Punnett squares

A 2 by 2 grid predicts that two Yy pea plants give green seeds one time in four.

Genotype and phenotype

Two words keep coming up. The genotype is the pair of alleles an organism carries, written with letters: a capital for the dominant allele and a small letter for the recessive one. The phenotype is what you can actually observe. A pea plant with yellow seeds might be YY or Yy, two genotypes with the same phenotype, because the dominant Y hides y. A plant with green seeds must be yy. When both alleles are the same (YY or yy) the organism is homozygous for that gene; when they differ (Yy) it is heterozygous.

Drawing the square

The British geneticist Reginald Punnett devised a simple grid for crosses. Write the possible gametes of one parent along the top and those of the other parent down the side. Each parent puts one of its two alleles into each gamete, with equal chance. Fill each box with the allele from its row and the allele from its column. Every box is an equally likely offspring. For Yy × Yy the boxes are YY, Yy, yY and yy. Yy and yY are the same genotype reached two ways, so the genotype ratio is 1 YY : 2 Yy : 1 yy, and the phenotype ratio is 3 yellow : 1 green. That is Mendel's 3:1.

A square gives chances, not guarantees. Each box is a probability of one in four for every single offspring, just as a coin has a one in two chance of heads on every toss. Two rules of probability do most of the work. The product rule: for independent events that must both happen, multiply. The chance that three offspring of Yy × Yy all have yellow seeds is ¾ × ¾ × ¾ = 27/64. The sum rule: for outcomes that cannot happen together, add. The chance of a Yy offspring is ¼ (Yy) + ¼ (yY) = ½.

The test cross

How can you tell a YY plant from a Yy plant when both have yellow seeds? Cross it with a green yy plant. A YY parent can only give Y, so all the offspring are Yy and yellow. A Yy parent gives Y or y equally, so about half the offspring are green. Breeders still use this test cross. Set parent 1 to Yy and parent 2 to yy in the square to see the 1:1.

Two genes at once, and other patterns

Mendel also followed seed colour and seed shape together. Their alleles are passed on independently, so a YyRr plant makes four kinds of gametes (YR, Yr, yR, yr) in equal numbers, and YyRr × YyRr needs a 4 by 4 square of 16 boxes. The phenotypes come out 9 yellow round : 3 yellow wrinkled : 3 green round : 1 green wrinkled. You can check the 9 with the product rule: ¾ yellow × ¾ round = 9/16.

Not every gene follows simple dominance. In snapdragons, red × white gives pink offspring because neither allele fully masks the other; this is incomplete dominance, and pink × pink gives red, pink and white in a 1:2:1 ratio. In codominance both alleles show at once. The ABO blood groups combine both ideas: the A and B alleles are codominant, so a person with one of each has type AB, while the O allele (written i) is recessive to both. Genes on the X chromosome add one more twist. A male has only one X, so a single recessive allele there, such as one for red-green colour blindness, shows up in him, which is why such traits are more common in males.