In this table, the total frequency is just the frequency of each parental mating pair taken from the parental mating frequency table. We now need to partition this total frequency of each parental mating into the frequencies of the three progeny genotypes produced. Let's look at an example. Parents with AA and Aa genotypes will produce progeny with two genotypes: half AA and half Aa (you can use a Punnett square to show this is true). Therefore, the AA × Aa parental matings, which have a total frequency of 2XY under random mating, are expected to produce (½)2XY = XY of each of AA and Aa progeny. The same logic applies to all of the other parental matings. Notice that each row in the offspring genotype frequency table sums to the total frequency of each parental mating.
The columns in the offspring genotype frequency table are the basis of the final step. The sum of each column gives the total frequencies of each progeny genotype expected in generation t + 1. Let's take the sum of each column, again expressed in the currency of genotype frequencies, and then simplify the algebra to see whether Hardy and Weinberg were correct.
(2.4)
So, we have proved that progeny genotype and allele frequencies are identical to parental genotype and allele frequencies over one generation or that f(A)t = f(A)t + 1. The major conclusion here is that genotype frequencies remain constant over generations as long as the assumptions of Hardy–Weinberg are met. In fact, we have just proved that under Mendelian heredity, genotype and allele frequencies should not change over time unless one or more of our assumptions is not met. This simple model of expected genotype frequencies has profound conclusions. In fact, Hardy–Weinberg expected genotype frequencies serve as one of the most basic tools to test for the action of biological processes that alter genotype and allele frequencies.
You might wonder whether Hardy–Weinberg applies to loci with more than two alleles. For the last point in this section, let's explore that question. With three alleles at one locus (allele frequencies symbolized by p, q, and r), Hardy–Weinberg expected genotype frequencies are p2 + q2 + r2 + 2pq + 2pr + 2qr = 1. These genotype frequencies are obtained by expanding (p + q + r)2, a method that can be applied to any number of alleles at one locus. In general, expanding the squared sum of the allele frequencies will show:
the frequency of any homozygous genotype is the squared frequency of the single allele that composes the genotype ([allele frequency]2);
the frequency of any heterozygous genotype is twice the product of the two allele frequencies that comprise the genotype (2[allele 1 frequency][allele 2 frequency]), and
there are as many homozygous genotypes as there are alleles and heterozygous genotypes where N is the number of alleles.
Do you think it would be possible to prove Hardy–Weinberg for more than two alleles at one locus? The answer is absolutely, yes. This would just require constructing larger versions of the parental genotype mating table and expected offspring frequency table as we did for two alleles at one locus.
2.4 Applications of Hardy–Weinberg
Estimate the frequency of an observed genotype in a forensic DNA typing case.
Test the null hypothesis that observed and expected genotype frequencies are identical.
Use Hardy–Weinberg to compare two genetic models for observed phenotypes.
In the previous two sections, we established the Hardy–Weinberg expectations for genotype frequencies. In this section, we will examine three ways that expected genotype frequencies are employed in practice. The goal of this section is to become familiar with realistic applications as well as hypothesis tests that compare observed and Hardy–Weinberg expected genotype frequencies. In this process, we will also look at a specific method to account for sampling error (see Appendix).
Forensic DNA profiling
Our first application of Hardy–Weinberg can be found in newspapers on a regular basis and commonly dramatized on television. A terrible crime has been committed. Left at the crime scene was a biological sample that law enforcement authorities use to obtain a multilocus genotype or DNA profile. A suspect in the crime has been identified and subpoenaed to provide a tissue sample for DNA profiling. The DNA profile from the suspect and from the crime scene are identical. The DNA profile is shown in Table 2.2. Should we conclude that the suspect left the biological sample found at the crime scene?
To answer this critical question, we will employ Hardy–Weinberg to predict the expected frequency of the DNA profile or genotype. Just because two DNA profiles match, there is not necessarily strong evidence that the individual who left the evidence DNA and the suspect are the same person. It is possible that there are actually two or more people with identical DNA profiles. Hardy–Weinberg and Mendel's second law will serve as the bases for us to estimate just how frequently a given DNA profile should be observed. Then, we can determine whether two unrelated individuals sharing an identical DNA profile is a likely occurrence.
Table 2.2 An example DNA profile for three STR (“simple tandem repeat”) loci commonly used in human forensic cases. Locus names refer to the human chromosome (e.g. D3 = third chromosome) and chromosome region where the SRT locus is found. The allele states are the numbers of repeats at that locus (see Box 2.1).
| Locus | D3S1358 |
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