Local Yield and Compliance in Active Cell Monolayers

The rheology of biological tissue plays an important role in many processes, from organ formation to cancer invasion. Here, we use a multiphase field model of motile cells to simulate active microrheology within a tissue monolayer. When unperturbed, the tissue exhibits a transition between a solidlike state and a fluidlike state tuned by cell motility and deformability—the ratio of the energetic costs of steric cell-cell repulsion and cell-edge tension. When perturbed, solid tissues exhibit local yield-stress behavior, with a threshold force for the onset of motion of a probe particle that vanishes upon approaching the solid-to-liquid transition. This onset of motion is qualitatively different in the low and high deformability regimes. At high deformability, the tissue is amorphous when solid, it responds compliantly to deformations, and the probe transition to motion is smooth. At low deformability, the monolayer is more ordered translationally and stiffer, and the onset of motion appears discontinuous. Our results suggest that cellular or nanoparticle transport in different types of tissues can be fundamentally different and point to ways in which it can be controlled.

Figure 1

A typical velocity-force curve for a hard probe particle in a glassy tissue [$(d,\mathrm{Pe})=(3.0,0.1)$] showing three dynamical regimes [from left to right: caging (brown), stick-slip (green), and moving freely (purple)]. Upper inset: the relative strength of the velocity fluctuations decreases above the critical force. Lower inset: a snapshot of the simulation setup.

Figure 2

Probe velocity $v_b$ versus applied force $F$ from varying cell motility and deformability. (a) Velocity-force curves at high deformability ($d=3.0$) versus Pe. Inset: the threshold force $F_c$ versus Pe. Circles denote a glassy state with a subdiffusive MSD, whereas squares indicate a liquid system with a diffusive MSD. (b) Velocity-force curves at $\mathrm{Pe}=0.1$ for various deformability ($d=0.3$, 0.75, 1.5, 3.0, and 6.0). Inset: $F_c$ versus $d$. All points correspond to the tissue being in a glassy state. (c)–(f) Normalized histograms of $v_b$ for $d=3.0$ and 0.3 for forces $F$ just below and above $F_c$. The values above $F_c$ are chosen as close as possible to give comparable values of $F/F_c$.

Figure 3

Quantifying cell deformation around the probe. (a) Schematics explaining the radial compression field $\mathcal{C}(\mathit{r})$, computed based on the alignment of the cell compression axis ${\widehat{\mathit{n}}}_i^c$ with the vector ${\widehat{\mathit{u}}}_i$ pointing from the center of the probe (blue) to that of the cell (yellow). (b) Heat map of $\mathcal{C}(\mathit{r})$ for $d=3.0$, $\mathrm{Pe}=0.1$, and $F=0.93F_c$, showing the axis of cell elongation ${\widehat{\mathit{n}}}_i^e$. (c) Profiles of $\mathcal{C}(\mathit{r})$ along the probe’s center in the $x$ direction, i.e., $⟨\mathcal{C}(x,y=y_b)⟩$, for $d=0.3$ and 3.0 with $\mathrm{Pe}=0.1$. (d) Absolute values of the minima and maxima of $⟨\mathcal{C}(x,y=y_b)⟩$ versus $F$ for parameters as in (c).

Figure 4

Reduction in the number of disclinations $N_d$ within the system over time at different deformability $d$. (a) Example time series of $N_d$ for $d=0.3$ and 3.0. Snapshots show the Delaunay triangulation of the system at $t{D}_r=0$ and 200. Fivefold and sevenfold disclinations are shown as blue and red circles, respectively, and the probe particle marked in gray. (b) and (c) Time series showing the percentage change in $N_d$ over time at various $F$ for (b) $d=0.3$ and (c) $d=3.0$.