Two traits at once: the dihybrid cross

One trait at a time is not much. Mendel went further: two traits followed simultaneously, and a new ratio appears.

The dihybrid cross

Mendel crossed peas differing in two traits: shape (round R dominant / wrinkled r) and colour (yellow Y dominant / green y).

P    :  round yellow  (R//R , Y//Y)   ×   wrinkled green  (r//r , y//y)
            |
            v
F1   :  100 % round yellow            all (R//r , Y//y)
            |
            v
F2   :  four phenotypes, in the ratio   9 : 3 : 3 : 1

On the 556 seeds Mendel actually counted:

   round yellow      315        theoretical  312.8      (9/16)
   round green       108        theoretical  104.2      (3/16)
   wrinkled yellow   101        theoretical  104.2      (3/16)
   wrinkled green     32        theoretical   34.8      (1/16)
                     ---
                     556

The agreement is striking. It is even so good that the statistician Ronald Fisher suspected in 1936 that the data had been tidied up — a debate still open, which has taken nothing away from the validity of the laws themselves.

The second law: independent assortment

Law of independent assortment. The alleles of two different genes are transmitted independently of each other.

The 9 : 3 : 3 : 1 ratio follows by simply multiplying probabilities:

   shape :  3/4 round,  1/4 wrinkled
   colour:  3/4 yellow, 1/4 green

   round AND yellow    :  3/4 × 3/4 = 9/16
   round AND green     :  3/4 × 1/4 = 3/16
   wrinkled AND yellow :  1/4 × 3/4 = 3/16
   wrinkled AND green  :  1/4 × 1/4 = 1/16

This is exactly the computation for two independent events in probability. On this point, Mendelian genetics is simply an exercise in counting.

The exception: linked genes

The second law holds only if the two genes lie on different chromosomes — or far apart on the same one.

genes on DIFFERENT chromosomes    ->  independence,  ratio 9:3:3:1
genes CLOSE on the SAME chromosome -> they travel TOGETHER
                                      -> the 9:3:3:1 ratio is distorted

Mendel was lucky: the seven traits he studied fall across the pea's seven chromosome pairs, or are too far apart to stay linked. With a different choice he would never have found his second law.

What separates two linked genes is crossing over in meiosis. The closer two genes are, the more rarely an exchange separates them — and that measurable frequency gives their distance along the chromosome. This is how the first genetic maps were built, from 1913 onwards.

Summary

• A dihybrid cross follows two traits; the F2 gives the ratio 9 : 3 : 3 : 1.
• Second law: alleles of different genes are transmitted independently.
• The ratio is obtained by multiplying the probabilities for each trait.
• Independence assumes genes on different chromosomes.
• Linked genes (close on the same chromosome) distort the ratio.
• Crossing-over frequency measures the distance between genes: genetic maps.