Nanomechanics of microtubules

How cytoskeletal crosstalk makes cells move: Bridging cell-free and cell studies

Cell migration is a fundamental process for life and is highly dependent on the dynamical and mechanical properties of the cytoskeleton. Intensive physical and biochemical crosstalk among actin, microtubules, and intermediate filaments ensures their coordination to facilitate and enable migration. In this review, we discuss the different mechanical aspects that govern cell migration and provide, for each mechanical aspect, a novel perspective by juxtaposing two complementary approaches to the biophysical study of cytoskeletal crosstalk: live-cell studies (often referred to as top-down studies) and cell-free studies (often referred to as bottom-up studies). We summarize the main findings from both experimental approaches, and we provide our perspective on bridging the two perspectives to address the open questions of how cytoskeletal crosstalk governs cell migration and makes cells move.

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.

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 coupling coordinates microtubule growth

During mitosis, kinetochore-attached microtubules form bundles (k-fibers) in which many filaments grow and shorten in near-perfect unison to align and segregate each chromosome. However, individual microtubules grow at intrinsically variable rates, which must be tightly regulated for a k-fiber to behave as a single unit. This exquisite coordination might be achieved biochemically, via selective binding of polymerases and depolymerases, or mechanically, because k-fiber microtubules are coupled through a shared load that influences their growth. Here, we use a novel dual laser trap assay to show that microtubule pairs growing in vitro are coordinated by mechanical coupling. Kinetic analyses show that microtubule growth is interrupted by stochastic, force-dependent pauses and indicate persistent heterogeneity in growth speed during non-pauses. A simple model incorporating both force-dependent pausing and persistent growth speed heterogeneity explains the measured coordination of microtubule pairs without any free fit parameters. Our findings illustrate how microtubule growth may be synchronized during mitosis and provide a basis for modeling k-fiber bundles with three or more microtubules, as found in many eukaryotes.

Mechanical coupling coordinates microtubule growth

During mitosis, kinetochore-attached microtubules form bundles (k-fibers) in which many filaments grow and shorten in near-perfect unison to align and segregate each chromosome. However, individual microtubules grow at intrinsically variable rates, which must be tightly regulated for a k-fiber to behave as a single unit. This exquisite coordination might be achieved biochemically, via selective binding of polymerases and depolymerases, or mechanically, because k-fiber microtubules are coupled through a shared load that influences their growth. Here, we use a novel dual laser trap assay to show that microtubule pairs growing in vitro are coordinated by mechanical coupling. Kinetic analyses show that microtubule growth is interrupted by stochastic, force-dependent pauses and indicate persistent heterogeneity in growth speed during non-pauses. A simple model incorporating both force-dependent pausing and persistent growth speed heterogeneity explains the measured coordination of microtubule pairs without any free fit parameters. Our findings illustrate how microtubule growth may be synchronized during mitosis and provide a basis for modeling k-fiber bundles with three or more microtubules, as found in many eukaryotes.

The cyto-linker and scaffolding protein "plectin" mis-localization leads to softening of cancer cells

Discovering tools to prevent cancer progression requires understanding the fundamental differences between normal and cancer cells. More than a decade ago, atomic force microscopy (AFM) revealed cancer cells' softer body compared to their healthy counterparts. Here, we investigated the mechanism underlying the softening of cancerous cells in comparison with their healthy counterparts based on AFM high resolution stiffness tomography and 3D confocal microscopy. We showed microtubules (MTs) network in invasive ductal carcinoma cell cytoskeleton is basally located and segmented for around 400 nm from the cell periphery. Additionally, the cytoskeleton scaffolding protein plectin exhibits a mis-localization from the cytoplasm to the surface of cells in the carcinoma which justifies the dissociation of the MT network from the cell's cortex. Furthermore, the assessment of MTs' persistence length using a worm-like-chain (WLC) model in high resolution AFM images showed lower persistence length of the single MTs in ductal carcinoma compared to that in the normal state. Overall, these tuned mechanics support the invasive cells to ascertain more flexibility under compressive forces in small deformations. These data provide new insights into the structural origins of cancer aids in progression.

Filament displacement image analytics tool for use in investigating dynamics of dense microtubule networks

The fate and motion of cells is influenced by a variety of physical characteristics of their microenvironments. Traditionally, mechanobiology focuses on external mechanical phenomena such as cell movement and environmental sensing. However, cells are inherently dynamic, where internal waves and internal oscillations are a hallmark of living cells observed under a microscope. We propose that these internal mechanical rhythms provide valuable information about cell health. Therefore, it is valuable to capture the rhythms inside cells and quantify how drugs or physical interventions affect a cell's internal dynamics. One of the key dynamical entities inside cells is the microtubule network. Typically, microtubule dynamics are measured by end-protein tracking. In contrast, this paper introduces an easy-to-implement approach to measure the lateral motion of the microtubule filaments embedded within dense networks with (at least) confocal resolution image sequences. Our tool couples the computer vision algorithm Optical Flow with an anisotropic, rotating Laplacian of Gaussian filtering to characterize the lateral motion of dense microtubule networks. We then showcase additional image analytics used to understand the effect of microtubule orientation and regional location on lateral motion. We argue that our tool and these additional metrics provide a fuller picture of the active forcing environment within cells.

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.

Physically informed data-driven modeling of active nematics

A continuum description is essential for understanding a variety of collective phenomena in active matter. However, building quantitative continuum models of active matter from first principles can be extremely challenging due to both the gaps in our knowledge and the complicated structure of nonlinear interactions. Here, we use a physically informed data-driven approach to construct a complete mathematical model of an active nematic from experimental data describing kinesin-driven microtubule bundles confined to an oil-water interface. We find that the structure of the model is similar to the Leslie-Ericksen and Beris-Edwards models, but there are appreciable and important differences. Rather unexpectedly, elastic effects are found to play no role in the experiments considered, with the dynamics controlled entirely by the balance between active stresses and friction stresses.

Nanomechanical Signatures in Glioma Cells Depend on CD44 Distribution in IDH1 Wild-Type but Not in IDH1R132H Mutant Early-Passage Cultures

Atomic force microscopy (AFM) recently burst into biomedicine, providing morphological and functional characteristics of cancer cells and their microenvironment responsible for tumor invasion and progression, although the novelty of this assay needs to coordinate the malignant profiles of patients' specimens to diagnostically valuable criteria. Applying high-resolution semi-contact AFM mapping on an extended number of cells, we analyzed the nanomechanical properties of glioma early-passage cell cultures with a different IDH1 R132H mutation status. Each cell culture was additionally clustered on CD44+/- cells to find possible nanomechanical signatures that differentiate cell phenotypes varying in proliferative activity and the characteristic surface marker. IDH1 R132H mutant cells compared to IDH1 wild-type ones (IDH1wt) characterized by two-fold increased stiffness and 1.5-fold elasticity modulus. CD44+/IDH1wt cells were two-fold more rigid and much stiffer than CD44-/IDH1wt ones. In contrast to IDH1 wild-type cells, CD44+/IDH1 R132H and CD44-/IDH1 R132H did not exhibit nanomechanical signatures providing statistically valuable differentiation of these subpopulations. The median stiffness depends on glioma cell types and decreases according to the following manner: IDH1 R132H mt (4.7 mN/m), CD44+/IDH1wt (3.7 mN/m), CD44-/IDH1wt (2.5 mN/m). This indicates that the quantitative nanomechanical mapping would be a promising assay for the quick cell population analysis suitable for detailed diagnostics and personalized treatment of glioma forms.

Thermal Treatment Impact on the Mechanical Properties of Mg3Si2O5(OH)4 Nanoscrolls

A group of phyllosilicate nanoscrolls conjoins several hydrosilicate layered compounds with a size mismatch between octahedral and tetrahedral sheets. Among them, synthetic Mg₃Si₂O₅(OH)₄ chrysotile nanoscrolls (obtained via the hydrothermal method) possess high thermal stability and mechanical properties, making them prospective composite materials fillers. However, accurate determination of these nano-objects with Young's modulus remains challenging. Here, we report on a study of the mechanical properties evolution of individual synthetic phyllosilicate nanoscrolls after a series of heat treatments, observed with an atomic force microscopy and calculated using the density functional theory. It appears that the Young's modulus, as well as shear deformation's contribution to the nanoscrolls mechanical behavior, can be controlled by heat treatment. The main reason for this is the heat-induced formation of covalent bonding between the adjacent layers, which complicate the shear deformation.

Measurement of the Persistence Length of Cytoskeletal Filaments using Curvature Distributions

Cytoskeletal filaments such as microtubules and actin filaments play important roles in the mechanical integrity of cells and the ability of cells to respond to their environment. Measuring the mechanical properties of cytoskeletal structures is crucial for gaining insight into intracellular mechanical stresses and their role in regulating cellular processes. One of the ways to characterize these mechanical properties is by measuring their persistence length, the average length over which filaments stay straight. There are several approaches in the literature for measuring filament deformations, such as Fourier analysis of images obtained using fluorescence microscopy. Here, we show how curvature distributions can be used as an alternative tool to quantify bio-filament deformations, and investigate how the apparent stiffness of filaments depends on the resolution and noise of the imaging system. We present analytical calculations of the scaling curvature distributions as a function of filament discretization, and test our predictions by comparing Monte Carlo simulations to results from existing techniques. We also apply our approach to microtubules and actin filaments obtained from in vitro gliding assay experiments with high densities of non-functional motors, and calculate the persistence length of these filaments. The presented curvature analysis is significantly more accurate compared to existing approaches for small data sets, and can be readily applied to both in vitro or in vivo filament data through the use of the open-source codes we provide.

Linking path and filament persistence lengths of microtubules gliding over kinesin

Microtubules and kinesin motor proteins are involved in intracellular transports in living cells. Such intracellular material transport systems can be reconstructed for utilisation in synthetic environments, and they are called molecular shuttles driven by kinesin motors. The performance of the molecular shuttles depends on the nature of their trajectories, which can be characterized by the path persistence length of microtubules. It has been theoretically predicted that the path persistence length should be equal to the filament persistence length of the microtubules, where the filament persistence length is a measure of microtubule flexural stiffness. However, previous experiments have shown that there is a significant discrepancy between the path and filament persistence lengths. Here, we showed how this discrepancy arises by using computer simulation. By simulating molecular shuttle movements under external forces, the discrepancy between the path and filament persistence lengths was reproduced as observed in experiments. Our close investigations of molecular shuttle movements revealed that the part of the microtubules bent due to the external force was extended more than it was assumed in the theory. By considering the extended length, we could elucidate the discrepancy. The insights obtained here are expected to lead to better control of molecular shuttle movements.

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.

Dynamics of exciton polaron in microtubule

In this paper, we study the dynamical properties of the exciton-polaron in the microtubule. The study was carried out using a unitary transformation and an approximate diagonalization technique. Analytically, the modeling of exciton-polaron dynamics in microtubules is presented. From this model, the ground state energy, mobility, and entropy of the exciton-polaron are derived as a function of microtubule's parameters. Numerical results show that, depending on the three vibrational modes (protofilament, helix, antihelix) in MTs, exciton-polaron energy is anisotropic and is more present on the protofilament than the helix and absent on the antihelix. Taking into account the variation of the protofilament vibrations by fixing the helix vibrations, exciton-polaron moves between the 1st and 2nd protofilaments. It is seen that the variation of the two vibrations induces mobility of the quasiparticle between the 1st and 15th protofilament. This result points out the importance of helix vibrations on the dynamics of quasiparticles. It is observed that the mobility of the exciton polaron and the entropy of the system are strongly influenced by the vibrations through the protofilament and helix. The effects of the one through the antihelix is negligible. The entropy of the system is similar to that of mobility. Confirming that the quasiparticles move in the protofilament faster than in the helix.

Microtubule Preparation for Investigation with High-Speed Atomic Force Microscopy

High-speed atomic force microscopy (AFM) is a versatile method that can visualize proteins and protein systems on the nanometer scale and at a temporal resolution of 100 ms. The application to microtubules can not only reveal structural information with single-tubulin resolution but can also extract mechanical information and allows to study single motor proteins walking on microtubules, among others. This chapter provides a step-by-step guide from microtubule polymerization to successful observation with high-speed AFM.

Biomechanics of cells and subcellular components: A comprehensive review of computational models and applications

Cells are a fundamental structural, functional and biological unit for all living organisms. Up till now, considerable efforts have been made to study the responses of single cells and subcellular components to an external load, and understand the biophysics underlying cell rheology, mechanotransduction and cell functions using experimental and in silico approaches. In the last decade, computational simulation has become increasingly attractive due to its critical role in interpreting experimental data, analysing complex cellular/subcellular structures, facilitating diagnostic designs and therapeutic techniques, and developing biomimetic materials. Despite the significant progress, developing comprehensive and accurate models of living cells remains a grand challenge in the 21st century. To understand current state of the art, this review summarises and classifies the vast array of computational biomechanical models for cells. The article covers the cellular components at multi-spatial levels, that is, protein polymers, subcellular components, whole cells and the systems with scale beyond a cell. In addition to the comprehensive review of the topic, this article also provides new insights into the future prospects of developing integrated, active and high-fidelity cell models that are multiscale, multi-physics and multi-disciplinary in nature. This review will be beneficial for the researchers in modelling the biomechanics of subcellular components, cells and multiple cell systems and understanding the cell functions and biological processes from the perspective of cell mechanics.

Physically-based structural modeling of a typical regenerative tissue analog bridges material macroscale continuum and cellular microscale discreteness and elucidates the hierarchical characteristics of cell-matrix interaction

This paper presents a comprehensive physically-based structural modelling for the passive and active biomechanical processes in a typical engineered tissue - namely, cell-compacted collagen gel. First, it introduces a sinusoidal curve analog for quantifying the mechanical response of the collagen fibrils and a probability distribution function of the characteristic crimp ratio for taking into account the fibrillar geometric entropic effect. The constitutive framework based on these structural characteristics precisely reproduces the nonlinearity, the viscoelasticity, and fairly captures the Poisson effect exhibiting in the macroscale tensile tests; which, therefore, substantially validates the structural modelling for the analysis of the cell-gel interaction during collagen gel compaction. Second, a deterministic molecular clutch model specific to the interaction between the cell pseudopodium and the collagen network is developed, which emphasizes the dependence of traction force on clutch number altering with the retrograde flow velocity, actin polymeric velocity, and the deformation of the stretched fibril. The modelling reveals the hierarchical features of cellular substrate sensing, i.e. a biphasic traction force response to substrate elasticity begins at the level of individual fibrils and develops into the second biphasic sensing by means of the fibrillar number integration at the whole-cell level. Singular in crossing the realms of continuum and discrete mechanics, the methodologies developed in this study for modelling the filamentous materials and cell-fibril interaction deliver deep insight into the temporospatially dynamic 3D cell-matrix interaction, and are able to bridge the cellular microscale and material macroscale in the exploration of related topics in mechanobiology.

Growth rate-dependent flexural rigidity of microtubules influences pattern formation in collective motion

Background

Microtubules (MTs) are highly dynamic tubular cytoskeleton filaments that are essential for cellular morphology and intracellular transport. In vivo, the flexural rigidity of MTs can be dynamically regulated depending on their intracellular function. In the in vitro reconstructed MT-motor system, flexural rigidity affects MT gliding behaviors and trajectories. Despite the importance of flexural rigidity for both biological functions and in vitro applications, there is no clear interpretation of the regulation of MT flexural rigidity, and the results of many studies are contradictory. These discrepancies impede our understanding of the regulation of MT flexural rigidity, thereby challenging its precise manipulation.

Results

Here, plausible explanations for these discrepancies are provided and a new method to evaluate the MT rigidity is developed. Moreover, a new relationship of the dynamic and mechanic of MTs is revealed that MT flexural rigidity decreases through three phases with the growth rate increases, which offers a method of designing MT flexural rigidity by regulating its growth rate. To test the validity of this method, the gliding performances of MTs with different flexural rigidities polymerized at different growth rates are examined. The growth rate-dependent flexural rigidity of MTs is experimentally found to influence the pattern formation in collective motion using gliding motility assay, which is further validated using machine learning.

Conclusion

Our study establishes a robust quantitative method for measurement and design of MT flexural rigidity to study its influences on MT gliding assays, collective motion, and other biological activities in vitro. The new relationship about the growth rate and rigidity of MTs updates current concepts on the dynamics and mechanics of MTs and provides comparable data for investigating the regulation mechanism of MT rigidity in vivo in the future.

Graphic Abstract

Supplementary Information

The online version contains supplementary material available at 10.1186/s12951-021-00960-y.

A finite element model of an osteoblast to quantify the transduction of exogenous forces to cellular components

Encouraged by recent advances of biophysical and biochemical assays we introduce a 3D finite element model of an osteoblast, seeking an analogue between exogenous forces and intracellularly activated sensory mechanisms. The cell was reverse engineered and the dimensions of the internal cellular structures were based on literature data. The model was verified and validated against atomic force microscopy experiments and four loading scenarios were considered. The stress distributions developing on the main cellular components were calculated along with their corresponding strain values. The nucleus and mitochondria exhibited similar loading trends, with the mitochondria being stressed by an order of magnitude higher than the nucleus (e.g. 1.4 vs. 0.16 MPa). Equivalent stiffness was determined to increase by almost 50%, from the apex to the cell's periphery, as was the cell's elasticity, which was lowest when the load was exerted directly above the nucleus. The assessment of how extrinsic loads are propagated to a cell's internal structures is inherently a problem of high complexity. The findings presented in this study can provide important insight into biophysical and biochemical responses elicited in cells through mechanical stimulus. This was evident in both the nuclear and mitochondrial loading and would stipulate the important contribution of even more accurate models in the interpretation of cellular events. One Sentence Summary: The results of this numerical biomechanical study demonstrated that even minor extrinsic loads irrespective of the application site, are transduced by a fraction of the cytoskeleton to its internal structure (primarily to its mitochondria and secondary to the cell's nucleus), indicating mechanical stimulus as the dominant pathway to cell expression.

Cells in Slow Motion: Apparent Undercooling Increases Glassy Behavior at Physiological Temperatures

Solvent conditions are unexpectedly sufficient to drastically and reversibly slow down cells. In vitro on the molecular level, protein-solvent interactions drastically change in the presence of heavy water (D₂ O) and its stronger hydrogen bonds. Adding D₂ O to the cell medium of living cells increases the molecular intracellular viscosity. While cell morphology and phenotype remain unchanged, cellular dynamics transform into slow motion in a changeable manner. This is exemplified in the slowdown of cell proliferation and migration, which is caused by a reversible gelation of the cytoplasm. In analogy to the time-temperature superposition principle, where temperature is replaced by D₂ O, an increase in viscosity slows down the effective time. Actin networks, crucial structures in the cytoplasm, switch from a power-law-like viscoelastic to a more rubber-like elastic behavior. The resulting intracellular resistance and dissipation impair cell movement. Since cells are highly adaptive non-equilibrium systems, they usually respond irreversibly from a thermodynamic perspective. D₂ O induced changes, however, are fully reversible and their effects are independent of signaling as well as expression. The stronger hydrogen bonds lead to glass-like, drawn-out intramolecular dynamics, which may facilitate longer storage times of biological matter, for instance, during transport of organ transplants.

Progress in the mechanical modulation of cell functions in tissue engineering

In mammals, mechanics at multiple stages-nucleus to cell to ECM-underlie multiple physiological and pathological functions from its development to reproduction to death. Under this inspiration, substantial research has established the role of multiple aspects of mechanics in regulating fundamental cellular processes, including spreading, migration, growth, proliferation, and differentiation. However, our understanding of how these mechanical mechanisms are orchestrated or tuned at different stages to maintain or restore the healthy environment at the tissue or organ level remains largely a mystery. Over the past few decades, research in the mechanical manipulation of the surrounding environment-known as substrate or matrix or scaffold on which, or within which, cells are seeded-has been exceptionally enriched in the field of tissue engineering and regenerative medicine. To do so, traditional tissue engineering aims at recapitulating key mechanical milestones of native ECM into a substrate for guiding the cell fate and functions towards specific tissue regeneration. Despite tremendous progress, a big puzzle that remains is how the cells compute a host of mechanical cues, such as stiffness (elasticity), viscoelasticity, plasticity, non-linear elasticity, anisotropy, mechanical forces, and mechanical memory, into many biological functions in a cooperative, controlled, and safe manner. High throughput understanding of key cellular decisions as well as associated mechanosensitive downstream signaling pathway(s) for executing these decisions in response to mechanical cues, solo or combined, is essential to address this issue. While many reports have been made towards the progress and understanding of mechanical cues-particularly, substrate bulk stiffness and viscoelasticity-in regulating the cellular responses, a complete picture of mechanical cues is lacking. This review highlights a comprehensive view on the mechanical cues that are linked to modulate many cellular functions and consequent tissue functionality. For a very basic understanding, a brief discussion of the key mechanical players of ECM and the principle of mechanotransduction process is outlined. In addition, this review gathers together the most important data on the stiffness of various cells and ECM components as well as various tissues/organs and proposes an associated link from the mechanical perspective that is not yet reported. Finally, beyond addressing the challenges involved in tuning the interplaying mechanical cues in an independent manner, emerging advances in designing biomaterials for tissue engineering are also explored.

Graph prolongation convolutional networks: explicitly multiscale machine learning on graphs with applications to modeling of cytoskeleton

We define a novel type of ensemble graph convolutional network (GCN) model. Using optimized linear projection operators to map between spatial scales of graph, this ensemble model learns to aggregate information from each scale for its final prediction. We calculate these linear projection operators as the infima of an objective function relating the structure matrices used for each GCN. Equipped with these projections, our model (a Graph Prolongation-Convolutional Network) outperforms other GCN ensemble models at predicting the potential energy of monomer subunits in a coarse-grained mechanochemical simulation of microtubule bending. We demonstrate these performance gains by measuring an estimate of the Floating Point OPerations spent to train each model, as well as wall-clock time. Because our model learns at multiple scales, it is possible to train at each scale according to a predetermined schedule of coarse vs. fine training. We examine several such schedules adapted from the algebraic multigrid literature, and quantify the computational benefit of each. We also compare this model to another model which features an optimized coarsening of the input graph. Finally, we derive backpropagation rules for the input of our network model with respect to its output, and discuss how our method may be extended to very large graphs.

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.

Length dependence of the rigidity of microtubules in small networks

Microtubules often form sparse networks in eukaryotic cells which simultaneously contribute to shape maintenance and help establish overall cell layout. It is therefore important to quantify not only how these filaments function individually but also as a coupled network. We have developed a straightforward approach to assemble such networks de novo and we now use it to measure microtubule rigidity within small networks under controlled conditions. Our results suggest that microtubule rigidity increases with the contour length of the filament both for single microtubules and within small microtubule networks.

Individual tubular J-aggregates stabilized and stiffened by silica encapsulation

Amphiphilic cyanine dyes in aqueous solution self-assemble into J-aggregates with diverse structures. In particular, the dye 3,3′-bis(3-sulfopropyl)-5,5′,6,6′-tetrachloro-1,1′-dioctylbenzimida-carbo-cyanine (C8S3) forms micrometer long double walled tubular J-aggregates with a uniform outer diameter of 13 ± 0.5 nm. Interestingly, these J-aggregates exhibit strong exciton delocalization and migration, similar to natural light harvesting systems. However, their structural integrity and hence their optical properties are very sensitive to their chemical environment as well as to mechanical deformation, rendering detailed studies on individual tubular J-aggregates difficult. We addressed this issue and examined a previously published route for their chemical and mechanical stabilization by in situ synthesis of a silica coating that leaves their absorbance and emission unaltered in solution. Here, we demonstrate that the silica shell with a thickness of a few nanometers is able to stabilize the tubular J-aggregates of C8S3 against changes of pH of solutions down to values where pure aggregates are oxidized, against drying under ambient conditions, and even against the vacuum conditions within an electron microscope. Dried silica–covered aggregates are brittle, as demonstrated by manipulation with a scanning force microscope on a surface. Transmission electron microscope images confirm that the thickness of the coatings is homogeneous and uniform with a thickness of less than 5 nm; scanning TEM energy dispersive X-ray spectroscopy confirms the chemical composition of the shell as SiO2; and electron energy loss spectra could be recorded across a single freely suspended aggregate. Such a silica shell may not only serve for stabilization but also could be the base for further functionalization of the aggregates by either chemical attachment of other units on top of the shell or by inclusion during the synthesis.

When Stiffness Matters: Mechanosensing in Heart Development and Disease

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

Third-order shear deformation beam model for flexural waves and free vibration of pipes

A third-order shear deformation beam model is proposed to analyze dynamic behavior of straight hollow cylinders of annular cross-section, in which shear stress vanishes on the inner and outer surfaces of the pipe. Shear deformation, warping, and rotational inertia of cross-section are all considered, and the shear correction factor is not needed. A single governing differential equation is derived for analyzing flexural wave propagation and free vibration of straight pipe-beams. The phase and group speeds of flexural waves propagating in pipes are determined for acoustic and optical modes. The dispersion of flexural waves is analyzed. The frequency equations are obtained explicitly for pipe-beams with ten typical boundary conditions including clamped, pinned, guided, and free ends. The natural frequencies of clamped-free, clamped-clamped, and pinned-pinned pipe-beams are evaluated for the first four vibration modes. A comparison of this paper's numerical results of the natural frequencies with the previous ones is made and turns out the effectiveness of the suggested method. The influences of the pipe's thickness and length on the natural frequencies and mode shapes for a cantilever pipe are presented.

Curvature Regularization near Contacts with Stretched Elastic Tubes

Inserting a rigid object into a soft elastic tube produces conformal contact between the two, resulting in contact lines. The curvature of the tube walls near these contact lines is often large and is typically regularized by the finite bending rigidity of the tube. Here, it is demonstrated using experiments and a Föppl-von Kármán-like theory that a second, independent, mechanism of curvature regularization occurs when the tube is axially stretched. In contrast with the effects of finite bending rigidity, the radius of curvature obtained increases with the applied stretching force and decreases with sheet thickness. The dependence of the curvature on a suitably rescaled stretching force is found to be universal, independent of the shape of the intruder, and results from an interplay between the longitudinal stresses due to the applied stretch and hoop stresses characteristic of curved geometry. These results suggest that curvature measurements can be used to infer the mechanical properties of stretched tubular structures.

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.

Label-free Imaging and Bending Analysis of Microtubules by ROCS Microscopy and Optical Trapping

Mechanical manipulation of single cytoskeleton filaments and their monitoring over long times is difficult because of fluorescence bleaching or phototoxic protein degradation. The integration of label-free microscopy techniques, capable of imaging freely diffusing, weak scatterers such as microtubules (MTs) in real-time, and independent of their orientation, with optical trapping and tracking systems, would allow many new applications. Here, we show that rotating-coherent-scattering microscopy (ROCS) in dark-field mode can also provide strong contrast for structures far from the coverslip such as arrangements of isolated MTs and networks. We could acquire thousands of images over up to 30 min without loss in image contrast or visible photodamage. We further demonstrate the combination of ROCS imaging with fast and nanometer-precise 3D interferometric back-focal-plane tracking of multiple beads in time-shared optical traps using acoustooptic deflectors to specifically construct and microrheologically probe small microtubule networks with well-defined geometries. Thereby, we explore the frequency-dependent elastic response of single microtubule filaments between 0.5 Hz and 5 kHz, which allows for investigating their viscoelastic response up to the fourth-order bending mode. Our spectral analysis reveals constant filament stiffness at low frequencies and frequency-dependent stiffening following a power law ∼ω^p with a length-dependent exponent p(L). We find further evidence for the dependence of the MT persistence length on the contour length L, which is still controversially debated. We could also demonstrate slower stiffening at high frequencies for longer filaments, which we believe is determined by the molecular architecture of the MT. Our results shed new light on the nanomechanics of this essential, multifunctional cytoskeletal element and pose new questions about the adaptability of the cytoskeleton.

An estimate to the first approximation of microtubule rupture force

Microtubule mechanical properties are essential for understanding basic cellular processes, including cell motility and division, but the forces that result in microtubule rupture or breakage have not yet been measured directly. These forces are essential to understand the mechanical properties of the cytoskeleton and responses by cells to both normal conditions and stress caused by injury or disease. Here we estimate the force required to rupture a microtubule by analyzing kinesin-14 Ncd motor-induced microtubule breakage in ensemble motility assays. We model the breakage events as caused by Ncd motors pulling or pushing on single microtubules that are clamped at one end by other motors attached to the glass surface. The number of pulling or pushing Ncd motors is approximated from the length of the microtubule bound to the surface and the forces produced by the pulling or pushing motors are estimated from forces produced by the Ncd motor in laser-trap assays, reported by others. Our analysis provides an estimate, to the first approximation, of ~ 500 pN for the minimal force required to rupture a 13-pf microtubule. The value we report is close to the forces estimated from microtubule stretching/fragmentation experiments and overlaps with the forces applied by AFM in microtubule indentation assays that destabilize microtubules and break microtubule protofilaments. It is also consistent with the forces required to disrupt protein noncovalent bonds in force spectroscopy experiments. These findings are relevant to microtubule deformation and breakage caused by cellular tension in vivo.

Electromechanical vibration of microtubules and its application in biosensors

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

Mechanical and thermodynamic properties of Aβ42, Aβ40, and α-synuclein fibrils: a coarse-grained method to complement experimental studies

We perform molecular dynamics simulation on several relevant biological fibrils associated with neurodegenerative diseases such as Aβ₄₀, Aβ₄₂, and α-synuclein systems to obtain a molecular understanding and interpretation of nanomechanical characterization experiments. The computational method is versatile and addresses a new subarea within the mechanical characterization of heterogeneous soft materials. We investigate both the elastic and thermodynamic properties of the biological fibrils in order to substantiate experimental nanomechanical characterization techniques that are quickly developing and reaching dynamic imaging with video rate capabilities. The computational method qualitatively reproduces results of experiments with biological fibrils, validating its use in extrapolation to macroscopic material properties. Our computational techniques can be used for the co-design of new experiments aiming to unveil nanomechanical properties of biological fibrils from a point of view of molecular understanding. Our approach allows a comparison of diverse elastic properties based on different deformations , i.e., tensile (Y _L), shear (S), and indentation (Y _T) deformation. From our analysis, we find a significant elastic anisotropy between axial and transverse directions (i.e., Y _T > Y _L) for all systems. Interestingly, our results indicate a higher mechanostability of Aβ₄₂ fibrils compared to Aβ₄₀, suggesting a significant correlation between mechanical stability and aggregation propensity (rate) in amyloid systems. That is, the higher the mechanical stability the faster the fibril formation. Finally, we find that α-synuclein fibrils are thermally less stable than β-amyloid fibrils. We anticipate that our molecular-level analysis of the mechanical response under different deformation conditions for the range of fibrils considered here will provide significant insights for the experimental observations.

Structural Changes in Tubulin Sheets Caused by Immobilization on Solid Supports

In the presence of zinc, the protein tubulin assembles into two-dimensional sheets that are a useful model system for the study of both tubulin and microtubule structure. Tubulin sheets present an ideal protein structure for study with atomic force microscopy because they contain a two-dimensional crystalline protein lattice and retain many of the structural features of tubulin and microtubules. However, high-resolution imaging requires nonperturbative immobilization onto an appropriate imaging substrate. In this report, several substrates commonly used for scanning probe microscopy are evaluated for their ability to effectively immobilize tubulin sheets: mica, gold, highly ordered pyrolytic graphite, and carbon-coated electron microscopy grids. We hypothesize that the different intermolecular interactions presented by these substrates will affect the morphology of adsorbed tubulin sheets as well as the amount of other contaminating adsorbates. Tubulin sheets were successfully imaged on all of these substrates and structural characterization is reported. The most consistent results were obtained on carbon-coated electron microscopy grids, which preserved fine structural features of the sheets and had the least amount of contamination from the adsorption of unpolymerized tubulin. Images of tubulin sheets obtained with atomic force microscopy also compare favorably with published electron micrographs of sheets produced using similar procedures. This work demonstrates the importance of assessing substrate effects when studying two-dimensional protein crystals and identifies suitable substrates for immobilizing tubulin sheets.

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.

Biomechanical Characterization at the Cell Scale: Present and Prospects

The rapidly growing field of mechanobiology demands for robust and reproducible characterization of cell mechanical properties. Recent achievements in understanding the mechanical regulation of cell fate largely rely on technological platforms capable of probing the mechanical response of living cells and their physico–chemical interaction with the microenvironment. Besides the established family of atomic force microscopy (AFM) based methods, other approaches include optical, magnetic, and acoustic tweezers, as well as sensing substrates that take advantage of biomaterials chemistry and microfabrication techniques. In this review, we introduce the available methods with an emphasis on the most recent advances, and we discuss the challenges associated with their implementation.

Shear Deformation Dissipates Energy in Biofilaments

Thermally fluctuating biofilaments possessing porous structures or viscoelastic properties exhibit energy losses from internal friction as well as external friction from drag. Prior models for internal friction account for energy dissipation solely from the dynamic bending of filaments. In this paper, we present a new energy dissipation model that captures the important effects of dynamic shear in addition to bending. Importantly, we highlight that shear-induced friction plays a major role in energy dissipation for shorter filaments and for shorter wavelengths (larger wavenumbers). The new model exhibits coupled shear-bending energy relaxation on two distinct time scales in lieu of a single time scale predicted by bending alone. We employ this model to interpret results from prior experiments on the internal friction of thermally fluctuating chromosomes and the drag-induced friction of thermally fluctuating microtubules. The examples confirm the energy relaxation on two time scales associated with internal friction and on two length scales associated with external friction. Overall, this new model that accounts for shear deformation yields superior estimates of energy dissipation for fluctuating biofilaments.

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.

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.

Elastic, dipole-dipole interaction and viscosity impact on vibrational properties of anisotropic hexagonal microtubule lattice

The paper investigates microtubules lattice properties taking into consideration elastic, dipole-dipole interaction of tubulins and viscosity. A microtubule is modeled as a system of bound tubulins, forming a skewed hexagonal two-dimensional lattice. Wave frequencies and group velocities have been calculated. Calculations have been performed for various directions of wave front propagation: helix, along the protofilament, and anti-helix. Three different wave polarization directions have been considered. It has been shown that the direction of the wave polarization influences the frequency and wave group velocity values in the lattice considerably. The impact of dipole-dipole interaction greatly depends on the direction of the wave polarization; thus, it is only moderate for the longitudinally (LA) polarized waves while it is sufficient for the transversely (TA), and out-of-plane (ZA) polarized waves. Moreover dipole-dipole interaction may result in the waves which are able to cause the rupture of microtubules. With viscosity considered, lattice oscillations become harmonically damped only over a certain wavelength range when longitudinal polarization occurs. Out of this range as well as for the other polarization directions, lattice deviations from equilibrium are dampened exponentially. Taking viscosity into consideration also results in a noticeable decrease in frequency and increase in the group wave velocity when the waves are longitudinally polarized. Reverse wave domains which may be associated with a possible phenomenon of negative refraction have been determined for hexagonal microtubule lattice.

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.

Elastic moduli of biological fibers in a coarse-grained model: crystalline cellulose and β-amyloids

We study the mechanical response of cellulose and β-amyloid microfibrils to three types of deformation: tensile, indentational, and shear. The cellulose microfibrils correspond to the allomorphs Iα or Iβ whereas the β-amyloid microfibrils correspond to the polymorphs of either two- or three-fold symmetry. This response can be characterized by three elastic moduli, namely, Y_L, Y_T, and S. We use a structure-based coarse-grained model to analyze the deformations in a unified manner. We find that each of the moduli is almost the same for the two allomorphs of cellulose but Y_L is about 20 times larger than Y_T (140 GPa vs. 7 GPa), indicating the existence of significant anisotropy. For cellulose we note that the anisotropy results from the involvement of covalent bonds in stretching. For β-amyloid, the sense of anisotropy is opposite to that of cellulose. In the three-fold symmetry case, Y_L is about half of Y_T (3 vs. 7) whereas for two-fold symmetry the anisotropy is much larger (1.6 vs. 21 GPa). The S modulus is derived to be 1.2 GPa for three-fold symmetry and one half of it for the other symmetry and 3.0 GPa for cellulose. The values of the moduli reflect deformations in the hydrogen-bond network. Unlike in our theoretical approach, no experiment can measure all three elastic moduli with the same apparatus. However, our theoretical results are consistent with various measured values: typical Y_L for cellulose Iβ ranges from 133 to 155 GPa, Y_T from 2 to 25 GPa, and S from 1.8 to 3.8 GPa. For β-amyloid, the experimental values of S and Y_T are about 0.3 GPa and 3.3 GPa respectively, while the value of Y_L has not been reported.