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Fig. 4 | BMC Biology

Fig. 4

From: Pivoting of microtubules driven by minus-end-directed motors leads to spindle assembly

Fig. 4

Theoretical model and solutions for MT dynamics. a Scheme of the model. Each MT (gray rod) is freely joint to its respective SPB (gray sphere). Orientations of two MTs are represented with unit vectors \( {\hat{\mathbf{r}}}_1 \) and \( {\hat{\mathbf{r}}}_2 \) respectively, while the SPBs are at fixed points separated by the distance dSPB. Motor proteins (green springs with rest length y0) can attach to and detach from MTs with rates kon and koff, respectively, and when attached, their elongation is y. In the Cartesian coordinates, the SPBs are at points (0, 0, 0) and (0, 0, dSPB). The MT orientations are described by the polar angles θ1 and θ2 and by the azimuthal angles φ1 and φ2 for the first and the second MT, respectively. b A sample path representing a bundling event. Top, polar angles denoted with a blue and a red line for the first and second MT, respectively. In the antiparallel configuration, θ1 = 0°, θ2 = 180° (see a for parametrization). Middle, number of attached motors. Bottom, distance between the two closest points on the MTs. Shaded regions: the search (dark gray) and the aligning phase (light gray); white region represents the bundled state. Simulation is performed with parameter values R1,2 = 1.5 μm, dSPB = 2 μm, nMT = 2, and other values from Table 1. c Illustrations of search (top), aligning (middle), and bundled state (bottom) of MTs (gray rods). To illustrate the motor distribution, the position of each motor (green) is randomly generated using a normal distribution around their mean position with the steady state variance, which is calculated from the MT orientations. The small gray spheres represent the SPBs and the large translucent gray sphere represents the nuclear envelope. The images are taken from Additional file 5: Movie S4, which is produced using the same data as in b. Time is given in min:s

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