On the atomistic-based continuum viscoelastic constitutive relations for axonal microtubules

Mechanical response of brain's interior during traumatic brain injury is primarily governed by the cytoskeleton (CSK) and occurs over multiple length scales starting from the axonal substructure level. The axonal cytoskeleton can be viewed as a nanofiber reinforced nanocomposite structure where nano-fibrous microtubules (MTs) are arranged in staggered arrays and cross-linked by Tau proteins. Each MT is made of thirteen laterally connected protofilaments (PFs), each of which is formed via linear polymerization of αβ-heterodimer protein called tubulin. Recent studies suggest that the unique viscoelastic nature of axons governs the damage during traumatic brain injury. To understand how the internal substructures of axon influences the viscoelastic mechanical behavior of axon from a theoretical perspective, the viscoelastic properties of MTs need to be properly described. Since viscosity is a bulk property, the measurement methods are fairly consistent. On the other hand, the reported experimentally measured elastic properties of MTs vary by several orders of magnitude due to limitations of experimental tools. Alternatively, many have attempted to determine MT properties using theoretical and computational methods at different length scales ranging between the atomistic and the continuum level. The atomistic approaches capture the dynamics and interactions of a material at the atomic or atomic cluster level but these methods are computationally expensive and can model only a very small physical scale. On the other hand, the continuum theories lack finer scale details. Here, we present an atomistic-based continuum viscoelastic constitutive relation for microtubules (MTs) based on the interatomic potential for proteins and continuum homogenization method. The interaction potential includes both van der Waals and electrostatic interactions between the protein molecules. The calculated Young's modulus of 3.385 GPa agrees reasonably well with the range of experimentally measured value without any parameter fitting. We have then investigated the viscoelastic response of MT based on the estimated viscosity using atomistic simulation and evaluated Young's modulus using our method. The current theory suggests that MT behaves like a viscoelastic material when applied loading rate is extremely high, otherwise it acts like an elastic solid material.