Mechanism underlying dynamic scaling properties observed in the contour of spreading epithelial monolayer

We found evidence of dynamic scaling in the spreading of Madin-Darby canine kidney (MDCK) cell monolayer, which can be characterized by the Hurst exponent $\alpha =0.86$ and the growth exponent $\beta =0.73$, and theoretically and experimentally clarified the mechanism that governs the contour shape dynamics. Dynamic scaling refers to the roughness of the surface scales, both spatially and temporally. During the spreading of the monolayer, it is known that so-called leader cells generate the driving force and lead the other cells. Our time-lapse observations of cell behavior showed that these leader cells appeared at the early stage of the spreading and formed the monolayer protrusion. Informed by these observations, we developed a simple mathematical model that included differences in cell motility, cell-cell adhesion, and random cell movement. The model reproduced the quantitative characteristics obtained from the experiment, such as the spreading speed, the distribution of the increment, and the dynamic scaling law. Analysis of the model equation shows that the model can reproduce different scaling laws from $(\alpha =0.5,\beta =0.25)$ to $(\alpha =0.9,\beta =0.75)$, where the exponents $\alpha$ and $\beta$ are determined by two dimensionless quantities determined by the microscopic cell behavior. From the analytical result, parameter estimation from the experimental results was achieved. The monolayer on the collagen-coated dishes showed a different scaling law, $\alpha =0.74,\beta =0.68$, suggesting that cell motility increased ninefold. This result was consistent with the assay of the single-cell motility. Our study demonstrated that the dynamics of the contour of the monolayer were explained by the simple model, and we propose a mechanism that exhibits the dynamic scaling property.

Figure 1

Overview of the experimental method and measurement procedures. (a) Observation of MDCK migration. MDCK cells were seeded in the region confined by the PDMS sheet. After cells reached confluence, the PDMS sheet was removed and time-lapse observations were performed. (b) Measurement of the contour shape. The MDCK monolayer was visualized by CellTracker green, and the microscopy images were transformed into binarized images. The distances $D\left(\theta \right)$ from the center to the edge of the monolayer were measured. The orientation of the points along the contour was defined as $\theta (0<\theta <2\pi )$. The angle $\varphi =l/\langle D\rangle \left(t\right)$ was defined from the unit of measurement $l$. $\mathrm{Scale}\mathrm{bar}=1000\mu \text{m}$.

Figure 2

Dynamic scaling law in the spreading of MDCK monolayer, obtained from experimental observations. (a) Contour of the MDCK monolayer, as it evolved from the initial smooth shape to the rough shape. (b) Time evolution of the average values of $D\left(\theta \right)$. Dots indicate the measured values, with the fitted line shown as dashed ($n=6$). (c) Time evolution of $w(l_{\text{max}},t)$, the standard deviation of $D\left(\theta \right)$ ($n=6$). (d) Measurement of the Hurst exponent. The plot shows the experimental results at different time points, and the black dashed line is the fitted line for $2\mathrm{\Delta }x\le l\le 10\mathrm{\Delta }x\left(\mu \text{m}\right)$ at 18 h. From this, the Hurst exponent was calculated as $\alpha =0.858$. (e) Log-log plot of $w(l_{\text{max}},t)$ and $t$. Dots represent the measured values, and the black dashed line is the fitted line ($n=6$). The growth exponent was calculated as $\beta =0.725$.

Figure 3

Behavior of the cells at the edge of the monolayer. (a) Cell tracking at the monolayer edge. Nuclei were visualized with Hoechst33342 (left panels, blue signals). Cells that remained near the edge from $t=1$ to $t=18$ are indicated by red circles. Those cells that dropped and intercalated from the edge by $t=18$ are indicated by green and yellow circles, respectively. Among 45 cells initially located at the edge, 19 cells dropped, while 11 cells divided (right panel). Those 11 cells newly intercalated to the edge, and therefore 77% of the edge cells at $t=18$ are originated from the cells at the edge at $t=1$. Scale bar = $100\mu \text{m}$. (b) Snapshot of the edge at $t=2$. Large cells that stretch their lamellipodia were observed (indicated with yellow arrows). The upper panel is a phase-contrast image, and the lower panel is a fluorescent image; the nuclei and F-actin are represented in blue and red, respectively. Scale bar = $20\mu \text{m}$. (c) Behavior of leader cells during collective cell migration. Circles indicate leader cells, which were distinguished by their locations and large sizes. Lines in the middle panel show the trajectory of the cells from $t=1$ to $t=10$ and in the bottom panel show that from $t=10$ to $t=18$. Scale bar = $200\mu \text{m}$.

Figure 4

Numerical simulation of the monolayer spreading. (a) Diagram of the mathematical model. Circles represent the cells at the edge, with the dark circles representing leader cells, and the light circles follower cells. The coordinates of the cell center are denoted by ${\mathbf{r}}_{\mathit{i}}$. Cells at the edge have different types of motility ${\mathbf{M}}_{\mathit{i}}$, and interact with neighboring cells through cell-cell adhesion ${\mathbf{T}}_{\mathit{i}}$. [(b)–(e)] Results from the numerical model corresponding to Figs. 2 to 2. (b) Time evolution of the average of the distance $D\left(\theta \right)$. (c) Time evolution of the standard deviation of $D\left(\theta \right)$ (global roughness). (d) Log-log plot of the local roughness $w(l,t)$ against $l$. (e) Log-log plot of the global roughness $w(l_{\text{max}},t)$ against $t$. Parameters: $v_l=94\left[\mu \text{m}{\text{h}}^{-1}\right], v_f=2\left[\mu \text{m}{\text{h}}^{-1}\right], p_l=0.1, \rho =6.67\left[{\text{h}}^{-1}\right]$, and $\sigma =3\left[\mu \text{m}{\text{h}}^{-1/2}\right]$.

Figure 5

Parameter dependency of the Hurst and growth exponents. (a) Comparison of numerically and analytically derived $w^2(l,t)$. Log-log plots of $w^2(l,t)$ against $l$ at $t=18$ (left panel) and of $w^2(l_{\text{max}},t)$ against $t$ (right panel). The light-gray lines indicate the numerically obtained values with different sample paths, dots indicate the average of $w(l,t)$ for different paths, and the black line indicates analytically obtained values. (b) Numerically obtained parameter dependency of the Hurst exponent $\alpha$ on $c$ and $\rho t$. (c) The parameter dependency of the growth exponent $\beta$ on $\rho ct$ and $\rho t$. Dots indicate the numerically obtained growth exponents, and the line indicates the plot of (15).

Figure 6

The experimental result with $100\mathrm{mg}/\mathrm{L}$ type I collagen-coated dish is also reproduced by the model (3). (a) Experimental results. Time evolution of averaged $D\left(\theta \right)$ (left panel), log-log plot of the local roughness $w(l,t)$ and $l$ (middle panel), and log-log plot of $w(l_{\text{max}},t)$ and $t$ (right panel). Dots indicate the experimentally obtained data and the dashed line is the fitted line for the data. (b) Numerical results corresponding to panels (a). Parameters: $v_l=126\left[\mu \text{m}{\text{h}}^{-1}\right], v_f=17.87\left[\mu \text{m}{\text{h}}^{-1}\right], p_l=0.1, \rho =6.67\left[{\text{h}}^{-1}\right]$, and $\sigma =28.8\left[\mu \text{m}{\text{h}}^{-1/2}\right]$. (c) Single-cell movement assay. The trajectories of the cell nuclei are indicated by the colored lines in the uncoated glass dish (left panel) and the collagen-coated dish (middle panel). Right panel shows the velocity of the cells; $6.36\mu \mathrm{m}/\mathrm{h}$ in the uncoated glass dish and $30.1\mu \mathrm{m}/\mathrm{h}$ in the collagen-coated dish. *** $p$-value = $3.68\times {10}^{-9}$; scale bar = $100\mu \text{m}$.