Figure 1.
Backward propagating wave and its effect on the physical properties measured.
(A) Backward propagating waves were observed for various cellular physical properties during in vitro wound healing assays. Regardless of the property measured, a spatial pulse-like profile was observed. For example, for a specific time point t (blue) cells near the wound edge do not accelerate, the pulse is maximized for deeper cells and then decreases for cell located farther from the wound edge. With time (t+1, red), the pulse's maximum is propagating farther back from the front. Not only that the pulse response maximum is located farther in response to the wound edge (dt<dt+Δt), but also the wave propagates backward even faster than the actual speed of the wound edge; vedgeΔt+dt<dt+Δt. Note that throughout the text, the waves were recorded in relation to the (advancing) leading edge. (B–E) The physical traits measured and their alternation in response to the peak in the backward propagating wave. t, t+Δt correspond to time points before and after the pulse's peek approaches a cell (a pulse profile is sketched below for illustration). (B) Acceleration, local temporal derivative of speed. Upon acceleration wave, the cells migrate faster. (C) Strain rate, local spatial derivative of speed. This is an implicit measure for cell stretching/deformation. Upon strain rate wave, the cells elongate. (D) Directionality, ratio between the magnitude of the local velocity component toward- and parallel- to the wound edge. Upon directionality wave, the cells migrate with enhanced directionality. (E) Coordination, the fraction of cells that migrate as clusters with coordinated trajectories. Upon coordination wave, more adjacent cells coordinate their trajectories.
Figure 2.
Wave of coordination of DA3 tumor cells.
The wound healing process was divided into three phases [29]: Phase 1 – free front propagation until first contact between cells from the opposing fronts of the wound. Phase 2 – front matching until full closure of the voids. Phase 3 – post wound closure. (A) Visualization of the wave of coordination during the 3 phases of the healing process: cell clusters migrating with coordinated trajectories are overlaid on the initial time frame of each healing phase. Raw zoomed-in images of each healing phase are found below. (B–C) Waves of coordination for control (B) and HGF/SF-treated (C) cells. Histograms of coordination, measured by the fraction of cells that move in coordinated clusters, in relation to the distance from the edge, accumulated over all experiments. For all following analysis only Phase 1 is considered. (D–G) Spatiotemporal wave of coordination. Coordination was measured in high temporal resolution by clustering a grid of short (72.5 minutes or 5 frames long) trajectories using the same clustering algorithm as in (B–C). The average coordination at time (t) and distance (d) from the wound edge is shown in color code for every bin (t,d) in the map. (D) and (F): spatiotemporal maps (kymographs) for control (D) and HGF/SF-treated (F) cells. In both cases a wave of coordination is observed, and is significantly amplified as a response to HGF/SF treatment. (E) and (G): the average coordination for four 100-minute time intervals of the spatiotemporal maps (D) and (F) respectively. Different time is due to variance in initial wound width and healing rate.
Figure 3.
HGF/SF-induced waves of acceleration and stretching.
(A–D) Spatiotemporal maps (kymographs) of acceleration (A and C) and strain rate (B and D), calculated for each agent as explained in the text and in Supporting Text SI4 in Text S1. The x-axis represents the time in minutes and the y-axis represents the distance from the wound edge in microns. Each element (t,d) in the map shows the average acceleration (in A and C) and strain rate (in B and D) measured at time (t) for all the agents of a layer of width Δd = 12.4 µm (10 pixels), located at a distance (d) from the wound edge, over a time interval Δt = 14.5 minutes (1 frame). (E) and (F): the average coordination for four 100-minute time intervals of the spatiotemporal maps in (C) and (D), respectively. These figures illustrate the highly coinciding waves of increasing motility (acceleration) and stretching (strain rate).
Figure 4.
HGF/SF-induced wave of directionality.
(A, C) Spatiotemporal maps (kymographs) of the average directionality of DA3 tumor cells, in response to HGF/SF treatment (C), in comparison with the control results (A). The x-axis represents the time measured in minutes and the y-axis represents the distance from the wound edge in microns (see the text and Figure 2). (B, D) The average directionality for four 100-minute time intervals of the spatiotemporal maps in (A) and (C), respectively. (E–F) HGF/SF enhances persistent migration of DA3 cells. (E) Average persistent migration as function of distance from the wound edge, with and without HGF/SF. In response to HGF/SF, cells migrate with higher persistence. Each treatment plot was composed of 3,000–4,000 distinct trajectories extracted throughout Phase 1. (F) Full distributions of trajectories' persistence, accumulated over all experiments.
Figure 5.
A simplified model to test the hypothesis that strain rate triggers cellular directional response.
(A) A qualitative description by a simple model of the relations between velocity, acceleration, strain rate, and directionality. Simulated particle velocity in the direction of the wound V+(t) (solid purple line), acceleration (dashed line), and linker attachment probability in the direction of the wound ρ(t) (solid yellow line). (B) Strain rate triggers a directional response (model calculation). Calculated ratio of the directional velocity parallel (V∥) to the directional velocity toward The wound (V+) as the acceleration wave propagates, for increasing sharpness of the wave (decreasing σ in Supporting Text S1), corresponding to a wave that is either sharper or with same σ but with larger overall peak acceleration leading to higher final velocity. Both give higher strain-rate and higher directionality according to our proposed relation of directionality on linker occupation (purple curve versus yellow curve as control). (C–D) Experimental results: scatter plot comparison of directionality: V+ vs. V∥. Each dot in the scatter plot represents an element (t,d) in the two Corresponding spatiotemporal maps. The results presented here were accumulated over all available experiments (N = 5 for HGF/SF treated cells, N = 6 for control cells). HGF/SF-treated DA3 cells migrate in an enhanced directional manner (D), compared to control cells (C), similarly to the corresponding theoretical purple and yellow plot in (B). (E) Morphology of single DA3 cells as function of time and distance from the wound. Average cell area (left panel) and eccentricity (elongation, right panel) as function of time. Cells stretch to become larger and more elongated as the acceleration and strain-rate wave traverses the monolayer. When the wave passes to deeper cells, enhanced directionality is lost (B) theoretically, (C) and (D) experimentally), but the cells keep maintaining their elongated morphology. (F) Subjective single cells observations served as another indication for the validity of the experimental and theoretic results. Visualization of manual cell tracking (each cell marker with a different color) show that cells elongate to the direction of the wound edge followed by migration in a directional manner upon arrival of the waves. Time is in the format hh:mm. The corresponding video is freely available at “The Cell: an Image Library”, http://www.cellimagelibrary.org/images/46351.
Figure 6.
Association among the waves for DA3 tumor cells exposed to HGF/SF.
(A, C): The association between the waves of acceleration and directionality; (B, D): the association between the waves of directionality and coordination. (A) Scatter plot comparison of directionality vs. acceleration. Each dot in a scatter plot represents an element (t,d) in the two corresponding spatiotemporal maps, accumulated over all the experimental replicates (N = 5). (B) Similar to (A) but for the coordination vs. directionality. (C) Cross correlation between acceleration and directionality. The graph shows the Pearson correlation for different time shifts computed between the spatiotemporal maps of acceleration and directionality accumulated for all the experiments. (D) Cross correlation between the spatiotemporal maps of directionality and coordination accumulated for all the experiments. Inset: cross correlation between a spatiotemporal map of directionality and coordination demonstrating about 30 minutes time shift for a specific experiment.
Figure 7.
Applying the same analysis and comparing the results to DA3 cells yielded similar directionality (A, C), and coordination (B, D) for the control case and in response to HGF/SF. (E) HGF/SF enhances persistent migration of MDCK cells (left – mean persistence as a function of the distance from the wound edge, right – full distribution of trajectories' persistent migration), accumulated over all experiments (N = 5 for HGF/SF treated cells, N = 5 for control cells). (F–G) Association between the waves, accumulated over all experiments (N = 5 for HGF/SF treated cells, N = 5 for control cells). (F) Acceleration and directionality, optimal delay was 217 minutes. Inset: association when considering the optimal delay. (G) Directionality and coordination. All phenomena observed in DA3 cells were evident also in MDCK cells, but with a lesser extent. We hypothesize that the tighter adhesions between MDCK cells [53] reduces the above mentioned phenomena by limiting efficient acceleration of cells in the wound's direction.
Figure 8.
Overview of the wound healing spatiotemporal dynamics.
(A) Snapshots from a wound healing assay of DA3 cells treated with HGF/SF. The color bands are used to visualize the locations of elevated cellular acceleration and stretching (green), directionality (purple) and their overlay (cyan), and were calculated from the experimental data by segmenting the corresponding kymographs. d1<d2<d3 represent the location of currently accelerating cells demonstrating the wave's backward propagation. The full video is freely available at “The Cell: an Image Library”, http://www.cellimagelibrary.org/images/45355. (B) Analysis of a wound healing assay of DA3 cells treated with HGF/SF. Strain rate, directionality and coordination were recorded as function from the wound edge for 3 time intervals: 0–100, 100–200 and 200–300 minutes. Note that strain rate (green) precedes directionality (blue) and coordination (red). (C) The general schema: acceleration and stretching (morphological deformation) followed by increased directionality and enhanced coordination. (D) Single vs. Group HGF/SF-Induced migration. Sketch of a different phenomenon observed in single vs. collective migration as a response to HGF/SF. Upon treatment, non-confluent cells accelerate, spread and scatter, whereas confluent monolayers accelerate, deform to a more elongated morphology and amplify intercellular coordination. This transition from low- to high-coordination in the single-cell and collective settings is of great interest for future studies.