Figure 1.
Crawling trajectories of PMG and MG5 cells.
A) Snapshot images showing the typical shapes of crawling PMG cells (left) and MG5 cell line (right), B) four crawling trajectories (red, PMG, total duration 56 and 25 hrs; blue, MG5, total duration 12 and 18 hrs), C) Log-log plot of mean-squared displacements vs. time interval (n = 8 for each plot), D) Blown up image showing a sequence of small zigzag turns [boxed area in (B)]. The green dots are the centroid positions and the red dots mark a turning event. In (C), the slopes of the red and blue dotted lines are 1 and 2, respectively.
Table 1.
Summary of the characteristic values of zigzag turns.
Figure 2.
Zigzag preference of crawling PMG and MG5 cells.
A) Typical return maps of turning angles (red, PMG; blue, MG5), B) Histograms of the zigzag preference
(n = 8 for each case), C) Auto-correlation functions of
and their corresponding fits to an exponential function, D) and E) Auto-correlation functions of the instantaneous direction of movement for PMG and MG5 cells, respectively (n = 8 for each case). F) Histograms of the two time constants obtained by fitting the curves in (D) and (E) to
.
cos
in (D) and (E) are the ensemble time average over the entire observation duration of the inner-product between the two directional unit vectors separated by
.
Figure 3.
Centroid trajectories of the crawling model cells for different parameter values of .
The other parameter values were fixed as follows:
m/s,
1/s,
/s,
,
1/(s
m),
1/
m,
1/s,
1/
m
,
,
,
m. Each frame is 1000
1000
m
and includes 200000 iteration steps.
Figure 4.
Long-range directional persistence and zigzag turns of the crawling trajectories of a mathematical model cell.
A) Mean square displacements vs. time for (red),
(blue),
(violet), and
(green). The cyan and black dotted lines have a slope of 1 and 2, respectively. B) Close-up view of the green highlighted segment in Fig. 3 (
). Some snapshot images of the crawling cell are superimposed on the trajectory. The red (blue) boundary is the moving front (trailing edge) where
(
). The inset plots the instantaneous local curvature along the centroid trajectory. Local maxima and minima are marked by red dots, which correspond to the turning points (black dots) along the centroid trajectory. C) Return map of the turning angle (
). The zigzag preference
. D) Auto-correlation function of the sequence of turning angles (
). The blue dotted line is an exponential function fit with a decay time constant of 0.705. E) Auto-correlation functions of the instantaneous direction of movement for
(red),
(blue),
(violet), and
(green). F) Two time constants obtained by fitting
cos
to
. The error bars represent the standard deviation based on 10 different trajectories obtained with a different initial condition.
Figure 5.
Various properties of the model cell trajectories:
A) mean zigzag preference factor, B) mean turning angle, C) mean inter-turn time interval, and D) mean inter-turn distance. The error bars indicates the s.d. for 10 different trials with a different initial condition.