The Hardy-Weinberg Principle

Consider an ‘ideal’ population in which there is an autosomal locus with two alleles, A and a, that have frequencies of p and q, respectively. These are the only alleles found at this locus, so that p + q = 100%, or 1. The frequency of each genotype in the population can be determined by construction of a Punnett square, which shows how the different genes can combine (Figure 8.1).

FIGURE 8.1 Punnett square showing allele frequencies and resulting genotype frequencies for a two-allele system in the first generation.

From Figure 8.1, it can be seen that the frequencies of the different genotypes are:

If there is random mating of sperm and ova, the frequencies of the different genotypes in the first generation will be as shown. If these individuals mate with one another to produce a second generation, Punnett square can again be used to show the different matings and their frequencies (Figure 8.2).

FIGURE 8.2 Punnett square showing frequencies of the different matings in the second generation.

From Figure 8.2 the total frequency for each genotype in the second generation can be derived (Table 8.1). This shows that the relative frequency or proportion of each genotype is the same in the second generation as in the first. In fact, no matter how many generations are studied, the relative frequencies will remain constant. The actual numbers of individuals with each genotype will change as the population size increases or decreases, but their relative frequencies or proportions remain constant. This is the fundamental tenet of the Hardy-Weinberg principle. When studies confirm that the relative proportions of each genotype remain constant with frequencies of p², 2pq, and q², then that population is said to be in Hardy-Weinberg equilibrium for that particular genotype.

Factors that Can Disturb Hardy-Weinberg Equilibrium

So far, this relates to an ‘ideal’ population. By definition such a population is large and shows random mating with no new mutations and no selection for or against any particular genotype. For some human characteristics, such as neutral genes for blood groups or enzyme variants, these criteria can be fulfilled. However, several factors can disturb Hardy-Weinberg equilibrium, either by influencing the distribution of genes in the population or by altering the gene frequencies. These factors include:

Small Population Size

In a large population, the numbers of children produced by individuals with different genotypes, assuming no alteration in fitness for any particular genotype, will tend to balance out, so that gene frequencies remain stable. However, in a small population it is possible that by random statistical fluctuation one allele could be transmitted to a high proportion of offspring by chance, resulting in marked changes in allele frequency from one generation to the next, so that Hardy-Weinberg equilibrium is disturbed. This is known as random genetic drift. If one allele is lost altogether, it is said to be extinguished and the other allele is described as having become fixed (Figure 8.3).

FIGURE 8.3 Possible effects of random genetic drift in large and small populations.

Gene Flow (Migration)

If new alleles are introduced into a population as a consequence of migration, with later intermarriage, a change in the relevant allele frequencies will result. This slow diffusion of alleles across racial or geographical boundaries is known as gene flow. The most widely quoted example is the gradient shown by the incidence of the B blood group allele throughout the world (Figure 8.4). This allele is thought to have originated in Asia and spread slowly westward as a result of admixture through invasion.

FIGURE 8.4 Distribution of blood group B throughout the world.

(From Mourant AE, Kopéc AC, Domaniewska-Sobczak K 1976 The distribution of the human blood groups and other polymorphisms, 2nd ed. Oxford University Press, London, with permission.)

Validity of Hardy-Weinberg Equilibrium

It is relatively simple to establish whether a population is in Hardy-Weinberg equilibrium for a particular trait if all possible genotypes can be identified. Consider a system with two alleles, A and a, with three resulting genotypes, AA, Aa/aA, and aa. Among 1000 individuals selected at random, the following genotype distributions are observed:

From these figures, the incidence of the A allele (p) equals [(2 × 800) + 185]/2000 = 0.8925 and the incidence of the a allele (q) equals [185 + (2 × 15)]/2000 = 0.1075.

Now consider what the expected genotype frequencies would be if the population were in Hardy-Weinberg equilibrium, and compare these with the observed values:

These observed and expected values correspond closely and formal statistical analysis with a χ² test would confirm that the observed values do not differ significantly from those expected if the population is in equilibrium.

Next consider a different system with two alleles, B and b. Among 1000 randomly selected individuals the observed genotype distributions are:

From these values, the incidence of the B allele (p) equals [(2 × 430) + 540]/2000 = 0.7 and the incidence of the b allele (q) equals [540 + (2 × 30)]/2000 = 0.3.

Using these values for p and q, the observed and expected genotype distributions can be compared:

These values differ considerably, with an increased number of heterozygotes at the expense of homozygotes. Such deviation from Hardy-Weinberg equilibrium should prompt a search for factors that could result in increased numbers of heterozygotes, such as heterozygote advantage or negative assortative mating—i.e., the attraction of opposites!

Despite the number of factors that can disturb Hardy-Weinberg equilibrium, most populations are in equilibrium for most genetic traits, and significant deviations from expected genotype frequencies are unusual.

Applications of Hardy-Weinberg Equilibrium

Why are Some Genetic Disorders More Common than Others?

It follows that if a gene has a high mutation rate, the disease incidence may be relatively high. However, factors other than the mutation rate and biological fitness may be involved, as mentioned previously. These are now considered in the context of population size.

Autosomal Dominant Inheritance

For an AD disorder, the simplest approach is to compare the observed numbers of affected offspring born to affected parents with what would be expected based on the disease penetrance (i.e., 50% if penetrance is complete). A χ² test can be used to see whether the observed and expected numbers differ significantly. Care must be taken to ensure that a bias is not introduced by excluding parents who were ascertained through an affected child.

Autosomal Recessive Inheritance

For disorders thought to follow AR inheritance, formal segregation analysis is much more difficult. This is because some couples who are both carriers will by chance not have affected children, therefore not feature in ascertainment. To illustrate this, consider 64 possible sibships of size 3 in which both parents are carriers, drawn from a large hypothetical population (Table 8.5). The sibship structure shown in Table 8.5 is that which would be expected, on average.

In this population, on average, 27 of the 64 sibships will not contain any affected individuals. This can be calculated simply by cubing —i.e., × × = . Therefore, when the families are analyzed, these 27 sibships containing only healthy individuals will not be ascertained—referred to as incomplete ascertainment. If this is not taken into account, a falsely high segregation ratio of 0.43 will be obtained instead of the correct value of 0.25.

Mathematical methods have been devised to cater for incomplete ascertainment, but analysis is usually further complicated by problems associated with achieving full or complete ascertainment. In practice ‘proof’ of AR inheritance requires accurate molecular or biochemical markers for carrier detection. Affected siblings (especially when at least one is female) born to unaffected parents usually suggests AR inheritance, but somatic and germline parental mosaicism (p. 121), non-paternity, and other possibilities need to be considered. There are some good examples of conditions originally reported to follow AR inheritance but subsequently shown to be dominant with germline or somatic mosaicism; for example, osteogenesis imperfecta and pseudoachondroplasia. However, a high incidence of parental consanguinity undoubtedly provides strong supportive evidence for AR inheritance, as first noted by Bateson and Garrod in 1902 (pp. 7, 113).

Recombination Fraction

The recombination fraction, usually designated as θ (Greek theta), is a measure of the distance separating two loci, or more precisely an indication of the likelihood that a cross-over will occur between them. If two loci are not linked then θ equals 0.5 because, on average, genes at unlinked loci will segregate together during 50% of all meioses. If θ equals 0.05, this means that on average the syntenic alleles will segregate together 19 times out of 20 (i.e., a crossover will occur between them during, on average, only 1 in 20 meioses).

Centimorgans

The unit of measurement for genetic linkage is known as a map unit or centimorgan (cM). If two loci are 1 cM apart, a crossover occurs between them, on average, only once in every 100 meioses (i.e., θ = 0.01). Centimorgans are a measure of the genetic, or linkage, distance between two loci. This is not the same as physical distance, which is measured in base pairs (kb – kilobases: 1000 base pairs: Mb – megabases: 1,000,000 base pairs).

The human genome has been estimated by recombination studies to be about 3000 cM in length in males. Because the physical length of the haploid human genome is approximately 3 × 10⁹ bp, 1 cM corresponds to approximately 10⁶ bp (1 Mb or 1000 kb). However, the relationship between linkage map units and physical length is not linear. Some chromosome regions appear to be particularly prone to recombination—so-called ‘hotspots’—and recombination occurs less often during meiosis in males than in females, in whom the genome ‘linkage’ length has been estimated to be 4200 cM. Generally, in humans one or two recombination events take place between each pair of homologous chromosomes in meiosis I, with a total of ~40 across the entire genome. Recombination events are rare close to the centromeres but relatively common in telomeric regions.

Linkage Analysis

Linkage analysis has proved invaluable for mapping genes (see Chapter 5). It is based on studying the segregation of the disease with polymorphic markers from each chromosome—preferably in large families. Eventually a marker will be identified that co-segregates with the disease more often than would be expected by chance (i.e., the marker and disease locus are linked). The mathematical analysis tends to be very complex, particularly if many closely adjacent markers are being used, as in multipoint linkage analysis. However, the underlying principle is relatively straightforward and involves the use of likelihood ratios, the logarithms of which are known as LOD scores (logarithm of the odds).

LOD Scores

When studying the segregation of alleles at two loci that could be linked, a series of likelihood ratios is calculated for different values of the recombination fraction (θ), ranging from θ = 0 to θ = 0.5. The likelihood ratio at a given value of θ equals the likelihood of the observed data, if the loci are linked at recombination value of θ, divided by the likelihood of the observed data if the loci are not linked (θ = 0.5). The logarithm to the base 10 of this ratio is known as the LOD score (Z)—i.e., LOD (θ) = log₁₀ [Lθ/L(0.5)]. Logarithms are used because they allow results from different families to be added together.

For example, when a research paper reports that linkage of a disease with a DNA marker has been identified with a LOD score (Z) of 4 at recombination fraction (θ) 0.05, this means that the results, in the families studied, indicate that it is 10,000 (10⁴) times more likely that the disease and marker loci are closely linked (i.e., 5 cM apart) than that they are not linked. It is generally agreed that a LOD score of +3 or more is confirmation of linkage. This would yield a ratio of 1000 to 1 in favor of linkage; however, because there is a prior probability of only 1 in 50 that any two given loci are linked, a LOD score of +3 means that the overall probability that the loci are linked is approximately 20 to 1—i.e., [1000 × ]:1. The importance of taking prior probabilities into account in probability theory is discussed in the section on Bayes’ theorem (p. 339).

A ‘Simple’ Example

Consider a three-generation family in which several members have an AD disorder (Figure 8.6). A and B are alleles at a locus that is being tested for linkage to the disease locus.

FIGURE 8.6 Three-generation pedigree showing segregation of an autosomal dominant disorder and alleles (A and B) at a locus that may or may not be linked to the disease locus.

To establish whether it is likely that these two loci are linked, the LOD score is calculated for various values of θ. The value of θ that gives the highest LOD score is taken as the best estimate of the recombination fraction. This is known as a maximum likelihood method.

To demonstrate the underlying principle, the LOD score is calculated for a value of θ equal to 0.05. If θ equals 0.05 then the loci are linked, in which case the disease gene and the B marker must be on the same chromosome in II2, as both of these characteristics have been inherited from the mother. Thus in II2 the linkage phase is known: the disease allele and the B allele are in coupling. Therefore the probability that III1 will be affected and will also inherit the B marker equals 0.95 (i.e., 1 – θ). A similar result is obtained for the remaining three members of the sibship in generation III, giving a value for the numerator of (0.95)⁴. If the loci are not linked, the likelihood of observing both the disease and marker B in III1 equals 0.5. A similar result is obtained for his three siblings, giving a value for the denominator of (0.5)⁴.

Therefore the LOD score for this family, given a value of θ = 0.05, equals log₁₀ 0.95⁴/0.5⁴ = log₁₀ 13.032 = 1.12. For a value of θ = 0, the LOD score equals log₁₀ 1⁴/0.5⁴ = log₁₀ 16 = 1.20. For a value of θ = 0.1, the LOD score equals log₁₀ 0.9⁴/0.5⁴ = log₁₀ 10.498 = 1.02. The highest LOD score is obtained for a value of θ equals 0, which is consistent with the fact that if the disease and marker loci are linked then no recombination has occurred between the two loci in members of generation III.

To confirm linkage other families would have to be studied by pooling all the results until a LOD score of +3 or greater was obtained. A LOD score of −2 or less is taken as proof that the loci are not linked. This less stringent requirement for proof of non-linkage (i.e., a LOD score of −2 compared with +3 for proof of linkage) is due to the high prior probability of that any two loci are not linked.

Multipoint Linkage Analysis

Two-point linkage analysis is often used to map a disease locus to a specific chromosome region. This gives a rather rough or ‘coarse’ indication of the location of the disease locus. The next step often involves multipoint linkage analysis using a series of polymorphic markers that are known to map to the disease region. This process allows fine tuning of the probable position of the disease locus within the rough interval defined by the small number of polymorphic marker loci.

Using this approach the results of linkage studies with the various markers are analyzed by a computer program that calculates the overall likelihood of the position of the disease locus in relation to the marker loci. The results are presented in the form of a likelihood ratio known as a location score. This is calculated for different positions of the disease locus and a graph is drawn up of location score against map distance (Figure 8.7). On this graph the peaks represent possible positions of the disease locus, with the tallest peak being the most probable location. The troughs represent the positions of the polymorphic marker loci.

FIGURE 8.7 Multipoint linkage analysis. A, B, and C represent the known linkage relationships of three polymorphic marker loci. X, Y, and Z represent in descending order of likelihood the probable position of the disease locus.

Multipoint linkage analysis is used to define the smallest possible interval in which a disease locus is located, so that physical mapping methods can then be applied to isolate the disease gene (see Chapter 5).

Autozygosity Mapping

This ingenious form of linkage analysis has been used to map many rare AR disorders. Autozygosity occurs when individuals are homozygous at particular loci by descent from a common ancestor. In an inbred pedigree containing two or more children with a rare AR disorder, it is very likely that the children will be homozygous not only at the disease locus but also at closely linked loci. In other words, all affected relatives in an inbred family will be homozygous for markers within the region surrounding the disease locus. Thus a search can be made for shared areas of homozygosity in affected relatives using highly polymorphic markers such as microsatellites (p. 69). In a pedigree with a relatively large number of affected individuals, only a small number of shared homozygous regions will be identified; one of these can be expected to harbor the relevant disease locus, which can then be isolated using physical mapping strategies.

Autozygosity mapping can be applied in both small inbred families (Figure 8.8) and in genetic isolates (p. 133) with a shared common genetic ancestry (e.g., the Old Order Amish). It is a particularly powerful technique in large inbred families in which more than one branch has affected individuals. Several of the genes that cause AR sensorineural hearing loss have been mapped in this way, as well as a number of skeletal dysplasias and primary microcephalies, for example.

FIGURE 8.8 Autozygosity mapping in a family with spondylocostal dysostosis. The father of individual I₁ is the brother of I₂’s grandfather. The region of homozygosity is defined by markers D15S155 and D15S127. A mutation in the MESP2 gene was subsequently shown to be the cause of spondylocostal dysostosis in this pedigree.

Linkage Disequilibrium

Linkage disequilibrium is defined formally as the association of two alleles at linked loci more frequently than would be expected by chance, and is also referred to as allelic association. The concept and the term relate to the study of diseases in populations rather than families. In the latter, an association between specific alleles and the disease in question holds true only within an individual family; in a separate affected family a different pattern of alleles, or markers, at the same locus may show association with the disease—because the alleles themselves are polymorphic.

The rationale for studying allelic association in populations is based on the assumption that a mutation occurred in a founder case some generations previously and is still causative of the disease. If this is true, the pattern of markers in a small region close to the mutation will have been maintained and thus constitutes what is termed the founder haplotype. The underlying principles used in mapping are the same as those for linkage analysis in families, the difference being the degree of relatedness of the individuals under study. In the pedigree shown in Figure 8.6, support was obtained for linkage of the disease gene with the B marker allele. Assume that further studies confirm linkage of these loci and that the A and B alleles have an equal frequency of 0.5. It would be reasonable to expect that the disease gene would be in coupling with allele A in approximately 50% of families and with allele B in the remaining 50%. If, however, the disease allele was found to be in coupling exclusively with one particular marker allele, this would be an example of linkage disequilibrium.

The demonstration of linkage disequilibrium in a particular disease suggests that the mutation causing the disease has occurred relatively recently and that the marker locus studied is very closely linked to the disease locus. There may be pitfalls, however, in interpreting haplotype data that suggest linkage disequilibrium. Other possible reasons for linkage disequilibrium include: (1) the rapid growth of genetically isolated populations leading to large regions of allelic association throughout the genome; (2) selection, whereby particular alleles enhance or diminish reproductive fitness; and (3) population admixture, where population subgroups with different patterns of allele frequencies are combined into a single study. Allowance for the latter problem can be made by using family-based controls and analyzing the transmission of alleles using a method called the transmission/disequilibrium test. This uses the fact that transmitted and non-transmitted alleles from a given parent are paired observations, and examines the preferential transmission of one allele over the other in all heterozygous parents. The technique has been applied, amongst others, to studies based on sibling pairs that are discordant for the disease or condition under study.

AD Disorders

If everyone with an AD disorder were successfully encouraged not to reproduce, the incidence of that disorder would decline rapidly, with all future cases being the result only of new mutations. This would have a particularly striking effect on the incidence of relatively mild conditions such as familial hypercholesterolemia, in which genetic fitness is close to 1.

Alternatively, if successful treatment became available for all patients with a serious AD disorder that at present is associated with a marked reduction in genetic fitness, there would be an immediate increase in the frequency of the disease gene followed by a more gradual leveling off at a new equilibrium level. If, at one time, all those with a serious AD disorder died in childhood (f = 0), then the incidence of affected individuals would be 2µ. If treatment raised the fitness from 0 to 0.9, the incidence of affected children in the next generation would rise to 2µ due to new mutations plus 1.8µ inherited, which equals 3.8µ. Eventually a new equilibrium would be reached, by which time the disease incidence would have risen tenfold to 20µ. This can be calculated relatively easily with the formula µ = [I(1 − f)]/2 (p. 133), which can also be expressed as I = 2µ/(1 – f). The net result would be that the proportion of affected children who died would be lower (from 100% down to 10%), but the total number affected would be much greater, although the actual number who died from the disease would remain unchanged at 2µ.

Autosomal Recessive Disorders

In contrast to an AD disorder, artificial selection against an AR condition will have only a very slow effect.

The reason for this difference is that in AR conditions most of the genes in a population are present in healthy heterozygotes who would not be affected by eugenic measures. It can be shown that if there is complete selection against an AR disorder, so that no homozygotes reproduce, the number of generations (n) required for the allele frequency to change from q₀ to q_n equals 1/q_n − 1/q₀. Therefore, for a condition with an incidence of approximately 1 in 2000 and an allele frequency of roughly 1 in 45, if all affected patients refrained from reproduction then it would take more than 500 years (18 generations) to reduce the disease incidence by half and more than 1200 years (45 generations) to reduce the gene frequency by half, assuming an average generation time of 27 years.

Now consider the opposite situation, where selection operating against a serious AR disorder is relaxed because of improvement in medical treatment. More affected individuals will reach adult life and transmit the mutant allele to their offspring. The result will be that the frequency of the mutant allele will increase until a new equilibrium is reached. Using the formula µ = I(1 − f), it can be shown that, when the new equilibrium is eventually reached, an increase in fitness from 0 to 0.9 will have resulted in a tenfold increase in the disease incidence.

X-Linked Recessive Disorders

When considering the effects of selection against these disorders, it is necessary to take into account the fact that a large proportion of the relevant genes are present in entirely healthy female carriers, who are often unaware of their carrier status. For a very serious condition, such as Duchenne muscular dystrophy (p. 307), with fitness equal to 0 in affected males, selection will have no effect unless female carriers choose to limit their families. If all female carriers opted not to have any children, the incidence would be reduced by two-thirds (i.e., from 3µ to µ).

A much more plausible possibility is that effective treatment will be forthcoming for these disorders. This will result in a steady increase in the disease incidence. For example, an increase in fitness from 0 to 0.5 will lead to a doubling of the disease incidence by the time a new equilibrium has been established. This can be calculated using the formula µ = [I^M(1 − f)]/3 (p. 133).