Flocking transitions in confluent tissues

Abstract

Collective cell migration in dense tissues underlies important biological processes, such as embryonic development, wound healing and cancer invasion. While many aspects of single cell movements are now well established, the mechanisms leading to displacements of cohesive cell groups are still poorly understood. To elucidate the emergence of collective migration in mechanosensitive cells, we examine a self-propelled Voronoi (SPV) model of confluent tissues with an orientational feedback that aligns a cell's polarization with its local migration velocity. While shape and motility are known to regulate a density-independent liquid-solid transition in tissues, we find that aligning interactions facilitate collective motion and promote solidification, with transitions that can be predicted by extending statistical physics tools such as effective temperature to this far-from-equilibrium system. In addition to accounting for recent experimental observations obtained with epithelial monolayers, our model predicts structural and dynamical signatures of flocking, which may serve as gateway to a more quantitative characterization of collective motility.

Fig. 1

Four distinct dynamical phases. Strong alignment interaction yield solid-like (a) and liquid-like (b) flocking states. For weak polar coupling between cells the system is either in a stationary solid (c) or stationary liquid (d) phase. The heat map represents the cosine of the angle of the instantaneous velocity field with respect the horizontal axis: a uniform color indicates coherent migration in a given direction.

Fig. 2

Schematic representation of the model. (a) Each cell is a polygon obtained by the Voronoi tessellation of initially random cell positions r_i, characterized by the area A_i and the perimeter P_i of the polygon. The cell experiences a force F_i = −∇E due to its neighbors and an internal propulsive force f_sⁱ along the direction n_i of its polarization (Eq. 2). (b) An active orientation mechanism reorients each cell’s propulsive force towards its migration velocity over a characteristic response time τ = J⁻¹ (Eq. 3).

Fig. 3

Phase diagram. (a) Different phases in the (p₀,J) plane. The solid/liquid transition line (red circles) is obtained from the vanishing of D_eff and the flocking transition line (green circles) corresponds to the peak in the susceptibility χ_φ. The dashed blue curve is the theoretical prediction J_c(v₀,p₀) given in (4). The black squares (the dashed line is a guide to the eye) are the estimate for J_flock(p₀) in terms of the numerically calculated cage lifetime τ_cage at J = 0. The vertical dashed black line marks the transition to a gas-like state, observed for p₀ ≳ 4.2, where flocking cannot occur. (b) The mean square displacement for J = 2.0 for a range of p₀ ∈ [3.4,4] across the liquid/solid transition (curves from red to violet). (c) The susceptibility χ_φ for p₀ = 3.1 (blue symbols, solid) and p₀ = 3.7 (red symbols, liquid) and sizes N = 100, 400, 1600, 3200, diamonds, squares, circles, and triangles, respectively.

Fig. 4

Identifying the caging timescale in the absence of alignment. (a) The neighbors mean-square separation MSS_nn(t) as a function of the the time increment t becomes constant as p₀ is decreased across the liquid-solid transition, showing the onset of caging. (b) The inverse cage lifetime ${\tau }_{\mathit{\text{cage}}}^{-1}$ at J = 0 as a function of p₀ calculated as described in the text. The vertical line denotes the critical value $p_0^{\ast }$ of the J = 0 rigidity transition, while the horizontal dotted line is the asymptotic value ${\tau }_{\mathit{\text{free}}}^{-1}$ attained by ${\tau }_{\mathit{\text{cage}}}^{-1}$ in the gas phase.

Fig. 5

Structural and dynamical anisotropy of the flocking liquid. (a) Radial distribution function g(r_‖,r_⊥) for a flocking liquid (p₀ = 3.5, J = 2), evaluated along (g(0,r_⊥), red solid line) and perpendicular (g(r_‖,0), blue dotted line) the direction of flocking. The inset shows an intensity plot of the 2D g(r), with the horizontal axis chosen along the flocking direction. (b) Order parameter P (blue squares) as a function of J for p₀ = 3.7. In the non-flocking state P = 0, while at the flocking transition, identified by the peak in the susceptibility χ (orange triangles), P becomes negative, indicating a tendency of the cells to elongate in the direction normal to that of mean motion. Inset: representative snapshot of a flocking liquid (J = 2, p₀ = 3.7); the small arrows indicate the orientation of the principal axis of each cell and the large arrow is in the direction of flocking. (c,d) Maps of the displacements Δr_i averaged over a time τ_α = 10² for (b) J = 0 and (c) J = 2 in a system of 4900 cells. Ellipses are guides to the eye for highlighting the anisotropy of the collective rearrangements in the flocking state. The red arrow indicates the average migration direction. (e,f) Spatial correlations C(x_‖,0) (red triangles) and C(0,x_⊥) (blue circles) along axes longitudinal (x_‖) and perpendicular (x_⊥) to the direction of mean motion of a given sample for J = 0 (e) and J = 2 (f), averaged over 10² samples (see ESI for details).