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

Microstructure-based modeling of primary cilia mechanics

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

Mechanical 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.

Inference of the force pattern acting on a semiflexible filament from shape analysis

The study of the active forces acting on semiflexible filaments networks such as the cytoskeleton requires noninvasive tools able to explore the deformation of single filaments in their natural environment. We propose here a practical method based on the solution of the hydrodynamic beam equation in the presence of transverse forces. We found that the derivative of the local curvature presents discontinuities that match the location of the applied forces, in contrast to the smooth curvature function obtained for the case of compressing longitudinal forces. These patterns can be easily appreciated in a kymograph of the curvature, which also reflects the temporal behavior of the forces. We assessed the method performance with numerical simulations describing the deformation of single microtubules provoked by the action of intracellular active forces.

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.

Developing Hybrid All-Atom and Ultra-Coarse-Grained Models to Investigate Taxol-Binding and Dynein Interactions on Microtubules

Simulating the conformations and functions of biological macromolecules by using all-atom (AA) models is a challenging task due to expensive computational costs. One possible strategy to solve this problem is to develop hybrid all-atom and ultra-coarse-grained (AA/UCG) models of the biological macromolecules. In the AA/UCG scheme, the interest regions are described by AA models, while the other regions are described in the UCG representation. In this study, we develop the hybrid AA/UCG models and apply them to investigate the conformational changes of microtubule-bound tubulins. The simulation results of the hybrid models elucidated the mechanism of why the taxol molecules selectively bound microtubules but not tubulin dimers. In addition, we also explore the interactions of the microtubules and dyneins. Our study shows that the hybrid AA/UCG model has great application potential in studying the function of complex biological systems.

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.

Approximate simulation of cortical microtubule models using dynamical graph grammars

Dynamical graph grammars (DGGs) are capable of modeling and simulating the dynamics of the cortical microtubule array (CMA) in plant cells by using an exact simulation algorithm derived from a master equation; however, the exact method is slow for large systems. We present preliminary work on an approximate simulation algorithm that is compatible with the DGG formalism. The approximate simulation algorithm uses a spatial decomposition of the domain at the level of the system's time-evolution operator, to gain efficiency at the cost of some reactions firing out of order, which may introduce errors. The decomposition is more coarsely partitioned by effective dimension (d= 0 to 2 or 0 to 3), to promote exact parallelism between different subdomains within a dimension, where most computing will happen, and to confine errors to the interactions between adjacent subdomains of different effective dimensions. To demonstrate these principles we implement a prototype simulator, and run three simple experiments using a DGG for testing the viability of simulating the CMA. We find evidence indicating the initial formulation of the approximate algorithm is substantially faster than the exact algorithm, and one experiment leads to network formation in the long-time behavior, whereas another leads to a long-time behavior of local alignment.

Structural specializations of the sperm tail

Sperm motility is crucial to reproductive success in sexually reproducing organisms. Impaired sperm movement causes male infertility, which is increasing globally. Sperm are powered by a microtubule-based molecular machine-the axoneme-but it is unclear how axonemal microtubules are ornamented to support motility in diverse fertilization environments. Here, we present high-resolution structures of native axonemal doublet microtubules (DMTs) from sea urchin and bovine sperm, representing external and internal fertilizers. We identify >60 proteins decorating sperm DMTs; at least 15 are sperm associated and 16 are linked to infertility. By comparing DMTs across species and cell types, we define core microtubule inner proteins (MIPs) and analyze evolution of the tektin bundle. We identify conserved axonemal microtubule-associated proteins (MAPs) with unique tubulin-binding modes. Additionally, we identify a testis-specific serine/threonine kinase that links DMTs to outer dense fibers in mammalian sperm. Our study provides structural foundations for understanding sperm evolution, motility, and dysfunction at a molecular level.

The rate of microtubule breaking increases exponentially with curvature

Microtubules, cylindrical assemblies of tubulin proteins with a 25 nm diameter and micrometer lengths, are a central part of the cytoskeleton and also serve as building blocks for nanobiodevices. Microtubule breaking can result from the activity of severing enzymes and mechanical stress. Breaking can lead to a loss of structural integrity, or an increase in the numbers of microtubules. We observed breaking of taxol-stabilized microtubules in a gliding motility assay where microtubules are propelled by surface-adhered kinesin-1 motor proteins. We find that over 95% of all breaking events are associated with the strong bending following pinning events (where the leading tip of the microtubule becomes stuck). Furthermore, the breaking rate increased exponentially with increasing curvature. These observations are explained by a model accounting for the complex mechanochemistry of a microtubule. The presence of severing enzymes is not required to observe breaking at rates comparable to those measured previously in cells.

Strain stiffening of Ndc80 complexes attached to microtubule plus ends

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

Viscoelastic damage evaluation of the axon

In this manuscript, we have studied the microstructure of the axonal cytoskeleton and adopted a bottom-up approach to evaluate the mechanical responses of axons. The cytoskeleton of the axon includes the microtubules (MT), Tau proteins (Tau), neurofilaments (NF), and microfilaments (MF). Although most of the rigidity of the axons is due to the MT, the viscoelastic response of axons comes from the Tau. Early studies have shown that NF and MF do not provide significant elasticity to the overall response of axons. Therefore, the most critical aspect of the mechanical response of axons is the microstructural topology of how MT and Tau are connected and construct the cross-linked network. Using a scanning electron microscope (SEM), the cross-sectional view of the axons revealed that the MTs are organized in a hexagonal array and cross-linked by Tau. Therefore, we have developed a hexagonal Representative Volume Element (RVE) of the axonal microstructure with MT and Tau as fibers. The matrix of the RVE is modeled by considering a combined effect of NF and MF. A parametric study is done by varying fiber geometric and mechanical properties. The Young’s modulus and spacing of MT are varied between 1.5 and 1.9 GPa and 20–38 nm, respectively. Tau is modeled as a 3-parameter General Maxwell viscoelastic material. The failure strains for MT and Tau are taken to be 50 and 40%, respectively. A total of 4 RVEs are prepared for finite element analysis, and six loading cases are inspected to quantify the three-dimensional (3D) viscoelastic relaxation response. The volume-averaged stress and strain are then used to fit the relaxation Prony series. Next, we imposed varying strain rates (between 10/sec to 50/sec) on the RVE and analyzed the axonal failure process. We have observed that the 40% failure strain of Tau is achieved in all strain rates before the MT reaches its failure strain of 50%. The corresponding axonal failure strain and stress vary between 6 and 11% and 5–19.8 MPa, respectively. This study can be used to model macroscale axonal aggregate typical of the white matter region of the brain tissue.

Three-dimensional structure of kinetochore-fibers in human mitotic spindles

During cell division, kinetochore microtubules (KMTs) provide a physical linkage between the chromosomes and the rest of the spindle. KMTs in mammalian cells are organized into bundles, so-called kinetochore-fibers (k-fibers), but the ultrastructure of these fibers is currently not well characterized. Here, we show by large-scale electron tomography that each k-fiber in HeLa cells in metaphase is composed of approximately nine KMTs, only half of which reach the spindle pole. Our comprehensive reconstructions allowed us to analyze the three-dimensional (3D) morphology of k-fibers and their surrounding MTs in detail. We found that k-fibers exhibit remarkable variation in circumference and KMT density along their length, with the pole-proximal side showing a broadening. Extending our structural analysis then to other MTs in the spindle, we further observed that the association of KMTs with non-KMTs predominantly occurs in the spindle pole regions. Our 3D reconstructions have implications for KMT growth and k-fiber self-organization models as covered in a parallel publication applying complementary live-cell imaging in combination with biophysical modeling (Conway et al., 2022). Finally, we also introduce a new visualization tool allowing an interactive display of our 3D spindle data that will serve as a resource for further structural studies on mitosis in human cells.

Actomyosin contractility and buckling of microtubules in nucleation, growth and disassembling of focal adhesions

Building up and maintenance of cytoskeletal structure in living cells are force-dependent processes involving a dynamic chain of polymerization and depolymerization events, which are also at the basis of cells’ remodelling and locomotion. All these phenomena develop by establishing cell–matrix interfaces made of protein complexes, known as focal adhesions, which govern mechanosensing and mechanotransduction mechanisms mediated by stress transmission between cell interior and external environment. Within this framework, by starting from a work by Cao et al. (Biophys J 109:1807–1817, 2015), we here investigate the role played by actomyosin contractility of stress fibres in nucleation, growth and disassembling of focal adhesions. In particular, we propose a tensegrity model of an adherent cell incorporating nonlinear elasticity and unstable behaviours, which provides a new kinematical interpretation of cellular contractile forces and describes how stress fibres, microtubules and adhesion plaques interact mechanobiologically. The results confirm some experimental evidences and suggest how the actomyosin contraction level could be exploited by cells to actively control their adhesion, eventually triggering cytoskeleton reconfigurations and migration processes observed in both physiological conditions and diseases.

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.

Controlling the Rigidity of Kinesin-Propelled Microtubules in an In Vitro Gliding Assay Using the Deep-Sea Osmolyte Trimethylamine N-Oxide

The biomolecular motor protein kinesin and its associated filamentous protein microtubule have been finding important nanotechnological applications in the recent years. Rigidity of the microtubules, which are propelled by kinesin motors in an in vitro gliding assay, is an important metric that determines the success of utilization of microtubules and kinesins in various applications, such as transportation, sensing, sorting, molecular robotics, etc. Therefore, regulating the rigidity of kinesin-propelled microtubules has been critical. In this work, we report a simple strategy to regulate the rigidity of kinesin-propelled microtubules in an in vitro gliding assay. We demonstrate that rigidity of the microtubules, propelled by kinesins in an in vitro gliding assay, can be modulated simply by using the natural osmolyte trimethylamine N-oxide (TMAO). By varying the concentration of TMAO in the gliding assay, the rigidity of microtubules can be modulated over a wide range. Based on this strategy, we are able to reduce the persistence length of microtubules, a measure of microtubule rigidity, ∼8 fold by using TMAO at the concentration of 1.5 M. Furthermore, we found that the decreased rigidity of the kinesin-propelled microtubules can be restored upon elimination of TMAO from the in vitro gliding assay. Alteration in the rigidity of microtubules is accounted for by the non-uniformity of the force applied by kinesins along the microtubules in the presence of TMAO. This work offers a facile strategy to reversibly regulate the rigidity of kinesin-propelled microtubules in situ, which would widen the applications of the biomolecular motor kinesin and its associated protein microtubule in various fields.

Local tissue heterogeneity may modulate neuronal responses via altered axon strain fields: insights about innervated joint capsules from a computational model

In innervated collagenous tissues, tissue scale loading may contribute to joint pain by transmitting force through collagen fibers to the embedded mechanosensitive axons. However, the highly heterogeneous collagen structures of native tissues make understanding this relationship challenging. Recently, collagen gels with embedded axons were stretched and the resulting axon signals were measured, but these experiments were unable to measure the local axon strain fields. Computational discrete fiber network models can directly determine axon strain fields due to tissue scale loading. Therefore, this study used a discrete fiber network model to identify how heterogeneous collagen networks (networks with multiple collagen fiber densities) change axon strain due to tissue scale loading. In this model, a composite cylinder (axon) was embedded in a Delaunay network (collagen). Homogeneous networks with a single collagen volume fraction and two types of heterogeneous networks with either a sparse center or dense center were created. Measurements of fiber forces show higher magnitude forces in sparse regions of heterogeneous networks and uniform force distributions in homogeneous networks. The average axon strain in the sparse center networks decreases when compared to homogeneous networks with similar collagen volume fractions. In dense center networks, the average axon strain increases compared to homogeneous networks. The top 1% of axon strains are unaffected by network heterogeneity. Based on these results, the interaction of tissue scale loading, collagen network heterogeneity, and axon strains in native musculoskeletal tissues should be considered when investigating the source of joint pain.

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.

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.

The multi-scale architecture of mammalian sperm flagella and implications for ciliary motility

Motile cilia are molecular machines used by a myriad of eukaryotic cells to swim through fluid environments. However, available molecular structures represent only a handful of cell types, limiting our understanding of how cilia are modified to support motility in diverse media. Here, we use cryo-focused ion beam milling-enabled cryo-electron tomography to image sperm flagella from three mammalian species. We resolve in-cell structures of centrioles, axonemal doublets, central pair apparatus, and endpiece singlets, revealing novel protofilament-bridging microtubule inner proteins throughout the flagellum. We present native structures of the flagellar base, which is crucial for shaping the flagellar beat. We show that outer dense fibers are directly coupled to microtubule doublets in the principal piece but not in the midpiece. Thus, mammalian sperm flagella are ornamented across scales, from protofilament-bracing structures reinforcing microtubules at the nano-scale to accessory structures that impose micron-scale asymmetries on the entire assembly. Our structures provide vital foundations for linking molecular structure to ciliary motility and evolution.

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.

Single-molecule stretching experiments of flexible (wormlike) chain molecules in different ensembles: Theory and a potential application of finite chain length effects to nick-counting in DNA

We propose a formalism for deriving force-elongation and elongation-force relations for flexible chain molecules from analytical expressions for their radial distribution function, which provides insight into the factors controlling the asymptotic behavior and finite chain length corrections. In particular, we apply this formalism to our previously developed interpolation formula for the wormlike chain end-to-end distance distribution. The resulting expression for the asymptotic limit of infinite chain length is of similar quality to the numerical evaluation of Marko and Siggia's variational theory and considerably more precise than their interpolation formula. A comparison to numerical data suggests that our analytical finite chain length corrections achieve a comparable accuracy. As an application of our results, we discuss the possibility of inferring the time-dependent number of nicks in single-molecule stretching experiments on double-stranded DNA from the accompanying changes in the effective chain length.

Drops and fibers - how biomolecular condensates and cytoskeletal filaments influence each other

The cellular cytoskeleton self-organizes by specific monomer-monomer interactions resulting in the polymerization of filaments. While we have long thought about the role of polymerization in cytoskeleton formation, we have only begun to consider the role of condensation in cytoskeletal organization. In this review, we highlight how the interplay between polymerization and condensation leads to the formation of the cytoskeleton.

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.

Photo-regulated trajectories of gliding microtubules conjugated with DNA

We regulate the persistency in motion of kinesin-driven microtubules (MTs) simply using a photoresponsive DNA (pDNA) and ultraviolet (UV)-visible light. The path persistence length of MTs, which is a measure of the persistency in their motion, increases and decreases upon illuminating the MTs with UV and visible light respectively. Moreover, pDNA is found to work as a shield for MTs against damage under UV irradiation.

Integrating Chemistry and Mechanics: The Forces Driving Axon Growth

The brain is our most complex organ. During development, neurons extend axons, which may grow over long distances along well-defined pathways to connect to distant targets. Our current understanding of axon pathfinding is largely based on chemical signaling by attractive and repulsive guidance cues. These cues instruct motile growth cones, the leading tips of growing axons, where to turn and where to stop. However, it is not chemical signals that cause motion-motion is driven by forces. Yet our current understanding of the mechanical regulation of axon growth is very limited. In this review, I discuss the origin of the cellular forces controlling axon growth and pathfinding, and how mechanical signals encountered by growing axons may be integrated with chemical signals. This mechanochemical cross talk is an important but often overlooked aspect of cell motility that has major implications for many physiological and pathological processes involving neuronal growth.

Interplay between Convective and Viscoelastic Forces Controls the Morphology of In Vitro Paclitaxel-Stabilized Microtubules

Microtubules (MTs) are self-assembling, high-aspect-ratio tubular nanostructures formed from the polymerization of tubulin protein. MTs are capable of globally assembling into optically birefringent morphologies, but there is disagreement on the mechanisms driving this behavior. We investigated the temporal evolution of paclitaxel (PTX)-stabilized MT solutions under a range of in vitro conditions. Significant morphological differences were observed in the polymerized PTX-MT solutions as a consequence of varying the orientation of the reaction vessel (vertical vs. horizontal), the type of heating source (hot plate vs. incubator), the incubation time, and the concentration of PTX (high vs. low). The most robust birefringent patterns were found only in vertically oriented cuvettes that were heated asymmetrically on a hot plate, suggesting dependence upon a convective flow, which we confirmed with a combination of optical and thermal imaging. Higher concentrations of PTX led to denser PTX-MT domain formation and brighter birefringence, due to more complete polymerization. Combining our experimental observations, we conclude that birefringent patterns arise principally through a combination of convective and viscoelastic forces, and we identify the sequence of dynamical stages through which they evolve.

Diverse balances of tubulin interactions and shape change drive and interrupt microtubule depolymerization

Microtubules are stiff biopolymers that self-assemble via the addition of GTP-tubulin (αβ-dimer bound to GTP), but hydrolysis of GTP- to GDP-tubulin within the tubules destabilizes them toward catastrophically-fast depolymerization. The molecular mechanisms and features of the individual tubulin proteins that drive such behavior are still not well-understood. Using molecular dynamics simulations of whole microtubules built from a coarse-grained model of tubulin, we demonstrate how conformational shape changes (i.e., deformations) in subunits that frustrate tubulin-tubulin binding within microtubules drive depolymerization of stiff tubules via unpeeling "ram's horns" consistent with experiments. We calculate the sensitivity of these behaviors to the length scales and strengths of binding attractions and varying degrees of binding frustration driven by subunit shape change, and demonstrate that the dynamic instability and mechanical properties of microtubules can be produced based on either balanced or imbalanced strengths of lateral and vertical binding attractions. Finally, we show how catastrophic depolymerization can be interrupted by small regions of the microtubule containing undeformed dimers, corresponding to incomplete lattice hydrolysis. The results demonstrate a mechanism by which microtubule rescue can occur.

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.

From isolated structures to continuous networks: A categorization of cytoskeleton-based motile engineered biological microstructures

As technology at the small scale is advancing, motile engineered microstructures are becoming useful in drug delivery, biomedicine, and lab-on-a-chip devices. However, traditional engineering methods and materials can be inefficient or functionally inadequate for small-scale applications. Increasingly, researchers are turning to the biology of the cytoskeleton, including microtubules, actin filaments, kinesins, dyneins, myosins, and associated proteins, for both inspiration and solutions. They are engineering structures with components that range from being entirely biological to being entirely synthetic mimics of biology and on scales that range from isotropic continuous networks to single isolated structures. Motile biological microstructures trace their origins from the development of assays used to study the cytoskeleton to the array of structures currently available today. We define 12 types of motile biological microstructures, based on four categories: entirely biological, modular, hybrid, and synthetic, and three scales: networks, clusters, and isolated structures. We highlight some key examples, the unique functionalities, and the potential applications of each microstructure type, and we summarize the quantitative models that enable engineering them. By categorizing the diversity of motile biological microstructures in this way, we aim to establish a framework to classify these structures, define the gaps in current research, and spur ideas to fill those gaps. This article is categorized under: Nanotechnology Approaches to Biology > Nanoscale Systems in Biology Nanotechnology Approaches to Biology > Cells at the Nanoscale Biology-Inspired Nanomaterials > Protein and Virus-Based Structures Therapeutic Approaches and Drug Discovery > Emerging Technologies.

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.

Mechanics of the Microtubule Seam Interface Probed by Molecular Simulations and in Vitro Severing Experiments

Microtubules (MTs) are structural components essential for cell morphology and organization. It has recently been shown that defects in the filament's lattice structure can be healed to create stronger filaments in a local area and ultimately cause global changes in MT organization and cell mobility. The ability to break, causing a defect, and heal appears to be a physiologically relevant and important feature of the MT structure. Defects can be created by MT severing enzymes and are target sites for complete severing or for healing by newly incorporated dimers. One particular lattice defect, the MT lattice ''seam" interface, is a location often speculated to be a weak site, a site of disassembly, or a target site for MT binding proteins. Despite seams existing in many MT structures, very little is known about the seam's role in MT function and dynamics. In this study, we probed the mechanical stability of the seam interface by applying coarse-grained indenting molecular dynamics. We found that the seam interface is as structurally robust as the typical lattice structure of MTs. Our results suggest that, unlike prior results that claim the seam is a weak site, it is just as strong as any other location on the MT, corroborating recent mechanical measurements.

The Role of Structural Polymorphism in Driving the Mechanical Performance of the Alzheimer's Beta Amyloid Fibrils

Alzheimer's Disease (AD) is related with the abnormal aggregation of amyloid β-peptides Aβ₁₋₄₀ and Aβ₁₋₄₂, the latter having a polymorphic character which gives rise to U- or S-shaped fibrils. Elucidating the role played by the nanoscale-material architecture on the amyloid fibril stability is a crucial breakthrough to better understand the pathological nature of amyloid structures and to support the rational design of bio-inspired materials. The computational study here presented highlights the superior mechanical behavior of the S-architecture, characterized by a Young's modulus markedly higher than the U-shaped architecture. The S-architecture showed a higher mechanical resistance to the enforced deformation along the fibril axis, consequence of a better interchain hydrogen bonds' distribution. In conclusion, this study, focusing the attention on the pivotal multiscale relationship between molecular phenomena and material properties, suggests the S-shaped Aβ₁₋₄₂ species as a target of election in computational screen/design/optimization of effective aggregation modulators.

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.

Distinct relaxation timescales of neurites revealed by rate-dependent indentation, relaxation and micro-rheology tests

Although the dynamic response of neurites is believed to play crucial roles in processes like axon outgrowth and formation of the neural network, the dynamic mechanical properties of such protrusions remain poorly understood. In this study, by using AFM (atomic force microscopy) indentation, we systematically examined the dynamic behavior of well-developed neurites on primary neurons under different loading modes (step loading, oscillating loading and ramp loading). Interestingly, the response was found to be strongly rate-dependent, with an apparent initial and long-term elastic modulus around 800 and 80 Pa, respectively. To better analyze the measurement data and extract information of key interest, the finite element simulation method (FEM) was also conducted where the neurite was treated as a viscoelastic solid consisting of multiple characteristic relaxation times. It was found that a minimum of three relaxation timescales, i.e. ∼0.01, 0.1 and 1 seconds, are needed to explain the observed relaxation curve as well as fit simulation results to the indentation and rheology data under different loading rates and driving frequencies. We further demonstrated that these three characteristic relaxation times likely originate from the thermal fluctuations of the microtubule, membrane relaxation and cytosol viscosity, respectively. By identifying key parameters describing the time-dependent behavior of neurites, as well as revealing possible physical mechanisms behind, this study could greatly help us understand how neural cells perform their biological duties over a wide spectrum of timescales.

Multiscale modelling of the human lumbar facet capsular ligament: analysing spinal motion from the joint to the neurons

Due to its high level of innervation, the lumbar facet capsular ligament (FCL) is suspected to play a role in low back pain (LBP). The nociceptors in the lumbar FCL may experience excessive deformation and generate pain signals. As such, understanding the mechanical behaviour of the FCL, as well as that of its underlying nerves, is critical if one hopes to understand its role in LBP. In this work, we constructed a multiscale structure-based finite-element (FE) model of a lumbar FCL on a spinal motion segment undergoing physiological motions of flexion, extension, ipsilateral and contralateral bending, and ipsilateral axial rotation. Our FE model was created for a generic FCL geometry by morphing a previously imaged FCL anatomy onto an existing generic motion segment model. The fibre organization of the FCL in our models was subject-specific based on previous analysis of six dissected specimens. The fibre structures from those specimens were mapped onto the FCL geometry on the motion segment. A motion segment model was used to determine vertebral kinematics under specified spinal loading conditions, providing boundary conditions for the FCL-only multiscale FE model. The solution of the FE model then provided detailed stress and strain fields within the tissue. Lastly, we used this computed strain field and our previous studies of deformation of nerves embedded in fibrous networks during simple deformations (e.g. uniaxial stretch, shear) to estimate the nerve deformation based on the local tissue strain and fibre alignment. Our results show that extension and ipsilateral bending result in largest strains of the lumbar FCL, while contralateral bending and flexion experience lowest strain values. Similar to strain trends, we calculated that the stretch of the microtubules of the nerves, as well as the forces exerted on the nerves' membrane are maximal for extension and ipsilateral bending, but the location within the FCL of peak microtubule stretch differed from that of peak membrane force.

Microtubule lattice plasticity

In classical microtubule dynamic instability, the dynamics of the built polymer depend only on the nucleotide state of its individual tubulin molecules. Recent work is overturning this view, pointing instead towards lattice plasticity, in which the fine-structure and mechanics of the microtubule lattice are emergent properties that depend not only on the nucleotide state of each tubulin, but also on the nucleotide states of its neighbours, on its and their isotypes, and on interacting proteins, drugs, local mechanical strain, post translational modifications, packing defects and solvent conditions. In lattice plasticity models, the microtubule is an allosteric molecular collective that integrates multiple mechanochemical inputs and responds adaptively by adjusting its conformation, stiffness and dynamics.

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.

Multiscale FEM simulations of cross-linked actin network embedded in cytosol with the focus on the filament orientation

The present contribution focuses on the application of the multiscale finite element method to the modeling of actin networks that are embedded in the cytosol. These cell components are of particular importance with regard to the cell response to external stimuli. The homogenization strategy chosen uses the Hill-Mandel macrohomogeneity condition for bridging 2 scales: the macroscopic scale that is related to the cell level and the microscopic scale related to the representative volume element. For the modeling of filaments, the Holzapfel-Ogden β-model is applied. It provides a relationship between the tensile force and the caused stretches, serves as the basis for the derivation of the stress and elasticity tensors, and enables a novel finite element implementation. The elements with the neo-Hookean constitutive law are applied for the simulation of the cytosol. The results presented corroborate the main advantage of the concept, namely, its flexibility with regard to the choice of the representative volume element as well as of macroscopic tests. The focus is particularly placed on the study of the filament orientation and of its influence on the effective behavior.

The long persistence length of model tubules

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

Effect of Nucleotide State on the Protofilament Conformation of Tubulin Octamers

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

Microtubule seams are not mechanically weak defects

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