Phase transition in the collective migration of tissue cells: Experiment and model

We have recorded the swarming-like collective migration of a large number of keratocytes (tissue cells obtained from the scales of goldfish) using long-term videomicroscopy. By increasing the overall density of the migrating cells, we have been able to demonstrate experimentally a kinetic phase transition from a disordered into an ordered state. Near the critical density a complex picture emerges with interacting clusters of cells moving in groups. Motivated by these experiments we have constructed a flocking model that exhibits a continuous transition to the ordered phase, while assuming only short-range interactions and no explicit information about the knowledge of the directions of motion of neighbors. Placing cells in microfabricated arenas we found spectacular whirling behavior which we could also reproduce in simulations.

Figure 1

Phase contrast images showing the typical behavior of cells for three different densities. (a) 1.8, (b) 5.3, (c) 14.7 $\mathrm{cells}∕100\times 100\mathrm{\mu }{\mathrm{m}}^2$. We observed that as cell density increases cell motility undergoes collective ordering. The speed of single cells is higher than that of cells moving in coherent groups. Scale bar $200\mathrm{\mu }\mathrm{m}$. (d)–(f) Velocity of cells. Scale bar $50\mathrm{\mu }\mathrm{m}∕\min$. Reference 17 contains corresponding videos.

Figure 2

Order parameter $\overline{V}$ is shown as a function of normalized cell density. Cell density was normalized with the maximal observed density of $2.5\times {10}^{-3}\mathrm{cells}∕\mathrm{\mu }{\mathrm{m}}^2$ and error bars indicate the standard error of the density and order parameter.

Figure 3

Computer simulations were performed of the system described in the text for different densities. These simulations showed a transition to the ordered phase similar to that seen in experiments. (a) Typical behavior of cells is shown for three different values of the normalized number density $\overline{\rho }=\rho ∕{\rho }_{\max }$, with ${\rho }_{\max }\approx 2$, which is approximately the density where gaps disappear, and the cells reach tight packing in simulations. (b) As cells moving in different directions come into contact adhesive intercellular forces act to align ${\mathbf{n}}_i$ and ${\mathbf{n}}_j$, resulting in an effective averaging of self-driving directions. For videos of simulation runs see Ref. 17.

Figure 4

The average value and standard error of $\overline{V}$ is shown as a function of the normalized number density (a) $\overline{\rho }=\rho ∕{\rho }_{\max }$ and (b) noise $\eta$. Each data point was obtained from at least $10$ independent simulation runs with $N=1000$, with (a) $\eta =0.6$, ${\rho }_{\max }\approx 2$ and (b) $\rho =0.6$. The insets show the dependence of $\ln \overline{V}$ on (a) $[\rho -{\rho }_c(\eta \left)\right]∕{\rho }_c$ and (b) $\left[{\eta }_c\right(\rho )-\eta ]∕{\eta }_c$, the slope of the fitted lines can be associated with the critical exponents $\delta$ and $\beta$. The large scaling regimes and the similarity of the numerical values obtained for the exponents with those found for other models indicate the existence of a continuous phase transition in both $\rho$ and $\eta$.

Figure 5

(Color online) (a) Experimental snapshot of a circulating cell group with the velocities of cells in the corner of a $2\times 2{\mathrm{mm}}^2$ square shaped microfabricated arena. (b) Simulations of model cells confined to a square arena show the emergence of circular motion over a wide range of model parameters. Video showing the circular motion of cells in the whole arena as well as in simulations are available in Ref. 17.