After examining each characteristic individually, Mendel also studied them in pairs. For example, he looked at seed shape (round vs. wrinkled) and seed color (yellow vs. green). Each of these characteristics studied individually had shown the typical 3:1 phenotypic ratio. When Mendel crossed true-breeding plants that produced yellow, round seeds with true breeding plants that produced green, wrinkled seeds, he observed the following results:

The F1 and F2 generations were identical to those shown above even if the parents were yellow, wrinkled and green, round.

How can we explain the phenotypic ratio in this case? Well, it's fairly straightforward if we consider it as two monohybrid crosses done separately:

• Yellow (G) is dominant to green (g), therefore the F1's are all yellow, and the F2's are 3/4 yellow, 1/4 green
• Round (W) is dominant to wrinkled (w), therefore the F1's are all round, and the F2's are 3/4 round, 1/4 wrinkled

According to the Product Law, the probability of getting a yellow round plant is equal to the product of the probability of getting a yellow-seeded plant (3/4) and the probability of getting a round-seeded plant (3/4).

The Product Law can be used to explain the observations for the other phenotypes as well. Take a few moments and work them out.

The fact that each trait in the dihybrid cross holds to its monohybrid probabilities strongly suggests that the two characteristics do not affect each other and are therefore independent. This led Mendel to formulate a fourth postulate.

Mendel's Fourth Postulate:

• During gamete formation, segregating pairs of genetic factors assort independently of each other.

This simply means that the factors for seed color, when they segregate during gamete formation, do not affect the segregation of the factors for seed shape, and vice versa. The entire process is random.

There are two good methods for determining the theoretical outcome of any cross. One of these is the Punnett square. Let's use the Punnett square to reexamine the example of the dihybrid cross. In the dihybrid F1 plants (GgWw), the possible combinations of alleles in the gametes are GW, Gw, gW, and gw. Setting this up in the Punnet square:

It is easy to see that there are sixteen allele combinations in the Punnett square (although some are duplicates). Check each genotype, determine the phenotype, and work out the phenotypic ratio. You should find that it matches the ratio that Mendel observed, as shown above. Each of the fractions in the ratio is the probability that that phenotype will occur. For example, there is a 3/16 probability that any offspring will have green, round seeds.

The other method for determining the outcome of a cross is the forked line method. This method takes advantage of the Product Law, and exploits the fact that a cross can be broken down into a set of monohybrid crosses. The forked line method is especially useful when three or more characteristics are crossed simultaneously. For example, a trihybrid cross would require a Punnett square with 64 spaces! (If you don't believe me, try it for yourself.) This gets to be confusing, and its easier to keep things straight using the forked line method, because it deals with phenotypes rather than genotypes.

Here's how the method works (using the dihybrid cross as an example). In this case, letters will be used to designate phenotypes rather than alleles, so G means a yellow-seeded plant, g means a green-seeded plant, W means a round-seeded plant, and w means a wrinkled-seeded plant. First, we'll consider the first characteristic, seed color. In a monohybrid cross (Gg x Gg) involving seed color, the offspring have a 3/4 probability of having yellow seeds, and a 1/4 probability of having green seeds, as shown in the diagram below. Next, we'll consider seed shape. For each of the color phenotypes, there's a 3/4 probability that the seeds will be round, and a 1/4 probability that the seeds will be wrinkled. These probabilities are filled in along forked lines from the seed color probabilities. To determine the overall probability of a particular phenotype, simply multiply (i.e. use the product law) all of the probabilities along a particular line.

For more characteristics (i.e. a trihybrid cross), simply brach out from each point in the second column.

The forked line method can also be used to determine the probabilities of particular phenotypes, but this gets more complicated, because each characteristic will have three possibilities (homozygous dominant, heterozygous, homozygous recessive) instead of two (dominant trait, recessive trait).

Dihybrid Test Cross

As with the monohybrid cross, with some of the phenotypes it seems impossible to tell what the genotype is. Yellow, round-seeded plants (GGWW, GgWW, GGWw, or GgWw?); yellow, wrinkled seeded plants (GGww or Ggww?); and green, round-seeded plants (ggWw or ggWW?) are examples of such phenotypes. As with the monohybrid cross, the way to tell determine the genotype of a plant in question is to do a test cross. In this case, the plant to be tested will be crossed with a double homozygous recessive, ggww.

Using a yellow, round-seeded plant as a test subject, work out all of the possible results of a test cross.