Motility-driven glass and jamming transitions in biological tissues

Abstract

Cell motion inside dense tissues governs many biological processes, including embryonic development and cancer metastasis, and recent experiments suggest that these tissues exhibit collective glassy behavior. To make quantitative predictions about glass transitions in tissues, we study a self-propelled Voronoi (SPV) model that simultaneously captures polarized cell motility and multi-body cell-cell interactions in a confluent tissue, where there are no gaps between cells. We demonstrate that the model exhibits a jamming transition from a solid-like state to a fluid-like state that is controlled by three parameters: the single-cell motile speed, the persistence time of single-cell tracks, and a target shape index that characterizes the competition between cell-cell adhesion and cortical tension. In contrast to traditional particulate glasses, we are able to identify an experimentally accessible structural order parameter that specifies the entire jamming surface as a function of model parameters. We demonstrate that a continuum Soft Glassy Rheology model precisely captures this transition in the limit of small persistence times, and explain how it fails in the limit of large persistence times. These results provide a framework for understanding the collective solid-to-liquid transitions that have been observed in embryonic development and cancer progression, which may be associated with Epithelial-to-Mesenchymal transition in these tissues.

FIG. 1

Analysis of glassy behavior. (A) The mean-squared displacement of cell centers for D_r = 1 and v₀ = 0.1 and various values of p₀ (bottom to top: p₀ = 3.5, 3.65, 3.7, 3.75, 3.8, 3.85) show the onset of dynamical arrest as p₀ is changed indicating a glass transition. The dashed lines indicate a slope of 2(ballistic) and 1(diffusive) on log-log plot. (B) The self-intermediate scattering function at the same values of p₀ shown in (A) shows the emergence of caging behavior at the glass transition. (C) The effective self-diffusivity as function of p₀ at v₀ = 0.1. At the glass transition D_eff becomes nonzero. (D)The cell displacement map in SPV model for a fluid state very close to the glass transition (p₀ = 3.8, v₀ = 0.1 and D_r = 1) over a time window t = 10⁴ corresponding to the structural relaxation at which F_s(t) ≈ 1 /2.

FIG. 2

(A) Glassy phase diagram for confluent tissues as function of cell motility v₀ and target shape index p₀ at fixed D_r = 1. Blue data points correspond to solid-like tissue with vanishing D_eff ; orange points correspond to flowing tissues (finite D_eff). The dynamical glass transition boundary also coincides with the locations in phase space where the structural order parameter $q=\langle p/\sqrt{a}\rangle =3.81$ (dashed line). In the solid phase, q ≈ 3.81 and q > 3.81 in the fluid phase. (B) Instantaneous tissue snapshots show the difference in cell shape across the transition. Cell tracks also show dynamical arrest due to caging in the solid phase and diffusion in the fluid phase.

FIG. 3

(A) The glass transition in v₀ – p₀ phase space shifts as the persistence time changes. Lines represent the glass transition identified by the structural order parameter q = 3.81. The phase boundary collapse to a single point at $p_0^{\ast }=3.81$, regardless of D_r, in the limit v₀ → 0. (B) The glass transition in p₀ – D_r phase space shifts as a function of v₀ (from top to bottom: v₀ = 0.02, 0.08, 0.14, 0.2, 0.26) For large v₀ there is a crossover in the behavior at D_r ~ μK_AA₀ = 1, as discussed in the main text. (C) The phase boundary between solid and fluid as function of motility v₀, persistence 1/D_r and p₀ which is tuned by cell-cell adhesion can be organized into a schematic 3D phase diagram. Red lines on the surface correspond to iso-v₀ contours and blue lines correspond to iso-D_r contours.

FIG. 4

(A–C) Instantaneous cell displacements at p₀ = 3.65 and v₀ = 0.5. They are different from the displacements shown in Fig. 1(D) which are averaged over the structural relaxation timescale. (A) At the lowest value of D_r = 0.01, the cells are able to flow by collectively displacing along the ‘soft’ modes of the system (Appendix. B 1). (B) At D_r = 0.1, collective displacements are less pronounced. (C) For D_r = 1 and larger, the displacements appear disordered and uncorrelated.

FIG. 5

Comparison between SPV glass transition and an analytic prediction based on a Soft Glass Rheology (SGR) continuum model. The dashed line corresponds to an SGR prediction with no fit parameter based on previously measured vertex model trap depths [8]. Data points correspond to SPV simulations with D_r = 10⁻³ and where we have defined $T_{\mathit{eff}}=c{v}_0^2$ with c = 0.1 as the best-fit normalization parameter. Blue points correspond to simulations which are solid-like, with D_eff < 10⁻³, and the boundary of these points define the observed SPV glass transition line. (Inset) L₂ difference between SPV glass transition line (at best-fit value of c) and the predicted SGR transition line at various values of D_r. The SGR prediction based on localized T1 trap depths works well in the high D_r limit, but not in the low D_r limit.

FIG. 6

Cell centers positions are specified by vectors $\left\{\overrightarrow{r}\right\}$. They form a Delaunay triangulation (black lines). Its dual is the Voronoi tessellation (red lines), with vertices given by $\{\overrightarrow{h}\}$.

FIG. 7

Comparison between glass transition boundaries obtained using shape order parameter (red line) and D_eff (blue squares and orange circles).