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Cell shape and intercellular adhesion regulate mitotic spindle orientation

Cell division orientation plays an essential role in tissue morphogenesis and cell fate decision. Recent studies showed that either cell shape or adhesion geometry can regulate the orientation of mitotic spindles and thereby the cell division orientation. However, how they together regulate the spindle orientation remains largely unclear. In this work, we use a general computational model to investigate the competitive mechanism of determining the spindle orientation between cell shape and intercellular adhesion in epithelial cells. We find the spindle orientation is dominated by the intercellular adhesion when the cell shape anisotropy is small, but dominated by the cell shape when the shape anisotropy is large. A strong adhesion and moderate adhesive size can ensure the planar division of epithelial cells with large apico-basal elongation. We also find the spindle orientation could be perpendicular to the adhesive region when only one side of the cell is adhered to an E-cadherin–coated matrix. But after the cell is compressed, the spindle orientation is governed by the cell shape and the spindle will be parallel to the adhesive region when the cell shape anisotropy is large. Finally, we demonstrate the competition between cell shape and tricellular junctions can also effectively regulate the spindle orientation.


Microtubules interact with chromosomes
Since the diameter of chromosome arms is around 1µm [20], only a fraction of microtubules can be blocked by the chromosomes, and most of microtubules can bypass the chromosomes. Microtubules blocked by the chromosome or close to the chromosome can be bound by the chromokinsin on arms [4,26] or the kinetochore. Considering these factors, we assume all microtubules cannot be blocked by the chromosomes in our 2D model, but the microtubules across the chromosome can be bound by the chromokinsin at a rate of k b,c or the kinetochore at a rate of k b,k (Inset 4 of Figure S1).
The force generated by the chromokinesin on the microtubule is also velocity-dependent [4] as f + c = f + 0 (1 − v + /v + 0 ), where f + 0 and v + 0 are the stall force and the unloaded velocity of kinesin; v + is the walking velocity of the motor, i.e., the component in the microtubule direction of the relative velocity between the centrosome and the chromosome. It should be also noted that the velocity is positive when the microtubule is elongating, while it is negative when the microtubule is shortening. In addition, since the microtubules bound by the chromokinesin are also applied a pushing force, the microtubules can also be buckled when the force is larger than the Euler buckling force. Therefore, the pushing force generated by the chromokinesin is given as Here L is not the total length of the microtubule, but the length between the centrosome and the binding site. The binding chromokinesin can unbind from the microtubule as well. The unbinding rate is also load-dependent [4] as where f + u is a characteristic force representing the sensitivity of the unbinding rate to the load, and k + 0 is the unloaded unbinding rate of kinesin. 4 Extra nucleation Figure S1: The computational model for the mitotic spindle. Microtubules can be nucleated either from the centrosome, or near the chromosomes, or from existing microtubules as branches in the spindle region. Considering the microtubules nucleated near the chromosomes are converged to the spindle pole by minus end-directed motor proteins [10], and the branching microtubules have the same polarity as their mother microtubules [24], we assume all microtubules are nucleated from the centrosomes for simplicity (Inset 1). Microtubules can form bundles at the kinetochore through the mechanism that the minus-end-directed molecular motor kinesin-14 cooperates with the plus-end tracker EB1 to guide the growing plus end along existing microtubules [21] (Inset 2). Through polymerization or depolymerization of microtubules or various molecular motors, the cell cortex (Inset 3), chromosomes (Inset 4), antiparallel microtubules (Inset 5), or cytoplasm (Inset 6) can generate pushing forces or pulling forces on microtubules. Black arrows indicate the directions of forces on the microtubule. These forces can drive the movements of centrosomes and chromosomes.
We define the central region of the chromosome (∼0.2 of the chromosome length) as the kinetochore region. The microtubules encountering the kinetochores can attach to the kinetochores, and generate pulling forces due to the depolymerization of microtubules [1] (Inset 4 of Figure S1). The pulling force is both microtubule length-dependent [11] and velocity-dependent [5]. Therefore, we assume the pulling force is where l is the length of the microtubule; f k,0 is the force per length of the microtubule; and v − equals the component in the microtubule direction of the relative velocity between the centrosome and the chromosome. The velocity is negative when Figure S2: The microtubule distribution in the simulation. (A) The real mitotic spindle observed by confocal microscope in MDCK [12] (B) A snapshot of a simulated spindle similar to the real spindle in (A). (C) The probability density distribution of microtubules nucleated from the right centrosome in (B). Scale bar: 5µm the microtubule is elongating, while it is positive when the microtubule is shortening. Similarly, the binding of kinetochore can also unbind at a rate of k u,k , which is load-dependent [4,14,18] as Eq. 4. After unbinding, the microtubule become depolymerization state.

Molecular motors cross-link antiparallel microtubules
Some molecular motors, such as kinesin-5, Ncd and dynein, can serve as cross-linkers to link a pair of antiparallel microtubules, and generate interactional forces on the microtubules (Inset 5 in Figure S1) [17,18,25]. We consider both dynein and kinesin here, and they can generate pulling force and pushing force, respectively. The two crossing microtubules from different centrosomes can be bound by dynein or kinesin at the rates of k − b,r and k + b,r . The forces generated by motors are also velocity-dependent as Eqs. 3 and S1. Therefore, we need to get the velocity of the motor bound on the p th microtubule from i th centrosome and the q th microtubule from j th centrosome. Based on the theorem of velocity composition of a particle, the absolute velocity of the motor is Here, the left and right are the absolute velocity of the motor computed by p th and q th microtubule, respectively. The first two items represent the transport velocity of the motor; V i and V j are the velocity vectors of the centrosomes; φ p and φ q are the angular velocities of the microtubules; L p and L q are the lengths of the microtubules from the centrosome to the linked point; and r * p and r * q are the unit normal vectors of the microtubules (rotating 90 degrees counterclockwise from the microtubule growing direction). In the third item, v p and v q are the relative velocity between the motor and the microtubules; and r p and r q are the unit vectors of the microtubule growing directions. The force on the motor can be decomposed into the microtubule direction and its normal direction as Here, are the forces generated by the motor walking along the microtubules, where the plus sign represents kinesin and minus sign represents dynein. F p and F q are the forces in the normal direction, which can drive the rotary movement of the microtubules. Thus, they equal the drag force, F p = φ p ξl p L p and F q = φ q ξl q L q , where φ p and φ q are the angular velocities of the microtubules; ξ is the rotary resistance coefficient of the microtubule with unit length; l p and l q are the total lengths of the microtubules; L p or L q are the microtubule lengths between the force-application point and the centrosomes.
Therefore, we can solve the four equations (Eqs. S4 and S5 can be decomposed into x-and y-directions) to obtain four unknowns, φ p , φ q , v p and v q . The motor forces along the microtubules can be obtained based on the motor velocity v p and v q . The cross-linked microtubules can change their direction at the angular velocity φ p , φ q . It also should be noted that if the pushing force generated by kinesin is larger than the Euler buckling force, the microtubule is also buckled and the force equals the buckling force. The length used in the Euler buckling formula is the length of the microtubule from the centrosome to the linked point (L p or L q ). In addition, the cross-linker can also unbind from the microtubules at the load-dependent rates (Eqs. 4 and S2).

Anti-overlapping force
The centrosomes and chromosomes cannot overlap with each other, and also cannot penetrate the cell cortex. To avoid the overlap or penetration, we define the short-range repulsive force between any two objects (centrosomes, chromosomes or the cell cortex) as Here d is the least distance between the two objects, including the centrosomes, chromosomes and the cell cortex. Besides, since the microtubules are very thin, they can provide axial forces, but transverse forces are negligible. The number of microtubules is large, and thus the complexity of computation will be largely increased if the deformation of microtubules is considered. Therefore, we ignore the colliding between microtubules, and the elastic deformation of microtubules except the bulking. We only consider the state, direction and length of each microtubule, instead of the detailed shape.

Motions of centrosomes and chromosomes
Now we have various forces on the microtubules ( Figure S1). All of microtubules can transfer their axial forces to the centrosome. Therefore, we can give the force balance equation of one centrosome as Here, the first four items represent the forces generated by microtubules nucleated from this centrosome; the fifth item represents the anti-overlapping forces generated by all other centrosomes, all chromosomes and the cortex; the sixth item represents the viscous drag on the centrosome generated by the cytoplasm, where ξ p is the viscous drag coefficient and V is the velocity of the centrosome. It should be noted that we have neglected the inertial forces due to the low Reynolds number of this system. We assume a chromosome is a rigid rod, and its central region is the kinetochore ( Figure S1). Microtubules attaching the chromosomes can apply forces on the chromosomes. Similarly, the force balance equations of one chromosome are given as Here r b is the vector pointing from the chromosome center to the binding site; V c is the centroid velocity of the chromosome; and Θ is the direction angle of the chromosome. For simplicity, we define the circumcircle of chromosome as its boundary here to compute the minimal distance of the anti-overlapping force, thus the anti-overlapping forces have no torque applied on the chromosome.

The cell shape polarity and the tricellular junction polarity
The polarity direction and anisotropy of the cell shape and tricellular junctions are obtained based on the same method in Ref. [2]. Specifically, the inertia matrix of the cell shape is constructed as Here, the cell outline is discretized by pixels. n sh is the number of the pixels, and r i is the vector pointing from the barycentre of the cell to each pixel. The anisotropy of the cell shape can be defined as where λ 1 and λ 2 are the eigenvalues of S sh (0 < λ 1 < λ 2 ). The eigenvector corresponding to the eigenvalue λ 2 is the direction of the cell shape polarity, and then the direction angle θ shape can be obtained. Similarly, the direction and anisotropy of tricellular junctions can also be obtained by their inertia matrix Here, n tcj is the number of the tricellular junctions, and u j is the unit vectors pointing from the barycentre of the cell to each tricellular junction. The anisotropy of the tricellular junctions can also be defined by the eigenvalues of S tcj as where 0 < λ tcj 1 < λ tcj 2 . The polarity direction (θ T CJ ) of the tricellular junctions can also be obtained from the eigenvector corresponding to λ tcj 2 .

Simulation step and method
Monte Carlo simulation is performed in MATLAB to simulate the motion and formation of mitotic spindles ( Figure S3A). Initially, the positions of centrosomes and chromosomes are randomly given in the cell. The initial number of the microtubules on each centrosome is 50, and these microtubules are evenly distributed in all directions. During each step, a certain number of new microtubules are generated according to the nucleation rate in a step (k * 0 /∆t). If the nucleation rate in a step is not an integer, a random number in (0, 1) uniformly is generated, and compared with its decimal part. If the random number is less than the decimal part, the nucleation number equals the nucleation rate rounded up, otherwise, the nucleation number equals the nucleation rate rounded down. For each microtubule, we record its state (as shown in Fig S3b), length and direction. We change all microtubule states based on given method ( Figure S3C), subsequently change the lengths and directions of all microtubules, delete the completely shrinking microtubules, and compute the resultant force and instantaneous velocity of each centrosome and chromosome (Eqs. S7∼S9). Finally, we change the positions of all centrosomes and chromosomes according to their velocities by p t+∆t = p t + v∆t. The system is iteratively solved.
All microtubule states in the simulations are shown in Figure S3B, and their switches are shown in Figure S3C. A new microtubule is growing freely (1-1), its length increases at the growing speed v 1 , and its direction keeps unchanged. The freely growing microtubule (1-1) can reach the cortex (3-1), or switch to freely shrinking state (2-1) through catastrophe (k 2 ). The freely shrinking microtubule (2-1) decreases its length at the shrinking speed v 2 , and also keeps its direction. The freely shrinking microtubule (2-1) can also switch to freely growing state (1-1) through rescue (k 1 ), or completely depolymerizate (delete). The microtubule blocked by the cortex does not change its length and direction, and can switch to freely shrinking state (2-1) through catastrophe (k * 2 ). All free microtubules (1-1,2-1,3-1) can be bound by chromokinesin (k b,a ) or kinetochore (k b,k ) if they cross with one of chromosomes, or be bound by the cross-linker (k ± b,r ) if they cross with other microtubules nucleated from the other centrosome. Accordingly, when the microtubule is bound by the chromosome or crosslinker [(1,2,3)-(3,4,5)], it changes their length as growing or shrinking as usual, but adapts its direction to keep the binding site fixed. When the microtubule is bound by cortical dynein or kinetochore [4-(2,6)], it adapts its length and direction to keep the end fixed. Microtubules can recover free if the motor unbinds (k u,a , k u,k , k ± u,r , or k − u ) or the microtubule shrinks its length to leave the crossing site. It should be noted that we assume each microtubule can only be bound by one molecular motor for simplicity, because the force will be very complex if the microtubule is simultaneously bound by more than one of chromokinesins, kinetochores, cross-linkers and cortical dyneins. Four switches, including catastrophe, rescue, binding and unbinding, are random events (colorful arrows in Figure S3C), while the reaching is a deterministic event (black arrows in Figure S3C). Each random event has its rate (collectively denoted as k for the moment). To determine if the random event occurs in the time step, a random δ in (0, 1) uniformly is generated for each event. If δ ≤ 1 − e −k∆t , the event occurs in the step, otherwise, it does not occur [18]. The parameters used in the simulations are summarized in Table S1 unless otherwise indicated. Table S1: Parameters used in the simulations.

Parameter Description
Geometric parameters R cell = 20µm Diameter of the cell. λ = 1 ∼ 2 The aspect ratio of the cell. It is defined as the ratio of the long axis to the short axis. L ch = 2.5µm Length of the chromosome. L kine = 0.5µm Size of the kinetochore.
Characteristic force represents the sensitivity of the unbinding rate of kinesins to the load [4,19].
Characteristic force represents the sensitivity of the unbinding rate of dyneins or kinetochores to the load [16,19].

Supplemental Results and Discussion
In the main text, we investigated the competition between cell shape and intercellular adhesion (bilateral and unilateral) determines spindle orientation in the side view. We also mentioned another cell shape, ellipse, observed when the tissue was stretched [30]. Although the elliptical shape seems impossible to appear in the two cases above, we also perform simulations in the elliptical cells as a discussion on the influence of cell shape. Firstly, we assume the elliptical cell adheres to neighbouring cells in the tissue as Figure 3 (i.e., bilateral adhesion), and find the results are totally similar to the stadium shape ( Figure  S10). However, under the same condition, the critical aspect ratio in elliptical cells is larger ( Figure S13), which indicates the ellipse shape can ensure more planar division than the stadium shape. This is probably because the two poles of the elliptical short axis are curving outward, and thus have a smaller repulsion on the spindle than the straight lines of the stadium shape. Secondly, we also assume the elliptical cell has the unilateral adhesion, i.e., adheres to the E-cadherin-coated matrix as Figures 4 and 5. The results are also totally similar to the stadium shape, but the phenomenon of double platforms cannot be observed apparently (Figures S11 and S12). The reason may be that the larger curvature at two poles of the elliptical long axis can limit the spindle orientation more strongly. Therefore, the spindle is little slanted in the same cases. Taken together, the elliptical shape has a greater influence on the spindle orientation than the stadium shape, and the influence of unilateral adhesion is total weaker than the bilateral adhesion in all cases ( Figure S13).
In addition, in the case of unilateral adhesion, the adhesive length was considered increasing with the compression. We also consider this in the case of bilateral adhesion, and find the results are similar to the fixed adhesive length in both cell shapes ( Figure S11). Compared to the cases of fixed adhesive length of 12µm, the critical aspect ratio is larger in the stadium cell with varying adhesive length, but is smaller in the elliptical cell with varying adhesive length ( Figure S13). This is obviously because the stadium cell has a larger contact length with the substrate than the elliptical cell under the current assumption of the adhesion condition.      The spindle orientation is plotted as the function of the aspect ratio of the cell shape with different adhesive lengths. (E) The curves of spindle orientation versus cell aspect ratio can also be fitted by using Eq. 1, and then the critical aspect ratios λ T can be obtained, and are the functions of adhesive length and strength.

Supporting References
When the adhesion length is large, the spindle always parallels to the substrate independent of the cell shape (indicated by white).
SBF SBV SUF SUV EBF EBV EUF EUV Figure S13: Regulating the spindle orientation via various cell shapes and adhesion geometries. The cells of stadium shape (S) or elliptical shape (E) adhere to neighbouring cells in the tissue (bilateral adhesion, B) or to E-cadherin-coated matrix under compression (unilateral adhesion, U). The adhesive length is fixed (F), or varying with the elongation of the cell (V). In each case, the spindle orientation is plotted against the cell aspect ratio. To compare them, we give the adhesive strength in all cases as k=11 and the adhesive length in the fixed cases as L=12µm.