A Micromechanically Based Anisotropic Constitutive Model for the Microtubule Wall

Abstract

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.