Surface deflection of a microtubule loaded by a concentrated radial force

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.

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.

FREE VIBRATION OF MICROTUBULES AS ELASTIC SHELL MODEL BASED ON MODIFIED COUPLE STRESS THEORY

In this study, first, the thin cylindrical shell theory was derived from the modified couple stress theory and, afterwards, the vibration of protein microtubules (MTs) was investigated using the developed model. In order to model protein MTs more precisely, the cylindrical micro-shell model was used. Also, to take account of small size effects, equations of motion were obtained on the basis of the modified couple stress theory. For this purpose, first, using Hamilton's principle, vibration equations of cylindrical shell with boundary conditions were derived from the modified couple stress theory. Finally, the effects of size parameters, MT dimensions, and the medium surrounding on the axial and circumferential vibration frequency of the MT were examined. It should be noted that the results obtained from the cylindrical micro-shell model, unlike those from the beam model, have lower dependency on MT length, but they have extreme dependency on MT thickness and radius. In the end, it is worth noting that the model developed in this study can predict experimental results with greater precision compared to classic models. In other words, this model narrows the gap existing between experimental results and previous models and theories.

Coupled oscillations of a protein microtubule immersed in cytoplasm: an orthotropic elastic shell modeling

Revealing vibration characteristics of sub-cellular structural components such as membranes and microtubules has a principal role in obtaining a deeper understanding of their biological functions. Nevertheless, limitations and challenges in biological experiments at this scale necessitates the use of mathematical and computational models as an alternative solution. As one of the three major cytoskeletal filaments, microtubules are highly anisotropic structures built from tubulin heterodimers. They are hollow cylindrical shells with a ∼ 25 nm outer diameter and are tens of microns long. In this study, a mechanical model including the effects of the viscous cytosol and surrounding filaments is developed for predicting the coupled oscillations of a single microtubule immersed in cytoplasm. The first-order shear deformation shell theory for orthotropic materials is used to model the microtubule, whereas the motion of the cytosol is analyzed by considering the Stokes flow. The viscous cytosol and the microtubule are coupled through the continuity condition across the microtubule-cytosol interface. The stress and velocity fields in the cytosol induced by vibrating microtubule are analytically determined. Finally, the influences of the dynamic viscosity of the cytosol, filament network elasticity, microtubule shear modulus, and circumferential wave-number on longitudinal, radial, and torsional modes of microtubule vibration are elucidated.