Modelling cell motility and chemotaxis with evolving surface finite elements

Abstract

We present a mathematical and a computational framework for the modelling of cell motility. The cell membrane is represented by an evolving surface, with the movement of the cell determined by the interaction of various forces that act normal to the surface. We consider external forces such as those that may arise owing to inhomogeneities in the medium and a pressure that constrains the enclosed volume, as well as internal forces that arise from the reaction of the cells' surface to stretching and bending. We also consider a protrusive force associated with a reaction-diffusion system (RDS) posed on the cell membrane, with cell polarization modelled by this surface RDS. The computational method is based on an evolving surface finite-element method. The general method can account for the large deformations that arise in cell motility and allows the simulation of cell migration in three dimensions. We illustrate applications of the proposed modelling framework and numerical method by reporting on numerical simulations of a model for eukaryotic chemotaxis and a model for the persistent movement of keratocytes in two and three space dimensions. Movies of the simulated cells can be obtained from http://homepages.warwick.ac.uk/∼maskae/CV_Warwick/Chemotaxis.html.

Figure 1.

Snapshots of the triangulation during a simulation of cell motility using a quadratic triangulated approximation to a surface. (Online version in colour.)

Figure 2.

Snapshots of the mesh during two simulations of cell motility using a scheme that moves the nodes only in the normal direction and the scheme that includes a tangential velocity. The large deformations result in meshes that are not suitable for computation with the former approach necessitating a remeshing step, while with the latter approach the mesh quality remains good throughout. (a) Movement in normal direction only; (b) tangential redistribution scheme. (Online version in colour.)

Figure 3.

Centroid trajectories of five cells migrating in the absence of a chemoattractant under different geometric evolution laws. In both cases, we see motion in a straight line for short times punctuated by sharp changes in direction corresponding to pseudopod splitting/decay. (a) Spider plots of cell centroid trajectories over the interval [0, 1.], under conserved surface tension evolution; for parameter values see table 1 and the text. (b) Spider plots of cell centroid trajectories over the interval [0, 1.], under a combination of surface tension and elastic evolution; for parameter values see table 1 and the text.

Figure 4.

Centroid trajectories of five cells migrating leftwards in the presence of a linear chemoattractant gradient under conserved surface tension evolution with varying signal strength (ρ).

Figure 5.

Response to a changing chemotactic signal. Initially, there is no signal with arrows indicating the time at which a signal is introduced and the signal direction. Note the two figures are not on the same scale and the cells have the same enclosed volume. (a) Chemotactic motion of a cell under conserved surface tension evolution, for parameter values, see table 1 and text. Cell outlines shown every 0.1 units of computational time over the interval [0, 1.8]. (b) Chemotactic motion of a cell under a combination of surface tension and elastic evolution with volume conservation, for parameter values, see table 1 and text. Cell outlines shown every 0.075 units of computational time over the interval [0, 1.725]. (Online version in colour.)

Figure 6.

Response to a changing chemotactic signal. In this example, the direction of the chemotactic signal is changed by 180°. We observe that the cell turns through 180° and successfully responds to the changing signal, migrating up the new chemotactic gradient. Cell outlines shown every 0.1 units of computational time over the interval [0, 1]. (Online version in colour.)

Figure 7.

Undirected migration (i.e. migration in the absence of a chemoattractant) in the presence of obstacles of a cell under conserved surface tension evolution; for parameter values see table 1 and text. (Online version in colour.)

Figure 8.

Undirected migration in the presence of obstacles of a cell under a combination of conserved surface tension and elastic evolution; for parameter values see table 1 and text. (Online version in colour.)

Figure 9.

Migration in the absence of a chemoattractant of a three-dimensional cell under conserved surface tension evolution; for parameter values see table 4 and text. (Online version in colour.)

Figure 10.

Initial position (at t = 0 right-hand cell) and persistent keratocyte-like migration of cells (at t = 5). The parameter k₂ = 0.6, 0.8, 1.0, 1.2, 1.4, 1.6, 1.8 reading from right to left for the seven polarized (left hand) cells (cf. equation (5.1)) with the remaining parameters given in table 6. (a) Activator (a₁) concentrations; (b) substrate (a₂) concentrations. (Online version in colour.)

Figure 11.

The speed of the cell centroid and aspect ratio both versus time of the cells shown in figure 10. We observe a positive relationship between aspect ratio and speed. (Online version in colour.)

Figure 12.

Asymmetry measures versus time of the cells shown in figure 10. We observe larger deviations from reflection symmetry in the cells travelling at slower speeds. As the cells attain persistent shapes, the values converge to a steady state. (Online version in colour.)

Figure 13.

Initial position (at t = 0 right-hand cell) and persistent keratocyte like migration of cells (at t = 5). The parameter k₂ = 0.6, 0.7, 0.8, 0.9, 1.0 reading from right to left for the five polarized (left hand) cells (cf. equation (5.1)), for the remaining parameter values, see table 8. (a) Activator (a₁) concentrations; (b) substrate (a₂) concentrations. (Online version in colour.)

Figure 14.

The speed of the cell centroid and cell surface area both versus time of the cells shown in figure 13. We observe a positive relationship between surface area and speed. (Online version in colour.)