Simulation of cyclic dynein-driven sliding, splitting, and reassociation in an outer doublet pair

Abstract

A regular cycle of dynein-driven sliding, doublet separation, doublet reassociation, and resumption of sliding was previously observed by Aoyama and Kamiya in outer doublet pairs obtained after partial dissociation of Chlamydomonas flagella. In the work presented here, computer programming based on previous simulations of oscillatory bending of microtubules was extended to simulate the cycle of events observed with doublet pairs. These simulations confirm the straightforward explanation of this oscillation by inactivation of dynein when doublets separate and resumption of dynein activity after reassociation. Reassociation is augmented by a dynein-dependent "adhesive force" between the doublets. The simulations used a simple mathematical model to generate velocity-dependent shear force, and an independent elastic model for adhesive force. Realistic results were obtained with a maximum adhesive force that was 36% of the maximum shear force. Separation between a pair of doublets is the result of a buckling instability that also initiates a period of uniform sliding that enlarges the separation. A similar instability may trigger sliding initiation events in flagellar bending cycles.

Figure 1

Panel A shows a sequence of images covering one cycle of oscillation of example A, a typical simulation of a doublet pair with doublet B straight and stationary. Images 0 and 10 should be identical. Association, propagation, and dissociation phases are indicated by the letters A, P, and D, respectively. Panel B shows a similar sequence for example B, in which doublet B has a fixed curvature of 0.05 rad μm⁻¹. The parameters used for these models are given in Table 1. For example A, the time interval between images is 14.9 ms and the frequency is 6.7 s⁻¹. For example B, the time interval between images is 9.8 ms and the frequency is 10.2 s⁻¹.

Figure 2

Positions of transition points on doublet A, plotted as a function of time, for simulation examples A and B. The positions of association transitions, at the basal end of a separated region, are shown as open circles. The positions of dissociation points, at the distal end of a separated region, are shown as solid circles. The solid vertical lines demarcate the A, P, and D phases for one cycle of oscillation. These plots can be compared with Fig. 4 of Aoyama and Kamiya (1).

Figure 3

Shear displacement in segment 98 of 100, plotted against time, for typical cycles of examples A (open circles; cycle period 0.149 s) and B (solid circles; cycle period 0.98 s). For examples A and B, a least-squares linear fit gives a velocity of 13.53 μm s⁻¹ and 13.67 μm s⁻¹, respectively. Shear resulting from the fixed curvature of doublet B has been subtracted from the shear values for example B.

Figure 4

Effects of varying the unloaded sliding velocity, V₀, generated by the active sliding system. Except for variations in V₀, all other parameters are the same as given in Table 1 for example A. (A) Propagation velocities for the association (upper points) and dissociation (lower points) transition points. Solid circles are from simulations with the non(steady-state) dynein model; open circles used the steady-state dynein model. The line through the dissociation point velocities is the calculated relationship between dissociation velocity and sliding velocity obtained from experimental data, with K_m for dissociation velocity vs. [ATP] of 0.107 mM (1) and K_m for V₀ vs. [ATP] of 0.177 mM (17), assuming [ATP] = 0.5 mM for the standard conditions where V₀ = 14 μm s⁻¹. This line diverges only slightly from the linear relationship expected if K_m is the same for both variables, so that the dissociation rate is proportional to V₀. (B) Cycle frequency (solid circles) and final shear attained in the distal associated region, at the end of phase D (open circles). Note that the results for V₀ = 18 and 24 μm s⁻¹, at the right ends of the curves, are beyond the range accessible in the experiments (1).

Figure 5

Panel A shows a sequence of images covering one cycle of oscillation of example A with V₀ reduced to 1.0 μm s⁻¹. The time interval between images is 38.5 ms and the frequency is 2.6 s⁻¹. Panel B shows a sequence of images covering one cycle of oscillation of example A with V₀ increased to 48 μm s⁻¹. Association, propagation, and dissociation phases are indicated by the letters A, P, and D, respectively. The time interval between images is 12.2 ms and the frequency is 8.2 s⁻¹.

Figure 6

Panel A shows a sequence of images covering one cycle of oscillation of example A with E_N reduced to 2 × 10⁻⁶ pN nm⁻². Association, propagation, and dissociation phases are indicated by the letters A, P, and D, respectively. The time interval between images is 23.3 ms and the frequency is 4.3 s⁻¹. Panel B shows a sequence of images covering one cycle of oscillation of example A with E_N increased to 0.019 pN nm⁻². The time interval between images is 27.4 ms and the frequency is 3.65 s⁻¹.