Torsional elastic deformations of microtubules within continuous sheet model

Curved crease origami and topological singularities at a cellular scale enable hyper-extensibility of Lacrymaria olor

Eukaryotic cells undergo dramatic morphological changes during cell division, phagocytosis and motility. Fundamental limits of cellular morphodynamics such as how fast or how much cellular shapes can change without harm to a living cell remain poorly understood. Here we describe hyper-extensibility in the single-celled protist Lacrymaria olor, a 40 μm cell which is capable of reversible and repeatable extensions (neck-like protrusions) up to 1500 μm in 30 seconds. We discover that a unique and intricate organization of cortical cytoskeleton and membrane enables these hyper-extensions that can be described as the first cellular scale curved crease origami. Furthermore, we show how these topological singularities including d-cones and twisted domain walls provide a geometrical control mechanism for the deployment of membrane and microtubule sheets as they repeatably spool thousands of time from the cell body. We lastly build physical origami models to understand how these topological singularities provide a mechanism for the cell to control the hyper-extensile deployable structure. This new geometrical motif where a cell employs curved crease origami to perform a physiological function has wide ranging implications in understanding cellular morphodynamics and direct applications in deployable micro-robotics.

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.

Torsional behavior of axonal microtubule bundles

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

Methods for modeling cytoskeletal and DNA filaments

This review summarizes the models that researchers use to represent the conformations and dynamics of cytoskeletal and DNA filaments. It focuses on models that address individual filaments in continuous space. Conformation models include the freely jointed, Gaussian, angle-biased chain (ABC), and wormlike chain (WLC) models, of which the first three bend at discrete joints and the last bends continuously. Predictions from the WLC model generally agree well with experiment. Dynamics models include the Rouse, Zimm, stiff rod, dynamic WLC, and reptation models, of which the first four apply to isolated filaments and the last to entangled filaments. Experiments show that the dynamic WLC and reptation models are most accurate. They also show that biological filaments typically experience strong hydrodynamic coupling and/or constrained motion. Computer simulation methods that address filament dynamics typically compute filament segment velocities from local forces using the Langevin equation and then integrate these velocities with explicit or implicit methods; the former are more versatile and the latter are more efficient. Much remains to be discovered in biological filament modeling. In particular, filament dynamics in living cells are not well understood, and current computational methods are too slow and not sufficiently versatile. Although primarily a review, this paper also presents new statistical calculations for the ABC and WLC models. Additionally, it corrects several discrepancies in the literature about bending and torsional persistence length definitions, and their relations to flexural and torsional rigidities.

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

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

Wave propagation in protein microtubules modeled as orthotropic elastic shells including transverse shear deformations

Wave propagation along the microtubules is one of the issues of major concern in various microtubule cellular functions. In this study, the general wave propagation behavior in protein microtubules is investigated based on a first-order shear deformation shell theory for orthotropic materials, with particular emphasis on the role of strongly anisotropic elastic properties of microtubules. According to experimental observation, the first-order shear deformation theory is used for the modeling of microtubule walls. A general displacement representation is introduced and a type of coupled polynomial eigenvalue problem is developed. Numerical examples describe the effects of shear deformation and rotary inertia on wave velocities in orthotropic microtubules. Finally, the influences of the microtubule shear modulus, axial external force, effective thickness and material temperature dependency on wave velocities along the microtubule protofilaments, helical pathway and radial directions are elucidated. Most results presented in the present investigation have been absent from the literature for the wave propagation in microtubules.