This section in each chapter contains a few solved problems that show how to approach the problem sets that follow. The purpose of the problem sets is to challenge your understanding of the genetic principles learned in the chapter. The best way to demonstrate an understanding of a subject is to be able to use that knowledge in a real or simulated situation. Be forewarned that there is no machine-like way of solving these problems. The three main resources at your disposal are the genetic principles just learned, logic, and trial and error.

Here is some general advice before beginning. First, it is absolutely essential to read and understand all of the problem. Most of the problems use data taken from research that somebody actually carried out: ask yourself why the research might have been initiated and what was the probable goal. Find out exactly what facts are provided, what assumptions have to be made, what clues are given in the problem, and what inferences can be made from the available information. Second, be methodical. Staring at the problem rarely helps. Restate the information in the problem in your own way, preferably using a diagrammatic representation or flowchart to help you think out the problem. Good luck.

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SOLVED PROBLEM 1. Crosses were made between two pure lines of rabbits that we can call A and B. A male from line A was mated with a female from line B, and the F₁ rabbits were subsequently intercrossed to produce an F₂. Three-fourths of the F₂ animals were discovered to have white subcutaneous fat, and one-fourth had yellow subcutaneous fat. Later, the F₁ was examined and was found to have white fat. Several years later, an attempt was made to repeat the experiment by using the same male from line A and the same female from line B. This time, the F₁ and all the F₂ (22 animals) had white fat. The only difference between the original experiment and the repeat that seemed relevant was that, in the original, all the animals were fed fresh vegetables, whereas in the repeat, they were fed commercial rabbit chow. Provide an explanation for the difference and a test of your idea.

Solution

The first time that the experiment was done, the breeders would have been perfectly justified in proposing that a pair of alleles determine white versus yellow body fat because the data clearly resemble Mendel’s results in peas. White must be dominant, and so we can represent the white allele as W and the yellow allele as w. The results can then be expressed as follows:

No doubt, if the parental rabbits had been sacrificed, one parent (we cannot tell which) would have been predicted to have white fat and the other yellow. Luckily, the rabbits were not sacrificed, and the same animals were bred again, leading to a very interesting, different result. Often in science, an unexpected observation can lead to a novel principle, and, rather than moving on to something else, it is useful to try to explain the inconsistency. So why did the 3:1 ratio disappear? Here are some possible explanations.

First, perhaps the genotypes of the parental animals had changed. This type of spontaneous change affecting the whole animal, or at least its gonads, is very unlikely, because even common experience tells us that organisms tend to be stable to their type.

Second, in the repeat, the sample of 22 F₂ animals did not contain any yellow fat simply by chance (“bad luck”). This explanation, again, seems unlikely, because the sample was quite large, but it is a definite possibility.

A third explanation draws on the principle that genes do not act in a vacuum; they depend on the environment for their effects. Hence, the formula “genotype + environment = phenotype” is a useful mnemonic. A corollary of this formula is that genes can act differently in different environments; so

genotype 1 + environment 1 = phenotype 1

and

genotype 1 + environment 2 = phenotype 2

In the present problem, the different diets constituted different environments, and so a possible explanation of the results is that the homozygous recessive w/w produces yellow fat only when the diet contains fresh vegetables. This explanation is testable. One way to test it is to repeat the experiment again and use vegetables as food, but the parents might be dead by this time. A more convincing way is to breed several of the white-fatted F₂ rabbits from the second experiment. According to the original interpretation, some of them should be heterozygous, and, if their progeny are raised on vegetables, yellow fat should appear in Mendelian proportions. For example, if a cross happened to be W/w and w/w, the progeny would be white fat and yellow fat.

If this outcome did not happen and no progeny having yellow fat appeared in any of the matings, we would be forced back to the first or second explanation. The second explanation can be tested by using larger numbers, and if this explanation doesn’t work, we are left with the first explanation, which is difficult to test directly.

As you might have guessed, in reality, the diet was the culprit. The specific details illustrate environmental effects beautifully. Fresh vegetables contain yellow substances called xanthophylls, and the dominant allele W gives rabbits the ability to break down these substances to a colorless (“white”) form. However, w/w animals lack this ability, and the xanthophylls are deposited in the fat, making it yellow. When no xanthophylls have been ingested, both W/ – and w/w animals end up with white fat.

SOLVED PROBLEM 2. Phenylketonuria (PKU) is a human hereditary disease resulting from the inability of the body to process the chemical phenylalanine, which is contained in the protein that we eat. PKU is manifested in early infancy and, if it remains untreated, generally leads to mental retardation. PKU is caused by a recessive allele with simple Mendelian inheritance.

A couple intends to have children but consult a genetic counselor because the man has a sister with PKU and the woman has a brother with PKU. There are no other known cases in their families. They ask the genetic counselor to determine the probability that their first child will have PKU. What is this probability?

Solution

What can we deduce? If we let the allele causing the PKU phenotype be p and the respective normal allele be P, then the sister and brother of the man and woman, respectively, must have been p/p. To produce these affected persons, all four grandparents must have been heterozygous normal. The pedigree can be summarized as follows:

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When these inferences have been made, the problem is reduced to an application of the product rule. The only way in which the man and woman can have a PKU child is if both of them are heterozygotes (it is obvious that they themselves do not have the disease). Both the grandparental matings are simple Mendelian monohybrid crosses expected to produce progeny in the following proportions:

We know that the man and the woman are normal, and so the probability of each being a heterozygote is 2/3 because, within the P/– class, 2/3 are P/p and 1/3 are P/P.

The probability of both the man and the woman being heterozygotes is 2/3 × 2/3 = 4/9. If both are heterozygous, then one-quarter of their children would have PKU, and so the probability that their first child will have PKU is 1/4 and the probability of their being heterozygous and of their first child’s having PKU is 4/9 × 1/4 = 4/36 = 1/9, which is the answer.

SOLVED PROBLEM 3. A rare human disease is found in a family as shown in the accompanying pedigree.

Solution

1 × 4: Must be A/a ♀ × a/Y ♂.

2 × 3: Must be a/Y ♂ × A/a ♀ (same as 1 × 4).

2 × 8: Must be a/Y ♂ × a/a ♀ (all progeny normal).

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