In contrast to a monohybrid cross, a dihybrid cross is a cross between F₁ offspring (first-generation offspring) of two individuals that differ in two traits of particular interest. For example, BB × bb (see the Punnett square). Example: B = brown. b = blue. BB = Dark brown. Bb = Brown (not blue). bb = Blue.

A dihybrid cross is often used to test for dominant and recessive genes in two separate characteristics. Such a cross has a variety of uses in Mendelian genetics.

For example: RRyy/rrYY or RRYY/rryy parents result in F₁ offspring that are heterozygous for both R and Y (RrYy).^[1]

Meiosis (cell reduction) is the cellular process of gamete creation. It is where sperm and eggs get the unique set of genetic information that will be used in the development and growth of the offspring of the mating. The rules of meiosis, as they apply to the dihybrid, are codified in Mendel's first law and Mendel's second law, which are also called the Law of Segregation and the Law of Independent Assortment, respectively.

For genes on separate chromosomes, each allele pair shows independent segregation. If the first filial generation (F₁ generation) produces four offspring, the second filial generation, which occurs by crossing the members of the first filial generation, shows a phenotypic (appearance) ratio of 9:3:3:1.

The dihybrid cross illustrates the law of independent assortment. The separation of gene pairs in a given pair of chromosomes and the distribution of the genes to gametes during meiosis is entirely independent of the distribution of other gene pairs in other pairs of chromosome.

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 F₁ offspring are therefore all RrYy, and are all round and yellow. In forming the F₂ 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 F₂ 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

(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.

Mendel's Law of Independent Assortment

To this point we have followed the expression of only one gene. Mendel also performed crosses in which he followed the segregation of two genes. These experiments formed the basis of his discovery of his second law, the law of independent assortment. First, a few terms are presented.
Dihybrid cross - a cross between two parents that differ by two pairs of alleles (AABB x aabb)
Dihybrid- an individual heterozygous for two pairs of alleles (AaBb)
Again a dihybrid cross is not a cross between two dihybrids. Now, let's look at a dihybrid cross that Mendel performed.
Parental Cross: Yellow, Round Seed x Green, Wrinkled Seed
F₁ Generation: All yellow, round
F₂ Generation: 9 Yellow, Round, 3 Yellow, Wrinkled, 3 Green, Round, 1 Green, Wrinkled
At this point, let's diagram the cross using specific gene symbols.

Choose Symbol	Seed Color: Yellow = G; Green = g
	Seed Shape: Round = W; Wrinkled = w

The dominance relationship between alleles for each trait was already known to Mendel when he made this cross. The purpose of the dihybrid cross was to determine if any relationship existed between different allelic pairs.
Let's now look at the cross using our gene symbols.