Regulating positioning and orientation of mitotic spindles via cell size and shape

Proper location of the mitotic spindle is critical for chromosome segregation and the selection of the cell division plane. However, how mitotic spindles sense cell size and shape to regulate their own position and orientation is still largely unclear. To investigate this question systematically, we used a general model by considering chromosomes, microtubule dynamics, and forces of various molecular motors. Our results show that in cells of various sizes and shapes, spindles can always be centered and oriented along the long axis robustly in the absence of other specified mechanisms. We found that the characteristic time of positioning and orientation processes increases with cell size. Spindles sense the cell size mainly by the cortical force in small cells and by the cytoplasmic force in large cells. In addition to the cell size, the cell shape mainly influences the orientation process. We found that more slender cells have a faster orientation process, and the final orientation is not necessarily along the longest axis but is determined by the radial profile and the symmetry of the cell shape. Finally, our model also reproduces the separation and repositioning of the spindle poles during the anaphase. Therefore, our work provides a general tool for studying the mitotic spindle across the whole mitotic phase.

Figure 1

Schematic diagram of the mechanical model. The schematic diagram of the mitotic spindle is shown at the center, and five force-generation mechanisms are illustrated, including (1) the pushing force generated by hindering the microtubule polymerization, (2) the pulling force generated by cortical dyneins or kinetochores, (3) the pushing force generated by cortical kinesins or chromosome kinesins, (4) the pulling force generated by cytoplasmic dyneins, and (5) the pushing force generated by cross-linkers. Bottom: Chromosomes are assumed as an $\mathsf{X}$-shaped rigid body with a tilt angle ${75}^{\circ }$ of the chromosome arms. The length of the chromosome is $L_c$. The width of the chromosome is defined as $W_c=0.1L_c$. The size of the kinetochore is defined as a circle with a diameter of $d_k=0.18L_c$ (red dot), where the microtubule tips can be bound by the kinetochore.

Figure 2

Self-assembly, positioning, and orientation of the mitotic spindle with three chromosomes and two centrosomes. (a) Screenshots of the simulation (see also Movie 1 in the Supplemental Material [57]). Initially, the centrosomes are located at ($-5,-3.5$) and ($-2,-5$); the chromosomes are located at (1.7,0), ($-1.7,0$), and (0,$-3$); and their orientations are 2.26 rad, 3.05 rad, and 0.44 rad. *The line width represents the number of microtubules overlapping at the kinetochore. (b) The schematic diagram illustrates the indicators of the spindle position $d_s$, spindle orientation ${\alpha }_s$, spindle length $L_s$, chromosome position $d_c$, and chromosome orientation ${\alpha }_c$. Two examples (solid lines and dashes, respectively) show the time evolution of the (c) spindle position, (d) spindle orientation, (e) spindle length, (f) chromosome orientation, and (g) chromosome position with the same initial conditions.

Figure 3

Self-assembly, positioning, and orientation of the mitotic spindle with two centrosomes and one chromosome. (a) Screenshots of the simulation at the beginning and end (see also Movie 3 in the Supplemental Material [57]). Initially, the two centrosomes are located at ($-10.2,-1.3$) and (2.2,5.3), the chromosome is located at ($-2,2$), and its orientation is 0.5 rad. The time evolution of the (b) spindle position and (c) orientation (20 simulations) is shown, which can be fitted by $d_s=a{e}^{-t/{\tau }_d}$ and ${\alpha }_s=b{e}^{-t/{\tau }_{\alpha }}$ (blue thick line) to obtain the characteristic time, ${\tau }_d$ and ${\tau }_{\alpha }$ (red dashed). From 100 groups of random initial conditions, the spindles can always be (d) positioned at the cell center and (e) oriented along the long axis of the cell.

Figure 4

Influence of the number, distribution, and composition of cortical microtubules on the positioning and orientation of mitotic spindles. (a) The characteristic time of the positioning and orientation vs. the nucleation rate ($\mathrm{mean}\pm \mathrm{SE}$ from 20 simulations). (b) The time-averaged numbers of the total cortical microtubules and the pulling cortical microtubules with the various fraction of the pulling microtubules, $p^{-}$, which is changed by the binding rate of dynein, $k_b^{-}$. The characteristic time of (c) positioning and (d) orientation are the functions of the fraction of the pulling microtubules, $p^{-}$, and the slipping drag coefficient, $\xi$, obtained by averaging 50 simulations for each case. The triangle indicates the values used in other sections. The white dash marks the cases that are most efficient. The black area in the lower left corner means the process is failed. The other settings are the same as those used in Fig. 3. (e) The schematic of the positioning mechanism. The results in (c) can be summarized as four regimes, relying on the slipping speed and the major force on the microtubules. The green arrows on the microtubules (four insets) indicate the major force (pulling or pushing). The big arrows (four insets) indicate the total force on the spindle. (f) Positioning the two poles, respectively, can provide a torque to orient the spindle.

Figure 5

The influence of cell size on the positioning and orientation of the spindles. The (a) position and (b) orientation of the mitotic spindles [indicated by $d_s$ and ${\alpha }_s$ defined in Fig. 2] in cells of various sizes are plotted as functions of time. Both the position and orientation are normalized by their initial values. (c) Characteristic time of the positioning and orientation vs. the cell size ($\mathrm{mean}\pm \mathrm{SE}$ from 50 simulations). (d) The contribution of cortical and cytoplasmic forces to the total force which drives the spindle motion in cells of various sizes (averaged over time for each case). The transition point is around 135 $\mu \mathrm{m}$ (dashed line) with our parameters, corresponding to that in (c).

Figure 6

The influence the of cell shape on the positioning and orientation of the spindles. (a) Top:) The schematic diagram of the aspect ratio of elliptical cells, ${\lambda }_0$. Bottom: The characteristic time of positioning and orientation vs. the aspect ratio ${\lambda }_0$ of elliptical cells ($\mathrm{mean}\pm \mathrm{SE}$ from 50 simulations). The other settings are the same as those used in Fig. 3. The curve of ${\tau }_{\alpha }$ can be fitted by an exponential decay function (black dashed curve). (b) The schematic diagrams of the radial profile of elliptical cells, $\lambda \left(\alpha \right)$. [(c)–(j)] The cellular radial profile $\lambda$ and the probability density function (PDF) of the stable spindle orientation are the functions of angle $\alpha$ in various elliptical cells. The stable spindle orientations are obtained from the simulations that 100 initial positions of centrosomes and chromosomes are given randomly. Here, we consider eight shapes, including [(c)–(e)] three equilateral polygons and [(f)–(j)] elongated along different directions. The red double arrows indicate the elongation direction. (k) The two selections of the stable spindle orientation in elliptical cells, the local maximum $\lambda$, and the symmetric axis.

Figure 7

Comparison of the rule-based predictions with the experiment results in Ref. [21]. Based on the rule of spindle orientation we proposed above [Fig. 6], we plot the radial profile $\lambda$ (inner blue solid line) of each shape in the polar coordinate system together with the cell shape (black solid line). The radial profile $\lambda$ has been rotated ${90}^{\circ }$ to indicate the division-axis orientation as the experiments. Experiments results show the spindle tends to orient along the local maximum of $\lambda$ and $\lambda >1$ (red dash). If the peak value of $\lambda$ is bigger, then the probability that the spindle is along this direction can be larger. In the meanwhile, division plane also tends to orientate the symmetric axis of the cell (blue dash). In some shapes (indicated by asterisk), there are some ranges that $\lambda$ almost keeps constant and equals 1. In this case, the orientation is randomly distributed in these ranges like round or square cells. The experimental results are in good agreement with the prediction of the rule.

Figure 8

Simulations of cell division during the mitotic anaphase. (a) Snapshots of simulation results for three cases, including the (1) normal symmetric division (see also Movie 4 in the Supplemental Material [57]); (2) the symmetric division with proximity of the two cell membranes (see also Movie 5 in the Supplemental Material [57]); and (3) the asymmetric division (see also Movie 6 in the Supplemental Material [57]). To achieve the asymmetric division, a higher binding rate of dynein $k_b^{-}=0.06$ on the right half cortex is given to deflect the spindle ($\sim 3\mu \mathrm{m}$). (b) Schematic diagram of the cell shape during the cytoplasmic division. The radius of contractile ring $r_s$ decreases linearly over time. If the cell membranes are not proximate to each other in the division plane, then $r_s\equiv r$. Otherwise, a critical angle $\gamma =3\pi /2$ is defined as the onset of the proximity, and then $r$ is constant. The volume of the whole cell is assumed constant. (c) Schematic diagrams of symmetric and asymmetric elongation of the cell membrane. (d) The distances between the spindle poles are the functions of time for the four cases. [(e)–(g)] The $x$ coordinates of the spindle poles and the centers of the two spherical caps are the functions of time for the (e) symmetric division, (f) asymmetric elongation, and (g) asymmetric division.