To solve the problem of finding the proportions of the genotypes AABbCCDd, AaBBCcDd, AaBbCcDd, and aabbccdd when a hybrid (AaBbCcDd) is selfed, we can follow these steps:
### Step 1: Understand the Cross
We are dealing with a tetrahybrid cross involving four genes (A, B, C, D) with two alleles each (dominant and recessive). The parent genotype is AaBbCcDd, which is heterozygous for all four traits.
### Step 2: Determine the Gametes
When AaBbCcDd is selfed (crossed with itself), we need to determine the possible gametes produced by this hybrid. Each gene can produce two types of gametes:
- For A: A or a
- For B: B or b
- For C: C or c
- For D: D or d
Thus, the total number of different gametes produced can be calculated as:
\[ 2^n \]
where \( n \) is the number of heterozygous gene pairs. Here, \( n = 4 \):
\[ 2^4 = 16 \text{ different gametes} \]
### Step 3: Set Up the Punnett Square
To find the proportions of the offspring genotypes, we can set up a Punnett square with 16 squares (since there are 16 gametes). Each square will represent a possible genotype of the offspring.
### Step 4: Count the Genotypes
Now we need to count how many times each genotype appears in the Punnett square:
1. **AABbCCDd**: This genotype can occur when we have AA from A, Bb from B, CC from C, and Dd from D. The probability of this genotype is \( \frac{1}{16} \).
2. **AaBBCcDd**: This genotype can occur when we have Aa from A, BB from B, Cc from C, and Dd from D. The probability of this genotype is \( \frac{3}{16} \).
3. **AaBbCcDd**: This genotype is the original hybrid and can occur in multiple combinations. The probability of this genotype is \( \frac{6}{16} \).
4. **aabbccdd**: This genotype can occur when we have aa from A, bb from B, cc from C, and dd from D. The probability of this genotype is \( \frac{1}{16} \).
### Step 5: Calculate the Proportions
Now we can summarize the proportions of each genotype:
- AABbCCDd: 1
- AaBBCcDd: 3
- AaBbCcDd: 6
- aabbccdd: 1
### Final Proportions
Thus, the final proportions of the genotypes when AaBbCcDd is selfed are:
- AABbCCDd : AaBBCcDd : AaBbCcDd : aabbccdd = 1 : 3 : 6 : 1
