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 **product** of their independent occurrence.
Consider two characters, seed color and seed shape. As
previously shown, Y
dominates y to
determine seed color, and R
factor for "*round*" dominates the **r** factor
for "*wrinkled*" to determine seed shape. He then proceeded
to test his hypothesis experimentally.

The P (**Parental**) 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
with an equal 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, the expected phenotypic ratios of the two traits together can be calculated**algebraically
**as a **binomial
distribution**:

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 (

Alternatively, recall that the phenotypic ratio expected for either character is 3:1, either 3 "Y" : 1 "y", or 3 "R" : 1 "R". Then, the expected phenotypic ratios of the two traits together can be calculated

(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 some but not all of the**"***7 choose 2***"
= ****(7)(7-1)/(2) ****= 21** possible dihybrid crosses with seven characters. He
performed several trihybrid crosses as well.

Homework:

(1) Repeat the analysis above with 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 some but not all of the

Homework:

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

(2) Predict the

Figure ©2002 by Griffiths et al.; all text material ©2016 by Steven M. Carr