Anisotropic elastic properties of microtubules

Microstructure-based modeling of primary cilia mechanics

A primary cilium, made of nine microtubule doublets enclosed in a cilium membrane, is a mechanosensing organelle that bends under an external mechanical load and sends an intracellular signal through transmembrane proteins activated by cilium bending. The nine microtubule doublets are the main load-bearing structural component, while the transmembrane proteins on the cilium membrane are the main sensing component. No distinction was made between these two components in all existing models, where the stress calculated from the structural component (nine microtubule doublets) was used to explain the sensing location, which may be totally misleading. For the first time, we developed a microstructure-based primary cilium model by considering these two components separately. First, we refined the analytical solution of bending an orthotropic cylindrical shell for individual microtubule, and obtained excellent agreement between finite element simulations and the theoretical predictions of a microtubule bending as a validation of the structural component in the model. Second, by integrating the cilium membrane with nine microtubule doublets and simulating the tip-anchored optical tweezer experiment on our computational model, we found that the microtubule doublets may twist significantly as the whole cilium bends. Third, besides being cilium-length-dependent, we found the mechanical properties of the cilium are also highly deformation-dependent. More important, we found that the cilium membrane near the base is not under pure in-plane tension or compression as previously thought, but has significant local bending stress. This challenges the traditional model of cilium mechanosensing, indicating that transmembrane proteins may be activated more by membrane curvature than membrane stretching. Finally, we incorporated imaging data of primary cilia into our microstructure-based cilium model, and found that comparing to the ideal model with uniform microtubule length, the imaging-informed model shows the nine microtubule doublets interact more evenly with the cilium membrane, and their contact locations can cause even higher bending curvature in the cilium membrane than near the base.

Mechanical communication within the microtubule through network-based analysis of tubulin dynamics

The identification of the mechanisms underlying the transfer of mechanical vibrations in protein complexes is crucial to understand how these super-assemblies are stabilized to perform specific functions within the cell. In this context, the study of the structural communication and the propagation of mechanical stimuli within the microtubule (MT) is important given the pivotal role of the latter in cell viability. In this study, we employed molecular modelling and the dynamical network analysis approaches to analyse the MT. The results highlight that β -tubulin drives the transfer of mechanical information between protofilaments (PFs), which is altered at the seam due to a different interaction pattern. Moreover, while the key residues involved in the structural communication along the PF are generally conserved, a higher diversity was observed for amino acids mediating the lateral communication. Taken together, these results might explain why MTs with different PF numbers are formed in different organisms or with different β -tubulin isotypes.

Microtubules as a signal hub for axon growth in response to mechanical force

Microtubules are highly polar structures and are characterized by high anisotropy and stiffness. In neurons, they play a key role in the directional transport of vesicles and organelles. In the neuronal projections called axons, they form parallel bundles, mostly oriented with the plus-end towards the axonal termination. Their physico-chemical properties have recently attracted attention as a potential candidate in sensing, processing and transducing physical signals generated by mechanical forces. Here, we discuss the main evidence supporting the role of microtubules as a signal hub for axon growth in response to a traction force. Applying a tension to the axon appears to stabilize the microtubules, which, in turn, coordinate a modulation of axonal transport, local translation and their cross-talk. We speculate on the possible mechanisms modulating microtubule dynamics under tension, based on evidence collected in neuronal and non-neuronal cell types. However, the fundamental question of the causal relationship between these mechanisms is still elusive because the mechano-sensitive element in this chain has not yet been identified.

Mechanical fatigue testing in silico: Dynamic evolution of material properties of nanoscale biological particles

Biological particles have evolved to possess mechanical characteristics necessary to carry out their functions. We developed a computational approach to "fatigue testing in silico", in which constant-amplitude cyclic loading is applied to a particle to explore its mechanobiology. We used this approach to describe dynamic evolution of nanomaterial properties and low-cycle fatigue in the thin spherical encapsulin shell, thick spherical Cowpea Chlorotic Mottle Virus (CCMV) capsid, and thick cylindrical microtubule (MT) fragment over 20 cycles of deformation. Changing structures and force-deformation curves enabled us to describe their damage-dependent biomechanics (strength, deformability, stiffness), thermodynamics (released and dissipated energies, enthalpy, and entropy) and material properties (toughness). Thick CCMV and MT particles experience material fatigue due to slow recovery and damage accumulation over 3-5 loading cycles; thin encapsulin shells show little fatigue due to rapid remodeling and limited damage. The results obtained challenge the existing paradigm: damage in biological particles is partially reversible owing to particle's partial recovery; fatigue crack may or may not grow with each loading cycle and may heal; and particles adapt to deformation amplitude and frequency to minimize the energy dissipated. Using crack size to quantitate damage is problematic as several cracks might form simultaneously in a particle. Dynamic evolution of strength, deformability, and stiffness, can be predicted by analyzing the cycle number (N) dependent damage, [Formula: see text] , where α is a power law and Nf is fatigue life. Fatigue testing in silico can now be used to explore damage-induced changes in the material properties of other biological particles. STATEMENT OF SIGNIFICANCE: Biological particles possess mechanical characteristics necessary to perform their functions. We developed "fatigue testing in silico" approach, which employes Langevin Dynamics simulations of constant-amplitude cyclic loading of nanoscale biological particles, to explore dynamic evolution of the mechanical, energetic, and material properties of the thin and thick spherical particles of encapsulin and Cowpea Chlorotic Mottle Virus, and the microtubule filament fragment. Our study of damage growth and fatigue development challenge the existing paradigm. Damage in biological particles is partially reversible as fatigue crack might heal with each loading cycle. Particles adapt to deformation amplitude and frequency to minimize energy dissipation. The evolution of strength, deformability, and stiffness, can be accurately predicted by analyzing the damage growth in particle structure.

Microstructure-Based Modeling of Primary Cilia Mechanics

A primary cilium, made of nine microtubule doublets enclosed in a cilium membrane, is a mechanosensing organelle that bends under an external mechanical load and sends an intracellular signal through transmembrane proteins activated by cilium bending. The nine microtubule doublets are the main load-bearing structural component, while the transmembrane proteins on the cilium membrane are the main sensing component. No distinction was made between these two components in all existing models, where the stress calculated from the structural component (nine microtubule doublets) was used to explain the sensing location, which may be totally misleading. For the first time, we developed a microstructure-based primary cilium model by considering these two components separately. First, we refined the analytical solution of bending an orthotropic cylindrical shell for individual microtubule, and obtained excellent agreement between finite element simulations and the theoretical predictions of a microtubule bending as a validation of the structural component in the model. Second, by integrating the cilium membrane with nine microtubule doublets, we found that the microtubule doublets may twist significantly as the whole cilium bends. Third, besides being cilium-length-dependent, we found the mechanical properties of the cilium are also highly deformation-dependent. More important, we found that the cilium membrane near the base is not under pure in-plane tension or compression as previously thought, but has significant local bending stress. This challenges the traditional model of cilium mechanosensing, indicating that transmembrane proteins may be activated more by membrane curvature than membrane stretching. Finally, we incorporated imaging data of primary cilia into our microstructure-based cilium model, and found that comparing to the ideal model with uniform microtubule length, the imaging-informed model shows the nine microtubule doublets interact more evenly with the cilium membrane, and their contact locations can cause even higher bending curvature in the cilium membrane than near the base.

SIGNIFICANCE

Factors regulating the mechanical response of a primary cilium to fluid flow remain unclear. Modeling the microtubule doublet as a composite of two orthotropic shells and the ciliary axoneme as an elastic shell enclosing nine such microtubule doublets, we found that the length distribution of microtubule doublets (inferred from cryogenic electron tomography images) is the primary determining factor in the bending stiffness of primary cilia, rather than just the ciliary length. This implies ciliary-associated transmembrane proteins may be activated by membrane curvature changes rather than just membrane stretching. These insights challenge the traditional view of ciliary mechanosensation and expands our understanding of the different ways in which cells perceive and respond to mechanical stimuli.

Strain stiffening of Ndc80 complexes attached to microtubule plus ends

In the mitotic spindle, microtubules attach to chromosomes via kinetochores. The microtubule-binding Ndc80 complex is an integral part of kinetochores, and is essential for kinetochores to attach to microtubules and to transmit forces from dynamic microtubule ends to the chromosomes. The Ndc80 complex has a rod-like appearance with globular domains at its ends that are separated by a long coiled coil. Its mechanical properties are considered important for the dynamic interaction between kinetochores and microtubules. Here, we present a novel method that allows us to time trace the effective stiffness of Ndc80 complexes following shortening microtubule ends against applied force in optical trap experiments. Applying this method to wild-type Ndc80 and three variants (calponin homology (CH) domains mutated or Hec1 tail unphosphorylated, phosphorylated, or truncated), we reveal that each variant exhibits strain stiffening; i.e., the effective stiffness increases under tension that is built up by a depolymerizing microtubule. The strain stiffening relation is roughly linear and independent of the state of the microtubule. We introduce structure-based models that show that the strain stiffening can be traced back to the specific architecture of the Ndc80 complex with a characteristic flexible kink, to thermal fluctuations of the microtubule, and to the bending elasticity of flaring protofilaments, which exert force to move the Ndc80 complexes. Our model accounts for changes in the amount of load-bearing attachments at various force levels and reproduces the roughly linear strain stiffening behavior, highlighting the importance of force-dependent binding affinity.

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.

Shape multistability in flexible tubular crystals through interactions of mobile dislocations

Crystalline sheets rolled up into cylinders occur in diverse biological and synthetic systems, including carbon nanotubes, biofilaments of the cellular cytoskeleton, and packings of colloidal particles. In this work, we show, computationally, that such tubular crystals can be programmed with reconfigurable shapes, due to motions of defects that interrupt the periodicity of the crystalline lattice. By identifying and exploiting stable patterns of these defects, we cause tubular crystals to relax into desired target geometries, a design principle that could guide the creation of versatile colloidal analogues to nanotubes. Our results suggest routes to tunable and switchable material properties in ordered, soft materials on deformable surfaces.

We study avenues to shape multistability and shape morphing in flexible crystalline membranes of cylindrical topology, enabled by glide mobility of dislocations. Using computational modeling, we obtain states of mechanical equilibrium presenting a wide variety of tubular crystal deformation geometries, due to an interplay of effective defect interactions with out-of-tangent-plane deformations that reorient the tube axis. Importantly, this interplay often stabilizes defect configurations quite distinct from those predicted for a two-dimensional crystal confined to the surface of a rigid cylinder. We find that relative and absolute stability of competing states depend strongly on control parameters such as bending rigidity, applied stress, and spontaneous curvature. Using stable dislocation pair arrangements as building blocks, we demonstrate that targeted macroscopic three-dimensional conformations of thin crystalline tubes can be programmed by imposing certain sparse patterns of defects. Our findings reveal a broad design space for controllable and reconfigurable colloidal tube geometries, with potential relevance also to architected carbon nanotubes and microtubules.

Mechanical Properties of Isolated Primary Cilia Measured by Micro-tensile Test and Atomic Force Microscopy

Mechanotransduction is a well-known mechanism by which cells sense their surrounding mechanical environment, convert mechanical stimuli into biochemical signals, and eventually change their morphology and functions. Primary cilia are believed to be mechanosensors existing on the surface of the cell membrane and support cells to sense surrounding mechanical signals. Knowing the mechanical properties of primary cilia is essential to understand their responses, such as sensitivity to mechanical stimuli. Previous studies have so far conducted flow experiments or optical trap techniques to measure the flexural rigidity EI (E: Young’s modulus, I: second moment of inertia) of primary cilia; however, the flexural rigidity is not a material property of materials and depends on mathematical models used in the determination, leading to a discrepancy between studies. For better characterization of primary cilia mechanics, Young’s modulus should be directly and precisely measured. In this study, the tensile Young’s modulus of isolated primary cilia is, for the first time, measured by using an in-house micro-tensile tester. The different strain rates of 0.01–0.3 s⁻¹ were applied to isolated primary cilia, which showed a strain rate–dependent Young’s modulus in the range of 69.5–240.0 kPa on average. Atomic force microscopy was also performed to measure the local Young’s modulus of primary cilia, showing the Young’s modulus within the order of tens to hundreds of kPa. This study could directly provide the global and local Young’s moduli, which will benefit better understanding of primary cilia mechanics.

MODELING OF FLAGELLUM BEHAVIOR AND TWO-DIMENSIONAL SPERM CELL MOTILITY WITHIN THE CONTEXT OF FLUID–SOLID INTERACTIONS

In this study, the flagellar motility of a swimmer microorganism as a model of a human sperm cell, inside a two-dimensional channel as a model of the female reproductive tract containing a viscous fluid, is numerically investigated. The Navier–Stokes equations governing the fluid are coupled with the equations governing the models flagellum via applying a fluid–solid interaction approach and then solved using the finite element method. To stimulate the flagellum to move, a prescribed sinusoidal waveform is applied to it. The strain induced by this waveform along the flagellum initiates a continuous interaction between the flagellum and the fluid. The simulations are validated using data available in the literature. A very good agreement is seen between them. The results show that by decreasing the Young modulus of the flagellum as well as increasing the fluid viscosity, the swimming velocity of the model significantly decreases. It is found that for lower Young modulus of the flagellum, the effect of the fluid viscosity on the flagellar deformation is stronger. It is also found that for higher amplitude of the waveform applied to stimulate the flagellum, both the swimming velocity of the model and the average work rate are greater. Moreover, it is found that in a channel with a smaller height, the model swims at a higher speed and with a higher average work rate.

Primary cilia have a length-dependent persistence length

The fluctuating position of an optically trapped cilium tip under untreated and Taxol-treated conditions was used to characterize mechanical properties of the cilium axoneme and its basal body by combining experimental, analytical, and computational tools. We provide, for the first time, evidence that the persistence length of a ciliary axoneme is length-dependent; longer cilia are stiffer than shorter cilia. We demonstrate that this apparent length dependence can be understood by a combination of modeling axonemal microtubules as anisotropic elastic shells and including actomyosin-driven stochastic basal body motion. Our results also demonstrate the possibility of using observable ciliary dynamics to probe interior cytoskeletal dynamics. It is hoped that our improved characterization of cilia will result in deeper understanding of the biological function of cellular flow sensing by this organelle.

Microtubule self-healing and defect creation investigated by in-line force measurements during high-speed atomic force microscopy imaging

Microtubules are biopolymers composed of tubulin and play diverse roles in a wide variety of biological processes such as cell division, migration and intracellular transport in eukaryotic cells. To perform their functions, microtubules are mechanically stressed and, thereby, susceptible to structural defects. Local variations in mechanical properties caused by these defects modulate their biological functions, including binding and transportation of microtubule-associated proteins. Therefore, assessing the local mechanical properties of microtubules and analyzing their dynamic response to mechanical stimuli provide insight into fundamental processes. It is, however, not trivial to control defect formation, gather mechanical information at the same time, and subsequently image the result at a high temporal resolution at the molecular level with minimal delay. In this work, we describe the so-called in-line force curve mode based on high-speed atomic force microscopy. This method is directly applied to create defects in microtubules at the level of tubulin dimers and monitor the following dynamic processes around the defects. Furthermore, force curves obtained during defect formation provide quantitative mechanical information to estimate the bonding energy between tubulin dimers.

Building Blocks of the Outer Membrane: Calculating a General Elastic Energy Model for β-Barrel Membrane Proteins

The outer membranes of Gram negative bacteria are the first points of contact these organisms make with their environment. Understanding how composition determines the mechanical properties of this essential barrier is of paramount importance. Therefore, we developed a new computational method to measure the elasticity of transmembrane proteins found in the outer membrane. Using all-atom molecular dynamics simulations of these proteins, we apply a set of external forces to mechanically stress the transmembrane β-barrels. Our results from four representative β-barrels show that outer membrane proteins display elastic properties that are approximately 70 to 190 times stiffer than neat lipid membranes. These findings suggest that outer membrane β-barrels are a significant source of mechanical stability in bacteria. Our all-atom approach further reveals that resistance to radial stress is encoded by a general mechanism that includes stretching of backbone hydrogen bonds and tilting of β-strands with respect to the bilayer normal. This computational framework facilitates an increased theoretical understanding of how varying lipid and protein amounts affect the mechanical properties of the bacterial outer membrane.

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.

Dynamics of microtubules: highlights of recent computational and experimental investigations

Microtubules are found in most eukaryotic cells, with homologs in eubacteria and archea, and they have functional roles in mitosis, cell motility, intracellular transport, and the maintenance of cell shape. Numerous efforts have been expended over the last two decades to characterize the interactions between microtubules and the wide variety of microtubule associated proteins that control their dynamic behavior in cells resulting in microtubules being assembled and disassembled where and when they are required by the cell. We present the main findings regarding microtubule polymerization and depolymerization and review recent work about the molecular motors that modulate microtubule dynamics by inducing either microtubule depolymerization or severing. We also discuss the main experimental and computational approaches used to quantify the thermodynamics and mechanics of microtubule filaments.

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

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

Optomechanical proposal for monitoring microtubule mechanical vibrations

Microtubules provide the mechanical force required for chromosome separation during mitosis. However, little is known about the dynamic (high-frequency) mechanical properties of microtubules. Here, we theoretically propose to control the vibrations of a doubly clamped microtubule by tip electrodes and to detect its motion via the optomechanical coupling between the vibrational modes of the microtubule and an optical cavity. In the presence of a red-detuned strong pump laser, this coupling leads to optomechanical-induced transparency of an optical probe field, which can be detected with state-of-the art technology. The center frequency and line width of the transparency peak give the resonance frequency and damping rate of the microtubule, respectively, while the height of the peak reveals information about the microtubule-cavity field coupling. Our method opens the new possibilities to gain information about the physical properties of microtubules, which will enhance our capability to design physical cancer treatment protocols as alternatives to chemotherapeutic drugs.

Single microtubules and small networks become significantly stiffer on short time-scales upon mechanical stimulation

The transfer of mechanical signals through cells is a complex phenomenon. To uncover a new mechanotransduction pathway, we study the frequency-dependent transport of mechanical stimuli by single microtubules and small networks in a bottom-up approach using optically trapped beads as anchor points. We interconnected microtubules to linear and triangular geometries to perform micro-rheology by defined oscillations of the beads relative to each other. We found a substantial stiffening of single filaments above a characteristic transition frequency of 1–30 Hz depending on the filament’s molecular composition. Below this frequency, filament elasticity only depends on its contour and persistence length. Interestingly, this elastic behavior is transferable to small networks, where we found the surprising effect that linear two filament connections act as transistor-like, angle dependent momentum filters, whereas triangular networks act as stabilizing elements. These observations implicate that cells can tune mechanical signals by temporal and spatial filtering stronger and more flexibly than expected.

Deformation pattern in vibrating microtubule: Structural mechanics study based on an atomistic approach

The mechanical properties of microtubules are of great importance for understanding their biological function and for applications in artificial devices. Although microtubule mechanics has been extensively studied both theoretically and experimentally, the relation to its molecular structure is understood only partially. Here, we report on the structural analysis of microtubule vibration modes calculated by an atomistic approach. Molecular dynamics was applied to refine the atomic structure of a microtubule and a C_α elastic network model was analyzed for its normal modes. We mapped fluctuations and local deformations up to the level of individual aminoacid residues. The deformation is mode-shape dependent and principally different in α-tubulins and β-tubulins. Parts of the tubulin dimer sequence responding specifically to longitudinal and radial stress are identified. We show that substantial strain within a microtubule is located both in the regions of contact between adjacent dimers and in the body of tubulins. Our results provide supportive evidence for the generally accepted assumption that the mechanics of microtubules, including its anisotropy, is determined by the bonds between tubulins.

Tubulin acetylation protects long-lived microtubules against mechanical ageing

Long-lived microtubules endow the eukaryotic cell with long-range transport abilities. While long-lived microtubules are acetylated on Lys40 of α-tubulin (αK40), acetylation takes place after stabilization and does not protect against depolymerization. Instead, αK40 acetylation has been proposed to mechanically stabilize microtubules. Yet how modification of αK40, a residue exposed to the microtubule lumen and inaccessible to microtubule-associated proteins and motors, could affect microtubule mechanics remains an open question. Here we develop FRET-based assays that report on the lateral interactions between protofilaments and find that αK40 acetylation directly weakens inter-protofilament interactions. Congruently, αK40 acetylation affects two processes largely governed by inter-protofilament interactions, reducing the nucleation frequency and accelerating the shrinkage rate. Most relevant to the biological function of acetylation, microfluidics manipulations demonstrate that αK40 acetylation enhances flexibility and confers resilience against repeated mechanical stresses. Thus, unlike deacetylated microtubules that accumulate damage when subjected to repeated stresses, long-lived microtubules are protected from mechanical ageing through their acquisition of αK40 acetylation. In contrast to other tubulin post-translational modifications that act through microtubule-associated proteins, motors and severing enzymes, intraluminal acetylation directly tunes the compliance and resilience of microtubules.

Feeling force: physical and physiological principles enabling sensory mechanotransduction

Organisms as diverse as microbes, roundworms, insects, and mammals detect and respond to applied force. In animals, this ability depends on ionotropic force receptors, known as mechanoelectrical transduction (MeT) channels, that are expressed by specialized mechanoreceptor cells embedded in diverse tissues and distributed throughout the body. These cells mediate hearing, touch, and proprioception and play a crucial role in regulating organ function. Here, we attempt to integrate knowledge about the architecture of mechanoreceptor cells and their sensory organs with principles of cell mechanics, and we consider how engulfing tissues contribute to mechanical filtering. We address progress in the quest to identify the proteins that form MeT channels and to understand how these channels are gated. For clarity and convenience, we focus on sensory mechanobiology in nematodes, fruit flies, and mice. These themes are emphasized: asymmetric responses to applied forces, which may reflect anisotropy of the structure and mechanics of sensory mechanoreceptor cells, and proteins that function as MeT channels, which appear to have emerged many times through evolution.

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.

Structure and function of the elastic organ in the tibia of a tenebrionid beetle

Many insects have a pair of claws on the tip of each foot (tarsus and pretarsus). The movement of the pretarsal claws is mediated by a long apodeme that originates from the claw retractor muscles in the femur. It is generally accepted that the pulling of the apodeme by the muscles flexes the claws to engage with a rough surface of a substrate, and the flexed claws return to their initial position by passive elastic forces within the tarso-pretarsal joint. We found that each tibia of the tenebrionid beetle Zophobas atratus had a chordal elastic organ that tied the apodeme to the distal end of the tibia and assisted the pulled apodeme to return smoothly. The elastic body of the elastic organ consists of a bundle of more than 1000 thin fibrils (0.3-1.5 μm in diameter) with a hairy yarn-shaped structure made by assemblies of intricately interwoven microfibers. Both ends of the fibrillar elastic body were supported by clusters of columnar cells. Ablation of the elastic organ often disturbed the rapid and smooth return of claws from a flexed position when the tarsal segments were forced to curve in order to increase the friction between the apodeme and surrounding tissues in the segments. The result suggests that rapid claw disengagement is an important step in each cycle of leg movements, and the elastic organ may have evolved to assist the reliable detachment of claws that engage tightly with the substrate when climbing or traversing inverted surfaces.

Mechanical aspects of microtubule bundling in taxane-treated circulating tumor cells

Microtubules play an important role in many cellular processes, including mitotic spindle formation and cell division. Taxane-based anticancer treatments lead to the stabilization of microtubules, thus preventing the uncontrolled proliferation of tumor cells. One of the striking physical features of taxane-treated cells is the localization of their microtubules, which can be observed via fluorescent microscopy as an intense fluorescent band and are referred to as a microtubule bundle. With the recent advances in capturing and analyzing tumor cells circulating in a patient's blood system, there is increasing interest in using these cells to examine a patient's response to treatment. This includes taxanes that are used routinely in clinics to treat prostate, breast, lung, and other cancers. Here, we have used a computational model of microtubule mechanics to investigate self-arrangement patterns of stabilized microtubules, which allowed for the identification of specific combinations of three physical parameters: microtubule stiffness, intracellular viscosity, and cell shape, that can prevent the formation of microtubule bundles in cells with stabilized microtubules, such as taxane-treated cells. We also developed a method to quantify bundling in the whole microtubule aster structure and a way to compare the simulated results to fluorescent images from experimental data. Moreover, we investigated microtubule rearrangement in both suspended and attached cells and showed that the observed final microtubule patterns depend on the experimental protocol. The results from our computational studies can explain the heterogeneous bundling phenomena observed via fluorescent immunostaining from a mechanical point of view without relying on heterogeneous cellular responses to the microtubule-stabilizing drug.

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.

Mechanotransduction of ultrasound is frequency dependent below the cavitation threshold

This study provides evidence that low-intensity ultrasound directly affects nuclear processes, and the magnitude of the effect varies with frequency. In particular, we show that the transcriptional induction of first load-inducible genes, which is independent of new protein synthesis, is frequency dependent. Bovine chondrocytes were exposed to low-intensity (below the cavitational threshold) ultrasound at 2, 5 and 8 MHz. Ultrasound elevated the expression of early response genes c-Fos, c-Jun and c-Myc, maximized at 5 MHz. The phosphorylated ERK inhibitor PD98059 abrogated any increase in c-series gene expression, suggesting that signaling occurs via the MAPPK/ERK pathway. However, phosphorylated ERK levels did not change with ultrasound frequency, indicating that processes downstream of ERK phosphorylation (such as nuclear transport and chromatin reorganization) respond to ultrasound with frequency dependence. A quantitative, biphasic mathematical model based on Biot theory predicted that cytoplasmic and nuclear stress is maximized at 5.2 ± 0.8 MHz for a chondrocyte, confirming experimental measurements.

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.

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.

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.

Coarse-grained mechanochemical model for simulating the dynamic behavior of microtubules

Modeling the structure and mechanics of microtubules, which play significant roles in various physiological functions of cells, has long been a fascinating issue. In this paper, a coarse-grained mechanochemical model is presented to study the mechanical-chemical coupling and dynamic attributes of microtubules. The interactions among tubulins are taken into account from the molecular basis. This model is used to characterize the conformations of sheet-ended microtubules, to analyze the distributions of interaction energy, and further to simulate the radial indentation process of a microtubule. This method also works for investigating the dynamic properties of microtubules, e.g., their assembly, growth, deformation, and structural evolution for different conditions. This study is helpful for understanding the structure-mechanics-function relationship of microtubules and lays a foundation for further investigation of their dynamic behavior.

Spindle pole mechanics studied in mitotic asters: dynamic distribution of spindle forces through compliant linkages

During cell division, chromosomes must faithfully segregate to maintain genome integrity, and this dynamic mechanical process is driven by the macromolecular machinery of the mitotic spindle. However, little is known about spindle mechanics. For example, spindle microtubules are organized by numerous cross-linking proteins yet the mechanical properties of those cross-links remain unexplored. To examine the mechanical properties of microtubule cross-links we applied optical trapping to mitotic asters that form in mammalian mitotic extracts. These asters are foci of microtubules, motors, and microtubule-associated proteins that reflect many of the functional properties of spindle poles and represent centrosome-independent spindle-pole analogs. We observed bidirectional motor-driven microtubule movements, showing that microtubule linkages within asters are remarkably compliant (mean stiffness 0.025 pN/nm) and mediated by only a handful of cross-links. Depleting the motor Eg5 reduced this stiffness, indicating that Eg5 contributes to the mechanical properties of microtubule asters in a manner consistent with its localization to spindle poles in cells. We propose that compliant linkages among microtubules provide a mechanical architecture capable of accommodating microtubule movements and distributing force among microtubules without loss of pole integrity-a mechanical paradigm that may be important throughout the spindle.

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.

Cancer physics: diagnostics based on damped cellular elastoelectrical vibrations in microtubules

This paper describes a proposed biophysical mechanism of a novel diagnostic method for cancer detection developed recently by Vedruccio. The diagnostic method is based on frequency selective absorption of electromagnetic waves by malignant tumors. Cancer is connected with mitochondrial malfunction (the Warburg effect) suggesting disrupted physical mechanisms. In addition to decreased energy conversion and nonutilized energy efflux, mitochondrial malfunction is accompanied by other negative effects in the cell. Diminished proton space charge layer and the static electric field around the outer membrane result in a lowered ordering level of cellular water and increased damping of microtubule-based cellular elastoelectrical vibration states. These changes manifest themselves in a dip in the amplitude of the signal with the fundamental frequency of the nonlinear microwave oscillator-the core of the diagnostic device-when coupled to the investigated cancerous tissue via the near-field. The dip is not present in the case of healthy tissue.

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.

Complexity of the tensegrity structure for dynamic energy and force distribution of cytoskeleton during cell spreading

Cytoskeleton plays important roles in intracellular force equilibrium and extracellular force transmission from/to attaching substrate through focal adhesions (FAs). Numerical simulations of intracellular force distribution to describe dynamic cell behaviors are still limited. The tensegrity structure comprises tension-supporting cables and compression-supporting struts that represent the actin filament and microtubule respectively, and has many features consistent with living cells. To simulate the dynamics of intracellular force distribution and total stored energy during cell spreading, the present study employed different complexities of the tensegrity structures by using octahedron tensegrity (OT) and cuboctahedron tensegrity (COT). The spreading was simulated by assigning specific connection nodes for radial displacement and attachment to substrate to form FAs. The traction force on each FA was estimated by summarizing the force carried in sounding cytoskeletal elements. The OT structure consisted of 24 cables and 6 struts and had limitations soon after the beginning of spreading by declining energy stored in struts indicating the abolishment of compression in microtubules. The COT structure, double the amount of cables and struts than the OT structure, provided sufficient spreading area and expressed similar features with documented cell behaviors. The traction force pointed inward on peripheral FAs in the spread out COT structure. The complex structure in COT provided further investigation of various FA number during different spreading stages. Before the middle phase of spreading (half of maximum spreading area), cell attachment with 8 FAs obtained minimized cytoskeletal energy. The maximum number of 12 FAs in the COT structure was required to achieve further spreading. The stored energy in actin filaments increased as cells spread out, while the energy stored in microtubules increased at initial spreading, peaked in middle phase, and then declined as cells reached maximum spreading. The dynamic flows of energy in struts imply that microtubules contribute to structure stabilization.

Structural basis of interprotofilament interaction and lateral deformation of microtubules

The diverse functions of microtubules require stiff structures possessing sufficient lateral flexibility to enable bending with high curvature. We used cryo-electron microscopy to investigate the molecular basis for these critical mechanical properties. High-quality structural maps were used to build pseudoatomic models of microtubules containing 11-16 protofilaments, representing a wide range of lateral curvature. Protofilaments in all these microtubules were connected primarily via interprotofilament interactions between the M loops, and the H1'-S2 and H2-S3 loops. We postulate that the tolerance of the loop-loop interactions to lateral deformation provides the capacity for high-curvature bending without breaking. On the other hand, the local molecular architecture that surrounds these connecting loops contributes to the overall rigidity. Interprotofilament interactions in the seam region are similar to those in the normal helical regions, suggesting that the existence of the seam does not significantly affect the mechanical properties of microtubules.

Mechanical properties of a complete microtubule revealed through molecular dynamics simulation

Microtubules (MTs) are the largest type of cellular filament, essential in processes ranging from mitosis and meiosis to flagellar motility. Many of the processes depend critically on the mechanical properties of the MT, but the elastic moduli, notably the Young's modulus, are not directly revealed in experiment, which instead measures either flexural rigidity or response to radial deformation. Molecular dynamics (MD) is a method that allows the mechanical properties of single biomolecules to be investigated through computation. Typically, MD requires an atomic resolution structure of the molecule, which is unavailable for many systems, including MTs. By combining structural information from cryo-electron microscopy and electron crystallography, we have constructed an all-atom model of a complete MT and used MD to determine its mechanical properties. The simulations revealed nonlinear axial stress-strain behavior featuring a pronounced softening under extension, a possible plastic deformation transition under radial compression, and a distinct asymmetry in response to the two senses of twist. This work demonstrates the possibility of combining different levels of structural information to produce all-atom models suitable for quantitative MD simulations, which extends the range of systems amenable to the MD method and should enable exciting advances in our microscopic knowledge of biology.

Bending of the "9+2" axoneme analyzed by the finite element method

Many data demonstrate that the regulation of the bending polarity of the "9+2" axoneme is supported by the curvature itself, making the internal constraints central in this process, adjusting either the physical characteristics of the machinery or the activity of the enzymes involved in different pathways. Among them, the very integrated Geometric Clutch model founds this regulation on the convenient adjustments of the probability of interaction between the dynein arms and the beta-tubulin monomers of the outer doublet pairs on which they walk. Taking into consideration (i) the deviated bending of the outer doublets pairs (Cibert, C., Heck, J.-V., 2004. Cell Motil. Cytoskeleton 59, 153-168), (ii) the internal tensions of the radial spokes and the tangential links (nexin links, dynein arms), (iii) a theoretical 5 microm long proximal segment of the axoneme and (iv) the short proximal segment of the axoneme, we have reevaluated the adjustments of these intervals using a finite element approach. The movements we have calculated within the axonemal cylinder are consistent with the basic hypothesis that found the Geometric Clutch model, except that the axonemal side where the dynein arms are active increases the intervals between the two neighbor outer doublet pairs. This result allows us to propose a mechanism of bending reversion of the axoneme, involving the concerted ignition of the molecular engines along the two opposite sides of the axoneme delineated by the bending plane.