Torsional behavior of axonal microtubule bundles

Abstract

Axonal microtubule (MT) bundles crosslinked by microtubule-associated protein (MAP) tau are responsible for vital biological functions such as maintaining mechanical integrity and shape of the axon as well as facilitating axonal transport. Breaking and twisting of MTs have been previously observed in damaged undulated axons. Such breaking and twisting of MTs is suggested to cause axonal swellings that lead to axonal degeneration, which is known as "diffuse axonal injury". In particular, overstretching and torsion of axons can potentially damage the axonal cytoskeleton. Following our previous studies on mechanical response of axonal MT bundles under uniaxial tension and compression, this work seeks to characterize the mechanical behavior of MT bundles under pure torsion as well as a combination of torsional and tensile loads using a coarse-grained computational model. In the case of pure torsion, a competition between MAP tau tensile and MT bending energies is observed. After three turns, a transition occurs in the mechanical behavior of the bundle that is characterized by its diameter shrinkage. Furthermore, crosslink spacing is shown to considerably influence the mechanical response, with larger MAP tau spacing resulting in a higher rate of turns. Therefore, MAP tau crosslinking of MT filaments protects the bundle from excessive deformation. Simultaneous application of torsion and tension on MT bundles is shown to accelerate bundle failure, compared to pure tension experiments. MAP tau proteins fail in clusters of 10-100 elements located at the discontinuities or the ends of MT filaments. This failure occurs in a stepwise fashion, implying gradual accumulation of elastic tensile energy in crosslinks followed by rupture. Failure of large groups of interconnecting MAP tau proteins leads to detachment of MT filaments from the bundle near discontinuities. This study highlights the importance of torsional loading in axonal damage after traumatic brain injury.

Figure 1

Schematic of a neuron cell along with dendrites and its axon. (a) A zoomed-in view of the axon initial segment showing the bundled MTs. MT filaments, represented by helical structures of heterodimers (blue and green beads), are bundled by MAP tau crosslinks (red). Discontinuities are randomly placed along the length of the filaments. (b) A zoomed-in view of two MT filaments and one MAP tau protein, showing a detailed schematic of the bead-spring model. Bundle filaments are represented by a network of beads (blue) connected by spring elements of axial constant k^MT (dark blue). Torsional springs (white), representing bending stiffness of k_b, were also included for each MT element. Crosslinks (red) are modeled as spring elements with axial spring constant k^CL. (Not drawn to scale.) To see this figure in color, go online.

Figure 2

Schematic of the mechanical model used to apply torsional forces. (a) The applied torque is a linear function of x, parallel to the long axis of the bundle. (b) Cross-section of the bundle with orthoradial forces f₁, f₂, and f₃, proportional to the corresponding radii R₁, R₂, and R₃, which are initially 45 nm, $45\sqrt{3}$ nm ≈ 77.0, and 90 nm, respectively. To see this figure in color, go online.

Figure 3

Progressive twisting of MT bundle. (a) 0 turn; (b) 0.83 turn; (c) 1.7 turns; (d) 2.7 turns; (e) 4 turns; (f) 5.1 turns; (g) 7 turns; and (h) >8 turns. Shrinking of the bundle happens at three turns (d and e), and after seven turns (g), the bundle radius is significantly reduced. To see this figure in color, go online.

Figure 4

Energy distribution in the case of pure torsion for seven bundle turns. Energy proportion is defined as the ratio of each type of energy to the total energy of the bundle. MT filament bending and tensile energies, MAP tau protein tensile energy, steric energy, and kinetic energy are given. Gradual increase of steric energy is observed, exceeding MAP tau tensile energy at three bundle turns and reaching MT bending energy at seven turns. Kinetic energy and MT tensile energy do not play a significant role, because the movement of beads is limited in torsion experiments. To see this figure in color, go online.

Figure 5

Rate of turns (number of turns per microseconds) as a function of average crosslink spacing under a fixed torque magnitude of 10 nN.nm. Four different spacings are considered: 25, 50, 75, and 100 nm. A linear trend line is shown (f(x) = 0.0096x + 0.675 and R² = 0.952), suggesting a proportionality between rate of bundle turns and MAP tau spacing. To see this figure in color, go online.

Figure 6

Energy distribution in a bundle relaxation experiment. The MT bundle is subjected to a torsional loading up to one turn and then the loading is removed at 1 μm. A steady state is reached at 2 μs. Release of the torsional load results in a sharp decrease in MT bending energy and a rapid increase in the proportion of MAP tau tensile energy. This suggests that MAP tau proteins try to recover the initial configuration of the bundle. Steric energy proportion remains roughly constant, suggesting that MT filaments spacing stays approximately constant during relaxation. To see this figure in color, go online.

Figure 7

Mechanism of MT failure under the combination of a tensile force of F = 5 nN and the torsion of seven turns. (a–c) Detachment of the end of one of the filaments. (d–f) Detachment of the filament near a discontinuity. (g) Number of failed MAP tau proteins in time. (Curves) Different rates of turns, i.e., number of turns per microsecond. Higher twisting rate increases the number of failed MAP tau proteins and leads to earlier failure, compared to pure stretching. Number of failed MAP tau proteins increases in a stepwise fashion, implying that MAP tau proteins gradually store tensile energy until they reach their critical strain and fail in groups. To see this figure in color, go online.

Figure 8

Effect of torsion on bundle failure under a F = 5 nN tensile force. (a) Axial coordinates of failed MAP tau proteins (in fraction of total length) as a function of number of bundle turns. Groups of crosslinks fail locally in certain regions of the bundle. A cluster of 30 MAP tau proteins fails just before one turn. After two turns, two groups of 20 and 15 MAP tau proteins break; after four turns, another set of 10 MAP tau proteins fails. Clusters of failed MAP tau proteins are located near discontinuities of MT filaments. (b) Radial position of failed tau proteins before two bundle turns. Most MAP tau failures are seen at the periphery of the bundle. (Open circle) The bundle. To see this figure in color, go online.

Figure 9

Stretching of pretwisted MT bundles. Each of the six bundles is twisted by a certain number of turns before being stretched (represented by six colors in legend). (Dashed lines) Point at which an MT filament is ejected from the bundle (see Fig. 7, a–f). Final number of failed MAP tau increases with the number of turns of the bundle before stretching. To see this figure in color, go online.

Figure 10

Effect of torsion on energy distribution under a F = 5 nN tensile force. (a) Energy distribution in the case of pure tension. A steady state is rapidly reached within the first microsecond. (b) Energy distribution in the case of combined torsional and tensile loading conditions. Addition of torsion influences the steady-state behavior, postponing the time at which energies stabilize, changing the steady-state values. The role of MT tensile energy is decreased, while steric energy role is becoming more significant due to torsion. To see this figure in color, go online.