Calculating Gene Frequencies Help (page 2)
Autosomal Loci with Two Alleles
- Dominant and Recessive Autosomal Alleles. A dominant phenotype may have either of two genotypes, AA or Aa, but we have no way (other than by laboriously testcrossing each dominant phenotype) of distinguishing how many are homozygous or heterozygous in our sample. The only phenotype whose genotype is known for certain is the recessive (aa). If the population is in equilibrium, then we can obtain an estimate of q (the frequency of the recessive allele) from q2 (the frequency of the recessive genotype or phenotype). Then the frequency of the dominant allele is
- p = 1 – q or p = 1 – √q2
- Codominant Autosomal Alleles. When codominant alleles are present in a two-allele system, each genotype has a distinctive phenotype. The numbers of each allele in both homozygous and heterozygous conditions may be counted in a sample of individuals from the population and expressed as a percentage of the total number of alleles in the sample. If the sample is representative of the entire population (containing proportionately the same numbers of genotypes as found in the entire population) then we can obtain an estimate of the allelic frequencies in the gene pool. Given a sample of N individuals of which D are homozygous for one allele (A1A1), H are heterozygous (A1A2), and R are homozygous for the other allele (A2A2), then N = D + H + R. Since each of the N individuals is diploid at this locus, there are 2N alleles represented in the sample. Each A1A1 genotype has two A1 alleles. Heterozygotes have only one A1 allele. Letting p represent the frequency of the A1 allele and q the frequency of the A2 allele, we have
This allows the translation of phenotype frequencies into gene (allele) frequencies.
- Sex-Influenced Traits. The expression of dominance and recessive relationships may be markedly changed in some genes when exposed to different environmental conditions, most notable of which are the sex hormones. In sex-influenced traits, the heterozygous genotype usually will produce different phenotypes in the two sexes, making the dominance and recessive relationships of the alleles appear to reverse themselves. We shall consider only those sex-influenced traits whose controlling genes are on autosomes. Determination of allelic frequencies must be indirectly made in one sex by taking the square root of the frequency of the recessive phenotype (q = √q2). A similar approach in the opposite sex should give an estimate of the alternate allele, p. Corroboration of sex influence is obtained if these estimates of p and q made in different sexes add close to one.
Autosomal Loci with Multiple Alleles
If we consider three alleles, A, a', and a, with the dominance hierarchy A > a' > a, occurring in the gene pool with respective frequencies p, q, and r, then random mating will generate progeny with the following frequencies:
For ease in calculation of a given allelic frequency, it may be possible to group the phenotypes of the population into just two types.
EXAMPLE 9.3 In a multiple allelic system where A > a' > a, we could calculate the frequency of the top dominant allele A by considering the dominant phenotype (A) in contrast to all other phenotypes produced by alleles at this locus. The latter group may be considered to be produced by an allele ax, which is recessive to A.
p = frequency of allele A, q = frequency of allele ax
q2 = frequency of phenotypes other than A
q = √q2
p = 1 – q = frequency of gene A
Many multiple allelic series involve codominant relationships such as (A1 = A2) > a, with respective frequencies p, q, and r. More genotypes can be phenotypically recognized in codominant systems than in systems without codominance.
The use of this formula in calculating multiple allelic frequencies is presented in Solved Problem 9.7. The multiple allelic problems in this chapter will be mainly concerned with three alleles.
- Dominant and Recessive Alleles. Since each male possesses only one sex-linked allele, the frequency of a sex-linked trait among males is a direct measure of the allelic frequency in the population, assuming that the allelic frequencies thus determined are representative of the allelic frequencies among females as well.
- Codominant Alleles. Data from both males and females can be used in the direct computation of sex-linked codominant allelic frequencies. Bear in mind that in organisms with an X-Y mechanism of sex determination, the heterozygous condition can only appear in females. Males are hemizygous for sex-linked genes.
EXAMPLE 9.4 In domestic cats, black melanin pigment is deposited in the hair by a sex-linked gene; its alternative allele produces yellow hair. Random inactivation of one of the X chromosomes occurs in each cell of female embryos. Heterozygous females are thus genetic mosaics, having patches of all-black and all-yellow hairs called tortoise-shell pattern. Since only one sex-linked allele is active in any cell, the inheritance is not really codominant, but the genetic symbolism used is the same as that for codominant alleles.
Testing a Locus for Equilibrium
In cases where dominance is involved, the heterozygous class is indistinguishable phenotypically from the homozygous dominant class. Hence, there is no way of checking the Hardy-Weinberg expectations against observed sample data unless the dominant phenotypes have been genetically analyzed by observation of their progeny from test crosses. Only when codominant alleles are involved can we easily check our observations against the expected equilibrium values through the chi-square test.
The number of variables in chi-square tests of Hardy-Weinberg equilibrium is not simply the number of phenotypes minus 1 (as in chi-square tests of classical Mendelian ratios). The number of observed variables (number of phenotypes = k) is further restricted by testing their conformity to an expected Hardy-Weinberg frequency ratio generated by a number of additional variables (number of alleles, or allelic frequencies = r).We have (k – 1) degrees of freedom in the number of phenotypes, (r – 1) degrees of freedom in establishing the frequencies for the r alleles. The combined number of degrees of freedom is (k – 1) – (r – 1) = k – r. Even in most chi-square tests for equilibrium involving multiple alleles, the number of degrees of freedom is the number of phenotypes minus the number of alleles.
Practice problems for these concepts can be found at:
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