Random mating is a critical assumption of the Hardy–Weinberg law. The assumption of random mating is met if all individuals in the population are equally likely as a choice when a mate is chosen. However, if a relative, a neighbor, or a phenotypically similar individual is a more likely mate than a random individual, then the assumption of random mating has been violated. Populations that are not random mating will not exhibit exact Hardy–Weinberg proportions for the genotypes at some or all genes. Three types of bias in mate choice that violate the assumption of random mating are assortative mating, isolation by distance, and inbreeding.

Assortative mating occurs if individuals choose mates based on resemblance to themselves. Positive assortative mating occurs when similar types mate; for example, if tall individuals preferentially mate with other tall individuals and short individuals mate with other short individuals. In these cases, genes controlling the difference in height will not follow the Hardy–Weinberg law. Rather, we’d expect to see an excess of homozygotes for the “tall” alleles among the progeny of tall mating pairs and an excess of homozygotes for “short” alleles among the progeny of short mating pairs. In humans, there is positive assortative mating for height.

Negative assortative or disassortative mating occurs when unlike individuals mate—that is, when opposites attract. One example of negative assortative mating is provided by the self-incompatibility, or S, locus in plants such as Brassica (broccoli and its relatives). There are numerous alleles at the S locus, S₁, S₂, S₃, and so forth. The stigma of a plant will not be receptive to pollen that carries either of its own two alleles (Figure 18-10). For example, the stigma of an S₁/S₂ heterozygote will not allow pollen grains carrying either an S₁ or S₂ allele to germinate and fertilize its ovules, although pollen grains carrying the S₃ or S₄ alleles can do so. This mechanism blocks self-fertilization, thereby enforcing cross-pollination. The S locus violates the Hardy–Weinberg law since homozygous genotypes at S are not formed.

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A second example of negative assortative mating is provided by the major histocompatibility complex (MHC), which is known to influence mate choice in vertebrates. MHC affects body odor in mice and rats, providing a basis for mate choice. In what are known as the “sweaty T-shirt experiments,” researchers asked a group of men to wear T-shirts for two days. Then they asked a group of women to smell the T-shirts and rate them for “pleasantness.” Women preferred the scent of men whose MHC haplotypes were different from their own. Data from the human HapMap project have since confirmed that American couples are significantly more heterozygous at the MHC than expected by chance. The MHC plays a central role in our immune response to pathogens, and heterozygotes may be more resistant to pathogens. Therefore, our offspring benefit if we mate disassortatively with respect to our MHC genotype. This mechanism may explain why the SNP in the MHC gene HLA-DQA1 that we discussed above does not follow the Hardy–Weinberg law among residents of Tuscany. Look back at Table 18-2 and you will notice that there are more heterozygotes than expected, 55 versus 42. Tuscans appear to be practicing disassortative mating with respect to this SNP.

Another form of bias in mate choice arises from the amount of geographic distance between individuals. Individuals are more apt to mate with a neighbor than another member of their species on the opposite side of the continent—that is, individuals can show isolation by distance. As a consequence, allele and genotype frequencies often differ between fish in separate lakes or between pine trees in different regions of a continent. Species or populations exhibiting such patterning of genetic variation are said to show population structure. A species can be divided into a series of subpopulations such as frogs in different ponds or people in different cities.

If a species has population structure, the proportion of homozygotes will be greater species-wide than expected under the Hardy–Weinberg law. Consider a hypothetical example of a species of wild sunflowers distributed across Kansas with a gradient in the frequency of the A allele from 0.9 near Kansas City to 0.1 near Elkhart (Figure 18-11a). We sample 100 sunflower plants from each of these two cities plus 100 from Hutchinson, in the middle of the state, and we calculate allele frequencies. Each city represents a subpopulation. For any of the three cities, the Hardy–Weinberg law works fine. For example, in Elkhart, we expect Nq² = 100 × (0.9)² = 81 a/a homozygotes, and that is what we observe. However, statewide, we’d predict Nq² = 300 × (0.5)² = 75 a/a homozygotes, yet we observed 107. Because of population structure, there are more homozygous sunflower plants than expected.

Here is a real example of population structure from our own species. In Africa, the FY^null allele of the Duffy blood group shows a gradient with a low frequency in eastern and northern Africa, moderate frequency in southern Africa, and high frequency across central Africa (Figure 18-11b). This allele is rare outside of Africa. Because of this gradient, we cannot use overall allele frequencies in Africa to calculate genotype frequencies using the Hardy–Weinberg law. Later in the chapter and in Chapter 20, we will discuss the relationship between FY^null and malaria.

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The third type of bias in mating is inbreeding, or mating between relatives. Long before anyone knew about deleterious recessive alleles, some societies recognized that disorders such as muteness, deafness, and blindness were more frequent among the children of marriages between relatives. Accordingly, brother–sister and first-cousin marriages were either outlawed or discouraged. Nevertheless, many famous individuals have married a cousin, including Charles Darwin, Albert Einstein, J. S. Bach, Edgar Allan Poe, Jesse James, and Queen Victoria. As we will see, the offspring of marriages between relatives are at higher risk of having an inherited disorder.

Progeny of inbreeding are more likely to be homozygous at any locus than progeny of non-inbred matings. Thus, they are more likely to be homozygous for deleterious recessive alleles. For this reason, inbreeding can lead to a reduction in vigor and reproductive success called inbreeding depression. However, inbreeding can have advantages too. Many plant species are highly self-pollinating and highly inbred. These include the model plant Arabidopsis, a successful weed, and the productive cereal crops rice and wheat. Since most plant species bear male and female organs on the same individual, self-pollination can be accomplished more easily than outcrossing. Another advantage of self-pollination is that when a single seed is dispersed to a new location, the plant that grows from the seed has a ready mate—itself, enabling a new population to be established from a single seed. Finally, if an individual plant has a beneficial combination of alleles at different loci, then inbreeding preserves that combination. In selfing plant species, benefits such as these offer advantages that outweigh the cost associated with inbreeding depression.

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Inbreeding increases the risk that an individual will be homozygous for a recessive deleterious allele and exhibit a genetic disease. The amount that risk increases depends on two factors: (1) the frequency of the deleterious allele in the population and (2) the degree of inbreeding. To measure the degree of inbreeding, geneticists use the inbreeding coefficient (F), which is the probability that two alleles in an individual trace back to the same copy in a common ancestor. Let’s first consider how to calculate F using pedigrees and then examine how F can be used to determine the increase in risk of inheriting a recessive disease condition.

Consider a simple pedigree for a mating between half-sibs, individuals who have one parent in common (Figure 18-12a). In the figure, B and C are half-sibs who have the same mother, A, but different fathers; B and C have a daughter, I. Notice that there is a closed loop from I through B and A and back to I through C. The presence of a closed loop in the pedigree informs us that I is inbred. The two copies of the gene in A are colored blue and pink—the blue from A’s father and pink from her mother. As drawn, I has inherited the pink copy both through her father (B) and her mother (C). Since I’s two copies of the gene trace back to the same copy in her grandmother, her two copies are identical by descent (IBD). More generally, if the two copies of a gene in an individual trace back to the same copy in an ancestor, then the copies are IBD. We’d like a way to calculate the probability that I’s two alleles will be IBD. This probability is the inbreeding coefficient for I, which is in symbol form as F_I.

First, since we are only interested in tracing the path of IBD alleles, we can simplify the pedigree to contain only the individuals in the closed loop and still follow the transmission of any IBD alleles (Figure 18-12b). Also, since the sex of the individual doesn’t matter, we use circles for both sexes. The alleles transmitted with each mating are labeled w, x, y, and z. We use “~” to symbolize IBD. We’d like to calculate the probability that w and x are IBD, but let’s take this calculation step by step. First, what is the probability that x and y are IBD or, symbolically, what is P(x ~ y)? This is the probability that C transmits the copy inherited from A to I, which is 1/2, or P(x ~ y) = 1/2. Similarly, the probability that B transmits the copy inherited from A to I is 1/2, or P(w ~ z) = 1/2.

Now we need to calculate the probability that z and y are IBD. There are two ways that z and y can be IBD. The first way is when z and y are both the same copy (both pink or both blue). This happens 1/2 of the time since 1/4 of the time they are both blue and 1/4 both pink. The second way is when z and y are different copies (one pink and the other blue) but individual A was inbred. If individual A is inbred, then there is a probability that her two copies of the gene are IBD. The probability that A’s two copies are IBD is the inbreeding coefficient of A, F_A. The probability that z and y are different copies (one pink, the other blue) is 1/2. So, the probability that z and y are different copies that are IBD is 1/2 multiplied by the inbreeding coefficient (F_A) to give F_A. Altogether, the probability that z and y are IBD is the probability that they are the same copy (1/2) plus the probability that they are different copies that are IBD ( F_A). Symbolically, we write

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P(x ~ y), P(w ~ z), and P(z ~ y) are independent probabilities, so we can use the product rule and put it all together to obtain

In the analysis of inbred pedigrees, we can substitute the value of F_A into the equation above if it is known. Otherwise, we can assume F_A is zero if there is no information to suggest that individual A is inbred. In the current example, if we assume F_A = 0, then

This calculation tells us that the offspring of half-sib matings will be homozygous for alleles that are IBD for at least 1/8 of their genes. It could be more than 1/8 if F_A is greater than zero. Additional inbred pedigrees and a general formula for calculating F can be found in Box 18-2.

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When there is inbreeding in a population, the random-mating assumption of Hardy–Weinberg will be violated. However, Hardy–Weinberg can be modified to correct the predicted genotypic proportions for different degrees of inbreeding using F, the mean inbreeding coefficient for the population. The modified Hardy–Weinberg frequencies are

These modified Hardy–Weinberg proportions make intuitive sense, showing how inbreeding reduces the frequency of heterozygotes by 2pqF and adds half this amount to each of the homozygous classes. With these modified Hardy–Weinberg equations, you’ll also notice that when there is no inbreeding (F = 0), you regain standard Hardy–Weinberg genotypic frequencies, and when there is complete inbreeding (F = 1), you get f_A/A = p and f_a/a = q.

How much does inbreeding increase the risk that offspring will exhibit a recessive disease condition? Table 18-3 shows the inbreeding coefficients for offspring of some different inbred matings and the predicted number of homozygous recessives for different frequencies (q) of the recessive allele. When q = 0.01, there is a 7-fold (7.19/1.0) increase in homozygous recessive offspring for first-cousin matings as compared to matings between unrelated individuals. The increase in risk jumps 13-fold (3.36/0.25) when q = 0.005 and 63-fold (0.63/0.01) when q = 0.001. In other words, the degree of risk jumps dramatically for rare alleles. Brother–sister and parent–offspring matings are the riskiest: when q = 0.001, they show a 250-fold (2.51/0.01) greater risk compared to matings between unrelated individuals.

The impact of inbreeding on the frequency of genetic disorders in human populations can be seen in Figure 18-13. Children of marriages of first cousins show about a twofold higher frequency of disorders as compared to children of unrelated parents. Historical records suggest that the risks of inbreeding were understood long before the field of genetics existed.

Population size is a major factor contributing to the level of inbreeding in populations. In small populations, individuals are more likely to mate with a relative than in large ones. The phenomenon is seen in small human populations such the one on the Tristan de Cunha Islands in the South Atlantic, which has fewer than 300 people. Let’s look at the effect of population size on the overall level of inbreeding in a population as measured by F.

Consider a population with F_t being the level of inbreeding at generation t. To form an individual in the next generation t + 1, we select the first allele from the gene pool. Suppose the population size is N. After the first allele is selected, the probability that the second allele we pick will be exactly the same copy is 1/2N and the inbreeding coefficient for this individual is 1.0. The probability that the second allele we pick will be a different copy from the first allele is 1 − 1/2N and the level of inbreeding for the resulting individual would be F_t, the average inbreeding coefficient for the initial population at generation t. The level of inbreeding in the next generation is the sum of these two possible outcomes or

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This equation informs us that F will increase over time as a function of population size. When N is large, F increases slowly over time. When N is small, F increases rapidly over time. For example, suppose F_t in the initial population is 0.1 and N = 10,000. Then F_t+1 would be 0.10005, just a slightly higher value. However, if N = 10, then F_t+1 would be 0.145, a much higher value. We can also use this equation recursively to calculate F_t+2 by using F_t+1 in place of F_t on the right side. The result with N = 10 and F_{t = 0.1} would be F_t+2 = 0.188. The effects of population size on inbreeding in populations are further explored in Box 18-3.

A consequence of the increased inbreeding is that individuals in small populations are more likely to be homozygous for deleterious alleles just as the offspring of first-cousin marriages are more likely to be homozygous for such alleles. This effect is seen in ethnic groups that live in small, reproductively isolated communities. For example, a form of dwarfism in which affected individuals have six fingers occurs at a frequency of more than 1 in 200 among a population of about 13,000 Amish in Lancaster County, Pennsylvania, although its frequency in the general U.S. population is only 1 in 60,000.

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