Review the following concepts if needed:
 HardyWeinberg Equilibrium for Genetics
 Calculating Gene Frequencies for Genetics
 Natural Selection and Evolution for Genetics
Population Genetics and Evolution Practice Problems
Practice 1
In a population gene pool, the alleles A and a are at initial frequencies p and q, respectively. Prove that the gene frequencies and the zygotic frequencies do not change from generation to generation as long as the HardyWeinberg conditions are maintained.
Solution 1
Zygotic frequencies generated by random mating are
p^{2} (AA) + 2 pq(Aa) + q^{2}(aa) =1
All of the gametes of AA individuals and half of the gametes of heterozygotes will bear the dominant allele A. Then the frequency of A in the gene pool of the next generation is
p^{2} + pq = p^{2} + p (1 – p) = p^{2} + p – p^{2} = p
Thus, each generation of random mating under HardyWeinberg conditions fails to change either the allelic or zygotic frequencies.
Practice 2
Prove the HardyWeinberg law by finding the frequencies of all possible kinds of matings and from these generating the frequencies of genotypes among the progeny using the symbols shown below.
Solution 2
There are six kinds of matings (ignoring malefemale differences) that are easily generated in a mating table.
The matings AA × Aa occur with the frequency 4p^{3}q. Half the offspring from this mating are expected to be AA [(1/2) (4p^{3}q) = 2 p^{3}q], and half are expected to be Aa (again with the frequency 2p^{3}q). Similar reasoning generates the frequencies of genotypes among the progeny shown in the following table.
Practice 3
At what allelic frequency does the homozygous recessive genotype (aa) become twice as frequent as the heterozygous genotype Aa in a HardyWeinberg population?
Solution 3
Let q = frequency of recessive allele, p = frequency of dominant allele. The frequency of homozygous recessives (q^{2}) is twice as frequent at heterozygotes (2pq) when
Therefore, either q = 0 (which is obviously an incorrect solution), or
Proof:
Practice 4
White wool is dependent upon a dominant allele B and black wool upon its recessive allele b. Suppose that a sample of 900 sheep of the Rambouillet breed in Idaho gave the following data: 891 white and 9 black. Estimate the allelic frequencies.
Solution 4
p^{2} (BB) + 2pq (Bb) + q^{2} (bb) = 1.0
If we assume the population is in equilibrium, we can take the square root of that percentage of the population that is of the recessive genotype (phenotype) as our estimator for the frequency of the recessive allele.
Since p + q = 1, the frequency of allele B is 0.9.
Practice 5
In Shorthorn cattle, the genotype C^{R}C^{R} is phenotypically red, C^{R}C^{W} is roan (a mixture of red and white), and C^{W}C^{W} is white. (a) If 108 red, 48 white, and 144 roan animals were found in a sample of Shorthorns from the central valley of California, calculate the estimated frequencies of the C^{R} allele and the C^{W} allele in the gene pool of the population. (b) If this population is completely panmictic, what zygotic frequencies would be expected in the next generation? (c) How does the sample data in part (a) compare with the expectations for the next generation in part (b)? Is the population represented in part (a) in equilibrium?
Solution 5
 Recall that panmixis is synonymous with random mating. We will let the frequency of the C^{R} allele be represented by p = 0.6, and the frequency of the C^{W} allele be represented by q = 0.4. Then according to the Hardy Weinberg law, we would expect as genotypic frequencies in the next generation
 In a sample of size 300 we would expect 0.36(300) = 108 C^{R}C^{R} (red), 0.48(300) = 144 C^{R}C^{W} (roan), and 0.16(300) = 48 C^{W}C^{W} (white). Note that these figures correspond exactly to those of our sample. Since the genotypic and gametic frequencies are not expected to change in the next generation, the original population must already be in equilibrium.
First, let us calculate the frequency of the C^{R} allele. There are 108 red individuals each carrying two C^{R} alleles; 2 × 108 = 216 C^{R} alleles. There are 144 roan individuals each carrying only one C^{R} allele; 1 × 144 = 144 C^{R} alleles. Thus, the total number of C^{R} alleles in our sample is 216 + 144 = 360. Since each individual is a diploid (possessing two sets of chromosomes, each bearing one of the alleles at the locus under consideration), the total number of alleles represented in this sample is 300 × 2 = 600. The fraction of all alleles in our sample of type C^{R} becomes 360/600 = 0.6 or 60%. The other 40%of the alleles in the gene pool must be of type C^{W}. We can arrive at this estimate for C^{W} by following the same procedure as above. There are 48 × 2 = 96 C^{W} alleles represented in the homoygotes and 144 in the heterozygotes; 96 + 144 = 240; 240/600 = 0.4 or 40% C^{W} alleles.
p^{2} = (0.6)^{2} = 0/36C^{R}C^{R}: 2pq = 2 (0.6) (0.4) = 0.48C^{R}C^{W}: q^{2} = (0.4)2 = 0.16C^{W}C^{W}
Practice 6
In the human population, an index finger shorter than the ring finger is thought to be governed by a sexinfluenced gene that appears to be dominant in males and recessive in females. A sample of the males in this population was found to contain 120 short and 210 long index fingers. Calculate the expected frequencies of long and short index fingers in females of this population.
Solution 6
Since the dominance relationships are reversed in the two sexes, let us use all lowercase letters with superscripts to avoid confusion with either dominance or codominance symbolism.
Let p = frequency of s1 allele, q = frequency of s^{2} allele. p^{2} (s^{1}s^{1}) + 2pq (s^{1}s^{2}) + q^{2} (s^{2}s^{2}) = 1.0. Inmales, the allele for long finger s2 is recessive. Then
In females, short index finger is recessive. Then p^{2} = (0.2)^{2} = 0.04 or 4% of the females of this population will probably be short fingered. The other 96% should possess long index fingers.
Practice 7
The ABO blood group system is governed by a multiple allelic system in which some codominant relationships exist. Three alleles, I^{A}, I^{B}, and i, form the dominance hierarchy (I^{A} = I^{B}) > i. (a) Determine the genotypic and phenotypic expectations for this blood group locus from a population in genetic equilibrium. (b) Derive a formula for use in finding the allelic frequencies at the ABO blood group locus. (c) Among New York Caucasians, the frequencies of the ABO blood groups were found to be approximately 49% type O, 36% type A, 12% type B, and 3% type AB. What are the allelic frequencies in this population? (d) Given the population in part (c) above, what percentage of type A individuals are probably homozygous?
Solution 7
 Let p = frequency of I^{A} allele, q = frequency of I^{B} allele, r = frequency of i allele. The expansion of (p + q + r)^{2} yields the zygotic ratio expected under random mating.
 Let A, B, and O represent the phenotypic frequencies of blood groups A, B, and O, respectively. Solving for the frequency of the recessive allele i,
 Frequency of allele i
Solving for the frequency of the I^{A} allele,
Solving for the frequency of the I^{B} allele q = 1 – p – r. Or, following the method for obtaining the frequency of the I^{A} allele,
Presenting the solutions in a slightly different form,
Frequency of I^{B} allele
Frequency of allelle I^{A}
Check: p + q + r = 0.22 + 0.08 + 0.70 = 1.00
Thus, 48/356 = 0.135 or 13.5% of all group A individuals in this population are expected to be homozygous.
Practice 8
White eye color in Drosophila is governed by a sexlinked recessive gene w and wildtype (red) eye color is produced by its dominant allele w^{+}. A laboratory population of Drosophila was found to contain 170 redeyed males and 30 whiteeyed males. (a) Estimate the frequency of the w^{+} allele and the w allele in the gene pool. (b) What percentage of the females in this population would be expected to be whiteeyed?
Solution 8

Thus, 30 of the 200 X chromosomes in this sample carry the recessive allele w.
 Since females possess two X chromosomes (hence two alleles), their expectations may be calculated in the same manner as that used for autosomal genes.
p^{2} (w^{+}w^{+}) + 2pq (w^{+}w) + q^{2} (ww) = 1.0 or 100% of the females
q^{2} = (0.15)^{2} = 0.0225 or 2:25% of all females in the population are expected to be whiteeyed.
Practice 9
The genetics of coat colors in cats was presented in Example 9.4: C^{B}C^{B} females or C^{B}Y males are black, C^{Y}C^{Y} females or C^{Y}Y males are yellow, C^{B}C^{Y} female are tortoiseshell (blotches of yellow and black). A population of cats in London was found to consist of the following phenotypes:
Determine the allelic frequencies using all of the available information.
Solution 9
The total number of C^{B} alleles in this sample is 311 + 2(277) + 54 = 919.The total number of alleles (X chromosomes) in this sample is 353 + 2(338) = 1029. Therefore, the frequency of the C^{B} allele is 919/1029 = 0.893. The frequency of the C^{Y} allele would then be 1 – 0.893 = 0.107.
Practice 10
A human serum protein called haptoglobin has two major electrophoretic variants produced by a pair of codominant alleles Hp1 and Hp2. A sample of 100 individuals has 10 Hp^{1}/Hp^{1}; 35 Hp^{1}/Hp^{2}, and 55 Hp^{2}/Hp^{2}. Are the genotypes in this sample conforming to the frequencies expected for a HardyWeinberg population within statistically acceptable limits?
Solution 10
First, we must calculate the allelic frequencies.
Let
From these gene (allelic) frequencies we can determine the genotypic frequencies expected according to the HardyWeinberg equation.
Converting these genotypic frequencies to numbers based on a total sample size of 100, we can do a chisquare test.
This is not a significant x^{2} value, and we may accept the hypothesis that this sample (and hence presumably the population from which it was drawn) is conforming to the equilibrium distribution of genotypes.
Practice 11
One of the "breeds" of poultry has been largely built on a singlegene locus, that for "frizzled" feathers. The frizzled phenotype is produced by the heterozygous genotype M^{N}M^{F}. One homozygote M^{F}M^{F} produces extremely frizzled birds called "woolies." The other homozygous genotype MNMN has normal plumage. A sample of 1000 individuals of this "breed" in the United States contained 800 frizzled, 150 normal, and 50 wooly birds. Is this population in equilibrium?
Solution 11
Let
Chisquare test for conformity to equilibrium expectations gives the following results:
This highly significant chisquare value will not allow us to accept the hypothesis of conformity with equilibrium expectations. The explanation for the large deviation from the equilibrium expectations is twofold. Much artificial selection (by people) is being practiced. The frizzled heterozygotes represent the "breed" type and are kept for show purposes as well as for breeding by bird fanciers. Such breeders dispose of (cull) many normal and wooly types. Natural selection is also operative on the wooly types because they tend to lose their feathers (loss of insulation) and eat more feed just to maintain themselves, are slower to reach sexual maturity, and lay fewer eggs than do the normal birds.
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