N

Continuing branch

Incoming branch

Figure 7.10 Haploid reproduction with the possibility of coalescence and natural selection events. Each haploid lineage replicates itself and is included in the next generation if there are no coalescence events (solid lines). A lineage makes an extra copy of itself (dashed line) and has the potential to displace one copy of another lineage. If the lineage making the extra copy of itself (open circle) has a higher-fitness haplotype than a randomly chosen lineage (blue circle), then it will displace the lineage of the lower fitness haplotype. Therefore, the outcome of a lineage-duplication event that may result in natural selection depends on the haplotype states of the specific lineages involved. The solid lines are continuing branches and the dashed line is an incoming branch. Compare with Figures 3.23 and 5.12.

In Fig. 7.10, the possible action of natural selection is shown by the dotted line. If the lineage represented by the open circles has a higher-fitness haplotype, then it will displace the lineage of the lower-fitness haplotype (closed circles). This displacement event is analogous to growth in the population size of the fitter haplotype given that the total population size is constant.

We can treat the dual continuing/incoming branching process as two independent parts of the overall coalescence process. When two independent processes are operating, the coalescence model is based on waiting for any event to occur and then deciding which type of event happened. When events are independent but mutually exclusive, the probability of each event is added over all possible events to obtain the total chance that an event occurs. As was done for both migration and mutation, we assume that coalescent and natural selection events are rare or that Ne is large and the selection coefficient is small. This assumption makes sure that natural selection and coalescence events are mutually exclusive and that when an event does occur going back in time it is either coalescence or natural selection.

Natural selection depends on the chance that the fitter haplotype makes an incoming branch that displaces a lineage bearing a less-fit haplotype. Twice the rate of natural selection events is

(o is pronounced "sigma") where 2Nes is the expected number of natural selection events that will occur for a single lineage during one unit of continuous time (see section 5.5 for a fuller explanation of such a rate in the context of mutation rates) and the fitter haplotype has a relative fitness of 1 + s compared to a relative fitness of 1 for the less-fit haplotype. When s is of the order of-then the rate of natural selection

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