## N

probability results from the assumption that all lineages in any given generation have an equal chance of being chosen as a common ancestor working back in time from the present. In the basic coalescent model where alleles are selectively neutral, each lineage within a generation has an equal and constant probability of becoming an ancestral lineage when working back in time to find the most recent common ancestor (MRCA). From the time-forward perspective, with neutral evolution each lineage has an equal probability of being sampled and represented in the next generation. In general, with selective neutrality the haplotype of a lineage does not influence its sampling properties.

Natural selection violates the basic assumption that all lineages have equal and constant probabilities of coalescence. When natural selection operates, some lineages tend to increase in frequency over time whereas other lineages tend to decrease in frequency over time due to fitness differences among the lineages caused by differences in haplotype relative fitnesses. These changes in the frequencies of lineage copies translate into probabilities of coalescence that change over time as well. A lineage bearing a haplotype favored by selection will increase over time and therefore have a decreasing probability of coalescence working back in time from the present. Similarly, a lineage bearing a lower-fitness haplotype will decrease in frequency over time and therefore have an increasing probability of coalescence moving back in time. Thus, natural selection presents a fundamental contradiction to the sampling process built into the genealogical branching model.

Directional selection and the ancestral selection graph

Fortunately, there is a clever and relatively simple way to modify the genealogical branching model to accommodate directional natural selection (Neuhauser & Krone 1997; Neuhauser 1999). This modification relies on treating coalescence and natural selection as distinct processes that can both possibly occur working back in time from the present toward the MRCA in the past exactly as was done to combine coalescence with migration or mutation. The first step in including natural selection in the genealogical branching model is to alter slightly our view of the sampling process. Figure 7.10 shows five lineages of 2N total across the span of one generation. If there is selective neutrality, then moving forward in time each lineage is sampled once to found the next generation. This is equivalent to the absence of a coalescence event moving backward in time. Alternatively, if there is natural selection operating then lineages with one haplotype are favored and will increase in frequency over time.

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