Quantifying stretching and rearrangement in epithelial sheet migration

MCF10A cells were cultured in DMEM/F12 media supplemented with 5% horse serum, 10 μg ml⁻¹ insulin (Invitrogen), 10 ng ml⁻¹ EGF (Peprotech, Rocky Hill, NJ), 0.5 μg ml⁻¹ hydrocortisone and 100 ng ml⁻¹ cholera toxin (both Sigma, St Louis, MO). The cells were kept in a humidified atmosphere at 37 °C and 5% CO₂. For the migration assay, cells were plated in 12 well glass bottomed plates coated with collagen IV (10 μg ml⁻¹ for approximately 3 days at 4 °C) and allowed to culture overnight in DMEM/F12 containing 1% horse serum (unless otherwise noted to be 5% horse serum). Figure 1 provides an illustration of the plating technique. Cells were then imaged for 24 h on an incubated microscope kept at 37 °C and 5% CO₂ (Zeiss Observer.Z1, Zeiss, Goettingen, Germany). Phase contrast images of the cells were acquired at 2 min intervals using a 10 ×  objective.

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PIV analysis is performed using the MatPIV MATLAB toolbox (J Kristian Sveen: http://folk.uio.no/jks/matpiv/, GNU general public license). We use multiple iterations of interrogation window sizes, starting with two iterations of 64 × 64 pixel windows (41.6 μm × 41.6 μm) and finishing with two iterations of 32 × 32 pixel windows (20.8 μm × 20.8 μm). At each interrogation, 50% overlap is used and the windows are cross-correlated. After the final iteration, outliers are detected using a signal-to-noise filter. Vectors with a signal-to-noise ratio less than 1.3 are replaced by linearly interpolated values. In the case of movies involving a leading edge of the cell sheet, the edge of the sheet is segmented and the detected edge is used to trim the PIV field to remove the zero velocity contribution of the empty space from any further metrics.

We track individual nucleoli using an algorithm adapted from hydrodynamic turbulence studies, implemented in Matlab and available online [33]. We begin by calculating each data set's mean image, which we take as the background. Wherever an individual image is sufficiently darker than the background, over a region of appropriate size, the algorithm identifies a dot (possible nucleolus), taking for its position the intensity-weighted centroid of the region. These dots are tracked with a predictive three-frame best-estimate algorithm [34], yielding the location of each dot over a range of frames. Because we are interested in the motion of cells, we then seek groups of dots that remain within a set distance of each other and replace those individuals with their centroid, approximating the motion of a cell's nucleus. We typically choose parameters that identify many dots, since we want to include as many real cells as possible and false positives produce short tracks with little statistical weight. In one example frame we hand-counted 674 dots in 372 cells, whereas our automated algorithm identified 817 dots, reducing that count to 504 after grouping. Automated tracking of dots in phase-contrast images gives us access to the individual motion of many more living cells than would be practical when counting by hand.

For each of the circular monolayers of cells, four fields of view were recorded. In two of the image sequences, the field of view is entirely filled with cells and will be referred to as an image of the 'bulk'. In the other two image sequences, referred to as the 'edge' movies, the field of view includes the leading edge of the cells and the empty space in front of the monolayer. Our initial analysis focuses on the comparison of behaviors between these two sets of movies with no further spatial distinction.

To probe the differences seen between edge and bulk migration behavior in more detail, we perform a segmentation of the leading edge to obtain an outline of the monolayer over time. This segmented edge is then fit to a circle, allowing us to find an effective radius and center of the monolayer as a function of time. Each cell's position is converted into the radial distance from the center of the monolayer. The radial distance of a given cell from the center, r, normalized by the total radius of the monolayer, R, allows us to compare cell motion to location within the sheet.

Working from PIV velocity fields, we calculate FTLEs, as shown in figure 3. FTLEs are calculated by numerically advecting 'virtual' tracers through the measured, time-varying flow field. We initially seed tracers on the PIV grid but subsequently allow them to move off of the grid. We follow the tracer motion for a deformation time of 80 min. Because FTLEs asymptotically approach Lyapunov exponents, our results are not sensitive to the deformation time if it is reasonably long [25]. At the end of the deformation, we calculate the logarithm of the largest eigenvalue of the Cauchy–Green strain tensor for each cluster of four tracers that were initially neighbors; this is the local FTLE value. By measuring one FTLE value for each point on the PIV grid, we obtain the full FTLE field [26].

To quantify rearrangements within the sheet, we look at non-affine motion. Specifically, we measure relative motion at the single cell level—for each cell we look at the relative displacements of its neighbors within a 40 μm radius. We then calculate the quantity $D_{\min ,i}^2$ [28, 29] which quantifies the non-affine deformation of the N neighbors (indexed by j) of cell i:

$$\begin{eqnarray*}&&D_{{\rm min},i}^2 = \min \left\{ {\frac{1}{N}\sum\limits_j {\left[ {\Delta \vec d_{ij} - E_i \vec d_{ij} } \right]^2 } } \right\},\end{eqnarray*}$$

where the vector $\skew3\vec{d}_{ij}$ is the relative initial position of cells i and j, $\Delta \skew3\vec{d}_{ij}$ is their relative displacement and the minimization is over all potential strain tensors E_i. In other words, we calculate an affine deformation field (E_i) that best describes the actual cell motion. If the motion around cell i were perfectly affine, $D_{\min ,i}^2$ would be zero; $D_{\min ,i}^2$ is larger for greater departures from purely affine motion. $D_{\min ,i}^2$ thus gives us the relative strength of non-affine motion, or plastic rearrangements, in the neighborhood of cell i.

Cells were added to collagen-coated coverslips in such a way as to form a roughly circular monolayer of cells (see figure 1 for a schematic of the assay). Time-lapse images were acquired at multiple locations of the dot: some within the bulk, and others at the leading edge. Both PIV and tracking of cell nuclei show significantly higher speeds for cells near the leading edge of the monolayer, as shown in figure 2(a). That the tracking shows a slight increase in mean speed compared to PIV is to be expected from the properties of the methods. PIV gives a smoothed velocity field that will suppress the contribution of isolated fast motion to the overall mean speed. It also measures all motion between two frames, including membrane and cytoplasm fluctuations in addition to the nuclear motion; this randomly directed motion may lead to smaller average speeds. Nuclear tracking, however, follows each cell's nucleus individually and retains information about isolated outliers. The two methods capture different aspects of the system and thus provide slightly different information.

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In the case of PIV, dividing cells and the few visible dead cells are included in the velocity information. In contrast, tracking does not include dead cells and is only able to measure dividing cells while the tracked features (nucleoli) are visible as during the very beginning and end of the division process. In a subset of movies, cell divisions were tracked and showed no correlation with changes in either speed metric (data not shown).

Despite the agreement between all experiments on the overall trend between edge and bulk, the experiments still exhibit a wide variety in speeds from day to day. As shown in figure 2(b), the mean speeds from one experiment are significantly higher. Although all three experiments show a significant increase in that day's edge speed compared to the same day's bulk speed, the bulk cells from experiment C move as quickly as the edge cells from experiments A and B. The variation in speed could result from sensitivity to initial conditions such as small differences in the coating of a microscope slide. The authors' experience with migration assays suggests that small variations in these parameters and variation in speeds are frequently observed. The results shown in figure 2(b) emphasize that it is important to be careful when comparing experiments performed on different days, and suggest the need for additional quantitative metrics when comparing the dynamics of epithelial sheet migration.

Figure 3 shows an example snapshot of FTLE values. We find that the spatial average of FTLEs is negative in all cases, and that it approaches zero as the deformation time T increases. Negative FTLEs imply that the flow is not chaotic, and that neighboring regions tend to separate only slowly as time passes. Slow separation is consistent with the fact that the majority of our observations show bulk cells surrounded by many neighbors, so their relative motion is highly constrained. However, the resolution of our data prevents the FTLE calculations from accounting for length scales smaller than the PIV grid; the flow may be chaotic at these small scales.

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By fixing the deformation time at T = 80 min we can consider FTLE fields in more detail, as shown in figure 4. Distributions of FTLE magnitude show that local values can vary substantially from the mean, occasionally being greater than zero. Positive FTLEs are more common in data sets recorded at the advancing edge of a cell culture than in the bulk, as indicated both by distributions and by snapshots—the edge experiences large stretching. This observation is borne out in FTLE statistics conditioned on the normalized radius r/R (as shown in figure 6(c)).

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Many simple flows are chaotic, regardless of the flow speed [25, 26], so the lack of chaotic motion is not solely due to the relatively slow speed of the migrating sheet. In fact, the chaos measured in the sheet is changed with modifications to the migration environment, as shown by the 5% serum data in figure 4. Serum contains many compounds, including growth factors and other stimulants, which can change the sheet's mode of migration. The change in migration activity is seen in the shift toward more chaotic motion in the bulk FTLE distributions (figure 4(a)) and even more dramatically in the distribution of edge FTLE values (figure 4(b)). Although in all cases the mean FTLE remains negative and thus not chaotic, in the spatial information provided by the FTLE field we find instances of positive values, i.e., chaotic motion. Although the speed of the sheet was moderately increased by the addition of serum (figure 6(b)), the relative quantity of positive values was many fold increased in migration at higher concentrations of serum (figure 4(c)). The more active motion provided by the stimulation at 5% serum is reflected in the increased instances of chaotic motion.

$D_{\min}^2$ is a spatially and temporally heterogeneous quantity: 'hot spots' of plastic rearrangements appear and die out after the rearrangement is complete. Figure 5(a) shows these hot spots in one frame of a bulk movie. We look at the differences in $D_{\min }^2$ between the edge and bulk in figure 5(b). $D_{\min }^2$ is normalized by the mean-squared displacement, so that the magnitude of $D_{\min }^2$ is not dependent on the macroscopic flow speed. Clearly, cells in the bulk undergo more plastic rearrangements than on the edge, and this trend holds for different experiments.

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To investigate this result further, we also look at the probability density of $D_{\min }^2$ values for several different experiments, shown in figure 5(c). $D_{\min }^2$ has units of area, so we have translated the x-axis into the corresponding length. We see two characteristic peaks in the distribution, one near 2 μm and one near 10 μm. The peak at longer lengths grows for the bulk datasets, suggesting that the primary mode of rearrangements changes in the bulk. The length scale associated with this peak is comparable to the cell size, suggesting that these are actual position-switching deformations.

We also stitch together data from bulk and edge to tease out the dependence of $D_{\min }^2$ on distance from the center, as shown in figure 6(d). The basic trend shows that non-affine motion increases into the bulk and eventually plateaus. Since the cell density changes from bulk to edge, we next consider the dependence of $D_{\min }^2$ on density.

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After measuring consistent bulk versus edge trends in our metrics and yet seeing a clear variability between experiments, we look to the original images for variation in factors other than speed. We find apparent differences in the edge densities between experiments, prompting the comparison of our metrics to the local cell density. We calculate local cell density by counting the number of neighboring nuclei contained within a 40 μm radius of the cell nuclear location.

As seen in figure 6(a), there is a clear distinction between the local neighbor count profiles for each experiment. These density differences are accompanied by variation in the radial speed profiles as seen in figure 6(b). Within each day of experiments, four migration assays were performed. The density and speed trends are consistent across all four assays on a single day—speed decreases as density increases. For two of the days, a clear decrease in number density is seen near the leading edge of the migrating cells. This is accompanied by a smooth speed profile: a nearly constant speed in the center that quickly increases to faster speeds at (the lower density) edge. For the third day (experiment C in figure 6), all four migration assays show a density profile that differs from the other two experiments. For these monolayers, a peak in density is found ∼0.1R away from the leading edge. This different density profile is accompanied by a different speed profile: faster overall, and lacking a constant speed in the center, as seen in figure 6(b). This set of experiments also shows a perturbation in stretching (figure 6(c)) and rearrangements (figure 6(d)) near the relative position of the density perturbation.

Intrigued by the differences near this perturbation, we further investigate the relationship between our metrics and the local number density. To avoid including edge effects in our investigation of number density, we restrict our analysis to bulk movies. The number of neighbors within a radius of 40 μm of each cell is used as a measure of the local cell density. Figure 7(a) shows the relationship between cell speed and number density for three characteristic bulk movies. In all cases, although the magnitude of the speed varies from experiment to experiment, speed is a consistent function of density: cells in higher density regions are slower on average.

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Although we investigated the relationship between FTLE values and local cell density, no strong trends were seen. As the FTLE analysis captures long time deformation and stretching, it is not surprising that it does not show a significant relationship to short time parameters such as the local cell density. Further, because the stretching measured by the FTLE values is relative between cells, the dependence of average speed on local cell density does not imply a FTLE dependence on density. After all, even if all the cells are moving faster, they need not be separating more quickly. The FTLE analysis is very similar to the affine motion of the cells [35], which does not have a clear relationship with density.

We also measure the non-affine component of the motion using the parameter $D_{\min }^2$. For each movie, for all frames and all cells, we determine $D_{\min }^2$ as a function of the local density, as shown in figure 7(b). $D_{\min }^2$ is larger for higher densities—in a consistent manner between experiments. This relationship points to the source of the difference in $D_{\min }^2$ for the edge and bulk data. The consistency between experiments is reassuring, but the implications of the trend are very intriguing. Non-affine motion increases as the density increases. This is the opposite trend one would see for a jammed system of soft particles [36]. In such jammed particulate systems, the probability of plastic deformations increases with decreasing density, as more voids are created. In contrast, as cells get denser (more jammed), they rearrange more, suggesting active, individualistic motion. This active motion is a known feature of both healthy and cancerous cells: cells in epithelial sheets actively generate traction forces on the underlying substrate [10–14] and tumor cells are known to actively infiltrate surrounding tissues [9]. Cells are able to take advantage of forces generated by their cytoskeletal structures to actively migrate using structures such as lamellipodia and filopodia [4]; these same protrusive and contractive forces may contribute to cell rearrangements within a migrating sheet.