  Recent microbial experiments suggest that enhanced genetic drift at the
frontier of a two-dimensional range expansion can cause genetic sectoring
patterns with fractal domain boundaries. Here, we propose and analyze a simple
model of asexual biological evolution at expanding frontiers to explain these
neutral patterns and predict the effect of natural selection. Our model
attributes the observed gradual decrease in the number of sectors at the
leading edge to an unbiased random walk of sector boundaries. Natural selection
introduces a deterministic bias in the wandering of domain boundaries that
renders beneficial mutations more likely to escape genetic drift and become
established in a sector. We find that the opening angle of those sectors and
the rate at which they become established depend sensitively on the selective
advantage of the mutants. Deleterious mutations, on the other hand, are not
able to establish a sector permanently. They can, however, temporarily "surf"
on the population front, and thereby reach unusual high frequencies. As a
consequence, expanding frontiers are susceptible to deleterious mutations as
revealed by the high fraction of mutants at mutation-selection balance.
Numerically, we also determine the condition at which the wild type is lost in
favor of deleterious mutants (genetic meltdown) at a growing front. Our
prediction for this error threshold differs qualitatively from existing
well-mixed theories, and sets tight constraints on sustainable mutation rates
for populations that undergo frequent range expansions.
