Mechanics of Spreading Colonies.

We begin our analysis by studying the spreading dynamics and the mechanical properties of the tissue in situations that closely correspond to the experimental configuration in refs. 11 and 15. Namely, we simulate the growth of colonies in an unbounded geometry on a substrate, initialized either from a single cell or from a group of cells located in a small region of the substrate. Fig. 2 and Movie S1 show snapshots of a colony grown from an initial 50 cells, spreading over the substrate. The colony spreads cohesively and assumes the shape of a disk. Fig. 3A shows the cell density and the velocity field for this simulation. Simulations initialized from a single cell and from 500 cells are presented in Movies S2 and S3, respectively, and show similar qualitative behavior.

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Fig. 2.

Snapshots of a growing colony at different time points, representative of the different regimes of spreading dynamics, with the simulation time and the number of cells indicated. The large tension in the tissue can lead to holes, which form and subsequently close again (Movie S1).

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Fig. 3.

(A) Cell density (color coded) and cell velocity (vectors) in the spreading colony at . (B) Traction force map in the colony at . The vector field represents the local traction forces exerted on the substrate, whereas the color code represents the y component of this force. The traction forces are computed by calculating the vector sum of the friction forces and motility forces. Motility forces align with the velocity field and dominate over substrate friction forces at the time point presented. This leads to tension in the colony. Movie S6 shows the evolution of traction forces in the spreading colony with time.

To probe the mechanical state inside the tissue, we use a similar approach to that used in refs. 11 and 33. Using the traction forces exerted on the substrate, we reconstruct the stress field in the tissue. In Fig. 3B, the traction force map of the colony is shown at a time point corresponding to Fig. 3A. The traction forces felt by the substrate are computed as the vector sum of the motility forces and background friction forces, as both the motility forces and the background friction imposed in our simulation are balanced by the substrate in reality. From Fig. 3B, it is apparent that although the traction force field is highly disordered, velocity–motility coupling leads to global alignment of motility forces with the cell velocity. As a result, the y component of the traction force has predominantly negative values in the upper half of the colony and predominantly positive values in the lower half of the colony. This, in turn, gives rise to tension in the center of the colony.

In general, it is not possible to reconstruct all three components of the stress tensor in two dimensions from the traction force field without making detailed assumptions on the rheology of the tissue. However, the 1D stress map can be determined by averaging the traction forces in the colony in one direction. Fig. 4 shows the temporal evolution of this 1D pressure field in the tissue, as the colony expands. The stress determined in this manner is independent of tissue rheology and can be compared directly with the experimental results obtained by Trepat et al. (11). The corresponding profiles for different initializations are presented in Figs. S1 and S2. Note that by averaging over a finite segment through the tissue in the x direction as in ref. 11, rather than over the entire colony, the tension profile presented in Fig. 4 is enhanced and exhibits large regions of tension even for very large colony sizes, as shown in Fig. S3. This is because parts of the colony close to the edge that are under pressure do not contribute to the stress in the center. However, the integral of the traction forces over the entire y direction is guaranteed to vanish only if the whole colony in the x direction is taken into account.

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Fig. 4.

One-dimensional pressure field in the simulation presented in Fig. 2 as a function of time. Positive values (red) represent pressure and negative values (blue) correspond to tension. The temporal evolution of the stress field is characterized by a phase during which the tissue is under tension due to radial motility forces followed by a final phase where the colony is under pressure.

The result presented in Fig. 4 is representative of the generic picture for the evolution of the stress field within the model tissue. It can be categorized as follows: After a short-lived transient phase where the colony is small and where motility forces are oriented in random directions, the velocity field aligns the motility forces radially and gives rise to tension throughout the colony. This phase is similar to the experimentally observed stress profile reported in ref. 11. As the colony continues to grow, cells in regions close to the edge collectively move at the maximum velocity possible from the motility force. This results in a small velocity gradient close to the edge of the colony and in turn leads to a buildup of pressure in peripheral regions of the tissue, as cells continue to divide. In the subsequent phase, as cell division dominates over expansion, it eventually leads to a pressure-limited, high cell density state in the bulk of the tissue.

The collective motility of cells and the tension it generates can lead to an inversion of the density profile, where cell density becomes lowest in the center of the tissue as recently reported experimentally (34). Very large tension can even lead to the formation of holes in the colony. Whether this is a physiological parameter regime can be told only by experiments. Holes appearing in some published material (15) suggest that this is possible. Movies S4 and S5 present simulations with an increased division rate . In these simulations, holes do not arise in the tissue and the tension profile in the colony is somewhat less pronounced (Fig. S4).

Note that the mechanism of velocity–motility coupling we suggest here does not always give rise to tension within spreading colonies. Rather, we find tension in a parameter regime, where growth pressure is small compared with motility forces. Nevertheless, even for parameters where the entire colony is under pressure, coordinated motility can enhance spreading speed if the cell division rate is low and the friction with the substrate is high.

Determinants of Spreading Speed and Wound Healing Efficiency.

To obtain a better understanding of colony expansion, we analyze the spreading process after perturbing important model parameters. In particular, we are interested in the role of the global coordination of motility forces. In Fig. 5, the square root of the area covered by the colony is plotted as a function of time for different parameters. Whereas small colonies expand exponentially in time, limited by cell division, for large colonies a constant speed of expansion arises from friction with the substrate, which balances the combination of motility and expansion forces in the tissue. For large colonies, cell division is no longer limiting for colony growth, but the maximum spreading speed is the main indicator of spreading efficiency.

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Fig. 5.

Square root of the area covered by the colony as a function of time. Large colonies radially expand at a constant velocity, which leads to a quadratic dependence of the covered area on time. The parameters varied in the subplots with the rest of the parameters identical to the standard tissue, from the fastest to the slowest spreading speed, are given by (A) the degree of coupling between cell velocity and motility given by the ratios , 10.0, 1.0, and 0.1, with fixed; (B) substrate frictions , 1.0, 4.0, and 10.0; and (C) rates of division for cells surpassing the size threshold, , 1.0, 0.2, and 0.1.

Fig. 5A shows the effect of coupling between cell velocity and motility forces, as given by the ratio . Indeed, we find that a positive coupling given by significantly enhances the asymptotic spreading velocity. This can be understood as follows: In the absence of velocity–motility coupling, cellular motility forces point in random directions and contribute to the pressure in the tissue. Cells in the bulk can divide only by pushing layers of cells at the edge over the substrate. The balance of bulk pressure and the substrate friction force determines the spreading velocity. The presence of velocity–motility coupling alleviates this problem. Instead of being passively pushed over the substrate by expansion pressure and hindering growth of the colony, cells close to the tissue edge collectively move toward the edge and thereby create the space required for cell division and reducing pressure in the sheet. On the other hand for , velocity–motility coupling counteracts the expansion of the colony (Fig. 5A).

The effect of substrate friction on spreading is illustrated in Fig. 5B. It is apparent that friction with the substrate plays an important role in the speed of the wound healing process. For fixed motility forces, substrate friction determines the maximum spreading velocity. Finally, in Fig. 5C, the division rate for cells surpassing the size threshold is varied. Increasing this rate leads to faster spreading of small colonies, where expansion is limited by the rate of cell division. On the other hand, the asymptotic spreading velocity for large colonies in the motility-dominated regime remains unchanged.

Analysis of Spreading Dynamics.

As shown in our simulation, velocity–motility coupling can enhance the spreading velocity of monolayered tissues and give rise to tension within much of the expanding colonies. In this section, we address two questions related to these dynamics: First, in which parameter regime does coordination of motility forces provide an advantage for spreading tissues and is this the case for real monolayered epithelial sheets? Second, our simulation shows a transition from large-scale tension to pressure in the bulk of spreading colonies as the colony expands. At what size does this transition take place and at what colony radius do we expect this transition for biological tissues?

For the first of these questions, we study a simple one-dimensional model and compare the spreading speed of a colony spreading purely due to tissue expansion pressure with that of a colony spreading via optimally aligned motility forces. By comparing the respective spreading velocities for large colonies, we obtain a condition that gives the parameter regime in which motility-based spreading is advantageous compared with pressure-based expansion. We first consider purely pressure-based spreading: As a simplification, we assume incompressibility and a dependence of cell division on pressure that can be described by a first-order expansion (30, 35). Hence, the continuity equation is given by , where κ and are expansion parameters. Furthermore, we neglect effects due to the internal viscosity of the tissue, assuming that the dominant mode of dissipation is friction with the substrate, in which case the force balance condition is given by , where ξ is a friction coefficient. With the edges of the tissue located at , the boundary conditions impose a vanishing stress at the tissue edges . For symmetry reasons the velocity vanishes at the origin . The spreading velocity of the colony is given by the cell velocity at the tissue edge . For large colonies , the spreading velocity converges to .

On the other hand, in the case of colony expansion based on an optimal alignment of motility forces in the absence of expansion pressure, the maximum spreading speed is simply given by the maximum velocity of an individual cell. Note that consistent with this assumption, experimentally, the spreading speed of epithelial sheets is comparable to the velocity of single motile cells, μm/h (7).

For spreading based on motility alignment to be advantageous, must be large compared with the maximum pressure-based spreading velocity . For real tissues, many parameters can be inferred only indirectly. To estimate the friction coefficient ξ, we use the force balance condition . From the work of Trepat et al. (11), we know the typical magnitude of traction forces on the substrate, Pa. As a length scale, we use the typical size of such stress patches, μm. Together with the typical cell velocity , this yields a friction coefficient Pa h/μm². Furthermore, we assume a typical cell division rate of one cell division per day based on ref. 11. From these estimates, we obtain the condition Pa on the expansion pressure of the tissue for motility-based spreading to be advantageous. There are no precise experimental values for this expansion pressure available. The analysis of Trepat et al. (11), however, did not detect a pressure contribution in the stress profile of the colony, suggesting that is significantly smaller than the condition derived above. Hence, unless the expansion pressure in epithelial sheets is on the order of several kilopascals, a mechanism for the global alignment of motility forces provides a distinct advantage for spreading and wound healing.

We now turn to the second question regarding the transition from bulk tension to pressure. One of the conclusions of our work is that the spreading regime observed in ref. 11, in which the colony is under tension, is a transient effect. Both for very small and for very large colonies, the tissue is under pressure. The size, below which the colony is under pressure, depends on initial conditions and on the details of the coupling mechanism. On the other hand, the colony size above which the expanding tissue is under pressure arises when exponential cell division overtakes the linear growth in colony diameter. Hence, the motility-based spreading velocity μm/h must be compared with the spreading speed of an exponentially growing colony of diameter d with a uniform cell division rate . Once again, assuming a division rate of about one cell division per day, , we expect the transition from global tension to pressure to take place around a colony diameter given by mm. We conclude from this analysis that even colonies several millimeters in size can be under tension, consistent with the observations of Trepat et al. (11). Applying this argument to the simulation presented in Fig. 2, we measure the effective cell division rate for small colonies, . The asymptotic spreading velocity can be determined from or can alternatively be measured from Fig. 6, . Together this yields a diameter (in simulation units) for the onset of pressure within the colony, consistent with the results of Fig. 4.

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Fig. 6.

Evolution of the edge of spreading tissues at different time points after removal of a hard wall in the y direction that was in place during initialization. Both tissues have identical parameters except that the adhesion force is set to zero in the tissue on the right-hand side, . Whereas the adhesive tissue shows pronounced finger-like protrusions, the tissue without adhesion spreads much more uniformly, consistent with our analysis (Movies S7 and S8). The parameters deviating from the standard tissue are given by , , , , and .

Undulation Instability of the Tissue Edge.

The coupling between cell velocity and motility forces in our simulations can give rise to finger-like protrusions of the tissue edge. To mimic wound healing experiments, the simulation was initialized by growing the tissue in a long narrow compartment with a periodic boundary condition in one direction and a hard wall constraining it in the other direction. After setting the cell velocities to zero and converting the cells into the nonmotile state, the simulation was started by replacing the hard walls with periodic boundary conditions and increasing the compartment size, effectively creating a wound that can be closed by the tissue from both sides. An example of a simulation is shown in Fig. 6 for the model with (Fig. 6A) and without (Fig. 6B) adhesion. For the adhesive case, the straight boundary of the adhesive tissue is clearly unstable, leading to pronounced finger-like protrusions. A linear stability analysis of a continuum version of our model in the absence of cell division shows that a planar tissue interface is unstable in the presence of velocity–motility coupling and demonstrates that the absence of adhesion (which appears as a decrease of surface tension) tends to stabilizes the interface.

Collective Motion in the Bulk.

Experiments often display large-scale velocity correlations in the bulk of spreading epithelial tissues (8, 9). Our model is able to capture these correlations as shown in Fig. 7A where we present snapshots of the velocity field computed using our model for three different densities. These snapshots clearly show large-scale collective motion in the form of swirls. To quantify this motion, we compute the velocity correlation function. Before calculating this correlation function, we subtract the mean velocity vector across the field of view, which is always orders of magnitude smaller than the mean squared velocity, and calculate the autocorrelation function of the difference . Then the velocity correlation function at a distance r is given bywhere brackets denote an average over all directions, making the correlation function depend solely on the scalar distance r. In Fig. 7B we plot as a function of r for four different densities, including the ones used in Fig. 7A. On short length scales, the velocity correlation function decays exponentially with a single characteristic length scale that increases for increasing cell density (Fig. 7B, Inset). The length scales extracted from the fits are 1.2, 1.3, 1.45, and 1.8, for, respectively, densities 1, 1.25, 1.5, and 2 of the reference tissue density. These should be compared with a characteristic cell length of 0.7 for the reference tissue. Thus, our length scales are comparable to the experiments in Angelini et al. (8), which reported decay lengths of ∼3 cell lengths.

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Fig. 7.

(A) Snapshots of the velocity field for low, intermediate, and high confluent densities. The typical size of the swirl patterns increases with cell density. Movie S9 shows this effect. (B) Velocity correlation function for different densities, at a ratio . We set the density at confluence to 1. The correlation length of the velocity correlation function increases with cell density. Left Inset shows a zoom on the region of the minima. Right Inset shows the decay of the velocity correlation function on a logarithmic scale. (C) Average cell speed as a function of density. This velocity decreases significantly with increasing density. (D) Phase diagram of aligned vs. swirling motion for different degrees of velocity–motility coupling and for different cell densities. The line indicates the critical cell density for the phase transition.

As in the experimental results of ref. 8, does not decay monotonically but shows a small negative minimum (Fig. 7B, Inset). This negative value indicates the presence of swirls due to the antiparallel velocity vectors on opposite sides of the swirl pattern and its location determines the length scale of the swirl (8). This length scale increases for increasing density, consistent with the experiments. Moreover, the average speed of cells (plotted in Fig. 7C) decreases consistently for increasing densities, matching the experimental results of ref. 9.

Above a critical density, a phase transition occurs and unconstrained cells start to move together at a uniform velocity, in agreement with the results obtained by Szabo et al. (19), or alternatively organize in a solid-like phase as in ref. 17. To gain a better understanding of this phase transition, we have investigated its dependence on cell density and the degree of coupling between velocity and motility given by . A tissue is considered “uniformly moving” if the order parameter for the alignment of the speed is at least 0.5. As shown in our phase diagram (Fig. 7D), decreasing the degree of coupling or decreasing cell density favors disorganized swirling motion. For higher degrees of coupling , the critical density for the phase transition to uniform motion decreases.