Microtubule organization within mitotic spindles revealed by serial block face scanning electron microscopy and image analysis

In readiness for cell division, eukaryotic cells build a mitotic spindle. This miniature machine, made of microtubules (MTs) and associated proteins, maneuvers the duplicated chromosomes and segregates them so that the two daughter cells each receive a complete copy of the genome. Understanding how this MT-based structure is organized and how it works is a major theme in cell biology (Petry, 2016).

In human cells, the mitotic spindle is a fusiform structure, which at metaphase is virtually spherical. MTs radiate from the two spindle poles, and belong to one of three classes: (1) astral MTs, which extend from the poles back towards the plasma membrane; (2) interpolar MTs, which run from one pole towards the other pole; and (3) kinetochore MTs, which are attached to the kinetochore on the chromosome (Mastronarde et al., 1993; McDonald et al., 1992; Helmke et al., 2013). Kinetochore MTs are bundled together to form kinetochore fibers (K-fibers) and, at metaphase, 20–40 MTs constitute a single K-fiber bundle (Booth et al., 2011, 2013; Sikirzhytski et al., 2014; McEwen et al., 1997). A network of crosslinking material, termed the ‘mesh’, is thought to hold kinetochore MTs together throughout their length (Hepler et al., 1970; Nixon et al., 2015). So far one protein complex containing TACC3, clathrin and ch-TOG (also known as CKAP5) has been identified as being a component of the mesh (Booth et al., 2011; Nixon et al., 2015).

The MTs of the mitotic spindle are densely packed and so resolving their organization in 3D is challenging. Advances in super-resolution imaging and expansion microscopy have improved the view of mitotic MTs that can be achieved by using light microscopy (Mikhaylova et al., 2015; Chozinski et al., 2016), yet the gold standard method is still electron microscopy (EM). Reconstructions of entire spindles in 3D using serial sectioning and transmission electron microscopy (TEM) have told us much about mitotic spindle structure (Mastronarde et al., 1993; McDonald et al., 1992; Redemann et al., 2016 preprint). However, these methods are extremely laborious, technically challenging, and prone to error. The advent of automated EM methods, such as serial block face scanning electron microscopy (SBF-SEM) has promised to speed up the analysis of complex cellular structures in 3D (Hughes et al., 2014). The spatial resolution of SBF-SEM is reported to be ∼10 nm, making it ideal for the study of MTs (25 nm diameter) in the mitotic spindle. Whether this resolution can be reached routinely for such analysis is unclear.

The first aim of the present study was to apply SBF-SEM to the problem of resolving MTs in the mitotic spindle. The resulting datasets are large and require robust statistical methods for analysis, and our second aim was, therefore, to provide computational tools to aid understanding of such datasets. The resulting workflow was then used to further understand how TACC3 levels influence MT organization in mitotic spindles.

We began by finding the optimal conditions for imaging MTs in mitotic cells by SBF-SEM. These conditions, described in the Materials and Methods section, allowed us to image a volume containing almost the entire spindle and to visualize bundles of MTs in the spindle running from the pole to the chromosomes (Fig. 1). The resulting datasets could be segmented, rendered in 3D and analyzed computationally (Fig. 1A and Materials and Methods). This workflow resulted in the visualization of a 3D rendering of the mitotic spindle (Fig. 1B,C). Even under these optimized conditions, we were working at the resolution limit of the instrument and this hampered our ability to resolve every single MT in the mitotic spindle (see below). Where single MTs could be discerned, we were unable to resolve the ends of MTs (Fig. 1B). Such resolution is possible with TEM and tomography; however, those methods are incredibly labor intensive. While SBF-SEM has limitations, our simple and rapid workflow allows the accurate mapping of mitotic spindle architecture sampled beyond the resolution limit of light microscopy and also permits much higher throughput.

Since the images were captured at the resolution limit of the instrument, a key question is whether or not we have single-MT resolution. To look at this, we focused on astral MTs, which are known, from serial section TEM experiments, to exist as single, isolated MTs (McDonald et al, 1992). Fig. 2A shows that clear density, attributable to individual astral MTs, can be identified in our images and easily segmented. This observation indicates that the method has single-MT resolution. We expected the detection of single MTs in regions of low MT density to be easier than discerning single MTs in the main body of the spindle. To examine MT detection in regions of higher density, we counted MTs that terminated at kinetochores and compared these numbers with previous observations undertaken with TEM (McEwen et al., 1998; 1997). On average, we resolved a median of six MTs terminating at the kinetochore [interquartile range (IQR)=5 (4–9); n=4 cells] (Fig. 2B–D; Movie 1). This varied somewhat between datasets, from four [IQR=3.5 (1.5–5)] to eight [IQR=4 (6–10)], but is fewer than the 20–23 MTs reported to be present at metaphase by serial section TEM (McEwen et al., 1998, 1997). Comparing the number of MTs at a given kinetochore with the number terminating at its sister, however, gave broadly comparable numbers (Fig. 2D). Assuming that the number of MTs at kinetochores does not vary substantially, this suggests that local differences in MT detection underlie the variability within a dataset. These data indicate that in higher density regions of the spindle, we do not have single-MT resolution. We conclude that detection of single MTs is possible, but that MT detection is not uniform throughout the volume. The sample of MTs that can be resolved is nonetheless substantial and sufficient for further analysis.

The large-scale high-resolution view of MTs resulting from our workflow was well suited for the development of geometric models to describe mitotic spindle organization. These methods and associated code are detailed in the Materials and Methods section, and an overview is presented here.

First, the volumetric density of MTs in the spindle could be determined. As an input, we take the segmented MTs from SBF-SEM data, and this is used to calculate the total volume of MTs. This value is then normalized to the volume of spindle, which is visible in the dataset (Fig. 3A). This normalization step is essential because at the magnification necessary to resolve MTs, the spindle volume is often incomplete. To test the sensitivity of this volume density calculation, we compared cells treated with cold (4°C) versus warm (37°C) media. When cells are treated with cold medium, any MTs which are not stably attached at both ends become depolymerized leaving only the K-fibers intact (Rieder, 1981). In HeLa cells, and in two different cell lines derived from glioblastoma patients, we measured a decrease in MT density to approximately one-third of the volume recorded in control cells in warm medium (Fig. 3B).

Second, the spatial organization of MTs in the spindle is analyzed. The aim is to determine the amount of order – or degree of alignment – for MTs in the mitotic spindle. To do this, an idealized spindle is generated computationally using the coordinates of the segmented MTs and simple geometric principles (Fig. 4 and Materials and Methods section). The measured angle between each MT segment and its idealized counterpart is calculated [θ⁽ⁱ⁾ in Fig. 4A]. For an ideal mitotic spindle, in which all microtubules are aligned, all angles will equal 0°. For real mitotic spindles, the angles deviate considerably from the ideal. Therefore, the magnitude of the angle deviations from the idealized spindle indicate a departure from perfect alignment. For example, the microtubules in the control GFP-expressing cell shown throughout this paper have a median angle of 11.2° [IQR=14.8° (5.2–20.0°)] (Fig. 5A). Previously, we found that overexpression of TACC3 caused changes in the degree of alignment of MTs (Nixon et al., 2015). These conclusions were based on tomography of a single thin section cut orthogonally to the K-fiber axis. Our new statistical method meant that this comparison could now be done on a near-global and whole-spindle scale. To look deeper into this phenomenon, we wanted to increase or decrease the TACC3 levels on the spindle and examine changes in the amount of MT order. Therefore, in addition to GFP-expressing control cells, we processed cells conditionally overexpressing GFP–TACC3 and also cells where the endogenous TACC3 was depleted by use of a shRNA but also expressed a mutated form (S558A) of GFP–TACC3 that was refractory to the shRNA. This mutant cannot bind clathrin and therefore cannot localize to the mitotic spindle (Hood et al., 2013). These three genotypes were examined in both warm and cold conditions (Fig. 5C; Fig. S1). Deviations from the ideal were compared using Π(Θ₁, Θ₂) for all combinations of data (see Materials and Methods section for further details). This comparison of angle deviations from these datasets showed that mitotic spindles in cells with more TACC3 had less order than controls under warm conditions (Fig. 5A,C). Similar deviations were seen in the cells expressing TACC3(S558A) in place of endogenous TACC3. This analysis extends, and is in agreement with, our previous conclusion that MT order in mitotic spindles is influenced by TACC3 levels on the mitotic spindle (Nixon et al., 2015). The comparison between warm- and cold-treated samples revealed that MTs in cells expressing TACC3(S558A) in place of endogenous TACC3 were more disorganized after cold treatment. Interestingly, the cold-treated cells overexpressing TACC3 did not show the same phenotype. The amount of order for MTs in this condition was similar to cold-treated GFP-expressing cells, which may suggest that elevated TACC3 levels protect MTs against cold-induced changes in spatial organization.

SBF-SEM datasets also allowed the segmentation of kinetochores, and therefore the calculation of their volume (Fig. 6). The average volume of a kinetochore was 8.17×10⁶±0.3×10⁶ nm³ (mean±s.e.m.). The surface area of the outer face of the kinetochore, as measured by serial section TEM is 0.16±0.05 μm² (McEwen et al., 1998), which would mean a thickness of 50 nm. This is very similar to the thickness in our segmented images, indicating agreement and accurate quantification. Since the pairing of kinetochores can be easily determined in the dataset, we tested the idea that sister kinetochores have similar volumes and that kinetochore volumes vary across the dataset. A plot of sister kinetochore volumes is shown in Fig. 6C. One-way ANOVA indicated that there were significant differences between pairs of kinetochores [F(33,34)=1.918, mean squared error (MSE)=4.6×10^–12, P=0.032]. Randomization of the data and retesting by one-way ANOVA gave further confidence to this result (Fig. 6D). These findings suggest that large kinetochores are paired with large kinetochores and small kinetochores with small kinetochores.

In addition to the mitotic spindle and associated chromosomes, the membranes of mitotic cells could also be rendered in three dimensions. This allowed us to explore the relationship between intracellular membranes and the mitotic spindle. We found that the mitotic spindle was situated within an ‘exclusion zone’ that was largely free of membranes and organelles (Fig. 7; Fig. S2, Movie 2). Outside of this zone, the endoplasmic reticulum (ER) was densely packed and mitochondria were distributed within the folds of ER. These observations agree with previous work that described the organization of ER membranes in mitotic cells (Puhka et al., 2007, 2012) and with more recent work that proposed that, although mitosis in human cells is open, the spindle is encapsulated by membrane (Schweizer et al., 2015). The purpose of this encapsulation is to exclude organelles and concentrate the proteins necessary to build a mitotic spindle rapidly (Schweizer et al., 2015).

While studying the organization of MTs in the mitotic spindle, we also had the opportunity to look for the presence of ‘bridging fibers’. These small bundles of MTs have been proposed to span from one K-fiber emanating from one pole, across the sister kinetochores and to reach the K-fiber that emanates from the opposing pole (Kajtez et al., 2016; Tolić and Pavin, 2016). Close inspection of our segmented datasets showed the presence of structures which may correspond to bridging fibers. Examples are shown in Movie 3 and a single example is shown in Fig. 8. We could not find evidence for bridging fibers associated with every K-fiber pair, although this is probably because our coverage is incomplete rather than an indication that there is not a one-to-one association of bridging fibers with each K-fiber pair.

SBF-SEM is commonly used to examine cellular and multicellular structures on the scale of tens to hundreds of micrometers. Whether or not this mode of imaging could discern small subcellular structures such as MTs (25 nm diameter) in complex networks such as the mammalian mitotic spindle was untested. Our results indicate that it is possible to visualize a large sample of MTs in the mitotic spindle of a human cell. The resulting datasets were complex and required robust analytical tools, which were developed and presented as part of this paper. We applied our workflow to test the influence of the levels of TACC3 at the mitotic spindle on the organization of MTs on a whole-spindle scale.

The influence of TACC3 on mitotic spindle structure has previously been examined by using two extreme methods. In the first, individual high magnification views of subsections of the spindle were obtained by means of electron tomography (Nixon et al., 2015). Second, low spatial resolution views of whole spindles were captured by light microscopy (Hood et al., 2013; Gergely et al., 2000). What was missing was a technique that combines the advantages of both methods to visualize MT order on the scale of the whole-mitotic spindle. Super-resolution methods cannot currently fulfill this gap, and the present study shows that SBF-SEM can, and that it does so at an appropriate scale to answer questions about MT organization. Moreover, the automation of the image capture and data processing means that it is practical to analyze spindles in this way. While the resolution of this method is lower than TEM and electron tomography, its power lies in the higher throughput and lower requirement for technical expertise. SBF-SEM is not suitable for answering questions that absolutely require single MT resolution and, particularly, the resolution of MT ends (McIntosh et al., 2013); however, it can compete with TEM and electron tomography. We showed that 35 pairs of kinetochores could be easily segmented from a single SBF-SEM dataset, a number sufficient to test whether or not sister kinetochores were of a comparable size. Capturing a similar dataset by serial section TEM would be extremely challenging and laborious, with no clear resolution advantage in this case.

Our analytical workflow allowed us to assess MT density and MT order. While it is possible to approximate the volume of MTs in cells using light microscopy, the higher resolution of SBF-SEM provides more precise quantification of the MT volume in a mitotic cell. Determining the order of MTs in a dense array is not currently possible by light microscopy, even in super-resolution modes, and the advantage of SBF-SEM, here is therefore clear. The simple description that we developed was based on comparison with an idealized spindle. We found that ellipsoids were a good model for angular comparison, out-performing other geometric shapes and comparison regimes based on near-neighbors or on the spindle axis (see Materials and Methods).

Our data indicate that too much TACC3 (or too little) affects MT order in mitotic spindles. Previous work has shown that mitotic progression is slowed in cells overexpressing TACC3 or in cells depleted of TACC3 through expression of siRNA (Nixon et al., 2015; Gergely et al., 2003; Lin et al., 2010). The current picture is that the major function of TACC3 in mitotic spindles at metaphase is as part of a TACC3–ch-TOG–clathrin complex that is assembled upon Aurora A-mediated phosphorylation, rather than as a MT plus-tip protein. The TACC3–ch-TOG–clathrin complex is an important component of the ‘mesh’ that is thought to maintain the integrity of the K-fiber MTs. The present data indicate that the sensitivity of mitosis to TACC3 levels is because of changes in the composition of the mesh that, in turn, alters K-fiber structure and organization; it strongly suggests that these changes prevent the chromosome movements required for high-fidelity mitosis. However, we cannot exclude the possibility that other functions of TACC3 contribute to MT order or to mitotic progression.

These 3D views of mitotic cells also allowed us to visualize the spindle exclusion zone and putative ‘bridging fibers’ of the mitotic spindle. Clearly distinguishing bona fide bridging fibers from other microtubules is challenging, even at this resolution. Some examples presented here clearly look like bridging fibers that run in parallel to the respective sister K-fibers (Kajtez et al., 2016; Tolić and Pavin, 2016). Others highlighted in Movie 3 were less clear. If these structures are not bridging fibers, they could be interpolar MTs or K-fiber MTs, which are laterally or merotelically attached to kinetochores (Cimini et al., 2001). A formal test to determine whether these MT bundles are associated with the sister K-fibers will require TEM analysis.

Analysis of MT order by SBF-SEM requires operating at the resolution limit of the instrument. We optimized the staining of the sample, section depth, the magnification and accelerating voltage/beam energy to be able to visualize MTs. This protocol allowed us to sample a large fraction of MTs in the spindle, but fell short of a complete view of the mitotic spindle for three reasons. First, the spindle volume was incomplete since the magnification required to see MTs resulted in an xy area of 9.2×9.2 μm, although this limitation can be circumvented by stitching fields-of-view for each slice. Second, we could only resolve single MTs in parts of the dataset; however, in denser regions, we were unable to resolve all MTs. For example, we found fewer than the expected numbers of MTs terminating at each kinetochore. Third, the section depth of 60 nm resulted in missing MT information. The step size matched the beam penetration depth, because heavily stained objects such as membranes could be accurately followed with no under- or over-sampling. However, thin objects such as MTs that are weakly stained appear discontinuous, presumably because they are only visible at the upper surface of the block. Note that the spindle exclusion zone actually improves MT visualization because the organelle-free zone does not interfere with MT detection and segmentation. Thinner sectioning, possibly using focused ion beam SEM, would improve visualization further. For these reasons, one caveat associated with the method presented here is that the analysis is performed on only a subset of the MTs in the mitotic spindle. We believe that this subset is a good representation of the mitotic spindle: it was sufficient to show differences in MT density and changes in MT order in the mitotic spindle. Future directions for better detection of MTs in mitotic cells include improved SEM detector technology and high-pressure freezing as a fixation method. It remains to be seen whether any of these innovations would allow full reconstruction of the mitotic spindle. While improvements in detection may allow greater coverage, the protocol presented here is sufficient for large-scale analysis of MT order in human cells; and in higher throughput than traditional EM methods.

HeLa cells that overexpress TACC3 upon doxycycline induction were as described previously (Nixon et al., 2015). Cells were maintained in Dulbecco’s modified Eagle’s medium (DMEM) plus 10% fetal bovine serum (FBS) and 1% penicillin-streptomycin in a humidified incubator at 37°C and 5% CO₂. The cell culture medium was further supplemented with G418 (300 μg ml^–1) for the parental HeLa TetOn cells, and with G418 and Hygromycin B (200 μg ml^–1) for the HeLa TetOn cells stably transfected with the TACC3 plasmid. Glioblastoma cell lines were cultured in neurobasal medium, supplemented with B27, N2, human fibroblast growth factor (40 ng ml^–1) and human epidermal growth factor (40 ng ml^–1).

For experiments to express GFP–TACC3(S558A), we used a plasmid (pBrain-GFP-TACC3(S558A)KDP-shTACC3) which knocks down the expression of endogenous TACC3 and simultaneously expresses GFP–TACC3(S558A), which is refractory to the shRNA (Gutiérrez-Caballero et al., 2015). Cells were transfected using GeneJuice, according to manufacturer's instructions.

Three HeLa conditions were studied: first, parental HeLa TetOn cells transiently transfected with pEGFP-C1 only as a control; second, the inducible GFP–TACC3-expressing HeLa TetOn stable cell line; third, parental HeLa TetOn cells transiently transfected with the pBrain-GFP-TACC3(S558A)KDP-shTACC3 construct. Cells were synchronized by thymidine-Ro3306 treatment (Nixon et al., 2015) and grown on gridded dishes (MatTek) to allow correlation between light and electron microscopy images. Synchronization using thymidine-Ro3306 is preferred to nocodazole or similar drugs, which perturb microtubules. For cold treatment of mitotic cells, cells were imaged to ensure the cells were at metaphase, and the growth medium was removed and replaced with ice-cold growth medium and incubated on ice for 10 min. Medium was then replaced with cold fixative.

Fluorescence imaging was performed using a Nikon Eclipse Ti-U microscope with standard filter sets for visualization of GFP. Light and epifluorescence micrographs were acquired by using a Photometrics Myo camera, with 20× air and 60×oil objectives, and NIS Elements acquisition software. Cells were imaged in a 37°C temperature-controlled chamber (OKOlab). Chemical fixation, processing and sectioning for TEM, was performed as described previously (Booth et al., 2013). To prepare samples for SBF-SEM, the cells were also processed in a correlative manner to ensure only metaphase cells expressing the protein of interest were chosen. Processing was as described in Booth et al., 2013 with the following extra steps to better visualize ultrastructural components of the cell by SEM. Formaldehyde (0.5%) and glutaraldehyde (3%) fixative solution was prepared in phosphate buffer (0.1 mol l^–1), with 0.1% tannic acid and 3% sucrose. Two osmication steps were performed. First, a 1 h reduced osmium step, where 2% OsO₄ was prepared in 1.5% potassium ferrocyanide solution, in phosphate buffer. After five 3-min ddH₂O washes, 0.1% thiocarbohydrazide (TCH) was added for 30 min at 60°C as a mordant. After another five 3-min ddH₂O washes, OsO₄ (2% in ddH₂O) was applied for 30 min. The cells were again washed in water as before, then stained overnight using 1% uranyl acetate in ddH₂O at 4°C. The following day, water washes were repeated before incubating with lead aspartate at 60°C for 30 min. The cells were then dehydrated using graded ethanol. Cells were infiltrated using Epon 812 Hard premixed resin kit (TAAB). Once fully polymerized (48 h at 60°C), the dish and coverslip were removed and the cell located. This area was then excised using a junior hacksaw and razor blades to leave an ∼2 mm³ piece of resin containing the cell. This cube of resin was mounted using Silver Dag onto a cryo pin which fits into the chuck of the 3View platform (Agar Scientific). The area around the cell was fine-trimmed using fresh glass knives, this produced an ∼100 μm² block face. Once trimmed, the block was coated with a silver paint using an eyelash, before a 15 nm gold–palladium coating was evaporated onto the block surface, again to reduce surface charging (Quorum Technologies, East Sussex, UK). The cell was imaged using the Gatan 3View System and Digital Micrograph software (Gatan, Abingdon,UK) and FEI Quanta 250 (FEI, Eindhoven, Netherlands). When aligned, the 3View machine was allowed to cut-and-image the block surface, with 100 nm z-slices, initially optimizing the contrast and magnification for that particular block, before leaving the platform to run until the cell was fully sectioned and imaged. A 768×768 pixel image, 60 nm sections, 60 μs pixel dwell time, 2.1 keV and 21,000× magnification (12 nm pixel size, 9.216×9.216 μs) was optimal for imaging the spindle zone. These conditions needed to be fine-tuned for the individual sample, and were a compromise between reduced levels of sample charging, magnification and resolution.

All 3D segmentation and surface rendering was performed with Amira software (Visualization Sciences Group, FEI). The voxel size in nm was entered upon loading of the file into Amira. MTs in SBF-SEM volume stacks were segmented manually in Amira by tracing all the MTs visible in each slice throughout the entire volume, this segmentation was rendered to generate a 3D surface. In addition, where membranes, chromosomes, kinetochores and centrioles were visible, these were segmented in a semi-automated manner, using pixel density to segment objects in 3D. The signal-to-noise ratio of segmented MTs was 3.5±0.4 (mean±s.e.m., n=1373 images), i.e. lower than electron tomographic data, which hampered automatic tracing (Weber et al., 2012). The segmentation files (*.am files) were processed in ImageJ/FIJI to create binary images that could be fed into Igor Pro 7.0. Code for analysis was custom-written in Igor Pro 7.0 and can be found at https://github.com/quantixed/VolumeFinder. For volume analysis, the image stacks are interpreted and pixels above threshold are counted for each slice of the stack. A 3D convex hull is drawn around all points to measure the volume for normalization. For ellipsoid comparison, described in detail below, the images are skeletonized in ImageJ/FIJI. This process converts each MT segment into a 1 pixel thick object. Each object is read by Igor and converted to a unique 2D vector by fitting a line y=a+bx to the xy coordinates of the object. All objects lie in the plane of the section/image and so a 2D vector is sufficient. The resulting straight lines are used for mathematical modeling. The modeling and comparison was performed in Igor Pro7.0 for code, see https://github.com/quantixed/VolumeFinder. To ensure reproducibility, and for error-checking, the modeling and analysis was independently rewritten in R using the xyz coordinates of straight lines as an input. This code is also available at https://github.com/quantixed/VolumeFinder, where a detailed walkthrough document is also provided.

A more detailed description follows; in summary, straight lines (representing MT segments from the SBF-SEM dataset) were compared to computer-generated counterparts that were placed in the ideal location. All straight lines were translated and rotated such that the spindle was vertical with the poles aligned on the z-axis and the center of the spindle at the origin. The midpoint of each straight line was used to make an ellipsoid that passes through this point and through both spindle poles. The tangent of a section through this ellipsoid at the midpoint was used for comparison with the original straight line. This was done after putting the straight lines and the idealized counterparts back in their original location, and projecting both segments onto a single xy-plane. We developed this analysis method because the spindle is a fusiform structure, and therefore comparison of each MT segment to the spindle axis gave a broad range of deviations that made comparisons between conditions impossible (Fig. S3). Angle comparisons between each MT segment and its neighboring segments were not representative since MT density was not uniform throughout the volume (Fig. S3). The calculations for ellipse comparison rely on simple geometric principles and make use of the specified locations of the two spindle poles. This is possible because the force balance in the metaphase spindle ensures a symmetrical arrangement: the chromosomes are situated approximately at the equator, and the spindle poles are equidistant from the metaphase plate. A more simplified model, with a straight-line or diamond-shaped spindle, was less realistic than ellipses (Fig. S3). More complex curves are likely to yield a better fit, and are possibly justified in light of data showing that MTs do not necessarily extend from the spindle pole to kinetochore (Sikirzhytski et al., 2014; Elting et al., 2014); however, the inclusion of more parameters is not desirable.

We observe a collection of straight lines, i∈{1, 2,n}, denoted ⁠. Each line may be summarized by its two endpoints; and ⁠; switching the labels of endpoints and results in the same line object. The section Translation and rotation justifies a particular choice of and ⁠.

Line locations and orientations are determined from a stack of images collected at increasing depth through the cell with each line segment being extracted from a single image within this stack as described under Image analysis. The co-ordinate axes are aligned such that the image planes are perpendicular to the z-direction.

The formula given in (Eqn 1) describes a spheroid centered at the origin and aligned with the co-ordinate axes. However, while we expect our observed lines to be related these trajectories, we do not expect them to be centered and aligned. It is possible to transform the model to match the location and alignment of the observed data. We choose instead to transform the observed data for later ease of interpretation and presentation.

Each observed line-segment is first translated so that the origin of the spheroid whose tangent-plane contains it is centered at the origin and then rotated so that this spheroid is aligned with the coordinate axes. This process begins by identification of two fixed points, and ⁠, which after translation and rotation will become (0,0, b) and (0,0, –b) respectively. The location of these fixed points can be determined directly from the images themselves, in the case where the centrosomes are visible. Where centrosomes are not directly visible, they were inferred using the orientation of observed lines and the assumption that and are the spindle poles from which all MTs should originate.

The translation shifts the center of the straight line joining and to the origin by subtracting the center of the spheroid from each observation. Rotation about the line through the origin, which is perpendicular to both the z-axis and the line-segment connecting to ⁠, by the angle required to align the image of that line-segment with the z-axis is then carried out. We allow to denote the shifted and rotated line segment obtained by transforming in this way, with end points and ⁠.

Under our model, we expect observed MT trajectories to initiate at (0, 0,±b) and to terminate at the plane z=0. We orient each trajectory so that it is consistent with this. This allows us to specify that is the end of each line segment closest to a fixed point and the end closest to the plane that models the metaphase plate.

Our interest is focused on those MTs leading from the spindle pole to the metaphase plate and so our analysis is restricted to observed lines for which the z-coordinate lies always in [−b,+b].

We wish to compare the direction of observed lines ⁠, conveniently denoted ⁠, with those consistent with the spheroid model. The first step is to determine the tangent to the geodesic path that passes through the fixed points (0,0, b), (0,0, –b) and the midpoint of the observed line ⁠, given by ⁠, on the surface of the spheroid.

We thank Heiko Wurdak (Leeds Institute of Cancer and Pathology, University of Leeds, UK) for the gift of glioblastoma cell lines.