Modeling of Noisy Spindle Dynamics Reveals Separable Contributions to Achieving Correct Orientation

Abstract

The mechanisms by which the mammalian mitotic spindle is guided to a predefined orientation through microtubule-cortex interactions have recently received considerable interest, but there has been no dynamic model that describes spindle movements toward the preferred axis in human cells. Here, we develop a dynamic model based on stochastic activity of cues anisotropically positioned around the cortex of the mitotic cell and we show that the mitotic spindle does not reach equilibrium before chromosome segregation. Our model successfully captures the characteristic experimental behavior of noisy spindle rotation dynamics in human epithelial cells, including a weak underlying bias in the direction of rotation, suppression of motion close to the alignment axis, and the effect of the aspect ratio of the interphase cell shape in defining the final alignment axis. We predict that the force exerted per cue has a value that minimizes the deviation of the spindle from the predefined axis. The model has allowed us to systematically explore the parameter space around experimentally relevant configurations, and predict the mechanistic function of a number of established regulators of spindle orientation, highlighting how physical modeling of a noisy system can lead to functional biological understanding. We provide key insights into measurable parameters in live cells that can help distinguish between mechanisms of microtubule and cortical-cue interactions that jointly control the final orientation of the spindle.

Introduction

During tissue development and homeostasis, geometry and position of the constituent cells is important for tissue function (1–3). Cell positioning defects can arise from errors in the process of cell division as the position of the mitotic spindle determines the position of the cleavage plane, and thus determines the size, content, and neighborhood of the two daughter cells (reviewed in Chin et al. (4)). Thus, in growing or repairing polarized tissue, the orientation of the spindle must be tightly controlled to ensure that each daughter cell is positioned correctly and receives the correct complement of polarity proteins to maintain the function and morphology of the tissue (5,6). Therefore, knowledge of the mechanisms that orient the spindle is vital to understanding the causes of many developmental disorders, cancers, and age-related tissue pathologies.

While the majority of microtubules that emanate from each spindle pole overlap to form the basic structure of the spindle, astral microtubules grow from each pole in all directions (reviewed in Tamura and Draviam (7)). Astral microtubules have long been implicated in spindle centering (8) through forces generated by cytoplasmic motors (9) and compressive forces transmitted to the poles (3,10–13). The dynamics of spindle positioning in the single-cell stage of the Caenorhabditis elegans embryo have been modeled, with temporal regulation of cortical pulling forces proposed to explain the centering and subsequent displacement of the spindle, giving rise to an asymmetric first division (14).

In addition to spindle position, correct spindle orientation is critical to ensure robust development. In cells in culture that have no predefined polarity, it has been proposed that the final spindle orientation is determined probabilistically by the distribution of cell-cortex bound structures, generally referred to as cortical cues (15), which form at the site of focal adhesions upon entry into mitosis. In the first model of spindle orientation in human cells, it is suggested that these cues interact with astral microtubules by exerting a pulling force, resulting in a torque that rotates the spindle. Growing cells on geometrically defined micropatterns of the extracellular matrix protein fibronectin controls the localization of focal adhesions to such an extent that the probability distribution of the final spindle orientation can be predicted based on a quantitative model (16). In this model, the angular distribution of cortical cues defines a potential energy landscape, which determines the probability distribution of the spindle angle at equilibrium in the presence of a specified level of Gaussian white noise. The distribution calculated is in dynamic equilibrium in the sense that the overall distribution should not change over time, although spindles are not fixed at the minimum in the potential.

Epithelial cells in culture tend to align their division plane with the short axis of the cell shape during interphase (17). Thus one can equate the preferred spindle axis, corresponding to the minimum of the potential landscape, with the long axis of the cell. In a recent experimental work in human epithelial cells (18), expressing a tubulin marker, the dynamics of spindle rotation were followed and the rotation was found to be noisy, with a small bias in the direction of movement being sufficient to achieve satisfactory final alignment. The dynamic measurements have highlighted additional complexities in spindle movements (18), requiring the model of cortical-cue-driven orientation to be reformulated in terms of the separable contributions to spindle behavior.

The potential landscape model (16) predicts the equilibrium orientation distribution but not the rotation dynamics, and is in essence a static model. It is important to emphasize that, while implicit in this static model is angular diffusion on a potential gradient, there is only one free parameter—an effective Boltzmann parameter—which is fitted to the final angular distribution of spindles. This single parameter is an aggregate of every process by which spindle rotation is affected and cannot distinguish a defect in final spindle orientation caused by a perturbation to microtubule dynamics, cortical cue positioning, force exertion, or duration of mitosis. Therefore, it is highly challenging to unambiguously infer the underlying mechanistic cause for a failure in spindle orientation exclusively on the basis of the final distribution of spindle orientations at the point of cell division.

To understand the time-dependent spindle orientation behavior, we develop a dynamic model of spindle rotation from nuclear envelope breakdown (NEBD) to anaphase onset. We identify the region in parameter space that recapitulates the behavior of unperturbed cells in culture and find that the noisy behavior and two regimes of spindle rotation arise from a weak, orientation-dependent bias in the direction of rotation. By exploring the sensitivity of the observations to parameter perturbations, the dynamic model is able to predict the mechanistic effect of a protein knockdown or drug treatment condition, and thus separate the potential contribution of factors that govern the final orientation of the mitotic spindle. We discuss the implications of our results in the context of recent experimental studies of spindle orientation lesions.

Materials And Methods

See the Supporting Material for Materials and Methods.

Results

Time-lapse imaging illustrates the strengths of a dynamic model in explaining spindle movements

Our starting point for the dynamic model is the angular distribution of cortical cues on the cortex of the rounded mitotic cell, which, as in the static model, interact in a first-order manner with astral microtubules (see Fig. S1 in the Supporting Material). In effect, the memory of the interphase cell shape and the long axis must be encoded in this angular distribution. To create a model where dynamics are integral to the behavior, we assume that each cortical cue has a binary activity profile; inactive cues switch on with a rate proportional to the angular density of microtubules at the cue position, $k_{\text{on}}=\kappa \left(\partial \beta /\partial \theta \right)$, while active cues switch off with a constant rate k_off. While active, cues exert a constant force on the spindle pole that was nearest at the time of activation. The magnitude and duration of the force is condensed into a single parameter, the cue impulse, f_cue. Full details of symbols, the dynamic model, and simulations are provided in Appendix A.

Stochastic cue activity provides two sources of noise in the number of active cues per frame and the total impulse exerted by a cue before it is deactivated. Additionally, random Brownian noise affects spindle rotation, which we model as Gaussian white noise of variance, σ²_noise. We considered the number of times a cue becomes active during a window of time and the impulse exerted by the active cue, as random variables. Thus for a given spindle orientation, the angular displacement is modeled as a random distribution (see Appendix B) characterized by the mean and standard deviation. The spindle displacement is expressed probabilistically as a stochastic matrix, allowing the evolution of the spindle orientation distribution to be calculated over time (Fig. S2). This approach allows the dynamics to be calculated at very short times where the magnitude of spindle rotation is small and then extrapolated to experimentally observable timescales where the spindle angle can change continuously throughout the frame interval. Predictions for the rotation bias and speed for a given frame interval were calculated from the stochastic matrix, and were found to be in good agreement with simulated spindles. In the following sections, spindle properties are calculated from the stochastic matrix theory (Appendix B); where individual spindle traces are used, they have been generated from the simulations outlined in Appendix A.

For there to be a defined preferential orientation toward which spindles preferentially rotate, the angular distribution of cortical cues must have some anisotropy. We quantify the anisotropy using a two-dimensional order parameter that varies from 0 for purely isotropic cues to 1 for cues positioned on the long axis (19):

$$\rho =2\langle {\text{cos}}^2{\theta }_{\text{cue}}\rangle -1.$$ (1)

We create a standard anisotropic distribution of cortical cues by uniformly positioning a number of cues around the cortex and adding additional polar cues within 10° of a predetermined preferred axis, defined as 0°. In the following analysis, the terms “long axis” and “preferred orientation” are used interchangeably, and all spindle orientations are measured relative to this axis. The degree of anisotropy can be increased by reducing the number of isotropic cues and increasing the number of polar cues, such that the total number remains constant (see Fig. S3). The initial cue distribution we use has an order parameter of 0.08.

Parameter space exploration

We first wish to find biologically relevant parameter configurations, which recapitulate the dynamic behavior observed experimentally. To constrain the parameters of the dynamic model, we obtained a large experimental data set (n_total = 358, n_{aspr > 1.4} = 139) of unperturbed cells from which to extract the characteristic behaviors we aim to understand. We imaged fields of cells at 3 min intervals and analyzed the time-lapse movies using our previously developed SPINDLE3D software (see Materials and Methods in the Supporting Material), to identify the characteristic experimental behavior that simulations must capture. All spindle angles are measured relative to the measured interphase long axis of the cell, on the interval [−90,90]°, with zero corresponding to alignment with the long axis. We also refer to the spindle angle as the deviation from the long axis. The long axis is defined by fitting an ellipse to the boundary of the interphase cell shape 15 min before NEBD. The initial spindle angle is measured in the first frame after NEBD when a bipolar spindle is first detected. The final spindle angle, also referred to as the division angle, is the angle of the spindle relative to the long axis in the first frame of anaphase, determined from separation of sister chromatids at the spindle equator. From this experimental dataset, we chose measurable properties that characterize the spindle behavior—the average division angle, the absolute frame-to-frame rotation speed 45° from the long axis, and the bias in the direction of rotation 45° from the long axis—with which to constrain the parameter space of the dynamic model.

We formed a grid of points covering two orders of magnitude of values of the cue activation rate, the impulse exerted per active cue, and the random Brownian noise. The duration of mitosis was approximated by a Gaussian distribution N(μ = 30 min, σ = 2.5 min) and a frame interval of 3 min was used, matching experimental conditions. The initial spindle orientation was sampled from a uniform distribution, U(0°,90°), consistent with experimental observation (Fig. S4 A), which essentially means that the spindle forms at a random orientation. Fig. 1 A shows the points that produce output values close to those observed experimentally, which exist in a narrow corridor ranging from low Brownian noise, where a small number of cortical cues exert a large force (left side of Fig. 1 A), to a regime where a large number of cues exert weak forces and stochasticity is provided by the Brownian noise term (right side of Fig. 1 A). We choose a point in the middle between these two limiting cases as a standard configuration (κ₀, f₀, σ₀) to represent the unperturbed system for further investigation (black square in Fig. 1 A).

Figure 1.

The dynamic model recapitulates characteristic spindle rotation in cells. (A) Parameter space exploration showing (on-rate, cue force, random noise) triplets, which give behavior quantitatively similar to experiments (circles). (Black square) Parameter set chosen as standard for further study. (B) Traces generated using the parameters (lower), compared with experimental data (upper). (C–G) Comparison of spindle dynamics between experiment and simulation: final spindle orientation at anaphase onset (C); orientation dependence of directional bias (D) and frame-to-frame speed (E); final orientation as a function of initial orientation (F); and time until first long axis alignment (within 15°) as a function of initial orientation (G). Error bars show SEM, and the shaded area in (G) shows standard deviation. (H) Probability of spindle being within 30° of long axis alignment over time, depending on the initial orientation. (Vertical dashed line) Average time to anaphase onset. To see this figure in color, go online.

Simulated individual spindle traces bear a strong resemblance to experimental traces, with noisy rotation toward the long axis (Fig. 1 B). The dynamic behavior is compared between simulation and in vivo measurement in Fig. 1, C–G, showing close agreement. The final angular deviation of the spindle from the long axis has a probability distribution peaked around zero (Fig. 1 C), although a significant fraction of spindles do not reach alignment with the long axis, in vivo and in simulations.

Alignment of the spindle with the long axis is achieved by a relatively small bias in the direction of rotation, with spindles slightly more likely to rotate toward the long axis than away (Fig. 1 D). We quantify this directional bias with a value between −1 (all spindles rotating clockwise, or increasing phi) and +1 (all spindles rotating anti-clockwise, or decreasing phi), with a value of zero corresponding to spindles equally likely to rotate in either direction, that is, no bias in direction. The bias has a roughly sinusoidal shape and reaches zero when the spindle is aligned with the long axis, or is approximately perpendicular. The speed of spindle rotation (calculated through the frame-to-frame angular displacement) was found to decrease when the spindle was close to the long axis (p = 8 × 10⁻⁷) (Fig. 1 E), which suggests there are two regimes of spindle rotation (18), reducing the likelihood of spindles rotating out of alignment. Orientation-dependent bias in rotation direction and rotation speed suppression both arise naturally in the parameter configurations identified in Fig. 1 A; the suppression of rotation at the long axis is a direct consequence of the variation of bias with orientation (see the Supporting Material). The sinusoidal variation in bias is reproduced in silico (Fig. 1 D); the magnitude of the reduction in speed is determined primarily by the anisotropy in the angular distribution of cortical cues, without the need to invoke additional tethering or anchoring once the spindle reaches the long axis.

The initial orientation strongly affects the probability of successful final alignment, with spindles that begin >60° from alignment being less likely to end within 20° of the long axis (Fig. 1 F). This failure to align arises because distant spindles require a longer period of time to reach the long axis (Fig. 1 G) and so are less likely to do so successfully before anaphase onset. This implies that the model of dynamic equilibrium in a potential energy landscape does not fully describe spindle dynamics, because the probability of dividing at a certain orientation is not only a function of cortical cue locations but also of initial orientation and time of anaphase onset. This behavior is captured by the dynamic model, which has important implications for investigating the functional roles of proteins involved in spindle orientation. A defect may arise either from an inability to sense the preferred orientation correctly, or from a problem in rotating the spindle successfully within the time constraint of the duration of mitosis.

The complexity of spindle dynamics means that a perturbation to the system, such as the knocking down of a protein important for spindle movement, may not result straightforwardly in a change in the final division angle. For example, in previous work the depletion of MCAK, a kinesin required for microtubule shrinkage, resulted in reduced rotation speeds but did not adversely affect the final spindle alignment and division angle (18). The good quantitative agreement between in vivo measurement and theory means that the dynamic model is a useful tool to understand spindle behavior and investigate the mechanisms that are perturbed in the disease state. Therefore, we next use our dynamic physical model to understand the relative importance of system parameters.

Importance of initial orientation and duration of mitosis

In light of the observation that spindles originating further from alignment take longer to reach the long axis, and are generally less successful in doing so, we next asked how the duration of mitosis influences the likelihood of alignment. It is interesting to note that the bias is very small when the spindle is close to orthogonal to the preferred axis, that is, as far as possible from alignment. Therefore, the typical alignment time depends strongly on the initial angular deviation. An important point is that while the spindle assembly checkpoint delays anaphase until all chromosomes are correctly attached to the spindle, there has been no report of a corresponding orientation checkpoint in mammalian cells, and it is therefore important that the typical alignment time is shorter than time until anaphase onset in order that the majority of spindles reach the correct orientation.

Fig. 1 H shows the probability that a cell has reached alignment as a function of time after NEBD (x axis), for different starting orientations, using the parameter set identified in Fig. 1 A. For nonaligned starting orientations, the probability of having reached alignment increases over time. However, spindles that start close to the preferred axis align faster than those that start further away. While this is a fairly intuitive result, it highlights the subtle effect of changing the duration of mitosis. Experimentally, HeLa cells have a duration of mitosis of ∼30 min with a skew-normal distribution (20), which we approximate with a Gaussian distribution. For spindles that start close to the long axis (30–40°), it is very likely that the spindle will reach alignment before anaphase. In contrast, for starting deviations of 70° or greater, it is much more likely that a cell will divide before the spindle has had sufficient time to reach alignment. Moreover, a weak perturbation will affect spindles with large initial deviations more strongly than those beginning close to the long axis.

Sensitivity analysis to compare the significance of parameters controlling the final orientation of the spindle

As described above, protein depletion or drug treatment conditions that impair the function of a particular molecular pathway will not always have a direct and unambiguously interpretable effect on the final division angle. To determine the extent to which various input parameters affected the dynamic movements of the spindle, we studied the effect of changing each input parameter in turn by a factor of 2 relative to the unperturbed control configuration. Fig. 2 A represents the parameter sensitivity as a heatmap, where the size and color of each circle denotes the relative change in the output property. The properties shown from left to right are the average final deviation from the long axis, the directional bias, the absolute frame-to-frame rotation speed, and the magnitude-of-rotation suppression at the long axis compared to spindles 45° from alignment. The final two columns show the final deviation for spindles that begin within 30° of the long axis (representing retention at the long axis) and outside 60° (representing ability to rotate toward the long axis), respectively. Differences in these two values demonstrate the importance of taking into account the initial spindle orientation. Together, the sensitivity measurements provide a fingerprint of the phenotypic effect of a perturbation, which can be compared with experimental protein depletion outcome in cells. It is important to note that the average final deviation increases for all the perturbations shown in Fig. 2 A, meaning the division angle distribution alone is unable to distinguish the affected parameter.

Figure 2.

Sensitivity analysis of system parameters. (A) Phenotypic effect of perturbing the input parameters. Each row represents perturbing a parameter by a factor of 2; each column shows the change in an experimentally measurable output relative to the unperturbed control point. (Left to right) Outputs are average division angle relative to the long axis, direction bias, rotation speed 45° from alignment, magnitude of rotation suppression at the long axis, average division angle for spindles that begin within 30° of long axis, and average division angle for spindles which begin >60° away from the long axis. The perturbations considered are decreased cue activation, decreased force per cue, increased noise, decreased force and noise, decreased polarization of angular cue distribution, and decreased mitosis duration. (B) Increasing the cue distribution order parameter as a proxy for increasing aspect ratio. (Upper panel) Increasing cue order gives an increase in the rotation speed of unaligned spindles (circles), and a decrease in speed once aligned (squares). (Central panel) The final deviation from the long axis decreases as the cue order increases. (Lower panel) Directional bias at 45° increases with cue order. (C) A sweep of the force exerted per cue, one order of magnitude above and below the estimated control value, reveals a finite window of force for optimal long-axis alignment. To see this figure in color, go online.

Fig. 2 B shows the effect of changing the degree of polarization in the angular cue distribution, characterized by the order parameter. The rotation speed (upper panel) for spindles far from alignment (>30°, blue circles) increases and the speed close to the long axis (<30°, black squares) shows a small decrease. Overall alignment with the long axis is improved for highly polarized distributions (central panel, Fig. 2 B), both by increasing the speed of rotation when far from the long axis and by reducing the rate of rotation once alignment has been achieved. A likely candidate for an experimental change in the cue order parameter is the aspect ratio of the interphase cell shape, which is studied below.

Force exerted by cortical cues determines alignment success

The sensitivity analysis revealed that the average force exerted by each cue does not have a straightforward effect on the probability of alignment. To investigate this further, we varied the force exerted per cue over two orders of magnitude, holding the other parameters, including the random noise, constant (Fig. 2 C). Interestingly, as the force is increased above the standard value, f₀, alignment success decreases as the effect of a single cue activation is sufficient to move the spindle away from the long axis (bottom panel). This gives rise to an optimal cue force at which spindle deviation from the long axis at the point of division is minimal. The magnitude of the cue force has two opposing effects: it must be large enough to overcome the noise in the system and rotate the spindle to the long axis, but if it is too large then fluctuations in the stochastic cue activity prevent the spindle remaining aligned. Thus, there exists a window of cue-force magnitude values where the spindle rotation speed is fast enough to reach alignment before anaphase, but the force is small enough that the system is not dominated by the stochasticity of the cue activity.

Importance of direction bias and rotation speed in distinguishing physical mechanisms that control final spindle alignment

Finally, we take a panel of experimental perturbations that are known to have an effect on the final orientation of the spindle and examine their dynamic phenotypes. The perturbations studied are RNAi-mediated depletion of LGN; treatment with a microtubule pausing agent, 2ME2 (18); comparing low-aspect ratio cells with high; and synthetically halving the duration of mitosis. The consequence of LGN depletion and 2ME2 treatment in disrupting the spindle alignment with the long axis of the cell has been presented previously (18); here we provide, to our knowledge, a novel interpretation in terms of the dynamic model. Fig. 3 A shows the dynamic fingerprint for RNA_i-mediated depletion of LGN, and for treatment with the drug 2ME2 compared with the respective control-treated cells. Also shown in Fig. 3 A is data for low-aspect ratio control cells (<1.3) compared with more elongated cells, and an artificial shortening of the mitosis duration, constructed by considering only the first half of the experimental spindle angle data. The phenotypic maps can be compared with the sensitivity maps of Fig. 2 A to predict the parameter(s) that are perturbed by the treatment. Comparing low-aspect ratio cells with high suggests that the interphase cell shape is linked to the polarization in the angular distribution of cue locations, as may be expected. Intuitively, shortening the mitosis duration does not affect the speed of rotation or bias, but adversely affects the alignment success of spindles originating >60° from the long axis. This also suggests that the system parameters do not change significantly throughout mitosis. LGN depletion and 2ME2 treatment have similar phenotypes, and are considered in more detail in Fig. 3 B. As with the sensitivity analysis (Fig. 2 A), the change in final spindle deviation is similar for all perturbations considered, and therefore measuring only the distribution of final spindle orientation yields very little insight into the physical mechanisms being disrupted.

Figure 3.

Interpreting the consequence of molecular defects. (A) Heatmap showing the effect of dynamic measurements from a range of experimental treatments. From top to bottom, the experimental perturbations are: LGN protein depletion, 2ME2 drug treatment, low aspect ratio unperturbed control cells (Aspr), and artificially halved mitosis duration (see text). (B) Decreasing cue force (Δ_force) and decreasing cue activation (Δ_on-rate) have similar phenotypic fingerprints (Fig. 2A). Simultaneously considering the changes in bias (y axis) and speed (x axis) can help to distinguish these perturbations relative to the estimated control point (solid triangle). Decreasing the force (open circles) is predicted to affect the speed to a greater extent than the bias—the LGN depletion phenotype (solid circle) is best explained by this perturbation. In contrast, treatment with 2ME2 (solid square) decreases the directional bias more strongly than the speed, consistent with a decrease in the cue activation rate (open squares). To see this figure in color, go online.

As described above, it is not trivial to unambiguously infer the parameter that has been affected for LGN-depleted or 2ME2-treated cells because the fingerprints for cortical-cue force and activation rate are similar. By considering the effect on the bias and speed together, we can predict the source of the perturbation. As shown in Fig. 3 B, reducing the force exerted per cue affects the rotation speed to a greater degree than the directional bias; whereas reducing the cue activation rate affects the bias more strongly than the speed. Analyzed in this way, we predict that LGN depletion (Fig. 3 B, red circle) results in a reduction in the force exerted by the cortical cues, while 2ME2 treatment affects the on-rate. Consistent with this interpretation, LGN depletion does not affect the final deviation of spindles starting close to alignment (near dev, Fig. 3 A), in agreement with the force perturbation in Fig. 2 A. Thus, the dynamic model allows us to distinguish between mechanisms that jointly govern the movements of the spindle toward its predefined final alignment axis.

Discussion

In this article, we have constructed a dynamic model of mitotic spindle rotation, considering the physical interaction between the astral microtubules of the spindle and stochastically active cortical cues to understand spindle behavior in a time-dependent manner. The model captures the characteristic dynamic behavior of the spindle—weak bias in the direction of rotation, strong dependence on initial orientation—and can be directly compared with experimental data from time-lapse movies of human epithelial cells to reveal the mechanistic function of the molecular players that control spindle orientation.

We have simulated the behavior for the case where each cortical cue exerts a fixed force and the preferred axis is defined by the anisotropy in the angular distribution of the cues. Equivalently, there may instead be anisotropy in the cue activity, either through the force exerted or the activation rate. It has been reported that cell stretching can change the preferred axis during mitosis (21), suggesting that a modulation of activity by retraction fiber tension can override the previous axis of anisotropy. A microtubule length dependence could be added to the activation rate, through the astral microtubule density (see Eq. A2 in Appendix A), or exerted force terms, by assuming an exponential distribution of lengths. This might be expected to reduce the speed of rotation once the spindle is aligned, because distant cortical cues will contribute less to spindle rotation. The potential for cortical cues to modulate microtubule length is discussed further below.

Our model highlights the importance of the initial spindle orientation in determining the likelihood of successful alignment. We showed experimentally how the final deviation depends strongly on the initial angle (Fig. 1 F), in agreement with in silico predictions. Therefore, in cell types where correct orientation is critically important, centrosome positioning may be employed before nuclear envelope breakdown to preorient the spindle and ensure that the orientation process is fast enough (3).

In our model, we consider only cortical forces acting on astral microtubules to generate a systematic torque on the spindle. Additionally, cytoplasmic motors may pull on the spindle with a force that is a function of the microtubule length (3,12). In this way, an asymmetry in astral microtubule length would generate rotation. However, in the conditions we have considered in human cells, where the spindle is in the center of a rounded cell, cytoplasmic forces will be absorbed into the random noise term.

We found that there is an optimal value of the cue force to obtain good alignment with the long axis, above which stochasticity in cue activity causes the spindle to be rotated away from the long axis, a feature that is not predicted by the static model. Experimental work has previously suggested that force-generating dynein must be recruited to the cortex at the appropriate level to result in successful orientation (22). The experimental rotation speed is consistent with a cue force close to this optimal value. This is an illustration of the complex phenotypic output of a perturbation; increases in activity and not just protein knockdowns can have deleterious effects. By performing a corresponding sweep in the value of the Brownian random noise, we found that reducing the Brownian noise increased the alignment time and typical deviation for spindles, which started far from alignment (Fig. S5 B). Potentially, some noise is beneficial in moving spindles away from perpendicular orientations where the net cue torque is small, sacrificing perfect alignment with the preferred axis in favor of achieving reasonable alignment for the majority of cells within the tight time constraint before anaphase onset. In support of this hypothesis, such sacrifice in perfect alignment with the preferred axis is in fact observed in vivo tissues where the spindle is oriented within 40° of the predefined axis (23,24).

By comparing the shift in dynamic properties relative to unperturbed cells, we are able to predict how the system parameters are changed when an experimental perturbation is applied. This dynamic fingerprint can distinguish among changes to the activation rate, the force exerted per cue, the Brownian noise, and the angular distribution of cortical cues—something that the static final spindle angle distribution cannot. We inferred that depletion of LGN caused the cue force to be reduced, consistent with a role for LGN in recruiting dynein (a minus-end directed force-generating motor protein), to the cortex to interact with astral microtubules (18,25–27). Treatment with 2ME2 led to a reduction in the cue activation rate, as may be expected from a drug that suppresses microtubule dynamics (28). The ability to predict the mechanistic effect of a genetic knockdown is a significant strength of the dynamic model.

For cells in culture, the precise distribution of cortical cues is not known and is instead approximated as a polarized nonuniform distribution characterized by a two-dimensional order parameter. In this respect, measuring the spindle dynamics of cells cultured on adhesive micropatterns may allow more reproducible shapes and cue distributions to be studied. As described above, a proportion of the Brownian random noise arises from astral microtubules, which are not interacting with the cortical cues. These microtubules are thought to exert a compressional pushing force when they are sufficiently long to reach the cortex, and have been implicated in maintaining the position of the spindle at the center of the cell (10). Mathematically there is no distinction between a pulling force and the absence of a pushing force.

Without a precise biochemical mechanism for the MT-cue interaction, it is interesting to speculate that a cue may operate by stalling or depolymerizing microtubules, thereby removing the pushing force at specific nonuniform locations on the cortex. The cue activity lifetime would then be defined by the time for the depolymerizing microtubule to be rescued and reach the cortex again, which might be significantly greater than the lifetime expected for a direct molecular interaction. Also, a single cue would be able to have a simultaneous effect on many microtubules, increasing the potential total force per cue beyond the pN range typical of MT-mediated forces. An alternative mechanism by which cortical cues could modulate the astral microtubule pushing force is by influencing the local membrane rigidity (29,30). In this scenario, an active cue reduces the membrane stiffness such that an elongating astral microtubule deforms the cortex to a greater degree and results in a reduced compressive force being transmitted to the spindle pole. Bundling of microtubules may also be a mechanism by which the lifetime or force produced by a cortical interaction can be increased beyond those of a single, dynamically unstable microtubule.

Regardless of the precise mechanism of action of cortical cues, the processes of spindle orientation and spindle location or centering are inescapably linked, and future work to combine the two phenomena in a single model will shed further light on the mechanisms at work. We have presented a dynamic framework for studying the interplay of factors that determine the degree of alignment of the mitotic spindle to a cell-polarity-defined preferred axis. Our approach of quantitative live-cell imaging combined with theory and simulation will find use in investigating the mechanics of spindle orientation and the protein regulators involved in cells grown in culture, as well as cells grown on adhesive micropatterns and within the context of tissues and metastatic cells.

Author Contributions

A.M.C., V.M.D., and A.M.D. designed research and wrote the article; A.M.C. developed theory and simulations; and A.M.C. and R.S. performed experimental work.

Editor: Douglas Robinson.

Appendix A: Detailed Description of Dynamic Model

The dynamic model makes the basic assumption that spindle orientation is controlled, at least in part, by cortical cues that are fixed in number and position over time. This proposal for how the cues influence the spindle is through an interaction with the astral microtubules, which are connected to one spindle pole or the other. More generally, we assume that the cues exert a pulling force on the nearest spindle pole, and thereby generate a torque on the spindle, without specifying the precise mechanism of this action.

Typically we represent stochastic cue deactivation with an exponentially distributed lifetime τ_cue = (k_off)⁻¹. An active cue exerts a fixed pulling force on the spindle pole that was nearest at the time of activation along the line between the cue and the pole, while an inactive cue does not influence the spindle. The pole being influenced by the cue is chosen as the nearest to the cue at the time of activation; the same pole is influenced until the cue deactivates, even if the other pole becomes closer during this time. Cue switching and rotation of the spindle is calculated over a very small time increment, typically 0.2 s. The precise value of the time increment chosen does not influence the results observed as long as the increment is much shorter than any characteristic timescale in the system.

The angular impulse acting on the spindle in each increment is calculated by summing the force vectors of active cues combined with a random noise torque, and each cue has a probability of switching per time increment. In the low Reynolds number, viscous regime within the cell and the angular displacement per time step is proportional to the calculated torque, and so we define the force exerted per cue and noise directly in terms of the resultant angular velocity of the spindle, incorporating the rotational friction coefficient of the spindle into the force and noise terms. Therefore it is important to note that if the rotational drag coefficient were to change, through either the shape of the spindle or the viscosity of the cytoplasm, the cue force and noise terms would need scaling accordingly. Also, while the magnitude of the force exerted by a cue is constant, the value of the torque will depend on this position of the spindle as well as the configuration of active cues.

When the cell rounds for mitosis, we make the assumption that the angular position of each cortical cue from the center of the cell does not change (see Fig. S2). The angles of the cortical cues are shown in Fig. S5, with an axis of anisotropy that defines the preferred spindle axis.

A cue is activated by interacting with a microtubule, therefore we assume that the rate of an off-cue switching on is proportional to the angular density of astral microtubules,

$$k_{\text{on}}={\kappa }_{\text{on}}{\frac{d\beta }{d\theta }|}_{{\theta }_i},$$ (A1)

where, from geometric considerations,

$${\frac{d\beta }{d\theta }|}_{{\theta }_i}=\frac{1-a\text{cos}\left({\theta }_i-\phi \right)}{1-2a\text{cos}\left({\theta }_i-\phi \right)+a^2}.$$ (A2)

The partial differential captures the change in angle from the spindle pole due to a small movement around the cortex, and hence reflects the density of microtubules experienced locally at cortex angle ϑ_i. Equivalently, locations on the cortex that are further from the spindle pole have a lower density of microtubules. The angles β, φ, and ϑ are defined as shown in Fig. S2 B. The effective on-rate varies from being maximal where the microtubules are densest close to a spindle pole, to a minimum when they are perpendicular to the spindle angle. Where the value of an on-rate is referred to explicitly, for example when using the analytical approximation described below, we refer to the prefactor κ. We assume that the rate of switching off is constant per time step, which means that the lifetime of an active cue has an exponential probability distribution with a mean duration τ_cue = 1/k_off.

For our simulations, we define a small and fixed time step, t_inc, during which the probability of transitions and the movement of the spindle is evaluated. The time step typically chosen is 0.2 s, orders of magnitude smaller than our typical imaging time step for experimental comparison. The method of small frame increments was chosen above a purely Monte Carlo approach because the on-rates and thus the probabilities of cue transitions are functions of this spindle orientation, which varies continuously.

While a cue is active, it exerts a fixed force along the line between the cue and one spindle pole. The pole being influenced by the cue is chosen as the nearest to the cue at the time of activation; the same pole is influenced until the cue deactivates, even if the other pole becomes closer during this time. Thus while the magnitude of the force exerted by a cue is constant, the value of the torque will depend on this position of the spindle as well as the configuration of active cues. The total vector force on each pole is summed and the torque calculated using the vector product with the axis of the spindle. Gaussian white noise is added to the torque before it is used to update the spindle position.

In updating the spindle orientation for the next time point, we assume that the angular velocity is constant throughout the short time step. At each time point, the process is repeated: within a time step, there is a probability that each active cue will be deactivated. Correspondingly, each inactive cue has a probability of switching on that is also a function of the angular density of microtubules (see Fig. S1).

Appendix B: Representing Spindle Movement as a Gaussian Random Variable

In this Appendix, we calculate the mean and variance of the random variable that describes the frame-to-frame spindle rotation. Because the angular inertia of the spindle is an unknown quantity, we make the assumption that rotation rates are in the viscous regime and express the impulse exerted per cue directly in terms of the resulting angular displacement divided by the spindle half-length, a. It is therefore important to note the perturbations that change the spindle size or rotational drag coefficient will scale the effective cue impulse parameter accordingly. We first calculate the expectations of mean and variance of the spindle displacement for a small patch of the cortex an angle θ from the spindle, and then sum over the whole cortex using standard rules for the combination of random variables.

If astral microtubules are assumed to emanate from each spindle pole, uniformly in all directions, the angular density of microtubules for the cortex at angle θ is given by

$${\rho }_{\text{MT}}\left(\theta \right)=\frac{d\beta }{d\theta }=\frac{1-a\text{cos}\theta }{1-2a\text{cos}\theta +a^2}.$$ (B1)

The equation above for astral microtubule density does not include a microtubule length-dependence factor; this could be incorporated by using the distance from pole to cortex, |l|, and assuming an exponential length distribution.

At a point on the cortex an angle θ from the spindle angle, the vector from spindle pole to cortex is given by (see Fig. S2 B)

$$\begin{array}{c}\mathbf{l}=\mathbf{r}-\mathbf{a}\\ \left(\begin{array}{c}{l}_x\\ l_y\end{array}\right)=\left(\begin{array}{c}\text{cos}\theta -a\\ \text{sin}\theta \end{array}\right).\end{array}$$

Only when the force vector is perpendicular to the spindle axis is the full cue impulse, f_cue, imparted to the spindle. For the cortex at angle θ, the attenuated angular impulse, α, is calculated using the geometry shown in Fig. S2 B as

$$\alpha =f_{\text{cue}}\text{sin}\beta =\frac{f_{\text{cue}}\text{sin}\theta }{\sqrt{1+a^2-2a\text{cos}\theta }}.$$ (B2)

The number of cue contributions from the patch of cortex has a Poisson distribution with mean λ, where λ is the product of the number of cortical cues in the patch, the frame interval, the activation rate, and the density of astral microtubules:

$$\lambda =n_{\text{cue}}\left(\psi \right)t_{\text{frame}}{\kappa }_{\text{on}}{\rho }_{\text{MT}}\left(\psi \right).$$

For combination of random variables, we can use the Gaussian approximation of a Poisson distribution. The cortical cues at angle θ each exert an impulse α on the spindle, meaning the distribution of angular impulse from the patch of cortex is given by

$$\text{Spindle}\text{displacement}=N\left(\mu =\alpha \lambda ,{\sigma }^2={\alpha }^2\lambda +t_{\text{frame}}{\sigma }_{\text{noise}}^2\right).$$

The mean and variance of the total spindle displacement distribution is found by integrating over the whole cortex:

$$\begin{array}{l}{\mu }_{\text{tot}}\left(\phi \right)=f_{\text{cue}}{t}_{\text{frame}}{\kappa }_{\text{on}}\underset{\theta =-\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{\overset{\theta =\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{\int }}{n}_{\text{cue}}\left(\theta +\phi \right)\frac{\text{sin}\theta \left(1-a\text{cos}\theta \right)}{{\left(1+a^2-2a\text{cos}\theta \right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}d\theta ,\\ {\sigma }_{\text{tot}}^2\left(\phi \right)=t_{\text{frame}}{\sigma }_{\text{noise}}^2+f_{\text{cue}}{t}_{\text{frame}}{\kappa }_{\text{on}}\underset{\theta =-\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{\overset{\theta =\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{\int }}{n}_{\text{cue}}\left(\theta +\phi \right)\frac{{\text{sin}}^2\theta \left(1-a\text{cos}\theta \right)}{{\left(1+a^2-2a\text{cos}\theta \right)}^2}d\theta .\end{array}$$ (B3)

For discrete cue locations,

$$n_{\text{cue}}\left(\psi \right)=\sum _j\delta \left(\psi -{\psi }_j\right),$$

Giving

$${\mu }_{\text{tot}}\left(\phi \right)=t_{\text{frame}}{\kappa }_{\text{on}}\sum _j\frac{\text{sin}{\theta }_j\left(1-a\text{cos}{\theta }_j\right)}{{\left(1+a^2-2a\text{cos}{\theta }_j\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}},$$

where θ_j = ψ_j - φ, and

$${\sigma }_{\text{tot}}^2\left(\phi \right)=t_{\text{frame}}{\sigma }_{\text{noise}}^2+t_{\text{frame}}{\kappa }_{\text{on}}\sum _j\frac{{\text{sin}}^2{\theta }_j\left(1-a\text{cos}{\theta }_j\right)}{{\left(1+a^2-2a\text{cos}{\theta }_j\right)}^2}.$$ (B4)

Thus, from the angular positions of the cortical cues relative to the spindle, we calculate the mean and variance of the rotational displacement. We express the result as a stochastic matrix, M,

$$M_{ij}=P\left(\phi \left(t+t_{\text{inc}}\right)={\phi }_j|\phi \left(t\right)={\phi }_i\right),$$

breaking the interval [−90,90]° into 720 bins. The stochastic matrix allows the evolution of a spindle angle distribution to be calculated over time, and allows the dynamics to be calculated at a very short timescale, t_step, and extrapolated to give the spindle displacements at typical frame interval, t_frame, by raising the matrix to the appropriate power:

$${\mathbf{M}}_{\mathbf{f}\mathbf{r}\mathbf{a}\mathbf{m}\mathbf{e}}={\mathbf{M}}^{t_{\text{frame}}/t_{\text{inc}}}.$$

Rotation statistics such as the bias in the direction of rotation and the absolute speed of rotation are calculated directly from the probability distribution of spindle displacements as a function of angle, i.e., the columns of M_frame:

$$M_{\text{frame}}^i=P\left(\phi \left(t+t_{\text{frame}}\right)|\phi \left(t\right)={\phi }_i\right).$$

A stochastic matrix has at least one eigenvalue of value 1, whose corresponding eigenvectors describe the equilibrium spindle orientation distributions (such that the distribution will be unchanged after multiplication by M). Furthermore, we can specify a starting angle distribution, v₀, and using the matrix calculate the distribution at the time of division as

$${\mathbf{v}}_{\mathbf{d}\mathbf{i}\mathbf{v}}={\mathbf{M}}^{DM/t_{\text{inc}}}{\mathbf{v}}_{\mathbf{0}}.$$

A probabilistic duration of mitosis is incorporated by summing over the duration distribution:

$${\mathbf{v}}_{\mathbf{d}\mathbf{i}\mathbf{v}}\left(\phi \right)=\sum _{x}p\left(DM=x\right){\left(\mathbf{M}\right)}^{\raisebox{1ex}{$x$}\!\left/ \!\raisebox{-1ex}{$t_{\text{frame}}$}\right.}{\mathbf{v}}_{\mathbf{0}}.$$ (B5)

As of this writing, the force is used only to calculate the torque acting around the center of the spindle; lateral forces are assumed not to displace the spindle away from the center of the cell. If a length-dependent pushing force was incorporated for astral microtubules not interacting with cortical cues, models for spindle orientation and spindle positioning could be combined, to evaluate the extent to which a defect in spindle centering could also induce misorientation.

In our study the number of cues was kept fixed at 60, a value that was chosen to be slightly smaller than that used by Théry et al. (16) in modeling the division angle of cells adhered to fibronectin patterns. The important variable in determining the dynamic behavior is the number of active cues per frame, which is determined by the total number of cues, N, the cue activation rate, κ, and the frame interval. Therefore, to a good approximation, an increase in the number of cues is equivalent to an increase in the activation rate. A secondary effect of increasing the cue number not considered in this study is that the cell shape is more accurately represented by the distribution of cues.

A striking observation from experimental measurements is the high level of noise present in spindle rotation. For cells of uncontrolled shape in culture, alignment with the long axis is accomplished with a relatively small bias in the direction of rotation. Regarding the origin of the high level of noise in this system, we propose three sources that contribute to the high level of noise. The first proposed source is the stochasticity in the cue activity, meaning that the number of cues acting to move the spindle within a frame interval is stochastic, and so even if the number of cues pulling towards the preferred axis outnumber those pulling away, there is a nonzero probability that the spindle will move away from alignment in a given frame. This source of noise was considered in our analytical approximation of the dynamic model. The second noise source arises from a probabilistic lifetime of cue activity. In our system, we assumed that there was a single rate-limiting step for cue activation and deactivation, giving an exponential active cue lifetime distribution. This adds a further level of uncertainty to the total torque and is calculated in Appendix C. The third source of noise is a cue-extrinsic Brownian noise term, defined as a Gaussian whose width is determined by the magnitude of all the additional torques acting on the spindle. As well as molecular collisions, this term includes fluctuations in the forces exerted on the spindle by astral microtubules that are pushing against the cortex rather than interacting with a cue.

Appendix C: Incorporating Stochastic Cue Activity Lifetimes

We wish to calculate the probability distribution of the total impulse produced by a Poisson distribution of active cues. Each active cue produces a constant force for a duration of time defined probabilistically by an exponential distribution. This means that the impulse exerted by each cue is a value sampled from the same exponential distribution, so that the total impulse from k cues is a hypoexponential distribution, H(f,k), with mean fk and variance f²k. In our notation, H(f,k) represents the sum of k exponential distributions, each with mean f.

If we now consider the case when the number of active cues has a Poisson distribution with parameter λ, the overall distribution is a mixture of hypoexponential distributions, where the mixture coefficients are the Poisson probabilities for the number of active cues. The distribution itself is nontrivial, but the mean and standard deviation can be calculated:

$$\begin{array}{l}\langle X\rangle =\underset{0}{\overset{\infty }{\int }}xp\left(x\right)dx,\\ =\underset{0}{\overset{\infty }{\int }}x\sum _{j=0}^{\infty }{C}_{j}H\left(f,j\right)dx,\\ =\sum _{j=0}^{\infty }{C}_j\underset{0}{\overset{\infty }{\int }}xH\left(f,j\right)dx,\\ =\sum _{j=0}^{\infty }{C}_j\langle H\left(f,j\right)\rangle ,\\ =\sum _{j=0}^{\infty }{C}_{j}fj,\\ =f\langle C\rangle ,\\ =f\lambda .\end{array}$$

In the equations above, C_j are the Poisson probabilities P(number of active cues = j). Therefore, the mean impulse is unchanged by incorporating exponential activity lifetimes.

Similarly for the variance,

$$\begin{array}{l}\langle X^2\rangle =\underset{0}{\overset{\infty }{\int }}{x}^{2}p\left(x\right)dx,\\ =\sum _{j=0}^{\infty }{C}_j\underset{0}{\overset{\infty }{\int }}{x}^{2}H\left(f,j\right)dx,\\ =\sum _{j=0}^{\infty }{C}_j\langle H^2\left(f,j\right)\rangle ,\\ =\sum _{j=0}^{\infty }{C}_j\left(j+j^2\right)f^2,\\ =f^2\sum _{j=0}^{\infty }{C}_{j}j+f^2\sum _{j=0}^{\infty }{C}_j{j}^2,\\ =f^2\lambda +f^2\left(\lambda +{\lambda }^2\right),\\ =f^2\left({\lambda }^2+2\lambda \right).\end{array}$$

Where above we have used the definition of variance, we now have

$$\begin{array}{l}\text{Var}\left[C\right]=\langle C^2\rangle -{\langle C\rangle }^2,\\ \lambda =\sum _j{C}_j{j}^2-{\lambda }^2,\\ \sum _j{C}_j{j}^2=\lambda +{\lambda }^2.\end{array}$$

Therefore, the variance of the overall distribution is

$$\begin{array}{l}\text{Var}\left[X\right]=\langle X^2\rangle -{\langle X\rangle }^2,\\ \text{Var}\left[X\right]=f^2\left({\lambda }^2+2\lambda \right)-{\left(f\lambda \right)}^2,\\ \text{Var}\left[X\right]=2f^2\lambda .\end{array}$$

The variance of the total impulse is increased by a factor of 2 due to the exponential lifetime of cue activity, compared with fixed cue impulses and only Poisson variance in cue number.