Macroscopic Model of Substrate-Based Cell Motility

Abstract

Eukaryotic cells moving in response to chemical or mechanical stimuli play a fundamental role in tissue growth, wound healing and the immune response. In addition, cell migration is essential for understanding several life-threatening pathologies. At the developmental stage, dysfunction of motility can result in certain disabilities, while in the developed organism it may result in cardiovascular diseases. Motility dysfunction is also involved in cancer growth, especially during metastasis. Beyond such obvious biological and medical relevance, cell motility is also a fascinating example of a self-organized and self-propelled system within the realm of physics.

After a brief overview of the processes involved in cellular motility, the experimental facts, and previous modeling approaches, we survey recent advances in model development that enabled capturing numerous features observed for moving cells. The model is based on the phase field method, which is especially suited to treat the moving and deformable boundaries involved in both individual and collective cell motility. We begin with a didactic introduction to a simple model for the movement of an individual cell and then describe how incrementally complexity can be added: the main components involved in cell motility—the deformability of the cell, the acto-myosin dynamics, the adhesion both to the substrate and to other cells, as well as the substrate deformations—can all be accounted for. Finally we discuss how the model can be generalized to describe the interactions and the collective movement of many, self-organized cells.

Throughout we discuss how the model, at the different stages, can describe a large part of the known features of cell motility: the subcritical onset of motion, the shape diversity of moving cells, and the occurrence of steady motion, stick-slip motion with concomitant shape oscillations, as well as of more complex dynamical modes like bipedal motion. We also investigate how cells navigate on substrates with patterned adhesion properties and modulated stiffness, situations currently under technological development to collect or sort cells. For multiple cells, the model is able to predict that collective cell migration emerges spontaneously as a result of inelastic collision-type interactions of cells. Finally, we discuss possible future extensions of the modeling framework.

Appendix

1.1.1 Numerical Methods

To solve the phase field model numerically, especially in the multiple cell case, we developed a highly parallel algorithm implemented on GPUs using CUDA. To avoid (slow) copying between GPU and CPU memory, the algorithm exclusively runs on the GPU (except for the output of data). The code can handle an arbitrary number of cells in single or double precision. Since the phase field keeps track of the cells’ boundaries, the problem can be solved on a square domain with periodic boundary conditions. In this case, a pseudo-spectral approach based on the fast Fourier transform (FFT) is the most efficient [131]. Pseudo-Spectral Code To simplify the notations, we illustrate the pseudo-spectral algorithm for an example reaction-diffusion system in one spatial dimension. An extension of the algorithm to higher spatial dimensions is straightforward. We assume a finite domain 0 ≤ x < L and a system of two coupled equations

$$ \displaystyle\begin{array}{rcl} \partial _{t}a\left (t,x\right )& =& D_{a}\partial _{x}^{2}a\left (t,x\right ) + f\left (a\left (t,x\right ),b\left (t,x\right )\right ),{}\end{array}$$

(1.46)

$$\displaystyle\begin{array}{rcl} \partial _{t}b\left (t,x\right )& =& D_{b}\partial _{x}^{2}b\left (t,x\right ) + g\left (a\left (t,x\right ),b\left (t,x\right )\right ),{}\end{array}$$

(1.47)

with periodic boundary conditions

$$\displaystyle\begin{array}{rcl} a\left (t,0\right )& =& a\left (t,L\right ),\quad b\left (t,0\right ) = b\left (t,L\right ).{}\end{array}$$

(1.48)

The time domain is discretized with time step Δ t and the spatial domain is discretized with step size $\varDelta x = L/N$, where N is the number of grid points. The discretized fields a _j, k and b _j, k are then defined as

$$\displaystyle\begin{array}{rcl} a_{j,k}& =& a\left (t_{0} + j\varDelta t,k\varDelta x\right ),\quad b_{j,k} = b\left (t_{0} + j\varDelta t,k\varDelta x\right ),{}\end{array}$$

(1.49)

where j enumerates time steps (with t ₀ the initial time) and k enumerates the spatial grid points.

In Fourier space (k-space), the wave vectors are then given by k _n  = n Δ k with $\varDelta k = \frac{2\pi } {L}$, and the complex exponential in the Fourier transform becomes

$$\displaystyle\begin{array}{rcl} \exp \left (i\,k_{n}\,k\varDelta x\right ) =\exp \left (2\pi i\frac{nk} {N} \right ).& &{}\end{array}$$

(1.50)

The FFT of a _j, k with respect to the spatial grid points k is defined as

$$\displaystyle\begin{array}{rcl} \hat{a}_{j,n} = \mathcal{F}\left [a_{j,k}\right ] =\sum _{ k=0}^{N-1}\exp \left (2\pi i\frac{nk} {N} \right )a_{j,k},& &{}\end{array}$$

(1.51)

and the inverse FFT is given by

$$\displaystyle\begin{array}{rcl} a_{j,k} = \mathcal{F}^{-1}\left [\hat{a}_{ j,n}\right ] = \frac{1} {N}\sum _{n=0}^{N-1}\exp \left (-2\pi i\frac{nk} {N} \right )\hat{a}_{j,n}.& &{}\end{array}$$

(1.52)

In spectral methods, it is especially easy to solve the linear part of the equations. In the given example, the discrete equivalent of the second order space derivative $\partial _{x}^{2}a\left (t,x\right )$ is easily computed: first, a _j, k is transformed to k-space by the forward FFT, then multiplied by − k _n ², and finally transformed back to coordinate space [23]

$$\displaystyle\begin{array}{rcl} \partial _{x}^{2}a\left (t,x\right ) \Leftrightarrow \mathcal{F}^{-1}\left [-k_{ n}^{2}\hat{a}_{ j,n}\right ] = -\frac{1} {N}\sum _{n=0}^{N-1}\frac{4\pi ^{2}n^{2}} {L^{2}} \exp \left (-2\pi i\frac{nk} {N} \right )\hat{a}_{j,n}.& &{}\end{array}$$

(1.53)

For the time stepping of the reaction-diffusion system Eqs. (1.46) and (1.47), we employ the operator-split method [34]:

$$\displaystyle\begin{array}{rcl} a_{j+1,k}& =& \mathcal{F}^{-1}\left \{\exp \left (-\varDelta tD_{ a}k_{n}^{2}\right )\left [\mathcal{F}\left (a_{ j,n} +\varDelta tf\left (a_{j,n},b_{j,n}\right )\right )\right ]\right \},{}\end{array}$$

(1.54)

$$\displaystyle\begin{array}{rcl} b_{j+1,k}& =& \mathcal{F}^{-1}\left \{\exp \left (-\varDelta tD_{ b}k_{n}^{2}\right )\left [\mathcal{F}\left (b_{ j,n} +\varDelta tg\left (a_{j,k},b_{j,n}\right )\right )\right ]\right \},{}\end{array}$$

(1.55)

where we used the reversed version of $\exp \left (\varDelta t\alpha \right ) = 1 +\varDelta t\alpha + \mathcal{O}\left (\varDelta t^{2}\right )$ to improve numerical stability [33]. Technically, one calculates an Euler step in real space, transforms to k-space, multiplies by the time evolution operator, and transforms back. The method just described is called pseudo-spectral, since the nonlinear part of the equations (contained in f, g) is evaluated in coordinate space, i.e. not in Fourier space where it would give rise to computationally expensive convolutions.

Combining Scalar to Complex Fields In case of dealing with many equations, the performance of the algorithm can be further improved by exploiting the fact that the FFT of two real fields can be computed more efficiently as a single FFT of a complex field [131]. We hence combine the real fields a _j, n and b _j, n in a single complex field z via

$$\displaystyle\begin{array}{rcl} z_{j,n}& =& a_{j,n} + ib_{j,n}.{}\end{array}$$

(1.56)

Note that the properties of the FFT imply $\hat{z}_{j,n+N} =\hat{ z}_{j,n}$, and especially

$$\displaystyle\begin{array}{rcl} \hat{z}_{j,n}^{{\ast}} =\sum _{ k=0}^{N-1}\exp \left (-2\pi i\frac{nk} {N} \right )\left (a_{j,k} - ib_{j,k}\right ),& &{}\end{array}$$

(1.57)

where z ^∗ denotes the complex conjugate of z. In turn, Eq. (1.57) implies

$$\displaystyle\begin{array}{rcl} \hat{z}_{j,N-n}^{{\ast}} =\sum _{ k=0}^{N-1}\exp \left (2\pi i\frac{nk} {N} \right )\left (a_{j,k} - ib_{j,k}\right ).& &{}\end{array}$$

(1.58)

Consequently, using Eqs. (1.57) and (1.58), we can obtain the FFT of a _j, n as

$$\displaystyle\begin{array}{rcl} \frac{1} {2}\left (\hat{z}_{j,n} +\hat{ z}_{j,N-n}^{{\ast}}\right ) =\sum _{ k=0}^{N-1}\exp \left (2\pi i\frac{nk} {N} \right )a_{j,k} =\hat{ a}_{j,k},& &{}\end{array}$$

(1.59)

and analogously, the FFT of b _j, n as

$$\displaystyle\begin{array}{rcl} \frac{1} {2}\left (\hat{z}_{j,n} -\hat{ z}_{j,N-n}^{{\ast}}\right ) = i\sum _{ k=0}^{N-1}\exp \left (2\pi i\frac{nk} {N} \right )b_{j,k} = i\hat{b}_{j,k}.& &{}\end{array}$$

(1.60)

We combine the equations for a _j+1, k and b _j+1, k , Eqs. (1.54) and (1.55), respectively, into a single equation for $z_{j+1,k} = a_{j+1,k} + ib_{j+1,k}$. Using the properties (1.59) and (1.60), this equation can be entirely expressed in terms of z:

$$\displaystyle\begin{array}{rcl} z_{j+1,k}& =& \mathcal{F}^{-1}\left [c_{ n}\hat{w}_{j,n} + d_{n}\hat{w}_{j,N-n}^{{\ast}}\right ],{}\end{array}$$

(1.61)

where we introduced the abbreviations

$$\displaystyle\begin{array}{rcl} \hat{w}_{j,k}& =& \mathcal{F}\left [z_{j,n} +\varDelta tf\left (\mathrm{Re}\left (z_{j,n}\right ),\mathrm{Im}\left (z_{j,n}\right )\right ) + i\varDelta tg\left (\mathrm{Re}\left (z_{j,n}\right ),\mathrm{Im}\left (z_{j,n}\right )\right )\right ],{}\end{array}$$

(1.62)

and

$$\displaystyle\begin{array}{rcl} c_{n}& =& \frac{1} {2}\left (\exp \left (-\varDelta tD_{a}k_{n}^{2}\right ) +\exp \left (-\varDelta tD_{ b}k_{n}^{2}\right )\right ),{}\end{array}$$

(1.63)

$$\displaystyle\begin{array}{rcl} d_{n}& =& \frac{1} {2}\left (\exp \left (-\varDelta tD_{a}k_{n}^{2}\right ) -\exp \left (-\varDelta tD_{ b}k_{n}^{2}\right )\right ).{}\end{array}$$

(1.64)

Compared to Eqs. (1.54)–(1.55), which require four real FFTs (two forwards and two backwards), having expressed the algorithm in terms of a single complex field, Eq. (1.61), requires only two complex FFTs (one forwards and one backwards).

Additional Details on the Implementation of the Algorithm In case of the single cell model, the phase field ρ and the concentration of the adhesive bonds A, as well as the two components of the actin polarization, p _x and p _y , are combined in two complex fields, respectively. For simulations of multiple cells, additional phase fields are combined as complex fields in pairs. For maximum performance, all arrays except the substrate displacement field u were merged in a single array. Using the CUDA library FFT (CUFFT), this allows to compute all FFTs with a single customized batch FFT. The substrate displacement field u is treated by a separate CUDA kernel, which calculates the spatial derivatives of the traction field in finite difference approximation.

As already discussed in Sect. 1.4.1.1, sums over phase fields, as e.g. $\sum _{j=1}^{N}\rho _{j}^{2}$ and $\sum _{j=1}^{N}\nabla \rho _{j}$, have to be calculated for the cells’ interactions. These, as well as the cells’ volumes and centers of mass are computed with customized CUDA kernels. The center of mass of each cell is determined as described in [14] to track the position of each cell over time, from which its velocity is readily determined in finite difference approximation. Mean velocities as well as order parameters over time are obtained by summing over all cells. The time-averaged mean velocity was determined by a subsequent averaging over all time steps for sufficiently long time intervals to diminish the contribution of initial transients. We typically use a resolution of 512 × 512 up to 2048 × 2048 Fourier modes for a square periodic domain size of $L = \text{100\textendash 200}$. For comparison, the initial cell radii typically used were $r_{0} = \text{10\textendash 15}$.

1.1.2 Derivation of the Equation for the Elastic Displacements

The stress tensor for an isotropic homogeneous incompressible visco-elastic solid (often called Kelvin–Voigt material) is given by [90]

$$\displaystyle\begin{array}{rcl} \sigma _{ik} =\tilde{ G}\left (u_{i,k} + u_{k,i}\right ) +\tilde{\eta } \left (\dot{u}_{i,k} +\dot{ u}_{k,i}\right ) - p\delta _{ik}\,,& &{}\end{array}$$

(1.65)

where $u_{i} = u_{i}\left (x,y,z;t\right ),\,i \in \left \{x,y,z\right \}$ are the components of the displacement field. $p = p\left (x,y,z;t\right )$ is the pressure field ensuring the incompressibility. $\tilde{G}$, $\tilde{\eta }$ are the shear modulus and viscosity, respectively. Assuming overdamped motion ($\ddot{u}_{i} = 0$) the force balance $\nabla \cdot \sigma$, i.e. $\sigma _{ik,k} = 0$, yields

$$\displaystyle\begin{array}{rcl} \tilde{G}\nabla ^{2}\mathbf{u} +\tilde{\eta } \nabla ^{2}\dot{\mathbf{u}}& =& \nabla p\,, \\ \nabla \cdot \mathbf{u}& =& 0\,.{}\end{array}$$

(1.66)

We assume the deformable elastic body to be periodic in the x- and y-direction (with period L). At the lower boundary (z = 0), we assume vanishing displacements, corresponding to the elastic body sticking on a non-deformable surface,

$$\displaystyle\begin{array}{rcl} \mathbf{u}\left (x,y,z = 0,t\right ) = 0\,.& &{}\end{array}$$

(1.67)

At the upper boundary (z = H), the cell exerts a two-dimensional traction force T = (T _x , T _y , 0), but zero normal force on the elastic body

$$\displaystyle\begin{array}{rcl} \sigma _{xz}\left (x,y,z = H,t\right )& =& T_{x}\left (x,y,t\right )\,, \\ \sigma _{yz}\left (x,y,z = H,t\right )& =& T_{y}\left (x,y,t\right )\,, \\ \sigma _{zz}\left (x,y,z = H,t\right )& =& 0\,. {}\end{array}$$

(1.68)

Nonlinearities arising from the free boundary at z = H are neglected. Eqs. (1.66) are equivalent to a biharmonic equation for $\mathbf{w} =\tilde{ G}\mathbf{u} +\tilde{\eta }\dot{ \mathbf{u}}$ and Laplace’s equation for p. After Fourier transforming in x- and y-direction and introducing the wavenumber $k^{2} = k_{x}^{2} + k_{y}^{2}$, these equations become

$$\displaystyle\begin{array}{rcl} \partial _{z}^{4}\mathbf{w} - 2k^{2}\partial _{ z}^{2}\mathbf{w} + k^{4}\mathbf{w}& =& 0\,,{}\end{array}$$

(1.69)

$$\displaystyle\begin{array}{rcl} \partial _{z}^{2}p - k^{2}p& =& 0\,.{}\end{array}$$

(1.70)

Six out of the necessary 14 boundary conditions for Eq. (1.69) are given as before by $\mathbf{w}\left (x,y,z = 0,t\right ) = 0$ and Eq. (1.68). The remaining eight boundary conditions are obtained by evaluating Eq. (1.66) at the boundaries.

The assumption of a vertical substrate layer height H much smaller than its horizontal extension L, H ≪ L, allows a long wavelength expansion (k _x , k _y  ≪ 1∕H) of the solution to Eq. (1.69). We will keep terms up to second order in k _x , k _y , corresponding to retaining spatial derivatives up to second order of the traction force T. Finally, integrating the result over z from z = 0 to z = H leads to Eq. (1.18) given in Sect. 1.2.4.

In case of an inhomogeneous substrate stiffness, $\tilde{G} =\tilde{ G}\left (x,y,z\right )$, one can easily see that the force balance, $\sigma _{ik,k} = 0$, creates a plethora of additional terms in Eq. (1.66), involving all kinds of first order derivatives of $\tilde{G}$. However, we can neglect these terms if the long wavelength expansion is truncated at the lowest order and assume no dependence on the vertical direction, $\tilde{G} =\tilde{ G}\left (x,y\right )$. Hence for simplicity, in all computations involving a space-dependent substrate stiffness G we used Eq. (1.18) in the limit h → 0 [note that the parameters in that equation are related to the ones introduced here via $h = \frac{H^{2}} {12}$, $G = \frac{2\tilde{G}} {\xi H}$ and $\eta = \frac{2\tilde{\eta }} {\xi H}$ with $\xi$ the efficiency of traction force transmission, see Eq. (1.19)] and substituted $G \rightarrow G\left (x,y\right )$.

1.1.3 Characterizing the Cell’s Velocity and Shape

To extract the velocity, the center-of-mass position x ^c is determined according to

$$\displaystyle{ x_{i}^{c} =\int x_{ i}\,\rho (x,y)dxdy,\,\,\,i = 1,2\,. }$$

(1.71)

The aspect ratio was determined via the corresponding 2×2 variance matrix I _ij

$$\displaystyle{ I_{ij} =\int (x_{i} - x_{i}^{c})(x_{ j} - x_{j}^{c})\rho (x,y)dxdy\,. }$$

(1.72)

Its eigenvalues $\lambda _{1,2}$ were calculated and their ratio

$$\displaystyle{ h = \sqrt{\lambda _{1 } /\lambda _{2}} }$$

(1.73)

is a measure for the aspect ratio of the cell. For a cell moving in x-direction (i = 1), the off-diagonal elements vanish, $I_{12} = I_{21} = 0$, and the aspect ratio is simply given by $h = \sqrt{I_{22 } /I_{11}}$. Since a circular shape has aspect ratio h = 1, in Fig. 1.7 we trace h − 1.

The asymmetry of moving cells, namely the deviation from reflection symmetry, can be described via the skewness tensor G _ijk

$$\displaystyle{ G_{ijk} =\int (x_{i} - x_{i}^{c})(x_{ j} - x_{j}^{c})(x_{ k} - x_{k}^{c})\rho (x,y)dxdy\,. }$$

(1.74)

Obviously, for an ellipse G _ijk  = 0. For a cell with an asymmetric shape moving in x-direction (i, j, k = 1), only 4 elements of the skewness tensor are non-zero: G ₁₁₁ ≠ 0, and $G_{122} = G_{212} = G_{221}\neq 0$. Hence one can define the following relative asymmetry measures

$$\displaystyle\begin{array}{rcl} \zeta & =& \frac{\vert G_{111}\vert ^{1/3}} {\sqrt{I_{11 } + I_{22}}}\,,{}\end{array}$$

(1.75)

$$\displaystyle\begin{array}{rcl} \eta & =& \frac{\vert G_{122}\vert ^{1/3}} {\sqrt{I_{11 } + I_{22}}}\,.{}\end{array}$$

(1.76)

For an ellipse, ζ, η = 0; for the asymmetric moving cells obtained by our simulations one has ζ, η ≠ 0, see Fig. 1.7.

1.1.4 Supplementary Movies

• 1. Bistability I (ch1_video1.mpg) http://www.physik.tu-berlin.de/~jakob/movies_small//ch1_video1.mpg. The movie shows a round spreading cell that is perturbed by adding a small polarization pointing to the right. The perturbation is insufficient to set the cell into motion. Cf. also Fig. 1.4.

• 2. Bistability II (ch1_video2.mpg) http://www.physik.tu-berlin.de/~jakob/movies_small//ch1_video2.mpg. The movie shows a round spreading cell that is perturbed by adding a small polarization pointing to the right. This perturbation leads to the onset of motion. Cf. also Fig. 1.4.

• 3. Bipedal motion (ch1_video3.avi) http://www.physik.tu-berlin.de/~jakob/movies_small//ch1_video3.avi. Bipedal motion is an overall straight motion concomitant with out-of-phase oscillations of the lower and upper halves of the cell. Shown is the shape of the cell and the substrate displacement field (Fig. 1.12a–d).

• 4. A cell bounces off a step in adhesion (ch1_video4.avi) http://www.physik.tu-berlin.de/~jakob/movies_small//ch1_video4.avi. The motion of a cell on a substrate where the adhesive strength is modulated by a step in the rate of adhesion formation a ₀, corresponding to a varying density of the adhesive ligands (a ₀ = 0. 2 in the blue region, a ₀ = 0. 01 in the black one; G = 0. 15 everywhere). The cell prefers to stay on the highly adhesive region and is reflected from the low-adhesion region (Fig. 1.13a).

• 5. Motion of cells on striped adhesive patterns I (ch1_video5.avi) http://www.physik.tu-berlin.de/~jakob/movies_small//ch1_video5.avi. The alternating stripes have a value of high (a ₀ = 0. 15, blue) and no adhesiveness (a ₀ = 0, black); G = 0. 2. The cell positions itself symmetrically and moves parallel to the stripes in a steady fashion (Fig. 1.13b top).

• 6. Motion of cells on striped adhesive patterns II (ch1_video6.avi) http://www.physik.tu-berlin.de/~jakob/movies_small//ch1_video6.avi. Overall low adhesiveness (a ₀ = 0. 0015, blue) and no adhesiveness (a ₀ = 0, black); G = 0. 1. After moving initially along the stripes, the cell turns and moves perpendicular to the stripes in a stick-slip fashion (Fig. 1.13b bottom).

• 7. Durotaxis I (ch1_video7.avi) http://www.physik.tu-berlin.de/~jakob/movies_small//ch1_video7.avi. A cell moving in a linear gradient in the substrate’s modulus G

  [varying along the y-direction from G = 0 (black) at the bottom to G = 0. 4 (blue) at the top]. Independently of the initial conditions, the cell approaches a trajectory with an optimal value of G (Fig. 1.14e).

• 8. Durotaxis II (ch1_video8.avi) http://www.physik.tu-berlin.de/~jakob/movies_small//ch1_video8.avi. A cell moving in a linear gradient in the substrate’s modulus G [varying along the y-direction from G = 0 (black) at the bottom to G = 0. 4 (blue) at the top]. Independently of the initial conditions, the cell approaches a trajectory with an optimal value of G (Fig. 1.14e).

• 9. Inelastic collision of cells (ch1_video9.avi) http://www.physik.tu-berlin.de/~jakob/movies_small//ch1_video9.avi. A strongly inelastic collision of two canoe-shaped cells, leading to an effective alignment of the directions of motion (Fig. 1.15a).

• 10. Elastic collision of cells (ch1_video10.avi) http://www.physik.tu-berlin.de/~jakob/movies_small//ch1_video10.avi. An almost elastic collision of two fan-shaped cells (Fig. 1.15b).

• 11. Transition from stationary to moving cells (ch1_video11.avi) http://www.physik.tu-berlin.de/~jakob/movies_small//ch1_video11.avi. Initially, only few cells move, while cells which adhere strongly to the substrate (those with green spots inside) are stationary. Repeated collisions between moving and stationary cells set all cells into motion (Fig. 1.16a–d).

• 12. Transition from moving to stationary cells (ch1_video12.avi) http://www.physik.tu-berlin.de/~jakob/movies_small//ch1_video12.avi. Initially moving cells come to rest and collect in stationary clusters (Fig. 1.16e–h).

• 13. Translational collective migration (ch1_video13.avi) http://www.physik.tu-berlin.de/~jakob/movies_small//ch1_video13.avi. Alignment of propagation directions due to collisions between cells in a domain with periodic boundary conditions. Cells do not adhere to each other (Fig. 1.17a–d).

• 14. Rotational collective migration (ch1_video14.avi) http://www.physik.tu-berlin.de/~jakob/movies_small//ch1_video14.avi. Emergence of rotational collective motion in a circular confined domain (in the red region, the rate of nonlinear adhesive bond formation to the substrate is reduced by a factor of 9) (Fig. 1.17e–h).

• 15. Suppression of rotational collective migration by cell–cell adhesion (ch1_video15.avi) http://www.physik.tu-berlin.de/~jakob/movies_small//ch1_video15.avi. Adhesion between cells prevents the emergence of collective rotational motion (Fig. 1.17j–l).

• 16. Clustering of cells (ch1_video16.avi) http://www.physik.tu-berlin.de/~jakob/movies_small//ch1_video16.avi. Clustering of cells due to strong cell–cell adhesion forces. A few cells are motile and the cluster changes its shape in time. Cells leave and join the cluster (Fig. 1.18a).

• 17. Traveling band of cells (ch1_video17.avi) http://www.physik.tu-berlin.de/~jakob/movies_small//ch1_video17.avi. A band of cells strongly adhering to each other is moving in a domain with periodic boundary conditions (Fig. 1.18b).

• 18. Cells competing for voids (ch1_video18.avi) http://www.physik.tu-berlin.de/~jakob/movies_small//ch1_video18.avi. Confined high-density state without cell–cell adhesion. Cells compete for voids, thereby moving slowly through a “crowded environment” in a random walk fashion (Fig. 1.18d).