Microtubule elasticity: connecting all-atom simulations with continuum mechanics

Multiscale Computational Analysis of the Effect of Taxol on Microtubule Mechanics

Microtubules (MTs) are widely recognized as targets for cancer therapies. They are directly related to unique mechanical properties, closely dependent on MT architecture and tubulin molecular features. Taxol is known to affect tubulin interactions resulting in the stabilization of the MT lattice, and thus the hierarchical organization stability, mechanics, and function. A deeper understanding of the molecular mechanisms through which taxol modulates intertubulin interactions in the MT lattice, and consequently, its stability and mechanical response is crucial to characterize how MT properties are regulated by environmental factors, such as interacting ligands. In this study, a computational analysis of the effect of taxol on the MT was performed at different scales, combining molecular dynamics simulation, dynamical network analysis, and elastic network modeling. The results show that the taxol-induced conformational differences at the M-loop region increase the stability of the lateral interactions and the amount of surface in contact between laterally coupled tubulins. Moreover, the conformational rearrangements in the taxane binding site result in a different structural communication pattern. Finally, the different conformation of the tubulin heterodimers and the stabilized lateral interactions resulted in a tendency toward higher deformation of the vibrating MT in the presence of taxol. Overall, this work provides additional insights into taxol-induced stabilization and relates the conformational changes at the tubulin level to the MT mechanics. Besides providing useful insights into taxol effect on MT mechanics, a methodological framework that could be used to characterize the effects of other MT stabilizing agents is presented.

Effect of the Rho-Kinase/ROCK Signaling Pathway on Cytoskeleton Components

The mechanical properties of cells are important in tissue homeostasis and enable cell growth, division, migration and the epithelial-mesenchymal transition. Mechanical properties are determined to a large extent by the cytoskeleton. The cytoskeleton is a complex and dynamic network composed of microfilaments, intermediate filaments and microtubules. These cellular structures confer both cell shape and mechanical properties. The architecture of the networks formed by the cytoskeleton is regulated by several pathways, a key one being the Rho-kinase/ROCK signaling pathway. This review describes the role of ROCK (Rho-associated coiled-coil forming kinase) and how it mediates effects on the key components of the cytoskeleton that are critical for cell behaviour.

Molecular mechanism of claudin-15 strand flexibility: A computational study

Claudins are one of the major components of tight junctions that play a key role in the formation and maintenance of the epithelial barrier function. Tight junction strands are dynamic and capable of adapting their structure in response to large-scale tissue rearrangement and cellular movement. Here, we present molecular dynamics simulations of claudin-15 strands of up to 225 nm in length in two parallel lipid membranes and characterize their mechanical properties. The persistence length of claudin-15 strands is comparable with those obtained from analyses of freeze-fracture electron microscopy. Our results indicate that lateral flexibility of claudin strands is due to an interplay of three sets of interfacial interaction networks between two antiparallel double rows of claudins in the membranes. In this model, claudins are assembled into interlocking tetrameric ion channels along the strand that slide with respect to each other as the strands curve over submicrometer-length scales. These results suggest a novel molecular mechanism underlying claudin-15 strand flexibility. It also sheds light on intermolecular interactions and their role in maintaining epithelial barrier function.

From System Modeling to System Analysis: The Impact of Resolution Level and Resolution Distribution in the Computer-Aided Investigation of Biomolecules

The ever increasing computer power, together with the improved accuracy of atomistic force fields, enables researchers to investigate biological systems at the molecular level with remarkable detail. However, the relevant length and time scales of many processes of interest are still hardly within reach even for state-of-the-art hardware, thus leaving important questions often unanswered. The computer-aided investigation of many biological physics problems thus largely benefits from the usage of coarse-grained models, that is, simplified representations of a molecule at a level of resolution that is lower than atomistic. A plethora of coarse-grained models have been developed, which differ most notably in their granularity; this latter aspect determines one of the crucial open issues in the field, i.e. the identification of an optimal degree of coarsening, which enables the greatest simplification at the expenses of the smallest information loss. In this review, we present the problem of coarse-grained modeling in biophysics from the viewpoint of system representation and information content. In particular, we discuss two distinct yet complementary aspects of protein modeling: on the one hand, the relationship between the resolution of a model and its capacity of accurately reproducing the properties of interest; on the other hand, the possibility of employing a lower resolution description of a detailed model to extract simple, useful, and intelligible information from the latter.

Pulling the springs of a cell by single-molecule force spectroscopy

The fundamental unit of the human body comprises of the cells which remain embedded in a fibrillar network of extracellular matrix proteins which in turn provides necessary anchorage the cells. Tissue repair, regeneration and reprogramming predominantly involve a traction force mediated signalling originating in the ECM and travelling deep into the cell including the nucleus via circuitry of spring-like filamentous proteins like microfilaments or actin, intermediate filaments and microtubules to elicit a response in the form of mechanical movement as well as biochemical changes. The 'springiness' of these proteins is highlighted in their extension-contraction behaviour which is manifested as an effect of differential traction force. Atomic force microscope (AFM) provides the magic eye to visualize and quantify such force-extension/indentation events in these filamentous proteins as well as in whole cells. In this review, we have presented a summary of the current understanding and advancement of such measurements by AFM based single-molecule force spectroscopy in the context of cytoskeletal and nucleoskeletal proteins which act in tandem to facilitate mechanotransduction.

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.

Linearized Bayesian inference for Young's modulus parameter field in an elastic model of slender structures

The deformations of several slender structures at nano-scale are conceivably sensitive to their non-homogenous elasticity. Owing to their small scale, it is not feasible to discern their elasticity parameter fields accurately using observations from physical experiments. Molecular dynamics simulations can provide an alternative or additional source of data. However, the challenges still lie in developing computationally efficient and robust methods to solve inverse problems to infer the elasticity parameter field from the deformations. In this paper, we formulate an inverse problem governed by a linear elastic model in a Bayesian inference framework. To make the problem tractable, we use a Gaussian approximation of the posterior probability distribution that results from the Bayesian solution of the inverse problem of inferring Young's modulus parameter fields from available data. The performance of the computational framework is demonstrated using two representative loading scenarios, one involving cantilever bending and the other involving stretching of a helical rod (an intrinsically curved structure). The results show that smoothly varying parameter fields can be reconstructed satisfactorily from noisy data. We also quantify the uncertainty in the inferred parameters and discuss the effect of the quality of the data on the reconstructions.

Mechanics and kinetics of dynamic instability

During dynamic instability, self-assembling microtubules (MTs) stochastically alternate between phases of growth and shrinkage. This process is driven by the presence of two distinct states of MT subunits, GTP- and GDP-bound tubulin dimers, that have different structural properties. Here, we use a combination of analysis and computer simulations to study the mechanical and kinetic regulation of dynamic instability in three-dimensional (3D) self-assembling MTs. Our model quantifies how the 3D structure and kinetics of the distinct states of tubulin dimers determine the mechanical stability of MTs. We further show that dynamic instability is influenced by the presence of quenched disorder in the state of the tubulin subunit as reflected in the fraction of non-hydrolysed tubulin. Our results connect the 3D geometry, kinetics and statistical mechanics of these tubular assemblies within a single framework, and may be applicable to other self-assembled systems where these same processes are at play.

Microtubule Simulations Provide Insight into the Molecular Mechanism Underlying Dynamic Instability

The dynamic instability of microtubules (MTs), which refers to their ability to switch between polymerization and depolymerization states, is crucial for their function. It has been proposed that the growing MT ends are protected by a "GTP cap" that consists of GTP-bound tubulin dimers. When the speed of GTP hydrolysis is faster than dimer recruitment, the loss of this GTP cap will lead the MT to undergo rapid disassembly. However, the underlying atomistic mechanistic details of the dynamic instability remains unclear. In this study, we have performed long-time atomistic molecular dynamics simulations (1 μs for each system) for MT patches as well as a short segment of a closed MT in both GTP- and GDP-bound states. Our results confirmed that MTs in the GDP state generally have weaker lateral interactions between neighboring protofilaments (PFs) and less cooperative outward bending conformational change, where the difference between bending angles of neighboring PFs tends to be larger compared with GTP ones. As a result, when the GDP state tubulin dimer is exposed at the growing MT end, these factors will be more likely to cause the MT to undergo rapid disassembly. We also compared simulation results between the special MT seam region and the remaining material and found that the lateral interactions between MT PFs at the seam region were comparatively much weaker. This finding is consistent with the experimental suggestion that the seam region tends to separate during the disassembly process of an MT.

Atomistic to continuum model for studying mechanical properties of RNA nanotubes

With rapid advancements in the emerging field of RNA nanotechnology, its current and potential applications, new important problems arise in our quest to better understand properties of RNA nanocomplexes. In this paper, our focus is on the modeling of RNA nanotubes which are important for many biological processes. These RNA complexes are also important for human beings, with their theurapeutical and biomedical applications discussed vigorously in the literature over the recent years. Here, we develop a continuum model of RNA nanotubes, originally obtained from self assembly of RNA building blocks in the molecular dynamics simulation. Based on the finite element method, we calculate the elastic properties of these nanostructures and provide a relationship between stress and strain induced in the RNA nanotube. We also analyze the variations in the displacement vector along the assembly axis for RNA nanotubes of different sizes. In particular, we show that oscillations in the amplitudes of strains and displacements significantly differ for such RNA nanotubes. These findings are discussed in the context of atomistic simulations and experimental results in this field.

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.

Damage and Failure of Axonal Microtubule under Extreme High Strain Rate: An In-Silico Molecular Dynamics Study

As a major cytoskeleton element of the axon, the breaking of microtubules (MTs) has been considered as a major cause of the axon degeneration. High strain rate loading is considered as one of the key factors in microtubule breaking. Due to the small size of microtubule, the real-time behavior of microtubule breaking is hard to capture. This study employs fully-atomistic molecular dynamics (MD) simulation to determine the failure modes of microtubule under different loadings conditions such as, unidirectional stretching, bending and hydrostatic expansion. For each loading conditions, MT is subjected to extreme high strain rate (108-109 s-1) loading. We argue that such level of high strain rate may be realized during cavitation bubble implosion. For each loading type, we have determined the critical energy for MT rupture. The associated rupture mechanisms are also discussed. We observed that the stretching has the lowest energy barrier to break the MT at the nanosecond time scale. Moreover, the breakage between the dimers starts at ~16% of total strain when stretched, which is much smaller compared to the reported strain-at-failure (50%) for lower strain rate loading. It suggests that MT fails at a significantly smaller strain states when loaded at higher strain rates.

The long persistence length of model tubules

Young's elastic modulus and the persistence length are calculated for a coarse-grained model of tubule forming polymers. The model uses a wedge shaped composite of particles that previously has been shown to self-assemble into tubules. These calculations demonstrate that the model yields very large persistence lengths (corresponding to 78-126 μm) that are comparable to that observed in experiments for the microtubule lengths accessible to the calculations. The source for the stiffness is the restricted rotation of the monomer due to the excluded volume interactions between bonded macromolecular monomers as well as the binding between monomers. For this reason, large persistence lengths are common in tubule systems with a macromolecule as the monomer. The persistence length increases linearly with increased binding strength in the filament direction. No dependence in the persistence length is found for varying the tubule pitch for geometries with the protofilaments remaining straight.

Effect of Nucleotide State on the Protofilament Conformation of Tubulin Octamers

At the molecular level, the dynamic instability (random growth and shrinkage) of the microtubule (MT) is driven by the nucleotide state (GTP vs GDP) in the β subunit of the tubulin dimers at the MT cap. Here, we use large-scale molecular dynamics (MD) simulations and normal-mode analysis (NMA) to characterize the effect of a single GTP cap layer on tubulin octamers composed of two neighboring protofilaments (PFs). We utilize recently reported high-resolution structures of dynamic MTs to simulate a GDP octamer both with and without a single GTP cap layer. We perform multiple replicas of long-time atomistic MD simulations (3 replicas, 0.3 μs for each replica, 0.9 μs for each octamer system, and 1.8 μs total) of both octamers. We observe that a single GTP cap layer induces structural differences in neighboring PFs, finding that one PF possesses a gradual curvature, compared to the second PF which possesses a kinked conformation. This results in either curling or splaying between these PFs. We suggest that this is due to asymmetric strengths of longitudinal contacts between the two PFs. Furthermore, using NMA, we calculate mechanical properties of these octamer systems and find that octamer system with a single GTP cap layer possesses a lower flexural rigidity.

Microtubule seams are not mechanically weak defects

Microtubule rigidity is important for many cellular functions to support extended structures and rearrange materials within the cell. The arrangement of the tubulin dimers within the microtubule can be altered to affect the protofilament number and the lattice type. Prior electron microscopy measurements have shown that when polymerized in the presence of a high concentration of NaCl, microtubules were more likely to be ten protofilaments with altered intertubulin lattice types. Specifically, such high-salt microtubules have a higher percentage of seam defects. Such seams have long been speculated to be a mechanically weak location in the microtubule lattice, yet no experimental evidence supported this claim. We directly measured the persistence length of freely fluctuating filaments made either with high salt or without. We found that the microtubules made with high salt were more flexible, by a factor of 2, compared to those polymerized the same way without salt present. The reduced persistence length of the high-salt microtubules can be accounted for entirely by a smaller cross-sectional radius of these microtubules, implying that the mixed lattice interactions have little effect on the bending rigidity. Our results suggest that the microtubule seam is not weaker than the typical lattice structure as previously speculated from structural studies.

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.

Catastrophic depolymerization of microtubules driven by subunit shape change

Microtubules exhibit a dynamic instability between growth and catastrophic depolymerization. GTP-tubulin (αβ-dimer bound to GTP) self-assembles, but dephosphorylation of GTP- to GDP-tubulin within the tubule results in destabilization. While the mechanical basis for destabilization is not fully understood, one hypothesis is that dephosphorylation causes tubulin to change shape, frustrating bonds and generating stress. To test this idea, we perform molecular dynamics simulations of microtubules built from coarse-grained models of tubulin, incorporating a small compression of α-subunits associated with dephosphorylation in experiments. We find that this shape change induces depolymerization of otherwise stable systems via unpeeling "ram's horns" characteristic of microtubules. Depolymerization can be averted by caps with uncompressed α-subunits, i.e., GTP-rich end regions. Thus, the shape change is sufficient to yield microtubule behavior.

Data-driven reduced-order model of microtubule mechanics

A beam element is constructed for microtubules based upon data reduction of the results from atomistic simulation of the carbon backbone chain of αβ-tubulin dimers. The database of mechanical responses to various types of loads from atomistic simulation is reduced to dominant modes. The dominant modes are subsequently used to construct the stiffness matrix of a beam element that captures the anisotropic behavior and deformation mode coupling that arises from a microtubule's spiral structure. In contrast to standard Euler-Bernoulli or Timoshenko beam elements, the link between forces and node displacements results not from hypothesized deformation behavior, but directly from the data obtained by molecular scale simulation. Differences between the resulting microtubule data-driven beam model (MTDDBM) and standard beam elements are presented, with a focus on coupling of bending, stretch, shear deformations. The MTDDBM is just as economical to use as a standard beam element, and allows accurate reconstruction of the mechanical behavior of structures within a cell as exemplified in a simple model of a component element of the mitotic spindle.

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.

Non-equilibrium assembly of microtubules: from molecules to autonomous chemical robots

Biological systems have evolved to harness non-equilibrium processes from the molecular to the macro scale. It is currently a grand challenge of chemistry, materials science, and engineering to understand and mimic biological systems that have the ability to autonomously sense stimuli, process these inputs, and respond by performing mechanical work. New chemical systems are responding to the challenge and form the basis for future responsive, adaptive, and active materials. In this article, we describe a particular biochemical-biomechanical network based on the microtubule cytoskeletal filament - itself a non-equilibrium chemical system. We trace the non-equilibrium aspects of the system from molecules to networks and describe how the cell uses this system to perform active work in essential processes. Finally, we discuss how microtubule-based engineered systems can serve as testbeds for autonomous chemical robots composed of biological and synthetic components.

Morphological basis for a tonotopic design of an insect ear

The tonotopically organized hearing organs of bushcrickets provide the opportunity for a detailed correlation of morphological and structural properties within hearing organs that are needed to establish tonotopic gradients. In the present study of a tonotopic insect hearing organ, we combine mechanical measurements of sound-induced hearing organ motion and detailed anatomical investigations to explore the anatomical basis of tonotopy. We compare mechanical data of frequency responses along the auditory organ to several anatomical parameters. Low frequency responses are related to larger organ and cap cell size in the proximal part of the hearing organ while in the distal part of the organ, small organ and cap cell size is related to high-frequency representation. However, the correlation between organ and cap cell size with continuous frequency representation along the organ is not very tight. Instead, the height of the organ and the corresponding length of the sensory dendrites are best correlated to tonotopic frequency representation. The sensory dendrite contains a ciliary root with a pronounced cross-banding of electron-dense material that should be important for the stiffness of the dendrite. The geometry of surrounding structures like the hemolymph channel and the acoustic trachea as well as the extension of the tectorial membrane are not correlated to the tonotopy. We provide evidence that tonotopy in the bushcricket hearing organ may depend on the size of ciliary structures. In particular, the ciliary root of the sensory cells is a likely cellular basis of tonotopy.

Mechanisms of kinetic stabilization by the drugs paclitaxel and vinblastine

Chemotherapeutic agents that target microtubule dynamics promote a universal phenotype of kinetic stabilization. Integrated computational modeling and fluorescence microscopy identify the fundamental kinetic and thermodynamic mechanisms that result in kinetic stabilization, specifically by the drugs paclitaxel and vinblastine.

Microtubule-targeting agents (MTAs), widely used as biological probes and chemotherapeutic drugs, bind directly to tubulin subunits and “kinetically stabilize” microtubules, suppressing the characteristic self-assembly process of dynamic instability. However, the molecular-level mechanisms of kinetic stabilization are unclear, and the fundamental thermodynamic and kinetic requirements for dynamic instability and its elimination by MTAs have yet to be defined. Here we integrate a computational model for microtubule assembly with nanometer-scale fluorescence microscopy measurements to identify the kinetic and thermodynamic basis of kinetic stabilization by the MTAs paclitaxel, an assembly promoter, and vinblastine, a disassembly promoter. We identify two distinct modes of kinetic stabilization in live cells, one that truly suppresses on-off kinetics, characteristic of vinblastine, and the other a “pseudo” kinetic stabilization, characteristic of paclitaxel, that nearly eliminates the energy difference between the GTP- and GDP-tubulin thermodynamic states. By either mechanism, the main effect of both MTAs is to effectively stabilize the microtubule against disassembly in the absence of a robust GTP cap.

Invited review: Microtubule severing enzymes couple atpase activity with tubulin GTPase spring loading

Microtubules are amazing filaments made of GTPase enzymes that store energy used for their own self-destruction to cause a stochastically driven dynamics called dynamic instability. Dynamic instability can be reproduced in vitro with purified tubulin, but the dynamics do not mimic that observed in cells. This is because stabilizers and destabilizers act to alter microtubule dynamics. One interesting and understudied class of destabilizers consists of the microtubule-severing enzymes from the ATPases Associated with various cellular Activities (AAA+) family of ATP-enzymes. Here we review current knowledge about GTP-driven microtubule dynamics and how that couples to ATP-driven destabilization by severing enzymes. We present a list of challenges regarding the mechanism of severing, which require development of experimental and modeling approaches to shed light as to how severing enzymes can act to regulate microtubule dynamics in cells. © 2016 Wiley Periodicals, Inc. Biopolymers 105: 547-556, 2016.

Multi scale modeling of 2450MHz electric field effects on microtubule mechanical properties

Microtubule (MT) rigidity and response to 2450MHz electric fields were investigated, via multi scale modeling approach. For this purpose, six systems were designed and simulated to consider all types of feasible interactions between α and β monomers in MT, by using all atom molecular dynamics method. Subsequently, coarse grain modeling was used to design different lengths of MT. Investigation of effects of external 2450MHz electric field on MT showed MT less rigidity in the presence of such field, which may perturb its functions. Moreover, an additional computational setup was designed to study effects of 2450MHz field on MT response to AFM tip. It was found, more tip velocity led to MT faster transformation and less time was required to change MT elastic response to plastic one, applying constant radius. Moreover it was observed smaller tip caused to increase required time to change MT elastic response to plastic one, considering constant velocity. Furthermore, exposing MT to 2450MHz field led to no significant changes in MT response to AFM tip, but quick change in MT elastic response to plastic one.

Maintenance of electrostatic stabilization in altered tubulin lateral contacts may facilitate formation of helical filaments in foraminifera

Microtubules in foraminiferan protists (forams) can convert into helical filament structures, in which longitudinal intraprotofilament interactions between tubulin heterodimers are thought to be lost, while lateral contacts across protofilaments are still maintained. The coarse geometric features of helical filaments are known through low-resolution negative stain electron microscopy (EM). In this study, geometric restraints derived from these experimental data were used to generate an average atomic-scale helical filament model, which anticipated a modest reorientation in the lateral tubulin heterodimer interface. Restrained molecular dynamics (MD) simulations of the nearest neighbor interactions combined with a Genalized Born implicit solvent model were used to assess the lateral, longitudinal, and seam contacts in 13-3 microtubules and the reoriented lateral contacts in the helical filament model. This electrostatic analysis suggests that the change in the lateral interface in the helical filament does not greatly diminish the lateral electrostatic interaction. After longitudinal dissociation, the 13-3 seam interaction is much weaker than the reoriented lateral interface in the helical filament model, providing a plausible atomic-detail explanation for seam-to-lateral contact transition that enables the transition to a helical filament structure.

The effects of external electric fields of 900 MHz and 2450 MHz frequencies onαβ-tubulin dimer stabilized by paclitaxel: Molecular dynamics approach

Using molecular dynamics simulation method, the effects of external electric fields of 900[Formula: see text]MHz and 2450 frequencies on [Formula: see text]-tubulin dimer stabilized by paclitaxel, have been modeled. Due to this purpose, two systems, (A) [Formula: see text]-tubulin dimer and (B) [Formula: see text]-tubulin dimer stabilized by paclitaxel, were exposed to an external electric field of 0.01[Formula: see text]V/nm with frequency values of 900[Formula: see text]MHz and 2450[Formula: see text]MHz. It was found that application of these fields, which are in the range of cell phone and microwave frequencies, increased the flexibility of each system. Since paclitaxel, as chemotherapy drug, is used to increase the rigidity of dimer, application of such fields may disturb the effect of paclitaxel on the dimer. Consequently, negative side effects on the chemotherapy process may be observed.

Relating nucleotide-dependent conformational changes in free tubulin dimers to tubulin assembly

The complex dynamic behavior of microtubules (MTs) is believed to be primarily due to the αβ-tubulin dimer architecture and its intrinsic GTPase activity. Hence, a detailed knowledge of the conformational variations of isolated α-GTP-β-GTP- and α-GTP-β-GDP-tubulin dimers in solution and their implications to interdimer interactions and stability is directly relevant to understand the MT dynamics. An attempt has been made here by combining molecular dynamics (MD) simulations and protein-protein docking studies that unravels key structural features of tubulin dimer in different nucleotide states and correlates their association to tubulin assembly. Results from simulations suggest that tubulin dimers and oligomers attain curved conformations in both GTP and GDP states. Results also indicate that the tubulin C-terminal domain and the nucleotide state are closely linked. Protein-protein docking in combination with MD simulations suggest that the GTP-tubulin dimers engage in relatively stronger interdimer interactions even though the interdimer interfaces are bent in both GTP and GDP tubulin complexes, providing valuable insights on in vitro finding that GTP-tubulin is a better assembly candidate than GDP-tubulin during the MT nucleation and elongation processes.

Multiscale method for modeling binding phenomena involving large objects: application to kinesin motor domains motion along microtubules

Many biological phenomena involve the binding of proteins to a large object. Because the electrostatic forces that guide binding act over large distances, truncating the size of the system to facilitate computational modeling frequently yields inaccurate results. Our multiscale approach implements a computational focusing method that permits computation of large systems without truncating the electrostatic potential and achieves the high resolution required for modeling macromolecular interactions, all while keeping the computational time reasonable. We tested our approach on the motility of various kinesin motor domains. We found that electrostatics help guide kinesins as they walk: N-kinesins towards the plus-end, and C-kinesins towards the minus-end of microtubules. Our methodology enables computation in similar, large systems including protein binding to DNA, viruses, and membranes.

Molecular control of stress transmission in the microtubule cytoskeleton

In this article, we will summarize recent progress in understanding the mechanical origins of rigidity, strength, resiliency and stress transmission in the MT cytoskeleton using reconstituted networks formed from purified components. We focus on the role of network architecture, crosslinker compliance and dynamics, and molecular determinants of single filament elasticity, while highlighting open questions and future directions for this work.

Fast and anisotropic flexibility-rigidity index for protein flexibility and fluctuation analysis

Protein structural fluctuation, typically measured by Debye-Waller factors, or B-factors, is a manifestation of protein flexibility, which strongly correlates to protein function. The flexibility-rigidity index (FRI) is a newly proposed method for the construction of atomic rigidity functions required in the theory of continuum elasticity with atomic rigidity, which is a new multiscale formalism for describing excessively large biomolecular systems. The FRI method analyzes protein rigidity and flexibility and is capable of predicting protein B-factors without resorting to matrix diagonalization. A fundamental assumption used in the FRI is that protein structures are uniquely determined by various internal and external interactions, while the protein functions, such as stability and flexibility, are solely determined by the structure. As such, one can predict protein flexibility without resorting to the protein interaction Hamiltonian. Consequently, bypassing the matrix diagonalization, the original FRI has a computational complexity of O(N(2)). This work introduces a fast FRI (fFRI) algorithm for the flexibility analysis of large macromolecules. The proposed fFRI further reduces the computational complexity to O(N). Additionally, we propose anisotropic FRI (aFRI) algorithms for the analysis of protein collective dynamics. The aFRI algorithms permit adaptive Hessian matrices, from a completely global 3N × 3N matrix to completely local 3 × 3 matrices. These 3 × 3 matrices, despite being calculated locally, also contain non-local correlation information. Eigenvectors obtained from the proposed aFRI algorithms are able to demonstrate collective motions. Moreover, we investigate the performance of FRI by employing four families of radial basis correlation functions. Both parameter optimized and parameter-free FRI methods are explored. Furthermore, we compare the accuracy and efficiency of FRI with some established approaches to flexibility analysis, namely, normal mode analysis and Gaussian network model (GNM). The accuracy of the FRI method is tested using four sets of proteins, three sets of relatively small-, medium-, and large-sized structures and an extended set of 365 proteins. A fifth set of proteins is used to compare the efficiency of the FRI, fFRI, aFRI, and GNM methods. Intensive validation and comparison indicate that the FRI, particularly the fFRI, is orders of magnitude more efficient and about 10% more accurate overall than some of the most popular methods in the field. The proposed fFRI is able to predict B-factors for α-carbons of the HIV virus capsid (313 236 residues) in less than 30 seconds on a single processor using only one core. Finally, we demonstrate the application of FRI and aFRI to protein domain analysis.

Influence of GHz electric fields on the mechanical properties of a microtubule

The effects of external GHz electric fields on the mechanical properties of a microtubule (MT) have been modeled through the application of a molecular dynamics simulation method. To explore the properties of the MT, two different systems each consisting of a pair of dimers were exposed to an 0.03 V/nm electric field with a frequency ranging between 1 to 10 GHz. It was found that the Young's modulus of each system, which is related to the flexibility of the MT, was lower at some frequencies and higher at others in comparison with normal biological conditions. Hence, the application of such an electric field with a frequency in this range may affect MT function, which could have positive or negative effects on cell health. Positive effects include its potential use in cancer treatment, where the application of such a field could lead to a decrease in MT rigidity, similar to the effect of Taxol on MTs. Negative effects include unwanted changes to the mechanical properties of MTs (e.g., disturbing the cell division process and in turn increasing the risk of cancer) upon the application of such a field.

Statistical mechanics provides novel insights into microtubule stability and mechanism of shrinkage

Microtubules are nano-machines that grow and shrink stochastically, making use of the coupling between chemical kinetics and mechanics of its constituent protofilaments (PFs). We investigate the stability and shrinkage of microtubules taking into account inter-protofilament interactions and bending interactions of intrinsically curved PFs. Computing the free energy as a function of PF tip position, we show that the competition between curvature energy, inter-PF interaction energy and entropy leads to a rich landscape with a series of minima that repeat over a length-scale determined by the intrinsic curvature. Computing Langevin dynamics of the tip through the landscape and accounting for depolymerization, we calculate the average unzippering and shrinkage velocities of GDP protofilaments and compare them with the experimentally known results. Our analysis predicts that the strength of the inter-PF interaction (E(s)(m)) has to be comparable to the strength of the curvature energy (E(b)(m)) such that E(s)(m) - E(b)(m) ≈ 1kBT, and questions the prevalent notion that unzippering results from the domination of bending energy of curved GDP PFs. Our work demonstrates how the shape of the free energy landscape is crucial in explaining the mechanism of MT shrinkage where the unzippered PFs will fluctuate in a set of partially peeled off states and subunit dissociation will reduce the length.

Tubulin bond energies and microtubule biomechanics determined from nanoindentation in silico

Microtubules, the primary components of the chromosome segregation machinery, are stabilized by longitudinal and lateral noncovalent bonds between the tubulin subunits. However, the thermodynamics of these bonds and the microtubule physicochemical properties are poorly understood. Here, we explore the biomechanics of microtubule polymers using multiscale computational modeling and nanoindentations in silico of a contiguous microtubule fragment. A close match between the simulated and experimental force–deformation spectra enabled us to correlate the microtubule biomechanics with dynamic structural transitions at the nanoscale. Our mechanical testing revealed that the compressed MT behaves as a system of rigid elements interconnected through a network of lateral and longitudinal elastic bonds. The initial regime of continuous elastic deformation of the microtubule is followed by the transition regime, during which the microtubule lattice undergoes discrete structural changes, which include first the reversible dissociation of lateral bonds followed by irreversible dissociation of the longitudinal bonds. We have determined the free energies of dissociation of the lateral (6.9 ± 0.4 kcal/mol) and longitudinal (14.9 ± 1.5 kcal/mol) tubulin–tubulin bonds. These values in conjunction with the large flexural rigidity of tubulin protofilaments obtained (18,000–26,000 pN·nm²) support the idea that the disassembling microtubule is capable of generating a large mechanical force to move chromosomes during cell division. Our computational modeling offers a comprehensive quantitative platform to link molecular tubulin characteristics with the physiological behavior of microtubules. The developed in silico nanoindentation method provides a powerful tool for the exploration of biomechanical properties of other cytoskeletal and multiprotein assemblies.

External electric field effects on the mechanical properties of the αβ-tubulin dimer of microtubules: a molecular dynamics study

The mechanical properties of the αβ-tubulin dimer of microtubules was modeled by using the molecular dynamics (MD) simulation method. The effect on the mechanical properties of the dimer of the existence and nonexistence of an applied electric field, either constant or periodic, was studied. Since there are charged or polar groups in the dimer structure, the electric field can interact with the dimer. The elastic constant and Young's modulus of the dimer were decreased when the dimer was exposed to a constant electric field of 0.03 V/nm. Furthermore, applying an oscillating electric field in the 1 GHz range to the dimer increased the elastic constant and Young's modulus of the dimer. These parameters were related to dimer rigidity and, consequently, in this frequency range, the application of electric fields may affect the function of microtubules.

Effects of microtubule mechanics on hydrolysis and catastrophes

We introduce a model for microtubule (MT) mechanics containing lateral bonds between dimers in neighboring protofilaments, bending rigidity of dimers, and repulsive interactions between protofilaments modeling steric constraints to investigate the influence of mechanical forces on hydrolysis and catastrophes. We use the allosteric dimer model, where tubulin dimers are characterized by an equilibrium bending angle, which changes from 0° to 22° by hydrolysis of a dimer. This also affects the lateral interaction and bending energies and, thus, the mechanical equilibrium state of the MT. As hydrolysis gives rise to conformational changes in dimers, mechanical forces also influence the hydrolysis rates by mechanical energy changes modulating the hydrolysis rate. The interaction via the MT mechanics then gives rise to correlation effects in the hydrolysis dynamics, which have not been taken into account before. Assuming a dominant influence of mechanical energies on hydrolysis rates, we investigate the most probable hydrolysis pathways both for vectorial and random hydrolysis. Investigating the stability with respect to lateral bond rupture, we identify initiation configurations for catastrophes along the hydrolysis pathways and values for a lateral bond rupture force. If we allow for rupturing of lateral bonds between dimers in neighboring protofilaments above this threshold force, our model exhibits avalanche-like catastrophe events.

Perspective: Coarse-grained models for biomolecular systems

By focusing on essential features, while averaging over less important details, coarse-grained (CG) models provide significant computational and conceptual advantages with respect to more detailed models. Consequently, despite dramatic advances in computational methodologies and resources, CG models enjoy surging popularity and are becoming increasingly equal partners to atomically detailed models. This perspective surveys the rapidly developing landscape of CG models for biomolecular systems. In particular, this review seeks to provide a balanced, coherent, and unified presentation of several distinct approaches for developing CG models, including top-down, network-based, native-centric, knowledge-based, and bottom-up modeling strategies. The review summarizes their basic philosophies, theoretical foundations, typical applications, and recent developments. Additionally, the review identifies fundamental inter-relationships among the diverse approaches and discusses outstanding challenges in the field. When carefully applied and assessed, current CG models provide highly efficient means for investigating the biological consequences of basic physicochemical principles. Moreover, rigorous bottom-up approaches hold great promise for further improving the accuracy and scope of CG models for biomolecular systems.

Self-assembly of chiral tubules

The efficient and controlled assembly of complex structures from macromolecular building blocks is a critical open question in both biological systems and nanoscience. Using molecular dynamics simulations we study the self-assembly of tubular structures from model macromolecular monomers with multiple binding sites on their surfaces [Cheng et al., Soft Matter, 2012, 8, 5666-5678]. In this work we add chirality to the model monomer and a lock-and-key interaction. The self-assembly of free monomers into tubules yields a pitch value that often does not match the chirality of the monomer (including achiral monomers). We show that this mismatch occurs because of a twist deformation that brings the lateral interaction sites into alignment when the tubule pitch differs from the monomer chirality. The energy cost for this deformation is small as the energy distributions substantially overlap for small differences in the pitch and chirality. In order to control the tubule pitch by preventing the twist deformation, the interaction between the vertical surfaces must be increased without resulting in kinetically trapped structures. For this purpose, we employ lock-and-key interactions and obtain good control of the self-assembled tubule pitch. These results explain some fundamental features of microtubules. The vertical interaction strength is larger than the lateral in microtubules because this yields a controlled assembly of tubules with the proper pitch. We also generally find that the control of the assembly into tubules is difficult, which explains the wide range of pitch values and protofilament numbers observed in microtubule assembly.

Multiscale multiphysics and multidomain models--flexibility and rigidity

The emerging complexity of large macromolecules has led to challenges in their full scale theoretical description and computer simulation. Multiscale multiphysics and multidomain models have been introduced to reduce the number of degrees of freedom while maintaining modeling accuracy and achieving computational efficiency. A total energy functional is constructed to put energies for polar and nonpolar solvation, chemical potential, fluid flow, molecular mechanics, and elastic dynamics on an equal footing. The variational principle is utilized to derive coupled governing equations for the above mentioned multiphysical descriptions. Among these governing equations is the Poisson-Boltzmann equation which describes continuum electrostatics with atomic charges. The present work introduces the theory of continuum elasticity with atomic rigidity (CEWAR). The essence of CEWAR is to formulate the shear modulus as a continuous function of atomic rigidity. As a result, the dynamics complexity of a macromolecular system is separated from its static complexity so that the more time-consuming dynamics is handled with continuum elasticity theory, while the less time-consuming static analysis is pursued with atomic approaches. We propose a simple method, flexibility-rigidity index (FRI), to analyze macromolecular flexibility and rigidity in atomic detail. The construction of FRI relies on the fundamental assumption that protein functions, such as flexibility, rigidity, and energy, are entirely determined by the structure of the protein and its environment, although the structure is in turn determined by all the interactions. As such, the FRI measures the topological connectivity of protein atoms or residues and characterizes the geometric compactness of the protein structure. As a consequence, the FRI does not resort to the interaction Hamiltonian and bypasses matrix diagonalization, which underpins most other flexibility analysis methods. FRI's computational complexity is of O(N(2)) at most, where N is the number of atoms or residues, in contrast to O(N(3)) for Hamiltonian based methods. We demonstrate that the proposed FRI gives rise to accurate prediction of protein B-Factor for a set of 263 proteins. We show that a parameter free FRI is able to achieve about 95% accuracy of the parameter optimized FRI. An interpolation algorithm is developed to construct continuous atomic flexibility functions for visualization and use with CEWAR.

Mechanical properties of doubly stabilized microtubule filaments

Microtubules are cytoskeletal filaments responsible for cell morphology and intracellular organization. Their dynamical and mechanical properties are regulated through the nucleotide state of the tubulin dimers and the binding of drugs and/or microtubule-associated proteins. Interestingly, microtubule-stabilizing factors have differential effects on microtubule mechanics, but whether stabilizers have cumulative effects on mechanics or whether one effect dominates another is not clear. This is especially important for the chemotherapeutic drug Taxol, an important anticancer agent and the only known stabilizer that reduces the rigidity of microtubules. First, we ask whether Taxol will combine additively with another stabilizer or whether one stabilizer will dominate another. We call microtubules in the presence of Taxol and another stabilizer, doubly stabilized. Second, since Taxol is often added to a number of cell types for therapeutic purposes, it is important from a biomedical perspective to understand how Taxol added to these systems affects the mechanical properties in treated cells. To address these questions, we use the method of freely fluctuating filaments with our recently developed analysis technique of bootstrapping to determine the distribution of persistence lengths of a large population of microtubules treated with different stabilizers, including Taxol, guanosine-5' [(α, β)-methyleno] triphosphate, guanosine-5'-O-(3-thiotriphosphate), tau, and MAP4. We find that combinations of these stabilizers have novel effects on the mechanical properties of microtubules.

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.

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.

Nucleotide-dependent lateral and longitudinal interactions in microtubules

Microtubule (MT) stability is related to the hydrolysis of the guanosine triphosphate nucleotide (NT) bound to β-tubulin. However, the molecular mechanism by which the NT state influences the stability of the contacts in the MT lattice remains elusive. Here, we present large-scale atomistic simulations of different tubulin aggregates, including individual dimers, short protofilaments, a small lattice patch, and a piece of the MT lattice with two infinite protofilaments in both NT states. Together with a coarse-grained (CG) analysis of the fluctuations, these simulations highlight several regions of the protein where local changes are induced by the NT state or by the lateral and longitudinal contacts in the aggregates. Additionally, the CG analysis provides an indication of how the structural changes affect the bonds between the proteins. The results suggest a consistent picture of a possible molecular mechanism by which the NT state induces changes in the H1-S2 loop and more stable longitudinal bonds, both of which locate the H1-S2 and M-loop in more favorable positions to form lateral contacts.

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.