Physical model of the dynamic instability in an expanding cell culture - PubMed (original) (raw)
Physical model of the dynamic instability in an expanding cell culture
Shirley Mark et al. Biophys J. 2010.
Abstract
Collective cell migration is of great significance in many biological processes. The goal of this work is to give a physical model for the dynamics of cell migration during the wound healing response. Experiments demonstrate that an initially uniform cell-culture monolayer expands in a nonuniform manner, developing fingerlike shapes. These fingerlike shapes of the cell culture front are composed of columns of cells that move collectively. We propose a physical model to explain this phenomenon, based on the notion of dynamic instability. In this model, we treat the first layers of cells at the front of the moving cell culture as a continuous one-dimensional membrane (contour), with the usual elasticity of a membrane: curvature and surface-tension. This membrane is active, due to the forces of cellular motility of the cells, and we propose that this motility is related to the local curvature of the culture interface; larger convex curvature correlates with a stronger cellular motility force. This shape-force relation gives rise to a dynamic instability, which we then compare to the patterns observed in the wound healing experiments.
Copyright (c) 2010 Biophysical Society. Published by Elsevier Inc. All rights reserved.
Figures
Figure 1
(a) Schematic representation of the mapping of the expanding rectangular culture (left), of initial width w, where the local cell motility is indicated by the shaded arrows. The leader cells are indicated by shading at the tips of the columns of cells (i.e., fingers) (11). This system is mapped in the model to a dynamic contour in the x,y plane, where the local velocity (solid arrows) depends on the local shape (curvature). The height of the fingers h is given by the maximal y coordinate. (b) The force-curvature rule that was used to describe the shape-dependence of the cell motility, as given in Eq. 4. The linear approximation used in the linear-stability analysis is indicated by the dashed line (Eq. 8). (c) A typical dispersion curve ω(q) for the linearized model, indicating the unstable mode q < _q_c and the most unstable model _q_max.
Figure 2
(a) Deterministic evolution (ν = 0) of the contour from an initial Gaussian perturbation of height 1 _μ_m, at different times (each contour is 5 h apart), using β = 0.1. (b) Using the same parameters as in panel a but with random cellular noise ν, at times 5–40 h (each contour is 5 h apart). (c) Roughening cycles of the contour from simulation in panel b. (d) Fingers split at the same range of heights, for both tensions β = 0.1, 0.2 (solid and shaded lines, respectively). The different lines give different realizations of the random noise. (Inset) Absolute location of the tip-splitting for β = 0.1, 0.2 (solid and dashed lines, respectively).
Figure 3
Qualitative comparison between the calculated evolution of the fingers (right) and the experiment (left). Lengths are in _μ_m, and the times are in hours. The simulations were done using β = 0.1.
Figure 4
(a) Evolution of the increase in the mean width of the expanding cell culture. Measurements for a rectangular culture of initial width 200 _μ_m are given in circles. Solid lines are the results of our simulations for different realizations of the cellular noise, and different surface tension: β = 0.05, 0.1, 0.2 (dashed, shaded, and solid lines, respectively). (b) A log-log plot of the data in panel a. The solid lines are two simulations, and the straight lines are fits to a power-law behavior, with slopes of 1.7.
Figure 5
(a) Calculated average cell density ρ as a function of time, for a 200-_μ_m and 100-_μ_m initial culture-width (solid and shaded lines), compared to the observed data (squares). The case of no proliferation is given by the dashed line. (b) The corresponding growth of the mean culture-width, corresponding to the cases shown in panel a.
Figure 6
(a) Evolution of the position of fingertips; points, experimental observations of leader cells (3), lines, using our model for different tensions (β = 0.05, 0.1,and 0.2, shaded, dashed, and solid lines, respectively). (b and c) Evolution of the tip velocity and curvature for the finger shown in the dashed line in panel a. Note that the initial time t = 0 is shifted in panel a to compare to the experimental data, which begins only when the finger is distinguishable for the first time.
Figure 7
(a) Calculated expansion of an initially circular culture, in the presence of cellular noise. (b) An initially elliptic culture expands and develops fingers even in the absence of noise. In both plots the time between contours is 5 h, and the lengths are in _μ_m.
References
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