Reproduction-time statistics and segregation patterns in growing populations

Pattern formation in microbial colonies of competing strains under purely space-limited population growth has recently attracted considerable research interest. We show that the reproduction time statistics of individuals has a significant impact on the sectoring patterns. Generalizing the standard Eden growth model, we introduce a simple one-parameter family of reproduction time distributions indexed by the variation coefficient $\delta \in [0,1]$, which includes deterministic ($\delta =0$) and memory-less exponential distribution ($\delta =1$) as extreme cases. We present convincing numerical evidence and heuristic arguments that the generalized model is still in the Kardar-Parisi-Zhang (KPZ) universality class, and the changes in patterns are due to changing prefactors in the scaling relations, which we are able to predict quantitatively. With the example of Saccharomyces cerevisiae, we show that our approach using the variation coefficient also works for more realistic reproduction time distributions.

Figure 1

Fluorescent images of colonies of (a) E. coli and (b) S. cerevisiae. The scaling properties of both patterns are believed to be in the KPZ universality class, and the differences are due to microscopic details of the mode of reproduction and shape of the microorganisms. The images have been taken with permission from [15], copyright (2007) National Academy of Sciences, USA.

Figure 2

A smaller variation coefficient $\delta$ in reproduction times [see (4) and (6)] leads to more regular growth, smoother domain boundaries, and finer sectors. Shown are simulated circular populations with (a) $\delta =1$ and (b) $\delta =0.1$. Both colonies have an initial radius of $r_0=50$, and they are grown up to simulation time $t=50$, leading to final radii of approximately 120 (a) and 95 (b). The different colors denote cell types 1 and 2.

Figure 3

Populations in a linear geometry with periodic boundary conditions in lateral direction with (a) $\delta =1$ and (b) $\delta =0.1$. Both populations have lateral width $L=300$, and the colonies are grown to a simulation time $t\approx 50$, leading to heights of approximately 70 (a) and 40 (b). The different colors denote cell types 1 and 2.

Figure 4

Family-Vicsek scaling (12) of the surface roughness $S\left(t\right)$. The data collapse under rescaling with $\alpha =1/2$ and $z=3/2$ occurs in a scaling window which is narrower for small $\delta$ due to intrinsic correlations. The different symbols correspond to different values of $\delta$, and the color represents system size, $L=1500$ (green/light gray) and $L=4000$ (blue/dark gray). The dashed lines indicate the expected slope $\beta =1/3$. The data for $L=1500$ has been averaged over 100 samples and for $L=4000$ over 30 samples. The error bars are comparable to the size of the symbols.

Figure 5

The height-height correlation function $C(l,t)$ for $L=4000$ at $t=11000$ for various values of $\delta$. The data has been averaged over 30 samples, and the error bars are comparable to the size of the symbols. The dashed lines indicate the expected slope $1/2$.

Figure 6

Dependence of the KPZ parameters $D/\left(2\nu \right)$ on $\delta$. Data are obtained from (15) by fitting the prefactor of the power law in Fig. 5, using a proportionality constant very close to 1 [cf. derivation (B11) in the Appendix]. The data are in good agreement with the prediction (21) with fitted parameters $\epsilon \approx 0.42$ and $D/\left(2\nu \right)(\delta =1)\approx 1.1$.

Figure 7

Scaling behavior of the mean square displacement $M\left(h\right)$ (24). The system size is $L=1000$ and the data is averaged over 500 samples, and the error bars are comparable to the size of the symbols. (a) Data collapse of the normalized quantity $M\left(h\right)/{\sigma }_{\delta }^2$ as a function of height $h$ for several values of $\delta$. The values in the normalization ${\sigma }_{\delta }^2$ are taken from the best fit shown as full line in (b). Each curve follows a power law with exponent $4/3$, the line corresponding to $h^{4/3}$ is shown as comparison. (b) The prefactor ${\sigma }_{\delta }^2$, where the data are best fits according to (24). The solid line used for the collapse in (a) follows the prediction ${({\delta }^2+{\epsilon }^2)}^{4/3}$ with fitted $\epsilon =0.40$, which is compatible with the fit in Fig. 6.

Figure 8

The sector boundary $X\left(h\right)$ behaves like a fractional Brownian motion. (a) The standardized probability density function (pdf) of $X\left(h\right)$ as a function of the rescaled argument $x/\left({\sigma }_{\delta }{h}^{2/3}\right)$ for different heights $h$ and values of $\delta$. The black solid parabola is the pdf of a standard Gaussian on a logarithmic scale. (b) The covariance function $\langle X(h+\Delta h)X\left(h\right)\rangle$ shows the behavior (25), which is plotted as the solid black curve. After rescaling we get a data collapse as a function of $h/\Delta h$, which agrees well with the prediction if $h$ is sufficiently large and the flat boundary conditions become irrelevant. Data are averages over 1000 realizations and the error bars are comparable to the size of the symbols.

Figure 9

The average number of sector boundaries $\langle N\left(h\right)\rangle$ follow a power law (26) with exponent $-2/3$, which is indicated by the full line. The data are plotted for a system size $L=1500$ and various values of $\delta$ (see legend), and collapse on the function $h^{-2/3}$ when rescaled by $L/\sqrt{4\pi {\sigma }_{\delta }^2}$. Data are averages over 500 realizations, and the error bars are comparable to the size of the symbols.

Figure 10

Comparison of realistic reproduction times with the $\delta$ model. (a) The probability density functions of reproduction times of mother cells $T_r$ (full orange line/light gray) and daughter cells $T_m+T_r$ (dashed orange line/light gray) with normalized mean compared to $T$ from (4) with corresponding $\delta$ (blue/dark gray). (b) The prefactor of the mean square displacement ${\sigma }_{\delta }^2$ as introduced in Eq. (24) and Fig. 7. The data correspond to reproduction times $T_r$ for all cells (denoted M), $T_m+T_r$ for all cells (denoted D), and the most realistic mixed model (denoted M and D) as explained in the text. All these cases are consistent with previous results from Fig. 7.