Motility-Driven Glass and Jamming Transitions in Biological Tissues

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 model that simultaneously captures polarized cell motility and multibody 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 solidlike state to a fluidlike 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.

Popular Summary

Cell motion inside dense tissues governs many biological processes, including embryonic development and cancer metastasis. Although recent experiments have suggested that these tissues exhibit a jamming transition from a fluidlike to a solidlike state, it remains unclear how cell motility and cell-cell interactions control this transition. Using a new model that captures both of these effects, we demonstrate that the jamming transition is controlled by single-cell speed, the persistence of single-cell motion, and a mechanical parameter that characterizes the competition between cell-cell adhesion and cortical tension. We also identify an experimentally accessible structural order parameter—the cell shape index—that specifies the entire transition surface.

We consider a model for a tissue without gaps between cells, where each cell is centered at a point and cell shapes are given by the Voronoi tessellation of the entire set of points. We vary model parameters and analyze the mean-squared displacements of cell trajectories. We find that in some parameter regimes the cells can move far from their starting positions and the tissue behaves as a fluid, and in other parameter regimes the cells rattle in a “cage” formed by their neighbors and the tissue behaves as a solid. This signature is associated with the so-called “jamming transition” that also occurs in polymers and foams. We discover that the transition occurs at a particular value of the cell shape index; this finding provides biologists and biophysicists with a new, simple way to determine the mechanical properties of a tissue just by imaging cell membranes. This prediction has been verified in tissues cultures from human asthma patients, and our theory makes additional predictions that could help explain aspects of the epithelial-to-mesenchymal transition (i.e., the process that allows cells to escape from a solid mass like a tumor).

We expect that our findings will motivate future studies taking into account additional cellular processes such as frictional forces between cells and cell division.

Figure 1

Analysis of glassy behavior. (a) The mean-squared displacement of cell centers for $D_r=1$ and $v_0=0.1$ and various values of $p_0$ (bottom to top: $p_0=3.5$, 3.65, 3.7, 3.75, 3.8, 3.85) show the onset of dynamical arrest as $p_0$ is changed indicating a glass transition. The dashed lines indicate a slope of 2 (ballistic) and 1 (diffusive) on a log-log plot. (b) The self-intermediate scattering function at the same values of $p_0$ shown in (a) shows the emergence of caging behavior at the glass transition. (c) The effective self-diffusivity as a function of $p_0$ at $v_0=0.1$. At the glass transition $D_{\mathrm{eff}}$ becomes nonzero. (d) The cell displacement map in the SPV model for a fluid state very close to the glass transition ($p_0=3.8$, $v_0=0.1$, and $D_r=1$) over a time window $t={10}^4$ corresponding to the structural relaxation at which $F_s(t)\approx 1/2$.

Figure 2

(a) Glassy phase diagram for confluent tissues as a function of cell motility $v_0$ and target shape index $p_0$ at fixed $D_r=1$. Blue data points correspond to solidlike tissue with vanishing $D_{\mathrm{eff}}$; orange points correspond to flowing tissues (finite $D_{\mathrm{eff}}$). The dynamical glass transition boundary also coincides with the locations in phase space where the structural order parameter $q=⟨P/\sqrt{A}⟩=3.81$ (dashed line). In the solid phase, $q\approx 3.81$, 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. Simulations videos of typical fluid and solid phases are included in Supplemental Materials [42].

Figure 3

(a) The glass transition in $v_0-p_0$ 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 $p_0^{*}=3.81$, regardless of $D_r$, in the limit $v_0\to 0$. (b) The glass transition in $p_0-D_r$ phase space shifts as a function of $v_0$ (from top to bottom: $v_0=0.02$, 0.08, 0.14, 0.2, 0.26). For large $v_0$ there is a crossover in the behavior at $D_r\sim \mu K_A{A}_0=1$, as discussed in the main text. (c) The phase boundary between solid and fluid as a function of motility $v_0$, persistence $1/D_r$, and $p_0$, which is tuned by cell-cell adhesion, can be organized into a schematic 3D phase diagram. Red lines on the surface correspond to iso-$v_0$ contours and blue lines correspond to iso-$D_r$ contours.

Figure 4

(a)–(c) Instantaneous cell displacements at $p_0=3.65$ and $v_0=0.5$. They are different from the displacements shown in Fig. 1, which are averaged over the structural relaxation time scale. (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 pp2-s1). (b) At $D_r=0.1$, collective displacements are less pronounced. (c) For $D_r=1$ and larger, the displacements appear disordered and uncorrelated.

Figure 5

Comparison between SPV glass transition and an analytic prediction based on a soft glass rheology (SGR) continuum model. The dashed line corresponds to a SGR prediction with no fit parameter based on previously measured vertex model trap depths [27]. Data points correspond to SPV simulations with $D_r=10^{-3}$ and where we define $T_{\mathrm{eff}}=c{v}_0^2$ with $c=0.1$ as the best-fit normalization parameter. Blue points correspond to simulations that are solidlike, with $D_{\mathrm{eff}}<10^{-3}$, and the boundary of these points defines the observed SPV glass transition line. Inset: $L_2$ 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.

Figure 6

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

Figure 7

Comparison between glass transition boundaries obtained using shape order parameter (red line) and $D_{\mathrm{eff}}$ (blue squares and orange circles).