Stability analysis of a hybrid cellular automaton model of cell colony growth

Cell colonies of bacteria, tumor cells, and fungi, under nutrient limited growth conditions, exhibit complex branched growth patterns. In order to investigate this phenomenon we present a simple hybrid cellular automaton model of cell colony growth. In the model the growth of the colony is limited by a nutrient that is consumed by the cells and which inhibits cell division if it falls below a certain threshold. Using this model we have investigated how the nutrient consumption rate of the cells affects the growth dynamics of the colony. We found that for low consumption rates the colony takes on an Eden-like morphology, while for higher consumption rates the morphology of the colony is branched with a fractal geometry. These findings are in agreement with previous results, but the simplicity of the model presented here allows for a linear stability analysis of the system. By observing that the local growth of the colony is proportional to the flux of the nutrient we derive an approximate dispersion relation for the growth of the colony interface. This dispersion relation shows that the stability of the growth depends on how far the nutrient penetrates into the colony. For low nutrient consumption rates the penetration distance is large, which stabilizes the growth, while for high consumption rates the penetration distance is small, which leads to unstable branched growth. When the penetration distance vanishes the dispersion relation is reduced to the one describing Laplacian growth without ultra-violet regularization. The dispersion relation was verified by measuring how the average branch width depends on the consumption rate of the cells and shows good agreement between theory and simulations.

Figure 1

Cell colony plots after 300 cell generations for nutrient consumption rates: (a) $k=0.01$, (b) $k=0.1$ and (c) $k=0.2$. The other parameters were fixed at $\tau =10$, $c_n=0.1$ and grid size $N=800$. Empty CA elements are colored white, inactive cells are colored gray, and active cells are colored black. This shows that the colony morphology depends on the nutrient consumption rate, where a high consumption rate gives rise to a fractal colony morphology. The insets in (b) and (c) show a log-log plot of the density-density correlation function, which show that at small length scales $C\left(r\right)\sim r^{-\alpha }$ in the fractal growth regime.

Figure 2

The nutrient profile plotted in the moving $\xi$ frame for $k=0.03$, $c_n=0.1$, and $\tau =10$ in (i) the inactive part, (ii) the active part, and (iii) outside the colony. The circles represent the radial nutrient profile from a simulation and the solid line is the analytic nutrient profile (10).

Figure 3

This figure shows the structure of the interface. It is assumed that the curve $c_{\delta }(x,\xi )=c_n$ is given by displacing the inteface by $-d$ in the $\xi$ direction.

Figure 4

This plot shows the dispersion relation (15) for $k=0.03,0.1,0.2$ and $\tau =10$, $c_n=0.1$. It can be seen that both the fastest growing mode and its growth rate depend on the consumption rate $k$.

Figure 5

The circle around which the average branch width was measured.

Figure 6

The average branch width as a function of the consumption rate $k$. The circles show the average result from simulations and the solid line represents the analytic result. The error bars show the standard deviation of the simulation results.