Length-dependence of flexural rigidity as a result of anisotropic elastic properties of microtubules

Li, C.; Ru, C. Q.; Mioduchowski, A.

doi:10.1016/j.bbrc.2006.08.153

Cited by 56 publications

(37 citation statements)

References 30 publications

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“…The observed EI length dependence indicates that the isotropic Euler beam is not an appropriate model for the kinocilium. as an Euler-Bernoulli beam leads to underestimates of EI for shorter MTs (Li et al, 2006). A Timoshenko beam model of an MT predicts an EI length dependence that matches experimental results, provided the shear modulus is five to six orders of magnitude lower than the longitudinal Young's modulus (Shi et al, 2008).…”

Section: Euler-bernoulli Vs Timoshenko Beam Analysismentioning

confidence: 69%

“…Indeed, atomic force microscopy shows that MTs are highly anisotropic because the longitudinal bond strength between tubulin dimers along the protofilaments is greater than the lateral bond strength between protofilaments (Kis et al, 2002). Computational studies in which the MT was modeled as a twodimensional orthotropic shell or a one-dimensional Timoshenko beam predict a length dependence of EI that results from an extremely low shear modulus relative to the longitudinal Young's modulus (E) (Li et al, 2006;Pampaloni et al, 2006;Shi et al, 2008). Use of the Euler-Bernoulli model, which does not account for shear deformation, leads to an underestimate of EI for shorter MTs (Li et al, 2006).…”

Section: Introductionmentioning

confidence: 99%

“…Computational studies in which the MT was modeled as a twodimensional orthotropic shell or a one-dimensional Timoshenko beam predict a length dependence of EI that results from an extremely low shear modulus relative to the longitudinal Young's modulus (E) (Li et al, 2006;Pampaloni et al, 2006;Shi et al, 2008). Use of the Euler-Bernoulli model, which does not account for shear deformation, leads to an underestimate of EI for shorter MTs (Li et al, 2006). The Timoshenko beam theory, which can account for shear deformation, was applied to MTs, yielding results comparable to that of the more complicated orthotropic shell model, which includes transverse shearing (Gu et al, 2009).…”

Section: Introductionmentioning

confidence: 99%

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Biomechanics of hair cell kinocilia: experimental measurement of kinocilium shaft stiffness and base rotational stiffness with Euler–Bernoulli and Timoshenko beam analysis

Spoon

Grant

2011

Journal of Experimental Biology

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the OL movement is transmitted to the mechanotransducing part of the hair bundle. Previous computational studies have shown that the tall and compliant kinocilia of extrastriolar bundles increase the hair cells' operating range (range of deflections over which vestibular sensory cell can encode) to several microns compared with that of striolar bundles, which is ~0.1m (Nam et al., 2005). Knowing the mechanical properties of the kinocilium will further our understanding of how these hair cells operate.Kinocilia are composed of a microtubular core, a 9+2 axoneme, composed of nine peripheral doublet microtubules encircling two central single tubules. This configuration is common to motile cilia and eukaryotic flagella (Fig.1A) but, despite their common structure, kinocilia are thought to be non-motile, unlike flagella, as they lack the inner arms of the motor protein dynein, which are essential for motility (Kikuchi et al., 1989). A limited number of spontaneous flagella-like oscillations have been observed in ampullary kinocilia but were limited in number and suggested to result from tissue Accepted 23 November 2010 SUMMARY Vestibular hair cell bundles in the inner ear contain a single kinocilium composed of a 9+2 microtubule structure. Kinocilia play a crucial role in transmitting movement of the overlying mass, otoconial membrane or cupula to the mechanotransducing portion of the hair cell bundle. Little is known regarding the mechanical deformation properties of the kinocilium. Using a force-deflection technique, we measured two important mechanical properties of kinocilia in the utricle of a turtle, Trachemys (Pseudemys) scripta elegans. First, we measured the stiffness of kinocilia with different heights. These kinocilia were assumed to be homogenous cylindrical rods and were modeled as both isotropic Euler-Bernoulli beams and transversely isotropic Timoshenko beams. Two mechanical properties of the kinocilia were derived from the beam analysis: flexural rigidity (EI) and shear rigidity (kGA). The Timoshenko model produced a better fit to the experimental data, predicting EI10,400pNm 2 and kGA247pN. Assuming a homogenous rod, the shear modulus (G1.9kPa) was four orders of magnitude less than Young's modulus (E14.1MPa), indicating that significant shear deformation occurs within deflected kinocilia. When analyzed as an Euler-Bernoulli beam, which neglects translational shear, EI increased linearly with kinocilium height, giving underestimates of EI for shorter kinocilia. Second, we measured the rotational stiffness of the kinocilium insertion () into the hair cell's apical surface. Following BAPTA treatment to break the kinocilial links, the kinocilia remained upright, and  was measured as 177±47pNmrad -1 . The mechanical parameters we quantified are important for understanding how forces arising from head movement are transduced and encoded by hair cells.

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Section: Euler-bernoulli Vs Timoshenko Beam Analysismentioning

confidence: 69%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Biomechanics of hair cell kinocilia: experimental measurement of kinocilium shaft stiffness and base rotational stiffness with Euler–Bernoulli and Timoshenko beam analysis

Spoon

Grant

2011

Journal of Experimental Biology

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“…Furthermore, MTs are considered as self-assembling biological nanotubes that are essential for cell motility, building the cytoskeleton, cell division and intracellular transport. The average Young's modulus of a microtubules is $2.0 GPa [2][3][4][5]. Among the three types of cytoskeletal filaments, microtubules are the most rigid.…”

Section: Introductionmentioning

confidence: 99%

Bending analysis of microtubules using nonlocal Euler–Bernoulli beam theory

Cívalek

Demir

2011

Applied Mathematical Modelling

315

131

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a b s t r a c tIn this paper, elastic beam model using nonlocal elasticity theory is developed for the bending analysis of microtubules (MTs) based on the Euler-Bernoulli beam theory. The size effect is taken into consideration using the Eringen's non-local elasticity theory. The derivation of governing equation of bending from shear and moment resultants of the beam and stress-strain relationship of the one-dimensional nonlocal elasticity model is presented. The model is then applied on the studies of static analysis of microtubules using the method of differential quadrature (DQ). After the developed DQ method is numerically validated, detailed numerical analyses about the effects of boundary conditions and load types are conducted and the influence of nonlocal parameter on the static response of MTs is discussed. It is hoped that the results in the manuscript may present a benchmark in the study of bending in microtubules.

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“…Microtubules (MTs), the stiffest elements of the axonal cytoskeleton, are responsible for various essential biological functions such as axonal transport of cargos (9,10). MTs are long and hollow cylinders, made up of 13 parallel longitudinally oriented protofilaments composed of polymerized a-and b-tubulin heterodimers (11)(12)(13)(14). Axonal MTs are found to be arranged in organized polarized arrays (15,16), with an average length of 4 mm (17).…”

Section: Introductionmentioning

confidence: 99%

Torsional Behavior of Axonal Microtubule Bundles

Lazarus

Soheilypour

Mofrad

2015

Biophysical Journal

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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.

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Length-dependence of flexural rigidity as a result of anisotropic elastic properties of microtubules

Cited by 56 publications

References 30 publications

Biomechanics of hair cell kinocilia: experimental measurement of kinocilium shaft stiffness and base rotational stiffness with Euler–Bernoulli and Timoshenko beam analysis

Biomechanics of hair cell kinocilia: experimental measurement of kinocilium shaft stiffness and base rotational stiffness with Euler–Bernoulli and Timoshenko beam analysis

Bending analysis of microtubules using nonlocal Euler–Bernoulli beam theory

Torsional Behavior of Axonal Microtubule Bundles

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