selection graph is used to describe the outcome of the three processes since it explicitly shows possible natural selection events. Natural selection events result in an addition of branches and thereby serve to visualize selection events that are not apparent on a genealogy alone. When branching occurs due to a natural selection event (going back in time), the resulting branch is called the incoming branch to represent a possible lineage displacement. The lineage that the incoming branch splits off from is called the continuing branch. The incoming branch coalesces with a randomly chosen lineage at a later time determined by the waiting time distribution and assumes the state of the branch where it coalesces. Refer again to Fig. 7.10 to see the distinction between incoming and continuing branches. Even though natural selection events make more branches, the coalescence process is faster and will eventually result in coalescence to the MRCA (the coalescence rate is proportional to k2 whereas the selection rate is proportional to k).

Figure 7.11 shows one outcome of the coalescence-natural-selection-mutation process. Figure 7.11a shows the events that occurred working back in time from six lineages in the present. The first potential natural selection event (labeled #3 because it is the last event when working forward in time) occurred on lineage 2, causing a branching event. The incoming branch eventually coalesced with the lineage that is ancestral to lineages 3 and 4. The second potential natural selection event caused a branching event on the lineage that is ancestral to lineages 5 and 6. That incoming branch coalesced again with the same lineage after a short time. The final potential natural selection event caused a branch from one of the two internal lineages near the MRCA that coalesced with the other lineage present near the MRCA.

The actual outcome of these three selection events can only be determined once the state of the MRCA and the fitness of the two haplotypes have been assigned. In Fig. 7.11b, the ancestor is assigned a state of A which is also assumed to be the higher fitness haplotype. Then moving forward in time on the ancestral selection graph, the outcome of each instance of natural selection can be determined. For selection event #1, the incoming branch has a haplotype of a due to the mutation while the continuing branch has the ancestral A haplotype. Since A is more fit, the state of the continuing branch is kept and the state of the incoming branch discarded. (This is exactly like the open circles being less fit and the closed circles more fit in Fig. 7.10.) At the second selection event (#2), the incoming branch has a state of A and the continuing branch has a state of a due to a mutation. Here the incoming branch displaces the continuing branch and the lineage has a state of A thereafter. The incoming branch has a state of A and the continuing branch a state of a at the final selection event (#3). This results in displacement of the continuing branch.

The genealogy that results from after resolving the potential natural selection events is shown in Fig. 7.11c. Given the ancestral state and high-fitness haplotype, selection events #1 and #2 had no impact on the branching pattern of the tree. In contrast, selection event #3 caused a change in the branching pattern that moved the coalescence point of lineage 2 from the continuing branch to the coalescence point of the incoming branch. This reflects the fact that after natural selection acted, lineage 2 was identical by descent to a different lineage. This change in the branching pattern causes the total branch length of the tree to be slightly longer than it was without natural selection. In this case, the height of the tree is not changed by natural selection.

Problem box 7.2 Resolving possible selection events on an ancestral selection graph

Use Fig. 7.11b and trace the lineage states forward in time, assuming that the state of the MRCA is an a haplotype and that A is the fitter haplotype. Are the resulting lineage states in the present the same as originally given in the figure? Has the height of the genealogy changed?

As another exercise to test your knowledge of the ancestral selection graph, resolve the genealogy in Fig. 7.11b using the a allele as the state of the MRCA, alternatively assuming that the A and then the a allele is the fitter haplotype.

The conclusions to be drawn from the ancestral selection graph with two alleles are straightforward. Weak to moderate directional natural selection tends to have only a minor impact on average times to coalescence. Stated another way, the action of directional natural selection does not greatly alter the average times to coalescence compared with a strictly neutral genealogy with the same number of lineages. When the selection coefficient and the mutation rate are approximately equal, the mean time to the MRCA is shortened slightly (Neuhauser & Krone 1997; see also Przeworski et al. 1999). However, the difference in the average coalescence times with directional selection and with strict neutrality is slight given the wide variation in coalescence times due to finite sampling. Strong natural selection for advantageous alleles or selection against deleterious mutations is expected to reduce the total height of genealogical trees because of lineages bearing states that are strongly disadvantageous (see Charlesworth et al. 1993, 1995).

Genealogies and balancing selection

Natural selection where heterozygotes have the highest fitness, also called balancing selection, can also be incorporated in genealogical branching models (Hudson & Kaplan 1988; Kaplan et al. 1988; Nordborg 1997; see Hudson 1990). Earlier in Chapter 6, we saw that balancing selection is expected to maintain both alleles at a diallelic locus segregating

Interact box 7.6 Coalescent genealogies with directional selection

The coalescent process can be simulated for directional selection via an ancestral selection graph. The simulation displays a genealogy based on parameter values for the number of lineages (n:), the strength of natural selection (S:), and the mutation rate (U:). The strength of selection parameter corresponds to the rate at which incoming branches are formed on the genealogy. Mutation events will change allelic states of branches in the genealogy. The blue dots in the genealogy (labeled Phony? in the legend) are incoming branches. The button at the top right of the genealogy animation labeled Show mutations can be used to show mutations by coloring the branches. Clicking this button again when it is labeled Show types displays the allelic states of each node in the genealogy.

Click the Recalc button to begin a new simulation. The resulting genealogy can be viewed as an animation going back in time (click Animation tab at top left and use playback controls at bottom) or as a genealogy (click Trees tab at top left).

Set the selection parameter to 1.0 and view a few resulting genealogies. Then set the selection parameter to 5.0 and view numerous genealogies. How do the genealogies compare under weak or strong selection? Rerun the simulation a few times while trying out a range of values for the selection strength. Be sure to view multiple replicate genealogies for each set of parameter values.

in the population at equilibrium. The two allele frequencies at equilibrium will depend on the selection coefficients against the two homozygous genotypes. Since the haploid genealogical model does not have diploid genotypes nor sexual reproduction, we will need to take an alternative approach rather than specifying multiple genotype fitness values.

Balancing selection is a special case of natural selection because it works counter to the fixation and loss due to genetic drift. In a genealogical branching model genetic drift is represented by the process of coalescence. So to approximate the overall effect of balancing selection we need a process that will delay coalescence to the same degree that selection favors heterozygotes in the diploid selection model. This same overall result can be obtained by modeling balancing selection along the lines of population structure with two demes. Although it sounds like an odd approach, population structure and balancing selection have similar effects for different reasons. In structured populations, two lineages cannot coalesce unless they are in different demes (refer back to Fig. 4.16). Gene-flow events that move lineages into different demes therefore tend to delay coalescence events. Using this same logic, we can model balancing selection in a single panmictic population as a process where there are two lineage types. A switching process akin to gene flow (or mutation) changes lineage types at random while the coalescence process operates at the same time. If two lineages must be of the same type to coalesce, then the switching process will prevent coalescence among the lineages that are of different types.

Let the two lineage types be A and B and their respective frequencies in the population be p and q so that p + q = 1 (see Fig. 7.12). Every generation, lineages of one type may switch to the other type with rate ||. Twice the expected number of the 2N total lineages in the population that switch types each generation is then v = 4N|. The expected number of lineages switching each generation serves as a surrogate for the strength of balancing selection since frequency switching will let lineages escape coalescence. Using this switching rate, the expected waiting time until an A lineage switches to a B lineage is

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