposing forces and repairing spindle architecture. Molecular imaging and biophysical analysis suggest that transport is powered by dynein pulling on minus ends of severed microtubules. NuMA and dynein/dynactin are specifically enriched at new minus ends within seconds, reanchoring minus ends to the spindle and delivering them to poles. This force on minus ends represents a newly uncovered chromosome transport mechanism that is independent of plus end forces at kinetochores and is well suited to robustly maintain spindle mechanical integrity.     

Electric field-directed fibroblast locomotion involves cell surface molecular reorganization and is calcium independent

Directional cellular locomotion is thought to involve localized intracellular calcium changes and the lateral transport of cell surface molecules. We have examined the roles of both calcium and cell surface glycoprotein redistribution in the directional migration of two murine fibroblastic cell lines, NIH 3T3 and SV101. These cell types exhibit persistent, cathode directed motility when exposed to direct current electric fields. Using time lapse phase contrast microscopy and image analysis, we have determined that electric field-directed locomotion in each cell type is a calcium independent process. Both exhibit cathode directed motility in the absence of extracellular calcium, and electric fields cause no detectable elevations or gradients of cytosolic free calcium. We find evidence suggesting that galvanotaxis in these cells involves the lateral redistribution of plasma membrane glycoproteins. Electric fields cause the lateral migration of plasma membrane concanavalin A receptors toward the cathode in both NIH 3T3 and SV101 fibroblasts. Exposure of directionally migrating cells to Con A inhibits the normal change of cell direction following a reversal of electric field polarity. Additionally, when cells are plated on Con A- coated substrata so that Con A receptors mediate cell-substratum adhesion, cathode-directed locomotion and a cathodal accumulation of Con A receptors are observed. Immunofluorescent labeling of the fibronectin receptor in NIH 3T3 fibroblasts suggests the recruitment of integrins from large clusters to form a more diffuse distribution toward the cathode in field-treated cells. Our results indicate that the mechanism of electric field directed locomotion in NIH 3T3 and SV101 fibroblasts involves the lateral redistribution of plasma membrane glycoproteins involved in cell-substratum adhesion.