Assume that gene A controls the conversion of a white pigment, P0 into another white pigment, P1 the dominant allele A specifies an enzyme necessary for this conversion, and the recessive allele a specifies a defective enzyme without biochemical function. Gene B controls the conversion of the white pigment, P1 into a pink pigment, P2 the dominant allele, B, produces the enzyme necessary for this conversion, and the recessive allele, b, produces a defective enzyme. The dominant allele, C, of the third gene specifies an enzyme that converts the pink pigment, P2 into a red pigment, P3 its recessive allele, c, produces an altered enzyme that cannot carry out this conversion. The dominant allele, D, of a fourth gene produces a polypeptide that completely inhibits the function of enzyme C; that is, it blocks the reaction P2-P3 Its recessive allele, d, produces a defective polypeptide that does not block this reaction. Assume that flower color is determined solely by these four genes and that they assort independently. In the F2 of a cross between plants of the genotype AA bb CC DD and plants of the genotype aa BB cc dd, what proportion of the plants will have (a) red flowers? (b) pink flowers? (c) white flowers?

In the presence of the recessive alleles a, b or c, the three reactions or some of them might not occur ( the X represents the inhibition of the reaction):

Po (white 0) --X--> P1 (White 1) --X--> P2 (Pink)--X--> P3 (Red)

a b c

In the presence of allele D, the last reaction is not possible, because D inhibits the action of C

P2 (Pink) --X-->P3 (Red)

C + D

So, in order to get Red flowers we must have the next genotype: A-B-C-dd (The three reactions can occur, and D must be absent)
To get Pink flowers we must have the next genotypes: A-B-C-D- (D Inhibits C, hence it inhibits the third reaction) or A-B-ccD- or A-B-ccdd (cc does not allow the third reaction to happen, independently of the D gene presence
To get White flowers we must have the next genotypes: aaB-C-D-,aabbC-D-, aaB-ccD-, aabbccD-, aabbccdd, aaB-C-dd, aaB-ccdd, aabbC-dd, A-bbC-D-, A-bbccD-, A-bbC-dd, A-bbccdd (aa must be present to get a white0 flower, or bb must be present to get a white1 flower)

In the proposed problem, we have that the parental are AAbbCCDD and aaBBccdd, so we can tell that the whole F1 progeny will be heterozygous for the four genes, this is AaBbCcDd.

Knowing that these four genes assort independently, then we can infer that the F2 will have the next genotypic proportions for each gene:

AA + Aa=3/4
aa= 1/4
BB + bb= 3/4
bb = 1/4
CC + Cc = 3/4
cc= 1/4
DD + Dd =3/4
dd= 1/4

So, to get each of the mentioned genotypes for each trait, we must multiply their respective genotypic proportions, like this:

Red: A- x B- x C- x dd: 3/4 x 3/4 x 3/4 x 1/4 = 27/256

Pink: A- x B- x C- x D-: 3/4 x 3/4 x 3/4 x 3/4 = 81/256

A- x B- x cc x D-: 3/4 x 3/ x 1/4 x 3/4 = 27/256

A- x B- x cc x dd: 3/4 x 3/4 x 1/4 x 1/4= 9/256

By summing these proportions, we can get the proportion of all the pink flowers: 27/256 + 81/256 + 27/256 + 9/256= 117/256

White: To get the proportion of white flowers, you can repeat the same multiplications as before for each possible white genotype and then sum the results, as we did for pink flowers. Or you can extract from the total number of flowers (256), the amounts of red and pink flowers (27 and 117). This is: 256 - (27 + 117) = 112.

Then 112/256 is the proportion of white flowers.