Figure 1.
The cartoon depiction of the dynein motor molecules (red) is for visualization purposes only. Individual dynein molecules are not modeled computationally, only the pulling force they produce. Microtubule thickness is greatly exaggerated in the diagram. The centrosome (green) is merely a marker in the diagram; the centrosome in the model is identified with the common anchoring point of the microtubules.
Figure 2.
Centrosome reorientation in the model.
Dynamics of the centrosome orientation in a T cell developing sequentially two synapses is shown. The insets are computer-generated snapshots of the actual numerical model cell. The graphic conventions are the same as in Figure 1. Flattened interfaces with target cells are also depicted. The centrosome is initially pointing down (orientation −90°), and the first synapse develops on the cell equator (orientation 0°). The evolution of the centrosome orientation with time is shown by the blue plot. Note the oscillations following the stabilization of the equatorial position of the centrosome. After this (at t = 10 min) the model cell is set to develop the second synapse. In one version of the simulation (green plot), the second synapse develops on the top of the cell, and the centrosome rapidly migrates to it. In the alternative branch of the simulation (red plot), the second synapse develops on the bottom of the cell. In this case, the centrosome does not leave its position near the middle of the first synapse (red line). Both of the alternative centrosome positions seen at the end of this graph persist for much longer than plotted. Pulling force density, 40 pN/µm; microtubule length, 16 µm; effective cytoplasm viscosity, 2 pN s/µm2.
Figure 3.
Quantitative analysis of centrosome reorientation.
(A) The time it takes the model centrosome to reorient by one-half of the initial angular separation, as a function of this initial separation, plotted for the indicated values of the microtubule length. The segments of the broken lines connect the points corresponding to the actual simulation results; where the segments are dashed, it indicates that they connect two data points between which a data point is missing because the half-reorientation could not be achieved. Pulling force density, 40 pN/µm; effective cytoplasm viscosity, 2 pN s/µm2. (B) Qualitatively different predictions obtained with the different microtubule length and initial angular separation between the centrosome and the middle of the synapse. Regions in the two-dimensional parameter space are color-coded and numbered. In region 1, the complete reorientation is achieved. In region 2, the reorientation is “jammed” at around 30° of remaining angular separation. In region 3, the reorientation is “jammed” at the characteristic angular separation of 100°. In region 4, reorientation does not commence because the microtubules are too short to contact the synapse. In region 5, complete reorientation is achieved after a catastrophic stability loss of the “locked” configuration of antiparallel microtubules overlapping at the synapse. In region 6, the same happens but the final reorientation is as incomplete as in region 2. In region 7, the “locked” overlapping configuration is stable and no reorientation occurs. Pulling force density, 40 pN/µm; effective cytoplasm viscosity, 2 pN s/µm2. (C) Effect of microtubule dynamic instability on the stability of the “locked” configuration such as predicted in region 7 of (B). Angular position of the centrosome is plotted vs. time as predicted by the purely deterministic model analyzed throughout the paper (black curve) and with an additional assumption of stochastic microtubule dynamic instability (colored curves). The three stochastic simulations are independent (in the sense of pseudo-random number generation on a computer) repetitions of a simulation which was otherwise set up the same way as the deterministic one. The angle plotted is defined as the angle formed by the lines drawn from the nucleus center to the centrosome and to the middle of the synapse. The deterministic prediction is that the centrosome, having started facing the opposite side of the cell from the synapse, will not be able to reorient to the synapse. The stochastic predictions differ between runs: one is similar to the deterministic prediction, in the other two the centrosome was able to reorient. Pulling force density, 40 pN/µm; microtubule length (starting microtubule length in stochastic simulations), 21.5 µm; effective cytoplasm viscosity, 2 pN s/µm2.
Figure 4.
Oscillations of the centrosome within the synaptic area.
(A) Graphs of the model cell structure at the indicated time points. (B) The oscillating microtubule system shown in projection onto the synaptic plane. The parts that are in contact with the synaptic surface and are experiencing the pulling are highlighted in red. Pulling force density, 20 pN/µm; microtubule length, 16 µm; effective cytoplasm viscosity, 2 pN s/µm2.
Figure 5.
Typical trajectories of centrosomes oscillating within a synaptic area.
(A) A centrosome trajectory in projection onto the synaptic area, with color denoting the height above it and arrows, the direction. The directions of axes are as indicated in Figure 4. Pulling force density, 40 pN/µm; microtubule length, 16 µm; effective cytoplasm viscosity, 2 pN s/µm2. (B) Positions of the centrosome along the two horizontal axes and its vertical position plotted vs. time. Note the phase shift between the oscillations along the x and y axes that leads to gyrations visible in (A), and apparent beats. Pulling force density, 40 pN/µm; microtubule length, 16 µm; effective cytoplasm viscosity, 2 pN s/µm2. (C) Effect of microtubule dynamic instability and of an annular shape of the pulling surface on the pattern of oscillations. Position of the centrosome is plotted vs. time as predicted by the purely deterministic model with the disk-shaped pulling surface, as analyzed throughout the paper (black curve), and with stochastic microtubule dynamic instability and annular pulling surface (colored curves). The two stochastic simulations are independent in the sense of pseudo-random number generation on a computer. The stochastic predictions differ between runs but preserve the characteristic features of the deterministic one. Pulling force density, 20 pN/µm in the deterministic simulation and 36 pN/µm in the stochastic simulations. Microtubule length (starting microtubule length in stochastic simulations) was 16 µm, effective cytoplasm viscosity, 2 pN s/µm2.
Figure 6.
Dependence of the oscillations within the synaptic area on the pulling force density.
(A–C) The three types of oscillations that are predicted correspondingly with low, intermediate, and high values of the pulling force density. The centrosome trajectory is plotted in the x and z coordinates that are the same as in Figure 4 (x parallel and z perpendicular to the synapse). In (A), the pulling force density f = 100 pN/µm, in (B), f = 143 pN/µm, and in (C), f = 200 pN/µm. Microtubule length, 16 µm; effective cytoplasm viscosity, 2 pN s/µm2. (D) The mean period of oscillations parallel and perpendicular to the synapse, as a function of the pulling force density. The error bars are S.E. (insignificant in size for most data points). Microtubule length, 16 µm; effective cytoplasm viscosity, 2 pN s/µm2. (E) The mean (solid line) and the characteristic minimum and maximum (dashed lines) of the centrosome distance from the synapse, as a function of the pulling force density. The minimum and maximum attained during each period were averaged over many periods to obtain the values of the minimum and maximum that are characteristic of the given force density. The error bars in this plot show the standard error associated with the statistical estimation of the characteristic minimum and maximum values. Microtubule length, 16 µm; effective cytoplasm viscosity, 2 pN s/µm2. (F) The peak deviation of the centrosome from the midpoint (amplitude) in oscillations parallel to the synapse (x) vs. the centrosome distance from the synapse z at the moment when the peak deviation was achieved. The datapoints are plotted for the indicated values of the pulling force density. Microtubule length, 16 µm; effective cytoplasm viscosity, 2 pN s/µm2.
Figure 7.
Oscillations of the centrosome between two synaptic areas.
(A) Graphs of the model cell structure. The angle between the two synaptic planes is indicated by the red arc and equals 144° in this simulation. Pulling force density, 40 pN/µm; microtubule length, 16 µm; effective cytoplasm viscosity, 2 pN s/µm2. (B) Trajectories of the centrosome predicted for the indicated angles between the two synaptic planes. An excerpt from the experimental trajectory extracted from the supplementary video to the cited paper [4] is also shown (dashed). The illustration in (A) corresponds to the red theoretical curve. Pulling force density, 40 pN/µm; microtubule length, 16 µm; effective cytoplasm viscosity, 2 pN s/µm2.