## N

To loss

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Initial allele frequency

Figure 3.14 Average time that an allele segregates, takes to reach fixation, or takes to reach loss depending on its initial frequency when under the influence of genetic drift alone. Alleles remain segregating (persist) for an average of 2.8N generations when their initial frequency is 1/2. Fixation or loss takes up to an average of 4N generations when alleles are initially very rare or nearly fixed, respectively. Since these are average times, alleles in individual populations experience longer and shorter fixation, loss, and segregation times. Time is scaled in multiples of the population size.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Initial allele frequency

Figure 3.14 Average time that an allele segregates, takes to reach fixation, or takes to reach loss depending on its initial frequency when under the influence of genetic drift alone. Alleles remain segregating (persist) for an average of 2.8N generations when their initial frequency is 1/2. Fixation or loss takes up to an average of 4N generations when alleles are initially very rare or nearly fixed, respectively. Since these are average times, alleles in individual populations experience longer and shorter fixation, loss, and segregation times. Time is scaled in multiples of the population size.

no matter what the population size is. The population size plays a role only in the absolute average number of generations that will elapse.

If you have worked your way through this section you deserve congratulations for your persistence. The basis of the diffusion equation is definitely more abstract than the basis of Markov chains, but the overall results provided by the two models are very similar. Those who would like to learn more about the diffusion equation, its assumptions, and how it can be extended to include processes such as mutation, migration, and natural selection along with genetic drift, can consult Roughgarden (1996) and Rice (2004).

3.3 Effective population size

• Defining genetic populations.

• Census and effective population size.

• Example of bottleneck and harmonic mean to demonstrate effective population size and census size.

• Effective population size due to unequal sex ratio and variation in family size.

Up to this point we have used the term population size without much fanfare to indicate how many individuals a population contains. We now need to focus additional attention on the idea of population size. The number of individuals in a population seems like a straightforward quantity that can be determined easily. In the context of the Wright-Fisher model the population size is an unambiguous quantity. Unfortunately, in most biological populations it is difficult or impossible to determine the number of gametes that contribute to the next generation. We need another way to define the size of populations.

The definition of the population size in population genetics relies on the dynamics of genetic variation in the population. This definition means that the size of a population is defined by the way genetic variation in the population behaves. The notion that "if it walks like a duck and quacks like a duck, it probably is a duck" is also applied to the size of populations. The size of a population depends on how genetic variation changes over time. If a population shows allele frequencies changing slowly over time under the exclusive influence of genetic drift, then the population has the dynamics associated with relatively large size. It "quacks" like a big population. In the same way, a population with a large number of individuals might show rapid genetic drift, indicating it is really a small population from the perspective of genetic variation. It looks big but its "quack" gives it away as a small population.

Making a distinction between the dynamics of genetic variation in a population and the number of individuals in a population suggests that there are really two types of population size. One is the head count of individuals in a population, called the census population size, symbolized by N. The other is the genetic size of a population. This genetic size is determined by comparing the rate of genetic drift in an actual population with the rate of genetic drift in an ideal population meeting the assumptions of the Wright-Fisher model. The population size in the model that produces that same rate of genetic drift as seen in an actual population is the genetic size of the actual population. In comparing an actual population with an ideal model population, we are asking about the overall genetic effects of the census size. Thus, we also recognize the effective population size, Ne, as the size of an ideal population that experiences as much genetic drift as an actual population regardless of its census size. This concept was originally introduced by Sewall Wright (1931), who is shown in Fig. 3.15. An approximate way to think of the difference between the two population sizes is that the census size is the total number of individuals and the effective size is the number of individuals that actually contribute gametes to the next generation. We will refine this definition throughout this rest of the chapter.

Census population size (N) The number of individuals in a population; the head count size of a population. Effective population size (Ne) The size of an ideal Wright-Fisher population that maintains as much genetic variation or experiences as much genetic drift as an actual population regardless of census size.

Let's examine several biological phenomena that cause effective population size and census population size to be different. This will help to illustrate the effective population size and make its definition more intuitive.

Actual populations often fluctuate in the number of individuals present over time. A classic example is rabbit/lynx population cycles due to predator/ prey dynamics, where census population sizes of both species fluctuate over a fairly wide range on about a 10 year cycle. Another category of example

Figure 3.15 Sewall Wright (1889-1988) with a guinea pig in an undated photograph taken during his years as a professor at the University of Chicago. Starting in 1912 and throughout his career, Wright studied the genetic basis of coat colors and physiological traits in guinea pigs. Wright, along with J.B.S. Haldane and R.A. Fisher, established many of the early expectations of population genetics using mathematical analyses. Many of the conceptual frameworks in population genetics today were originated by Wright, especially those related to consanguineous mating, genetic drift, and structured populations. An often retold (although mythical) story was that Wright, who would sometimes carry a guinea pig with him, would on occasion absent-mindedly employ the animal to erase the chalk board while lecturing. Provine's (1986) biography details Wright's manifold contributions to population genetics and his interactions with other major figures such as Fisher. Photograph courtesy of Special Collections Research Center, University of Chicago Library. Archival Photofiles, Series 1, Wright, Sewall, Informal #2.

is the establishment of a new population by a small number of individuals, called a founder event. One well-documented founder event was the introduction of European starlings in the New World. These birds, now very common throughout North America, can all be traced to approximately 15 pairs that survived from a larger group released in New York's Central Park in 1890. What is now a very large population descended from a sample of 60 alleles in the small number of founding individuals, which were sampled from a very large population of birds in Europe.

To model the genetic effects of this type of fluctuation in population size over time, suppose a population starts out with 100 individuals, experiences a reduction in size to 10 individuals for one generation, and then recovers to 100 individuals in the third generation (Fig. 3.16). (Recall from earlier in the chapter that this situation violates the constant population size assumption of the Wright-Fisher model of genetic drift.) This will cause an increased chance of fixation or loss of alleles (variance in allele frequency will increase) and thereby increase the rate of genetic drift in that one generation. But what is the effective size of this population after it recovers to 100 individuals? We can estimate the effect of fluctuations in populations on the overall effective size using the harmonic mean:

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