We have dealt mainly with only two genes, but the same principles hold for more than two genes. Consider the following cross:
A/a ; B/b ; C/c ; D/d ; E/e × a/a ; B/b; c/c ; D/d ; e/e
a. What proportion of progeny will phenotypically resemble (1) the first parent, (2) the second parent, (3) either parent, and (4) neither parent?
b. What proportion of progeny will be genotypically the same as (1) the first parent, (2) the second parent, (3) either parent, and (4) neither parent? Assume independent assortment.
To determine the proportion of progeny with specific phenotypes and genotypes, we can use the principles of Mendelian genetics and Punnett squares.
a. Proportions of progeny phenotypically resembling each parent:
(1) The first parent: To have the phenotype of the first parent (A/a ; B/b ; C/c ; D/d ; E/e), all offspring must inherit at least one copy of each allele from the first parent.
The possible combinations for offspring to have the same phenotype as the first parent are: A/a ; B/b ; C/c ; D/d ; E/e or A/a ; B/b ; c/c ; D/d ; E/e or A/a ; B/b ; C/c ; d/d ; E/e or A/a ; B/b ; C/c ; D/d ; e/e.
So, in each combination, the proportion of inheriting the corresponding allele from the first parent is 1/2.
Thus, the proportion of progeny phenotypically resembling the first parent is (1/2) * (1/2) * (1/2) * (1/2) * (1/2) = 1/32.
(2) The second parent: To have the phenotype of the second parent (a/a ; B/b ; c/c ; D/d ; e/e), all offspring must inherit at least one copy of each allele from the second parent.
The possible combinations for offspring to have the same phenotype as the second parent are: a/a ; B/b ; c/c ; D/d ; e/e.
The proportion of inheriting the corresponding allele from the second parent is 1/1 for each allele.
Thus, the proportion of progeny phenotypically resembling the second parent is (1/1) * (1/2) * (1/1) * (1/2) * (1/2) = 1/8.
(3) Either parent: To have the phenotype resembling either parent, the offspring can inherit at least one copy of each allele from either parent.
The proportion of inheriting the corresponding allele from either parent is 1/2 for each allele.
Thus, the proportion of progeny phenotypically resembling either parent is (1/2) * (1/2) * (1/2) * (1/2) * (1/2) + (1/1) * (1/2) * (1/1) * (1/2) * (1/2) = 9/32.
(4) Neither parent: To have a phenotype that does not resemble either parent, the offspring must inherit a different combination of alleles.
The proportion of inheriting the corresponding allele from different combinations is (1/2) * (1/2) * (1/2) * (1/2) * (1/2) = 1/32.
b. Proportions of progeny genotypically resembling each parent:
The principles of independent assortment state that each gene segregates independently during gamete formation. Therefore, the proportion of genotypically resembling each parent is the same as the proportion of phenotypically resembling each parent.
Therefore, the proportions of progeny genotypically resembling each parent are the same as those determined in part (a):
(1) The first parent: 1/32
(2) The second parent: 1/8
(3) Either parent: 9/32
(4) Neither parent: 1/32.
To answer these questions, let's break down the cross and use the principles of Mendelian genetics.
Given cross:
Parent 1: A/a; B/b; C/c; D/d; E/e
Parent 2: a/a; B/b; c/c; D/d; e/e
a. Proportion of progeny phenotypically resembling:
(1) the first parent:
To resemble the first parent, the offspring must inherit the dominant allele at each gene from the first parent. The first parent is heterozygous at all genes. For each gene, the probability of inheriting the dominant allele is 1/2.
So, the probability of inheriting A/a from the first parent is 1/2, B/b is 1/2, C/c is 1/2, D/d is 1/2, and E/e is 1/2.
Multiplying these probabilities together, we get (1/2) x (1/2) x (1/2) x (1/2) x (1/2) = 1/32.
Therefore, 1/32 of the progeny will phenotypically resemble the first parent.
(2) the second parent:
To resemble the second parent, the offspring must inherit the recessive allele at each gene from the second parent. The second parent is homozygous recessive at all genes. Therefore, the probability of inheriting the recessive allele from the second parent is 1.
So, the probability of inheriting a/a from the second parent is 1, B/b is 1/2, c/c is 1, D/d is 1, and e/e is 1.
Multiplying these probabilities together, we get 1 x (1/2) x 1 x 1 x 1 = 1/2.
Therefore, 1/2 of the progeny will phenotypically resemble the second parent.
(3) either parent:
To resemble either parent, the offspring can inherit the respective alleles from either parent.
Using the probabilities we calculated above, the proportion of progeny resembling either parent is the sum of the probabilities from (1) and (2):
1/32 + 1/2 = 17/32.
Therefore, 17/32 of the progeny will phenotypically resemble either parent.
(4) neither parent:
To not resemble either parent, the offspring must inherit different allele combinations at each gene, compared to both parents.
The probability of inheriting a different allele combination at each gene from both parents is calculated by subtracting the probability of resembling either parent from 1:
1 - 17/32 = 15/32.
Therefore, 15/32 of the progeny will phenotypically resemble neither parent.
b. Proportion of progeny genotypically the same as:
(1) the first parent:
To be genotypically the same as the first parent, the offspring must inherit the exact same allele combination at each gene.
Since the first parent is heterozygous at all genes, there are two possible allele combinations for each gene: A or a, B or b, C or c, D or d, E or e.
Therefore, the proportion of progeny genotypically the same as the first parent is 1/2 x 1/2 x 1/2 x 1/2 x 1/2 = 1/32.
(2) the second parent:
To be genotypically the same as the second parent, the offspring must inherit the exact same allele combination at each gene.
Since the second parent is homozygous recessive at all genes, there is only one possible allele combination for each gene: a, B, c, D, e.
Therefore, the proportion of progeny genotypically the same as the second parent is 1 x 1/2 x 1 x 1 x 1 = 1/2.
(3) either parent:
To be genotypically the same as either parent, the offspring can inherit the respective allele combinations from either parent.
Using the probabilities we calculated above, the proportion of progeny genotypically the same as either parent is the sum of the probabilities from (1) and (2):
1/32 + 1/2 = 17/32.
(4) neither parent:
To not be genotypically the same as either parent, the offspring must inherit different allele combinations at each gene, compared to both parents.
The probability of inheriting a different allele combination at each gene from both parents is calculated by subtracting the probability of being genotypically the same as either parent from 1:
1 - 17/32 = 15/32.
Therefore, 15/32 of the progeny will genotypically be different from both parents.
Please note that these calculations assume independent assortment, meaning that the genes are not physically linked on the same chromosome.