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.     

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. © 2012 Wiley Periodicals, Inc. Biopolymers 99: 282–291, 2013.     

Gating mechanisms of mechanosensitive channels of large conductance, I: a continuum mechanics-based hierarchical framework

A hierarchical simulation framework that integrates information from molecular dynamics (MD) simulations into a continuum model is established to study the mechanical response of mechanosensitive channel of large-conductance (MscL) using the finite element method (FEM). The proposed MD-decorated FEM (MDeFEM) approach is used to explore the detailed gating mechanisms of the MscL in Escherichia coli embedded in a palmitoyloleoylphosphatidylethanolamine lipid bilayer. In Part I of this study, the framework of MDeFEM is established. The transmembrane and cytoplasmic helices are taken to be elastic rods, the loops are modeled as springs, and the lipid bilayer is approximated by a three-layer sheet. The mechanical properties of the continuum components, as well as their interactions, are derived from molecular simulations based on atomic force fields. In addition, analytical closed-form continuum model and elastic network model are established to complement the MDeFEM approach and to capture the most essential features of gating. In Part II of this study, the detailed gating mechanisms of E. coli-MscL under various types of loading are presented and compared with experiments, structural model, and all-atom simulations, as well as the analytical models established in Part I. It is envisioned that such a hierarchical multiscale framework will find great value in the study of a variety of biological processes involving complex mechanical deformations such as muscle contraction and mechanotransduction.     

Radial compression of microtubules and the mechanism of action of taxol and associated proteins

Microtubules (MTs) are hollow cylindrical polymers composed of alphabeta-tubulin heterodimers that align head-to-tail in the MT wall, forming linear protofilaments that interact laterally. We introduce a probe of the interprotofilament interactions within MTs and show that this technique gives insight into the mechanisms by which MT-associated proteins (MAPs) and taxol stabilize MTs. In addition, we present further measurements of the mechanical properties of MT walls, MT-MT interactions, and the entry of polymers into the MT lumen. These results are obtained from a synchrotron small angle x-ray diffraction (SAXRD) study of MTs under osmotic stress. Above a critical osmotic pressure, P(cr), we observe rectangular bundles of MTs whose cross sections have buckled to a noncircular shape; further increases in pressure continue to distort MTs elastically. The P(cr) of approximately 600 Pa provides, for the first time, a measure of the bending modulus of the interprotofilament bond within an MT. The presence of neuronal MAPs greatly increases P(cr), whereas surprisingly, the cancer chemotherapeutic drug taxol, which suppresses MT dynamics and inhibits MT depolymerization, does not affect the interprotofilament interactions. This SAXRD-osmotic stress technique, which has enabled measurements of the mechanical properties of MTs, should find broad application for studying interactions between MTs and of MTs with MAPs and MT-associated drugs.     

Buckling Behavior of Individual and Bundled Microtubules

As the major structural constituent of the cytoskeleton, microtubules (MTs) serve a variety of biological functions that range from facilitating organelle transport to maintaining the mechanical integrity of the cell. Neuronal MTs exhibit a distinct configuration, hexagonally packed bundles of MT filaments, interconnected by MT-associated protein (MAP) tau. Building on our previous work on mechanical response of axonal MT bundles under uniaxial tension, this study is focused on exploring the compression scenarios. Intracellular MTs carry a large fraction of the compressive loads sensed by the cell and therefore, like any other column-like structure, are prone to substantial bending and buckling. Various biological activities, e.g., actomyosin contractility and many pathological conditions are driven or followed by bending, looping, and buckling of MT filaments. The coarse-grained model previously developed in our lab has been used to study the mechanical behavior of individual and bundled in vivo MT filaments under uniaxial compression. Both configurations show tip-localized, decaying, and short-wavelength buckling. This behavior highlights the role of the surrounding cytoplasm and MAP tau on MT buckling behavior, which allows MT filaments to bear much larger compressive forces. It is observed that MAP tau interconnections improve this effect by a factor of two. The enhanced ability of MT bundles to damp buckling waves relative to individual MT filaments, may be interpreted as a self-defense mechanism because it helps axonal MTs to endure harsher environments while maintaining their function. The results indicate that MT filaments in a bundle do not buckle simultaneously implying that the applied stress is not equally shared among the MT filaments, that is a consequence of the nonuniform distribution of MAP tau proteins along the bundle length. Furthermore, from a pathological perspective, it is observed that axonal MT bundles are more vulnerable to failure in compression than tension.     

Effects of the cross-linkers on the buckling of microtubules in cells

In cells, the protein cross-linkers lead to a distinct buckling behavior of microtubules (MTs) different from the buckling of individual MTs. This paper thus aims to examine this issue via the molecular structural mechanics (MSM) simulations. The transition of buckling responses was captured as the two-dimensional-linkers were replaced by the three-dimensional (3D) ones. Then, the effects of the radial orientation and the axial density of the 3D-linkers were examined, showing that more uniform distribution of the radial orientation leads to the higher critical load with 3D buckling modes, while the inhomogeneity of the axial density results in the localized buckling patterns. The results demonstrated the important role of the cross-linker in regulating MT stiffness, revealed the physics of the experimentally observed localized buckling and these results will pave the way to a new multi-component mechanics model for whole cells.     

A Highlights from MBoC Selection: MAP65/Ase1 promote microtubule flexibility

Microtubules (MTs) are dynamic cytoskeletal elements involved in numerous cellular processes. Although they are highly rigid polymers with a persistence length of 1–8 mm, they may exhibit a curved shape at a scale of few micrometers within cells, depending on their biological functions. However, how MT flexural rigidity in cells is regulated remains poorly understood. Here we ask whether MT-associated proteins (MAPs) could locally control the mechanical properties of MTs. We show that two major cross-linkers of the conserved MAP65/PRC1/Ase1 family drastically decrease MT rigidity. Their MT-binding domain mediates this effect. Remarkably, the softening effect of MAP65 observed on single MTs is maintained when MTs are cross-linked. By reconstituting physical collisions between growing MTs/MT bundles, we further show that the decrease in MT stiffness induced by MAP65 proteins is responsible for the sharp bending deformations observed in cells when they coalign at a steep angle to create bundles. Taken together, these data provide new insights into how MAP65, by modifying MT mechanical properties, may regulate the formation of complex MT arrays.     

Modulating Microtubules: A Molecular Perspective on the Effects of Tail Modifications

Microtubules (MTs), an essential component of the eukaryotic cytoskeleton, are a lattice of polymerized tubulin dimers and are crucial for various cellular processes. The genetic and chemical diversity of tubulin and their disordered tails gives rise to a “tubulin code”. The functional role of tubulin post-translational modifications (PTMs), which contribute to the chemical diversity of the tubulin code, is gradually being unraveled. However, variation in the length and spatial organization of tubulin poly-modifications leads to an enormous combinatorial PTM space, which is difficult to study experimentally. Hence, the impact of the combinatorial tubulin PTM space on the biophysical properties of tubulin tails and their interactions with other proteins remains elusive.Here, we combine all-atom and coarse-grained molecular dynamics simulations to elucidate the biophysical implications of the large combinatorial tubulin PTM space in the context of an MT lattice. We find that tail–body interactions are more dominant in the tubulin dimer than in an MT lattice, and are more significant for the tails of α compared with β tubulin. In addition, polyglutamylation, but not polyglycylation, expands the dimensions of the tubulin tails. Polyglutamylation also leads to a decrease in the diffusion rate of MT-associated protein EB1 on MTs, while polyglycylation often increases diffusion rate. These observations are generally not sensitive to the organization of the polymodifications. The effect of PTMs on MT charge density and tail dynamics are also discussed. Overall, this study presents a molecular quantification of the biophysical properties of tubulin tails and their polymodifications, and provides predictions on the functional importance of tubulin PTMs.     

Geometrical nonlinear elasticity of axon under tension: A coarse-grained computational study

Axon bundles cross-linked by microtubule (MT) associate proteins and bounded by a shell skeleton are critical for normal function of neurons. Understanding effects of the complexly geometrical parameters on their mechanical properties can help gain a biomechanical perspective on the neurological functions of axons and thus brain disorders caused by the structural failure of axons. Here, the tensile mechanical properties of MT bundles cross-linked by tau proteins are investigated by systematically tuning MT length, axonal cross-section radius, and tau protein spacing in a bead-spring coarse-grained model. Our results indicate that the stress-strain curves of axons can be divided into two regimes, a nonlinear elastic regime dominated by rigid-body like inter-MT sliding, and a linear elastic regime dominated by affine deformation of both tau proteins and MTs. From the energetic analyses, first, the tau proteins dominate the mechanical performance of axons under tension. In the nonlinear regime, tau proteins undergo a rigid-body like rotating motion rather than elongating, whereas in the nonlinear elastic regime, tau proteins undergo a flexible elongating deformation along the MT axis. Second, as the average spacing between adjacent tau proteins along the MT axial direction increases from 25 to 125 nm, the Young’s modulus of axon experiences a linear decrease whereas with the average space varying from 125 to 175 nm, and later reaches a plateau value with a stable fluctuation. Third, the increment of the cross-section radius of the MT bundle leads to a decrease in Young’s modulus of axon, which is possibly attributed to the decrease in MT numbers per cross section. Overall, our research findings offer a new perspective into understanding the effects of geometrical parameters on the mechanics of MT bundles as well as serving as a theoretical basis for the development of artificial MT complexes potentially toward medical applications.     

Torsional Behavior of Axonal Microtubule Bundles

Axonal microtubule (MT) bundles crosslinked by microtubule-associated protein (MAP) tau are responsible for vital biological functions such as maintaining mechanical integrity and shape of the axon as well as facilitating axonal transport. Breaking and twisting of MTs have been previously observed in damaged undulated axons. Such breaking and twisting of MTs is suggested to cause axonal swellings that lead to axonal degeneration, which is known as “diffuse axonal injury”. In particular, overstretching and torsion of axons can potentially damage the axonal cytoskeleton. Following our previous studies on mechanical response of axonal MT bundles under uniaxial tension and compression, this work seeks to characterize the mechanical behavior of MT bundles under pure torsion as well as a combination of torsional and tensile loads using a coarse-grained computational model. In the case of pure torsion, a competition between MAP tau tensile and MT bending energies is observed. After three turns, a transition occurs in the mechanical behavior of the bundle that is characterized by its diameter shrinkage. Furthermore, crosslink spacing is shown to considerably influence the mechanical response, with larger MAP tau spacing resulting in a higher rate of turns. Therefore, MAP tau crosslinking of MT filaments protects the bundle from excessive deformation. Simultaneous application of torsion and tension on MT bundles is shown to accelerate bundle failure, compared to pure tension experiments. MAP tau proteins fail in clusters of 10–100 elements located at the discontinuities or the ends of MT filaments. This failure occurs in a stepwise fashion, implying gradual accumulation of elastic tensile energy in crosslinks followed by rupture. Failure of large groups of interconnecting MAP tau proteins leads to detachment of MT filaments from the bundle near discontinuities. This study highlights the importance of torsional loading in axonal damage after traumatic brain injury.     

Molecular modeling study on the differential microtubule-stabilizing effect in singly- and doubly-bonded complexes with peloruside A and paclitaxel

Microtubules (MT) are dynamic cytoskeletal components that play a crucial role in cell division. Disrupting MT dynamics by MT stabilizers is a widely employed strategy to control cell proliferation in cancer therapy. Most MT stabilizers bind to the taxol (TX) site located at the luminal interface between protofilaments, except laulimalide and peloruside A (PLA), which bind to an interfacial pocket on outer MT surface. Cryo-electron microscopy MTs reconstructions have shown differential structural effects on the MT lattice in singly- and doubly-bonded complexes with PLA, TX, and PLA/TX, as PLA is able to revert the lattice heterogeneity induced by TX association leading to more regular MT assemblies. In this work, fully-atomistic molecular dynamics simulations were employed to examine the single and double association of MT stabilizers to reduced MT models in the search for structural and energetic evidence that could be related to the differential regularization and stabilization effects exerted by PLA and TX on the MT lattice. Our results revealed that the double association of PLA/TX (a) strengthens the lateral contact between tubulin dimers compared to singly-bonded complexes, (b) favors a more parallel arrangement between tubulin dimers, and (c) induces a larger restriction in the interdimeric conformational motion increasing the probability of finding structures consistent with 13-protofilaments arrangements. These results and are valuable to increase understanding about the molecular mechanism of action of MT stabilizers, and could account for an overstabilization of MTs in doubly-bonded complexes compared to singly-bonded systems.     

Structural studies of N-terminal mutants of Connexin 32 using (1)H NMR spectroscopy

Microtubules (MTs) control cell replication, material transport and motion in eukaryotic cells, but MT role in several pathologies is still unknown. These functions are related to the MT physico-chemical properties and MT formation mode starting from tubulin molecules. This study describes a new method, based on the computer aided analysis of the electron paramagnetic resonance (EPR) spectra of selected spin probes to obtain structural and dynamical information on tubulins and MTs and the kinetics of MTs formation promoted by guanosine-5'-triphosphate (GTP). It was found that tubulin and MTs avoid radical quenching caused by ethylene glycol tetraacetic acid (EGTA). MT formation showed different kinetics as a function of tubulin concentration. At 5 mg/mL of tubulin, MTs were formed in 8 min. These results are also useful for getting information on MT-drug interactions.     

Cytoplasmic dynein is involved in the retention of microtubules at the centrosome in interphase cells

Cytoplasmic dynein is known to be involved in the establishment of radial microtubule (MT) arrays. During mitosis, dynein activity is required for tethering of the MTs at the spindle poles. In interphase cells, dynein inhibitors induce loss of radial MT organization; however, the exact role of dynein in the maintenance of MT arrays is unclear. Here, we examined the effect of dynein inhibitors on MT distribution and the centrosome protein composition in cultured fibroblasts. We found that while these inhibitors induced rapid ( t _1/2 ∼ 20 min) loss of radial MT organization, the levels of key centrosomal proteins or the rates of MT nucleation did not change significantly in dynein-inhibited cells, suggesting that the loss of dynein activity does not affect the structural integrity of the centrosome or its capacity to nucleate MTs. Live observations of the centrosomal activity showed that dynein inhibition enhanced the detachment of MTs from the centrosome. We conclude that the primary role of dynein in the maintenance of a radial MT array in interphase cells consists of retention of MTs at the centrosome and hypothesize that dynein has a role in the MT retention, separate from the delivery to the centrosome of MT-anchoring proteins.     

Microtubular Protofilament Analysis Based on Molecular Level Tubulin Interaction

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

Molecular structural mechanics model for the mechanical properties of microtubules

The aim of this paper was to develop a structural mechanics (SM) model for the microtubules (MTs) in cells. The technique enables one to study the configuration effect on the mechanical properties of MTs and enjoys greatly improved computational efficiency as compared with molecular dynamics simulations. The SM model shows that the Young’s modulus has nearly a constant value around 0.83 GPa, whereas the shear modulus, two orders of magnitude lower, varies considerably with the protofilament number \(N\) and helix-start number \(S\) . The dependence of the bending stiffness and persistence length on the MT length and protofilament number \(N\) is also examined and explained based on the continuum mechanics theories. Specifically, the SM model is found to be in good agreement with available simulation and experiment results, showing its robustness in studying the static deformation of MTs and the potential for characterizing the buckling and vibration of MTs as well as the mechanical behaviour of intermediate and actin filaments.     

Crosslinkers and motors organize dynamic microtubules to form stable bipolar arrays in fission yeast

Microtubule (MT) nucleation not only occurs from centrosomes, but also in large part from dispersed nucleation sites. The subsequent sorting of short MTs into networks like the mitotic spindle requires molecular motors that laterally slide overlapping MTs and bundling proteins that statically connect MTs. How bundling proteins interfere with MT sliding is unclear. In bipolar MT bundles in fission yeast, we found that the bundler ase1p localized all along the length of antiparallel MTs, whereas the motor klp2p (kinesin-14) accumulated only at MT plus ends. Consequently, sliding forces could only overcome resistant bundling forces for short, newly nucleated MTs, which were transported to their correct position within bundles. Ase1p thus regulated sliding forces based on polarity and overlap length, and computer simulations showed these mechanisms to be sufficient to generate stable bipolar bundles. By combining motor and bundling proteins, cells can thus dynamically organize stable regions of overlap between cytoskeletal 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.     

Nonlocal Mechanism of Self-Organization and Centering of Microtubule Asters

Fragments of fish melanophore cells can form and center aggregates of pigment granules by dynein-motor-driven transport along a self-organized radial array of microtubules (MTs). We present a quantitative model that describes pigment aggregation and MT-aster self-organization and the subsequent centering of both structures. The model is based on the observations that MTs are immobile and treadmill, while dynein-motor-covered granules have the ability to nucleate MTs. From assumptions based on experimental observations, we derive partial integro-differential equations describing the coupled granule–MT interaction. We use scaling arguments and perturbation theory to study the model in two limiting cases. The model analysis explains the mechanism of aster self-organization as a positive feedback loop between motor aggregation at the MT minus ends and MT nucleation by motors. Furthermore, the centering mechanism is explained by the spontaneous nucleation of MTs throughout the cytosol which acts as a volume sensing tool. Numerical simulations lend additional support to the analysis. The model sheds light on role of polymer dynamics and polymer–motor interactions in cytoskeletal organization.