Axonal transport: how high microtubule density can compensate for boundary effects in small-caliber axons

Abstract

Long-distance intracellular axonal transport is predominantly microtubule-based, and its impairment is linked to neurodegeneration. In this study, we present theoretical arguments that suggest that near the axon boundaries (walls), the effective viscosity can become large enough to impede cargo transport in small (but not large) caliber axons. Our theoretical analysis suggests that this opposition to motion increases rapidly as the cargo approaches the wall. We find that having parallel microtubules close enough together to enable a cargo to simultaneously engage motors on more than one microtubule dramatically enhances motor activity, and thus minimizes the effects of any opposition to transport. Even if microtubules are randomly placed in axons, we find that the higher density of microtubules found in small-caliber axons increases the probability of having parallel microtubules close enough that they can be used simultaneously by motors on a cargo. The boundary effect is not a factor in transport in large-caliber axons where the microtubule density is lower.

Figure 1

(A) Cargos being hauled by motors along MTs inside an axon. The spherical cargo has a radius a in a cylindrical axon of radius R. The center of the cargo is displaced a distance b from the axon’s axis. The distance h is the closest approach of the cargo to the axon wall. Two scenarios are shown. The top shows a single motor hauling a cargo along an MT. The key point of this paper is that the enhanced viscosity encountered by the cargo near the wall of the axon can be overcome by having multiple motors hauling the cargo along closely spaced parallel microtubules as shown at the bottom. (B) Correction factor K from Eq. 2 for a sphere of radius 250 nm for two axon diameters (k = 0.1 and k = 0.5) as a function of the cargo-wall distance h. As a comparison, we have also shown the correction factor obtained from the Faxén formula (blue dotted line) for the same parameters. Far away from the wall in a large axon (magenta dashed curve) our theory agrees well with Faxén’s law. Note that for a relatively large cargo (250 nm radius) in a relatively small axon (k = 0.5, corresponding to a 1 μm diameter axon, solid line), the “edge” effect, represented by the correction factor, extends over the width of the axon, even when the cargo is far away from the wall. (C) Comparison of the theoretical correction factor K given by Eq. 2 to experimental data (82) for a sphere moving through a fluid-filled cylinder for various values of k = a/R. To see this figure in color, go online.

Figure 2

Effective viscosity of the medium in the presence of the macromolecules of length L as a function of the ratio of the cargo-cylinder surface-to-surface distance h to the radius a of the cargo. (A) Axon diameter D = 300 nm; (B) D = 400 nm; (C) D = 500 nm; and (D) D = 800 nm. Irrespective of the caliber D, there is dramatic increase in the effective viscosity when this ratio h/a is of the order of or less than one. Polymer concentration is fixed at 4.17% excluded volume. For L = 20, 100, and 400 nm, this excluded volume corresponds to concentrations of 16.5, 3.31, and 0.827 μM, respectively. To see this figure in color, go online.

Figure 3

Cargo run length (i.e., cargo travel distance) along two parallel microtubules as a function of viscosity for a cargo (radius a = 100 nm) driven by a maximum of (A) N = 2 motors, (B) N = 3 motors, (C) N = 4 motors, and (D) N = 5 motors. For each number of (maximally engaged) motors, we investigated different microtubule-microtubule spacings d. The two microtubules were separated by center-to-center distances of d = 50 nm (upper curve); d = 75 nm (middle curve); and d = 275 nm (lower curve). To see this figure in color, go online.

Figure 4

Cargo run length (i.e., cargo travel distance) as a function of the microtubule center-to-center separation for different numbers of motors. There are two microtubules. N is the number of motors on the cargo. Here the cargo radius is a = 100 nm, and the viscosity is 100 times that of water. In the simulations, a single motor binds to one of the microtubules and the system then evolves. The additional (free) motors can bind to either microtubule, governed by the prescribed binding rules implemented in the simulation (see the Supporting Material). To see this figure in color, go online.

Figure 5

Probability of an MT having a neighbor within a given distance d versus MT density, where d = 75, 100, and 200 nm. Note that to have at least an 80% chance of having a pair of MTs with a separation of 75 nm (blue dashed line), the axon needs to have an MT density of ∼ 100 MTs/μm². To see this figure in color, go online.