Computational model for cell migration in three-dimensional matrices - PubMed (original) (raw)

Computational model for cell migration in three-dimensional matrices

Muhammad H Zaman et al. Biophys J. 2005 Aug.

Abstract

Although computational models for cell migration on two-dimensional (2D) substrata have described how various molecular and cellular properties and physiochemical processes are integrated to accomplish cell locomotion, the same issues, along with certain new ones, might contribute differently to a model for migration within three-dimensional (3D) matrices. To address this more complicated situation, we have developed a computational model for cell migration in 3D matrices using a force-based dynamics approach. This model determines an overall locomotion velocity vector, comprising speed and direction, for individual cells based on internally generated forces transmitted into external traction forces and considering a timescale during which multiple attachment and detachment events are integrated. Key parameters characterize cell and matrix properties, including cell/matrix adhesion and mechanical and steric properties of the matrix; critical underlying molecular properties are incorporated explicitly or implicitly. Model predictions agree well with experimental results for the limiting case of migration on 2D substrata as well as with recent experiments in 3D natural tissues and synthetic gels. Certain predicted features such as biphasic behavior of speed with density of matrix ligands for 3D migration are qualitatively similar to their 2D counterparts, but new effects generally absent in 2D systems, such as effects due to matrix sterics and mechanics, are now predicted to arise in many 3D situations. As one particular sample manifestation of these effects, the optimal levels of cell receptor expression and matrix ligand density yielding maximal migration are dependent on matrix mechanical compliance.

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Figures

FIGURE 1

FIGURE 1

Random walk in 3D. The model predicts that the migration of cells in a three-dimensional matrix is similar to a random walk in three dimensions. The raw (displacement) data for a cell moving in three dimensions is shown in the figure.

FIGURE 2

FIGURE 2

Force-velocity curves for cells on a 2D substrate. Results are obtained using a 2D version of the model presented in text, ignoring steric resistance and assuming receptor-ligand interaction on a flat 2D surface. The value for _β_f is varied between 0 and 1 and the magnitude of _F_protrusion is varied between 100 pN and 100 _μ_N (28,29). The value of _n_f = 0.9 × N. This behavior of cell speed as a function of mean detachment force is very similar to results obtained through experiments of cell migration on 2D surfaces (32).

FIGURE 3

FIGURE 3

Cell speed as a function of receptor-ligand adhesivity at the front of the cell. The biphasic behavior suggests that at extremes of adhesivity the cell shows very little motility. The maximum speed is obtained at an intermediate adhesivity and increases with the increase in the degree of asymmetry of the cell. The number of receptors is the same as given in Table 1. The value of _β_f is normalized so that the maximum adhesivity for the parameters listed in Table 1 is 1. The figure is a summation over many individual simulations, and each point represents the averaged behavior in an ensemble of simulations with a given value of _β_f.

FIGURE 4

FIGURE 4

Cell speed as a function of asymmetry of the cell. The speed of the cell decreases as ψ increases. The ligand density used is equal to the one reported in Table 1.

FIGURE 5

FIGURE 5

Cell speed as a function of mean detachment force. A biphasic relationship suggests that no net cell migration is observed when either there is no protrusion, or the very small individual protrusions are balanced by retractions, such that the time-averaged protrusive force is very small, and the cell is unable to move due to very strong bonds at the front and the rear of the cell.

FIGURE 6

FIGURE 6

Cell speed as a function of the number of available ligands. The cell speed varies with the number of available ligands, first increasing with an increase in the total number of available ligands and then decreasing as further increase in the number corresponds to steric hindrance to the cell movement.

FIGURE 7

FIGURE 7

3D plot of cell speed as a function of matrix stiffness, ligand density, and available receptors. (A) The cell speed is plotted simultaneously as a function of matrix stiffness (varying by one order of magnitude in arbitrary units) and number of available receptors (varying by two orders of magnitude). The highest speed occurs at maximum receptors and intermediate stiffness. Lower stiffness values correspond to lower traction and lower speeds, whereas at very high stiffness the traction forces are high and the cells are unable to detach. (B) The cell speed is plotted simultaneously as a function of matrix stiffness and number of available ligands. The maximum speed occurs at intermediate stiffness and intermediate ligand concentration. Higher ligand concentrations correspond to steric resistance, whereas very low ligand concentrations correspond to negligible traction.

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