Pinning-depinning transition in a stochastic growth model for the evolution of cell colony fronts in a disordered medium

We study a stochastic lattice model for cell colony growth, which takes into account proliferation, diffusion, and rotation of cells, in a culture medium with quenched disorder. The medium is composed of sites that inhibit any possible change in the internal state of the cells, representing the disorder, as well as by active medium sites that do not interfere with the cell dynamics. By means of Monte Carlo simulations we find that the velocity of the growing interface, which is taken as the order parameter of the model, strongly depends on the density of active medium sites (${\rho }_A$). In fact, the model presents a (continuous) second-order pinning-depinning transition at a certain critical value of ${\rho }_A^{\mathrm{crit}}$, such as, for ${\rho }_A>{\rho }_A^{\mathrm{crit}}$, the interface moves freely across the disordered medium, but for ${\rho }_A<{\rho }_A^{\mathrm{crit}}$ the interface becomes irreversible pinned by the disorder. By determining the relevant critical exponents, our study reveals that within the depinned phase the interface can be rationalized in terms of the Kardar-Parisi-Zhang universality class, but when approaching the critical threshold, the nonlinear term of the Kardar-Parisi-Zhang equation tends to vanish and then the pinned interface belongs to the quenched Edwards-Wilkinson universality class.

Figure 1

(a) Representation of the proliferation cycle as considered in the model: a cell of type $I_1$ becomes an $I_2$ cell with probability $P_{I_1}$, a cell of type $I_2$ evolves into a double $M$ cell with probability $P_{I_2}$, and a double cell of type $M$ splits out into two $I_1$ cells with probability $P_M$. (b) Snapshot configuration as obtained for ${\rho }_A=0.9$ showing the different cell states: $I_1$ cells (black squares), $I_2$ cells (violet triangles), and $M$ cells (green circles). Blocked medium sites are shown as empty circles. Typical snapshot configurations as obtained for (c) ${\rho }_A=0.9$ showing a interface with KPZ behavior since ${\rho }_A>{\rho }_A^{\mathrm{crit}}$; (e) ${\rho }_A^{\mathrm{crit}}=0.74$, just at the critical point when the interface belongs to QEW universality class; and (g) ${\rho }_A=0.7$, the interface is pinned since ${\rho }_A<{\rho }_A^{\mathrm{crit}}$. Cells are classified according to free cells (turquoise circles) and pinned cells (black triangles), regardless of cell type. Blocked medium sites are shown as empty circles. Typical multi-valuated interfaces for ${\rho }_A=0.9$ (d), ${\rho }_A^{\mathrm{crit}}=0.74$ (f), and ${\rho }_A=0.7$ (h), as obtained from the snapshots of panels (c), (e), and (g), respectively. More details in the text.

Figure 2

(a) Plots of the average position of the growth interface $\langle h\rangle$ versus time ($t$) as obtained for different values of the density of active medium sites ($0.75\le {\rho }_A\le 1$), as indicated. For those values of ${\rho }_A$ we do not observe any pinned interface. (b) Velocity of the growth interface ($v$) as a function of the density of active medium sites (${\rho }_A$). Data for ${\rho }_A\ge 0.75$ obtained from panel (a) by using Eq. (10). Note that the interface becomes pinned at ${\rho }_A^{\mathrm{crit}}\sim 0.74$ (vertical dashed line). (c) Log-log plot of $v$ as a function of $\frac{({\rho }_A-{\rho }_A^{*})}{{\rho }_A^{*}}$ as obtained for different trial values of the critical density of active medium sites ${\rho }_A^{*}$. From the straight line (dashed line) we determine the critical concentration ${\rho }_A^{*}={\rho }_A^{\mathrm{crit}}=0.740\left(2\right)$ and the exponent $\theta =0.21\left(3\right)$. For all figures the results are averaged over a number of $\sim 1000$ different individual realizations for a system size $D=512$. More details in the text.

Figure 3

(a) Log-log plots of the velocity of the growing interface ($v$) as a function of time ($t$) as obtained for different values of the density of active medium sites (${\rho }_A=1,0.8,0.75,0.74,0.7325,0.73,0.7275,0.72,0.71,0.7,0.69,0.68$, and 0.67, from top to down). Data obtained from measurements of the average position of the growth interface $\langle h\rangle$ [according to Eq. (10)] for a system size $D=512$. For ${\rho }_A\ge 0.75$ the interfaces grow at constant speed (after a short transient, i.e., for $t\ge 300$), whereas for ${\rho }_A\le 0.7325$, the interfaces become pinned for a finite time. The critical active medium density is estimated as ${\rho }_A^{\mathrm{crit}}=0.74\left(1\right)$. A broader time window is shown for ${\rho }_A=0.74$ and 0.75 in order to confirm the long-time behavior near criticality. From the best fit of the data for ${\rho }_A^{\mathrm{crit}}=0.74$ we obtain $q=0.17\left(1\right)$, indicated by the solid line. The straight dashed-line indicates $v^{*}=0.01$ used to determine $t^{*}$ (see the inset). Inset: Log-log plot of the cutoff time of the velocity ($t^{*}$) as a function of $\frac{|{\rho }_A-{\rho }_{A^{\mathrm{crit}}}|}{{\rho }_{A^{\mathrm{crit}}}}$. From the best fit of the data we obtain ${\nu }_t=1.32\left(5\right)$. (b) Log-lineal plot of $v$ as a function of time ($t$) as measured within the subcritical phase (${\rho }_A<{\rho }_A^{\mathrm{crit}}$), as indicated. Inset: Log-log plot of the characteristic pinning time (${\xi }_t$) as a function of $\frac{|{\rho }_A-{\rho }_{A^{\mathrm{crit}}}|}{{\rho }_{A^{\mathrm{crit}}}}$. From the best fit of the data we obtain ${\nu }_t=1.36\left(2\right)$. (c) Scaling plots of the data already shown in (a) according to Eqs. (25) and (27) and obtained by taking $q=0.17$ and ${\nu }_t=1.32$. For all figures the results are obtained by starting with flat interfaces and averaging over a number of $\sim 1000$ different individual realizations and by considering all the interfaces (pinned and free). More details in the text.

Figure 4

(a) Log-log plots of $v$ as a function of time ($t$) at the critical active medium density ${\rho }_A^{\mathrm{crit}}=0.74$, as measured for different lattice sizes ($16\le D\le 512$), as indicated. The straight dashed-line indicates $v^{*}=0.01$ used to determine $t^{*}$ (see the inset). Inset: Log-log plot of the cutoff time of the velocity ($t^{*}$) as a function of $1/D$. From the best fit of the data we obtain $z=1.2\left(2\right)$. (b) Scaling plots of the data already shown in (a) according to Eqs. (25) and (5) and obtained by taking $q=0.17$ and $z=1.2$. The results are obtained by starting with flat interfaces and averaging over a number of $\sim 1000$ different individual realizations and by considering all the interfaces (pinned and free). More details in the text.

Figure 5

Pinning probability ($p_{\mathrm{pin}}$) as a function of the density of active medium sites (${\rho }_A$) for different lattice sizes $D$, as indicated. Results are averaged over a number of $\sim 5.000$ different individual realizations, each one until a fixed time $t=20.000$ MCS. Inset: Plot of the density of active medium sites for which $p_{\mathrm{pin}}=0.5 \left[{\rho }_A^{50\%}\left(D\right)\right]$ as a function of $1/D$. From the best fit of the data we obtain ${\rho }_A^{\mathrm{crit}}=0.741\left(3\right)$ and ${\nu }_{\parallel }=0.98\left(2\right)$. More details in the text.

Figure 6

Log-log plots of the interface width ($w$) versus time ($t$) as obtained for different lattice sizes $D$ and ${\rho }_A=0.9$. From the best fit of the data for $D=512$ we obtain the growing exponent $\beta =0.30\left(3\right)$ (dashed line) by considering $t>50$ MCS. Left-hand side inset: Log-log plot of interface width during the saturation regime ($w_{\mathrm{sat}}$) as a function of the lattice size $D$ (data obtained from the main figure for $t\gg t_x$). From the fit of the data (continuous line) we obtain the roughness exponent $\alpha =0.47\left(4\right)$. Results are obtained by starting with flat interfaces and averaging over a number of $\sim 1000$ different individual realizations. Right-hand side inset: Log-log plots of the structure factor $S\left(k\right)$ versus $k$ as obtained during the saturation regime of the interface. Data obtained for different lattice sizes ($D=64,128,256,512$) and ${\rho }_A=0.9$. The dashed line is the best fit of the data with slope $-(2\alpha +1)=-1.89\left(5\right)$ [10], which give us $\alpha =0.45\left(5\right)$. More details in the text.

Figure 7

(a) Log-log plots of the interface width ($w$) versus time ($t$) as obtained for different lattice sizes $D$ at the critical point, e.g., ${\rho }_A^{\mathrm{crit}}=0.74$. From the best fit of the data for $D=512$ (dashed line) we obtain the growing exponent $\beta =0.83\left(3\right)$ by considering $t>50$ MCS. Results obtained by starting with flat interfaces and averaging over a number of $\sim 1000$ different individual realizations and by considering all the interfaces (pinned and free). Inset: Log-log plot of interface width during the saturation regime ($w_{\mathrm{sat}}$) as a function of the lattice size $D$ (data obtained from the main figure for $t\gg t_x$). From the fit of the data (continuous line) we obtain the roughness exponent $\alpha =1.02\left(3\right)$. (b) Log-log plots of the scaled width, $\frac{w}{D^{\alpha }}$, versus $\frac{t}{D^z}$, as it follows from Eq. (7), and applied to the data already shown in Fig. 7. The data collapse is achieved by using the exponents $\alpha =1.02$ and $z=\alpha /\beta =1.23$. More details in the text.

Figure 8

(a) Velocity ($v$) of the growing interface as a function of the squared tilt of the starting interface ($m^2$) for different values of the density of active medium sites (${\rho }_A$), as indicated, and $D=512$. Results obtained averaging over a number of $\sim 1000$ different individual realizations. (b) The nonlinear term of the KPZ equation ($\lambda$) as a function of ${\rho }_A$. Note that $\lambda \to 0$ when ${\rho }_A\to {\rho }_A^{\mathrm{crit}}\sim 0.74$ (vertical dashed line). Data obtained from Fig. 8. Inset: Log-log plot of $\lambda$ versus $\frac{({\rho }_A-{\rho }_{A^{\mathrm{crit}}})}{{\rho }_{A^{\mathrm{crit}}}}$ near the transition point (${\rho }_A\le 0.8$). From the best fit of the data we obtain $\varphi =0.20\left(3\right)$. (c) Scaling plots of the data already shown in (a) near the critical point, that is for ${\rho }_A\le 0.8$, according to Eq. (20), and obtained by using $\theta =0.21$ and $\varphi =0.20$. More details in the text.