Hardy Weinberg equilibrium

Under certain conditions, the genotype frequencies within a given population will follow a predictable pattern. To illustrate this point, we will use the example of the scarlet tiger moth Panaxia dominula. In this species a one locus/two allele system generates three alternative wing patterns that vary in the amount of white spotting on the black forewings and in the amount of black marking on the predominantly red hindwings. Since these patterns correspond to homozygous dominant, heterozygous and homozygous recessive genotypes, the allele frequencies at this locus can be calculated from phenotypic data. We will refer to the two relevant alleles as A and a. Because this is a diploid species, each individual has two alleles at this locus. The two homozygote genotypes are therefore AA and aa and the heterozygote genotype is Aa. Recall from Chapter 2 that allele frequencies are calculations that tell us how common an allele is within a population. In a two-allele system such as that which determines the scarlet tiger moth wing genotypes, the frequency of the dominant allele (A) is conventionally referred to as p, and the frequency of the recessive allele (a) is conventionally referred to as q. Because there are only two alleles at this locus, p+q= 1.

Genotype frequencies, which refer to the proportions of different genotypes within a population (in this case AA, Aa and aa), must also add up to 1.0. If we know the frequencies of the relevant alleles, we can predict the frequency of each genotype within a population provided that a number of assumptions about that population are met. These include:

• There is random mating within the population (panmixia). This occurs if mating is equally likely between all possible male--female combinations.

• No particular genotype is being selected for.

• The effects of migration or mutation on allele frequencies are negligible.

• The size of the population is effectively infinite.

• The alleles segregate following normal Mendelian inheritance.

If these conditions are more or less met, then a population is expected to be in Hardy-Weinberg equilibrium (HWE). The genotype frequencies of such a population can be calculated from the allele frequencies because the probability of an individual having an AA genotype depends on how likely it is that one A allele will unite with another A allele, and under HWE this probability is the square of the frequency of that allele (p2). Similarly, the probability of an individual having an aa genotype will depend on how likely it is that an a allele will unite with another a allele, and under HWE this probability is the square of the frequency of that allele (q2). Finally, the probability of two gametes yielding an Aa individual will depend on how likely it is that either an A allele from the male parent will unite with an a allele from the female parent (creating an Aa individual), or that an a allele from the male parent will unite with an A allele from the female parent (creating an aA individual). Since there are two possible ways that a heterozygote individual can be created, the probability of this occurring under HWE is 2pq.

The genotype frequencies in a population that is in HWE can therefore be expressed as:

The various frequencies of heterozygotes and homozygotes under HWE are shown in Figure 3.2, and examples are calculated in Box 3.2.

Figure 3.2 The combinations of homozygote and heterozygote frequencies that can be found in populations that are in HWE. Note that the frequency of heterozygotes is at its maximum when p = q=0.5. When the allele frequencies are between 1/3 and 2/3, the genotype with the highest frequency will be the heterozygote. Adapted from Hartl and Clark (1989)

Figure 3.2 The combinations of homozygote and heterozygote frequencies that can be found in populations that are in HWE. Note that the frequency of heterozygotes is at its maximum when p = q=0.5. When the allele frequencies are between 1/3 and 2/3, the genotype with the highest frequency will be the heterozygote. Adapted from Hartl and Clark (1989)

Table 3.1 Data from a collection of 1612 scarlet tiger moths

Phenotype individuals genotype A alleles a alleles

White spotting 1469 AA 1469 x 2 = 2938 -Intermediate 138 Aa 138 138 Little spotting 5 aa -- 5 x 2 = 10

Box 3.2 Calculating Hardy-Weinberg equilibrium

Table 3.1 is an actual data set on scarlet tiger moths that was collected by the geneticist E.B. Ford. The data in Table 3.1 tell us that in this sample there is a total of 2(1612) = 3224 alleles at this particular locus. Of these, 3076 are A alleles (2938+138) and 148 are a alleles (138+10), therefore the frequency p of the A allele in this population is:

and the frequency q of the a allele can be calculated as either: q = 148/3224 = 0.046

If we know p and q, then we can calculate the frequencies of AA (p2), Aa (2pq) and aa (q2) that would be expected if the population is in HWE as follows:

p2 = (0.954)2 = 0.9101 2pq = 2(0.954)(0.046) = 0.0878 q2 = (0.046)2 = 0.002

We now need to calculate the number of moths in this population that would have each genotype if this population is in HWE. We can do this by multiplying the total number of moths (1612) by each genotype frequency:

AA = (0.9101)( 1612) = 1467 Aa = (0.0878)(1612) = 142 aa = (0.002)(1612) = 3

Therefore the Hardy--Weinberg ratio expressed as the number of individuals with each genotype is 1467:142:3. This is very close to the actual ratio of genotypes within the population (from Table 3.1) of 1469:138:5.

We can check whether or not there is a significant difference between the observed and expected genotype frequencies by using a chi-squared (X2) test. This is based on the difference between the observed (O) number of genotypes and the number that would be expected (E) under the HWE, and is calculated as:

The x2 value of the scarlet tiger moth example is:

X2 = (1469 - 1467)2/1467 + (138 - 142)2/142 + (5 - 3)2/3 = 0.0027 + 0.11 + 1.33 1 . 44

The number of degrees of freedom (d.f.) is determined as 3 (the number of genotypes) minus 1 (because the total number was used) minus 1 (the number of alleles), which leaves d.f. = 1. By using a statistical table, we learn that a x value of 1.44, in conjunction with 1 d.f., leaves us with a probability of P = 0.230. This means that there is no significant difference between the observed genotype frequencies in the scarlet tiger moth population and the genotype frequencies that are expected under the HWE. We would conclude, therefore, that this population is in HWE.

Despite the fairly rigorous set of criteria that are associated with HWE, many large, naturally outbreeding populations are in HWE because in these populations the effects of mutation and selection will be small. There are also many populations that are not expected to be in HWE, including those that reproduce asexually. A deviation from HWE may also be an unexpected result, and when this happens researchers will try to understand why, because this may tell us something quite interesting about either the locus in question (e.g. natural selection) or the population in question (e.g. inbreeding). First, however, we must ensure that an unexpected result is not attributable to human error. Deviations from HWE may result from improper sampling. The ideal population sample size is often at least 30--40, although this will depend to some extent on the variability of the loci that are being characterized. Inadequate sampling will lead to flawed estimates of allele frequencies and is therefore one reason why conclusions about HWE may be unreliable.

Another possible source of error is to inadvertently sample from more than one population. We noted earlier that identifying population boundaries is often problematic. If genetic data from two or more populations that have different allele frequencies are combined then a Wahlund effect will be evident, which means that the proportion of homozygotes will be higher in the aggregrate sample than it would be if the populations were analysed separately. This could lead us to conclude erroneously that a population was not in HWE, whereas if the data had been analysed separately then we may have found two or more populations that were in HWE. An example of this was found in a study of a diving water beetle (Hydroporus glabriusculus) that lives in fenland habitats in eastern England. An allozyme study of apparent populations revealed significant heterozygosity deficits (Bilton, 1992), but it was only after conducting a detailed study of the beetle's ecology that the author of this study realized that each body of water actually harbours multiple populations that seldom interbreed. This population subdivision meant that samples pooled from a single water body represented multiple populations, and therefore the heterozygosity deficits could be explained by the Wahlund effect.

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