Computational modeling of axonal microtubule bundles under tension

Abstract

Microtubule bundles cross-linked by tau protein serve a variety of neurological functions including maintaining mechanical integrity of the axon, promoting axonal growth, and facilitating cargo transport. It has been observed that axonal damage in traumatic brain injury leads to bundle disorientation, loss of axonal viability, and cognitive impairment. This study investigates the initial mechanical response of axonal microtubule bundles under uniaxial tension using a discrete bead-spring representation. Mechanisms of failure due to traumatic stretch loading and their impact on the mechanical response and stability are also characterized. This study indicates that cross-linked axonal microtubule bundles in tension display stiffening behavior similar to a power-law relationship from nonaffine network deformations. Stretching of cross-links and microtubule bending were the primary deformation modes at low stresses. Microtubule stretch was negligible up to tensile stresses of ∼1 MPa. Bundle failure occurred by failure of cross-links leading to pull-out of microtubules and loss of bundle integrity. This may explain the elongation, undulation, and delayed elasticity of axons following traumatic stretch loading. More extensively cross-linked bundles withstood higher tensile stresses before failing. The bundle mechanical behavior uncovered by these computational techniques should guide future experiments on stretch-injured axons.

Figure 1

Schematic of a neuron with an axon and dendrites. Microtubules, represented by thick, black lines, are bundled by MAP proteins. Bundle morphology is dependent on the bundling MAP. Right inset: Axon showing microtubules cross-linked by tau protein (red). Left inset: Dendrite showing microtubules cross-linked by MAP2 (blue). (Not drawn to scale).

Figure 2

Schematic of the bundle filaments represented by a network of beads connected by spring elements. Angular springs representing bending stiffness are also included between microtubule elements (not shown). (Not drawn to scale).

Figure 3

Example simulation geometry generated using in-house code. Beads are drawn with a radius of 12.5 nm to show microtubules at the correct width (the physical radii of beads are not important except in steric repulsion). Elements between microtubule beads are not shown. Bundle length is 8 μm, and microtubules are placed with a 20 nm edge-to-edge spacing. Note the single randomly placed discontinuity in each row and the random distribution of cross-links at a specified average spacing (25 nm average shown).

Figure 4

Steady-state bundle stress-strain curves in tension. Error bars representing the standard error are shown. Curves are shown for 25 nm (solid line), 50 nm (dashed line), 75 nm (dash dotted line), and 100 nm (dotted line) average cross-link spacing. The stress-strain response of continuous bundles (without discontinuities) is shown as the gray dashed line, where the entirety of the load is carried by microtubule stretching. Strain stiffening behavior of the bundle response is evident due to nonaffine network deformations.

Figure 5

Steady-state energy sharing curves of elastic energy storage modes. Curves are shown for 25 nm (solid line), 50 nm (dashed line), 75 nm (dash dotted line), and 100 nm (dotted line) average cross-link spacing. From left to right: percentage of energy stored in microtubule bending, percentage of energy stored in microtubule stretching, percentage of energy stored in cross-link stretching. Microtubule stretching contributes <5% of the elastic energy storage up to a tensile stress of 100 kPa.

Figure 6

Bundle strain dynamics at 50 MPa tensile stress. Curves are shown for 25 nm (solid line), 50 nm (dashed line), 75 nm (dash dotted line), and 100 nm (dotted line) average cross-link spacing. Steady-state bundle strains are reached in 1–2 μs.

Figure 7

Catastrophically failing microtubule bundle showing microtubule pull-out. Average cross-link spacing is 50 nm, and the applied tensile stress is 300 MPa. Element failure was monitored, indicating only failure of cross-link elements. Tightening of the bundle toward the centerline is also evident. Simulation times are 0 μs (top), 1.25 μs (middle), and 5 μs (bottom). (Scale bar = 1 μm).

Figure 8

Steady-state stress-strain curves of bundles with microtubule and cross-link critical strain failure criteria enforced. Strain values at the initiation of failure are indicated by an x. Curves are shown for 25 nm (solid line), 50 nm (dashed line), 75 nm (dash dotted line), and 100 nm (dotted line) average cross-link spacings. Lower cross-link densities resulted in catastrophic bundle failure at lower stress and strain values.

Figure 9

Sensitivity plots of the steady-state bundle response to system parameters. System parameters that are varied include microtubule bending stiffness (solid line), microtubule elastic modulus (dashed line), cross-link elastic modulus (dash dotted line), and cross-link length (dotted line). Response sensitivity is monitored for percent change in bundle strain (top left), percentage of energy stored in microtubule bending (top right), percentage of energy stored in microtubule stretching (bottom left), and percentage of energy stored in cross-link stretching (bottom right). The model shows relatively low sensitivity to modest changes in system parameters, lending confidence to the conclusions.