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

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

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

Deformation pattern in vibrating microtubule: Structural mechanics study based on an atomistic approach

The mechanical properties of microtubules are of great importance for understanding their biological function and for applications in artificial devices. Although microtubule mechanics has been extensively studied both theoretically and experimentally, the relation to its molecular structure is understood only partially. Here, we report on the structural analysis of microtubule vibration modes calculated by an atomistic approach. Molecular dynamics was applied to refine the atomic structure of a microtubule and a C_α elastic network model was analyzed for its normal modes. We mapped fluctuations and local deformations up to the level of individual aminoacid residues. The deformation is mode-shape dependent and principally different in α-tubulins and β-tubulins. Parts of the tubulin dimer sequence responding specifically to longitudinal and radial stress are identified. We show that substantial strain within a microtubule is located both in the regions of contact between adjacent dimers and in the body of tubulins. Our results provide supportive evidence for the generally accepted assumption that the mechanics of microtubules, including its anisotropy, is determined by the bonds between tubulins.

Vibration of bioliquid-filled microtubules embedded in cytoplasm including surface effects using modified couple stress theory

This paper aims to investigate vibrational behavior of bioliquid-filled microtubules (MTs) embedded in cytoplasm considering surface effects. The interactions between the MT, considered as an orthotropic beam within the framework of Euler-Bernoulli beam (EBB) and Timoshenko beam (TB) models, and its surrounding elastic media are simulated by Pasternak foundation model. The modified couple stress theory (MCST) is applied so as to consider the small scale effects while motion equations are derived using energy method and Hamilto's principle for both EBB and TB models. Finally, an analytical method is employed to obtain the frequency of a bioliquid-filled MT, and therefore frequency-response curves are plotted to investigate the influences of small scale parameter, mass density of bioliquid, surface layer and surrounding elastic medium graphically. The results indicate that bioliquid and surface layers play a key role on the frequency of MTs and that the frequency of MTs is decreased with increasing of the mass density of the bioliquid. Vibration analysis of MTs is being considered as a vital problem since MTs look like the nervous system of the biological cells and transmit vibrational signals. It should be noted that the results of this work are hoped to be of use in advanced medical applications especially in the forthcoming use of MTs in transporters for bio-nanosensors.

Variational principles for buckling of microtubules modeled as nonlocal orthotropic shells

A variational principle for microtubules subject to a buckling load is derived by semi-inverse method. The microtubule is modeled as an orthotropic shell with the constitutive equations based on nonlocal elastic theory and the effect of filament network taken into account as an elastic surrounding. Microtubules can carry large compressive forces by virtue of the mechanical coupling between the microtubules and the surrounding elastic filament network. The equations governing the buckling of the microtubule are given by a system of three partial differential equations. The problem studied in the present work involves the derivation of the variational formulation for microtubule buckling. The Rayleigh quotient for the buckling load as well as the natural and geometric boundary conditions of the problem is obtained from this variational formulation. It is observed that the boundary conditions are coupled as a result of nonlocal formulation. It is noted that the analytic solution of the buckling problem for microtubules is usually a difficult task. The variational formulation of the problem provides the basis for a number of approximate and numerical methods of solutions and furthermore variational principles can provide physical insight into the problem.

A higher-order mathematical modeling for dynamic behavior of protein microtubule shell structures including shear deformation and small-scale effects

Microtubules in mammalian cells are cylindrical protein polymers which structurally and dynamically organize functional activities in living cells. They are important for maintaining cell structures, providing platforms for intracellular transport, and forming the spindle during mitosis, as well as other cellular processes. Various in vitro studies have shown that microtubules react to applied mechanical loading and physical environment. To investigate the mechanisms underlying such phenomena, a mathematical model based on the orthotropic higher-order shear deformation shell formulation and Hamilton's principle is presented in this paper for dynamic behavior of microtubules. The numerical results obtained by the proposed shell model are verified by the experimental data from the literature, showing great consistency. The nonlocal elasticity theory is also utilized to describe the nano-scale effects of the microtubule structure. The wave propagation and vibration characteristics of the microtubule are examined in the presence and absence of the cytosol employing proposed formulations. The effects of different system parameters such as length, small scale parameter, and cytosol viscosity on vibrational behavior of a microtubule are elucidated. The definitions of critical length and critical viscosity are introduced and the results obtained using the higher order shell model are compared with those obtained employing a first-order shear deformation theory. This comparison shows that the small scale effects become important for higher values of the wave vector and the proposed model gives more accurate results for both small and large values of wave vectors. Moreover, it is shown that for higher circumferential wave number, the torsional wave velocity obtained by the higher-order shell model tend to be higher than the one predicted by the first-order shell model.