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Research Papers

A Micromechanically Based Anisotropic Constitutive Model for the Microtubule Wall

[+] Author and Article Information
Melis Arslan

 Department of Civil and Environmental Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139; Centre des Matériaux, Mines Paris, Paristech, CNRS UMR 7633, BP 87, F-91003 Evry Cedex, France

Mary C. Boyce

 Department of Mechanical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139

The thickness of the microtubule wall could be intuitively thought of as being (25–15)/2 = 5 nm, which is the dimer thickness. This physical thickness does not represent the lateral interactions between the dimers. The lateral interactions occur at thicknesses down to 1 nm–2.5 nm [7].

J. Appl. Mech 79(2), 021002 (Feb 09, 2012) (7 pages) doi:10.1115/1.4005548 History: Received October 29, 2010; Revised April 26, 2011; Posted January 30, 2012; Published February 08, 2012; Online February 09, 2012

Microtubules serve as one of the structural components of the cell and govern several important cellular functions including mitosis and vesicular transport. Microtubules are comprised of tubulin subunits formed by α and β tubulin dimers arranged in a cylindrical hollow tube with diameter ∼20 nm. The tube is typically comprised of 13 or 14 protofilaments extending axially and staggered to give a spiral configuration. The longitudinal bonds between the tubulin dimers are much stiffer and stronger than the lateral bonds. This gives a highly anisotropic structure and mechanical properties of the microtubule. In this work, the aim is to define a complete set of effective anisotropic elastic properties of the tube wall that capture the atomistic interactions. A seamless microtubule wall is represented as a two dimensional triangulated lattice of dimers from which a representative volume element is defined. A harmonic potential is adapted for the dimer–dimer interactions. Estimating the lattice elastic constants and following the methodology from the analysis of the mechanical behavior of the triangulated spectrin network of the red blood cell membrane (Arslan and Boyce, 2006, “Constitutive Modeling of the Finite Deformation Behavior of Membranes Possessing a Triangulated Network Microstructure,” ASME J. Appl. Mech., 73 , pp. 536–543), a general anisotropic hyperelastic strain energy function is formulated and used to define the effective anisotropic continuum level constitutive model of the mechanical behavior of the microtubule wall. In particular, the role of the anisotropic microstructure resulting from the different lattice bond lengths and bond stiffnesses is examined to explain nature’s optimization of microstructural orientation in providing a high axial stiffness combined with low shear stiffness.

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Copyright © 2012 by American Society of Mechanical Engineers
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Figures

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Figure 1

Unrolling the microtubule, the 3D hollow tube that the dimers form provides a 2D lattice. Each lattice configuration (A and B) is shown with their corresponding RVE.

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Figure 2

The RVE for the seamless B-lattice depicting the corresponding spring constants for the constituent bonds; kA , kB , and kC . The skew angle, φ, is depicted in the figure.

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Figure 3

Schematic of the intraprotofilament interactions

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Figure 4

Binding free energy for the A and B lattice structures recreated using the data from [15] depicting the subunit rise for each lattice structure

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Figure 5

The longitudinal and shear stiffness in the transformed axis system for different transformation angles

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Figure 6

The tension-shear coupling components in the transformed axis system for different transformation angles

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Figure 7

The RVE is shown depicting the constituent bond lengths (LA , LB , LC ), constituent angles (θA , θB , θC ), the crosslink coordinates (a, b, c) and the torsional spring constants (kTA , kTB , kTC )

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