Mendel and Punnett Squares

Gregor Mendel crossed pea plants with purple flowers and pea plants with white flowers, and every offspring had purple flowers. When those purple offspring fertilized themselves, white flowers came back in about one plant in four.

The white seemed to vanish for a generation, then return. Mendel counted 224 white-flowered plants among 929, about one in four. Where was the white hiding, and how could anyone predict “one in four”?

Explore: cross two plants and count the offspring

In the simulation you pick the genotypes of two parent pea plants for one gene or for two genes. You see their Punnett square and the predicted ratio. Then you “plant” 10, 100 or 1,000 offspring, each chosen at random, and compare your counts with the prediction. All counts in the simulation are made by random draws, not by real plants.

Genes One gene (flower color) Two genes (seed shape and color)

Parent 1

Parent 2

Plant Plant 10 Plant 100 Plant 1,000 Resample Reset

In this sketch: complete dominance, two alleles per gene, genes sort independently, and every box of the square is equally likely. The offspring are random draws by the computer, not real plants.

A Punnett square gives probabilities, not certainties, so real counts scatter around its ratios. Expected counts are what the ratio predicts, not a promise.

Try this.

1. Cross two Pp plants. Before you plant, predict how many white plants you will get out of 10. Plant 10 and compare. Repeat three times.
2. Now plant 100, then 1,000. Predict first: will the share of white plants move closer to one in four, or stay just as scattered?
3. Cross PP with pp. Predict the offspring, then test.
4. Switch to two genes and cross two RrYy plants. Predict which seed type will be rarest, then plant 1,000 and check.

Explain: genes, alleles and the hidden white

A gene is a section of DNA that is a basic unit of inheritance. Many genes code for a protein. An allele is one version of a gene. Pea plants have a flower-color gene with a purple allele and a white allele.

For most genes, your body cells carry two alleles, one from each parent. Peas are the same. We write alleles as letters: P for purple and p for white.

• A dominant allele shows its trait even when only one copy is present. We write it as a capital letter.
• A recessive allele shows its trait only when both copies are recessive. We write it as a small letter.
• Homozygous means the two alleles are the same, like PP or pp.
• Heterozygous means the two alleles are different, like Pp.

Your genotype is the pair of alleles you carry, such as Pp. Your phenotype is the trait that shows, such as purple flowers. A phenotype is anything you can observe or measure, including chemical traits, such as whether your body makes a certain enzyme.

Genotype	Homozygous or heterozygous?	Phenotype
PP	homozygous dominant	purple
Pp	heterozygous	purple
pp	homozygous recessive	white

Now the hidden white makes sense. Mendel’s first-generation plants were all Pp. They looked purple, but each one still carried a white allele.

When a plant makes sex cells by meiosis, the two alleles of each gene separate. Each sex cell gets one allele. This is called the law of segregation. You can read how meiosis does this in that lesson.

Probability is the chance that something will happen, from 0 (never) to 1 (always). For a Pp plant, each sex cell has a 50 percent chance of carrying P and a 50 percent chance of carrying p. It is like a coin toss. The analogy breaks in one place: a coin has no connection to the parent, while the allele in the sex cell really came from the parent’s own pair.

Explain: building a Punnett square

A Punnett square is a grid that shows every way one sex cell from each parent can combine. Put one parent’s possible sex cells across the top and the other parent’s down the side. Fill each box with one allele from the top and one from the side.

Here is the cross of two Pp plants. Because both parents are heterozygous for one gene, this is a monohybrid cross.

	P (1/2)	p (1/2)
P (1/2)	PP	Pp
p (1/2)	Pp	pp

Each box has a 1 in 4 chance, because 1/2 × 1/2 = 1/4.

• Genotype ratio: 1 PP : 2 Pp : 1 pp, or 1:2:1.
• Phenotype ratio: 3 purple : 1 white, or 3:1, because PP and Pp both look purple.

A dihybrid cross follows two genes at once. Mendel studied plants that were heterozygous for seed shape (R round, r wrinkled) and seed color (Y yellow, y green). If the two genes sort into sex cells independently, getting R or r does not affect whether a sex cell gets Y or y. An RrYy plant then makes four kinds of sex cells, RY, Ry, rY and ry, each with a 1 in 4 chance. This is the law of independent assortment. It holds well for genes on different chromosomes.

	RY	Ry	rY	ry
RY	RRYY	RRYy	RrYY	RrYy
Ry	RRYy	RRyy	RrYy	Rryy
rY	RrYY	RrYy	rrYY	rrYy
ry	RrYy	Rryy	rrYy	rryy

Count the 16 boxes by phenotype:

round yellow : round green : wrinkled yellow : wrinkled green = 9 : 3 : 3 : 1

Each box is one of 16 equally likely outcomes. It is not a promise about the next 16 seeds. Four seeds from two Pp plants could include no white plants at all. The chance of that is 3/4 × 3/4 × 3/4 × 3/4, about 32 percent.

Elaborate: Mendel’s pea counts

Mendel let hybrid plants fertilize themselves for seven traits and counted the second generation. For each trait, the first trait listed is the one every first-generation plant showed.

Trait	Dominant count	Recessive count	Total	Recessive share (%)
Seed shape: round vs wrinkled (seeds)	5,474	1,850	7,324	25.3
Seed color: yellow vs green (seeds)	6,022	2,001	8,023	24.9
Flower color: purple vs white (plants)	705	224	929	24.1
Pod form: inflated vs constricted (plants)	882	299	1,181	25.3
Unripe pod color: green vs yellow (plants)	428	152	580	26.2
Flower position: along stem vs at tip (plants)	651	207	858	24.1
Stem length: long vs short (plants)	787	277	1,064	26.0

Counts are from Mendel’s paper, printed in Brno in 1866 (English translation, Experiments in Plant Hybridization). Each row is one experiment on one trait, with seeds or plants counted as labelled. The paper does not give a harvest year for each row. The recessive share was calculated here as the recessive count divided by the total.

In his dihybrid experiment, Mendel counted 556 seeds from 15 plants: 315 round yellow, 108 round green, 101 wrinkled yellow and 32 wrinkled green. A 9:3:3:1 ratio predicts about 313, 104, 104 and 35.

1. Which two traits had the largest samples? How close were their recessive shares to 25 percent?
2. Which trait had the smallest sample, and how far was its share from 25 percent?
3. In the dihybrid count, which category was furthest from its prediction, and by how many seeds?

Design problem

You have a purple-flowered pea plant. You do not know whether it is PP or Pp. Plan a cross that would tell you, and decide how many offspring you need.

One way to do it

Cross the purple plant with a white plant (pp). This is a test cross. A PP parent can only give P, so every offspring would be purple. A Pp parent gives p half the time, so about half the offspring would be white.

• Independent variable: the unknown genotype of the purple parent.
• Dependent variable: the number of purple and white offspring.
• Constants: the same white parent line, the same soil, water, light and growing time.
• Control of pollen: remove the purple flower’s stamens before they ripen, brush on pollen from the white plant, and cover the flower so insects cannot bring other pollen.
• Sample size: grow at least 20 offspring. With 20 offspring from a Pp parent, the chance of getting no white plants by bad luck is under one in a million. With only 2 offspring it is 1 in 4.

Even one white offspring means the purple parent is Pp. All purple in a large sample means it is very likely PP.

Evaluate: check your thinking

The white flowers were never gone. Mendel’s purple first-generation plants were Pp, so each carried a hidden white allele. When they fertilized themselves, each offspring had a 1 in 4 chance of getting two white alleles, and about one plant in four did.

Science practices you used: making predictions before testing, using a Punnett square as a model and naming its limits, calculating proportions from real counts, and judging how sample size affects a result.