Mathematical modelling of microtubule-tau protein transients: Insights into the superior mechanical behavior of axon

Jiayu Wu,Shanqing Li,Kunjie Fan,Kin-Nam Lee,Bing Li,Long-Yuan Li,Hong Yuan

doi:10.1016/j.apm.2019.02.030

Abstract

• The theory of shear lag model is extended. • Viscoelastic shear lag model is developed to illustrate the mechanical behavior of axon under transient tensile loading. • The paper explains how the vulnerable axons protect themselves from overall damage over the course of evolution. • The paper explains how axons achieve an outstanding mechanical balance of high stiffness and toughness. Axons with staggered microtubules cross-linked by tau protein possess a remarkable mechanical balance of high specific stiffness and toughness. Owing to their viscoelastic nature, axons exhibit stress rate-dependent mechanical behavior, which is relevant to their selective vulnerability to damage in traumatic brain injury. A Kelvin–Voigt viscoelastic shear lag model is developed to elucidate the mechanical responses of axons under transient tensile force. Analytical closed-form expressions are derived to characterize the relative sliding, stress transfer and failure mechanism between microtubule and tau protein while the axon is stretched transiently. The results from the theoretical solutions elucidate how the MT-tau interface length and stress rate affect the mechanical responses of axon. It is found that axonal failure mechanism may be different under different loading conditions. Long microtubules are more vulnerable to rupture at high stress rate, yet short microtubules are likely to detach from microtubule bundles under large deformations. In the view of multi-level failure of axon, it is illustrated how the vulnerable axons protect themselves from overall damage, and how the axon can simultaneously achieve an outstanding mechanical balance of high specific stiffness and toughness.

Highlights

Staggered biocomposites are found to exhibit a remarkable mechanical balance of fracture toughness versus stiffness and strength [1, 2]
Peter et al [16] used a discrete bead-spring model to simulate the biomechanical behavior of axonal microtubule bundle under uniaxial tension, and they assumed that both MT and tau protein are linear elastic materials
Ahmadzadeh et al [10] verified that Kelvin viscoelastic model can efficiently characterize the viscoelastic mechanical behavior of axon, but only MT failure is investigated in their model without consideration of tau protein breaking

Summary

Introduction

Staggered biocomposites are found to exhibit a remarkable mechanical balance of fracture toughness versus stiffness and strength [1, 2]. Tolomeo et al [15] reported that cross-linking proteins provided shear resistance between MTs. Peter et al [16] used a discrete bead-spring model to simulate the biomechanical behavior of axonal microtubule bundle under uniaxial tension, and they assumed that both MT and tau protein are linear elastic materials. Shamloo et al [17] considered MTs as a large number of discrete masses and modelled tau protein as Kelvin-Voigt element so that viscoelastic model can be employed to simulate the transient response of axonal MTs under sudden forces. Ahmadzadeh et al [10] verified that Kelvin viscoelastic model can efficiently characterize the viscoelastic mechanical behavior of axon, but only MT failure is investigated in their model without consideration of tau protein breaking. How does the axon simultaneously achieve an outstanding mechanical balance of high specific stiffness and toughness?

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