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Chapter 14: Problem 7

The genotype of \(\mathrm{F}_{1}\) individuals in a tetrahybrid cross is AaBbCcDd. Assuming independent assortment of these four genes, what are the probabilities that \(\mathrm{F}_{2}\) offspring will have the following genotypes? $$\begin{array}{ll}{\text { (a) }} & {\text { abbccdd }} \\ {\text { (b) } A a B b C c D d} & {\text { (e) AaBBCCdd }} \\ {\text { (c) } A A B B C C D D}\end{array}$$

Short Answer

Expert verified
P(abbccdd) = 1/256, P(AaBbCcDd) = 1/16, P(AaBBCCdd) = 1/128

Step by step solution

01

Calculate probability for a single trait

Consider each gene independently for simplicity. For any gene, the probability of each genotype can be determined using a Punnett square for one gene pair. Each pair follows a ratio of 1:2:1 for AA, Aa, and aa respectively, when Aa x Aa: \( P(\text{AA}) = 1/4 \), \( P(\text{Aa}) = 1/2 \), \( P(\text{aa}) = 1/4 \).
02

Probability of genotype 'abbccdd'

Since each gene independently assorts, calculate the probability of each part of the genotype being homozygous recessive (aa) or homozygous dominant (AA): \( P(aabbccdd) = P(aa) \times P(bb) \times P(cc) \times P(dd) \) Which gives: \( P(aa) = 1/4 \) \( P(bb) = 1/4 \) \( P(cc) = 1/4 \) \( P(dd) = 1/4 \) Thus, \( P(aabbccdd) = \left( \frac{1}{4} \right) \times \left( \frac{1}{4} \right) \times \left( \frac{1}{4} \right) \times \left( \frac{1}{4} \right) = \frac{1}{256} \).
03

Probability of genotype 'AaBbCcDd'

Similarly, calculate for heterozygous genotype, each using the same independent assortment principle: \( P(AaBbCcDd) = P(Aa) \times P(Bb) \times P(Cc) \times P(Dd) \) Which gives: \( P(Aa) = 1/2 \) \( P(Bb) = 1/2 \) \( P(Cc) = 1/2 \) \( P(Dd) = 1/2 \) Thus, \( P(AaBbCcDd) = \left( \frac{1}{2} \right) \times \left( \frac{1}{2} \right) \times \left( \frac{1}{2} \right) \times \left( \frac{1}{2} \right) = \frac{1}{16} \).
04

Probability of genotype 'AaBBCCdd'

Calculate for mixed homozygous and heterozygous traits: \( P(AaBBCCdd) = P(Aa) \times P(BB) \times P(CC) \times P(dd) \) Which gives: \( P(Aa) = 1/2 \) \( P(BB) = 1/4 \) \( P(CC) = 1/4 \) \( P(dd) = 1/4 \) Thus, \( P(AaBBCCdd) = \left( \frac{1}{2} \right) \times \left( \frac{1}{4} \right) \times \left( \frac{1}{4} \right) \times \left( \frac{1}{4} \right)= \frac{1}{128} \).
05

Probability of genotype 'AABBCCDD'

Calculate for homozygous dominant genotype: \( P(AABBCCDD) = P(AA) \times P(BB) \times P(CC) \times P(DD) \) Which gives: \( P(AA) = 1/4 \) \( P(BB) = 1/4 \) \( P(CC) = 1/4 \) \( P(DD) = 1/4 \) Thus, \( P(AABBCCDD) = \left( \frac{1}{4} \right) \times \left( \frac{1}{4} \right) \times \left( \frac{1}{4} \right) \times \left( \frac{1}{4} \right)= \frac{1}{256} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Independent Assortment
Independent assortment is a fundamental principle of genetics discovered by Gregor Mendel. It states that the alleles for different genes segregate, or separate, independently of one another during the formation of gametes. This means that the inheritance of an allele for one trait does not influence the inheritance of an allele for another trait. For example, in a tetrahybrid cross involving four genes, the allele a from the gene A/a independently sorts from the allele b from gene B/b, and so forth. This independence leads to the potential combinations in offspring being a product of the individual probabilities for each gene.
Punnett Square
A Punnett square is a diagram that is used to predict the genotype and phenotype combinations in offspring from a particular genetic cross. Named after Reginald C. Punnett, this tool helps visualize how alleles from parent organisms can combine in their progeny. In a tetrahybrid cross, where we analyze four different genes, a Punnett square can quickly become complex. However, by considering each gene independently, we can simplify the process. For each gene pair (like Aa x Aa), the square shows ratios such as 1:2:1 for AA, Aa, and aa, respectively. From these, the probabilities of combinations for multiple genes can be identified and calculated.
Mendelian Inheritance
Mendelian inheritance is the set of primary principles relating to the transmission of hereditary characteristics from parent organisms to their offspring. These principles were first described by Gregor Mendel in the 19th century and include concepts like segregation, dominance, recessiveness, and independent assortment. Tetrahybrid crosses, which involve four distinct traits, rely heavily on these principles. For instance, each trait follows the segregation principle where each allele separates into different gametes. By employing Mendel鈥檚 principles, we can predict the possible outcomes and their probabilities in the F2 generation.
Genetic Ratios
Genetic ratios are the expected frequencies of different genotypes or phenotypes in the offspring, resulting from particular genetic crosses. In a tetrahybrid cross, each gene independently follows a specific ratio according to Mendelian inheritance. When examining individual genes from a cross Aa x Aa, the ratios are typically 1:2:1 for genotypes AA, Aa, and aa, respectively. For more complicated crosses, the overall genetic ratios for the offspring are determined by multiplying the individual probabilities of these ratios. This multiplication yields insights like the probability of observing a genotype like AABBCCDD in the F2 generation.

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Most popular questions from this chapter

SCIENTIFIC INQUIRY You are handed a mystery pea plant with tall stems and axial flowers and asked to determine its genotype as quickly as possible. You know that the allele for tall stems \((T)\) is dominant to that for dwarf stems \((t)\) and that the allele for axial flowers \((A)\) is dominant to that for terminal flowers (a). (a) Identify all the possible genotypes for your mystery plant. (b) Describe the one cross you would do, out in your garden, to determine the exact genotype of your mystery plant. (c) While waiting for the results of your cross, you predict the results for each possible genotype listed in part a. Explain how you do this and why this is not called "performing a cross." a Explain how the results of your cross and your predictions will help you learn the genotype of your mystery plant.

DRAW IT A pea plant heterozygous for inflated pods (Ii) is crossed with a plant homozygous for constricted pods (ii). Draw a Punnett square for this cross to predict genotypic and phenotypic ratios. Assume that pollen comes from the ii plant.

EVOLUTION CONNECTION Over the past half century, there has been a trend in the United States and other developed countries for people to marry and start families later in life than did their parents and grandparents. What effects might this trend have on the incidence (frequency) of late-acting dominant lethal alleles in the population?

A man with type A blood marries a woman with type \(\mathrm{B}\) blood. Their child has type O blood. What are the genotypes of these three individuals? What genotypes, and in what frequencies, would you expect in future offspring from this marriage?

Hemochromatosis is an inherited disease caused by a recessive allele. If a woman and her husband, who are both carriers, have three children, what is the probability of each of the following? (a) All three children are of normal phenotype. (b) One or more of the three children have the disease. (c) All three children have the disease. (d) At least one child is phenotypically normal. (Note: It will help to remember that the probabilities of all possible outcomes always add up to 1.)
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