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The following problem is from Schaum's Outlines on Genetics, 5th edition, by Elrod and Stansfield. I'm having some trouble solving it. It is found under a section entitled "Interactions with Three or More Factors" in the problems at the end of the chapter on Epistasis.

Problem 4.30 (pg. 112): If a pure-white onion strain is crossed to a pure-yellow strain, the F$_2$ ratio is 12 white : 3 red : 1 yellow. If another pure-white onion is crossed to a pure-red onion, the F$_2$ ratio is 9 red : 3 yellow : 4 white. (a) What percentage of the white F$_2$ from the second mating would be homozygous for the yellow allele? (b) If the white F$_2$ (homozygous for the yellow allele) of part (a) is crossed to the pure-white parent of the first mating mentioned at the beginning of this problem, determine the F$_1$ and F$_2$ phenotypic expectations.

In the first crossing, it looks like the type of interaction is one of dominant epistasis between two loci. So, we could call the alleles for the first locus W and w and the alleles for the second locus A and a. Then W-A- and W-aa would have a white phenotype, wwA- would have a red phenotype, and wwaa would have a yellow phenotype. This would be consistent with the ratios provided in the problem.

In order to explain the 9 red : 3 yellow : 4 white, a third locus can be considered with alleles B and b. Proceeding as above, where we had the locus for W or w epistatic with respect to the locus for A and a, we could have the same locus for W or w epistatic with respect to the locus for B and b. This time, wwB- and wwbb would be white, W-B- would be red, and W-bb would be yellow. This would be consistent with the 9 red : 3 yellow : 4 white ratio.

This is where I get stuck. If the locus for W and w is epistatic in a dominant way with respect to one locus and in a recessive way with respect to another locus, then all onions would be white. Therefore, I must have done something wrong. Am I on the right track?

This isn't really a homework problem, despite the "homework" tag.

Here are the answers provided in the text for the problem:

Problem 4.30 (pg. 115): (a) 25% (b) F$_1$: all white; F$_2$ : 52 white : 9 red : 3 yellow

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Is it necessary to have atleast one common loci for the two crosses? I fail to see that mentioned in the question. –  Satwik Pasani Jun 6 at 10:55
    
@SatwikPasani: I've added the answers in the text for the problem to the original question (I apologize, it would have been better to do this when the question was first posted). –  Andrew Jun 6 at 16:31

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up vote 3 down vote accepted

After reading Satwik Pasani's comment to the original problem, I tried using the hypostatic locus as the common locus in the epistatic relationships, instead of the locus causing the onions to be white. The following solution fits the answers supplied in the book.

Three loci can be used to determine the colors of the onion, the alleles of which can be denoted by C, D, and E. Suppose that C and D are each epistatic to E, such that the phontypes of the onion are given as:

cc---- white
--D--- white
C-ddE- red
C-ddee yellow

Then the first crossing in the problem can be given by CCDDEE x CCddee. The second crossing could be ccddee x CCddEE.

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