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
©2013 by Steven M. Carr