Range Expansion of Heterogeneous Populations

Risk spreading in bacterial populations is generally regarded as a strategy to maximize survival. Here, we study its role during range expansion of a genetically diverse population where growth and motility are two alternative traits. We find that during the initial expansion phase fast-growing cells do have a selective advantage. By contrast, asymptotically, generalists balancing motility and reproduction are evolutionarily most successful. These findings are rationalized by a set of coupled Fisher equations complemented by stochastic simulations.

Figure 1

Segregation patterns emerging from the stochastic simulation of a range expansion dynamics starting from a genetically diverse inoculum. (a) Local average motility $⟨\epsilon ⟩$ with blue (dark gray) signifying a low, yellow (light gray) a medium, and red (medium gray) a high motility. The front is dominated by genotypes with a motility close to $\epsilon =0.5$ (generalists). (b) Local genetic diversity with blue (dark gray) indicating a homogeneous and red (medium gray) a heterogeneous population. Genetic diversity is rapidly lost during the range expansion process. The population remains genetically diverse only close to the inoculum and at sector boundaries. Stochastic simulations were run on a hexagonal lattice with $601\times 695$ sites, with carrying capacity $\mathrm{\Omega }=100$, initial number of genotypes $G=900$, and dimensionless time $t=490$.

Figure 2

(a) Decrease of genetic diversity $H_f(t)$ in the front region for one and two spatial dimensions. After a short transient period, genetic diversity decreases rapidly. While for $d=1$ genetic diversity is quickly lost, $H_f(t)=1$, it decreases only slowly in $d=2$ due to the formation of homogeneous sectors. (b) Probability to find a certain motility at a large time $t=220$. The population is dominated by individuals with an approximately equal probability to migrate or reproduce. The histograms were averaged over $10^3$ sample runs with $\mathrm{\Omega }=100$.

Figure 3

(a) To investigate which genotypes are selected by the evolutionary dynamics at specific times, we numerically solved the Fisher equations, Eqs. (1a) and (1b), for various values of $\delta$ (as indicated in the graph) and computed the average motility $⟨\epsilon ⟩$ in the population. (b) The solid (dashed) line illustrates the analytical result for the genotype ${\epsilon }^{*}$ with the optimal front velocity in one (two) spatial dimensions, [Eq. (2)]. Triangles indicate the average genotype $⟨\epsilon ⟩$ obtained from the numerical solution of Eqs. (1), evaluated at a large time $t=3000$ for $d=1$ and at $t=1100$ for $d=2$, respectively.

Figure 4

(a) Comparison of $⟨\epsilon ⟩(t)$ between stochastic simulations with $\mathrm{\Omega }=100$ and simulations of Eqs. (1), both for $\delta =1$ and $d=2$. The measured value ${\epsilon }^{*}$ for $\mathrm{\Omega }=100$ from (c) and the analytic value from Eq. (2) are indicated. [(b), (c)] Comparison of $⟨\epsilon ⟩$ as obtained from stochastic simulations at a large time $t=220$ with ${\epsilon }^{*}$ as function of $\mathrm{\Omega }$. These quantities differ by a small constant value $\mathrm{\Delta }$ indicated in the graph. Stochastic simulation results in (a)–(c) were averaged over at least 500 sample runs.