What are the forces that modulate the amount of genetic variation in a population? How do new alleles enter the gene pool? What forces remove alleles from the gene pool? How can genetic variants be recombined to create novel combinations of alleles? Answers to these questions are at the heart of understanding the process of evolution. In this section, we will examine the roles of mutation, migration, recombination, genetic drift (chance), and selection in sculpting the genetic composition of populations.

Mutation is the ultimate source of all genetic variation. In Chapter 16, we discussed the molecular mechanisms that underlie small-scale mutations such as point mutations, indels, and changes in the number of repeat units in microsatellites. Population geneticists are particularly interested in the mutation rate, which is the probability that a copy of an allele changes to some other allelic form in one generation. The mutation rate is typically symbolized by the Greek letter µ. As we will see below, if we know the mutation rate and the number of nucleotide differences between two sequences, then we can estimate how long ago the two sequences diverged.

How can geneticists estimate the mutation rate? Geneticists can estimate mutation rates by starting with a single homozygous individual and following the pedigree of its descendants for several generations. Then they can compare the DNA sequence of the founding individual to the DNA sequences of the descendants several generations later and record any new mutations that have occurred. The number of observed mutations per genome per generation provides an estimate of the rate. Because one is looking for rather rare events, it is necessary to sequence billions of nucleotides to find just a few SNP mutations. In 2009, the SNP mutation rate for a part of the human Y chromosome was estimated by this approach to be 3.0 × 10⁻⁸ mutations/nucleotide/generation, or about one mutation every 30 million bp. If we extrapolate to the entire human genome (3 billion bp), then each of us has inherited 100 new mutations from each of our parents. Luckily, the vast majority of mutations are not detrimental since they occur in regions of the genome that are not critical.

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Table 18-5 lists the mutation rates for SNPs and microsatellites in several model organisms. The SNP mutation rate is several orders of magnitude lower than the microsatellite rate. Their higher mutation rate and greater variation make microsatellites particularly useful in population genetics and DNA forensics. The SNP mutation rate per generation appears to be lower for unicellular organisms than for large multicellular organisms. This difference can be explained at least partially by the number of cell divisions per generation. There are about 200 cell divisions from zygote to gamete in humans but only 1 in E. coli. If the human rate is divided by 200, then the rate per cell division in humans is remarkably close to the rate in E. coli.

Other than mutation, the only other means for new variation to enter a population is through migration or gene flow, the movement of individuals (or gametes) between populations. Most species are divided into a set of small local populations or subpopulations. Physical barriers such as oceans, rivers, or mountains may reduce gene flow between subpopulations, but often some degree of gene flow occurs despite such barriers. Within subpopulations, an individual may have a chance to mate with any other member of the opposite sex; however, individuals from different subpopulations cannot mate unless there is migration.

Isolated subpopulations tend to diverge as each accumulates its own unique mutations. Gene flow limits genetic divergence between subpopulations. One of the genetic consequences of migration is genetic admixture, the mix of genes that results when individuals have ancestry from more than one subpopulation. This phenomenon is common in human populations. It is readily observed in South Africa, where migrants from around the world were brought together. As shown in Figure 18-16, the genomes of South Africans of mixed ancestry are complex and include parts from the indigenous people of southern Africa plus contributions of migrants from western Africa, Europe, India, East Asia, and other regions.

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Recombination is a critical force sculpting patterns of genetic variation in populations. In this case, alleles are not gained or lost; rather, recombination creates new haplotypes. Let’s look at how this works. Consider linked loci A and B. There could be a population in which only two haplotypes are found at generation t₀: AB and ab. Suppose an individual in this population is heterozygous for these two haplotypes:

If a crossover occurs in this individual, then gametes with two new haplotypes, Ab and aB, could be formed and enter the population in generation t₁.

Thus, recombination can create variation that takes the form of new haplotypes. The new haplotypes can have unique properties that alter protein function. For example, suppose an amino acid variant in a protein on one haplotype increases the enzyme activity of the protein twofold and a second amino acid variant on another haplotype also increases activity twofold. A recombination event that combines these two variants would yield a protein with fourfold higher activity.

Let’s now consider the observed and expected frequencies of the four possible haplotypes for two loci, each with two alleles. Linked loci, A and B, have alleles A and a and B and b with frequencies p_A, p_a, p_B, and p_b, respectively. The four possible haplotypes are AB, Ab, aB, and ab with observed frequencies P_AB, P_Ab, P_aB, and P_ab. At what frequency do we expect to find each of these four haplotypes? If there is a random relationship between the alleles at the two loci, then the frequency of any haplotype will be the product of the frequencies of the two alleles that compose that haplotype:

For example, suppose that the frequency of each of the alleles is 0.5; that is, p_A = p_a = p_B = p_b = 0.5. When we sample the gene pool, the probability of drawing a chromosome with an A allele is 0.5. If the relationship between the alleles at locus A and the alleles at locus B is random, then the probability that the selected chromosome has the B allele is also 0.5. Thus, the probability that we draw a chromosome with the AB haplotype is

If the association between the alleles at two loci is random as just described, then the two loci are said to be at linkage equilibrium. In this case, the observed and expected frequencies will be the same. Figure 18-17a diagrams a case of two loci at linkage equilibrium.

If the association between the alleles at two loci is nonrandom, then the loci are said to be in linkage disequilibrium (LD). In this case, a specific allele at the first locus is associated with a specific allele at the second locus more often than expected by chance. Figure 18-17b diagrams a case of complete LD between two loci. The A allele is always associated with the B allele, while the a allele is always associated with the b allele. There are no chromosomes with haplotypes Ab or aB. In this case, the observed and expected frequencies will not be the same.

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We can quantify the level of LD between two loci as the difference (D) between the observed frequency of a haplotype and the expected frequency given a random association among alleles at the two loci. If both loci involved have just two alleles, then

In Figure 18-17a, D = 0 since there is no LD, and in Figure 18-17b, D = 0.25, which is greater than 0, indicating the presence of LD.

How does LD arise? Whenever a new mutation occurs at a locus, the mutation appears on a single specific chromosome and so it is instantly linked to (or associated with) the specific alleles at any neighboring loci on that chromosome. Consider a population in which there are just two haplotypes: AB and Ab. If a new mutation (a) arises at the A locus on a chromosome that already possesses the b allele at the B locus, then a new ab haplotype would be formed. Over time, this new ab haplotype might rise in frequency in the population. Other chromosomes in the population would possess the AB or Ab haplotypes at these two loci, but no chromosomes would possess aB. Thus, the loci would be in LD. Migration can also cause LD when one subpopulation possesses only the AB haplotype and another only the ab haplotype. Any migrants between the subpopulations would give rise to LD within the subpopulation that receives the migrants.

LD between two loci will decline over time as crossovers between them randomize the relationship between their alleles. The rate of decline in LD depends on the rate at which crossing over occurs. The frequency of recombinants (RF) between the two loci among the gametes that form the next generation (see Chapter 4) provides an estimate of recombination rate, which in population genetics is symbolized by the lowercase letter r. If D₀ is the value for linkage disequilibrium between two loci in the current generation, then the value in the next generation (D₁) is given by this equation:

In other words, linkage disequilibrium as measured by D declines at a rate of (1 − r) per generation. When r is small, D declines slowly over time. When r is at its maximum (0.5), then D declines by 1/2 each generation.

Since LD decays as a function of time and the recombination fraction, population geneticists can use the level of LD between a mutation and the loci surrounding it to estimate the time in generations since the mutation first arose in the population. Older mutations have little LD with neighboring loci, while recent mutations show a high level of LD with neighboring loci. If you look again at Figure 18-14, you’ll notice that there is considerable LD between SNP2 in G6PD and the neighboring SNPs. SNP2 encodes the amino acid change of valine to methionine in the A⁻ allele that confers resistance to malaria. Population geneticists have used LD at G6PD to estimate that the A⁻ allele arose about 10,000 years ago. Malaria is not thought to have been prevalent in Africa until then. Thus, the A⁻ arose by random mutation but was maintained in the population because it provided protection against malaria.

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The Hardy–Weinberg law tells us that allele frequencies remain the same from one generation to the next in an infinitely large population. However, actual populations of organisms in nature are finite rather than infinite in size. In finite populations, allele frequencies may change from one generation to the next as the result of chance (sampling error) when gametes are drawn from the gene pool to form the next generation. Change in allele frequencies between generations due to sampling error is called random genetic drift or just drift for short.

Let’s consider a simple but extreme case—a population composed of a single heterozygous (A/a) individual (N = 1) at generation t₀. We will allow self-fertilization. In this case, the gene pool can be described as having two alleles, A and a, each present at a frequency of p = q = 0.5. The size of the population remains the same, N = 1, in the subsequent generation, t₁. What is the probability that the allele frequencies will change (“drift”) to p = 1 and q = 0 at generation t₁? In other words, what is the probability that the population will become fixed for the A allele, so that it consists of a single homozygous A/A individual? Since N = 1, we need to draw just two gametes from the gene pool to form a single individual. The probability of drawing two A’s is p² = 0.5² = 0.25. Thus, 25 percent of the time this population will “drift” away from the initial allele frequencies and become fixed for the A allele after just one generation.

What happens if we increase the population size to N = 2 and the initial gene pool still has p = q = 0.5? The allele frequencies will change to p = 1 and q = 0 in the next generation only if the population consists of two A/A individuals. For this to happen, we need to draw four A alleles, each with a probability of p = 0.5, so the probability that the next generation will have p = 1 and q = 0.0 is p⁴ = (0.5)⁴ = 0.0625, or just over 6 percent. Thus, an N = 2 population is less likely to drift to fixation of the A allele than an N = 1 population. More generally, the probability of a population drifting to the fixation of the A allele in a single generation is p^2N, and thus this probability gets progressively smaller as the population size (N) gets larger. Drift is a weaker force in large populations.

Drift means any change in allele frequencies due to sampling error, not just loss or fixation of an allele. In a population of N = 500 with two alleles at a frequency of p = q = 0.5, there are 500 copies of A and 500 copies of a. If the next generation has 501 copies of A (p = 0.501) and 499 copies of a (q = 0.499), then there has been genetic drift, albeit a very modest level of drift. A general formula for calculating the probability of observing a specific number of copies of an allele in the next generation, given the frequencies in the current generation, is presented in Box 18-4.

When drift is operating in a finite population, one can calculate the probabilities of different outcomes, but one cannot accurately predict the specific outcome that will occur. The process is like rolling dice. At any locus, drift can continue from one generation to the next until one allele has become fixed. Also, in a particular population, the frequency of the A allele may increase from generation t₀ to t₁ but then decrease from generation t₁ to t₂. Drift does not proceed in a specific direction toward loss or fixation of an allele.

Figures 18-18a and 18-18b show computer-simulated random trials (rolls of the dice) for six populations of size N = 10 and N = 500. Each population starts having two alleles at a frequency of p = q = 0.5, then the random trials proceed for 30 generations. First, notice the randomness of the process from one generation to the next. For example, the frequency of A in the population depicted by the yellow line in Figure 18-18a bounces up and down from one generation to the next, hitting a low of p = 0.15 at t₁₆ but then rebounding to p = 0.75 at t₃₀. Second, whether N = 10 or N = 500, notice that no two populations have exactly the same trajectory. Drift is a random process, and we are not likely to observe exactly the same outcome with different populations over many generations except when N is very small. Third, notice that when N = 10, the populations became fixed (either p = 1 or p = 0) before generation 20 in five of the six trials. However, when N = 500, the populations retained both alleles in all six trials even after 30 generations.

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In addition to population size, the fate of an allele is determined by its frequency in the population. Specifically, the probability that an allele will drift to fixation in a future generation is equal to its frequency in the present generation. An allele that is at a frequency of 0.5 has a 50:50 chance of fixation or loss from the population in a future generation. You can see the effect of allele frequency on the fate of an allele in Figure 18-18c. For ten populations with an initial frequency of p = 0.1, eight populations experienced the loss of the A allele, one its fixation, and one population retained both alleles after 30 generations. That’s very close to the expectation that A will go to fixation 10 percent of the time when p = 0.1.

The fact that the frequency of an allele is equal to its probability of fixation means that most newly arising mutations will ultimately be lost from a population because of drift. The initial frequency of a new mutation in the gene pool is

If N is even modestly large, such as 10,000, then the probability that a new mutation will ultimately reach fixation is extremely small: 1/2N = 1/20,000 = 5 × 10⁻⁵. The probability that a new mutation will ultimately be lost from the population is

which is close to 1.0 in large populations. It is 0.99995 in a population of 10,000.

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Figure 18-19a shows a graphical representation of the fate of new mutations in a population. The x-axis represents time and the y-axis the number of copies of an allele. The black lines show the fate of most new mutations. They appear and then are soon lost from the population. The colored lines show the few “lucky” new mutations that become fixed. From population genetic theory, it can be shown that the average time required for a lucky mutation to become fixed is 4N generations. Figure 18-19b shows a population that is 1/2 the size of the population in Figure 18-19a. Thus, 4N generations is 1/2 as long and the lucky new mutations are fixed more rapidly.

An important consequence of drift is that slightly deleterious alleles can be brought to fixation or advantageous alleles lost by this random process. Consider a new allele that arises in a population and endows the individual carrying it with a stronger immune system. This individual can pass the advantageous allele to his or her offspring, but those offspring might die before reproducing because of a random event such as being struck by lightning. Or if the individual carrying the favorable allele is heterozygous, he or she may pass only the less favorable allele to his or her offspring by chance.

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In calculating the probabilities of different outcomes under genetic drift, we are assuming that the A and a alleles do not confer differences in viability or reproductive success to the individuals that carry them. We assume that A/A, A/a, and a/a individuals are all equally likely to survive and reproduce. In this case, A and a would be termed neutral alleles (or variants) relative to each other. Change in the frequencies of neutral alleles over time due to drift is called neutral evolution. The process of neutral evolution is the foundation for the molecular clock, the constant rate of substitution of newly arising allelic variants for preexisting ones over long periods (Box 18-5). Neutral evolution is distinct from Darwinian evolution, in which favorable alleles rise in frequency because the individuals that carry them leave more offspring. We will discuss Darwinian evolution in the next section of this chapter and in Chapter 20.

Up until now, we have been considering drift in the context of populations that remain the same size from one generation to the next. In reality, populations often contract or expand in size over time. For example, a new population of much smaller size can suddenly form when a relatively small number of the members of a population migrate to a new location and establish a new population. The migrants, or “founders,” of the new population may not carry all the alleles present in the original population, or they may carry the same alleles but at different frequencies. Genetic drift caused by random sampling of the original population to create the new population is known as the founder effect. One of many founder events in human history occurred when people crossed the Bering land bridge from Asia to the Americas during the ice age about 15,000 to 30,000 years ago. As a result, genetic diversity among Native Americans is lower than among people in other regions of the world (Figure 18-20).

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Population size can also change within a single location. A period of one or several consecutive generations of contraction in population size is known as a population bottleneck. Bottlenecks occur in natural populations because of environmental fluctuations such as a reduction in the food supply or increase in predation. The gray wolf, American bison, bald eagle, California condor, whooping crane, and many whale species are some familiar examples of species that have experienced recent bottlenecks because of hunting by humans or encroachment by humans on their habitat. The reduction in population size during a bottleneck increases the level of drift in a population. As explained earlier in the chapter, the level of inbreeding in populations is also dependent on population size. Thus, bottlenecks also cause an increase in the level of inbreeding.

The California condor presents a remarkable example of a bottleneck. This species was once wide ranging but in the 1980s declined to a breeding population of only 14 captive birds. The population is now above 400 individuals, but the average heterozygosity in the genome decreased by 8 percent during the initial bottleneck. Furthermore, a deleterious recessive allele for a lethal form of dwarfism occurs at a frequency of about 9 percent among the surviving animals, presumably as a result of drift from a lower frequency in the pre-bottleneck population. To manage these problems, conservation biologists set up matings of captive animals to minimize further inbreeding and to purge deleterious alleles from the population.

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Box 18-6 discusses the well-characterized bottleneck that occurred during the domestication of crop species. This bottleneck explains why our crop plants possess much less genetic diversity than their wild ancestors.

So far, we have considered how new alleles enter a population through mutation and migration and how these alleles can become fixed in (or lost from) a population by random drift. But mutation, migration, and drift cannot explain why organisms seem so well adapted to their environments. They cannot explain adaptations, features of an organism’s form or physiology that allow it to better cope with the environmental conditions under which it lives. To explain the origin of adaptations, Charles Darwin, in 1859 in his historic book The Origin of Species, proposed that adaptations arise through the action of another process, which he called “natural selection.” In this section, we will explore the role of natural selection in modulating genetic variation within populations. Later, in Chapter 20, we will consider the effects of natural selection on the evolution of genes and traits over extended periods.

Let’s define natural selection as the process by which individuals with certain heritable features are more likely to survive and reproduce than are other individuals that lack these features. As outlined by Darwin, the process works like this. In each generation more offspring are produced than can survive and reproduce in the environment. Nature has a mechanism (mutation) to generate new heritable forms or variants. Individuals with particular variants of some features are more likely to survive and reproduce. Individuals with features that enhance their ability to survive and reproduce will transmit these features to their offspring. Over time, these features will rise in frequency in the population. Thus, populations will change over time (evolve) as the environment (nature) favors (selects) features that enhance the ability to survive and reproduce. This is Darwin’s theory of evolution by means of natural selection.

Darwinian evolution is often described using the phrase “survival of the fittest.” This phrase can be misleading. An individual who is physically strong, resistant to disease, and lives a long life but has no offspring is not fit in the Darwinian sense. Darwinian fitness refers to the ability to survive and reproduce. It considers both viability and fecundity. One measure of Darwinian fitness is simply the number of offspring that an individual has. This measure is called absolute fitness, and we will symbolize it with an uppercase W. For an individual with no offspring, W equals 0, for an individual with one offspring, W equals 1, for an individual with two offspring, W equals 2, and so forth. W is also the number of alleles at a locus that an individual contributes to the gene pool.

Absolute fitness confounds population size and differences in reproductive success among individuals. Population geneticists are primarily interested in the latter, and so they use a measure called relative fitness (symbolized by a lowercase w), which is the fitness of an individual relative to some other individual, usually the most fit individual in the population. If individual X has two offspring and the most fit individual, Y, has 10 offspring, then the relative fitness of X is w = 2/10 = 0.2. The relative fitness of Y is w = 10/10 = 1. For every 10 alleles Y contributes to the next generation, X will contribute 2.

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The concept of fitness applies to genotypes as well as to individuals. The absolute fitness for the A/A genotype (W_A/A) is the average number of offspring left by individuals with that genotype. If we know the absolute fitnesses for all genotypes at a locus, we can calculate the relative fitnesses for each of the genotypes.

Let’s now look at how allele frequencies can change over time when different genotypes have different fitnesses; that is, when natural selection is at work. Below are the fitnesses and genotype frequencies for the three genotypes at the A locus in a population. In this case, A is a favored dominant allele since the fitnesses of the A/A and A/a individuals are the same and superior to the fitness of the a/a individuals. We are assuming that this population follows the Hardy–Weinberg law, with p = 0.1 and q = 0.9.

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The relative contribution of each genotype to the gene pool is determined by the product of its fitness and its frequency. The more fit and the higher the frequency of a genotype, the more it contributes.

The relative contributions do not sum to 1, so we need to rescale them by dividing each by the sum of all three (0.595) to get the expected frequencies of the genotypes that contribute to the gene pool.

Using these expected genotype frequencies and the Hardy–Weinberg law, we can calculate the frequencies of the alleles in the next generation:

and

The difference between p′ and p (Δp = p′ − p) is 0.17 − 0.1 = 0.07, so we conclude that the A allele has climbed 7 percent in one generation due to natural selection. Box 18-7 presents the standard equations for calculating changes in allele frequencies over time due to natural selection.

We could go through this process recursively, using the allele frequencies from the first generation to calculate those in the second generation, then using those from the second to calculate the third, and so forth. If we then plotted p by time measured in number of generations (t), we’d have a picture of the tempo with which allele frequencies change under the force of natural selection. Figure 18-21 shows such a plot for both a favored dominant and a favored recessive allele. The dominant allele rises rapidly to start but then hits a plateau and only slowly approaches fixation. Once the favored dominant allele is at a high frequency, the unfavored recessive allele occurs mostly in heterozygotes and rarely as homozygotes with reduced fitness, so selection is ineffective at purging it from the population. The favored recessive behaves in the opposite manner—it rises slowly in frequency at first since a/a homozygotes with enhanced fitness are rare but proceeds more rapidly to fixation later. Since the heterozygous class has reduced fitness, the unfavored dominant allele can eventually be purged from the population.

Natural selection can operate in several different ways. Directional selection, which we have been discussing, moves the frequency of an allele in one direction until it reaches fixation or loss. Directional selection can be either positive or purifying. Positive selection works to bring a new, favorable mutation or allele to a higher frequency. This type of selection is at work when new adaptations evolve. A selective sweep occurs when a favorable allele reaches fixation. Directional selection can also work to remove deleterious mutations from the population. This form of selection is called purifying selection, and it prevents existing adaptive features from being degraded or lost. Selection does not always proceed directionally until loss or fixation of an allele. If the heterozygous class has a higher fitness than either of the homozygous classes, then natural selection will favor the maintenance of both alleles in the population. In this case, the locus is under balancing selection and natural selection will move the population to an equilibrium point at which both alleles are maintained in the population (see Chapter 20).

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The different forms of selection each leave a distinct signature on the DNA sequence near the target locus in a population. For example, positive selection can be detected in DNA sequences by its effects on genetic diversity and linkage disequilibrium. Figure 18-22 shows schematic haplotypes before and after an episode of positive selection. In the panel showing the haplotypes before selection, the bracketed region has many polymorphisms and multiple haplotypes. However, after selection, there is only a single haplotype in this region and thus no polymorphism. When selection is applied to the target site (shown in red), the target and neighboring sites can all be swept to fixation before recombination breaks up the haplotype in which the favorable mutation first occurred. The result is lower diversity and higher LD near the target. As distance from the target increases, there is more opportunity for recombination, and so diversity goes gradually back up.

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Figure 18-23 shows the pattern of diversity in the region surrounding the SLC24A5 gene in humans. This gene influences the deposition of the melanin in the skin. When people migrated from Africa to Europe, a selective sweep at SLC24A5 caused a loss of all diversity at this locus. As a consequence, there is a single allele and a single haplotype at this locus in Europe. The single allele that was selected for in Europe produces lighter skin color. Moving away from the gene in either direction, the number of haplotypes rises in European populations since recombination disrupted the linkage disequilibrium between SLC25A5 and more distance sites. Light skin may be adaptive in northern latitudes. People are able to synthesize vitamin D, but to do so they need to absorb UV radiation through the skin. In the equatorial latitudes, people are exposed to high levels of UV light and can synthesize vitamin D even with heavily pig-mented skin. At more distance from the equator, people are exposed to less UV light, and lighter skin color may facilitate vitamin D synthesis at these latitudes.

Table 18-6 lists a few of the genes that show evidence for natural selection in modern humans. These genes fall into a few basic categories. One group strengthens resistance to pathogens. The genes G6PD, FY^null, and Hb (hemoglobin B, the sickle-cell-anemia gene) all help adapt humans to the threat of malaria. Figure 18-11b shows that the frequency of FY^null is highest in central Africa. Central Africa also has the highest prevalence of malaria, suggesting that selection has driven FY^null to its highest frequency in the region where selection pressure is greatest. Recently, medical geneticists have uncovered the gene CCR5 (chemokine receptor 5), having an allele (CCR5-Δ32) that provides resistance to AIDS. This allele is now a target of natural selection. As long as there are pathogens, natural selection will continue to operate in human populations.

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Another group of selected genes in Table 18-6 adapts people to regional diets. Before 10,000 years ago, all humans were hunter–gatherers. More recently, most humans switched to agricultural foods, but there are regional differences in diet. In northern Europe and parts of Africa, milk products are a substantial part of the diet. In most populations, the lactase enzyme for digesting milk sugar (lactose) is expressed during childhood but is switched off in adults. In parts of Europe and Africa where adults drink milk, however, special alleles of the lactase gene that continue to express the lactase enzyme during adulthood have risen in frequency due to natural selection. Finally, Table 18-6 includes some genes for physiological adaptations to climate. Among these are the genes for skin pigmentation such as SLC24A5, discussed above.

Whereas directional selection causes a loss of genetic variation in the region surrounding the target locus, balancing selection can prevent the loss of diversity by random genetic drift, leading to regions of unusually high genetic diversity in the genome. One region of high genetic diversity surrounds the major histocompatibility complex (MHC) gene complex on chromosome 6. Figure 18-24 shows a distinct spike in the number of SNPs at the MHC. This complex includes the human leukocyte antigen (HLA) genes, which are involved in immune system recognition of (and response to) pathogens. Balancing selection is one hypothesis proposed to explain the high diversity observed at the MHC. Since heterozygotes have two alleles, they may be resistant to a greater repertoire of pathogen types, giving heterozygotes a fitness advantage.

Finally, selection can be imposed by an agent other than nature. Humans have imposed selection in the process of domesticating and improving cultivated plants and animals. This form of selection is called artificial selection. In this case, individuals with traits that humans prefer contribute more alleles to the gene pool than individuals with unfavored traits. Over time, the alleles that confer the favored traits rise in frequency in the population. The many breeds of dogs and dairy cows and varieties of garden vegetables and cereal crops are all the products of artificial selection.

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We have considered the forces that regulate variation in populations individually. Let’s now consider the opposing effects of mutation and drift, the former adding variation and the latter removing it from populations. When these two forces are in balance, a population can reach an equilibrium at which the loss and gain of variation are equal. We will use heterozygosity (H) as a measure of variation. Remember that H will be near 0 when a population is near fixation for a single allele (low variation), and H approaches 1 when there are many alleles of equal frequency (high variation).

Let’s use H with a “hat,” , as the symbol for the equilibrium value of H. To find , we start with two mathematical equations: one equation that relates change in H to population size (drift) and another equation that relates change in H to the mutation rate. We can then set these equations equal to each other and solve for .

First, we need an equation for the decline in variation (H) between generations as a function of population size (drift). We developed such an equation in Box 18-3 when discussing inbreeding:

This equation applies to the effects of drift as well as those of inbreeding. From this equation, it follows that the change in H between generations due to drift is

Second, we need an equation for the increase in variation, as measured by H, between generations due to mutation. Any new mutation will increase heterozygosity at a rate proportional to the frequency of homozygotes in the population (1 − H) times the rate at which mutation converts them to heterozygotes (2μ). (The 2 is necessary because there are two alleles that could mutate in a diploid.) Thus, the change in H between generations due to mutation is

When the population reaches an equilibrium, the loss of heterozygosity by drift will be equal to the gain from mutation. Thus, we have

which can be rewritten as

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This equation gives the equilibrium value of when the loss by drift and gain by mutation are balanced. This equation applies only to neutral variation; that is, we are assuming selection is not at work. We are also assuming that each new mutation yields a unique allele.

Expressions such as this are useful when we have estimates for two of the variables and would like to know the third. For example, nucleotide diversity (H at the nucleotide level) for noncoding sequences, which are largely neutral, is about 0.0013 in humans, and μ for humans is 3 × 10⁻⁸ (see Table 18-5). Using these values and solving the equation above for N yields an estimate of the human population size of 10,498 humans. This estimate is far below the 7.2 billion of us alive today. What’s up? This is an estimate for the equilibrium value. Modern humans are a young group, only about 150,000 years old. Over the last 150,000 years, our population has grown dramatically as we filled the globe, but mutation is a slow process, so genetic diversity has not kept up and the human population is not at equilibrium. The population size of 10,498 represents an estimate of our historical size, or how many breeding members there were about 150,000 years ago.

Allelic frequencies may also reach a stable equilibrium when the introduction of new alleles by repeated mutation is balanced by their removal by natural selection. This balance probably explains the persistence of genetic diseases as low-level polymorphisms in human populations. New deleterious mutations are constantly arising spontaneously. These mutations may be completely recessive or partly dominant. Selection removes them from the population, but there is an equilibrium between their appearance and removal.

Let’s begin with the simplest case—the frequency for a deleterious recessive when an equilibrium is reached between mutation and selection. For this purpose, it is convenient to express the relative fitnesses in terms of the selection coefficient (s), which is the selective disadvantage of (or loss of fitness in) a genotype:

Then, as shown in Box 18-8, the equation for equilibrium frequency of a deleterious recessive allele is

This equation shows that the frequency at equilibrium depends on the ratio μ/s. When the mutation rate for A → a gets larger and the selective disadvantage smaller, then the equilibrium frequency ( ) of a recessive deleterious allele will rise. As an example, a recessive lethal allele (s = 1) that arises by mutation from the wild-type allele at the rate of μ = 10⁻⁶ will have an equilibrium frequency of 10⁻³.

Let’s consider the equilibrium between selection and mutation for the slightly more complicated case of a partially dominant deleterious allele—that is, an allele with some deleterious effect in heterozygotes as well as its effect in homozygotes. We’ll define h as the degree of dominance of the deleterious allele. When h is 1, the deleterious allele is fully dominant, and when h is 0, the deleterious allele is fully recessive. Then, the fitnesses are

where a is a partially dominant deleterious allele. A derivation similar to the one in Box 18-8 gives us

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Here is an example. If μ = 10 ⁶ and the lethal allele is not totally recessive but causes a 5 percent reduction in fitness in heterozygotes (s = 1.0, h = 0.05), then

This result is smaller by two orders of magnitude than the equilibrium frequency for the purely recessive case described above. In general, then, we can expect deleterious, completely recessive alleles to have frequencies much higher than those of partly dominant alleles because the recessive alleles are protected in heterozygotes.