Two-Stage Structural and Slowing-Down Percolation Transitions in the Densifying Cancer Cell Monolayer

We experimentally demonstrate the two-stage structural and slowing-down percolating transitions, followed by the confluent transition in the densifying cancer cell monolayers from the dilute state, and investigate their impacts on collective cell dynamics. It is found that cells aggregate into clusters at low cell density. With increasing cell number density, the structural percolation through the formation of a large cell cluster percolating through the space precedes the dynamical percolation transition of forming a percolating cluster of slow cell elements. Both percolating transitions exhibit scale-free scaling behaviors of cluster size distributions and fractal structures, similar to those of the universality class of 2D nonequilibrium systems governed by percolation theory. Dynamically, at low cell density, cell aggregation enhances cooperative motion. The structural percolation leads to slower motion, especially with stronger suppression for the high-frequency modes in the turbulent-like velocity power spectra. The following slowing-down percolation associated with the onset of cell crowding in regions occupied by cells further enhances dynamical slowing-down, and suppresses the increasing trend of dynamical heterogeneity and the steepening of the power spectrum of motion, until their reversions after the confluent transition.

Figure 1

(a) Phase contrast images of CC monolayers at $N$ (averaged number density of CCs in the FOV) $=150$, 300, 500, and $700\mathrm{cell}/{\mathrm{mm}}^2$, respectively. Yellow curves indicate CC cluster boundaries. (b) Images showing SS and FS clusters color coded by green and red, respectively, enlarged from the boxed region of (a) at different $N$. White regions are the regions without cells.

Figure 2

(a) Temporal evolution of $N$. (b) Fractions of total areas occupied by CCs ($F_{\mathrm{CC}}$), slow sites ($F_{\mathrm{SS}}$), and fast sites ($F_{\mathrm{FS}}$) in the FOV versus $N$. (c) Water fall plots of $P_{\mathrm{CC}}$, $P_{\mathrm{SS}}$, and $P_{\mathrm{FS}}$, the probability distribution functions of the areas $A$ of single CC, SS, and FS clusters, respectively, obtained from 1 hr images after each $N$. The gray lines show examples of the best power-law fit with scaling exponent $\alpha$. (d) Scaling exponents $\alpha$ of $P_{\mathrm{CC}}$, $P_{\mathrm{SS}}$, and $P_{\mathrm{FS}}$ versus $N$.

Figure 3

(a) Mean displacement over 20 min of single CC elements ($⟨D⟩$), mean area of a single cluster of the fastest 20% elements of each image frame ($⟨A_{20\%}⟩$), mean areas of single CC, FS, and SS clusters ($⟨A_{\mathrm{CC}}⟩$, $⟨A_{\mathrm{FS}}⟩$, and $⟨A_{\mathrm{SS}}⟩$), and the average cell number density $n$ in CC clusters, versus $N$. (b) Frequency power spectra $S(f)$ of Lagrangian velocity $V_x$ along $x$ direction of all CC elements after reaching each $N$. (c) Mean square displacements (MSD) over time interval $\tau$ obtained from Lagrangian trajectories of CC elements. The two gray lines with scaling exponent $=1$ for $\tau >40\min$ are used as references to indicate transition from super diffusion to normal diffusion. The insets of (b) and (c) show the scaling exponents $\beta$ and $\gamma$ of $S(f)$ and MSD($\tau$) versus $N$, respectively.