Vvf J

The ratio of the frequencies (q/p and p/q) serves to adjust the exponent for the relative frequencies of the two lineage types.

Next we need to express the waiting time until a coalescence event, keeping in mind that lineages can only coalesce if they are of the same type. Given that there are 2Np lineages of type A and 2Nq lineages of

Lineage type A

Lineage type B

Figure 7.12 A genealogy where balancing natural selection is modeled by type switching. Every generation, lineages of one type (here A and B) may switch to the other type with rate |i. Twice the expected number of the 2N total lineages in the population that switch types each generation is then v = 4N|i. Since lineages can only coalesce when they are of the same type, type switching increases the average time to coalescence. This is analogous to natural selection favoring heterozygotes because overdominance also extends the segregation times of alleles. Genealogical trees that result from balancing selection modeled as type switching tend to have longer branches compared to genealogies that result from genetic drift or directional natural selection.

type B and coalescence events are mutually exclusive, the expected waiting time until a coalescence event is


If switching events and coalescence events are all mutually exclusive, the individual exponents can be added together to obtain the total waiting time to any event:

drawing a random number between zero and one and comparing it with the cumulative total of the chance of each event divided by the total probability of all events.

Genealogical trees that result from balancing selection modeled as type switching tend to have longer branches compared with genealogies that result from genetic drift alone (Fig. 7.12). This is due to the increase in average waiting times between coalescence events caused by lineage type switching. The final two lineages in particular are expected to take a long time to coalesce since they must switch to the same type. The results of two allele balancing selection are qualitatively similar to genealogies with long waiting times for the last two lineages expected with subdivided populations. If mutation is also operating along with balancing selection, then genealogies with longer branches would also accumulate more mutations since the number of mutation events is proportional to the total branch length of a genealogy. Lineages of the two different types, in particular, are expected to have more mutational changes between them than would be expected in a genealogy under the basic neutral coalescent model.

An additional model of balancing selection exists for populations with more than two alleles and equivalent (overdominant) fitness values for the possible heterozygous genotypes (Vekemans & Slatkin 1994; Uyenoyama 1997). Such multi-allelic balancing selection is distinct from balancing selection with only two alleles, resulting in genealogies that have long coalescence times for the lineages near the present (or shorter times to coalescence further back in time nearer the MRCA) compared to neutral genealogies. Classic examples of multi-allelic balancing selection are the many alleles found at single self-incompatibility loci in some plants (e.g. Schierup et al. 1998).

Chapter 8 on molecular evolution further considers the consequences of natural selection on the height and shape of genealogies in the context of comparisons designed to test whether or not a genealogy is different than expected by the processes of drift and mutation alone.

coalescence < t)

Given that an event has occurred with a known waiting time, the type of event can be determined by

Chapter 7 review

• Mean fitness in a population can be viewed as a graph of average fitness by genotype or allele frequencies called a fitness surface. Natural selection acts as an uphill climber on a fitness surface, moving genotype frequencies uphill based on the slope at the current genotype frequencies. Fitness surfaces will have multiple peaks and valleys if there is dominance or epistasis. When fitness depends on three alleles at one locus, the results of natural selection depend on initial genotype frequencies in the population if there is strong over- and underdominance. The net balance of recombination and natural selection may result in equilibria that do not correspond to mean fitness maxima. Recombination works toward gametic equilibrium irrespective of mean fitness and may act in opposition to natural selection.

Although the viability model of fitness is used as a standard, changes in genotype frequencies and their equilibria are often distinct when fitness is defined as differential fecundity or carrying capacity, or when fitness values vary in time and space. When both natural selection and genetic drift are acting, selection is strong relative to genetic drift when 4Nes is much greater than one, selection is weak relative to genetic drift when 4Nes is much less than one, and when 4Nes is approximately one then selection and drift are about the same strength.

When both natural selection and mutation are acting, deleterious alleles will be maintained in a population at a level that increases with the mutation rate but decreases with consanguineous mating and the selection coefficient against the allele when homozygous.

In genealogical branching models, directional selection can be modeled as an ancestral selection graph. Weak directional selection does not greatly alter the total branch length nor the total height of genealogical trees on average. In genealogical branching models, balancing selection can be modeled as a lineage type switching process similar to gene flow or mutation. With two alleles, balancing selection lengthens the average time to coalescence for the final two lineages since they have to switch to the same type to coalesce. With three or more alleles, balancing selection will tend to increase coalescence times and lengthen terminal branches in a genealogical tree.

Further reading

A classic and approachable treatment of two-locus selection that features fitness surfaces can be found in:

Lewontin RC and White MJD. 1960. Interaction between inversion polymorphisms of two chromosome pairs in the grasshopper, Moraba scurra. Evolution 14: 116-29.

For case studies, perspective, and basic theory relating to genotype interactions at two or more loci that influence fitness see chapters in:

Wolf JB, Brodie ED III, and Wade MJ (eds). 2000. Epistasis and the Evolutionary Process. Oxford University Press, Oxford.

Extensions of the basic viability natural selection model that account for biological variations such as spatial and temporal variation in fitness, fitness tradeoffs, competition, and predation can be found in:

Roff DA. 2001. Life History Evolution. Sinauer Associates,

Sunderland, MA Roughgarden J. 1996. Theory of Population Genetics and Evolutionary Ecology: an Introduction. Prentice Hall, Upper Saddle River, NJ.

A genealogical model of directional natural selection where haplotypes are neutral when rare and become favored after reaching a threshold frequency can be found in:

Przeworski M, Coop G, and Wall JD. 2005. The signature of positive selection on standing genetic variation. Evolution 59: 2312-23.

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