EXPLORABLES

This explorable is about pattern formation in a model system for the growth of a bacterial population in a petri dish in which the bacterial population is made up of a mix of two or three mutant strains that have identical fitness and reproduction properties. In the model system, an initially well mixed drop of bacteria with an equal amounts of every mutant strain is positioned in the center of the petri dish. After that, the bacteria start replicating, the population expands radially and a pattern will emerge.

Although the initial population is mixed, the radial growth will yield radial segments that are uniform, consisting of only one of the mutant strains, as opposed to a radially growing population that remains well mixed. This phenomenon can also be observed experimentally and is discussed in the 2007 paper Genetic drift at expanding frontiers promotes gene segregation by O. Hallatschek et al.

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## This is how it works

In the model, the two-dimensional square petri dish is initially empty, except for a central circular patch that contains an initial population of bacteria of two or three mutant strains (choose the number system using the radiobox).

Let's assume that we have a system with two mutants $$A$$ and $$B$$. Every location $$\mathbf{r}=(x,y)$$ (a pixel in the simulation) contains a discrete number of bacteria, say $$N_a (\mathbf{r})$$ bacteria of mutant $$A$$ and $$N_b(\mathbf{r})$$ of mutant $$B$$.

At every location $$\mathbf{r}$$ the number of bacteria cannot exceed a maximum value $$M$$ which is the local capacity that can be adjusted with a slider. When $$N_a (\mathbf{r})+N_a (\mathbf{r})< M$$, mutants at $$\mathbf{r}$$ reproduce at a reproduction rate (that can also be adjusted with a slider) until the location is filled.

In addition to local reproduction mutants can have offspring that immediately move into a random neighboring locations $$\mathbf{r}\rightarrow \mathbf{r}+\delta\mathbf{r}$$ if there's still room. The rate at which this happens can be adjusted with the diffusion slider.

## Explanation

Why do these uniform segments emerge? Why do we not observe a homogeneously, well mixed radially growing patch? This is what happens: Because of fluctuations at the egde of the colony mutants of one type will pioneer to a neighboring site where they start reproducing, having a head start compared to other mutants. Although other mutants may also jump to that location at a later time and before the capacity is reached, the pioneer mutant has a bit of a head start and a higher likelihood of filling this or the next neighboring patch. Eventually the mutants segregate into uniform radial segments.

## References

• O. Hallatschek, P. Hersen, S. Ramanathan, and D. R. Nelson, Genetic drift at expanding frontiers promotes gene segregation, PNAS, 104, 19926-19930 (2007)