Motor Protein Accumulation on Antiparallel Microtubule Overlaps

Abstract

Biopolymers serve as one-dimensional tracks on which motor proteins move to perform their biological roles. Motor protein phenomena have inspired theoretical models of one-dimensional transport, crowding, and jamming. Experiments studying the motion of Xklp1 motors on reconstituted antiparallel microtubule overlaps demonstrated that motors recruited to the overlap walk toward the plus end of individual microtubules and frequently switch between filaments. We study a model of this system that couples the totally asymmetric simple exclusion process for motor motion with switches between antiparallel filaments and binding kinetics. We determine steady-state motor density profiles for fixed-length overlaps using exact and approximate solutions of the continuum differential equations and compare to kinetic Monte Carlo simulations. Overlap motor density profiles and motor trajectories resemble experimental measurements. The phase diagram of the model is similar to the single-filament case for low switching rate, while for high switching rate we find a new (to our knowledge) low density-high density-low density-high density phase. The overlap center region, far from the overlap ends, has a constant motor density as one would naïvely expect. However, rather than following a simple binding equilibrium, the center motor density depends on total overlap length, motor speed, and motor switching rate. The size of the crowded boundary layer near the overlap ends is also dependent on the overlap length and switching rate in addition to the motor speed and bulk concentration. The antiparallel microtubule overlap geometry may offer a previously unrecognized mechanism for biological regulation of protein concentration and consequent activity.

Figure 1

Model and results overview. (A) Schematic of the model of motor motion on an antiparallel microtubule overlap. Two filaments (green and blue) are modeled as one-dimensional lattices with their plus-ends oppositely oriented. The filaments are labled R (L) if the plus end is pointing to the right (left). Motors (red) bind to empty lattice sites with rate $k_{\text{on}}c$ and unbind with rate $k_{\text{off}}$. Bound motors step toward the MT plus-end with rate v (if the adjacent site toward the MT plus-end is empty) or switch to the other MT with rate s (if the corresponding site on the adjacent MT is empty). At MT minus ends, motors are inserted at rate αv. At MT plus ends, motors are removed at rate βv. (B) Simulated experimental images made from our kMC model. (Green) Motor density. (Red) Overlap region. Scale bar, 5 μm. Simulations used to generate these images used the reference parameter set (Table 1) and the indicated bulk motor concentrations. (C) Simulated kymograph made from our kMC model with motor spatial position on the horizontal axis and time increasing downwards. Horizontal scale bar, 10 μm. Vertical scale bar, 5 s. The simulations used to generate the kymograph used the reference parameter set and 0.5 nM bulk motor concentration. To see this figure in color, go online.

Figure 2

Nonequilibrium phases. (Left) Motor density ${\rho }_R\left(x\right)$ on MT with rightward-moving motors. (Right) Total motor density ${\rho }_R\left(x\right)+{\rho }_L\left(x\right)$ on both MTs in the overlap. The parameters comprise the reference parameter set of Table 1 with a bulk motor concentration of 200 nM, except for the switching rates, which are 0.5 s⁻¹ for the LHLH curve and 0.1 s⁻¹ for all other curves and the boundary conditions, as noted below. (LHLH) low density-high density-low density-high density coexistence (α = 0.3, β = 0.9, orange). (H_n) High-density center-minimum phase ($\alpha =0.95$, $\beta =0.99$, light blue). (H_x) High-density center-maximum phase (α = 0.6, β = 0.95, dark blue). (LH_x) Low-density/high-density center-maximum coexistence (α = 0.3, β = 0.95, green). (M) Meissner phase α = 0.6, β = 0.4, yellow). (L_n) Low-density center-minimum phase (α = 0.3, β = 0.4, black). (LH_n) Low-density/high-density center-minimum coexistence (α = 0.1, β = 0.95, magenta). (L_x) Low-density center-maximum phase (α = 0.05, β = 0.1, red). To see this figure in color, go online.

Figure 3

Phase diagrams. (Left) Low switching rate (0.1 s⁻¹); (right) high switching rate (0.5 s⁻¹). Solid lines indicate phase boundaries, and dotted lines indicate boundaries between center-maximum (x) and center-minimum (n) overlap motor density profiles. The bulk motor concentration is 200 nM, the motor speed is 5 μm s⁻¹, and other parameters are the reference values (Table 1).

Figure 4

Motor density profiles for the reference parameter set. (Left) Motor density ${\rho }_R\left(x\right)$ on MT with rightward-moving motors. (Right) Total motor density ${\rho }_R\left(x\right)+{\rho }_L\left(x\right)$ on both MTs in the overlap. Parameters comprise the reference parameter set of Table 1 with the bulk motor concentrations indicated in the legend. To see this figure in color, go online.

Figure 5

Motor density at the center of the overlap. (Left) Variation with bulk motor concentration; (right) variation with motor speed. Points indicate simulation results and dashed lines theoretical prediction from Eq. S31. The red line shows the value of the Langmuir density ${\rho }_0$ for the same parameters. To see this figure in color, go online.

Figure 6

Dependence of density profiles on motor speed. (Left) Motor density ${\rho }_R\left(x\right)$ on MT with rightward-moving motors. (Right) Total motor density ${\rho }_R\left(x\right)+{\rho }_L\left(x\right)$ on both MTs in the overlap. Varying motor speed can significantly alter the motor density at the center of the overlap. These simulations use the large-S parameter set of Table S1 with bulk motor concentration c = 200 nM and the motor speeds indicated in the legend. To see this figure in color, go online.

Figure 7

Control of boundary layer length. (Left) Boundary layer length as a fraction of total overlap length, shown as a function of bulk motor concentration and motor speed. (Right) Boundary layer length as a function of overlap length for the reference parameter set and varying bulk motor concentration. The boundary layer length is the distance at each end of the overlap where significant motor accumulation occurs and is determined by Eq. S30. To see this figure in color, go online.