Pivot-and-bond model explains microtubule bundle formation

During mitosis, bundles of microtubules form a spindle, but the physical mechanism of bundle formation is still not known. Here we show that random angular movement of microtubules around the spindle pole and forces exerted by passive cross-linking proteins are sufficient for the formation of stable microtubule bundles. We test these predictions by experiments in wild-type and ase1Δ fission yeast cells. 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.


Introduction
During mitosis the cell forms a spindle, a complex self-organized molecular machine composed of bundles of microtubules (MTs), which segregates the chromosomes into two daughter cells (1-3). MTs are thin stiff filaments that typically extend in random directions from two spindle poles (4). MTs that extend from the same pole can form parallel bundles, whereas MTs originating from opposite spindle poles form anti-parallel bundles (5-7).
Stability of MT bundles is ensured by cross-linking proteins, which bind along the MT lattice, (11,12)).
Spindle self-organization was studied in different biological systems and several theoretical models were proposed. Formation of antiparallel bundles of MTs in somatic cells of higher eukaryotes was investigated by computer simulations, which include MTs that grow in random directions from two spindle poles and motor proteins that link them (13). Further, several studies have explored the forces generated in the antiparallel overlaps in vitro (14-16) and in Drosophila embryo (17)(18)(19). Spindle formation was studied in Xenopus eggs, using the "slide and cluster" (20,21) and liquid crystal models (22,23). In budding yeast, it is suggested that MTs growing in arbitrary directions from the opposite spindle poles can change their direction due to minus end directed kinesin-14 motors bound to both MTs and get aligned, forming anti-parallel bundles (24). During spindle positioning, myosin motors walking along actin cables accelerate pivoting of astral MTs when they search for cortical anchor sites (25). Studies in fission yeast have shown that passive (thermal) pivoting motion of MTs around the spindle pole body accelerates kinetochore capture (26)(27)(28), together with dynamic instability of MTs (29). MT rotational motion about a pivot at the SPB was also included in the model for spindle formation (30) and in vitro studies (31). However, observation of the dynamics of bundle formation in vivo and a corresponding physical model are required to understand the formation and stability of MT bundles.
In this paper, we combine experiments and theory to explore the formation of parallel MT bundles. We introduce the pivot-and-bond model for the formation of parallel MT bundles, which includes random angular motion of MTs around the spindle pole (26), along with the attractive forces exerted by cross-linking proteins. The model predicts faster bundle formation if MTs diffuse faster and the density of cross-linking proteins is higher, which we tested experimentally. We conclude that the angular motion drives the alignment of MTs, which in turn allows the cross-linking proteins to connect the MTs into a stable bundle.

Experimentally observed bundle formation
The process of MT bundle formation can be observed experimentally in the fission yeast Schizosaccharomyces pombe because of a small number of MTs in the spindle. At the onset of mitosis, two spindle pole bodies nucleate MTs that form the spindle and additional MTs grow from the spindle pole bodies performing angular motion (26). In our experiments, we observed that MTs growing at an angle with respect to the spindle eventually join the bundle of spindle MTs (Supporting Fig. 1a, Supporting Movie 1). Such events are also accompanied by an increase in the tubulin-GFP signal intensity in the spindle, suggesting an increase in the number of MTs in the spindle and arguing against the scenario in which one of the MTs depolymerized (Fig. 1b). Additionally, we used cells with GFP-labeled Mal3, a protein that binds to the growing end of the MT (32). Here we observed MT bundling at a finer time resolution (Supporting Fig. 1c in Supporting Note) and the increased Mal3 signal in the spindle after bundling (Fig. 1d). Aside from MTs joining the already formed spindle, we also observed bundling between pairs of MTs which were both freely pivoting (see Supporting   Fig. 1a). We did not observe un-bundling events after the bundles were formed. In all cases, MTs performed angular motion around the spindle pole body, which allowed them to approach each other and form a bundle.

Theory
To explore the physical principles underlying the formation and stability of MT bundles, we introduce the pivot-and-bond model (Fig. 2a). We describe two MTs as thin rigid rods of fixed length with one end freely joint at the spindle pole body, based on experimental observations both in vivo (26,27) and in vitro (33). The orientation of the first MT at time t is described by a unit vector ( )r t (Fig. 2b). The orientation of the unit vector changes as where the vector ω denotes angular velocity of the MT. The other MT has a fixed orientation along the z-axis in the direction of unit vector . z In the overdamped limit, the angular friction is balanced by the torque, , T experienced by the MT: .
Here, γ denotes the angular drag coefficient. We calculate the total torque as ( ) , where the first and the second term represent the deterministic and the stochastic components, respectively. If the noise is caused by thermal fluctuations, as in fission yeast (26), is a 3-dimensional Gaussian white noise, where i-th and j-th components for times t and ' t obey Dirac delta function and δ ij is the Kronecker delta function. The magnitude of the noise is related to the angular drag coefficient, following the equipartition theorem, as , with B k T being the Boltzmann constant multiplied by the temperature. We introduce the In our model, the torque τ in equation (3) is the consequence of forces exerted by crosslinking proteins connecting both MTs. If we denote the positions along the MTs as =r r r and =ẑ z z respectively, the torque contribution from cross-linking proteins is with dN being the number of cross-linking proteins connecting the MT elements [ ] where we used ( ) × − = × r z r r z and allowed the fixed MT to span the entire positive z-axis.
When the total number of cross-linking proteins is large we can use the mean field limit and consider them continually distributed along the MT. In this limit, the cross-linking protein density is given by: Here, the currents describe the redistribution of cross-linking proteins along the MTs,

Results
To obtain the time course of the MT orientation, we parameterize the orientation of the MT given by the unit vector by ( ) ( ), sin cos ,sin sin ,cos r where θ and ϕ denote the polar and azimuthal angle, respectively. In this parameterization, the equation of motion for the polar angle reads ( ).
The normalized potential describing the interaction between the MTs, ( ), where τ denotes the magnitude of the torque and θ cot is the spurious drift term (35)   By numerically solving equation (7) for the polar angle, for a large initial angle, we found that the MT performs random movement and spans a large space (Fig. 2c). However, the movement can become abruptly constrained in the vicinity of angle zero. These small angles correspond to a bundled state. Our numerical solutions also show that, in rare cases, constrained MT movement in the vicinity of angle zero can suddenly switch back to free random movement (Fig. 2d). The constrained movement near angle zero is a consequence of short range attractive forces exerted by the cross-linkers that accumulate in larger densities when MTs are in close proximity (compare green and magenta lines in Fig. 2e). The density is constant up to 0 r because in that region, the cross-linkers can always attach in a relaxed configuration, while for , > 0 r r the cross-linkers will always be under tension and their density will drop off dramatically as r increases further (see illustration in Fig. 2e and Supporting Note 1). Our numerically obtained time courses that correspond to the MT bundling are similar to those from experiments (compare Figs. 2c and 2f).
To systematically explore the formation of MT bundles and their stability, we first examine the normalized potential describing the interaction between the MTs. The shape of the normalized potential for different MT lengths and nucleoplasm cross-linker concentrations is shown in Fig. 3a and there are enough cross-linking proteins in the nucleoplasm (see Fig. 3c).
Finally, we calculate how the MT bundling time depends on the parameters of the system. In the case of an isotropic distribution of initial MT orientations, we calculate the bundling time as  Table 1). We observed that the bundling time increases with MT length (Fig. 3e), and that the bundling time is significantly longer in ase1Δ cells (compare green and magenta line in Fig 3e). We normalized the bundling time by the diffusion constant and found a weak dependence on MT length, but a significant increase in ase1Δ cells compared to wild type (inset of Fig 3e, for theory see Supporting Note 1, for diffusion measurements see Supporting Fig. 2). The theory reproduces the weak dependence on MT length and implies that the deletion of ase1 decreases the effective cross-linker concentration roughly five fold.
In conclusion, our work implies that only passive processes, thermally driven motion    Table 1.