The HardyWeinberg Theorem describes what happens to gene frequencies when no evolutionary forces act on phenotypes. This assumes that no selection, migration, mutation, or genetic drift alters the gene frequencies from generation to generation. Another important assumption is that individuals in the population breed randomly with respect to genotype and phenotype. In a randomly mating population gametes combine in proportion to the frequency of each gametic type taking part in the union. For a single locus with two alternative alleles, A which occurs at a frequency of p, and a which occurs at frequency q (or 1p) the proportion of genotypes is given by multiplying the frequency of each gametic type:
Genotype  AA 
Aa 
aA 
aa 
frequency  p * p 
p * q 
q * p 
q * q 
or the more familiar: p * p + 2 * p* q + q * q = 1. These calculations describe how gametes pair up randomly during fertilization. Another expression of random mating occurs at the level of the phenotype. If the frequency of each phenotype is: p2, 2pq, and q2, and phenotypes pair up randomly we would expect to see the following crosses to occur with the following frequencies:
AA  Aa  aa  
AA  p2 * p2  2pq * 2pq  p2 * q2 
Aa  p2 * 2pq  2pq * 2pq  q2 * 2pq 
aa  p2 * q2  2pq * q2  q2 * q2 
Tom Smith observed the following frequencies of matings in Pyrenestes ostrinus: Smales X Sfemales = 34, Sm X Lf = 14, Lm X Sf = 14, Lm X Lf = 6. A few simple computations are in order to learn how the birds are breeding. The birds may mate randomly, assortitively by phenotype (like breeds with like) or perhaps by disassortative mating (birds seek out a more dissimilar partner). While we cannot distinguish between all the genotypic classes in P. ostrinus, we can ask whether the phenotypes are breeding randomly. What frequency of matings would we expect by chance? We need to compute the following items.
frequency of smallbilled males  (34+14)/(34+14+14+6) = 48/68 = 0.71 
frequency of smallbilled females  (34+14)/(34+14+14+6) = 48/68 = 0.71 
frequency of largebilled males  (14+6)/(34+14+14+6) = 22/68 = 0.29 
frequency of largebilled females  (14+6)/(34+14+14+6) = 22/68 = 0.29 
What is the probability that a smallbilled male pairs randomly with a smallbilled female? The probability that a small breeds with small is given by multiplying the overall frequency of each in the population:
0.71 * 0.71 = 0.54 and,
we would expect to see a total of 68 * 0.54 = 34.3.
By the same logic, we can compute our random expectations for the other three kinds of matings to derive an expected number of matings if the birds were randomly mating. By inspection alone we can see that the observed and expected random frequencies are nearly identical. We could carry out a formal test, the Chisquare, which is based on observed versus expected frequencies. I leave this as an excercise for the reader.


For a single locus with three alleles, the formula for frequency of each genotypic class is slightly more complicated as we add a few more heterozygous classes and one homozygyous class.
Genotype   
 
   
 
 
frequency  p * p 
q * q 
r * r 
2 * p * q 
2 * q * r 
2 * p * r 
The two allele and multiallele HardyWeinberg Law really only implies that gametes achieve union randomly with respect to genotype. Given the observed Ams gene frequencies in isopods, what is the frequency of observed mating phenotypes that we would expect under random mating?