Dihybrid crosses and independent assortment

Two traits at once

A seed can be round or wrinkled, and it can be yellow or green. Are the two traits passed on together, or does each travel by itself? Mendel followed both traits in the same cross. A cross with two pairs of contrasting traits is a dihybrid cross.

He crossed a plant with round yellow seeds (RRYY) with a plant with wrinkled green seeds (rryy). Round (R) and yellow (Y) are dominant.

• The F1 plants were all RrYy, with round yellow seeds.
• The F1 plants pollinated themselves. The F2 plants showed four kinds of seeds in the ratio 9 : 3 : 3 : 1.

Two of the four kinds are new combinations: round green seeds and wrinkled yellow seeds. They did not occur in the parents.

Building the Punnett square

An RrYy plant makes four kinds of gametes: RY, Ry, rY and ry. They are equally common, one quarter each. The four gametes of each parent make a square of 4 × 4 = 16 boxes.

Count the colours in the square. There are 9 round yellow, 3 round green, 3 wrinkled yellow and 1 wrinkled green. The 16 boxes hold 9 different genotypes. Four of them have two heterozygous genes (RrYy), and one box each has RRYY, RRyy, rrYY and rryy.

Each gene on its own

Look at the shape of the seed alone. There are 12 round seeds (9 + 3) and 4 wrinkled seeds (3 + 1). That is 3 : 1, the ratio of a monohybrid cross. Look at the colour alone: 12 yellow and 4 green, again 3 : 1.

So each pair of alleles behaves as if the other pair were not there. This is the law of independent assortment: when two pairs of traits are combined in a hybrid, the segregation of one pair is independent of the other pair.

A shortcut with probability

You do not need the 16 boxes. The chance of two independent events together is the product of their chances.

• Round and yellow: $\tfrac{3}{4} \times \tfrac{3}{4} = \tfrac{9}{16}$.
• Round and green: $\tfrac{3}{4} \times \tfrac{1}{4} = \tfrac{3}{16}$.
• Wrinkled and green: $\tfrac{1}{4} \times \tfrac{1}{4} = \tfrac{1}{16}$.

This is the forked line method. It also gives the ratio of three or four genes with little work. For n heterozygous genes on different chromosomes:

• The number of kinds of gamete is $2^n$.
• The number of boxes in the Punnett square is $4^n$.
• The number of kinds of genotype in the F2 is $3^n$.
• The share of the F2 with every trait dominant is $(\tfrac{3}{4})^n$.

You should see that the chance of seeing every dominant trait falls quickly as more genes are added. For n = 2 it is 9 out of 16. For n = 4 it is only about 32 per cent.

The dihybrid test cross

To find out whether a round yellow plant is RrYy, cross it with a double recessive plant (rryy). An RrYy plant gives four kinds of gamete, and rryy gives only ry. The offspring come in four classes, 1 round yellow : 1 round green : 1 wrinkled yellow : 1 wrinkled green. The ratio 1 : 1 : 1 : 1 shows that the genes assort independently.

When the law does not hold

Independent assortment is true only when the two genes lie on different chromosomes, or far apart on one chromosome. Genes that lie close together on the same chromosome travel together. This is linkage, and the next course studies it.

Try it in Python

The program lists the gametes of RrYy and counts the phenotypes of the 16 boxes.

Practise with new numbers

Common mistakes

• Writing the gametes of RrYy as Rr and Yy. A gamete carries one allele of each gene: RY, Ry, rY or ry.
• Writing 9 : 3 : 3 : 1 for every dihybrid cross. It is the ratio of RrYy × RrYy only. The test cross gives 1 : 1 : 1 : 1.
• Using the product rule for events that depend on each other. The rule works because the genes assort independently.
• Forgetting that there are two ways to get a mixed phenotype. Round green and wrinkled yellow both have 3 parts out of 16.

Remember these

• The F2 of RrYy × RrYy is 9 round yellow : 3 round green : 3 wrinkled yellow : 1 wrinkled green.
• The law of independent assortment holds for genes on different chromosomes.
• A dihybrid test cross gives 1 : 1 : 1 : 1.
• n heterozygous genes give $2^n$ gametes, $4^n$ boxes and $3^n$ genotypes.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. List the gametes of a plant with the genotype TtGg, where T is tall and G is green pods.

2. In the F2 of RrYy × RrYy, what is the chance that a seed is wrinkled and yellow? How many such seeds do you expect among 800 seeds?

3. A plant with round yellow seeds is crossed with a plant with wrinkled green seeds. The offspring are 1 : 1 : 1 : 1 for the four kinds. What is the genotype of the round yellow plant?

4. How many boxes does the Punnett square have for a cross of two plants that are each heterozygous for three genes on different chromosomes?

5. In which case does the law of independent assortment fail?

References

• NCERT (2023). Biology, Textbook for Class XII, chapter 4, section 4.1.
• Clark M. A., Douglas M. and Choi J. (2018). Biology 2e, OpenStax, chapter 12.3.