Pivot-and-bond model explains microtubule bundle formation

During mitosis, microtubules form a spindle, which is responsible for proper segregation of the genetic material. A common structural element in a mitotic spindle is a parallel bundle, consisting of two or more microtubules growing from the same origin and held together by cross-linking proteins. An interesting question is what are the physical principles underlying the formation and stability of such microtubule bundles. Here we show, by introducing the pivot-and-bond model, that random angular movement of microtubules around the spindle pole and forces exerted by cross-linking proteins can explain the formation of microtubule bundles as observed in our experiments. The model predicts that stable parallel bundles can form in the presence of either passive crosslinkers or plus-end directed motors, but not minus-end directed motors. In the cases where bundles form, the time needed for their formation depends mainly on the concentration of cross-linking proteins and the angular diffusion of the microtubule. In conclusion, the angular motion drives the alignment of microtubules, which in turn allows the cross-linking proteins to connect the microtubules into a stable bundle.

Figure 1

Formation of MT bundles in S. pombe cells. (a) Time-lapse images and the corresponding drawings showing the formation of a parallel MT bundle at the lower spindle pole body in an S. pombe cell expressing tubulin-GFP and Sid4-GFP. (b) Measurement of the tubulin-GFP signal intensity of MTs before bundling (green curve in the graph, measured along the green line in the inset) and after bundling (magenta curve in the graph, measured along the magenta line in the inset). The measurements were done on the first and the last image in panel (a), respectively. After the contact between two MTs, the intensity of the tubulin-GFP signal increases. (c) Time-lapse images and the corresponding drawings showing a MT joining the bundle of spindle MTs in an S. pombe cell expressing tubulin-GFP and Sid4-GFP. (d) Measurements of the tubulin-GFP signal intensity of the spindle and MT before bundling (green curve in the graph, measured along the green line in the inset) and after bundling (magenta curve in the graph, measured along the magenta line in the inset). The measurements were done on the first and the last image in panel (c), respectively. After the bundling event, the intensity of the tubulin-GFP signal of the spindle increases. (e) Time-lapse images and the corresponding drawings showing a MT joining the bundle of spindle MTs in an S. pombe cell expressing Mal3-GFP and Sid4-GFP. (f) Measurements of the Mal3-GFP signal intensity of the spindle and MT before bundling (green curve in the graph, measured along the green line in the inset) and after bundling (magenta curve in the graph, measured along the magenta line in the inset). The measurements were done on the first and the last image in panel (e), respectively. After the bundling event, the intensity of the tubulin-GFP signal of the spindle increases. Scale bars in (a), (c), and (e) are $1\mu \mathrm{m}$. In the drawings, microtubule orientations from the previous images are marked with white dashed lines.

Figure 2

Scheme of the model and numerical solutions. (a) Cartoon representation of the bundling process. Left, a MT (green rod) pivots around the spindle pole (gray ball). Right, when the MTs come into close proximity, cross-linking proteins (grey springs) attach and cause the MTs to from a bundle. (b) Scheme of the model. The orientations of two MTs are represented by the unit vectors $\widehat{\mathrm{r}}$ and $\widehat{\mathrm{z}}$. Cross-linking proteins attach to and detach from MTs at rates $k_a$ and $k_d$, respectively. The elongation of the attached cross-linking protein is denoted $\mathrm{y}$ and their relaxed length is ${\mathrm{y}}_0$. The angle between the MTs is denoted $\theta$. (c) Density of passive crosslinkers. The inset shows the positions (drawn to scale) of crosslinkers that correspond to the points denoted on the main graph. Angle between the MTs is $\theta =0.3\mathrm{rad}$. (d) Effective torque caused by passive crosslinkers as a function of the angle between the MTs. (e) A sample path for the starting angle $\theta =1\mathrm{rad}$, which shows a bundling event (around the $1\min$ mark). The crosslinker concentration is $c_0=60\mu {\mathrm{m}}^{-1}$. (f) A sample path for the starting angle $\theta =0.01\mathrm{rad}$, which shows an unbundling event (around the $1\min$ mark). (g) A sample of MT angle time series obtained using light microscopy on cells with the Mal3-GFP label. Green line corresponds to the segment of the sample path in the shaded area in (e). Unless stated otherwise, calculations are done with $c_0=10\mu {\mathrm{m}}^{-1},R=1\mu \mathrm{m}$ and other parameters shown in Table 1.

Figure 3

Solutions of the model in the adiabatic approximation. (a) The effective potential as a function of the polar angle, for angles between $0\mathrm{rad}$ and $0.4\mathrm{rad}$. The inset shows the potential for all angles. Dashed square in the inset represents the region plotted in the large image. Values of the parameters used are given in the legend. (b) Stability diagram for plus-end directed motors. The gray area represents the region where bundles are stable, i.e., bundling is more probable than unbundling. (c) Logarithmic plot of the bundling (black) and unbundling (green) times as functions of MT length. The crosslinker concentration is $c_0=30\mu {\mathrm{m}}^{-1}$. (d) Bundling (black) and unbundling (green) times as functions of crosslinker concentration. The MT length is $R=1\mu \mathrm{m}$. Arrow represents the parameter values shown with arrows in (b). All calculations done with parameter values from Table 1.

Figure 4

Solutions of the model for motors. (a) Density of plus-end directed motors. The inset shows the positions (drawn to scale) of motors that correspond to the points denoted on the main graph. Angle between the MTs is $\theta =0.3\mathrm{rad}$. (b) Effective torque caused by plus-end directed motors as a function of the angle between the MTs. The inset shows the potential. (c) Stability diagram for plus-end directed motors. The gray area represents the region where bundles are stable, i.e., bundling is more probable than unbundling. (d) Density of minus-end directed motors. The inset shows the positions (drawn to scale) of motors that correspond to the points denoted on the main graph. Angle between the MTs is $\theta =0.3\mathrm{rad}$. (e) Effective torque caused by minus-end directed motors as a function of the angle between the MTs. The inset shows the potential. All calculations are done with $c_0=10\mu {\mathrm{m}}^{-1},R=1\mu \mathrm{m}$ and other parameters shown in Table 1.