Oscillation modes of microtubules

Brain Microtubule Electrical Oscillations-Empirical Mode Decomposition Analysis

Microtubules (MTs) are essential cytoskeletal polymers of eukaryote cells implicated in various cell functions, including cell division, cargo transfer, and cell signaling. MTs also are highly charged polymers that generate electrical oscillations that may underlie their ability to act as nonlinear transmission lines. However, the oscillatory composition and time-frequency differences of the MT electrical oscillations have not been identified. Here, we applied the Empirical Mode Decomposition (EMD) to bovine brain MT sheet recordings to determine the number and fundamental frequencies of the Intrinsic Modes Functions (IMF) and evaluate their energetic contribution to the electrical signal. As previously reported, raw signals were obtained from cow brain MTs (Cantero et al. Sci Rep 6:27143, 2016), sampled, filtered, and subjected to signal decomposition from representative experiments. Filtered signals (200 Hz) allowed us to identify either six or seven IMFs. The reconstructed tracings faithfully resembled the original signals, with identifiable frequency peaks. To extend the analysis to obtain time-frequency information and the energy implicated in each IMF, we applied the Hilbert-Huang Transform (HHT) and the Continuous Wavelet Transform (CWT) to the same samples. The analyses disclosed the presence of more fundamental frequency peaks than initially reported and evidenced the advantages and disadvantages of each transform. The study indicates that the EMD is a robust approach to quantifying signal decomposition of brain MT oscillations and suggests novel similarities with human brain wave electroencephalogram (EEG) recordings. The evidence points to the potentially fundamental role of MT oscillations in brain electrical activity.

Coarse-Grained Simulation of Mechanical Properties of Single Microtubules With Micrometer Length

Microtubules are one of the most important components in the cytoskeleton and play a vital role in maintaining the shape and function of cells. Because single microtubules are some micrometers long, it is difficult to simulate such a large system using an all-atom model. In this work, we use the newly developed convolutional and K-means coarse-graining (CK-CG) method to establish an ultra-coarse-grained (UCG) model of a single microtubule, on the basis of the low electron microscopy density data of microtubules. We discuss the rationale of the micro-coarse-grained microtubule models of different resolutions and explore microtubule models up to 12-micron length. We use the devised microtubule model to quantify mechanical properties of microtubules of different lengths. Our model allows mesoscopic simulations of micrometer-level biomaterials and can be further used to study important biological processes related to microtubule function.

When Stiffness Matters: Mechanosensing in Heart Development and Disease

During embryonic morphogenesis, the heart undergoes a complex series of cellular phenotypic maturations (e.g., transition of myocytes from proliferative to quiescent or maturation of the contractile apparatus), and this involves stiffening of the extracellular matrix (ECM) acting in concert with morphogenetic signals. The maladaptive remodeling of the myocardium, one of the processes involved in determination of heart failure, also involves mechanical cues, with a progressive stiffening of the tissue that produces cellular mechanical damage, inflammation, and ultimately myocardial fibrosis. The assessment of the biomechanical dependence of the molecular machinery (in myocardial and non-myocardial cells) is therefore essential to contextualize the maturation of the cardiac tissue at early stages and understand its pathologic evolution in aging. Because systems to perform multiscale modeling of cellular and tissue mechanics have been developed, it appears particularly novel to design integrated mechano-molecular models of heart development and disease to be tested in ex vivo reconstituted cells/tissue-mimicking conditions. In the present contribution, we will discuss the latest implication of mechanosensing in heart development and pathology, describe the most recent models of cell/tissue mechanics, and delineate novel strategies to target the consequences of heart failure with personalized approaches based on tissue engineering and induced pluripotent stem cell (iPSC) technologies.

Microtubule Polymerization and Cross-Link Dynamics Explain Axonal Stiffness and Damage

Axonal damage is a critical indicator for traumatic effects of physical impact to the brain. However, the precise mechanisms of axonal damage are still unclear. Here, we establish a mechanistic and highly dynamic model of the axon to explore the evolution of damage in response to physical forces. Our axon model consists of a bundle of dynamically polymerizing and depolymerizing microtubules connected by dynamically detaching and reattaching cross-links. Although the probability of cross-link attachment depends exclusively on thermal fluctuations, the probability of detachment increases in the presence of physical forces. We systematically probe the landscape of axonal stretch and stretch rate and characterize the overall axonal force, stiffness, and damage as a direct result of the interplay between microtubule and cross-link dynamics. Our simulations reveal that slow loading is dominated by cross-link dynamics, a net reduction of cross-links, and a gradual accumulation of damage, whereas fast loading is dominated by cross-link deformations, a rapid increase in stretch, and an immediate risk of rupture. Microtubule polymerization and depolymerization decrease the overall axonal stiffness, but do not affect the evolution of damage at timescales relevant to axonal failure. Our study explains different failure mechanisms in the axon as emergent properties of microtubule polymerization, cross-link dynamics, and physical forces. We anticipate that our model will provide insight into causal relations by which molecular mechanisms determine the timeline and severity of axon damage after a physical impact to the brain.

Electromechanical vibration of microtubules and its application in biosensors

An electric field (EF) has the potential to excite the vibration of polarized microtubules (MTs) and thus enable their use as a biosensor for the biophysical properties of MTs or cells. To facilitate the development, this paper aims to capture the EF-induced vibration modes and the associated frequency for MTs. The analyses were carried out based on a molecular structural mechanics model accounting for the structural details of MTs. Transverse vibration, radial breathing vibration and axial vibration were achieved for MTs subject to a transverse or an axial EF. The frequency shift and stiffness alteration of MTs were also examined due to the possible changes of the tubulin interactions in physiological or pathological processes. The strong correlation achieved between the tubulin interaction and MT vibration excited by EF provides a new avenue to a non-contacting technique for the structural or property changes in MTs, where frequency shift is used as a biomarker. This technique can be used for individual MTs and is possible for those in cells when the cytosol damping on MT vibrations is largely reduced by the unique features of MT-cytosol interface.

Microtubules soften due to cross-sectional flattening

We use optical trapping to continuously bend an isolated microtubule while simultaneously measuring the applied force and the resulting filament strain, thus allowing us to determine its elastic properties over a wide range of applied strains. We find that, while in the low-strain regime, microtubules may be quantitatively described in terms of the classical Euler-Bernoulli elastic filament, above a critical strain they deviate from this simple elastic model, showing a softening response with increasingdeformations. A three-dimensional thin-shell model, in which the increased mechanical compliance is caused by flattening and eventual buckling of the filament cross-section, captures this softening effect in the high strain regime and yields quantitative values of the effective mechanical properties of microtubules. Our results demonstrate that properties of microtubules are highly dependent on the magnitude of the applied strain and offer a new interpretation for the large variety in microtubule mechanical data measured by different methods.

Looking "Under the Hood" of Cellular Mechanotransduction with Computational Tools: A Systems Biomechanics Approach across Multiple Scales

Signal modulation has been developed in living cells throughout evolution to promote utilizing the same machinery for multiple cellular functions. Chemical and mechanical modules of signal transmission and transduction are interconnected and necessary for organ development and growth. However, due to the high complexity of the intercommunication of physical intracellular connections with biochemical pathways, there are many missing details in our overall understanding of mechanotransduction processes, i.e., the process by which mechanical signals are converted to biochemical cascades. Cell-matrix adhesions are mechanically coupled to the nucleus through the cytoskeleton. This modulated and tightly integrated network mediates the transmission of mechanochemical signals from the extracellular matrix to the nucleus. Various experimental and computational techniques have been utilized to understand the basic mechanisms of mechanotransduction, yet many aspects have remained elusive. Recently, in silico experiments have made important contributions to the field of mechanobiology. Herein, computational modeling efforts devoted to understanding integrin-mediated mechanotransduction pathways are reviewed, and an outlook is presented for future directions toward using suitable computational approaches and developing novel techniques for addressing important questions in the field of mechanotransduction.

The axoneme, a biological template to design a swell energy recovery system

The axonemal micro-machinery, the axial skeleton and actuator of cilia and flagella of eukaryotic cells, is able to bend and twist and generates wave trains. We already demonstrated that it can be the template to construct an active trunk robot (Cibert 2013 Bioinspir. Biomim. 8 026006). The work presented in this paper describes how the axonemal model can also be useful to design worm shaped devices to convert swell energy into usable energy such as electricity. Using dynamic simulation we have designed three models either very close to the biological machinery or being a reasonable interpretation of its functioning principles. Their main advantage as compared with another existing device, Pelamis, is the additional twist degree-of-freedom that gives interesting properties to the system.

Electrical Oscillations in Two-Dimensional Microtubular Structures

Microtubules (MTs) are unique components of the cytoskeleton formed by hollow cylindrical structures of αβ tubulin dimeric units. The structural wall of the MT is interspersed by nanopores formed by the lateral arrangement of its subunits. MTs are also highly charged polar polyelectrolytes, capable of amplifying electrical signals. The actual nature of these electrodynamic capabilities remains largely unknown. Herein we applied the patch clamp technique to two-dimensional MT sheets, to characterize their electrical properties. Voltage-clamped MT sheets generated cation-selective oscillatory electrical currents whose magnitude depended on both the holding potential, and ionic strength and composition. The oscillations progressed through various modes including single and double periodic regimes and more complex behaviours, being prominent a fundamental frequency at 29 Hz. In physiological K(+) (140 mM), oscillations represented in average a 640% change in conductance that was also affected by the prevalent anion. Current injection induced voltage oscillations, thus showing excitability akin with action potentials. The electrical oscillations were entirely blocked by taxol, with pseudo Michaelis-Menten kinetics and a KD of ~1.29 μM. The findings suggest a functional role of the nanopores in the MT wall on the genesis of electrical oscillations that offer new insights into the nonlinear behaviour of the cytoskeleton.

Down-regulation of Stathmin Is Required for the Phenotypic Changes and Classical Activation of Macrophages

Macrophages are important cells of innate immunity with specialized capacity for recognition and elimination of pathogens and presentation of antigens to lymphocytes for adaptive immunity. Macrophages become activated upon exposure to pro-inflammatory cytokines and pathogenic stimuli. Classical activation of macrophages with interferon-γ (IFNγ) and lipopolysaccharide (LPS) triggers a wide range of signaling events and morphological changes to induce the immune response. Our previous microtubule (MT) proteomic work revealed that the stathmin association with MTs is considerably reduced in activated macrophages, which contain significantly more stabilized MTs. Here, we show that there is a global decrease in stathmin levels, an MT catastrophe protein, in activated macrophages using both immunoblotting and immunofluorescent microscopy. This is an LPS-specific response that induces proteasome-mediated degradation of stathmin. We explored the functions of stathmin down-regulation in activated macrophages by generating a stable cell line overexpressing stathmin-GFP. We show that stathmin-GFP overexpression impacts MT stability, impairs cell spreading, and reduces activation-associated phenotypes. Furthermore, overexpressing stathmin reduces complement receptor 3-mediated phagocytosis and cellular activation, implicating a pivotal inhibitory role for stathmin in classically activated macrophages.

A higher-order mathematical modeling for dynamic behavior of protein microtubule shell structures including shear deformation and small-scale effects

Microtubules in mammalian cells are cylindrical protein polymers which structurally and dynamically organize functional activities in living cells. They are important for maintaining cell structures, providing platforms for intracellular transport, and forming the spindle during mitosis, as well as other cellular processes. Various in vitro studies have shown that microtubules react to applied mechanical loading and physical environment. To investigate the mechanisms underlying such phenomena, a mathematical model based on the orthotropic higher-order shear deformation shell formulation and Hamilton's principle is presented in this paper for dynamic behavior of microtubules. The numerical results obtained by the proposed shell model are verified by the experimental data from the literature, showing great consistency. The nonlocal elasticity theory is also utilized to describe the nano-scale effects of the microtubule structure. The wave propagation and vibration characteristics of the microtubule are examined in the presence and absence of the cytosol employing proposed formulations. The effects of different system parameters such as length, small scale parameter, and cytosol viscosity on vibrational behavior of a microtubule are elucidated. The definitions of critical length and critical viscosity are introduced and the results obtained using the higher order shell model are compared with those obtained employing a first-order shear deformation theory. This comparison shows that the small scale effects become important for higher values of the wave vector and the proposed model gives more accurate results for both small and large values of wave vectors. Moreover, it is shown that for higher circumferential wave number, the torsional wave velocity obtained by the higher-order shell model tend to be higher than the one predicted by the first-order shell model.

Electro-acoustic behavior of the mitotic spindle: a semi-classical coarse-grained model

The regulation of chromosome separation during mitosis is not fully understood yet. Microtubules forming mitotic spindles are targets of treatment strategies which are aimed at (i) the triggering of the apoptosis or (ii) the interruption of uncontrolled cell division. Despite these facts, only few physical models relating to the dynamics of mitotic spindles exist up to now. In this paper, we present the first electromechanical model which enables calculation of the electromagnetic field coupled to acoustic vibrations of the mitotic spindle. This electromagnetic field originates from the electrical polarity of microtubules which form the mitotic spindle. The model is based on the approximation of resonantly vibrating microtubules by a network of oscillating electric dipoles. Our computational results predict the existence of a rapidly changing electric field which is generated by either driven or endogenous vibrations of the mitotic spindle. For certain values of parameters, the intensity of the electric field and its gradient reach values which may exert a not-inconsiderable force on chromosomes which are aligned in the spindle midzone. Our model may describe possible mechanisms of the effects of ultra-short electrical and mechanical pulses on dividing cells--a strategy used in novel methods for cancer treatment.

Analysis of wave propagation in orthotropic microtubules embedded within elastic medium by Pasternak model

Microtubules are embedded within elastic medium in living cells, where they perform a wide variety of functions; in cell motility and division, in organelle transport, and in cell organization. Waves propagate along microtubules in performing their physiological functions, so, wave propagation along microtubules has been the topic of research in the past decade. In the present article, the wave propagation in microtubules embedded in the elastic medium has been investigated on the basis of orthotropic-Pasternak model. We considered microtubules as orthotropic elastic shell and its surrounding elastic matrix as Pasternak foundation. We found that the flexural rigidity of microtubules has been increased with the stiffening of the elastic medium. Moreover, we observed that due to the mechanical coupling of microtubules with the elastic medium, their radial wave velocity has increased considerably as compared to other two wave velocities, i.e., longitudinal wave velocity and torsional wave velocity. The effect of foundation parameters H and G is more pronounced on radial wave velocity, to a lesser extent on torsional wave velocity and least even negligible on longitudinal wave velocity.

FINITE ELEMENT MODELING OF THE THERMAL FLUCTUATIONS OF A SINGLE ANISOTROPIC POLYMER

Thermal fluctuations of microtubules (MTs) and other cytoskeletal filaments govern to a great extent the complex rheological properties of the cytoskeleton in eukaryotic cells. In recent years, much effort has been put into capturing the dynamics of these fluctuations by means of analytical and numerical models. These attempts have been very successful for, but also remain limited to, isotropic polymers. To correctly interpret experimental work on (strongly) anisotropic semiflexible polymers, there is a need for a numerical modeling tool that accurately captures the dynamics of polymers with anisotropic material properties. In the current study, we present a finite element (FE) framework for simulating the thermal dynamics of a single anisotropic semiflexible polymer. First, we demonstrated the accuracy of our framework by comparison of the simulated mean square displacement (MSD) of the end-to-end distance with analytical predictions based on the worm-like chain model. Then, we implemented a transversely isotropic material model, characteristic for biopolymers such as MTs, and studied the persistence length for various ratios between the longitudinal shear modulus, G12, and corresponding Young's modulus, E1. Finally, we put our findings in context by addressing a recent experimental work on grafted transversely isotropic MTs. In that research, a simplified static mechanical model was used to deduce a very high level of MT anisotropy to explain the observation that the persistence length of grafted MTs increases as contour length increases. We showed, by means of our FE framework, that the anisotropic properties cannot account for the reported length-dependent persistence length.

Bending, twisting and beating trunk robot bioinspired from the '3 + 0' axoneme

The axoneme is the skeleton and motor axis of flagella and cilia in eukaryotic organisms. Basically it consists of a series of longitudinal fibers (outer doublets of microtubules) that design a cylinder and whose sliding, due to the coordinated activities of dedicated molecular motors (the dynein arms), is converted into a bending because outer doublets pairs are stabilized by elastic links (the nexine molecules). In spite of these interesting mechanical properties, mechanical and robotics engineers have never considered this amazing molecular machinery as a model. The aim of this paper is to propose the robotic design and the kinematic modeling of the '3 + 0' axoneme that makes motile the flagellum of Diplauxis hatti, the simplest that exists. The model that we propose bends and twists and combines the two movements. It is able to propagate wave trains that could be involved in the development of biomimetic actuators of various mechanisms such as (sub)aquatic robotic propellers as well as robotic trunks.

Microtubule circumferential vibrations in cytosol

Microtubules are key components of the cytoskeleton and perform a variety of functions, including chromosome movement during cell division, intracellular transport of materials, movement of organelles and intracellular tracking. A combination of essential and up-to-date methods is needed for investigating the biology of microtubules and understanding the mechanisms of microtubule-drug interaction. Coupled cytosol-microtubule mechanical vibrations of microtubules are studied in this article. Such investigations provide helpful insights on the functional mechanisms of microtubules and their interactions with other proteins and drugs. The viscous cytosol and the microtubule are coupled through the continuity condition across the microtubule-cytosol interface. The stress field in the cytosol induced by vibrating microtubule is analytically determined and the coupled circumferential vibrations of the cytosol-microtubule system are investigated by developing a coupled polynomial eigenvalue problem. Finally, the variations of vibration frequencies of a coupled system with cytosol dynamic viscosity, and microtubule circumferential Young's modulus are examined. Furthermore, the validity of the present analysis is confirmed by comparing the results with those obtained from the literature.

Coupled oscillations of a protein microtubule immersed in cytoplasm: an orthotropic elastic shell modeling

Revealing vibration characteristics of sub-cellular structural components such as membranes and microtubules has a principal role in obtaining a deeper understanding of their biological functions. Nevertheless, limitations and challenges in biological experiments at this scale necessitates the use of mathematical and computational models as an alternative solution. As one of the three major cytoskeletal filaments, microtubules are highly anisotropic structures built from tubulin heterodimers. They are hollow cylindrical shells with a ∼ 25 nm outer diameter and are tens of microns long. In this study, a mechanical model including the effects of the viscous cytosol and surrounding filaments is developed for predicting the coupled oscillations of a single microtubule immersed in cytoplasm. The first-order shear deformation shell theory for orthotropic materials is used to model the microtubule, whereas the motion of the cytosol is analyzed by considering the Stokes flow. The viscous cytosol and the microtubule are coupled through the continuity condition across the microtubule-cytosol interface. The stress and velocity fields in the cytosol induced by vibrating microtubule are analytically determined. Finally, the influences of the dynamic viscosity of the cytosol, filament network elasticity, microtubule shear modulus, and circumferential wave-number on longitudinal, radial, and torsional modes of microtubule vibration are elucidated.

Mechano-electrical vibrations of microtubules--link to subcellular morphology

Spontaneous mechanical oscillations were predicted and experimentally proven on almost every level of cellular structure. Besides morphogenetic potential of oscillatory mechanical force, oscillations may drive vibrations of electrically polar structures or these structures themselves may oscillate on their own natural frequencies. Vibrations of electric charge will generate oscillating electric field, role of which in morphogenesis is discussed in this paper. This idea is demonstrated in silico on the conformation of two growing microtubules.

Wave propagation in protein microtubules modeled as orthotropic elastic shells including transverse shear deformations

Wave propagation along the microtubules is one of the issues of major concern in various microtubule cellular functions. In this study, the general wave propagation behavior in protein microtubules is investigated based on a first-order shear deformation shell theory for orthotropic materials, with particular emphasis on the role of strongly anisotropic elastic properties of microtubules. According to experimental observation, the first-order shear deformation theory is used for the modeling of microtubule walls. A general displacement representation is introduced and a type of coupled polynomial eigenvalue problem is developed. Numerical examples describe the effects of shear deformation and rotary inertia on wave velocities in orthotropic microtubules. Finally, the influences of the microtubule shear modulus, axial external force, effective thickness and material temperature dependency on wave velocities along the microtubule protofilaments, helical pathway and radial directions are elucidated. Most results presented in the present investigation have been absent from the literature for the wave propagation in microtubules.

Anisotropic elastic network modeling of entire microtubules

Microtubules are supramolecular structures that make up the cytoskeleton and strongly affect the mechanical properties of the cell. Within the cytoskeleton filaments, the microtubule (MT) exhibits by far the highest bending stiffness. Bending stiffness depends on the mechanical properties and intermolecular interactions of the tubulin dimers (the MT building blocks). Computational molecular modeling has the potential for obtaining quantitative insights into this area. However, to our knowledge, standard molecular modeling techniques, such as molecular dynamics (MD) and normal mode analysis (NMA), are not yet able to simulate large molecular structures like the MTs; in fact, their possibilities are normally limited to much smaller protein complexes. In this work, we developed a multiscale approach by merging the modeling contribution from MD and NMA. In particular, MD simulations were used to refine the molecular conformation and arrangement of the tubulin dimers inside the MT lattice. Subsequently, NMA was used to investigate the vibrational properties of MTs modeled as an elastic network. The coarse-grain model here developed can describe systems of hundreds of interacting tubulin monomers (corresponding to up to 1,000,000 atoms). In particular, we were able to simulate coarse-grain models of entire MTs, with lengths up to 350 nm. A quantitative mechanical investigation was performed; from the bending and stretching modes, we estimated MT macroscopic properties such as bending stiffness, Young modulus, and persistence length, thus allowing a direct comparison with experimental data.

Prediction of bending stiffness and deformed shape of non-axially compressed microtubule by a semi-analytical approach

The bending stiffness of a microtubule is one of the most important parameters needed in the analysis of microtubule deformation. In this study, a semi-analytical approach is developed to predict the bending stiffness and deformed shape of a non-axially compressed microtubule in an explicit closed form. By using the solution presented in this paper and the experimentally observed values given in the literature, both the deformed configuration and bending stiffness of a single microtubule are determined. The proposed method is validated by comparing the obtained results with available data in the literature. The comparison shows that the present semi-analytical formulation provides the same accuracy with reduced numerical effort.

Torsional elastic deformations of microtubules within continuous sheet model

This paper develops a rigorous analysis of the microtubule elastic deformations in terms of the torsional degrees of freedom using the helix-based cylindrical structure of this biopolymer. Methods of differential geometry and the theory of elasticity are employed in our analysis. We find equilibrium conditions and constitutive equations in the linear regime. We estimate the value of torsional rigidity for microtubules based on their structure and some experimentally known elastic properties. The paper concludes with the derivation of a bulk modulus formula for a microtubule in solution. Both the entropy change and the fluctuation of the twist angle are obtained.

On the octagonal structure of the nuclear pore complex: insights from coarse-grained models

The basic structure of the nuclear pore complex (NPC), conserved across almost all organisms from yeast to humans, persists in featuring an octagonal symmetry involving the nucleoporins that constitute the NPC ring. In this article, we seek to understand and evaluate the potential biomechanical reasons for this eightfold symmetry. Our analytical investigation shows that the eightfold symmetry maximizes the bending stiffness of each of the eight NPC spokes while our computational analyses identify the most likely deformation modes, frequencies, and associated kinetic energies of the NPC. These modes have energies close to other published findings using membrane analysis of the nuclear membrane pore opening, and deformation states in agreement with experimental observations. A better understanding of NPC mechanics is essential for characterizing the nucleocytoplasmic transport, which has a central importance in cell biology.

Temperature-dependent elasticity of microtubules

Central to the biological function of microtubules is their ability to modify their length which occurs by addition and removal of subunits at the ends of the polymer, both in vivo and in vitro. This dynamic behavior is strongly influenced by temperature. Here, we show that the lateral interaction between tubulin subunits forming microtubule is strongly temperature dependent. Microtubules deposited on prefabricated substrates were deformed in an atomic force microscope during imaging, in two different experimental geometries. Microtubules were modeled as anisotropic, with the Young's modulus corresponding to the resistance of protofilaments to stretching and the shear modulus describing the weak interaction between the protofilaments. Measurements involving radial compression of microtubules deposited on flat mica confirm that microtubule elasticity depends on the temperature. Bending measurements performed on microtubules deposited on lithographically fabricated substrates show that this temperature dependence is due to changing shear modulus, implying that the lateral interaction between the protofilaments is strongly determined by the temperature. These measurements are in good agreement with previously reported measurements of the disassembly rate of microtubules, demonstrating that the mechanical and dynamic properties of microtubules are closely related.

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.

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.

Surface deflection of a microtubule loaded by a concentrated radial force

Microtubules are hollow cylindrical filaments of a eukaryotic cytoskeleton which are sensitive to externally applied radial forces due to their low circumferential elastic modulus. In this work, an orthotropic elastic shell model for microtubules is used to study the surface radial deflection of a microtubule loaded by a concentrated radial force generated by either a single molecular motor or a radial indentation tip. Our results show that the maximum surface radial deflection of a microtubule generated by a concentrated radial force of a few pN can be as large as a few nanometers (a significant fraction of the radius of microtubules), which could cause significant surface morphological non-uniformity of the microtubule. In contrast, radial indentation under a much larger compressive force, which can be as large as a few hundreds of pN, will cause hardening of the circumferential elastic modulus almost equal to the longitudinal modulus of microtubules. In this case, our results show that a microtubule can withstand a concentrated radial compressive force as large as a few hundreds of pN, with a maximum radial deflection not more than a few nanometers, in good agreement with recent experiments on radial indentation of microtubules. These results offer useful data and new insights into the basic understanding of elastic interaction between microtubules and molecular motors and radial indentation of microtubules.

Micromechanical properties of tobacco mosaic viruses

A tobacco mosaic virus (TMV) subject to local forces can be viewed as an uniform beam with local loads. We used a custom built Atomic Force Microscope (AFM) to determine the curvature induced in the TMV by concentrated load or by distributed forces. Local forces were created by the AFM tip. Distributed forces were applied to the virus via the surface tension of receding droplets. The experimental results of both methods can be described when we attribute a Young modulus of 6 +/- 3 GPa to the virus. Our value is about five times larger than published data. We compare our results to the literature and work out possible error sources in our experiment and in published one.

Orthotropic elastic shell model for buckling of microtubules

In view of the fact that microtubules exhibit strong anisotropic elastic properties, an orthotropic elastic shell model for microtubules is developed to study buckling behavior of microtubules. The predicted critical pressure is found to agree well with recent unexplained experimental data on pressure-induced buckling of microtubules [Needleman, Phys. Rev. Lett. 93, 198104 (2004); Biophys. J. 89, 3410 (2005)] which are lower than that predicted by the isotropic shell model by four orders of magnitude. General buckling behavior of microtubules under axial compression or radial pressure is studied. The results show that the isotropic shell model greatly overestimates the bucking loads of microtubules, except columnlike axially compressed buckling of long microtubules (of length-to-diameter ratio larger than, say, 150). In particular, the present results also offer a plausible explanation for the length dependency of flexibility of microtubules reported in the literature.

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.

Thermal fluctuations of grafted microtubules provide evidence of a length-dependent persistence length

Microtubules are hollow cylindrical structures that constitute one of the three major classes of cytoskeletal filaments. On the mesoscopic length scale of a cell, their material properties are characterized by a single stiffness parameter, the persistence length l(p). Its value, in general, depends on the microscopic interactions between the constituent tubulin dimers and the architecture of the microtubule. Here, we use single-particle tracking methods combined with a fluctuation analysis to systematically study the dependence of l(p) on the total filament length L. Microtubules are grafted to a substrate with one end free to fluctuate in three dimensions. A fluorescent bead is attached proximally to the free tip and is used to record the thermal fluctuations of the microtubule's end. The position distribution functions obtained with this assay allow the precise measurement of l(p) for microtubules of different contour length L. Upon varying L between 2.6 and 47.5 mum, we find a systematic increase of l(p) from 110 to 5,035 mum. At the same time we verify that, for a given filament length, the persistence length is constant over the filament within the experimental accuracy. We interpret this length dependence as a consequence of a nonnegligible shear deflection determined by subnanometer relative displacement of adjacent protofilaments. Our results may shine new light on the function of microtubules as sophisticated nanometer-sized molecular machines and give a unified explanation of seemingly uncorrelated spreading of microtubules' stiffness previously reported in literature.