Typical simulated population dynamics in the “stochastic tunneling” regime, with neutral single mutants,

*N*μ ≪ 1, and

. Light shading shows

, dark shading shows

, and solid shading shows

. [Square roots are shown to make low-frequency lineages more visible and because this is the natural scale for measuring the variance of the frequency of rare mutants ().] The inset is a magnified view of the last few thousand generations before

*AB* takes over the population, starting from the birth of the first successful single mutant. We see that the time to cross the valley is dominated by the waiting time for an unusually large single-mutant lineage that drifts to size

*n*_{Ab} over ∼

*n*_{Ab} generations. While this lineage drifts, many small

*aB* and

*AB* lineages arise and go extinct. The largest

*aB* lineage drifts to a size

*n*_{aB} ∼

*N*μ

*n*_{Ab} over ∼

*n*_{aB} generations, recombining many times with the

*Ab* lineage and producing the successful

*AB* lineage.

## PubMed Commons