Fig 1.
Virtual marker flows for two test cases w.r.t different degrees of regularization: strongly regularized (A, D), weakly regularized (B, E), and non-regularized (C, E). In panel (A)–(C), a stretching effect is illustrated as gray area. Furthermore, mapping violations are highlighted as red lines (F).
Fig 2.
Algorithm to compute regularized flows.
The algorithm input consists of three parameters r, σn, and N regarding the Gaussian process regression, a regularization parameter λ and the initial (segmented) contours γ0, …, γK−1. The regularized flow is described by the output variables Γ0, …, ΓK−1 denoting smooth contours evaluated at a finite number of coordinate markers .
Fig 3.
Algorithm to compute curvature, local motion and local dispersion.
The two regularization parameters λglo and λloc are used in the RegFlow routine (see Fig 2) to determine the global (strongly regularized) and local (weakly regularized) flow. The output comprises the smoothed contours Γ0, …, ΓK−1 as well as geometric quantities of interest such as curvature, local motion, and local dispersion.
Fig 4.
Cell trajectory of a persistently moving amoebae.
(A) Fluorescence image with closed string of M = 400 equidistant nodes resulting from the segmentation process; shown is only every fifth segmentation point (blue points). (B) Smooth representation Φk of the cell contour (orange line) obtained by spatial Gaussian regression on the segmentation points. Every fifth cell contour is displayed as dashed gray line. (C) Entire cell track of K = 500 cell contours (only every fifth shown). (D) Global trace of the cell track (gray area) and the trajectory of the center of mass of the contour (solid line, color coded as in panel (C)). The initial contour is shown as dashed black line and the final contour as dashed gray line.
Fig 5.
Impact of regularization on the distribution of virtual markers for (A) no regularization (λ = 0), (B) weak regularisation with λ = 0.1, and (C) strong regularisation with λ = 1000, illustrated on two frames (roughly 20s apart for illustration purpose). Using the strongly regularised so-called coordinate markers as a means to map local characteristics into a kymograph, the lower panel shows the local motion (D) and curvature (E). The local motion is defined by the magnitude of each mapping vector, which are determined based on the weakly regularised marker flow. All panels correspond to the persistently motile cell of Fig 4.
Fig 6.
Comparison of different cell tracks of Dictyostelium discoideum: persistently motile (left), weakly motile (middle) and almost stationary (right). The corresponding kymographs contain information on the local dispersion (left), local motion (middle) and curvature (right). For details see text.
Fig 7.
From the local dispersion kymograph to expanding areas and expansion events.
(A) Local dispersion of a persistently motile cell as in Figs 4 and 6 (left-hand side). (B) Discretised local dispersion with different areas of activity: high (dark red), medium (light red), low (white) expanding activity, and low (white), medium (light blue), high (dark blue) contracting activity. Local maxima of positive local dispersion are depicted as black dots. Areas of medium and high expanding (C) and contracting (D) activity mapped back on the trace of the cell track.
Fig 8.
Expanding and contracting areas with corresponding expansion events.
Illustrative sequence of the contour dynamics for 96s ≤ t ≤ 144s (left), and 316s ≤ t ≤ 350s (right) based on the cell track shown in Fig 7. Features with high and medium expanding activities are shown in dark and light red, respectively. Features with high and medium contracting activities are shown in dark and light blue, respectively. All patterns shown possess a minimal growth time of 3s. The black dots show local maxima of the local dispersion in areas of medium and high expanding activity.
Fig 9.
Statistical analysis of example cell track.
(A) Local dispersion kymograph with thresholds as in Fig 7. (B) Area growth of cell segments which are part of identified expansions (red) and contractions (blue) of high and middle intensity. (C) Number of expanding and contracting areas with high intensity with respect to time. The time with > 2 features are colored gray to highlight the change in activity in the different phases.
Fig 10.
Statistical analysis of example cell track.
Distributions of local dispersion (A), local motion (B) and curvature (C) inside expanding and contracting patterns with high intensity. (D) Circular histogram displaying the angle, where high expanding activity appears along the cell contour. (E) Correlation between area and growth time of identified patterns. (F) Histograms of growth times of expansions and contractions with high intensity.
Fig 11.
Application to other kinds of cell motility.
Displayed are three tracks for embryonic killifish cells (A-C) and keratocytes (D-E), and their corresponding local dispersion kymographs. Protrusive migration can be seen in (A, C), circular waves around the cell border in (B, C, F), and a steady persistent translation of the contour in (D, E).
Fig 12.
Underlying distributions are based on averaging over several cell tracks. (A) Circular histogram displaying the position, where expansions and contractions events appear along the cell contour. (B) Distribution of local motion of expansion and contractions events above a threshold ±1/3. (C) Simulated data obtained from self-excited Poisson point processes (so-called Hawkes processes) on the unit circle. Afterward, we obtained regions of high (red) and low (blue) intensity w.r.t. to a clustering algorithm. (D) Continuous kymograph obtained from a regression model (e.g. Gaussian process regression) based on sampled magnitudes at event locations shown in (C) from the distribution in (B).