Mendel's explanation of the results of a dihybrid cross

Given
the
principles revealed in a monohybrid
cross, Mendel hypothesized that the result of two
characters segregating simultaneously (a dihybrid cross) would
be the statistical product of their independent occurrence.
Consider two characters, seed color and seed shape. As
previously shown, Y
dominates y to
determine seed color. Mendel also showed that the "round" R factor dominates the
"wrinkled" r factor
to determine seed shape. He then proceeded to test his
hypothesis experimentally.

The P cross is between true-breeding lines of wrinkled yellow peas (rrYY) and round green peas (RRyy). The F_{1} offspring
are therefore all RrYy,
and are all round and
yellow. In forming the F_{2} plants, the alleles at the two
loci segregate
independently. That is, the chance of getting an R allele and a Y allele is 1/2 x 1/2, of getting an
R and a y 1/2 x 1/2, and so on.
Thus, all four possible diallelic combinations occur in
equal proportions, and each has a probability of 1/4. The same is true
for both parents. Given four possible gamete types in each
parent, there are 4 x 4 = 16 possible F_{2} combinations, and the
probability of any particular dihybrid type is 1/4 x 1/4 = 1/16. The
phenotypes and phenotypic ratios of these 16 genotype can be
determined by inspection of the diagram above, called a Punnet Square after the
geneticist who first used it.

Alternatively, recall that the phenotypic ratio expected for either character is 3:1, either 3 "Y" : 1 "y", or 3 "R" : 1 "R". Then, algebra tells us that

That is, we expect a characteristic 9:3:3:1 phenotypic ratio of round-yellow : wrinkled-yellow : round-green : wrinkled-green pea seeds.

To predict the genotypic ratios, recall that for each gene the ratio is 1:2:1 :: AA:Aa:aa . Then, algebraically

The P cross is between true-breeding lines of wrinkled yellow peas (rrYY) and round green peas (RRyy). The F

Alternatively, recall that the phenotypic ratio expected for either character is 3:1, either 3 "Y" : 1 "y", or 3 "R" : 1 "R". Then, algebra tells us that

(3Y + 1y) x (3R + 1r) = 9YR + 3Yr + 3Ry + 1 ry

That is, we expect a characteristic 9:3:3:1 phenotypic ratio of round-yellow : wrinkled-yellow : round-green : wrinkled-green pea seeds.

To predict the genotypic ratios, recall that for each gene the ratio is 1:2:1 :: AA:Aa:aa . Then, algebraically

(1YY + 2Yy +
1yy) x (1RR +
2Rr + 1rr) = 1 YYRR + 2 YYRr + 1 YYrr + 2YyRR + 4YyRr +
2 yyRR + 1yyRR + 2yyRr + 1yyrr

That is, we expect a
characteristic 1:2:1:2:4:2:1:2:1 ratio of the
nine possible genotypes. These nine genotypes can be
grouped into four phenotypes, for example 1 YYRR + 2 YYRr + 2 YyRR +
4 YyRr = 9Y-R- round, yellow peas. The ratio
of these phenotypes is of course 9:3:3:1.

Mendel reported the results of several dihybrid crosses, of which (7)(7-1)/2 = 21 are possible with seven characters. He performed several trihybrid crosses as well.

Homework:

(1) Repeat the analysis above for a cross of RRYY x rryy.

(2) Predict the phenotypic and genotypic ratios of a trihybrid cross. Pea plants may be tall or short: use T for the tall allele, which is dominant to the t allele for short plants. How would you diagram such a cross?

Mendel reported the results of several dihybrid crosses, of which (7)(7-1)/2 = 21 are possible with seven characters. He performed several trihybrid crosses as well.

Homework:

(1) Repeat the analysis above for a cross of RRYY x rryy.

(2) Predict the phenotypic and genotypic ratios of a trihybrid cross. Pea plants may be tall or short: use T for the tall allele, which is dominant to the t allele for short plants. How would you diagram such a cross?

Figure ©2002 by Griffiths et al.; all text material ©2014 by Steven M. Carr