Genealogies of rapidly adapting populations

Abstract

The genetic diversity of a species is shaped by its recent evolutionary history and can be used to infer demographic events or selective sweeps. Most inference methods are based on the null hypothesis that natural selection is a weak or infrequent evolutionary force. However, many species, particularly pathogens, are under continuous pressure to adapt in response to changing environments. A statistical framework for inference from diversity data of such populations is currently lacking. Towards this goal, we explore the properties of genealogies in a model of continual adaptation in asexual populations. We show that lineages trace back to a small pool of highly fit ancestors, in which almost simultaneous coalescence of more than two lineages frequently occurs. Whereas such multiple mergers are unlikely under the neutral coalescent, they create a unique genetic footprint in adapting populations. The site frequency spectrum of derived neutral alleles, for example, is nonmonotonic and has a peak at high frequencies, whereas Tajima's D becomes more and more negative with increasing sample size. Because multiple merger coalescents emerge in many models of rapid adaptation, we argue that they should be considered as a null model for adapting populations.

Fig. 1.

A shows a maximum-likelihood tree of influenza nucleotide sequences (HA segment) sampled in Asia in 2009 (subtype H3N2) produced using Fasttree (13). B shows a tree drawn from a simulation of our model of adapting populations. Both trees often branch very unevenly, with almost all descendants on the left-most branch. Although approximate multiple mergers are common in both trees, the influenza tree does not display the uniformly long terminal branches that we observe in simulations. This could be caused by heterogeneous sampling of influenza. Trees are drawn with Figtree (http://tree.bio.ed.ac.uk/software/figtree/).

Fig. 2.

Ancestral lineages in evolving populations. The figure shows the fitness distribution of the population, translating to higher fitness with velocity , at two time points. Randomly sampled individuals (green, blue, and violet dots in the later population) tend to come from the center of the distribution, whereas ancestors tend to be among the fittest in the population. The ancestral lineages wiggle because of mutations that randomly perturb their fitness. Simultaneously, lineages move to the high fitness edge, where they are likely to meet and coalesce. The fittest individuals are typically at above the mean fitness.

Fig. 3.

The distribution of pair coalescence times (proportional to heterozygosity) in a model of rapidly adapting populations. After rescaling time by , curves for different N and s collapse onto a single master curve. This collapse shows that is the timescale of coalescence. After a delay, , is exponentially distributed, which is apparent from the Inset showing the cumulative distribution . An exponential is indicated as a black dashed line. Different line styles correspond to (solid), (dashed), and (dotted), whereas the mutation rate is . For each parameter combination, random pairs are sampled at 10,000 time points generations apart. Fig. S1 shows the corresponding distributions of T₃ and T₄.

Fig. 4.

SFS of (derived) neutral alleles in rapidly adapting populations is nonmonotonic, with peaks at low frequencies and near fixation. The asymptotic behavior of the SFS at low and high derived frequencies is shown as dashed black lines. The solid black line is the SFS of the BSC simulated using Eqs. 7 and 8 with averaged over 10,000 runs. Line styles and parameters are as in Fig. 3.

Fig. 5.

A shows the distribution of the log-fitness of all CAs and all MRCAs compared with the average distribution, , of log-fitness in the population (described in the text). These distributions were measured in forward simulations with , and . (B) SFS of derived neutral alleles in a background selection scenario with deleterious mutations of effect s. As the ratio is varied while keeping constant, the SFS interpolates between the expectation for the Kingman’s coalescent and the BSC ().