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Some results typical for sampling from these microcentrifuge tube populations are given in Table 3.1. The results have several striking patterns. First, micro-centrifuge tube "allele" frequencies fluctuate quite a bit in the samples. In some cases the results are 0.50/0.50 like in the ancestral population, but the results range from 0.35/0.65 in the sample of 20 to 0.0/1.0 in the sample of four. This latter result is called fixation or loss, since one allele composed the entire sample (its frequency went to 1) and the other allele was not sampled at all (its frequency went to zero). Second, the amount of fluctuation in the allele frequencies appears to be related to the size of the sample that was taken from the original population. The samples of four had greater fluctuations in allele frequency including a case of fixation and loss. The samples of 20 deviated somewhat less from the original allele frequencies of 0.50/0.50 and in those 10 trials no fixation/loss events were observed.

Compare these micro-centrifuge tube sampling results with what would be expected in an infinite population with p = q = 0.5. In the infinite population there would not be a sample of four or 20 drawn to found the next generation, the entire population would be used to found the next generation. Within the bounds of the micro-centrifuge tube population analogy, that would be like taking the entire beaker and just pouring it into another beaker to found the next generation: the allele frequencies would remain identical to the original frequency. This would also mean that if all other assumptions of Hardy-Weinberg were met, genotype frequencies would also remain constant (1/4 AA, 1/2 Aa, and 1/4aa with p = q = 0.5).

Now return to the finite micro-centrifuge tube populations with a sample of two individuals, or four gametes, drawn to found the population in the next generation. What are the chances that this next generation will consist of only AA genotypes? This is the same as asking what is the probability of sampling four blues or clear tubes in a handful of four. Since drawing one clear or blue tube has an independent probability of V2, the probability of getting four is (V2)4 = 1/i6. The same result can be seen from the perspective of genotypes by asking what are the chances of founding a population with two homozygous genotypes. If the source population is in Hardy-Weinberg equilibrium then 1/4 of all genotypes are one of the two homozygotes. The chance of drawing two identical homozygous genotypes is the product of their independent probabilities, or (1/4)2 = 1/l6.

The micro-centrifuge tube populations are a low-tech demonstration that genotype and allele frequencies fluctuate from one generation to the next due to small samples, or sampling error, in a process called genetic drift. The amount of genetic drift increases as the size of the sample used to found the next generation decreases. Another way to restate the population size assumption of Hardy-Weinberg is to say instead that Hardy-Weinberg assumes that there is very little or no genetic drift occurring.

Sampling error The difference between the value found in a finite sample from a population and the true value in the population.

Genetic drift Random changes in allele frequency from one generation to the next in biological populations due to the finite samples of individuals, gametes, and ultimately alleles that contribute to the next generation. The amount of genetic drift increases as the size of the sample used to found the next generation decreases. Stochastic process A process where individual outcomes are dictated by chance but the average of a large number of outcomes can be described as a probability distribution based on initial conditions. Wright-Fisher model A simplified version of the biological life cycle where all sampling to found the next generation occurs from an infinite pool of gametes built from equal contributions of all individuals. This approximation is commonly employed to model genetic drift.

To extend and generalize the model of genetic drift started with the micro-centrifuge tube populations, a model of the biological process of reproduction is helpful. To do this, let's consider the process of reproduction in populations. During reproduction, individual adult organisms produce gametes. These gametes are exchanged with mates and fuse to form zygotes, and these zygotes develop into a new generation of adult organisms (Fig. 3.2). This schematic of the biological life cycle is called the Wright-Fisher model of sampling (introduced by Sewall Wright (1931) and Ronald A. Fisher (1999; originally published in 1930)). It is not completely biologically realistic. There is obviously not an infinite number of gametes in any real population and sampling events can take place at many points during the life history of a population of organisms. But it allows the process of genetic drift to be reduced to a point that it can be modeled in a simple fashion. The Wright-Fisher model makes assumptions identical to those of Hardy-Weinberg (see section 2.2), with the exception that the population is finite rather than approaching infinite. Particularly critical assumptions include:

• generations are discrete and do not overlap, equivalent to adults that reproduce synchronously but only once during their lifetime;

• the numbers of females and males are equal;

• the size of the population (N individuals) remains constant through time; and

• all individuals are equal in their production of gametes and all gametes are equally viable, equivalent to no natural selection.

These assumptions reduce the complexity of sampling error in biological populations, concentrating all sampling into a single step as an approximation. This simplification approximates the genetic drift that occurs in biological populations. For example, sampling at several points in the life cycle can be equivalent in its effect on allele frequencies to the same total amount of sampling at a single point in the life cycle. As will be shown later in this chapter, sampling events may occur at many stages of the life

Figure 3.2 The Wright-Fisher model of genetic drift uses a simplified view of biological reproduction where all sampling occurs at one point: sampling 2N gametes from an infinite gamete pool. In this case N diploid individuals (N/2 of each sex) generate an infinite pool of gametes where allele frequencies are perfectly represented, and a finite sample of 2N alleles is drawn from this gamete pool to form N new diploid individuals in the next generation. Genetic drift takes place only in the random sample of 2N gametes to form the next generation. Major assumptions include non-overlapping generations, equal fitness of all individuals, and constant population size through time. The model can easily be adjusted for haploid individuals or loci by assuming 2N individuals or sampling N gametes to form the next generation.

Figure 3.2 The Wright-Fisher model of genetic drift uses a simplified view of biological reproduction where all sampling occurs at one point: sampling 2N gametes from an infinite gamete pool. In this case N diploid individuals (N/2 of each sex) generate an infinite pool of gametes where allele frequencies are perfectly represented, and a finite sample of 2N alleles is drawn from this gamete pool to form N new diploid individuals in the next generation. Genetic drift takes place only in the random sample of 2N gametes to form the next generation. Major assumptions include non-overlapping generations, equal fitness of all individuals, and constant population size through time. The model can easily be adjusted for haploid individuals or loci by assuming 2N individuals or sampling N gametes to form the next generation.

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