Two traits and the 9:3:3:1 ratio

Two genes at once

Mendel also followed two traits together. Pea seeds can be:

• round (R) or wrinkled (r), with round dominant;
• yellow (Y) or green (y), with yellow dominant.

A plant now carries four letters, two for each gene, such as RrYy. Each gamete carries one letter of each gene: one of R or r, and one of Y or y.

The gametes from product

itertools.product makes every way to pick one item from each list:

It prints [('R', 'Y'), ('R', 'y'), ('r', 'Y'), ('r', 'y')]. Joining each pair gives the four gametes RY, Ry, rY and ry. For a plant RrYy, the first two letters are g[:2] and the last two are g[2:]:

This is Mendel's law of independent assortment in one line: the shape letter and the colour letter are chosen separately.

The 16-box square

Each of four gametes from one parent can meet each of four from the other: $4 \times 4 = 16$ boxes. The child's genotype sorts each gene on its own, so R comes before r and Y before y:

The phenotype of two traits

Counting the 16 boxes gives 9 round yellow, 3 round green, 3 wrinkled yellow and 1 wrinkled green. That is the 9:3:3:1 ratio. Mendel counted $315$, $108$, $101$ and $32$ seeds, out of $556$, very close to it.

Why 9:3:3:1?

Each gene alone gives 3:1. Round is $\dfrac{3}{4}$ and yellow is $\dfrac{3}{4}$, and the genes are chosen separately, so round and yellow together is $\dfrac{3}{4} \times \dfrac{3}{4} = \dfrac{9}{16}$. Round and green is $\dfrac{3}{4} \times \dfrac{1}{4} = \dfrac{3}{16}$, and so on. The program and the multiplication agree.

The test cross

A round yellow plant may be RRYY, RRYy, RrYY or RrYy. You cannot tell by looking. The test cross crosses it with rryy, which shows every hidden small letter:

• RrYy with rryy gives all four kinds, 4 boxes each: 1:1:1:1;
• RRYy with rryy gives round yellow and round green, half and half.

The program can try every candidate and print what each would give.

Practise with new numbers

Try it in Python

The program lists the gametes of RrYy, fills the 16 boxes, counts the four phenotypes from the most common down, and then does the test cross with rryy.

Common mistakes

• Taking two letters of the same gene into a gamete. A gamete has one shape letter and one colour letter, never Rr.
• Sorting the whole genotype at once. sorted("RrYy") mixes the genes into RYry. Sort each gene on its own.
• Reading the wrong slice. g[:2] is the shape gene and g[2:] is the colour gene. Swap them and round and yellow get muddled.
• Expecting 1:1:1:1 from every cross. That comes from a test cross of RrYy. Two RrYy plants give 9:3:3:1.

Remember these

• product("Rr", "Yy") gives the four gametes of RrYy: RY, Ry, rY and ry.
• Four gametes meet four gametes in 16 boxes; sort each gene of the child separately.
• RrYy with RrYy gives 9:3:3:1; each gene alone still gives 3:1.
• A test cross with rryy reveals the hidden letters of a dominant-looking plant.

Check yourself

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

1. What does list(product("Rr", "yy")) give, and how many different gametes is that?

2. Why are there 16 boxes in the square for RrYy with RrYy?

3. What fraction of the seeds is wrinkled and green, and why is it the smallest group?

4. A round yellow plant crossed with rryy gives only round yellow and wrinkled yellow seeds. What is its genotype?

5. Why does child sort each gene separately?