Trackosome: a computational toolbox to study the spatiotemporal dynamics of centrosomes, nuclear envelope and cellular membrane

Mitosis is a highly regulated stage of the cell cycle in which multiple subcellular structures take part in a complex chain of events that culminate in chromosome segregation. As cells prepare to enter mitosis, adhesion complexes disassemble (Dao et al., 2009) and the cytoskeleton reorganizes (Matthews et al., 2012; Mchedlishvili et al., 2018). At the same time, duplicated centrosomes need to migrate along the nuclear envelope (NE) so that a bipolar spindle can form (Tanenbaum and Medema, 2010). This process requires the activity of multiple players, such as microtubule-associated motors kinesin-5 (Whitehead et al., 1996) and dynein (Raaijmakers et al., 2012), but also actin (Cao et al., 2010) and myosin II (Rosenblatt et al., 2004). How the dynamic changes in all these events are coordinated in space and time to ensure efficient centrosome separation and spindle assembly remains unknown.

Recent advances in live-cell imaging and image analysis techniques have made it possible to access the subcellular environment and quantitatively examine its underlying mechanisms (Gerlich et al., 2001; Otsuka et al., 2018). Commercial imaging software such as Imaris and Metamorph are equipped with automatic particle tracking functions, which have been used to track centrosome pairs in three dimensions (Collins et al., 2014; De Simone et al., 2016; Yamashita et al., 2015). The open-source freeware ImageJ (Eliceiri et al., 2012) with its particle-tracking plugin Trackmate (Tinevez et al., 2017) has also been used for centrosome tracking in several studies (Boudreau et al., 2019; Mahen, 2018; Stiff et al., 2020). However, these generic tracking tools are not fine-tuned for the appearance and motion dynamics of centrosome pairs, which limits their tracking performance and often requires exhaustive parameter optimization, particularly when high-quality videos are not available. Moreover, when studying the dynamics of spindle formation, it is often necessary to analyze centrosome movement in reference to the cellular and nuclear membrane (and, therefore, in a non-canonical coordinate system). To the best of our knowledge, the available computational tools do not directly allow analysis of the coordinated changes between different structures in specific subcellular frames of reference (such as the nuclear membrane).

The NE also exhibits rich spatiotemporal dynamics, displaying measurable oscillations that correlate with the forces exerted by chromatin, nuclear lamina and the cytoskeleton (Chu et al., 2017; Schreiner et al., 2015; Stephens et al., 2017) and strongly influence nuclear functions (Jahed and Mofrad, 2019; Stephens et al., 2019). One way to study this complex interplay between cytoskeletal forces imposed on the nucleus and the resistive forces triggered by chromatin and nuclear lamina is by measuring the dynamics of NE membrane fluctuations (Chu et al., 2017; Hampoelz et al., 2011; Schreiner et al., 2015), which allow the distinction between thermally driven and active fluctuations (Chu et al., 2017). Importantly, during mitotic entry, chromosomes condense (Antonin and Neumann, 2016) and the nuclear lamina disassembles (Georgatos et al., 1997). How these events affect the pattern of NE oscillations and impact other aspects of early spindle assembly remains to be determined. Current methods used to calculate NE fluctuations were developed under the assumption that each point of the membrane oscillates radially around its time-averaged position (Almonacid et al., 2019; Blanchoud et al., 2010; Caragine et al., 2018; Chu et al., 2017; Schreiner et al., 2015). However, describing the membrane deformations as a radial displacement is a coarse approximation that can lead to erroneous results. Also, to the best of our knowledge, there are no available toolboxes for calculating these membrane fluctuations.

Driven by these computational limitations and the need to better characterize the crosstalk between subcellular structures during mitotic entry, we developed the open-source software Trackosome. This novel computational tool enables a quantitative analysis of the spatiotemporal dynamics of three cellular components: centrosomes, NE and cellular membrane. Trackosome relies on live-cell imaging datasets, where the structures of interest are independently tagged. The tool has two modules: (1) the Centrosome Dynamics module, used for tracking the centrosomes (or other subcellular organelles) in 3D and study their spatiotemporal relations with the nucleus and cell membrane; (2) the Nuclear Envelope Fluctuations module, used to reconstruct, measure and analyze the dynamic fluctuations of the nuclear membrane (or other membranes) in 2D. The accurate 3D reconstruction of the centrosome trajectories relative to the nucleus and cell membranes (in ellipsoidal coordinates) allowed us to unravel and quantify a significant correlation between centrosome trajectories that has not previously been characterized. In addition, the Nuclear Envelope Fluctuations module allowed us to observe and quantify distinct patterns of membrane oscillations in 2D in different stages of the cell cycle.

Trackosome is made publically available (open-source) as a platform-independent MATLAB toolbox and can be downloaded from https://github.com/Trackosome.

Tracking and trajectory analysis of centrosomes is performed in the Trackosome toolbox through the Centrosome Dynamics module. This module has a graphical user interface (Fig. 1A; Fig. S1, Movies 1-3) that provides useful feedback about the automatic tracking status, and allows manual or semi-automatic corrections whenever needed. The accuracy of the tracking algorithm was tested and validated in synthetic data, created with imposed controlled conditions. The videos generated had two centrosome-like objects moving in 3D with biologically realistic dynamics (coordinates taken from real centrosome trajectories), in a noisy environment. We set three levels of tracking difficulty by varying the signal-to-noise ratio (Fig. 1B). The coordinates obtained by the Trackosome algorithm were then compared with the original coordinates used to create the videos (Fig. 1C). Our tool was able to track the centrosomes with high fidelity even in highly noisy environments (signal-to-noise ratio of 0.7) where the particles were almost indistinguishable from the background. For all videos analyzed, the root mean squared error associated with the tracking (Fig. 1C) was at the subpixel level (1 pixel=0.189 µm).

We then recorded cells seeded on line micropatterns to normalize cell and nuclear shape (Versaevel et al., 2012). Note, however, that Trackosome does not require cells to be seeded in such micropatterns and can be used in unconstrained conditions (Fig. S3A-D). Taking advantage of the precision provided by Trackosome, we quantified specific spatiotemporal relations between three cellular structures during mitotic entry, namely centrosome pair, nuclear membrane and cell membrane. We chose this stage of the cell cycle as it involves extensive dynamic reorganization of the entire cell (Champion et al., 2017), providing a good benchmark to test the ability of Trackosome to detect these dynamic changes. Here, Trackosome was able to reconstruct in 3D the surfaces of the membranes, together with the trajectories of centrosomes (Fig. 2A,B). Based on their relative positions, the software was able to output different quantitative metrics, such as the distance and angles between centrosomes (Fig. 2C,D), the eccentricity of the nuclear and cellular membranes (Fig. 2E) and the angles between the major axes of the nucleus, cell and centrosomes (Fig. 2F). These metrics describe the intracellular reorganization that occurs as cells enter mitosis, such as movement of the centrosomes to opposite sides of the nucleus (Fig. 2B-D) and nucleus-centrosome axis reorientation (Fig. 2B,F), which we previously described (Nunes et al., 2020).

HeLa or U2OS cells labeled with histone H2B-GFP/α-tubulin-RFP/SiR-actin or EB3-GFP/Lifeact-mCherry/SiR-DNA, respectively, were tracked until nuclear envelope breakdown (NEB). During this stage, nuclear shape remained approximately constant and correlated with the labeled chromatin, allowing us to define the median nucleus membrane (Fig. 2G, grey mask) based on the 3D reconstructions of the NE. The centrosomes, on the other hand, exhibited complex trajectories that resembled a search or adaptive path around the nucleus (Fig. 2G). For this reason, to infer the coordination of movement between centrosomes, their trajectories were analyzed using the nucleus as a reference. An ellipsoid was fitted to the median nuclear membrane, setting the frame of reference for the new coordinate system (Fig. S3F-H). Each point of the centrosome trajectory was defined by a latitude φ, longitude λ and height h; the respective ellipsoidal velocities φ′, λ′, h′ and velocity norm |v|; and the ellipsoidal accelerations φ″, λ″, h″ and acceleration norm |a|. We calculated the correlation matrix of these features for each centrosome pair (Fig. 2H), revealing which components were temporally correlated between the two trajectories. Our results indicate a considerable degree of similarity and synchrony among trajectory pairs (Fig. 2I), with correlation values as high as 0.7, in contrast to a previous report (Waters et al., 1993). By calculating the median of the absolute correlation matrices obtained, we produced a map of the most consistent correlations (Fig. 2J). Interestingly, the positions of the centrosomes, φ, λ, h, and the acceleration norm |a|, showed significant correlations. The latter is particularly relevant because the acceleration of a particle reflects the force applied to it, suggesting a synchronous variation of the forces applied to both centrosomes, probably driven by kinesin-5 (Whitehead et al., 1996) or dynein (Raaijmakers et al., 2012). It is worth emphasizing that if the movement of centrosomes had been described in Cartesian coordinates (i.e. ignoring the nuclear surface constraint), the observed correlations would not be evident.

The dynamic morphology of the NE was analyzed using the Membrane Fluctuations module of Trackosome (Fig. S2; Movies 4 and 5). The membrane oscillations were determined by calculating the orthogonal displacement of each point of the membrane with respect to its medial position (Fig. S3I,J). Our new method does not rely on prior assumptions regarding the nucleus shape. We believe that this approach leads to a more realistic description of the membrane displacements than the radial displacement approximations (with respect to the centroid of the membrane) usually described in the literature (Almonacid et al., 2019; Blanchoud et al., 2010; Caragine et al., 2018; Chu et al., 2017; Schreiner et al., 2015). The radial displacements are particularly flawed for an irregularly shaped nucleus with wide fluctuations (Fig. 3A), which generally limits its use to cells in interphase. Here, we were able to quantify and compare the nuclear deformations for cells in interphase and early mitosis.

As before, we recorded cells seeded on line micropatterns. From our analysis, it is possible to confirm that cells in interphase present subtle but measurable nuclear membrane movements. As a negative control, fluctuation measurements were made for interphase cells fixed with formaldehyde, which showed a significant decrease in undulations upon fixation (Fig. 3B,C).

During mitotic entry, chromosomes condense (Antonin and Neumann, 2016) and the nuclear lamina disassembles (Georgatos et al., 1997). Whether these events change nuclear behavior remains to be determined. We analyzed the nuclear membrane fluctuations of cells using Trackosome (Fig. 3D-G). After segmenting and registering (centering on centroid) the NE (Fig. 3E), the fluctuations (u) of each frame were calculated assuming normal displacements (Fig. 3A, top). The values obtained were concatenated frame-by-frame and filtered in time and space with a 2D Gaussian kernel. The resulting fluctuations map reveals how the envelope curvatures are dominated by specific wavelengths (Fig. 3F). These wavelengths are determined by calculating the spatial Fourier transform (FT) of the fluctuations, u_f, for each frame and then obtaining the maximum magnitude of each wavelength across frames (Fig. 3G). Calculating the majorant for each frequency component u_f has the advantage of highlighting the most well-defined spatial frequencies, even if they occur for a limited number of frames. On the other hand, averaging the FT curves of all frames would attenuate these components, and the magnitude of the final FT peaks would not correspond to the magnitude of the actual membrane waves that originated them, thus providing a less intuitive readout.

The FT curve allowed us to identify and quantify the most prominent spatial frequencies and obtain relevant information about the dynamics of NE fluctuation. For the cell in prophase exemplified in Fig. 3D-G, the highest FT peak, with a wavelength of 25 µm (half of the nucleus perimeter), was the result of the large membrane displacement seen around the 15 and 40 µm arc landmarks in Fig. 3E,F. As prophase progressed, exertion of compression forces in these two separate points made the nuclear membrane wobble, with a wavelength that was half of the membrane perimeter (Fig. S4A). In addition, the FT analysis allowed the identification of membrane ‘wrinkles’ with spatial signatures defined by wavelengths of approximately 10, 5 and 4 µm (Fig. 3F,G). Trackosome captures these spatial signatures with high accuracy, even for curvatures with amplitudes at the subpixel level under noisy backgrounds (Fig. S4B-I).

To help uncover the processes behind these observed spatial frequencies, membrane fluctuations were calculated for cells in early mitosis and interphase (Fig. 4A,B). We calculated the majorant u_f for all cells (as in Fig. 3G) and then the median curve across trials for each group. Our results showed a significant difference in the FT curves of the two groups (Fig. 4B). In interphase, the magnitude of the fluctuations was very low through the entire spectrum of wavelengths, indicating that the membrane barely deviated from its basal position. Still, the u_f magnitude was above the noise floor established by the formaldehyde curve. This behavior changed in prophase, as there was an increase in the fluctuation amplitude for the entire range of wavelengths. This increase was particularly significant for the low frequency components, reflecting the occurrence of nucleus-wide deformations as described above.

Fluctuations of the NE are known to depend on the cytoskeleton and, in particular, on the microtubule (MT) network (Chu et al., 2017; Hampoelz et al., 2011). To assess whether Trackosome was sensitive enough to detect these changes, we treated cells with low doses of nocodazole (NOC) to suppress the polymerization of MTs without affecting overall cell structure. Interphase cells treated with NOC had an FT curve similar to the control curve obtained for interphase cells treated with DMSO only (Fig. 4A,C). On the other hand, nuclear fluctuations of mitotic cells were significantly altered with the addition of NOC. The FT obtained for this experimental condition showed a decrease in the low-frequency peak compared with the control group. This suggests that disruption of the MT cytoskeleton led to a decreased nuclear compression, which is in agreement with reports that the MTs induce large-scale deformations of the NE (Hampoelz et al., 2011; Schreiner et al., 2015). The frequency components beyond the peak were also severely attenuated, which means that the shorter membrane wrinkles were also mitigated. Most of these effects can be qualitatively evaluated by examining the nuclei images directly (Fig. 4A), but Trackosome provides a precise quantification.

Biological membranes exhibit passive fluctuations, which are thermally excited at physiological temperatures. The equilibrium properties of these undulating systems can be modeled by Helfrich-type models (Helfrich, 1978). To evaluate the spectral dependencies in line with Helfrich theory, we calculated the average FT of the squared fluctuations for each cell, <u_f²>, and then the median curve across cells for each group. The curves of the fluctuations obtained for the NOC and control groups showed regions with distinct spectral dependencies (Fig. 4D). For frequencies above 1.5 µm⁻¹ (wavelengths below 0.7 µm), the FT curves followed a f⁻⁴ power law, consistent with what is expected for thermally driven fluctuations at short wavelengths (Brandt et al., 2011). At lower frequencies, each group followed a different power law, ranging from close to f⁰ for formaldehyde, up to f^−1.5 for the control group in mitosis. These can be attributed to different nonequilibrium active forces governing each system. It is important to note that in Helfrich theory these spectral dependencies are defined in terms of the adimensional wavenumber q, and not spatial frequency. However, for the frequency ranges marked by the solid lines in Fig. 4D, the slopes were equivalent in both scales (Fig. S4J-L).

Finally, we compared the fluctuation amplitude [in terms of log root mean square (RMS) of u_t as in Fig. 3C, right] for all groups (Fig. 4E). NOC led to a significant decrease in the fluctuation amplitude during prophase. Importantly, this decrease was not caused by DMSO, because prophase cells in the presence of DMSO exhibited higher fluctuations than the untreated prophase cells. Also, the fluctuations obtained for the interphase cells were similar among the different groups and the range of amplitudes was consistent with those already described in the literature (Schreiner et al., 2015).

Overall, our results indicate that nuclear membrane fluctuations increase as cells transition from interphase to mitosis and that interfering with the MT network significantly reduces the large scale deformations of the nucleus during this stage.

Recent developments in light microscopy have generated extensive datasets that allow unprecedented access to subcellular events with high spatiotemporal resolution. Here, we report on Trackosome, a new open-source software that enables an automated quantitative analysis of the spatiotemporal dynamics of subcellular structures even in conditions of low signal-to-noise ratio. During the transition from G2 to mitosis, chromosomes condense (Antonin and Neumann, 2016), cells round up (Matthews et al., 2012) and centrosomes separate (Whitehead et al., 1996). In this context, the extent of centrosome separation at the time of NEB remains a matter of debate. Although some reports show that cells can enter mitosis with unseparated centrosomes (Kaseda et al., 2012; Whitehead et al., 1996), others indicate that centrosomes are fully separated at NEB (Magidson et al., 2011; Mardin et al., 2013; Nunes et al., 2020). These discrepancies highlight the need to determine carefully how the events leading up to mitosis are coordinated to ensure efficient spindle assembly. By using the different modules available on Trackosome, we were able to correlate centrosome movement accurately and centrosome-nucleus axis reorientation during mitotic entry. Our results, using cells seeded on fibronectin micropatterns, indicate that centrosomes are fully separated before NEB and that their movement is coordinated to optimize their positioning on opposite sides of the nucleus (Magidson et al., 2011), in contrast to previous observations (Waters et al., 1993). This coordination was evidenced using elliptical coordinates to describe the centrosome trajectories around the nucleus. Interestingly, the high correlation scores between movement components of both centrosomes strongly indicate a mechanical coupling between the two structures. This coupling could be provided by specific pools of dynein on the cell cortex (Kotak et al., 2012; Woodard et al., 2010) or at the NE (Bolhy et al., 2011; Splinter et al., 2012), which are known to generate pulling forces on centrosomal microtubules to position asters (Laan et al., 2012). Determining the exact nature of this mechanical coupling between the centrosomes during mitotic entry will be of interest in the future.

The mechanical properties of the nucleus depend on the chromatin condensation state (Stephens et al., 2017, 2018) and the nuclear lamina (Lammerding et al., 2006). Given that Lamin A levels (Chu et al., 2017; Moir et al., 2000) and chromatin compaction (Hinde et al., 2012) change throughout the cell cycle, it is possible that the mechanical properties of the nucleus change accordingly. In agreement, NE fluctuations, which reflect forces imposed on the nucleus, were shown to vary depending on the cell cycle stage (Chu et al., 2017). However, whether the mechanical properties of the nucleus change during the transition from G2 to mitosis, when mitotic chromosomes condense and the nuclear lamina disassembles, remains unclear. Here, we report on a new tool for analyzing NE fluctuations during mitotic entry that measures orthogonal displacements relative to the medial position of the nuclear membrane. We reveal that chromosome condensation, together with microtubules, triggers significant changes in the spatial pattern of NE fluctuations as cells prepare to enter mitosis. To the best of our knowledge, it is the first time that the transition from G2 to prophase has been characterized in terms of NE fluctuations. In the future, it will be interesting to determine how these fluctuations reflect the mechanical properties of the nucleus.

The novel algorithms compiled in the computational toolbox Trackosome provide a reliable and accurate instrument with subpixel precision to help uncover new elements in the spatiotemporal dynamics of subcellular structures. Importantly, Trackosome is not restricted to centrosomes and the nucleus or cell membrane; its algorithms can be used to track other subcellular structures, providing these structures can be adequately segmented from 3D live imaging data (in its current version, Trackosome can simultaneously track two particles and two membranes sets). Moreover, this toolbox has the potential to be adapted to other experimental conditions such as the study of cell migration and cell polarity, where the capacity to analyze dynamic datasets with high accuracy is highly relevant. Being specifically developed in MATLAB and made publically available as open-source software allows other researchers to easily modify Trackosome and integrate additional functions or metrics to suit their needs.

Cell lines were cultured in Dulbecco's modified Eagle medium (DMEM; Life Technologies) supplemented with 10% fetal bovine serum (FBS; Life Technologies) and grown in a 37°C humidified incubator with 5% CO₂. The HeLa POM121-3×GFP/H2B-mCherry cell line was a kind gift from the laboratory of Katharine Ullman (University of Utah, Salt Lake City, UT, USA). The HeLa cell line expressing histone H2B-GFP/mRFP-α-tubulin was generated in our laboratory using lentiviral vectors. All cell lines were authenticated in the laboratory of origin. All cell lines used in this study are routinely tested for mycoplasma contamination and always found to be negative.

Micropatterns to control individual cell shape and adhesion pattern were produced as previously described (Azioune et al., 2009). Briefly, glass coverslips (22×22 mm No. 1.5; VWR) were activated with plasma (Zepto Plasma System, Diener Electronic) for 1 min and incubated with 0.1 mg/ml of PLL(20)-g[3,5]-PEG(2) (SuSoS) in 10 mM HEPES at pH 7.4, for 1 h at room temperature. After rinsing and air-drying, the coverslips were placed on a synthetic quartz photomask (Delta Mask), previously activated with deep-UV light (PSD-UV, Novascan Technologies) for 5 min. MiliQ water (3 µl) was used to seal each coverslip to the mask. The coverslips were then irradiated through the photomask with the UV lamp for 5 min. Afterwards, coverslips were incubated with 25 μg/ml of fibronectin (Sigma-Aldrich) and 5 μg/ml of Alexa Fluor 546 or 647-conjugated fibrinogen (ThermoFisher Scientific) in 100 mM NaHCO₃ at pH 8.6, for 1 h at room temperature. Cells were seeded at a density of 50,000 cells/coverslip and allowed to spread for 10-15 h before imaging. Non-attached cells were removed by changing the medium 2-5 h after seeding.

For time-lapse microscopy, 12-24 h before the experiments, 5×10⁴ cells were seeded on coverslips coated with FBN (25 μg/ml; F1141, Sigma). Prior to each experiment, cell culture medium was changed from DMEM with 10% FBS to Leibovitz-L15 medium (Life Technologies) supplemented with 10% FBS and Antibiotic-Antimycotic 100× (AAS; Life Technologies). When SiR dyes were used, they were added to the culture medium 30-60 min before acquisition (20 nM SiR-tubulin or 10 nM SiR-DNA; Spirochrome). Where stated, nocodazole was added to the cells 30 min before the experiment (20 nM; Sigma-Aldrich). Live-cell imaging was performed using temperature-controlled Nikon TE2000 microscopes equipped with a modified Yokogawa CSU-X1 spinning-disc head (Yokogawa Electric), an electron multiplying iXon+ DU-897 EM-CCD camera (Andor) and a filter-wheel. Three laser lines were used for excitation at 488, 561 and 647 nm. For nuclear membrane fluctuations, an oil-immersion 100×1.4 NA Plan-Apo DIC (Nikon) was used. All the remaining experiments were done with an oil-immersion 60×1.4 NA Plan-Apo DIC (Nikon). Image acquisition was controlled by NIS Elements AR software. For centrosome tracking, 17-21 z-stacks with a 0.5 µm separation were collected every 20 s or, to analyze centrosome correlations, every 10 s. For NE fluctuation measurements, a single z-stack was collected every 100 ms.

Trackosome is made publically available (open-source) as a platform-independent MATLAB toolbox and can be downloaded from https://github.com/Trackosome.

A 3D Laplacian of Gaussian filter is applied to the video of centrosomes to highlight centrosome-like blobs. The user must then select the approximate position of the two centrosomes in an XY maximum projection of the first frame. These are the seeding points for the tracking loop that follows. For each frame, two 3D regions of interest (ROIs) are centered at the coordinates of the centrosomes obtained for the previous frame (for the first frame, it uses the coordinates selected by the user). The ROIs are large enough to encompass the centrosome displacements between frames. This ensures that the ROIs, centered in the coordinates of the centrosomes at the previous frame, still contain the centrosomes in the current frame. The ROIs are thresholded to isolate regions of high intensity. Then, each ROI is iteratively shortened, one-dimension at a time, until it confines a high-intensity blob with the approximate size of the centrosome. Once each centrosome is enclosed by a 3D mask, the centroids are found by projecting the masked intensities along the three axes and fitting a Gaussian curve to each intensity profile – the coordinates of the centrosome correspond to the peaks of the fitted Gaussian curves (Fig. S3E).

The images from each video (nucleus and cellular membrane) are filtered with a 4D (three spatial dimensions and one temporal dimension) average filter. Then, they are binarized using Otsu's method and submitted to an iterative set of morphological operations intended to create a closed, smooth, binary volume. The membrane of each structure corresponds to the outer pixels of their binary volume. The major axes of the cell and nucleus are obtained by calculating the principal components of the binary volumes for each frame.

The correlations among displacements of the centrosomes were evidenced by defining their movements as two ellipsoidal trajectories surrounding the approximately ellipsoidal nucleus (Fig. S3F-H). The nucleus was centered to ensure a common centroid in each frame. The median nucleus was obtained by calculating the median binary volume across time. The videos were analyzed until NE breakdown. An ellipsoid was fitted to the median nucleus, setting the coordinate system used to define the ellipsoidal trajectories. The trajectories of the centrosomes were normalized with regards to the centroid of the nucleus for each frame. The transformed trajectories were converted to ellipsoidal coordinates, using the fitted ellipsoid as a referential of the coordinate system. The trajectories were also characterized by the norm of the ellipsoidal velocity and acceleration, and by their decomposition along the latitude, longitude and altitude directions at each point. Once all the features of both centrosome trajectories were calculated, the correlation matrix was directly obtained. We only included videos with long and changing centrosome trajectories, because short monotonic trajectories would invariantly have high, but misleading, correlation coefficients. We also excluded videos with a nonstationary nucleus because a moving referential would lead to fallacious correlation values.

We consider that the points of the model-free membrane move in a direction perpendicular to its surface. This is an important novelty, compared with the frequently used methodology that relies on the, much easier to calculate, but flawed, radial direction (see Fig. 3A). We define the fluctuations as being the distance from each point of the median membrane to the membrane at a given frame, along a direction normal to the median membrane. The membranes are segmented for all frames and then centered, assuring a common centroid among frames. To calculate the basal position (the reference membrane) of the NE, we calculate the median projection of the centered frames and segment the resulting membrane. Each point of the reference membrane is associated with a normal vector that defines the direction of the membrane displacements. The fluctuations are then measured by calculating the distance from the reference membrane to the membrane of each frame, along the directions defined by the normal vectors (see Fig. S3I,J).

The authors would like to thank Margarida Dantas for the critical reading of the manuscript and Helder Maiato for providing access to microscopes and equipment. The authors would like to thank Katharine Ullman for the HeLa POM121-3×GFP/H2B-mCherry.

The peer review history is available online at https://jcs.biologists.org/lookup/doi/10.1242/jcs.252254.reviewer-comments.pdf