Deriving Genotypic and Allelic Frequencies
Hardy-Weinberg Equilibrium
Darwin's Theory of Natural Selection
Population and Evolutiionary Genetics Overheads
The Hardy-Weinberg Law
The unifying concept of population genetics is the Hardy-Weinberg Law (named after the two scientists who simultaneously discovered the law). The law predicts how gene frequencies will be transmitted from generation to generation given a specific set of assumptions. Specifically,
- If an infinitely large, random mating population is free from outside evolutionaryforces (i.e. mutation, migration and natural selection),
- then the gene frequencies will not change over time and the frequencies in the next generation will be p2 for the AA genotype, 2pq for the Aa genotype and q2 for the aa genotype.
Let's examine the assumptions and conclusions in more detail starting first with the assumptions.
Infinitely large population - No such population actually exists, but does this necessarily negate the Hardy-Weinberg Law? NO!! The effect that is of concern is genetic drift. Genetic drift is a change in gene frequency that is the result of chance deviation from expected genotypic frequencies. This is a problem in small population, but is minimal in moderate sized or larger populations.
Random mating - Random mating refers to matings in a population that occur in proportion to their genotypic frequencies. For example, if the genotypic frequencies in a population are MM=0.83, MN=0.16 and NN=0.01 then we would expect that 68.9% (0.83 x 0.83 X 100) of the matings would occur between MM individuals. If a significant deviation from this expected value occurred, then random mating did not happen in this population. If significant deviations occurred in the other matings (for example MM x MN or MN x NN), again the assumption of random mating will have been violated.
In humans, at least, for many traits such as blood type, random mating will occur. Individuals do not consciously select a mate according to blood type. But for other traits, such as intelligence or physical stature, this is the case. For these traits the population is not random mating. But this does not preclude the analysis of the population for those traits at which random mating is occurring.
No evolutionary forces affecting the population - These forces may or may not be at work on a population, and we will discuss them in more detail later. As with random mating, some loci may be more affected by these forces. For these loci this assumption will be violated, whereas at those loci not affected by these forces this assumption will not be violated.
The two conclusion of the Hardy-Weinberg Law can be mathematically demonstrated in the following table. If p equals the frequency of allele A and q is the frequency of allele a, union of gametes would occur as follows:
| p | q | |
|---|---|---|
| p | p2 | pq |
| q | pq | q2 |
One of the predictions of the Hardy-Weinberg Law refers to the genotypic frequencies after one generation of random mating. In the above table the genotypic frequencies for AA is p^2, the genotypic frequency for Aa is 2pq and the genotypic frequency for aa will be q^2. These are the values that are predicted by the law.
The second prediction is that the frequencies of the two alleles will remain the same from generation to generation. The following is a mathematical proof of the second prediction. To determine the allelic frequency, they can be derived from the genotypic frequencies as shown above.
p = f(AA) + ½f(Aa) (substitute from the above table)
p = p2 + ½(2pq) (factor out p and divide)
p = p(p + q) (p + q =1; therefore q =1 - p; make this substitution)
p = p [p + (1 - p)] (subtract and multiply)
p = p
Therefore, gene frequencies do not change in one generation. Analogous calculations would also show that the frequency of the a (q) allele would not change in one generation. In absence of any factors that change the allelic frequencies, the genotypic and allelic frequencies will remain the same from generation to generation. These two conclusions have been demonstrated experimentally to be valid and form the basis upon which all further population and evolutionary genetics research is based.
Copyright © 1997. Phillip McClean